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- Author or Editor: Arthur Z. Loesch x
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Abstract
Resonant interactions between a finite-amplitude marginally unstable and two neutral baroclinic waves are investigated in a quasi-geostrophic two-layer model on the beta plane. The three wave-amplitudes are taken of comparable magnitude, a scaling assumption which permits the problem to be carried out only near the minimum critical shear required for instability. It is shown, using a numerical approach, that in the absence of dissipation the finite-amplitude state exhibits a long-period oscillation of the individual wave-amplitudes and of the total energy in the wave field. The neutral waves alter the phase shift between the unstable wave in the upper and lower layer and, therefore, affect the energy exchange with the mean flow. In the region F<10.5, where F represents the internal rotational Froude number, depending on the initial conditions, it is possible to transfer within an oscillation all of the energy available in the mean field to one of the neutral waves. In the region F>10.5, irrespective of the initial conditions, all of the available energy is transferred to both neutral waves.
Abstract
Resonant interactions between a finite-amplitude marginally unstable and two neutral baroclinic waves are investigated in a quasi-geostrophic two-layer model on the beta plane. The three wave-amplitudes are taken of comparable magnitude, a scaling assumption which permits the problem to be carried out only near the minimum critical shear required for instability. It is shown, using a numerical approach, that in the absence of dissipation the finite-amplitude state exhibits a long-period oscillation of the individual wave-amplitudes and of the total energy in the wave field. The neutral waves alter the phase shift between the unstable wave in the upper and lower layer and, therefore, affect the energy exchange with the mean flow. In the region F<10.5, where F represents the internal rotational Froude number, depending on the initial conditions, it is possible to transfer within an oscillation all of the energy available in the mean field to one of the neutral waves. In the region F>10.5, irrespective of the initial conditions, all of the available energy is transferred to both neutral waves.
Abstract
Spacial and temporal changes in the mean field, due to the presence of finite-amplitude marginally un-stable and two neutral baroclinic waves, which form a resonant triad, are investigated in a quasi-geostrophic, inviscid, two-layer model on the beta plane. It is shown that the triad introduces, in each layer, mean horizontal Reynolds stresses which are responsible for a continuous north-south redistribution of the mean momentum in a vertical column. In the region π2/2½<F<10.5, where F represents the internal rotational Froude number, the time change of the mean zonal velocity in the upper layer is affected mainly by the divergence of the mean horizontal Reynolds stresses. In the region Fgt;10.5, it is affected strongly by both the horizontal component of the mean meridional circulation and the divergence of the mean horizontal Reynolds stresses. In the lower layer, in both regions of F, the time change of the mean zonal velocity is affected mainly by the horizontal component of the mean meridional circulation.
Abstract
Spacial and temporal changes in the mean field, due to the presence of finite-amplitude marginally un-stable and two neutral baroclinic waves, which form a resonant triad, are investigated in a quasi-geostrophic, inviscid, two-layer model on the beta plane. It is shown that the triad introduces, in each layer, mean horizontal Reynolds stresses which are responsible for a continuous north-south redistribution of the mean momentum in a vertical column. In the region π2/2½<F<10.5, where F represents the internal rotational Froude number, the time change of the mean zonal velocity in the upper layer is affected mainly by the divergence of the mean horizontal Reynolds stresses. In the region Fgt;10.5, it is affected strongly by both the horizontal component of the mean meridional circulation and the divergence of the mean horizontal Reynolds stresses. In the lower layer, in both regions of F, the time change of the mean zonal velocity is affected mainly by the horizontal component of the mean meridional circulation.
Abstract
Finite-amplitude stability characteristics of Rossby wave flow are investigated in the context of the inviscid barotropic model on a beta plane. It is shown that a superimposed disturbance, unstable in the linear sense, grows as long as it lags the basic Rossby wave. However, when the disturbance becomes sufficiently large, it alters the phase and the amplitude of the Rossby wave flow. The phase correction of the Rossby wave is to the west and, in time, large enough to reverse the phase relation between the disturbance and the basic wave. At the time when the two become in phase, the growth of the disturbance is halted and subsequently, when the disturbance leads the Rossby wave, the disturbance slowly decays out. The basic Rossby wave equilibrates with a phase and amplitude which differ from their initial values.
Abstract
Finite-amplitude stability characteristics of Rossby wave flow are investigated in the context of the inviscid barotropic model on a beta plane. It is shown that a superimposed disturbance, unstable in the linear sense, grows as long as it lags the basic Rossby wave. However, when the disturbance becomes sufficiently large, it alters the phase and the amplitude of the Rossby wave flow. The phase correction of the Rossby wave is to the west and, in time, large enough to reverse the phase relation between the disturbance and the basic wave. At the time when the two become in phase, the growth of the disturbance is halted and subsequently, when the disturbance leads the Rossby wave, the disturbance slowly decays out. The basic Rossby wave equilibrates with a phase and amplitude which differ from their initial values.
Abstract
Spectral energetics associated with an N-wave system composed of a marginally unstable and N/1 neutral waves, interacting resonantly in triad configurations, are examined in the context of the two-layer baroclinic model. The unstable mode is found to dominate all allowable spectra at small initial energy levels. At larger initial energy levels this dominance persists only in the case of largest allowable spectra; otherwise, resonant interaction mechanism shifts the dominance to the neutral part of the spectrum. The shift occurs at lower initial energy levels and to a broader spectral range at higher values of internal rotational Froude number.
Abstract
Spectral energetics associated with an N-wave system composed of a marginally unstable and N/1 neutral waves, interacting resonantly in triad configurations, are examined in the context of the two-layer baroclinic model. The unstable mode is found to dominate all allowable spectra at small initial energy levels. At larger initial energy levels this dominance persists only in the case of largest allowable spectra; otherwise, resonant interaction mechanism shifts the dominance to the neutral part of the spectrum. The shift occurs at lower initial energy levels and to a broader spectral range at higher values of internal rotational Froude number.
Abstract
Dynamics of N ≥ 5 resonantly interacting baroclinic waves, one of which is marginally unstable and the remaining N − 1 are neutral, are investigated in a quasi-geostrophic inviscid two-layer model. Numerical solutions of equations governing the long time evolution of the wave amplitudes predominantly yield a separation between instability and interactions in period and characteristics of amplitude oscillations. The instability is felt on a longer time scale and via the marginal wave. For certain choices of initial conditions and/or neutral waves the separation breaks down in favor of one of the following. (i) resonant-like behavior of all amplitudes, (ii) a multi-time scale evolution of the amplitudes, or (iii) a very long-period, complex finestructure evolution of the individual amplitudes. The strongest energy exchange between the wave field and the mean field occurs whenever the separation takes place.
Abstract
Dynamics of N ≥ 5 resonantly interacting baroclinic waves, one of which is marginally unstable and the remaining N − 1 are neutral, are investigated in a quasi-geostrophic inviscid two-layer model. Numerical solutions of equations governing the long time evolution of the wave amplitudes predominantly yield a separation between instability and interactions in period and characteristics of amplitude oscillations. The instability is felt on a longer time scale and via the marginal wave. For certain choices of initial conditions and/or neutral waves the separation breaks down in favor of one of the following. (i) resonant-like behavior of all amplitudes, (ii) a multi-time scale evolution of the amplitudes, or (iii) a very long-period, complex finestructure evolution of the individual amplitudes. The strongest energy exchange between the wave field and the mean field occurs whenever the separation takes place.
Abstract
Nonlinear evolution of baroclinic waves in the presence of surface topography is investigated in an Eady model modified to include Ekman dissipation and sloping horizontal boundaries. The topographic form drag competes with baroclinicity for the control of amplitude evolution and propagation characteristics of the various disturbance modes. The effectiveness of the topography to phase lock and equilibrate a given mode versus that of baroclinicity to propagate and vacillate that mode depends on the topographic height, its zonal structure, and the level of supercriticality.
When topography is sinusoidal and of the same wavelength as the baroclinically most unstable mode, it induces a short-period amplitude modulation whose envelope represents the baroclinic evolution. The dynamics of this modulation is explored analytically. When the sinusoidal topography is of a longer zonal wavelength, a “mixed” wave state persists with amplitude dominance shifting between the topographic mode and the baroclinic mode, depending on the relative strength of form drag and baroclinicity. When topography is a superposition of the most unstable wavelength and a longer one, amplitude dominance and phase locking tend to shift to the mode associated with the “taller” mountain harmonic. When the harmonic structure of topography corresponds to that observed at 35°N, amplitude dominance is concentrated in the longest seven wavelengths. Model results are also compared, in appropriate parameter limits, with near-resonance asymptotic results and annulus experiments.
Abstract
Nonlinear evolution of baroclinic waves in the presence of surface topography is investigated in an Eady model modified to include Ekman dissipation and sloping horizontal boundaries. The topographic form drag competes with baroclinicity for the control of amplitude evolution and propagation characteristics of the various disturbance modes. The effectiveness of the topography to phase lock and equilibrate a given mode versus that of baroclinicity to propagate and vacillate that mode depends on the topographic height, its zonal structure, and the level of supercriticality.
When topography is sinusoidal and of the same wavelength as the baroclinically most unstable mode, it induces a short-period amplitude modulation whose envelope represents the baroclinic evolution. The dynamics of this modulation is explored analytically. When the sinusoidal topography is of a longer zonal wavelength, a “mixed” wave state persists with amplitude dominance shifting between the topographic mode and the baroclinic mode, depending on the relative strength of form drag and baroclinicity. When topography is a superposition of the most unstable wavelength and a longer one, amplitude dominance and phase locking tend to shift to the mode associated with the “taller” mountain harmonic. When the harmonic structure of topography corresponds to that observed at 35°N, amplitude dominance is concentrated in the longest seven wavelengths. Model results are also compared, in appropriate parameter limits, with near-resonance asymptotic results and annulus experiments.
Abstract
Nonlinear dynamics of unstable baroclinic disturbances are examined in the context of the Eady model modified by Ekman dissipation at the lower boundary while the upper boundary remains stress-free. Three approaches are used: the asymptotic approach which pivots about the constraints of strong bottom dissipation and weak supercriticality, the ad hoc approach which neglects wave-wave interactions by truncating the wave field to a single wave, and the spectral numerical approach.
The time evolution of the disturbance is generally characterized by a “single hump” pattern consisting of a growth stage to a maximum amplitude followed by a decay stage. During the decay stage, the spectral solution develops an amplitude vacillation which, for most parameter settings, becomes chaotic in nature and persists at a mean level substantially below the “hump” maximum and of the order of the initial amplitude. The exceptions are moderate or long waves in a strongly viscous fluid, for which the vacillation decays yielding a wave free final state, and short waves in a strongly viscous and weakly stratified fluid, for which both the initial “hump” and the subsequent vacillation are of the order of the initial amplitude. In the weak viscosity limit, a different kind of vacillation also appears during the early evolution stage of the disturbance. The asymptotic and the ad hoc solutions qualitatively capture the growth and the decay of the disturbance but fail to predict its vacillatory final state.
Abstract
Nonlinear dynamics of unstable baroclinic disturbances are examined in the context of the Eady model modified by Ekman dissipation at the lower boundary while the upper boundary remains stress-free. Three approaches are used: the asymptotic approach which pivots about the constraints of strong bottom dissipation and weak supercriticality, the ad hoc approach which neglects wave-wave interactions by truncating the wave field to a single wave, and the spectral numerical approach.
The time evolution of the disturbance is generally characterized by a “single hump” pattern consisting of a growth stage to a maximum amplitude followed by a decay stage. During the decay stage, the spectral solution develops an amplitude vacillation which, for most parameter settings, becomes chaotic in nature and persists at a mean level substantially below the “hump” maximum and of the order of the initial amplitude. The exceptions are moderate or long waves in a strongly viscous fluid, for which the vacillation decays yielding a wave free final state, and short waves in a strongly viscous and weakly stratified fluid, for which both the initial “hump” and the subsequent vacillation are of the order of the initial amplitude. In the weak viscosity limit, a different kind of vacillation also appears during the early evolution stage of the disturbance. The asymptotic and the ad hoc solutions qualitatively capture the growth and the decay of the disturbance but fail to predict its vacillatory final state.
Abstract
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Abstract
Finite-amplitude dynamics of baroclinic waves are examined in the presence of asymmetric Ekman pumping at the lower and upper boundaries. Both asymptotic and spectral numerical methods are employed. The resulting amplitude equations yield time evolutions that can lead to an eventual equilibration, a regular perpetual vacillation or a chaotic vacillation depending on the actual values of the supercriticality, the dissipation and the stratification parameters and the fundamental zonal wavenumber. Within the limits of strong bottom dissipation and weak supercriticality, the system always eventually equilibrates and the asymptotic results compare favorably with the numerical results. The vacillation is most likely to occur when the bottom dissipation is weak, supercriticality is strong or the viscous asymmetry is high. Vacillatory final states are possible for up to an order of magnitude larger bottom dissipation than predicted by the symmetric configuration, provided the top dissipation is small in comparison.
Abstract
Finite-amplitude dynamics of baroclinic waves are examined in the presence of asymmetric Ekman pumping at the lower and upper boundaries. Both asymptotic and spectral numerical methods are employed. The resulting amplitude equations yield time evolutions that can lead to an eventual equilibration, a regular perpetual vacillation or a chaotic vacillation depending on the actual values of the supercriticality, the dissipation and the stratification parameters and the fundamental zonal wavenumber. Within the limits of strong bottom dissipation and weak supercriticality, the system always eventually equilibrates and the asymptotic results compare favorably with the numerical results. The vacillation is most likely to occur when the bottom dissipation is weak, supercriticality is strong or the viscous asymmetry is high. Vacillatory final states are possible for up to an order of magnitude larger bottom dissipation than predicted by the symmetric configuration, provided the top dissipation is small in comparison.
Abstract
Nonlinear dynamics of equatorial waves, interacting resonantly in coupled triad configurations which form closed systems, are investigated in the context of the divergents β-plane model. Closure is attained by demanding that spatial structures of the modes obey the atmospheric constraints. At larger fluid depths the wave systems are relatively small and concentrated at the smaller wavenumbers; at small depths the systems are larger and spread more widely in the wavenumber domain. Strong energy transfers in a system are consistently associated with modes characterized by the maximum frequency in individual triads. The lower frequency modes are energetically less active, especially when their frequencies are much less than and amplitudes greater than those of the maximum frequency modes in the same triads.
Abstract
Nonlinear dynamics of equatorial waves, interacting resonantly in coupled triad configurations which form closed systems, are investigated in the context of the divergents β-plane model. Closure is attained by demanding that spatial structures of the modes obey the atmospheric constraints. At larger fluid depths the wave systems are relatively small and concentrated at the smaller wavenumbers; at small depths the systems are larger and spread more widely in the wavenumber domain. Strong energy transfers in a system are consistently associated with modes characterized by the maximum frequency in individual triads. The lower frequency modes are energetically less active, especially when their frequencies are much less than and amplitudes greater than those of the maximum frequency modes in the same triads.
