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Under quasigeostrophic (QG) dynamics, the presence of a sharp transition in the stratification profile leads to the formation of comparably sharp vertical gradients of buoyancy. When the QG potential vorticity is assumed to be confined to the jump region, as in surface quasigeostrophy (SQG), the sharp gradients are present initially in the buoyancy field. If smoothly varying initial conditions are considered instead, jump-scale features nevertheless emerge after a few turnover times. Inspection of the omega equation reveals that vertical velocity is vertically smoother than buoyancy. As such, the vertical velocity cannot compensate for the sharpness of the stratification jump in the buoyancy equation. Consequently, buoyancy evolves differently above and below the model tropopause, quickly generating sharp vertical gradients. This is confirmed by numerical simulations. The introduction of this small scale, h, characterizing the tropopause implies a larger Froude number, thereby undermining the validity of the quasigeostrophic approximation. For fixed h, scale analysis gives a characteristic horizontal velocity, U, above which not only does QG break down, but statically unstable conditions also develop. Using typical atmospheric values for the Brunt-Väisälä frequency, N = 0.01 s-1, and the jump width, h = 100 m, we argue that U must be less than about 1 ms-1 for static stability to hold (and smaller still for quasigeostrophy to be formally valid). Therefore, quasigeostrophic dynamics are consistent only with very weak near-tropopause flows and thus can hardly account for the observed wind profiles (e.g., the Nastrom and Gage spectral break). We also find that initially smooth flows exhibit secondary roll-up of filaments and shallow slopes near the model tropopause, reminiscent of SQG dynamics. These flows, however, are not SQG-like in the sense they have non-vanishing vertical velocities at the tropopause. [ABSTRACT FROM AUTHOR]