On the Movement of a Discontinuous Surface in the Free Atmosphere

Abstract

Consider a discontinuous surface given by the equation where x, y, z denote the cartesian coordinatcs and t the time. The free atmosphere is divided into two strata by the above discontinuous surface, the upper stratum of the atmosphere being expressed by the suffix u and the lower stratum by the suffix d.
The fundamental equations of motion and continuity are, in customary notation, λ being the coriolian parameter. Here the approximation as is adopted, which gives rise to no serious error in the present problem. By the relations the equations of motion may be transformed as ollo ws: in the upper stratum, and in the lower stratum.
Denoting the jump of physical quantity along the discontinuous surface by Δο={ρ_??_(H)-ρu(H)}, Δ(ρ_??_), etc. the following relations are readily derivable: The above three equations determine the motion of the discontinuous surface.
Let the x-axis make an angle ϑ with Δ(ρ_??_) and an angle θ with Grad H, then Put and we get the result: Generally speaking the variation of ψ is so small that we may safely put as ψ=const.
In this case From the above equations the following relations are noticeable: The present result is also essentially the same as Ertel's equation derived from his theory of singular advection. Take ξ-axis along the direction of Grad H, then The above equation, with the initial condition of (H=-tanδ•ξat t=0), may be integrated as follows: which means the motion of the discontinuous surface with the velocity g/2λsin 2ψ•tanδ, the inclination of the surface being invariable. The above result, in a special case of ψ=π/2, reduces to the stationary discontinuous surface by M. Margules. In the last part ot the paper some numerical examples are given.