Figure 2: The three steps used in the ALE method. Step 1: layer thickness is computed using a Lagrangian approach based on physics. Step 2: If the layer becomes too thin, the layer thickness is increased, i.e. the coordinate is moved. Step 3: Volume flux across the moving interface is adjusted to maintain total volume.
MIXED LAYER AND COORDINATE CONTROL
The mixed layer model will not be discussed in much detail here. Fluid exchanges between layers take place for non-zero values of wej in (5). Computation of this term involves mixed layer dynamics as well numerical remapping, i.e the ALE method. The computation is done as a sum
wej=δj1wk + wsj + wdj + wnj = wpj + wnj (6)
where the first term is only active for the top layer. It represents entrainment and detrainment by classical Kraus-Turner mixed layer dynamics (Kraus and Turner, 1967). The second term due to interfacial stress, wsj is positive for a layer in case its thickness becomes less that a minimum depth Hmin,j, which is either a preset value or determined by a bulk Richardson number (McCreary and Kundu, 1988, 1989). Similarly, wdj becomes negative for layers exceeding a maximum thickness Hmax,j when the upper ocean is heated or fresh water is added.
The ALE method will remap layers when the methods based on physical conditions are insufficient to maintain a numerical acceptable vertical structure, e.g. too thin (or too thick) layers. This correction is mainly used in areas with prolonged upwelling or downwelling. Basically the local vertical coordinate, relative to the surrounding layers, becomes a z-coordinate. The method is outlined in Fig. 2.
An example of how the ALE method works can be illustrated using a one-dimensional mixed layer model (i.e. single column model). The ocean is chosen to be 150m deep with 5 layers.
The initial thickness of each layer from top to bottom were chosen to be 10m, 10m, 10m, 20m and 100m, respectively. The initial temperatures are 10.0, 7.5, 5.0, 2.5 and 0.0℃, while salinity was 35 psu for all layers.
The model is started from rest. Wind forcing is increased from zero to 0.1N/m2 over 10 days. The net heat flux is sinusoidal with annual period and an amplitude of 200W/m2. The heat flux is zero initially, followed by the cooling phase.
Figure 3 shows the solution after 1 year of integration. During the first month of the integration, the mixed layer deepens and entrains water due to wind forcing. Additional cooling of the mixed layer takes place during the next 50 days. After day 80 convection becomes active and creates a homogeneous temperature profile after day 100. Notice that the vertical coordinate at this time is essentially constant, i.e. a z coordinate.
