Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids

Abstract

The nonlinear solutions of the two-dimensional Boussinesq equations describing internal waves in rotating stratified fluids were obtained as group invariant solutions. The latter nonlinear solutions correspond to the rotation transformation preserving the form of the original nonlinear equations of motion. It is shown that the obtained class of exact solutions can be associated with the spherically pulsating patterns observed in uniformly stratified fluids. It is also shown that the obtained rotationally symmetric solutions are bounded functions that can be visualized as spinning patterns in stratified fluids. It is also shown that the rotational transformation provides the energy conservation law together with other conservation laws for which the spinning phenomena is observed. The effects of nonlinearity and the Earth's rotation on such a phenomenon are also discussed.

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

“…The similarity to the 3D case, in which mechanical oscillators act as point sources rather than as line sources (and so the generated gravity wave disturbances have a conical shape, containing almost perfectly circular phase lines in planes of constant height) can be found in our earlier work [28], where such gravity waves were linked to spinning patterns in the atmosphere.…”

Nonlinear Analysis of Perturbed Rotating Whirlpools in the Ocean and Atmosphere

“…Internal waves are suspected to play an important role in the dynamics of the ocean, especially in affecting the large-scale general circulation model, and they are responsible for a large fraction of mixing and energy exchange in the deep ocean. The exact solutions of the essential equations describing the internal gravity waves are only obtained for special cases ( [2]; [12]; [14]; [13]; [16]; [18]; [24]). This paper is devoted to analytical and numerical study of the existence of small perturbations of a certain class of stationary exact solutions to the nonlinear governing equations in a cylindrical wave field.…”

“…Our model is considered in a rotating reference frame within a cylindrical basin with radius r 0 and of the depth H with z ∈ [0, H] . As has been discussed in [19], [24] and [20], this allows us to associate the model with Kelvin waves that are known to play an important role in meteorology [24], climate variability models [47], the general atmospheric circulation model [21]; [31]; [32] and weather prediction (see e.g., [4]; [44]). Particularly, the existence of the Kelvin wave relies on (i) a gravity and stable stratification N 2 > 0 for sustaining a gravitational oscillation, (ii) significant Coriolis acceleration, and (iii) the presence of vertical boundaries or the equator.…”

Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets

“…It is therefore potentially unstable, for it can release the stored potential energy by means of an instability that would cause the density surfaces to flatten out. In this case, vertical shear of the mean flow would decrease, and perturbations would gain kinetic energy (see also [16], [17], [18]).…”