thetis package

Submodules

thetis.assembledschur module

class thetis.assembledschur.AssembledSchurPC

Bases: firedrake.matrix_free.preconditioners.PCBase

Preconditioner for the Schur complement, where the preconditioner matrix is assembled by explicitly matrix multiplying A10*Minv*A10. Here: A01, A10 are the assembled sub-blocks of the saddle point system. The form

of this system needs to be supplied in the appctx to the solver as appctx[‘a’]

Minv is the inverse of the mass-matrix which is assembled as

assemble(v*u*dx, inverse=True), i.e. the element-wise inverse, where v and u are the test and trial of the 00 block. This gives the exact inverse of the mass matrix for a DG discretisation.

Create a PC context suitable for PETSc.

Matrix free preconditioners should inherit from this class and implement:

• initialize()
• update()
• apply()
• applyTranspose()

apply(pc, X, Y)

applyTranspose(pc, X, Y)

initialize(pc)

update(pc)

view(pc, viewer=None)

thetis.callback module

Defines custom callback functions used to compute various metrics at runtime.

class thetis.callback.CallbackManager

Bases: collections.defaultdict

Stores callbacks in different categories and provides methods for evaluating them.

Create callbacks and register them under 'export' mode

cb1 = VolumeConservation3DCallback(...)
cb2 = TracerMassConservationCallback(...)
cm = CallbackManager()
cm.add(cb1, 'export')
cm.add(cb2, 'export')

Evaluate callbacks, calls evaluate() method of all callbacks registered in the given mode.

cm.evaluate('export')

add(callback, mode)

Add a callback under the given mode

Parameters:

• callback – a DiagnosticCallback object
• mode (str) – register callback under this mode

evaluate(mode, index=None)

Evaluate all callbacks registered under the given mode

Parameters:

• mode (str) – evaluate all callbacks under this mode
• index (int) – if provided, sets the export index. Default behavior is to append to the file or stream.

class thetis.callback.DetectorsCallback(solver_obj, detector_locations, field_names, name, detector_names=None, **kwargs)

Bases: thetis.callback.DiagnosticCallback

Callback that writes the specified fields interpolated in the specified detector locations

Parameters:

• solver_obj – Thetis solver object
• detector_locations – List of x, y locations in which fields are to be interpolated.
• field_names – List of fields to be interpolated.
• name – Unique name for this callback and its associated set of detectors. This determines the name of the output h5 file (prefixed with diagnostic_).
• detector_names – List of names for each of the detectors. If not provided automatic names of the form ‘detectorNNN’ are created where NNN is the (0-padded) detector number.
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

name

variable_names

class thetis.callback.DiagnosticCallback(solver_obj, array_dim=1, attrs=None, outputdir=None, export_to_hdf5=True, append_to_log=True, hdf5_dtype='d')

Bases: object

A base class for all Callback classes

Parameters:

• solver_obj – Thetis solver object
• outputdir (str) – Custom directory where hdf5 files will be stored. By default solver’s output directory is used.
• array_dim (int) – Dimension of the output array. 1 for scalars.
• attrs (dict) – Additional attributes to be saved in the hdf5 file.
• export_to_hdf5 (bool) – If True, diagnostics will be stored in hdf5 format
• append_to_log (bool) – If True, callback output messages will be printed in log
• hdf5_dtype – Precision to use in hdf5 output: ‘d’ for double precision (default), and ‘f for single precision

evaluate(index=None)

Evaluates callback and pushes values to log and hdf file (if enabled)

message_str(*args)

A string representation.

Parameters:args – If provided, these will be the return value from __call__().

name

The name of the diagnostic

push_to_hdf5(time, args, index=None)

Append values to HDF5 file.

Parameters:

• time – time stamp of entry
• args – the return value from __call__().

push_to_log(time, args)

Push callback status message to log

Parameters:

• time – time stamp of entry
• args – the return value from __call__().

set_write_mode(mode)

Define whether to create a new hdf5 file or append to an exisiting one

Parameters:mode (str) – Either ‘create’ (default) or ‘append’

variable_names

Names of all scalar values

class thetis.callback.DiagnosticHDF5(filename, varnames, array_dim=1, attrs=None, comm=<mpi4py.MPI.Intracomm object>, new_file=True, dtype='d')

Bases: object

A HDF5 file for storing diagnostic time series arrays.

Parameters:

• filename (str) – Full filename of the HDF5 file.
• varnames – List of variable names that the diagnostic callback provides
• array_dim (int) – Dimension of the output array. 1 for scalars.
• attrs (dict) – Additional attributes to be saved in the hdf5 file.
• comm – MPI communicator
• new_file (bool) – Define whether to create a new hdf5 file or append to an existing one (if any)

export(time, variables, index=None)

Appends a new entry of (time, variables) to the file.

The HDF5 is updated immediately.

Parameters:

• time (float) – time stamp of entry
• variables (tuple of float) – values of entry
• index (int) – If provided, defines the time index in the file

class thetis.callback.MinMaxConservationCallback(minmax_callback, solver_obj, **kwargs)

Bases: thetis.callback.DiagnosticCallback

Base class for callbacks that check conservation of a minimum/maximum

Parameters:

• minmax_callback – Python function that takes the solver object as an argument and returns a (min, max) value tuple
• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

message_str(*args)

variable_names = ['min_value', 'max_value', 'undershoot', 'overshoot']

class thetis.callback.ScalarConservationCallback(scalar_callback, solver_obj, **kwargs)

Bases: thetis.callback.DiagnosticCallback

Base class for callbacks that check conservation of a scalar quantity

Creates scalar conservation check callback object

Parameters:

• scalar_callback – Python function that takes the solver object as an argument and returns a scalar quantity of interest
• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

message_str(*args)

variable_names = ['integral', 'relative_difference']

class thetis.callback.TracerMassConservationCallback(tracer_name, solver_obj, **kwargs)

Bases: thetis.callback.ScalarConservationCallback

Checks conservation of total tracer mass

Parameters:

• tracer_name – Name of the tracer. Use canonical field names as in FieldDict.
• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

name = 'tracer mass'

class thetis.callback.TracerOvershootCallBack(tracer_name, solver_obj, **kwargs)

Bases: thetis.callback.MinMaxConservationCallback

Checks overshoots of the given tracer field.

Parameters:

• tracer_name – Name of the tracer. Use canonical field names as in FieldDict.
• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

name = 'tracer overshoot'

class thetis.callback.VolumeConservation2DCallback(solver_obj, **kwargs)

Bases: thetis.callback.ScalarConservationCallback

Checks conservation of 2D volume (integral of water elevation field)

Parameters:

• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

name = 'volume2d'

class thetis.callback.VolumeConservation3DCallback(solver_obj, **kwargs)

Bases: thetis.callback.ScalarConservationCallback

Checks conservation of 3D volume (volume of 3D mesh)

Parameters:

• solver_obj – Thetis solver object
• **kwargs –

  any additional keyword arguments, see DiagnosticCallback

name = 'volume3d'

thetis.callback.timed_region()

Log.Event(type cls, name, klass=None)

thetis.callback.timed_stage()

Log.Stage(type cls, name)

thetis.configuration module

Utility function and extensions to traitlets used for specifying Thetis options

class thetis.configuration.ABCMetaHasTraits(name, bases, classdict)

Bases: abc.ABCMeta, traitlets.traitlets.MetaHasTraits

Combined metaclass of ABCMeta and MetaHasTraits

Finish initializing the HasDescriptors class.

class thetis.configuration.BoundedFloat(default_value=traitlets.Undefined, bounds=None, **kwargs)

Bases: traitlets.traitlets.Float

info()

validate(obj, proposal)

class thetis.configuration.BoundedInteger(default_value=traitlets.Undefined, bounds=None, **kwargs)

Bases: traitlets.traitlets.Int

info()

validate(obj, proposal)

class thetis.configuration.FiredrakeCoefficient(default_value=traitlets.Undefined, allow_none=False, read_only=None, help=None, config=None, **kwargs)

Bases: traitlets.traitlets.TraitType

Declare a traitlet.

If allow_none is True, None is a valid value in addition to any values that are normally valid. The default is up to the subclass. For most trait types, the default value for allow_none is False.

Extra metadata can be associated with the traitlet using the .tag() convenience method or by using the traitlet instance’s .metadata dictionary.

default_value = None

default_value_repr()

info_text = 'a Firedrake Constant or Function'

validate(obj, value)

class thetis.configuration.FiredrakeConstant(default_value=traitlets.Undefined, allow_none=False, read_only=None, help=None, config=None, **kwargs)

Bases: traitlets.traitlets.TraitType

Declare a traitlet.

If allow_none is True, None is a valid value in addition to any values that are normally valid. The default is up to the subclass. For most trait types, the default value for allow_none is False.

Extra metadata can be associated with the traitlet using the .tag() convenience method or by using the traitlet instance’s .metadata dictionary.

default_value = None

default_value_repr()

info_text = 'a Firedrake Constant'

validate(obj, value)

class thetis.configuration.FiredrakeScalarExpression(default_value=traitlets.Undefined, allow_none=False, read_only=None, help=None, config=None, **kwargs)

Bases: traitlets.traitlets.TraitType

Declare a traitlet.

If allow_none is True, None is a valid value in addition to any values that are normally valid. The default is up to the subclass. For most trait types, the default value for allow_none is False.

Extra metadata can be associated with the traitlet using the .tag() convenience method or by using the traitlet instance’s .metadata dictionary.

default_value = None

default_value_repr()

info_text = 'a scalar UFL expression'

validate(obj, value)

class thetis.configuration.FiredrakeVectorExpression(default_value=traitlets.Undefined, allow_none=False, read_only=None, help=None, config=None, **kwargs)

Bases: traitlets.traitlets.TraitType

Declare a traitlet.

If allow_none is True, None is a valid value in addition to any values that are normally valid. The default is up to the subclass. For most trait types, the default value for allow_none is False.

Extra metadata can be associated with the traitlet using the .tag() convenience method or by using the traitlet instance’s .metadata dictionary.

default_value = None

default_value_repr()

info_text = 'a vector UFL expression'

validate(obj, value)

class thetis.configuration.FrozenConfigurable(*args, **kwargs)

Bases: thetis.configuration.OptionsBase, traitlets.config.configurable.Configurable

A Configurable class that only allows adding new attributes in the class definition or when self._isfrozen is False.

class thetis.configuration.FrozenHasTraits(*args, **kwargs)

Bases: thetis.configuration.OptionsBase, traitlets.traitlets.HasTraits

class thetis.configuration.NonNegativeFloat(default_value=traitlets.Undefined, allow_none=False, **kwargs)

Bases: traitlets.traitlets.Float

info()

validate(obj, proposal)

class thetis.configuration.NonNegativeInteger(default_value=traitlets.Undefined, allow_none=False, **kwargs)

Bases: traitlets.traitlets.Int

info()

validate(obj, proposal)

class thetis.configuration.OptionsBase

Bases: object

Abstract base class for all options classes

name

Human readable name of the configurable object

update(options)

Assign options from another container

Parameters:options – Either a dictionary of options or another HasTraits object

class thetis.configuration.PETScSolverParameters(trait=None, traits=None, default_value=traitlets.Undefined, **kwargs)

Bases: traitlets.traitlets.Dict

PETSc solver options dictionary

Create a dict trait type from a Python dict.

The default value is created by doing dict(default_value), which creates a copy of the default_value.

trait : TraitType [ optional ]

The specified trait type to check and use to restrict contents of the Container. If unspecified, trait types are not checked.

traits : Dictionary of trait types [ optional ]

A Python dictionary containing the types that are valid for restricting the content of the Dict Container for certain keys.

default_value : SequenceType [ optional ]

The default value for the Dict. Must be dict, tuple, or None, and will be cast to a dict if not None. If trait is specified, the default_value must conform to the constraints it specifies.

default_value = None

info_text = 'a PETSc solver options dictionary'

validate(obj, value)

class thetis.configuration.PairedEnum(values, paired_name, default_value=traitlets.Undefined, **kwargs)

Bases: traitlets.traitlets.Enum

A enum whose value must be in a given sequence.

This enum controls a slaved option, with default values provided here.

Parameters:

• values – iterable of (value, HasTraits class) pairs. The HasTraits class will be called (with no arguments) to create default values if necessary.
• paired_name – trait name this enum is paired with.
• default_value – default value.

info()

class thetis.configuration.PositiveFloat(default_value=traitlets.Undefined, allow_none=False, **kwargs)

Bases: traitlets.traitlets.Float

info()

validate(obj, proposal)

class thetis.configuration.PositiveInteger(default_value=traitlets.Undefined, allow_none=False, **kwargs)

Bases: traitlets.traitlets.Int

info()

validate(obj, proposal)

thetis.configuration.attach_paired_options(name, name_trait, value_trait)

Attach paired options to a Configurable object.

Parameters:

• name – the name of the enum trait
• name_trait – the enum trait (a PairedEnum)
• value_trait – the slaved value trait.

thetis.configuration.rst_all_options(cls, nspace=0, prefix=None)

Recursively generate rST for a provided Configurable class.

Parameters:

• cls – The Configurable class.
• nspace – Indentation level.
• prefix – Prefix to use for new traits.

thetis.coordsys module

Generic methods for converting data between different spatial coordinate systems. Uses pyproj library.

class thetis.coordsys.VectorCoordSysRotation(source_sys, target_sys, x, y)

Bases: object

Rotates vectors defined in source_sys coordinates to a different coordinate system.

Parameters:

• source_sys – pyproj coordinate system where (x, y) are defined in
• target_sys – target pyproj coordinate system
• x – x coordinate
• y – y coordinate

thetis.coordsys.convert_coords(source_sys, target_sys, x, y)

Converts coordinates from source_sys to target_sys

This function extends pyproj.transform method by handling NaNs correctly.

Parameters:

• source_sys – pyproj coordinate system where (x, y) are defined in
• target_sys – target pyproj coordinate system
• x (float or numpy.array_like) – x coordinate
• y (float or numpy.array_like) – y coordinate

thetis.coordsys.get_vector_rotation_matrix(source_sys, target_sys, x, y, delta=None)

Estimate rotation matrix that converts vectors defined in source_sys to target_sys.

Assume that we have a vector field defined in source_sys: vectors located at (x, y) define the x and y components. We can then rotate the vectors to represent x2 and y2 components of the target_sys by applying a local rotation:

R, theta = get_vector_rotation_matrix(source_sys, target_sys, x, lat)
v_xy = numpy.array([[v_x], [v_y]])
v_new = numpy.matmul(R, v_xy)
v_x2, v_y2 = v_new

thetis.coupled_timeintegrator module

Time integrators for solving coupled 2D-3D system of equations.

class thetis.coupled_timeintegrator.CoupledLeapFrogAM3(solver)

Bases: thetis.coupled_timeintegrator.CoupledTimeIntegrator

Leap-Frog Adams-Moulton 3 time integrator for coupled 2D-3D problem

This is an ALE time integrator. Implementation follows the SLIM time integrator by Karna et al (2013)

Karna, et al. (2013). A baroclinic discontinuous Galerkin finite element model for coastal flows. Ocean Modelling, 61(0):1-20. http://dx.doi.org/10.1016/j.ocemod.2012.09.009

advance(t, update_forcings=None, update_forcings3d=None)

Advances the equations for one time step

Parameters:

• t (float) – simulation time
• update_forcings – Optional user-defined function that takes simulation time and updates time-dependent boundary conditions of the 2D equations.
• update_forcings3d – Optional user defined function that updates boundary conditions of the 3D equations

integrator_2d

alias of DIRK22

integrator_3d

alias of LeapFrogAM3

integrator_vert_3d

alias of BackwardEuler

class thetis.coupled_timeintegrator.CoupledTimeIntegrator(solver)

Bases: thetis.coupled_timeintegrator.CoupledTimeIntegratorBase

Base class of mode-split time integrators that use 2D, 3D and implicit 3D time integrators.

Parameters:solver – FlowSolver object

initialize()

Assign initial conditions to all necessary fields

Initial conditions are read from fields dictionary.

integrator_2d

time integrator for 2D equations

integrator_3d

time integrator for explicit 3D equations (momentum, tracers)

integrator_vert_3d

time integrator for implicit 3D equations (vertical diffusion)

set_dt(dt, dt_2d)

Set time step for the coupled time integrator

Parameters:

• dt (float) – Time step. This is the master (macro) time step used to march the 3D equations.
• dt_2d (float) – Time step for 2D equations. For consistency dt_2d must be an integer fraction of dt. If 2D solver is implicit set dt_2d equal to dt.

class thetis.coupled_timeintegrator.CoupledTimeIntegratorBase(solver)

Bases: thetis.timeintegrator.TimeIntegratorBase

Base class for coupled 2D-3D time integrators

Provides common functionality for updating diagnostic fields etc.

Parameters:solver – FlowSolver object

class thetis.coupled_timeintegrator.CoupledTwoStageRK(solver)

Bases: thetis.coupled_timeintegrator.CoupledTimeIntegrator

Coupled time integrator based on SSPRK(2,2) scheme

This ALE time integration method uses SSPRK(2,2) scheme to advance the 3D equations and a compatible implicit Trapezoid method to advance the 2D equations. Backward Euler scheme is used for vertical diffusion.

advance(t, update_forcings=None, update_forcings3d=None)

Advances the equations for one time step

Parameters:

• t (float) – simulation time
• update_forcings – Optional user-defined function that takes simulation time and updates time-dependent boundary conditions of the 2D equations.
• update_forcings3d – Optional user defined function that updates boundary conditions of the 3D equations

compute_mesh_velocity(istage)

Computes mesh velocity for stage i

Must be called after updating the 2D mode.

Parameters:istage (int) – stage of the Runge-Kutta iteration

integrator_2d

alias of ESDIRKTrapezoid

integrator_3d

alias of SSPRK22ALE

integrator_vert_3d

alias of BackwardEuler

store_elevation(istage)

Store current elevation field for computing mesh velocity

Must be called before updating the 2D mode.

Parameters:istage (int) – stage of the Runge-Kutta iteration

thetis.coupled_timeintegrator.timed_region()

Log.Event(type cls, name, klass=None)

thetis.coupled_timeintegrator.timed_stage()

Log.Stage(type cls, name)

thetis.equation module

Implements Equation and Term classes.

class thetis.equation.Equation(function_space)

Bases: object

Implements an equation, made out of terms.

Parameters:function_space – the FunctionSpace the solution belongs to

SUPPORTED_LABELS = frozenset({'implicit', 'explicit', 'source', 'nonlinear'})

Valid labels for terms, indicating how they should be treated in the time integrator.

source

The term is a source term, i.e. does not depend on the solution.

explicit

The term should be treated explicitly

implicit

The term should be treated implicitly

nonlinear

The term is nonlinear and should be treated fully implicitly

add_term(term, label)

Adds a term in the equation

Parameters:

• term – Term object to add_term
• label (string) – Assign a label to the term. Valid labels are given by SUPPORTED_LABELS.

jacobian(label, solution, solution_old, fields, fields_old, bnd_conditions)

Returns an UFL form of the Jacobian by summing up all the Jacobians of the terms.

Sign convention: all terms are assumed to be on the right hand side of the equation: d(u)/dt = term.

Parameters:

• label – string defining the type of terms to sum up. Currently one of ‘source’|’explicit’|’implicit’|’nonlinear’. Can be a list of multiple labels, or ‘all’ in which case all defined terms are summed.
• solution – solution Function of the corresponding equation
• solution_old – a time lagged solution Function
• fields – a dictionary that provides all the remaining fields that the term depends on. The keys of the dictionary should standard field names in field_metadata
• fields_old – Time lagged dictionary of fields
• bnd_conditions – A dictionary describing boundary conditions. E.g. {3: {‘elev_2d’: Constant(1.0)}} replaces elev_2d function by a constant on boundary ID 3.

label_term(term, label)

Assings a label to the given term(s).

Parameters:

• term – Term object, or a tuple of terms
• label – string label to assign

mass_term(solution)

Returns default mass matrix term for the solution function space.

Returns:UFL form of the mass term

residual(label, solution, solution_old, fields, fields_old, bnd_conditions)

Returns an UFL form of the residual by summing up all the terms with the desired label.

Sign convention: all terms are assumed to be on the right hand side of the equation: d(u)/dt = term.

Parameters:

• label – string defining the type of terms to sum up. Currently one of ‘source’|’explicit’|’implicit’|’nonlinear’. Can be a list of multiple labels, or ‘all’ in which case all defined terms are summed.
• solution – solution Function of the corresponding equation
• solution_old – a time lagged solution Function
• fields – a dictionary that provides all the remaining fields that the term depends on. The keys of the dictionary should standard field names in field_metadata
• fields_old – Time lagged dictionary of fields
• bnd_conditions – A dictionary describing boundary conditions. E.g. {3: {‘elev_2d’: Constant(1.0)}} replaces elev_2d function by a constant on boundary ID 3.

select_terms(label)

Generator function that selects terms by label(s).

label can be a single label (e.g. ‘explicit’), ‘all’ or a tuple of labels.

class thetis.equation.Term(function_space)

Bases: object

Implements a single term of an equation.

Note

Sign convention: all terms are assumed to be on the right hand side of the equation: d(u)/dt = term.

Parameters:function_space – the FunctionSpace the solution belongs to

jacobian(solution, solution_old, fields, fields_old, bnd_conditions)

Returns an UFL form of the Jacobian of the term.

Parameters:

• solution – solution Function of the corresponding equation
• solution_old – a time lagged solution Function
• fields – a dictionary that provides all the remaining fields that the term depends on. The keys of the dictionary should standard field names in field_metadata
• fields_old – Time lagged dictionary of fields
• bnd_conditions – A dictionary describing boundary conditions. E.g. {3: {‘elev_2d’: Constant(1.0)}} replaces elev_2d function by a constant on boundary ID 3.

residual(solution, solution_old, fields, fields_old, bnd_conditions)

Returns an UFL form of the term.

Parameters:

• solution – solution Function of the corresponding equation
• solution_old – a time lagged solution Function
• fields – a dictionary that provides all the remaining fields that the term depends on. The keys of the dictionary should standard field names in field_metadata
• fields_old – Time lagged dictionary of fields
• bnd_conditions – A dictionary describing boundary conditions. E.g. {3: {‘elev_2d’: Constant(1.0)}} replaces elev_2d function by a constant on boundary ID 3.

thetis.equation.timed_region()

Log.Event(type cls, name, klass=None)

thetis.equation.timed_stage()

Log.Stage(type cls, name)

thetis.exporter module

Routines for handling file exports.

class thetis.exporter.ExportManager(outputdir, fields_to_export, functions, field_metadata, export_type='vtk', next_export_ix=0, verbose=False)

Bases: object

Helper object for exporting multiple fields simultaneously

from .field_defs import field_metadata
field_dict = {'elev_2d': Function(...), 'uv_3d': Function(...), ...}
e = exporter.ExportManager('mydirectory',
                           ['elev_2d', 'uv_3d', salt_3d'],
                           field_dict,
                           field_metadata,
                           export_type='vtk')
e.export()

Parameters:

• outputdir (string) – directory where files are stored
• fields_to_export (list of strings) – list of fields to export
• functions – dict that contains all existing Function s
• field_metadata – dict of all field metadata. See field_defs
• export_type (str) – export format, either ‘vtk’ or ‘hdf5’
• next_export_ix (int) – index for next export (default 0)
• verbose (bool) – print debug info to stdout

export()

Export all designated functions to disk

Increments export index by 1.

export_bathymetry(bathymetry_2d)

Special function to export 2D bathymetry data to disk

Bathymetry does not vary in time so this only needs to be called once.

Parameters:bathymetry_2d – 2D bathymetry Function

set_next_export_ix(next_export_ix)

Set export index to all child exporters

class thetis.exporter.ExporterBase(filename, outputdir, next_export_ix=0, verbose=False)

Bases: object

Base class for exporter objects.

Parameters:

• filename (string) – output file name (without directory)
• outputdir (string) – directory where file is stored
• next_export_ix (int) – set the index for next output
• verbose (bool) – print debug info to stdout

export(function)

Exports given function to disk

set_next_export_ix(next_export_ix)

Sets the index of next export

class thetis.exporter.HDF5Exporter(function_space, outputdir, filename_prefix, next_export_ix=0, verbose=False)

Bases: thetis.exporter.ExporterBase

Stores fields in disk in native discretization using HDF5 containers

Create exporter object for given function.

Parameters:

• function_space (FunctionSpace) – space where the exported functions belong
• outputdir (string) – directory where outputs will be stored
• filename_prefix (string) – prefix of output filename. Filename is prefix_nnnnn.h5 where nnnnn is the export number.
• next_export_ix (int) – index for next export (default 0)
• verbose (bool) – print debug info to stdout

export(function)

Export function to disk.

Increments export index by 1.

Parameters:function – Function to export

export_as_index(iexport, function)

Export function to disk using the specified export index number

Parameters:

• iexport (int) – export index >= 0
• function – Function to export

gen_filename(iexport)

Generate file name ‘prefix_nnnnn.h5’ for i-th export

Parameters:iexport (int) – export index >= 0

load(iexport, function)

Loads nodal values from disk and assigns to the given function

Parameters:

• iexport (int) – export index >= 0
• function – target Function

class thetis.exporter.VTKExporter(fs_visu, func_name, outputdir, filename, next_export_ix=0, project_output=False, coords_dg=None, verbose=False)

Bases: thetis.exporter.ExporterBase

Class that handles Paraview VTK file exports

Parameters:

• fs_visu – function space where input function will be cast before exporting
• func_name – name of the function
• outputdir – output directory
• filename – name of the pvd file
• next_export_ix (int) – index for next export (default 0)
• project_output (bool) – project function to output space instead of interpolating
• coords_dg (bool) – Discontinuous coordinate field. Needed to avoid allocating new coordinate field in case of discontinuous export functions.
• verbose (bool) – print debug info to stdout

export(function)

Exports given function to disk

set_next_export_ix(next_export_ix)

Sets the index of next export

thetis.exporter.get_visu_space(fs)

Returns an appropriate VTK visualization space for a function space

Parameters:fs – function space Returns:function space for VTK visualization

thetis.exporter.is_2d(fs)

Tests wether a function space is 2D or 3D

thetis.exporter.timed_region()

Log.Event(type cls, name, klass=None)

thetis.exporter.timed_stage()

Log.Stage(type cls, name)

thetis.field_defs module

Definitions and meta data of fields

thetis.field_defs.field_metadata = {'baroc_head_3d': {'shortname': 'Baroclinic head', 'name': 'Baroclinic head', 'unit': 'm', 'filename': 'BaroHead3d'}, 'bathymetry_2d': {'shortname': 'Bathymetry', 'name': 'Bathymetry', 'unit': 'm', 'filename': 'bathymetry2d'}, 'bathymetry_3d': {'shortname': 'Bathymetry', 'name': 'Bathymetry', 'unit': 'm', 'filename': 'bathymetry3d'}, 'bottom_drag_2d': {'shortname': 'Bottom drag coefficient', 'name': 'Bottom drag coefficient', 'unit': '', 'filename': 'BottomDrag2d'}, 'bottom_drag_3d': {'shortname': 'Bottom drag coefficient', 'name': 'Bottom drag coefficient', 'unit': '', 'filename': 'BottomDrag3d'}, 'buoy_freq_3d': {'shortname': 'Buoyancy frequency squared', 'name': 'Buoyancy frequency squared', 'unit': 's-2', 'filename': 'BuoyFreq3d'}, 'coriolis_2d': {'shortname': 'Coriolis parameter', 'name': 'Coriolis parameter', 'unit': 's-1', 'filename': 'coriolis_2d'}, 'coriolis_3d': {'shortname': 'Coriolis parameter', 'name': 'Coriolis parameter', 'unit': 's-1', 'filename': 'coriolis_3d'}, 'density_3d': {'shortname': 'Density', 'name': 'Water density', 'unit': 'kg m-3', 'filename': 'Density3d'}, 'eddy_diff_3d': {'shortname': 'Eddy diffusivity', 'name': 'Eddy diffusivity', 'unit': 'm2 s-1', 'filename': 'EddyDiff3d'}, 'eddy_visc_3d': {'shortname': 'Eddy Viscosity', 'name': 'Eddy Viscosity', 'unit': 'm2 s-1', 'filename': 'EddyVisc3d'}, 'elev_2d': {'shortname': 'Elevation', 'name': 'Water elevation', 'unit': 'm', 'filename': 'Elevation2d'}, 'elev_3d': {'shortname': 'Elevation', 'name': 'Water elevation', 'unit': 'm', 'filename': 'Elevation3d'}, 'elev_cg_2d': {'shortname': 'Elevation', 'name': 'Water elevation CG', 'unit': 'm', 'filename': 'ElevationCG2d'}, 'elev_cg_3d': {'shortname': 'Elevation', 'name': 'Water elevation CG', 'unit': 'm', 'filename': 'ElevationCG3d'}, 'eps_3d': {'shortname': 'TKE dissipation rate', 'name': 'TKE dissipation rate', 'unit': 'm2 s-2', 'filename': 'TurbEps3d'}, 'h_elem_size_2d': {'shortname': 'Horizontal element size', 'name': 'Element size in horizontal direction', 'unit': 'm', 'filename': 'h_elem_size_2d'}, 'h_elem_size_3d': {'shortname': 'Horizontal element size', 'name': 'Element size in horizontal direction', 'unit': 'm', 'filename': 'h_elem_size_3d'}, 'hcc_metric_3d': {'shortname': 'HCC metric', 'name': 'HCC mesh quality', 'unit': '-', 'filename': 'HCCMetric3d'}, 'int_pg_3d': {'shortname': 'Int. Pressure gradient', 'name': 'Internal pressure gradient', 'unit': 'm s-2', 'filename': 'IntPG3d'}, 'len_3d': {'shortname': 'Turbulent length scale', 'name': 'Turbulent length scale', 'unit': 'm', 'filename': 'TurbLen3d'}, 'max_h_diff': {'shortname': 'Maximum horizontal diffusivity', 'name': 'Maximum stable horizontal diffusivity', 'unit': 'm2 s-1', 'filename': 'MaxHDiffusivity3d'}, 'parab_visc_3d': {'shortname': 'Parabolic Viscosity', 'name': 'Parabolic Viscosity', 'unit': 'm2 s-1', 'filename': 'ParabVisc3d'}, 'psi_3d': {'shortname': 'Turbulence psi variable', 'name': 'Turbulence psi variable', 'unit': '', 'filename': 'TurbPsi3d'}, 'salt_3d': {'shortname': 'Salinity', 'name': 'Water salinity', 'unit': 'psu', 'filename': 'Salinity3d'}, 'shear_freq_3d': {'shortname': 'Vertical shear frequency squared', 'name': 'Vertical shear frequency squared', 'unit': 's-2', 'filename': 'ShearFreq3d'}, 'smag_visc_3d': {'shortname': 'Smagorinsky viscosity', 'name': 'Smagorinsky viscosity', 'unit': 'm2 s-1', 'filename': 'SmagViscosity3d'}, 'split_residual_2d': {'shortname': 'Momentum residual', 'name': 'Momentum eq. residual for mode splitting', 'unit': 'm s-2', 'filename': 'SplitResidual2d'}, 'temp_3d': {'shortname': 'Temperature', 'name': 'Water temperature', 'unit': 'C', 'filename': 'Temperature3d'}, 'tke_3d': {'shortname': 'Turbulent Kinetic Energy', 'name': 'Turbulent Kinetic Energy', 'unit': 'm2 s-2', 'filename': 'TurbKEnergy3d'}, 'uv_2d': {'shortname': 'Depth averaged velocity', 'name': 'Depth averaged velocity', 'unit': 'm s-1', 'filename': 'Velocity2d'}, 'uv_3d': {'shortname': 'Horizontal velocity', 'name': 'Horizontal velocity', 'unit': 'm s-1', 'filename': 'Velocity3d'}, 'uv_bottom_2d': {'shortname': 'Bottom velocity', 'name': 'Bottom velocity', 'unit': 'm s-1', 'filename': 'BottomVelo2d'}, 'uv_bottom_3d': {'shortname': 'Bottom velocity', 'name': 'Bottom velocity', 'unit': 'm s-1', 'filename': 'BottomVelo3d'}, 'uv_dav_2d': {'shortname': 'Depth averaged velocity', 'name': 'Depth averaged velocity', 'unit': 'm s-1', 'filename': 'DAVelocity2d'}, 'uv_dav_3d': {'shortname': 'Depth averaged velocity', 'name': 'Depth averaged velocity', 'unit': 'm s-1', 'filename': 'DAVelocity3d'}, 'uv_mag_3d': {'shortname': 'Velocity magnitude', 'name': 'Magnitude of horizontal velocity', 'unit': 'm s-1', 'filename': 'VeloMag3d'}, 'uv_p1_3d': {'shortname': 'P1 Velocity', 'name': 'P1 projection of horizontal velocity', 'unit': 'm s-1', 'filename': 'VeloCG3d'}, 'v_elem_size_2d': {'shortname': 'Vertical element size', 'name': 'Element size in vertical direction', 'unit': 'm', 'filename': 'VElemSize2d'}, 'v_elem_size_3d': {'shortname': 'Vertical element size', 'name': 'Element size in vertical direction', 'unit': 'm', 'filename': 'VElemSize3d'}, 'w_3d': {'shortname': 'Vertical velocity', 'name': 'Vertical velocity', 'unit': 'm s-1', 'filename': 'VertVelo3d'}, 'w_mesh_3d': {'shortname': 'Mesh velocity', 'name': 'Mesh velocity', 'unit': 'm s-1', 'filename': 'MeshVelo3d'}, 'w_mesh_surf_2d': {'shortname': 'Surface mesh velocity', 'name': 'Surface mesh velocity', 'unit': 'm s-1', 'filename': 'SurfMeshVelo3d'}, 'w_mesh_surf_3d': {'shortname': 'Surface mesh velocity', 'name': 'Surface mesh velocity', 'unit': 'm s-1', 'filename': 'SurfMeshVelo3d'}, 'wind_stress_3d': {'shortname': 'Wind stress', 'name': 'Wind stress', 'unit': 'Pa', 'filename': 'wind_stress_3d'}, 'z_bottom_2d': {'shortname': 'Bottom cell z coordinates', 'name': 'Bottom cell z coordinates', 'unit': 'm', 'filename': 'ZBottom2d'}, 'z_coord_3d': {'shortname': 'Z coordinates', 'name': 'Mesh z coordinates', 'unit': 'm', 'filename': 'ZCoord3d'}, 'z_coord_ref_3d': {'shortname': 'Z coordinates', 'name': 'Static mesh z coordinates', 'unit': 'm', 'filename': 'ZCoordRef3d'}}

Dictionary that contains the meta data of each field.

Required meta data entries are:

• name: human readable description
• shortname: description used in visualization etc
• unit: SI unit of the field
• filename: filename for output files

The naming convention for field keys is snake_case: field_name_3d

thetis.firedrake module

thetis.implicitexplicit module

Implicit-explicit time integrators

class thetis.implicitexplicit.IMEXEuler(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_dirk={})

Bases: thetis.implicitexplicit.IMEXGeneric

Forward-Backward Euler

dirk_class

alias of BackwardEuler

erk_class

alias of ERKEuler

class thetis.implicitexplicit.IMEXGeneric(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_dirk={})

Bases: thetis.timeintegrator.TimeIntegrator

Generic implementation of Runge-Kutta Implicit-Explicit schemes

Derived classes must define the implicit dirk_class and explicit erk_class Runge-Kutta time integrator classes.

This method solves the linearized equations: All implicit terms are fed to the implicit solver, while all the other terms are fed to the explicit solver. In case of non-linear terms proper linearization must defined in the equation using the two solution functions (solution, solution_old)

advance(t, update_forcings=None)

Advances equations for one time step.

dirk_class

Implicit DIRK class

erk_class

Explicit Runge-Kutta class

get_final_solution()

Evaluates the final solution.

initialize(solution)

Assigns initial conditions to all required fields.

set_dt(dt)

Update time step

Parameters:dt (float) – time step

solve_stage(i_stage, t, update_forcings=None)

Solves i-th stage

update_solver()

Create solver objects

class thetis.implicitexplicit.IMEXLPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_dirk={})

Bases: thetis.implicitexplicit.IMEXGeneric

SSP-IMEX RK scheme (20) in Higureras et al. (2014)

CFL coefficient is 2.0

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

dirk_class

alias of DIRKLPUM2

erk_class

alias of ERKLPUM2

class thetis.implicitexplicit.IMEXLSPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_dirk={})

Bases: thetis.implicitexplicit.IMEXGeneric

SSP-IMEX RK scheme (17) in Higureras et al. (2014)

CFL coefficient is 2.0

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

dirk_class

alias of DIRKLSPUM2

erk_class

alias of ERKLSPUM2

class thetis.implicitexplicit.IMEXMidpoint(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_dirk={})

Bases: thetis.implicitexplicit.IMEXGeneric

Implicit-explicit midpoint scheme (1, 2, 2) from Ascher et al. (1997)

Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037

dirk_class

alias of ESDIRKMidpoint

erk_class

alias of ERKMidpoint

thetis.implicitexplicit.timed_region()

Log.Event(type cls, name, klass=None)

thetis.implicitexplicit.timed_stage()

Log.Stage(type cls, name)

thetis.interpolation module

Methods for interpolating data from structured data sets on Thetis fields.

Simple example of an atmospheric pressure interpolator:

def to_latlon(x, y, positive_lon=False):
    # Converts mesh (x,y) points to coordinates used in the atm data
    lon, lat = coordsys_spcs.spcs2lonlat(x, y)
    if positive_lon and lon < 0.0:
        lon += 360.
    return lat, lon


class WRFInterpolator(object):
    # Interpolates WRF atmospheric model data on 2D fields
    def __init__(self, function_space, atm_pressure_field, ncfile_pattern, init_date):
        self.atm_pressure_field = atm_pressure_field

        # object that interpolates forcing data from structured grid on the local mesh
        self.grid_interpolator = NetCDFLatLonInterpolator2d(function_space, to_latlon)
        # reader object that can read fields from netCDF files, applies spatial interpolation
        self.reader = NetCDFSpatialInterpolator(self.grid_interpolator, ['prmsl'])
        # object that can find previous/next time stamps in a collection of netCDF files
        self.timesearch_obj = NetCDFTimeSearch(ncfile_pattern, init_date, NetCDFTimeParser)
        # finally a linear intepolator class that performs linar interpolation in time
        self.interpolator = LinearTimeInterpolator(self.timesearch_obj, self.reader)

    def set_fields(self, time):
        # Evaluates forcing fields at the given time
        pressure = self.interpolator(time)
        self.atm_pressure_field.dat.data_with_halos[:] = pressure

Usage:

atm_pressure_2d = Function(solver_obj.function_spaces.P1_2d, name='atm pressure')
wrf_pattern = 'forcings/atm/wrf/wrf_air.2016_*_*.nc'
wrf_atm = WRFInterpolator(
    solver_obj.function_spaces.P1_2d,
    wind_stress_2d, atm_pressure_2d, wrf_pattern, init_date)
simulation_time = 3600.
wrf_atm.set_fields(simulation_time)

class thetis.interpolation.FileTreeReader

Bases: object

Abstract base class of file tree reader object

class thetis.interpolation.GridInterpolator(grid_xyz, target_xyz, fill_mode=None, fill_value=nan, normalize=False)

Bases: object

A reuseable griddata interpolator object.

Usage:

interpolator = GridInterpolator(source_xyz, target_xyz)
vals = interpolator(source_data)

Example:

x0 = np.linspace(0, 10, 10)
y0 = np.linspace(5, 10, 10)
X, Y = np.meshgrid(x, y)
x = X.ravel(); y = Y.ravel()
data = x + 25.*y
x_target = np.linspace(1, 10, 20)
y_target = np.linspace(5, 10, 20)
interpolator = GridInterpolator(np.vstack((x, y)).T, np.vstack((target_x, target_y)).T)
vals = interpolator(data)

Based on http://stackoverflow.com/questions/20915502/speedup-scipy-griddata-for-multiple-interpolations-between-two-irregular-grids

Parameters:

• grid_xyz – Array of source grid coordinates, shape (npoints, 2) or (npoints, 3)
• target_xyz – Array of target grid coordinates, shape (n, 2) or (n, 3)

class thetis.interpolation.LinearTimeInterpolator(timesearch_obj, reader)

Bases: object

Interpolates time series in time

User must provide timesearch_obj that finds time stamps from a file tree, and a reader that can read those time stamps into numpy arrays.

Previous/next data sets are cached in memory to avoid hitting disk every time.

Parameters:

• timesearch_obj – TimeSearch object
• reader – FileTreeReader object

class thetis.interpolation.NetCDFLatLonInterpolator2d(function_space, to_latlon)

Bases: thetis.interpolation.SpatialInterpolator2d

Interpolates netCDF data on a local 2D unstructured mesh

The intepolator is constructed for a single netCDF file that defines the source grid. Once the interpolator has been constructed, data can be read from any file that uses the same grid.

This routine returns the data in numpy arrays.

Usage:

Parameters:

• function_space – target Firedrake FunctionSpace
• to_latlon – Python function that converts local mesh coordinates to latitude and longitude: ‘lat, lon = to_latlon(x, y)’

interpolate(nc_filename, variable_list, itime)

Interpolates data from a netCDF file onto Firedrake function space.

Parameters:

• nc_filename (str) – netCDF file to read
• variable_list – list of netCDF variable names to read
• itime (int) – time index to read

Returns:

list of numpy.arrays corresponding to variable_list

class thetis.interpolation.NetCDFSpatialInterpolator(grid_interpolator, variable_list)

Bases: thetis.interpolation.FileTreeReader

Wrapper class that provides FileTreeReader API for grid interpolators

class thetis.interpolation.NetCDFTimeParser(filename, time_variable_name='time', allow_gaps=False)

Bases: thetis.interpolation.TimeParser

Describes the time stamps stored in a netCDF file.

Construct a new object by scraping data from the given netcdf file.

Parameters:

• filename (str) – name of the netCDF file to read
• time_variable_name (str) – name of the time variable in the netCDF file (default: ‘time’)
• allow_gaps (bool) – if False, an error is raised if time step is not constant.

find_time_stamp(t, previous=False)

get_end_time()

get_start_time()

scalars = {'days': 86400.0, 'seconds': 1.0}

class thetis.interpolation.NetCDFTimeSearch(file_pattern, init_date, netcdf_class, *args, **kwargs)

Bases: thetis.interpolation.TimeSearch

Finds a nearest time stamp in a collection of netCDF files.

find(simulation_time, previous=False)

Find file that contains the given simulation time

Parameters:

• simulation_time (float) – simulation time in seconds
• previous (bool) – if True finds previous existing time stamp instead of next (default False).

Returns:

(filename, time index, simulation time) of found data

simulation_time_to_datetime(t)

class thetis.interpolation.NetCDFTimeSeriesInterpolator(ncfile_pattern, variable_list, init_date, time_variable_name='time', scalars=None, allow_gaps=False)

Bases: object

Reads and interpolates scalar time series from a sequence of netCDF files.

Parameters:

• ncfile_pattern (str) – file search pattern, e.g. “mydir/foo_*.nc”
• variable_list – list if netCDF variable names to read
• init_date (datetime.datetime) – simulation start time
• scalars – (optional) list of scalars; scale output variables by

a factor.

class thetis.interpolation.NetCDFTimeSeriesReader(variable_list, time_variable_name='time')

Bases: thetis.interpolation.FileTreeReader

A simple netCDF reader that returns a time slice of the given variable.

This class does not interpolate the data in any way. Useful for interpolating time series.

class thetis.interpolation.SpatialInterpolator(function_space, to_latlon)

Bases: object

Abstract base class for spatial interpolators that read data from disk

Parameters:

• function_space – target Firedrake FunctionSpace
• to_latlon – Python function that converts local mesh coordinates to latitude and longitude: ‘lat, lon = to_latlon(x, y)’

interpolate(filename, variable_list, itime)

Interpolates data from the given file at given time step

class thetis.interpolation.SpatialInterpolator2d(function_space, to_latlon)

Bases: thetis.interpolation.SpatialInterpolator

Abstract spatial interpolator class that can interpolate onto a 2D Function

Parameters:

• function_space – target Firedrake FunctionSpace
• to_latlon – Python function that converts local mesh coordinates to latitude and longitude: ‘lat, lon = to_latlon(x, y)’

interpolate(filename, variable_list, time)

Calls the interpolator object

class thetis.interpolation.TimeParser

Bases: object

Abstract base class for time definition objects.

Defines the time span that a file (or data set) covers and provides a time index search routine.

find_time_stamp(t, previous=False)

Given time t, returns index of the next (previous) time stamp

raises IndexError if t is out of range, i.e. t > self.get_end_time() or t < self.get_start_time()

get_end_time()

Returns the last time stamp in the file/data set

get_start_time()

Returns the first time stamp in the file/data set

class thetis.interpolation.TimeSearch

Bases: object

Base class for searching nearest time steps in a file tree or database

find(time, previous=False)

Find a next (previous) time stamp from a given time

Parameters:

• time (float) – input time stamp
• previous (bool) – if True, look for last time stamp before requested time. Otherwise returns next time stamp.

Returns:

a (filename, time_index, time) tuple

thetis.limiter module

Slope limiters for discontinuous fields

class thetis.limiter.VertexBasedP1DGLimiter(p1dg_space, time_dependent_mesh=True)

Bases: firedrake.slope_limiter.vertex_based_limiter.VertexBasedLimiter

Vertex based limiter for P1DG tracer fields, see Kuzmin (2010)

Note

Currently only scalar fields are supported

Kuzmin (2010). A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. Journal of Computational and Applied Mathematics, 233(12):3077-3085. http://dx.doi.org/10.1016/j.cam.2009.05.028

Parameters:p1dg_space – P1DG function space

apply(field)

Applies the limiter on the given field (in place)

Parameters:field – Function to limit

compute_bounds(field)

Re-compute min/max values of all neighbouring centroids

Parameters:field – Function to limit

thetis.limiter.assert_function_space(fs, family, degree)

Checks the family and degree of function space.

Raises AssertionError if function space differs. If the function space lies on an extruded mesh, checks both spaces of the outer product.

Parameters:

• fs – function space
• family (string) – name of element family
• degree (int) – polynomial degree of the function space

thetis.limiter.timed_region()

Log.Event(type cls, name, klass=None)

thetis.limiter.timed_stage()

Log.Stage(type cls, name)

thetis.log module

Loggers for Thetis

Creates two logger instances, one for general model output and one for debug, warning, error etc. messages.

To print to the model output stream, use print_output().

Debug, warning etc. messages are issued with debug(), info(), warning(), error(), critical() methods.

thetis.momentum_eq module

3D momentum equation for hydrostatic Boussinesq flow.

The three dimensional momentum equation reads

(1)\[\frac{\partial \textbf{u}}{\partial t} + \nabla_h \cdot (\textbf{u} \textbf{u}) + \frac{\partial \left(w\textbf{u} \right)}{\partial z} + f\textbf{e}_z\wedge\textbf{u} + g\nabla_h \eta + g\nabla_h r = \nabla_h \cdot \left( \nu_h \nabla_h \textbf{u} \right) + \frac{\partial }{\partial z}\left( \nu \frac{\partial \textbf{u}}{\partial z}\right)\]

where \(\textbf{u}\) and \(w\) denote horizontal and vertical velocity, \(\nabla_h\) is the horizontal gradient, \(\wedge\) denotes the cross product, \(g\) is the gravitational acceleration, \(f\) is the Coriolis frequency, \(\textbf{e}_z\) is a vertical unit vector, and \(\nu_h, \nu\) stand for horizontal and vertical viscosity. Water density is given by \(\rho = \rho'(T, S, p) + \rho_0\), where \(\rho_0\) is a constant reference density. Above \(r\) denotes the baroclinic head

(2)\[r = \frac{1}{\rho_0} \int_{z}^\eta \rho' d\zeta.\]

The internal pressure gradient is computed as a separate diagnostic field:

(3)\[\mathbf{F}_{pg} = g\nabla_h r.\]

In the case of purely barotropic problems the \(r\) and \(\mathbf{F}_{pg}\) fields are omitted.

When using mode splitting we split the velocity field into a depth average and a deviation, \(\textbf{u} = \bar{\textbf{u}} + \textbf{u}'\). Following Higdon and de Szoeke (1997) we write an equation for the deviation \(\textbf{u}'\):

(4)\[\frac{\partial \textbf{u}'}{\partial t} = + \nabla_h \cdot (\textbf{u} \textbf{u}) + \frac{\partial \left(w\textbf{u} \right)}{\partial z} + f\textbf{e}_z\wedge\textbf{u}' + g\nabla_h r = \nabla_h \cdot \left( \nu_h \nabla_h \textbf{u} \right) + \frac{\partial }{\partial z}\left( \nu \frac{\partial \textbf{u}}{\partial z}\right)\]

In (4) the external pressure gradient \(g\nabla_h \eta\) vanishes and the Coriolis term only contains the deviation \(\textbf{u}'\). Advection and diffusion terms are not changed.

Higdon and de Szoeke (1997). Barotropic-Baroclinic Time Splitting for Ocean Circulation Modeling. Journal of Computational Physics, 135(1):30-53. http://dx.doi.org/10.1006/jcph.1997.5733

class thetis.momentum_eq.MomentumEquation(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.equation.Equation

Hydrostatic 3D momentum equation (4) for mode split models

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

class thetis.momentum_eq.MomentumTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.equation.Term

Generic term for momentum equation that provides commonly used members and mapping for boundary functions.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

class thetis.momentum_eq.HorizontalAdvectionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Horizontal advection term, \(\nabla_h \cdot (\textbf{u} \textbf{u})\)

The weak form reads

\[\int_\Omega \nabla_h \cdot (\textbf{u} \textbf{u}) \cdot \boldsymbol{\psi} dx = - \int_\Omega \nabla_h \boldsymbol{\psi} : (\textbf{u} \textbf{u}) dx + \int_{\mathcal{I}_h \cup \mathcal{I}_v} \textbf{u}^{\text{up}} \cdot \text{jump}(\boldsymbol{\psi} \textbf{n}_h) \cdot \text{avg}(\textbf{u}) dS\]

where the right hand side has been integrated by parts; \(:\) stand for the Frobenius inner product, \(\textbf{n}_h\) is the horizontal projection of the normal vector, \(\textbf{u}^{\text{up}}\) is the upwind value, and \(\text{jump}\) and \(\text{avg}\) denote the jump and average operators across the interface.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.VerticalAdvectionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Vertical advection term, \(\partial \left(w\textbf{u} \right)/(\partial z)\)

The weak form reads

\[\int_\Omega \frac{\partial \left(w\textbf{u} \right)}{\partial z} \cdot \boldsymbol{\psi} dx = - \int_\Omega \left( w \textbf{u} \right) \cdot \frac{\partial \boldsymbol{\psi}}{\partial z} dx + \int_{\mathcal{I}_{h}} \textbf{u}^{\text{up}} \cdot \text{jump}(\boldsymbol{\psi} n_z) \text{avg}(w) dS\]

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.HorizontalViscosityTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Horizontal viscosity term, \(- \nabla_h \cdot \left( \nu_h \nabla_h \textbf{u} \right)\)

Using the symmetric interior penalty method the weak form becomes

\[\begin{split}\int_\Omega \nabla_h \cdot \left( \nu_h \nabla_h \textbf{u} \right) \cdot \boldsymbol{\psi} dx =& -\int_\Omega \nu_h (\nabla_h \boldsymbol{\psi}) : (\nabla_h \textbf{u})^T dx \\ &+ \int_{\mathcal{I}_h \cup \mathcal{I}_v} \text{jump}(\boldsymbol{\psi} \textbf{n}_h) \cdot \text{avg}( \nu_h \nabla_h \textbf{u}) dS + \int_{\mathcal{I}_h \cup \mathcal{I}_v} \text{jump}(\textbf{u} \textbf{n}_h) \cdot \text{avg}( \nu_h \nabla_h \boldsymbol{\psi}) dS \\ &- \int_{\mathcal{I}_h \cup \mathcal{I}_v} \sigma \text{avg}(\nu_h) \text{jump}(\textbf{u} \textbf{n}_h) \cdot \text{jump}(\boldsymbol{\psi} \textbf{n}_h) dS\end{split}\]

where \(\sigma\) is a penalty parameter, see Epshteyn and Riviere (2007).

Epshteyn and Riviere (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843-872. http://dx.doi.org/10.1016/j.cam.2006.08.029

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.VerticalViscosityTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Vertical viscosity term, \(- \frac{\partial }{\partial z}\left( \nu \frac{\partial \textbf{u}}{\partial z}\right)\)

Using the symmetric interior penalty method the weak form becomes

\[\begin{split}\int_\Omega \frac{\partial }{\partial z}\left( \nu \frac{\partial \textbf{u}}{\partial z}\right) \cdot \boldsymbol{\psi} dx =& -\int_\Omega \nu \frac{\partial \boldsymbol{\psi}}{\partial z} \cdot \frac{\partial \textbf{u}}{\partial z} dx \\ &+ \int_{\mathcal{I}_h} \text{jump}(\boldsymbol{\psi} n_z) \cdot \text{avg}\left(\nu \frac{\partial \textbf{u}}{\partial z}\right) dS + \int_{\mathcal{I}_h} \text{jump}(\textbf{u} n_z) \cdot \text{avg}\left(\nu \frac{\partial \boldsymbol{\psi}}{\partial z}\right) dS \\ &- \int_{\mathcal{I}_h} \sigma \text{avg}(\nu) \text{jump}(\textbf{u} n_z) \cdot \text{jump}(\boldsymbol{\psi} n_z) dS\end{split}\]

where \(\sigma\) is a penalty parameter, see Epshteyn and Riviere (2007).

Epshteyn and Riviere (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843-872. http://dx.doi.org/10.1016/j.cam.2006.08.029

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.PressureGradientTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Internal pressure gradient term, \(g\nabla_h r\)

where \(r\) is the baroclinic head (2). Let \(s\) denote \(r/H\). We can then write

\[F_{IPG} = g\nabla_h((s -\bar{s}) H) + g\nabla_h\left(\frac{1}{H}\right) H^2\bar{s} + g s_{bot}\nabla_h h\]

where \(\bar{s},s_{bot}\) are the depth average and bottom value of \(s\).

If \(s\) belongs to a discontinuous function space, the first term is integrated by parts. Its weak form reads

\[\int_\Omega g\nabla_h((s -\bar{s}) H) \cdot \boldsymbol{\psi} dx = - \int_\Omega g (s -\bar{s}) H \nabla_h \cdot \boldsymbol{\psi} dx + \int_{\mathcal{I}_h \cup \mathcal{I}_v} g (s -\bar{s}) H \boldsymbol{\psi} \cdot \textbf{n}_h dx\]

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.CoriolisTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Coriolis term, \(f\textbf{e}_z\wedge \bar{\textbf{u}}\)

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.BottomFrictionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Quadratic bottom friction term, \(\tau_b = C_D \| \textbf{u}_b \| \textbf{u}_b\)

The weak formulation reads

\[\int_{\Gamma_{bot}} \tau_b \cdot \boldsymbol{\psi} dx = \int_{\Gamma_{bot}} C_D \| \textbf{u}_b \| \textbf{u}_b \cdot \boldsymbol{\psi} dx\]

where \(\textbf{u}_b\) is reconstructed velocity in the middle of the bottom element:

\[\textbf{u}_b = \textbf{u}\Big|_{\Gamma_{bot}} + \frac{\partial \textbf{u}}{\partial z}\Big|_{\Gamma_{bot}} h_b,\]

\(h_b\) being half of the element height. For implicit solvers we linearize the stress as \(\tau_b = C_D \| \textbf{u}_b^{n} \| \textbf{u}_b^{n+1}\)

The drag is computed from the law-of-the wall

\[C_D = \left( \frac{\kappa}{\ln (h_b + z_0)/z_0} \right)^2\]

where \(z_0\) is the bottom roughness length, read from z0_friction field. The user can override the \(C_D\) value by providing quadratic_drag_coefficient field.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.LinearDragTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Linear drag term, \(\tau_b = D \textbf{u}_b\)

where \(D\) is the drag coefficient, read from linear_drag_coefficient field.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.SourceTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_nonlinear_equations=True, use_lax_friedrichs=True, use_bottom_friction=False)

Bases: thetis.momentum_eq.MomentumTerm

Generic momentum source term

The weak form reads

\[F_s = \int_\Omega \sigma \cdot \boldsymbol{\psi} dx\]

where \(\sigma\) is a user defined vector valued Function.

This term also implements the wind stress, \(-\tau_w/(H \rho_0)\). \(\tau_w\) is a user-defined wind stress Function wind_stress. The weak form is

\[F_w = \int_{\Gamma_s} \frac{1}{\rho_0} \tau_w \cdot \boldsymbol{\psi} dx\]

Wind stress is only included if vertical viscosity is provided.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_nonlinear_equations (bool) – If False defines the linear shallow water equations
• use_bottom_friction (bool) – If True includes bottom friction term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.momentum_eq.InternalPressureGradientCalculator(fields, options, bnd_functions, solver_parameters=None)

Bases: thetis.momentum_eq.MomentumTerm

Computes the internal pressure gradient term, \(g\nabla_h r\)

where \(r\) is the baroclinic head (2).

If \(r\) belongs to a discontinuous function space, the term is integrated by parts:

\[\int_\Omega g \nabla_h r \cdot \boldsymbol{\psi} dx = - \int_\Omega g r \nabla_h \cdot \boldsymbol{\psi} dx + \int_{\mathcal{I}_h \cup \mathcal{I}_v} g \text{avg}(r) \text{jump}(\boldsymbol{\psi} \cdot \textbf{n}_h) dx\]

Note

Due to the Term sign convention this term is assembled on the right-hand-side.

Parameters:

• solver – class`FlowSolver object
• solver_parameters (dict) – PETSc solver options

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

solve()

Computes internal pressure gradient and stores it in int_pg_3d field

thetis.options module

Thetis options for the 2D and 3D model

All options are type-checked and they are stored in traitlets Configurable objects.

class thetis.options.CommonModelOptions(*args, **kwargs)

Bases: thetis.configuration.FrozenConfigurable

Options that are common for both 2d and 3d models

atmospheric_pressure

cfl_2d

cfl_3d

check_volume_conservation_2d

A boolean (True, False) trait.

coriolis_frequency

element_family

An enum whose value must be in a given sequence.

export_diagnostics

A boolean (True, False) trait.

fields_to_export

An instance of a Python list.

fields_to_export_hdf5

An instance of a Python list.

horizontal_velocity_scale

horizontal_viscosity

horizontal_viscosity_scale

lax_friedrichs_velocity_scaling_factor

linear_drag_coefficient

log_output

A boolean (True, False) trait.

manning_drag_coefficient

momentum_source_2d

name = 'Model options'

no_exports

A boolean (True, False) trait.

output_directory

A trait for unicode strings.

polynomial_degree

quadratic_drag_coefficient

simulation_end_time

simulation_export_time

timestep

use_grad_depth_viscosity_term

A boolean (True, False) trait.

use_grad_div_viscosity_term

A boolean (True, False) trait.

use_lax_friedrichs_velocity

A boolean (True, False) trait.

use_nonlinear_equations

A boolean (True, False) trait.

verbose

An int trait.

volume_source_2d

wind_stress

class thetis.options.CrankNicolsonTimestepperOptions2d(*args, **kwargs)

Bases: thetis.options.SemiImplicitTimestepperOptions2d

Options for 2d Crank-Nicolson time integrator

implicitness_theta

use_semi_implicit_linearization

A boolean (True, False) trait.

class thetis.options.EquationOfStateOptions(*args, **kwargs)

Bases: thetis.configuration.FrozenHasTraits

Base class of equation of state options

name = 'Equation of State'

class thetis.options.ExplicitTimestepperOptions(*args, **kwargs)

Bases: thetis.options.TimeStepperOptions

Options for explicit time integrator

use_automatic_timestep

A boolean (True, False) trait.

class thetis.options.ExplicitTimestepperOptions2d(*args, **kwargs)

Bases: thetis.options.ExplicitTimestepperOptions

Options for 2d explicit time integrator

solver_parameters

PETSc solver options dictionary

class thetis.options.ExplicitTimestepperOptions3d(*args, **kwargs)

Bases: thetis.options.ExplicitTimestepperOptions

Base class for all 3d time stepper options

solver_parameters_2d_swe

PETSc solver options dictionary

solver_parameters_momentum_explicit

PETSc solver options dictionary

solver_parameters_momentum_implicit

PETSc solver options dictionary

solver_parameters_tracer_explicit

PETSc solver options dictionary

solver_parameters_tracer_implicit

PETSc solver options dictionary

class thetis.options.GLSModelOptions(*args, **kwargs)

Bases: thetis.options.TurbulenceModelOptions

Options for Generic Length Scale turbulence model

apply_defaults(closure_name)

Applies default parameters for given closure name

Parameters:closure_name (string) – name of the turbulence closure model

Sets default values for parameters p, m, n, schmidt_nb_tke, schmidt_nb_psi, c1, c2, c3_plus, c3_minus, f_wall, k_min, psi_min

c1

A float trait.

c2

A float trait.

c3_minus

A float trait.

c3_plus

A float trait.

closure_name

An enum whose value must be in a given sequence.

cmu0

compute_c3_minus

A boolean (True, False) trait.

compute_cmu0

A boolean (True, False) trait.

compute_kappa

A boolean (True, False) trait.

compute_len_min

A boolean (True, False) trait.

compute_psi_min

A boolean (True, False) trait.

diff_min

eps_min

f_wall

A float trait.

galperin_lim

k_min

kappa

A float trait.

len_min

limit_eps

A boolean (True, False) trait.

limit_len

A boolean (True, False) trait.

limit_len_min

A boolean (True, False) trait.

limit_psi

A boolean (True, False) trait.

m

A float trait.

n

A float trait.

name = 'Generic Lenght Scale turbulence closure model'

p

A float trait.

psi_min

ri_st

A float trait.

schmidt_nb_psi

schmidt_nb_tke

stability_function_name

An enum whose value must be in a given sequence.

visc_min

class thetis.options.LinearEquationOfStateOptions(*args, **kwargs)

Bases: thetis.options.EquationOfStateOptions

Linear equation of state options

alpha

A float trait.

beta

A float trait.

name = 'Linear Equation of State'

rho_ref

s_ref

th_ref

A float trait.

class thetis.options.ModelOptions2d(*args, **kwargs)

Bases: thetis.options.CommonModelOptions

Options for 2D depth-averaged shallow water model

name = 'Depth-averaged 2D model'

timestepper_options

A trait whose value must be an instance of a specified class.

The value can also be an instance of a subclass of the specified class.

Subclasses can declare default classes by overriding the klass attribute

timestepper_type

A enum whose value must be in a given sequence.

This enum controls a slaved option, with default values provided here.

Parameters:

• values – iterable of (value, HasTraits class) pairs. The HasTraits class will be called (with no arguments) to create default values if necessary.
• paired_name – trait name this enum is paired with.
• default_value – default value.

use_wetting_and_drying

A boolean (True, False) trait.

wetting_and_drying_alpha

class thetis.options.ModelOptions3d(*args, **kwargs)

Bases: thetis.options.CommonModelOptions

Options for 3D hydrostatic model

check_salinity_conservation

A boolean (True, False) trait.

check_salinity_overshoot

A boolean (True, False) trait.

check_temperature_conservation

A boolean (True, False) trait.

check_temperature_overshoot

A boolean (True, False) trait.

check_volume_conservation_3d

A boolean (True, False) trait.

constant_salinity

constant_temperature

equation_of_state_options

A trait whose value must be an instance of a specified class.

The value can also be an instance of a subclass of the specified class.

Subclasses can declare default classes by overriding the klass attribute

equation_of_state_type

A enum whose value must be in a given sequence.

This enum controls a slaved option, with default values provided here.

Parameters:

• values – iterable of (value, HasTraits class) pairs. The HasTraits class will be called (with no arguments) to create default values if necessary.
• paired_name – trait name this enum is paired with.
• default_value – default value.

horizontal_diffusivity

lax_friedrichs_tracer_scaling_factor

momentum_source_3d

name = '3D hydrostatic model'

salinity_source_3d

smagorinsky_coefficient

solve_salinity

A boolean (True, False) trait.

solve_temperature

A boolean (True, False) trait.

temperature_source_3d

timestep_2d

timestepper_options

A trait whose value must be an instance of a specified class.

The value can also be an instance of a subclass of the specified class.

Subclasses can declare default classes by overriding the klass attribute

timestepper_type

A enum whose value must be in a given sequence.

This enum controls a slaved option, with default values provided here.

Parameters:

• values – iterable of (value, HasTraits class) pairs. The HasTraits class will be called (with no arguments) to create default values if necessary.
• paired_name – trait name this enum is paired with.
• default_value – default value.

turbulence_model_options

A trait whose value must be an instance of a specified class.

The value can also be an instance of a subclass of the specified class.

Subclasses can declare default classes by overriding the klass attribute

turbulence_model_type

A enum whose value must be in a given sequence.

This enum controls a slaved option, with default values provided here.

Parameters:

• values – iterable of (value, HasTraits class) pairs. The HasTraits class will be called (with no arguments) to create default values if necessary.
• paired_name – trait name this enum is paired with.
• default_value – default value.

use_ale_moving_mesh

A boolean (True, False) trait.

use_baroclinic_formulation

A boolean (True, False) trait.

use_bottom_friction

A boolean (True, False) trait.

use_implicit_vertical_diffusion

A boolean (True, False) trait.

use_lax_friedrichs_tracer

A boolean (True, False) trait.

use_limiter_for_tracers

A boolean (True, False) trait.

use_limiter_for_velocity

A boolean (True, False) trait.

use_parabolic_viscosity

A boolean (True, False) trait.

use_quadratic_density

A boolean (True, False) trait.

use_quadratic_pressure

A boolean (True, False) trait.

use_smagorinsky_viscosity

A boolean (True, False) trait.

use_smooth_eddy_viscosity

A boolean (True, False) trait.

use_turbulence

A boolean (True, False) trait.

use_turbulence_advection

A boolean (True, False) trait.

vertical_diffusivity

vertical_velocity_scale

vertical_viscosity

class thetis.options.PacanowskiPhilanderModelOptions(*args, **kwargs)

Bases: thetis.options.TurbulenceModelOptions

Options for Pacanowski-Philander turbulence model

alpha

exponent

max_viscosity

name = 'Pacanowski-Philander turbulence closure model'

class thetis.options.PressureProjectionTimestepperOptions2d(*args, **kwargs)

Bases: thetis.options.TimeStepperOptions

Options for 2d pressure-projection time integrator

implicitness_theta

picard_iterations

solver_parameters_momentum

PETSc solver options dictionary

solver_parameters_pressure

PETSc solver options dictionary

use_semi_implicit_linearization

A boolean (True, False) trait.

class thetis.options.SemiImplicitTimestepperOptions2d(*args, **kwargs)

Bases: thetis.options.TimeStepperOptions

Options for 2d explicit time integrator

solver_parameters

PETSc solver options dictionary

class thetis.options.SemiImplicitTimestepperOptions3d(*args, **kwargs)

Bases: thetis.options.ExplicitTimestepperOptions3d

Class for all 3d time steppers that have a configurable semi-implicit 2D solver

implicitness_theta_2d

class thetis.options.SteadyStateTimestepperOptions2d(*args, **kwargs)

Bases: thetis.options.TimeStepperOptions

Options for 2d steady state solver

solver_parameters

PETSc solver options dictionary

class thetis.options.TimeStepperOptions(*args, **kwargs)

Bases: thetis.configuration.FrozenHasTraits

Base class for all time stepper options

name = 'Time stepper'

class thetis.options.TurbulenceModelOptions(*args, **kwargs)

Bases: thetis.configuration.FrozenHasTraits

Abstract base class for all turbulence model options

name = 'Turbulence closure model'

thetis.physical_constants module

Default values for physical constants and parameters

thetis.rungekutta module

Implements Runge-Kutta time integration methods.

The abstract class AbstractRKScheme defines the Runge-Kutta coefficients, and can be used to implement generic time integrators.

class thetis.rungekutta.AbstractRKScheme

Bases: object

Abstract class for defining Runge-Kutta schemes.

Derived classes must define the Butcher tableau (arrays a, b, c) and the CFL number (cfl_coeff).

Currently only explicit or diagonally implicit schemes are supported.

a

Runge-Kutta matrix \(a_{i,j}\) of the Butcher tableau

b

weights \(b_{i}\) of the Butcher tableau

c

nodes \(c_{i}\) of the Butcher tableau

cfl_coeff

CFL number of the scheme

Value 1.0 corresponds to Forward Euler time step.

class thetis.rungekutta.BackwardEuler(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.BackwardEulerAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.BackwardEulerAbstract

Bases: thetis.rungekutta.AbstractRKScheme

Backward Euler method

a = [[1.0]]

b = [1.0]

c = [1.0]

cfl_coeff = inf

class thetis.rungekutta.CrankNicolsonAbstract

Bases: thetis.rungekutta.AbstractRKScheme

Crack-Nicolson scheme

a = [[0.0, 0.0], [0.5, 0.5]]

b = [0.5, 0.5]

c = [0.0, 1.0]

cfl_coeff = inf

class thetis.rungekutta.CrankNicolsonRK(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.CrankNicolsonAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRK22(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRK22Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRK22Abstract

Bases: thetis.rungekutta.AbstractRKScheme

2-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method

This method has the Butcher tableau

\[\begin{split}\begin{array}{c|cc} \gamma & \gamma & 0 \\ 1 & 1-\gamma & \gamma \\ \hline & 1/2 & 1/2 \end{array}\end{split}\]

with \(\gamma = (2 + \sqrt{2})/2\).

From DIRK(2,3,2) IMEX scheme in Ascher et al. (1997)

Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037

a = [[1.7071067811865475, 0], [-0.70710678118654746, 1.7071067811865475]]

b = [-0.70710678118654746, 1.7071067811865475]

c = [1.7071067811865475, 1]

cfl_coeff = inf

gamma = 1.7071067811865475

class thetis.rungekutta.DIRK23(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRK23Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRK23Abstract

Bases: thetis.rungekutta.AbstractRKScheme

2-stage, 3rd order Diagonally Implicit Runge Kutta method

This method has the Butcher tableau

\[\begin{split}\begin{array}{c|cc} \gamma & \gamma & 0 \\ 1-\gamma & 1-2\gamma & \gamma \\ \hline & 1/2 & 1/2 \end{array}\end{split}\]

with \(\gamma = (3 + \sqrt{3})/6\).

From DIRK(2,3,3) IMEX scheme in Ascher et al. (1997)

Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037

a = [[0.78867513459481275, 0], [-0.57735026918962551, 0.78867513459481275]]

b = [0.5, 0.5]

c = [0.78867513459481275, 0.21132486540518725]

cfl_coeff = inf

gamma = 0.78867513459481275

class thetis.rungekutta.DIRK33(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRK33Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRK33Abstract

Bases: thetis.rungekutta.AbstractRKScheme

3-stage, 3rd order, L-stable Diagonally Implicit Runge Kutta method

From DIRK(3,4,3) IMEX scheme in Ascher et al. (1997)

Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037

a = [[0.4358665215, 0, 0], [0.28206673925000003, 0.4358665215, 0], [1.208496649153235, -0.6443631706532353, 0.4358665215]]

b = [1.208496649153235, -0.6443631706532353, 0.4358665215]

b1 = 1.208496649153235

b2 = -0.6443631706532353

c = [0.4358665215, 0.71793326075, 1]

cfl_coeff = inf

gamma = 0.4358665215

class thetis.rungekutta.DIRK43(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRK43Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRK43Abstract

Bases: thetis.rungekutta.AbstractRKScheme

4-stage, 3rd order, L-stable Diagonally Implicit Runge Kutta method

From DIRK(4,4,3) IMEX scheme in Ascher et al. (1997)

Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037

a = [[0.5, 0, 0, 0], [0.16666666666666666, 0.5, 0, 0], [-0.5, 0.5, 0.5, 0], [1.5, -1.5, 0.5, 0.5]]

b = [1.5, -1.5, 0.5, 0.5]

c = [0.5, 0.6666666666666666, 0.5, 1.0]

cfl_coeff = inf

class thetis.rungekutta.DIRKGeneric(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.RungeKuttaTimeIntegrator

Generic implementation of Diagonally Implicit Runge Kutta schemes.

All derived classes must define the Butcher tableau coefficients a, b, c.

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

get_final_solution()

Assign final solution to self.solution

initialize(init_cond)

Assigns initial conditions to all required fields.

solve_stage(i_stage, t, update_forcings=None)

Solve i-th stage and assign solution to self.solution.

solve_tendency(i_stage, t, update_forcings=None)

Evaluates the tendency of i-th stage.

update_solution(i_stage)

Updates solution to i_stage sub-stage.

Tendencies must have been evaluated first.

update_solver()

Create solver objects

class thetis.rungekutta.DIRKLPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRKLPUM2Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRKLPUM2Abstract

Bases: thetis.rungekutta.AbstractRKScheme

DIRKLPUM2, 3-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method

From IMEX RK scheme (20) in Higureras et al. (2014).

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

a = [[0.18181818181818182, 0, 0], [0.2662337662337662, 0.18181818181818182, 0], [0.3412042502951594, 0.34710743801652894, 0.18181818181818182]]

b = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]

c = [0.18181818181818182, 0.44805194805194803, 0.8701298701298701]

cfl_coeff = 4.34

class thetis.rungekutta.DIRKLSPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.DIRKLSPUM2Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.DIRKLSPUM2Abstract

Bases: thetis.rungekutta.AbstractRKScheme

DIRKLSPUM2, 3-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method

From IMEX RK scheme (17) in Higureras et al. (2014).

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

a = [[0.18181818181818182, 0, 0], [0.44372294372294374, 0.18181818181818182, 0], [0.44004329004329007, 0.19090909090909092, 0.18181818181818182]]

b = [0.43636363636363634, 0.2, 0.36363636363636365]

c = [0.18181818181818182, 0.6255411255411255, 0.8127705627705628]

cfl_coeff = 4.34

class thetis.rungekutta.ERKEuler(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.ERKGeneric, thetis.rungekutta.ForwardEulerAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ERKGeneric(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.RungeKuttaTimeIntegrator

Generic explicit Runge-Kutta time integrator.

Implements the Butcher form. All terms in the equation are treated explicitly.

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

get_final_solution(additive=False)

Assign final solution to self.solution

If additive=False, will overwrite solution function, otherwise will add to it.

initialize(solution)

Assigns initial conditions to all required fields.

solve_stage(i_stage, t, update_forcings=None)

Solve i-th stage and assign solution to self.solution.

solve_tendency(i_stage, t, update_forcings=None)

Evaluates the tendency of i-th stage

update_solution(i_stage, additive=False)

Computes the solution of the i-th stage

Tendencies must have been evaluated first.

If additive=False, will overwrite solution function, otherwise will add to it.

update_solver()

class thetis.rungekutta.ERKGenericShuOsher(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.timeintegrator.TimeIntegrator

Generic explicit Runge-Kutta time integrator.

Implements the Shu-Osher form.

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

advance(t, update_forcings=None)

Advances equations for one time step.

initialize(solution)

solve_stage(i_stage, t, update_forcings=None)

Solve i-th stage and assign solution to self.solution.

update_solver()

class thetis.rungekutta.ERKLPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.ERKGeneric, thetis.rungekutta.ERKLPUM2Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ERKLPUM2Abstract

Bases: thetis.rungekutta.AbstractRKScheme

ERKLPUM2, 3-stage, 2nd order Explicit Runge Kutta method

From IMEX RK scheme (20) in Higureras et al. (2014).

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

a = [[0, 0, 0], [0.5, 0, 0], [0.5, 0.5, 0]]

b = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]

c = [0, 0.5, 1.0]

cfl_coeff = 2.0

class thetis.rungekutta.ERKLSPUM2(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.ERKGeneric, thetis.rungekutta.ERKLSPUM2Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ERKLSPUM2Abstract

Bases: thetis.rungekutta.AbstractRKScheme

ERKLSPUM2, 3-stage, 2nd order Explicit Runge Kutta method

From IMEX RK scheme (17) in Higureras et al. (2014).

Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011

a = [[0, 0, 0], [0.8333333333333334, 0, 0], [0.4583333333333333, 0.4583333333333333, 0]]

b = [0.43636363636363634, 0.2, 0.36363636363636365]

c = [0, 0.8333333333333334, 0.9166666666666666]

cfl_coeff = 1.2

class thetis.rungekutta.ERKMidpoint(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.ERKGeneric, thetis.rungekutta.ERKMidpointAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ERKMidpointAbstract

Bases: thetis.rungekutta.AbstractRKScheme

a = [[0.0, 0.0], [0.5, 0.0]]

b = [0.0, 1.0]

c = [0.0, 0.5]

cfl_coeff = 1.0

class thetis.rungekutta.ESDIRKMidpoint(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.ESDIRKMidpointAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ESDIRKMidpointAbstract

Bases: thetis.rungekutta.AbstractRKScheme

a = [[0.0, 0.0], [0.0, 0.5]]

b = [0.0, 1.0]

c = [0.0, 0.5]

cfl_coeff = 1.0

class thetis.rungekutta.ESDIRKTrapezoid(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.ESDIRKTrapezoidAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ESDIRKTrapezoidAbstract

Bases: thetis.rungekutta.AbstractRKScheme

a = [[0.0, 0.0], [0.5, 0.5]]

b = [0.5, 0.5]

c = [0.0, 1.0]

cfl_coeff = inf

class thetis.rungekutta.ForwardEulerAbstract

Bases: thetis.rungekutta.AbstractRKScheme

Forward Euler method

a = [[0]]

b = [1.0]

c = [0]

cfl_coeff = 1.0

class thetis.rungekutta.ImplicitMidpoint(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.DIRKGeneric, thetis.rungekutta.ImplicitMidpointAbstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.ImplicitMidpointAbstract

Bases: thetis.rungekutta.AbstractRKScheme

Implicit midpoint method, second order.

This method has the Butcher tableau

\[\begin{split}\begin{array}{c|c} 0.5 & 0.5 \\ \hline & 1.0 \end{array}\end{split}\]

a = [[0.5]]

b = [1.0]

c = [0.5]

cfl_coeff = inf

class thetis.rungekutta.RungeKuttaTimeIntegrator(equation, solution, fields, dt, solver_parameters={})

Bases: thetis.timeintegrator.TimeIntegrator

Abstract base class for all Runge-Kutta time integrators

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• solver_parameters (dict) – PETSc solver options

advance(t, update_forcings=None)

Advances equations for one time step.

get_final_solution(additive=False)

Evaluates the final solution

solve_stage(i_stage, t, update_forcings=None)

Solves a single stage of step from t to t+dt. All functions that the equation depends on must be at right state corresponding to each sub-step.

class thetis.rungekutta.SSPRK33(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.rungekutta.ERKGenericShuOsher, thetis.rungekutta.SSPRK33Abstract

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

class thetis.rungekutta.SSPRK33Abstract

Bases: thetis.rungekutta.AbstractRKScheme

3rd order Strong Stability Preserving Runge-Kutta scheme, SSP(3,3).

This scheme has Butcher tableau

\[\begin{split}\begin{array}{c|ccc} 0 & \\ 1 & 1 \\ 1/2 & 1/4 & 1/4 & \\ \hline & 1/6 & 1/6 & 2/3 \end{array}\end{split}\]

CFL coefficient is 1.0

a = [[0, 0, 0], [1.0, 0, 0], [0.25, 0.25, 0]]

b = [0.16666666666666666, 0.16666666666666666, 0.6666666666666666]

c = [0, 1.0, 0.5]

cfl_coeff = 1.0

thetis.rungekutta.butcher_to_shuosher_form(a, b)

Converts Butcher tableau to Shu-Osher form.

The Shu-Osher form of a s-stage scheme is defined by two s+1 by s+1 arrays \(\alpha\) and \(\beta\):

\[\begin{split}u^{0} &= u^n \\ u^{(i)} &= \sum_{j=0}^s \alpha_{i,j} u^{(j)} + \sum_{j=0}^s \beta_{i,j} F(u^{(j)}) \\ u^{n+1} &= u^{(s)}\end{split}\]

The Shu-Osher form is not unique. Here we construct the form where beta values are the diagonal entries (for DIRK schemes) or sub-diagonal entries (for explicit schemes) of the concatenated Butcher tableau [\(a\); \(b\)].

For more information see Ketchelson et al. (2009) http://dx.doi.org/10.1016/j.apnum.2008.03.034

thetis.rungekutta.timed_region()

Log.Event(type cls, name, klass=None)

thetis.rungekutta.timed_stage()

Log.Stage(type cls, name)

thetis.shallowwater_eq module

Depth averaged shallow water equations

Equations

The state variables are water elevation, \(\eta\), and depth averaged velocity \(\bar{\textbf{u}}\).

Denoting the total water depth by \(H=\eta + h\), the non-conservative form of the free surface equation is

(5)\[\frac{\partial \eta}{\partial t} + \nabla \cdot (H \bar{\textbf{u}}) = 0\]

The non-conservative momentum equation reads

(6)\[\frac{\partial \bar{\textbf{u}}}{\partial t} + \bar{\textbf{u}} \cdot \nabla\bar{\textbf{u}} + f\textbf{e}_z\wedge \bar{\textbf{u}} + g \nabla \eta + \nabla \left(\frac{p_a}{\rho_0} \right) + g \frac{1}{H}\int_{-h}^\eta \nabla r dz = \nabla \cdot \big( \nu_h ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T )\big) + \frac{\nu_h \nabla(H)}{H} \cdot ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T ),\]

where \(g\) is the gravitational acceleration, \(f\) is the Coriolis frequency, \(\wedge\) is the cross product, \(\textbf{e}_z\) is a vertical unit vector, \(p_a\) is the atmospheric pressure at the free surface, and \(\nu_h\) is viscosity. Water density is given by \(\rho = \rho'(T, S, p) + \rho_0\), where \(\rho_0\) is a constant reference density.

Above \(r\) denotes the baroclinic head

\[r = \frac{1}{\rho_0} \int_{z}^\eta \rho' d\zeta.\]

In the case of purely barotropic problems the \(r\) and the internal pressure gradient are omitted.

If the option ModelOptions.use_nonlinear_equations is False, we solve the linear shallow water equations (i.e. wave equation):

(7)\[\frac{\partial \eta}{\partial t} + \nabla \cdot (h \bar{\textbf{u}}) = 0\]

(8)\[\frac{\partial \bar{\textbf{u}}}{\partial t} + f\textbf{e}_z\wedge \bar{\textbf{u}} + g \nabla \eta = \nabla \cdot \big( \nu_h ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T )\big) + \frac{\nu_h \nabla(H)}{H} \cdot ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T ).\]

In case of a 3D problem with mode splitting, we use a simplified 2D system that contains nothing but the rotational external gravity waves:

(9)\[\frac{\partial \eta}{\partial t} + \nabla \cdot (H \bar{\textbf{u}}) = 0\]

(10)\[\frac{\partial \bar{\textbf{u}}}{\partial t} + f\textbf{e}_z\wedge \bar{\textbf{u}} + g \nabla \eta = \textbf{G},\]

where \(\textbf{G}\) is a source term used to couple the 2D and 3D momentum equations.

Boundary Conditions

All boundary conditions are imposed weakly by providing external values for \(\eta\) and \(\bar{\textbf{u}}\).

Boundary conditions are set with a dictionary that defines all prescribed variables at each open boundary. For example, to assign elevation and volume flux on boundary 1 we set

swe_bnd_funcs = {}
swe_bnd_funcs[1] = {'elev':myfunc1, 'flux':myfunc2}

where myfunc1 and myfunc2 are Constant or Function objects.

The user can provide \(\eta\) and/or \(\bar{\textbf{u}}\) values. Supported combinations are:

• unspecified : impermeable (land) boundary, implies symmetric \(\eta\) condition and zero normal velocity
• 'elev': elevation only, symmetric velocity (usually unstable)
• 'uv': 2d velocity vector \(\bar{\textbf{u}}=(u, v)\) (in mesh coordinates), symmetric elevation
• 'un': normal velocity (scalar, positive out of domain), symmetric elevation
• 'flux': normal volume flux (scalar, positive out of domain), symmetric elevation
• 'elev' and 'uv': water elevation and 2d velocity vector
• 'elev' and 'un': water elevation and normal velocity
• 'elev' and 'flux': water elevation and normal flux

The boundary conditions are assigned to the FlowSolver2d or FlowSolver objects:

solver_obj = solver2d.FlowSolver2d(...)
...
solver_obj.bnd_functions['shallow_water'] = swe_bnd_funcs

Internally the boundary conditions passed to the Term.residual() method of each term:

adv_term = shallowwater_eq.HorizontalAdvectionTerm(...)
adv_form = adv_term.residual(..., bnd_conditions=swe_bnd_funcs)

Wetting and drying

If the option ModelOptions.use_wetting_and_drying is True, then wetting and drying is included through the formulation of Karna et al. (2011).

The method introduces a modified bathymetry \(\tilde{h} = h + f(H)\), which ensures positive total water depth, with \(f(H)\) defined by

\[f(H) = \frac{1}{2}(\sqrt{H^2 + \alpha^2} - H),\]

introducing a wetting-drying parameter \(\alpha\), with dimensions of length. This results in a modified total water depth \(\tilde{H}=H+f(H)\).

The value for \(\alpha\) is specified by the user through the option ModelOptions.wetting_and_drying_alpha, in units of meters. The default value for ModelOptions.wetting_and_drying_alpha is 0.5, but the appropriate value is problem specific and should be set by the user.

An approximate method for selecting a suitable value for \(\alpha\) is suggested by Karna et al. (2011). Defining \(L_x\) as the horizontal length scale of the mesh elements at the wet-dry front, it can be reasoned that \(\alpha \approx |L_x \nabla h|\) yields a suitable choice. Smaller \(\alpha\) leads to a more accurate solution to the shallow water equations in wet regions, but if \(\alpha\) is too small the simulation will become unstable.

When wetting and drying is turned on, two things occur:

1. All instances of the height, \(H\), are replaced by \(\tilde{H}\) (as defined above);
2. An additional displacement term \(\frac{\partial \tilde{h}}{\partial t}\) is added to the bathymetry in the free surface equation.

The free surface and momentum equations then become:

(11)\[\frac{\partial \eta}{\partial t} + \frac{\partial \tilde{h}}{\partial t} + \nabla \cdot (\tilde{H} \bar{\textbf{u}}) = 0,\]

(12)\[\frac{\partial \bar{\textbf{u}}}{\partial t} + \bar{\textbf{u}} \cdot \nabla\bar{\textbf{u}} + f\textbf{e}_z\wedge \bar{\textbf{u}} + g \nabla \eta + g \frac{1}{\tilde{H}}\int_{-h}^\eta \nabla r dz = \nabla \cdot \big( \nu_h ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T )\big) + \frac{\nu_h \nabla(\tilde{H})}{\tilde{H}} \cdot ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T ).\]

class thetis.shallowwater_eq.BaseShallowWaterEquation(function_space, bathymetry, options)

Bases: thetis.equation.Equation

Abstract base class for ShallowWaterEquations, ShallowWaterMomentumEquation and FreeSurfaceEquation.

Provides common functionality to compute time steps and add either momentum or continuity terms.

add_continuity_terms(*args)

add_momentum_terms(*args)

residual_uv_eta(label, uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions)

class thetis.shallowwater_eq.ShallowWaterEquations(function_space, bathymetry, options)

Bases: thetis.shallowwater_eq.BaseShallowWaterEquation

2D depth-averaged shallow water equations in non-conservative form.

This defines the full 2D SWE equations (5) - (6).

Parameters:

• function_space – Mixed function space where the solution belongs
• bathymetry (Function or Constant) – bathymetry of the domain
• options – AttrDict object containing all circulation model options

mass_term(solution)

residual(label, solution, solution_old, fields, fields_old, bnd_conditions)

class thetis.shallowwater_eq.ModeSplit2DEquations(function_space, bathymetry, options)

Bases: thetis.shallowwater_eq.BaseShallowWaterEquation

2D depth-averaged shallow water equations for mode splitting schemes.

Defines the equations (9) - (10).

Parameters:

• function_space – Mixed function space where the solution belongs
• bathymetry (Function or Constant) – bathymetry of the domain
• options – AttrDict object containing all circulation model options

add_momentum_terms(*args)

residual(label, solution, solution_old, fields, fields_old, bnd_conditions)

class thetis.shallowwater_eq.ShallowWaterMomentumEquation(u_test, u_space, eta_space, bathymetry, options)

Bases: thetis.shallowwater_eq.BaseShallowWaterEquation

2D depth averaged momentum equation (6) in non-conservative form.

Parameters:

• u_test – test function of the velocity function space
• u_space – velocity function space
• eta_space – elevation function space
• bathymetry (Function or Constant) – bathymetry of the domain
• options – AttrDict object containing all circulation model options

residual(label, solution, solution_old, fields, fields_old, bnd_conditions)

class thetis.shallowwater_eq.FreeSurfaceEquation(eta_test, eta_space, u_space, bathymetry, options)

Bases: thetis.shallowwater_eq.BaseShallowWaterEquation

2D free surface equation (5) in non-conservative form.

Parameters:

• eta_test – test function of the elevation function space
• eta_space – elevation function space
• u_space – velocity function space
• function_space – Mixed function space where the solution belongs
• bathymetry (Function or Constant) – bathymetry of the domain
• options – AttrDict object containing all circulation model options

mass_term(solution)

residual(label, solution, solution_old, fields, fields_old, bnd_conditions)

class thetis.shallowwater_eq.ShallowWaterTerm(space, bathymetry=None, options=None)

Bases: thetis.equation.Term

Generic term in the shallow water equations that provides commonly used members and mapping for boundary functions.

get_bnd_functions(eta_in, uv_in, bnd_id, bnd_conditions)

Returns external values of elev and uv for all supported boundary conditions.

Volume flux (flux) and normal velocity (un) are defined positive out of the domain.

get_total_depth(eta)

Returns total water column depth

wd_bathymetry_displacement(eta)

Returns wetting and drying bathymetry displacement as described in: Karna et al., 2011.

class thetis.shallowwater_eq.ShallowWaterMomentumTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterTerm

Generic term in the shallow water momentum equation that provides commonly used members and mapping for boundary functions.

class thetis.shallowwater_eq.ShallowWaterContinuityTerm(eta_test, eta_space, u_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterTerm

Generic term in the depth-integrated continuity equation that provides commonly used members and mapping for boundary functions.

class thetis.shallowwater_eq.HUDivTerm(eta_test, eta_space, u_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterContinuityTerm

Divergence term, \(\nabla \cdot (H \bar{\textbf{u}})\)

The weak form reads

\[\int_\Omega \nabla \cdot (H \bar{\textbf{u}}) \phi dx = \int_\Gamma (H^* \bar{\textbf{u}}^*) \cdot \text{jump}(\phi \textbf{n}) dS - \int_\Omega H (\bar{\textbf{u}}\cdot\nabla \phi) dx\]

where the right hand side has been integrated by parts; \(\textbf{n}\) denotes the unit normal of the element interfaces, and \(\text{jump}\) and \(\text{avg}\) denote the jump and average operators across the interface. \(H^*, \bar{\textbf{u}}^*\) are values at the interface obtained from an approximate Riemann solver.

If \(\bar{\textbf{u}}\) belongs to a discontinuous function space, the form on the right hand side is used.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.ContinuitySourceTerm(eta_test, eta_space, u_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterContinuityTerm

Generic source term in the depth-averaged continuity equation

The weak form reads

\[F_s = \int_\Omega S \phi dx\]

where \(S\) is a user defined scalar Function.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.HorizontalAdvectionTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Advection of momentum term, \(\bar{\textbf{u}} \cdot \nabla\bar{\textbf{u}}\)

The weak form is

\[\int_\Omega \bar{\textbf{u}} \cdot \nabla\bar{\textbf{u}} \cdot \boldsymbol{\psi} dx = - \int_\Omega \nabla_h \cdot (\bar{\textbf{u}} \boldsymbol{\psi}) \cdot \bar{\textbf{u}} dx + \int_\Gamma \text{avg}(\bar{\textbf{u}}) \cdot \text{jump}(\boldsymbol{\psi} (\bar{\textbf{u}}\cdot\textbf{n})) dS\]

where the right hand side has been integrated by parts; \(\textbf{n}\) is the unit normal of the element interfaces, and \(\text{jump}\) and \(\text{avg}\) denote the jump and average operators across the interface.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.HorizontalViscosityTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Viscosity of momentum term

If option ModelOptions.use_grad_div_viscosity_term is True, we use the symmetric viscous stress \(\boldsymbol{\tau}_\nu = \nu_h ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T )\). Using the symmetric interior penalty method the weak form then reads

\[\begin{split}\int_\Omega -\nabla \cdot \boldsymbol{\tau}_\nu \cdot \boldsymbol{\psi} dx =& \int_\Omega (\nabla \boldsymbol{\psi}) : \boldsymbol{\tau}_\nu dx \\ &- \int_\Gamma \text{jump}(\boldsymbol{\psi} \textbf{n}) \cdot \text{avg}(\boldsymbol{\tau}_\nu) dS - \int_\Gamma \text{avg}(\nu_h)\big(\text{jump}(\bar{\textbf{u}} \textbf{n}) + \text{jump}(\bar{\textbf{u}} \textbf{n})^T\big) \cdot \text{avg}(\nabla \boldsymbol{\psi}) dS \\ &+ \int_\Gamma \sigma \text{avg}(\nu_h) \big(\text{jump}(\bar{\textbf{u}} \textbf{n}) + \text{jump}(\bar{\textbf{u}} \textbf{n})^T\big) \cdot \text{jump}(\boldsymbol{\psi} \textbf{n}) dS\end{split}\]

where \(\sigma\) is a penalty parameter, see Epshteyn and Riviere (2007).

If option ModelOptions.use_grad_div_viscosity_term is False, we use viscous stress \(\boldsymbol{\tau}_\nu = \nu_h \nabla \bar{\textbf{u}}\). In this case the weak form is

\[\begin{split}\int_\Omega -\nabla \cdot \boldsymbol{\tau}_\nu \cdot \boldsymbol{\psi} dx =& \int_\Omega (\nabla \boldsymbol{\psi}) : \boldsymbol{\tau}_\nu dx \\ &- \int_\Gamma \text{jump}(\boldsymbol{\psi} \textbf{n}) \cdot \text{avg}(\boldsymbol{\tau}_\nu) dS - \int_\Gamma \text{avg}(\nu_h)\text{jump}(\bar{\textbf{u}} \textbf{n}) \cdot \text{avg}(\nabla \boldsymbol{\psi}) dS \\ &+ \int_\Gamma \sigma \text{avg}(\nu_h) \text{jump}(\bar{\textbf{u}} \textbf{n}) \cdot \text{jump}(\boldsymbol{\psi} \textbf{n}) dS\end{split}\]

If option ModelOptions.use_grad_depth_viscosity_term is True, we also include the term

\[\boldsymbol{\tau}_{\nabla H} = - \frac{\nu_h \nabla(H)}{H} \cdot ( \nabla \bar{\textbf{u}} + (\nabla \bar{\textbf{u}})^T )\]

as a source term.

Epshteyn and Riviere (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843-872. http://dx.doi.org/10.1016/j.cam.2006.08.029

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.ExternalPressureGradientTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

External pressure gradient term, \(g \nabla \eta\)

The weak form reads

\[\int_\Omega g \nabla \eta \cdot \boldsymbol{\psi} dx = \int_\Gamma g \eta^* \text{jump}(\boldsymbol{\psi} \cdot \textbf{n}) dS - \int_\Omega g \eta \nabla \cdot \boldsymbol{\psi} dx\]

where the right hand side has been integrated by parts; \(\textbf{n}\) denotes the unit normal of the element interfaces, \(n^*\) is value at the interface obtained from an approximate Riemann solver.

If \(\eta\) belongs to a discontinuous function space, the form on the right hand side is used.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.CoriolisTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Coriolis term, \(f\textbf{e}_z\wedge \bar{\textbf{u}}\)

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.LinearDragTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Linear friction term, \(C \bar{\textbf{u}}\)

Here \(C\) is a user-defined drag coefficient.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.QuadraticDragTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Quadratic Manning bottom friction term \(C_D \| \bar{\textbf{u}} \| \bar{\textbf{u}}\)

where the drag term is computed with the Manning formula

\[C_D = g \frac{\mu^2}{H^{1/3}}\]

if the Manning coefficient \(\mu\) is defined (see field manning_drag_coefficient). Otherwise \(C_D\) is taken as a constant (see field quadratic_drag_coefficient).

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.BottomDrag3DTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Bottom drag term consistent with the 3D mode, \(C_D \| \textbf{u}_b \| \textbf{u}_b\)

Here \(\textbf{u}_b\) is the bottom velocity used in the 3D mode, and \(C_D\) the corresponding bottom drag. These fields are computed in the 3D model.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.MomentumSourceTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Generic source term in the shallow water momentum equation

The weak form reads

\[F_s = \int_\Omega \boldsymbol{\tau} \cdot \boldsymbol{\psi} dx\]

where \(\boldsymbol{\tau}\) is a user defined vector valued Function.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.WindStressTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Wind stress term, \(-\tau_w/(H \rho_0)\)

Here \(\tau_w\) is a user-defined wind stress Function.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

class thetis.shallowwater_eq.AtmosphericPressureTerm(u_test, u_space, eta_space, bathymetry=None, options=None)

Bases: thetis.shallowwater_eq.ShallowWaterMomentumTerm

Atmospheric pressure term, \(\nabla (p_a / \rho_0)\)

Here \(p_a\) is a user-defined atmospheric pressure Function.

residual(uv, eta, uv_old, eta_old, fields, fields_old, bnd_conditions=None)

thetis.solver module

Module for three dimensional baroclinic solver

class thetis.solver.FlowSolver(mesh2d, bathymetry_2d, n_layers, options=None, extrude_options=None)

Bases: thetis.utility.FrozenClass

Main object for 3D solver

Example

Create 2D mesh

from thetis import *
mesh2d = RectangleMesh(20, 20, 10e3, 10e3)

Create bathymetry function, set a constant value

fs_p1 = FunctionSpace(mesh2d, 'CG', 1)
bathymetry_2d = Function(fs_p1, name='Bathymetry').assign(10.0)

Create a 3D model with 6 uniform levels, and set some options (see ModelOptions3d)

solver_obj = solver.FlowSolver(mesh2d, bathymetry_2d, n_layers=6)
options = solver_obj.options
options.element_family = 'dg-dg'
options.polynomial_degree = 1
options.timestepper_type = 'SSPRK22'
options.timestepper_options.use_automatic_timestep = False
options.solve_salinity = False
options.solve_temperature = False
options.simulation_export_time = 50.0
options.simulation_end_time = 3600.
options.timestep = 25.0

Assign initial condition for water elevation

solver_obj.create_function_spaces()
init_elev = Function(solver_obj.function_spaces.H_2d)
coords = SpatialCoordinate(mesh2d)
init_elev.project(2.0*exp(-((coords[0] - 4e3)**2 + (coords[1] - 4.5e3)**2)/2.2e3**2))
solver_obj.assign_initial_conditions(elev=init_elev)

Run simulation

solver_obj.iterate()

See the manual for more complex examples.

Parameters:

• mesh2d – Mesh object of the 2D mesh
• bathymetry_2d (2D Function) – Bathymetry of the domain. Bathymetry stands for the mean water depth (positive downwards).
• n_layers (int) – Number of layers in the vertical direction. Elements are distributed uniformly over the vertical.
• options (ModelOptions3d instance) – Model options (optional). Model options can also be changed directly via the options class property.

M_modesplit = None

Mode split ratio (int)

add_callback(callback, eval_interval='export')

Adds callback to solver object

Parameters:

• callback – DiagnosticCallback instance
• eval_interval (str) – Determines when callback will be evaluated, either ‘export’ or ‘timestep’ for evaluating after each export or time step.

assign_initial_conditions(elev=None, salt=None, temp=None, uv_2d=None, uv_3d=None, tke=None, psi=None)

Assigns initial conditions

Parameters:

• elev (scalar 2D Function, Constant, or an expression) – Initial condition for water elevation
• salt (scalar 3D Function, Constant, or an expression) – Initial condition for salinity field
• temp (scalar 3D Function, Constant, or an expression) – Initial condition for temperature field
• uv_2d (vector valued 2D Function, Constant, or an expression) – Initial condition for depth averaged velocity
• uv_3d (vector valued 3D Function, Constant, or an expression) – Initial condition for horizontal velocity
• tke (scalar 3D Function, Constant, or an expression) – Initial condition for turbulent kinetic energy field
• psi (scalar 3D Function, Constant, or an expression) – Initial condition for turbulence generic lenght scale field

callbacks = None

CallbackManager object that stores all callbacks

compute_dt_2d(u_scale)

Computes maximum explicit time step from CFL condition.

\[\Delta t = \frac{\Delta x}{U}\]

Assumes velocity scale \(U = \sqrt{g H} + U_{scale}\) where \(U_{scale}\) is estimated advective velocity.

Parameters:u_scale (float or Constant) – User provided maximum advective velocity scale

compute_dt_diffusion(nu_scale)

Computes maximum explicit time step for horizontal diffusion.

\[\Delta t = \alpha \frac{(\Delta x)^2}{\nu_{scale}}\]

where \(\nu_{scale}\) is estimated diffusivity scale.

compute_dt_h_advection(u_scale)

Computes maximum explicit time step for horizontal advection

\[\Delta t = \frac{\Delta x}{U_{scale}}\]

where \(U_{scale}\) is estimated horizontal advective velocity.

Parameters:u_scale (float or Constant) – User provided maximum horizontal velocity scale

compute_dt_v_advection(w_scale)

Computes maximum explicit time step for vertical advection

\[\Delta t = \frac{\Delta z}{W_{scale}}\]

where \(W_{scale}\) is estimated vertical advective velocity.

Parameters:w_scale (float or Constant) – User provided maximum vertical velocity scale

compute_dx_factor()

Computes normalized distance between nodes in the horizontal direction

The factor depends on the finite element space and its polynomial degree. It is used to compute maximal stable time steps.

compute_dz_factor()

Computes a normalized distance between nodes in the vertical direction

The factor depends on the finite element space and its polynomial degree. It is used to compute maximal stable time steps.

compute_mesh_stats()

Computes number of elements, nodes etc and prints to sdtout

create_equations()

Creates all dynamic equations and time integrators

create_fields()

Creates all fields

create_function_spaces()

Creates function spaces

Function spaces are accessible via function_spaces object.

dt = None

Time step

dt_2d = None

Time of the 2D solver

export()

Export all fields to disk

Also evaluates all callbacks set to ‘export’ interval.

export_initial_state = None

Do export initial state. False if continuing a simulation

fields = None

FieldDict that holds all functions needed by the solver object

function_spaces = None

AttrDict that holds all function spaces needed by the solver object

iterate(update_forcings=None, update_forcings3d=None, export_func=None)

Runs the simulation

Iterates over the time loop until time options.simulation_end_time is reached. Exports fields to disk on options.simulation_export_time intervals.

Parameters:

• update_forcings – User-defined function that takes simulation time as an argument and updates time-dependent boundary conditions of the 2D system (if any).
• update_forcings_3d – User-defined function that takes simulation time as an argument and updates time-dependent boundary conditions of the 3D equations (if any).
• export_func – User-defined function (with no arguments) that will be called on every export.

load_state(i_export, outputdir=None, t=None, iteration=None)

Loads simulation state from hdf5 outputs.

This replaces assign_initial_conditions() in model initilization.

This assumes that model setup is kept the same (e.g. time step) and all pronostic state variables are exported in hdf5 format. The required state variables are: elev_2d, uv_2d, uv_3d, salt_3d, temp_3d, tke_3d, psi_3d

Currently hdf5 field import only works for the same number of MPI processes.

Parameters:

• i_export (int) – export index to load
• outputdir (string) – (optional) directory where files are read from. By default options.output_directory.
• t (float) – simulation time. Overrides the time stamp stored in the hdf5 files.
• iteration (int) – Overrides the iteration count in the hdf5 files.

mesh = None

3D :class`Mesh`

mesh2d = None

2D :class`Mesh`

options = None

Dictionary of all options. A ModelOptions3d object.

print_state(cputime)

Print a summary of the model state on stdout

Parameters:cputime (float) – Measured CPU time

set_time_step()

Sets the model the model time step

If the time integrator supports automatic time step, and ModelOptions3d.timestepper_options.use_automatic_timestep is True, we compute the maximum time step allowed by the CFL condition. Otherwise uses ModelOptions3d.timestep.

Once the time step is determined, will adjust it to be an integer fraction of export interval options.simulation_export_time.

thetis.solver.timed_region()

Log.Event(type cls, name, klass=None)

thetis.solver.timed_stage()

Log.Stage(type cls, name)

thetis.solver2d module

Module for 2D depth averaged solver

class thetis.solver2d.FlowSolver2d(mesh2d, bathymetry_2d, options=None)

Bases: thetis.utility.FrozenClass

Main object for 2D depth averaged solver

Example

Create mesh

from thetis import *
mesh2d = RectangleMesh(20, 20, 10e3, 10e3)

Create bathymetry function, set a constant value

fs_p1 = FunctionSpace(mesh2d, 'CG', 1)
bathymetry_2d = Function(fs_p1, name='Bathymetry').assign(10.0)

Create solver object and set some options

solver_obj = solver2d.FlowSolver2d(mesh2d, bathymetry_2d)
options = solver_obj.options
options.element_family = 'dg-dg'
options.polynomial_degree = 1
options.timestepper_type = 'CrankNicolson'
options.simulation_export_time = 50.0
options.simulation_end_time = 3600.
options.timestep = 25.0

Assign initial condition for water elevation

solver_obj.create_function_spaces()
init_elev = Function(solver_obj.function_spaces.H_2d)
coords = SpatialCoordinate(mesh2d)
init_elev.project(exp(-((coords[0] - 4e3)**2 + (coords[1] - 4.5e3)**2)/2.2e3**2))
solver_obj.assign_initial_conditions(elev=init_elev)

Run simulation

solver_obj.iterate()

See the manual for more complex examples.

Parameters:

• mesh2d – Mesh object of the 2D mesh
• bathymetry_2d (Function) – Bathymetry of the domain. Bathymetry stands for the mean water depth (positive downwards).
• options (ModelOptions2d instance) – Model options (optional). Model options can also be changed directly via the options class property.

add_callback(callback, eval_interval='export')

Adds callback to solver object

Parameters:

• callback – DiagnosticCallback instance
• eval_interval (string) – Determines when callback will be evaluated, either ‘export’ or ‘timestep’ for evaluating after each export or time step.

assign_initial_conditions(elev=None, uv=None)

Assigns initial conditions

Parameters:

• elev (scalar Function, Constant, or an expression) – Initial condition for water elevation
• uv (vector valued Function, Constant, or an expression) – Initial condition for depth averaged velocity

callbacks = None

CallbackManager object that stores all callbacks

compute_time_step(u_scale=Constant(FiniteElement('Real', None, 0), 15))

Computes maximum explicit time step from CFL condition.

\[\Delta t = \frac{\Delta x}{U}\]

Assumes velocity scale \(U = \sqrt{g H} + U_{scale}\) where \(U_{scale}\) is estimated advective velocity.

Parameters:u_scale (float or Constant) – User provided maximum advective velocity scale

create_equations()

Creates shallow water equations

create_exporters()

Creates file exporters

create_function_spaces()

Creates function spaces

Function spaces are accessible via function_spaces object.

create_timestepper()

Creates time stepper instance

dt = None

Time step

export()

Export all fields to disk

Also evaluates all callbacks set to ‘export’ interval.

export_initial_state = None

Do export initial state. False if continuing a simulation

fields = None

FieldDict that holds all functions needed by the solver object

function_spaces = None

AttrDict that holds all function spaces needed by the solver object

initialize()

Creates function spaces, equations, time stepper and exporters

iterate(update_forcings=None, export_func=None)

Runs the simulation

Iterates over the time loop until time options.simulation_end_time is reached. Exports fields to disk on options.simulation_export_time intervals.

Parameters:

• update_forcings – User-defined function that takes simulation time as an argument and updates time-dependent boundary conditions (if any).
• export_func – User-defined function (with no arguments) that will be called on every export.

load_state(i_export, outputdir=None, t=None, iteration=None)

Loads simulation state from hdf5 outputs.

This replaces assign_initial_conditions() in model initilization.

This assumes that model setup is kept the same (e.g. time step) and all pronostic state variables are exported in hdf5 format. The required state variables are: elev_2d, uv_2d

Currently hdf5 field import only works for the same number of MPI processes.

Parameters:

• i_export (int) – export index to load
• outputdir (string) – (optional) directory where files are read from. By default options.output_directory.
• t (float) – simulation time. Overrides the time stamp stored in the hdf5 files.
• iteration (int) – Overrides the iteration count in the hdf5 files.

options = None

Dictionary of all options. A ModelOptions2d object.

print_state(cputime)

Print a summary of the model state on stdout

Parameters:cputime (float) – Measured CPU time

set_time_step(alpha=0.05)

Sets the model the model time step

If the time integrator supports automatic time step, and ModelOptions2d.timestepper_options.use_automatic_timestep is True, we compute the maximum time step allowed by the CFL condition. Otherwise uses ModelOptions2d.timestep.

Parameters:alpha (float) – CFL number scaling factor

thetis.solver2d.timed_region()

Log.Event(type cls, name, klass=None)

thetis.solver2d.timed_stage()

Log.Stage(type cls, name)

thetis.stability_functions module

Implements turbulence closure model stability functions.

\[\begin{split}S_m &= S_m(\alpha_M, \alpha_N) \\ S_\rho &= S_\rho(\alpha_M, \alpha_N)\end{split}\]

where \(\alpha_M, \alpha_N\) are the normalized shear and buoyancy frequency

\[\begin{split}\alpha_M &= \frac{k^2}{\varepsilon^2} M^2 \\ \alpha_N &= \frac{k^2}{\varepsilon^2} N^2\end{split}\]

The following stability functions have been implemented

• Canuto A
• Canuto B
• Kantha-Clayson
• Cheng

References:

Umlauf, L. and Burchard, H. (2005). Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Continental Shelf Research, 25(7-8):795–827. http://dx.doi.org/10.1016/j.csr.2004.08.004

Burchard, H. and Bolding, K. (2001). Comparative Analysis of Four Second-Moment Turbulence Closure Models for the Oceanic Mixed Layer. Journal of Physical Oceanography, 31(8):1943–1968. http://dx.doi.org/10.1175/1520-0485(2001)031

Umlauf, L. and Burchard, H. (2003). A generic length-scale equation for geophysical turbulence models. Journal of Marine Research, 61:235–265(31). http://dx.doi.org/10.1357/002224003322005087

class thetis.stability_functions.StabilityFunction(lim_alpha_shear=True, lim_alpha_buoy=True, smooth_alpha_buoy_lim=True, alpha_buoy_crit=-1.2)

Bases: object

Base class for all stability functions

Parameters:

• lim_alpha_shear (bool) – limit maximum \(\alpha_M\) values (see Umlauf and Burchard (2005) eq. 44)
• lim_alpha_buoy (bool) – limit minimum (negative) \(\alpha_N\) values (see Umlauf and Burchard (2005))
• smooth_alpha_buoy_lim (bool) – if \(\alpha_N\) is limited, apply a smooth limiter (see Burchard and Bolding (2001) eq. 19). Otherwise \(\alpha_N\) is clipped at minimum value.
• alpha_buoy_crit (float) – parameter for \(\alpha_N\) smooth limiter

compute_c3_minus(c1, c2, ri_st)

Compute c3_minus parameter from c1, c2 and stability functions.

c3_minus is solved from equation

\[Ri_{st} = \frac{s_m}{s_h} \frac{c2 - c1}{c2 - c3_minus}\]

where \(Ri_{st}\) is the steady state gradient Richardson number. (see Burchard and Bolding, 2001, eq 32)

compute_cmu0()

Computes the paramenter c_mu_0 from stability function parameters

Umlauf and Buchard (2005) eq A.22

compute_kappa(sigma_psi, n, c1, c2)

Computes von Karman constant from the Psi Schmidt number.

n, c1, c2 are GLS model parameters.

from Umlauf and Burchard (2003) eq (14)

eval_funcs(alpha_buoy, alpha_shear)

Evaluate (unlimited) stability functions

from Burchard and Petersen (1999) eqns (30) and (31)

Parameters:

• alpha_buoy – normalized buoyancy frequency \(\alpha_N\)
• alpha_shear – normalized shear frequency \(\alpha_M\)

evaluate(shear2, buoy2, k, eps)

Evaluates stability functions. Applies limiters on alpha_buoy and alpha_shear.

Parameters:

• shear2 – \(M^2\)
• buoy2 – \(N^2\)
• k – turbulent kinetic energy
• eps – TKE dissipation rate

get_alpha_buoy_min()

Compute minimum alpha buoy

from Umlauf and Buchard (2005) table 3

get_alpha_buoy_smooth_min(alpha_buoy)

Compute smoothed alpha_buoy minimum

from Burchard and Petersen (1999) eq (19)

Parameters:alpha_buoy – normalized buoyancy frequency \(\alpha_N\)

get_alpha_shear_max(alpha_buoy, alpha_shear)

Compute maximum alpha shear

from Umlauf and Buchard (2005) eq (44)

Parameters:

• alpha_buoy – normalized buoyancy frequency \(\alpha_N\)
• alpha_shear – normalized shear frequency \(\alpha_M\)

class thetis.stability_functions.StabilityFunctionCanutoA(lim_alpha_shear=True, lim_alpha_buoy=True, smooth_alpha_buoy_lim=True, alpha_buoy_crit=-1.2)

Bases: thetis.stability_functions.StabilityFunction

Canuto et al. (2001) version A stability functions

Parameters are from Umlauf and Buchard (2005), Table 1

class thetis.stability_functions.StabilityFunctionCanutoB(lim_alpha_shear=True, lim_alpha_buoy=True, smooth_alpha_buoy_lim=True, alpha_buoy_crit=-1.2)

Bases: thetis.stability_functions.StabilityFunction

Canuto et al. (2001) version B stability functions

Parameters are from Umlauf and Buchard (2005), Table 1

class thetis.stability_functions.StabilityFunctionKanthaClayson(lim_alpha_shear=True, lim_alpha_buoy=True, smooth_alpha_buoy_lim=True, alpha_buoy_crit=-1.2)

Bases: thetis.stability_functions.StabilityFunction

Kantha and Clayson (1994) quasi-equilibrium stability functions

Parameters are from Umlauf and Buchard (2005), Table 1

class thetis.stability_functions.StabilityFunctionCheng(lim_alpha_shear=True, lim_alpha_buoy=True, smooth_alpha_buoy_lim=True, alpha_buoy_crit=-1.2)

Bases: thetis.stability_functions.StabilityFunction

Cheng et al. (2002) quasi-equilibrium stability functions

Parameters are from Umlauf and Buchard (2005), Table 1

thetis.stability_functions.compute_normalized_frequencies(shear2, buoy2, k, eps)

Computes normalized buoyancy and shear frequency squared.

\[\begin{split}\alpha_M &= \frac{k^2}{\varepsilon^2} M^2 \\ \alpha_N &= \frac{k^2}{\varepsilon^2} N^2\end{split}\]

From Burchard and Bolding (2001).

Parameters:

• shear2 – \(M^2\)
• buoy2 – \(N^2\)
• k – turbulent kinetic energy
• eps – TKE dissipation rate

thetis.timeintegrator module

Generic time integration schemes to advance equations in time.

class thetis.timeintegrator.CrankNicolson(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, theta=0.5, semi_implicit=False)

Bases: thetis.timeintegrator.TimeIntegrator

Standard Crank-Nicolson time integration scheme.

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• theta (float) – Implicitness parameter, default 0.5
• semi_implicit (bool) – If True use a linearized semi-implicit scheme

advance(t, update_forcings=None)

Advances equations for one time step.

cfl_coeff = inf

initialize(solution)

Assigns initial conditions to all required fields.

update_solver()

Create solver objects

class thetis.timeintegrator.ForwardEuler(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={})

Bases: thetis.timeintegrator.TimeIntegrator

Standard forward Euler time integration scheme.

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options

advance(t, update_forcings=None)

Advances equations for one time step.

cfl_coeff = 1.0

initialize(solution)

Assigns initial conditions to all required fields.

update_solver()

class thetis.timeintegrator.LeapFrogAM3(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.timeintegrator.TimeIntegrator

Leap-Frog Adams-Moulton 3 ALE time integrator

Defined in (2.27)-(2.30) in [1]; (2.21)-(2.22) in [2]

[1] Shchepetkin and McWilliams (2005). The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modelling, 9(4):347-404. http://dx.doi.org/10.1016/j.ocemod.2013.04.010

[2] Shchepetkin and McWilliams (2009). Computational Kernel Algorithms for Fine-Scale, Multiprocess, Longtime Oceanic Simulations, 14:121-183. http://dx.doi.org/10.1016/S1570-8659(08)01202-0

Parameters:

• equation (Equation object) – equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

advance(t, update_forcings=None)

Advances equations for one time step.

cfl_coeff = 1.5874

correct()

Correction step from \(t_{n}\) to \(t_{n+1}\)

Let \(M_n\) denote the mass matrix at time \(t_{n}\). The correction step is

\[M_{n+1} T_{n+1} = M_{n} T_{n} + \Delta t L_{n+1/2}\]

This is Euler ALE step: LHS is evaluated in \(\Omega_{n+1}\), RHS in \(\Omega_n\).

eval_rhs()

initialize(solution)

Assigns initial conditions to all required fields.

predict()

Prediction step from \(t_{n-1/2}\) to \(t_{n+1/2}\)

Let \(M_n\) denote the mass matrix at time \(t_{n}\). The prediction step is

\[\begin{split}T_{n-1/2} &= (1/2 - 2\gamma) T_{n-1} + (1/2 + 2 \gamma) T_{n} \\ M_n T_{n+1/2} &= M_n T_{n-1/2} + \Delta t (1 - 2\gamma) M_n L_{n}\end{split}\]

This is computed in a fixed mesh: all terms are evaluated in \(\Omega_n\).

class thetis.timeintegrator.PressureProjectionPicard(equation, equation_mom, solution, fields, dt, bnd_conditions=None, solver_parameters={}, solver_parameters_mom={}, theta=0.5, semi_implicit=False, iterations=2)

Bases: thetis.timeintegrator.TimeIntegrator

Pressure projection scheme with Picard iteration for shallow water equations

Parameters:

• equation (Equation object) – free surface equation
• equation_mom (Equation object) – momentum equation
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• solver_parameters_mom (dict) – PETSc solver options for velocity solver
• theta (float) – Implicitness parameter, default 0.5
• semi_implicit (bool) – If True use a linearized semi-implicit scheme
• iterations (int) – Number of Picard iterations

advance(t, updateForcings=None)

Advances equations for one time step.

cfl_coeff = 1.0

initialize(solution)

Assigns initial conditions to all required fields.

update_solver()

Create solver objects

class thetis.timeintegrator.SSPRK22ALE(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={}, terms_to_add='all')

Bases: thetis.timeintegrator.TimeIntegrator

SSPRK(2,2) ALE time integrator for 3D fields

The scheme is

\[\begin{split}u^{(1)} &= u^{n} + \Delta t F(u^{n}) \\ u^{n+1} &= u^{n} + \frac{\Delta t}{2}(F(u^{n}) + F(u^{(1)}))\end{split}\]

Both stages are implemented as ALE updates from geometry \(\Omega_n\) to \(\Omega_{(1)}\), and \(\Omega_{n+1}\).

Parameters:

• equation (Equation object) – equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options
• terms_to_add ('all' or list of 'implicit', 'explicit', 'source'.) – Defines which terms of the equation are to be added to this solver. Default ‘all’ implies [‘implicit’, ‘explicit’, ‘source’].

advance(t, update_forcings=None)

Advances equations for one time step.

cfl_coeff = 1.0

initialize(solution)

Assigns initial conditions to all required fields.

prepare_stage(i_stage, t, update_forcings=None)

Preprocess stage i_stage.

This must be called prior to updating mesh geometry.

solve_stage(i_stage)

Solves i-th stage

stage_one_prep()

Preprocess first stage: compute all forms on the old geometry

stage_one_solve()

First stage: solve \(u^{(1)}\) given previous solution \(u^n\).

This is a forward Euler ALE step between domains \(\Omega^n\) and \(\Omega^{(1)}\):

\[\int_{\Omega^{(1)}} u^{(1)} \psi dx = \int_{\Omega^n} u^n \psi dx + \Delta t \int_{\Omega^n} F(u^n) \psi dx\]

stage_two_prep()

Preprocess 2nd stage: compute all forms on the old geometry

stage_two_solve()

2nd stage: solve \(u^{n+1}\) given previous solutions \(u^n, u^{(1)}\).

This is an ALE step:

\[\begin{split}\int_{\Omega^{n+1}} u^{n+1} \psi dx &= \int_{\Omega^n} u^n \psi dx \\ &+ \frac{\Delta t}{2} \int_{\Omega^n} F(u^n) \psi dx \\ &+ \frac{\Delta t}{2} \int_{\Omega^{(1)}} F(u^{(1)}) \psi dx\end{split}\]

class thetis.timeintegrator.SteadyState(equation, solution, fields, dt, bnd_conditions=None, solver_parameters={})

Bases: thetis.timeintegrator.TimeIntegrator

Time integrator that solves the steady state equations, leaving out the mass terms

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• bnd_conditions (dict) – Dictionary of boundary conditions passed to the equation
• solver_parameters (dict) – PETSc solver options

advance(t, update_forcings=None)

Advances equations for one time step.

cfl_coeff = inf

initialize(solution)

Assigns initial conditions to all required fields.

update_solver()

Create solver objects

class thetis.timeintegrator.TimeIntegrator(equation, solution, fields, dt, solver_parameters={})

Bases: thetis.timeintegrator.TimeIntegratorBase

Base class for all time integrator objects that march a single equation

Parameters:

• equation (Equation object) – the equation to solve
• solution – Function where solution will be stored
• fields (dict of Function or Constant objects) – Dictionary of fields that are passed to the equation
• dt (float) – time step in seconds
• solver_parameters (dict) – PETSc solver options

set_dt(dt)

Update time step

class thetis.timeintegrator.TimeIntegratorBase

Bases: object

Abstract class that defines the API for all time integrators

Both TimeIntegrator and CoupledTimeIntegrator inherit from this class.

advance(t, update_forcings=None)

Advances equations for one time step

Parameters:

• t (float) – simulation time
• update_forcings – user-defined function that takes the simulation time and updates any time-dependent boundary conditions

initialize(init_solution)

Initialize the time integrator

Parameters:init_solution – initial solution

thetis.timeintegrator.timed_region()

Log.Event(type cls, name, klass=None)

thetis.timeintegrator.timed_stage()

Log.Stage(type cls, name)

thetis.timezone module

Timezone definitions and conversion methods

class thetis.timezone.FixedTimeZone(offset, name)

Bases: pytz._FixedOffset

Class that represents a fixed time zone defined by UTC offset in hours.

arg int offset: timezone UTC offset in hours arg str name: timezone name

tzname(dt)

thetis.timezone.datetime_to_epoch(t)

Convert python datetime object to epoch time stamp.

thetis.timezone.epoch_to_datetime(t)

Convert python datetime object to epoch time stamp.

thetis.tracer_eq module

3D advection diffusion equation for tracers.

The advection-diffusion equation of tracer \(T\) in conservative form reads

(13)\[\frac{\partial T}{\partial t} + \nabla_h \cdot (\textbf{u} T) + \frac{\partial (w T)}{\partial z} = \nabla_h \cdot (\mu_h \nabla_h T) + \frac{\partial}{\partial z} \Big(\mu \frac{T}{\partial z}\Big)\]

where \(\nabla_h\) denotes horizontal gradient, \(\textbf{u}\) and \(w\) are the horizontal and vertical velocities, respectively, and \(\mu_h\) and \(\mu\) denote horizontal and vertical diffusivity.

class thetis.tracer_eq.TracerEquation(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.equation.Equation

3D tracer advection-diffusion equation (13) in conservative form

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

class thetis.tracer_eq.TracerTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.equation.Term

Generic tracer term that provides commonly used members and mapping for boundary functions.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

get_bnd_functions(c_in, uv_in, elev_in, bnd_id, bnd_conditions)

Returns external values of tracer and uv for all supported boundary conditions.

Volume flux (flux) and normal velocity (un) are defined positive out of the domain.

Parameters:

• c_in – Internal value of tracer
• uv_in – Internal value of horizontal velocity
• elev_in – Internal value of elevation
• bnd_id (int) – boundary id
• bnd_conditions – dict of boundary conditions: {bnd_id: {field: value, ...}, ...}

class thetis.tracer_eq.HorizontalAdvectionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.tracer_eq.TracerTerm

Horizontal advection term \(\nabla_h \cdot (\textbf{u} T)\)

The weak formulation reads

\[\int_\Omega \nabla_h \cdot (\textbf{u} T) \phi dx = -\int_\Omega T\textbf{u} \cdot \nabla_h \phi dx + \int_{\mathcal{I}_h\cup\mathcal{I}_v} T^{\text{up}} \text{avg}(\textbf{u}) \cdot \text{jump}(\phi \textbf{n}_h) dS\]

where the right hand side has been integrated by parts; \(\mathcal{I}_h,\mathcal{I}_v\) denote the set of horizontal and vertical facets, \(\textbf{n}_h\) is the horizontal projection of the unit normal vector, \(T^{\text{up}}\) is the upwind value, and \(\text{jump}\) and \(\text{avg}\) denote the jump and average operators across the interface.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.tracer_eq.VerticalAdvectionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.tracer_eq.TracerTerm

Vertical advection term \(\partial (w T)/(\partial z)\)

The weak form reads

\[\int_\Omega \frac{\partial (w T)}{\partial z} \phi dx = - \int_\Omega T w \frac{\partial \phi}{\partial z} dx + \int_{\mathcal{I}_v} T^{\text{up}} \text{avg}(w) \text{jump}(\phi n_z) dS\]

where the right hand side has been integrated by parts; \(\mathcal{I}_v\) denotes the set of vertical facets, \(n_z\) is the vertical projection of the unit normal vector, \(T^{\text{up}}\) is the upwind value, and \(\text{jump}\) and \(\text{avg}\) denote the jump and average operators across the interface.

In the case of ALE moving mesh we substitute \(w\) with \(w - w_m\), \(w_m\) being the mesh velocity.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.tracer_eq.HorizontalDiffusionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.tracer_eq.TracerTerm

Horizontal diffusion term \(-\nabla_h \cdot (\mu_h \nabla_h T)\)

Using the symmetric interior penalty method the weak form becomes

\[\begin{split}\int_\Omega \nabla_h \cdot (\mu_h \nabla_h T) \phi dx =& -\int_\Omega \mu_h (\nabla_h \phi) \cdot (\nabla_h T) dx \\ &+ \int_{\mathcal{I}_h\cup\mathcal{I}_v} \text{jump}(\phi \textbf{n}_h) \cdot \text{avg}(\mu_h \nabla_h T) dS + \int_{\mathcal{I}_h\cup\mathcal{I}_v} \text{jump}(T \textbf{n}_h) \cdot \text{avg}(\mu_h \nabla \phi) dS \\ &- \int_{\mathcal{I}_h\cup\mathcal{I}_v} \sigma \text{avg}(\mu_h) \text{jump}(T \textbf{n}_h) \cdot \text{jump}(\phi \textbf{n}_h) dS\end{split}\]

where \(\sigma\) is a penalty parameter, see Epshteyn and Riviere (2007).

Epshteyn and Riviere (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843-872. http://dx.doi.org/10.1016/j.cam.2006.08.029

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.tracer_eq.VerticalDiffusionTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.tracer_eq.TracerTerm

Vertical diffusion term \(-\frac{\partial}{\partial z} \Big(\mu \frac{T}{\partial z}\Big)\)

Using the symmetric interior penalty method the weak form becomes

\[\begin{split}\int_\Omega \frac{\partial}{\partial z} \Big(\mu \frac{T}{\partial z}\Big) \phi dx =& -\int_\Omega \mu \frac{\partial T}{\partial z} \frac{\partial \phi}{\partial z} dz \\ &+ \int_{\mathcal{I}_{h}} \text{jump}(\phi n_z) \text{avg}\Big(\mu \frac{\partial T}{\partial z}\Big) dS + \int_{\mathcal{I}_{h}} \text{jump}(T n_z) \text{avg}\Big(\mu \frac{\partial \phi}{\partial z}\Big) dS \\ &- \int_{\mathcal{I}_{h}} \sigma \text{avg}(\mu) \text{jump}(T n_z) \cdot \text{jump}(\phi n_z) dS\end{split}\]

where \(\sigma\) is a penalty parameter, see Epshteyn and Riviere (2007).

Epshteyn and Riviere (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843-872. http://dx.doi.org/10.1016/j.cam.2006.08.029

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.tracer_eq.SourceTerm(function_space, bathymetry=None, v_elem_size=None, h_elem_size=None, use_symmetric_surf_bnd=True, use_lax_friedrichs=True)

Bases: thetis.tracer_eq.TracerTerm

Generic source term

The weak form reads

\[F_s = \int_\Omega \sigma \phi dx\]

where \(\sigma\) is a user defined scalar Function.

Parameters:

• function_space – FunctionSpace where the solution belongs
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size
• use_symmetric_surf_bnd (bool) – If True, use symmetric surface boundary condition in the horizontal advection term

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

thetis.turbulence module

Generic Length Scale Turbulence Closure model

This model solves two dynamic equations, for turbulent kinetic energy (TKE, \(k\)) and one for an additional variable, the generic length scale \(\psi\) [1]:

(14)\[\frac{\partial k}{\partial t} + \nabla_h \cdot (\textbf{u} k) + \frac{\partial (w k)}{\partial z} = \frac{\partial}{\partial z}\left(\frac{\nu}{\sigma_k} \frac{\partial k}{\partial z}\right) + P + B - \varepsilon\]

(15)\[\frac{\partial \psi}{\partial t} + \nabla_h \cdot (\textbf{u} \psi) + \frac{\partial (w \psi)}{\partial z} = \frac{\partial}{\partial z}\left(\frac{\nu}{\sigma_\psi} \frac{\partial \psi}{\partial z}\right) + \frac{\psi}{k} (c_1 P + c_3 B - c_2 \varepsilon f_{wall})\]

with the production \(P\) and buoyancy production \(B\) are

\[\begin{split}P &= \nu M^2 \\ B &= -\mu N^2\end{split}\]

where \(M\) and \(N\) are the shear and buoyancy frequencies

\[\begin{split}M^2 &= \left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2 \\ N^2 &= -\frac{g}{\rho_0}\frac{\partial \rho}{\partial z}\end{split}\]

The generic lenght scale variable is defined as

\[\psi = (c_\mu^0)^p k^m l^n\]

where \(p, m, n\) are parameters and \(c_\mu^0\) is an empirical constant.

The parameters \(c_1,c_2,c_3,f_{wall}\) depend on the chosen closure. The parameter \(c_3\) takes two values: \(c_3^-\) in stably stratified regime, and \(c_3^+\) in unstably stratified cases.

Turbulent length scale \(l\), and the TKE dissipation rate \(\varepsilon\) are obtained diagnostically as

\[\begin{split}l &= (c_\mu^0)^3 k^{3/2} \varepsilon^{-1} \\ \varepsilon &= (c_\mu^0)^{3+p/n} k^{3/2 + m/n} \psi^{-1/n}\end{split}\]

Finally the vertical eddy viscosity and diffusivity are also computed diagnostically

\[\begin{split}\nu &= \sqrt{2k} l S_m \\ \mu &= \sqrt{2k} l S_\rho\end{split}\]

Stability functions \(S_m\) and \(S_\rho\) are defined in [2] or [3]. Implementation follows [4].

Supported closures

The GLS model parameters are controlled via the GLSModelOptions class.

The parameters can be accessed from the solver object:

solver = FlowSolver(...)
solver.options.turbulence_model_type = 'gls'  # activate GLS model (default)
turbulence_model_options = solver.options.turbulence_model_options
turbulence_model_options.closure_name = 'k-omega'
turbulence_model_options.stability_function_name = 'CB'
turbulence_model_options.compute_c3_minus = True

Currently the following closures have been implemented:

• \(k-\varepsilon\) model

  This closure is obtained with \(p=3, m=3/2, n=-1\), resulting in \(\psi=\varepsilon\). To use this option set closure_name = k-epsilon

• \(k-\omega\) model

  This closure is obtained with \(p=-1, m=1/2, n=-1\), resulting in \(\psi=\omega\). To use this option set closure_name = k-omega

• GLS model A

  This closure is obtained with \(p=2, m=1, n=-2/3\), resulting in \(\psi=\omega\). To use this option set closure_name = gen

The following stability functions have been implemented

• Canuto A [3]

  To use this option set closure_name = CA

• Canuto B [3]

  To use this option set closure_name = CB

• Kantha-Clayson [2]

  To use this option set closure_name = KC

• Cheng [6]

  To use this option set closure_name = CH

See stability_functions for more information.

[1] Umlauf, L. and Burchard, H. (2003). A generic length-scale equation for

geophysical turbulence models. Journal of Marine Research, 61:235–265(31). http://dx.doi.org/10.1357/002224003322005087

[2] Kantha, L. H. and Clayson, C. A. (1994). An improved mixed layer model for

geophysical applications. Journal of Geophysical Research: Oceans, 99(C12):25235–25266. http://dx.doi.org/10.1029/94JC02257

[3] Canuto et al. (2001). Ocean Turbulence. Part I: One-Point Closure Model -

Momentum and Heat Vertical Diffusivities. Journal of Physical Oceanography, 31(6):1413-1426. http://dx.doi.org/10.1175/1520-0485(2001)031

[4] Warner et al. (2005). Performance of four turbulence closure models

implemented using a generic length scale method. Ocean Modelling, 8(1-2):81–113. http://dx.doi.org/10.1016/j.ocemod.2003.12.003

[5] Umlauf, L. and Burchard, H. (2005). Second-order turbulence closure models

for geophysical boundary layers. A review of recent work. Continental Shelf Research, 25(7-8):795–827. http://dx.doi.org/10.1016/j.csr.2004.08.004

[6] Cheng et al. (2002). An improved model for the turbulent PBL.

J. Atmos. Sci., 59:1550-1565. http://dx.doi.org/10.1175/1520-0469(2002)059%3C1550:AIMFTT%3E2.0.CO;2

[7] Burchard and Petersen (1999). Models of turbulence in the marine

environment - a comparative study of two-equation turbulence models. Journal of Marine Systems, 21(1-4):29-53. http://dx.doi.org/10.1016/S0924-7963(99)00004-4

class thetis.turbulence.BuoyFrequencySolver(rho, n2, n2_tmp, relaxation=1.0, minval=1e-12)

Bases: object

Computes buoyancy frequency squared form the given horizontal velocity field.

\[N^2 = -\frac{g}{\rho_0}\frac{\partial \rho}{\partial z}\]

Parameters:

• rho (Function) – water density field
• n2 (Function) – \(N^2\) field
• n2_tmp (Function) – temporary field
• relaxation (float) – relaxation coefficient for mixing old and new values N2 = relaxation*N2_new + (1-relaxation)*N2_old
• minval (float) – minimum value for \(N^2\)

solve(init_solve=False)

Computes buoyancy frequency

Parameters:init_solve (bool) – Set to True if solving for the first time, skips relaxation

class thetis.turbulence.GLSVerticalDiffusionTerm(function_space, schmidt_nb, bathymetry=None, v_elem_size=None, h_elem_size=None)

Bases: thetis.tracer_eq.VerticalDiffusionTerm

Vertical diffusion term where the diffusivity is replaced by viscosity/Schmidt number.

Parameters:

• function_space – FunctionSpace where the solution belongs
• schmidt_nb – the Schmidt number of TKE or Psi
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.turbulence.GenericLengthScaleModel(solver, k_field, psi_field, uv_field, rho_field, l_field, epsilon_field, eddy_diffusivity, eddy_viscosity, n2, m2, options=None)

Bases: thetis.turbulence.TurbulenceModel

Generic Length Scale turbulence closure model implementation

Parameters:

• solver – FlowSolver object
• k_field (Function) – turbulent kinetic energy (TKE) field
• psi_field (Function) – generic length scale field
• uv_field (Function) – horizontal velocity field
• rho_field (Function) – water density field
• l_field (Function) – turbulence length scale field
• epsilon_field (Function) – TKE dissipation rate field
• eddy_diffusivity (Function) – eddy diffusivity field
• eddy_viscosity (Function) – eddy viscosity field
• n2 (Function) – field for buoyancy frequency squared
• m2 (Function) – field for vertical shear frequency squared
• options – GLS model options

initialize()

Initializes fields

postprocess()

Updates all diagnostic variables that depend on the turbulence state variables \(k,\psi\).

To be called after updating the turbulence PDEs.

preprocess(init_solve=False)

Computes all diagnostic variables that depend on the mean flow model variables.

To be called before updating the turbulence PDEs.

class thetis.turbulence.P1Average(p0, p1, p1dg)

Bases: object

Takes a discontinuous field and computes a P1 field by averaging around nodes

Source must be either a P0 or P1DG Function. The averaging operation is both mass conservative and positivity preserving.

Parameters:

• p0 – P0 function space
• p1 – P1 function space
• p1dg – P1DG function space

apply(source, solution)

Averages discontinuous Function source on P1 Function solution

update_volumes()

Computes nodal volume of the P1 and P1DG function function_spaces

This must be called when the mesh geometry is updated

class thetis.turbulence.PacanowskiPhilanderModel(solver, uv_field, rho_field, eddy_diffusivity, eddy_viscosity, n2, m2, options=None)

Bases: thetis.turbulence.TurbulenceModel

Gradient Richardson number based model by Pacanowski and Philander (1981).

Given the gradient Richardson number \(Ri\) the eddy viscosity and diffusivity are computed diagnostically as

\[\begin{split}\nu &= \frac{\nu_{max}}{(1 + \alpha Ri)^n} \\ \mu &= \frac{\nu}{1 + \alpha Ri}\end{split}\]

where \(\nu_{max},\alpha,n\) are constant parameters. In unstably stratified cases where \(Ri<0\), value \(Ri=0\) is used.

Pacanowski and Philander (1981). Parameterization of vertical mixing in numerical models of tropical oceans. Journal of Physical Oceanography, 11(11):1443-1451. http://dx.doi.org/10.1175/1520-0485(1981)011%3C1443:POVMIN%3E2.0.CO;2

Parameters:

• solver – FlowSolver object
• uv_field (Function) – horizontal velocity field
• rho_field (Function) – water density field
• eddy_diffusivity (Function) – eddy diffusivity field
• eddy_viscosity (Function) – eddy viscosity field
• n2 (Function) – field for buoyancy frequency squared
• m2 (Function) – field for vertical shear frequency squared
• options – model options

initialize()

Initializes fields

postprocess()

Updates all diagnostic variables that depend on the turbulence state variables.

To be called after evaluating the equations.

preprocess(init_solve=False)

Computes all diagnostic variables that depend on the mean flow model variables.

To be called before updating the turbulence PDEs.

class thetis.turbulence.PsiEquation(function_space, gls_model, bathymetry=None, v_elem_size=None, h_elem_size=None)

Bases: thetis.equation.Equation

Generic length scale equation (15) without advection terms.

Advection of \(\psi\) is implemented using the standard tracer equation.

Parameters:

• function_space – FunctionSpace where the solution belongs
• gls_model – GenericLengthScaleModel object
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size

class thetis.turbulence.PsiSourceTerm(function_space, gls_model, bathymetry=None, v_elem_size=None, h_elem_size=None)

Bases: thetis.tracer_eq.TracerTerm

Production and destruction terms of the Psi equation (15)

\[F_\psi = \frac{\psi}{k} (c_1 P + c_3 B - c_2 \varepsilon f_{wall})\]

To ensure positivity we use Patankar-type time discretization: all source terms are treated explicitly and sink terms are treated implicitly. To this end the buoyancy production term \(c_3 B\) is split in two:

\[F_\psi = \frac{\psi^n}{k^n} (c_1 P + B_{source}) + \frac{\psi^{n+1}}{k^n} (B_{sink} - c_2 \varepsilon f_{wall})\]

with \(B_{source} \ge 0\) and \(B_{sink} < 0\).

Also implements Neumann boundary condition at top and bottom [7]

\[\left( \frac{\nu}{\sigma_\psi} \frac{\psi}{z} \right)\Big|_{\Gamma_b} = n_z \frac{\nu}{\sigma_\psi} (c_\mu^0)^p k^m \kappa^n (z_b + z_0)^{n-1}\]

where \(z_b\) is the distance from boundary, and \(z_0\) is the roughness length.

Parameters:

• function_space – FunctionSpace where the solution belongs
• gls_model – GenericLengthScaleModel object
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.turbulence.ShearFrequencySolver(uv, m2, mu, mv, mu_tmp, relaxation=1.0, minval=1e-12)

Bases: object

Computes vertical shear frequency squared form the given horizontal velocity field.

\[M^2 = \left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\]

Parameters:

• uv (Function) – horizontal velocity field
• m2 (Function) – \(M^2\) field
• mu (Function) – field for x component of \(M^2\)
• mv (Function) – field for y component of \(M^2\)
• mu_tmp (Function) – temporary field
• relaxation (float) – relaxation coefficient for mixing old and new values M2 = relaxation*M2_new + (1-relaxation)*M2_old
• minval (float) – minimum value for \(M^2\)

solve(init_solve=False)

Computes buoyancy frequency

Parameters:init_solve (bool) – Set to True if solving for the first time, skips relaxation

class thetis.turbulence.SmoothVerticalGradSolver(source, solution)

Bases: object

Computes vertical gradient in a smooth(er) way.

The source is first interpolated on P0 field. The gradient is computed as \(G = (G_{P0} + G_{P1DG})/2\).

Parameters:

• source – A Function or expression to differentiate.
• solution – A Function where the solution will be stored.

solve()

Computes the gradient

class thetis.turbulence.TKEEquation(function_space, gls_model, bathymetry=None, v_elem_size=None, h_elem_size=None)

Bases: thetis.equation.Equation

Turbulent kinetic energy equation (14) without advection terms.

Advection of TKE is implemented using the standard tracer equation.

Parameters:

• function_space – FunctionSpace where the solution belongs
• gls_model – GenericLengthScaleModel object
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size

class thetis.turbulence.TKESourceTerm(function_space, gls_model, bathymetry=None, v_elem_size=None, h_elem_size=None)

Bases: thetis.tracer_eq.TracerTerm

Production and destruction terms of the TKE equation (14)

\[F_k = P + B - \varepsilon\]

To ensure positivity we use Patankar-type time discretization: all source terms are treated explicitly and sink terms are treated implicitly. To this end the buoyancy production term \(B\) is split in two:

\[F_k = P + B_{source} + \frac{k^{n+1}}{k^n}(B_{sink} - \varepsilon)\]

with \(B_{source} \ge 0\) and \(B_{sink} < 0\).

Parameters:

• function_space – FunctionSpace where the solution belongs
• gls_model – GenericLengthScaleModel object
• bathymetry (3D Function or Constant) – bathymetry of the domain
• v_elem_size – scalar Function that defines the vertical element size
• h_elem_size – scalar Function that defines the horizontal element size

residual(solution, solution_old, fields, fields_old, bnd_conditions=None)

class thetis.turbulence.TurbulenceModel

Bases: object

Base class for all vertical turbulence models

initialize()

Initialize all turbulence fields

postprocess()

Updates all diagnostic variables that depend on the turbulence state variables.

To be called after updating the turbulence PDEs.

preprocess(init_solve=False)

Computes all diagnostic variables that depend on the mean flow model variables.

To be called before updating the turbulence PDEs.

class thetis.turbulence.VerticalGradSolver(source, solution, solver_parameters=None)

Bases: object

Computes vertical gradient in the weak sense.

Parameters:

• source – A Function or expression to differentiate.
• solution – A Function where the solution will be stored. Must be in P0 space.
• solver_parameters (dict) – PETSc solver options

solve()

Computes the gradient

thetis.turbulence.set_func_max_val(f, maxval)

Sets a minimum value to a Function

thetis.turbulence.set_func_min_val(f, minval)

Sets a minimum value to a Function

thetis.turbulence.timed_region()

Log.Event(type cls, name, klass=None)

thetis.turbulence.timed_stage()

Log.Stage(type cls, name)

thetis.utility module

Utility functions and classes for 3D hydrostatic ocean model

class thetis.utility.ALEMeshUpdater(solver)

Bases: object

Class that handles vertically moving ALE mesh

Mesh geometry is updated to match the elevation field (solver.fields.elev_2d). First the discontinuous elevation field is projected to continuous space, and this field is used to update the mesh coordinates.

This class stores the reference coordinate field and keeps track of the updated mesh coordinates. It also provides a method for computing the mesh velocity from two adjacent elevation fields.

Parameters:solver – FlowSolver object

compute_mesh_velocity_begin()

Stores the current 2D elevation state as the “old” field

compute_mesh_velocity_finalize(c=1.0)

Computes mesh velocity from the elevation difference

Stores the current 2D elevation state as the “new” field, and computes w_mesh using the given time step factor c.

initialize()

Set values for initial mesh (elevation at rest)

update_elem_height()

Updates vertical element size fields

update_mesh_coordinates()

Updates 3D mesh coordinates to match current elev_2d field

elev_2d is first projected to continous space

class thetis.utility.AttrDict(*args, **kwargs)

Bases: dict

Dictionary that provides both self[‘key’] and self.key access to members.

http://stackoverflow.com/questions/4984647/accessing-dict-keys-like-an-attribute-in-python

class thetis.utility.DensitySolver(salinity, temperature, density, eos_class)

Bases: object

Computes density from salinity and temperature using the equation of state.

Water density is defined as

\[\rho = \rho'(T, S, p) + \rho_0\]

This method computes the density anomaly \(\rho'\).

Density is computed point-wise assuming that temperature, salinity and density are in the same function space.

Parameters:

• salinity (Function) – water salinity field
• temperature (Function) – water temperature field
• density (Function) – water density field
• eos_class (EquationOfState) – equation of state that defines water density

solve()

Compute density

class thetis.utility.DensitySolverWeak(salinity, temperature, density, eos_class)

Bases: object

Computes density from salinity and temperature using the equation of state.

Water density is defined as

\[\rho = \rho'(T, S, p) + \rho_0\]

This method computes the density anomaly \(\rho'\).

Density is computed in a weak sense by projecting the analytical expression on the density field.

Parameters:

• salinity (Function) – water salinity field
• temperature (Function) – water temperature field
• density (Function) – water density field
• eos_class (EquationOfState) – equation of state that defines water density

ensure_positive_salinity()

make sure salinity is not negative

some EOS depend on sqrt(salt).

solve()

Compute density

class thetis.utility.ElementContinuity(horizontal, vertical)

Bases: tuple

A named tuple describing the continuity of an element in the horizontal/vertical direction.

The field value is one of “cg”, “hdiv”, or “dg”.

horizontal

Alias for field number 0

vertical

Alias for field number 1

class thetis.utility.EquationOfState

Bases: object

Base class of all equation of state objects

compute_rho(s, th, p, rho0=0.0)

Compute sea water density.

Parameters:

• s (float or numpy.array) – Salinity expressed on the Practical Salinity Scale 1978
• th (float or numpy.array) – Potential temperature in Celsius, referenced to pressure p_r = 0 dbar.
• p (float or numpy.array) – Pressure in decibars (1 dbar = 1e4 Pa)
• rho0 (float) – Optional reference density. If provided computes \(\rho' = \rho(S, Th, p) - \rho_0\)

Returns:

water density

Return type:

float or numpy.array

All pressures are gauge pressures: they are the absolute pressures minus standard atmosperic pressure 10.1325 dbar.

eval(s, th, p, rho0=0.0)

Compute sea water density.

class thetis.utility.ExpandFunctionTo3d(input_2d, output_3d, elem_height=None)

Bases: object

Copy a 2D field to 3D

Copies a field from 2D mesh to 3D mesh, assigning the same value over the vertical dimension. Horizontal function spaces must be the same.

U = FunctionSpace(mesh, 'DG', 1)
U_2d = FunctionSpace(mesh2d, 'DG', 1)
func2d = Function(U_2d)
func3d = Function(U)
ex = ExpandFunctionTo3d(func2d, func3d)
ex.solve()

Parameters:

• input_2d (Function) – 2D source field
• output_3d (Function) – 3D target field
• elem_height – scalar Function in 3D mesh that defines the vertical element size. Needed only in the case of HDiv function spaces.

solve()

class thetis.utility.FieldDict(*args, **kwargs)

Bases: thetis.utility.AttrDict

AttrDict that checks that all added fields have proper meta data.

Values can be either Function or Constant objects.

class thetis.utility.FrozenClass

Bases: object

A class where creating a new attribute will raise an exception if _isfrozen == True

class thetis.utility.JackettEquationOfState

Bases: thetis.utility.EquationOfState

Equation of State according of Jackett et al. (2006) for computing sea water density.

(16)\[\rho = \rho'(T, S, p) + \rho_0\]

\(\rho'(T, S, p)\) is a nonlinear rational function.

Jackett et al. (2006). Algorithms for Density, Potential Temperature, Conservative Temperature, and the Freezing Temperature of Seawater. Journal of Atmospheric and Oceanic Technology, 23(12):1709-1728. http://dx.doi.org/10.1175/JTECH1946.1

a = array([ 9.99840854e+02, 7.34716259e+00, -5.32112318e-02, 3.64924391e-04, 2.58805710e+00, -6.71682828e-03, 1.92032021e-03, 1.17982637e-02, 9.89202193e-08, 4.69966428e-06, -2.58621871e-08, -3.29214140e-12])

b = array([ 1.00000000e+00, 7.28152101e-03, -4.47872655e-05, 3.38510030e-07, 1.36512024e-10, 1.76321267e-03, -8.80665833e-06, -1.88326894e-10, 5.74637767e-06, 1.47162755e-09, 6.71032463e-06, -2.44616980e-17, -9.15344176e-18])

compute_rho(s, th, p, rho0=0.0)

Compute sea water density.

Parameters:

• s (float or numpy.array) – Salinity expressed on the Practical Salinity Scale 1978
• th (float or numpy.array) – Potential temperature in Celsius, referenced to pressure p_r = 0 dbar.
• p (float or numpy.array) – Pressure in decibars (1 dbar = 1e4 Pa)
• rho0 (float) – Optional reference density. If provided computes \(\rho' = \rho(S, Th, p) - \rho_0\)

Returns:

water density

Return type:

float or numpy.array

All pressures are gauge pressures: they are the absolute pressures minus standard atmosperic pressure 10.1325 dbar.

eval(s, th, p, rho0=0.0)

class thetis.utility.LinearEquationOfState(rho_ref, alpha, beta, th_ref, s_ref)

Bases: thetis.utility.EquationOfState

Linear Equation of State for computing sea water density

\[\rho = \rho_{ref} - \alpha (T - T_{ref}) + \beta (S - S_{ref})\]

Parameters:

• rho_ref (float) – reference density
• alpha (float) – thermal expansion coefficient
• beta (float) – haline contraction coefficient
• th_ref (float) – reference temperature
• s_ref (float) – reference salinity

compute_rho(s, th, p, rho0=0.0)

Compute sea water density.

Parameters:

• s (float or numpy.array) – Salinity expressed on the Practical Salinity Scale 1978
• th (float or numpy.array) – Potential temperature in Celsius
• p (float or numpy.array) – Pressure in decibars (1 dbar = 1e4 Pa)
• rho0 (float) – Optional reference density. If provided computes \(\rho' = \rho(S, Th, p) - \rho_0\)

Returns:

water density

Return type:

float or numpy.array

Pressure is ingored in this equation of state.

eval(s, th, p, rho0=0.0)

class thetis.utility.Mesh3DConsistencyCalculator(solver_obj)

Bases: object

Computes a hydrostatic consistency criterion metric on the 3D mesh.

Let \(\Delta x\) and \(\Delta z\) denote the horizontal and vertical element sizes. The hydrostatic consistency criterion (HCC) can then be expressed as

\[R = \frac{|\nabla h| \Delta x}{\Delta z} < 1\]

where \(\nabla h\) is the bathymetry gradient (or gradient of the internal horizontal facet).

Violating the hydrostatic consistency criterion leads to internal pressure gradient errors. In practice one can violate the \(R < 1\) condition without jeopardizing numerical stability; typically \(R < 5\). Mesh consistency can be improved by coarsening the vertical mesh, refining the horizontal mesh, or smoothing the bathymetry.

For a 3D prism, let \(\delta z_t,\delta z_b\) denote the maximal \(z\) coordinate difference in the surface and bottom facets, respectively, and \(\Delta z\) the height of the prism. We can then compute \(R\) for the two facets as

\[\begin{split}R_t &= \frac{\delta z_t}{\Delta z} \\ R_b &= \frac{\delta z_b}{\Delta z}\end{split}\]

For a straight prism we have \(R = 0\), and \(R = 1\) in the case where the highest bottom node is at the same level as the lowest surface node.

Parameters:solver_obj – FlowSolver object

solve()

Compute the HCC metric

class thetis.utility.ParabolicViscosity(uv_bottom, bottom_drag, bathymetry, nu, solver_parameters={})

Bases: object

Computes parabolic eddy viscosity profile assuming log layer flow

\[\nu = \kappa u_{bf} \frac{(-z)(h + z_0 + z)}{h + z_0}\]

with

\[u_{bf} = \sqrt{C_D} |\mathbf{u}_b|\]

Parameters:

• uv_bottom (3D Function) – bottom velocity
• bottom_drag (3D Function) – bottom drag field
• bathymetry (3D Function) – bathymetry field
• nu (3D Function) – eddy viscosity field
• solver_parameters (dict) – PETSc solver options

solve()

Computes viscosity and stores it in nu field

class thetis.utility.SmagorinskyViscosity(uv, output, c_s, h_elem_size, max_val, min_val=1e-10, weak_form=True, solver_parameters={})

Bases: object

Computes Smagorinsky subgrid scale horizontal viscosity

This formulation is according to Ilicak et al. (2012) and Griffies and Hallberg (2000).

\[\nu = (C_s \Delta x)^2 |S|\]

with the deformation rate

\[\begin{split}|S| &= \sqrt{D_T^2 + D_S^2} \\ D_T &= \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \\ D_S &= \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\end{split}\]

\(\Delta x\) is the horizontal element size and \(C_s\) is the Smagorinsky coefficient.

To match a certain mesh Reynolds number \(Re_h\) set \(C_s = 1/\sqrt{Re_h}\).

Ilicak et al. (2012). Spurious dianeutral mixing and the role of momentum closure. Ocean Modelling, 45-46(0):37-58. http://dx.doi.org/10.1016/j.ocemod.2011.10.003

Griffies and Hallberg (2000). Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale eddy-permitting ocean models. Monthly Weather Review, 128(8):2935-2946. http://dx.doi.org/10.1175/1520-0493(2000)128%3C2935:BFWASL%3E2.0.CO;2

Parameters:

• uv_3d (3D vector Function) – horizontal velocity
• output (3D scalar Function) – Smagorinsky viscosity field
• c_s (float or Constant) – Smagorinsky coefficient
• h_elem_size (3D scalar Function or Constant) – field that defines the horizontal element size
• max_val (float) – Maximum allowed viscosity. Viscosity will be clipped at this value.
• min_val (float) – Minimum allowed viscosity. Viscosity will be clipped at this value.
• weak_form (bool) – Compute velocity shear by integrating by parts. Necessary for some function spaces (e.g. P0).
• solver_parameters (dict) – PETSc solver options

solve()

Compute viscosity

class thetis.utility.SubFunctionExtractor(input_3d, output_2d, boundary='top', elem_facet=None, elem_height=None)

Bases: object

Extract a 2D sub-function from a 3D function in an extruded mesh

Given 2D and 3D functions,

U = FunctionSpace(mesh, 'DG', 1)
U_2d = FunctionSpace(mesh2d, 'DG', 1)
func2d = Function(U_2d)
func3d = Function(U)

Get surface value:

ex = SubFunctionExtractor(func3d, func2d,
    boundary='top', elem_facet='top')
ex.solve()

Get bottom value:

ex = SubFunctionExtractor(func3d, func2d,
    boundary='bottom', elem_facet='bottom')
ex.solve()

Get value at the top of bottom element:

ex = SubFunctionExtractor(func3d, func2d,
    boundary='bottom', elem_facet='top')
ex.solve()

Parameters:

• input_3d (Function) – 3D source field
• output_2d (Function) – 2D target field
• boundary (str) – ‘top’|’bottom’ Defines whether to extract from the surface or bottom 3D elements
• elem_facet (str) – ‘top’|’bottom’|’average’ Defines which facet of the 3D element is extracted. The ‘average’ computes mean of the top and bottom facets of the 3D element.
• elem_height – scalar Function in 2D mesh that defines the vertical element size. Needed only in the case of HDiv function spaces.

solve()

class thetis.utility.SumFunction

Bases: object

Helper class to keep track of sum of Coefficients.

Initialize empty sum.

get operation returns Constant(0)

add(coeff)

Adds a coefficient to self

get_sum()

Returns a sum of all added Coefficients

class thetis.utility.VelocityMagnitudeSolver(solution, u=None, w=None, min_val=1e-06, solver_parameters={})

Bases: object

Computes magnitude of (u[0],u[1],w) and stores it in solution

Parameters:

• solution (Function) – scalar field for velocity magnitude scalar Function
• u (Function) – horizontal velocity
• w (Function) – vertical velocity
• min_val (float) – minimum value of magnitude. Minimum value of solution will be clipped to this value
• solver_parameters (dict) – PETSc solver options

If u is None computes magnitude of (0,0,w).

If w is None computes magnitude of (u[0],u[1],0).

solve()

Compute the magnitude

class thetis.utility.VerticalIntegrator(input, output, bottom_to_top=True, bnd_value=Constant(FiniteElement('Real', None, 0), 5), average=False, bathymetry=None, elevation=None, solver_parameters={})

Bases: object

Computes vertical integral (or average) of a field.

Parameters:

• input – 3D field to integrate
• output – 3D field where the integral is stored
• bottom_to_top – Defines the integration direction: If True integration is performed along the z axis, from bottom surface to top surface.
• bnd_value – Value of the integral at the bottom (top) boundary if bottom_to_top is True (False)
• average – If True computes the vertical average instead. Requires bathymetry and elevation fields
• bathymetry – 3D field defining the bathymetry
• elevation – 3D field defining the free surface elevation
• solver_parameters (dict) – PETSc solver options

solve()

Computes the integral and stores it in the output field.

class thetis.utility.VerticalVelocitySolver(solution, uv, bathymetry, boundary_funcs={}, solver_parameters={})

Bases: object

Computes vertical velocity diagnostically from the continuity equation

Vertical velocity is obtained from the continuity equation

(17)\[\frac{\partial w}{\partial z} = -\nabla_h \cdot \textbf{u}\]

and the bottom impermeability condition (\(h\) denotes the bathymetry)

\[\begin{split}\textbf{n}_h \cdot \textbf{u} + w n_z &= 0 \quad \forall \mathbf{x} \in \Gamma_{b} \\ \Leftrightarrow w &= -\nabla_h h \cdot \mathbf{u} \quad \forall \mathbf{x} \in \Gamma_{b}\end{split}\]

\(w\) can be solved with the weak form

\[\begin{split}\int_{\Gamma_s} w n_z \varphi dS + \int_{\mathcal{I}_h} \text{avg}(w) \text{jump}(\varphi n_z) dS - \int_{\Omega} w \frac{\partial \varphi}{\partial z} dx = \\ \int_{\Omega} \mathbf{u} \cdot \nabla_h \varphi dx - \int_{\mathcal{I}_h \cup \mathcal{I}_v} \text{avg}(\mathbf{u}) \cdot \text{jump}(\varphi \mathbf{n}_h) dS - \int_{\Gamma_s} \mathbf{u} \cdot \varphi \mathbf{n}_h dS\end{split}\]

where the \(\Gamma_b\) terms vanish due to the bottom impermeability condition.

Parameters:

• solution – w Function
• uv – horizontal velocity Function
• bathymetry – bathymetry Function
• boundary_funcs (dict) – boundary conditions used in the 3D momentum equation. Provides external values of uv (if any).
• solver_parameters (dict) – PETSc solver options

solve()

Compute w

thetis.utility.beta_plane_coriolis_function(latitude, out_function, y_offset=0.0)

Interpolates beta plane Coriolis function to a field

Parameters:

• latitude (float) – latitude in degrees
• out_function – Function where to interpolate
• y_offset (float) – offset (y - y_0) used in Beta-plane approximation. A constant in mesh coordinates.

thetis.utility.beta_plane_coriolis_params(latitude)

Computes beta plane parameters \(f_0,\beta\) based on latitude

Parameters:latitude (float) – latitude in degrees Returns:f_0, beta Return type:float

thetis.utility.comp_tracer_mass_3d(scalar_func)

Computes total tracer mass in the 3D domain

Parameters:scalar_func – scalar Function to integrate

thetis.utility.comp_volume_2d(eta, bath)

Computes volume of the 2D domain as an integral of the elevation field

thetis.utility.comp_volume_3d(mesh)

Computes volume of the 3D domain as an integral

thetis.utility.compute_baroclinic_head(solver)

Computes the baroclinic head \(r\) from the density field

\[r = \frac{1}{\rho_0} \int_{z}^\eta \rho' d\zeta.\]

thetis.utility.compute_bottom_drag(h_b, drag)

Computes bottom drag coefficient (Cd) from the law-of-the wall

\[C_D = \left( \frac{\kappa}{\ln (h_b + z_0)/z_0} \right)^2\]

Parameters:

• h_b (Function) – the height above bed where the bottom velocity is evaluated in the law-of-the-wall fit
• drag (Function) – field where C_D is stored

thetis.utility.compute_bottom_friction(solver, uv_3d, uv_bottom_2d, z_bottom_2d, bathymetry_2d, bottom_drag_2d)

Updates bottom friction related fields for the 3D model

Parameters:

• solver – FlowSolver object
• uv_3d (3D vector Function) – horizontal velocity
• uv_bottom_2d (2D vector Function) – 2D bottom velocity field
• z_bottom_2d (2D scalar Function) – Bottom element z coordinate
• bathymetry_2d (2D scalar Function) – Bathymetry field
• bottom_drag_2d (2D scalar Function) – Bottom grad field

thetis.utility.compute_boundary_length(mesh2d)

Computes the length of the boundary segments in given 2d mesh

thetis.utility.compute_elem_height(zcoord, output)

Computes the element height on an extruded mesh.

Parameters:

• zcoord (Function) – field that contains the z coordinates of the mesh
• output (Function) – field where element height is stored

thetis.utility.create_directory(path, comm=<mpi4py.MPI.Intracomm object>)

Create a directory on disk

Raises IOError if a file with the same name already exists.

thetis.utility.element_continuity(ufl_element)

Return an ElementContinuity instance with the continuity of a given element.

Parameters:ufl_element – The UFL element to determine the continuity of. Returns:A new ElementContinuity instance.

thetis.utility.extrude_mesh_sigma(mesh2d, n_layers, bathymetry_2d, z_stretch_fact=1.0, min_depth=None)

Extrudes a 2d surface mesh with bathymetry data defined in a 2d field.

Generates a uniform terrain following mesh.

Parameters:

• mesh2d – 2D mesh
• n_layers – number of vertical layers
• bathymetry – 2D Function of the bathymetry (the depth of the domain; positive downwards)

thetis.utility.get_facet_mask(function_space, mode='geometric', facet='bottom')

Returns the top/bottom nodes of extruded 3D elements.

Parameters:

• function_space – Firedrake FunctionSpace object
• mode (str) – ‘topological’, to retrieve nodes that lie on the facet, or ‘geometric’ for nodes whose basis functions do not vanish on the facet.
• facet (str) – ‘top’ or ‘bottom’

Note

The definition of top/bottom depends on the direction of the extrusion. Here we assume that the mesh has been extruded upwards (along positive z axis).

thetis.utility.get_horizontal_elem_size_2d(sol2d)

Computes horizontal element size from the 2D mesh

Parameters:sol2d – 2D Function where result is stored

thetis.utility.get_horizontal_elem_size_3d(sol2d, sol3d)

Computes horizontal element size from the 2D mesh, then copies it on a 3D field

Parameters:

• sol2d – 2D Function for the element size field
• sol3d – 3D Function for the element size field

thetis.utility.get_zcoord_from_mesh(zcoord, solver_parameters={})

Evaluates z coordinates from the 3D mesh

Parameters:zcoord – scalar Function where coordinates will be stored

thetis.utility.select_and_move_detectors(mesh, detector_locations, detector_names=None, maximum_distance=0.0)

Select those detectors that are within the domain and/or move them to the nearest cell centre within the domain

Parameters:

• mesh – Defines the domain in which detectors are to be located
• detector_locations – List of x, y locations
• detector_names – List of detector names (optional). If provided, a list of selected locations and a list of selected detector names are returned, otherwise only a list of selected locations is returned
• maximum_distance – Detectors whose initial locations is outside the domain, but for which the nearest cell centre is within the specified distance, are moved to this location. By default a maximum distance of 0.0 is used, i.e no detectors are moved.

thetis.utility.tensor_jump(v, n)

Jump term for vector functions based on the tensor product

\[\text{jump}(\mathbf{u}, \mathbf{n}) = (\mathbf{u}^+ \mathbf{n}^+) + (\mathbf{u}^- \mathbf{n}^-)\]

This is the discrete equivalent of grad(u) as opposed to the vectorial UFL jump operator ufl.jump() which represents div(u).

thetis.utility.timed_region()

Log.Event(type cls, name, klass=None)

thetis.utility.timed_stage()

Log.Stage(type cls, name)

Module contents

thetis.timed_region()

Log.Event(type cls, name, klass=None)

thetis.timed_stage()

Log.Stage(type cls, name)