Feb 26: Coastal-Trapped Waves (2)

Free Barotropic Shelf Waves

Starting from the linear, frictionless, barotropic momentum and continuity equations, making the rigid-lid approximation, introducing a depth-integrated velocity (transport) streamfunction

and assuming a continental shelf with h = h(x),we obtain a vorticity equation for streamfunction

Further assuming propagating waves traveling in the negative y direction, , we obtain a 2^nd-order differential equation for the across-shelf modal structure:

which you may recognize as defining a Sturm-Liouville eigenvalue problem:

Solve subject to boundary conditions . There are non-trivial solutions for an infinite set of discrete eigenvalues, , with corresponding eigenfunctions, . (See e.g. http://en.wikipedia.org/wiki/Sturm-Liouville_theory.)

A consequence of this is that the eigenfunctions are orthogonal and have the property that they form an orthonormal basis set so that any general solution of the governing equation can be represented as a weighted linear combination of the eigenfunctions: (much as in Fourier series).

In the special case of an exponential depth profile

the coefficients become constant and a simple solution is obtained for the modal structure on the sloping shelf . The solution exists only for a certain relationship between frequency and wave-number given by:

i.e. the dispersion relation.

This has the property that for showing phase advances toward negative y, whereas for phase advances toward positive y. This shows the waves propagate phase with the coast on the right (left) in the northern (southern) hemisphere.

Boundary/matching conditions

In the deep water off the continental shelf, x > L, the bottom depth is constant (i.e. h_x = 0) and the governing equation is simply

, with solution:

The boundary condition at the foot of the continental shelf is a matching condition for the sea level and across-shelf velocity of the general solutions on the shelf, , and in deep water, .

Matching u implies we need to match

which shows it is sufficient to match the shelf and deep modes because h is continuous.

We naturally expect the sea level to vary like:

In which case the along-shelf momentum equation is:

from which we can argue that the sea level will match provided the across-shelf gradients of the mode structures match, i.e.

This condition will also match along-shelf velocity in addition to sea level and cross-shelf velocity.

Matching the mode structures themselves gives:

so , while matching the cross-shelf gradient gives:

These matching conditions give:

This transcendental equation has an infinite number of discrete solutions that can be viewed graphically:

For large kL the solutions approach the vertical asymptotes of the tangent function at

For long waves

which are non-dispersive, but the speed decreases with increasing mode number (because k becomes larger),

whereas in the case :

the waves are strongly dispersive with frequency decreasing with increasingly shorter waves.

On the basis of these limiting behaviors we can qualitatively sketch the dispersion relation for this class of waves:

Exercises for further study:

Differentiate w with respect to l to derive a expression for the group velocity:

Find the frequency at which c_g = 0 and show that this is always less the f.

Show that the wavenumber, l, for which c_g = 0 increases with increasing mode number, n.

The restoring force that sustains these waves is the vortex stretching associated with displacement of the water column across the sloping continental shelf.

Conservation of depth averaged vorticity in the absence of forcing or friction is , where is relative vorticity.

[Sketch vortex line for across-shelf displacement, and implied phase propagation.]

(If we hadn’t made the rigid lid approximation we would have a mode n=0 barotropic Kelvin wave.)

(Beware the limit because as wavelength becomes small we begin to violate the separation of scales L and wavelength that is the basis of the rigid lid approximation.)

The property that the set of wave modes form an orthonormal basis set onto which any linear dynamics can be projected is an extremely useful concept for understanding the coastal ocean response to dynamics.

Wind-forced long shelf waves

Consider now the question of how winds might generate these coastally trapped waves.

We will see that the essential features of the wind-forced CTW problem can be described by considering the influence of the along-shelf component of the wind and in the limit of long-wave (low frequency) wave dynamics.

The spatial scale of wind variability is typically many times the scale of the width of continental shelf (1000s vs. 100s of km). In our assumed coordinate system of x across-shelf and across-shelf and y along-shelf, this implies L_y >> L_x or that l << k which is the long-wave limit considered earlier:

Small l implies small w so we also have a separation in time scales:

The order of magnitude of terms in the continuity equation relate along-shelf and across-shelf components with respect to the corresponding length scales:

The left-hand-side of the across-shelf momentum equation has terms that scale as:

so the time derivative term can be safely neglected and the across-shelf momentum balance is essentially geostrophic.

The along-shelf momentum equation has LHS terms of relative magnitude:

For wind forcing to be significant then in the along-shelf momentum equation we must have:

We must assume but in the across-shelf momentum equation it follows that:

so wind forcing is not significant in the across-shelf momentum balance.

We see immediately that it is the along-shelf component of the wind that is the most effective in generating low (sub-inertial) frequency current and sea level variability in the coastal ocean.

(This is because the along-shelf wind efficiently drives coherent across-shelf Ekman transport over an extensive along-shelf distance, and the compensating flow across isobaths is significant in the potential vorticity dynamics.)

For now we will neglect the role of bottom friction (since we have typically seen that is takes many wavelengths for bottom friction to decrease the CTW energy).

The important dynamics are then described by the equations:

The continuity equation still allows us to seek solutions expressed in terms of a transport streamfunction.

The wind-forced vorticity equation in the long-wave limit is:

What we will see is that we can obtain a solution expressed in terms of a sum of the modes that are solutions to the unforced problem, i.e.

We pursue a solution by separation of variables and find

only has a solution if

which shows we need solutions of the form

where the free mode structures satisfy