Recently, the widely studied effect of bottom topography on oceanic Rossby waves has received a great deal of attention. This renewed interest follows the observation of baroclinic Rossby waves by satellite altimeter, which indicated a significant influence of topographic features on wave propagation (Chelton and Schlax 1996). In particular, it has been suggested that topographic effects may cause the observed enhancement of the phase speed of the waves compared with that predicted by the simple, flat-bottom theory [although mean-flow effects have a primary importance; see Killworth et al. (1997)].
Recent studies have been devoted to slowly varying topography (Killworth and Blundell 1999), ridges (Tailleux and McWilliams 2000), as well as to more general situations (Reznik and Tsybaneva 1999; Bobrovich and Tsybanev 1999), and focused primarily on one-dimensional topographies. By contrast, two-dimensional topographies are considered in Vanneste (2000). This paper investigates (barotropic) quasigeostrophic motion over small-scale, periodic topography: using a multiple-scale (or homogenization) technique, the large-scale motion is shown to evolve according to an averaged quasigeostrophic equation in which the effect of topography is represented by a time-convolution term. Although Vanneste (2000) concentrates mainly on the enhancement of dissipation that is caused by the small-scale topography, it is clear from the averaged equation that topography affects wave propagation in a more complex way and, in particular, that it perturbs the frequency of planetary Rossby waves. The aim of the present note is to demonstrate this explicitly.
To this end, we investigate the impact of small-scale topography on Rossby wave propagation using a highly idealized model in which the topography consists of cylindrical seamounts separated by distances large compared to their radii. Admittedly, the model and asymptotic regime considered are not very realistic; however, they allow the rigorous derivation of simple analytical results in a problem whose general treatment is fairly difficult. Since the purpose of this note is mainly illustrative, further assumptions are made in order to achieve maximum simplicity: the model is barotropic and neglects viscous dissipation and large-scale variations of the topography. These assumptions can easily be relaxed; in particular, it would be straightforward to examine the effect of baroclinicity by applying our method to a multilayer model.
In the plethoric literature on topographic effects, a variety of asymptotic regimes have been investigated [see Reznik and Tsybaneva (1999) for a discussion]. Here, we consider a small-scale topography that is steep, that is, such that the associated potential-vorticity gradient is much larger than potential-vorticity gradient associated with the β effect. This assumption ensures that the topography has an effect on the large-scale flow of the same order as the β effect and thus modifies the Rossby wave dispersion relation at leading order (Vanneste 2000). In contrast, most earlier studies (Thomson 1975; Prahalad and Sengupta 1986) assume a much shallower topography; this allows standard techniques for the study of waves in random media to be employed [see, e.g., Mysak (1978) and references therein] but only leads to small changes in the dispersion relation that affect wave propagation only for distances and times much longer than the wavelength and period. Notable exceptions are the studies of localization (Sengupta et al. 1992; Klyatskin 1996), as well as the recent papers by Reznik and Tsybaneva (1999) and Bobrovich and Tsybanev (1999). However, the high anisotropy of the (one-dimensional) topography considered in these papers makes their theory and results markedly different from those presented here.
The plan of this note is as follows. In section 2, the dispersion relation for large-scale Rossby waves in the presence of topography is derived using a homogenization approach.1 The particular case of well-separated cylindrical seamounts is examined in section 3 where a dispersion relation valid for random distributions of seamounts is obtained perturbatively. The cases of seamounts with fixed height and with normally distributed heights is analyzed in detail in section 4. A similar qualitative conclusion is drawn in both cases: in addition to enhancing the dissipation of the Rossby waves, the topography induces a change in their frequency. In absolute value this change is a decrease (increase) when the flat-bottom, Rossby wave frequency is smaller (larger) than a suitably defined topographic frequency. Some remarks conclude the note in section 5.
2. Homogenization and dispersion relation
In general (2.3) needs to be solved numerically and the dependence of wi on ω cannot be expressed in closed form; thus, the solution of (2.4) for ω must rely on an extensive iterative procedure that requires the solution of the partial differential equation (2.3) at each iteration. Analytical progress can nevertheless be made by considering topographies consisting of well-separated, isolated features.
3. Random array of cylindrical seamounts
Difficulties in solving (2.3) for wi arise because the coefficients in this equation depend on space through h. The problem is significantly simplified for a piecewise topography: in this case, wi is harmonic (i.e.,
Here, since our objective is mainly to gain qualitative insight into the Rossby wave frequency change due to topography, we concentrate on an asymptotic limit that allows the derivation of simple analytical results. The limit we consider is that of well-separated seamounts; more precisely, we study cylindrical seamounts with radius a whose centers are separated by a distance d and we assume that a/d ≪ 1. In this limit, studied by Maxwell for the conductivity problem, the interaction between seamounts is neglected so that (2.3) is solved (analytically) for an isolated seamount in an infinite domain. Interestingly, the limit allows a simple treatment of the random case in which the radius of the seamounts a, their height Ht, and the position of their centers are distributed randomly.
4. Rossby wave frequency change
a. Single-height seamounts
Equation (4.1) becomes invalid for |ωr| ≈ |ωt| as a result of a resonance between Rossby and topographic waves. A nonzero damping δ ≠ 0 smoothes this resonance; it can easily be shown that the transition between the increase and decrease of the Rossby wave frequency then occurs for |ωr| =
Figures 1 and 2 illustrate the discussion above: they displays ωr and Re(ω(1)) as functions of the wavenumber k1. Without loss of generality, we have taken λ = β = 1 (corresponding to a choice of space and time units); we have also chosen k2 = 1. Figure 1 is obtained for a height of the topography such that ωt = 0.25. The results demonstrate the change in the sign of Re(ω(1)) that occurs for |ωr| ≈ |ωt| = 0.25. They also show the resonance phenomenon that appears in the absence of Ekman friction, that is, for r = 0, and its smoothing for r ≠ 0, here for r = 0.15. Figure 2 is obtained for ωt = 0.5: this is greater than the maximum Rossby wave frequency, so there is no sign change for Re(ω(1)), which is always positive (corresponding to a slow down of the Rossby waves), and there is no resonance phenomenon. Note that the imaginary part of ω(1) (not shown) is always negative, as is expected for an additional damping. This can be established directly from (3.6) or more generally from (2.4)–(2.5) for arbitrary topographies (Vanneste 2000).
b. Random height seamounts
Figures 3 and 4 show the real and imaginary parts of ω(1) calculated from (4.3) for λ = β = k2 = 1. In Fig. 3 the topography is such that ωσ = 0.25; it follows that the transition in the sign of the (real) frequency change occurs for ωr = 0.2355. The damping introduced by the topography is seen to be significant since |Im(ω(1))| > |Re(ω(1))| except for large values of k1. Note that the frequency shift Re(ω(1)) decreases rather slowly for large k1 (it is then proportional to 1/ωr). In Fig. 4, ωσ = 0.5; since |ωr| < x0|ωσ| for all k1, the frequency shift is always positive, corresponding to a slowdown of the Rossby wave.
5. Concluding remarks
In this note, a homogenization technique is used to derive the dispersion relation for large-scale Rossby waves in the presence of a steep small-scale topography. The frequency change induced by topography is computed explicitly for sparsely distributed cylindrical seamounts. The assumption α ≪ 1 of a low density of seamounts allows the derivation of an analytical expression for the frequency change; it also implies that this change is small [O(α)]. It should be emphasized, however, that the homogenization procedure, and in particular the dispersion relation (2.4), can be employed for dense topographies, although in this case numerical computations similar to those used in different contexts (e.g., Sang and Acrivos 1983) will be necessary to evaluate sij(ω) from (2.5) and to compute the frequency. Importantly, a dense topography will lead to a frequency change of the same order as the flat-bottom Rossby wave frequency itself.
Our approach relies on a spatial averaging procedure and thus centers on large-scale waves. Consequently, the topographic waves, whose (small) scale is fixed by that of the topography, do not appear explicitly in the dispersion relation (2.4)—only the coupling between topography, β effect and Ekman friction has an impact on the large scales. It is therefore not surprising that, even for dense topography, the large-scale restoring mechanism provided by the β effect (or equivalently by a large-scale topography) is required for the existence of waves.3 This contrasts our study with the work of Jansons and Johnson (1988) who focused on small-scale topographic modes supported by arrays of seamounts.
The main qualitative result of our work concerns the sign of the Rossby wave frequency shift. A general rule appears to be that topography “pushes away” the Rossby wave frequency from a typical topographic frequency; that is, the difference between the Rossby wave frequency and the topographic wave frequency increases as a consequence of the interaction between the two types of waves. This result, which is consistent with the general picture of linear wave interaction in stable systems (Craik 1985), is likely to hold in more general situations, for example, for baroclinic flows. Of course, the question of what the relevant topographic frequency is precisely for realistic topographies remains open. However, in realistic situations, it is likely that this topographic frequency is larger than the Rossby wave frequency and one may expect small-scale topography to cause a slowdown of the waves’ phase speed, which should be contrasted with the phase speed increase observed by Chelton and Schlax (1996).
Another result of this work is the finiteness of the damping induced by topography in the presence of infinitesimal Ekman friction. This phenomenon, which is associated with the possible resonance between topographic and Rossby waves, is analogous to the finite damping that appears for infinitesimally damped harmonic oscillators forced by a continuous spectrum of frequencies (Landau and Lifschitz 1976, sec. 26): topographic waves play the role of oscillators, while a Rossby wave provides the external forcing. The analogy has a twist, however, since in the Rossby wave case the forcing has a single frequency whereas the continuous frequency spectrum is associated with the oscillators because of the continuous distribution of seamount heights.
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The dispersion relation could also be deduced from the more general result of Vanneste (2000).
The limiting process implicitly assumes that α ≪ δ ≪ 1.