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

\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_x = -\partial_x p \ \ \ \ \ (1)

\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_y = \rho - \partial_y p \ \ \ \ \ (2)

\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) \rho = 0 \ \ \ \ \ (3)

\displaystyle \partial_x u_x + \partial_y u_y = 0 \ \ \ \ \ (4)

where {u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2} is the velocity field, {p: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}} is the pressure field, and {\rho: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}} 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 {D_t := \partial_t + u_x \partial_x + u_y \partial_y}, one can abbreviate these equations as

\displaystyle D_t u_x = -\partial_x p

\displaystyle D_t u_y = \rho - \partial_y p

\displaystyle D_t \rho = 0

\displaystyle \partial_x u_x + \partial_y u_y = 0.

One can eliminate the role of the pressure {p} by working with the vorticity {\omega := \partial_x u_y - \partial_y u_x}. A standard calculation then leads us to the equivalent “vorticity-stream” formulation

\displaystyle D_t \omega = \partial_x \rho

\displaystyle D_t \rho = 0

\displaystyle \omega = \partial_x u_y - \partial_y u_x

\displaystyle \partial_x u_y + \partial_y u_y = 0

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

\displaystyle u_x := -\partial_y \Delta^{-1} \omega; \quad u_y = \partial_x \Delta^{-1} \omega.

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

\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p

\displaystyle \nabla \cdot u = 0

where now the velocity field {u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3} and pressure field {p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}} are over the three-dimensional domain {{\bf R}^3}. If one expresses {{\bf R}^3} in polar coordinates {(z,r,\theta)} then one can write the velocity vector field {u} in these coordinates as

\displaystyle u = u^z \frac{d}{dz} + u^r \frac{d}{dr} + u^\theta \frac{d}{d\theta}.

If we make the axial symmetry assumption that these components, as well as {p}, do not depend on the {\theta} variable, thus

\displaystyle \partial_\theta u^z, \partial_\theta u^r, \partial_\theta u^\theta, \partial_\theta p = 0,

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

\displaystyle \tilde D_t \omega = \frac{1}{r^4} \partial_z \rho \ \ \ \ \ (5)

\displaystyle \tilde D_t \rho = 0 \ \ \ \ \ (6)

\displaystyle \omega = \frac{1}{r} (\partial_z u^r - \partial_r u^z) \ \ \ \ \ (7)

\displaystyle \partial_z(ru^z) + \partial_r(ru^r) = 0 \ \ \ \ \ (8)

where {\tilde D_t := \partial_t + u^z \partial_z + u^r \partial_r} is the modified material derivative, and {\rho} is the field {\rho := (r u^\theta)^2}. This is almost identical with the Boussinesq equations except for some additional powers of {r}; thus, the intuition is that the Boussinesq equations are a simplified model for axially symmetric Euler flows when one stays away from the axis {r=0} and also does not wander off to {r=\infty}.

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 {r=1}, 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

\displaystyle \partial_t u + \nabla_u u = - \mathrm{grad}_g p

\displaystyle \mathrm{div}_g u = 0

of the Euler equations to an arbitrary Riemannian manifold {(M,g)} (ignoring any issues of boundary conditions for this discussion), where {u: {\bf R} \rightarrow \Gamma(TM)} is a time-dependent vector field, {p: {\bf R} \rightarrow C^\infty(M)} is a time-dependent scalar field, and {\nabla_u} is the covariant derivative along {u} using the Levi-Civita connection {\nabla}. In Penrose abstract index notation (using the Levi-Civita connection {\nabla}, and raising and lowering indices using the metric {g = g_{ij}}), the equations of motion become

\displaystyle \partial_t u^i + u^j \nabla_j u^i = - \nabla^i p \ \ \ \ \ (9)

 

\displaystyle \nabla_i u^i = 0;

in coordinates, this becomes

\displaystyle \partial_t u^i + u^j (\partial_j u^i + \Gamma^i_{jk} u^k) = - g^{ij} \partial_j p

\displaystyle \partial_i u^i + \Gamma^i_{ik} u^k = 0 \ \ \ \ \ (10)

where the Christoffel symbols {\Gamma^i_{jk}} are given by the formula

\displaystyle \Gamma^i_{jk} := \frac{1}{2} g^{il} (\partial_j g_{lk} + \partial_k g_{lj} - \partial_l g_{jk}),

where {g^{il}} is the inverse to the metric tensor {g_{il}}. If the coordinates are chosen so that the volume form {dg} is the Euclidean volume form {dx}, thus {\mathrm{det}(g)=1}, then on differentiating we have {g^{ij} \partial_k g_{ij} = 0}, and hence {\Gamma^i_{ik} = 0}, and so the divergence-free equation (10) simplifies in this case to {\partial_i u^i = 0}. The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

\displaystyle \frac{1}{2} \int_M |u|_g^2\ dg = \frac{1}{2} \int_M g_{ij} u^i u^j\ dg

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

The specific four-dimensional manifold in question is the space {{\bf R} \times {\bf R}^+ \times {\bf R}/{\bf Z} \times {\bf R}/{\bf Z}} with metric

\displaystyle dx^2 + dy^2 + y^{-1} dz^2 + y dw^2

and solutions to the Boussinesq equation on {{\bf R} \times {\bf R}^+} 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 {\rho} appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric {g}). I am writing the details below the fold (partly for my own benefit).

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