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- Author or Editor: John Miles x
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Abstract
A shallow, rotating layer of fluid that supports Rossby waves is subjected to turbulent friction through an Ekman layer at the bottom and is driven by a wave that exerts a shear stress on the upper boundary and for which the phase approximate that of a Rossby wave. The steady-state response may be either 1) a single wave that is synchronous with the driving wave or 2) a resonant triad of waves, one of which is synchronous with the driving wave. The triadic solutions constitute a one-parameter family, of which not more than one member is stable. The evolution equations for the slowly varying complex amplitudes of the responding waves, the fixed points of which correspond to the solutions 1) and 2), are established. The stability of these fixed points, and hence the stability boundaries for 1) and 2), are determined. There are no Hopf bifurcations of the fixed-point solutions, and the evolution equations apparently do not admit periodic (limit cycle), multiply periodic or chaotic solutions.
Abstract
A shallow, rotating layer of fluid that supports Rossby waves is subjected to turbulent friction through an Ekman layer at the bottom and is driven by a wave that exerts a shear stress on the upper boundary and for which the phase approximate that of a Rossby wave. The steady-state response may be either 1) a single wave that is synchronous with the driving wave or 2) a resonant triad of waves, one of which is synchronous with the driving wave. The triadic solutions constitute a one-parameter family, of which not more than one member is stable. The evolution equations for the slowly varying complex amplitudes of the responding waves, the fixed points of which correspond to the solutions 1) and 2), are established. The stability of these fixed points, and hence the stability boundaries for 1) and 2), are determined. There are no Hopf bifurcations of the fixed-point solutions, and the evolution equations apparently do not admit periodic (limit cycle), multiply periodic or chaotic solutions.
Abstract
It is shown that the evolution equations for a triad of weakly damped, resonantly interacting waves are isomorphic to the corresponding equations for undamped waves (and therefore may be integrated in term of elliptic functions) if the damping coefficient is the same for each member of the triad. This condition is satisfied for topographic Rossby waves for which dissipation is through a turbulent Ekman layer and the wavelengths are small compared with the Rossby radius of deformation.
Abstract
It is shown that the evolution equations for a triad of weakly damped, resonantly interacting waves are isomorphic to the corresponding equations for undamped waves (and therefore may be integrated in term of elliptic functions) if the damping coefficient is the same for each member of the triad. This condition is satisfied for topographic Rossby waves for which dissipation is through a turbulent Ekman layer and the wavelengths are small compared with the Rossby radius of deformation.
Abstract
Janssen’s model for the effect of gustiness on the mean energy transfer from a turbulent wind to a surface wave is simplified through a Gauss–Hermite approximation to his probability integral. Two examples are considered.
Abstract
Janssen’s model for the effect of gustiness on the mean energy transfer from a turbulent wind to a surface wave is simplified through a Gauss–Hermite approximation to his probability integral. Two examples are considered.
Abstract
The conventional parameterization of diffusion in geophysical fluid dynamics, which replaces v∇2 u in the Navier-Stokes equations by [AH (δ x 2 + δ y 2) + AV δ z 2]u, where AH and AV are eddy viscosities, rests on the hypothesis that the Reynolds-stress tensor τij is a symmetric, transversely isotropic function of the mean velocity-gradient tensor. This implies that τij depends on both the mean rate of strain and the mean vorticity and that the kinetic energy of the turbulent fluctuations may be negative. The most general transversely isotropic relation that can be derived from the Boussinesq closure hypothesis, which separates the isotropic and deviatoric parts of. τij and excludes the vorticity, comprises three parameters. It is impossible to obtain the conventional parameterization through any choice of these parameters, but it is possible to obtain an equivalent parameterization if the hydrostatic and quasigeostrophic approximations are invoked.
Abstract
The conventional parameterization of diffusion in geophysical fluid dynamics, which replaces v∇2 u in the Navier-Stokes equations by [AH (δ x 2 + δ y 2) + AV δ z 2]u, where AH and AV are eddy viscosities, rests on the hypothesis that the Reynolds-stress tensor τij is a symmetric, transversely isotropic function of the mean velocity-gradient tensor. This implies that τij depends on both the mean rate of strain and the mean vorticity and that the kinetic energy of the turbulent fluctuations may be negative. The most general transversely isotropic relation that can be derived from the Boussinesq closure hypothesis, which separates the isotropic and deviatoric parts of. τij and excludes the vorticity, comprises three parameters. It is impossible to obtain the conventional parameterization through any choice of these parameters, but it is possible to obtain an equivalent parameterization if the hydrostatic and quasigeostrophic approximations are invoked.
Abstract
A zonal-wind configuration in which wind speed is proportional to pressure-altitude and stability is proportional to the square of the density is posed. A solution to the eigenvalue problem governing the stability of this configuration with respect to small disturbances is obtained in terms of hypergeometric functions. It is proved that one and only one (exponentially) unstable mode exists for each point in a wavelength, wind-shear plane.
Abstract
A zonal-wind configuration in which wind speed is proportional to pressure-altitude and stability is proportional to the square of the density is posed. A solution to the eigenvalue problem governing the stability of this configuration with respect to small disturbances is obtained in terms of hypergeometric functions. It is proved that one and only one (exponentially) unstable mode exists for each point in a wavelength, wind-shear plane.
Abstract
The baroclinic instability problem is reformulated to include diffusion of both heat and momentum through conduction and viscosity. A priori arguments suggest that the effects of heat conduction should dominate those of viscosity in the critical layer, where local wind speed equals wave speed and the adiabatic model for small disturbances is not uniformly valid. An asymptotic solution of the singular perturbation problem (based on the hypothesis that the Peclet and Reynolds numbers tend to infinity) supports this conjecture but also implies that the effects of diffusion on baroclinic instability are negligible insofar as the critical layer is within the geostrophic regime of the mean flow. This last condition is satisfied for the disturbances of principal meteorological interest.
Abstract
The baroclinic instability problem is reformulated to include diffusion of both heat and momentum through conduction and viscosity. A priori arguments suggest that the effects of heat conduction should dominate those of viscosity in the critical layer, where local wind speed equals wave speed and the adiabatic model for small disturbances is not uniformly valid. An asymptotic solution of the singular perturbation problem (based on the hypothesis that the Peclet and Reynolds numbers tend to infinity) supports this conjecture but also implies that the effects of diffusion on baroclinic instability are negligible insofar as the critical layer is within the geostrophic regime of the mean flow. This last condition is satisfied for the disturbances of principal meteorological interest.
Abstract
Abstract not available.
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Abstract
An approximate solution of the eigenvalue problem governing the stability of the zonal wind with respect to small disturbances of long wavelength is developed for profiles with strong, positive-definite vertical shear. It is found that certain disturbances, characterized by positive phase velocities, appear to be stable on the basis of a first approximation but are unstable in higher approximations. The results, together with the previously established instability for short wavelengths and/or weak vertical shear, support the conjecture that typical zonal-wind configurations are unstable with respect to small disturbances of almost all wavelengths at almost all windspeeds.
Abstract
An approximate solution of the eigenvalue problem governing the stability of the zonal wind with respect to small disturbances of long wavelength is developed for profiles with strong, positive-definite vertical shear. It is found that certain disturbances, characterized by positive phase velocities, appear to be stable on the basis of a first approximation but are unstable in higher approximations. The results, together with the previously established instability for short wavelengths and/or weak vertical shear, support the conjecture that typical zonal-wind configurations are unstable with respect to small disturbances of almost all wavelengths at almost all windspeeds.
Abstract
No abstract available.
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Abstract
A variational principle and an associated integral invariant are constructed for two-dimensional (non-divergent) waves of permanent form in a Rossby β-plane. A solitary-wave solution is obtained, and it is shown that the effects of cubic nonlinearity may be comparable with those of quadratic nonlinearity and may limit the amplitude of the wave.
Abstract
A variational principle and an associated integral invariant are constructed for two-dimensional (non-divergent) waves of permanent form in a Rossby β-plane. A solitary-wave solution is obtained, and it is shown that the effects of cubic nonlinearity may be comparable with those of quadratic nonlinearity and may limit the amplitude of the wave.
