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.