6.7. Stommel box model for ocean heat transport#
Recall from Fig. 6.5 that the ocean makes a non-trivial contribution to the global heat transport, particularly in the subtropics. Unlike the atmosphere, most of the oceanic heat transport is associated with the mean circulation, rather than eddies. In particular, currents that develop on the western sides of ocean basins make a major contribution to the heat flux. These western boundary current include the Gulf Stream in the North Atlantic and the Kuroshio in the North Pacific.

Fig. 6.22 Mean air-sea heat flux. Data from Yu and Weller, 2007. Positive values indicate heat transfer from the atmosphere to the ocean.#
As warm water flows polewards, it encounters cold air and heat is transferred from the ocean to the atmosphere. Fig. 6.22 shows the annual average net surface heat flux. The western boundary currents are highlighted as regions with large positive heat fluxes, indicating a large amount of heat going from the ocean to the atmosphere. Instantaneous heat fluxes can be much larger. For example, field work conducted in the Gulf Stream during the winter found ocean surface temperatures of 19-20

Fig. 6.23 Cartoon of the global meridional overturning circulation.#
When the near-surface water cools, it becomes more dense. Eventually the water becomes dense enough to sink into the interior of the ocean, where it returns to lower latitudes and gradually upwells. This circulation is called the Thermohaline Circulation, or THC, because both temperature (thermo) and salt (haline) control the density of seawater[1].
In this section, we will develop a box model for the Thermohaline Circulation and use it to investigate some important features of the large-scale ocean circulation and heat transport. In one of the practical sessions, we will construct a computer code to solve the box model equations numerically and we will use this to study the time evolution of the system in more detail. In the next two sections we will describe the physical mechanisms underlying the box model to help put it the model in context and to understand the parameter dependence of each of the terms.

Fig. 6.24 Sketch of the Stommel box model for ocean heat transport.#
The model that we will use is sketched in Fig. 6.24. This is often called the `Stommel model’ since it was first proposed by Henry Stommel, a pioneering physical oceanographer at the Woods Hole Oceanographic Institute in Massachusetts, USA. Like the atmosphere box model that we saw in the last section, the Stommel model has two boxes - one for low latitudes (denoted with “l”) and one for high latitudes (denoted wth “h”). Each box represents the mean ocean properties (averaged in longitude and depth) in that region. As before, the high latitude box accounts for the ocean at high latitudes in the northern and southern hemispheres.
In the ocean, the density of seawater depends on the water temperature, salt content (or salinity), and pressure. The density is a nonlinear function of temperature and salinity, and the equation relating density to temperature, salinity, and pressure is called the equation of state. Since we are considering the depth-averaged water properties, we won’t explicitly consider the pressure dependence. To make the model simpler, we will use a linear approximation to the equation of state in the following form:
where
The arrows at the top of each box indicate fluxes between the ocean and atmosphere. The salt content in the ocean changes due to evaporation and precipitation. Although the rate of evaporation depends on the temperature of the atmosphere and the ocean (and the wind speed), we will assume that the net freshwater flux is constant and we will denote this by
We will assume that the rate of heat transfer between the ocean and atmosphere is proportional to the difference between the ocean and atmosphere temperature in the high latitude or low latitude box. We will denote the temperate in the atmosphere above the low and high latitude boxes by
where the subscript
where
The last term that we need to specify is the transfer of water between the two boxes. This flux is intended to represent the Thermohaline Circulation (THC), which as we have said carries warm water from low to high latitudes near the surface, while returning at depth. The term
where
The heat flux carried out of a box is proportional to the product of the circulation strength (
Putting this together, the equations for the Stommel model can be written:
where
In the practical project, we will solve the equations for the Stommel model numerically. Here, we will look for steady state solutions in the limit when the ocean rapidly relaxes towards the atmospheric temperature. Specifically, let’s consider the limit when
Taking the time derivative of Eq. (6.30) with
Combining these two equations, we find that
We can write this as a single equation for
where
Steady state solutions satisfy a quadratic equation where the right hand side of Eq. (6.34) is zero. Solutions to this quadratic equation are
An interesting feature of the Stommel model is that it has multiple equilibria. In fact, another pair of equilibria emerge if we allow