Fig. 2.23 Sketch of a model basin (in nondimensional length), subjected to the air-sea heat flux q(y), and with the sea surface temperature profile T(y).

In the following analysis, we define heat flux into the ocean as positive. Assuming that the heat flux into the ocean is q (in W/m2), the corresponding buoyancy flux is Fb = agq/p0cp, where a is the thermal expansion coefficient, and cp is the specific heat under constant pressure. Although /f qdxdy = 0 for a steady state, // Fbdxdy = 0, because a = a(T) is a function of temperature. For simplicity, we assume that sea surface temperature is a linear function of y, T = T0(1 - y), and the thermal expansion coefficient is a linear function of temperature, a = a0T = a0T0 (1 - y). Then we have the total buoyancy gain and loss:

It is clear that buoyancy flux is not balanced, even if the heat flux is balanced for a steady state. Similarly, if we treat the air-sea flux at a station in terms of a one-dimensional model, then the annual mean buoyancy flux is non-zero, even though the annual mean heat flux is zero for a steady state. In particular, it is important to note that although buoyancy has been widely used as a tool in diagnosing the structure of the thermal circulation and its energetics, its success is limited to the case when the equation of state is linear. For the case with a nonlinear equation of state, buoyancy is not a conserved quantity, and using buoyancy transport as a tool in diagnosing the circulation in the ocean may introduce some artifacts; therefore, such a method should be used with caution.

2.7.5 Balance of buoyancy in a model with a nonlinear equation of state

To show the potential problems of using buoyancy as a diagnostic tool, we examine the imbalance of buoyancy from the models. For the simple model in Figure 2.23 and the world's oceans, the buoyancy imbalance is listed in Table 2.9, in which the imbalance of the model is defined as

Table 2.9. Buoyancy flux balance for three idealized model oceans and the world's oceans

Buoyancy Unit

One-dimensional model (Fig. 2.22) Nondimensional unit

World oceans

108m4/s3

Buoyancy q = profile

Gain Loss

Unbalance

5/48

q = 0.5 — y q = cos (ny) Annual mean Monthly mean

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