We now turn to the wave–vortex interactions that go together with refraction by a single vortex. Here we follow [16] and truncate the wavepacket trajectory by putting irrotational wave sources and sinks a finite distance L apart. Moreover, we consider a steady wavetrain generated by the source “loudspeaker” and absorbed by the sink
“anti-loudspeaker”. The finite wavetrain allows us to see the effect of pseudomo- mentum changes at O(). This set-up is described in the somewhat busy Fig. 5.7.
The figure shows the loudspeakers and the vortex at a distance D from the steady wavetrain. The circumferential basic velocity with magnitude U is indicated by˜ the dashed line. The loudspeakers are slightly angled because of the mean flow
13The azimuthal symmetry of this flow induces a further ray invariance, namely l x−ky. However, this “angular momentum” invariant is not needed here.
U(r)~ D
RV
B A R
R
Fig. 5.7 A steady wavetrain travels left to right from the wavemaker A to the wave absorber B.
The vortex refracts the wavetrain and there is a net recoil force RA+RB in the y-direction on the loudspeakers. Concomitant is an effective remote recoil force RV felt by the vortex such that RA+RB+RV =0. The recoil is mediated by a leftward material displacement of the vortex at O(a2)due to the Bretherton return flow
refraction and the wave crests are counter-rotating relative to the vortex. One can see by inspection that the y-component of the intrinsic group velocity at the wavemaker A on the left cancels the y-component of the basic velocity there and vice versa at the wave absorber B on the right.
Now, at the irrotational loudspeakers we have FL =F and RA,B = −
A,B
FLd xd y (5.86)
are the respective recoil forces exerted on the wavemaker and wave absorber (we use H =1 for simplicity). There is an equal-and-opposite push in the x-direction and a net recoil in the negative y-direction due to the refraction, i.e.
RA+RB+RV =0 where RV =(0,RV) with RV >0, (5.87) say. The net recoil−RV equals the net pseudomomentum generation per unit time due to the refraction.
How does the mean flow impulse change? The impulse plus pseudomomentum conservation law (5.56) for a steady wavetrain yields
d(I+P) dt =dI
dt =
FLd xd y= −(RA+RB)=RV. (5.88) This shows that the mean flow impulse should change in order to compensate for the net recoil in the y-direction exerted on the loudspeakers. The total impulse due to a
single vortex with nonzero net circulation depends on the origin of the coordinate system, butchanges in the impulse due to movement of the vortex are coordinate- independent. In particular, the mean flow impulse is
I=
(y,−x)qLd xd y=(Y,−X) (5.89) if(X,Y)are the coordinates of the vortex centroid. Therefore (5.88) implies
dY
dt =0 and d X
dt = −RV
. (5.90)
This is a surprising result because it means that the vortex must move to the left in Fig. 5.7. Where does the velocity field come from that achieves this material displacement?
The answer comes from the wave–mean response at O(a2)to the presence of the finite wavetrain. In fact, to compute this response at leading order it is not necessary to include the vortex. Without the vortex there is no tilt in the loudspeakers and the wave crests do not rotate. The Lagrangian mean flow in the presence of the irrotational steady wavetrain is then determined from
∇ ãuL =0 and qL =0 ⇒ ∇ìuL =∇ìp. (5.91) These equations express that uL is the least-squares projection ofponto the space of non-divergent vector fields. The pseudomomentum curl is positive above the wavetrain and negative below, so uL is the incompressible flow described by a vortex couple with precisely this curl. Away from the wavetrain uL = u and the flow resembles the standard dipole flow familiar from elementary fluid mechanics.
The streamlines of this dipole flow are indicated by the solid lines in Fig. 5.7; we suggested the name “Bretherton flow” for this characteristic mean flow response to a wavepacket, because its description goes back to [8].
Crucially, the Bretherton flow points backward, i.e. in the negative x-direction, at the vortex location above the wavetrain. Therefore the Bretherton flow does indeed push the vortex to the left and it has been checked in [16] that it does so with pre- cisely the right magnitude to be consistent with (5.90).
We called this action-at-a-distance of the wavetrain on the vortex “remote recoil”
in order to stress the non-local nature of this wave–vortex interaction. After all, the waves and the vortex do not overlap in physical space. The term “recoil” is also apt because the movement of the vortex is consistent with the action of a compact body force on the vortex with net integral equal to RV. Such a force would be relevant in a parametrization problem in which the small-scale wavetrain is not explicitly resolved but modelled. This force will produce positive vorticity to the left of the vortex and negative vorticity to its right, which would lead to a movement of the vortex centroid to the left as required. In this case the vorticity moves although the fluid particles do not.
Finally, it can be shown that the remote recoil idea remains valid at O(2)if the loudspeakers recede to infinity and the pseudomomentum generation is due to the weak O(2)net scattering of the waves into the lee of the vortex.
So whilst the set-up in Fig. 5.7 is certainly very special it is not artificial; the recoil is real.