Discrete enstrophy conservation

Vorticity Term with ENS scheme (ln_dynvor_ens=.true.)

In the ENS scheme, the vorticity term is descretized as follows:

$$\left\{ \begin{aligned}+\frac{1}{e_{1u}} & \overline{q}^{ i} &... ...{2u} e_{3u}\; u \right) } } }^{ i+1/2, j} \end{aligned} \right.$$

The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=0). Indeed, using the symmetry or skew symmetry properties of the operators (Eqs (4.12) and (4.11)), it can be shown that:

$$\int_D {q \;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0$$ (C.14)

where $dv=e_1 e_2 e_3 \; di dj dk$ is the volume element. Indeed, using (C.13), the discrete form of the right hand side of (C.14) can be transformed as follow:

$$\int_D q \; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times \left( e_3 q \; \textbf{k} \times \textbf{U}_h \right)\; dv$$

$$\overline {\;\cdot \;} \delta \delta_{i+1/2} \left[ {\overline a^{ i}} \right] = \overline {\delta_i \left[ a \right]}^{ i+1/2} \chi$$

$$\qquad {\begin{array}{*{20}l} &\equiv \sum\limits_{i,j,k} q \le... ...,j} \right] \right\} && \end{array} } \allowdisplaybreaks\allowdisplaybreaks \qquad {\begin{array}{*{20}l} &\equiv \sum\limits_{i,j,k} - \frac... ...1t} e_{2t} e_{3t}^{} \chi}}^{ i+1/2,j+1/2} \quad \equiv 0 && \end{array} }$$

The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=0.

Vorticity Term with EEN scheme (ln_dynvor_een=.true.)

With the EEN scheme, the vorticity terms are represented as:

$$\left\{ { \begin{aligned}+q e_3 v &\equiv +\frac{1}{e_{1u} ... ...e_{3u} u \right)^{i+i_p}_{j+j_p-1/2} \end{aligned} } \right.$$

where the indices $i_p$ and $k_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:

$$_i^j \mathbb{Q}^{i_p}_{j_p} = \frac{1}{12} \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right)$$ (C.16)

This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$).

Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2}$, similar manipulation can be done for the 3 others. The discrete form of the right hand side of (C.14) applied to this triad only can be transformed as follow:

$$\int_D {q \;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv}$$

$$\equiv \sum\limits_{i,j,k} {q} \biggl\{ \;\; \delta_{i+1/2} \left[ - ... ...} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] \;\;\biggr\}$$

$$\equiv \sum\limits_{i,j,k} \biggl\{ \delta_i [q] {{^i_j}\mathbb{Q}^{+1... ...j}} + \delta_j [q] {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\}$$

$$...$$

$$Demonstation to be done...$$

$$...$$

$$\equiv \frac{1} {2} \sum\limits_{i,j,k} \biggl\{ \delta_i \Bigl[ \left( ... ...{+1/2}_{+1/2}} \right)^2 \Bigr]\; \overline{\overline {V}}^{ i+1/2,j} \biggr\}$$

$$\equiv - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2... ...] + \delta_{j+1/2} \left[ \overline{\overline {V}}^{ i+1/2,j} \right] \biggr\}$$

$$\equiv \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \; \overline{\overline{ b_t^{} \chi}}^{ i+1/2, j+1/2}$$

$$\equiv 0$$