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Research Papers: Fundamental Issues and Canonical Flows

# Energy Balance Associated With a Mixing Process at the Interface of a Two-Layer Longitudinal Atmospheric Model

[+] Author and Article Information
Ranis N. Ibragimov, Humberto Villasenor

Department of Mathematics,
University of Texas at Brownsville,
Brownsville, TX 78520

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 17, 2013; final manuscript received January 21, 2014; published online May 6, 2014. Assoc. Editor: Ye Zhou.

J. Fluids Eng 136(7), 071202 (May 06, 2014) (8 pages) Paper No: FE-13-1165; doi: 10.1115/1.4026857 History: Received March 17, 2013; Revised January 21, 2014

## Abstract

Several examples illustrating the energy balance associated with a mixing process at the interface of a planar dynamical model describing two-phase perfect fluid circulating around a circle with a sufficiently large radius within a central gravitational field are presented. The model is associated with the spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around the Earth. The fluid is supposed to be irrotational and pressure on a outer layer is constant. It is postulated that the total fluid depth is small compared to the radius of the circle and the gravity vector is directed to the center of the circle. Under these assumptions, this problem can be associated with a spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around equatorial plane. The model is the subject to the rigid lid approximation to the external boundary conditions for the outer fluid layer. One of the novelties in this work is the derivation of the nonlinear shallow water model by means of the average velocity. This introduction simplifies essentially further potential studies of mixing criteria associated with nonlinear mathematical models representing shallow water equations.

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## Figures

Fig. 1

Definition of variables for a two-dimensional free-boundary model for a nonstationary longitudinal motion for a two phase atmospheric circulation around the earth

Fig. 2

Schematic showing the boundary conditions associated with the trivial stationary solution represented by a whirlpool

Fig. 3

Mixing criteria Δ+PE/Δ-KE < 1 at different values of parameter Γ1 when μ = 0.8 and the other parameters were taken as in Table 1

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