where
01
f dz
and h is the height of the domain Q (h = 100 km for numerical experiments
in Chapter 6). If 01 = 250000 (see [1]), c = 2.852108 x 10-9.
Finally, using the transformation O(x, y, z) = j(z) + O(x, y, z), we have the
boundary condition: 4 = 0 on 0Q.
Since E = Vq and 0 = ( + 0, Equation (3.2) becomes
e AV)+ UA4'+ua +VJ =0,
at OZ
where A denotes the Laplacian operator defined by A = V V, and az denotes the
derivative of a with respect to z.
Then our model is the partial differential equation with the boundary condi-
tion as follows:
PDE: A+ 'AO + -_ + =0
at e e 9z e
BC: P = 0 on O
Domain: Q = {(x, y, z) : 0 < z < Dz, Ix < Dx, yl < Dy},
where D,, Dy, and D, are prescribed values. In our computation, we take D, =
100 km. For reference,
Qi = {(x,y, z) : lx< 50 km, |y| < 50 km, 0 < z < 100 km}. (3.5)
3.2 Discretized Equations
We demonstrate the discretization process for a uniform mesh in a rectangular
coordinate system. Consider the differential equation
A9- aO a- + (3.6)
At e E 95z e E