39.7.2.8 Cabbeling

The isoneutral diffusion piece $\nabla \theta \cdot \vec{F}_{I}(\theta)$ of cabbeling is given by

$$\frac{ ison\_fe_{1,i,k,j} \; \delta_{x} t_{i,k,j,1,\tau-1} + ison\_fe_{1,i-1,k,j} \; \delta_{x} t_{i-1,k,j,1,\tau-1} }{2}$$ ISO_i,k,j =

$$\frac{ ison\_fn_{1,i,k,j} \; \delta_{y} t_{i,k,j,1,\tau-1} + ison\_fn_{1,i,k,j-1} \; \delta_{y} t_{i,k,j-1,1,\tau-1} }{2}$$ +

$$\frac{ ison\_fb_{1,i,k,j} \; \delta_{z} t_{i,k,j,1,\tau-1} + ison\_fb_{1,i,k-1,j} \; \delta_{z} t_{i,k-1,j,1,\tau-1} }{2},$$ + (39.110)

where the isoneutral diffusion flux components are those for temperature, and they are defined by

$$ison\_fe_{n,i,k,j} diff\_fe^{iso}_{i,k,j}$$ = (39.111)

$$ison\_fn_{n,i,k,j} diff\_fn^{iso}_{i,k,j}$$ = (39.112)

$$ison\_fb_{n,i,k,j} diff\_fb^{iso}_{i,k,j} + K33_{i,k,j}\delta_{z}t_{i,k,j,n,\tau-1}.$$ = (39.113)

The diffusive fluxes $diff\_fe^{iso}, diff\_fn^{iso}, diff\_fb^{iso}$and diagonal diffusion tensor component K33 are defined in Section 34.1. The nonlinear equation of state piece $\rho_{a b} V^{a} V^{b}$ is given by

$$(\rho_{\theta \theta})_{i,k,j} - 2 (\rho_{\theta s})_{i,k,j} \; \... ...i,k,j} \left( \frac{ (\rho_{\theta})_{i,k,j} }{(\rho_{s})_{i,k,j}} \right)^{2}.$$ NONLIN_i,k,j = (39.114)

The cabbeling term is given by the product of these two terms

cabbel_i,k,j = -ISO_i,k,j * NONLIN_i,k,j, (39.115)

where the minus sign accounts for the minus sign used for computing the discretized flux components.