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The Euler equations for three-dimensional incompressible inviscid fluid flow are

where is the velocity field, and is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as or formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

where is the active scalar, and is the velocity field. The SQG equations are often used as a toy model for the 3D Euler equations, as they share many of the same features (e.g. vortex stretching); see this paper of Constantin, Majda, and Tabak for more discussion (or this previous blog post).

I recently found a more direct way to connect the two equations. We first recall that the Euler equations can be placed in *vorticity-stream* form by focusing on the vorticity . Indeed, taking the curl of (1), we obtain the vorticity equation

while the velocity can be recovered from the vorticity via the Biot-Savart law

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field that is being transported by a divergence-free vector field , which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field that is being transported (in the Lie derivative sense) by a divergence-free vector field , which is a linear function of the vector field as per (5). However, the system (4), (5) is in three dimensions whilst (2), (3) is in two spatial dimensions, the dynamical field is a scalar field for SQG and a vector field for Euler, and the relationship between the velocity field and the dynamical field is given by a zeroth order Fourier multiplier in (3) and a order operator in (5).

However, we can make the two equations more closely resemble each other as follows. We first consider the generalisation

where is an invertible, self-adjoint, positive-definite zeroth order Fourier multiplier that maps divergence-free vector fields to divergence-free vector fields. The Euler equations then correspond to the case when is the identity operator. As discussed in this previous blog post (which used to denote the inverse of the operator denoted here as ), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

the Kelvin circulation theorem, and conservation of helicity

Also, if we require to be divergence-free at time zero, it remains divergence-free at all later times.

Let us consider “two-and-a-half-dimensional” solutions to the system (6), (7), in which do not depend on the vertical coordinate , thus

and

but we allow the vertical components to be non-zero. For this to be consistent, we also require to commute with translations in the direction. As all derivatives in the direction now vanish, we can simplify (6) to

where is the two-dimensional material derivative

Also, divergence-free nature of then becomes

In particular, we may (formally, at least) write

for some scalar field , so that (7) becomes

The first two components of (8) become

which rearranges using (9) to

Formally, we may integrate this system to obtain the transport equation

Finally, the last component of (8) is

At this point, we make the following choice for :

where is a real constant and is the Leray projection onto divergence-free vector fields. One can verify that for large enough , is a self-adjoint positive definite zeroth order Fourier multiplier from divergence free vector fields to divergence-free vector fields. With this choice, we see from (10) that

so that (12) simplifies to

This implies (formally at least) that if vanishes at time zero, then it vanishes for all time. Setting , we then have from (10) that

and from (11) we then recover the SQG system (2), (3). To put it another way, if and solve the SQG system, then by setting

then solve the modified Euler system (6), (7) with given by (13).

We have , so the Hamiltonian for the modified Euler system in this case is formally a scalar multiple of the conserved quantity . The momentum for the modified Euler system is formally a scalar multiple of the conserved quantity , while the vortex stream lines that are preserved by the modified Euler flow become the level sets of the active scalar that are preserved by the SQG flow. On the other hand, the helicity vanishes, and other conserved quantities for SQG (such as the Hamiltonian ) do not seem to correspond to conserved quantities of the modified Euler system. This is not terribly surprising; a low-dimensional flow may well have a richer family of conservation laws than the higher-dimensional system that it is embedded in.

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.

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