Applications of Lie Group Analysis in Geophysical Fluid Dynamics

“…This section provides a concise introduction to the calculus of Lie group analysis which represents a simplified version of the overview of basic concepts of Lie symmetry groups outlined in Ibragimov & Ibragimov, [8].…”

Section: Preliminariesmentioning

confidence: 99%

Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid

Ibragimov

Ibragimov²

2013

Math. Model. Nat. Phenom.

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Abstract. Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained by the Kelvin's conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.

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“…By introducing a stream function ψ for the disturbance velocities, i.e., 16) and taking the curl of Eqs. (1.13) and (1.14), one arrives at the vorticity equation in the limiting case R e → ∞ :…”

Section: Introductionmentioning

confidence: 99%

“…In terms of group theoretical modeling, the dynamics of the system in the question can often be described by means of the formal Lagrangian and the adjoint equations [22], [16]. The construction of conservation laws can be deduced from the Noether's theorem [27] that manifests a connection between symmetries and conservation laws for variational problems and thus provides a simpler procedure for construction of conservation laws for equations with known symmetries.…”

Section: Introductionmentioning

confidence: 99%

Conserved Vectors for a Model of Nonlinear Atmospheric Flows Around The Rotating Spherical Surface

Araslanov

Galiakberova

Ibragimov

et al. 2013

Math. Model. Nat. Phenom.

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Abstract. We derive the conserved vectors for the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the equations in question can be used to model zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. As a particualr example, the conserved densities are analyzed by visualizing the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.

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Applications of Lie Group Analysis in Geophysical Fluid Dynamics

Cited by 19 publications

References 0 publications

Laplace Transformation of Lie Class ω = 1 Overdetermined Systems

Laplace Transformation of Lie Class ω = 1 Overdetermined Systems

Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid

Conserved Vectors for a Model of Nonlinear Atmospheric Flows Around The Rotating Spherical Surface

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