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## 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

The previously known analytic solution for the unbounded plane Couette flow [i.e., a mean flow *U*(*y*)=*Sy*, *S* constant] is extended by 1) inclusion of the beta effect, and 2) more general initial conditions. It is shown that the beta effect and sidewall boundaries in latitude both have little or no effect on the physics of the waves. For large times, as already known, the continuous spectrum always decays away algebraically with time. It is shown, however, that before the final decay, the continuous spectrum may grow rapidly for a finite time interval if the latitudinal length scale of the initial perturbation is small in comparison to the zonal scale.

## Abstract

The previously known analytic solution for the unbounded plane Couette flow [i.e., a mean flow *U*(*y*)=*Sy*, *S* constant] is extended by 1) inclusion of the beta effect, and 2) more general initial conditions. It is shown that the beta effect and sidewall boundaries in latitude both have little or no effect on the physics of the waves. For large times, as already known, the continuous spectrum always decays away algebraically with time. It is shown, however, that before the final decay, the continuous spectrum may grow rapidly for a finite time interval if the latitudinal length scale of the initial perturbation is small in comparison to the zonal scale.

## Abstract

With the simplifying assumption that the mean zonal wind is a function of latitude only, numerical and analytical methods are applied to study the effects of critical latitudes (where the Doppler-shifted frequency is 0) on planetary waves. On the midlatitude beta-plane, it is shown that the modes divide into two limiting classes. The low-order, vertically propagating modes are confined to that side of the critical latitude where the mean winds are westerly as found by Dickinson (1968b). The high-order modes, although vertically trapped, are indifferent to the singularity and oscillate sinusoidally on both sides of the critical latitude as if it were not present. On the sphere, there is also a third class of low-order modes which are latitudinally trapped near the pole where the winds are easterly and also are unaffected by the critical latitude. Numerical studies show that it is the location of the critical latitude far more than the intensity or shape of the winds that controls the dynamics of the low-order, vertically propagating modes.

The most striking conclusion on the equatorial beta-plane is that sufficiently strong linear shear, although stable by conventional criterion, makes the Kelvin wave unstable. Together with the transparency of the high-order global modes, this shows that strong baroclinity may drastically alter the behavior of waves with critical latitudes, from that predicted by the barotropic or near-barotropic models so widely applied to critical latitudes in the past.

## Abstract

With the simplifying assumption that the mean zonal wind is a function of latitude only, numerical and analytical methods are applied to study the effects of critical latitudes (where the Doppler-shifted frequency is 0) on planetary waves. On the midlatitude beta-plane, it is shown that the modes divide into two limiting classes. The low-order, vertically propagating modes are confined to that side of the critical latitude where the mean winds are westerly as found by Dickinson (1968b). The high-order modes, although vertically trapped, are indifferent to the singularity and oscillate sinusoidally on both sides of the critical latitude as if it were not present. On the sphere, there is also a third class of low-order modes which are latitudinally trapped near the pole where the winds are easterly and also are unaffected by the critical latitude. Numerical studies show that it is the location of the critical latitude far more than the intensity or shape of the winds that controls the dynamics of the low-order, vertically propagating modes.

The most striking conclusion on the equatorial beta-plane is that sufficiently strong linear shear, although stable by conventional criterion, makes the Kelvin wave unstable. Together with the transparency of the high-order global modes, this shows that strong baroclinity may drastically alter the behavior of waves with critical latitudes, from that predicted by the barotropic or near-barotropic models so widely applied to critical latitudes in the past.

## Abstract

It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.

## Abstract

It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.

## Abstract

The analytical and numerical methodology of Boyd (1978) is applied to observed atmospheric waves. It is found that the structure and vertical wavelength of the stratospheric Kelvin wave of 15-day period and the tropospheric Kelvin wave of 40–50 day period are both negligibly affected by even the strongest shear. In contrast, the shear of the quasi-biennial oscillation can decrease the wavelength of the stratospheric *n*=0 mixed Rossby-gravity wave of 5-day period by 60% and produce changes of 50–100% in wave fluxes and velocities. The structure of synoptic-scale easterly waves (*n*=1 Rossby waves of 5-day period) is not drastically altered by shear, but the wavelength is tripled. This makes it unlikely that one can construct a quantitative wave-CISK theory of this mode without including latitudinal shear.

## Abstract

The analytical and numerical methodology of Boyd (1978) is applied to observed atmospheric waves. It is found that the structure and vertical wavelength of the stratospheric Kelvin wave of 15-day period and the tropospheric Kelvin wave of 40–50 day period are both negligibly affected by even the strongest shear. In contrast, the shear of the quasi-biennial oscillation can decrease the wavelength of the stratospheric *n*=0 mixed Rossby-gravity wave of 5-day period by 60% and produce changes of 50–100% in wave fluxes and velocities. The structure of synoptic-scale easterly waves (*n*=1 Rossby waves of 5-day period) is not drastically altered by shear, but the wavelength is tripled. This makes it unlikely that one can construct a quantitative wave-CISK theory of this mode without including latitudinal shear.

## Abstract

Using a simple separable model in which the mean wind *U*(*y*) is assumed to be a function of latitude only, those effects of latitudinal shear which do not depend on the vanishing of *U*(*y*) are examined for planetary waves in the middle atmosphere (the stratosphere and mesosphere).

First, it is shown that for nonsingular wind profiles the WKB method and ray tracing may be inaccurate for meridional shear. It is both physically and mathematically preferable to interpret the results of more complex models in terms of vertically propagating modes since the amplitude of the waves as a function of latitude is determined primarily by the modal structure rather than by variations of the mean wind or the refractive index.

Second, it is demonstrated that westerly planetary gravity waves, which are vertically trapped as shown by Charney and Drazin (1961), are also latitudinally trapped near the pole where the mean winds are easterly. In consequence, such waves, which form the quasi-stationary spectrum of the summer hemisphere, are unaffected by the critical latitude in the subtropics of the winter hemisphere. The physical implications of these and other findings are discussed.

## Abstract

Using a simple separable model in which the mean wind *U*(*y*) is assumed to be a function of latitude only, those effects of latitudinal shear which do not depend on the vanishing of *U*(*y*) are examined for planetary waves in the middle atmosphere (the stratosphere and mesosphere).

First, it is shown that for nonsingular wind profiles the WKB method and ray tracing may be inaccurate for meridional shear. It is both physically and mathematically preferable to interpret the results of more complex models in terms of vertically propagating modes since the amplitude of the waves as a function of latitude is determined primarily by the modal structure rather than by variations of the mean wind or the refractive index.

Second, it is demonstrated that westerly planetary gravity waves, which are vertically trapped as shown by Charney and Drazin (1961), are also latitudinally trapped near the pole where the mean winds are easterly. In consequence, such waves, which form the quasi-stationary spectrum of the summer hemisphere, are unaffected by the critical latitude in the subtropics of the winter hemisphere. The physical implications of these and other findings are discussed.

## 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

For linearized hydrostatic waves on a spherical earth with a zonal mean wind which is a function of latitude and pressure I derive, without further approximations, expressions for the vertical and meridional energy fluxes in terms of the meridional heat flux and the vertical and meridional fluxes of zonal momentum. Using these expressions, I prove that in the absence of critical surfaces, dissipation, thermal heating and nonharmonic time dependence, that the waves and mean flow do not interact: the wave Reynold's stresses are exactly balanced by a mean meridional circulation whose streamfunction is simply the meridional beat flux divided by the static stability. In the presence of dissipation, thermal heating or transience, 1 am able to express the net forcing of the mean blow by the waves as expressions which are explicitly proportional to the coefficients of dissipation and heating and to the imaginary part of the phase speed. My work significantly extends earlier theorems on the noninteraction of waves with the zonally averaged flow and on the interrelationships of wave fluxes proved by Eliassen and Palm, Charney and Drazin, and Holton because my theorems eliminate some important restrictive assumptions and include all these previous results as special cases.

## Abstract

For linearized hydrostatic waves on a spherical earth with a zonal mean wind which is a function of latitude and pressure I derive, without further approximations, expressions for the vertical and meridional energy fluxes in terms of the meridional heat flux and the vertical and meridional fluxes of zonal momentum. Using these expressions, I prove that in the absence of critical surfaces, dissipation, thermal heating and nonharmonic time dependence, that the waves and mean flow do not interact: the wave Reynold's stresses are exactly balanced by a mean meridional circulation whose streamfunction is simply the meridional beat flux divided by the static stability. In the presence of dissipation, thermal heating or transience, 1 am able to express the net forcing of the mean blow by the waves as expressions which are explicitly proportional to the coefficients of dissipation and heating and to the imaginary part of the phase speed. My work significantly extends earlier theorems on the noninteraction of waves with the zonally averaged flow and on the interrelationships of wave fluxes proved by Eliassen and Palm, Charney and Drazin, and Holton because my theorems eliminate some important restrictive assumptions and include all these previous results as special cases.