Baroclinic Rossby waves in the ocean: normal modes, phase speeds and instability
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1 Baroclinic Rossby waves in the ocean: normal modes, phase speeds and instability J. H. LaCasce, University of Oslo J. Pedlosky, Woods Hole Oceanographic Institution P. E. Isachsen, Norwegian Meteorological Institute Following the seminal work of Anderson and Gill (1975), baroclinic Rossby waves are thought to mediate oceanic adjustment to changes in large scale forcing. An alteration in the winds is communicated throughout an ocean basin by a series of Rossby waves originating at the eastern boundary. The waves affect the general circulation with the corresponding baroclinic structure, with the result that the circulation becomes ever more surface-trapped. Figure 1: Global sea surface height from April, The white curve indicates the trough of a westward-propagating wave. From Chelton and Schlax, There have been numerous oceanic observations of westward phase propagation consistent with baroclinic Rossby waves. Perhaps the most spectacular came with the advent of sea surface height (SSH) measurement from satellite. Time sequences of SSH reveal large scale features propagating west across most ocean basins (Fig. 1). The phase speeds are similar to those of first mode baroclinic Rossby waves, and are largest at low latitudes. However, the phase speeds are faster than in classical theory, by roughly a factor of two at mid-latitudes and still more at high latitudes (Chelton and Schlax, 1996). This finding prompted a number of theoretical works, exploring Rossby wave phase speed enhancement by mean flow advection (e.g. Killworth et al., 1997), potential vorticity homogenization (e.g. Dewar, 1998), bottom topography (Tailleux and McWilliams, 2000) and lateral boundaries (LaCasce, 2000). 1
2 The last-mentioned work involved the solution of the quasi-geostrophic (QG) potential vorticity equation in a single layer, with a free surface: t ( 2 ψ Fψ) + β x ψ = 0 (1) where ψ is the streamfunction, β the derivative of the Coriolis parameter and F is a Froude number. The solution was obtained in a rectangular basin, with the streamfunction equal to an unknown but time-varying constant on the boundaries. The study did not produce faster Rossby waves, but it did reveal a previously unknown type of Rossby basin normal mode. This mode involves an interaction between Rossby waves in the interior and oscillations on the boundaries. Rather than having small horizontal scales (as occurs in the absence of boundary oscillations), the modes have very large scales, similar to the structures seen in the Pacific (Fig. 1). The modes (simultaneously discovered by Cessi and Primeau, 2001) have time scales of years and are thus potentially important for climatic variability. However, by virtue of their scale and baroclinicity, these waves are potentially baroclinically unstable. Such instability is precluded in a system with a single fluid layer with a free surface, so LaCasce and Pedlosky (2004) studied the corresponding wave problem with two fluid layers: t q T + β x ψ T = (h 2 h 1 ) u T q T u B q T u T q B (2) where: t q B + β x ψ B = u B q B h 1 h 2 u T q T (3) ψ T = ψ 1 ψ 2, q T ( 2 F)ψ T ψ B = h 1 ψ 1 + h 2 ψ 2, q B 2 ψ B are the streamfunctions and potential vorticities for the baroclinic and barotropic modes, respectively, and h 1 and h 2 are the layer depths. The authors found that baroclinic Rossby waves are always unstable, producing deformation scale, barotropic eddies. In fact, the main question is simply whether a wave, originating at the eastern boundary, can cross the basin before it breaks up. The time for a baroclinic Rossby wave to cross scales as the basin width, L, divided by the baroclinic long wave speed: T R = L βl D 2 where and L D is the deformation radius. The unstable growth time on the other hand scales as: T G = L D U 2
3 where U is the scale of the velocity shear. The ratio of these scales is: Z UL βl D 3 The waves can cross when Z is small. The strong dependence of Z on the deformation radius suggests that the latitudinal boundary between propagating and disintegrating waves is sharp. The authors found that waves north of about 20 degrees should break up. Figure 2: The baroclinic streamfunction from a simulations with a QG numerical model. The model was initialized with a baroclinic basin mode and Z = 0.5. From LaCasce and Pedlosky, An idealized example is shown in Fig. (2), obtained from a numerical simulation with a two layer QG basin code. The model was initialized with a mode 4 basin mode, and Z = 0.5. Waves form in the east and propagate westward, but disintegrate into small scale eddies about half-way across the basin. The preceding QG analysis assumed a constant deformation radius and thus is not strictly applicable to a Pacific-sized basin. So Isachsen et al. (2007) extended the analysis to baroclinic Rossby waves in large domains. Their theory was based on an expansion which they termed locally quasi-geostrophic (LQG). In this, the wave basic wave obeys the long wave equation: t Φ T β F(y) x Φ T = 0. (4) Note this is equation (1), except that the relative vorticity is neglected and the Froude number is allowed to vary with latitude. The unstable perturbations on the other hand are assumed to be quasi-geostrophic, and obey equations like those in (2-3) above. The LQG analysis of Isachsen et al. (2007) produced results consistent with the earlier QG analysis. 3
4 Figure 3: A baroclinic Rossby wave originating near the eastern boundary of a large basin. The solution was obtained using the ROMS primitive equation model. The left panels show the interface height at an early and a later time, and the right panels show the barotropic streamfunction. From Isachsen et al., The authors also conducted a series of numerical tests, using the ROMS ocean model (Shchepetkin and McWilliams, 2005). An example is shown in Fig. (3). A baroclinic wave was initialized on the right side of the basin and allowed to propagate westward. The wave bends, reflecting the faster phase speeds at low latitudes. The wave crosses in tact below degrees, but breaks up north of there. The numerical results supported the results of the LQG analysis. A consequence of the break-up at higher latitudes is that the SSH field no longer reflects baroclinic Rossby waves but rather the deformation scale, barotropic eddies. These features also propagate westward, but at faster speeds than their baroclinic counterparts. Analysis of the wave in Fig. 3 revealed a phase speed roughly twice the baroclinic long wave speed, and increasing to values up to four times the long wave speed at high latitudes. Thus the instability mechanism could explain several features from the satellite observations. The SSH fields suggest an absence of large scale propagation outside low latitudes (Fig. 1). These fields also suggest a change in the dominant eddy scale at latitudes higher than about 20 degrees (Stammer, 1997). In addition, instability would account for the excessive phase speeds. The instability also has several interesting consequences. For one, it appears unlikely that baroclinic basin modes can be important for coherent climate variation. Outside the tropics, they should break into smaller scale eddies, so the signal should be more chaotic. Secondly, the instability should greatly impact oceanic adjustment, as envisioned by Anderson and Gill (1975). If the baroclinic modes are unable to propagate in tact across the 4
5 basin, they are less likely to alter the higher baroclinic structure in the general circulation. That suggests more barotropic flows at higher latitudes. Such questions remain to be explored. References Anderson, D.L.T. and A.E. Gill, Spin-up of a stratified ocean with application to upwelling. Deep-Sea Res., 22, Cessi, P. and F. Primeau, Dissipative selection of low-frequency modes in a reduced-gravity basin. J. Phys. Oceanogr., 31, Chelton, D. B. and M. G. Schlax, Global observations of oceanic Rossby waves. Science, 272, Dewar, W.K., On too fast baroclinic planetary waves in the general circulation. J. Phys. Oceanogr., 28, Isachsen, P.E., J.H. LaCasce and J. Pedlosky, Rossby wave instability and apparent phase speeds in large ocean basins. J. Phys. Oceanogr., 37, Killworth, P.D., D. B. Chelton and R. A. de Szoeke, The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr., 27, LaCasce, J.H., Baroclinic Rossby waves in a square basin. J. Phys. Oceanogr., 30, LaCasce, J.H. and J. Pedlosky, The instability of Rossby basin modes and the oceanic eddy field. J. Phys. Oceanogr. 34, Pedlosky, J., Geophysical Fluid Dynamics. Springer, 728 pp. Shchepetkin, A. F. and J. C. McWilliams, The Regional Ocean Modeling System: A split-explicit, free-surface, topography following coordinates ocean model. Ocean Mod., 9, Stammer, D., Global characteristics of ocean variability estimated from regional TOPEX/POSEIDON altimeter measurements. J. Phys. Oceanogr., 27, Tailleux, R. and J.C. McWilliams, The effect of bottom pressure decoupling on the speed of extratropical, baroclinic Rossby waves. J. Phys. Oceanogr., 31,
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