Summary of a Theoretical Investigation into the Factors Controlling the Instability of Long Waves in Zonal Currents

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Abstract
It is assumed that in horizontally uniform zonal currents certain long waves can grow in amplitude whereby the kinetic energy associated with the growth is drawn from the potential energy of the basic current system. This restriction as to the energy redistribution permits to treat the simultaneous physical changes of state as polytropic, which simplifies considerably the integration of the relevant equations of motion. It is shown that the amplitude growth, at a prescribed zonal dimension of the perturbation (wave-length), is primarily controlled by the value of a non-dimensional quantity, viz. the Richardson number (= static stability divided by the square of the vertical shear of wind), in a similar manner as this number controls the time variation of turbulence to the extent of its being a result of energy redistribution. The derived criterion for instability predicts that all but the very long waves in the middle latitude westerlies are unstable. The intensity of the absolute circulation, in addition to the vertical shear, has an important bearing on the considered phenomenon of instability. This is discussed with regard to the nonequilibrium position of a cold anticyclonic vortex centered at the pole. The stability criterion is also scrutinized as to its validity at tropical latitudes. DOI: 10.1111/j.2153-3490.1949.tb01273.x