Waves in the Ocean and Atmosphere

Frontmatter

Lecture 1. Introduction

Abstract

A course on wave motions for oceanographers and meteorologists has (at least) two purposes.

Joseph Pedlosky

Lecture 2. Kinematic Generalization

Abstract

Suppose the medium is not homogeneous. For example, gravity waves impinging on a beach see of varying depth as the waves run up the beach, acoustic waves see fluid of varying pressure and temperature as they propagate vertically, etc. Then a pure plane wave in which all attributes of the wave are constant in space (and time) will not be a proper description of the wave field. Nevertheless, if the changes in the background occur on scales that are long and slow compared to the wavelength and period of the wave, a plane wave representation may be locally appropriate (Fig. 2.1). Even in a ho¬mogeneous medium, the wave might change its length if the wave is a superposition of plane waves (as we shall see later).

Joseph Pedlosky

Lecture 3. Equations of Motion; Surface Gravity Waves

Abstract

For a rotating stratified fluid, the general equations of motion can be written as:

1.

Momentum equation:

$$ \rho \left[ {\frac{{d\vec u}}{{dt}} + 2\Omega \times \vec u} \right] = - \nabla p + \mu {\nabla ^2}\vec u + k\nabla (\nabla \cdot \vec u)\;(if\;\mu \;cons\tan t,\;k\;is\sec ondvisosity) $$

(3.1)

 

2.

Mass conservation:

$$ \frac{{\partial \rho }}{{\partial t}} + \nabla \cdot (\rho \vec u) = 0\;;and $$

(3.2)

 

3.

Thermodynamic energy equation:

$$ \frac{{ds}}{{dt}} = H $$

where s is specific entropy and H is the nonreversible heat addition. This can be rewritten, assuming that s is a thermodynamic function of p and ρ,

$$ {c_p}\frac{{dT}}{{dt}} - \frac{{\alpha T}}{{\rho t}}\frac{{dp}}{{dt}} = \Phi + \frac{k}{\rho }{\nabla ^2}T + Q + k{(\nabla \cdot \vec u)^2} \equiv H $$

(3.3)

 

where s is specific entropy and H is the nonreversible heat addition.

Joseph Pedlosky

Lecture 4. Fields of Motion in Gravity Waves and Energy

Abstract

Now that we have the dispersion relation, i.e., the dependence of frequency on wave number (we define the magnitude, K, of the wave vector K to be the wave number), we can ask what the fluid motion is in the wave field.

Joseph Pedlosky

Lecture 5. The Initial Value Problem

Abstract

It is not easy to see how a uniform or nearly uniform wave train can realistically emerge from some general initial condition or from a realistic forcing unless the initial condition or the forcing is periodic. That turns out not to be the case, and the ideas we have so far developed about group velocity and energy propagation turn out to be invaluable in getting to the heart of the general question of wave signal propagation. Indeed, it is the very dispersive nature of the wave physics (i.e., the dependence of the phase speed on the wave number) that is responsible for the emergence of locally nearly periodic solutions. This can be seen by examining the solution to the general initial value problem. This was first done by Cauchy in 1816. It was also solved at the same time by Poisson. The problem was considered so difficult at that time that the solution was in response to a prize offering of the Paris Academie (French Academy of Sciences). Now it is a classroom exercise.

Joseph Pedlosky

Lecture 6. Discussion of Initial Value Problem (Continued)

Zusammenfassung

We have seen that the initial spectrum of the waves, which is initially localized in space, gets strung out with time so that at time t, each wave number appears at x = c _g (k)t. We might expect that the energy, if conserved, would also be distributed by wave number, so that the amount of energy at wave number k in the original spectrum at wave number k would also be found at the position x = c _g (k)t for a large enough time. This is as if the original disturbance is composed of an infinite number of packets of constant wave number, each of which moves away from the origin of the disturbance with its own group velocity. Each satchel of energy moves with the group velocity (Fig. 6.1).

Joseph Pedlosky

Lecture 7. Internal Gravity Waves

Zusammenfassung

In both the atmosphere and the ocean, the fluid is density stratified, i.e., ρ = ρ(z) (it is also a function of horizontal coordinates and time) so that usually dense fluid underlies lighter fluid. This stratification supports a new class of waves called internal waves. Internal waves are designated as such, because the vertical structure of the waves is oscillatory in z (contrast with the surface gravity wave) and most of the vertical displacement occurs within the fluid as opposed to the upper boundary, as in the gravity wave example we have just studied.

Joseph Pedlosky

Lecture 8. Internal Waves, Group Velocity and Reflection

Zusammenfassung

The rather unusual dispersion relation and the nonintuitive relation between group velocity and the wave vector lead to some very unusual physical consequences.

Joseph Pedlosky

Lecture 9. WKB Theory for Internal Gravity Waves

Zusammenfassung

The buoyancy frequency is never really constant. Indeed, in the ocean there is a significant variation of N from top to bottom. Figure 9.1 (next page) from the Levitus Atlas (1982) shows the distribution of N of the zonally averaged global ocean.

Joseph Pedlosky

Lecture 10. Vertical Propagation of Waves: Steady Flow and the Radiation Condition

Zusammenfassung

There are numerous situations in which fluid flows over an obstacle, say a mountain in the atmosphere, a sea mount, or a ridge in the ocean, and we would imagine that internal gravity waves, if the fluid is stratified, would be generated. Such situations are of interest in their own right, but additionally they force us to carefully examine the radiative properties of the waves, which must be understood, sometimes, to actually solve the problem.

Joseph Pedlosky

Lecture 11. Rotation and Potential Vorticity

Abstract

For motions whose time scales are of the order of a day or greater, or more precisely when the frequency of the wave motion is of the order of the Coriolis parameter or less, the effects of the Earth’s rotation can no longer be ignored. Such waves are evident in both oceanic and atmospheric observational spectra. Figure 11.1 taken from the article of Garrett and Munk (1979) shows a power spectrum of vertical displacement of an isotherm. We see a great deal of variance at frequencies less than N (as we might expect) with a peak near the Coriolis frequency f= 2Ω; sin θ.

Joseph Pedlosky

Lecture 12. Large-Scale Hydrostatic Motions

Abstract

For many motions in both the ocean and the atmosphere, the horizontal scale far exceeds the vertical scale of the motion. For example, motions in the ocean occurring in the thermocline will have a vertical scale of a kilometer or less, while the horizontal scales might be of the order of hundreds of kilometers. Motions in the ocean induced by traveling meteorological systems will have such large scales. If the motion has such disparate scales in the vertical and horizontal, we can expect important influences on the dynamics. First of all, we would expect that the vertical velocity will be small compared with the horizontal velocity, since the motion consists of nearly flat trajectories. That in turn could mean that the vertical acceleration is small. Such dynamical consequences often allow simplifications to our treatment of the physics, and we are always looking for such simplifications so that we can make progress with more difficult problems; not just make life easier for ourselves.

Joseph Pedlosky

Lecture 13. Shallow Water Waves in a Rotating Fluid; Poincaré and Kelvin Waves

Abstract

We now examine the nature of the waves which serve, among other things, to sculpt the geostrophic final state from an arbitrary initial state. These waves, as we noted earlier, have no potential vorticity, because in the simple models we are examining, the conservation of pv is simply the statement:

$$\frac{{\partial q}}{{\partial t}} = 0$$

(13.1a)

$$q = \varsigma - \eta \frac{f}{D}$$

(13.1b)

Joseph Pedlosky

Lecture 14. Rossby Waves

Abstract

When we consider waves of large enough scale, the sphericity of the Earth can no longer be ignored. Rossby was the first to point out that the most significant effect of the Earth’s sphericity is that it rendered the Coriolis parameter f = 2Ω; sin θa function of latitude. Since the large scale motions in the ocean are nearly horizontal, the only component of the Coriolis acceleration that really matters is the one involving the horizontal velocities, and therefore only the local vertical component of the Coriolis parameter is dynamically significant. Otherwise, for scales that are large but still sub-planetary, a Cartesian coordinate system can be used to obtain at least a qualitatively correct view of the dynamics. Such an approximation in which the variation of the Coriolis parameter with latitude is treated but in which the geometry is otherwise Cartesian is called the beta-plane approximation, and we shall use it without a detailed justification. The student is referred to Pedlosky (1987) for a careful derivation. In this course, we will use the heuristic approach outlined above.

Joseph Pedlosky

Lecture 15. Rossby Waves (Continued), Quasi-Geostrophy

Abstract

For the Poincaré wave, ω≥ f and so the wave motion is not in geostrophic balance, while for the Rossby wave,

$$\omega \le \beta {L_d} = \beta {c_0}/f$$

(15.1)

so that

$$\omega /f = \frac{{\beta {L_d}}}{f} \le \frac{{\beta L}}{f}$$

(15.2)

where L is the scale of the motion. Thus, for Rossby waves, the frequency is less than f so that in the x-momentum equation, for example,

$$\overbrace {{u_t}}^1 - \overbrace {fv}^2 = - g{\eta _x}$$

(15.3)

term (1) will be less than term (2) by the order of ω/f. The velocity will be in approximate geostrophic balance to that order. This is similar to the hydrostatic approximation in which the vertical pressure gradient can be calculated as if the fluid were at rest, even though it is motion, because the vertical accelerations are very small when the aspect ratio D/L of the motion is small. Here the horizontal pressure gradient is given by the Coriolis acceleration as if there were no acceleration of the relative velocity, i.e., as if the flow were uniform in space and time even though it is not because that acceleration is very small compared to the Coriolis acceleration.

Joseph Pedlosky

Lecture 16. Energy and Energy Flux in Rossby Waves

Abstract

In discussing the energy and its flux for Rossby waves, we encounter the problem that the natural definition of the energy flux at the lowest order pu. is horizontally non-divergent and therefore has no effect on the wave energy. To discuss the real energy flux, one has to include the divergent, non-geostrophic O(ε) part of the velocity field as well as the pressure contribution at this order. This would be a messy business, and what is worse is that the solution of the quasi-geostrophic potential vorticity equation doesn’t give us these quantities as part of the solution. Is there a way we can describe the energy flux entirely within the quasi-geostrophic framework? The answer is yes, and it follows from a direct consideration of the linear quasi-geostrophic equation. First, though, let us orient the y-axis in the direction of the gradient of the ambient potential vorticity, VQ, and call the magnitude of the gradient β for obvious reasons. As long as the gradient is a constant, there is no loss of generality. It will be up to the student to try to generalize these results when the gradient is not constant. The linear qgpve is

$${\textstyle{\partial \over {\partial t}}}[{\nabla ^2}\psi - {a^2}\psi ] + \beta {\textstyle{{\partial \psi } \over {\partial x}}} = 0$$

(16.1)

Joseph Pedlosky

Lecture 17. Laplace Tidal Equations and the Vertical Structure Equation

Abstract

Let’s return to the linearized wave equations before the gravity waves are filtered out by the quasi-geostrophic approximation. What we will see now is that the analysis of the homogeneous model can be carried over, in important cases, to the motion of a stratified fluid. A vertical modal decomposition can be done for these cases, and we will be able to show that the equations for each vertical mode are analogous to the equations for the single layer. Exactly what that relationship is will be the subject of our development that follows.

Joseph Pedlosky

Lecture 18. Equatorial Beta-Plane and Equatorial Waves

Abstract

The equator is a special region dynamically, most obviously because there the vertical component of the Earth’s rotation vanishes. It turns out to be, in consequence, a region in which certain linear waves have unusually strong signals and are involved in some important atmospheric and oceanic phenomena such as the Quasi-Biennial Oscillation in the atmosphere and the El Nino (ENSO) phenomenon in the ocean (and atmosphere). Good, useful references that describe in detail those phenomena are Andrews et al. (1987) for the former and Philander (1990) for the latter.

Joseph Pedlosky

Lecture 19. Stratified Quasi-Geostrophic Motion and Instability Waves

Abstract

We return from our brief visit to the equator and investigate the low frequency motions in mid-latitudes that occur in a stratified fluid. The motion we consider will be in near (quasi-)geostrophic balance, but we will develop the equations in an informal, heuristic way, leaning heavily on the formal analysis of Lecture 15. We will also employ the beta-plane approximation so that we are assuming that two parameters, ε = U/f ₀ L, b = βL /f ₀, are both small. That being the case, the lowest order balances in the horizontal momentum equation imply that

$$u = - {\textstyle{{Py} \over {{\rho _0}{f_0}}}}\bar = - {\psi _y}$$

(19.1a)

$$v = {\textstyle{{{p_x}} \over {{\rho _0}{f_0}}}}\bar = {\psi _x}$$

(19.1b)

Joseph Pedlosky

Lecture 20. Energy Equation and Necessary Conditions for Instability

Abstract

To get a better feeling for where the source of the instability is, it is useful to develop an equation for the perturbation energy for waves in the presence of a mean flow that contains both horizontal and vertical shear. This entire subject is enormous, and we will only scratch the surface in our discussion. The text by Gill (1982) and Pedlosky (1987) contain ample discussion for further reading.

Joseph Pedlosky

Lecture 21. Wave-Mean Flow Interaction

Abstract

We have been considering the dynamics of waves in this course and have remarked several times on the linearization restriction we have normally placed on the dynamics to make progress, and we have skirted rather completely the role of nonlinearity on the dynamics of the waves themselves. It is a difficult subject.

Joseph Pedlosky

Backmatter