The Modified Quasi-geostrophic Barotropic Models Based on Unsteady Topography

water equations based on barotropic fluids. In the paper, to discuss the irregular topography with different magnitudes, especially considering the condition of the vast terrain, some modified quasi-geostrophic barotropic models were obtained. The unsteady terrain is more suitable to describe the motion of the fluid state of the earth because of the change of global climate and environment, so the modified models are more rational potential vorticity equations. If we do not consider the influence of topography and other factors, the models degenerate to the general quasi-geostrophic barotropic equations in the previous studies.


Introduction
In recent decades, many scholars have conducted extensive research on large-scale atmospheric and oceanic dynamics. Among them, the topographic effect has great influences on the dynamic mechanism of potential vorticity equations (Pedlosky, 1974;Collings, 1980;Pedlosky, 1980;Treguier, 1989). In the atmosphere, the topographic height plays an importance role in atmospheric cyclone and anticyclone changes. Luo (1990) pointed out the topographic effect was also an important factor of forming atmospheric blocking, and the global atmospheric circulation would even get affected by the topographic effect. Lu (1987) discussed the effects of topographic height and shaped on Rossby wave activation and the influences of topographic south-north and east-west slopes on waveform and energy propagation. Chen (1998) and Jiang (2000) derived the quasi-geostrophic potential vorticity equation with large-scale topography, friction, and heating under the barotropic model, and the large-scale effects of Qinghai-Tibet Plateau on atmosphere were discussed. In addition, the oceanic topography is very complex, such as The North West Shelf of Australia (Holloway, 1997), Portugal Shelf Sea Area (Sherwin, 2002), etc. The relationship between topography and ocean circulation was pointed out in the literature (Roslee et al., 2017b;Kamsani, 2017;La, 1990;Marshall, 1995;Alvarez, 1994;Sou, 1996). Cessi (1986) discussed the important role of topography in ocean circulation. Holloway (1992) introduced the interaction of eddies with seafloor topography and argued that ocean circulations would be significant interaction between turbulent vortices and topography rather than gravity wave drag. Then, general expressions for the eddy-topographic force, eddy viscosity, and stochastic backscatter, as well as a residual Jacobian term, are derived for barotropic flow over mean topography by Frederiksen (1999). All the above researches, Actually the topography height also changes with time in the earth fluid. Changes in topography can lead to tsunamis, floods and natural disasters (Abdullah, 2017;Elfithri, 2017;Rahim, 2017). Yang (2011Yang ( ,2012 and Song (2012Song ( , 2013 discussed topography changes over time discussed topography changes over time t, the influence of the nonlinear long wave amplitude and waveform. Da (2013) discussed the shallow water equation forms when underlying surface slowly changes with time and obtained the vorticity equation with an underlying surface. This model considered the actual circumstances that the topography changes with time-space (Erfen et al., 2017;Roslee, 2017a) . In this paper, we discuss different magnitude topography under spatial-temporal variable and obtain some modified models, which have important effects on the future discussion about the waveform changes of nonlinear long waves. This paper is organized as follows: In Section 2, starting from the rotating shallow water equation set, we give the bottom topography which is not smooth boundary and simplify the equation set with unsteady topography. Section 3 is given scale analysis and perturbation methods; then we obtain new equation set by topographic conditions with different magnitudes. Later we derive the equations with different orders and get some modified models in Section 4. Finally, make the relevant conclusions in Section 5.

Basic equations
Assuming the static equilibrium condition, the fluid can be regarded as barotropic, incompressible, frictionless state. The upper boundary height is ( , , ) h x y t , and , free surface pressure intensity is constant. The basic equation set can be written as (Pedlosky, 1987) Eqs. (5) and (6) indicate that the horizontal pressure gradient under the barotropic model can be expressed by the gradient of the free surface gravitational potential (Taharin & Roslee, 2017).
We assume that the preliminary horizontal velocity is independent of the z , Eqs. (1a) and (1b)  assuming that the free surface height H is constant when the fluid is static (Pedlosky, 1987).

Scale analysis, perturbation methods
Making dimensionless analysis on the Eqs.

Derive the barotropic models
Making classified discussion on the magnitude of λ.

λ ~ 1 magnitude
The scale of the ϕ B is consistent with the small amplitude function ϕ, most of the topography parameters following with this situation in the real world.   λ R is the Rossby radius of deformation, Eq. (25) is a modified model.

λ ~ 10 magnitude
Large-scale atmospheric motion of large topography is suitable for such conditions (L ~ 10 6 m, U ~ 10m/s, f 0 ~ 10 -4 s -1 ). For example, a case study of Tibetan Plateau topography, the height is 3 4 ×10 3 m approximately which is suit for the situation.   (Liu, 1991) Eq. (40) is a classics dynamics model used by the large-scale atmospheric and oceanic motions.

Conclusion
(a). Under the unsteady topography, some new modified models (25), (35) are derived. These models meet the general rule that the topography changes with time in reality. When the topography has nothing to do with the time, Eq. (35) degenerates into a dynamics model (36), Eq. (25) degenerates into Eq. (38) which is a quasi-geostrophic barotropic model under the spatial topography.
(b). The modified models under the topographic effect with different magnitudes are presented, we can see the pattern under the condition of large terrain, which is the improvement of the model. After the above-modified models are given, we will also derive the mathematical model for Rossby wave in the further study, and the further exploration of the large-scale factual influences of topography on atmosphere and ocean are required.