P. Z.-G. acknowledges support from the MOVAC (Grant 200800050084028 from the Ministerio de Medio Ambiente, y Medio Rural y Marino of Spain), DEVIAJE (CGL2009-06944), and COMETH (CGL2012-30641) projects. G.K.V. was funded by the National Science Foundation under Grant AGS-1144302. We are grateful to Peter Haynes and the anonymous reviewers for their insightful comments, which greatly improved the scientific depth of our manuscript. We credit a reviewer with pointing out the relation between our work and that of Robinson.
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An important qualitative difference with observations is that the high tropopause is observed over the region of baroclinic eddy generation (the region with surface westerlies and large eddy heat flux) in our model, rather than being confined to the tropics as observed. A possible explanation for this difference is that the jet maximum is extratropical in our model, while in observations the tropical forcing might displace the jet and tropopause break equatorward.
Because the stratification typically increases with height, a spurious increase in our estimate for tropopause height will tend to bias the tropospheric average stratification to larger values, which would produce some semblance of agreement with the radiative constraint even when the mean state does not change. The reverse is true for the marginal criticality prediction that the tropopause height scales with the isentropic slope: since this slope flattens with height, the mean tropospheric isentropic slope will decrease when using a larger tropopause height, producing a fictitious anticorrelation.
Despite their comparable values of tightness, the vertical heat transport is much larger for our loose run than for the loose limit of the Newtonian cooling model, with the vertical heat transport only weakening by a factor of 1.5 when going from the tight to the loose limit, compared to a factor of 7.5 in the Newtonian cooling model. As a result, the adjustment to marginal criticality might not be as prevalent for the loose limit of this model as for the Newtonian heating model. Because radiative time scales are very long in the upper troposphere in the gray radiation model, tightness in this model is not as strongly tied to the strength of the lower-troposphere vertical heat flux (always large) as in the Newtonian cooling model.