Double Diffusion Mixing (key_ zdfddm)

!-----------------------------------------------------------------------
&namzdf_ddm    !   double diffusive mixing parameterization             ("key_zdfddm")
!-----------------------------------------------------------------------
   rn_avts     = 1.e-4     !  maximum avs (vertical mixing on salinity)
   rn_hsbfr    = 1.6       !  heat/salt buoyancy flux ratio
/

Options are defined through the namzdf_ddm namelist variables. Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa. The former condition leads to salt fingering and the latter to diffusive convection. Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean. Merryfield et al. [1999] include a parameterisation of such phenomena in a global ocean model and show that it leads to relatively minor changes in circulation but exerts significant regional influences on temperature and salinity. This parameterisation has been introduced in zdfddm.F90 module and is controlled by the key_ zdfddm CPP key.

Diapycnal mixing of S and T are described by diapycnal diffusion coefficients

$$% \label{Eq_zdfddm_Kz} A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}$$

$$A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}$$

where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, and $o$ by processes other than double diffusion. The rates of double-diffusive mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline contraction (see §5.8.1). To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981):

$$A_f^{vS} = \begin{cases}\frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } 0 &\text{otherwise} \end{cases}$$ (10.23)

$$A_f^{vT} = 0.7 A_f^{vS} / R_\rho$$ (10.24)

Figure 10.4: From Merryfield et al. [1999] : (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy curves denote $A^{\ast v} = 10^{-3} m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4} m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation. The latter is not implemented in NEMO.

The factor 0.7 in (10.24) reflects the measured ratio $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy flux of salt ($e.g.$, McDougall and Taylor [1984]). Following Merryfield et al. [1999], we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4} m^2.s^{-1}$.

To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by Federov (1988) is used:

$$A_d^{vT} = \begin{cases}1.3635 \exp{\left( 4.6 \exp{ \left[ -0.54 ( ... ...t)} &\text{if $0<R_\rho < 1$ and $N^2>0$ } 0 &\text{otherwise} \end{cases}$$ (10.25)

$$A_d^{vS} = \begin{cases}A_d^{vT} \left( 1.85 R_{\rho} - 0.85 \right) &\t... ...text{if $ 0 < R_\rho<0.5$ and $N^2>0$ } 0 &\text{otherwise} \end{cases}$$ (10.26)

The dependencies of (10.23) to (10.26) on $R_\rho$ are illustrated in Fig. 10.4. Implementing this requires computing $R_\rho$ at each grid point on every time step. This is done in eosbn2.F90 at the same time as $N^2$ is computed. This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).