Introduction

In the previous chapters we discussed various equilibrium statistical theories. These statistical theories are developed for the idealized inviscid unforced geophysical flows. However, as we have discussed in Section 10.4, virtually all practical geophysical flows are subject to both forcing and dissipation. For instance, the earth's atmosphere is subject to intense random small-scale forcing from convective storms, and the ocean is subject to forcing from unresolved baroclinic instability processes on a small length scale. Thus, a natural question to ask is whether the equilibrium statistical theories can be applied in a forced and damped environment. The purpose of this chapter is to address this question. More precisely, we want to provide answers to the applied issue (A-4) and theoretical issue (T-5) from Chapter 10.

As was discussed in Section 10.4, equilibrium statistical theories will not be able to approximate geophysical flows in a statistical sense for all forcing. This is not a surprise, since intuitively we could only expect equilibrium theories to succeed when the flow is near equilibrium. What we are interested in here are external forcing which is random and small scale in space and kicks in time. This kind of forcing mimics the small-scale random forcing in the atmosphere and ocean as discussed above and in Chapter 10. The “quasi-equilibrium” state of the geophysical flow is achieved if the inverse cascade of energy from the small scales, where the external forcing occurs, to the large scales (the scale characterized by the equilibrium statistical theory) is sufficiently weak.