Comments on ‘Zonal and eddy forms of available potential energy equations in pressure coordinates’ by G. J. Boer

Abstract

This note is concerned with the formula for diabatic flow in Boer's (1975) paper, where p , is the reference pressure p,(O,t) -%dOdu eT ae u is the area of the earth and 8, is the potential temperature at the top of the atmosphere. For the sake of simplicity, let us aasume that 8 is greater than the potential temperature at any point of the earth's surface so that the integration in (2) can be carried out without difficulty p , (0, t ) is proportional to the mass of the air above the isentropic surface with potential temperature 8. Boer (1975) quotes (1) from Smith (1969) who refers to Leibnitz's rule in order to prove (1).However, it can be shown that (1) is not valid for diabatic flow. Let us suppose that (1) is correct and let us look at a certain time to at the particles a t an isentropic surface with the potential temperature O0. For diabatic flow, 8 = dO/dt + O and the particles will leave the O0surface. Due to ( l ) , all these particles will keep the same reference pressure p,(O0, to). Therefore, all these particles will have the same potential temperature 8(t) for t > t o since p , Tellus XXVIII (1976), 4 (8, t ) is a unique function of 0 and time. Of course, 0 must not coincide with €lo. This is possible only if 8 does not depend on the coordinates z and y. In the general diabatic case, however, the particles will move to different @-surfaces and will have different reference pressures after some time, in contrast to (1).We may express dp,/dt aa a function of 8, p and p,:Integrating the equation of continuity (4) from 8 to 0, and over the area of the earth (8 = 0 for 8 =8,) we obtain -p d a + 8 -d a -0 sa" t 1 : It follows (7)It can be seen from (7) that (1) is not valid, since the first term on the right hand side of ( 7 ) depends on 0 and t only, whereas the second term depends also on z and y (except for 8 =8 (8, t)). Obviously, (1) is correct for adiabatic flow.

Sources and Sinks of Available Potential Energy in a Moist Atmosphere

“…From a physical point of view, this term can be interpreted as the amount of mechanical work necessary for reorganize the parcels in the reference state. Boer (1976; see also Egger 1976) shows that this term vanishes in the dry framework. However, Boer's proof uses the analytic expression of the reference state in a dry atmosphere and cannot be straightforwardly extended to the moist case.…”

The energetics of extratropical cyclones

“…1. Extreme values (open, hatched) from the six winter/spring cases in Table 4 DP integrated change of reference pressure [Egger, 1976;Smith et al, 1977], which is equal to zero when integrated over the entire atmosphere [Boer, 1976].…”

The time dependence of reference pressure in limited region available potential energy budget equations

“…Recently , Egger (1976), in comments on a paper by Boer (1975), has pointed out that in general dp,/dt and thus, the term in the APE equation in which dp,ldt is contained are not zero. In his response, Boer (1976) confirms Egger's conclusion and further shows that the dp,ldt term is zero only when averaged over the entire globe.…”