We want to thank Paul O'Gorman, Alan Plumb, and Isaac Held for helpful comments and discussions. This work was supported through NSF Award OCE-0849233.
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Equation (4) is commonly used to parameterize the eddy-driven overturning circulation in ocean models (Gent and McWilliams 1990). It is typically motivated in terms of a closure for the horizontal eddy buoyancy flux
Notice that z1 denotes the level that separates net warming and cooling in the global mean. Locally at the latitude y1, the change in sign of the diabatic forcing may occur at a slightly different level, implying that Ψ†(y1, z1) is not exactly the global maximum of Ψ†. Moreover, the representation of the heat transport by a residual overturning streamfunction in z coordinates is justified strictly only in the limit of nearly adiabatic, small-amplitude eddies (e.g., Plumb and Ferrari 2005). These approximations are adequate for the bulk scaling relations derived below but may require further consideration when deriving local arguments or when considering strongly inhomogeneous domains. A more general derivation, using isentropic coordinates, is presented in JF12.
The derivation of Eq. (32) in JF12 assumes that the upper-level radiative cooling scales as Q ~ τ−1Δθeq, where Δθeq denotes the variation of the radiative equilibrium potential temperature along an isentrope. Owing to the neutral restoring profile used in this study, we have that, in a domain-averaged sense, Q ~ τ−1∂zθH. Equation (9) is thus recovered by replacing Δθeq in Eq. (32) of JF12 with H∂zθ.
Thompson and Young (2007) compare their results to the scaling relation of Lapeyre and Held (2003), which attempts to generalize the results of Held and Larichev (1996) to the marginally critical limit in the two-layer model. Owing to the large difference between the behavior of the two-layer model and our continuously stratified model near marginal criticality, we here focus our comparison only on the strongly supercritical limit, where the scaling relation in Lapeyre and Held (2003) reduces to the one in Held and Larichev (1996).
Notice that for the simulations with varying f and β, τfric = 50 days was prescribed directly as an external parameter. For the simulations with varying thermal expansion coefficient of JF12, the frictional spindown time scale can be estimated from the vertical viscosity, which together with the no-slip bottom boundary condition generates a linear Ekman layer. As for the simulations with varying f and β, the frictional spindown time scale for the barotropic mode is found to be about 50 days (see Jansen 2012).
Notice that, even if eddies are adiabatic in the sense that
One possible way to argue that this limit could become relevant for the atmosphere is to consider latent heat release as an external forcing in a warm, moist, climate. One might argue that the latent heat release due to moist convection here acts as a restoring to a dry statically stable state. In how far dry dynamics are at all relevant in such a case, and in how far latent heat release can be reasonably thought of as an external diabatic forcing, however, remains questionable.