r"""
:math:`k-\varepsilon` closure parameters, states and computation functions.
This module contains the implementation of the :math:`k-\varepsilon` model described as a GLS
case as in [1]_ as a :class:`~closure.Closure` instance. The model was traduced from Frotran to JAX
with the work of Florian Lemarié and Manolis Perrot [2]_, the translation was done in part using the
work of Anthony Zhou, Linnia Hawkins and Pierre Gentine [3]_. The parameters of the closure are
available in the :class:`KepsParameters` class, the closure state in :class:`KepsState` class. The
function :attr:`keps_step` compute one time-step of the closure, which means that it computes the
eddy-diffusivity and viscosity. The module contains other functions that are used by this main one.
These classes and the function step can be obtained by the prefix :code:`tunax.closures.k_epsilon`
or directly by :code:`tunax.closures`.
References
----------
.. [1] L. Umlauf and H. Burchard. A generic length-scale equation for geophysical turbulence models
(2003). Journal of Marine Research 61 pp. 235-265. doi : `10.1357/002224003322005087
<https://www.semanticscholar.org/paper/
A-generic-length-scale-equation-for-geophysical-Umlauf-Burchard/
24fd6403615fc7a6c5d9b6156e4f1e8d4d280af2>`_.
.. [2] M. Perrot and F. Lemarié. Energetically consistent Eddy-Diffusivity Mass-Flux convective
schemes. Part I: Theory and Models (2024). url : `hal.science/hal-04439113
<https://hal.science/hal-04439113>`_.
.. [3] A. Zhou, L. Hawkins and P. Gentine. Proof-of-concept: Using ChatGPT to Translate and
Modernize an Earth System Model from Fortran to Python/JAX (2024). url :
`arxiv.org/abs/2405.00018 <https://arxiv.org/abs/2405.00018>`_.
"""
from __future__ import annotations
from typing import Tuple, cast
import equinox as eqx
import jax.numpy as jnp
from jax import lax
from tunax.case import CaseTracable
from tunax.space import Grid, State, ArrNz, ArrNzp1
from tunax.functions import FloatJax, tridiag_solve, add_boundaries
from tunax.closure import ClosureParametersAbstract, ClosureStateAbstract
[docs]
class KepsParameters(ClosureParametersAbstract):
r"""
Parameters and constants for :math:`k-\varepsilon`.
The first 19 attributes are the parameters of :math:`k-\varepsilon` that may be calibrated.
This class also contains some physical constants used in the closure computing. The next 16
attirbutes are some physical parameters for k-epsilon that can be modified but that are not
specially supposed to be modified. The last 19 attributes are the one for the stability function
that are computed from the parameters of :math:`k-\varepsilon`. The constructor of the class
takes as parameters the 19 parameters of the closure and the 16 physical parameters, but not the
last 19 stability functions for parameters.
Attributes
----------
c1 : float, default=5.
:math:`k-\varepsilon` parameter :math:`c_1` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c2 : float, default=0.8
:math:`k-\varepsilon` parameter :math:`c_2` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c3 : float, default=1.968
:math:`k-\varepsilon` parameter :math:`c_3` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c4 : float, default=1.136
:math:`k-\varepsilon` parameter :math:`c_4` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c5 : float, default=0.
:math:`k-\varepsilon` parameter :math:`c_5` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c6 : float, default=0.4
:math:`k-\varepsilon` parameter :math:`c_6` for the dissipation of the corelation tensor
pressure/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cb1 : float, default=5.95
:math:`k-\varepsilon` parameter :math:`c_{b1}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cb2 : float, default=.6
:math:`k-\varepsilon` parameter :math:`c_{b2}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cb3 : float, default=1.
:math:`k-\varepsilon` parameter :math:`c_{b3}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cb4 : float, default=0.
:math:`k-\varepsilon` parameter :math:`c_{b4}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cb5 : float, default=0.3333
:math:`k-\varepsilon` parameter :math:`c_{b5}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
cbb : float, default=.72
:math:`k-\varepsilon` parameter :math:`c_{bb}` for the dissipation of the corelation tensor
buoyancy/velocity (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c_mu0 : float, default=0.5477
:math:`k-\varepsilon` parameter :math:`c_\mu^0` which links the mixing length to the
dissipation (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
sig_k : float, default=1.
:math:`k-\varepsilon` parameter :math:`\sigma_k` Schmit number for the dissipation of TKE
(Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
sig_eps : float, default=1.3
:math:`k-\varepsilon` parameter :math:`\sigma_\varepsilon` Schmit number for the dissipation
of :math:`\varepsilon` (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c_eps1 : float, default=1.44
:math:`k-\varepsilon` parameter :math:`c_{\varepsilon 1}` correction of the
:math:`\varepsilon` equation (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c_eps2 : float, default=1.92
:math:`k-\varepsilon` parameter :math:`c_{\varepsilon 2}` correction of the
:math:`\varepsilon` equation (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c_eps3m : float, default=-0.4
:math:`k-\varepsilon` parameter :math:`c_{\varepsilon 3}^-` correction of the
:math:`\varepsilon` equation (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
c_eps3p : float, default=1.
:math:`k-\varepsilon` parameter :math:`c_{\varepsilon 3}^+` correction of the
:math:`\varepsilon` equation (Umlauf and Burchard notations) :math:`[\text{dimensionless}]`.
chk_grav : float, default=1400.
Charnock coefficient times gravity :math:`[\text{dimensionless}]`.
galp: float, default=0.53
Parameter for Galperin mixing length limitation :math:`[\text{dimensionless}]`.
z0s_min : float, default=1e-2
Minimal surface roughness length :math:`[\text m]`.
z0b_min : float, default=1e-4
Minimal bottom roughness length :math:`[\text m]`.
z0b : float, default=1e-14
Bottom roughness length :math:`[\text m]`.
akt_min : float, default=1e-5
Minimal and initialization value of eddy-diffusivity
:math:`\left[\text m^2 \cdot \text s^{-1} \right]`.
akv_min : float, default=1e-4
Minimal and initialization value of eddy-viscosity
:math:`\left[\text m^2 \cdot \text s^{-1} \right]`.
tke_min : float, default=1e-6
Minimal and initialization value of turbulent kinetic energy (TKE)
:math:`\left[\text m^3 \cdot \text s^{-2} \right]`.
eps_min : float, default=1e-12
Minimal and initialization value of TKE dissipation
:math:`\left[\text m^2 \cdot \text s^{-3} \right]`.
c_mu_min : float, default=0.1
Minimal and initialization value of :math:`c_\mu` in GLS formalisim
:math:`[\text{dimensionless}]`.
c_mu_prim_min : float, default=0.1
Minimal and initialization value of `c_\mu'` in GLS formalisim
:math:`[\text{dimensionless}]`.
dir_sfc: bool, default=False
Apply a Dirichlet boundary condition at the surface for TKE, else
apply a Neumann boundary condition.
dir_btm: bool, default=True
Apply a Dirichlet boundary condition at the bottom for TKE, else
apply a Neumann boundary condition.
gls_p : float, default=3
GLS coefficient :math:`p` to define :math:`k-\varepsilon` :math:`[\text{dimensionless}]`.
gls_m : float, default=1.5
GLS coefficient :math:`m` to define :math:`k-\varepsilon` :math:`[\text{dimensionless}]`.
gls_n : float, default=-1
GLS coefficient :math:`n` to define :math:`k-\varepsilon` :math:`[\text{dimensionless}]`.
sf_d0 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_d1 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_d2 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_d3 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_d4 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_d5 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_n0 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_n1 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_n2 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_nb0 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_nb1 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
sf_nb2 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am0 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am1 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am2 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am3 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am4 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am5 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
lim_am6 : float (not a parameter, computed from the above attributes)
Limitation coefficient for :math:`k-\varepsilon` computed from the parameters
:math:`[\text{dimensionless}]`.
"""
# k-epsilon coefficients (Umlauf and Burchard notations)
c1: float
c2: float
c3: float
c4: float
c5: float
c6: float
cb1: float
cb2: float
cb3: float
cb4: float
cb5: float
cbb: float
c_mu0: float
sig_k: float
sig_eps: float
c_eps1: float
c_eps2: float
c_eps3m: float
c_eps3p: float
# physical constants
chk_grav: float
galp: float
z0s_min: float
z0b_min: float
z0b: float
akt_min: float
akv_min: float
tke_min: float
eps_min: float
c_mu_min: float
c_mu_prim_min: float
# physical case_tracable
dir_sfc: bool
dir_btm: bool
# GLS coefficient for k-epsilon
gls_p: float
gls_m: float
gls_n: float
# limitation coefficients
sf_d0: float
sf_d1: float
sf_d2: float
sf_d3: float
sf_d4: float
sf_d5: float
sf_n0: float
sf_n1: float
sf_n2: float
sf_nb0: float
sf_nb1: float
sf_nb2: float
lim_am0: float
lim_am1: float
lim_am2: float
lim_am3: float
lim_am4: float
lim_am5: float
lim_am6: float
def __init__(
self,
c1: float = 5.,
c2: float = .8,
c3: float = 1.968,
c4: float = 1.136,
c5: float = 0.,
c6: float = .4,
cb1: float = 5.95,
cb2: float = .6,
cb3: float = 1.,
cb4: float = 0.,
cb5: float = 0.3333,
cbb: float = 0.72,
c_mu0: float = 0.5477,
sig_k: float = 1.,
sig_eps: float = 1.3,
c_eps1: float = 1.44,
c_eps2: float = 1.92,
c_eps3m: float = -.4,
c_eps3p: float = 1.,
chk_grav: float = 1400.,
galp: float = .53,
z0s_min: float = 1e-2,
z0b_min: float = 1e-2,
z0b: float = 1e-14,
akt_min: float = 1e-5,
akv_min: float = 1e-4,
tke_min: float = 1e-6,
eps_min: float = 1e-12,
c_mu_min: float = .1,
c_mu_prim_min: float = .1,
dir_sfc: bool = False,
dir_btm: bool = True,
gls_p: float = 3,
gls_m: float = 1.5,
gls_n: float = -1
):
# initialisation of parameters
self.c1 = c1
self.c2 = c2
self.c3 = c3
self.c4 = c4
self.c5 = c5
self.c6 = c6
self.cb1 = cb1
self.cb2 = cb2
self.cb3 = cb3
self.cb4 = cb4
self.cb5 = cb5
self.cbb = cbb
self.c_mu0 = c_mu0
self.sig_k = sig_k
self.sig_eps = sig_eps
self.c_eps1 = c_eps1
self.c_eps2 = c_eps2
self.c_eps3m = c_eps3m
self.c_eps3p = c_eps3p
self.chk_grav = chk_grav
self.galp = galp
self.z0s_min = z0s_min
self.z0b_min = z0b_min
self.z0b = z0b
self.akt_min = akt_min
self.akv_min = akv_min
self.tke_min = tke_min
self.eps_min = eps_min
self.c_mu_min = c_mu_min
self.c_mu_prim_min = c_mu_prim_min
self.dir_sfc = dir_sfc
self.dir_btm = dir_btm
self.gls_p = gls_p
self.gls_m = gls_m
self.gls_n = gls_n
# stability function for limitation coefficients
a1 = .66666666667 - .5*self.c2
a2 = 1 - .5*self.c3
a3 = 1 - .5*self.c4
a5 = .5 - .5*self.c6
nn = .5*self.c1
nb = self.cb1
ab1 = 1 - self.cb2
ab2 = 1 - self.cb3
ab3 = 2*(1 - self.cb4)
ab5 = 2*self.cbb*(1-self.cb5)
sf_d0 = 36*nn*nn*nn*nb*nb
sf_d1 = 84*a5*ab3*nn*nn*nb + 36*ab5*nn*nn*nn*nb
sf_d2 = 9*(ab2*ab2 - ab1*ab1)*nn*nn*nn - 12*(a2*a2 - 3*a3*a3)*nn*nb*nb
sf_d3 = 12*a5*ab3*(a2*ab1 - 3*a3*ab2)*nn + 12*a5*ab3*(a3*a3 - a2*a2)*nb + \
12*ab5*(3*a3*a3 - a2*a2)*nn*nb
sf_d4 = 48*a5*a5*ab3*ab3*nn + 36*a5*ab3*ab5*nn*nn
sf_d5 = 3*(a2*a2 - 3*a3*a3)*(ab1*ab1 - ab2*ab2)*nn
sf_n0 = 36*a1*nn*nn*nb*nb
sf_n1 = -12*a5*ab3*(ab1+ab2)*nn*nn + 8*a5*ab3*(6*a1-a2-3*a3)*nn*nb + 36*a1*ab5*nn*nn*nb
sf_n2 = 9*a1*(ab2*ab2 - ab1*ab1)*nn*nn
sf_nb0 = 12*ab3*nn*nn*nn*nb
sf_nb1 = 12*a5*ab3*ab3*nn*nn
sf_nb2 = 9*a1*ab3*(ab1 - ab2)*nn*nn + (6*a1*(a2 - 3*a3) - 4*(a2*a2 - 3*a3*a3))*ab3*nn*nb
lim_am0 = sf_d0*sf_n0
lim_am1 = sf_d0*sf_n1 + sf_d1*sf_n0
lim_am2 = sf_d1*sf_n1 + sf_d4*sf_n0
lim_am3 = sf_d4*sf_n1
lim_am4 = sf_d2*sf_n0
lim_am5 = sf_d2*sf_n1 + sf_d3*sf_n0
lim_am6 = sf_d3*sf_n1
self.sf_d0 = sf_d0
self.sf_d1 = sf_d1
self.sf_d2 = sf_d2
self.sf_d3 = sf_d3
self.sf_d4 = sf_d4
self.sf_d5 = sf_d5
self.sf_n0 = sf_n0
self.sf_n1 = sf_n1
self.sf_n2 = sf_n2
self.sf_nb0 = sf_nb0
self.sf_nb1 = sf_nb1
self.sf_nb2 = sf_nb2
self.lim_am0 = lim_am0
self.lim_am1 = lim_am1
self.lim_am2 = lim_am2
self.lim_am3 = lim_am3
self.lim_am4 = lim_am4
self.lim_am5 = lim_am5
self.lim_am6 = lim_am6
[docs]
class KepsState(ClosureStateAbstract):
r"""
Define the state of the water column for the :math:`k-\varepsilon` model.
The first initilisation is done from the minimal values of the different variables given in an
instance of :class:'KepsParameters`.
Parameters
----------
grid : Grid
cf. attribute.
keps_params : KepsParameters
Used to define the initialization values of the variables.
Attributes
----------
grid : Grid
Geometry of the water column, should be the same than for the :class:`~space.State` instance
used in the model.
akt : float :class:`~jax.Array` of shape (nz+1)
Eddy-diffusivity on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
akv : float :class:`~jax.Array` of shape (nz+1)
Eddy-viscosity on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
tke : float :class:`~jax.Array` of shape (nz+1)
Turbulent kinetic energy (TKE) denoted :math:`k` on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]`.
eps : float :class:`~jax.Array` of shape (nz+1)
TKE dissipation denoted :math:`\varepsilon` on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
c_mu : float :class:`~jax.Array` of shape (nz+1)
:math:`c_\mu` in GLS formalisim on the interfaces of the cells
:math:`[\text{dimensionless}]`.
c_mu_prim : float :class:`~jax.Array` of shape (nz+1)
:math:`c_\mu'` in GLS formalisim on the interfaces of the cells
:math:`[\text{dimensionless}]`.
"""
grid: Grid
akt: ArrNzp1
akv: ArrNzp1
tke: ArrNzp1
eps: ArrNzp1
c_mu: ArrNzp1
c_mu_prim: ArrNzp1
def __init__(self, grid: Grid, keps_params: KepsParameters) -> None:
self.grid = grid
nz = grid.nz
self.akt = jnp.full(nz+1, keps_params.akt_min)
self.akv = jnp.full(nz+1, keps_params.akv_min)
self.tke = jnp.full(nz+1, keps_params.tke_min)
self.eps = jnp.full(nz+1, keps_params.eps_min)
self.c_mu = jnp.full(nz+1, keps_params.c_mu_min)
self.c_mu_prim = jnp.full(nz+1, keps_params.c_mu_prim_min)
[docs]
def keps_step(
state: State,
keps_state: KepsState,
dt: float,
keps_params: KepsParameters,
case_tracable: CaseTracable
) -> KepsState:
r"""
Run one time-step of the :math:`k-\varepsilon` closure.
The purpose of this function is to get the eddy-diffusivity and eddy-viscosity at the next time-
step. It works in 3 steps
1. The Brunt–Väisälä frequency and the shear is computed from the :code:`state` and the boundary
conditions are computed.
2. The equations on :math:`k` and :math:`\varepsilon` are solved and their values are computed
for the next time step.
3. The eddy-diffusivity and viscosity are computed as diagnostic variables and the
:math:`keps_state` is updated.
Parameters
----------
state : State
Current state of the water column.
keps_state : KepsState
Current state of the water column for the variables used by :math:`k-\varepsilon`.
dt : float
Time-step of the forward model :math:`[\text s]`.
keps_params: KepsParameters
Values of the parameters used by :math:`k-\varepsilon` (time-independant).
case_tracable : CaseTracable
Physical parameters and forcings of the model run.
Returns
-------
keps_state : KepsState
State of the water column for the variables used by :math:`k-\varepsilon` at the next
time-step.
"""
akt = keps_state.akt
akv = keps_state.akv
tke = keps_state.tke
eps = keps_state.eps
c_mu = keps_state.c_mu
c_mu_prim = keps_state.c_mu_prim
u = state.u
v = state.v
hz = state.grid.hz
# prognostic computations
_, bvf = compute_eos(state, case_tracable)
shear2 = compute_shear(state, u, v)
tke_sfc_bc, tke_btm_bc, eps_sfc_bc, eps_btm_bc = compute_tke_eps_bc(
tke, hz, keps_params, case_tracable
)
# integrations
tke_new = advance_turb(
akt, akv, tke, tke, eps, c_mu, c_mu_prim, bvf, shear2, hz, dt,
tke_sfc_bc, tke_btm_bc, eps_sfc_bc, eps_btm_bc, keps_params, True
)
eps_new = advance_turb(
akt, akv, tke, tke_new, eps, c_mu, c_mu_prim, bvf, shear2, hz, dt,
tke_sfc_bc, tke_btm_bc, eps_sfc_bc, eps_btm_bc, keps_params, False
)
# diagnostic variables
akt_new, akv_new, eps_new, c_mu_new, c_mu_prim_new = compute_diag(
tke_new, eps_new, bvf, shear2, keps_params
)
keps_state = eqx.tree_at(lambda t: t.akv, keps_state, akv_new)
keps_state = eqx.tree_at(lambda t: t.akt, keps_state, akt_new)
keps_state = eqx.tree_at(lambda t: t.tke, keps_state, tke_new)
keps_state = eqx.tree_at(lambda t: t.eps, keps_state, eps_new)
keps_state = eqx.tree_at(lambda t: t.c_mu, keps_state, c_mu_new)
keps_state = eqx.tree_at(lambda t: t.c_mu_prim, keps_state, c_mu_prim_new)
return keps_state
[docs]
def compute_eos(state: State, case: CaseTracable) -> Tuple[ArrNzp1, ArrNz]:
r"""
Compute density anomaly and Brunt–Väisälä frequency.
Prognostic computation via linear Equation Of State (EOS) :
:math:`\rho = \rho_0(1-\alpha (T-T_0) + \beta (S-S_0))`
:math:`N^2 = - \dfrac g {\rho_0} \partial_z \rho`
Parameters
----------
state : State
Current state of the water column.
case : CaseTracable
Physical parameters and forcings of the model run.
Returns
-------
bvf : float :class:`~jax.Array` of shape (nz+1)
Brunt–Väisälä frequency squared :math:`N^2` on cell interfaces
:math:`\left[\text s^{-2}\right]`.
rho : Float[Array, 'nz']
Density anomaly :math:`\rho` on cell interfaces
:math:`\left[\text {kg} \cdot \text m^{-3}\right]`
Raises
------
ValueError
If the value of case.eos_tracers is not one of {'t', 's', 'ts', 'b'}.
"""
rho0 = case.rho0
match case.eos_tracers:
case 't':
rho = rho0 * (1. - case.alpha*(jnp.array(state.t)-case.t_rho_ref))
case 's':
rho = rho0 * (1. + case.beta*(jnp.array(state.s)-case.s_rho_ref))
case 'ts':
rho = rho0 * (1. - case.alpha*(jnp.array(state.t)-case.t_rho_ref) + \
case.beta*(jnp.array(state.s)-case.s_rho_ref))
case 'b':
rho = rho0*(1-jnp.array(state.b)/case.grav)
case _:
raise ValueError("The attribute Case.eos_tracers must be one of {'t', 's', 'ts', 'b'}.")
cff = 1./(state.grid.zr[1:]-state.grid.zr[:-1])
bvf_in = - cff*case.grav/rho0 * (rho[1:]-rho[:-1])
bvf = add_boundaries(0., bvf_in, cast(float, bvf_in[-1]))
return rho, bvf
[docs]
def compute_shear(
state: State,
u_np1: ArrNz,
v_np1: ArrNz
) -> ArrNzp1:
r"""
Compute shear production term for TKE equation.
The prognostic equations are
:math:`S_h^2 = \partial_Z U^n \cdot \partial_z U^{n+1/2}`
where :math:`U^{n+1/2}` is the mean between :math:`U^n` and :math:`U^{n+1}`.
Parameters
----------
state : State
Current state of the water column.
u_np1 : float :class:`~jax.Array` of shape (nz)
Zonal velocity on the center of the cells at the next time step
:math:`\left[\text m \cdot \text s^{-1}\right]`.
v_np1 : float :class:`~jax.Array` of shape (nz)
Meridional velocity on the center of the cells at the next time step
:math:`\left[\text m \cdot \text s^{-1}\right]`.
Returns
-------
shear2 : float :class:`~jax.Array` of shape (nz+1)
Shear production squared :math:`S_h^2` on cell interfaces
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
"""
u_n = state.u
v_n = state.v
zr = state.grid.zr
cff = 1.0 / (zr[1:] - zr[:-1])**2
du = 0.5*cff * (u_np1[1:]-u_np1[:-1]) * (u_n[1:]+u_np1[1:]-u_n[:-1]-u_np1[:-1])
dv = 0.5*cff * (v_np1[1:]-v_np1[:-1]) * (v_n[1:]+v_np1[1:]-v_n[:-1]-v_np1[:-1])
shear2_in = du + dv
return add_boundaries(0., shear2_in, 0.)
[docs]
def compute_tke_eps_bc(
tke: ArrNzp1,
hz: ArrNz,
keps_params: KepsParameters,
case_tracable: CaseTracable
) -> Tuple[FloatJax, FloatJax, FloatJax, FloatJax]:
r"""
Compute top and bottom boundary conditions for TKE and :math:`\varepsilon`.
Parameters
----------
tke : float :class:`~jax.Array` of shape (nz+1)
Turbulent kinetic energy (TKE) denoted :math:`k` on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]`.
hz : float :class:`~jax.Array` of shape (nz)
Thickness of cells from deepest to shallowest :math:`[\text m]`.
keps_params : KepsParameters
Values of the parameters used by :math:`k-\varepsilon`.
case_tracable : CaseTracable
Physical parameters and forcings of the model run.
Returns
-------
tke_sfc_bc : float
TKE value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-3}\right]`).
tke_btm_bc : float
TKE value for bottom boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-3}\right]`).
eps_sfc_bc : float
:math:`\varepsilon` value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-4}\right]`).
eps_btm_bc : float
:math:`\varepsilon` value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-4}\right]`).
Note
----
The kind of boundary conditions between Neumann and Dirichlet are register in the parameters
:code:`keps_params`.
"""
# constants
rp, rm, rn = keps_params.gls_p, keps_params.gls_m, keps_params.gls_n
c_mu0 = keps_params.c_mu0
cm0inv2 = 1./c_mu0**2
vkarmn = case_tracable.vkarmn
chk = keps_params.chk_grav/case_tracable.grav
sig_eps = keps_params.sig_eps
# velocity scales
ustar2_sfc = jnp.sqrt(case_tracable.ustr_sfc**2 + case_tracable.vstr_sfc**2)
ustar2_bot = jnp.sqrt(case_tracable.ustr_btm**2 + case_tracable.vstr_btm**2)
# TKE Dirichlet boundary condition
tke_sfc_dir = jnp.maximum(keps_params.tke_min, cm0inv2*ustar2_sfc)
tke_btm_dir = jnp.maximum(keps_params.tke_min, cm0inv2*ustar2_bot)
# TKE Neumann boundary condition
tke_sfc_neu = 0.0
tke_btm_neu = 0.0
# epsilon surface conditions
z0_s = jnp.maximum(keps_params.z0s_min, chk*ustar2_sfc)
lgthsc = vkarmn*(0.5*hz[-1] + z0_s)
tke_sfc = 0.5*(tke[-1]+tke[-2])
eps_sfc_dir = jnp.maximum(keps_params.eps_min, c_mu0**rp * lgthsc**rn * tke_sfc**rm)
eps_sfc_neu = cast(float, -rn*vkarmn/sig_eps * c_mu0**(rp+1) * tke_sfc**(rm+.5) * lgthsc**rn)
# epsilon bottom conditions
z0b = jnp.maximum(keps_params.z0b, keps_params.z0b_min)
lgthsc = vkarmn *(0.5*hz[0] + z0b)
tke_btm = 0.5*(tke[0]+tke[1])
eps_btm_dir = jnp.maximum(keps_params.eps_min, c_mu0**rp * lgthsc**rn * tke_btm**rm)
eps_btm_neu = cast(float, -rn*vkarmn/sig_eps * c_mu0**(rp+1) * tke_btm**(rm+.5) * lgthsc**rn)
tke_sfc_bc = jnp.where(keps_params.dir_sfc, tke_sfc_dir, tke_sfc_neu)
tke_btm_bc = jnp.where(keps_params.dir_btm, tke_btm_dir, tke_btm_neu)
eps_sfc_bc = jnp.where(keps_params.dir_sfc, eps_sfc_dir, eps_sfc_neu)
eps_btm_bc = jnp.where(keps_params.dir_btm, eps_btm_dir, eps_btm_neu)
return tke_sfc_bc, tke_btm_bc, eps_sfc_bc, eps_btm_bc
[docs]
def advance_turb(
akt: ArrNzp1,
akv: ArrNzp1,
tke: ArrNzp1,
tke_np1: ArrNzp1,
eps: ArrNzp1,
c_mu: ArrNzp1,
c_mu_prim: ArrNzp1,
bvf: ArrNzp1,
shear2: ArrNzp1,
hz: ArrNz,
dt: float,
tke_sfc_bc: FloatJax,
tke_btm_bc: FloatJax,
eps_sfc_bc: FloatJax,
eps_btm_bc: FloatJax,
keps_params: KepsParameters,
do_tke: bool
) -> ArrNzp1:
r"""
Integrate TKE or :math:`\varepsilon` quantities.
First the shear and buoyancy production are computed, then they are used in the building of the
tridiagonal problem, the boundary conditions are then added and finally the tridiagonal problem
is solved.
Parameters
----------
akt : float :class:`~jax.Array` of shape (nz+1)
Current eddy-diffusivity on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
akv : float :class:`~jax.Array` of shape (nz+1)
Current eddy-viscosity on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
tke : float :class:`~jax.Array` of shape (nz+1)
Current turbulent kinetic energy (TKE) denoted :math:`k` on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]`.
tke_np1 : float :class:`~jax.Array` of shape (nz+1)
Turbulent kinetic energy (TKE) denoted :math:`k` on the interfaces of the cells at next step
(usefull only for :math:`\varepsilon` integration)
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]`.
eps : float :class:`~jax.Array` of shape (nz+1)
Current TKE dissipation denoted :math:`\varepsilon` on the interfaces of the cells
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
c_mu : float :class:`~jax.Array` of shape (nz+1)
Current :math:`c_\mu` in GLS formalisim on the interfaces of the cells
:math:`[\text{dimensionless}]`.
c_mu_prim : float :class:`~jax.Array` of shape (nz+1)
Current :math:`c_\mu'` in GLS formalisim on the interfaces of the cells
:math:`[\text{dimensionless}]`.
bvf : float :class:`~jax.Array` of shape (nz+1)
Current Brunt–Väisälä frequency squared :math:`N^2` on cell interfaces
:math:`\left[\text s^{-2}\right]`.
shear2 : float :class:`~jax.Array` of shape (nz+1)
Current shear production squared :math:`S_h^2` on cell interfaces
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
hz : float :class:`~jax.Array` of shape (nz)
Thickness of cells from deepest to shallowest :math:`[\text m]`.
dt : float
Time-step of the forward model :math:`[\text s]`.
tke_sfc_bc : float
TKE value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-3}\right]`).
tke_btm_bc : float
TKE value for bottom boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-3}\right]`).
eps_sfc_bc : float
:math:`\varepsilon` value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-4}\right]`).
eps_btm_bc : float
:math:`\varepsilon` value for surface boundary condition (Dirichlet
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]` or Neumann
:math:`\left[\text m ^3 \cdot \text s ^{-4}\right]`).
keps_params : KepsParameters
Values of the parameters used by :math:`k-\varepsilon`.
do_tke : bool
If :code:`True` solve the equation for TKE, else for :math:`\varepsilon`.
Returns
-------
vec : float :class:`~jax.Array` of shape (nz+1)
TKE or :math:`\varepsilon` at next step (depending on :code:`do_tke`).
"""
# fill the matrix off-diagonal terms for the tridiagonal problem
cff = -0.5*dt
ak_vec = jnp.where(do_tke, akv/keps_params.sig_k, akv/keps_params.sig_eps)
a_in = cff*(ak_vec[1:-1]+ak_vec[:-2]) / hz[:-1]
c_in = cff*(ak_vec[1:-1]+ak_vec[2:]) / hz[1:]
# shear and buoyancy production
s_prod_tke = akv[1:-1]*shear2[1:-1]
b_prod_tke = -akt[1:-1]*bvf[1:-1]
s_prod_eps = keps_params.c_eps1*c_mu[1:-1]*tke[1:-1]*shear2[1:-1]
b_prod_eps = -c_mu_prim[1:-1] * tke[1:-1] * (keps_params.c_eps3m*jnp.maximum(bvf[1:-1], 0) + \
keps_params.c_eps3p*jnp.minimum(bvf[1:-1], 0))
s_prod = jnp.where(do_tke, s_prod_tke, s_prod_eps)
b_prod = jnp.where(do_tke, b_prod_tke, b_prod_eps)
# diagonal and f term
cff = 0.5*(hz[:-1] + hz[1:])
f_tke_in = lax.select(
b_prod+s_prod > 0, cff*(tke[1:-1]+dt*(b_prod+s_prod)), cff*(tke[1:-1]+dt*s_prod)
)
f_eps_in = lax.select(
b_prod+s_prod > 0, cff*(eps[1:-1]+dt*(b_prod+s_prod)), cff*(eps[1:-1]+dt*s_prod)
)
f_in = jnp.where(do_tke, f_tke_in, f_eps_in)
b_tke_in = lax.select(
(b_prod + s_prod) > 0, cff*(1. + dt*eps[1:-1]/tke[1:-1]) - a_in - c_in,
cff*(1. + dt*(eps[1:-1] - b_prod)/tke[1:-1]) - a_in - c_in
)
b_eps_in = lax.select(
(b_prod + s_prod) > 0,
cff*(1. + dt*keps_params.c_eps2*eps[1:-1]/tke_np1[1:-1]) - a_in - c_in,
cff*(1. + dt*keps_params.c_eps2*eps[1:-1]/tke_np1[1:-1] - dt*b_prod/eps[1:-1]) - a_in - c_in
)
b_in = jnp.where(do_tke, b_tke_in, b_eps_in)
# surface boundary condition
dir_sfc = keps_params.dir_sfc
a_sfc = cast(float, jnp.where(dir_sfc, 0., -0.5*(ak_vec[-1] + ak_vec[-2])))
b_sfc = cast(float, jnp.where(dir_sfc, 1., 0.5*(ak_vec[-1] + ak_vec[-2])))
sfc_bc = jnp.where(do_tke, tke_sfc_bc, eps_sfc_bc)
f_sfc = cast(float, jnp.where(dir_sfc, sfc_bc, hz[-1]*sfc_bc))
# bottom boundary condition
dir_btm = keps_params.dir_btm
b_btm = cast(float, jnp.where(dir_btm, 1., -0.5*(ak_vec[0] + ak_vec[1])))
c_btm = cast(float, jnp.where(dir_btm, 0., 0.5*(ak_vec[0] + ak_vec[1])))
btm_bc = jnp.where(do_tke, tke_btm_bc, eps_btm_bc)
f_btm = cast(float, jnp.where(dir_btm, btm_bc, hz[0]*btm_bc))
# vectors rassembly
a = add_boundaries(0., a_in, a_sfc)
b = add_boundaries(b_btm, b_in, b_sfc)
c = add_boundaries(c_btm, c_in, 0.)
f = add_boundaries(f_btm, f_in, f_sfc)
# solve tridiagonal problem
f = tridiag_solve(a, b, c, f)
vec_min = jnp.where(do_tke, keps_params.tke_min, keps_params.eps_min)
vec = jnp.maximum(f, vec_min)
return vec
[docs]
def compute_diag(
tke: ArrNzp1,
eps: ArrNzp1,
bvf: ArrNzp1,
shear2: ArrNzp1,
keps_params: KepsParameters
) -> Tuple[ArrNzp1, ArrNzp1, ArrNzp1, ArrNzp1, ArrNzp1]:
r"""
Computes the diagnostic variables of :math:`k-\varepsilon` closure.
This function first apply the Galperin limitation, then it computes :math:`c_\mu'` and
:math:`c_\mu` with the stability function, and finally it computes the eddy-diffusivity and
viscosity with these variables.
Parameters
----------
tke : float :class:`~jax.Array` of shape (nz+1)
Turbulent kinetic energy (TKE) denoted :math:`k` on the interfaces of the cells at next step
:math:`\left[\text m ^2 \cdot \text s ^{-2}\right]`.
eps : float :class:`~jax.Array` of shape (nz+1)
TKE dissipation denoted :math:`\varepsilon` on the interfaces of the cells at next step
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
bvf : float :class:`~jax.Array` of shape (nz+1)
Current Brunt–Väisälä frequency squared :math:`N^2` on cell interfaces
:math:`\left[\text s^{-2}\right]`.
shear2 : float :class:`~jax.Array` of shape (nz+1)
Current shear production squared :math:`S_h^2` on cell interfaces
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
keps_params : KepsParameters
Values of the parameters used by :math:`k-\varepsilon`.
Returns
-------
akt : float :class:`~jax.Array` of shape (nz+1)
Eddy-diffusivity on the interfaces of the cells at next step
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
akv : float :class:`~jax.Array` of shape (nz+1)
Eddy-viscosity on the interfaces of the cells at next step
:math:`\left[\text m ^2 \cdot \text s ^{-1}\right]`.
eps : float :class:`~jax.Array` of shape (nz+1)
TKE dissipation denoted :math:`\varepsilon` on the interfaces of the cells at next step
:math:`\left[\text m ^2 \cdot \text s ^{-3}\right]`.
c_mu : float :class:`~jax.Array` of shape (nz+1)
:math:`c_\mu` in GLS formalisim on the interfaces of the cells at next step
:math:`[\text{dimensionless}]`.
c_mu_prim : float :class:`~jax.Array` of shape (nz+1)
:math:`c_\mu'` in GLS formalisim on the interfaces of the cells at next step
:math:`[\text{dimensionless}]`.
"""
# parameters
akv_min = keps_params.akv_min
akt_min = keps_params.akt_min
eps_min = keps_params.eps_min
sf_d0 = keps_params.sf_d0
sf_d1 = keps_params.sf_d1
sf_d2 = keps_params.sf_d2
sf_d3 = keps_params.sf_d3
sf_d4 = keps_params.sf_d4
sf_d5 = keps_params.sf_d5
sf_n0 = keps_params.sf_n0
sf_n1 = keps_params.sf_n1
sf_n2 = keps_params.sf_n2
sf_nb0 = keps_params.sf_nb0
sf_nb1 = keps_params.sf_nb1
sf_nb2 = keps_params.sf_nb2
lim_am0 = keps_params.lim_am0
lim_am1 = keps_params.lim_am1
lim_am2 = keps_params.lim_am2
lim_am3 = keps_params.lim_am3
lim_am4 = keps_params.lim_am4
lim_am5 = keps_params.lim_am5
lim_am6 = keps_params.lim_am6
c_mu0 = keps_params.c_mu0
rp, rm, rn = keps_params.gls_p, keps_params.gls_m, keps_params.gls_n
e1 = 3 + rp/rn
e2 = 1.5 + rm/rn
e3 = -1/rn
# minimum value of alpha_n to ensure that alpha_m is positive
alpha_n_min = 0.5*(- (sf_d1 + sf_nb0) + jnp.sqrt((sf_d1 + sf_nb0)**2 - \
4.0*sf_d0*(sf_d4 + sf_nb1))) / (sf_d4 + sf_nb1) # CHECK
# Galperin limitation : l <= l_li
l_lim = keps_params.galp*jnp.sqrt(2.0*tke[1:-1] / jnp.maximum(1e-14, bvf[1:-1]))
# limitation (use MAX because rn is negative)
cff = c_mu0**rp * l_lim**rn * tke[1:-1]**rm
eps = eps.at[1:-1].set(jnp.maximum(eps[1:-1], cff))
epsilon = c_mu0**e1 * tke[1:-1]**e2 * eps[1:-1]**e3
epsilon = jnp.maximum(epsilon, eps_min)
# compute alpha_n and alpha_m
cff = (tke[1:-1] / epsilon)**2
alpha_m = cff*shear2[1:-1]
alpha_n = cff*bvf[1:-1]
# limitation of alpha_n and alpha_m
alpha_n = jnp.minimum(jnp.maximum(0.73*alpha_n_min, alpha_n), 1e10)
alpha_m_max = (lim_am0 + lim_am1*alpha_n + lim_am2*alpha_n**2 + lim_am3*alpha_n**3) / \
(lim_am4 + lim_am5*alpha_n + lim_am6*alpha_n**2)
alpha_m = jnp.minimum(alpha_m, alpha_m_max)
# compute stability functions
denom = sf_d0 + sf_d1*alpha_n + sf_d2*alpha_m + sf_d3*alpha_n*alpha_m + sf_d4*alpha_n**2 + \
sf_d5*alpha_m**2
cff = 1./denom
c_mu_in = cff*(sf_n0 + sf_n1*alpha_n + sf_n2*alpha_m)
c_mu = add_boundaries(keps_params.c_mu_min, c_mu_in, keps_params.c_mu_min)
c_mu_prim_in = cff*(sf_nb0 + sf_nb1*alpha_n + sf_nb2*alpha_m)
c_mu_prim = add_boundaries(keps_params.c_mu_prim_min, c_mu_prim_in, keps_params.c_mu_prim_min)
# finalize the computation of akv and akt
cff = tke[1:-1]**2 / epsilon
akt_in = jnp.maximum(cff*c_mu_prim[1:-1], akt_min)
akv_in = jnp.maximum(cff*c_mu[1:-1], akv_min)
akt_btm = cast(float, jnp.maximum(1.5*akt_in[0] - 0.5*akt_in[1], akt_min))
akt_sfc = cast(float, jnp.maximum(1.5*akt_in[-1] - 0.5*akt_in[-2], akt_min))
akv_btm = cast(float, jnp.maximum(1.5*akv_in[0] - 0.5*akv_in[1], akv_min))
akv_sfc = cast(float, jnp.maximum(1.5*akv_in[-1] - 0.5*akv_in[-2], akv_min))
akt = add_boundaries(akt_btm, akt_in, akt_sfc)
akv = add_boundaries(akv_btm, akv_in, akv_sfc)
return akt, akv, eps, c_mu, c_mu_prim