Abstract
The research of gravity solitary waves movement is of great significance to the study of ocean and atmosphere. Baroclinic atmosphere is a complex atmosphere, and it is closer to the real atmosphere. Thus, the study of gravity waves in complex atmosphere motion is becoming increasingly essential. Deriving fractional partial differential equation models to describe various waves in the atmosphere and ocean can open up a new window for us to understand the fluid movement more deeply. Generally, the time fractional equations are obtained to reflect the nonlinear waves and few space-time fractional equations are involved. In this paper, using multiscale analysis and perturbation method, from the basic dynamic multivariable equations under the baroclinic atmosphere, the integer order mKdV equation is derived to describe the gravity solitary waves which occur in the baroclinic atmosphere. Next, employing the semi-inverse and variational method, we get a new model under the Riemann-Liouville derivative definition, i.e., space-time fractional mKdV (STFmKdV) equation. Furthermore, the symmetry analysis and the nonlinear self-adjointness of STFmKdV equation are carried out and the conservation laws are analyzed. Finally, adopting the exp(-Φ(ξ)) method, we obtain five different solutions of STFmKdV equation by considering the different cases of the parameters (η,σ). Particularly, we study the formation and evolution of gravity solitary waves by considering the fractional derivatives of nonlinear terms.
Highlights
With the intercross and penetration of different knowledge, Rossby solitary waves have been applied to many fields successfully, such as physical oceanography, atmospheric science, hydraulic engineering, and communication engineering
First we introduce the definition of Riemann-Liouville fractional derivatives and Caputo fractional derivatives[41, 63]
We discuss the nonlinear selfadjointness of the space-time fractional mKdV (STFmKdV) equation and get the conservation vectors of the equation
Summary
With the intercross and penetration of different knowledge, Rossby solitary waves have been applied to many fields successfully, such as physical oceanography, atmospheric science, hydraulic engineering, and communication engineering. In this paper, starting with the basic equations adopting the Bousinesq approximation [9] and under the baroclinic atmospheric environment, using multiscale analysis and turbulence method, we get a new model (mKdV) to describe the Rossby solitary waves. In the field of oceanography, the fractional partial differential equation can better describe the generation and evolution of solitary waves, which is more favorable for us to study the theory of fluctuation. This paper is organized as follows: In Section 2, using multiscale analysis and turbulence method, from the basic dynamic multivariable equations under the baroclinic atmosphere [59, 60], the integer order mKdV equation is derived.
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