22.3.1 Advective velocities for T cells

Advective velocities are defined in a direction normal to T-cell faces. Refer to Figure 22.3a which illustrates a T-cell with all six advective velocities and their indices. Note that a T grid point is within the T-cell and there are four surrounding U points; one on each corner of the T-cell. A horizontal slice through the plane containing the grid points is depicted in Figure 22.3c showing the relation between indices in the central T-cell and four surrounding U-cells. Advective velocity on the eastern face of a T cell is a meridional weighted average of the zonal velocities at the vertices of the face. A similar relation holds for the advective velocity on the northern face of a T cell but it involves a zonal average of the meridional velocities. In both cases the averaging takes place within the plane of the face.

  

$$adv\_vet_{i,k,j} \frac{u_{i,k,j,1,\tau}\cdot \dyuj + u_{i,k,j-1,1,\tau}\cdot dyu_{jrow-1}}{2\cdot \dytj}$$ = (22.11)

$$adv\_vnt_{i,k,j} \csuj\;\frac{u_{i,k,j,2,\tau}\cdot dxu_i + u_{i-1,k,j,2,\tau}\cdot dxu_{i-1}}{2\cdot dxt_i}$$ = (22.12)

This form of the advective velocities is not arbitrary and comes from the condition that the work done by horizontal pressure forces must equal the work done by buoyancy. The details are given in Section A.2.4. Note that with non-uniform resolution, the denominator is not equal to the sum of the weights if the grid is constructed as:

$$\frac{dxt_i + dxt_{i+1}}{2}$$ dxu_i = (22.13)

$$\dyuj \frac{\dytj + dyt_{jrow+1}}{2}$$ = (22.14)

which was done in MOM 1 and earlier implementations^22.10. However, MOM allows grid construction^22.11 by

$$\frac{dxu_i + dxu_{i-1}}{2}$$ dxt_i = (22.15)

$$\dytj \frac{\dyuj + dyu_{jrow-1}}{2}$$ = (22.16)

which guarantees that the denominator always equals the sum of the weighting factors. This is presumed to allow more accurate advective velocities when the grid is stretched in the horizontal although differences should be of second order. Refer to Chapter 16 for a further discussion on non-uniform grids and advection.

From the incompressibility condition expressed by Equation (4.3), the advective velocity on the bottom face of each cell is defined as the vertical integral of the convergence of the horizontal advective velocities on each cell face from the surface down to the bottom face of any cell. The rigid lid assumption sets the advective velocity at the top face of the first cell to zero. If option implicit_free_surface is enabled, the advective velocity at the top face of the first cell is diagnosed from $\frac{1}{rho_\circ\cdot grav}\cdot\delta_\tau(ps_{i,jrow})$. If the option explicit_free_surface is enabled, then the vertical velocity is diagnosed as the convergence of the vertically integrated transport. As discussed in Section 7.2.3, this convergence includes both the time tendency of the free surface height as well as the input of fresh water.

For points within the ocean, the vertical advection velocity is calculated diagnostically as the advective velocity at the bottom face of each cell through the expression

 

$$adv\_vbt_{i,k,j} \frac{1}{\cstj}\cdot\sum_{m=1}^{k} \biggl( \delx(adv\_vet_{i-1,m,j}) + \dely(adv\_vnt_{i,m,j-1}) \biggr)\cdot dzt_m$$ =

$$\frac{1}{\cstj}\cdot\sum_{m=1}^{k} \biggl(\frac{adv\_vet_{i,m,j} - adv\_vet_{i-1,m,j}}{dxt_i} +$$ =

$$\hskip+6em\frac{adv\_vnt_{i,m,j} - adv\_vnt_{i,m,j-1}}{\dytj}\biggr)\cdot dzt_m$$ (22.17)

where refers to the latitude at point .