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The Stokes drift of the sum of waves

In Herterich and Hasselmann (1982, JPO), it is claimed that the ensemble-mean Stokes drift of a random field of surface waves is given by the sum of the Stokes drift of the individual components of the wave spectrum. There is even a paper by Jansons and Lythe (1998, Phys. Rev. Lett.) that claims that

“When there is more than one wave, the drift velocity is calculated by summing the contributions from each wave.”

Is this an incorrect generalization of Herterich and Hasselman’s claim?

Well to my surprise, it might not be. Take the most simple model of wave with the velocity field of the form

a cos(kx- ɷ t)

where a is the wave amplitude, k the wavenumber and ɷ the frequency. The calculation of the Stokes drift at the second-order in wave amplitude leads to a term proportional to the square of sin(kx- ɷ t) which is on average in time equals to 1/2.

Now if two waves of two different wavenumbers (k1 and k2) and periods (ɷ1 and ɷ2) are present, the calculation leads to two terms proportional to the square of each sin(kx- ɷ t), as well as two terms proportional to sin(k1x- ɷ1 t)*sin(k2x- ɷ2 t). The Stokes drift is then computing by averaging in time the above terms. The first two gives each the Stokes drift of the individual waves. What is then the Stokes drift associated with the next two terms, which represent the interaction between the two waves?

To my surprise, it seems that as long as the period over which the average is performed is much larger than any of the periods, then the Stokes drift associated with the interaction between the two waves vanishes and is exactly zero to the limit of infinite time. I deduced this by computing the time average of sin( ɷ1 t)*sin( ɷ2 t) for various values of ɷ1 and ɷ2. In the upper panel of Fig. 1 is shown the waves with periods of 10 and 25 units, respectively. In blue in the lower panel is plotted the term sin( ɷ1 t)*sin( ɷ2 t) and in black its time average with the period of average varying from 0 to the end of the time series.

The Stokes drift of two waves

Figure 1: Two sinusoids of different periods (upper panel), their product (blue, lower panel) and their running time average (black). The periods are 10 and 25 units.

First, I am surprised how periodic the product looks, although there is no reason why such product should be periodic. Is there one? Then, as the black curve shows, the longer the period of average, the smaller its contribution to the Stokes drift.

Is the case chosen particular? I tried many. In Fig. 2 are shown the interaction term and its running time average when the periods are 10.4 and 12.3 units, respectively:

The Stokes drift of two waves in other case

Figure 2: As in the lower panel of Fig. 1 except that the periods are 10.4 and 12.3 units respectively.

The story seems the same. The only cases where we start to get significant contribution is when one of the two periods is not small compared to the total length of the record. Which makes sense.

There is a detail I may underestimate: in front of the interaction term, there is a coefficient of the form ki/ɷj so that the interaction between waves i and j might become important if ki and ɷi are both large and kj and ɷj are both small.

If we forget the last point, this demonstration-less illustration suggests that the Stokes drift generated by the sum of any waves is the sum of the Stokes drift of the individual waves as long as the period over which the average is performed is long compared to any of the wave period.

Am I completely misleading myself? No! Eric simply noticed that the product of sinusoids are the sum of two and any long-time average will give zero. Simple! Thus, the above conclusion seems to be correct.

Following discussion

  • Francois:

    So you agree with the conclusion: the Stokes drift of a field composed of many waves is the sum of the Stokes drift of the individual waves, with the time average being infinite, or much longer than any of the wave periods?


  • Eric:

    Yes, given that we are looking at order epsilon**2 flow generated by an order epsilon wave field.

    How do things change when the Lagrangian flow we are looking for is the same order of magnitude as the wave (or other disturbance) field?

    A related problem is that averaging over a very long time will make sense in this context only in the limit of weak Lagrangian flow, so that the Lagrangian displacements over the averaging interval are still small compared to the wave-induced excursions. (I think this is at least implicit in the hybrid nature of the GLM theory. Maybe it is explicit.)


Important note from Longuet-Higgins (1969)

“The analysis of this paper has been carried through for motions that are assumed periodic in the first place. However the analysis is equally valid for quasi-periodic motions having a more or less broad spectrum, provided that meaningful time averages can be taken while a particle moves through a distance delta_x which is small compared to a typical length scale of the field of motion.”

So, as long as the period of average is such that the typical distance of parcels is smaller than the typical scale of the velocity field, then the formula given by Longuet-Higgins (1969) is correct.