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## Abstract

Via the method of multiple scales, it is shown that the time and space evolution of the envelope of wave packets of weakly nonlinear, strongly dispersive equatorial waves is governed by the Nonlinear Schrödinger equation. The diverse phenomena of this equation—envelope solitons, sideband instability, FPU recurrence and more—is briefly reviewed. Which of the alternatives occurs is determined largely by the relative signs of the coefficients of the dispersive and nonlinear terms in the Nonlinear Schrödinger equation which, for a given latitudinal mode, are functions of a single nondimensional parameter, the zonal wavenumber *k*. Gravity waves propagating toward the east form solitary waves and are subject to sideband instability only for large *k*. The mixed Rossby-gravity wave has solitons only in a range of intermediate *k*.

For waves with group velocities toward the west, the physics is much more complicated because these waves have an infinite number of second harmonic resonances and long wave/short wave resonances tucked into a finite interval of intermediate wavenumber. These two species of resonance form the topic of the two companion papers Boyd (1983a,b). One finds that for westward-propagating gravity waves, the resonances are very weak, and solitary waves occur more or less continuously within an intermediate range of wavenumber. For Rossby waves one can also state that solitons are forbidden for both large and small *k*, but the resonances completely dominate the intermediate range of wavenumbers so that the situation is very confused and complicated.

Together with earlier papers, this present work completes the description of the weakly nonlinear evolution of equatorial waves in a shallow water wave model.

## Abstract

Via the method of multiple scales, it is shown that the time and space evolution of the envelope of wave packets of weakly nonlinear, strongly dispersive equatorial waves is governed by the Nonlinear Schrödinger equation. The diverse phenomena of this equation—envelope solitons, sideband instability, FPU recurrence and more—is briefly reviewed. Which of the alternatives occurs is determined largely by the relative signs of the coefficients of the dispersive and nonlinear terms in the Nonlinear Schrödinger equation which, for a given latitudinal mode, are functions of a single nondimensional parameter, the zonal wavenumber *k*. Gravity waves propagating toward the east form solitary waves and are subject to sideband instability only for large *k*. The mixed Rossby-gravity wave has solitons only in a range of intermediate *k*.

For waves with group velocities toward the west, the physics is much more complicated because these waves have an infinite number of second harmonic resonances and long wave/short wave resonances tucked into a finite interval of intermediate wavenumber. These two species of resonance form the topic of the two companion papers Boyd (1983a,b). One finds that for westward-propagating gravity waves, the resonances are very weak, and solitary waves occur more or less continuously within an intermediate range of wavenumber. For Rossby waves one can also state that solitons are forbidden for both large and small *k*, but the resonances completely dominate the intermediate range of wavenumbers so that the situation is very confused and complicated.

Together with earlier papers, this present work completes the description of the weakly nonlinear evolution of equatorial waves in a shallow water wave model.

## Abstract

It is shown that resonant coupling between ultra long equatorial Rossby waves and packets of either short Rossby or short westward-traveling gravity waves is possible. Simple analytic formulas give the discrete value of the packet wave number *k*, for which the group velocity of the packet of meridional mode number *n* matches the group velocity of a nondispersive long Rossby wave of odd mode number *m*. The equations that describe the coupling are derived via the method of multiple scale and tables of the interaction coefficients are numerically calculated. For realistic parameter values, it appears this coupling could be important in the tropical ocean.

The principal physics of the coupled equations is threefold: 1) modulational or “side band” instability of plane waves, 2) instability of a short wave packet with respect to growing long waves if no long waves are initially present, and 3) solitary waves which consist of an envelope soliton of short waves of Nonlinear Schrödinger type traveling in conjunction with a unimontane soliton of Korteweg-deVries type as a single entity.

## Abstract

It is shown that resonant coupling between ultra long equatorial Rossby waves and packets of either short Rossby or short westward-traveling gravity waves is possible. Simple analytic formulas give the discrete value of the packet wave number *k*, for which the group velocity of the packet of meridional mode number *n* matches the group velocity of a nondispersive long Rossby wave of odd mode number *m*. The equations that describe the coupling are derived via the method of multiple scale and tables of the interaction coefficients are numerically calculated. For realistic parameter values, it appears this coupling could be important in the tropical ocean.

The principal physics of the coupled equations is threefold: 1) modulational or “side band” instability of plane waves, 2) instability of a short wave packet with respect to growing long waves if no long waves are initially present, and 3) solitary waves which consist of an envelope soliton of short waves of Nonlinear Schrödinger type traveling in conjunction with a unimontane soliton of Korteweg-deVries type as a single entity.

## Abstract

Simple, exact analytical conditions for second harmonic resonance between equatorial waves are derived. Such resonance can occur only between two Rossby waves or two westward travelling gravity waves. It is shown that regardless of whether the waves, are plane waves or localized wave packets, the physical consequence of the resonance is *instability* of the fundamental with a corresponding transfer of its energy to its second harmonic. The time scale of the instability is O(1/*I*ε) where ε is the amplitude of the fundamental and *I* the interaction coefficient, which is tabulated for various resonances. For reasonable parameter values, it appears that second harmonic resonance can be important in the tropical ocean. The *n* = 1 Rossby wave and all gravity waves propagating towards the east are immune to this instability, however, because they cannot satisfy the analytical conditions for second harmonic resonance. Besides these results for equatorial waves, a new approximate solution to the “resonant dyad” equations for an arbitrary initial wave packet is derived which is applicable to *any* second harmonic resonance, whether the waves are equatorial or not.

## Abstract

Simple, exact analytical conditions for second harmonic resonance between equatorial waves are derived. Such resonance can occur only between two Rossby waves or two westward travelling gravity waves. It is shown that regardless of whether the waves, are plane waves or localized wave packets, the physical consequence of the resonance is *instability* of the fundamental with a corresponding transfer of its energy to its second harmonic. The time scale of the instability is O(1/*I*ε) where ε is the amplitude of the fundamental and *I* the interaction coefficient, which is tabulated for various resonances. For reasonable parameter values, it appears that second harmonic resonance can be important in the tropical ocean. The *n* = 1 Rossby wave and all gravity waves propagating towards the east are immune to this instability, however, because they cannot satisfy the analytical conditions for second harmonic resonance. Besides these results for equatorial waves, a new approximate solution to the “resonant dyad” equations for an arbitrary initial wave packet is derived which is applicable to *any* second harmonic resonance, whether the waves are equatorial or not.

## Abstract

Boyd's previous work on equatorial Rossby solitary waves which derived the Korteweg-deVries equation using the method of multiple scales is here extended in several ways. First, the perturbation theory is carried, to the next highest order to (i) assess the accuracy and limitations of the zeroth-order theory and (ii) analytically explore solitons of moderate amplitude. Second, using the refined theory, it is shown that Rossby solitary waves will carry a region of closed recirculating fluid along with the wave as it propagates provided that the amplitude of the wave is greater than some (moderate) threshold. The presence of such closed “streaklines”, i.e., closed streamlines in a coordinate system *moving* with the wave, is an important property of modons in the theory of Flieri, McWilliams and others. The “closed-streakline” Rossby waves have many other properties in common with modons including (i) phase speed outside the linear range, (ii) two vortex centers of equal magnitude and opposite sign, (iii) vortex centers aligned due north-south, (iv) propagation east-west only and (v) a roughly circular shape for the outermost closed streakline, which bounds the region of recirculating fluid. Because of these similarities, it seems reasonable to use “equatorial modon” as a shorthand for “closed-streakline, moderate amplitude equatorial Rossby soliton,” but it should not be inferred that the relationship between midlatitude modons and equatorial solitary waves is fully understood or that all aspects of their behavior are qualitatively the same. Kindle's numerical experiments which showed that small amplitude Rossby solitons readily appear in El Niño simulations, suggest—but do not prove—that the very large El Niño of 1982 could have generated equatorial modons.

## Abstract

Boyd's previous work on equatorial Rossby solitary waves which derived the Korteweg-deVries equation using the method of multiple scales is here extended in several ways. First, the perturbation theory is carried, to the next highest order to (i) assess the accuracy and limitations of the zeroth-order theory and (ii) analytically explore solitons of moderate amplitude. Second, using the refined theory, it is shown that Rossby solitary waves will carry a region of closed recirculating fluid along with the wave as it propagates provided that the amplitude of the wave is greater than some (moderate) threshold. The presence of such closed “streaklines”, i.e., closed streamlines in a coordinate system *moving* with the wave, is an important property of modons in the theory of Flieri, McWilliams and others. The “closed-streakline” Rossby waves have many other properties in common with modons including (i) phase speed outside the linear range, (ii) two vortex centers of equal magnitude and opposite sign, (iii) vortex centers aligned due north-south, (iv) propagation east-west only and (v) a roughly circular shape for the outermost closed streakline, which bounds the region of recirculating fluid. Because of these similarities, it seems reasonable to use “equatorial modon” as a shorthand for “closed-streakline, moderate amplitude equatorial Rossby soliton,” but it should not be inferred that the relationship between midlatitude modons and equatorial solitary waves is fully understood or that all aspects of their behavior are qualitatively the same. Kindle's numerical experiments which showed that small amplitude Rossby solitons readily appear in El Niño simulations, suggest—but do not prove—that the very large El Niño of 1982 could have generated equatorial modons.

## Abstract

Using the method of strained coordinates, a uniformly valid approximation to the nonlinear equatorial Kelvin wave is derived. It is shown that nonlinear effects are negligible for the Kelvin waves associated with the Gulf of Guinea upwelling. The Kelvin waves involved in El Niño, however, are significantly distorted both in shape and speed. The leading edge is smoothed and expanded rather than steepened, but the trailing edge will form sharp fronts and eventually break.

## Abstract

Using the method of strained coordinates, a uniformly valid approximation to the nonlinear equatorial Kelvin wave is derived. It is shown that nonlinear effects are negligible for the Kelvin waves associated with the Gulf of Guinea upwelling. The Kelvin waves involved in El Niño, however, are significantly distorted both in shape and speed. The leading edge is smoothed and expanded rather than steepened, but the trailing edge will form sharp fronts and eventually break.

## Abstract

Using the method of multiple scales, I show that long, weakly nonlinear, equatorial Rossby waves are governed by either the Korteweg-deVries (KDV) equation (symmetric modes of odd mode number *n*) or the modified Korteweg-deVries (MKDV) equation. From the same localized initial conditions, the nonlinear and corresponding linearized waves evolve very differently. When nonlinear effects are neglected, the whole solution is an oscillatory wavetrain which decays algebraically in time so that the asymptotic solution as *t*→∝ is everywhere zero. The nonlinear solution consists of two parts: solitary waves plus an oscillatory tail. The solitary waves are horizontally localized disturbances in which nonlinearity and dispersion balance to create a wave of permanent form.

The solitary waves are important because 1) they have no linear counterpart and 2) they are the sole asymptotic solution as *t*→∝. The oscillatory wavetrain, which lags behind and is well-separated from the solitary waves for large time, dies out algebraically like its linear counterpart, but the leading edge decays faster, rather than slower, than the rest of the wavetrain. Graphs of explicit case studies, chosen to model impulsively excited equatorial Rossby waves propagating along the thermocline in the Pacific, illustrate these large differences between the linearized and nonlinear waves. The case studies suggest that Rossby solitary waves should be clearly identifiable in observations of the western Pacific.

## Abstract

Using the method of multiple scales, I show that long, weakly nonlinear, equatorial Rossby waves are governed by either the Korteweg-deVries (KDV) equation (symmetric modes of odd mode number *n*) or the modified Korteweg-deVries (MKDV) equation. From the same localized initial conditions, the nonlinear and corresponding linearized waves evolve very differently. When nonlinear effects are neglected, the whole solution is an oscillatory wavetrain which decays algebraically in time so that the asymptotic solution as *t*→∝ is everywhere zero. The nonlinear solution consists of two parts: solitary waves plus an oscillatory tail. The solitary waves are horizontally localized disturbances in which nonlinearity and dispersion balance to create a wave of permanent form.

The solitary waves are important because 1) they have no linear counterpart and 2) they are the sole asymptotic solution as *t*→∝. The oscillatory wavetrain, which lags behind and is well-separated from the solitary waves for large time, dies out algebraically like its linear counterpart, but the leading edge decays faster, rather than slower, than the rest of the wavetrain. Graphs of explicit case studies, chosen to model impulsively excited equatorial Rossby waves propagating along the thermocline in the Pacific, illustrate these large differences between the linearized and nonlinear waves. The case studies suggest that Rossby solitary waves should be clearly identifiable in observations of the western Pacific.

## Abstract

A series of high-resolution numerical experiments, augmented by theory, to further explore the dynamics of equatorial dipole vortices (Rossby solitary waves) is performed. When the amplitude is sufficiently large so that the vortices trap fluid internally, the solitary waves for a given phase speed are not unique. The potential vorticity–streak function (*q*–Ψ) relationship is everywhere linear for one branch, but highly nonlinear in the recirculation region for the second branch. Westward-traveling vortex pairs are highly unstable on the midlatitude beta plane, but the equatorial wave guide stabilizes vortex pairs that straddle the equator, even when given a strong initial tilt. As discovered by Williams and Wilson and explained theoretically by Boyd, the author confirms that higher latitudinal mode solitary waves are weakly nonlocal through radiation of sinusoidal Rossby waves of lower latitudinal mode number. The amplitude and wavelength of the radiation are in good agreement with nonlocal soliton theory.

It is sometimes said that the great discovery of the nineteenth century was that the equations of nature were linear, and the great discovery of the twentieth century is that they are not.—Thomas Körner,

Fourier Analysis(1988, p. 99)

## Abstract

A series of high-resolution numerical experiments, augmented by theory, to further explore the dynamics of equatorial dipole vortices (Rossby solitary waves) is performed. When the amplitude is sufficiently large so that the vortices trap fluid internally, the solitary waves for a given phase speed are not unique. The potential vorticity–streak function (*q*–Ψ) relationship is everywhere linear for one branch, but highly nonlinear in the recirculation region for the second branch. Westward-traveling vortex pairs are highly unstable on the midlatitude beta plane, but the equatorial wave guide stabilizes vortex pairs that straddle the equator, even when given a strong initial tilt. As discovered by Williams and Wilson and explained theoretically by Boyd, the author confirms that higher latitudinal mode solitary waves are weakly nonlocal through radiation of sinusoidal Rossby waves of lower latitudinal mode number. The amplitude and wavelength of the radiation are in good agreement with nonlocal soliton theory.

It is sometimes said that the great discovery of the nineteenth century was that the equations of nature were linear, and the great discovery of the twentieth century is that they are not.—Thomas Körner,

Fourier Analysis(1988, p. 99)

## Abstract

The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber *s* (always an integer) and Lamb’s parameter *ϵ*. An asymptotic approximation valid in the limit *s*
^{2} + *ϵ*
*ϵ* → ∞ for fixed *s*. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (*ϵ* + *s*
^{2})^{−1/4} rather than *ϵ*
^{−1/4} as in the classical equatorial beta-plane approximation.

## Abstract

The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber *s* (always an integer) and Lamb’s parameter *ϵ*. An asymptotic approximation valid in the limit *s*
^{2} + *ϵ*
*ϵ* → ∞ for fixed *s*. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (*ϵ* + *s*
^{2})^{−1/4} rather than *ϵ*
^{−1/4} as in the classical equatorial beta-plane approximation.

## Abstract

The effects of divergence on low-frequency Rossby wave propagation are examined by using the two-dimensional Wentzel–Kramers–Brillouin (WKB) method and ray tracing in the framework of a linear barotropic dynamic system. The WKB analysis shows that the divergent wind decreases Rossby wave frequency (for wave propagation northward in the Northern Hemisphere). Ray tracing shows that the divergent wind increases the zonal group velocity and thus accelerates the zonal propagation of Rossby waves. It also appears that divergence tends to feed energy into relatively high wavenumber waves, so that these waves can propagate farther downstream. The present theory also provides an estimate of a phase angle between the vorticity and divergence centers. In a fully developed Rossby wave, vorticity and divergence display a *π*/2 phase difference, which is consistent with the observed upper-level structure of a mature extratropical cyclone. It is shown that these theoretical results compare well with observations.

## Abstract

The effects of divergence on low-frequency Rossby wave propagation are examined by using the two-dimensional Wentzel–Kramers–Brillouin (WKB) method and ray tracing in the framework of a linear barotropic dynamic system. The WKB analysis shows that the divergent wind decreases Rossby wave frequency (for wave propagation northward in the Northern Hemisphere). Ray tracing shows that the divergent wind increases the zonal group velocity and thus accelerates the zonal propagation of Rossby waves. It also appears that divergence tends to feed energy into relatively high wavenumber waves, so that these waves can propagate farther downstream. The present theory also provides an estimate of a phase angle between the vorticity and divergence centers. In a fully developed Rossby wave, vorticity and divergence display a *π*/2 phase difference, which is consistent with the observed upper-level structure of a mature extratropical cyclone. It is shown that these theoretical results compare well with observations.

## Abstract

The evolution, both stable and unstable, of contrarotating vortex pairs (“modons”) perturbed by upper-surface and bottom Ekman pumping is investigated using a homogeneous model with a variable free upper-surface and bottom topography. The Ekman pumping considered here differs from the classical Ekman pumping in that the divergence-vorticity term in the vorticity equation, nonlinear and omitted in previous studies, is explicitly included. Under the influence of both nonlinear Ekman pumping and the beta term, eastward- and westward-moving modons behave very differently.

Eastward-moving modons are stable to the upper-surface perturbation but westward-moving modons are not. The latter move southwestward, triggering the tilt instability: the beta effect deepens the cyclones but weakens the anticyclone, and the vortex pair disperses into wave packets.

Eastward-moving modons are stable to bottom friction in the sense that they diminish in time gradually at a rate independent of the signs of the vortices. Westward-moving modons behave differently depending on the strength of bottom friction. Cyclones decay faster than anticyclones, triggering the tilt instability in westward-moving modons, but only if the bottom friction is very weak. For sufficiently strong bottom friction, in contrast, modons decay monotonically: the cyclones still decay faster than anticyclones, but no wave packets formed before the modons completely dissipate.

Westward-moving modons are *always* unstable to topographic forcing. Eastward-moving modons have varying behavior controlled by the height and width of the topography. Below a critical height, determined by the width, modons survive the topographic interaction: their trajectory meanders but the two contrarotating vortices always remain bound together after escaping the topography. Above the critical height, modons disassociate: the two vortices separate and disperse into wave packets. When the width of the topography is comparable to modon width, there exists a stable window within the unstable region of the topographic height in which the modons also survive the topographic encounter.

## Abstract

The evolution, both stable and unstable, of contrarotating vortex pairs (“modons”) perturbed by upper-surface and bottom Ekman pumping is investigated using a homogeneous model with a variable free upper-surface and bottom topography. The Ekman pumping considered here differs from the classical Ekman pumping in that the divergence-vorticity term in the vorticity equation, nonlinear and omitted in previous studies, is explicitly included. Under the influence of both nonlinear Ekman pumping and the beta term, eastward- and westward-moving modons behave very differently.

Eastward-moving modons are stable to the upper-surface perturbation but westward-moving modons are not. The latter move southwestward, triggering the tilt instability: the beta effect deepens the cyclones but weakens the anticyclone, and the vortex pair disperses into wave packets.

Eastward-moving modons are stable to bottom friction in the sense that they diminish in time gradually at a rate independent of the signs of the vortices. Westward-moving modons behave differently depending on the strength of bottom friction. Cyclones decay faster than anticyclones, triggering the tilt instability in westward-moving modons, but only if the bottom friction is very weak. For sufficiently strong bottom friction, in contrast, modons decay monotonically: the cyclones still decay faster than anticyclones, but no wave packets formed before the modons completely dissipate.

Westward-moving modons are *always* unstable to topographic forcing. Eastward-moving modons have varying behavior controlled by the height and width of the topography. Below a critical height, determined by the width, modons survive the topographic interaction: their trajectory meanders but the two contrarotating vortices always remain bound together after escaping the topography. Above the critical height, modons disassociate: the two vortices separate and disperse into wave packets. When the width of the topography is comparable to modon width, there exists a stable window within the unstable region of the topographic height in which the modons also survive the topographic encounter.