Feb 22: Coastal-Trapped Waves (1)

Begin from the linear, frictionless, barotropic momentum and continuity equations:

We will examine motions at sub-inertial ( w < f ) frequencies on a straight coast with water depth h that varies across the shelf, i.e. h=h(x).

First estimate the order of magnitude of sea level variability in the case of an approximate cross-shelf geostrophic balance:

Now consider the order of magnitude of the sea level term in continuity compared to the horizontal flow divergence in the case of time variability of order f^{-1 :}

This is the ratio of continental shelf length scale to the barotropic Rossby radius. This is small and argues that we can neglect sea level variability in the continuity equation. This is the so-called “rigid lid” approximation.

Equation (3) will be satisfied is we define the velocity in terms of a transport streamfunction:

where the subscripts now denote partial differentiation.

Form a vorticity equation by cross-differentiating the momentum equations to eliminate the pressure gradient terms, i.e. d/dx (2)-d/dy (1):

Further assume we are on an f-plane (constant Coriolis parameter), and that the depth h=h(x) is independent of the along-shelf direction y.

Seek solutions of the form of a propagating wave traveling in the positive y direction:

<< changed sign of w from classroom lecture

Substitute into the vorticity equation (a wave equation):

Some boundary conditions:

No velocity through the coast implies u = 0 at x = 0.

As we move offshore we require only that remain finite.

Suppose the bottom depth increases exponentially toward the continental shelf break:

Then the equation for the cross-shelf modal structure becomes:

Try a solution of the form

which we see satisfies the required boundary condition at the coast x=0.

which is the dispersion relation for barotropic shelf waves.

shows the phase of the waves always propagates in the positive y along-shelf direction.