Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current

This study examines the interactions between barotropic Rossby waves and a zonal current, with particular reference to the Antarctic Circumpolar Current (ACC). In the high latitude of the Southern Ocean, the effect, which provides the restoring mechanism for Rossby waves, is relatively weak, resulting in barotropic waves that are extremely long, with periods of the order of 1 week or longer. We model the interactions between the current and the waves using the linearized potential vorticity (PV) equation with the inclusion of background PV terms corresponding to a barotropic zonal flow of finite width. An analytical solution is found for the simplest, piecewise-linear flow and numerical solutions obtained for more realistic, smoothly varying flows. The results show that, in general, the long waves are not appreciably modified or reflected by the shear flow, except in the case where the wave is incident at an oblique angle. Wave reflection is also more pronounced for shorter waves, such as Rossby waves having eastward group velocity or the case where the wave frequency is just below the cutoff frequency.


Introduction
[2] Analysis of TOPEX/POSEIDON altimetry by Hughes [1995] has shown that it is possible for the Antarctic Circumpolar Current to act as a waveguide [Hughes, 1996] for baroclinic Rossby waves, in effect constraining the wave energy to within the confines of the shear flow.Such behavior is consistent with the theoretical predictions of Killworth [1979], who used the Wentzel, Kramers, Brillouin and Jeffreys (WKBJ) method to study this case.However, high-frequency wind stress forcing with periods of the order of a few days to weeks, such as that associated with storm systems, will generate barotropic Rossby waves.
[3] The propagation of barotropic Rossby waves provides the mechanism for the rapid adjustment of the ocean to high-frequency forcing, and the establishment of the Sverdrup balance between oceanic vorticity and wind stress.The dynamics of the waves are governed by the conservation of potential vorticity (PV), and as such they can be strongly affected by variations in the ambient PV such as those associated with bottom topography and mean flows.Furthermore, the interaction between Rossby waves and mean flows provides a mechanism for those flows to adjust to variations in wind stress, additional to and more rapid than the directly forced variations in the flow field.
[4] This study considers the interaction between barotropic Rossby waves and a barotropic shear flow, in the absence of bottom topography.In particular we calculate the diffracted wavefield when long or short barotropic Rossby waves are incident upon a barotropic shear flow.For a piecewise linear shear flow the diffracted wavefield can be determined analytically.For other shear flows, the diffracted field is determined numerically.
[5] Theoretical studies of the interaction of these barotropic waves with shear flows have tended to be of limited application with respect to the ACC.Mofjeld and Rattray [1975], Warn and Warn [1976], and Killworth and McIntyre [1985], have considered parameter regimes, valid in atmospheric applications, which permit the formation of critical layers and the transmission, reflection and absorption properties of such layers have been studied in detail.Studies of the ACC, such as Clarke [1982], typically give a core velocity of the order 0.3 m/s, whereas midlatitude, long, barotropic Rossby waves typically have phase velocities far in excess of this.This disparity excludes the formation of a critical layer.The stability and wave propagation characteristics of such flows have been considered by Drazin et al. [1982], who found analytical solutions of the governing equations and analyzed their asymptotic properties.Harlander et al. [2000] have analyzed the interaction of barotropic waves and the ACC using the WKBJ method and Frobenius series solutions.For the short waves considered in that work, the WKBJ method produces a reasonable approximation, but that approximation deteriorates if it is extended to long waves.
[6] The purpose of this study is to consider the diffraction of linear Rossby waves by a barotropic current.For long, oceanic barotropic Rossby waves the wavelength orthogonal to the axis of the current is at least as long as, and typically much longer than, the width of the current.
Consequently the predicted behavior of such waves incident on a zonal current with parameter values typical of the ACC differ markedly from that discussed by Harlander et al. [2000] which implicitly assumes that the background potential vorticity varies on a spatial scale much longer than the wavelength of a barotropic Rossby wave.
[7] The plan of the paper is as follows.Section 2 gives the formulation of the problem and derives the nondimensional equation which governs the wave propagation.This equation is solved for linear, sinusoidal and Gaussian background flow fields in section 3. We also consider the addition of localized, narrow jets to the background flow to produce the correct peak velocities and better model the filament structure of the ACC, and the possibility of wave trapping by linear and piecewise linear flows, of finite width, adjacent to a southern boundary.The scattering characteristics of each flow field are given in section 4. Finally, a summary and discussion of the results are presented in section 5.

Parameters and Equation of Motion
[8] We assume the ocean to be a Boussinesq fluid occupying ÀH z 0, on a midlatitude b plane, with uniform depth and Coriolis parameter given by f = f 0 + b y, where f 0 and b are constants and a right handed coordinate frame Oxyz has been introduced, with Ox directed eastward and Oy northward.Further, we assume that the flow is both barotropic and quasi-geostrophic and that there is a background steady zonal shear flow, U(y), of finite meridional extent in an otherwise quiescent ocean.The problem geometry is shown schematically in Figure 1.
[9] The components of the horizontal velocity field may be fully determined from a quasi-geostrophic stream function, F(x, y, t), and are given by We shall assume that the total stream function is composed of a component, F 0 (x, y), that represents the background zonal shear flow, and a component, f(x, y, t) representing the perturbation induced by a propagating linear barotropic Rossby wave.The total stream function, satisfies the quasi-geostrophic shallow water equation [Pedlosky, 1987, p. 99] where J(Á,Á) is the Jacobian, r 2 is the horizontal Laplacian and h B is the background potential vorticity field.Substituting (2) into (3) and retaining only those terms linear in f we find the perturbed stream function satisfies the equation where U = À@F 0 /@ y, is the steady stable zonal barotropic current, assumed to vary only with latitude.The system is forced by a monochromatic, harmonic, barotropic Rossby wave train incident from the north (see Figure 1), of the form We have chosen this form for the incident wave for convenience, since when nondimensionalizing (4), we wish to have k > 0. The angular frequency, w, and wave number components, k, l > 0 then satisfy the dispersion relation where r e = ffiffiffiffiffiffiffiffiffiffiffiffiffi gH=f 2 0 p is the external Rossby deformation radius.For a given frequency and propagation angle, the two roots of the dispersion relation correspond to long and short waves, with westward and eastward group velocities, respectively.In addition, we shall impose the radiation condition that as y !± 1 the scattered wavefield has group velocity directed away from the zonal current, and thus no additional source of energy is introduced.
[10] As our model, we consider the ocean to be situated on a midlatitude b plane at a latitude of 55°S, and of uniform depth 4000 m.This gives an external deformation radius of around 1650 km, and a cutoff period of approximately 6.7 days, below which no zonal propagating waves are supported.The physical characteristics that arise from the use of these parameters waves are summarized in Tables 1, 2, and 3. Clearly, for periods longer than ten days, the wavelength of a single long wave is greater than the total length of a latitude circle (about 23,000 km at 55°S).Such long waves therefore can be considered as nonphysical, and for such periods we shall consider only short wave dynamics.
[11] We model the ACC by a steady, zonal, barotropic current, of half-width y 0 = 500 km, Further, we assume U(y) has a maximum velocity, U 0 , obtained at the center of the flow, y = 0. Our barotropic model of the ACC is convenient for analysis, but cannot be considered overly realistic.In fact, in both theoretical and observational studies the flow speeds are seen to decay with depth [see, e.g., Killworth and Hughes, 2002].It is estimated [Marshall et al., 1993] that the vertically integrated velocity within the ACC is approximately 1400 m Â the surface flow speed.Although this figure is disputed [see Karsten and Marshall, 2002] we shall consider the net transport by our current to be given by 1400m Â Except where noted, we shall take the peak velocity, U 0 to give a total barotropic transport within the ACC of 100Sv, consistent with estimates of the net transport through Drake Passage [Cunningham et al., 2003].

Nondimensionalization
[12] For simplicity we shall nondimensionalize equation (4).Defining the nondimensional variables x ¼ kx; y ¼ ky; y 0 ¼ ky 0 and t ¼ wt; ð8Þ and substituting into equation (4) we find where the F is defined by F = f 2 /(gHk 2 ), B = b/wk and m( y) = U(y)k/w is the local 'Mach number', the ratio of the flow speed to zonal wave speed.
[13] It may be seen, from the symmetry of the problem and the form of time dependence of the incident wave, that the stream function may be written in the form where y( y) is the complex-valued amplitude function of the diffracted wave.Substituting (10) into equation ( 9) and using the dispersion relation ( 6) results in the ordinary differential equation where tan q = l/k and the prime denotes differentiation with respect to y.
[14] Equation ( 11) is the linear Schro ¨dinger equation, with the quotient representing the square of the local wave number, and is often solved using perturbation techniques, such as the WKBJ method.However, for such a perturbation solution to be valid we require a small parameter, , to appear naturally from the nondimensionalization and to multiply the leading derivative of y, reflecting the ratio of the long and short spatial scales on which the physical processes within the equation occur.[15] Only when this local wave number varies slowly (in the sense defined in section 10.1 of Bender and Orszag [1978]) on the short length scale is the WKBJ method applicable.This step is only justified for short Rossby waves, and thus the perturbation methods then applied in that work cannot be utilized for the long waves considered here.
[16] The remainder of this paper is concerned with the solutions of the nondimensionalized equation ( 11).
3. Solutions of the Linearized PV Equation 3.1.Piecewise Linear Shear [17] In general, solutions of equation ( 11) must be obtained numerically.The only exception known to the present authors is for the case of a piecewise linear shear, which is solved analytically by Drazin et al. [1982], for which the dimensional flow field is given by Imposing the condition that the net transport equals 100Sv, results in peak velocities of approximately U 0 = 0.15 m/s.Clarke [1982] records peak velocities within the ACC of order U 0 = 0.3 m/s.The addition of localized zonal jets to the background current can produce the correct peak velocities.
[18] Nondimensionally, the flow profile takes the form where m 0 = U 0 k/w and the local nondimensional shear, q, is defined by q = m 0 / y 0 .Substituting this form of the flow field into the Rayleigh-Kuo criterion for barotropic instability [Kuo, 1949]: shows that an eastward flow is stable throughout Ày 0 < y < y 0 but supports unstable barotropic modes centered around the edges of the flow at y = ± y 0 (where U 00 is unbounded and positive).Again, this is consistent with the large scale meanders of the current observed in the altimetry and float data, such as those observed by Gille [2003], although it is clear that much of the variability in the ACC's direction is a result of topographic steering, rather than instability.
[19] Following Drazin et al. [1982], the linearly independent analytic solutions may be written in terms of hypergeometric functions which, using the notation of section 13 of Abramowitz and Stegun [1965], are given by and where 1 F 1 and U are confluent hypergeometric functions of the first and second kind, respectively.By symmetry, then, the stream function within the flow region is given by where a 1 , a 2 , b 1 and b 2 are undetermined constants.In the northern ocean region, y > y 0 , the stream function comprises the incident wave and a reflected plane wave, and in the region y < À y 0 , there is only the transmitted plane wave.Thus, choosing the sign of the wave number such that the reflected and transmitted plane waves have group velocity directed away from the zonal channel, we have [20] Equations ( 18) and ( 19) give the stream function throughout the flow in terms of the unknown scattering coefficients, R, T, a 1 , a 2 , b 1 and b 2 .To determine these coefficients, we impose the condition that the velocity field and the total potential vorticity are continuous throughout the domain.Across the interfaces at y = 0, y = ± y 0 the former conditions gives us where [ ] denotes the jump across the interface.The exact form of the latter matching condition must be found by integrating equation ( 11) across the interface, in the manner of Rhines [1969], to obtain For piecewise linear velocity profiles, we require At each interface the discontinuity in the ambient potential vorticity field due to the discontinuous shear induces a compensatory jump in the PV field generated by the wave propagation.

Sinusoidal Shear
[21] In addition to the piecewise linear case considered above we also consider the sinusoidal flow profile given, in dimensional coordinates, by Throughout this region we have U 00 > 0, and thus, as in the piecewise linear case, the background flow is barotropicly stable, up to the edges of the flow.In the region, jyj < y 0 , the stream function satisfies the nondimensional equation For the parameter values given in section 2.1, it can be shown that y 0 2 ( 1, and thus the curvature term, m 00 , is dominant.Therefore equation ( 25) is qualitatively different the equation governing the piecewise linear flow profile considered above.As in that case, however, it is necessary to impose jump conditions across the interfaces y = ± y 0 , given by [22] Unlike the linear profile, there are no known analytical solutions of equation ( 25).However, the equation is readily solved numerically.We obtained a numerical solution, y 1 ( y), by specifying the boundary conditions and solving with a shooting method, using the basic NDSolve routine of Mathematica.A second, linearly independent solution is given by y 2 ( y) = Ày 1 (À y), which satisfies Thus we may write the stream function within the flow, À y 0 < y < y 0 , as Applying the boundary conditions in equations ( 20) and ( 21) to the stream functions given in ( 19) and ( 29) we obtain We may solve this set of simultaneous equations to obtain the scattering coefficients, T, R, a 1 and a 2 .

Narrow Gaussian Jets
[23] While we have modeled the ACC as a smoothly varying shear flow across its entire width, it is believed that much of the net transport within the current is carried by jetlike structures, located near the fronts that define current itself.Gille [1994] identified two main jets in the GEOSAT altimetry of the ACC, each between 35 and 50 km in width located at the sub-Antarctic Front and the Polar Front.To model the filament structure of the ACC, we shall also consider the scattering of waves by fine structures, in this case a Gaussian jet.In general, the nondimensional flow field is then given by the sum of an arbitrary number of jets of differing spatial scale and flow strength, i.e., where Y i is the y coordinate of the center of the ith jet, y 0i is its halfwidth and m 0i its peak velocity.However, due to the lack of interaction between such jets, when presenting the results we shall consider only a single jet.Again, there is no known analytical solution for this diffraction problem, but the numerical solution method outlined above is applicable.
Since the velocity fields are exponentially small away from the jets, the jump conditions at the extremities of the flow may be neglected.Unlike the two flow profiles considered above, narrow Gaussian jets (of width 100 km and less) do not satisfy b À U yy 6 ¼ 0; ð36Þ throughout the flow.It is therefore possible that the flows are subject to barotropic instability.Again, this is consistent with the observed large-scale meanders of the jets.

Analytical Solution
[24] For ready comparison with the results of Harlander et al. [2000], we shall also consider the possible existence of waves trapped against a southern boundary.In this case, we do not force the problem with an incident wave of known frequency and wave number, and the absence of such external length and timescales require us to nondimensionalize equation (4) in a different manner.Furthermore, purely for algebraic convenience, we shall move the origin so that Y = 0 corresponds to the southern boundary at y = Àr e .In summary, we write and r e is the external Rossby radius of deformation and where U 0 is the characteristic flow velocity at a distance r e from the southern boundary.Taking an ocean depth of 2000 m gives and a latitude of 55°S gives the following parameter values: and This shallower ocean results in an external deformation radius is approximately 1200 km, or approximately 10°of latitude.Again, for comparison with Harlander et al.
[2000], we shall allow our basic flow to be unbounded in the meridional direction, which may allow the flow to become both broader and faster than any terrestrial oceanic geophysical flow.The nondimensionalized equation of motion becomes We shall look for trapped modes of the form where m and w are the nondimensional wave number and frequency, respectively, and are to be determined.These modes must satisfy the nonnormal flow boundary condition at the coastline and decay away from the coast, i.e., [25] Restricting ourselves to a linear shear of the form we obtain the normal mode equation where and the nondimensional parameter b is given by b For all wave numbers, m there is a turning layer, where f (Y) = 0, located at [26] The form of equation ( 44) is identical to that solved in section 3.1, and thus admits the same pair of linearly independent solutions.However, it can be shown that the solution involving the Kummer function, 1 F 1 , grows exponentially in the far field, and is discarded.The solution that satisfies the radiation condition is therefore given by where a, b, c and d are as defined by (45).The existence of trapped modes is then reliant on the discovery of pairs of values of ( w,m) such that y(0) = 0.

WKB Solution
[27] Although there is no clearly defined scale separation parameter, in order to compare our results with those of Harlander et al. [2000] we shall obtain a WKB solution valid between the coast at Y = 0 and the turning layer, where the local wave number is zero.We shall compare that solution with the analytical solution obtained above.Although this step is not rigorously justified, it is well known that asymptotic methods frequently give accurate results far beyond parameter regimes in which they are strictly applicable.Applying the usual WKB ansatz gives a pair of linearly independent solutions of the form where F(Y) is given by For linear flows, the integral in (50) may be performed exactly using the trigonometric identities.Applying the boundary condition that y(0) = 0, it is clear that the WKB approximation for a trapped mode is given by This solution is singular at the location of the turning layer, and grows exponentially as Y ! 1.In order to obtain a bounded solution it is necessary to solve the governing equation approximately in the region of the turning layer, and match that inner solution to the WKB solution.
[28] Linearizing equation ( 44) around the location of the turning layer, Y = Y T = a 2 /b 2 , we obtain Neglecting those terms that are O(x 2 ), the general solution of equation ( 52) may be written as a linear combination of the Airy functions, Ai(jf 0 (Y T )j 1/3 x) and Bi(jf 0 (Y T )j 1/3 x).
[29] Both Airy functions are bounded and oscillatory for x < 0, and in the limit x !À1 they have the asymptotic behavior and as given by Abramowitz and Stegun [1965].Similarly, considering the asymptotic behavior of the WKB solution (51) as it approaches the turning layer we have and Combining ( 55) and ( 56) with (51) it can be seen that the WKB solution has asymptotic behavior which is precisely the correct form for matching with the inner, Airy function solutions given in ( 53) and ( 54).
Performing this matching, it is apparent that unless the coefficient of Bi is nonzero.Since Bi grows exponentially as x !+1, the condition in ( 58) is an eigenvalue equation for the existence of a trapped mode.
[30] The eigenvalue equation ( 58) is a necessary condition on the phase of the incoming wave for the wave amplitude to be attenuated exponentially within the turning layer.However, it is not a sufficient condition for such a mode to be trapped.For that to happen, it is also necessary for the turning layer to be sufficiently broad for the wave to be completely damped before the wave emerges from the far side.Typically, for short Rossby waves in geophysically realistic shear flows, the e-folding scale is comparable to the wavelength of free waves, and to the total width of the ACC, and so this secondary condition should not be neglected.

Linear Shear
[31] Figures 2a -2c show the reflection coefficient for long (westward) Rossby waves in the linear flow profile described in section 3.1, for net transports of 50, 100 and 200Sv, respectively.It can be immediately seen that except for grazing incidence (q ] 10°), there is very little wave scattering at all, for all periods.Compared to the wavelength of the incident waves, the ACC is relatively narrow and the variation in local wave number is small and thus the wave propagation is almost unaffected by the change in ambient PV field, for either short or long waves.A comparison of the scales for Figures 2a -2c show that the magnitude of the reflection coefficient is approximately linear in the transport (and hence in U 0 ).
[32] For the case of grazing incidence (q $ 1°), the same applies.Although the cross-flow component of the wavelength of the incident waves is smaller than the previous case, it is still on the order of the width of the current.For oceanographically realistic values of the transport (100-200 Sv) considerably less than 1% of the incident wave energy is reflected, and only then when the incidence is grazing and the wave period close to the cutoff.Thus, in general, the long waves that are responsible for the establishment of the Sverdrup balance are almost entirely unaffected by a linear flow of realistic oceanographic magnitude.
[33] For the case of short waves shown in Figures 3a -3c, there is a marked difference in the wave scattering.The width of the ACC is now comparable in size to the incident wavelength, and the variations in the local wave number act to attenuate the wavefield as it penetrates the shear layer, causing a greater reflection than in the previous cases.For periods slightly longer than the cutoff, the two real wave number solutions of ( 6) are approximately equal, and both propagating disturbance have similar wavelengths.In this case the similarity in lengths of the eastward and westward propagating waves results in similar scattering characteristics, as shown by the reflection.Conversely, for waves with a relatively long period, there is a correspondingly greater disparity between short and long wavelengths, and a larger difference in reflection between the short and long waves.Thus it is normal for short waves to be significantly attenuated even by relatively weak shear.

Sinusoidal Velocity Profile
[34] Figures 4a -4c and 5a -5c show the reflection coefficients for the sinusoidal profile of width 1000 km.These show trends qualitatively different from those from the  linear profile.As noted in section 3.2, the dominant term in the local wave number comes now from the curvature term, m 00 , in equation ( 11).These larger wave number variations over the entire width of the flow cause a far greater amount of reflection than the linear case, where the total curvature is localized (and expressed only in the jump conditions).The importance of the curvature is more apparent in the case shown in Figures 6a -6c, where the ACC is modeled as a narrower current, of width 500 km. Figure 5 also reveals regions of reduced reflection for grazing incidence, visible as a saddle on the contour plots.For those parameter values, the component of the wavelength in the cross-flow direction matches the width, and this resonance increases the proportion of wave energy that is transmitted through the current.
[35] It is clear then, that for oceanographically relevant regimes there is substantial reflection of incident long and short waves.Therefore, when considering the effect of the ACC on Rossby wave propagation, it is more relevant to consider the spatial variation of the mean flow field (and the meridional scale of that variation) than its strength, and thus the scattering properties cannot be naively parameterized in terms of net transport alone.

Gaussian Jets
[36] We finally consider the scattering by Gaussian jets, much narrower than the broad linear and sinusoidal flows.Observational studies, such as by Gille [1994] reveal the structure of the ACC to be composed of a relatively weak, broad mean flow with the narrower filament-like jets of greater peak velocity.Although our governing equation is not linear in U(y), we treat these two components separately, as numerical experiments determined that the interaction between the two structures within the flow did not qualitatively change the overall reflection coefficient.Our consideration of Gaussian flows is restricted to those considerably narrower than the ACC itself.For broader flows, the result are qualitatively similar to those for the sinusoidal flows considered above.
[37] Figure 6 and Figure 7 shows the reflection coefficient for wave of period 10 days, for a single jet of variable peak velocity and width (which corresponds to 4y 0i in equation ( 35)).It may be seen that the appreciable reflection of long waves occurs for grazing incidence only when the peak velocity is of the order 1 m/s or greater, and the meridional extent of the jet is of the order 100 km or greater.Such a jet is considerably stronger and wider than any recorded in observations, and qualitatively, the physics of such a jet are similar to the the broad sinusoidal flow in section 4.2.However, the regions of high curvature (and thus rapid variation of local wave number) are more narrow, and thus the total reflection is much less.For narrow, fast jets there is very little long-wave reflection for any incident angle.Figure 8 shows the reflection of short waves, of the same period, by a zonal jet.Again, the reflection is greater than for long waves, but there is still only appreciable reflection for very strong jets, and grazing incidence.[2000] demonstrated that an meridionally unbounded shear flow can trap a short Rossby wave against a coastline.Figure 9 shows the meridional structure of such a trapped mode in a linear shear with period 68 days, velocity gradient 0.5 m/s per 700 km for oceans of depth 2 km (directly comparable to Figure 4 of Harlander et al. [2000]) and the more realistic value of 4 km.Both waves have zonal wavelengths of approximately 1600 km.[39] The vertical dotted line denotes the location of the turning latitude.Firstly, it can immediately be seen that the different ocean depths do not qualitatively change the structure of the modes.It may also be seen that even though the turning latitude is located at approximately 58°S, it requires it further 10°of latitude before the waves are fully attenuated.
[40] For shorter waves, i.e., those of a longer period, this trapping distance is greatly reduced.Figure 10 shows the meridional structure of trapped modes of 6-month and 12-month periods, which have zonal wavelengths of 1070 km and 870 km, respectively.It is clear that for a wave to be completely attenuated by the ACC, whose width may be considered to be of the order of 1000 km to 1500 km.

Finite Shear
[41] Since the trapping scale is of vital importance in determining whether trapped modes exist we shall now consider trapped modes in piecewise linear shear flows of the type outlined in section 3.1.Figure 11 shows the meridional structure of a wave which satisfies the eigenvalue relationship (58), with period 68 days in a flow of width 1500 km.In each case, the narrowness of the turning layer prevents significant attenuation, resulting in a nonnegligible standing wave solution on the equatorial side of the ACC.This standing wave represents two waves, one propagating northward away from the ACC, and an incident wave from the north.Thus the solution cannot be thought of as representing a trapped wave.It is also notable that varying the peak flow speed (and hence the total transport), does not qualitatively change either the meridional structure or the amplitude of the northward propagating wave, only the eigenvalue (and hence the zonal wavelength).This is also true of the other waves considered here.[42] Figures 12 and 13 show the modal structures of trapped and nearly trapped waves in flows of transport 100 Sv.As can be clearly seen, the width of the flows (and therefore the width of the turning layer) is of greater importance than the peak velocity in determining both the zonal and meridional structure of any waves which may be trapped.

Conclusions
[43] When considering the dynamics of large-scale ocean circulation, it is primarily the long waves that are of interest.It is the long waves that establish the barotropic circulation and provide the mechanism for its large-scale adjustment.The dynamics of such waves is dominated throughout by the long wavelengths of the wave in the direction orthogonal to the axis of the jet.A long wave of period 7 to 10 days will have a wavelength of the order of many thousands of kilometers; at least as long as the width of the ACC, and typically much longer.Thus, except in the case of grazing incidence, the meridional component of the wavelength is itself greater than the width of the current.For oceanographically relevant currents (e.g., the ACC) the variation in local wave number due to the background velocity is negligible for long waves, and small for short waves.Only grazing incidence and short incident waves produce appreciable wave reflection.This difference between long and short wave behavior is a possible mechanism for the trapping of wave energy between the ACC and Antarctica.A long Rossby wave incident on the ACC from midlatitudes will pass through almost unattenuated.The rough coastline of Antarctica will scatter the incident wave into a combination of short and long waves, the greater reflection of short waves incident from the poleward side of the ACC will cause a proportion of the short wave energy to be localized between the southern edge of the current and Antarctica.
[44] While the effect of the background flows is usually small, the relatively small width of the current, once non-  dimensionalized, can cause the ambient PV gradient due to curvature (second derivative) of the velocity field to be large.In this case, the difference between short and long wave scattering is not marked.
[45] In this investigation we have also modeled the effect of the strong, narrow filament jets using Gaussian profile.The narrow jets have a high curvature, which can create a turning layer within the flow.However, this turning layer is confined to the jet and, by necessity, also narrow.Such a narrow turning layer will not appreciably attenuate the incident wave.Thus neither rapid, narrow jets or broad, slow flows produce appreciable reflection of long waves.
The maximal reflection occurring for moderate flows with nonzero curvature, of the order of magnitude of the ACC.In this case the curvature is conducive to the rapid formation of a turning layer, and the breadth of the flow means that the turning layer is sufficiently wide to appreciably damp the incident wave.It is clear then that when considering long barotropic wave scattering by mean flows the structure and shape of the flow profile are as important as the width, velocity and transport.[46] For short waves there is appreciable reflection for both narrow and broad flows.The shorter wavelengths mean that the width of a turning layer may be sufficient to attenuate the wave, and act as an barrier to wave transmission.This increased attenuation increases the degree to which incident short waves are reflected Broader, weaker, flows are more likely to possess trapped wave solutions, but these modes will be of shorter wavelengths than those trapped by narrower flows.South of the ACC, then it would be possible to observe short Rossby waves trapped against the coastline of Antarctica, particularly of a period in excess of 4 -6 months.However, it should be noted that forcing of that period tends to also excite baroclinic wave activity;  barotropic waves being excited preferentially only when the frequency is above the cutoff frequency for baroclinic waves.Furthermore, such long-period, short barotropic waves are far more readily affected by both coastal and bottom friction than long waves.
[47] Acknowledgments.The authors would like to acknowledge the contribution of two anonymous reviewers, whose insight and suggestions have greatly improved this work.G. Owen would like to acknowledge the support of institutional Leverhulme Trust research grant F/130/U and Natural Environment Research Council grant NER/T/S/2000/00585.

Figure 1 .
Figure 1.Idealized problem geometry showing the three flow profiles U(y) considered (schematically) as well as the orientation of the incident long Rossby wave.Parallel lines in the incident wave denote lines of constant phase, and the arrow denotes the direction of phase propagation.

Figure 2 .
Figure 2. Plot of the reflection coefficient jRj for long Rossby waves in a piecewise linear flow of width 1000 km and transport (a) 50 Sv, (b) 100 Sv, and (c) 200 Sv.

Figure 3 .
Figure 3. Plot of the reflection coefficient jRj for short Rossby waves in a piecewise linear flow of width 1000 km and transport (a) 50 Sv, (b) 100 Sv, and (c) 200 Sv.

Figure 4 .
Figure 4. Plot of the reflection coefficient jRj for long Rossby waves in a sinusoidal flow of width 500 km and transport (a) 50 Sv, (b) 100 Sv, and (c) 200 Sv.

Figure 5 .
Figure 5. Figure Plot of the reflection coefficient jRj for long Rossby waves in a sinusoidal flow of width 1000 km and transport (a) 50 Sv, (b) 100 Sv, and (c) 200 Sv.

Figure 7 .
Figure 7. (a and b) Plot of the reflection coefficient for a long Rossby wave, with incident angle q in a Gaussian jet of variable width (in km) and flow strength.

Figure 6 .
Figure 6.Plot of the reflection coefficient jRj for short Rossby waves in a sinusoidal flow of width 1000 km and transport (a) 50 Sv, (b) 100 Sv, and (c) 200 Sv.

Figure 8 .
Figure 8. (a and b) Plot of the reflection coefficient for a short Rossby wave, with incident angle q in a Gaussian jet of variable width (in km) and flow strength.

Figure 9 .
Figure 9. Meridional modal structure for trapped waves of period 68 days in an unbounded linear shear flow for depths of (a) 2 and (b) 4 km.

Figure 10 .
Figure 10.Meridional modal structure for trapped waves in an unbounded linear shear flow for (a) a period of 6 months and wavelength of 1070 km and (b) a period of 12 months and wavelength of 870 km.

Figure 11 .
Figure 11.Meridional modal structure for trapped waves of period 68 days in a piecewise linear shear flow of width 1500 km for (a) peak velocity of 0.05 m/s and zonal wavelength of 700 km, (b) peak velocity of 0.1 m/s and zonal wavelength of 820 km, and (c) peak velocity of 0.2 m/s and zonal wavelength of 1020 km.

Figure 12 .
Figure 12.Meridional modal structure for trapped waves of period 6 months, in a piecewise linear shear flow of transport 100 Sv for (a) flow width of 1000 km and zonal wavelength of 620 km, (b) flow width of 1200 km and zonal wavelength of of 520 km, and (c) flow width of 1500 km and zonal wavelength of 430 km.

Figure 13 .
Figure 13.Meridional modal structure for trapped waves of period 1 year in a piecewise linear shear flow of transport 100 Sv for (a) flow width of 800 km and zonal wavelength of 620 km, (b) flow width of 900 km and zonal wavelength of 530 km, and (c) flow width of 1000 km and zonal wavelength of 460 km.

Table 1 .
Wavelengths l, Phase Velocities c, and Group Velocities c g of Long and Short Zonal Rossby Waves at 55°S and a Depth of 4 km aValues are in km.

Table 3 .
Zonal Wavelengths (2p/k) of Short Rossby Waves at 55°S and a Depth of 4 km a aValues are in km.