35.2.2.2 Length scale based on baroclinic zone width

The length scale L in the scheme of VMHS is determined by the regional structure of the baroclinicity. Figure 34.1 illustrates the basics of the algorithm used to determine L (algorithm based on that used at the Hadley Centre).

• VMHS used a minimum L as the maximum of either the grid spacing $\Delta s$ or the first baroclinic Rossby radius (NH/f), where $\Delta s = max(\Delta x, \Delta y)$. For most global models employing this scheme, $\Delta s$ will be larger than the first baroclinic Rossby radius. Only for those cases in which the mesoscale eddies are partially resolved will there be overlap. The implementation in MOM simply sets the minimum length scale to $L_{min} = \Delta s = max(\Delta x, \Delta y)$.
• Generally, L will be larger than its minimum value. To determine its value, the spatial variability of the time scale T is used to determine L. L is proportional to the distance it takes to have the growth rate T^-1 become smaller than some rate $vmhs\_rate\_limit$. The MOM default value is $vmhs\_rate\_limit = 1.4 \times 10^{-6} sec^{-1}$. This value yields reasonable diffusivities using the Levitus (1982) dataset (Malcolm Roberts, personal communication). In words, the algorithm proceeds as follows (see Figure 34.1):

  1.

  If the growth rate at a grid point (x_i,y_j) is less than $vmhs\_rate\_limit$, then the length scale at this point is set to $L_{i,j} = max(\Delta x_{i}, \Delta y_{j})$. Point P1 in Figure 34.1 is an example of such a point.

  2.

  If the growth rate at a grid point (i,j) is greater than $vmhs\_rate\_limit$, then the ``width of the baroclinic zone'' is determined. Using point P2 in Figure 34.1 as an example, the distances
  

  distns = distn + dists (35.96)

  distew = diste + distw (35.97)

  $$\min(distns, distew)$$ zone = (35.98)

  
  

  are found using a search algorithm. The distance zone defines the ``width of the baroclinic zone'' so far as point point P2 is concerned. Additionally, the four ``dist'' distances ( distn, dists, diste, distw) have a maximum grid spacing of $ijvmhs\_maxlength$. $ijvmhs\_maxlength = 10$ is the default used in MOM (based on the same value used at the Hadley Centre).

  3.

  For point P2 in Figure 34.1, zone = distns. In this case, define the fraction
  

  $$frac = \frac{\min(distn,dists)}{\max(distn,dists)}.$$ (35.99)

  
  

  For point P2, frac = distn/dists. The length scale for point P2 is then defined by
  

  L_P2 = frac * zone

  $$distn \, \left( 1 + \frac{distn}{dists} \right).$$ = (35.100)

  
  

  Note that the length scale L_P2 is always larger than the grid scale $\Delta y$. If frac were instead defined by $frac= \min(distn,dists)/distns$, then L_P2 would simply equal $\min(distn,dists)$, which has a minimum value equal to the grid scale. Instead, the chosen algorithm gives a bit more weight to any of the points within a baroclinic zone, even if they are just a single point within the zone.