namespace mom_thickness_diffuse¶

Overview¶

Thickness diffusion (or Gent McWilliams) More…

namespace mom_thickness_diffuse {

// global functions

subroutine, public thickness_diffuse(
    h h,
    uhtr uhtr,
    vhtr vhtr,
    tv tv,
    dt dt,
    G G,
    GV GV,
    US US,
    MEKE MEKE,
    VarMix VarMix,
    CDp CDp,
    CS CS
    );

subroutine, public thickness_diffuse_init(
    Time Time,
    G G,
    GV GV,
    US US,
    param_file param_file,
    diag diag,
    CDp CDp,
    CS CS
    );

subroutine, public thickness_diffuse_get_kh(CS CS, KH_u_GME KH_u_GME, KH_v_GME KH_v_GME, G G);
subroutine, public thickness_diffuse_end(CS CS);

} // namespace mom_thickness_diffuse

Detailed Documentation¶

Thickness diffusion (or Gent McWilliams)

Thickness diffusion (aka Gent-McWilliams)¶

Thickness diffusion is implemented via along-layer mass fluxes

\[h^\dagger \leftarrow h^n - \Delta t \nabla \cdot ( \vec{uh}^* )\]

where the mass fluxes are cast as the difference in vector streamfunction

\[\vec{uh}^* = \delta_k \vec{\psi} .\]

The GM implementation of thickness diffusion made the streamfunction proportional to the potential density slope

\[\vec{\psi} = - \kappa_h \frac{\nabla_z \rho}{\partial_z \rho} = \frac{g\kappa_h}{\rho_o} \frac{\nabla \rho}{N^2} = \kappa_h \frac{M^2}{N^2}\]

but for robustness the scheme is implemented as

\[\vec{\psi} = \kappa_h \frac{M^2}{\sqrt{N^4 + M^4}}\]

since the quantity $\frac{M^2}{\sqrt{N^2 + M^2}}$ is bounded between $-1$ and $1$ and does not change sign if $N^2<0$.

Optionally, the method of Ferrari et al, 2010, can be used to obtain the streamfunction which solves the vertically elliptic equation:

\[\gamma_F \partial_z c^2 \partial_z \psi - N_*^2 \psi = ( 1 + \gamma_F ) \kappa_h N_*^2 \frac{M^2}{\sqrt{N^4+M^4}}\]

which recovers the previous streamfunction relation in the limit that $c \rightarrow 0$. Here, $c=\max(c_{min},c_g)$ is the maximum of either $c_{min}$ and either the first baroclinic mode wave-speed or the equivalent barotropic mode wave-speed. $N_*^2 = \max(N^2,0)$ is a non-negative form of the square of the Brunt-Vaisala frequency. The parameter $\gamma_F$ is used to reduce the vertical smoothing length scale. This elliptic form for $\psi$ is turned on with the logical KHTH_USE_FGNV_STREAMFUNCTION.

Thickness diffusivities are calculated independently at u- and v-points using the following expression

\[\kappa_h = \left( \kappa_o + \alpha_{s} L_{s}^2 < S N > + \alpha_{M} \kappa_{M} \right) r(\Delta x,L_d)\]

where $S$ is the isoneutral slope magnitude, $N$ is the square root of Brunt-Vaisala frequency, $\kappa_{M}$ is the diffusivity calculated by the MEKE parameterization (mom_meke module) and $r(\Delta x,L_d)$ is a function of the local resolution (ratio of grid-spacing, $\Delta x$, to deformation radius, $L_d$). The length $L_s$ is provided by the mom_lateral_mixing_coeffs module (enabled with USE_VARIABLE_MIXING=True and the term $<SN>$ is the vertical average slope times the Brunt-Vaisala frequency prescribed by Visbeck et al., 1996.

The result of the above expression is subsequently bounded by minimum and maximum values, including an upper diffusivity consistent with numerical stability ($\kappa_{cfl}$ is calculated internally).

\[\kappa_h \leftarrow \min{\left( \kappa_{max}, \kappa_{cfl}, \max{\left( \kappa_{min}, \kappa_h \right)} \right)} f(c_g,z)\]

where $f(c_g,z)$ is a vertical structure function. $f(c_g,z)$ is calculated in module mom_lateral_mixing_coeffs. If KHTH_USE_EBT_STRUCT=True then $f(c_g,z)$ is set to look like the equivalent barotropic modal velocity structure. Otherwise $f(c_g,z)=1$ and the diffusivity is independent of depth.

In order to calculate meaningful slopes in vanished layers, temporary copies of the thermodynamic variables are passed through a vertical smoother, function vert_fill_ts() :

\[\begin{split}\begin{eqnarray*} \left[ 1 + \Delta t \kappa_{smth} \frac{\partial^2}{\partial_z^2} \right] \theta & \leftarrow & \theta \\ \left[ 1 + \Delta t \kappa_{smth} \frac{\partial^2}{\partial_z^2} \right] s & \leftarrow & s \end{eqnarray*}\end{split}\]

Module mom_thickness_diffuse parameters¶

References¶

Ferrari, R., S.M. Griffies, A.J.G. Nurser and G.K. Vallis, 2010: A boundary-value problem for the parameterized mesoscale eddy transport. Ocean Modelling, 32, 143-156. http://doi.org/10.1016/j.ocemod.2010.01.004

Viscbeck, M., J.C. Marshall, H. Jones, 1996: On he dynamics of convective “chimneys” in the ocean. J. Phys. Oceangr., 26, 1721-1734. http://dx.doi.org/10.1175/1520-0485(1996)026%3C1721:DOICRI%3E2.0.CO;2

Global Functions¶

subroutine, public thickness_diffuse(
    h h,
    uhtr uhtr,
    vhtr vhtr,
    tv tv,
    dt dt,
    G G,
    GV GV,
    US US,
    MEKE MEKE,
    VarMix VarMix,
    CDp CDp,
    CS CS
    )

Calculates thickness diffusion coefficients and applies thickness diffusion to layer thicknesses, h. Diffusivities are limited to ensure stability. Also returns along-layer mass fluxes used in the continuity equation.

Parameters:

g	Ocean grid structure
gv	Vertical grid structure
us	A dimensional unit scaling type
h	Layer thickness [H ~> m or kg m-2]
uhtr	Accumulated zonal mass flux [L2 H ~> m3 or kg]
vhtr	Accumulated meridional mass flux [L2 H ~> m3 or kg]
tv	Thermodynamics structure
dt	Time increment [s]
meke	MEKE control structure
varmix	Variable mixing coefficients
cdp	Diagnostics for the continuity equation
cs	Control structure for thickness diffusion

subroutine, public thickness_diffuse_init(
    Time Time,
    G G,
    GV GV,
    US US,
    param_file param_file,
    diag diag,
    CDp CDp,
    CS CS
    )

Initialize the thickness diffusion module/structure.

Parameters:

time	Current model time
g	Ocean grid structure
gv	Vertical grid structure
us	A dimensional unit scaling type
param_file	Parameter file handles
diag	Diagnostics control structure
cdp	Continuity equation diagnostics
cs	Control structure for thickness diffusion

subroutine, public thickness_diffuse_get_kh(
    CS CS,
    KH_u_GME KH_u_GME,
    KH_v_GME KH_v_GME,
    G G
    )

Copies ubtav and vbtav from private type into arrays.

Parameters:

cs	Control structure for this module
g	Grid structure
kh_u_gme	interface height diffusivities at u-faces [L2 T-1 ~> m2 s-1]
kh_v_gme	interface height diffusivities at v-faces [L2 T-1 ~> m2 s-1]

subroutine, public thickness_diffuse_end(CS CS)

Deallocate the thickness diffusion control structure.

Parameters:

cs	Control structure for thickness diffusion