• Challenor, P. G., P. Cipollini, and D. Cromwell, 2001: Use of the 3D Radon transform to examine the properties of oceanic Rossby waves. J. Atmos. Oceanic Technol., 18 , 15581566.

    • Search Google Scholar
    • Export Citation
  • Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, 272 , 234238.

  • Fu, L. L., and D. B. Chelton, 2001: Large-scale ocean circulation. Satellite Altimetry and Earth Sciences, L. L. Fu, and A. Cazenave, Eds., Academic Press, 133–169.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 1999: The effect of bottom topography on the speed of long extratropical planetary waves. J. Phys. Oceanogr., 29 , 26892710.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 2003a: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part I: The local problem. J. Phys. Oceanogr., 33 , 784801.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 2003b: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part II: Ray propagation and comparison with observations. J. Phys. Oceanogr., 33 , 802821.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., D. B. Chelton, and R. A. de Szoeke, 1997: The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr., 27 , 19461966.

    • Search Google Scholar
    • Export Citation
  • Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn., 1 , 273302.

  • Tailleux, R., 2003: Comments on “The effect of bottom topography on the speed of long extratropical planetary waves.”. J. Phys. Oceanogr., 33 , 15361541.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2001: The effect of bottom pressure decoupling on the speed of extratropical, baroclinic Rossby waves. J. Phys. Oceanogr., 31 , 14611476.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2002: The energy propagation of long baroclinic Rossby waves over slowly varying topography. J. Fluid Mech., 473 , 295319.

    • Search Google Scholar
    • Export Citation
  • Welander, P., 1959: An advective model of the ocean thermocline. Tellus, 11 , 309318.

  • View in gallery

    The cumulative probability of the value of the wavenumber ratio l/k needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. Locations equatorward of 50° and poleward of 5° are considered. The squares show the locations of |l/k| = 1; the circles show |l/k| = 2. Only 14% of the ocean has |l/k| < 1; 23% has |l/k| < 2

  • View in gallery

    Contours of |l/k| needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. The dark gray region shows |l/k| < 1; the light gray region shows 1 < |l/k| < 2; unshaded regions have the ratio above 2 in magnitude. Other gray areas indicate shallow water or the omitted region within 5° of the equator

  • View in gallery

    Ray trajectories for rays started on a 5 × 5 grid at 5° lat and 10° lon intervals in the South Atlantic over a topography smoothed to 9°, with initial values of wavenumbers k, l needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. Dots show positions of the trajectory every 30 days; a vertical bar shows the termination of the ray when the east–west group velocity becomes too large to continue the integration. Not all rays are visible, as some had too short a duration to be plotted. Some western rays terminate in shallow water (marked with a D)

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  • 1 Southampton Oceanography Centre, Empress Dock, Southampton, United Kingdom
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Corresponding author address: Dr. Peter D. Killworth, Process Modelling Group, James Rennell Division, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: pki@soc.soton.ac.uk

Corresponding author address: Dr. Peter D. Killworth, Process Modelling Group, James Rennell Division, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: pki@soc.soton.ac.uk

1. Introduction

Tailleux (2003, henceforth T03) makes an interesting suggestion, basically that there is another relevant vertical mode (termed here pseudobarotropic) present in linear, long planetary wave motion in the presence of bottom topography, which was mentioned but little discussed by Killworth and Blundell (1999, henceforth KB99). They found that the presence of topography did not, overall, increase the speed of planetary waves over the flat-bottomed case.

Tailleux raises several points. He argues that

  1. the neglected mode is faster than the first baroclinic mode studied by KB99 and could perhaps account for the discrepancy between observed and theoretical planetary wave speeds,

  2. our continuous theory is at odds with results from two-layer models (and therefore the continuous theory must be wrong),

  3. ray theory and local calculations are methodologically not equivalent, and

  4. our statement that local topographic effects cancel when averaged over the basin scale is ambiguous.

In this response we shall mainly address point (i) above, since this is interesting and worthy of analysis beyond the simplified analysis by T03. We shall argue that the mode, first examined by Rhines (1970), (i) is ill-conditioned, possessing unrealistically fast phase and group velocities for low east–west wavenumber, and requiring what are probably unphysical orientations of the wavevector for realistic phase velocities; (ii) cannot propagate long distances over realistic topography, in particular becoming unphysical when the topography becomes locally flat; and (iii) would be hard to generate. Thus, although the mode clearly exists, it is not likely to provide the explanation for the “too fast” basinwide planetary waves observed remotely; instead, background mean flow (Killworth et al. 1997) or the combination of mean flow and topography (Killworth and Blundell 2003a,b, hereinafter KB03a,b) are considered more likely causes.

Our responses to the other, more discursive points, are given in the discussion.

2. Properties of the pseudobarotropic mode

Recall that the slowly varying Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) long-wave problem is described by Welander's (1959) formulation in terms of a single quantity, M (essentially proportional to the vertical velocity), whose structure is
MFλ,θ,ziϕλ,θωt
where the phase ϕ(λ, θ) is assumed to be a rapidly varying function of longitude λ and latitude θ, t is time, the frequency is ω, and z denotes the vertical coordinate, which is zero at the surface. The amplitude F is slowly varying horizontally and depends on z through the eigenvalue system:
i1520-0485-33-7-1542-e22
where a is the radius of the earth, f = 2Ω sinθ is the Coriolis parameter (Ω is the earth's rotation rate), and k and l are local horizontal wavenumbers given by
kϕλlϕθ
We assume a rigid lid for these calculations, which prompts a minor digression to deal with an issue raised by T03. He notes that the rigid-lid assumption has little effect on the baroclinic modes considered by KB99 but suggests that it has a somewhat stronger effect on the pseudobarotropic mode to be discussed here. This is true, but only where the mode would have unrealistically fast speeds. To see this, note that the free-surface condition can be written
i1520-0485-33-7-1542-e26
where C is the internal wave speed for the problem; this is related to the westward propagation speed by the ratio −β/f2, where β is the northward gradient of f. Suppose that there is a mode in which the westward propagation is (wildly) as much as 10 times as fast as the speeds computed by KB99 (i.e., C2 is of order 10 times a “normal” value of around 3 m s−1). Then, in (2.6), F/HFz at the surface is 0.002 so that the rigid-lid condition holds. The rigid-lid condition only fails in parameter ranges in which the flow is far faster (essentially near barotropic speeds, which are undetectable by satellite) than anything ever observed. This is shown in T03's Fig. 1 (obscured by his decision to plot 1/C2). Thus the rigid lid applies except in circumstances in which the wave speed would be so high as to be unphysical, and certainly not applicable to the issue of the observed propagation speeds.
We now return to the main discussion. Note that the N–S wavenumber l only appears in the bottom boundary condition. We restrict attention, as does T03, to long-wave dynamics, despite the large wavenumbers that will appear in discussions of this mode. The phase velocity of the waves is
i1520-0485-33-7-1542-e27
Killworth and Blundell (1999) show that there is a critical value μ1 of
i1520-0485-33-7-1542-e28
or, equivalently, a value k1, which is the lowest value of μ (or |k|) for which Fz(−H) = 0. The solutions they discussed and tabulated all involve bigger values of μ, and depend on the interlacing of zeros of F and Fz as μ increases. Those solutions can be numbered (T03 uses the same numbering for these) by the number of sign changes in the vertical possessed by the perturbation horizontal velocity, with the lowest (first baroclinic) mode having one sign change in the horizontal velocity perturbation.
Let us now examine possible solutions with 0 < μ < μ1. The two end limits are excluded on physical grounds:
  • When μ = 0, k = 0, F = −z and so α = H, cp = ∞. (This point is again somewhat obscured in T03, by his choice to plot 1/cp.)

  • When μ = μ1, Fz(−H) = 0 and so α = ∞, and cp/cpflat takes the largest limiting value discussed by KB99 (dependent upon position and stratification), where cpflat is the equivalent phase speed for a flat-bottomed ocean. This ratio of speeds is on the order of 2 for realistic stratifications. The condition of zero normal velocity at the bottom means that α = ∞ can only be achieved by l/k = ∞, which is clearly unphysical. [Indeed, Tailleux and McWilliams (2002) refer to the mode as “ill-behaved,” though this refers to the infinite phase speed limit.]

What can occur for values of μ between 0 and μ1? First, we show trivially that α increases monotonically with μ in this range. To see this, recast (2.2) for the variable A = −Fz/F so that A(−H) = α. Then
Azμ2A2A
which shows that A(−H) does indeed increase monotonically with μ. Hence H < α < ∞ for 0 < μ < μ1. In addition, A cannot reach zero until an infinity has been passed, so that Fz has no sign change for this range of wavenumber. The flow is pseudobarotropic in that the horizontal perturbation flow is in the same direction at all depths and can be properly termed the n = 0 mode, in agreement with T03.

But are any of the n = 0 solutions realistic? The problem is that, if α is required to be “large” (and positive) in some sense for the mode to exist (α must be greater than H), then this can be achieved in three ways: (1) if there is a sufficiently large poleward depth gradient; (2) if there is a sufficiently large zonal depth gradient, and the wavevector is in an appropriate direction to make α positive; or (3) there is a zonal depth gradient, and the ratio l/k multiplying it is sufficiently large and of the correct sign to make α of the required size. (There is a legitimate question as to what length scale the depth gradients should be computed over; however, KB03b show that results from a 1° and 9° smoothing yield very similar results.)

Possibility 1, a poleward depth gradient, is noted by KB99, although not pursued by them. As part of the more general problem (with mean flow and topographic gradients), KB03a examine this more carefully and note that most of the ocean does not possess large enough poleward gradients for α to exceed H. Thus zonal gradients would almost certainly be involved if α is to be large enough to see the n = 0 mode effectively, which in turn requires consideration of the wavenumber ratio l/k.

Observations of the direction of phase velocity (Challenor et al. 2001) show that, for the North Atlantic at least, the vast majority of the wave energy lies within a few degrees of pure westward propagation, especially when considering only waves whose westward phase velocity was between 0.5 and 3 times the linear theoretical speed. Only at high latitudes was there any noticeable deviation from pure westward propagation, and then only within about a 40° angle band. Converted into wavevector orientation, this indicates that the observed waves have |l/k| < 1.1 Their results also showed hints of southward propagation (cf. Glazman 2003, manuscript submitted to J. Geophys. Res.), indicating waves for which |l/k| ≫ 1, but these wavenumber combinations were already filtered out from the analyses that gave rise to the too-fast westward propagation of Chelton and Schlax (1996) and later papers.

This gives a useful benchmark against which to test the n = 0 mode. At each 1° square ocean point, using the dataset described by KB99,2 for an annual frequency we located the value of μ1 (and hence the east–west wavenumber k = k1) which made Fz(−H) = 0. Values of |k| < |k1| thus correspond to the n = 0 mode. Clearly, values very close to k1 require extremely large α and thus extremely large l/k. Similarly very small values of k, while not needing large values of α (i.e., the values are near H), yield unphysically large phase speeds. As a compromise, we selected k = 0.9k1, a value which yields a phase speed 10%–11% larger than the maximum range considered by KB99. Equation (2.2) is integrated to the floor, and the value of α is deduced to ensure satisfaction of the boundary condition (2.4). In turn, this gives the wavenumber ratio l/k from (2.5).

Figure 1 shows the cumulative fraction of required wavenumber ratios for the World Ocean in the latitude ranges between 5° and 50° in both hemispheres. The nonzero values at −20, and the nonunity values at +20, reflect the large ocean area in which the required ratios are, simply, very large. Only 14% of the area considered has the required l/k ratio under unity in modulus. Put another way, if the Challenor et al. (2001) observations in the North Atlantic about phase directionality hold in the other ocean basins, the n = 0 mode could not be observed in 86% of the ocean. Furthermore, Fig. 2 shows contours of the required wavenumber ratio, from which it is clear that the areas in which the n = 0 mode might be able to exist are also not very coherent over the World Ocean.

3. Propagation of the pseudobarotropic mode

Killworth and Blundell (1999) showed with ray theory that the n = 1 baroclinic mode could propagate large distances across a realistic ocean basin, essentially by slowly changing its properties as the topographic gradients changed. We now investigate whether the n = 0 pseudobarotropic mode can easily propagate.

It is easy to see in simple cases that the the n = 0 mode will not propagate well. Consider the case of latitude-independent topography H(λ). Then KB99 note that the phase is given exactly by
ϕλ,θ2θχλ
[Tailleux and McWilliams (2001) use a similar construct.] If we take N = const, the solution is
i1520-0485-33-7-1542-e32
The bottom boundary condition then becomes
i1520-0485-33-7-1542-e33

Consider what occurs as the topography flattens, Hλ → 0, for example, on the western side of a ridge, as the wave propagates westward. It is easy to show that the phase χ cannot grow faster than, or at the same rate as, Hλ decreases. Thus the rhs of (3.4) tends to zero on flat topography, so μ tends to zero also, resulting in infinite phase speed, infinite group velocity (for large l/k it can be shown that westward phase and group velocity are almost identical), zero wavenumber, and an infinite energy convergence.

This example was for the relatively unphysical problem of uniform buoyancy frequency and no north–south topographic gradient. A realistic buoyancy frequency can be considered by performing ray tracing experiments on the n = 0 mode in a similar fashion to KB99's treatment for the n = 1 mode using a 9° smoothed topography (KB03b show that local wave solutions at 1° and 9° smoothings are very similar). Unlike their calculations, the mode cannot be initiated on the eastern boundary (which appears locally flat to the initialization they used). Instead, we consider a variety of possible starting points for a ray tracing experiment. At each location, we select k = 0.9k1 as above, giving an n = 0 wave with a not unrealistically large phase velocity. The bottom boundary condition yields the north–south wavenumber l, and the ray calculation may be started.

Figure 3 shows a 5 × 5 grid of solutions, separated by 10° of longitude and 5° of latitude, from the South Atlantic (chosen for its relative lack of isolated topography), with dots marking each 30 days of trajectory, and a vertical bar marking where the trajectory ends when the solution cannot be found by the integration procedure. Two features are immediate. First, no ray propagates more than about 10° of longitude. This is comparable with the length scale over which Challenor et al. (2001) analyzed the observations and far smaller than the basin width used by Fu and Chelton (2001) when reporting propagation across the entire basin of too-fast modes. Second, the wavenumber ratios are large. The average ratio of |l/k| for the 25 rays is initially 65. This ratio grows along all trajectories. On most of the rays (save those near the western boundary), |k| eventually decreases along the trajectory as in the simple model above. One or both of k → 0 and α → 1 occurs, causing termination of the ray. A repeat of the calculation with k = 0.5k1 (i.e., a phase speed of order 3–4 times the flat-bottom speed, almost certainly too large) gives similar results. Thus the n = 0 solutions do not appear to be able to propagate realistically.

4. Generation of the pseudobarotropic mode

The mechanisms for generation of planetary waves are still unclear although direct wind forcing, thermal surface forcing, and eastern boundary upwelling are three likely candidates. The last of these (considered by KB99 for the n = 1 mode) could not produce waves with KB99's boundary condition of no flow normal to a depth contour, as noted earlier, since the bottom appears locally flat in such a case. Typically, west of the continental shelf–slope at the eastern side of an ocean basin, the topography becomes flat so that, even if upwelling could generate the n = 0 mode at the boundary, it would be unable to propagate far from there, as discussed above.

The alternative mechanisms would require rather specific (and high) ratios of north–south to east–west wavenumber in the wind stress curl or surface heating, from the estimates presented earlier. Equivalently, they would need a surface forcing moving predominantly north–south and, while this might happen locally (e.g., storm tracking), it is difficult to see how the mode could efficiently be forced.

5. Discussion

T03 has made four suggestions. His first, and main, suggestion is that the n = 0 mode is responsible for the speedup of long planetary waves. We have addressed his suggestion here, by examining the properties of this mode, and argue that it is hard for this mode to play a major role in what might be observed because (i) it requires a wavenumber ratio—alternatively, requires a direction of phase propagation—that is not observed by remote sensing (at the very least, the required ratios have been filtered out of the data that give rise to the too fast westward propagation results; (ii) it cannot propagate across realistic topography; and (iii) because its propagation direction is tilted north–south, the forcing mechanism for the mode remains unclear. However, we feel that it has been valuable to look at the properties of this mode in some detail, especially since KB99 only briefly discussed the mode. [If mean flow is included, none of the above conclusions about modal behavior apply anyway (cf. KB03a).]3

T03 made three other suggestions, which we have so far not considered.

His first [(ii) in the introduction] is that our continuous theory is at odds with results from two-layer models so that, by extension, the continuous theory must be wrong. We find this conclusion to be the wrong way round! Two-layer models are at best simplified models of a continuous system, and one should properly examine why a two-layer model fails, in some circumstances, to generate the same findings as a continuous model. In particular, in the full mean flow and topography case of KB03a, we show explicitly that there are results from layered models that cannot hold in the continuous problem. Thus, although two-layer models serve as a useful guide, they cannot be used to impute failings of continuous problems.

He adds to this comment an extra note in which he returns to the different bottom boundary condition considered by Tailleux and McWilliams (2001), of zero bottom pressure perturbation (which permits flow through the ocean floor since w is nonzero there). He connects this to the speedup issue (as did those authors). This is not the forum to discuss that in detail (cf. KB03a for an in-depth discussion of how close solutions using no normal flow at the bottom may or may not be to those of Tailleux and McWilliams), save to note that the results from including both mean flow and topographic gradient (KB03a,b) are in excellent agreement with data and do not require additional, nonpropagating, modes to be included.

His point (iii) is that ray theory and local calculations are methodologically not equivalent. At one level this is a truism. Clearly, any ray theoretic solution (at some horizontal location) is a solution of the local problem. However, T03 appears to be arguing that more general solutions of local problems will not necessarily agree well with ray theoretic solutions because parameters such as wavenumber orientation are generated automatically by ray theory but need to be supplied for local calculations. In our case, his statement does not appear to hold, as we show in later papers on the subject (KB03a,b); KB99 did not address local calculations. In KB03b, we made direct comparisons between local and ray calculations in the case of mean flow and topography. The agreement for basinwide modes is excellent (cf. their Fig. 4a in KB03a and Fig. 6b in KB03b). The only significant discrepancy concerns the n = 0 mode, which we have shown does not propagate effectively.

His final point (iv) is that he found our statement that local topographic effects cancel when averaged over the basin scale to be ambiguous. This is not so. Phase speeds in data are estimated over wide near-basin widths (Fu and Chelton 2001). KB03a show that averaging the local values of wave speed using just topographic gradients (no mean flow included) shows little if any net speedup across precisely the distances used by Fu and Chelton.

REFERENCES

  • Challenor, P. G., P. Cipollini, and D. Cromwell, 2001: Use of the 3D Radon transform to examine the properties of oceanic Rossby waves. J. Atmos. Oceanic Technol., 18 , 15581566.

    • Search Google Scholar
    • Export Citation
  • Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, 272 , 234238.

  • Fu, L. L., and D. B. Chelton, 2001: Large-scale ocean circulation. Satellite Altimetry and Earth Sciences, L. L. Fu, and A. Cazenave, Eds., Academic Press, 133–169.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 1999: The effect of bottom topography on the speed of long extratropical planetary waves. J. Phys. Oceanogr., 29 , 26892710.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 2003a: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part I: The local problem. J. Phys. Oceanogr., 33 , 784801.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. R. Blundell, 2003b: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part II: Ray propagation and comparison with observations. J. Phys. Oceanogr., 33 , 802821.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., D. B. Chelton, and R. A. de Szoeke, 1997: The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr., 27 , 19461966.

    • Search Google Scholar
    • Export Citation
  • Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn., 1 , 273302.

  • Tailleux, R., 2003: Comments on “The effect of bottom topography on the speed of long extratropical planetary waves.”. J. Phys. Oceanogr., 33 , 15361541.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2001: The effect of bottom pressure decoupling on the speed of extratropical, baroclinic Rossby waves. J. Phys. Oceanogr., 31 , 14611476.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2002: The energy propagation of long baroclinic Rossby waves over slowly varying topography. J. Fluid Mech., 473 , 295319.

    • Search Google Scholar
    • Export Citation
  • Welander, P., 1959: An advective model of the ocean thermocline. Tellus, 11 , 309318.

Fig. 1.
Fig. 1.

The cumulative probability of the value of the wavenumber ratio l/k needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. Locations equatorward of 50° and poleward of 5° are considered. The squares show the locations of |l/k| = 1; the circles show |l/k| = 2. Only 14% of the ocean has |l/k| < 1; 23% has |l/k| < 2

Citation: Journal of Physical Oceanography 33, 7; 10.1175/1520-0485(2003)033<1542:R>2.0.CO;2

Fig. 2.
Fig. 2.

Contours of |l/k| needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. The dark gray region shows |l/k| < 1; the light gray region shows 1 < |l/k| < 2; unshaded regions have the ratio above 2 in magnitude. Other gray areas indicate shallow water or the omitted region within 5° of the equator

Citation: Journal of Physical Oceanography 33, 7; 10.1175/1520-0485(2003)033<1542:R>2.0.CO;2

Fig. 3.
Fig. 3.

Ray trajectories for rays started on a 5 × 5 grid at 5° lat and 10° lon intervals in the South Atlantic over a topography smoothed to 9°, with initial values of wavenumbers k, l needed to produce a westward component of phase speed for the n = 0 mode that is 10% larger than the maximum discussed by KB99 for the n = 1 mode. Dots show positions of the trajectory every 30 days; a vertical bar shows the termination of the ray when the east–west group velocity becomes too large to continue the integration. Not all rays are visible, as some had too short a duration to be plotted. Some western rays terminate in shallow water (marked with a D)

Citation: Journal of Physical Oceanography 33, 7; 10.1175/1520-0485(2003)033<1542:R>2.0.CO;2

1

KB03a cautiously examined 19 orientations of wavevector, from nearly northward, through westward, to nearly southward. Their concentration on group velocity (which they found to be much better behaved than phase velocity) meant that by a cluster analysis they could identify dominant group velocity values, possessing small dependence on orientation.

2

Except that the buoyancy frequency was computed locally rather than as a basin average. The differences are small.

3

The possibility remains that the mode might be able to propagate basinwide (at unrealistically high speeds) under a free-surface condition, although even by T03's analysis this seems implausible.

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