A nonlinear Schrцdinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously
vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function
of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is
shown that the modulational instability criteria for the weakly vortical waves and potential Stokes waves on
deep water coincide. The effect of vorticity manifests itself in a shift of the wavenumber of high-frequency filling.
A special case of Gerstner waves with a zero coefficient at the nonlinear term in the NSE is noted.
The influence of nonlinear interaction of oppositely directed nonlinear waves in a shallow basin is studied theoretically and numerically within the nonlinear theory of shallow water. It is shown that this interaction leads to a change in the phase of propagation of the main wave, which is forced to propagate along the flow induced by the oncoming wave. The estimates of the undisturbed wave height at the time of interaction agree with the theoretical predictions. The phase shift during the interaction of undisturbed waves is sufficiently small, but becomes noticeable in the case of the propagation of breaking waves.
The nonlinear Schrödinger (NLS) equation describing the propagation of inhomogeneous vertical wave packets in an infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulation instability criteria of the considered weakly vortical waves and potential Stokes waves on deep water coincide. The effect of vorticity manifests itself in the shift of the wave number. A special case of Gerstner waves is noted, for which the coefficient of the nonlinear term in the NSE is zero.
Modifications of integral bubble and jet models including the pressure force are proposed. Exact solutions are found for the modified model of a stationary convective jet from a point source of buoyancy and momentum. The exact solutions are compared against analytical solutions of the integral models for a stationary jet that are based on the approximation of the vertical boundary layer. It is found that the modified integral models of convective jets retain the power-law dependences on the altitude for the vertical velocity and buoyancy obtained in classical models. For a buoyant jet in a neutrally stratified atmosphere, the inclusion of the pressure force increases the amplitude of buoyancy and decreases the amplitude of vertical velocity. The total amplitude change is about 10%. It is shown that in this model there is a dynamic invariant expressing the law of a uniform distribution of the potential and kinetic energy along the jet axis. For a spontaneous jet rising in an unstably stratified atmosphere, the inclusion of the pressure force retains the amplitude of buoyancy and increases the amplitude of vertical velocity by about 15%. It is shown that in the model of a spontaneous jet there is a dynamic invariant expressing the law of a uniform distribution of the available potential and kinetic energy along the jet axis. The results are of interest for the problems of anthropogenic pollution diffusion in the air and water environments and the formulation of models for statistical and stochastic ensembles of thermals in a mass-flux parameterization of turbulent moments.
The Monin–Obukhov similarity theory for the convective surface layer provides two limiting cases: a dynamic limit and the free-convection limit. The dynamic limit of the theory of the convective surface layer is defended as a flow with a logarithmic profile of wind and zero buoyancy flux at the underlying surface. The free-convection limit of the theory of the surface layer defended as a flow with zero wind velocity and a positive buoyancy flux at the underlying surface. The limits of the generalize Monin–Obukhov theory are able to describe the higher order turbulent moments. In this article, it is assumed that the convective surface layer consists of two sublayers: the lower one is a dynamic sublayer and the upper one is a forced convective sublayer. The turbulent moments of the both sublayers can be defined independently. Linear approximations for the turbulent moments of the vertical velocity and the potential temperature variance are proposed. The first-order expansion terms of them correspond to the free-convection limit of the Monin–Obukhov theory under no wind conditions. The second-order expansion allows describing the profiles of the turbulent moments under convectives conditions with a moderate wind. The comparison between the proposed approximations and the field data shows the correctness of the linear approximation within the forced convection range.
We study the runup of long solitary waves of different polarities on a beach in the case of composite bottom topography: a plane sloping beach transforms into a region of constant depth. We confirm that nonlinear wave deformation of positive polarity (wave crest) resulting in an increase in the wave steepness leads to a significant increase in the runup height. It is shown that nonlinear effects are most strongly pronounced for the runup of a wave with negative polarity (wave trough). In the latter case, the runup height of such waves increases with their steepness and can exceed the amplitude of the incident wave.
⎯Internal gravity wave (IGW) data obtained during the passage of atmospheric fronts over the Moscow region in June–July 2015 is analyzed. IGWs were recorded using a group of four microbarographs (developed at the Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) located at distances of 7 to 54 km between them. Regularities of variations in IGW parameters (spatial coherence, characteristic scales, propagation direction, horizontal propagation velocity, and amplitudes) before, during, and after the passage of an atmospheric front over the observation network, when the observation network finds itself inside the cyclone and outside the front, are studied. The results may be useful in studying the relationships between IGW effects in different physical fields at different atmospheric heights. It is shown that, within periods exceeding 30 min, IGWs are coherent between observation points horizontally spaced at distances of about 60 km (coherence coefficient is 0.6–0.9). It is also shown that there is coherence between wave fluctuations in atmospheric pressure and fluctuations in horizontal wind velocity within the height range 60–200 m. A joint analysis of both atmospheric pressure and horizontal wind fluctuations has revealed the presence of characteristic dominant periods, within which cross coherences between fluctuations in atmospheric pressure and wind velocity have local maxima. These periods are within approximate ranges of 20–29, 37–47, 62–72, and 100–110 min. The corresponding (to these dominant periods) phase propagation velocities of IGWs lie within an interval of 15–25 m/s, and the horizontal wavelengths vary from 52 to 99 km within periods of 35 to 110 min, respectively.