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5.2. A β-plane barotropic model on a two-dimensional double cyclic domain

The governing equations and boundary conditions of a β-plane barotropic model on a two-dimensional double cyclic domain are as follows:

When the domain is infinite, there exists a solitary vortex solution called the "modon" ([8], [9]). The solution is expressed as follows with plane-polar coordinates (r,θ) whose origin is at the center of the vortex:

where a and c are the radius and the propagation speed of the solitary vortex, respectively. Here, q= √(β/c), and k is determined by

which is equivalent to the dispersion relation.

The resulting program source code is given as plbaro-beta_abcn_test3.f90. The Adams-Bashforth scheme is utilized for the time integration of the governing equation, although the Crank-Nicolson scheme is adopted for the dissipation term. The main part of the time integration is as follows:

do it=1,nt
    ee_DVorDt = - ee_Jacobian_ee_ee(ee_StrFunc,ee_Vor)  &
                - Beta * ee_Dx_ee(ee_StrFunc)

    ee_Vor = ( (1.0D0 - ee_HVisc*delta_t/2)*ee_Vor    &
              + 3.0D0 * delta_t/2.0D0 * ee_DVorDt  &
              - 1.0D0 * delta_t/2.0D0 * ee_DVorDtB ) &
            /(1.0D0 + ee_HVisc*delta_t/2)

    ee_DVorDtB = ee_DVorDt
    ee_StrFunc = ee_LaplaInv_ee(ee_Vor)
    ...
enddo

The picture below shows the time development of an initially given modon solution with a=c=1. You can observe how a solitary vortex propagates without changing its shape.


[animation]

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