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## 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

Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the −8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant *t* = *t*
_{
β
} when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for *t* > *t*
_{
β
}, the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers„ study of Blumen, we find that the energy spectrum is more accurately approximated by the −8/3 power of the wavenumber, rather than by the *k*
^{−2} energy spectrum of a discontinuity, for less than two hours after the time of breaking.

We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at *t* = *t*
_{
β
} because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for *t* > *t*
_{β}, the characteristic equation has two roots. We show by explicit solution that there are actually three.

## Abstract

Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the −8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant *t* = *t*
_{
β
} when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for *t* > *t*
_{
β
}, the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers„ study of Blumen, we find that the energy spectrum is more accurately approximated by the −8/3 power of the wavenumber, rather than by the *k*
^{−2} energy spectrum of a discontinuity, for less than two hours after the time of breaking.

We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at *t* = *t*
_{
β
} because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for *t* > *t*
_{β}, the characteristic equation has two roots. We show by explicit solution that there are actually three.

## 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.

## 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

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

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

Some equatorially trapped motions cannot be modeled by the equatorial beta-plane. Our proof is a counter-example: if the zonal wavenumber *m* is large, barotropic Rossby-Haurwitz waves decay with latitude outside a narrow band about the equator and can be approximated by Hermite functions. The rather subtle effects of spherical geometry which create thew barotropic equatorial waves have important implications for short zonal wavelength motion in the tropics.

## Abstract

Some equatorially trapped motions cannot be modeled by the equatorial beta-plane. Our proof is a counter-example: if the zonal wavenumber *m* is large, barotropic Rossby-Haurwitz waves decay with latitude outside a narrow band about the equator and can be approximated by Hermite functions. The rather subtle effects of spherical geometry which create thew barotropic equatorial waves have important implications for short zonal wavelength motion in the tropics.

## 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

By using the method of multiple scales in height and a variety of methods in latitude, analytic solutions for equatorial waves in combined vertical and horizontal shear are derived. In contrast to the formulation of Andrews and McIntyre (1976b), latitudinal shear is incorporated at lowest order in the vertical shear expansion, showing that it is unnecessary to carry the calculation to first explicit order in Ri^{−½}, where Ri is the Richardson number. The multiple-scales approximation implies that, with the exception of the overall amplitude factor and arbitrary overall constant phase factor, all properties of the wave at the height *z*=*z*
_{0} are determined by the local wind profile, *V*(*y*)=*U*(*y*,*z*
_{0}). In consequence, understanding waves in two-dimensional shear reduces to the much simpler problem of solving the one-dimensional eigenvalue equation in latitude which is derived by assuming that the mean wind is *V*(*y*), a function of latitude only. This is done using ordinary perturbation theory, a non-perturbative analytic procedure and the Hermite spectral method for various classes of waves. Once its solutions are known, the overall amplitude factor may be found by using the wave action equation as shown in the text. When the method of multiple scales is invalid, as appears true of the tropical ocean, it is shown that Hermite spectral methods in latitude are much more accurate (at least in the absence of coastal boundaries) than the finite-difference methods used in the past. The techniques discussed here are applied to several classes of observed atmospheric equatorial waves in Part II (Boyd, 1978a).

## Abstract

By using the method of multiple scales in height and a variety of methods in latitude, analytic solutions for equatorial waves in combined vertical and horizontal shear are derived. In contrast to the formulation of Andrews and McIntyre (1976b), latitudinal shear is incorporated at lowest order in the vertical shear expansion, showing that it is unnecessary to carry the calculation to first explicit order in Ri^{−½}, where Ri is the Richardson number. The multiple-scales approximation implies that, with the exception of the overall amplitude factor and arbitrary overall constant phase factor, all properties of the wave at the height *z*=*z*
_{0} are determined by the local wind profile, *V*(*y*)=*U*(*y*,*z*
_{0}). In consequence, understanding waves in two-dimensional shear reduces to the much simpler problem of solving the one-dimensional eigenvalue equation in latitude which is derived by assuming that the mean wind is *V*(*y*), a function of latitude only. This is done using ordinary perturbation theory, a non-perturbative analytic procedure and the Hermite spectral method for various classes of waves. Once its solutions are known, the overall amplitude factor may be found by using the wave action equation as shown in the text. When the method of multiple scales is invalid, as appears true of the tropical ocean, it is shown that Hermite spectral methods in latitude are much more accurate (at least in the absence of coastal boundaries) than the finite-difference methods used in the past. The techniques discussed here are applied to several classes of observed atmospheric equatorial waves in Part II (Boyd, 1978a).

## 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.