H. v. Helmholtz introduced an important idea of air-mass ring into the field of dynamical meteorology and discussed the equilibrium condition of two air-mass rings in juxtaposition.
The author intends to develop this theory and to investigate the stability of the atmosphere in a rather generalized form.
The equation of pressure adopted by Helmholtz is as follows:
where θ is potential temperature, ω absolute angular velocity of the air, ω₀ angular velocity of the earth's rotation, ρ distance from the polar axis, r distance from the earth's center, G acceleration of gravity, a radius of the earth, and q, π are certain functions of pressure alone. C is an integrating constant and may be discarded in this problem.
Considering two points (r, ρ) and (r+Δr, ρ+Δρ) respectively, as in Fig. 1, the condition of stability is given schematically as follows.
If the pressure (in Helmholtz's sense) at (r, ρ) is larger than that at (r+Δr, ρ+Δρ), the atmosphere is unstable for the displacement from the former point to the latter, and stable for the reverse displacement. If the magnitude of pressure is reverse, the above condition of stability becomes also reverse.
Thus, the following relation may be adopted as the critical condition of stability:
pressure at (r, ρ)=pressure at (r+Δr, ρ+Δρ). (2)
Denoting the quantities at (r+Δr, ρ+Δρ) by θ+Δθ, ω+Δω and regarding Δ as very small, the following equation is obtained:
Here φ is latitude and Δr is replaced by Δz. Adopting the approximation that ω_??_ω₀, ρ_??_a cosφ, ρΔω=Δr, and taking the limit as Δ→0, the final result is as follows:
where tan α=dz/adφ. Eq. (4a) is an alternative form of Eq. (4b), and the both are essentially the same.
In case of α=90°, Eq. (4a) is reduced to the from:
which is the equation of the vertical stability of the atmosphere, while in case of α=0°, Eq. (4b) becomes:
which is the equation of the horizontal stability.
In case of G=dθ/dz=dθ/dφ=0, the above equations become: respectively, and they coincide with those derived from the circulation theorem of a homogeneous atmosphere (Astrophys. Norveg. 1, No.6, p. 218, 1936).