The previous considerations made clear that the exponential surge in packet-integr- ated pseudomomentum is compensated by the loss of impulse of the vortex cou- ple far away. Still, there is a lingering concern about the local structure of uL at the wavepacket. For instance, the exact GLM relation (5.16) for periodic zonally symmetric flows suggests that uL at the core of the wavepacket might make a large amplitude excursion because it might follow the local pseudomomentump1, which is growing exponentially in time. This is an important consideration, because a large uL might induce wave breaking or other effects.
We can study this problem easily in a simple two-dimensional set-up, brushing aside concerns that our two-dimensional theory may be misleading for the three- dimensional stratified case. In particular, we look at a wavepacket centred at the origin of an(x,y)coordinate system such that at t = 0 the pseudomomentum is p = (1,0)f(x,y)for some envelope function f that is proportional to the wave action density. This is the same wavepacket set-up as in Sect. 5.3.3. At all times the local Lagrangian mean flow at O(a2)induced by the wavepacket is the Bretherton flow, which by qL =0 is the solution of
uxL+vLy =0 and vLx −uLy =∇×p= −fy(x,y). (5.105) We imagine that the wavepacket is exposed to a pure straining basic flow U = (−x,+y), which squeezes the wavepacket in x and stretches it in y. We ignore intrinsic wave propagation relative to U , which implies that the wave action density f is advected by U , i.e. Dtf =0. We then obtain the refracted pseudomomentum as p=(α,0)f(αx,y/α) and ∇×p= −fy(αx,y/α). (5.106) Hereα = exp(t) ≥ 1 is the scale factor at time t ≥ 0 and (5.106) shows thatp1 grows exponentially whilst∇×pdoes not; in fact∇×pis materially advected by U , just as the wave action density f and unlike the pseudomomentum densityp. This is a consequence of the stretching in the transverse y-direction, which diminishes the curl because it makes the x-pseudomomentum vary more slowly in y. Thus whilst there is an exponential surge inp1there is none in∇×p.
In an unbounded domain we can go one step further and explicitly compute uL at the core of the wavepacket, say. We use Fourier transforms defined by
FT{f}(k,l)=
e−i[kx+l y]f(x,y)d xd y (5.107) and
f(x,y)= 1 4π2
e+i[kx+l y]FT{f}(k,l)dkdl. (5.108) The transforms of uL and ofp1are related by
FT{uL}(k,l)= l2
k2+l2FT{p1}(k,l). (5.109) This follows fromp=(p1,0)and the intermediate introduction of a stream func- tionψsuch that(uL, vL)=(−ψy,+ψx)and therefore∇2ψ = −p1y. The scale- insensitive pre-factor varies between zero and one and quantifies the projection onto non-divergent vector fields in the present case. This relation by itself does not rule
out exponential growth of uL in some proportion to the exponential growth ofp1. We need to look at the spectral support ofp1as the refraction proceeds.
We denote the initialp1forα=1 byp11and then the pseudomomentum for other values ofαispα1(x,y)=αp11(αx,y/α). The transform is found to be
FT{pα1}(k,l)=αFT{p11}(k/α, αl). (5.110) This shows that with increasingαthe spectral support shifts towards higher values of k and lower values of l. The value of uL at the wavepacket core x = y = 0 is the total integral of (5.109) over the spectral plane, which using (5.110) can be written as
uL(0,0)= 1 4π2
l2
k2+l2FT{pα1}(k,l)dkdl
= α 4π2
l2
α4k2+l2FT{p11}(k,l)dkdl (5.111) after renaming the dummy integration variables. This is as far as we can go without making further assumptions about the shape of the initial wavepacket.
For instance, if the wavepacket is circularly symmetric initially, thenp11depends only on the radius r =
x2+y2and FT{p11}depends only on the spectral radius κ =√
k2+l2. In this case (5.111) can be explicitly evaluated by integrating over the angle in spectral space and yields the simple formula
uL(0,0)= α
α2+1p11(0,0)= 1
α2+1pα1(0,0). (5.112) The pre-factor in the first expression has maximum value 1/2 at α = 1, which implies that the maximal Lagrangian mean velocity at the wavepacket core is the initial velocity, when the wavepacket is circular. At this initial time uL =0.5p1at the core and thereafter uL decays; there is no growth at all.
So this proves that there is no surge of local mean velocity even though there is a surge of local pseudomomentum. This simple example serves as a useful illustration of how misleading zonally symmetric wave–mean interaction theory can be when we try to understand more general wave–vortex interactions.
Finally, how about a wavepacket that is not circularly symmetric at t = 0?
The worst case scenario is an initial wavepacket that is long in x and narrow in y; this corresponds to values of α near zero and the second expression in (5.112) then shows that the mean velocity at the core is almost equal to the pseudomomentum. This scenario recovers the predictions of zonally symmetric theory.
The subsequent squeezing in x now amplifies the pseudomomentum and this leads to a transient growth of uL in proportion, at least whilst the wavepacket still has approximately the initial aspect ratio. However, eventually the aspect ratio
reverses and the wavepacket becomes short in x and wide in y; this corresponds toα much larger than unity. Eventuallyαbecomes large and uLdecays as 1/α=exp(−t).