Oscillatory nature and dissipation of of the internal waves energy spectrum in the deep ocean

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

“…In the simplest case when explicit viscosity and diffusion terms are ignored and the Boussinesq approximation is employed, the governing equations of uniformly stratified incompressible fluid are (Ibragimov [13,14]) oũ ot…”

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

“…However, it follows directly from [42] that the Casimir invariants are related to the symmetries of the Lagrangian representation. For visualization purposes, Figure 1 illustrates the results of numerical simulations for energy distribution in the wavenumber space between 20000 resonantly interacting internal waves at latitude 20 o North ( [24]). The numerical model used in ( [24]) is based on the governing equations (4.24) -(4.26) and represents the resonant triad simulator designed to study the energy exchange between arbitrarily large number of coupled resonant waves with frequencies spanning the range of possible frequencies, i.e., between a maximum of the buoyancy frequency and a minimum of the inertial frequency (two limiting cases of internal wave propagation) and mode numbers between n min = 1 and n max = 50.…”

“…Panel (a) shows the resonant wave distribution in (ω, n) −space. Cayan points are used to indicate waves generated resonantly from the primary wave (Level 0 waves), blue and red points represent Level 1 and level 2 waves generated from all the Level 0 and level 1 waves (see [24] for the nomenclature). Panel (b) shows the corresponding initial energy distribution.…”

“…The interested reader is addressed to the reference [24] for more details on the resonant triad simulator.…”

Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov׳s chemical source term

“…Similar to our approach in [5], since for RDE, the radial dimension is typically small compared to the azimuthal and axial dimension, we shall neglect the radial dependence in our modeling, which allows the RDE to be unrolled into two dimensional flow. Another similar approach for describing the dynamics of weakly nonlinear waves for compressible inviscous flows rotating in an annular ring has been developed in [9] and [8] for stratified and homogeneous fluids (see [7]; [6]; [10] ).…”