Geology Reference
In-Depth Information
3.4.3 Eigenvalue bounds.
For prescribed velocity profiles made up of simple straight lines and arcs,
complex eigenvalue relationships connecting c (that is, (
r
+ i
i
)/k) to g, , s, m
and the wavenumber k can be found, and the corresponding eigenfunctions can
be calculated as in the previous example. However, we choose to demonstrate
an alternative mathematical procedure: we obtain
upper and lower bounds
on
the eigenvalues c
without
restricting the form of the parallel flow U(y).
Let us introduce (y) = (c-U)F(y) and W = U - c, so that Equation 3.102
takes the form (W
2
F')' - k
2
W
2
F = 0. Multiplication by the
complex conjugate
F
*
and integration over (-H,0) leads to
W
2
{|F'|
2
+ k
2
|F|
2
} dy = F
*
(W
2
F')| = G|F
0
|
2
(3.111)
where all limits of integration at y = 0 and y = -H are henceforth omitted for
brevity. In Equation 3.111, we have set G = (G
r
+ i G
i
) = g + Tk
2
+ s -
mk
2
c
2
, taken Equations 3.110 and 3.103 in the form W
2
F' = GF and F(-H) = 0,
and written F
0
= F( 0 ) . Now , de f i ne Q = | F' |
2
+ k
2
|F|
2
> 0 so that real and
imaginary parts give
[(U - c
r
)
2
- c
i
2
] Q dy = G
r
|F
0
|
2
(3.112)
(U - c
r
) Q dy = G
i
|F
0
|
2
c
i
(3.113)
Expansion of Equation 3-112 using Equation 3-113 leads to
U
2
Q dy = (G
r
- c
r
G
i
/c
i
) |F
0
|
2
+ ( c
r
2
+ c
i
2
) Q dy (3.114)
assuming c
i
> 0. Next, let A and B denote the minimum and maximum of U(y)
in (-H,0). Then, the identity
0
(U-A)(U-B) Q dy
=
U
2
Q dy - (A+B) UQ dy + AB Q dy
(3.115)
on using Equations 3.113 and 3.114 leads to
0 [(c
r
2
+ c
i
2
) - (A+B)c
r
+ AB][mk
2
|F
0
|
2
/ + Qdy]
+[ g + s / + ( k
2
/ )(T - mAB)]|F
0
|
2
(3.116)
Note that each term in the bracketed quantity [ mk
2
|F
0
|
2
/ + Qdy] is positive
(again Q > 0 follows from an earlier definition). If g + s/ + ( k
2
/ )(T - mAB)
0, it follows that (c
r
2
+c
i
2
) - (A+B)c
r
+ AB 0, or
[c
r
- 1/2 (A+B)]
2
+ c
i
2
[1/2 (A+B)]
2
- AB = [1/2 (B-A)]
2
(3.117)
Thus, we have the general theorem:
the complex wave velocity c for any
unstable mode must lie inside the semicircle in the upper half of the complex c-
plane which has the range of U for diameter.
Quantitative results have been
obtained
without
knowing U(y), and
without
recourse to calculations for the
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