Upper Ocean 1-D Seasonal Models
File last modified 16 November 1998
19.2b The Physical Model: Internal Workings
Here we talk about those things that go on inside the model which adjust the distributions of properties, and that set the mixed layer depth. The internal workings are actually rather simple. After adding the forcing, we apply three criteria for vertical stability (i.e. whether water should mix vertically, and whether the mixed layer should deepen). After that, we apply advection diffusion (vertical advection and vertical diffusion) to the water column.
The first stability criterion, and the one that will prove the most important in our model, is static stability. In fact, it accounts for about 80% of the "action". Quite simply put, you cannot have denser water overlying lighter water. This means that you must have
Thus you go through your model domain (let "i" be your position index, with "i" increasing downward), you test to make sure that
and where you find this not to be the case, you then mix all the
cells above this depth (that is average them among themselves).
In general, what you should really do is to just mix the two cells
together, then start from the top of your model and do it again.
What happens in practice, however, is since all the heat exchange
(in particular cooling, which increases the density) takes place
at the top of the model, you will always find that the effect
of this instability is to mix all the way back to the top. So
you may as well do it the first time. Also, we tend to deal in
, which is the density anomaly (the density
minus 1), which works just the same.
The second stability criterion is the bulk Richardson Number stability. This arises due to the fact that if the mixed layer gets going too fast (i.e. the wind stress is allowed to accelerate it to too great a speed), it tends to "stumble" over itself. What actually happens is that if there is too much velocity shear at the base of the mixed layer, it will tend to mix downward. This effect, determined by field and laboratory experiments is such that the mixed layer deepens if the bulk Richardson number goes below a critical value
where g is the gravitational acceleration (9.8 m/s2),
h is the height (thickness) of the mixed layer, 
is the density contrast between the mixed layer and the water
below, and 
is the difference in horizontal
velocity between the mixed layer and the underlying water. This
effect tends to be important when the mixed layer becomes very
thin, because a thin mixed layer becomes easily accelerated by
wind stress, and the inverse quadratic nature of the dependence
makes for a strong damping. The relative activity of this process
is about 20% of the static instability.
The third stability criterion is based on the gradient Richardson number, and has the effect of stirring together layers where the velocity gradient becomes too great. You can think of this as the mixed layer "rubbing" against the water underneath it. This largely has the effect of blurring the transition between the mixed layer and the seasonal thermocline below, which would normally be rather sharp. Laboratory experiments indicate that there is a critical gradient Richardson number, below which stirring occurs:
This turns out to be a not very vigorous process, but becomes a little more important in the absence of any explicit turbulent vertical diffusion.
Now before we get down to actually running the model, we'll show the relative behavior of these profile adjustment processes. Keep in mind that there are no "free parameters" in this formulation: you cannot "dial" the behavior of the model by tuning these processes. They are fundamental physical processes constrained by laboratory observations. Below are plots of the relative activities of the three adjustment mechanisms during different parts of the annual cycle.
For plotting purposes, the static instability activity has been reduced by a factor of 5, so its role is actually 5 times greater than implied by the red curve. The lower figure shows the depth of the mixed layer during the model run, and the model was run for 3 years, starting in mid-March. Note that the static instability becomes extremely important during the cooling part of the seasonal cycle, and serves to steadily deepen the mixed layer from about October onward. At the end of the winter, when warming starts up, the static instability shuts off, since the water column is being heated from above. As it shuts down, you see the bulk Richardson instability (BRN) take over, spiking in spring, but maintaining a plateau during the summer months when the mixed layer is thin. The mixed layer depth is actually a little deceptive, because what really happens (and you can see the BRN adjustment responding) is that the early spring mixed layer starts off as instantaneously very shallow, but is rapidly destroyed by BRN processes, so that average mixed layer depth only gradually increases. The Gradient Richardson number instability is at its most active during the rapid spring shoaling of the mixed layer, for reasons related to the BRN effect.
The one variable, and unknown parameter in this model is the vertical mixing coefficient. It turns out that one is necessary, because if the model is run without vertical mixing, there is strong trapping of heat in the mixed layer, and it becomes extremely hot and thin in the summer time. Moreover, vertical mixing is needed to carry heat down into the seasonal thermocline and for ultimately eroding it during the autumn. We have no a priori way of determining the "right" vertical diffusivity, except by numerical experiment: we need to adjust it until we see the "right" behavior of the temperature field. Certainly, one might want to refine the formulation of the vertical diffusivity, but we choose in this case the very simple approach of a constant value. We have experimented with ones that are inversely proportional to some power of N2 or Rg (see class notes) but this actually yields worse results. We'll leave it for you to experiment with that one.
GoTo Next Section