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ArticleRossby wave instability and apparent phase speeds in large ocean basins(American Meteorological Society, 200705) Isachsen, P. E. ; LaCasce, Joseph H. ; Pedlosky, JosephThe stability of baroclinic Rossby waves in large ocean basins is examined, and the quasigeostrophic (QG) results of LaCasce and Pedlosky are generalized. First, stability equations are derived for perturbations on largescale waves, using the twolayer shallowwater system. These equations resemble the QG stability equations, except that they retain the variation of the internal deformation radius with latitude. The equations are solved numerically for different initial conditions through eigenmode calculations and time stepping. The fastestgrowing eigenmodes are intensified at high latitudes, and the slowergrowing modes are intensified at lower latitudes. All of the modes have meridional scales and growth times that are comparable to the deformation radius in the latitude range where the eigenmode is intensified. This is what one would expect if one had applied QG theory in latitude bands. The evolution of largescale waves was then simulated using the Regional Ocean Modeling System primitive equation model. The results are consistent with the theoretical predictions, with deformationscale perturbations growing at rates inversely proportional to the local deformation radius. The waves succumb to the perturbations at the mid to high latitudes, but are able to cross the basin at low latitudes before doing so. Also, the barotropic waves produced by the instability propagate faster than the baroclinic longwave speed, which may explain the discrepancy in speeds noted by Chelton and Schlax.

ArticleOn the weakly nonlinear Ekman layer : thickness and flux(American Meteorological Society, 200806) Pedlosky, JosephThe firstorder effects of nonlinearity on the thickness and frictionally driven flux in the Ekman layer are described for the case of an Ekman layer on a solid, flat plate driven by an overlying geostrophic flow as well as the Ekman layer on a free surface driven by a wind stress in the presence of a deep geostrophic current. In both examples, the fluid is homogeneous. Particular attention is paid to the effect of nonlinearity in determining the thickness of the Ekman layer in both cases. An analytical expression for the Ekman layer thickness as a function of Rossby number is given when the Rossby number is small. The result is obtained by insisting that the perturbation expansion of the Ekman problem in powers of the Rossby number remains uniformly valid. There are two competing physical effects. The relative vorticity of the geostrophic currents tends to reduce the width of the layer, but the vertical velocity induced in the layer can fatten or thin the layer depending on the sign of the vertical velocity. The regularized expansion is shown to give, to lowest order, expressions for the flux in agreement with earlier calculations.

ArticleThe skirted island : the effect of topography on the flow around planetary scale islands(Sears Foundation for Marine Research, 200909) Pedlosky, Joseph ; Iacono, Roberto ; Napolitano, Ernesto ; Helfrich, Karl R.The flow around planetary scale islands is examined when the island possesses a topographic skirt representing a steep continental shelf. The model is barotropic and governed by the shallow water equations and the motion is driven by a wind stress with a constant curl. The presence of the strong topographic "skirt" around the island vitiates the elegant Island Rule of Godfrey and the closed potential vorticity contours around the island produced by the topography allow a geostrophic, stationary mode to resonate with an amplitude that is limited only by dissipation. In the limit of weak forcing the outline of the outermost closed potential vorticity isoline essentially replaces the island shape and determines the flow beyond that contour. Stronger nonlinearity produces substantial changes in the flow pattern as well as the transports trapped on the closed contours and the transport between the island and the basin boundary. Laboratory experiments, numerical calculations and analytical results are presented describing the structure of the flow. A western standing meander at the edge of the island's topography involves a rapid change in the direction of flow and this feature, predicted by analytical and numerical calculations is confirmed in laboratory experiments. As the measure of nonlinearity is increased beyond a threshold that depends on the ratio of the inertial boundary layer thickness to the Munk layer thickness the flow becomes time dependent and a strong eddy field emerges. The transports on the closed contours and the interbasin exchange outside the closed potential vorticity contours show an enhancement over the linear analytical approximation as nonlinearity increases.

ArticleLateral coupling in baroclinically unstable flows(American Meteorological Society, 200806) Spall, Michael A. ; Pedlosky, JosephA twolayer quasigeostrophic model in a channel is used to study the influence of lateral displacements of regions of different sign mean potential vorticity gradient (Πy) on the growth rate and structure of linearly unstable waves. The mean state is very idealized, with a region of positive Πy in the upper layer and a region of negative Πy in the lower layer; elsewhere Πy is zero. The growth rate and structure of the model’s unstable waves are quite sensitive to the amount of overlap between the two regions. For large amounts of overlap (more than several internal deformation radii), the channel modes described by Phillips’ model are recovered. The growth rate decreases abruptly as the amount of overlap decreases below the internal deformation radius. However, unstable modes are also found for cases in which the two nonzero Πy regions are separated far apart. In these cases, the wavenumber of the unstable waves decreases such that the aspect ratio of the wave remains O(1). The waves are characterized by a largescale barotropic component that has maximum amplitude near one boundary but extends all the way across the channel to the opposite boundary. Near the boundaries, the wave is of mixed barotropic–baroclinic structure with crossfront scales on the order of the internal deformation radius. The perturbation heat flux is concentrated near the nonzero Πy regions, but the perturbation momentum flux extends all the way across the channel. The perturbation fluxes act to reduce the isopycnal slopes near the channel boundaries and to transmit zonal momentum from the region of Πy > 0 to the region on the opposite side of the channel where Πy < 0. These nonzero perturbation momentum fluxes are found even for a mean state that has no lateral shear in the velocity field.

ArticleInteraction of Ekman layers and islands(American Meteorological Society, 201305) Spall, Michael A. ; Pedlosky, JosephThe circulation induced by the interaction of surface Ekman transport with an island is considered using both numerical models and linear theory. The basic response is similar to that found for the interaction of Ekman layers and an infinite boundary, namely downwelling (upwelling) in narrow boundary layers and deformationscale baroclinic boundary layers with associated strong geostrophic flows. The presence of the island boundary, however, allows the pressure signal to propagate around the island so that the regions of upwelling and downwelling are dynamically connected. In the absence of stratification the island acts as an effective barrier to the Ekman transport. The presence of stratification supports baroclinic boundary currents that provide an advective pathway from one side of the island to the other. The resulting steady circulation is quite complex. Near the island, both geostrophic and ageostrophic velocity components are typically large. The density anomaly is maximum below the surface and, for positive wind stress, exhibits an anticyclonic phase rotation with depth (direction of Kelvin wave propagation) such that anomalously warm water can lie below regions of Ekman upwelling. The horizontal and vertical velocities exhibit similar phase changes with depth. The addition of a sloping bottom can act to shield the deep return flow from interacting with the island and providing mass transport into/out of the surface Ekman layer. In these cases, the required transport is provided by a pair of recirculation gyres that connect the narrow upwelling/downwelling boundary layers on the eastern and western sides of the island, thus directly connecting the Ekman transport across the island.

ArticleResponse to a steady poleward outflow. Part II : oscillations and eddies(American Meteorological Society, 200907) Durland, Theodore S. ; Spall, Michael A. ; Pedlosky, JosephA conceptually simple model is presented for predicting the amplitude and periodicity of eddies generated by a steady poleward outflow in a 1½layer βplane formulation. The prediction model is rooted in linear quasigeostrophic dynamics but is capable of predicting the amplitude of the β plume generated by outflows in the nonlinear range. Oscillations in the plume amplitude are seen to represent a nearzero group velocity response to an adjustment process that can be traced back to linear dynamics. When the plumeamplitude oscillations become large enough so that the coherent β plume is replaced by a robust eddy field, the eddy amplitude is still constrained by the plumeamplitude prediction model. The eddy periodicity remains close to that of the predictable, nearzero groupvelocity linear oscillations. Striking similarities between the patterns of variability in the model and observations south of Indonesia’s Lombok Strait suggest that the processes investigated in this study may play an important role in the generation of the observed eddy field of the IndoAustralian Basin.

ArticleWinddriven barotropic gyre I : circulation control by eddy vorticity fluxes to an enhanced removal region(Sears Foundation for Marine Research, 200403) FoxKemper, Baylor ; Pedlosky, JosephIt is well known that the barotropic, winddriven, singlegyre ocean model reaches an inertiallydominated equilibrium with unrealistic circulation strength when the explicit viscosity is reduced to realistically low values. It is shown here that the overall circulation strength can be controlled nonlocally by retaining thin regions of enhanced viscosity parameterizing the effects of increased mixing and topographic interaction near the boundaries. The control is possible even when the inertial boundary layer width is larger than the enhanced viscosity region, as eddy fluxes of vorticity from the interior transport vorticity across the mean streamlines of the inertial boundary current to the frictional region. In relatively inviscid calculations the eddies are the major means of flux across interior mean streamlines.

ArticleResponse to a steady poleward outflow. Part I : the linear, quasigeostrophic problem(American Meteorological Society, 200907) Durland, Theodore S. ; Pedlosky, Joseph ; Spall, Michael A.The response of a zonal channel to a uniform, switchedon but subsequently steady poleward outflow is presented. An eastward coastal current with a Kelvin wave’s crossshore structure is found to be generated instantly upon initiation of the outflow. The current is essentially in geostrophic balance everywhere except for the vicinity of the outflow channel mouth, where the streamlines must cross planetary vorticity contours to feed the current. The adjustment of this region generates a plume that propagates westward at Rossby wave speeds. The crossshore structure of the plume varies with longitude, and at any given longitude it evolves with time. The authors show that the plume evolution can be understood both conceptually and quantitatively as the westward propagation of the Kelvin current’s meridional spectrum, with each spectral element propagating at its own Rossby wave group velocity.

ArticleRossby waves with continuous stratification and bottom friction(American Meteorological Society, 20180917) Brink, Kenneth H. ; Pedlosky, JosephPublished observations of subinertial ocean current variability show that the vertical structure is often well described by a vertical mode that has a node of horizontal velocity at the bottom rather than the traditional node of vertical velocity. The theory of forced and free linear Rossby waves in a continuously stratified ocean with a sloping bottom and bottom friction is treated here to see if frictional effects can plausibly contribute to this phenomenon. For parameter values representative of the mesoscale, bottom dissipation by itself appears to be too weak to be an explanation, although caution is required because the present approach uses a linear model to address a nonlinear phenomenon. One novel outcome is the emergence of a shortwave, bottomtrapped, strongly damped mode that is present even with a flat bottom.

ArticleBaroclinic instability of timedependent currents(Cambridge University Press, 20030819) Pedlosky, Joseph ; Thomson, James M.The baroclinic instability of a zonal current on the betaplane is studied in the context of the twolayer model when the shear of the basic current is a periodic function of time. The basic shear is contained in a zonal channel and is independent of the meridional direction. The instability properties are studied in the neighbourhood of the classical steadyshear threshold for marginal stability. It is shown that the linear problem shares common features with the behaviour of the wellknown Mathieu equation. That is, the oscillatory nature of the shear tends to stabilize an otherwise unstable current while, on the contrary, the oscillation is able to destabilize a current whose timeaveraged shear is stable. Indeed, this parametric instability can destabilize a flow that at every instant possesses a shear that is subcritical with respect to the standard stability threshold. This is a new source of growing disturbances. The nonlinear problem is studied in the same near neighbourhood of the marginal curve. When the timeaveraged flow is unstable, the presence of the oscillation in the shear produces both periodic finiteamplitude motions and aperiodic behaviour. Generally speaking, the aperiodic behaviour appears when the amplitude of the oscillating shear exceeds a critical value depending on frequency and dissipation. When the timeaveraged flow is stable, i.e. subcritical, finiteamplitude aperiodic motion occurs when the amplitude of the oscillating part of the shear is large enough to lift the flow into the unstable domain for at least part of the cycle of oscillation. A particularly interesting phenomenon occurs when the timeaveraged flow is stable and the oscillating part is too small to ever render the flow unstable according to the standard criteria. Nevertheless, in this regime parametric instability occurs for ranges of frequency that expand as the amplitude of the oscillating shear increases. The amplitude of the resulting unstable wave is a function of frequency and the magnitude of the oscillating shear. For some ranges of shear amplitude and oscillation frequency there exist multiple solutions. It is suggested that the nature of the response of the finiteamplitude behaviour of the baroclinic waves in the presence of the oscillating mean flow may be indicative of the role of seasonal variability in shaping eddy activity in both the atmosphere and the ocean.

ArticleA note on interior pathways in the meridional overturning circulation(American Meteorological Society, 20180312) Pedlosky, JosephA simple oceanic model is presented for source–sink flow on the β plane to discuss the pathways from source to sink when transport boundary layers have large enough Reynolds numbers to be inertial in their dynamics. A representation of the flow as a Fofonoff gyre, suggested by prior work on inertial boundary layers and eddydriven circulations in twodimensional turbulent flows, indicates that even when the source and sink are aligned along the same western boundary the flow must intrude deep into the interior before exiting at the sink. The existence of interior pathways for the flow is thus an intrinsic property of an inertial circulation and is not dependent on particular geographical basin geometry.

ArticleRadiating instability of a meridional boundary current(American Meteorological Society, 200810) Hristova, Hristina G. ; Pedlosky, Joseph ; Spall, Michael A.A linear stability analysis of a meridional boundary current on the beta plane is presented. The boundary current is idealized as a constantspeed meridional jet adjacent to a semiinfinite motionless far field. The farfield region can be situated either on the eastern or the western side of the jet, representing a western or an eastern boundary current, respectively. It is found that when unstable, the meridional boundary current generates temporally growing propagating waves that transport energy away from the locally unstable region toward the neutral far field. This is the socalled radiating instability and is found in both barotropic and twolayer baroclinic configurations. A second but important conclusion concerns the differences in the stability properties of eastern and western boundary currents. An eastern boundary current supports a greater number of radiating modes over a wider range of meridional wavenumbers. It generates waves with amplitude envelopes that decay slowly with distance from the current. The radiating waves tend to have an asymmetrical horizontal structure—they are much longer in the zonal direction than in the meridional, a consequence of which is that unstable eastern boundary currents, unlike western boundary currents, have the potential to act as a source of zonal jets for the interior of the ocean.

ArticleBaroclinic flow around planetary islands in a double gyre : a mechanism for crossgyre flow(American Meteorological Society, 201005) Pedlosky, JosephA quasigeostrophic, twolayer model is used to study the baroclinic circulation around a thin, meridionally elongated island. The flow is driven by either buoyancy forcing or wind stress, each of whose structure would produce an antisymmetric doublegyre flow. The ocean bottom is flat. When the island partially straddles the intergyre boundary, fluid from one gyre is forced to flow into the other. The amount of the intergyre flow depends on the island constant, that is, the value of the geostrophic streamfunction on the island in each layer. That constant is calculated in a manner similar to earlier studies and is determined by the average, along the meridional length of the island, of the interior Sverdrup solution just to the east of the island. Explicit solutions are given for both buoyancy and winddriven flows. The presence of an island of nonzero width requires the determination of the baroclinic streamfunction on the basin’s eastern boundary. The value of the boundary term is proportional to the island’s area. This adds a generally small additional baroclinic intergyre flow. In all cases, the intergyre flow produced by the island is not related to topographic steering of the flow but rather the pressure anomaly on the island as manifested by the barotropic and baroclinic island constants. The vertical structure of the flow around the island is a function of the parameterization of the vertical mixing in the problem and, in particular, the degree to which long baroclinic Rossby waves can traverse the basin before becoming thermally damped.

ArticleBaroclinic instability over topography : unstable at any wave number(Sears Foundation for Marine Research, 20160102) Pedlosky, JosephThe instability of an inviscid, baroclinic vertically sheared current of uniform potential vorticity, flowing along a uniform topographic slope, becomes linearly unstable at all wave numbers if the flow is in the direction of propagation of topographic waves. The parameter region of instability in the plane of scaled topographic slope versus wave number then extends to arbitrarily large wave numbers at large slopes. The weakly nonlinear treatment of the problem reveals the existence of a nonlinear enhancement of the instability close to one of the two boundaries of this parametrically narrow unstable region. Because the domain of instability becomes exponentially narrow for large wave numbers, it is unclear how applicable the results of the asymptotic, weakly nonlinear theory are given that it must be limited to a region of small supercriticality. This question is pursued in that parameter domain through the use of a truncated model in which the approximations of weakly nonlinear theory are avoided. This more complex model demonstrates that the linearly most unstable wave in the narrow wedge in parameter space is nonlinearly stable and that the region of nonlinear destabilization is limited to a tiny region near one of the critical curves rendering both the linear and nonlinear growth essentially negligible.

Technical Report2007 program of studies : boundary layers(Woods Hole Oceanographic Institution, 200806) Cenedese, Claudia ; Whitehead, John A. ; Pedlosky, Joseph ; Lentz, Steven J.The topic of the Principal Lectures for the fortyninth year of the program was “Boundary Layers”. The subject centers around those problems in which the boundary conditions lead to a large gradient near the boundary. Nine of this year’s principal lectures were given by Joe Pedlosky and the tenth was given by Steve Lentz. The fluid mechanics of boundary layers was reviewed, first starting from its classical roots and then extending the concepts to the sides, bottoms, and tops of the oceans. During week four, a minisymposium on “Ocean Bottom and Surface Boundary Layers” gathered a number of oceanographers and meteorologists together to report recent advances. And, finally, Kerry Emanuel of MIT delivered the Sears Public Lecture to a packed hall in Clark 507. The title was “Divine Wind: The History and Sciences of Hurricanes.”

ArticleRole of eddy forcing in the dynamics of multiple zonal jets in a model of the North Atlantic(American Meteorological Society, 200906) Kamenkovich, Igor V. ; Berloff, Pavel S. ; Pedlosky, JosephMultiple zonal jets are observed in satellite data–based estimates of oceanic velocities, float measurements, and highresolution numerical simulations of the ocean circulation. This study makes a step toward understanding the dynamics of these jets in the real ocean by analyzing the vertical structure and dynamical balances within multiple zonal jets simulated in an eddyresolving primitive equation model of the North Atlantic. In particular, the authors focus on the role of eddy flux convergences (“eddy forcing”) in supporting the buoyancy and relative/potential vorticity (PV) anomalies associated with the jets. The results suggest a central role of baroclinic eddies in the barotropic and baroclinic dynamics of the jets, and significant differences in the effects of eddy forcing between the subtropical and subpolar gyres. Additionally, diabatic potential vorticity sources and sinks, associated with vertical diffusion, are shown to play an important role in supporting the potential vorticity anomalies. The resulting potential vorticity profile does not resemble a “PV staircase”—a distinct meridional structure observed in some idealized studies of geostrophic turbulence.

ArticleThe twolayer skirted island(Yale University, 20110930) Pedlosky, Joseph ; Iacono, Roberto ; Napolitano, Ernesto ; Spall, Michael A.The flow around a planetary scale island in a baroclinic ocean is examined when the island possesses a topographic skirt representing a steep continental slope and the ocean is modeled as a twolayer system in order to examine the role of stratification in the circulation. The study extends an earlier barotropic model of similar geometry and forcing to focus on the degree to which the topography, limited here to the lower of the two layers, affects the circulation and to what degree the circulation is shielded by stratification from the topographic effects noted in the simpler barotropic model. As in the barotropic model, the topography is steep enough to produce closed, ambient potential vorticity contours over the topography in the lower layer providing free "highways" for the deep flow in the presence of small forcing by the winddriven upper layer flow. The flow is very weak outside the region of closed contours but can become of the same order, if somewhat smaller, as the upper layer flow on those contours in the presence of even weak coupling to the upper layer. A series of models, analytical and numerical, are studied. Linear theory is applied to two configurations. The first consists of a long, meridionally oriented island with a topographic skirt in the lower layer. The lower layer flow is driven by a hypothesized frictional coupling between the two layers that depends on the circulation of the upper layer velocity on a circuit defined by the closed potential vorticity contours of the lower layer. The largest part of the driving flow is identical on both sides of the island and cancels in the contour integration. The major part of the residual forcing comes from relatively small but effective forcing on the semicircular tips of the topographic skirt. A circular island with a topographic skirt is also examined in which the coupling to the upper layer is stronger all around the island. Even in this case there is a delicate balance of the forcing of the lower layer on each side of the island. In all cases the flow on closed potential vorticity contours in the lower layer is much weaker than in the barotropic model but much stronger than in the flat region of the lower layer. A sequence of numerical calculations that both check and extend the analytic linear theory is presented demonstrating the subtlety of the force balances. Further nonlinear, eddycontaining experiments give a preview of the direction of future work.

ArticleCrossshelf and outofbay transport driven by an openocean current(American Meteorological Society, 20111101) Zhang, Yu ; Pedlosky, Joseph ; Flierl, Glenn R.This paper studies the interaction of an Antarctic Circumpolar Current (ACC)–like winddriven channel flow with a continental slope and a flatbottomed bayshaped shelf near the channel’s southern boundary. Interaction between the model ACC and the topography in the second layer induces local changes of the potential vorticity (PV) flux, which further causes the formation of a firstlayer PV front near the base of the topography. Located between the ACC and the firstlayer slope, the newly formed PV front is constantly perturbed by the ACC and in turn forces the firstlayer slope with its own variability in an intermittent but persistent way. The volume transport of the slope water across the firstlayer slope edge is mostly directly driven by eddies and meanders of the new front, and its magnitude is similar to the maximum Ekman transport in the channel. Near the bay’s opening, the effect of the topographic waves, excited by offshore variability, dominates the crossisobath exchange and induces a mean clockwise shelf circulation. The waves’ propagation is only toward the west and tends to be blocked by the bay’s western boundary in the narrowshelf region. The ensuing wave–coast interaction amplifies the wave amplitude and the crossshelf transport. Because the interaction only occurs near the western boundary, the shelf water in the west of the bay is more readily carried offshore than that in the east and the mean shelf circulation is also intensified along the bay’s western boundary.

ArticleThe nonlinear dynamics of timedependent subcritical baroclinic currents(American Meteorological Society, 200704) Flierl, Glenn R. ; Pedlosky, JosephThe nonlinear dynamics of baroclinically unstable waves in a timedependent zonal shear flow is considered in the framework of the twolayer Phillips model on the beta plane. In most cases considered in this study the amplitude of the shear is well below the critical value of the steady shear version of the model. Nevertheless, the timedependent problem in which the shear oscillates periodically is unstable, and the unstable waves grow to substantial amplitudes, in some cases with strongly nonlinear and turbulent characteristics. For very small values of the shear amplitude in the presence of dissipation an analytical, asymptotic theory predicts a selfsustained wave whose amplitude undergoes a nonlinear oscillation whose period is amplitude dependent. There is a sensitive amplitude dependence of the wave on the frequency of the oscillating shear when the shear amplitude is small. This behavior is also found in a truncated model of the dynamics, and that model is used to examine larger shear amplitudes. When there is a mean value of the shear in addition to the oscillating component, but such that the total shear is still subcritical, the resulting nonlinear states exhibit a rectified horizontal buoyancy flux with a nonzero time average as a result of the instability of the oscillating shear. For higher, still subcritical, values of the shear, a symmetry breaking is detected in which a second crossstream mode is generated through an instability of the unstable wave although this second mode would by itself be stable on the basic timedependent current. For shear values that are substantially subcritical but of order of the critical shear, calculations with a full quasigeostrophic numerical model reveal a turbulent flow generated by the instability. If the beta effect is disregarded, the inviscid, linear problem is formally stable. However, calculations show that a small degree of nonlinearity is enough to destabilize the flow, leading to large amplitude vacillations and turbulence. When the most unstable wave is not the longest wave in the system, a cascade up scale to longer waves is observed. Indeed, this classically subcritical flow shows most of the qualitative character of a strongly supercritical flow. This result supports previous suggestions of the important role of background time dependence in maintaining the atmospheric and oceanic synoptic eddy field.

ArticleTriad instability of planetary rossby waves(American Meteorological Society, 200708) Zhang, Yu ; Pedlosky, JosephThe triad instability of the largescale, firstmode, baroclinic Rossby waves is studied in the context of the planetary scale when the Coriolis parameter is to its lowest order varying with latitude. Accordingly, rather than remain constant as in quasigeostrophic theory, the deformation radius also changes with latitude, yielding new and interesting features to the propagation and triad instability processes. On the planetary scale, baroclinic waves vary their meridional wavenumbers along group velocity rays while they conserve both frequencies and zonal wavenumbers. The amplitudes of both barotropic and baroclinic waves would change with latitude along a ray path in the same way that the Coriolis parameter does if effects of the nonlinear interaction are ignored. The triad interaction for a specific triad is localized within a small latitudinal band where the resonance conditions are satisfied and quasigeostrophic theory is applicable locally. Using the growth rate from that theory as a measure, at each latitude along the ray path of the basic wave, a barotropic wave and a secondary baroclinic wave are picked up to form the most unstable triad and the distribution of this maximum growth rate is examined. It is found to increase southward under the assumption that triad interactions do not cause a noticeable decrease in the quantity of the basic wave’s amplitude divided by the Coriolis parameter. Different barotropic waves that maximize the growth rate at different latitudes have almost the same meridional length scale, on the order of the deformation radius. With many rays starting from different latitudes on the eastern boundary and with wavenumbers on each of them satisfying the nonormalflow condition, the resulting twodimensional distribution of the growth rate is a complicated function of the relative relations of zonal wavenumbers or frequencies on different rays and the orientation of the eastern boundary. In general, the growth rate is largest on rays originating to the north.