39.7.2.9 Thermobaricity and halobaricity

In the model, the horizontal hydrostatic pressure gradients are stored in an array $grad\_p_{i,k,j}$ (Section 22.9.1). These gradients are defined at the U-cell point, which is the northeast corner of a T-cell in the x-y plane. Since the diffusive flux components are defined at the T-cell faces, it is necessary to average the pressure gradients before multiplying by the tracer flux. Thereafter, an average over the faces of the T-cell is prescribed in order to get the thermobaric and halobaric contributions at the tracer point. For the vertical pressure gradient, the hydrostatic approximation yields $\partial_{z} p_{i,k,j} = - g \rho_{i,k,j}$. Within the Bousinessq limit, this pressure gradient is evaluated as $-g \rho_{o}$, where $\rho_{o} = 1.035 g/cm^{3}$. In symbols, this prescription yields for the contribution to thermobaricity and halobaricity

$$-(\rho_{p \theta})_{i,k,j} \left( \frac{ (grad\_p_{i,k,j,1} + grad\_p_{i,k,j-1,1}) \; ison\_fe_{1,i,k,j}}{4} \right.$$ thermo_i,k,j =

$$\frac{(grad\_p_{i-1,k,j,1} + grad\_p_{i-1,k,j-1,1}) \; ison\_fe_{1,i-1,k,j} }{4}$$ +

$$\frac{ (grad\_p_{i,k,j,2} + grad\_p_{i-1,k,j,2}) \; ison\_fn_{1,i,k,j} }{4}$$ +

$$\frac{ (grad\_p_{i,k,j-1,2} + grad\_p_{i-1,k,j-1,2}) \; ison\_fn_{1,i,k,j-1} }{4}$$ +

$$\left. g \; \rho_{o} \; \biggl[\frac{ ( ison\_fb_{1,i,k,j} + ison\_fb_{1,i,k-1,j} ) }{2} \biggr] \right),$$ - (39.116)

where the overall minus sign accounts for the minus sign used for computing the discretized flux components, relative to the fluxes in the continuum. The pressure derivatives of $\rho_{\theta}$ and $\rho_{s}$ are evaluated as

$$\partial_{p}\rho_{\theta} -(\rho_{o} \; g)^{-1} \;\partial_{z}\rho_{\theta}$$ = (39.117)

$$\partial_{p}\rho_{s} -(\rho_{o} \; g)^{-1} \; \partial_{z}\rho_{s}.$$ = (39.118)

Halobaricity is diagnosed from thermobaricity as

$$- \left( \frac{(\rho_{\theta})_{i,k,j} \; (\rho_{sp})_{i,k,j} } { (\rho_{s})_{i,k,j} \; (\rho_{\theta p})_{i,k,j} } \right) thermo_{i,k,j}.$$ halob_i,k,j = (39.119)