Source code for fridom.nonhydro.modules.closures.smagorinsky_lilly
import fridom.framework as fr
import fridom.nonhydro as nh
import numpy as np
[docs]
@fr.utils.jaxify
class SmagorinskyLilly(fr.modules.Module):
r"""
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 :math:`\boldsymbol{u}` and a scalar
tracer :math:`\phi` is given by:
.. math::
\Delta \boldsymbol{u} = \nabla \cdot \mathbf{\tau}
.. math::
\Delta \phi = \nabla \cdot \left( \kappa_t \nabla \phi \right)
with the stress tensor :math:`\boldsymbol{\tau}` and turbulent diffusivity
:math:`\kappa_t` given by:
.. math::
\mathbf{\tau} = \nu_t \mathbf{\Sigma}
.. math::
\nu_t = \nu_s + \nu_{\text{background}}
.. math::
\kappa_t = \frac{\nu_s}{\text{Pr}} + \nu_{\text{background}}
where :math:`\mathbf{\Sigma}` is the strain rate tensor given by:
.. math::
\mathbf{\Sigma} = \frac{1}{2} \left(
\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right)
and :math:`\nu_s` is the Smagorinsky viscosity given by:
.. math::
\nu_s = \left( C_s \sqrt[3]{\Delta V} \right)^2
|\mathbf{\Sigma}| \Gamma(\text{Ri})
where :math:`C_s` is the Smagorinsky constant, :math:`\Delta V` is the
grid cell volume, :math:`|\mathbf{\Sigma}|` is the magnitude of the strain
rate tensor, and :math:`\Gamma(\text{Ri})` is the stratification damping
factor given by:
.. math::
\Gamma(\text{Ri}) = \sqrt{1 - \min(\beta \text{Ri}, 1)}
where :math:`\beta` is the buoyancy multiplier and :math:`\text{Ri}` is the
resolved Richardson number given by:
.. math::
\text{Ri} = \frac{N^2}{|\mathbf{\Sigma}|}
where :math:`N^2` is the buoyancy frequency:
.. math::
N^2 = \partial_z b + N^2_{\text{background}}
where :math:`b` is the buoyancy field and :math:`N^2_{\text{background}}`
is the background buoyancy frequency. The magnitude of the strain rate
tensor is given by:
.. math::
|\mathbf{\Sigma}|^2 = \sum_{i=1}^3 \sum_{j=1}^3 \Sigma_{ij}^2
Parameters
----------
`background_viscosity` : `float`, (default=1.05e-6)
The background viscosity for velocity fields.
`background_diffusivity` : `float`, (default=1.46e-7)
The background diffusivity for tracer fields.
`turbulent_prandtl_number` : `float`, (default=1.0)
The turbulent Prandtl number.
`smagorinsky_constant` : `float`, (default=0.16)
The Smagorinsky constant.
`buoyancy_multiplier` : `float | None`, (default=None)
The buoyancy multiplier. If None, the buoyancy multiplier is set to
:math:`1 / \text{turbulent_prandtl_number}`.
"""
name = "Smagorinsky-Lilly"
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def __init__(self,
background_viscosity: float = 1.05e-6,
background_diffusivity: float = 1.46e-7,
turbulent_prandtl_number: float = 1.0,
smagorinsky_constant: float = 0.16,
buoyancy_multiplier: float | None = None
):
super().__init__()
self.background_viscosity = background_viscosity
self.background_diffusivity = background_diffusivity
self.turbulent_prandtl_number = turbulent_prandtl_number
self.smagorinsky_constant = smagorinsky_constant
self.buoyancy_multiplier = buoyancy_multiplier or 1 / turbulent_prandtl_number
return
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@fr.modules.module_method
def setup(self, mset: 'nh.ModelSettings') -> None:
super().setup(mset)
self.filter_width = self.grid.dV**(1/3)
return
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@fr.utils.jaxjit
def smagorinsky_lilly_operator(self, z: nh.State, dz: nh.State) -> nh.State:
self.mset
diff_mod = self.diff_module
ncp = fr.config.ncp
# Compute the velocity gradients
du = diff_mod.grad(z.u)
dv = diff_mod.grad(z.v)
dw = diff_mod.grad(z.w)
# Compute the strain rate tensor
s_11 = du[0]; s_12 = 0.5 * (du[1] + dv[0]); s_13 = 0.5 * (du[2] + dw[0])
s_21 = s_12 ; s_22 = dv[1] ; s_23 = 0.5 * (dv[2] + dw[1])
s_31 = s_13 ; s_32 = s_23 ; s_33 = dw[2]
# Compute the squared magnitude of the strain rate tensor
# ignore the different grid positions of each component of s here
sigma2 = ( (s_11**2 + s_22**2 + s_33**2)
+ 2 * (s_12**2 + s_13**2 + s_23**2) ).arr
# Compute the buoyancy frequency (also ignoring the grid position)
N2 = (diff_mod.diff(z.b, axis=2) + self.mset.N2).arr
# Set the buoyancy frequency to zero where it is negative
N2 = ncp.maximum(N2, 0.0)
# Compute the resolved Richardson number
with np.errstate(divide='ignore', invalid='ignore'):
Ri = N2 / sigma2
# Compute the stratification damping factor
gamma = ncp.sqrt(1 - ncp.minimum(self.buoyancy_multiplier * Ri, 1.0))
# set nan values to 0
gamma = ncp.nan_to_num(gamma, nan=0)
# Compute the smagorinsky viscosity
nu_s = ( (self.smagorinsky_constant * self.filter_width)**2
* ncp.sqrt(sigma2) * gamma )
# Compute the turbulent diffusivities
nu_t = nu_s + self.background_viscosity
kappa_t = ( nu_s / self.turbulent_prandtl_number
+ self.background_diffusivity )
# Compute the stress tensor
tau_11 = s_11 * nu_t; tau_12 = s_12 * nu_t; tau_13 = s_13 * nu_t
tau_21 = tau_12 ; tau_22 = s_22 * nu_t; tau_23 = s_23 * nu_t
tau_31 = tau_13 ; tau_32 = tau_23 ; tau_33 = s_33 * nu_t
# Compute the friction terms
dz.u += diff_mod.div((tau_11, tau_12, tau_13))
dz.v += diff_mod.div((tau_21, tau_22, tau_23))
dz.w += diff_mod.div((tau_31, tau_32, tau_33))
# Compute the mixing terms
for name, field in z.fields.items():
if not field.flags["ENABLE_MIXING"]:
# skip the fields that are not enabled for mixing
continue
# Compute the gradient of the field
df = diff_mod.grad(field)
# multiply the gradient with the turbulent diffusivity
df = tuple(d * kappa_t for d in df)
# Compute the divergence of the gradient
dz.fields[name] += diff_mod.div(df)
return dz
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def update(self, mz: fr.ModelState) -> fr.ModelState:
mz.dz = self.smagorinsky_lilly_operator(mz.z, mz.dz)
return mz
