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Transient growth in a shearing stratified atmosphere

The stability of fully compressible isothermal shear flows is studied in response to recent claims that such flows could be locally unstable to a linear instability. The behavior of the solution is characterized by the time-dependent Rayleigh quotient, which lies between the smallest and the largest eigenvalue. The imaginary part of the Rayleigh quotient (corresponding to a growth rate) is, for different initial conditions, always found to approach zero provided the flow is Rayleigh stable (angular momentum increasing outwards). The real part (corresponding to frequency) is usually growing linearly in time, which reflects the progressive increase in wavenumber for horizontally propagating sound waves.