SmagorinskyLilly

class fridom.nonhydro.modules.closures.smagorinsky_lilly.SmagorinskyLilly(background_viscosity: float = 1.05e-06, background_diffusivity: float = 1.46e-07, turbulent_prandtl_number: float = 1.0, smagorinsky_constant: float = 0.16, buoyancy_multiplier: float | None = None)

Bases: Module

A Smagorinsky-Lilly closure model for the non-hydrostatic model.

Description

The Smagorinsky-Lilly model is a subgrid-scale turbulence model proposed by Smagorinsky (1963) and Lilly (1962). This implementation is based on the implementation in Oceananigans (https://clima.github.io/OceananigansDocumentation/v0.91.5/appendix/library/#Oceananigans.TurbulenceClosures.SmagorinskyLilly-Union{Tuple{},%20Tuple{TD},%20Tuple{TD,%20Any}}%20where%20TD).

The closure for the velocity field \(\boldsymbol{u}\) and a scalar tracer \(\phi\) is given by:

\[\Delta \boldsymbol{u} = \nabla \cdot \mathbf{\tau}\]

\[\Delta \phi = \nabla \cdot \left( \kappa_t \nabla \phi \right)\]

with the stress tensor \(\boldsymbol{\tau}\) and turbulent diffusivity \(\kappa_t\) given by:

\[\mathbf{\tau} = \nu_t \mathbf{\Sigma}\]

\[\nu_t = \nu_s + \nu_{\text{background}}\]

\[\kappa_t = \frac{\nu_s}{\text{Pr}} + \nu_{\text{background}}\]

where \(\mathbf{\Sigma}\) is the strain rate tensor given by:

\[\mathbf{\Sigma} = \frac{1}{2} \left( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right)\]

and \(\nu_s\) is the Smagorinsky viscosity given by:

\[\nu_s = \left( C_s \sqrt[3]{\Delta V} \right)^2 |\mathbf{\Sigma}| \Gamma(\text{Ri})\]

where \(C_s\) is the Smagorinsky constant, \(\Delta V\) is the grid cell volume, \(|\mathbf{\Sigma}|\) is the magnitude of the strain rate tensor, and \(\Gamma(\text{Ri})\) is the stratification damping factor given by:

\[\Gamma(\text{Ri}) = \sqrt{1 - \min(\beta \text{Ri}, 1)}\]

where \(\beta\) is the buoyancy multiplier and \(\text{Ri}\) is the resolved Richardson number given by:

\[\text{Ri} = \frac{N^2}{|\mathbf{\Sigma}|}\]

where \(N^2\) is the buoyancy frequency:

\[N^2 = \partial_z b + N^2_{\text{background}}\]

where \(b\) is the buoyancy field and \(N^2_{\text{background}}\) is the background buoyancy frequency. The magnitude of the strain rate tensor is given by:

\[|\mathbf{\Sigma}|^2 = \sum_{i=1}^3 \sum_{j=1}^3 \Sigma_{ij}^2\]

Parameters

background_viscosityfloat, (default=1.05e-6)

The background viscosity for velocity fields.

background_diffusivityfloat, (default=1.46e-7)

The background diffusivity for tracer fields.

turbulent_prandtl_numberfloat, (default=1.0)

The turbulent Prandtl number.

smagorinsky_constantfloat, (default=0.16)

The Smagorinsky constant.

buoyancy_multiplierfloat | None, (default=None)

The buoyancy multiplier. If None, the buoyancy multiplier is set to \(1 / \text{turbulent_prandtl_number}\).

__init__(background_viscosity: float = 1.05e-06, background_diffusivity: float = 1.46e-07, turbulent_prandtl_number: float = 1.0, smagorinsky_constant: float = 0.16, buoyancy_multiplier: float | None = None) → None

Methods

__init__([background_viscosity, ...])
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.
smagorinsky_lilly_operator(z, dz)
start()	Start the module.
stop()	Stop the module.
update(mz)	Update the model state.

Attributes

diff_module	The differentiation module to be used by this module.
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.

Examples using fridom.nonhydro.modules.closures.SmagorinskyLilly

Convection and Closures

Convection and Closures

Rayleigh-Bénard Convection

Rayleigh-Bénard Convection

Rayleigh-Taylor Instability

Rayleigh-Taylor Instability

Symmetric Instability

Symmetric Instability

name = 'Smagorinsky-Lilly'

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

update(mz: ModelState) → ModelState

Update the model state.

Description

This method is called by the model at each time step. Child classes should overwrite this method to update the module. Make sure to decorate the method with the @module_method decorator.

Parameters

mzfr.ModelState

The model state at the current time step.

Returns

fr.ModelState

The updated model state.