Abstract
Let (Γ*aP/TP) − (Γ*/TE*) = kP and (Γ*aE/TE) − (Γ*/TE*) = kE, where suffix P denotes a plume element and E the surroundings of the element, T is absolute temperature, Γa* a modified adiabatic lapse rate, TE* an absolute virtual temperature of the surroundings, and Γ* the lapse rate of TE*. It is argued that the path of the plume element is unstable either if, in a sufficiently deep layer of the atmosphere surrounding the plume, kE < 0, or if kE ≥ 0 and kP < 0.
If kE ≥ 0 and kP = 0, the element is neutrally stable; if kE ≥ 0 and kP > 0, the element is stable. If kE ≥ 0 and kP is a positive constant, the plume element oscillates in the vertical with frequency N*/(2π), where N*2 = gΛP2kP, g is the gravitational acceleration, and ΛP is a constant. Also 0.32 < ΛP < 0.86 approximately.
The theory does not apply either if kE > 0 and a great deal of the gravitational potential energy is transformed into energy of internal gravity wives, or if the plume sheds a great deal of fluid at a sharp inversion.
