Ertelの水平安定度の式の吟味

抄録

Recently H. Ertel discussed the horizontal stability of zonal circulation and derived the equation⁽¹⁾:
Here f denotes the coriolian parameter 2ω sinφ, ρ₀ the density, P₀ the pressure, U₀ the east component of wind velocity and n the constant in polytropic change of state. b is the Lagrangian coordinate and directed towards the north pole.
In Ertel's theory is assumed the conservation of pressure in horizontal virtual displacement and this is the very reason why the term -U₀∂f/∂b appears in his equation of horizontal stability, since the pressure P₀ is connected with f and U₀ by the relation:
Thus Ertel's assumption of conservation of pressure in virtual displacement directly determines the law of change of coriolian parameter and relative wind velocity (geostrophic wind velocity) in horizontal virtual displacement. But this assumption is generally inacceptable on our rotating earth.
As already pointed out by v. Helmholtz in his theory of air mass ring, it is reasonable to assume the conservation of angular momentum and potential temperature in virtual displacement on the rotating earth. Therefore the pressure does not obey the law of conservation in horizontal displacement, as is the case with the problem of vertical stability. Following v. Helmholtz's theory the present author derived the equation of horizontal stability of the atmosphere as follows:
Here a is the radius of the earth, G the acceleration due to gravity, υλ the longitudinal component of velocity and Θ the potential temperature. In case of υ_λ_??_aωcosφ, the above equation becomes:
in which some terms are also found in Ertel's equation, but there exists an essential difference between the two equations.
H. Ertel applied his theory to the problem of horizontal stability of the actual atmosphere and concluded that the atmosphere is stable for horizontal displacement in a state of zonal circulation. This conclusion holds good also in the author's present paper, but in the substratosphere where the temperature gradient is reverse to that in the troposphere, horizontal instability occurs and predominant advection is expected.