26.2.1 Momentum equations

To account for a generalized variation in cell thickness, a factor of dhu_i,k,j must be included in horizontal diffusive and advective operators. The following equations should be compared with the corresponding set in Section 21.3.1:

  

$$ADV\_Ux_{i,k,j} \frac{1}{2\;\csuj \;dhu_{i,k,j}}\delx(adv\_fe_{i-1,k,j})$$ = (26.2)

$$ADV\_Uy_{i,k,j} \frac{1}{2\;\csuj \;dhu_{i,k,j}}\dely(adv\_fn_{i,k,j-1})$$ = (26.3)

$$DIFF\_Ux_{i,k,j} \frac{1}{\csuj \;dhu_{i,k,j}}\delx(\diff\_fe_{i-1,k,j})$$ = (26.4)

$$DIFF\_Uy_{i,k,j} \frac{1}{\csuj \;dhu_{i,k,j}}\dely(\diff\_fn_{i,k,j-1})$$ = (26.5)

where the horizontal viscous fluxes on U-cell faces become

$$\diff\_fe_{i,k,j} \frac{visc\_ceu_{i,k,j}}{\csuj}\;min(dhu_{i,k,j},dhu_{i+1,k,j})\;\delx(u_{i,k,j,n,\tau-1})$$ = (26.6)

$$\diff\_fn_{i,k,j} visc\_cnu_{i,k,j}\; \cstjp \;min(dhu_{i,k,j},dhu_{i,k,j+1})\; \dely(u_{i,k,j,n,\tau-1})$$ = (26.7)

Because the lateral boundary condition is no-flux for tracers but no-slip for velocity, the horizontal viscosity $\frac{1}{h}\nabla\cdot(h\;\nabla (\vec{u}))$ has zonal and meridional components given by

  

$${\cal VISC}^\lambda \frac{A_m}{dhu_{i,k,j}\;\mbox{$\cos^2\phi^U$ }}\;\delx(\zeta^{U\l... ...j}\;\mbox{$\cos\phi^U$ }}\;\dely( \mbox{$\cos\phi^T$ }\;\zeta^{U\phi}\;\dely u)$$ =

$$A_m\frac{1-\tan^2\phi^U}{ a^2}\; u \;-\; A_m\frac{2\sin\phi^U}{ a^2\; \cos^2\phi^U }\;\delx(\overline{v}^\lambda)$$ +

+ S(u) (26.8)

$${\cal VISC}^\phi \frac{A_m}{dhu_{i,k,j}\;\mbox{$\cos^2\phi^U$ }}\;\delx(\zeta^{U\l... ...j}\;\mbox{$\cos\phi^U$ }}\;\dely( \mbox{$\cos\phi^T$ }\;\zeta^{U\phi}\;\dely v)$$ =

$$A_m\frac{1-\tan^2\phi^U}{ a^2}\; v + A_m\frac{2\sin\phi^U}{ a^2\; \cos^2\phi^U }\;\delx(\overline{u}^\lambda)$$ +

$$S(v)\;.$$ + (26.9)

where A_m is the lateral viscosity coefficient^26.2. The above form differs from Bryan (1969) due to the inclusion of effective cell face heights $\zeta^{U\lambda}$ and $\zeta^{U\phi}$ and a sink term due to a no-slip lateral boundary condition given by

 

$$\zeta^{U\lambda}_{i,k,j} min(dhu_{i,k,j} ,\; dhu_{i+1,k,j})$$ = (26.10)

$$\zeta^{U\phi}_{i,k,j} min(dhu_{i,k,j} ,\; dhu_{i,k,j+1})$$ = (26.11)

$$S(\beta) - \frac{A_m }{dhu_{i,k,j}\; \mbox{$\cos^2\phi^U$ }\dxui}\bigl[\fr... ...ambda}}{\dxtip} + \frac{dhu_{i,k,j}-\zeta^{U\lambda}_{i-1}}{\dxti}\bigr]\;\beta$$ =

$$\frac{A_m}{dhu_{i,k,j}\; \mbox{$\cos\phi^U$ }\dyuj}\bigl[\frac{\m... ...frac{\mbox{$\cos\phi^T$ }(dhu_{i,k,j}-\zeta^{U\phi}_{j-1})}{\dytj}\bigr]\;\beta$$ - (26.12)

which is zero where U-cell thickness is constant (i.e. $dhu_{i,k,j} = \zeta^{U\lambda}$) within a vertical level but acts effectively as a bottom drag

$$S(\beta) \propto (A_m/\Delta^2 x^U)\;\beta \;\;\; \cdots \; assuming \;\;\dxui = \dyuj$$ (26.13)

where U-cell thickness varies within a vertical level. The constant of proportionality ( $(dhu_{i,k,j}-\zeta^{U\lambda})/dhu_{i,k,j}$) represents the fraction of cell face height over which a no-slip condition is applied.

There is no change in the form of the horizontal advective flux given by Equations (22.86) and (22.87) because advective velocities $adv\_vnt_{i,k,j}$ and $adv\_vet_{i,k,j}$ absorb a factor dhu_i,k,jand become advective transports given by Equations (26.40) and (26.41). Horizontal advective velocities for the U-cells are then given by Equations (22.22) and (22.21).

The only change in the vertical advective and vertical diffusive operators is to replace dzt_k in the vertical derivative with dhu_i,k,j. When defining vertical fluxes at the bottom cell faces, dhw_k in the vertical derivatives is replaced by $min\; (dhut_{i,k,j},\; dhut_{i+1,k,j},\; dhut_{i,k,j+1},\; dhut_{i+1,k,j+1})$.