Remote sensing of the sea surface has shown that Rossby waves (RWs), or planetary waves, are ubiquitous in the world ocean (*1*, *2*), although the precise mechanisms for their formation and propagation are still under debate (*3*–*5*). Recently, similar propagating signals have been observed in ocean color (*6*, *7*). Here, too, the mechanisms remain unclear; nutrient pumping into the photic layer, followed by uptake by phytoplankton, has been suggested (*8*), as has lateral advection by the propagating RWs (*7*). The phase difference between the color and surface-height signals serves as an excellent discriminator between mechanisms (*7*).

A recent suggestion (*9*) has added to the list. The idea is that color anomalies are not caused by chlorophyll itself, but as a sensor response to surface material accumulated at lateral convergence zones in the RW pattern (likened to a “hay rake”). This was tested in a region of strong color signal (*7*). Under this assumption, apparent highs of the color signal should be located within convergence zones. Certainly particles must congregate in stationary convergence zones, but the zones in RWs are propagating westward at an approximately steady rate. The behavior of trajectories can be hard to predict in nondivergent flows (*10*), is more complicated in simple divergent flows (*11*), and becomes still more so in eddying situations or fully three-dimensional flows (*12*, *13*). In particular, is there any reason to expect particle trajectories to congregate in propagating convergence zones in RWs?

The simplest problem to consider is one-dimensional, where a particle at position *x*(*t*) changes over time *t* according to where α > 0 is the amplitude of a wave propagating at unit speed and *ẋ* denotes the time derivative of the particle position. The maximum convergence, for positive α, occurs at *x* – *t* = (2*n* + 1)π for integer *n*. This can be solved analytically, but the solutions are not enlightening.

The long-time behavior can be derived easily by defining a coordinate *y* = *x* – *t* traveling with the wave, which gives For α < 1, there can be no steady solution, as particles cannot “keep up” with any convergence zone; *y* decreases monotonically with time. For α ≥ 1, there is a steady solution for large times, at *y* = π – sin^{–1}(α^{–1}). This only approaches π for large α, when the advection speed is considerably larger than the wave speed. Comparison with the example in figure 1 of (*9*) indicates that the amplitude of the east-west velocity, on the order of a few cm s^{–1}, is unlikely to be as large as the wave's phase velocity (on the order of 5 cm s^{–1}). In general, the particle cannot remain in the convergence regions (Fig. 1). This simple example hardly serves to prove whether particles behave similarly in RWs, but it is indicative (*10*).

The theory of RWs is well known (*14*). On a spherical earth, for a single normal mode or reduced-gravity one-layer geostrophic (long-wave) flow, the momentum and mass conservation equations are where *g* represents the gravitational acceleration, η represents the perturbation free surface height, Ω represents the Earth's rotation rate, *a* represents the Earth's radius, *t* represents time, λ represents longitude, and Θ represents latitude, with *u* and *v* representing the zonal and meridional velocity components. Longitude is typically measured from some eastern boundary. *H* represents an equivalent depth such that the internal wave speed *C* = (*gH*)^{1/2}.

The exact solution of these equations is an arbitrary function of two variables which represents a westward-propagating geostrophic RW. On a “beta-plane,” the westward phase speed, including the cos Θ factor as the meridians converge, is simply βC^{2}/*f* ^{2}, where *f* and β are the local Coriolis parameter and its northward gradient. A wave of this form is a good descriptor for the observed RWs (*2*, *4*, *9*). The geostrophic flow is aligned along phase lines which are normal to the direction of propagation (*14*).

Vertical gradients of the horizontal flow are small, so that turbulent mixing within a mixed layer can be ignored; particles move quasi-horizontally; particles at the cartesian position (*x* = *a*λ cos Θ, *y* = *a*Θ) move under the geostrophic velocities It can easily be shown— denoting (2Ω*a*^{2}/*C*^{2})λsin^{2}Θ + *t* by χ and defining μ = ln sin Θ—that so that *gF*/*C*^{2} behaves as a streamfunction for what is now an effectively divergence-free flow in the new, steadily westward propagating coordinates (χ,μ). These are exactly the equations studied in (*10*), which discusses solutions for such cases extensively; none of those solutions involves convergence of trajectories.

As an example, consider a simple plane wave propagating at some angle to due west: Here, ψ(Θ) is an arbitrary phase. It is straightforward to show from above that But this is just the nontemporal part of the phase ϕ. Thus, particles move along phase lines, approximately orthogonal to the line of propagation of the RW (*14*). Then, following the trajectory, the surface is simply η_{0} sin (constant + ω*t*), or simple harmonic motion. The particle positions can be obtained by solving the Θ equation, giving and then substituting for λ. Particle trajectories are then curves, repeating with frequency ω (Fig. 2). Replacing the constant amplitude η_{0} with one that varies with latitude has only a weak effect on trajectories. Note that trajectories too near the poles can achieve the unphysical sin Θ > 1; but poleward of about 45° geostrophic (long-wave) theory becomes dubious anyway.

The predicted lack of clustering of trajectories near convergences applies only for single monochromatic waves. Varieties of particle motion, including chaotic solutions, are possible for a linear superposition of quasi-geostrophic RWs (*15*); trajectories would also differ for nonlinear RWs, or if small-scale unstable features were included.

Nonetheless, the approximate coincidence of convergence zones and maximum southward (northward) component of velocity in the northern (southern) hemisphere is clear (*7*, *9*), at least for the area chosen in (9). Such coincidence indicates phase differences between sea surface and color signals of amplitude π/2. Phase patterns like this, as noted, serve as useful ways to discriminate between suggested mechanisms. If the color sensor is directly reading a measure of chlorophyll, the combination of north-south advection against an existing mean gradient of chlorophyll and a relaxation towards normality on a time scale short compared with the wave period gives a good fit to the observed phase and a reasonable, if slightly small, fit to the observed amplitude of the color signal (*7*). If, as suggested by Dandonneau *et al*. (*9*), the sensor registers particles, then the RW propagation would have to be more complex than it is currently believed to be in order to locate the particles at the convergence zones.