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B.a.iii. Diagnostic equation of pressure function
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Before making the finite difference equation,
(A.8)
are transformed as follows.
(definition of  is shown in appendix B.a.i ).
This equation is solved by using the dimension reduction method.
The finite difference form of the pressure equation can be written in
matrix form as follows.
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(B.19) |
where  are matrixes whose elements are finite
difference form of following terms.
 and
 are eigevalue and eigenvector of
 respectively.
By using the eigenvalue
matrix  and the eigenvector matrix  of
,
 .
Expanding  ,
(B.19) can be rewritten as follows.
Therefore,
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(B.20) |
The elements of matrix
which is required to derive eigenvalue
and eigenvector
are evaluated by finite form of
The second and forth order centered schemes are used because
the space differencing in the continuity equation is evaluated
by the forth order centered scheme while that in the pressure gradient
term is evaluated by the second order centered scheme.
Therefore is quintdiagonal matrix.
The element  is represented as follows.
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(B.21) |
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(B.22) |
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(B.23) |
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(B.24) |
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(B.25) |
The boundary conditions are  at the lower and upper boundary.
The horizontal dependent terms are expanding by using some eigenfunction.
In this model, we use Fourier series.
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(B.29) |
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