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).
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
Also, if we require to be divergence-free at time zero, it remains divergence-free at all later times.
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
Finally, the last component of (8) is
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
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.