Conservation Properties on Tracer Physics

The numerical schemes used for tracer subgridscale physics are written such that the heat and salt contents are conserved (equations in flux form). Since a flux form is used to compute the temperature and salinity, the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish. As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear.

Conservation of Tracers

constraint of conservation of tracers:

$$\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv$$

$$\equiv \sum\limits_{i,j,k} \biggl\{ \biggr. \delta_i \left[ A_u^{... ...v^{ lT} \frac{e_{1v} e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right]$$

$$\qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; + \delta_k... ..._{2t}} {e_{3t}} \delta_{k+1/2} \left[ T \right] \right] \biggr\} \quad \equiv 0$$

In fact, this property simply results from the flux form of the operator.

Dissipation of Tracer Variance

constraint on the dissipation of tracer variance:

$$\int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv$$

$$\equiv \sum\limits_{i,j,k} \; T \biggl\{ \biggr. \delta_i \left[ A_u^{ lT} \frac{e_{2u} e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] + \delta_j \left[ A_v^{ lT} \frac{e_{1v} e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] \quad$$

$$\biggl. + \delta_k \left[A_w^{ vT}\frac{e_{1t} e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] \biggr\}$$

$$\equiv - \sum\limits_{i,j,k} \biggl\{ \biggr. \quad A_u^{ lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u} e_{2u} e_{3u}$$

$$+ A_v^{ lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v} e_{2v} e_{3v}$$

$$\biggl. + A_w^{ vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w} e_{2w} e_{3w} \biggr\} \quad \leq 0$$