Planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin waves on the f-plane and β-plane in the presence of a uniform zonal flow

2021 ◽  
Author(s):  
Yair De-Leon ◽  
Chaim I. Garfinkel ◽  
Nathan Paldor
Keyword(s):  
Numerical Solutions ◽  
Bottom Topography ◽  
Phase Speed ◽  
Zonal Flow ◽  
Dispersion Curves ◽  
Kelvin Waves ◽  
Linear Wave ◽  
Hermite Function ◽  
Mean Flow ◽  
Zonal Wavenumber

<p>A linear wave theory of the Rotating Shallow Water Equations (RSWE) is developed in a channel on either the mid-latitude f-plane/β-plane or on the equatorial β-plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. We show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f-plane similar to the generation of these waves by PV sources such as the β–effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin-like waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the “no mean-flow” phase speeds does not account for the difference in phase speeds, and the meridional structure does not often oscillate across the channel but is trapped near one the channel's boundaries in mid latitudes or behaves as Hermite function in the case of an equatorial channel. The phase speed of Kelvin-like waves is modified by the presence of a mean flow compared to the classical gravity wave speed but their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wavenumber, and the convexity of the dispersion curves also changes with the zonal wavenumber. In some cases, the Kelvin-like dispersion curve crosses those of Poincaré modes, but it is not an evidence for the existence of instability since the Kelvin waves are not part of the solutions of an eigenvalue problem. </p>

scholarly journals The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow

2005 ◽  
Vol 35 (6) ◽  
pp. 1138-1142 ◽  
Author(s):  
John P. Boyd
Keyword(s):  
Numerical Solutions ◽  
Phase Speed ◽  
Nonlinear Behavior ◽  
Short Wave ◽  
Kelvin Waves ◽  
Jet Flows ◽  
Strong Nonlinearity ◽  
Kelvin Wave ◽  
Zonal Wavenumber ◽  
The Mean

Abstract The effects of latitudinal shear on equatorial Kelvin waves in the one-and-one-half-layer model are examined through a mixture of perturbation theory and numerical solutions. For waves proportional to exp(ikx), where k is the zonal wavenumber and x is longitude, earlier perturbation theories predicted arbitrarily large distortions in the limit k → ∞. In reality, the distortions are always finite but are very different depending on the sign of the equatorial jet. When the mean jet is westward, the Kelvin wave becomes very, very narrow. When the mean jet flows eastward, the Kelvin wave splits in two with peaks well off the equator and exponentially small amplitude at the equator itself. The phase speed is always a bounded function of k, asymptotically approaching a constant. This condition has important implications for the nonlinear behavior of Kelvin waves. Strong nonlinearity cannot be balanced by contracting longitudinal scale, as in the author’s earlier Korteweg–deVries theory for equatorial solitons: for sufficiently large amplitude, the Kelvin wave must always evolve to a front.

scholarly journals Equatorial Waves in Opposite QBO Phases

2011 ◽  
Vol 68 (4) ◽  
pp. 839-862 ◽  
Author(s):  
Gui-Ying Yang ◽  
Brian J. Hoskins ◽  
Julia M. Slingo
Keyword(s):  
Group Velocity ◽  
Lower Stratosphere ◽  
Phase Speed ◽  
Kelvin Waves ◽  
Western Hemisphere ◽  
Equatorial Waves ◽  
Vertical Wavelength ◽  
Quasi Biennial Oscillation ◽  
Zonal Wavenumber ◽  
The Waves

Abstract A methodology for identifying equatorial waves is used to analyze the multilevel 40-yr ECMWF Re-Analysis (ERA-40) data for two different years (1992 and 1993) to investigate the behavior of the equatorial waves under opposite phases of the quasi-biennial oscillation (QBO). A comprehensive view of 3D structures and of zonal and vertical propagation of equatorial Kelvin, westward-moving mixed Rossby–gravity (WMRG), and n = 1 Rossby (R1) waves in different QBO phases is presented. Consistent with expectation based on theory, upward-propagating Kelvin waves occur more frequently during the easterly QBO phase than during the westerly QBO phase. However, the westward-moving WMRG and R1 waves show the opposite behavior. The presence of vertically propagating equatorial waves in the stratosphere also depends on the upper tropospheric winds and tropospheric forcing. Typical propagation parameters such as the zonal wavenumber, zonal phase speed, period, vertical wavelength, and vertical group velocity are found. In general, waves in the lower stratosphere have a smaller zonal wavenumber, shorter period, faster phase speed, and shorter vertical wavelength than those in the upper troposphere. All of the waves in the lower stratosphere show an upward group velocity and downward phase speed. When the phase of the QBO is not favorable for waves to propagate, their phase speed in the lower stratosphere is larger and their period is shorter than in the favorable phase, suggesting Doppler shifting by the ambient flow and a filtering of the slow waves. Tropospheric WMRG and R1 waves in the Western Hemisphere also show upward phase speed and downward group velocity, with an indication of their forcing from middle latitudes. Although the waves observed in the lower stratosphere are dominated by “free” waves, there is evidence of some connection with previous tropical convection in the favorable year for the Kelvin waves in the warm water hemisphere and WMRG and R1 waves in the Western Hemisphere, which is suggestive of the importance of convective forcing for the existence of propagating coupled Kelvin waves and midlatitude forcing for the existence of coupled WMRG and R1 waves.

scholarly journals Rossby waves and two-dimensional turbulence in a large-scale zonal jet

1987 ◽  
Vol 183 ◽  
pp. 467-509 ◽  
Author(s):  
Theodore G. Shepherd
Keyword(s):  
Large Scale ◽  
Zonal Flow ◽  
Order Effect ◽  
Spatial Inhomogeneity ◽  
Mean Flow ◽  
Scale Separation ◽  
Zonal Flows ◽  
Beta Plane ◽  
Zonal Wavenumber ◽  
Scale Shear

The theory of homogeneous barotropic beta-plane turbulence is here extended to include effects arising from spatial inhomogeneity in the form of a zonal shear flow. Attention is restricted to the geophysically important case of zonal flows that are barotropically stable and are of larger scale than the resulting transient eddy field.Because of the presumed scale separation, the disturbance enstrophy is approximately conserved in a fully nonlinear sense, and the (nonlinear) wave-mean-flow interaction may be characterized as a shear-induced spectral transfer of disturbance enstrophy along lines of constant zonal wavenumber k. In this transfer the disturbance energy is generally not conserved. The nonlinear interactions between different disturbance components are turbulent for scales smaller than the inverse of Rhines's cascade-arrest scale κβ≡ (β0/2urms)½ and in this regime their leading-order effect may be characterized as a tendency to spread the enstrophy (and energy) along contours of constant total wavenumber κ ≡ (k2 + l2)½. Insofar as this process of turbulent isotropization involves spectral transfer of disturbance enstrophy across lines of constant zonal wavenumber k, it can be readily distinguished from the shear-induced transfer which proceeds along them. However, an analysis in terms of total wavenumber K alone, which would be justified if the flow were homogeneous, would tend to mask the differences.The foregoing theoretical ideas are tested by performing direct numerical simulation experiments. It is found that the picture of classical beta-plane turbulence is altered, through the effect of the large-scale zonal flow, in the following ways: (i) while the turbulence is still confined to KKβ, the disturbance field penetrates to the largest scales of motion; (ii) the larger disturbance scales K < Kβ exhibit a tendency to meridional rather than zonal anisotropy, namely towards v2 > u2 rather than vice versa; (iii) the initial spectral transfer rate away from an isotropic intermediate-scale source is significantly enhanced by the shear-induced transfer associated with straining by the zonal flow. This last effect occurs even when the large-scale shear appears weak to the energy-containing eddies, in the sense that dU/dy [Lt ] κ for typical eddy length and velocity scales.

scholarly journals Physics of tidal dissipation in early-type stars and white dwarfs: hydrodynamical simulations of internal gravity wave breaking in stellar envelopes

2020 ◽  
Vol 495 (1) ◽  
pp. 1239-1251 ◽  
Author(s):  
Yubo Su ◽  
Daniel Lecoanet ◽  
Dong Lai
Keyword(s):  
Wave Breaking ◽  
White Dwarfs ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Early Type ◽  
Mean Flow ◽  
Layer Width ◽  
Early Type Stars ◽  
Hydrodynamical Simulations

ABSTRACT In binaries composed of either early-type stars or white dwarfs, the dominant tidal process involves the excitation of internal gravity waves (IGWs), which propagate towards the stellar surface, and their dissipation via non-linear wave breaking. We perform 2D hydrodynamical simulations of this wave breaking process in a stratified, isothermal atmosphere. We find that, after an initial transient phase, the dissipation of the IGWs naturally generates a sharp critical layer, separating the lower stationary region (with no mean flow) and the upper ‘synchronized’ region (with the mean flow velocity equal to the horizontal wave phase speed). While the critical layer is steepened by absorption of these waves, it is simultaneously broadened by Kelvin–Helmholtz instabilities such that, in steady state, the critical layer width is determined by the Richardson criterion. We study the absorption and reflection of incident waves off the critical layer and provide analytical formulae describing its long-term evolution. The result of this study is important for characterizing the evolution of tidally heated white dwarfs and other binary stars.

scholarly journals The MJO as a Dispersive, Convectively Coupled Moisture Wave: Theory and Observations

2016 ◽  
Vol 73 (3) ◽  
pp. 913-941 ◽  
Author(s):  
Ángel F. Adames ◽  
Daehyun Kim
Keyword(s):  
Group Velocity ◽  
Rossby Wave ◽  
Linear Regression Analysis ◽  
Phase Speed ◽  
Wave Theory ◽  
Reanalysis Data ◽  
Horizontal Wind ◽  
Model Parameters ◽  
Linear Wave ◽  
Zonal Wavenumber

Abstract A linear wave theory for the Madden–Julian oscillation (MJO), previously developed by Sobel and Maloney, is extended upon in this study. In this treatment, column moisture is the only prognostic variable and the horizontal wind is diagnosed as the forced Kelvin and Rossby wave responses to an equatorial heat source/sink. Unlike the original framework, the meridional and vertical structure of the basic equations is treated explicitly, and values of several key model parameters are adjusted, based on observations. A dispersion relation is derived that adequately describes the MJO’s signal in the wavenumber–frequency spectrum and defines the MJO as a dispersive equatorial moist wave with a westward group velocity. On the basis of linear regression analysis of satellite and reanalysis data, it is estimated that the MJO’s group velocity is ~40% as large as its phase speed. This dispersion is the result of the anomalous winds in the wave modulating the mean distribution of moisture such that the moisture anomaly propagates eastward while wave energy propagates westward. The moist wave grows through feedbacks involving moisture, clouds, and radiation and is damped by the advection of moisture associated with the Rossby wave. Additionally, a zonal wavenumber dependence is found in cloud–radiation feedbacks that cause growth to be strongest at planetary scales. These results suggest that this wavenumber dependence arises from the nonlocal nature of cloud–radiation feedbacks; that is, anomalous convection spreads upper-level clouds and reduces radiative cooling over an extensive area surrounding the anomalous precipitation.

Dynamical Processes of Equatorial Atmospheric Angular Momentum

2006 ◽  
Vol 63 (2) ◽  
pp. 565-581 ◽  
Author(s):  
Steven B. Feldstein
Keyword(s):  
Angular Momentum ◽  
Surface Pressure ◽  
Time Derivative ◽  
Phase Speed ◽  
Wave Interaction ◽  
Friction Torque ◽  
Atmospheric Angular Momentum ◽  
Mean Flow ◽  
Dynamical Processes ◽  
Zonal Wavenumber

Abstract The dynamical processes that drive intraseasonal equatorial atmospheric angular momentum (EAAM) fluctuations are examined with the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data. The primary methodology involves the regression of relevant variables including the equatorial bulge, mountain, and friction torques, surface pressure, streamfunction, and outgoing longwave radiation, against the time derivative of the two components and the amplitude of the EAAM vector. The results indicate that the observed 10-day westward rotation of the EAAM vector corresponds to the propagation of a zonal wavenumber-1, antisymmetric, Rossby wave normal mode. Additional findings suggest that fluctuations in the amplitude of the EAAM vector are driven by poleward-propagating Rossby waves excited by the latent heating within equatorial mixed Rossby–gravity waves and also by wave–wave interaction among planetary waves. Both of these processes can induce surface pressure anomalies that amplify the EAAM vector via the equatorial bulge torque. The Antarctic and Greenland mountain torques were found to drive large fluctuations in the amplitude of the EAAM vector. Both the friction torque and wave–zonal-mean flow interaction were shown to dampen the EAAM amplitude fluctuations. A comparison of the EAAM dynamics in the atmosphere with that in an aquaplanet GCM suggests that the mountain torque also drives fluctuations in the phase speed of the atmospheric wave field associated with the EAAM vector, and it confines the wave–wave interaction to planetary scales.

Jet Interaction and the Influence of a Minimum Phase Speed Bound on the Propagation of Eddies

2013 ◽  
Vol 70 (8) ◽  
pp. 2614-2628 ◽  
Author(s):  
Amanda K. O'Rourke ◽  
Geoffrey K. Vallis
Keyword(s):  
Phase Speed ◽  
Zonal Flow ◽  
Separation Distance ◽  
Long Waves ◽  
Minimum Phase ◽  
Planetary Scale ◽  
Subtropical Jet ◽  
Zonal Wavenumber ◽  
Propagating Waves ◽  
Cospectral Analysis

Abstract The feedback between planetary-scale eddies and analogs of the midlatitude eddy-driven jet and the subtropical jet is investigated in a barotropic β-plane model. In the model the subtropical jet is generated by a relaxation process and the eddy-driven jet by an imposed wavemaker. A minimum zonal phase speed bound is proposed in addition to the established upper bound, where the zonal phase speed of waves must be less than that of the zonal mean zonal flow. Cospectral analysis of eddy momentum flux convergence shows that eddy activity is generally restricted by these phase speed bounds. The wavenumber-dependent minimum phase speed represents a turning line for meridionally propagating waves. By varying the separation distance between the relaxation and stirring regions, it is found that a sustained, double-jet state is achieved when either a critical or turning latitude forms in the interjet region. The interjet turning latitude filters eddies by zonal wavenumber such that shorter waves tend to be reflected off of the relaxed jet and are confined to the eddy-driven jet. The interjet region is transparent to long waves that act to blend the jets and may be associated with barotropic instability. The eddy-driven and relaxed jets tend to merge owing to the propagation of these long waves through the relaxed jet waveguide.

Quantifying Residual, Eddy, and Mean Flow Effects on Mixing in an Idealized Circumpolar Current

2017 ◽  
Vol 47 (8) ◽  
pp. 1897-1920 ◽  
Author(s):  
Phillip J. Wolfram ◽  
Todd D. Ringler
Keyword(s):  
High Performance ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Regime ◽  
Linear Waves ◽  
Parameter Ratio ◽  
Low Pass ◽  
Circumpolar Current

AbstractMeridional diffusivity is assessed for a baroclinically unstable jet in a high-latitude idealized circumpolar current (ICC) using the Model for Prediction across Scales Ocean (MPAS-O) and the online Lagrangian in Situ Global High-Performance Particle Tracking (LIGHT) diagnostic via space–time dispersion of particle clusters over 120 monthly realizations of O(106) particles on 11 potential density surfaces. Diffusivity in the jet reaches values of O(6000) m2 s−1 and is largest near the critical layer supporting mixing suppression and critical layer theory. Values in the vicinity of the shelf break are suppressed to O(100) m2 s−1 because of the presence of westward slope front currents. Diffusivity attenuates less rapidly with depth in the jet than both eddy velocity and kinetic energy scalings would suggest. Removal of the mean flow via high-pass filtering shifts the nonlinear parameter (ratio of the eddy velocity to eddy phase speed) into the linear wave regime by increasing the eddy phase speed via the depth-mean flow. Low-pass filtering, in contrast, quantifies the effect of mean shear. Diffusivity is decomposed into mean flow shear, linear waves, and the residual nonhomogeneous turbulence components, where turbulence dominates and eddy-produced filamentation strained by background mean shear enhances mixing, accounting for ≥80% of the total diffusivity relative to mean shear [O(100) m2 s−1], linear waves [O(1000) m2 s−1], and undecomposed full diffusivity [O(6000) m2 s−1]. Diffusivity parameterizations accounting for both the nonhomogeneous turbulence residual and depth variability are needed.

On a theory of amplitude vacillation in baroclinic waves

1977 ◽  
Vol 79 (2) ◽  
pp. 289-306 ◽  
Author(s):  
R. K. Smith
Keyword(s):  
Energy Source ◽  
Boundary Condition ◽  
Side Wall ◽  
Zonal Flow ◽  
Mean Flow ◽  
Viscous Effects ◽  
Baroclinic Waves ◽  
Wall Boundary Condition ◽  
Zonal Wavenumber ◽  
The Mean

This study contributes to the theory of amplitude vacillation for finite amplitude baroclinic waves in a two-layer, quasi-geostrophic, zonal flow as worked out by Pedlosky. In a recent paper the author has shown that Pedlosky omitted a certain side-wall boundary condition on the mean zonal flow. The neglect of this boundary condition results in an unspecified energy source at the side-wall boundaries and the physical problem is incorrectly posed.In this paper, Pedlosky's analysis is repeated but with the side-wall boundary condition included. It is shown that the side-wall energy source is negligible only when the zonal wavenumber of the disturbance is large compared with the meridional wavenumber, and not otherwise. Moreover, the energy conversions to and from mean zonal kinetic energy corresponding to Pedlosky's calculations and those given here have essential differences, although for fixed meridional wavenumber, these differences become less pronounced as the zonal wavenumber increases.It is also shown that, when the side-wall condition is included, the mean flow distortion associated with the wave is different in structure to that which occurs when the condition is omitted. However, as the total disturbance wavenumber a increases, the influence of the side wall on the mean flow structure is confined to a boundary layer of width comparable with the internal deformation radius 2½a−1.Even when the deformation radius is comparable with the channel width, the conclusions of Pedlosky (1972) concerning the existence of stable periodic solutions are correct, providing viscous effects are vanishingly small, and the criteria for stability of the steady solutions obtained herein are not significantly different from those given by Pedlosky. In this viscous regime, we have also studied the evolution to limit-cycle solutions.Evolution in the case where viscous effects are small on the time scale for the initial growth of an incipient wave, but not vanishingly small, will be discussed in a subsequent paper.

scholarly journals The Interaction of a Baroclinic Mean Flow with Long Rossby Waves

2005 ◽  
Vol 35 (5) ◽  
pp. 865-879 ◽  
Author(s):  
A. Colin de Verdière ◽  
R. Tailleux
Keyword(s):  
Doppler Shift ◽  
Rossby Waves ◽  
Numerical Solutions ◽  
Phase Speed ◽  
Standard Theory ◽  
Buoyancy Frequency ◽  
Implicit Assumption ◽  
Constant Sign ◽  
Mean Flow ◽  
The Mean

Abstract The effect of a baroclinic mean flow on long oceanic Rossby waves is studied using a combination of analytical and numerical solutions of the eigenvalue problem. The effect is summarized by the value of the nondimensional numberwhen the mean flow shear keeps a constant sign throughout the water column. Because previous studies have shown that no interaction occurs if the mean flow has the shape of the first unperturbed mode (the non–Doppler shift effect), an implicit assumption in the application of the present work to the real ocean is that the relative projections of the mean flow on the second and higher modes remain approximately constant. Because R2 is large at low latitudes between 10° and 30° (the southern branches of subtropical gyres or the regions of surface westward shear), the phase speed of the first mode is very slightly decreased from the no mean flow standard theory case. Between 30° and 40° latitudes (the northern branches of the subtropical gyres or the regions of surface eastward shear), R2 is O(10) and the westward phase speed can increase significantly (up to a factor of 2). At still higher latitudes when R2 is O(1) a critical transition occurs below which no discrete Rossby waves are found that might explain the absence of observations of zonal propagations at latitudes higher than 50°. Our case studies, chosen to represent the top-trapped and constant-sign shear profiles of observed mean flows, all show the importance of three main effects on the value of the first baroclinic mode Rossby wave speed: 1) the meridional gradient of the quantity N2/f (where N is the buoyancy frequency) rather than that of the potential vorticity fN2; 2) the curvature of the mean flow in the vertical direction, which appears particularly important to predict the sign of the phase speed correction to the no-mean-flow standard theory case: increase (decrease) of the westward phase speed when the surface-intensified mean flow is eastward (westward); and 3) a weighted vertical average of the mean flow velocity, acting as a Doppler-shift term, which is small in general but important to determine the precise value of the phase speed.

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