Available Potential Energy and the Maintenance of the General Circulation

The available potential energy of the atmosphere may be defined as the difference between the total potential energy and the minimum total potential energy which could result from any adiabatic redistribution of mass. It vanishes if the density stratification is horizontal and statically stable everywhere, and is positive otherwise. It is measured approximately by a weighted vertical average of the horizontal variance of temperature. In magnitude it is generally about ten times the total kinetic energy, but less than one per cent of the total potential energy. Under adiabatic flow the sum of the available potential energy and the kinetic energy is conserved, but large increases in available potential energy are usually accompanied by increases in kinetic energy, and therefore involve nonadiabatic effects. Available potential energy may be partitioned into zonal and eddy energy by an analysis of variance of the temperature field. The zonal form may be converted into the eddy form by an eddy-transport of sensible heat toward colder latitudes, while each form may be converted into the corresponding form of kinetic energy. The general circulation is characterized by a conversion of zonal available potential energy, which is generated by low-latitude heating and high-latitude cooling, to eddy available potential energy, to eddy kinetic energy, to zonal kinetic energy. I. The concept of available potential energy The strengths of the cyclones, anticyclones, and other systems which form the weather pattern are often measured in terms of the kinetic energy which they possess. Intensifying and weakening systems are then regarded as those which are gaining or losing kinetic energy. When such gains or losses occur, the source or sink of kinetic energy is a matter of importance. Under adiabatic motion, the total energy of the whole atmosphere would remain constant. The only sources or sinks for the The research resulting in this work has been sponsored by the Geophysics Research Directorate of the Air Force Cambridge Research Center, under Contract No. AF I9 (604)-1000. Tellus VII (1955). 2 2-5 0 4 2 2 3 kinetic energy of the whole atmosphere would then be potential energy and internal energy. In general the motion of the atmosphere is not adiabatic. The only nonadiabatic process which directly alters kinetic energy is friction, whch ordinarily generates internal energy while it destroys kinetic energy, but which may also, under suitable circumstances, change atmospheric kinetic energy into the kinetic and potential energy of ocean currents and ocean waves. The remaining nonadiabatic processes, includmg the release of latent energy, alter only the internal energy directly. Hence the only sources for the kinetic energy of the whole atmosphere are atmospheric potential energy and internal energy, while the environment may also act as a sink. EDWARD N. LORENZ I S 8 It is easily shown (cf. HAURWITZ 1941) that the potential and internal energies within a column extending to the top of the atmosphere bear a constant ratio to each other, to the extent that hydrostatic equilibrium prevails. Hence, net gains of kinetic energy occur in general at the expense of both potential and internal energy, in this same ratio. It is therefore convenient to treat potential and internal energy as if they were a single form of energy. The sum of the potential and internal energy has been called total potential energy by Margules Evidently the total potential energy is not a good measure of the amount of energy available for conversion into kinetic energy under adiabatic flow. Some simple cases will serve to illustrate this point. Consider first an atmosphere whose density stratification is everywhere horizontal. In this case, although total potential energy is plentiful, none at all is available for conversion into kinetic energy. Next suppose that a horizontally stratified atmosphere becomes heated in a restricted region. This heating adds total potential energy to the system, and also disturbs the stratification, thus creating horizontal pressure forces which may convert total potential energy into kinetic energy. But next suppose that a horizontally stratified atmosphere becomes cooled rather than heated. The cooling removes total potential energy from the system, but it still disturbs the stratification, thus creating horizontal pressure forces which may convert total potential energy into kinetic energy. Evidently removal of energy is sometimes as effective as addition of energy in making more energy available. We therefore desire a quantity which measures the energy available for conversion into kinetic energy under adiabatic flow. A quantity of this sort was discussed by MARGULES (1903) in his famous paper concerning the energy of storms. Margules considered a closed system possessing a certain distribution of mass. Under akabatic flow, the mass may become redistributed, with an accompanying change in total potential energy, and an equal and opposite change in kinetic energy. It the stratification becomes horizontal and statically stable, the total potential energy reaches its minimum possible value, and the gain of kinetic energy thus reaches its maxi(1 903). mum. This maximum gain of kinetic energy equals the maximum amount of total potential energy available for conversion into kinetic energy under any adiabatic redistribution of mass, and as such may be called the available potential energy. 1 Available potential energy in this sense can be defined only for a fixed mass of atmosphere which becomes redistributed within a fixed region. The storms with which Margules was primarily concerned do not consist of fixed masses within fixed regions, nor do any other systems having the approximate size of storms. It is perhaps for this reason that available potential energy has not become a more familiar quantity. It is in considering the general circulatioii that we deal with a fixed mass within a fixed region-the whole atmosphere. It is thus possible to define the available potential energy of the whole atmosphere as the lfference between the total potential energy of the whole atmosphere and the total potential energy which would exist if the mass were redistributed under conservation of potential teinperature to yield a horizontal stable stratifica-