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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 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.

## 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.

## Abstract

The equatorial soliton studies of Boyd are extended to include the effects of continuous vertical stratification. We use vertical profiles of density measured in the equatorial Pacific Ocean and an idealized profile.

The wavenumber intervals of existence/nonexistence of wavepacket solitons are similar to Boyd for the Rossby mode, less similar for the eastward propagating gravity mode, and least similar for the westward gravity mode. Competing resonances and vertical diffusion are discussed.

The observed stratification for the equatorial Pacific has a significant effect on solitons. Some of the most dramatic effects are for the KdV solitons of Boyd. We propose that the longitudinal variation of stratification may result in the existence of the antisymmetric (with respect to the equator) Rossby solitons in only the western Pacific Ocean. The nonlinear coefficients of the symmetric KdV Rossby soliton decrease monotonically eastward by approximately 30% across the Pacific Ocean.

## Abstract

The equatorial soliton studies of Boyd are extended to include the effects of continuous vertical stratification. We use vertical profiles of density measured in the equatorial Pacific Ocean and an idealized profile.

The wavenumber intervals of existence/nonexistence of wavepacket solitons are similar to Boyd for the Rossby mode, less similar for the eastward propagating gravity mode, and least similar for the westward gravity mode. Competing resonances and vertical diffusion are discussed.

The observed stratification for the equatorial Pacific has a significant effect on solitons. Some of the most dramatic effects are for the KdV solitons of Boyd. We propose that the longitudinal variation of stratification may result in the existence of the antisymmetric (with respect to the equator) Rossby solitons in only the western Pacific Ocean. The nonlinear coefficients of the symmetric KdV Rossby soliton decrease monotonically eastward by approximately 30% across the Pacific Ocean.