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- Author or Editor: John P. Boyd x
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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
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/Iε) 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/Iε) 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
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
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
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 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
The slow manifold of an inviscid five-mode model introduced by Lorenz is investigated. When the influence of the gravity modes on the Rossby modes is neglected, the analytical solution given by Lorenz and Krishnamurthy is generalized. When gravity-Rossby coupling is included, direct numerical solutions are computed by solving a nonlinear boundary value problem. In all cases, the slow manifold has gravity mode oscillations that mimic free gravity waves and whose amplitude is proportional to the exponential of the reciprocal of the Rossby number ε.
Abstract
The slow manifold of an inviscid five-mode model introduced by Lorenz is investigated. When the influence of the gravity modes on the Rossby modes is neglected, the analytical solution given by Lorenz and Krishnamurthy is generalized. When gravity-Rossby coupling is included, direct numerical solutions are computed by solving a nonlinear boundary value problem. In all cases, the slow manifold has gravity mode oscillations that mimic free gravity waves and whose amplitude is proportional to the exponential of the reciprocal of the Rossby number ε.
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
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” , p. 775)
Abstract
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” , p. 775)
Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
