Counter-propagating Rossby Waves Research Articles

Linear and nonlinear stability of a quasigeostrophic mixing layer subject to a uniform background shear

The aim of this work is to shed light by revisiting - through the kernel-wave (KW) perspective - the breakdown of a quasi-geostrophic (QG) mixing layer (or vortex strip/filament) in atmosphere under the influence of a background shear. The QG mixing layer is modelled with a family of quasi-Rayleigh velocity profiles in which the potential vorticity (PV) is constant in patches. In the KW perspective a counter-propagating Rossby wave (CRW) is created at each PV edge, i.e. the edge where a PV jump is located. The important parameters of our study are (i) the vorticity of the uniform shear m and (ii) the Rossby deformation radius Ld, which indicates how far the pressure perturbations can vertically propagate. While an adverse shear (m < 0) stabilizes the system, a favorable shear (m > 0) strengthens the instability. This is due to how the background shear affects the two uncoupled CRWs by shifting the optimal phase difference towards large (small) wavenumber when m < 0 (m > 0). As the QG environment is introduced a general weakening of the instability is noticed, particularly for m > 0...

Shear flow instabilities in shallow-water magnetohydrodynamics

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as Høiland’s growth-rate bound and Howard’s semi-circle theorem, are extended to this shallow-water system for quite general flow and field profiles. In the limit of long-wavelength disturbances, a generalisation of the asymptotic analysis of Drazin &amp; Howard (J. Fluid Mech., vol. 14, 1962, pp. 257–283) is performed, establishing that flows can be distinguished as either shear layers or jets. These possess contrasting instabilities, which are shown to be analogous to those of certain piecewise-constant velocity profiles (the vortex sheet and the rectangular jet). In both cases it is found that the magnetic field and stratification (as measured by the Froude number) are generally each stabilising, but weak instabilities can be found at arbitrarily large Froude number. With this distinction between shear layers and jets in mind, the results are extended numerically to finite wavenumber for two particular flows: the hyperbolic-tangent shear layer and the Bickley jet. For the shear layer, the instability mechanism is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification. For the jet, the competition between even and odd modes is discussed, together with the existence at large Froude number of multiple modes of instability.

On the nonnormal–nonlinear interaction mechanism between counter-propagating Rossby waves

The counter-propagating Rossby wave perspective to shear flow instability is extended here to the weakly nonlinear phase. The nonlinear action at a distance interaction mechanism between a pair of waves is identified and separated from the linear one. In the former, the streamwise velocity converges the far-field vorticity anomaly of the opposed wave, whereas in the latter, the cross-stream velocity advects the far-field mean vorticity. A truncated analytical model of two vorticity interfaces shows that higher harmonics generated by the nonlinear interaction act as a forcing on the nonnormal linear dynamics. Furthermore, an intrinsic positive feedback toward small-scale enstrophy results from the fact that higher harmonic pair of waves are generated in anti-phase configuration which is favored for nonnormal growth. Near marginal stability, the waves preserve their structure and numerical simulations of the weakly nonlinear interaction show wave saturation into finite amplitudes, in good agreement both with the fixed point solution of the truncated model, as well as with its corresponding weakly nonlinear Ginzburg–Landau amplitude equation.

Interaction between Rossby and Gravity Waves in a Simple Analytical Model

The interaction between Rossby and gravity waves is examined in a vertical-zonal two-dimensional model, in which the basic state has an upward gradient of buoyancy at a lower level and a downward gradient of horizontal vorticity at a higher level. Because of the gradients, there exist westward-propagating upper Rossby and westward- and eastward-propagating lower gravity waves, where the propagation is relative to the fluid. The initial value problem for the disturbance is analytically solved. The temporal evolution of the analytical solution from an initial value shows the following characteristics. Resonant interaction between the westward-propagating upper Rossby wave and the eastward-propagating lower gravity wave is possible, in the same way as between two counter-propagating Rossby waves in barotropic and baroclinic problems. On the boundary in the parameter space between the unstable region, where resonant exponential growth occurs, and the stable region, where the solution oscillates, the marginal solution grows as a linear function of time. As in other instability problems, the marginal linear growth in the present model is not trivial. For small horizontal wave numbers, the westward-propagating gravity wave makes a non-negligible contribution to the resonant interaction between the westward-propagating Rossby wave and the eastward-propagating gravity waves.

Counterpropagating Rossby waves in confined plane wakes

In the present work, we revisit the temporal and the spatio-temporal stability of confined plane wakes under the perspective of the counterpropagating Rossby waves (CRWs). Within the context of broken line velocity profiles, each vorticity discontinuity can be associated to a counterpropagating Rossby wave. In the case of a wake modeled by a broken line profile, the interaction of two CRWs is shown to originate in a shear instability. Following this description, we first recover the stability results obtained by Juniper [J. Fluid Mech. 590, 163-185 (2007)]10.1017/S0022112007007975 and Biancofiore and Gallaire [Phys. Fluids 23, 034103 (2011)]10.1063/1.3554764 by means of the classical normal mode analysis. In this manner, we propose an explanation of the stabilizing influence of the confinement on the temporal stability properties. The CRW description further allows us to propose a new interpretation of the counterintuitive spatio-temporal destabilization in wake flows at moderate confinement noticed by Juniper [J. Fluid Mech. 565, 171-195 (2006)]10.1017/S0022112006001558: it is well predicted by the mean group velocity of the uncoupled CRWs.

Baroclinic Waves with Parameterized Effects of Moisture Interpreted Using Rossby Wave Components

Abstract A theoretical framework is developed for the evolution of baroclinic waves with latent heat release parameterized in terms of vertical velocity. Both wave–conditional instability of the second kind (CISK) and large-scale rain approaches are included. The new quasigeostrophic framework covers evolution from general initial conditions on zonal flows with vertical shear, planetary vorticity gradient, a lower boundary, and a tropopause. The formulation is given completely in terms of potential vorticity, enabling the partition of perturbations into Rossby wave components, just as for the dry problem. Both modal and nonmodal development can be understood to a good approximation in terms of propagation and interaction between these components alone. The key change with moisture is that growing normal modes are described in terms of four counterpropagating Rossby wave (CRW) components rather than two. Moist CRWs exist above and below the maximum in latent heating, in addition to the upper- and lower-level CRWs of dry theory. Four classifications of baroclinic development are defined by quantifying the strength of interaction between the four components and identifying the dominant pairs, which range from essentially dry instability to instability in the limit of strong heating far from boundaries, with type-C cyclogenesis and diabatic Rossby waves being intermediate types. General initial conditions must also include passively advected residual PV, as in the dry problem.

Persistent Multiple Jets and PV Staircase

Abstract The persistence of multiple jets is investigated with a quasigeostrophic, two-layer, β-plane channel model. Linearly unstable normal modes are found to be capable of qualitatively describing the eddy fluxes of the nonlinear model. For a persistent double jet (PDJ) state, the most unstable normal mode has its largest amplitude located between the two jets, with a downshear tilt that acts to keep the jets separated. The opposite tilt occurs for a double jet state that is intermittent. An analysis of these normal modes, which utilized the concept of counterpropagating Rossby waves (CRWs), suggests that the downshear tilt in the interjet region hinges on the presence of critical latitudes only in the lower layer. This conclusion in turn suggests that the initial generation of the persistent jets requires L/Cgy &amp;lt; r−1, where L is the distance between the wave source (jet) and sink (interjet), Cgy is the meridional group velocity, and r is the linear damping rate. Similar CRW analysis for a conventional normal mode, which has its largest amplitude at the jet centers, suggests that the downshear tilt adjacent to the jet maxima is associated with the presence of critical latitudes only in the upper layer. The PDJ is found to be accompanied by potential vorticity (PV) staircases in the upper layer, characterized by a strong PV gradient at the jet centers and a broad region of homogenized PV between the jets. This PV mixing is realized through baroclinic waves that propagate slowly westward in the interjet region. Nonlinear evolution of the most unstable normal mode of the PDJ shows that northward heat flux by these waves is crucial for broadening the interjet PV mixing zone necessary for producing the PV staircase.

Linear amplification of marginally neutral baroclinic waves

Baroclinic wave development is investigated for unstable parallel shear flows in the limit of vanishing normal-mode growth rate. This development is described in terms of the propagation and interaction mechanisms of two coherent structures, called counter-propagating Rossby waves (CRWs). It is shown that, in this limit of vanishing normal-mode growth rate, arbitrary initial conditions produce sustained linear amplification of the marginally neutral normal mode (mNM). This linear excitation of the mNM is subsequently interpreted in terms of a resonance phenomenon. Moreover, while the mathematical character of the normal-mode problem changes abruptly as the bifurcation point in the dispersion diagram is encountered and crossed, it is shown that from an initial-value (CRW) viewpoint, this transition is smooth. Consequently, the resonance interpretation remains relevant (albeit for a finite time) for wavenumbers slightly different from the ones defining cut-off points. The results are further applied to a two-layer version of the classic Eady model in which the upper rigid lid has been replaced by a simple stratosphere.

Relating Overreflection and Wave Geometry to the Counterpropagating Rossby Wave Perspective: Toward a Deeper Mechanistic Understanding of Shear Instability

Abstract The two major approaches to understanding plane parallel shear instability—the counterpropagating Rossby wave (CRW) and overreflection (OR) approaches—illuminate very different fundamental aspects. This work examines how such seemingly different views can explain the same phenomena and why these differences arise, with the overall goal of deepening our understanding of shear instability. This is done by rationalizing the OR theory in terms of CRW thinking, using a generalization of the CRW approach to multiple infinitesimal potential vorticity (PV) kernels. First the mechanics of cross-shear wave propagation and the exponential decay in an evanescent region are qualitatively explained. Then the cross-shear behavior in different wave geometries (full, partial, and overreflection) is examined. It is found that overreflection is basically a mutual amplification between PV kernels, with the main kernels being at the two sides of the evanescent region, which forms on one side of the critical level. The nondimensional overreflection coefficient is obtained by assuming a balance between the mutual amplification on the two sides of the evanescent region, and the rate at which this amplification is carried away. The critical level is found to be the only point at which there is a mutual amplification between two adjacent kernels. A further examination of the sources of energy growth yields a generalized view of the restoking of the Orr mechanism, thought to be central to OR: growth depends explicitly on the shear, as in the Orr mechanism, while mutual PV kernel interactions are responsible for holding the perturbation at a configuration that allows it to contribute to growth (a restoking).

Waves in a Cloudy Vortex

Abstract This paper derives a system of equations that approximately govern small-amplitude perturbations in a nonprecipitating cloudy vortex. The cloud coverage can be partial or complete. The model is used to examine moist vortex Rossby wave dynamics analytically and computationally. One example shows that clouds can slow the growth of phase-locked counter-propagating vortex Rossby waves in the eyewall of a hurricane-like vortex. Another example shows that clouds can (indirectly) damp discrete vortex Rossby waves that would otherwise grow and excite spiral inertia–gravity wave radiation from a monotonic cyclone at high Rossby number.

Interpretation of discrete and continuum modes in a two-layer Eady model

ABSTRACTThe upper rigid lid of the conventional Eady model for baroclinic instability is replaced by a more realistic stratosphere with an increased buoyancy frequency and a different shear of the zonal wind. Previously reported results of a normalmode stability analysis are re-interpreted using the concept of interacting surface and tropopausePVanomalies, calledPV building blocks (PVBs). In this perspective, which directly relates to the counter-propagating Rossby wave formalism, the appearance of both the short-wave and the long-wave cut-off becomes physically transparent.New results include a discussion of the continuum modes in terms of interacting PVBs. Continuum modes are modal solutions to the inviscid linearised equations, specified by non-zero PV at one interior level (as well as non-zero PV at the surface and the tropopause). If the stratospheric zonal wind decreases with height, the continuum modes cause resonances at multiple (even stratospheric) levels. These resonant continuum modes may play an important role in the explanation of disturbance growth from initial conditions in which the discrete normal modes are neutral.

The counter-propagating Rossby-wave perspective on baroclinic instability. Part III: Primitive-equation disturbances on the sphere

SUMMARY Baroclinic instability of perturbations described by the linearized primitive equations, growing on steady zonal jets on the sphere, can be understood in terms of the interaction of pairs of counter-propagating Rossby waves (CRWs). The CRWs can be viewed as the basic components of the dynamical system where the Hamiltonian is the pseudoenergy and each CRW has a zonal coordinate and pseudomomentum. The theory holds for adiabatic frictionless flow to the extent that truncated forms of pseudomomentum and pseudoenergy are globally conserved. These forms focus attention on Rossby wave activity. Normal mode (NM) dispersion relations for realistic jets are explained in terms of the two CRWs associated with each unstable NM pair. Although derived from the NMs, CRWs have the conceptual advantage that their structure is zonally untilted, and can be anticipated given only the basic state. Moreover, their zonal propagation, phase-locking and mutual interaction can all be understood by ‘PV-thinking’ applied at only two ‘home-bases’— potential vorticity (PV) anomalies at one home-base induce circulation anomalies, both locally and at the other home-base, which in turn can advect the PV gradient and modify PV anomalies there. At short wavelengths the upper CRW is focused in the mid-troposphere just above the steering level of the NM, but at longer wavelengths the upper CRW has a second wave-activity maximum at the tropopause. In the absence of meridional shear, CRW behaviour is very similar to that of Charney modes, while shear results in a meridional slant with height of the air-parcel displacement-structures of CRWs in sympathy with basic-state zonal angular-velocity surfaces. A consequence of this slant is that baroclinically growing eddies (on jets broader than the Rossby radius) must tilt downshear in the horizontal, giving rise to up-gradient momentum fluxes that tend to accelerate the barotropic component of the jet.

The counter-propagating Rossby-wave perspective on baroclinic instability. Part IV: Nonlinear life cycles

Pairs of counter-propagating Rossby waves (CRWs) can be used to describe baroclinic instability in linearized primitive-equation dynamics, employing simple propagation and interaction mechanisms at only two locations in the meridional plane—the CRW ‘home-bases’. Here, it is shown how some CRW properties are remarkably robust as a growing baroclinic wave develops nonlinearly. For example, the phase difference between upper-level and lower-level waves in potential-vorticity contours, defined initially at the home-bases of the CRWs, remains almost constant throughout baroclinic wave life cycles, despite the occurrence of frontogenesis and Rossby-wave breaking. As the lower wave saturates nonlinearly the whole baroclinic wave changes phase speed from that of the normal mode to that of the self-induced phase speed of the upper CRW. On zonal jets without surface meridional shear, this must always act to slow the baroclinic wave. The direction of wave breaking when a basic state has surface meridional shear can be anticipated because the displacement structures of CRWs tend to be coherent along surfaces of constant basic-state angular velocity, U. This results in up-gradient horizontal momentum fluxes for baroclinically growing disturbances. The momentum flux acts to shift the jet meridionally in the direction of the increasing surface U, so that the upper CRW breaks in the same direction as occurred at low levels

The counter‐propagating Rossby‐wave perspective on baroclinic instability. I: Mathematical basis

AbstractIt is shown that Bretherton's view of baroclinic instability as the interaction of two counter‐propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home‐base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home‐bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self‐propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home‐bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home‐bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV‐thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super‐modal growth rate for a short period. A phase‐locking configuration (and normal‐mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft. Copyright © 2004 Royal Meteorological Society

The counter‐propagating Rossby‐wave perspective on baroclinic instability. II: Application to the Charney model

AbstractThe constant‐density Charney model describes the simplest unstable basic state with a planetary‐vorticity gradient, which is uniform and positive, and baroclinicity that is manifest as a negative contribution to the potential‐vorticity (PV) gradient at the ground and positive vertical wind shear. Together, these ingredients satisfy the necessary conditions for baroclinic instability. In Part I it was shown how baroclinic growth on a general zonal basic state can be viewed as the interaction of pairs of ‘counter‐propagating Rossby waves’ (CRWs) that can be constructed from a growing normal mode and its decaying complex conjugate. In this paper the normal‐mode solutions for the Charney model are studied from the CRW perspective.Clear parallels can be drawn between the most unstable modes of the Charney model and the Eady model, in which the CRWs can be derived independently of the normal modes. However, the dispersion curves for the two models are very different; the Eady model has a short‐wave cut‐off, while the Charney model is unstable at short wavelengths. Beyond its maximum growth rate the Charney model has a neutral point at finite wavelength (r=1). Thereafter follows a succession of unstable branches, each with weaker growth than the last, separated by neutral points at integer r—the so‐called ‘Green branches’. A separate branch of westward‐propagating neutral modes also originates from each neutral point. By approximating the lower CRW as a Rossby edge wave and the upper CRW structure as a single PV peak with a spread proportional to the Rossby scale height, the main features of the ‘Charney branch’ (0&lt;r&lt;1) can be deduced without prior calculation of the normal modes. Furthermore, CRWs from this branch are seen to make a smooth transition into the boundary and interior PV structure of the neutral modes appearing at r=1. The behaviour of the other branches and neutral points is essentially the same when viewed from the CRW perspective, but with cancelling interior PV structures reducing the self and mutual interaction of the CRWs. The underlying dynamics determining the nature of all the solutions is the difference in the scale‐dependence of PV inversion for boundary and interior PV anomalies, the Rossby‐wave propagation mechanism and the CRW interaction. The behaviour of the Charney modes and the first neutral branch, which rely on tropospheric PV gradients, are arguably more applicable to the atmosphere than modes of the Eady model where the positive PV gradient exists only at the tropopause. Copyright © 2004 Royal Meteorological Society

Counter-propagating Rossby waves in the barotropic Rayleigh model of shearinstability

Counter-propagating Rossby waves in the barotropic Rayleigh model of shearinstability

Counter‐propagating rossby waves in the barotropic rayleigh model of shear instability

AbstractWave development, using Rayleigh's 1880 model of barotropic, inviscid and incompressible flow, with a mean zonal wind which is linearly sheared between two edges, is described in terms of the interaction between two counter‐propagating Rossby waves (CRWs). Although the solutions described by this approach could also be described by a sum of the normal modes originally obtained by Rayleigh, we offer a CRW description of Rayleigh's model because it provides a useful pedagogical framework for illustrating, in a precise and quantifiable manner, the interacting Rossby wave view of instability. A CRW interpretation of a modified version of the Rayleigh model, consisting of a jet‐like flow featuring two strips of vorticity with opposite signs, is also given.

A Study of Finite Amplitude Barotropic Instability

We report cases with a series of disturbances tilted upshear along the shear zone during the Mei-Yu season. The scale involved is much smaller than the local Rossby radius of deformation; we hypothesize the relevance of barotropic instability, and have explored the nonlinear evolution of barotropic instability. The small and finite amplitude theories are reviewed; their relevance to the observations are briefly discussed. Eigenvalues of ideal three and four region models are calculated analytically. A new Fourier Chebyshev nondivergent barotropic model is constructed. With the initial value problem approach, our experiments on barotropic instability with a vorticity strip either have hyperbolic tangent or Bickerly jet type of wind profiles. We studied the time evolution of the shear layers in terms of the formation of fundamental eddies and successive pairing or merging of eddies. The mutual intensification of counter-propagating Rossby waves (vorticity gradient waves) across the vorticity strip, breakdown of the vorticity strip, local concentration of vorticity and vortex mergering processes were simulated. While the barotropic instability is an efficient way to concentrate vorticity in a small region, the later vortex merging can enlarge the vortex sizes but not the intensity of the resultant vortex. We propose that the concentration and the merging of vortices can create a favorable localized environment within the shear zone for the moist baroclinic processes to operate. As far as the intensity, shape and evolution of the individual vortices are concerned, they are very sensitive to initial background noise(1/100 of mean vorticity). In other words, there is no predictability in the nonlinear evolution. However, the maximum growth rate and the dominant wavelength of vortex can be predicted from the linear analysis. The Bickerly jet possesses higher growth rate than the hyperbolic tangent case on the f plane. We document chaotic behavior of sudden breakup of shear zone and associated vortex merging after a couple of regular cycles of wave-mean flow interaction with shear zone maintained. In the thirty-day integration, the vortices on the β plane become disorganized and scattered while they still remain well organized on the f plane.