4.6 Energetics of Growth Mechanism

Where does the energy for growth come from? While our model does not rigorously conserve energy, we may still consider the energetics of the natural system with true mechanical and thermal energy fluxes in both air and sea, closing the energy budget. The atmosphere gains energy from the ocean through surface heat flux and loses energy through surface windstress drag. The storage of energy in the atmosphere is small, so these two processes approximately balance. The ocean therefore ``sees'' the atmosphere as a device which converts thermal energy (from surface heat flux) into mechanical energy (via windstress).

Consider the entrainment-dominated SST parametrization of section 3.2.1. If the interface between two ocean layers with temperature difference $\Delta T$ is anomalously low by an amount $\Delta h$, that column of water has an extra amount of heat (thermal energy) per unit area of magnitude

$$E_{th} = C_p \rho \Delta h \Delta T$$

This heat is tied to an SST anomaly and so is accessible to the atmosphere through air-sea interaction. If a nearby column has the opposite perturbation $-\Delta h$, the atmosphere can be thought of as a heat engine which removes heat from the warm patch and supplies it to the cold patch, diverting some of that heat flux to do ``useful work'' (i.e., generate a windstress). This windstress can increase the kinetic (E_k) and gravitational potential energy (E_p) of the ocean. Since our anomalies are much larger than the oceanic Rossby radius, $E_p \gg E_k$ (Gill, 1982). The gravitational potential energy density of the above configuration, i.e., the amount of energy per unit area that must be imparted by the wind to lift an interface between fluids of density difference $\Delta \rho$ a height $\Delta h$ is

$$E_p = \frac{g}{2} \Delta \rho (\Delta h)^2 = \frac{g}{2} \varepsilon \rho \Delta T (\Delta h)^2$$

where $\varepsilon = \frac{\Delta \rho}{\rho \Delta T}$, equivalent to the coefficient of thermal expansion if salinity is constant. The thermal energy contained in this anomaly is much, much greater than the energy required to make it available:

$$\frac{E_{th}}{E_p} = \frac{C_p}{(g/2) \varepsilon \Delta h} \approx 1.6\cdot10^{5}$$

for $\Delta h$ = 50 m, $C_p = 4000 \mbox{ J kg}^{-1}\mbox{K}^{-1}$, and $\varepsilon = 10^{-4} \mbox{K}^{-1}$. So if the atmospheric heat engine is just .0006% efficient at converting the lateral thermal energy difference into windstress which further lifts the interface, the coupled wave can replenish its energy store.

We thus see that the energy for growth comes from the huge amount of thermal energy stored in the thermocline, which is usually unavailable to the ocean dynamics. But the application of windstress tilts the thermocline, turning vertical thermal gradients into horizontal gradients which the atmosphere can use in a heat-engine fashion to create a windstress which further tilts the thermocline. The atmosphere is a `catalyst', allowing the ocean to extract energy from the vertical stratification. An identical argument holds for the meridional-advection SST equation: the energy for growth is now extracted from the mean meridional SST gradient.