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## Abstract

Solutions are found for the transmission of 1½-layer subinertial Kelvin waves through an “ideal strait” with parallel channel walls and square corner mouths. This is shown to be a wave interference problem similar to the classical optics problem of multiple beam interference. The outlet of an ideal strait has the same reflection characteristics as the inlet, setting up internal reflections within the strait and a consequent interference condition that has several important consequences: energy transmission of a strait is a function of channel length as well as width; transmission is frequency dependent, and approaches 100% at low frequencies for all channel widths, as long as the dynamics remain linear and inviscid; and the amplitude of the pressure signal on the downstream side of the channel approaches zero as the transmission approaches 100%. Numerical solutions are found for transmission through a strait with coastal geometry corresponding to the 200-m isobath at Lombok Strait, and the results show that the ideal strait model with an appropriate effective width can accurately predict the transmission characteristics of a more complex strait. In particular, even though the minimum channel width is less than one-fifth of the local Rossby radius, the strait is shown to approach total transmission within the intraseasonal frequency band.

## Abstract

Solutions are found for the transmission of 1½-layer subinertial Kelvin waves through an “ideal strait” with parallel channel walls and square corner mouths. This is shown to be a wave interference problem similar to the classical optics problem of multiple beam interference. The outlet of an ideal strait has the same reflection characteristics as the inlet, setting up internal reflections within the strait and a consequent interference condition that has several important consequences: energy transmission of a strait is a function of channel length as well as width; transmission is frequency dependent, and approaches 100% at low frequencies for all channel widths, as long as the dynamics remain linear and inviscid; and the amplitude of the pressure signal on the downstream side of the channel approaches zero as the transmission approaches 100%. Numerical solutions are found for transmission through a strait with coastal geometry corresponding to the 200-m isobath at Lombok Strait, and the results show that the ideal strait model with an appropriate effective width can accurately predict the transmission characteristics of a more complex strait. In particular, even though the minimum channel width is less than one-fifth of the local Rossby radius, the strait is shown to approach total transmission within the intraseasonal frequency band.

## Abstract

The ventilated thermocline theory is adapted to include the interaction between an island and the wind-driven subtropical circulation. The primary goal is an understanding of the effect of this interaction on the Hawaiian Lee Countercurrent (HLCC), an eastward current crossing the Pacific west of Hawaii in the latitude band 18°–21°N. A two-and-one-half-layer model is used, with the model island located in midgyre and equatorward of the gyre center, simulating the Hawaiian Islands. The presence of the island creates two wakes. A transport wake directly west of the island is characterized by an alteration of the zonal transport caused by the diversion of the interior flow incident upon the east coast of the island into zonal jets extending westward from the northern and southern tips of the island. A potential vorticity (pv) wake embedded in the second-layer streamlines westward and equatorward of the island is characterized by an alteration of the ventilated pv signature and of the baroclinic nature of the flow. The effects of the two wakes combine for a significant impact on both the transport and the baroclinic structure of the modeled HLCC, indicating that effective modeling of the HLCC should include not only the forcing mechanism, but also the influence of the large-scale Sverdrup flow to the east as modified by the Hawaiian Islands. In particular, a zonal variation in baroclinic structure is predicted that is consistent with observations.

## Abstract

The ventilated thermocline theory is adapted to include the interaction between an island and the wind-driven subtropical circulation. The primary goal is an understanding of the effect of this interaction on the Hawaiian Lee Countercurrent (HLCC), an eastward current crossing the Pacific west of Hawaii in the latitude band 18°–21°N. A two-and-one-half-layer model is used, with the model island located in midgyre and equatorward of the gyre center, simulating the Hawaiian Islands. The presence of the island creates two wakes. A transport wake directly west of the island is characterized by an alteration of the zonal transport caused by the diversion of the interior flow incident upon the east coast of the island into zonal jets extending westward from the northern and southern tips of the island. A potential vorticity (pv) wake embedded in the second-layer streamlines westward and equatorward of the island is characterized by an alteration of the ventilated pv signature and of the baroclinic nature of the flow. The effects of the two wakes combine for a significant impact on both the transport and the baroclinic structure of the modeled HLCC, indicating that effective modeling of the HLCC should include not only the forcing mechanism, but also the influence of the large-scale Sverdrup flow to the east as modified by the Hawaiian Islands. In particular, a zonal variation in baroclinic structure is predicted that is consistent with observations.

## Abstract

The theoretical resonant excitation of equatorial inertia–gravity waves and mixed Rossby–gravity waves is examined. Contrary to occasionally published expectations, solutions show that winds that are broadband in both zonal wavenumber and frequency do not in general produce peaks in the wavenumber–frequency spectrum of sea surface height (SSH) at wavenumbers associated with vanishing zonal group velocity.

Excitation of total wave energy in inertia–gravity modes by broadband zonal winds is virtually wavenumber independent when the meridional structure of the winds does not impose a bias toward negative or positive zonal wavenumbers. With increasing wavenumber magnitude |*k*|, inertia–gravity waves asymptote toward zonally propagating pure gravity waves, in which the magnitude of meridional velocity *υ* becomes progressively smaller relative to the magnitude of zonal velocity *u* and pressure *p*. When the total wave energy is independent of wavenumber, this effect produces a peak in |*υ*|^{2} near the wavenumber where group velocity vanishes, but a trough in |*p*|^{2} (or SSH variance). Another consequence of the shift toward pure gravity wave structure is that broadband meridional winds excite inertia–gravity modes progressively less efficiently as |*k*| increases and *υ* becomes less important to the wave structure. Broadband meridional winds produce a low-wavenumber peak in total wave energy leading to a subtle elevation of |*p*|^{2} at low wavenumbers, but this is due entirely to the decrease in the forcing efficiency of meridional winds with increasing |*k*|, rather than to the vanishing of the group velocity. Physical conditions that might alter the above conclusions are discussed.

## Abstract

The theoretical resonant excitation of equatorial inertia–gravity waves and mixed Rossby–gravity waves is examined. Contrary to occasionally published expectations, solutions show that winds that are broadband in both zonal wavenumber and frequency do not in general produce peaks in the wavenumber–frequency spectrum of sea surface height (SSH) at wavenumbers associated with vanishing zonal group velocity.

Excitation of total wave energy in inertia–gravity modes by broadband zonal winds is virtually wavenumber independent when the meridional structure of the winds does not impose a bias toward negative or positive zonal wavenumbers. With increasing wavenumber magnitude |*k*|, inertia–gravity waves asymptote toward zonally propagating pure gravity waves, in which the magnitude of meridional velocity *υ* becomes progressively smaller relative to the magnitude of zonal velocity *u* and pressure *p*. When the total wave energy is independent of wavenumber, this effect produces a peak in |*υ*|^{2} near the wavenumber where group velocity vanishes, but a trough in |*p*|^{2} (or SSH variance). Another consequence of the shift toward pure gravity wave structure is that broadband meridional winds excite inertia–gravity modes progressively less efficiently as |*k*| increases and *υ* becomes less important to the wave structure. Broadband meridional winds produce a low-wavenumber peak in total wave energy leading to a subtle elevation of |*p*|^{2} at low wavenumbers, but this is due entirely to the decrease in the forcing efficiency of meridional winds with increasing |*k*|, rather than to the vanishing of the group velocity. Physical conditions that might alter the above conclusions are discussed.

## Abstract

In the 1970s and 1980s, there was considerable interest in near-equatorial variability at periods of days to weeks associated with oceanic equatorial inertia–gravity waves and mixed Rossby–gravity waves. At that time, the measurements available for studying these waves were much more limited than today: most of the available observations were from scattered island tide gauges and a handful of short mooring records. More than a decade of the extensive modern data record from the Tropical Atmosphere Ocean (TAO)/Triangle Trans-Ocean Buoy Network (TRITON) mooring array in the Pacific Ocean is used to reexamine the internal-wave climate in the equatorial Pacific, with a focus on interpretation of the zonal-wavenumber/frequency spectrum of surface dynamic height relative to 500 decibars at periods of 3–15 days and zonal wavelengths exceeding 30° of longitude. To facilitate interpretation of the dynamic height spectrum and identification of equatorial wave modes, the spectrum is decomposed into separate spectra associated with dynamic height fluctuations that are symmetric or antisymmetric about the equator. Many equatorial-wave meridional modes can be identified, for both the first and second baroclinic mode. Zonal-wavenumber/frequency spectra of the zonal and meridional wind stress components are also examined. The observed wind stress spectra are used with linear theory of forced equatorial waves to provide a tentative explanation for the zonal-wavenumber extent of the spectral peaks seen in dynamic height. Examination of the cross-equatorial symmetry properties of the wind stress suggests that virtually all of the large-scale equatorial inertia–gravity and mixed Rossby–gravity waves examined may be sensitive to both zonal and meridional wind stress.

## Abstract

In the 1970s and 1980s, there was considerable interest in near-equatorial variability at periods of days to weeks associated with oceanic equatorial inertia–gravity waves and mixed Rossby–gravity waves. At that time, the measurements available for studying these waves were much more limited than today: most of the available observations were from scattered island tide gauges and a handful of short mooring records. More than a decade of the extensive modern data record from the Tropical Atmosphere Ocean (TAO)/Triangle Trans-Ocean Buoy Network (TRITON) mooring array in the Pacific Ocean is used to reexamine the internal-wave climate in the equatorial Pacific, with a focus on interpretation of the zonal-wavenumber/frequency spectrum of surface dynamic height relative to 500 decibars at periods of 3–15 days and zonal wavelengths exceeding 30° of longitude. To facilitate interpretation of the dynamic height spectrum and identification of equatorial wave modes, the spectrum is decomposed into separate spectra associated with dynamic height fluctuations that are symmetric or antisymmetric about the equator. Many equatorial-wave meridional modes can be identified, for both the first and second baroclinic mode. Zonal-wavenumber/frequency spectra of the zonal and meridional wind stress components are also examined. The observed wind stress spectra are used with linear theory of forced equatorial waves to provide a tentative explanation for the zonal-wavenumber extent of the spectral peaks seen in dynamic height. Examination of the cross-equatorial symmetry properties of the wind stress suggests that virtually all of the large-scale equatorial inertia–gravity and mixed Rossby–gravity waves examined may be sensitive to both zonal and meridional wind stress.

## Abstract

Longuet-Higgins in 1964 first pointed out that the Rossby wave energy flux as defined by the pressure work is not the same as that defined by the group velocity. The two definitions provide answers that differ by a nondivergent vector. Longuet-Higgins suggested that the problem arose from ambiguity in the definition of energy flux, which only impacts the energy equation through its divergence. Numerous authors have addressed this issue from various perspectives, and we offer one more approach that we feel is more succinct than previous ones, both mathematically and conceptually. We follow the work described by Cai and Huang in 2013 in concluding that there is no need to invoke the ambiguity offered by Longuet-Higgins. By working directly from the shallow-water equations (as opposed to the more involved quasigeostrophic treatment of Cai and Huang), we provide a concise derivation of the nondivergent pressure work and demonstrate that the two energy flux definitions are equivalent when only the divergent part of the pressure work is considered. The difference vector comes from the nondivergent part of the *geostrophic* pressure work, and the familiar westward component of the Rossby wave group velocity comes from the divergent part of the geostrophic pressure work. In a broadband wave field, the expression for energy flux in terms of a single group velocity is no longer meaningful, but the expression for energy flux in terms of the divergent pressure work is still valid.

## Abstract

Longuet-Higgins in 1964 first pointed out that the Rossby wave energy flux as defined by the pressure work is not the same as that defined by the group velocity. The two definitions provide answers that differ by a nondivergent vector. Longuet-Higgins suggested that the problem arose from ambiguity in the definition of energy flux, which only impacts the energy equation through its divergence. Numerous authors have addressed this issue from various perspectives, and we offer one more approach that we feel is more succinct than previous ones, both mathematically and conceptually. We follow the work described by Cai and Huang in 2013 in concluding that there is no need to invoke the ambiguity offered by Longuet-Higgins. By working directly from the shallow-water equations (as opposed to the more involved quasigeostrophic treatment of Cai and Huang), we provide a concise derivation of the nondivergent pressure work and demonstrate that the two energy flux definitions are equivalent when only the divergent part of the pressure work is considered. The difference vector comes from the nondivergent part of the *geostrophic* pressure work, and the familiar westward component of the Rossby wave group velocity comes from the divergent part of the geostrophic pressure work. In a broadband wave field, the expression for energy flux in terms of a single group velocity is no longer meaningful, but the expression for energy flux in terms of the divergent pressure work is still valid.

## Abstract

A conceptually simple model is presented for predicting the amplitude and periodicity of eddies generated by a steady poleward outflow in a 1½-layer *β*-plane formulation. The prediction model is rooted in linear quasigeostrophic dynamics but is capable of predicting the amplitude of the *β* plume generated by outflows in the nonlinear range. Oscillations in the plume amplitude are seen to represent a near-zero group velocity response to an adjustment process that can be traced back to linear dynamics. When the plume-amplitude oscillations become large enough so that the coherent *β* plume is replaced by a robust eddy field, the eddy amplitude is still constrained by the plume-amplitude prediction model. The eddy periodicity remains close to that of the predictable, near-zero group-velocity linear oscillations.

Striking similarities between the patterns of variability in the model and observations south of Indonesia’s Lombok Strait suggest that the processes investigated in this study may play an important role in the generation of the observed eddy field of the Indo-Australian Basin.

## Abstract

A conceptually simple model is presented for predicting the amplitude and periodicity of eddies generated by a steady poleward outflow in a 1½-layer *β*-plane formulation. The prediction model is rooted in linear quasigeostrophic dynamics but is capable of predicting the amplitude of the *β* plume generated by outflows in the nonlinear range. Oscillations in the plume amplitude are seen to represent a near-zero group velocity response to an adjustment process that can be traced back to linear dynamics. When the plume-amplitude oscillations become large enough so that the coherent *β* plume is replaced by a robust eddy field, the eddy amplitude is still constrained by the plume-amplitude prediction model. The eddy periodicity remains close to that of the predictable, near-zero group-velocity linear oscillations.

Striking similarities between the patterns of variability in the model and observations south of Indonesia’s Lombok Strait suggest that the processes investigated in this study may play an important role in the generation of the observed eddy field of the Indo-Australian Basin.

## Abstract

The response of a zonal channel to a uniform, switched-on but subsequently steady poleward outflow is presented. An eastward coastal current with a Kelvin wave’s cross-shore structure is found to be generated instantly upon initiation of the outflow. The current is essentially in geostrophic balance everywhere except for the vicinity of the outflow channel mouth, where the streamlines must cross planetary vorticity contours to feed the current. The adjustment of this region generates a plume that propagates westward at Rossby wave speeds. The cross-shore structure of the plume varies with longitude, and at any given longitude it evolves with time. The authors show that the plume evolution can be understood both conceptually and quantitatively as the westward propagation of the Kelvin current’s meridional spectrum, with each spectral element propagating at its own Rossby wave group velocity.

## Abstract

The response of a zonal channel to a uniform, switched-on but subsequently steady poleward outflow is presented. An eastward coastal current with a Kelvin wave’s cross-shore structure is found to be generated instantly upon initiation of the outflow. The current is essentially in geostrophic balance everywhere except for the vicinity of the outflow channel mouth, where the streamlines must cross planetary vorticity contours to feed the current. The adjustment of this region generates a plume that propagates westward at Rossby wave speeds. The cross-shore structure of the plume varies with longitude, and at any given longitude it evolves with time. The authors show that the plume evolution can be understood both conceptually and quantitatively as the westward propagation of the Kelvin current’s meridional spectrum, with each spectral element propagating at its own Rossby wave group velocity.

## Abstract

Previously unaddressed aspects of how equatorial currents affect long Rossby wave phase speeds are investigated using solutions of the shallow-water equations linearized about quasi-realistic currents. Modification of the background potential vorticity (PV) gradient by curvature in the narrow equatorial currents is shown to play a role comparable to the Doppler shift emphasized by previous authors. The important variables are the meridional projections of mean-current features onto relevant aspects of the wave field. As previously shown, Doppler shifting of long Rossby waves is determined by the projection of the mean currents onto the wave’s squared zonal-velocity and pressure fields. PV-gradient modification matters only to the extent that it projects onto the wave field’s squared meridional velocity.

Because the zeros of an equatorial wave’s meridional velocity are staggered relative to those of the zonal velocity and pressure, and because the meridional scales of the equatorial currents are similar to those of the low-mode Rossby waves, different parts of the current system dominate the advective and PV-gradient modification effects on a single mode. Since the equatorial symmetry of classical equatorial waves alternates between symmetric and antisymmetric with increasing meridional mode number, the currents produce opposite effects on adjacent modes. Meridional mode 1 is slowed primarily by a combination of eastward advection by the Equatorial Undercurrent (EUC) and the PV-gradient decrease at the peaks of the South Equatorial Current (SEC). The mode-2 phase speed, in contrast, is increased primarily by a combination of westward advection by the SEC and the PV-gradient increase at the core of the EUC.

Perturbation solutions are carried to second order in *ϵ*, the Rossby number of the mean current, and it is shown that this is necessary to capture the full effect of quasi-realistic current systems, which are asymmetric about the equator. Equatorially symmetric components of the current system affect the phase speed at *O*(*ϵ*), but antisymmetric components of the currents and distortions of the wave structures by the currents do not influence the phase speed until *O*(*ϵ*
^{2}).

## Abstract

Previously unaddressed aspects of how equatorial currents affect long Rossby wave phase speeds are investigated using solutions of the shallow-water equations linearized about quasi-realistic currents. Modification of the background potential vorticity (PV) gradient by curvature in the narrow equatorial currents is shown to play a role comparable to the Doppler shift emphasized by previous authors. The important variables are the meridional projections of mean-current features onto relevant aspects of the wave field. As previously shown, Doppler shifting of long Rossby waves is determined by the projection of the mean currents onto the wave’s squared zonal-velocity and pressure fields. PV-gradient modification matters only to the extent that it projects onto the wave field’s squared meridional velocity.

Because the zeros of an equatorial wave’s meridional velocity are staggered relative to those of the zonal velocity and pressure, and because the meridional scales of the equatorial currents are similar to those of the low-mode Rossby waves, different parts of the current system dominate the advective and PV-gradient modification effects on a single mode. Since the equatorial symmetry of classical equatorial waves alternates between symmetric and antisymmetric with increasing meridional mode number, the currents produce opposite effects on adjacent modes. Meridional mode 1 is slowed primarily by a combination of eastward advection by the Equatorial Undercurrent (EUC) and the PV-gradient decrease at the peaks of the South Equatorial Current (SEC). The mode-2 phase speed, in contrast, is increased primarily by a combination of westward advection by the SEC and the PV-gradient increase at the core of the EUC.

Perturbation solutions are carried to second order in *ϵ*, the Rossby number of the mean current, and it is shown that this is necessary to capture the full effect of quasi-realistic current systems, which are asymmetric about the equator. Equatorially symmetric components of the current system affect the phase speed at *O*(*ϵ*), but antisymmetric components of the currents and distortions of the wave structures by the currents do not influence the phase speed until *O*(*ϵ*
^{2}).