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The control of gravity waves
in data assimilation
27 April 1999
By Adrian
Simmons
European Centre for Medium-Range Weather
Forecasts
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APPENDIX B . The Lamb Wave
The vertical structure and equivalent depth of the Lamb wave can be specified analytically. A temperature profile of the form:
|
(B.9) |
enables the term
in (17) and (18)
to be written:
|
(B.10) |
With
also varying in the vertical as 
, the temperature and surface-pressure equations, (19) and (20), can both be satisfied only if 
. Thus, if we write:
|
(B.11) |
|
(B.12) |
|
(B.13) |
and
|
(B.14) |
equations (17) to (20) reduce to the shallow water equations:
|
(B.15) |
|
(B.16) |
|
(B.17) |
The phase speed of the plane wave in the absence of rotation is thus
.Since
and 
, where 
is the specific heat at constant volume, the phase speed may
be written 
. This may be recognized as the speed
of sound in a gas of temperature 
.With height,
, as the vertical coordinate, the vertical
structure 
becomes 
. Wave energy density varies as 
. Horizontal winds and temperature thus increase exponentially with
increasing height, but wave energy density decreases away from the ground.
The vertical velocity vanishes identically. This may be seen by writing,
to first order in wave amplitude:
|
(B.18) |
The vertical velocity,
, is then given to this order by:
|
(B.19) |
and 
are given by:
|
(B.20) |
and
|
(B.21) |
Using (B.9) and (B.10), (B.21) becomes:
|
(B.22) |
and substituting (B.20) and (B.22) into (B.19):
|
(B.23) |
The right-hand side of (B.23) is equal to zero by virtue of equation (19).
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