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# geoflow

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# geoflow   { GEOFLOW.PDE

In its simplest form, the nonlinear steady-state quasi-geostrophic equation

is the coupled set:

q  = eps*del2(psi) + y                    (1)

J(psi,q) = F(x,y) - k*del2(psi)                 (2)

where psi     is the stream function

q       is the absolute vorticity

F       is a specified forcing function

eps and k are specified parameters

J       is the Jacobian operator:

J(a,b) = dx(a)*dy(b) - dy(a)*dx(b)

The single boundary condition is the one on psi stating that the closed

boundary C of the 2D area should be streamline:

psi = 0 on C.

In this test, the term k*del2(psi) in (2) has been replaced by (k/eps)*(q-y),

and a smoothing diffusion term damp*del2(q) has been added.

Only the natural boundary condition is needed for Q.

}

title 'Quasi-Geostrophic Equation, square, eps=0.005'

variables

psi

q

definitions

kappa = .05

epsilon = 0.005

koe = kappa/epsilon

size = 1.0

f = -sin(pi*x)*sin(pi*y)

damp =  1.e-3*koe

initial values

psi = 0.

q   = y

equations

psi: epsilon*del2(psi) - q = -y

q:   dx(psi)*dy(q) - dy(psi)*dx(q) + koe*q - damp*del2(q) = koe*y + f

boundaries

region 1

start(0,0) value(psi)=0 natural(q)=0

line to (1,0) to (1,1) to (0,1) to close

monitors

contour(psi)

contour(q)

plots

contour(psi) as "Potential"

contour(q)   as "Vorticity"

surface(psi) as "Potential"

surface(q)   as "Vorticity"

vector(-dy(psi),dx(psi)) as "Flow"

end