Mar 1: Coastal-Trapped Waves (3)
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Wind-forced long shelf waves
Assumptions:
·
linear,
frictionless, barotropic momentum
·
rigid-lid continuity, hence streamfunction solution
·
low
frequency, long waves
Scale analysis of the order
of magnitude of the acceleration, Coriolis and wind stress terms shows that
wind forcing in the along-shelf momentum equation is significant, but that the
across momentum balance is simply geostrophic.
The important dynamics are
thus described by:
(1)
for which we form a vorticity
equation
(2)
We seek a solution to this by
an expansion in the unforced barotropic shelf wave modes that are found by
solving (2) (for the case of zero forcing) by the method of separation of
variables of the form:
Substituting in to equation
(2) we obtain 
which can be rearranged to
show that 
and it follows that 
because the along-shelf wave propagation at speed cn
follows from 
.
The across-shelf modes
satisfy:
subject to boundary
conditions at the coast and offshore extent of the continental shelf.
The boundary condition u =
0 at the coast is effectively 
.
At the foot of the
continental shelf, x=L, beyond which h is constant, we have 
because in the
long-wave limit l tends to zero and the equation 
becomes 
so 
which can only be zero
for a finite solution in the semi-infinite deep ocean.
It is useful to determine the
orthogonality condition for these modes (which we know must exist because the
problem as posed is a member of the class of Sturm-Liouville problems which
have a set of eigenfunction solutions that form a orthonormal basis set).
We form the orthogonality
condition by multiplying the governing equation for one mode, 
, by another eigenfunction, 
, subtracting the reverse product, then integrating over the
domain and enlisting the boundary conditions.
which can be written:
Integrate with respect to x
across the shelf from 0 to L:
The first term is zero by
virtue of the boundary conditions. The second term will be zero if m = n
regardless of 
. But if 
then the integral itself must be zero. With an appropriate
choice of amplitude to normalize the , which is arbitrary, we have:
This defines the
orthonormality property.
This facilitates an analysis
of how the wind stress projects on to the free wave modes that form the
orthonormal basis set.
Substituting the summation
over modes
into (2) we get:
But 
, so
Multiply by, integrate from x = 0 to L, and use the
orthogonality relation:
where 
is the wind
coupling coefficients for mode m.
This shows that the Ekman
transport, 
, projects on to each mode generating a response that travels
along the coast, non-dispersively, at speed cm. The total
effect of the wind stress is the sum of all the individual mode responses.
We can solve this set of
first-order wave equations (FOWE) by the Method of Characteristics for
each mode m in turn.
Make a change of variables:
Then
which can be integrated with
respect to 
(holding 
constant) from 
to 
.
This says that at a fixed
value of , the solution depends on some “initial” condition at (some distance away
along the coast) and the action of the wind stress over the interval 
.
This is telling us that the
solution at a particular place and time is given by the F that
propagated into the domain at the origin of the integration plus the integrated
effect of the wind generating free waves at the coast.
Lines of constant are characteristics of
this simple first order wave equation.
In the case that the wind
stress is zero we have the very simple solution that
so we have the wave amplitude
from some prior time some distance upstream.
Which direction is
“upstream”, and how far?
Physically, we know that at
location yo at time to, the solution can
only depend on times in the past. Going back in time while holding constant we can only move
toward increasing y. Thus the information is propagating toward negative
y, i.e. with the coast on the right.
If a storm were limited in
spatial extent, but propagating at the same velocity cm as a
free wave mode, it could continually pump energy into mode m. If it were propagating at this speed but in
the wrong direction, its influence would be only transitory.
A storm that crosses the
coast over some limited time duration, will generate a train of CTWs that will
propagate away toward negative y. The relative projection onto each mode
depends on the wind-coupling coefficient bm. The full
solution is the summation over m of all the waves.
If friction were included in
the analysis the wave equation would be modified:
with amm being
the frictional decay constant and anm describing the
frictional coupling to other modes.
The essential features of the
coastal trapped wave phenomena described above still hold true when more
realistic bathymetry and stratification are included.
For realistic bathymetry the
term hx/h2 requires numerical solution of the
Sturm-Liouville equation.
For stratification that is a
function of depth only (i.e. no horizontal density gradients that would have an
associated mean baroclinic geostrophic flow), the eigenfunction problem becomes
one for across-shelf modal structures that are non-separable functions of depth
and across-shelf coordinate, F(x,z). However, in the long-wave limit,
the along-shelf propagation still separates into a set of FOWE with eigenvalues
cm associated to a set of Fm(x,z) by virtue
of an orthogonality condition.
The introduction of
stratification, with strength measured by the Burger number 
where N is an
approximate average Brunt-Vaisala frequency or buoyancy frequency defined by:
modifies the CTW dispersion
properties:
(The figures below show a different sign convention for wavenumber to that used in the notes about. The qualitative behavior is correct if the plots are reflected in the y-axis)
1.
There
is still a single infinite discrete set of wave modes for any choice of
bathymetry and stratification.
2.
Increasing
stratification, all else being equal, increases the wave frequency and makes
the modal structures more “horizontal”. (In the extreme limit of very large
Burger number, the modal structures fall flat and give the limiting case of
baroclinic Kelvin waves in a deep flat bottom ocean – the continental shelf
effectively vanishes as the scaled width L decreases).
3.
The
dispersion curves for all modes approach the same frequency as the wavelength
decreases, given by 
The short waves (large l) become bottom-trapped and evanescent, meaning
the eigenvales are no longer strictly real. The imaginary part of the
eigenvalue causes an exponential decay and wave amplitude is damped on the
scale of the (short) wavelength as
All low frequencies (a few to
several days) the weakly dispersive long wave behavior persists. Numerical
solutions to the eigenvalue problem can be obtained for arbitrary bathymetry
and assumed stratification N2(z), and the FOWE equations can
still be integrated as before.
This approach has been used
to model low-frequency variability in sea level and velocity on the continental
shelf.
The solution of the set of
first order wave equations for mode structures Fm and wave
speeds cm computed numerically for observed bathymetry and
stratification, and forced by observed winds, has been used my several
investigators to successfully model the low frequency sea level and velocity
variability on some US coasts (West Coast, and West Florida Shelf). See for
example the papers of:
Battisti, D. S., and B. M. Hickey (1984), Application of
Remote Wind-Forced Coastal Trapped Wave Theory to the
Chapman, D. C. (1987), Application of wind-forced, long,
coastal-trapped wave theory along the
Clarke, A. J., and S. Van Gorder (1986), A method for estimating wind-driven frictional, time-dependent, stratified shelf and slope water flow, Journal of Physical Oceanography, 16, 1013-1028. (pdf)