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

Correlation Between Three-dimensional Sine Sweep Dynamics Vibration Testing and Resonantly Interacting Internal Gravity Wave Field

“…As has been shown in Ibragimov & Ibragimov,[11], in two-dimensional case, the nonlinear model (2.1) -( 2.3) can be written in the form (A.1)-(A.3), which has a remarkable property to be self-adjoint. This property has been used in [11] to obtain a Lagrangian and construct new physically relevant conservation laws for internal waves in the ocean. As a particular example, it has been shown in [11] that using the dilation group of transformation and the variational derivative of the formal Lagrangian for Eqs.…”

“…This property has been used in [11] to obtain a Lagrangian and construct new physically relevant conservation laws for internal waves in the ocean. As a particular example, it has been shown in [11] that using the dilation group of transformation and the variational derivative of the formal Lagrangian for Eqs. (A.1) -(A.3) provides the mean perturbation energy for internal waves per unit volume in the form,…”

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

“…In our notation Q(s) = ∞ −∞ Q(η)e −isη dη is the Fourier transform of the initial condition for ψ (η, 0) which represents an amount of energy radiated along the internal planewave beam at the angle α in the positive or negative (depending on the sign of s) direction. As has been justified in [45] and illustrated in our previous work [23], the anisotropic property expressed (4.22) makes it possible to construct, via superposition of sinusoidal plane waves with wavenumbers inclined at the same angle α to the vertical, general plane-wave disturbances of frequency ω in the form of beams, that are uniform along the transverse direction ξ = x cos α + z sin α and have general profile along the invariant η. Furthermore, since Q (η) is an arbitrary wave amplitude of a superposition of plane waves, the direction of the beam propagation satisfying the initial condition ψ (η, 0) = Q (η) is determined uniquely.…”

“…It was shown in our previous work [23] that uni-directional internal wave beams, i.e., the nonlinear solution obeying the dispersion relation (4.23), can also be obtained as invariant solutions of nonlinear equations of motion. Namely, the invariant solution based on the dilation and translation transformations can be represented by uni-directional internal wave beams of the form 27) where the asterisk * denotes the complex conjugate and η = kx + mz = x sin α − z cos α is the invariant of the translational symmetries.…”

“…The analytic examples showing that the obtained generalized invariant solution (4.27) represents the uni-directional internal plane-wave beams satisfying the radiation condition (i.e., propagating in the interior of the ocean in a preferred direction away from the source) have been considered in our earlier work in Ref. [23].…”