As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with *Lagrangian space* and *Eulerian space* , we let be the space of volume-preserving, orientation-preserving maps from Lagrangian space to Eulerian space. Given a curve , we can define the *Lagrangian velocity field* as the time derivative of , and the *Eulerian velocity field* . The volume-preserving nature of ensures that is a divergence-free vector field:

If we formally define the functional

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

for some pressure field . As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

the rigid motion symmetries of Eulerian space give conservation of the total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

for any closed loop in , or equivalently pointwise conservation of the Lagrangian vorticity , where is the -form associated with the vector field using the Euclidean metric on , with denoting pullback by .

It turns out that one can generalise the above calculations. Given any self-adjoint operator on divergence-free vector fields , we can define the functional

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

for some pressure field , where in coordinates is with the usual summation conventions. (When , , and this term can be absorbed into the pressure , and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

If the operator commutes with rigid motions on , then we have conservation of total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

or pointwise conservation of the Lagrangian vorticity . These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions , when is the inverse derivative . The vorticity is a -form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write , then

Since is also divergence-free, we may therefore write

where the stream function is given by the formula

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As commutes with rigid motions, we also have (formally, at least) conservation of momentum

(which up to trivial transformations is also expressible in impulse form as , after integration by parts), and conservation of angular momentum

(which up to trivial transformations is ). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity , thus is transported by the flow (which is also evident from (3). In particular, all integrals of the form for a fixed function are conserved by the flow.

** — 1. Euler-Lagrange calculations — **

We now justify the claim that stationary points of the functional obey (2). We consider continuous deformations of the critical point , thus now depends on both and . We already have the Eulerian velocity field , which is related to the derivative of by the formula

similarly we may introduce a deformation field by

The vector field is divergence free and has to obey appropriate vanishing conditions at infinity, but is otherwise unconstrained. If we compute using the above two equations and the chain rule, we arrive at the “zero-curvature” condition

On the other hand, as is a critical point, we have

when . Differentiating under the integral sign and using the self-adjoint nature of , the left-hand side is

Inserting (4) and integrating by parts (and using the divergence-free nature of ), this expression can be rewritten as

Since is essentially an arbitrary divergence-free vector field, the expression inside parentheses must vanish, and the equation (2) follows.

## 2 comments

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6 March, 2014 at 8:43 pm

AnonymousAt the end of the second sentence, do you mean ‘these equations’ instead of ‘this equation’.

[Corrected, thanks - T.]25 June, 2014 at 1:48 am

Navier-Stokes Fluid Computers | Combinatorics and more[…] here is a follow up post on Tao’s blog (and a few more II, III), and a post on Shtetl […]