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 ${{\cal L} = ({\bf R}^n, \hbox{vol})}$ and Eulerian space ${{\cal E} = ({\bf R}^n, \eta, \hbox{vol})}$, we let ${M}$ be the space of volume-preserving, orientation-preserving maps ${\Phi: {\cal L} \rightarrow {\cal E}}$ from Lagrangian space to Eulerian space. Given a curve ${\Phi: {\bf R} \rightarrow M}$, we can define the Lagrangian velocity field ${\dot \Phi: {\bf R} \times {\cal L} \rightarrow T{\cal E}}$ as the time derivative of ${\Phi}$, and the Eulerian velocity field ${u := \dot \Phi \circ \Phi^{-1}: {\bf R} \times {\cal E} \rightarrow T{\cal E}}$. The volume-preserving nature of ${\Phi}$ ensures that ${u}$ is a divergence-free vector field:

$\displaystyle \nabla \cdot u = 0. \ \ \ \ \ (1)$

If we formally define the functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} |u(t,x)|^2\ dx dt = \frac{1}{2} \int_R \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx dt$

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

$\displaystyle [\partial_t + u \cdot \nabla] u = - \nabla p$

$\displaystyle \nabla \cdot u = 0$

for some pressure field ${p: {\bf R} \times {\cal E} \rightarrow {\bf R}}$. As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\cal E}} |u(t,x)|^2\ dx = \frac{1}{2} \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx;$

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

$\displaystyle \int_{{\cal E}} u(t,x)\ dx$

and total angular momentum

$\displaystyle \int_{{\cal E}} x \wedge u(t,x)\ dx;$

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

$\displaystyle \int_{\Phi(\gamma)} u^*$

for any closed loop ${\gamma}$ in ${{\cal L}}$, or equivalently pointwise conservation of the Lagrangian vorticity ${\Phi^* \omega = \Phi^* du^*}$, where ${u^*}$ is the ${1}$-form associated with the vector field ${u}$ using the Euclidean metric ${\eta}$ on ${{\cal E}}$, with ${\Phi^*}$ denoting pullback by ${\Phi}$.

It turns out that one can generalise the above calculations. Given any self-adjoint operator ${A}$ on divergence-free vector fields ${u: {\cal E} \rightarrow {\bf R}}$, we can define the functional

$\displaystyle J_A[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx dt;$

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

$\displaystyle [\partial_t + u \cdot \nabla] Au + (\nabla u) \cdot Au= - \nabla \tilde p \ \ \ \ \ (2)$

$\displaystyle \nabla \cdot u = 0$

for some pressure field ${\tilde p: {\bf R} \times {\cal E} \rightarrow {\bf R}}$, where ${(\nabla u) \cdot Au}$ in coordinates is ${\partial_i u_j Au_j}$ with the usual summation conventions. (When ${A=1}$, ${(\nabla u) \cdot Au = \nabla(\frac{1}{2} |u|^2)}$, and this term can be absorbed into the pressure ${\tilde p}$, and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx.$

If the operator ${A}$ commutes with rigid motions on ${{\cal E}}$, then we have conservation of total momentum

$\displaystyle \int_{{\cal E}} Au(t,x)\ dx$

and total angular momentum

$\displaystyle \int_{{\cal E}} x \wedge Au(t,x)\ dx,$

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

$\displaystyle \int_{\Phi(\gamma)} (Au)^*$

or pointwise conservation of the Lagrangian vorticity ${\Phi^* \theta := \Phi^* d(Au)^*}$. 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 ${n=2}$, when ${A}$ is the inverse derivative ${A = |\nabla|^{-1} = (-\Delta)^{-1/2}}$. The vorticity ${\theta = d(Au)^*}$ is a ${2}$-form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write ${u = (u_1,u_2)}$, then

$\displaystyle \theta = \partial_{x_1} |\nabla|^{-1} u_2 - \partial_{x_2} |\nabla|^{-1} u_1.$

Since ${u}$ is also divergence-free, we may therefore write

$\displaystyle u = (- \partial_{x_2} \psi, \partial_{x_1} \psi )$

where the stream function ${\psi}$ is given by the formula

$\displaystyle \psi = |\nabla|^{-1} \theta.$

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

$\displaystyle [\partial_t + u \cdot \nabla] \theta = 0 \ \ \ \ \ (3)$

$\displaystyle u = (-\partial_{x_2} |\nabla|^{-1} \theta, \partial_{x_1} |\nabla|^{-1} \theta).$

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

$\displaystyle \frac{1}{2} \int_{{\bf R}^2} u\cdot |\nabla|^{-1} u\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta \psi\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta |\nabla|^{-1} \theta\ dx$

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

$\displaystyle \int_{{\bf R}^2} Au\ dx$

(which up to trivial transformations is also expressible in impulse form as ${\int_{{\bf R}^2} \theta x\ dx}$, after integration by parts), and conservation of angular momentum

$\displaystyle \int_{{\bf R}^2} x \wedge Au\ dx$

(which up to trivial transformations is ${\int_{{\bf R}^2} \theta |x|^2\ dx}$). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity ${\Phi^* \theta}$, thus ${\theta}$ is transported by the flow (which is also evident from (3). In particular, all integrals of the form ${\int F(\theta)\ dx}$ for a fixed function ${F}$ are conserved by the flow.

— 1. Euler-Lagrange calculations —

We now justify the claim that stationary points of the functional ${J_A[\Phi]}$ obey (2). We consider continuous deformations ${s \mapsto \Phi(s)}$ of the critical point ${\Phi}$, thus ${\Phi}$ now depends on both ${s}$ and ${t}$. We already have the Eulerian velocity field ${u}$, which is related to the ${t}$ derivative of ${\Phi}$ by the formula

$\displaystyle \partial_t \Phi^i = u^i( \Phi );$

similarly we may introduce a deformation field ${v}$ by

$\displaystyle \partial_s \Phi^i = v^i( \Phi ).$

The vector field ${v}$ is divergence free and has to obey appropriate vanishing conditions at infinity, but is otherwise unconstrained. If we compute ${\partial_t \partial_s \Phi^i = \partial_s \partial_t \Phi^i}$ using the above two equations and the chain rule, we arrive at the “zero-curvature” condition

$\displaystyle \partial_t v^i + u^j \partial_j v^i = \partial_s u^i + v^j \partial_j u^i$

so in particular

$\displaystyle \partial_s u^i = \partial_t v^i + u^j \partial_j v^i - v^j \partial_j u^i. \ \ \ \ \ (4)$

On the other hand, as ${\Phi}$ is a critical point, we have

$\displaystyle \partial_s J_A[\Phi] = 0$

when ${s=0}$. Differentiating under the integral sign and using the self-adjoint nature of ${A}$, the left-hand side is

$\displaystyle \int_{\bf R} \int_{{\cal E}} (\partial_s u^i) A u^i\ dx dt.$

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

$\displaystyle \int_{\bf R} \int_{{\cal E}} v^i ( - \partial_t A u^i - u^j \partial_j Au^i - Au^j \partial_i u^j)\ dx dt.$

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