Physical interpretation of unstable modes of a linear shear flow in shallow water on an equatorial beta-plane (original) (raw)
Related papers
Effects of equatorially-confined shear flow on MRG and Rossby waves
Dynamics of Atmospheres and Oceans, 2022
Linear modal stability analysis of a mean zonal shear flow is carried out in the framework of rotating shallow water equations (RSWE), both under the β-plane approximation and in the full spherical coordinate system. Two base flows – equatorial easterly (EE) and westerly (EW) – with Gaussian profiles highly confined to small latitudes are analysed. At low Froude number, mixed Rossby-gravity (MRG) and Rossby waves are found to be particularly affected by shear, with prominent changes at higher wavenumbers. These waves become practically non-dispersive at large wavenumbers in EE. The perturbations are found to be more confined equatorially in EE than in EW with the degree of confinement being more pronounced in the β-plane system compared to the full spherical system. At high Froude number, the phase speeds are significantly larger in the β-plane system for all families of waves. Under the β-plane approximation, exponentially unstable modes can be excited, having negative (positive) phase speed in EE (EW). Strikingly, this flow is always neutrally stable with the full spherical system. This speaks for the importance of studying the whole spherical system even for equatorially confined shear.
Linear instability of uniform shear zonal currents on the β-plane
Journal of Marine Research, 2011
A unified formulation of the instability of a mean zonal flow with uniform shear is proposed, which includes both the coupled density front and the coastal current. The unified formulation shows that the previously found instability of the coupled density front on the f-plane has natural extension to coastal currents, where the instability exists provided that the net transport of the current is sufficiently small. This extension of the coupled front instability to coastal currents implies that the instability originates from the interaction between Inertia-Gravity waves and a vorticity edge wave and not from the interaction of the two edge waves that exist at the two free streamlines due to the Potential Vorticity jump there. The present study also extends these instabilities to the β-plane and shows that β slightly destabilizes the currents by adding instabilities in wavelength ranges that are stable on the f-plane but has little effect on the growthrates in wavelength ranges that are unstable on the f-plane. An application of the β-plane instability theory to the generation of rings in the retroflection region of the Agulhas Current yields a very fast perturbation growth of the scale of 1 day and this fast growth rate is consistent with the observation that at any given time, as many as 10 Agulhas rings can exist in this region.
Stable and unstable shear modes of rotating parallel flows in shallow water
Journal of Fluid Mechanics, 1987
This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial 8-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is $(u2+w2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh'u'). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability. A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously. For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.
Physics of Fluids, 2014
A detailed linear stability analysis of an easterly barotropic Gaussian jet centered at the equator is performed in the long-wave sector in the framework of one- and two-layer shallow-water models on the equatorial β-plane. It is shown that the dominant instability of the jet is due to phase-locking and resonance between Yanai waves, although the standard barotropic and baroclinic instabilities due to the resonance between Rossby waves are also present. In the one-layer case, this dominant instability has non-zero growth rate at zero wavenumber for high enough Rossby and low enough Burger numbers, thus reproducing the classical symmetric inertial instability. Yet its asymmetric counterpart has the highest growth rate. In the two-layer case, the dominant instability may be barotropic or baroclinic, the latter being stronger, with the maximum of the growth rate shifting towards smaller downstream wavenumbers as Rossby number increases at fixed Burger number, and given thickness and de...
Meteorology and Atmospheric Physics, 1993
In this study, the response of a dynamically unstable shear flow with a critical level to periodic forcing is presented. An energy argument is proposed to explain the upshear tilt of updrafts associated with disturbances in two-dimensional stably stratified flows. In a dynamically unstable flow, the energy equation requires an upshear tilt of the perturbation streamfunction and vertical velocity where U, is positive. A stability model is constructed using an iteration method. An upshear tilt of the vertical velocity and the streamfunction fields is evident in a dynamically unstable flow, which is required by energy conversion from the basic shear to the growing perturbation wave energy according to the energy argument. The momentum flux profile indicates that the basic flow is decreased (increased) above (below) the critical level. Thus, the shear instability tends to smooth the shear layer. Following the energy argument, a downshear tilt of the updraft is produced in an unstably stratified flow since the perturbation wave energy is negative. The wave energy budget indicates that the disturbance is caused by a thermal instability modified by a shear flow since the potential energy grows faster than the kinetic energy.
Geophysical Research Letters, 2009
The nonlinearity of Tropical Instability Waves (TIWs) is studied using shallow-water equations with various mean states from the equatorial Pacific Ocean. In the early linear stage, unstable TIWs, centered near 5°N with a wavelength about 1000 km and a period about one month, dominate the whole domain. However neutral Yanai waves with periods about 15-22 days emerge near the equator when the unstable TIWs grow into fully nonlinear vortices and begin to rotate, which stabilizes the mean states substantially. The strength of these Yanai waves are sensitive to the instability of the initial mean flow. Meanwhile the TIWs centered near 5°N are slowed down and weakened. The external forcing terms are found to be important for them to retain their dominance from 3°N to 7°N and also be able to suppress the late emerging Yanai waves if strong enough.
Local baroclinic instability of flow over variable topography
Journal of Fluid Mechanics, 1990
Local baroclinic instability is studied in a two-layer quasi-geostrophic model. Variable meridional bottom slope controls the local supercriticality of a uniform zonal flow. Solutions are found by matching pressure, velocity, and upper-layer vorticity across longitudes where the bottom slope changes abruptly so as to destabilize the flow in a central interval of limited zonal extent. In contrast to previous results from heuristic models, an infinite number of modes exist for arbitrarily short intervals. For long intervals, modal growth rates and frequencies approach the numerical and WKB results for the most unstable mode. For intervals of length comparable to and smaller than the wavelengths of unstable waves in the homogeneous problem, the WKB results lose accuracy. The modes retain large growth rates (about half maximum) for intervals as short as the internal deformation radius. Evidently, the deformation radius and not the homogeneous instability determines the fundamental scale...