Periodic solutions of the set of equations governing the nonadiabatic convection of dry isolated thermals

The changes with timet of a temperature deviation δT(t,α) and of a vertical velocityW _i(t,α) of an isolated dry thermal have been investigated theoretically. Solutions for the functionW _i(t, α) have been derived for stable and unstable environmental stratifications. Comparing these solutions with the corresponding ones for the rise of an adiabatic thermal yield some interesting conclusions. Firstly, there is the evident relation between the rate of entrainment of environmental air (expressed by the parameter α=(1/M _i) dM _i/dz whereM _i is the mass of the thermal) and the vertical velocity of the thermal: an increase in α decreases the velocity. Two similar thermals in stably stratified surroundings, one of them moving adiabatically (α=0) the other nonadiabatically (α>0), would rise for the same length of timet ₂=π/N, whereN is a typical Brunt-Väisälä frequency, but with different velocities and to different heights: the ascent timet ₂ depends only on environmental parameters. In an unstable stratification, the vertical non-adiabatic velocity of the thermal, instead of increasing without limit, tends towards a finite asymptotic velocity $$W_t (\infty ) = (\sqrt { - \mathcal{N}^2 } )/\alpha $$ the value of which depends upon both the stratification of the surroundings and upon the entrainment rate α. In a real atmosphere, where additional retarding forces exist, the motion will certainly be damped.