Finite difference approximations for the determination of dynamic instability

Abstract

Approximate forms of the vorticity and thermal equations are linearized and combined to yield a second-order partial differential equation for the amplitude of harmonic perturbations. Finite-difference approximations for the derivatives yield a homogeneous system of algebraic equations; and the condition that its determinants vanish for a non-trivial solution yields the “frequency” equation, which may be solved to give the phase velocities of the harmonic waves. Solutions are obtained for zonal currents in which the wind varies vertically and horizonally and for a variety of conditions with respect to grid distances, latitude and current width. Generally speaking, the computations showed that decreasing the latitude and shear and increasing the static stability were all destabilizing influences, not without some exceptions, however. In addition, very short waves were found to be stable; however, instability was found for very long waves, including a retrogressive unstable mode. Moreover, multiple unstable modes were found for many wavelengths. Calculations based on actual observations of the jet stream in December show it to be dynamically unstable, both baroclinically and barotropically, with one mode of maximum instability at a wavelength of about 3000 to 4000 km and a secondary maximum at about 10,000 km.