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B.a.iii. Diagnostic equation of the pressure function
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Before deriving the finite difference equation,
Equation (A.8)
is transformed as follows.
(The definition of
is described in Appendix B.a.i.).
This equation can be solved by using the dimension reduction
method. The finite difference form of the pressure equation can be
written in matrix form.
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(B.19) |
where
 are matrixes whose elements are in finite difference form for the following terms.
 and 
are the eigenvalue and eigenvector of
 .
By using the eigenvalue matrix
and the eigenvector matrix
of
,
 .
Expanding so
 ,
Equation (B.19) can be rewritten as follows.
Therefore,
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(B.20) |
The elements of matrix
, which are required to derive eigenvalue
and eigenvector
,
are evaluated in finite form as
Second- and fourth-order centered schemes are used since space
differencing in the continuity equation is evaluated by a fourth-order
centered scheme while space differencing in the pressure gradient term
is evaluated by a second-order centered scheme.
Therefore,
is a quint-diagonal matrix.
 , which is element of
, 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) |
Boundary conditions are  at the lower and upper boundary.
The horizontally-dependent terms are expanded by using an
eigenfunction. In this model, we use a Fourier series.
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(B.29) |
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