Lateral diffusion term (dynldf.F90)

!-----------------------------------------------------------------------
&namdyn_ldf    !   lateral diffusion on momentum
!-----------------------------------------------------------------------
   !                       !  Type of the operator :
   ln_dynldf_lap    =  .true.   !  laplacian operator
   ln_dynldf_bilap  =  .false.  !  bilaplacian operator
   !                       !  Direction of action  :
   ln_dynldf_level  =  .false.  !  iso-level
   ln_dynldf_hor    =  .true.   !  horizontal (geopotential)            (require "key_ldfslp" in s-coord.)
   ln_dynldf_iso    =  .false.  !  iso-neutral                          (require "key_ldfslp")
   !                       !  Coefficient
   rn_ahm_0_lap     = 40000.    !  horizontal laplacian eddy viscosity   [m2/s]
   rn_ahmb_0        =     0.    !  background eddy viscosity for ldf_iso [m2/s]
   rn_ahm_0_blp     =     0.    !  horizontal bilaplacian eddy viscosity [m4/s]
   rn_cmsmag_1      =     3.    !  constant in laplacian Smagorinsky viscosity
   rn_cmsmag_2      =     3     !  constant in bilaplacian Smagorinsky viscosity
   rn_cmsh          =     1.    !  1 or 0 , if 0 -use only shear for Smagorinsky viscosity
   rn_ahm_m_blp     =    -1.e12 !  upper limit for bilap  abs(ahm) < min( dx^4/128rdt, rn_ahm_m_blp)
   rn_ahm_m_lap     = 40000.    !  upper limit for lap  ahm < min(dx^2/16rdt, rn_ahm_m_lap)
/

Options are defined through the namdyn_ldf namelist variables. The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators. The coefficients may be constant or spatially variable; the description of the coefficients is found in the chapter on lateral physics (Chap.9). The lateral diffusion of momentum is evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression is the before velocity in time, except for the pure vertical component that appears when a tensor of rotation is used. This latter term is solved implicitly together with the vertical diffusion term (see §3)

At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied according to the user's choice (see Chap.8).

Iso-level laplacian operator (ln_dynldf_lap=true)

For lateral iso-level diffusion, the discrete operator is:

$$\left\{ \begin{aligned}D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\... ...left[ {A_f^{lm} \;e_{3f} \zeta } \right] \end{aligned} \right.$$

As explained in §2.5.2, this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.

Rotated laplacian operator (ln_dynldf_iso=true)

A rotation of the lateral momentum diffusion operator is needed in several cases: for iso-neutral diffusion in the $z$-coordinate (ln_dynldf_iso=true) and for either iso-neutral (ln_dynldf_iso=true) or geopotential (ln_dynldf_hor=true) diffusion in the $s$-coordinate. In the partial step case, coordinates are horizontal except at the deepest level and no rotation is performed when ln_dynldf_hor=true. The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on each momentum component. It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry. The resulting discrete representation is:

$$\begin{split}D_u^{l\textbf{U}} &= \frac{1}{e_{1u} e_{2u} \... ...delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} \end{split}$$ (6.26)

where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and the surface of computation ($z$- or $s$-surfaces). The way these slopes are evaluated is given in the lateral physics chapter (Chap.9).

Iso-level bilaplacian operator (ln_dynldf_bilap=true)

The lateral fourth order operator formulation on momentum is obtained by applying (6.26) twice. It requires an additional assumption on boundary conditions: the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, while the third derivative terms normal to the coast are set to zero (see Chap.8).