5.2 Lagrangian Mean Flow and Pseudomomentum
5.2.2 Pseudomomentum and the Circulation Theorem
The circulation around a closed material loop Cξ, say, is defined in a two- dimensional domain by
=
Cξu(x,t)ãd x=
Aξ∇×u d xd y. (5.9) The second form uses Stokes’s theorem and Aξ is the area enclosed by Cξ, i.e.
Cξ =∂Aξ. As written, the material loopCξ is formed by theactualpositions of a certain set of fluid particles. Under the assumption3that the map
x →x+ξ (5.10)
is smooth and invertible, we can associate with each such actual position also a meanposition of the respective particle, and the set of all mean positions then forms another closed loopC, say. In other words, we define the mean loopCvia
x ∈C ⇔ x+ξ(x,t)∈Cξ. (5.11)
This allows rewriting the contour integral in (5.9) in terms ofC, which mathemati- cally amounts to a variable substitution in the integrand. The only non-trivial step is the transformation of the line element d x, which is
d x→d(x+ξ)=d x+(d xã∇)ξ. (5.12) In index notation this corresponds to
d xi →d xi +ξi,jd xj. (5.13) This leads to
=
C(ui(x+ξ,t)+ξj,iuj(x+ξ,t))d xi (5.14) after renaming the dummy indices. The integration domain is now a mean material loop and therefore we can average (5.14) by simply averaging the factors multiply- ing the mean line element d x. The first term brings in the Lagrangian mean velocity and the second term serves as the definition of the pseudomomentum, i.e.
=
C(uL−p)ãd x where pi = −ξj,iuj(x+ξ,t) (5.15) is the GLM definition of the pseudomomentum vector; the minus sign is conven- tional and turns out to be convenient in wave applications. This exact kinematic relation shows that the mean circulation is due to a cooperation of uLandp, i.e. both the mean flow and the wave-related pseudomomentum contribute to the circulation.
3This can fail for large waves.
In perfect fluid flow the circulation is conserved by Kelvin’s theorem and hence =. Just asis constant becauseCξ follows the actual fluid flow we now also have that is constant becauseC follows the Lagrangian mean flow. This mean circulation conservation statement alone has powerful consequences if the flow is zonally periodic and the Eulerian-averaging operation consists of zonal averaging, which is the typical setup in atmospheric wave–mean interaction theory. In this peri- odic case a material line traversing the domain in the zonal x-direction qualifies as a closed loop for Kelvin’s circulation theorem. By construction,∂x(. . .)=0 for any mean field, and therefore a straight line in the zonal direction qualifies as a mean closed loop. The mean conservation theorem then implies theorem I of [2], i.e.
DLuL =DLp1, (5.16)
where p1is the zonal component ofp. This is an exact statement and its straight- forward extension to forced–dissipative flows constitutes the most general state- ment about so-called non-acceleration conditions, i.e. wave conditions under which the zonal mean flow is not accelerated. These are powerful statements, but their validity is restricted to the simple geometry of periodic flows combined with zonal averaging.
In order to exploit the mean form of Kelvin’s circulation theorem for more gen- eral flows, we need to derive its local counterpart in terms of vorticity or potential vorticity. Indeed, the mean circulation theorem implies a mean material conservation law for a mean PV by the same standard construction that yields (5.2) from Kelvin’s circulation theorem. Specifically, the invariance of in the second form in (5.9) for arbitrary infinitesimally small material areasAξ implies the material invariance of∇×u d xd y. The area element d xd y is not a material invariant in compressible shallow-water flow, but the mass element h d xd y is. Factorizing with h leads to
D Dt
∇×u h h d xd y
=0 ⇒ D Dt
∇×u h
=0, (5.17)
which is (5.2) for perfect flow. Mutatis mutandis, the same argument applied to (5.15) yields
qL = ∇×(uL−p)
h˜ and DLqL =0, (5.18)
provided the mean layer depth h is defined such that˜ h d xd y is the mean mass˜ element, which is invariant following uL. This is true ifh satisfies the mean conti-˜ nuity equation
DLh˜+ ˜h∇ ãuL =0. (5.19)
Unfortunately,h˜ =hLin general, which is a disadvantage of GLM theory. It can be shown thath˜(x,t)=h(x+ξ,t)J(x,t), where J =det(δi j+ξi,j)is the Jacobian of the map (5.10).
The mean circulation theorem is an exact statement, so in particular it is not limited to small wave amplitudes. It shows that the Lagrangian mean flow inherits a version of the constraints that Kelvin’s circulation theorem puts on strong wave–
vortex interactions. For example, in irrotational flows we have q =0 and therefore
qL =0 ⇒ ∇×uL =∇×p. (5.20)
This shows that if∇×pis uniformly bounded at O(a2)in time then so is∇×uL, which rules out strong interactions based on mean flow vorticity. Of course, even though uL andphave the same curl they can still be different vector fields. This can be either because∇ ãuL is markedly different from∇ ãpor because uL andp satisfy different boundary conditions at impermeable walls (see [2] for an example involving sound waves). Any strong wave–vortex interaction in the present case of irrotational flow must therefore involve wavelike behaviour of the mean flow itself, with significant values of∇ ãuL for instance.
If q =0, then (5.20) is replaced by
∇×uL =∇×p+ ˜hqL. (5.21)
This illustrates the scope for further changes in∇×uLdue to dilation effects medi- ated by variableh (i.e. vortex stretching) or due to material advection of different˜ values of qLinto the region of interest. The latter process requires the existence of a PV gradient, as discussed earlier. Obviously, any knowledge of bounds on changes inh and q˜ L can be converted into bounds on changes in∇×uL by using the exact (5.21) as a constraint.