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Peter D. Killworth and Jeffrey R. Blundell

Abstract

An eigenvalue problem for the dispersion relation for planetary waves in the presence of mean flow and bottom topographic gradients is derived, under the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) assumption, for frequencies that are low when compared with the inertial frequency. Examples are given for the World Ocean that show a rich variety of behavior, including no frequency (or latitudinal) cutoff, solutions trapped at certain depths, coalescence of waves, and a lack of dispersion for most short waves.

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Peter D. Killworth and Jeffrey R. Blundell

Abstract

The one-dimensional examples of the dispersion relation for planetary waves under the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) assumption given in Part I are extended to two dimensions and analyzed globally. The dispersion relations are complicated, and there is a nontrivial lower bound to the frequency given by the column maximum of what would be the local Doppler shift to the frequency. This generates short waves of a much higher frequency than would be expected from traditional theory; these waves can have larger phase velocities than long waves but do not appear to have faster group velocities. The longer waves possess phase speeds in excellent agreement with recent remotely sensed data. Waves cannot propagate efficiently across ocean basins, suggesting that mechanisms other than eastern boundary generation may be playing a role in the ubiquitous nature of planetary waves.

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Peter D. Killworth and Jeffrey R. Blundell

Abstract

The local response of an ocean with slowly varying mean flow, stratification, and topography to two sources of disturbance is examined, concentrating on whether the resulting surface elevations are observable. The first is the ocean response to surface forcing (Ekman pumping or buoyancy forcing). For typical amplitudes of random forcing, while much of the ocean response is small (surface elevations less than 1 mm), there are sufficient near resonances (or pseudoresonances involving a critical layer) to produce elevations of 1 cm or more in much of the ocean. The second source is baroclinic instability. The fastest linear growth rate, as well as those for specific wavelengths, is computed globally. Almost all of the ocean is baroclinically unstable, and the most unstable waves are found to possess a small wavelength (often less than 10 km) with a disturbance concentrated near the surface: e-folding times O(20 days) are frequently found. However, the phase speed for the disturbances is almost everywhere slower westward than free planetary waves with mean flow and topography. Since the free waves propagate at speeds similar to observations, instability may be a good source of variability but is probably not responsible directly for observed wave propagation.

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Peter D. Killworth and Jeffrey R. Blundell

Abstract

This paper examines how slowly varying topography induces changes in all aspects of long planetary wave propagation, including phase speed and surface signature, through steering effects. The approach introduces a method for the exact solution of the vertical topographic eigenvalue problem for arbitrary realistic stratification and ray theory in the horizontal. It is shown that, for observed stratifications, first internal mode topographic waves have phase speeds between about 0.4 and twice the local flat-bottom phase speed. Increases occur on the western and equatorward sides of hills. Focusing of ray trajectories and caustics are common features of the solutions. Despite a bias between slowdown and speedup, on average there is little speedup except in high latitudes (where long-wave theory is less applicable). Calculations are performed for five main ocean basins, assuming waves are generated at the eastern coastline, using smoothed topography. These calculations confirm the above findings: there are significant local effects on wave speed, but these largely cancel over the basin scale. Thus, topographic effects cannot explain recent observations, which demonstrate long planetary waves propagating about twice as fast as linear theory. The presence of mean flow, which induces changes to the planetary vorticity gradient, remains the prime candidate for the observed speedup.

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Peter D. Killworth and Jeffrey R. Blundell
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Peter D. Killworth and Jeffrey R. Blundell

Abstract

Propagating features and waves occur everywhere in the ocean. This paper derives a concise description of how such small-amplitude, large-scale oceanic internal disturbances propagate dynamically against a slowly varying background mean flow and stratification, computed using oceanic data. For a flat-bottomed ocean, assumed here, the linear internal modes, computed using the local stratification, form a useful basis for expanding the oceanic shear modes of propagation. Remarkably, the shear modal structure is largely independent of orientation of the flow. The resulting advective velocities, which are termed pseudovelocities, comprise background flow decomposed onto normal modes, and westward planetary wave propagation velocities. The diagonal entries of the matrix of pseudovelocities prove to be reasonably accurate descriptors of the speed and direction of propagation of the shear modes, which thus respond as if simply advected by this diagonal-entry velocity field. The complicated three-dimensional propagation problem has thus been systematically reduced to this simple rule.

The first shear mode is dominated by westward propagation, and possesses a midlatitude speed-up over the undisturbed linear first-mode planetary wave. The pseudovelocity for the second shear mode, in contrast, while still dominated by westward propagation at lower latitudes, shows a gyrelike structure at latitudes above 30°. This resembles in both shape and direction the geostrophic baroclinic flow between about 500- and 1000-m depth, but are much slower than the flow at these depths. Features may thus be able to propagate some distance around a subtropical or subpolar gyre, but not, in general, at the speed of the circulation.

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Peter D. Killworth and Jeffrey R. Blundell

Abstract

One of the most successful theories to date to explain why observed planetary waves propagate westward faster than linear flat-bottom theory predicts has been to include the effect of background baroclinic mean flow, which modifies the potential vorticity waveguide in which the waves propagate. (Barotropic flows are almost everywhere too small to explain the observed differences.) That theory accounted for most, but not all, of the observed wave speeds. A later attempt to examine the effect of the sloping bottom on these waves (without the mean flow effect) did not find any overall speedup. This paper combines these two effects, assuming long (geostrophic) waves and slowly varying mean flow and topography, and computes group velocities at each point in the global ocean. These velocities turn out to be largely independent of the orientation of the wave vector. A second speedup of the waves is found (over that for mean flow only). Almost no eastward-oriented group velocities are found, and so features that appear to propagate in the same sense as a subtropical gyre would have to be coupled with the atmosphere or be density compensated in some manner.

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Andrew Mc C. Hogg and Jeffrey R. Blundell

Abstract

The intrinsic variability of the Antarctic Circumpolar Current is investigated using an idealized wind-driven model. The model uses three quasigeostrophic layers, with steady wind stress forcing, and no diabatic effects. Despite the idealized nature of the model, the simulations display a robust mode of low-frequency variability in the flow. It is demonstrated that this variability is dependent upon the explicit simulation of the dynamics of mesoscale eddies. As such, the variability is sensitive to stratification, horizontal viscosity, bottom stress, and topography. The energetic balance of the variability is diagnosed, and a driving mechanism is proposed that involves positive feedback between the generation of eddies through baroclinic instability and the dynamics of the mean circulation.

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Peter D. Killworth and Jeffrey R. Blundell

Abstract

Ray theory is used to predict phase and group velocities for long planetary waves under realistic, albeit slowly varying, oceanic conditions. The results are compared with local theory using fields smoothed to the same amount (9° latitude/longitude) as well as those with much less smoothing (1°). The agreement is excellent, showing that local theory forms a good proxy for ray theory results. The predicted speeds agree well with observations of planetary waves deduced from sea surface height data. The theory uses purely baroclinic mean flow; the inclusion of barotropic flow has little effect except at high latitudes.

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Peter D. Killworth, Jeffrey R. Blundell, and William K. Dewar

Abstract

The linear stability of two-layer primitive equation ocean rings is considered in the case when the rings are wide compared with a deformation radius, as is usually observed. Asymptotic theory is developed to show the existence of solutions for arbitrarily wide rings, and these solutions can be followed as the rings are made successively narrower. An exponential cubic radial dependence is used for the mean flow, rather than the more usual Gaussian structure. There are two reasons: a Gaussian shape was fully discussed in a previous paper, and a Gaussian has exceptional properties, unlike other power laws. The specific cases of warm and cold Gulf Stream rings are considered in detail. The theory provides an accurate prediction of phase velocity and growth rate for cold rings and a reasonable prediction for warm rings. Solutions in the asymptotic regime have a larger growth rate than other (nonasymptotic) solutions for cold rings, but not for warm rings. Attention is given to the role of the mean barotropic circulation, which had been found in earlier work to have a strong effect on ring stability. There is still evidence for stabilization when the mean flow in the lower layer vanishes, but other features are also involved. In particular, the linear stability of a ring appears to be as sensitive to subtle shape details as it is to the sense of the deep flow. The authors generally find warm co-rotating rings with a cubic exponential form to be unstable, although somewhat less so than counterrotating rings.

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