22 Large-Scale Internal Waves: Vertical Normal Modes, Kelvin Waves

22.1 Motivation

Waves are the means by which the effects of time-varying forcing are propagated. Variable forcing on large horizontal scales excites waves with large horizontal scales as well. These waves are most easily understood using the concept of vertical normal modes. The two most important classes of large-scale waves are Kelvin waves and Rossby waves. And, it is the internal versions of these waves that are most readily observed, and on which we will focus our attention. To keep the complexity manageable we will ignore bottom topography.

22.2 Long waves: hydrostatic balance

The limiting case with which we are concerned here is that of long waves. Recall from the earlier discussion of simple gravity waves that in the long wave limit, the vertical momentum balance goes to the hydrostatic balance. That will be the case here as well. Recall also that the hydrostatic balance means the velocity is depth-independent within any homogeneous layer.

22.3 Vertical normal modes

In a stratified fluid, waves can in general propagate in all three dimensions. In fact, for small-scale internal waves, we imagined the ocean to extend to infinity in all three dimensions so that we could focus on this three-dimensional propagation. For large-scale waves this is not so appropriate. If the horizontal scale is large compared with the depth of the ocean, then the wave is effectively constrained to propagate only in the horizontal. Also, if the fluid is stratified into a number of discrete layers, there will be no continuous propagation in the vertical. Let’s use this special case to get a first understanding of vertical normal modes.

22.3.1 A layered ocean

If the ocean has only two layers, there would seem to be two general types of motion: either the layers move together, or they move in opposite directions. The situation is exactly analogous to the coupled oscillator problem. With two masses there are two normal modes, each with its own frequency of oscillation. Any free motion of the system can be expressed as some fixed linear combination of the two normal modes. In the absence of dissipation, the energy in each of the normal modes separately is constant. The two modes are completely uncoupled, or independent, even though the two masses are coupled together, and trade energy back and forth.

Similarly, in a two-layer fluid, there are two long-wave normal modes. The first, called the barotropic mode, is depth-independent; the two layers move in unison as if there were no stratification at all. The second, called the baroclinic mode, has a vertical average of zero. The two layers move in opposition, with a speed ratio inversely proportional to the ratio of their thicknesses. Each of these modes has a characteristic long gravity wave propagation speed (phase speed). In the absence of dissipation, the two modes are completely uncoupled and independent.

Now, if there are $n$ layers instead of 2, there will be $n$ modes. The “zeroth” mode will still be the barotropic mode, independent of depth. The remaining modes will be the first, second, … $n{-}1$th baroclinic modes. The higher the mode, the slower the corresponding horizontal gravity wave speed.

22.3.2 Continuous stratification: standing waves

Taking the limit of an infinite number of layers we arrive at continuous stratification, with an infinite number of baroclinic modes. For each mode, the mode number is the number of depths at which the horizontal velocity is zero.

The set of baroclinic normal modes for continuous stratification is analogous to the set of harmonics of a vibrating string. In particular, recall from basic physics that the standing wave on a string is the superposition of two propagating waves going in opposite directions. They always cancel at the boundaries, which is how the boundary conditions (no motion at the ends of the string) are met. Similarly, the vertical modes in the ocean, or standing waves in the vertical, are equivalent to pairs of opposed vertically propagating waves. The boundary condition to be met is zero vertical velocity at the top and bottom of the ocean. (Actually, the top boundary condition is not exactly zero vertical velocity, but it turns out that for the baroclinic modes this boundary condition is a very good approximation. It is known as the “rigid lid” approximation, and is almost always made.)

If the ocean were uniformly stratified—if the buoyancy frequency, $N$, were constant with depth—then the vertical normal modes for the horizontal velocity components and the pressure would be cosine functions, and the mode structure for the vertical velocity component would be sine functions. The integer number of half-wavelengths within the depth of the ocean would be the mode number. That is, if the ocean goes from $z = 0$ to $z = -D$, then for the $j$th mode, $u$, $v$, and $p$ vary as $\cos(j\pi z/D)$, and $w$ varies as $\sin(j\pi z/D)$. Note that extrema of $w$ coincide with zeros of $u$, $v$, and $p$.

Given typical stratification with a thermocline, the actual normal modes are something like highly distorted sines and cosines. The amplitudes are larger and the zero-crossings closer together in the highly stratified thermocline than in the weakly stratified deeper regions.

22.3.3 Phase speeds and equivalent depths

Each standing wave on a vibrating string corresponds to a particular frequency; each vertical normal mode in the ocean corresponds to a particular long gravity wave phase speed, $c_j$, where $j$ is the mode number. These speeds decrease with increasing wave number; in fact, after the first 3 or 4 modes, $c_j$ is very nearly proportional to $1/j$.

For the central equatorial Pacific (the example with which I am most familiar), $c_j$ for $j = 1\ldots4$ is 2.87, 1.80, 1.12, and 0.80 m/s.

Using a 2-layer model we can come up with a reasonable estimate of a first baroclinic mode phase speed. If the layer depths are $h_1$ and $h_2$, and the density difference between layers is $\Delta\rho$, then the phase speed is

Let $h_1 = 300\mathrm{m}$ and $h_2 = 4700\mathrm{m}$. Estimate $\Delta\rho/\rho$ as a coefficient of thermal expansion, $2 \times 10^{-4}$ per degree, times a temperature difference between layers of 15°C. The result is $c = 2.9$ m/s.

For a homogeneous fluid, the long gravity wave phase speed is $\sqrt{gh}$, where $h$ is the depth. This leads to the definition of an equivalent depth, $h_j$, for the $j$th baroclinic mode:

For the first baroclinic mode, the equivalent depth is around a meter; the propagation characteristics of a long wave with first baroclinic mode structure are the same as for a long wave in a one-meter-deep homogeneous ocean!

22.4 Kelvin Waves

For long waves with periods of several hours and longer, the Coriolis force becomes important, and the simple long wave theory that we studied at the beginning of the course must be modified. One of the simplest, and yet most important, of such modifications leads to the class of waves called Kelvin waves. They come in two varieties: coastal and equatorial. The two are conceptually identical, but the coastal variety is mathematically a little simpler, so we will use it to illustrate how Kelvin waves work.

Let the coordinate system be chosen so that the coast is a line of constant $y$, running East-West for example. Characteristics of Kelvin waves:

• $v = 0$
• $u$ is in geostrophic balance
• amplitude decays exponentially with distance from the coast
• $\omega/k = \sqrt{gh_j}$; they propagate as long gravity waves, exactly, except that
• they propagate only with the coast on the right in the northern hemisphere, on the left in the southern.

In other words, they propagate cyclonically around a basin, but not the other way.

Using the concept of the equivalent depth, and the first two characteristics of Kelvin waves listed above, we can show that the relevant dynamical equations are:

where $h$ is the equivalent depth and $\eta$ is the sea surface height perturbation for the ocean with that equivalent depth. The first two equations are horizontal components of the momentum balance, with the pressure gradient terms on the right hand side expressed in terms of the sea surface perturbation. The third equation comes from vertically integrating the continuity equation. Note that the first and third equations together do not involve rotation at all, and are just the equations needed for ordinary long gravity waves propagating in the $x$ direction. The second equation is the geostrophic balance. It constrains the system such that the behavior in $y$ must be exponential, instead of being constant as it would be if there were no rotation.

There are two solutions to this system: one growing exponentially away from the boundary, one decaying. The former becomes infinite in amplitude at infinite distance, and is therefore unphysical. The remaining solution (picking the horizontal phase arbitrarily) is:

In the last equation we introduced a new definition: $\lambda$ is the Rossby radius of deformation. Note that it has dimensions of length: it is the distance a particle would travel at the long gravity wave phase speed in the time an inertial oscillation rotates by one radian. Thus it is related to the relative importance of ordinary wave propagation versus the effects of the Coriolis force.

For the barotropic mode (depth-independent, so $h$ is around 4000 m), $c$ is about 200 m/s, so in mid-latitudes $\lambda$ is around 2000 km. In other words, rotation affects barotropic long waves only when they have rather large horizontal scales. For the first baroclinic mode, say with $h$ of 1 m, $\lambda$ is around 30 km in mid latitudes—rotation is important for long gravity waves with fairly small scales. And, baroclinic Kelvin waves decay within a few tens of km of the coast.

Equatorial Kelvin waves are symmetric about the equator. Their behavior in $y$ is Gaussian, with a typical scale of 350 km for the first baroclinic mode. They propagate only to the east. Consider a closed ocean basin spanning the equator: Kelvin waves propagate cyclonically around both the northern and the southern half-basins, with the equator acting something like a boundary between the half basins. If you were to excite an equatorial Kelvin wave, it would propagate eastward until it hit the eastern boundary. There it would split into two poleward-propagating coastal Kelvin waves, which would continue their cyclonic circuits of their half-basins until they met at the equator on the west side of the ocean. There they would combine to form an equatorial Kelvin wave, and the process would be repeated. Actually there is more to the story than this, because the poleward-propagating Kelvin waves would lose some of their energy to Rossby waves radiating westward—but that is a story for another time.

Although baroclinic equatorial Kelvin waves are prominent in theory and observations, coastal Kelvin waves are relatively more important in theory and models than in observations. The reason is that the ocean does not have vertical walls; bottom topography varies from zero depth to full ocean depth on an offshore scale similar to the baroclinic radius of deformation, so coastally-trapped waves tend to differ considerably from ideal Kelvin waves. Nevertheless, they propagate in the same direction as Kelvin waves and at similar speeds, so the Kelvin wave idealization remains a useful first approximation. For example, see Brink (1982).

Coastal topography is less important for basin-scale barotropic Kelvin waves because the barotropic radius of deformation is much larger than the offshore scale of topography. Barotropic tides have a Kelvin-wave-like character. Truly ideal Kelvin waves still are not to be expected, however, because coastlines are curved and the Coriolis parameter varies on the scale of the barotropic radius of deformation.

References

Brink, K. H., 1982. A comparison of long coastal trapped wave theory with observations off Peru, J. Phys. Oceanogr., 12, 897–913.

Notes by Eric Firing · Last updated 2011/11/27