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 Consider a tidal wave whose wavelength is very much less than the radius of the Earth. This corresponds to the limit
, where
is an azimuthal mode number. Suppose, however, that the ocean depth,
, is allowed to vary with position. Show that, in this limit, the Laplace tidal equations
are written
where
 Consider shortwavelength tidal waves in a region of the ocean that is sufficiently localized that it is a good approximation to treat
and
as constants. We can define local Cartesian coordinates,
,
,
, such that
,
,
and
. It follows that the
axis is directed southward, the
axis is directed eastward, and the
axis is directed vertically upward. Show that, when expressed in terms of this local coordinate system, the Laplace
tidal equations derived in the previous exercise reduce to
where
is a constant. In fact, the previous equations are the linearized equations of motion of a body of shallow
water confined to a tangent plane that touches the Earth at the angular coordinates
,
. This
plane is known as the
plane because, as a consequence of the Earth's diurnal rotation, it rotates about
at the angular velocity
.
 Demonstrate that the set of equations derived in the previous exercise can be written in the coordinatefree (in the

plane) form
where
.
Let
and
. Assuming that
and
are constants,
demonstrate that the previous equations are equivalent to
where
denotes a twodimensional Laplacian (in the

plane).
 Consider free (i.e.,
) planewave solutions to the
plane equations, derived in Exercise ii, of the form
Here,
, and
,
,
are constants. Assuming that
is
constant, show that
and
where
. This type of wave is known as a Poincaré wave.
 Suppose that the region
corresponds to an ocean of constant depth
, whereas the region
corresponds to land. Consider free solutions to the
plane equations in the region
. We
can trivially satisfy the constraint
by searching for solutions which are such that
for all
. Show that the most general such solution takes the form
where
, and
Here,
and
are arbitrary functions,
, and
.
These solutions are known as Kelvin waves. Deduce that Kelvin waves propagate
along coastlines, at the speed
, in such a manner as to keep the coastline to the right of the direction of propagation in the Earth's northern hemisphere, and to
the left in the southern hemisphere.
 We can take into account the latitude dependence of the parameter
by writing
, where
and
.
Let us assume that the ocean is of constant depth,
. Furthermore, let us search for an almost incompressible, free
solution of the Laplace tidal equations which is such that
and
. By eliminating
from the final two
plane equations,
show that
By searching for a wavelike solution of the previous equation of the form
,
deduce that
This is the dispersion relation of a socalled Rossby wave. Demonstrate that Rossby waves always travel with a
westward component of phase velocity. Finally, show that it is reasonable to neglect compression provided that
.
Next: Equilibrium of Compressible Fluids
Up: Terrestrial Ocean Tides
Previous: Hemispherical Ocean Tides
Richard Fitzpatrick
20160122