An experimental study of forced barotropic rossby waves

Abstract

Barotropic Rossby waves are studied in a homogeneous fluid contained in a rotating cylindrical annulus with a radially sloping bottom boundary. The waves are forced by a simple source-sink distribution which can be rotated differentially relative to the annulus. When the speed of the source-sink distribution is close to the phase speed for a free Rossby wave of a given mode, resonant amplification occurs. The experimental results are in qualitative agreement with the predictions of a simple linear theory, but certain systematic differences between theory and experiment were observed.     

Rossby wave resonance in the presence of a nonlinear critical layer

Abstract

The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.     

Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L _b , π/L _b ], where L _b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.     

Generation of baroclinic topographic waves by a tropical cyclone impacting a low-latitude continental shelf

Numerical model experiments have been performed to analyze the low-latitude baroclinic continental shelf response to a tropical cyclone. The theory of coastally trapped waves suggests that, provided appropriate slope, latitude, stratification and wind stress, bottom-intensified topographic Rossby waves can be generated by the storm. Based on a scale analysis, the Nicaragua Shelf is chosen to study propagating topographic waves excited by a storm, and a model domain is configured with simplified but similar geometry. The model is forced with wind stress representative of a hurricane translating slowly over the region at 6 km h⁻¹. Scale analysis leads to the assumption that baroclinic Kelvin wave modes have minimal effect on the low-frequency wave motions along the slope, and coastal-trapped waves are restricted to topographic Rossby waves. Analysis of the simulated motions suggests that the shallow part of the continental slope is under the influence of barotropic topographic wave motions and at the deeper part of the slope baroclinic topographic Rossby waves dominate the low-frequency motions. Numerical solutions are in a good agreement with theoretical scale analysis. Characteristics of the simulated baroclinic waves are calculated based on linear theory of bottom-intensified topographic Rossby waves. Simulated waves have periods ranging from 153 to 203 h. The length scale of the waves is from 59 to 87 km. Analysis of energy fluxes for a fixed volume on the slope reveals predominantly along-isobath energy propagation in the direction of the group velocity of a topographic Rossby wave. Another model experiment forced with a faster translating hurricane demonstrates that fast moving tropical cyclones do not excite energetic baroclinic topographic Rossby waves. Instead, robust inertial oscillations are identified over the slope.     

Dispersive effects of zonally varying topography on quasigeostrophic Rossby waves

Abstract

Dispersion of linear quasigeostrophic plane waves in a stratified ocean with bottom topography is discussed. Particular emphasis is given to cases for which zonal gradients in the sea floor height are important. As such, the relative importance of the topographic and planetary β-effects is strongly dependent on wave vector orientation. The magnitude of the topographic slopes considered is chosen such that these two effects (topographic and planetary β) are of comparable importance. In the interest of simplicity, stratification is taken to be independent of depth. The eigenvalue problem which must be solved to find the free modes of oscillation has already been treated in the literature (in fact, Charney and Flierl (1981) have treated the effects of a more realistic stratification). The aim of this note is to more fully expose, primarily by example, several dispersive properties of these free wave modes which have been largely overlooked.     

On isallobaric effect in a quasi-geostrophic topographic-drag problem

Abstract

Isallobaric effect of a slowly varying quasi-geostrophic flow represented by propagating waves may give rise to a mean steady topographic drag component which turns out to be the principal one when viscous effects are negligibly small. This drag component decreases, in contrast to the quasi-geostrophic component, when statistical properties of the topography become isotropic. When the phase velocity of the incident wave is much larger (smaller) than the phase speed of Rossby waves, the isallobaric drag becomes independent (dependent) on the sign of that velocity.     

Convection driven by centrifugal bouyancy in a rotating annulus

Abstract

Drift rates and amplitudes of convection columns driven by centrifugal bouyancy in a cylindrical fluid annulus rotating about a vertical axis have been measured by thermistor probes. Conical top and bottom boundaries of the annular fluid region are responsible for the prograde Rossby wave like dynamics of the convection columns. A constant positive temperature difference between the outer and the inner cylindrical boundaries is generated by the circulation of thermostatically controled water. Mercury and water have been used as converting fluids. The measurements extend the earlier visual observations of Busse and Carrigan (1974) and provide quantitative data for an eventual comparison with nonlinear theories of thermal Rossby waves. The measured drift frequencies are in general agreement with linear theory. Of particular interest is the decline of the amplitude of convection with increasing Rayleigh number in a region beyond the onset of convection.     

The generation of baroclinic rossby waves by meridional oscillations of a zonal wind stress

Abstract

This paper investigates the generation of linear baroclinic Rossby waves by meridional oscillations of a climatological zonal wind stress in a reduced gravity ocean bounded by an eastern coastline. Using a power series technique an analytical solution is derived for the interfacial displacement. It is found that for a given period of oscillation of the zonal winds, a finite number of propagating Rossby waves will be generated with frequencies equal to a harmonic of the forcing frequency. The number of propagating modes increases with increasing period of the wind stress. In addition to the propagating waves the complete solution for the interfacial displacement consists of a rapidly convergent infinite sum of evanescent terms. The displacement field is calculated for atmospheric forcing parameters typical of those found at mid-latitudes. Further, it is shown that a near resonant response can be generated using atmospheric parameter values typical of those found over the North Pacific.     

Resonance induced by Rossby Lee waves

Abstract

The mutual interaction of fields induced by spatially separated potential vorticity sources in a quasi-geostrophic barotropic flow is investigated using the weakly nonlinear approach. It is found that a powerful nonlinear response can be triggered by Rossby lee waves. This resonance phenomenon which dominates all other nonlinear corrections depends on certain global resonance conditions and on the change in the phase of the Rossby lee wave across the distance separating the sources. The response is particularly strong for topographic forcing possessing δ-function characterisitics.     

Barotropic instability of a boundary jet on a sloping bottom

Abstract

The stability of a shear flow on a sloping bottom in a homogeneous, rotating system was investigated by means of a laboratory experiment.

The basic flow was driven near a vertical wall of a circular container by a ring-shaped plate that contacted with a free surface of the working fluid and rotated relative to the fluid container. The velocity profile was asymmetric in the radial direction and had only one inflection point. The velocity profile was well expressed by a linear theory for the vertical shear layer.

The effect of the circular geometry was checked by comparing experimental results obtained in two fluid systems in which only the sign of the curvature was opposite and it was confirmed that circular geometry was not essential for the shear flow on the sloping bottom in this experiment.

It was found that the sloping bottom stabilizes the basic flow only when the drift direction of the topographic Rossby wave is opposite to that of the basic flow. The viscous dissipation in both the Ekman layer and the interior region was also important in determining the critical Rossby number.

The eddy fields caused by the instability can be classified into two types: One is the stationary eddy field in which a row of eddies moves along the basic flow without changing form. The other is the flow pattern in which eddies have finite life times and their configuration is not well organized. When the sloping bottom does not stabilize the basic flow, the former flow pattern is realized, otherwise the latter flow pattern appears.

The wave numbers of the eddies in the regular flow pattern were observed as a function of the Rossby number. The relation did not fit to linear preferred modes predicted by an eigenvalue problem.     

Barotropic and baroclinic processes in the transport variability of the Antarctic Circumpolar Current

Synoptic scale variability of the Southern Ocean wind field in the high-frequency range of barotropic Rossby waves results in transport variations of the Antarctic Circumpolar Current (ACC), which are highly coherent with the bottom pressure field all around the Antarctic continent. The coherence pattern, in contrast to the steady state ACC, is steered by the geostrophic f/h contours passing through Drake Passage and circling closely around the continent. At lower frequencies, with interannual and decadal periods, the correlation with the bottom pressure continues, but baroclinic processes gain importance. For periods exceeding a few years, variations of the ACC transport are in geostrophic balance with the pressure field associated with the baroclinic potential energy stored in the stratification, whereas bottom pressure plays a minor role. The low-frequency variability of the ACC transport is correlated with the baroclinic state variable in the entire Southern Ocean, mediated by baroclinic topographic–planetary Rossby waves that are not bound to f/h contours. To clarify the processes of wave dynamics and pattern correlation, we apply a circulation model with simplified physics (the barotropic–baroclinic-interaction model BARBI) and use two types of wind forcing: the National Centers for Environmental Prediction (NCEP) wind field with integrations spanning three decades and an artificial wind field constructed from the first three empirical orthogonal functions of NCEP combined with a temporal variability according to an autoregressive process. Experiments with this Southern Annular Mode type forcing have been performed for 1,800 years. We analyze the spin-up, trends, and variability of the model runs. Particular emphasis is placed on coherence and correlation patterns between the ACC transport, the wind forcing, the bottom pressure field and the pressure associated with the baroclinic potential energy. A stochastic dynamical model is developed that describes the dominant barotropic and baroclinic processes and represents the spectral properties for a wide range of frequencies, from monthly periods to hundreds of years.     

Solitary and cnoidal planetary waves

Abstract

A class of long planetary waves in a zonal channel analogous to the solitary and cnoidal waves of surface and internal gravity wave theory is discussed. On a mid-latitude β-plane, such waves exist as the result of divergence, non-uniform zonal velocity fields or bottom topography. In all cases studied the wave profile along the channel was found to satisfy the Korteweg-de Vries equation.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

Effects of wave-wave and wave-mean flow interactions on the evolution of a baroclinic wave

Abstract

The weakly nonlinear evolution of a free baroclinic wave in the presence of slightly supercritical, vertically sheared zonal flow and a forced stationary wave field that consists of a single zonal scale and an arbitrary number of meridional harmonics is examined within the context of the conventional two-layer model. The presence of the (planetary-scale) stationary wave introduces zonal variations in the supercriticality and is shown to alter the growth rate and asymptotic equilibrium of the (synoptic-scale) baroclinic wave via two distinct mechanisms: The first is due to the direct interaction of the stationary wave with the shorter synoptic wave (wave-wave mechanism), and the second is due to the interaction of the synoptic wave with that portion of the mean field that is corrected by the zonally rectified stationary wave fluxes (wave-mean mechanism). These mechanisms can oppose or augment each other depending on the amplitude and spatial structure of the stationary wave field. If the stationary wave field is confined primarily to the upper (lower) layer and consists of only the gravest cross-stream mode, conditions are favorable (unfavorable) for nonzero equilibrium of the free wave.

In addition to the time dependent heat flux generated by baroclinic growth of the free wave, its interaction with a stationary wave field consisting of two or more meridional harmonics generates time dependent heat fluxes that vary with period of the free wave. However, if the stationary wave field contains several meridional harmonics of sufficiently large amplitude, the free baroclinic wave is destroyed.     

The effects of vertical shear and stratification on stationary rossby waves

Abstract

The generation of stationary Rossby waves by sources of potential vorticity in a westerly flow is examined here in the context of a two-layer, quasi-geostrophic, β-plane model. The response in each layer consists of a combination of a barotropic Rossby wave disturbance that extends far downstream of the source, and a baroclinic disturbance which is evanescent or wave-like in character, depending on the shear and degree of stratification. Contributions from each of these modes in each layer are strongly dependent on the basic flows in each layer; the degree of stratification; and the depths of the two layers. The lower layer response is dominated by an evanescent baroclinic mode when the upper layer westerlies are much larger than those in the lower layer. In this case, weak stationary Rossby waves of large wavelengths are confined to the upper layer and the disturbance in the lower layer is confined to the source region.

Increasing the upper layer flow (with the lower layer flow fixed) increases the Rossby wavelength and decreases the amplitude. Decreasing the lower layer flow (with the upper layer flow fixed) decreases the wavelength and increases the amplitude. Stratification increases the contribution from the barotropic wave-like mode and causes the response to be confined to the lower layer.

The finite amplitude response to westerly flow over two sources of potential vorticity is also considered. In this case stationary Rossby waves induced by both sources interact to reinforce or diminish the downstream wave pattern depending on the separation distance of the sources relative to the Rossby wavelength. For fixed separation distance, enhancement of the downstreatm Rossby waves will only occur for a narrow range of flow variables and stratification.     

Topographic instability and multiple equilibria on an f-plane

Abstract

The flow of a two-layer flow in a rotating channel on an f-plane over topography with sinusoidal variation of height in a direction parallel to the flow is investigated. When the two layers flow in opposite directions a resonance is found when the topographic scale matches the free mode of the system. We examine the stability of the forced mode in the vicinity of this resonance by means of a perturbation expansion of the topographic height. Both subresonant and super-resonant instabilities are found and their equilibration is examined. For small values of the dissipation multiple equilibria are found. The topographic drag releases potential energy even when the flow is baroclinically stable.     

Scattering of rossby waves by partial barriers

Abstract

The scattering of Rossby waves by partial barriers is studied. It is found that only a small fraction of the wave energy will penetrate the gap between South America and Antarctica if the wave period is less than a month.     

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.     

Onset of an energy cascade and nonperiodic behaviour in the nonlinear propagation of MHD waves in the solar atmosphere

Abstract

We study the nonlinear stability of MHD waves propagating in a two-dimensional, compressible, highly magnetized, viscous plasma. These waves are driven by a weak, shear body force which could be imposed by large scale internal fluctuations present in the solar atmosphere.

The effects of anisotropic viscosity (leading to a cubic damping) and of the nonlinear coupling of the Alfven and the magnetoacoustic waves are analysed using Galerkin and multiple-scale analysis: the MHD equations are reduced to a set of nonlinear ordinary differential equations which is then suitably truncated to give a model dynamical system, representing the interaction of two complex Galerkin modes.

For propagation oblique to the background magnetic field, analytical integration shows that the low-wavenumber mode is physically unstable. For propagation parallel to the background magnetic field the high-wavenumber wave can undergo saddlenode bifurcations, in way that is similar to the van der Pol oscillator; these bifurcations lead to the appearance of a hysteresis cycle.

A numerical integration of the dynamical system shows that a sequence of Hopf bifurcations takes place as the Reynolds number is increased, up to the onset of nonperiodic behaviour. It also shows that energy can be transferred from the low- wavenumber to the high-wavenumber mode.     

Froude number effects on flow over topography

Abstract

The response or a depth independent two layer flow to an underlying topographic irregularity is studied for flows in which the square of the internal Froude number exceeds the Rossby number. Irrespective of the magnitude of the Rossby number, rotation is important for such flows. The flow generally adjusts so that the thickness of the lower layer is nearly constant. However small anomalies from the constant thickness are found to extend to very large distances from the topography when the Rossby number exceeds unity.