|July 4, 2004, 16:43||
simple 3-d simulation of atmosphere
Hello. I want to simulate the atmosphere on a global scale.
The equations I use are as follows:
Horizontal wind equations:
du/dt = -udu/dx - v du/dy - wdu/dp +fv -g dz/dx +Fx
dv/dt = -udv/dx - v dv/dy - wdv/dp -fu -g/dz/dy +Fy
Equation of continuity(used to predict vertical motion)
du/dx + dv/dy + dw/dp = 0
Temperature Forecast equation:
dT/dt = -u dT/dx - vdT/dy - w(dT/dp - RT/cp*p) + H/cp
dz/dp = -RT/pg
the reference for these are http://meted.ucar.edu/nwp/pcu1/ic2/frameset.htm (select Model Type/Math.Basis for NWP and page 2)
I use a box model with 100x100x20 boxes, where i store the quantities of p, u,v,w,T,rho, and so on.
The model is initialized with the Logarithmic law p=p0(1-az/T0)^(Mg/Ra), so i get pressure and also density as a function of height, and the temperature is made latitude dependent.
The boundary conditions are periodical in the x-direction while in the y and z direction I use a layer which doesnt get updated, but is only initialized to some sensible value, so that the finite differences work in the boundary points also.
I use finite differences to discretize the partial derivatives, i.e du/dx = (u2-u1)/(x2-x1), where u2 is the rightmost box and u1 the leftmost box, and x2-x1 is the length of the boxes in x*2 -> |u1|u|u2|
Then I go through each box and update the quantities of u,v,w,T. But here's the first problem, how should i update rho(density) and p(pressure)? Moreover, my model seems to blow up after only a few tens of iterations; the horizontal wind velocities of v and u seems to blow up and diverge after some time.
I did convert from x,y,p, to x,y,z coordinate system using the hydrostatic equation and used layers of unequal spacing in vertical directions, to get more accuracy at lower altitudes.
So my question is, are these equations enough to get something working or is there something missing? The model is hydrostatic, and the equation of continuity is used to predict only vertical motion, that is, if winds blows into a particular square from different directions, then the excess air starts moving upwards, since the model seems to assume incompressibility also. Does this model take into account the conservation of mass? Otherwise the wind equations looks pretty ok to me, they are a form of Navier-Stokes equation for incompressible flows, with some vertical motion included.
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