Nonlinear baroclinic equilibration in the presence of Ekman friction

The baroclinic adjustment hypothesis, e.g. as described in Vallis's textbook (sec. 12.6.2), states that unstable baroclinic flows adjust through a process of potential vorticity homogenization to a state that is close to neutral with respect to baroclinic instability. A question of some theoretical interest is whether or not there exist relatively simple forced-dissipative flows for which adjustment ideas can be shown to lead to qualitatively and quantitatively correct predictions of the equilibrated state. To answer this question, two explicit nonlinear theories for the equilibration of baroclinic waves in a two-layer fluid in a β-channel are tested by comparison with high-resolution numerical simulations. Predictions are tested for a range of parameters (β , κ), where the inverse criticality β measures the degree of instability, and the quasi-geostrophic Ekman number κ the strength of Ekman friction. The first theory, due to Warn, Gauthier and Pedlosky (WGP) is formally valid for marginally unstable waves in the absence of Ekman friction (κ=0). The second, due to Romea, is formally valid for non-zero κ, and for waves that are marginally stable with respect to a different criterion, which enters due to the dissipative destabilization of otherwise stable waves by Ekman friction. The predictions of the two theories are in conflict in the limit κ tends to zero. The results of the numerical experiments used to test the theories are as follows. When κ is slightly greater than zero it is found that the WGP theory accurately predicts the maximum wave amplitude attained during a baroclinic lifecycle across a significant range of parameter space (typical PV evolution during this period is shown in the right panels of the figure). By contrast accurate predictions of the long-time asymptotic wave amplitude are obtained only from Romea's theory, even in cases where WGP describes the initial behavior during the lifecycle accurately. The lower layer PV evolution during the frictional equilibration is markedly different from that predicted by WGP, and resembles that seen in the left panels of the figure. The results firstly indicate the importance of understanding the nonlinear equilibration mechanism of dissipatively destabilized waves. Secondly, it follows that baroclinic adjustment theories formulated from invisicid and frictionless stability criterion make demonstrably incorrect predictions for the equilibrated state, even in the limit of vanishing Ekman friction. Ref: Willcocks and Esler, 2012, J. Phys. Ocean, 42, 225–242.