SadournyAdvection

class fridom.shallowwater.modules.advection.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)	Advect the state vector.
advection(velocity, quantity)	Advect a quantity using the given velocity field.
disable()	Disable the module.
enable()	Enable the module.
is_enabled()	Whether the module is enabled or not.
reset()	Stop and start the module.
setup(mset[, setup_mode])	Set the module up.
start()	Start the module.
stop()	Stop the module.
update(mz)	Update the model state.

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.
is_setup	Whether the module is set up.
mset	The model settings.
name
required_halo	The required halo points for this module.
scaling	A scaling factor for the nonlinear terms (default: 1.0).

name = 'Sadourny Advection'

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

Advect the state vector.