5.3 PV Generation by Wave Breaking and Dissipation
5.3.3 A Wavepacket Life Cycle Experiment
We illustrate the results obtained so far by considering a simple life cycle of a wave- packet in shallow water. A wavepacket is a special case of a wavetrain, in which the wavenumber is essentially constant across a compact amplitude envelope; this is a single bullet of wave activity. In a wavepacket the amplitude varies more rapidly than the wavenumber vector and it has been known for a long time [8] that the
wavepacket scaling is less robust than the wavetrain scaling, i.e. a wavepacket will quickly evolve into a wavetrain such that wavenumber and amplitude again vary on the same scale. Nevertheless, wavepackets are convenient conceptual building blocks and they are helpful to understand both waves and wave–vortex interactions.
Now, the life cycle begins with a quiescent fluid in which at time t = 0 and location x = (0,0), say, a wavepacket is generated by an irrotational body force.
The wavepacket then travels to a new location where at time t =T it is dissipated by a momentum-conserving body force. Let the large wavenumber be k = (k,0) with k>0 and we use the conventionω >ˆ 0 as before. Then the pseudomomentum p=(p1,0)withp1>0 and the x-component of the ray tracing law (5.54) is
∂p1
∂t +c∂p1
∂x =F1. (5.62)
This uses c =√
g H =constant and U =0. It is easiest to envisage a process of quite rapid wave generation, i.e.F1acts impulsively at t=0 to produce the wave- packet in a short time interval such that the flux term is negligible during generation and therefore∂tp1=F1. Such rapid forcing is not essential, but it gives the easiest picture. For definiteness, we write
wave generation: F =δ(t)(f(x,y),0), (5.63) where f(x,y) ≥ 0 is a smooth envelope for the wavepacket centred around the origin; a Gaussian would do. Directly after the generation we havep1 = f(x,y). The curl of this pseudomomentum field is∇×p= −∂yf , which is positive to the left of the wavepacket and negative to the right. Here left and right are relative to the direction of p. This is the characteristic signature of∇×pfor a wavepacket.
For definiteness, we let f integrate to unity soP=(1,0)at this stage. The impulse I=0 because qL =0, of course.
After generation, the wavepacket propagates without change in shape in a straight line with constant speed c from left to right, i.e.,p1 = f(x−ct,y). BothI =0 andP = (1,0)remain constant during propagation. The subsequent dissipation process can be modelled either impulsively as well or by an exponential attenuation such that F1 = −αp1, whereα > 0 is an exponential damping rate. The latter option is familiar from linear wave dissipation mechanisms. Either way, the sum I+P=(1,0)remains constant because the dissipation is momentum conserving.
In other words, any pseudomomentum lost to dissipation is converted into impulse.
How does the mean flow react in detail to this wavepacket life cycle? We are really only interested in the vortical part of the mean flow response, but in shallow water it is easy enough to write down the complete mean flow equations at O(a2)for small-amplitude slowly varying wavetrains (e.g. [14]). In the present case they are
h˜t+H∇ ãuL =0 and uLt +g∇h˜=pt−1
2∇|u|2+FL−F. (5.64)
A simple derivation of (5.64) uses thatp≈hu/H andh˜ ≈h for slowly varying wavetrains; taking the curl of (5.64) recovers (5.52) at O(a2), in which uL =O(a2) andh˜ = H to sufficient approximation. The complete set (5.64) illustrates that the mean flow inherits a forced version of the modal structure of the linear equations, i.e.
there are two gravity-wave modes and one balanced, vortical mode. It also shows that the mean flow forcing can be viewed as being due to a combination of tran- sience, wavetrain inhomogeneity, external forcing, and dissipation.
During the rapid generation of the wavepacket the third and fourth forcing terms cancel and the first term dominates the second term because of the time derivative.
The same argument applies to the left-hand side of (5.64) and therefore we have h˜ = H and uL =p =(f,0)just after the impulsive wavepacket generation. This initial condition for the mean flow is a compact bullet of x-momentum centred at the origin.
The subsequent evolution of the mean flow consists of the evolution of this initial condition under the additional influence of the forcing terms. The upshot is that there is a persistent generation of weak O(a2)mean flow gravity waves during the propagation of the two-dimensional wavepacket [8].11 The vorticity of the mean flow remains bound to the wavepacket because of qL =0 and therefore∇×uL =
∇×p; a mean flow vorticity probe would detect a vorticity couple flanking the wavepacket, but this vorticity couple would move with the linear wave speed c, not the nonlinear advection speed.
After t =T dissipation becomes active; during dissipation FL is negligible and F = −αp, say. Then (5.62) leads top1= f(x−ct,y)exp(−α(t−T))for t ≥T . During this attenuation process we have
H qLt = −∇×F =α∇×p, (5.65)
which shows how∇×pis transferred into qLin a manner that is consistent with the conservation ofI+P =(1,0). The form of (5.65) indicates that qL evolves “as if” an effective mean force equal to minus the dissipation rate of pseudomomentum were acting on the mean flow; this effective force points in the same direction as p. After a long time t−T 1/αthe wavepacket is practically gone, and so is the wavelike part of the mean flow, which will propagate away from the dissipation site of the wavepacket. What remains behind afterp→0 is the vortical mean flow with vorticity∇×uL =H qL. This is a dipolar vortex couple withI=(1,0)that resembles a smeared-out version of the instantaneous curl of the wavepacket’s pseu- domomentum. The smearing-out is due to the advection of the wavepacket during dissipation, it can be reduced by making the dissipation more rapid.
11Actually, there is also a non-essential resonance effect because the wavepacket forcing terms move with speed c, which is the only available speed in the non-dispersive shallow-water equa- tions. This projects resonantly onto the mean flow gravity waves, which is also a reminder of the beginning of shock formation in shallow water. Still, for our purposes this resonance is artificial and we will not consider it. For example, by adding some dispersion to the equations (e.g. by adding Coriolis forces) this resonance would disappear.
The take-home message from this wavepacket life cycle is that irrotational wave generation changesPbut notI, that propagation through a quiescent medium pre- serves bothP andI, and that dissipation leads to a zero-sum transference ofP intoI. The lasting vortical mean flow response is described by qL and behaves in an easy, generic manner. The full mean flow response also contains parts to do with wavelike mean flow dynamics, which are not generic and complicated.
If the life cycle is repeated many times then qL would grow secularly at the dissipation site, which would in time lead to the spin-up of a substantial vortex couple there. This is a model example of a strong wave–vortex interaction.