35.1.5.3 Vertical isoneutral diffusion flux

The component of the isoneutral diffusive flux through the bottom face of a T cell $diff\_fbt^{iso}_{i,k,j}$ is broken into two parts: the K³³_i,k,j component and the off diagonal component $diff\_fbiso_{i,k,j}$ which contains K³¹_i,k,j and K³²_i,k,j pieces. The vertical diffusion operator term for tracers is also broken into two parts for isoneutral diffusion. First, the part containing $diff\_fbiso_{i,k,j}$ is solved explicitly with all other explicit components. Second, the part containing the K³³_i,k,j component is solved implicitly along with any other vertical diffusivity piece arising from dianeutral diffusion. The use of an implicit solver for the K³³_i,k,j term allows for steeper neutral direction slopes to be handled within the constraints of the linear diffusion equation stability (see Cox 1987 and Griff ies et al. 1997 for details). In the model, the vertical component is written

$$-F^{z}_{i,k,j} = K^{33}_{i,k,j}\delta_{z}T_{i,k,j} + \diff\_fbiso_{i,k,j} = K^{33}_{i,k,j}\delta_{z}T_{i,k,j}$$

$${1 \over 4 dxt_{i} \; dht_{i,k,j} } \; \sum_{ip=0}^{1} dxu_{i-1+i... ...lta^{(i-1+ip,k,j)}_{(i-1+ip,k+kr,j)} \; Ax^{(i,k+kr)}_{(i-1+ip,k+kr \vert i,k)}$$ +

$$\times \; Sx^{(i,k+kr)}_{(i-1+ip,k+kr \vert i,k)} \; \delta_{x}T_{i-1+ip,k+kr}$$

$${1 \over 4 \cos\phi^{T}_{j} dyt_{j} \; dht_{i,k,j} } \; \sum_{jq=... ...lta^{(i,k,j-1+jq)}_{(i,k+kr,j-1+jq)} \; Ay^{(k+kr,j)}_{(k+kr,j-1+jq \vert k,j)}$$ +

$$\times \; Sy^{(k+kr,j)}_{(k+kr,j-1+jq \vert k,j)} \; \delta_{y}T_{k+kr,j-1+jq}.$$ (35.8)

where the non-negative diagonal component to the small angle isoneutral diffusion tensor is given by

$$K^{33 \; small}_{i,k,j} {1 \over 4 dxt_{i} \; dht_{i,k,j} } \; \sum_{ip=0}^{1} dxu_{i-1+ip} \times$$ =

$$\sum_{kr=0}^{1} \Delta^{(i-1+ip,k,j)}_{(i-1+ip,k+kr,j)} \; Ax^{(i,k+kr)}_{(i-1+ip,k+kr \vert i,k)} \; (Sx^{(i,k+kr)}_{(i-1+ip,k+kr \vert i,k)})^{2}$$

$${1 \over 4 \cos\phi^{T}_{j} dyt_{j} \; dht_{i,k,j} } \; \sum_{jq=0}^{1} \cos\phi^{U}_{j-1+jq} dyu_{j-1+jq} \times$$ +

$$\sum_{kr=0}^{1} \Delta^{(i,k,j-1+jq)}_{(i,k+kr,j-1+jq)} \; Ay^{(k... ...j)}_{(k+kr,j-1+jq \vert k,j)} \; (Sy^{(k+kr,j)}_{(k+kr,j-1+jq \vert k,j)})^{2},$$ (35.9)

The contribution of this flux to the diffusion operator is given by

$$R^{z}(T)_{i,k,j} = \delta_{z} (\diff\_fbt^{iso}_{i,k-1,j}).$$ (35.10)

Note that certain irrelevant longitudinal i and latitudinal jlabels were omitted for brevity.