Conservation Properties on Tracers

All the numerical schemes used in NEMO are written such that the tracer content is conserved by the internal dynamics and physics (equations in flux form). For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) conserves the global variance of tracer. Nevertheless the other schemes ensure that the global variance decreases ($i.e.$ they are at least slightly diffusive). For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved to a very high accuracy.

Advection Term

conservation of a tracer, $T$:

$$\frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 T \right)}{\partial t} \;dv }=0$$

conservation of its variance:

$$\frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) = \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 T \right) }... ... \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$$

Whatever the advection scheme considered it conserves of the tracer content as all the scheme are written in flux form. Indeed, let $T$ be the tracer and $\tau_u$, $\tau_v$, and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is), the conservation of the tracer content due to the advection tendency is obtained as follows:

$$\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv$$

$$\equiv - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_t} \left( \delt... ...ght] \right) + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t$$

$$\equiv - \sum\limits_{i,j,k} \left\{ \delta_i \left[ U \;\tau_u \... ...delta_j \left[ V \;\tau_v \right] + \delta_k \left[ W \;\tau_w \right] \right\}$$

$$\equiv 0$$

The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme, $i.e.$ when $\tau_u= \overline T^{ i+1/2}$, $\tau_v= \overline T^{ j+1/2}$, and $\tau_w= \overline T^{ k+1/2}$. It can be demonstarted as follows:

$$\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 T \right) }... ...} \;dv } = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv$$

$$\equiv - \sum\limits_{i,j,k} T\; \left\{ \delta_i \left[ U \overline T^{... ... T^{ j+1/2} \right] + \delta_k \left[ W \overline T^{ k+1/2} \right] \right\}$$

$$\equiv + \sum\limits_{i,j,k} \left\{ U \overline T^{ i+1/2} \delta_{i... ...[ T \right] + W \overline T^{ k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\}$$

$$\equiv + \frac{1} {2} \sum\limits_{i,j,k} \Bigl\{ U \;\delta_{i+1/2} \le... ...elta_{j+1/2} \left[ T^2 \right] + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\}$$

$$\equiv - \frac{1} {2} \sum\limits_{i,j,k} T^2 \Bigl\{ \delta_i \left[ U \right] + \delta_j \left[ V \right] + \delta_k \left[ W \right] \Bigr\}$$

$$\equiv + \frac{1} {2} \sum\limits_{i,j,k} T^2 \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t} T }{\partial t} \Bigr\}$$

which is the discrete form of $\frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$.