Topographic Rossby Waves Research Articles

Observations of Topographic Rossby Waves on the Continental Margin off Nova Scotia

Abstract Bursts of topographic-Rossby-wave energy have been observed in data recorded by an array of current meters moored on the outer continental shelf and slope off Nova Scotia north of the Gulf Stream. The waves persist for three or four cycles during which both the period (10 to 23 days) and the amplitude of the oscillations are modulated. At least four different events have been observed over approximately one year and appear to be associated with warm eddies shed by the Gulf Stream. One particularly clear event, dealt with in detail, had a period of 21 days, an offshore scale of 175 km and an offshore phase speed of 8 km day−1. The array of moorings was too small to resolve accurately the longshore scale or propagation direction. The wave appears to be barotropic (nearly uniform in amplitude and phase) at the 1000 m isobath, but increasingly baroclinic in both cross-slope directions. Deep-water kinetic energy associated with the wave appears to be uniformly distributed over the upper portion of the...

Coastally trapped waves, baroclinic eddies, internal tides and oceanic fronts at the shelf break: their implications for exchanges between shelf and oceanic waters

The seaward edge of the continental shelf, or shelf break, is the locus of strong physical variability in the overlying waters. Near the shelf-break, surface tides scatter energy into internal modes that propagate both onshore and offshore and produce strong vertical shears. Atmospheric forcing generates subinertial-frequency topographic Rossby waves, which propagate parallel to the coastline and are strongly trapped near the shelf break. Relative to the sloping topography, wind-driven coastal upwelling generates prograde fronts, and river run-off generates retrograde fronts. Located near the shelf break, these fronts are boundaries between oceanic and coastal waters. Oceanic eddies impinge on, and move along, the shelf-break entraining coastal waters. Eddies may also be shed by shelf-break fronts. Submarine capes and canyons contort the shelf break and significantly modify the enumerated processes. Based on observational evidence from a few coastal regimes, the shelf break is a zone where several mesoscale and synoptic-scale processes operate and probably produce significant turbulent transfers.

Beta-dispersion of low-frequency Rossby waves

An investigation is made into the dispersion of oceanic internal Rossby waves at annual and semi-annual frequencies. Turning of the group velocity vector due to latitudinal variations in the radius of deformation cannot be neglected, particularly in basins as large as the Pacific. This turning allows disturbances to propagate from high lattitudes into the equatorial zone and distorts the solutions in the western part of the basin. For no mean flow, and a coastline aligned north-south, an almost exact focus of wave energy is found very close to the equator at a distance of just under πc/4ω from the eastern boundary, where c is the eigenspeed of a high-frequency internal wave mode, and ω is the angular frequency of the low-frequency wave being studied. The focus depends on a long meridional wavelength excited at the coast, and a frequency small compared with c/a, where a is the radius of the Earth. For the lowest baroclinic mode and waves of annual period, this distance is about 12 000 km. Equivalence of the ray theory and the theory of equatorial meridional modes is demonstrated for the simple cases where the latter applies. The effects of mean currents and irregular coastlines are examined. Barotropic mean currents may change the turning latitude and ray shapes, inducing critical layers and enhancing reflection. Baroclinic mean currents are seen to affect the rays by simply changing the speed in proportion to the depth of the thermocline. As long as the mean currents are geostrophically balanced, no “effective beta” term from variations in the thermocline depth appears, in contrast to the topographic Rossby wave problem.

On the Propagation of Free Topographic Rossby Waves near Continental Margins. Part 2: Numerical Model

Abstract Ou (1980, Part 1) presented analytical solutions of free topographic Rossby waves propagating in an infinite wedge filled with a uniformly stratified fluid. We present here in Part 2 a numerical model incorporating more realistic topography and bottom friction to simulate the propagation of these waves across continental margins. Although the analytical solutions in a wedge can explain many of the wave properties observed in our numerical model, new features are introduced that have practical importance. It is found that the replacement of the apex by a finite nonreflecting shelf introduces at the shelf break an antinode in the pressure field, causing the kinetic energy to drop rapidly across the shelf break onto the shelf. It is also found that the baroclinic fringe waves excited near the slope/rise junction can cause an amphidromic point to form for shorter waves and reverse the direction of phase propagation above it. This changes the sign of the Reynolds stress locally and might have importan...

On the Propagation of Free Topographic Rossby Waves near Continental Margins. Part 1: Analytical Model for a Wedge

Abstract An analytical model has been constructed to study the propagation of free waves of subinertial frequency in an infinite wedge filled with a uniformly stratified fluid. The problem is found to transform into the corresponding surface gravity wave problem in a nonrotating homogeneous fluid with the roles of the surface and bottom boundaries interchanged. Analytical solutions are thus available for waves that are either progressive or trapped in the cross-wedge direction, forming respectively continuous and discrete spectra in frequency space. The separation occurs when the nondimensional wave frequency a (scaled by the inertial frequency f) equals the Burger number S, defined here as (N/f) tanθ*, where N is the Brunt-Vaisala frequency and tanθ * is the bottom slope. Since an infinite wedge has no intrinsic length scale, the only relevant nondimensional parameters are the wave frequency σ and the Burger number S. Thus, stratification and bottom slope play the same dynamical role, and the analysis is...

Solitary Rossby waves over variable relief and their stability. Part I: The analytical theory

Nonlinear permanent form solutions have been found for the barotropic, quasi-geostrophic divergenceless vorticity equation describing large scale, rotating flows over zonal relief. In the linear limit these solutions are topographic Rossby waves. The analytical procedure is an expansion in two small dimensionless parameters, an amplitude parameter (the Rossby number) and the aspect ratio between North-South (cross-relief) and East-West length scales. Permanent form solutions exist when these two parameters, and the related effects of dispersion and nonlinearity, mutually balance. By the same expansion procedure, an analytical linearized stability theory has been formulated which proves the neutral stability of these solutions to infinitesimal, two-dimensional perturbations.

Solitary rossby waves over variable relief and their stability. Part II: numerical experiments

The barotropic, quasi-geostrophic vorticity equation describing large scale, rotating flows over zonal relief supports nonlinear permanent form solutions, namely nonlinear topographic Rossby waves. Through an analytical theory, these solutions have been shown to be neutrally stable to infinitesimal perturbations. Numerical algorithms, which necessarily truncate the infinite number of degrees of freedom of any continuum model to a finite number, are capable of reproducing the numerical equivalent of these form-preserving solutions. Moreover, these numerical solutions are shown to preserve their shape throughout the numerical experiment not only in the limit of small amplitude, but also for high amplitude (Rossby number → O(1)). Through numerical simulation, the stability analysis is carried far beyond the analytical limit of infinitesimal perturbations. The solutions maintain their stability in agreement with the analytical theory, up to perturbations having intensities almost of the same order as the solutions themselves. For higher-amplitude perturbations, the solutions break up and typical turbulent behavior ensues. The passage from wave-like to turbulent behavior, upon surpassing a critical perturbation value, can be observed in the sudden loss of phase locking of the permanent solution Fourier modes.

The Effect of Current Shear on Topographic Rossby Waves

Abstract Topographic Rossby waves are long-period waves which occur on continental slopes. In this paper we examine the effect of a mean current on these waves, when the current is directed along the isobaths and has a linear shear in the transverse direction. Solutions are obtained in terms of Whittaker functions and are applied to some data analyzed by Hamon et al. (1975), which has been interpreted as evidence of topographic Rossby waves by Garrett (1979). The modifications due to a linear friction law are also considered.

Mean flow generation by topographic Rossby waves

This paper makes use of the ease of modelling topographic Rossby waves in a laboratory context to investigate the ability of these waves to generate strong zonal mean flows when the geostrophic (f/H) contours are closed. A zonally travelling wave is forced in a narrow latitude band of a ‘polar beta plane’. Stronger signals occur when the motion of the driving is retrograde and at the phase speed of the gravest free modes. An important zonal westward mean flow occurs in the free interior while a compensating eastward jet is found at forced latitudes. The dependence of the mean flow strength upon the wave steepness indicates that genuine rectification processes are indeed taking place when the fluid is stirred by purely oscillating devices.This general tendency for topographic Rossby waves to transfer energy to zonal components is first analysed theoretically by investigating a side-band instability mechanism within an unforced fluid. Among the products of the interactions between a primary wave of wavenumberkand its side bands of wavenumberk± δk, the zonal flow is prominent. Wave steepnesses of order (|δk|/|k|)½only are required for zonal energy to grow whereas non-zonal components of scale longer or shorter than the primary wave need huge steepnesses [of order (|δk|/|k|−3/2] for amplification. This supplements the earlier notion that ‘nearly zonal’ waves may be generated by weak resonant interaction.For gentle driving certain classical aspects of Rossby wave propagation can be checked against the experiments. The linear theory provides also a convenient framework to discuss the meridional structure of the wave-induced Reynolds stress. For more energetic driving, a test of the potential vorticity mixing theory can be carried out and sheds further light upon the rectification mechanisms.

The Kinetic Energy on a Continental Shelf from Topographic Rossby Waves Generated off the Shelf

Abstract Over a deep barotropic ocean an isolated pressure cell, producing a wind stress curl, generates topographic Rossby waves which are incident on a continental shelf with a simple exponentially varying slope. It is shown that this energy appears as a distinct peak at low frequency on the energy spectrum on the shelf. A peak at 0.05 cycles per day (cpd) in the spectrum from data of Smith (1974) for the Oregon shelf is consistent with this effect. For the case where topography dominates the beta effect off the shelf, general equations are found to estimate the frequency and dominant wavelength of the energy peak. An interesting result of this analysis is that, for all parameters being equal, this frequency should be lower on an eastern shelf than a western shelf in the Northern Hemisphere due to the beta effect. The variation of the magnitude and frequency of this peak with the location on the shelf, the distance of the center of forcing from the shelf, and the scale of the pressure cell is then inves...

Low‐frequency current fluctuations in Lake Ontario, winter 1972–1973 (IFYGL)

An analysis is presented of the low‐frequency (ω ≲ 0.6 cycles per day) current fluctuations in Lake Ontario, using measurements made at 15 and 75 m during the 1972–1973 winter. The data, collected as part of the International Field Year for the Great Lakes (IFYGL), were obtained at nine stations: one near midlake and the other eight spaced around the lake, about 50 km apart and 12 km from shore. Let U denote the alongshore velocity (positive leaving the shore to the right) and V the cross‐shore velocity (positive offshore). It is found that (1) the standard deviation of V is approximately constant; (2) the standard deviation of U, which is about 2.4 times greater than that of V, may be depth independent and is less at the southern shore stations than at the shallower northern stations where the flow appears to be directly wind driven; (3) the velocity fluctuations are roughly aligned with the local orientation of the coastline; (4) the U fluctuations show much better mutual correlation than the V fluctuations, having an alongshore coherence length of about 250 km (nearly half the basin perimeter); (5) the U signal shows high coherence for periods of about 10–25 and 4–5 days; and (6) the computation of lagged correlations shows the U fluctuations to propagate in a counterclockwise sense from the northern shore at a rate of about 42 km d−1 (0.5 m s−1), which implies a period of about 13 days for a complete circuit around the lake. These results provide additional evidence for the existence of topographic Rossby waves (shelf waves) in Lake Ontario, the calculated period being consistent with results of previous studies which used data from the summer and fall of 1972.

Topographic Rossby Waves off East Australia: Identification and Role in Shelf Circulation

Hamon et al. (1975) analyzed two years of surface current data from ships' set along two tracks parallel to 560 km of the east Australian shelf and found a phase lag of 10 days between long (120 day) period current fluctuations 19 km offshore and 6.5 km offshore. this is explained in terms of the propagation characteristics of topographic waves on the continental shelf. It is shown that these current fluctuations can lead to a significant amount of coastal upwelling as they are damped out by bottom friction. The average circulation on the east Australian shelf is discussed. The simplest response to the known longshore pressure gradient would be a steady longshore current and upwelling circulation, but it is shown that this is incompatible with a heat budget for the water on the self. Hence something other than bottom friction is required to balance most of the longshore pressure gradient; the onshore momentum flux of the topographic waves identified in the current data is shown to be a likely candidate. The implications of this interpretation for the circulation in deeper water off the shelf are discussed.

Simulated Dynamic Balances for Mid-Ocean Mesoscale Eddies

Abstract Eulerian potential vorticity and heat balances calculated for a multi-layered quasi-geostrophic model are shown to be consistent with those inferred from the MODE data. Above the thermocline the balances are the result of the turbulent cascade of relative vorticity which tends to separate the first moments of the enstrophy and energy spectra. The vorticity balance is dominated by the passive advection of the small-scale contributions to the relative vorticity by the energy-containing scales. By either transferring to the Lagrangian frame or focusing on the energy-containing scales this advective effect is suppressed and one observes a nearly linear response by relative vorticity and vertical vortex stretching to changes in planetary vorticity. In the bottom layer there are topographic Rossby waves superposed on a time-mean flow along f/H contours. Both of these effects “detune” the bottom layers from those above the thermocline to inhibit the cascade to barotropic flow which otherwise occurs in q...

Low‐frequency motions on the Scotian shelf and slope

Abstract An examination of current‐meter data gathered in 1967/68 on the continental shelf and slope off Nova Scotia has shown that meteorological forcing is an important source of energy. The response of currents to wind forcing is concentrated in a frequency band of 2.5 to 7 days. Daily mean currents of up to 25 cm/s appear to be associated with wind‐stress events. The highest correlations between wind and current are for the alongshore components of these variables. Wind‐induced currents may have been responsible for an intrusion of slope water onto the shelf which was observed in hydrographic sections from October and December 1968. Long data series (up to 167 days) formed by patching together shorter records demonstrate the existence of distinct low‐frequency variability at periods greater than 10 days. Some aspects of these motions suggest the presence of topographic Rossby waves on the shelf and slope. However, spatial and temporal coverage of data are not sufficient to define the sources of this v...

Observations of Rossby waves near site D 1

A theory of Rossby waves makes a number of predictions about motions below the thermocline at Site D (39°10'N, 70°W). An experiment was made to test these predictions. Kinetic energy spectra show most of the energy is associated with periods of a week to a few months. The Reynolds' stress is negative for these periods. There is high coherence and on phase shift between 1000 m and 2000 m depths. Most of the wave-number estimates point to the south-west quadrant, consistent with westward propagation and with momentum flux into the Gulf Stream and energy flux out of it. There is enough consistency between statistics based on successive 8-month records to conclude that statistics based on a single 8-month record near Site D are meaningful. In an 8-month array of twenty-six current meters, for periods of a week to a month, the divergence is small compared to the vorticity, and the motion is transverse. The energy increases toward the bottom. The observed wave-numbers and frequencies fit the theoretical dispersion relationship satisfactorily. The kinetic energy of the fluctuations is much larger near the Gulf Stream than farther upslope. The vorticity is in quadrature with the upslope velocity. I conclude there is strong evidence that topographic Rossby wave mechanics are dominant below the thermocline on the continental rise north of the Gulf Stream.

A Note on the Theory of Low‐Frequency Waves in a Rotating Stratified Channel

It is found that the possible low‐frequency, quasigeostrophic motions in a rotating, stratified channel with a wave‐maker at one end include: (i) “standing waves” whose amplitudes are damped exponentially away from the forcing, and (ii) baroclinic internal Kelvin waves, trapped to the right‐hand wall when facing in the direction of phase propagation. The Kelvin waves are excited only if the wave‐maker transfers mean energy to the fluid. The standing waves, on the other hand, carry no energy and thus serve mainly to provide continuity between the wave‐maker and the fluid. When the bottom of the channel is inclined to the horizontal by a small angle θ, topographic oscillations are possible. These waves behave like topographic Rossby waves if the forcing frequency is greater than sN and if the ratio HN/fL is small, where s=tanθ, θ is the angle of the bottom slope, L is the width of the channel, H is the mean depth, f is the Coriolis parameter, and N is the Brunt‐Vaisala (or buoyancy) frequency. It is determined that topographic Rossby waves cannot exist in the channel if HN/fL⪞0.65. If the wave‐maker frequency is smaller than sN, and if HN/fL~1, the topographic oscillations become bottom‐trapped, decaying away from the bottom boundary in a distance ~λf/N, where λ is the horizontal wavelength. The phase and energy of the bottom‐trapped wave both move to the left of an observer who is facing shallow water. The Kelvin waves are basically unchanged when the bottom is inclined if their down‐channel wavelength is large relative to the width of the channel. The standing oscillations of the flat‐bottom case exist as complex‐horizontal‐wave‐number solutions to the topographic wave dispersion relation. Although these waves have propagating phase when s≠0, they are still trapped to the forcing, and do not transfer net energy from the wave‐maker to the fluid. All three modes are required to solve the general matching conditions for an arbitrary wave‐maker when the channel has a sloping bottom.

Coastal-Trapped Waves in a Continuously Stratified Ocean

The theory of coastal-trapped waves is extended to include the general features of continuous density stratification, variable bottom topography, and a finite coastal wall. In the limit of a vanishing coastal wall, topographic Rossby waves are the only class of sub-inertial frequency, trapped wave motion. The stratification effect on topographic Rossby waves depends on both the local baroclinic radius of deformation and the characteristic offshore length scale of the wave motion. For intermediate density stratification, long waves are nearly depth-independent in the shelf region, and are bottom-trapped in the slope region. The topographic Rossby waves reduce to the barotropic shelf waves and the bottom-trapped waves in the limits of small and large density stratification, respectively. In the general case of comparable influences from the coastal wall and bottom slope effects, baroclinic Kelvin waves and topographic Rossby waves are eigenmodes of the system. The eigenfunctions are modified from the elementary cases, which can be discerned by their structures along the coastal and bottom bound-aries. In particular, a resonance condition is suggested, i.e., the properties of a wavemode vary with the wavenumber and stratification. For example, mode 1 is a topographic Rossby wave for small wavenumbers and it is a baroclinic Kelvin wave for large wavenumbers. Also, the high-frequency cutoff found in the barotropic theories is lost.

The Transmission and Decay of Barotropic Topographic Rossby Waves Incident on a Continental Shelf

Abstract We know that long-period (>1 day) and long-wavelength (>100 km) topographical Rossby waves can be generated by a wind acting directly on a continental shelf (Adams and Buchwald, 1969). Here we examine the characteristics of them waves which can also be produced off the shelf by wind and current eddies and can propagate up to and onto the shelf. We use a shelf model which varies in depth in one direction only and assume that a shelf can be approximated by at most two breaks with the depth varying exponentially. We assume velocity-dependent bottom friction to determine the effect of frictional dissipation. The following results are derived by our analysis. The regression angle of scatter plots for topography-dominated waves should be small and the preponderant direction of the waves determined by the sign. The group velocity directed up the slope possesses an absolute maximum which occurs at a relatively short period. The ability of a wave moving up a slope to overcome friction correlates with this...

Evidence for bottom-trapped topographic Rossby waves from single moorings

Current-meter observations near 39°N, 70°W, on the continental rise, provide evidence that the motions with periods of 1 to 2 weeks are dominated by baroclinic topographic Rossby waves which decay upward from the bottom. Temperature and up-slope velocity are coherent and in quadrature at these frequencies, as predicted. The kinetic energy structure versus depth is consistent with horizontal wavelengths of 100 to 200 km. The spectra drop abruptly for periods shorter than a week, the shortest period the model says the slope and stratification around Site D can maintain. The principal axis of the velocity shifts from nearly perpendicular to the isobaths at 1-week period to nearly along the isobaths at long periods, in satisfactory quantitative agreement with the model.

The ‘sliced-cylinder’ laboratory model of the wind-driven ocean circulation. Part 1. Steady forcing and topographic Rossby wave instability

The nonlinear response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is re-examined here in part 1 for the case of strong steady forcing. Introduced by Pedlosky & Greenspan (1967), the model consists of a rapidly rotating right cylinder with a planar sloping bottom. The homogeneous contained fluid is driven by the slow rotation of the flat upper lid relative to the rest of the basin. Except in thin Ekman and Stewartson boundary layers on the solid surfaces of the basin, the horizontal flow in the interior and western boundary layer is constrained by the rapid rotation of the basin to be independent of depth. The model thus effectively simulates geophysical flows through the physical analogy between topographic vortex stretching in the laboratory model and the creation of relative vorticity in planetary flows by the β effect.As the forcing is increased, the flow in both the sliced-cylinder laboratory and numerical models first exhibits downstream intensification in the western boundary layer. At greater forcing, separation of the western boundary current occurs with the development of stationary topographic Rossby waves in the western boundary-layer transition regions. The observed flow ultimately becomes unstable when a critical Ekman-layer Reynolds number is exceeded. We first review and compare the experimental and numerical descriptions of this low-frequency instability, then present a simple theoretical model which successfully explains this observed instability in terms of thelocalbreakdown of the finite-amplitude topographic Rossby waves embedded in the western boundary current transition region. The inviscid stability analysis of Lorenz (1972) is extended to include viscous effects, with the consequence that dissipative processes in the sliced-cylinder problem (i.e. lateral and bottom friction) are shown to inhibit the onset of the instability until the topographic Rossby wave slope exceeds a finite critical value.