A Theorem on the Lapse-rate of Free Atmosphere

Abstract

In this paper an approximate formula is introduced, which describes the change of the lapse-rate.
Coordinates: x. y in horizontal, z in vertical plane.
Velocity components: u in x axis, in y axis and w in z. Temperature. Pressure: T, P.
Assumptions: Geostrophic wind is blowing.
Two equations of horizontal motion (geostrophic wind), Laplace's equation, the first Iaw of thermodynamics and the following equation are used:
This equation describes the change of the lapse-rate (∂T/∂_z) of a fluid element with time (t).
(The physical meaning of this equation needs to be examined. d/dt refers to one fluid particle, but ∂/∂_z must be calculated from (at least) two particles.
And so, it is necessary to consider the physical meaning of the notation d/d_t (∂T/∂_z)•.
As regards this, see also my paper in the previous number of this journal.)
If ∂/∂_z (dQ/d_t) is negligible, we get after some ealculations-
where Q is the heat added from outside of the air particle. ρ the density A is a constant of an individual particle, and not a universal one.
The formula states:- As regards one individual air element, the lapse-rate is proportional to its density.
Particle of geostrophic wind has const. density and const. ∂T/∂_z. It must be understood that the change of ∂T/∂_z is caused by non-adiabatic changes.