Equatorially trapped Rossby waves in the presence of meridionally sheared baroclinic flow in the Pacific Ocean

TOPEX/POSEIDON altimeter data are analyzed for the 8.5-year period November 1992 to May 2001 to investigate the sea surface height (SSH) and geostrophic velocity signatures of quasi-annual equatorially trapped Rossby waves in the Pacific. The latitudinal structures of SSH and both components of geostrophic velocity are found to be asymmetric about the equator across the entire Pacific with larger amplitude north of the equator. The westward phase speeds are estimated by several different methods to be in the range 0.5-0.6 m s⁻¹. These observed characteristics are inconsistent with the classical theory for first vertical, first meridional mode equatorially trapped Rossby waves, which predicts a phase speed of about 0.9 m s⁻¹ with latitudinally symmetric structures of SSH and zonal velocity and antisymmetric structure of meridional velocity. The observations are even less consistent with the latitudinal structures of SSH and geostrophic velocity components for other modes of the classical theory.The latitudinal asymmetries deduced here have also been consistently observed in past analyses of subsurface thermal data and altimeter data and have been variously attributed to sampling errors in the observational data, a superposition of multiple meridional Rossby wave modes, asymmetric forcing by the wind, and forcing by cross-equatorial southerly winds in the eastern Pacific. We propose a different mechanism to account for the observed asymmetric latitudinal structure of low-frequency equatorial Rossby waves. From the free-wave solutions of a simple 1.5-layer model, it is shown that meridional shears in the mean equatorial current system significantly alter the potential vorticity gradient in the central and eastern tropical Pacific. The observed asymmetric structures of sea surface height and geostrophic velocity components are found to be a natural consequence of the shear modification of the potential vorticity gradient. The mean currents also reduce the predicted westward phase speed of first meridional mode Rossby waves, improving consistency with the observations.     

Zonal Wavelengths of Planetary Rossby Waves Derived from Hydrographic Transects in the Northeast Atlantic Ocean?

Using hydrographic data of three extended zonal sections, which cover the upper 1000 dbar layer along 10°, 21°, and 32°N in the North-East Atlantic between 20° and 45°W, observational evidence is presented for zonal wavelengths of resonantly excited, first mode, long, baroclinic Rossby waves. The amplitudes of associated anomalies in the mass field decrease with increasing offshore distance. The associated zonal wavelengths reach several hundred kilometres and decrease with increasing latitude. Due to the Rossby dispersion, the detected wave patterns slowly propagate westward, somewhat faster in the south than in the north. The results obtained confirm the data sets remotely sensed by satellites, as well as the outcomes of analytical and numerical models.     

Alternating zonal flows in a two-layer wind-driven ocean

Alternating zonal flows in an idealized wind-driven double-gyre ocean circulation have been investigated using a two-layer shallow-water eddy-permitting numerical model. While the alternating zonal flows are found almost everywhere in the time-mean zonal velocity field, their meridional scales differ from region to region. In the subpolar western boundary region, where the energetic eddy activity induces quasi two-dimensional turbulence, the alternating zonal flows are generated by the inverse energy cascade and its arrest by Rossby waves, and the meridional scale of the flows corresponds well to the Rhines scale. In the eastern part of the basin, where barotropic basin modes are dominant, the zonal structure is formed through the nonlinear effect of the basin modes and is wider than the Rhines scale. Both effects are likely to form zonal structure between the two regions. These results show that Rossby basin modes become an important factor in the formation of alternating zonal flows in a closed basin in addition to the arrest of the inverse energy cascade by Rossby waves. The wind-driven general circulation associated with eddy activities plays an essential role in determining which mechanism of the alternating zonal flows is possible in each region.     

Diffraction of a long Kelvin wave at the apex angle formed by intersecting shores

This paper discusses the diffraction of a low-frequency Kelvin wave (having a frequency comparable to that of a baroclinic Rossby wave) at the apex angle. The problem is handled, with the Coriolis parameter being variable, by expanding into series over the minor parameter ε characteristic of the ratio between the deformation radius and the length of an incident wave travelling along a latitudinal shore. An analytical expression has been derived for a wave translating along meridional shore. It is demonstrated that when the latitudinal shore is located above the critical latitude, the energy flux through the horizontal cross-section transported by the wave coincides with the corresponding flux in the incident wave. If, on the other hand, it is situated below the critical latitude, then part of the energy is consumed by offshore Rossby waves. Translated by Vladimir A. Puchkin.     

Oceanic Rossby waves induced by the meridional shift of the ITCZ in association with ENSO events

This study investigated the eastern Pacific Intertropical Convergence Zone (ITCZ) as an atmospheric forcing to the ocean by using various observed and reanalysis data sets over 29 years. Climatologically, a zonal band of positive wind stress curl (WSC) with a 10° meridional width was exhibited along the ITCZ. A southward shift of the positive WSC band during the El Niño phase induced a negative (positive) WSC anomaly along the northern (southern) portion of the ITCZ, and vice versa during the La Niña phase. This meridional dipole accounted for more than 25 % of interannual variances of the WSC anomalies (WSCAs), based on analysis of the period 1993–2008. The negative (positive) WSCA in the northern portion of the ITCZ during the El Niño (La Niña) phase was collocated with a positive (negative) sea surface height anomaly (SSHA) that propagated westward as a Rossby wave all the way to the western North Pacific. This finding indicates that this off-equatorial Rossby wave is induced by the WSCA around the ITCZ. Our analysis of a 1.5-layer reduced gravity model revealed that the Rossby waves are mostly explained by wind stress forcing, rather than by reflection of an equatorial Kelvin wave on the eastern coastal boundary. The off-equatorial Rossby wave had the same SSHA polarity as the equatorial Kelvin wave, and generation of a phase-preserving Rossby wave without the Kelvin wave reflection was explained by meridional movement of the ITCZ. Thus, the ITCZ acts as an atmospheric bridge that connects the equatorial and off-equatorial oceanic waves.     

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.     

Observation of mixed Rossby-gravity waves in the Western equatorial Pacific

During the IOP (Intensive Observation Period) of TOGA/COARE (Tropical Ocean and Global Atmosphere/Coupled Ocean Atmosphere Response Experiment) from December 1992 to February 1993, four Japanese moored ADCPs (Acoustic Doppler Current Profilers) measured vertical profiles of three-component velocities at the stations 2S (2°S, 156°E), 2N (2°N, 156°E), 154E (0°N, 154°E) and 147E (0°N, 147°E). Power spectra of the surface current showed a pronounced peak having a period of around 14 days for both the zonal and meridional velocities at the stations 2S and 2N near the equator, and for only the meridional velocity at the equator. This 14-day phenomenon is considered to be a kind of equatorial wave of the first baroclinic mode, from a comparison of the result of the vertical mode analysis and the vertical distribution of the standard deviation of band-pass filtered velocity fluctuations. A dispersion relationship obtained from the horizontal mode analysis of this wave confirmed that the 14-day phenomenon is a mixed Rossby-gravity wave with the westward propagating phase speed and eastward propagating group velocity. From the cross-spectral analysis of velocity data, the average phase speed and wavelength of the wave were estimated as 3.64 m s⁻¹ and 3939 km, respectively, for station pair 2S∼147E. These values were in good agreement with the average phase speed and wavelength of 3.58 m s⁻¹ and 3836 km estimated from the dispersion curve and the observed period. A northerly wind burst blew over all the mooring sites during the middle of the observation period. The mixed Rossby-gravity wave, which is anti-symmetric for the zonal velocity about the equator, is likely to be forced by this northerly wind burst crossing the equator. Generation of the oceanic mixed Rossby-gravity wave of the first baroclinic mode is discussed in association with the atmospheric Rossby wave having the same period.     

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

Reflection and diffraction of internal solitary waves by a circular island

We have investigated the reflection and diffraction of first-mode and second-mode solitary waves by an island, using a three-dimensional nonhydrostatic numerical model. The model domain consists of a circular island 15 km in diameter in an ocean 300 m deep. We use prescribed density anomalies in an initially motionless ocean to produce highly energetic internal solitary waves; their subsequent propagation is subject to island perturbations with and without the effect of earth’s rotation. In addition to reflected waves, two wave branches pass around the island and reconnect behind it. Island perturbations to the first-mode and second-mode waves are qualitatively similar, but the latter is more profound because of the longer contact time and, in the presence of earth’s rotation, the scale compatibility between Rossby radius of the second baroclinic mode and the island diameter. Without earth’s rotation, reflected and diffracted waves are symmetrical relative to the longitudinal axis passing through the island center. With earth’s rotation, the current following the wave front veers to the right due to Coriolis deflection. For a westward propagating incoming wave, the deflection favors northward wave propagation in the region between the crossover point and the island, shifting the wave reconnection point behind the island northward. It also displaces the most visible part of the reflected waves to the southeast. In the presence of earth’s rotation, a second-mode incoming wave produces island-trapped internal Kelvin waves, which are visible after the passage of the wave front.     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

Analytic study to wave scattering by a general Homma island using the explicit modified mild-slope equation

In this paper, an exact analytic solution in terms of Taylor series to the explicit modified mild-slope equation (EMMSE) for wave scattering by a general Homma island is constructed and the convergence of the series solution is analyzed. To validate the new analytic solution, comparisons are made against the existing solutions including analytic solutions to both the long-wave equation and Helmholtz equation, approximate analytic solutions to the modified mild-slope equation, numerical solutions to the mild-slope equation and experimental solutions. Because of the use of the governing equation EMMSE together with mass-conserving matching conditions along the toe of the shoal, the present model is valid for not only waves in the whole spectrum from long waves to short waves but also bathymetries with the maximal seabed slope being as high as 4.27:1. Since the general Homma island is an extension of the original Homma island, the present solution can be very conveniently used to study the effects of bottom topography on combined refraction and diffraction. It is found that the larger the shoal size is, the more significant the wave amplification against the cylinder is.     

Investigation of conditions for the generation and propagation of low-frequency disturbances in the troposphere

Low-frequency disturbances responsible for the excitation of torsional oscillations—variations in the zonal mean flow intensity with a characteristic scale of 15–20 days—propagating along the meridian at mid and low latitudes of both hemispheres are investigated [1]. As data observed over the eastern parts of continents and the western parts of oceans are processed with the lag correlation statistics, traveling waves intersecting the eastern parts of continents from northwest to southeast and then returning to the north along the ocean coasts are identified. In this case, trains of anomalies oriented in the zonal direction periodically appear and are destructed in the western parts of continents. The simulation of the propagation of disturbances in the quasi-geostrophic approximation made it possible to explain the specific features of lag correlation statistics over continents by the dispersion of two-dimensional Rossby waves from traveling sources. The turnover of disturbances over Asia and wave trains to the west from the pole were reproduced. Torsional oscillations caused by the dispersion of two-dimensional Rossby waves have a characteristic form of inclined bands in the latitude-time diagram, whose steepness is controlled by the velocity of displacement of the vorticity source along the meridian.     

Frequency-wavenumber spectra of equatorial waves detected from satellite altimeter data

The sea surface height anomaly in the Pacific equatorial area was separated into equatorial modes using satellite altimeter data. The power-spectral density (PSD) was obtained for the east-west wavenumber and frequency for each separated mode. The PSD distribution was compared with the theoretical dispersion curve for the equatorial modes derived by Matsuno (1966). The first Rossby modes have a high-density distribution that is slightly lower than the theoretical dispersion curve, but the Kelvin mode and the higher Rossby modes have high-density distribution that almost matches each dispersion curve. Results of analyses of satellite observational data show that wave motion near the equator mainly shows characteristics of equatorial waves, especially for a intraseasonal time scale.     

Baroclinic seiches in rotating basins of variable depth in the case of two-layer density stratification

We solve a plane problem of linear baroclinic seiches in closed rotating basins of variable depth with two-layer density stratification. In the long-wave approximation, we get a boundary-value problem for a system of ordinary differential equations and propose a numerical procedure for finding internal seiches. The analytic solutions of the problem are obtained for a basin of constant depth. The numerical analysis of seiches is performed for the distributions of depth corresponding to the zonal and meridional sections of the Black Sea and model basins including the cases of a shelf zone and an underwater ridge. It is shown that the baroclinic seiches become more intense in shallow-water regions and that the intense longshore currents caused by Earth’s rotation are formed in the shelf zones and over the underwater ridges.     

Rhines效应对高度计观测的海洋中尺度涡空间分布特征的影响

Rhines效应是指Rossby波和大湍流（中尺度涡）相互作用,将涡动能量以波的形式传播出去,从而使中尺度涡发生形变,最终消亡的一种动力学机制。本文通过比较海洋里涡特征速度和Rossby长波波速的方法,研究了一种广义的Rhines效应对高度计观测的海洋中尺度涡空间分布特征的影响。结果显示,广义Rhines效应比只考虑行星涡度梯度的传统形式对中尺度涡的分布具有更显著的影响。大部分中尺度涡分布在涡特征速度（U_e）大于由广义Beta值计算的Rossby长波波速（U_cg）的区域。这些涡可以由动能反向串级过程获取能量,成长为振幅和空间尺度较大的涡。热带海域以外的“涡旋沙漠”区域,中尺度涡的数量稀少,强度很弱,大都分布于U_ecg的海域。广义Rhines效应可能是这些海域中尺度涡难以成长的动力学机制。     

On nonlinear planetary waves: A class of solutions missed by the traditional quasi-geostrophic approximation

Weakly nonlinear quasi-geostrophic planetary waves on a beta-plane and topographic waves over a linearly inclined bottom are examined by use of shallow water equations for a small beta parameter. Long solitary wave solutions missed by the use of the traditional quasi-geostrophic approximation are found in a channel ocean with neither a sheared current nor a curved (non-linearly inclined) bottom topography. The solutions are missed in the traditional approach because the irrotational motion associated with the geostrophic divergence is neglected by the quasi-geostrophic approximation. Another example which calls attention to the limitation of the traditional quasi-geostrophic approximation is the nonlinear evolution of divergent planetary eddies whose scale is much larger than the Rossby's radius of deformation. Some aspects of a new evolution equation are briefly discussed.     

Why was the 2008 Indian Ocean Dipole a short-lived event?

In this paper, the role of equatorial oceanic waves in affecting the evolution of the 2008 positive Indian Ocean Dipole (IOD) event was evaluated using available observations and output from a quasi-analytical linear wave model. It was found that the 2008 positive IOD was an early matured and abruptly terminated event: developed in April, matured in July, and diminished in September. During the development and the maturation of the 2008 positive IOD event, the wind-forced Rossby waves played a dominant role in generating zonal current anomalies in the western equatorial Indian Ocean, while a complex interplay between the wind-forced upwelling Kelvin waves and the eastern-boundary-generated Rossby waves accounted for most of the variability in the eastern basin. The latter induced eastward zonal current anomalies near the eastern boundary during the peak phase of the event. The 2008 positive IOD event was abruptly terminated in mid-July. We found that there were strong eastward zonal currents in mid-July, though the surface wind anomalies in the eastern basin continued to be westward (upwelling favorable). Our analysis shows that these eastward zonal currents mainly resulted from the easternboundary-generated upwelling Rossby waves, although the contribution from the wind-forced downwelling Kelvin waves was not negligible. These eastward zonal currents terminated the zonal heat advection and provided a favorable condition for surface heat flux to warm the eastern basin.     

On the modeling of wave propagation on non-uniform currents and depth

By transforming two different time-dependent hyperbolic mild slope equations with dissipation term for wave propagation on non-uniform currents into wave-action conservation equation and eikonal equation, respectively, shown are the different effects of dissipation term on the eikonal equation in the two different mild slope equations. The performances of intrinsic frequency and wave number are also discussed. Thus the suitable mathematical model is chosen in which the wave number vector and intrinsic frequency are expressed both more rigorously and completely. By using the perturbation method, an extended evolution equation, which is of time-dependent parabolic type, is developed from the time-dependent hyperbolic mild slope equation which exists in the suitable mathematical model, and solved by using the alternating direction implicit (ADI) method. Presented is the numerical model for wave propagation and transformation on non-uniform currents in water of slowly varying topography. From the comparisons of the numerical solutions with the theoretical solutions of two examples of wave propagation, respectively, the results show that the numerical solutions are in good agreement with the exact ones. Calculating the interactions between incident wave and current on a sloping beach [Arthur, R.S., 1950. Refraction of shallow water waves. The combined effects of currents and underwater topography. EOS Transactions, August 31, 549–552], the differences of wave number vector between refraction and combined refraction–diffraction of waves are discussed quantitatively, while the effects of different methods of calculating wave number vector on numerical results are shown.     

A Three-Point Method for Separating Incident and Reflected Waves over A Sloping Bed

This study presents a three-point method for separating incident and reflected waves to explain normally incident waves' propagating over a sloping bed. Linear wave shoaling is used to determine changes in wave amplitude and phase in response to variations of bathymetry. The wave reflection coefficient and incident amplitude are estimated from wave heights measured at three fixed wave gauges with unequal spacing. Sensitivity analysis demonstrates that the proposed method can predict the reflection and amplitude of waves over a sloping bed more accurately than the two-point method.     

A note on the accuracy of the mild-slope equation

The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom. As its name indicates, it is based on the assumption of a mild bottom slope. The purpose of this paper is to examine the accuracy of this equation as a function of the bottom slope. To this end a number of numerical experiments is carried out comparing solutions of the three-dimensional wave equation with solutions of the mild-slope equation.For waves propagating parallel to the depth contours it turns out that the mild-slope equation produces accurate results even if the bottom slope is of order 1. For waves propagating normal to the depth contours the mild-slope equation is less accurate. The equation can be used for a bottom inclination up to 1:3.