Upper Ocean 1-D Seasonal Models
File last modified 16 November 1998
19.2a The Physical Model: Boundary Conditions and Forcing
Our main goal in this section is to construct a simple physical model which simulates the temperature, mixed layer depth and vertical mixing as a function of time throughout the year. We will do this for a location near Bermuda (where we have time series data to work with), and run the model for several years. It turns out, as we'll discuss below, Bermuda is one of the few places where this kind of model works well. If we can get this right, then we can move on to modeling the gas concentration evolution in the water column.
Remember, all we want to do is set up a simple numerical laboratory to explore the behavior of gases (in the end oxygen) and learn about the magnitude of primary production and gas transport. We're not trying to do any ground-breaking physics here, just do the physics well enough to create a comfortable home for our gases. There are essentially three different things we need to do "well" in the model.
The temperature history will be critical for determining the dissolved gas solubilities, so we must be able to simulate the surface water temperature as a function of time. It is the surface water temperature which controls the solubility equilibrium dissolved gas concentrations, and hence the flux of gases (which are modeled as a function of concentration difference from equilibrium) between the ocean and atmosphere. The subsurface temperature distribution plays no direct role in controlling dissolved gas concentrations, but does greatly influence the temporal evolution of surface temperatures.
The mixed layer depth must be simulated, because the time-scale of gas exchange equilibrium is related to the mixed-layer depth: the deeper the mixed layer, the longer it takes to reach equilibrium for a given gas exchange rate. If the model mixed layer is too shallow, then the mixed layer will quickly reach equilibrium, and the gas exchange flux will shut off. Also, too small a part of the water column would be involved with the exchange process.
Vertical mixing (and in general vertical transport) is likely to be important. Although vertical mixing tends to be very small in the main thermocline (e.g. at 300 m depth), there is a lot of wind and thermal energy being put in at the ocean surface which will drive turbulent motions and hence mixing. Further, there is a net convergence of surface water in the area due to wind stress patterns. This is, in fact characteristic of all subtropical gyres. The net result is a down-welling of surface waters which will affect property distributions as well. Getting vertical transport right will be important not only for getting the temperature distributions and evolution right, it will play a role in the dissolved gas transport.
Actually, running such a model is relatively simple, providing you do your homework right and pay attention to the details. The model involves a vertical, one-dimensional domain which we must:
We need to keep track of a few things: temperature and salinity (they control the density of the water, which is important for regulating the mixed layer depth) and the horizontal velocities (which are also important for controlling mixed layer depth). For a more complete discussion, see Price et al (1987).
We start the model with some initial profile of temperature and salinity (and zero horizontal velocity): the exact starting profile doesn't matter too much, providing it isn't too far from what is observed. We then run the model by time stepping for several years. The model will "spin up" to some final, reproducible cycle over the course of a year or two. In a very general sense, how long you run a model, and how long you wait before you "take a picture" of its behavior depends on a compromise between waiting long enough to have it stabilize and "forget" initial conditions, and how soon it will begin to drift because of either imperfections in boundary conditions, unresolved physics, or imperfect forcing.
We'll start the model in mid-March (when the water column is at it's most uniform), spin the model up through the remainder of the first year and into the second, and treat our stable cycle as starting the second January of the model run. Here's how a model time step would look:
That's all there is to it. Now for those picky details.
We will apply climatological or "long term averaged" forcing to the model. You could go one step further and pick a series of years where you can get hold of the actual surface data (wind speeds, heat fluxes, etc.) but we want to look at the average behavior of the upper ocean, not some response that may be due to anomalous variations in forcing. As long as the data that we use to "calibrate" the model are averaged in a comparable way, then we are all set. Another problem with "actual" forcing data is that some attributes are so difficult to measure (for example E-P, wind stress curl) or so spatially variable, that you need the climatological data to make a reasonable estimate.
We will use mostly data from the Oberhuber climatology, which is a data product using the COADS data base. It can be accessed on the web from the LDEO climatology server at http://rainbow.ldeo.columbia.edu/AT-DataLibraryquery.html and specifying OBERHUBER for your search.
We need the surface heat flux. Heat flux from the surface most part of the ocean arises from a combination of three types of heat flux:
These three combine to largely cool the water in this part of the ocean. They are shown below (in the left hand figure)
The four solid lines on the left correspond to the sensible heat flux (cyan), latent heat flux (green), the outgoing long-wave radiation (blue), and the sum, the Total Heat Flux (yellow). Note that the heat fluxes are all negative (i.e. heat is going out of the water), and that the latent heat flux is dominant. The dashed line is the best sinusoidal fit to the sum, which seems to fit the curve quite well, and which we will use to drive our toy model:
where t is the time (say in seconds) and tyr is the number of seconds in a year. Note the phase shift, which puts the most negative heat flux in mid December.
The incoming radiative heating (red curve in the right hand figure above) we've fit with another sinusoid:
The zero subscript indicates that this is the irradiance at the sea surface. Note that the phase shift is such that the minimum radiative heating is close to December 21 (winter solstice), for obvious reasons.
However, while the latent, sensible and o.l.r. heat fluxes apply to very top of the ocean, the radiant heating is distributed over a vertical distance. As light penetrates the ocean, it is attenuated as a function of depth. This attenuation is the conversion from light to heat. The spectrum of the down-welling irradiance also changes with depth. The character of this attenuation is governed by a variety of factors, primarily biological in origin. There is a classification of water types by optical properties compiled by Jerlov. The region where we are modeling is a type Ia in the Jerlov classification. Paulson and Simpson (1977) have modeled the net down-welling irradiance as a double exponential function of depth, which for Jerlov type Ia waters is given by
the first term represents the long-wave component, which attenuates very rapidly, and is deposited in the upper-most meter or two. The second term represents the short-wave component, which is distributed over the top 25 to 50 m. Computationally, what we do is to use the above function at the beginning of the program to calculate the fraction of surface irradiance arriving at the top and the bottom of each cell in our model. Then we compute the difference in the two as being the fraction of surface irradiance to be deposited in each cell. This is then used during the simulation as a multiplicative factor, along with the heat capacity and mass of water in the model cell, to calculate the resultant temperature change during a time step with a given surface irradiance.
Finally, it's fair to ask the question can (should) the model work here? The answer is a very definite "maybe". What really happens when we look at the time evolution of a profile at a fixed location is that we are actually seeing water advecting in from some place upstream of our site. You can think of the water column moving by, so setting up forcing and expecting it to work may be a bit optimistic, since we maybe should be looking "upstream" of the site for the forcing. Actually, Bermuda may be one of the few places where it can work. The reason is that the flow lines of the circulation actually run parallel to the contours of heat flux. Below is a comparison of the upper water circulation from Worthington's analysis (upper panel) and the Oberhuber climatological surface heat flux for January (lower panel). We've drawn a "tail" on the site pointing upstream (red tail in the circulation, black tail in the heat flux map). What is very clearly evident from these maps is that the mean circulation parallels the heat flux contours, and that the an average water column tends to feel the same ocean forcing conditions over its pathway.
Worthington's analysis of the circulation for the upper water column.
Oberhuber's January climatology of the total heat flux.
As the wind blows on the surface of the ocean, it supplies momentum to the mixed layer. The momentum transfer to the sea surface is proportional to the square of the wind speed. This constant of proportionality is called the drag coefficient and is approximately 1.3 x 10-3, but increases for wind speeds above 10 m/s due to increasing roughness of the sea surface. Using the climatological winds, however, tends to underestimate the actual wind stress applied to the surface: because the wind is variable, and because the square of the mean wind speed is always less than the mean of the square wind speed. Thus it is better to use the wind stress climatologies. Here, however, one must be careful as to which climatology one selects, since they of variable quality. We'll use the Hellerman climatology (again from the LDEO site), where we have the zonal and meridional wind stress components:
Hellerman Wind Stress Climatology vs Month for Bermuda Area
Now this approach is not perfect either, since there may be short term variations in wind speed not accounted for in the construction of the climatologies, and because of the non-linear dependence of wind stress on wind speed, we may be still underestimating the stress. One approach is to add a random component to the wind, with an r.m.s. amplitude equal to the climatology's and a temporal decorrelation scale of about 3-4 days. The parameterization is sensitive to this time scale because the longer the winds are allowed to blow, the more influence they have. The 3-4 days is a typical time scale associated with large scale meteorological forcing (read that as storms). For our toy model, we will just use the steady winds. What seems to happen is that using the "stochastic winds" reduces the need for explicit turbulent diffusivity in the model (see Musgrave et. al, 1988).
Next, we need to apply Ekman pumping to the model. As we mentioned, there is a convergence of surface waters by wind stress patterns in the region, and this forces water downward at the surface. This flow does not simply extend all the way to the bottom, but because of the dynamical balance of the subtropical gyre, the vertical velocity attenuates downward. For the purposes of our toy model, we will use the Trenberth wind stress curl climatology (taken from the LDEO server) for this area, which gives us a seasonally dependent wind stress curl in nanodynes/cm3 (which in m.k.s. is in units of 10-8 N/m3) we've plotted it below (on the left side) in nanoNewtons/m3:
Now we can get from the wind stress curl to the rate of Ekman pumping by noting that
and in the plot above (right hand side) we've converted to m/y to give you a feeling for its significance. Thus over the course of the year, there is a net downward pumping of about 40 meters. Most of this occurs in the winter, when the mixed layer is deep or getting deeper, but there is a net downward flow of about 10-20 m/y during the rest of the year. The climatology shown above has no simple sinusoidal form, so we have to use linearly interpolated values as a function of year time.
GoTo Next Section