Barotropic Shear Research Articles

Momentum Flux, Flow Symmetry, and the Nonlinear Barotropic Governor

Abstract A number of idealized life-cycle simulations of baroclinically unstable waves are systematically analyzed to study the effects of eddy momentum flux and of zonal mean horizontal shear on the finite-amplitude evolution of the waves. Twenty-level quasigeostrophic and primitive equation models with channel geometry are numerically integrated with the most unstable linear normal mode as an initial condition. The flows are inviscid except for weak second-order horizontal diffusion. It is found that the finite-amplitude baroclinic waves are sensitively influenced by the vertically integrated eddy momentum flux of the normal mode via the large barotropic shear it spins up in the mean flow. This “barotropic governor” mechanism prevents the eddy from attaining all the available potential energy stored in the domain, leading to irreversible barotropic decay. Only in the purely baroclinic, f-plane, quasigeostrophic problem, where the vertically integrated eddy momentum flux identically vanishes due to symme...

Is the Midlatitude Zonal Flow Absolutely Unstable?

Abstract An analysis is performed of the growth and propagation of unstable baroclinic wave packets in relatively realistic midlatitude zonal currents. The absolute growth rates are calculated, incorporating the effects of both Ekman friction and barotropic shear. It is found that these effects do not alter earlier conclusions (based on simpler models)that a finite easterly low-level wind is required for absolute instability in models with continuous shear. Ekman friction exacerbates the situation, shifting the absolute instability threshold from essentially zero surface wind to negative surface wind. While absolute instability of the midlatitude terrestrial jets thus seems unlikely, the conclusion stands that the atmosphere is at least near the boundary of absolute instability, especially under conditions prevailing in the oceanic storm tracks. In consequence, slow-moving short waves with growth rates below the maximum normal-mode growth rate typically contribute more to the amplification of wave packets...

An Illustrative Model of Instabilities in Meridionally and Vertically Sheared Flows

An analytic model is formulated to study the characteristics of shear instabilities in meridionally and vertically sheared flows. The model is based on the quasigeostrophic equations in two layers. The layers are divided into sections of piecewise uniform potential vorticity. An algebraic dispersion relation is obtained for the complex phase speed c. The magnitude and the sign of the potential vorticity jumps, their meridional separation, the barotropic shear, and the wavenumber of the modes determine the stability of the system. Solutions describe not only pure baroclinic and barotropic instabilities, but also mixtures of these instabilities. The influences of linearly sheared barotropic flows on baroclinic instability are studied in detail, with an emphasis on the direction of vertically integrated momentum flux. The model's implications for the nonlinear life cycle of baroclinic waves are also discussed.

Nonlinear Equilibration of Two-Dimensional Eady Waves: A New Perspective

Abstract The equilibration of two-dimensional baroclinic waves differs fundamentally from equilibration in three dimensions because two-dimensional eddies cannot develop meridional temperature or velocity structure. It was shown in an earlier paper that frontogenesis together with diffusive mixing in a two-dimensional Eady wave brings positive potential vorticity (PV) anomalies deep into the atmosphere from both boundaries and allows the disturbance to settle into a steady state without meridional gradients. Here we depart from the earlier explanation of this equilibration and associate the PV intrusions with essentially the same kind of vortex “roll-up” that characterizes the evolution of barotropic shear layers. To avoid subgrid turbulence parameterizations and computational diffusion, the analogy is developed using Eady's generalized baroclinic instability problem. Eady's generalized model has two semi-infinite regions of large PV surrounding a layer of relatively small PV. Without boundaries, frontal ...

Isolated potential vorticity patches in quasi-geostrophic zonal shear flows

We consider the behavior of isolated uniform potential vorticity patches in a two-layer quasi-geostrophic model. The eddies are embedded in a zonal shear flow having uniform potential vorticity in each layer. This model illustrates many features of vortices in more complex geophysical systems. The form of steady vortex patches is determined by two opposite effects: (1) adjustment of the boundary of the vorticity patches to the deformation by the shearing zonal flow; (2) adjustment of the eddy stream function by the displacement of the boundary of the vorticity patches. The former effect is larger for smaller vorticity patches (compared with the Rossby deformation radius) where relative vorticity is strong, while the latter is more important in the case of larger vorticity patches, when vortex stretching effects are dominant. There are two transition points in the expression for the equilibrium form of the potential vorticity patches as functions of radius; one where the latter effects exceed the former for the linear shear component of the zonal flow and one for the parabolic component. The stability of the equilibrium forms is also investigated by numerical experiments using contour dynamics. It turns out that the vortex is generally unstable when its radius is larger than the transition points. A wide variety of behaviors occur, depending upon the background zonal velocity structure. Barotropic shear can tear apart the counter-rotating part of the vortex. Curvature in the flow causes the eddy to take on a triangular shape, with shedding of filaments from the peaks. Baroclinic shear can separate the upper and lower vortices and lead to propagation of the upper and lower pair in the meridional direction.

An analytic theory of tropical‐cyclone motion in a barotropic shear flow

AbstractThe recent analytic theory for the barotropic motion of an initially symmetric vortex on a beta plane at rest presented by Smith and Ulrich is extended to the case of a horizontal shear flow, enabling the precise effects of shear on vortex motion to be isolated. These effects are characterized by the contribution of the shear‐related terms in the vorticity equation to the evolution and structure of the wave‐number‐1 component of the vortex asymmetry. The analysis proceeds on the assumption that a general shear flow may be adequately approximated by the first three terms of a double Taylor‐series expansion about the initial vortex centre. The manner in which the linear and quadratic terms of this expansion affect the asymmetries is illustrated in detail for a zonal shear flow with a linear or quadratic variation in the meridional direction.For tropical‐cyclone‐scale vortices, the theory shows excellent agreement with equivalent numerical calculations for a period of one or two days. As well as providing insight into the effects of horizontal shear on tropical‐cyclone motion, the calculations have the potential to assist in the design of ‘bogus’ vortices for the initialization of dynamically based tropical‐cyclone forecast models.

An analytic theory of tropical-cyclone motion in a barotropic shear flow

An analytic theory of tropical-cyclone motion in a barotropic shear flow

The Palette of Fronts and Cyclones within a Baroclinic Wave Development

Abstract A study is undertaken of the concurrent development of fronts and cyclones within a growing baroclinic wave. The dynamical framework for the study is that of the semigeostrophic, adiabatic flow of a uniform potential vorticity atmosphere that is sandwiched between two horizontal plates on an f plane. A three-dimensional spectral model is used to examine the growth to finite amplitude of the normal modes of this system for basic states that comprise of a symmetric jetlike baroclinic flow with (and without) a superposition of a uniform barotropic shear of either cyclonic or anticyclonic vorticity. It is demonstrated that the presence of such lateral shear in the ambient environment exerts a profound influence upon the ensuing palette of fronts and cyclones. On the synoptic scale this flow component influences the scale and strength of the individual surface highs and lows, and also modifies the extent of their lateral drift away from, and their relative movement along, the baroclinic zone. Aloft, t...

Finite amplitude waves on barotropic shear layers and jets

Abstract We give a detailed derivation of the amplitude equations governing weak, slowly modulated, varicose wavetrains on a barotropic, triangular jet on a β-plane. As one might expect, at wavenumbers that are neutral by linear theory, well-behaved wavetrain solutions can be found. Such solutions are not unique. The properties of the wavetrain (in particular, the form of the nonlinear coefficient in the amplitude equation) depend upon assumptions that one makes about the structure of the wave field in the region far from the jet. Such assumptions turn out to be equivalent to assumptions made about the initial conditions of the problem. This link is established by a wave packet analysis that retains separate, nondimensional parameters for the amplitude of the wavetrain and the length scale of the modulation envelope. We briefly state similar results for sinuous waves on a triangular jet and for waves on a shear layer composed of a strip of uniform potential vorticity.

Orographic destabilization of a laterally sheared flow

It is shown that a topographically-induced instability can exist for a laterally sheared flow aligned along the contours of a two-dimensional ridge. The nature of the destabilization mechanism is explored within a quasi-geostrophic, f -plane framework for adiabatic flow of an incompressible stratified fluid occupying the half-space z > 0. For a basic state of uniform barotropic shear with (or without) the superposition of an uniform baroclinic component, the instability is shown to be engendered by the terrain shape, sustained by the lateral shear, and inhibited by the baroclinicity. A phenomenological interpretation in terms of the interaction of two “vortex-like” wave-trains located on the adjacent slopes of the terrain accounts neatly for the salient features of the instability. It is shown that there is a similar instability associated with the hybrid configuration of an enhanced baroclinic zone set alongside a terrain slope. Consideration is also given to the limitations of the diagnosed instability and to the possibility of its manifestation in various geophysical systems. In particular, it is suggested that some of the observed cyclogenetic activity along the eastern seaboard of north America and around the periphery of Antarctica could be related to terrain-induced instability effects. DOI: 10.1034/j.1600-0870.1991.t01-4-00007.x

Suppression of Baroclinic Instability in Horizontally Sheared Flows

Abstract Baroclinic instability in the presence of a meridionally sheared barotropic component to the basic flow is studied using a two-level model. The inclusion of a linear shear results in merldionally confined normal modes with growth rates much reduced compared to the unsheared case. Similar results hold for more complicated situations, such as a baroclinic jet with barotropic shear. The results are applied to life cycle calculations in which it is shown that the nonlinear decay phase of a baroclinic disturbance sets up barotropic shears whichtend to suppress further baroclinic developments. An example of the Southern Hemisphere winter mean flow is found to be strongly stabilized by virtue of a strong sheared barotropic part of the zonal flow.

On Radiating Waves Generated from Barotropic Shear Instability of a Western Boundary Current

Results from barotropic stability analyses of strong westerly jets generally tend to conclude that those waves that can extract energy from the shear of the jet are largely trapped by the jet. Therefore, the available shear energy of the flow cannot be transmitted by propagating waves away from the strong zone. We show here that the situation is different if the jet is in the meridional instead of the zonal direction. In this case, unstable waves are generated that are able to propagate energy eastward even in the presence of a realistic dissipation. It is then possible for the shear energy of a western boundary current to be transported large distances into the interior of the ocean.

On the stability of the Soya Warm Current

The stability of the Soya Warm Current is examined, in an attempt to explain the mechanism of the formation of the wave-like pattern seen in satellite infrared imagery in summer. A linear stability theory is applied to barotropic shear flows over a realistic bottom topography. Effects of bottom friction are also taken into consideration. For this current in summer, when volume transport is greatest, the possibility of barotropic instability is suggested. The most unstable waves obtained in this study have wavelengths of 60–80 km, periods of about 1. 5 days, and phase velocities of 45–55 cm sec−1, which is in good agreement with observations.

The stability of non-separable barotropic and baroclinic shear flows

The linear stability with respect to three-dimensional perturbations of unbounded barotropic and baroclinic shear flows depending linearly on both transverse coordinates is studied. The Boussinesq approximation is used, but the usual hydrostatic approximation in the vertical is relaxed. Dissipation effects are ignored. A baroclinic flow can always be destabilized by sufficiently large horizontal anticyclonic shear. The results are relevant for the stability of differential rotation in radiative stars and accretion disks.

Ageostrophic instabilities in a two fluid system with rotation

The stability of synoptic scale waves formed on a frontal surface is studied including nongeostrophic effects with the basic flow subjected to both vertical and horizontal shear. Spectral method is used to obtain the desired solutions. The stability characteristics of the developed unstable modes are presented as a function of shears of the basic flow. With the inclusion of barotropic shear the spectrum of instabilities increase. The lower speeded member of the mixed mode (gravitational-rotational) pair is influenced by the barotropic shear in the basic current and it appears at lower vertical shears. The structure of the height perturbations are utilized to distinguish the various unstable modes developed in the system together with their stability characteristics. This investigation has shown that the ageostrophic effects can be a significant factor in the development of synoptic scale waves on a frontal surface.

Stable and unstable Rossby waves in the North Pacific current as inferred from the mean stratification

We have determined free Rossby waves in the North Pacific Current by numerical methods. We have found only two stable solutions — the barotropic and first-order baroclinic Rossby shear modes. The influence of the current on the dispersion features of these waves is small for the barotropic shear mode, but is significant for the baroclinic shear mode. An explicit comparison of the dispersion relations for the baroclinic wave in case of vanishing and non-vanishing current is given. We have found at most one unstable solution per wave number. The unstable wave with largest growth rate has an e-folding time of 1.1 year. We have calculated vertical profiles of the stream function and the temperature for the various shear modes at various wave numbers. The temperature shear modes have been calculated for later usage in a Rossby wave model to be fitted to observed temperature data from the North Pacific Current area.

Numerical Study of an Interacting Rossby Wave and Barotropic Zonal Flow Near a Critical Level

This paper treats the initial value problem of a forced Rossby wave encountering a critical level in a barotropic zonal shear flow which can change in response to the wave momentum flux divergence. The main result of the calculation is that the shape of the zonal flow profile changes with time in such a way as to reduce the potential vorticity gradient (β−Uyy) to zero at the critical level. For this configuration the wave is totally reflected at the critical level and in the absence of dissipation no longer interacts with the zonal flow. Details of the evolution toward the steady state depend on the ratio of two time scales, one a measure of the wave amplitude and the other representing the time it takes for the wave momentum flux to be concentrated in a well-defined critical layer. The steady-state balance between wave and mean flow probably never occurs in the atmosphere because the time required to set it up is long compared to the expected time scale of natural variability of the zonal flow. More relevant to atmospheric flows is the fact that excursions of (β−Uyy) to negative values during the approach to a steady state are attended by over reflection of the incident wave and a temporary reversal of the wave momentum flux. After the first of these excursions, occurring on a time scale comparable to that required to set up a critical layer, the zonal flow is never far from the final equilibrium profile.

Numerical Study of the Unstable Modes of a Hyperbolic-Tangent Barotropic Shear Flow

Abstract An unbounded hyperbolic-tangent barotropic shear flow is assumed. The complex phase speeds and eigenfunction structure of unstable modes are studied numerically. A reversal of the planetary and shear vorticity gradient is required for these modes to exist. For sufficiently small wavenumber, the unstable modes, which can be associated with the modes of a discontinuous shear layer, are found on both sides of the neutral solution curve in wavenumber-shear parameter space. For sufficiently large wavenumbers, the unstable modes are contiguous to the neutral solution. For an intermediate range of wavenumbers, there are two sets of unstable modes (one contiguous to the neutral solution and one not); one set merges with the zero β unstable mode and one set vanishes for sufficiently small β (or large shear). Above some critical value of wavenumber, the set of modes at fixed wavenumber merging with the neutral solution is contiguous to the neutral boundary, whereas below the critical wavenumber value this ...

On the stability of a barotropic shear flow to nongeostrophic disturbances

The stability of a barotropic shear flow ?* = U tanh ( y */ L ) of a stably stratified fluid of depth H to three-dimensional disturbances, is investigated by a linear analysis. Both the basic-state and perturbed motion are hydrostatic and the Brunt-Vaisala frequency and Coriolis parameter are assumed constant. A neutral solution is found, under the condition that 1 - (U/cgi) 2 > 0, where c gi denotes the phase speed of a gravityinertia wave which is characterized by zonal and vertical wave numbers ?* and n ?/ H respectively. With the aid of this solution, some characteristics of the unstable waves are found by numerical methods. It is then shown that, for values of the parameters characteristic of large-scale mid-latitude flow, the basic current is stable to the internal baroclinic modes (n ? 0) but unstable to the barotropic mode ( n = 0) in the quasi-geostrophic range, R 0 = U/fL 1 - ??1 = 0.682, well-outside the quasi-geostrophic range. Characteristics of the instability are presented and discussed. DOI: 10.1111/j.2153-3490.1971.tb00575.x

On the stability of a barotropic shear flow to nongeostrophic disturbances

On the stability of a barotropic shear flow to nongeostrophic disturbances