The Boussinesq equations for inviscid, incompressible two-dimensional fluid flow in the presence of gravity are given by

where is the velocity field, is the pressure field, and is the density field (or, in some physical interpretations, the temperature field). In this post we shall restrict ourselves to formal manipulations, assuming implicitly that all fields are regular enough (or sufficiently decaying at spatial infinity) that the manipulations are justified. Using the material derivative , one can abbreviate these equations as

One can eliminate the role of the pressure by working with the *vorticity* . A standard calculation then leads us to the equivalent “vorticity-stream” formulation

of the Boussinesq equations. The latter two equations can be used to recover the velocity field from the vorticity by the Biot-Savart law

It has long been observed (see e.g. Section 5.4.1 of Bertozzi-Majda) that the Boussinesq equations are very similar, though not quite identical, to the three-dimensional inviscid incompressible Euler equations under the hypothesis of axial symmetry (with swirl). The Euler equations are

where now the velocity field and pressure field are over the three-dimensional domain . If one expresses in polar coordinates then one can write the velocity vector field in these coordinates as

If we make the axial symmetry assumption that these components, as well as , do not depend on the variable, thus

then after some calculation (which we give below the fold) one can eventually reduce the Euler equations to the system

However, this heuristic is not rigorous; the above calculations do not actually give an embedding of the Boussinesq equations into Euler. (The equations do match on the cylinder , but this is a measure zero subset of the domain, and so is not enough to give an embedding on any non-trivial region of space.) Recently, while playing around with trying to embed other equations into the Euler equations, I discovered that it is possible to make such an embedding into a *four*-dimensional Euler equation, albeit on a slightly curved manifold rather than in Euclidean space. More precisely, we use the Ebin-Marsden generalisation

of the Euler equations to an arbitrary Riemannian manifold (ignoring any issues of boundary conditions for this discussion), where is a time-dependent vector field, is a time-dependent scalar field, and is the covariant derivative along using the Levi-Civita connection . In Penrose abstract index notation (using the Levi-Civita connection , and raising and lowering indices using the metric ), the equations of motion become

in coordinates, this becomes

where the Christoffel symbols are given by the formula

where is the inverse to the metric tensor . If the coordinates are chosen so that the volume form is the Euclidean volume form , thus , then on differentiating we have , and hence , and so the divergence-free equation (10) simplifies in this case to . The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

and can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The specific four-dimensional manifold in question is the space with metric

and solutions to the Boussinesq equation on can be transformed into solutions to the Euler equations on this manifold. This is part of a more general family of embeddings into the Euler equations in which passive scalar fields (such as the field appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric ). I am writing the details below the fold (partly for my own benefit).

Firstly, it is convenient to transform the Euler equations on an arbitrary Riemannian manifold to a *covelocity* formulation, by introducing the covelocity -form , as this formulation allows one to largely avoid having to work with covariant derivatives or Christoffel symbols. Lowering indices in the Euler equation (9) then gives the system

Noting that , and introducing the modified pressure , we arrive at the system

As the Levi-Civita connection is torsion-free (or equivalently, one has the symmetry , we have , thus we arrive at the system

which is equivalent to (and thus embeddable in) the Euler equations. The advantage of this formulation is that the metric now plays no role whatsoever in the main equation (11), and only appears in (12) and (13). One can also interpret the expression as the Lie derivative of the covelocity along the velocity .

If one works in a coordinate system in which the volume form is Euclidean (that is to say, ), then the Riemannian divergence is the same as the ordinary divergence ; this can be seen either by viewing the divergence as the adjoint of the gradient operator with respect to the volume form, or else by differentiating the condition to conclude that , which implies that and hence . But actually, as already observed in my previous paper, one can replace with “for free”, even if one does not have the Euclidean volume form condition , if one is prepared to add an additional “dummy” dimension to the manifold . More precisely, if solves the system

then these fields obey the same system, and hence (since ) solve (11), (12), (13). Thus the above system is embeddable into the Euler equations in one higher dimension. To embed the Boussinesq equations into the four-dimensional Euler equations mentioned previously, it thus suffices to embed these equations into the system (14)–(16) for the three-dimensional manifold with metric

Let us more generally consider the system (14)–(16) under the assumption that splits as a product of two manifolds , with all data independent of the coordinates (but, for added flexibility, we do *not* assume that the metric on splits as the direct sum of metrics on and , allowing for twists and shears). This, if we use Roman indices to denote the coordinates and Greek indices to denote the coordinates (with summation only being restricted to these coordinates), and denote the inverse metric by the tensor with components , then we have

and then the system (14)–(16) in these split coordinates become

We can view this as a system of PDE on the smaller manifold , which is then embedded into the Euler equations. Introducing the material derivative , this simplifies slightly to

We substitute the third and fourth equations into the first, then drop the fourth (as it can be viewed as a definition of the field , which no longer plays any further role), to obtain

We can reverse the pressure modification by writing

to move some derivatives off of the covelocity fields and onto the metric, so that the system now becomes

At this point one can specialise to various special cases to obtain some possibly simpler dynamics. For instance, one could set to be flat (so that is constant), and set and to both vanish, then we obtain the simple-looking (but somewhat overdetermined) system

This is basically the system I worked with in my previous paper. For instance, one could set one of the components of , say to be identically , and to be an arbitrary divergence-free vector field for that component, then , and all the other components of are transported by this static velocity field, leading for instance to exponential growth of vorticity if has a hyperbolic fixed point and the initial data of the components of other than are generic. (Alas, I was not able to modify this example to obtain something more dramatic than exponential growth, such as finite time blowup.)

Alternatively, one can set to vanish, leaving one with

If consists of a single coordinate , then on setting , this simplifies to

If we take to be with the Euclidean metric , and set (so that has the metric (17)), then one obtains the Boussinesq system (1)–(3), giving the claimed embedding.

Now we perform a similar analysis for the axially symmetric Euler equations. The cylindrical coordinate system is slightly inconvenient to work with because the volume form is not Euclidean. We therefore introduce Turkington coordinates

to rewrite the metric as

so that the volume form is now Euclidean, and the Euler equations become (14)–(16). Splitting as before, with being the two-dimensional manifold parameterised by , and the one-dimensional manifold parameterised by , the symmetry reduction (18)–(21) gives us (26)–(29) as before. Explicitly, one has

Setting to eliminate the pressure , we obtain

Since , , , , and , we obtain the system (5)–(8).

Returning to the general form of (22)–(25), one can obtain an interesting transformation of this system by writing for the inverse of (caution: in general, this is *not* the restriction of the original metric on to ), and define the modified covelocity

then by the Leibniz rule

Replacing the covelocity with the modified covelocity, this becomes

We thus have the system

where

and so if one writes

we obtain

For each , we can specify as an arbitrary smooth function of space (it has to be positive definite to keep the manifold Riemannian, but one can add an arbitrary constant to delete this constraint), and as an arbitrary time-independent exact -form. Thus we obtain an incompressible Euler system with two new forcing terms, one term conming from passive scalars , and another term that sets up some rotation between the components , with the rotation speed determined by a passive scalar .

Remark 1As a sanity check, one can observe that one still has conservation of the kinetic energy, which is equal toand can be expressed in terms of and as

One can check this is conserved by the above system (mainly due to the antisymmetry of ).

As one special case of this system, one can work with a one-dimensional fibre manifold , and set and for the single coordinate of this manifold. This leads to the system

where is some smooth time-independent exact -form that one is free to specify. This resembles an Euler equation in the presence of a “magnetic field” that rotates the velocity of the fluid. I am currently experimenting with trying to use this to force some sort of blowup, though I have not succeeded so far (one would obviously have to use the pressure term at some point, for if the pressure vanished then one could keep things bounded using the method of characteristics).

Filed under: expository, math.AP Tagged: Boussinesq equations, incompressible Euler equations, Riemannian geometry