SadournyAdvection

class fridom.shallowwater.modules.sadourny_advection.SadournyAdvection

Bases: AdvectionBase

Advection scheme based on Sadourny [1975] that conserves the total energy.

The nonlinear advection terms for the shallow water equations are given by:

\[ \begin{align}\begin{aligned}\partial_t \boldsymbol{u} = - (\boldsymbol{u} + \boldsymbol{u}_b) \cdot \nabla \boldsymbol{u} = - \underset{\neg}{\boldsymbol{u}} \zeta - \frac{1}{2} \nabla \boldsymbol{u}^2 - \nabla \left( \boldsymbol{u_b} \cdot \boldsymbol{u} \right)\\\partial_t p = - \nabla \left\[ (\boldsymbol{u} + \boldsymbol{u}_b) p \right\]\\\partial_t C = - (\boldsymbol{u} + \boldsymbol{u}_b) \cdot \nabla C\end{aligned}\end{align} \]

where \(\boldsymbol{u_b}\) is a divergence free background flow that can be set with the background attribute of this module, \(\zeta\) is the relative vorticity, and \(C\) is a passive tracer. We express the rotational part of the momentum advection with the potential vorticity \(q\):

\[\underset{\neg}{\boldsymbol{u}} \zeta = \underset{\neg}{\boldsymbol{f_u}} q\]

with

\[\boldsymbol{f_u} = (c^2 + Ro~ p) \boldsymbol{u} ~, \quad q = \frac{\zeta}{c^2 + Ro~ p}\]

__init__() → None

Methods

__init__()
advect_state(z, dz)
advection(velocity, quantity)	Advect a quantity using the given velocity field.
disable()	Enabling the module means that it will be executed at each time step.
enable()	Enabling the module means that it will be executed at each time step.
is_enabled()	Return whether the module is enabled or not.
reset()	Stop and start the module.
setup(mset)	Start the module
start()	Start the module
stop()	Stop the module
update(mz)	Update the module

Attributes

background	The background state.
diff_module	The differentiation module to be used by this module.
disable_nonlinear	Whether to disable advection by the state vector itself.
grid	The grid of the model settings
info	Return a dictionary with information about the time stepper.
interp_module	The interpolation module to be used by this module.
mset	The model settings
name
required_halo
scaling	A scaling factor for the nonlinear terms (default: 1.0)

name = 'Sadourny Advection'

setup(mset: ModelSettings) → None

Start the module

Description

This method is called by the ModelSettings.setup() and sets the ModelSettings as well as the differentiation and interpolation modules.

advect_state(z: State, dz: State) → State