The Boundary Conditions at the Tropopause

In this paper the general properties of surfaces are investigated at which the derivatives of pressure, temperature, density and velocity may suffer discontinuities. The assumed continuity of the meteorological variables implies severe conditions on the properties of the surface, so that only two types are possible: either the surface moves at the speed of sound relative to the fluid or it is a material surface and moves with the fluid. The tropopause, like a front, is of the latter type. It is shown that the pressure gradient, all particle time derivatives and the velocity divergence are continuous. A knowledge of the discontinuity in temperature lapse rate determines the discontinuities in all the first derivatives of density and potential temperature. In general, the components of the discontinuities in the vertical wind shear parallel to the tropopause can be prescribed arbitrarily (the normal component is zero), but when the wind is geostrophic on both sides of the tropopause, these components too are determined by the jump in the temperature lapse rate. The importance of these considerations for the internal consistency of mathematical models in dynamical meteorology is discussed.