### Geometric Fluid Dynamics

In the 1750's, Leonhard Euler (1707-1783) derived the Euler equations for fluid dynamics, which are a set of partial differential equations describing the relation among the fluid density, velocity and pressure. Soon after Euler, Joseph-Louis Lagrange (1736-1813) ambitiously formulated the known mechanics at the time in a way that they can be understood as the results of the least action principle. While it is easy to translate between Newtonian mechanics and Lagrangian mechanics for a single particle, it was not obvious how to write the action for continuum mechanics including elasticity and fluid. For Lagrange's approach to work for continuum mechanics, the domain of the problem was therefore shifted to the Lagrangian coordinate which traces each particle in the material. An important byproduct of Lagrangian coordinate is that Euler's velocity vector field was represented as a differential 1-form. It is worth noting that though Lagrange's 1788 book was published much earlier than the time of Élie Cartan, much of the derivations involve exterior calculus.

Lagrange's differential form formulation allowed Augustin-Louis Cauchy (1789-1857) to find in 1815 the conservation of vorticity 2-form in barotropic fluids (i.e. Kelvin's circulation theorem rediscovered in 1869). Vorticity measures the local swirling motion of the fluid, and Cauchy's circulation theorem amounts to conservation of angular momentum. Later in 1858, Hermann von Helmholtz (1821-1894) solidified the theory of vorticity, vortex dynamics, and the conversion between vorticity and velocity through Helmholtz's decomposition invented for it.

Soon after Helmholtz's vortex theory, in 1859 Alfred Clebsch (1833-1872) came up with a fluid representation using Clebsch variables, which are multiple potential functions whose joint level-sets are vortex lines. Using Clebsch's representation, not only Cauchy-Helmholtz's conservation of vorticity 2-form can be translated into conservation laws of Clebsch variables, but also an Eulerian-coordinate action for the Euler Equations can be written in terms of Clebsch variables.

Though I assume the reader has a basic knowledge of exterior calculus, I will still go through the useful exterior calculus operators, identities, and their geometric meaning, as they may not be mentioned in all materials.

#### Differential Forms

On a manifold $$M$$ of dimension $$n$$, a differential $$k$$-form ($$0\leq k\leq n$$ integer) is a quantity that is to be integrated over a $$k$$-dimensional submanifold. For example, functions, which can be evaluated on points, are 0-forms. Mass densities, which are supposed to be evaluated over $$n$$-dimensional submanifolds measuring total mass, are $$n$$-forms. Fluxes are $$(n-1)$$-forms which can be evaluated on codimension-1 interfaces.

Notationwise, a $$k$$-form $$\alpha$$ (denoted as $$\alpha\in\Omega^k(M)$$) evaluated on a $$k$$-dimensional oriented submanifold $$\Gamma\subset M$$ is denoted as $\int_\Gamma\alpha.$ The integral sign $$\int_{(\cdot)}(\cdot)$$ is treated as a bilinear pairing between formal linear combinations of $$k$$-submanifolds and $$k$$-forms.

To arrive at the above formalism, differential forms are often viewed as alternating linear forms by the following construction. A $$k$$-form $$\alpha$$ at a point $$x\in M$$ can take $$k$$ tangent vectors $$X_1,\ldots,X_k\in T_xM$$ and output a real number $$\alpha\vert_x(X_1,\ldots,X_k)$$. This quantity is defined to measure the integral of $$\alpha$$ over a small piece of $$k$$-parallelepiped $$\Gamma^\varepsilon$$ centered at $$x$$ and spanned by $$\varepsilon X_1,\ldots \varepsilon X_k$$: $\alpha\vert_x(X_1,\ldots,X_k)\equiv {1\over \varepsilon^k}\lim_{\varepsilon\to 0^+}\int_{\Gamma^\varepsilon}\alpha.$ Due to the linearity of $$\int_{\Gamma^\varepsilon}\alpha$$ in $$\Gamma^\varepsilon$$, we see $$\alpha\vert_x(X_1,\ldots,X_k)$$ is linear in each $$X_i$$, and it is skew symmetric (alternating) for adapting the change in the orientation of $$\Gamma^{\varepsilon}$$. Conversely, if a skew $$k$$-linear form $$\alpha$$ is defined continuously over $$M$$, we can define its integral $$\int_\Gamma\alpha$$ by partitioning the $$\Gamma$$ into (essentially-)non-overlapping infinitesimal pieces of $$k$$-parallelepiped $$\{\Gamma^{(i)}\}_{i}$$, $$\Gamma = \sum_{i}\Gamma^{(i)}$$, where $$\Gamma^{(i)}$$ is centered at $$x^{(i)}\in\Gamma$$ spanned by $$\varepsilon X_1^{(i)},\ldots,\varepsilon X_k^{(i)}\in T_{x^{(i)}}\Gamma$$, and find the limit of the Riemann sum $\int_\Gamma\alpha = \lim_{\substack{{\rm partition}\\{\rm refines}}}\sum_i\varepsilon^k\alpha\vert_{x^{(i)}}\left(X_1^{(i)},\ldots,X_k^{(i)}\right).$

At this point, the only important operator on alternating linear forms is the wedge product $$\wedge$$. If $$\alpha$$ is a $$k$$-form and $$\beta$$ is an $$\ell$$-form, then $$\alpha\wedge\beta$$ is a $$(k+\ell)$$-form so that

• $$(\cdot)\wedge(\cdot)$$ is bilinear;
• $$\wedge$$ is associative;
• $$\alpha\wedge\beta = (-1)^{k\ell}\beta\wedge\alpha$$.

#### Pullbacks

Let $$M,N$$ be two manifolds, and suppose $$f\colon M\rightarrow N$$ is a differentiable map. Given a submanifold $$\Gamma\subset M$$, and a differential form $$\alpha\in\Omega^k(N)$$, we may integrate $$\alpha$$ over the image swept by $$f\vert_\Gamma$$, denoted by $$\int_{f\vert_\Gamma}\alpha$$. This integral is defined as $\int_{f\vert_\Gamma}\alpha := \int_\Gamma f^*\alpha$ where $$f^*\colon \Omega^k(N)\rightarrow\Omega^k(M)$$ is the pullback operator defined so that $(f^*\alpha)(X_1,\ldots,X_k) = \alpha\big(df(X_1),\ldots,df(X_k)\big).$ In multivariable calculus, this is recognized as the formula for a change of integration variable with $$df$$ being the Jacobian. But formally you can think of $$f^*$$ as the "adjoint operator" of $$f\vert_{(\cdot)}$$ in the bilinear pairing $$\int_{(\cdot)}(\cdot)$$.

One can check by definition that $$f^*$$ distributes over the wedge products. That is, $f^*(\alpha_1\wedge\cdots\wedge\alpha_m) = (f^*\alpha_1)\wedge\cdots\wedge(f^*\alpha_m).$

#### Interior Products

If the map $$f_t\colon M\rightarrow N$$ depends differentiably on one parameter $$t\in[0,\varepsilon]$$, then we define the extrusion of a $$k$$-dimensional submanifold $$\Gamma\subset M$$ as ${\rm ext}_f^\varepsilon(\Gamma) = f\vert_{\Gamma\times [0,\varepsilon]}$ the $$(k+1)$$-dimensional image swept by $$f_t\vert_\Gamma$$ for $$t\in[0,\varepsilon]$$. Given $$\alpha\in\Omega^{k+1}(N)$$, with $$\dot f$$ denoting the variation $${\partial\over\partial t}f$$, we consider ${d\over d\varepsilon}\bigg\vert_{\varepsilon=0}\int_{{\rm ext}_f^\varepsilon(\Gamma)}\alpha =: \int_{\Gamma}\iota_{\dot f}\alpha.$ The operator $$\iota_{\dot f}\colon \Omega^{k+1}(N)\rightarrow\Omega^{k}(M)$$, as the adjoint operator of the instantaneous extrusion, is called the interior product (with a builtin pullback). One can check that in the sense of alternating multilinear forms $\iota_{\dot f}\alpha(X_1,\ldots,X_k) = \alpha(\dot f,df(X_1),\ldots,df(X_k)).$ If $$X$$ is a vector field on $$M$$ (denoted as $$X\in\Gamma(TM)$$), we have a flow map $$\phi_t\colon M\rightarrow M$$ so that $$\phi_0 = {\rm id}_M$$ and $$\dot\phi = X\circ\phi$$. Then in this special case the interior product defined through this flow map is the more familiar interior product $$\iota_X\colon \Omega^{k+1}(M)\rightarrow\Omega^{k}(M)$$. In particular, it can be defined as a linear operator so that

• $$\iota_X\circ\iota_X = 0$$;
• $$\iota_X(\alpha\wedge\beta) = (\iota_X\alpha)\wedge\beta + (-1)^k\alpha\wedge(\iota_X\beta)$$ when $$\alpha$$ is a $$k$$-form;
• when $$\alpha$$ is a 1-form, $$\iota_X\alpha = \alpha(X)$$.

For example, if $$\det\in\Omega^n(M)$$ is the volume form, then $$\iota_X\det\in\Omega^{n-1}(M)$$ is the flux form, which can be seen directly from the extrusion picture.

#### Exterior Derivatives

The exterior derivative $$d\colon \Omega^k(M)\rightarrow\Omega^{k+1}(M)$$ is the adjoint operator of the boundary operator through the bilinear pairing $$\int_{(\cdot)}(\cdot)$$. That is, $\int_{\partial\Gamma}\alpha = \int_{\Gamma}d\alpha$ for $$(k+1)$$-dimensional submanifolds $$\Gamma$$ and $$k$$-forms $$\alpha$$.

In terms of alternating forms, $$d$$ is a linear operator defined so that

• $$d\circ d = 0$$;
• $$d(\alpha\wedge\beta) = (d\alpha)\wedge\beta + (-1)^{k}\alpha\wedge(d\beta)$$ when $$\alpha$$ is a $$k$$-form;
• when $$f$$ is a 0-form, $$df(X)$$ is the directional derivative of $$f$$ in the $$X$$ direction.

#### Lie Derivatives

Consider again the one-parameter family of maps $$f_t\colon M\rightarrow N$$ for $$t\in[0,\varepsilon]$$. The rate of change ${d\over dt}\bigg\vert_{t=0}\int_{f_t\vert_\Gamma}\alpha =: \int_\Gamma\mathscr{L}_{\dot f}\alpha$ for $$k$$-forms $$\alpha$$ and $$k$$-submanifolds $$\Gamma$$. The operator $$\mathscr{L}_{\dot f}$$ is called the Lie derivative. In other words, $$\mathscr{L}_{\dot f}\colon\Omega^k(N)\rightarrow\Omega^k(M)$$ is defined by $\mathscr{L}_{\dot f}\alpha := {\partial\over\partial t}\bigg\vert_{t=0}f_t^*(\alpha).$ Note that in the case of some senario where $$\alpha_t$$ also depends on $$t$$, we have ${\partial\over\partial t}\bigg\vert_{t=0}f_t^*(\alpha_t) = f^*({\dot\alpha})+\mathscr{L}_{\dot f}\alpha.$

Note that one has the Cartan's magic formula given by $\mathscr{L}_{\dot f} = d\circ \iota_{\dot f} + \iota_{\dot f}\circ d$ which is a direct consequence of the homotopy formula $\partial(\Gamma\times[0,t]) = (\partial\Gamma)\times[0,t]-\Gamma\times\partial[0,t].$

In the special case when $$f_t = \phi_t$$ is a flow map generated by a vector field $$X$$, we have the more familiar Lie derivative $$\mathscr{L}_X\colon\Omega^k(M)\rightarrow\Omega^k(M)$$. Besides the most commonly used Cartan's magic formula $$\mathscr{L}_X = d\circ\iota_X + \iota_X\circ d$$, the following properties are also useful:

• $$\mathscr{L}_X(\alpha\wedge\beta) = (\mathscr{L}_X\alpha)\wedge\beta + \alpha\wedge\mathscr{L}_X\beta$$;
• $$\mathscr{L}_X(\iota_Y\alpha) = \iota_{[X,Y]}\alpha + \iota_Y\mathscr{L}_X\alpha$$;
• $$d\circ\mathscr{L}_X = \mathscr{L}_X\circ d$$.

An example of the use of the Lie derivative is the definition of the divergence of a vector field $$X\in\Gamma(TM)$$. Divergence of a vector field is the rate of change of the volume flown by the vector field. This geometric description directly translates to $\mathscr{L}_X\det =: {\rm div}(X)\det,$ and by Cartan's formula we see the divergence theorem ${\rm div}(X)\det = d(\iota_X\det)$ where the right-hand side measures the total flux over a boundary.

#### Vector-valued Differential Forms

Suppose $$E$$ is a vector bundle over $$M$$. (In most cases $$E=TM$$.) A vector-valued differential form $$F\in\Omega^k(M)\otimes\Gamma(E)$$ is a alternating $$k$$-linear form which returns a tangent vector $$F\vert_x(X_1,\ldots,X_k)\in E_x$$ as its value. For example, if $$f\colon M\rightarrow N$$, then $$df\in\Omega^1(M)\otimes\Gamma(T_fN)$$ (where $$T_fN$$ is called the pullback bundle so that $$(T_fN)_x\cong T_{f(x)N}$$).

When it comes to vector-valued functions or forms, one has to be careful that one needs a connection $$\nabla$$ on the vector bundle in order to talk about their derivatives. $$\nabla$$ is an endomorphism-valued 1-form, i.e. $$\nabla_X$$ transforms a vector in $$E$$ linearly to another vector in the same fiber. An affine connection further satisfies $\nabla(fV) = df V + f\nabla V,\quad f\in\Omega^0(M), V\in\Gamma(E).$ With an affine connection given, one can define the exterior covariant derivative $$d^\nabla$$ as the counterpart of $$d$$ when applying to vector-valued forms. In particular, one has

• $$d^\nabla(F\wedge G) = d^\nabla F\wedge G + (-1)^kF\wedge d^\nabla G$$ when $$F$$ is a (vector-valued) $$k$$-form;
• $$d^\nabla d^\nabla F =: R^\nabla\wedge F$$
where $$R^\nabla$$ is an endomorphism-valued 2-form called the curvature tensor of the connection $$\nabla$$.

When $$E = TM$$, we come to the territory of Riemannian geometry. If each tangent space is equipped with a scalar product $$\langle\cdot,\cdot\rangle$$ called metric, then $$\nabla$$ is said to be (compatible with) metric if $d\langle X,Y\rangle = \langle\nabla X,Y\rangle + \langle X,\nabla Y\rangle.$ A metric connection implies that for vector-valued forms $$F,G$$ $d\langle F\wedge G\rangle = \langle d^\nabla F\wedge G\rangle + (-1)^k\langle F\wedge d^\nabla G\rangle$ where $$k$$ is the degree of the form $$F$$.

A special vector-valued 1-form is the identity 1-form $I\in \Omega^1(M)\otimes\Gamma(TM),\quad I(X) := X,\ \forall X\in\Gamma(TM).$ The torsion of an affine connection $$\nabla$$ on $$TM$$ is defined by $d^\nabla I\in \Omega^2(M)\otimes \Gamma(TM).$ There is a unique affine connection which is metric and free of torsion, and it is called the Levi-Civita connection. The curvature tensor associated with the Levi-Civita connection is called the Riemannian curvature.

Given a fluid domain $$M$$ (an $$n$$-dimensional manifold), a fluid state is described by

• mass density form $$\rho\in\Omega^n(M)$$,
• velocity $$u\in\Gamma(TM)$$,
• pressure $$p\colon M\rightarrow\Bbb R$$.

#### Continuity Equation

Given $$\rho$$ and $$u$$, the mass flux $$(n-1)$$-form is given by $$\iota_u\rho$$ (recall of the extrusion picture). Now given a $$n$$-dimensional region $$\Omega\subset M$$, the total mass in $$\Omega$$ is given by $$\int_\Omega\rho$$. The conservation of mass says that the rate of change of this mass is given by the total in-flow of mass from the boundary. Hence $${d\over dt}\int_\Omega\rho = -\int_{\partial M}\iota_u\rho$$. Therefore $$\dot\rho + d\iota_u\rho = 0$$ or equivalently $\dot\rho + \mathscr{L}_u\rho = 0.$

If $$M$$ is equipped with a volume form $$\det\in\Omega^n(M)$$, then we can define the mass density function $$q\colon M\rightarrow\Bbb R$$ such that $$\rho = q\det.$$ Then the above equation becomes $$\dot q\det + d(\iota_u q\det)=0$$, which implies $\dot q + {\rm div}(qu) = 0.$

It might be interesting to note that in terms of $$\rho$$ and the Lie derivative, we do not need a notion of volume or metric to talk about the continuity equation (whereas most of the introductory texts on continuity equation requires a normal vector on the interface). In fact, the finite volume numerical method is a discrete version where only a $$n$$-form $$\rho$$ and flux evaluation (in this case the operator $$\iota_u$$) is defined.

#### Momentum Equation

Suppose $$TM$$ is equipped with a connection $$\nabla$$, then we can define the acceleration of the fluid as $\dot u + \nabla_u u.$ (This is the rate of change of the vector $$u\circ\phi$$ where $$\phi$$ is the flow map generated by $$u$$.)

It is at the next stage of the derivation that requires some more structure of the fluid domain. The force that acts on a fluid particle is the pressure force, which is the sum of normal forces with strength $$p$$ acting from the boundary. To express such a force we need a metric to talk about normal. If a metric is given, we shall take $$\nabla$$ as the Levi-Civita connection.

Given a point $$x\in M$$, a test vector $$X\in T_xM$$, its neighborhood $$U\subset M$$ and an extension $$X\in\Gamma(TU)$$ with $$\nabla X\vert_x=0$$, Newton's law of motion says that \begin{align*} \int_U\langle X, \dot u+\nabla_uu\rangle\rho &= \oint_{\partial U}p\langle X,N\rangle\iota_N\det\\ &= \oint_{\partial U}p\iota_X\det\\ &= \int_{U} dp\iota_X\det + p{\rm div}(X)\det\\ &= \int_U -\iota_Xdp\det + p{\rm div}(X)\det\\ &= \int_U \langle X, -{\rm grad}\, p\rangle \det + p{\rm div}(X)\det. \end{align*} Now take the limit $$U\rightarrow x$$ with $${\rm div}(X)\vert_x = 0$$ we have $\dot u + \nabla_u u = -{{\rm grad}\, p\over q}.$

#### Equation of State

For barotropic fluids, the pressure and the density are related by a single functional relation given from thermodynamical law $p = F(q)$ for some $$F\colon \Bbb R\rightarrow \Bbb R$$. With such a function given, the Euler equations are completed as $\begin{cases} \dot\rho + \mathscr{L}_u\rho = 0\\ \dot u + \nabla_u u = -{1\over q}{\rm grad}\, p\\ p = W(q). \end{cases}$

A steady solution to the Euler equation is given by that $$q = q_0$$ is a constant, and $$u = 0$$. A linearized perturbed solution from this trivial solution gives the acoustic wave equation with sound speed given by $$\sqrt{W'(q_0)}$$. In a low Mach number flow, where $$|u|\ll \sqrt{W'(q_0)}$$, we may assume that $$q = q_0$$ remains constant. The continuity equation $$\dot q + {\rm div}(qu)=0$$ then implies that $${\rm div}(u)=0$$, i.e. the flow is incompressible.

The incompressible Euler equation is given by $\begin{cases} \dot u + \nabla_u u = -{\rm grad}\, p\\ {\rm div}(u) = 0 \end{cases}$

where we have replaced $$p/q_0$$ by a redefined $$p$$ (known as the enthalpy).

Given a velocity field $$u\in\Gamma(TM)$$ one may consider the associated velocity 1-form $\eta:=u^\flat = \langle u,I\rangle$ where $$I$$ is the identity vector-valued 1-form. Then from the Euler equation (let's consider the incompressible case) we have \begin{align*} \langle\dot u + \nabla_uu,I\rangle = -dp \end{align*} where $$\langle\dot u,I\rangle = \dot\eta$$ on the left hand side, and \begin{align*} \langle \nabla_uu,I\rangle &= \langle\iota_u\nabla u\wedge I\rangle = \iota_u\langle\nabla u\wedge I\rangle + \langle\nabla u,\iota_uI\rangle\\ &=\iota_u d\langle u,I\rangle + \tfrac{1}{2}d\langle u,u\rangle = \mathscr{L}_u\eta - \tfrac{1}{2}d|u|^2. \end{align*} Therefore we arrive at the incompressible Euler equation written in terms of velocity 1-form $\dot\eta + \mathscr{L}_u\eta = -d\left(p-\tfrac{1}{2}|u|^2\right),\quad {\rm div}(u)=0.$

#### On the Lagrangian Coordinate

What is remarkable about the velocity 1-form Euler equation is that the left-hand side is written in terms of the derivative $${\partial\over\partial t}+\mathscr{L}_u$$, which is just like what appears in the continuity equation. Consider the flow map $$\phi_t\colon \hat M\rightarrow M$$ where $$\hat M$$ is the material coordinate (Lagrangian coordinate), with $$\dot\phi_t = u_t\circ\phi_t$$. We can use the flow map to pullback the velocity 1-form $\hat\eta_t:=\phi_t^*\eta_t\in\Omega^1(\hat M).$ Then ${\partial\over\partial t}\hat\eta_t = \phi_t^*\left(\dot\eta + \mathscr{L}_u\eta\right) = -\phi_t^*d\left(p-\tfrac{1}{2}|u|^2\right).$ By calling $$\hat p = \phi_t^*\left(p-\tfrac{1}{2}|u|^2\right)$$, we find the pulled-back velocity 1-form is conserved up to an exact differential $\dot{\hat\eta}_t = -d\hat p.$ This motivates the consideration of an equivalence relation $$\eta\sim\tilde\eta$$ if and only if $$\eta-\tilde\eta$$ is exact. In Lagrange's formulation one sees that the equivalence class is conserved: ${d\over dt}[\hat\eta_t] = [0].$ One can also see from the velocity 1-form equation $\dot{[\eta]}+\mathscr{L}_u[\eta] = [0].$ Note that the operator $$\mathscr{L}_u$$ still depends on $$u$$ rather than just the equivalence class. This $$u$$ is nevertheless determined uniquely by the equivalence using the lemma below. Therefore, in an incompressible fluid the fluid state is characterized by $$[\eta]$$.

Lemma. Let $$f\in\Omega^n(M)$$ and $$g\in\Omega^{n-1}(\partial M)$$ satisfy the condition that for any connected component $$M_i$$ of $$M$$, $\int_{M_i}f = \oint_{\partial M_i}g.$ Then for each $$[\tilde\eta]\in\Omega^1(M)/d\Omega^0(M)$$ there exists a unique representative $$\eta=u^\flat\in[\tilde\eta]$$ so that $\begin{cases} d\iota_u\det = f&\mbox{in $$M$$}\\ j_{\partial M}^*\iota_u\det = g&\mbox{on $$\partial M$$}. \end{cases}$

#### Vorticity 2-form

If $$M$$ has trivial first cohomology (simply-connected), then $$\Omega^1(M)/d\Omega^0(M)$$ is isomorphic to exact 2-forms $$d\Omega^1(M)\subset\Omega^2(M)$$ by $$[\eta]\mapsto\omega = d\eta$$. The exact 2-form $$\omega = d\eta$$ is called the vorticity 2-form, and as a direct corollary of the velocity-1-form equation, $\dot\omega + \mathscr{L}_u\omega = 0$ or in Lagrangian coordinate $\dot{\hat \omega}_t = 0,\quad\hat\omega_t = \phi_t^*\omega_t$

#### Helmholtz's Vorticity Equation

When $$M$$ is 3-dimensional, we define the vorticity vector field $$w\in\Gamma(TM)$$ as the one so that $\omega = \iota_w\det.$ That is, $$w$$ is the vector field of which $$\omega$$ is the flux form. Now, under incompressible flow $$\left({\partial\over\partial t}+{\mathscr{L}_u}\right)\det = 0.$$ Applying $$\left({\partial\over\partial t}+{\mathscr{L}_u}\right)$$ on both sides of $$\omega = \iota_w\det$$ we get $0=\iota_{\dot w + [u,w]}\det.$ Therefore, using the torsion-freeness of $$\nabla$$ we have $\dot w + \nabla_uw - \nabla_w u = 0.$ If the fluid is barotropic but compressible, we still have $$\dot\omega + \mathscr{L}_u\omega = 0$$. However, to write it in terms of the vorticity vector field, we can no longer use $$\left({\partial\over\partial t}+{\mathscr{L}_u}\right)\det = 0.$$ Instead, we take advantage of the continuity equation $$\left({\partial\over\partial t}+{\mathscr{L}_u}\right)\rho = 0,$$ and apply $$\left({\partial\over\partial t}+{\mathscr{L}_u}\right)$$ on $$\omega = \iota_{w/q}\rho.$$ Hence by the same calculation we have $\tfrac{\partial}{\partial t}\left(\tfrac{w}{q}\right) + \nabla_u\left(\tfrac{w}{q}\right) - \nabla_{\frac{w}{q}}u=0.$

In Helmholtz's 1858 paper on vortex dynamics, he not only derived the vorticity equations but also investigates the geometry of vortices in a fluid.

Suppose $$M$$ is 3-dimensional. Then $$w\in T_x M$$ is called a vortex direction if $$\iota_w\omega = 0.$$ (Generically there is a unique one dimensional subspace in $$T_xM$$ that is the vortex direction when the dimension is 3.) A vortex line is an integral curve of the vortex direction.

A vortex tube in a 3-dimensional $$M$$ is a 3-dimensional submanifold $$j_{U}\colon U\hookrightarrow M$$ so that $j_{\partial U \setminus \partial M}^*\omega = 0,$ that is $$U$$ is a region where no vorticity passes through its boundary surface except for the boundary $$\partial M$$ of the domain. In other words, the vortex direction on $$\partial U$$ is always tangent to $$\partial U$$.

If the topology of a vortex tube is a solid cylinder or a solid torus, then the strength of the vortex tube $$\kappa$$ is given by the vorticity flux $$\kappa = \int_{\Sigma}\omega$$ of a cross section $$\Sigma$$ of $$U$$. One may check that $$\kappa$$ is independent of the choice of cross section $$\Sigma$$ using $$d\omega = 0$$ and $$j^*_{\partial U\setminus\partial M}\omega = 0.$$ In general, a vortex tube can take a more general topology, and the choice of cross section matters.

A cross section $$\Sigma$$ of $$U$$ is a submanifold of $$U$$ so that $$\partial\Sigma\subset\partial U\setminus\partial M.$$ Two cross sections $$\Sigma_1,\Sigma_2$$ necessarily evaluate the same vorticity if they co-border a volume of $$U.$$ Therefore the strength of a vortex tube is a functional on the relative homology $$H_2(U,\partial U\setminus\partial M)$$ $\kappa\in{\rm Hom}\big(H_2(U,\partial U\setminus\partial M);\Bbb R\big)$

Since vorticity tends to stretch concentrate under Euler fluid flow, one often finds $$\omega$$ is supported only in some thin vortex tubes, which is well-described as a tubular neighborhood of a space curve $$\Gamma.$$ In this case we call $$\Gamma$$ a vortex filament. Note the difference between a vortex filament and a vortex line: a vortex filament has a finite strength while a vortex line has none.

While many people at the time took the potential flow ansatz $$\eta = d\varphi$$ to find at best irrotational flow, Clebsch took its generalization: $\eta = \sum_{i=1}^m\lambda_i d\mu_i + d\varphi.$ Here $$\lambda_i,\mu_i,\varphi\colon M\rightarrow\Bbb R$$ are called Clebsch potentials or Clebsch variables. Under this Clebsch representation, the vorticity 2-form is given by $\omega = d\eta = \sum_{i=1}^m d\lambda_i\wedge d\mu_i.$ The insights Clebsch discovered are the following.

1. If the number of Clebsch variable pairs is $$m=1$$, then $$\omega = d\lambda\wedge d\mu.$$ In this case, the intersection of the level-sets of $$\lambda$$ and $$\mu$$ are vortex lines.
2. If $\begin{cases} {\partial\lambda_i\over\partial t} + \mathscr{L}_u\lambda_i = -{\partial\Pi(\vec\lambda,\vec\mu)\over\partial\mu_i}\\ {\partial\mu_i\over\partial t} + \mathscr{L}_u\mu_i = {\partial\Pi(\vec\lambda,\vec\mu)\over\partial\lambda_i} \end{cases}$ for any function $$\Pi\colon\Bbb R^{2m}\rightarrow\Bbb R,$$ then $$\eta = \sum_{i=1}^m\lambda_id\mu_i+d\varphi\in [\eta]$$ satisfies the incompressible Euler equation.
3. The above equation for the Clebsch variables is the Euler-Lagrange equation of the following action $S[\vec\lambda,\vec\mu,\varphi] = \int_0^T\int_M \Big(\sum_i\lambda_i\dot\mu_i + \dot\varphi + {1\over 2}|u|^2 + \Pi(\vec\lambda,\vec\mu)\Big)\,\det\, dt.$
The derivations of these items are straightforward. We will later cover a derivation for a generalization of Clebsch variables.

#### Prequantum Bundle

The above Clebsch representation has the following geometric generalization. The $$(2m+1)$$ Clebsch variables can be viewed as a map onto coordinates on a $$(2m+1)$$-dimensional manifold $$Q.$$ In Clebsch's original setting $$Q = \Bbb R^{2m+1}.$$ If $$(x_1,\ldots,x_m,y_1,\ldots,y_m,z)$$ are the coordinates on $$Q,$$ then we define a 1-form $$\vartheta\in\Omega^1(Q)$$ defined by $$\vartheta = \sum_{i=1}^m x_i dy_i + dz.$$ Now we can say that a Clebsch variable is a map $$\Psi\colon M\rightarrow Q$$ that represents the velocity 1-form as $\eta = \Psi^*\vartheta.$ In the theory, there is also a natural projection $$\pi\colon Q\rightarrow\Sigma$$ where $$\Sigma$$ is a $$(2m)$$-dimensional symplectic manifold. In Clebsch's setting $$\Sigma = \Bbb R^{2m}$$, $$\pi(\vec x,\vec y,z):=(\vec x,\vec y)$$, and the symplectic form is given by $$\sigma = \sum_{i=1}^m dx_i\wedge dy_i$$. Note that one has $d\vartheta = \pi^*\sigma.$ In addition, there is a vector field $$V = (\vec 0,\vec 0,1)$$ on $$Q$$, which is vertical (i.e. $$d\pi(V)=0$$) and $$\vartheta(V) = 1.$$ This vector field is also called a Reeb vector field in the context of contact geometry.

Next, given a Clebsch variable $$\Psi = (\vec\lambda,\vec\mu,\varphi)\colon M\rightarrow Q$$, define $s := \pi\circ\Psi = (\vec\lambda,\vec\mu).$ Then the vorticity is indeed \begin{align*} \omega &= d\eta = d\Psi^*\vartheta = \Psi^* d\vartheta\\ &=\Psi^*\pi^*\sigma = (\pi\circ\Psi)^*\sigma = s^*\sigma. \end{align*}

Definition. A fiber bundle $$Q\xrightarrow{\pi}\Sigma$$ with one-dimensional fibers over a symplectic manifold $$(\Sigma,\sigma)$$ is a prequantum bundle if there is a vertical vector field $$V\in\Gamma(TQ),$$ $$d\pi(V)=0,$$ and a 1-form $$\vartheta\in\Omega^1(Q)$$ so that $$d\vartheta = \pi^*\sigma$$ and $$\vartheta(V)=1.$$

The last notable structure is that the vertical vector field $$V$$ gives rise to a flow within each fiber $$\pi^{-1}(p\in\Sigma).$$ One may view the flow as a group action: let $$G$$ be a 1-dimensional Lie group, $$\mathfrak{g}\cong\Bbb R$$ be its Lie algebra, $$i\colon \Bbb R\xrightarrow{\cong}\mathfrak{g}$$ be the isomorphism; define the group action of $$\exp(it)\in G$$ on a fiber $$\pi^{-1}(p)$$ by flowing along the fiber with $$V$$ for time $$t$$. Of course for this group action to be well-defined the orbit should at least cover the fiber.

Note that the flow generated by $$V$$ leaves $$\vartheta$$ invariant. That is, $$\mathscr{L}_V\vartheta = 0.$$ This can be seen from the Cartan's formula $$\mathscr{L}_V\vartheta = d\iota_V\vartheta + \iota_Vd\vartheta = d(1) + \iota_V\pi^*\vartheta = 0$$ since $$d\pi(V)=0.$$ Because of this equivariance, we have the following formula for phase shifts in Clebsch variables. Suppose $$\Psi\colon M\rightarrow Q$$ and $$\eta = \Psi^*\vartheta.$$ Then for any $$\varphi\colon M\rightarrow\Bbb R$$ the phase shift $$\tilde\Psi = \exp(i\varphi)\cdot\Psi$$ gives $\tilde\eta = \tilde\Psi^*\vartheta = \eta + d\varphi.$ That is, moving around in the same equivalence class $$[\eta]\in \Omega^1(M)/d\Omega^0(M)$$ is exactly achieved by phase shifts. Let us then define an equivalence relation in the Clebsch variables $$\Psi\sim\tilde\Psi$$ if there exists $$\varphi\in\Omega^0(M)$$ so that $$\tilde\Psi = \exp(i\varphi)\cdot\Psi.$$ Each equivalence class $$[\Psi]$$ defines a unique equivalence class $$[\eta]$$ by $$\eta = \Psi^*\vartheta$$, which is well-defined. In particular, for each equivalence class $$[\tilde\Psi]$$ there is a unique (up to constant phase) $$\Psi\in[\tilde\Psi]$$ so that $$\eta = \Psi^*\vartheta$$ is divergence-free (and meet a given boundary condition).

#### Clebsch's Action

In terms of the prequantum bundle, the Clebsch action is written as $S[\Psi] = \int_0^T\int_M\left(\iota_{\dot\Psi}\vartheta + {1\over 2}|u|^2 + \Pi(\pi\circ\Psi)\right)\, \det\, dt,\quad \eta = \Psi^*\vartheta.$ Now take the variation with respect to $${\partial_\epsilon}\Psi = \mathring\Psi$$: \begin{align*} \mathring S = \iint\left(\mathscr{L}_{\partial_\epsilon}\iota_{\dot\Psi}\vartheta + \iota_u\mathring\eta + \iota_{\mathring s}d\Pi\right)\,\det\,dt. \end{align*} The term $$\mathscr{L}_{\partial_\epsilon}\iota_{\dot\Psi}\vartheta$$ can be worked out as follows: (we will drop the total time or space derivative terms since they are eventually under integral sign) \begin{align*} \mathscr{L}_{\partial_\epsilon}\iota_{\dot\Psi}\vartheta&=\iota_{\Psi_*(\underbrace{[\partial_\epsilon,\partial_t]}_{=0})}\vartheta + \iota_{\partial_t}\mathscr{L}_{\mathring\Psi}\vartheta\\ &=\iota_{\partial_t}d\iota_{\mathring\Psi}\vartheta + \iota_{\partial_t}\iota_{\mathring\Psi}d\vartheta\\ &=\mathscr{L}_{\partial t}\iota_{\mathring\Psi}\vartheta + \iota_{\partial_t}\iota_{\mathring\Psi}\pi^*\sigma\\ &=\iota_{\partial_t}\iota_{\mathring s}\sigma = \sigma(\mathring s,\dot s). \end{align*} The kinetic energy term \begin{align*} \int_M\iota_u\mathring\eta\, \det &= \int_M\iota_u d\left(\iota_{\mathring\Psi}\vartheta\right)\, \det + \int_M\iota_u\iota_{\mathring\psi}d\vartheta\, \det\\ &=\int_M-d(\iota_u \det)\iota_{\mathring\Psi}\vartheta + \int_{M}\iota_u\iota_{\mathring s}\sigma\,\det\\ &=\int_{M}-{\rm div}(u)\iota_{\mathring\Psi}\vartheta\, \det + \int_M\sigma(\mathring s,ds(u))\,\det. \end{align*} The last term $$\iota_{\mathring s}d\Pi = \sigma(\mathring s, {\rm sgrad}\, \Pi)$$ by the definition of symplectic gradient on $$\Sigma$$. Combining all terms, we have the variation of action $\mathring S = \iint \left(\sigma\Big(\mathring s, \dot s + ds(u) + {\rm sgrad}\,{\Pi}\Big) - {\rm div}(u)\iota_{\mathring\Psi}\vartheta \right)\,\det\,dt = 0$ for arbitrary $$\mathring s$$ and $$\iota_{\mathring\Psi}\vartheta$$ (the latter variation can be chosen arbitrarily independent of $$\mathring s$$ through the phase shift degree of freedom in $$\Psi$$). Therefore, the Euler-Lagrange equations are $\dot s + ds(u) = -{\rm sgrad}\,\Pi,\quad{\rm div}(u)=0.$ Finally, to arrive at the Euler equation, we compute $$[\dot\eta]$$: \begin{align*} \dot\eta &= d\iota_{\dot\Psi}\vartheta + \iota_{\dot\Psi}d\vartheta\\ &\in d\Omega^0(M) + \iota_{\dot\Psi}\pi^*\sigma\\ &=d\Omega^0(M) + \iota_{\dot s}\sigma\\ &=d\Omega^0(M) + \iota_{-ds(u)-{\rm sgrad}\,\Pi}\sigma\\ &=d\Omega^0(M) - \iota_us^*\sigma + d\Pi\\ &=d\Omega^0 - \iota_u d\eta = d\Omega^0 - \mathscr{L}_u\eta. \end{align*} Therefore we obtain the incompressible Euler equation $[\dot\eta]+\mathscr{L}_u[\eta] = 0,\quad{\rm div}(u) = 0.$