Iso/diapycnal 2nd Order Tracer Diffusive Operators

In z-coordinates

The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) curvilinear coordinate system in which the equations of the ocean circulation model are formulated, takes the following form [Redi, 1982]:

$$\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} ... ... {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ \end{array} }} \right]$$ (B.3)

where ($a_1$, $a_2$) are the isopycnal slopes in ( $\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials:

$$a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} ... ...o }{\partial j}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}$$

In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably [Cox, 1987]:

$$\begin{equation}{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\t... ...t[ A^{lT} \;\Re \cdot \left. \nabla \right\vert _z T \right]. \\ \end{equation}$$

Physically, the full tensor (B.3) represents strong isoneutral diffusion on a plane parallel to the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. However, the approximate `weak-slope' tensor (B.4a) represents strong diffusion along the isoneutral surface, with weak vertical diffusion - the principal axes of the tensor are no longer orthogonal. This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same form, (B.4), as (B.2), the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. Written out explicitly,

$$\begin{multline} D^T=\frac{1}{e_1 e_2 }\left\{ {\;\frac{\partial }{\partial i}... ...ight)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ \end{multline}$$

The isopycnal diffusion operator (B.4), (B.5) conserves tracer quantity and dissipates its square. The demonstration of the first property is trivial as (B.4) is the divergence of fluxes. Let us demonstrate the second one:

$$\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabl... ...iint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,$$

and since

the property becomes obvious.

In generalized vertical coordinates

Because the weak-slope operator (B.4), (B.5) is decoupled in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into generalized $s$-coordinates in the same way as (B.1) was transformed into (B.2). The resulting operator then takes the simple form

$$D^T = \left. \nabla \right\vert _s \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right\vert _s T \right] \\ \;\; \;\Re =\left( {{\begin{array}{*{20}c} 1 \hfill & 0 \hfill & {-r _... ... _2 } \hfill & {\varepsilon +r _1 ^2+r _2 ^2} \hfill \\ \end{array} }} \right),$$ (B.5)

where ($r_1$, $r_2$) are the isopycnal slopes in ( $\textbf{i}$, $\textbf{j}$) directions, relative to $s$-coordinate surfaces:

$$r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} ... ... }{\partial j}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.$$

To prove (B.7) by direct re-expression of (B.5) is straightforward, but laborious. An easier way is first to note (by reversing the derivation of (B.2) from (B.1) ) that the weak-slope operator may be exactly reexpressed in non-orthogonal $i,j,\rho$-coordinates as

$$D^T = \left. \nabla \right\vert _\rho \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right\vert _\rho T \right] \\ \;\; \;\Re =\left( {{\begin{array}{*{20}c} 1 \hfill & 0 \hfill &0 \hfi... ...0 \hfill \\ 0 \hfill & 0 \hfill & \varepsilon \hfill \\ \end{array} }} \right).$$ (B.6)

Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives (B.6) immediately.

Note that the weak-slope approximation is only made in transforming from the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates, (B.1) onto $s$-coordinates is exact, however steep the $s$-surfaces.