Nonlinear Wave Fields Research Articles

Reversible and Irreversible Finestructure

Abstract Various statistics of temperature profiles are examined in an attempt to distinguish irreversible structures due to mixing from reversible distortions induced by internal wave straining. Even if all the low gradient regions were the result of mixing events, an analysis of the profiles shows that such events are rare and most often incomplete. An upper bound on the mixing effectiveness is obtained; it increases as the vertical scale decreases. Taking next the opposite view that internal wave straining is the sole process, an analytic model is developed to calculate the probability density function of temperature gradients. The model considers the straining by a weakly nonlinear Gaussian internal wave field of a linear temperature profile. The nonlinearity of the field is essential to account for the skewness of the probability distributions. Comparisons with data are quite satisfactory at scales larger than ∼2 m, less so at smaller scales. We conclude that nonlinear effects are important; at scale...

Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves

We have derived an equation governing the evolution of a random field of nonlinear, deep-water, gravity waves by extending the approach used by Zakharov [1] for describing the deterministic system. This equation accounts for both the effects of inhomogeneity and the energy transfer mechanism associated with the homogeneous spectrum. The narrow-band limit of this equation is used to study the stability of a random wavetrain to two-dimensional deterministic perturbations. The effect of randomness is found to reduce the growth rate and the extent of the instability.

Development of generalized nonlinear behavior of elastic waveforms in an arbitrary inhomogeneous, lossy medium

A general method of developing the nonlinear wavefield for an arbitrary inhomogeneous medium incorporating energy loss mechanisms is presented. The equations of motion for an inhomogeneous medium with physical constants that are arbitrary functions of position, are modified by the incorporation of a generalized loss function, L. This acts as a sink function by introducing a dissipative force field in the force balance equations that accounts for losses in the medium. A general power series solution is developed and the nonlinear wavefield solution is presented that incorporates the effects of losses and higher order interactions in its functional form. The effect of loss on the dispersion relationship is seen as a frequency shift of the wave and loss effects introduce harmonics which lead to waveform distortion. The classical theory of attenuation is extended by this treatment. [Work supported by Navy Long-term Training Committee.

Diffusion of action in some nonlinear wave fields

The transfer of action among the modes in certain nonlinear random wave fields is described by a cubic Boltzmann equation. For two space scales, this equation reduces to a diffusion equation for action in k space.

Example of a completely integrable nonlinear wave field with nontrivial dynamics (lee model)

Example of a completely integrable nonlinear wave field with nontrivial dynamics (lee model)

The dispersion relation for a nonlinear random gravity wave field

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.

Energy transport in a nonlinear and inhomogeneous random gravity wave field

Certain tertiary resonant interactions of gravity waves which have been found previously can be obtained more easily by using a simple extension of Whitham's formalism. The contribution of these interactions to the total energy transfer in an inhomogeneous random field of gravity waves is calculated. It is found to be small for open-ocean waves, but to be of some importance for shallow-water waves, where topography or mean shear currents may produce strong inhomogeneities. The nonlinear splitting of the group velocity is found to be unimportant in wave fields with sufficiently smooth spectra.