The linear dynamics of symmetric and nonsymmetric perturbations in unbounded zonal inviscid flows with a constant vertical shear of velocity, when a fluid is incompressible and density is stably stratified along the vertical and meridional directions, is investigated. A small–Richardson number Ri ≲ 1 and large–Rossby number Ro ≳ 1 regime is considered, which satisfies the condition for symmetric instability. Specific features of this dynamics are closely related to the nonnormality of linear operators in shear flows and are well interpreted in the framework of the nonmodal approach by tracing the linear dynamics of spatial Fourier harmonics (Kelvin modes) of perturbations in time. The roles of stable stratification, the Coriolis parameter, and vertical shear in the dynamics of perturbations are analyzed. Classification of perturbations into two types or modes—vortex (i.e., quasigeostrophic balanced motions) and inertia–gravity wave—is made according to the value of potential vorticity. The emerging picture of the (linear) transient dynamics for these two modes at Ri ≲ 1 and Ro ≳ 1 indicates that vortex mode perturbations are able to gain basic flow energy and undergo exponential transient amplification and in this process generate inertia–gravity waves. Transient growth of the vortex mode and, consequently, the effectiveness of the wave generation both increase with decreasing Ri and increasing Ro. This linear coupling of perturbation modes is, in general, specific to shear flows but is not fully appreciated yet. A parallel analysis of the transient dynamics of nonsymmetric perturbations versus symmetric instability is also presented. It is shown that the nonnormality-induced transient growth of nonsymmetric perturbations can prevail over the symmetric instability for a wide range of Ri and Ro. The current analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations rather than by symmetric ones, as was accepted in earlier papers. It is noteworthy that the transient growth of perturbations is asymmetric in the wavenumber space—the constant phase plane of maximally amplified perturbations is inclined in a direction northeast to the zonal one and the inclination angle is different for different Ri and Ro.
SUMMARYThe linear dynamics of non-symmetric wave and vortex mode perturbations in spectrally stable geostrophic zonal flows with a constant horizontal shear along the meridional direction, when a fluid is incompressible and stratified, is investigated. Specific features of these dynamics are closely related to the non-normality of the linear operators governing perturbation evolution in shear flows and are well interpreted in the framework of the nonmodal approach-by tracing the linear dynamics of spatial Fourier harmonics of perturbations in time. If the Rossby number Ro < 1, there occurs an algebraic (asymptotically linear) amplification of non-symmetric shear internal waves. It is shown that at Ro > 1, in addition, there takes place an exponential/explosive transient growth of the waves that precedes the time interval of algebraic amplification. We also describe the evolution of pure vortex mode perturbations imposed initially on the mean flow. It is shown that at Ro ∼ 1, pure vortex (aperiodic) perturbations are able to gain basic flow energy and then generate non-symmetric shear internal waves. The studied linear phenomena are specific to geostrophic zonal flows and cast doubt on the filtering of fast wave perturbations in the traditional quasi-geostrophic models of geophysical hydrodynamics.
The linear mechanism of generation of gravity waves by potential vorticity (PV) disturbances in flows with constant horizontal and vertical shears is studied. The case of the initial singular distribution of PV, in which the PV is localized in one coordinate and is periodic with respect to other coordinates, is considered. In a stratified rotating medium, such a distribution induces a vortex wave (continuous mode), the propagation of which is accompanied by the emission of gravity waves. To find the emission characteristics, a linearized system of dynamical equations is reduced to wave equations with sources that are proportional to the initial distributions of PV. The asymptotic solutions of the equations are constructed for small Rossby numbers (horizontal shear) and large Richardson numbers (vertical shear). When passing through the inertial levels symmetrically located with respect to a vortex source, the behavior of the solutions for wave amplitudes radically changes. Directly in the vicinity of the source, the solutions are of monotonic character, corresponding to a quasigeostrophic vortex wave. At long distances from the source, the solutions oscillate. The horizontal momentum flux and the Eliassen–Palm flux are estimated using asymptotic solutions. It is found that, within the indicated range of both Rossby and Richardson numbers, these fluxes are exponentially small: that is, the emission of waves is weak.
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