2.1 A few words about differential geometry

In the present paper we prefer not starting with a barrage of mathematical definitions and we rather appeal to the reader’s intuition. For those hungry of precise definitions, more elaborate, but still quite elementary, material and guides to the literature are found in appendix B and its various subsections. For reasons explained in the Introduction, we feel that it is essential not to restrict our discussion to flat spaces. Otherwise we would have used a ‘half-way house’ approach where all the differential geometry is expressed in the standard language of vector operations, as done, for example in the paper of Larsson (Reference Larsson1996).

The concept of a differentiable manifold $M$ generalises to an arbitrary dimension $d$ that of a curve or a surface embedded in the 3D Euclidean space $\mathbb{R}^{3}$ . To achieve this in an intrinsic fashion without directly using Cartesian coordinates, the most common procedure makes use of collections of local charts, which are smooth bijections (one-to-one correspondences) with pieces of  $\mathbb{R}^{d}$ .

By taking infinitesimal increments near a point $a\in M$ , one obtains tangent vectors, which are in the $d$ -dimensional tangent space $TM_{a}$ , a generalisation of the tangent line to a curve and the tangent plane to a surface. The union of all these tangent vectors $\bigcup _{a\in M}TM_{a}$ , denoted $TM$ , is called the tangent bundle.

As for ordinary vector spaces, one can define the dual of the tangent bundle, noted $T^{\ast }M$ , which can be constructed through linear forms, called 1-forms or cotangent vectors, acting on vectors of the tangent bundle $TM$ . The set of all these cotangent vectors is called the cotangent bundle, noted $T^{\ast }M$ . Similarly, $p$ -forms, where $p$ is an integer, are skew-symmetric $p$ -linear forms over the tangent bundle $TM$ . Note that in a flat (Euclidean) space $\mathbb{R}^{d}$ with coordinates $x=\{x_{i}\}$ , where $i=1,\ldots ,d$ , a 1-form is simply an expression $\sum _{i}a_{i}(x)\,\text{d}x_{i}$ , which depends linearly on the infinitesimal increments $\text{d}x_{i}$ . It is also interesting to note that 1-forms were in common use in fluid mechanics in the works of D’Alembert, Euler and Lagrange more than a century before vectors were commonly used, say, in the lectures of Gibbs.

An important operator on $p$ -forms is the exterior derivative, $\text{d}$ , which linearly maps $p$ -forms to $(p+1)$ -forms (see § B.8). An explicit definition of $\text{d}$ is not very helpful to build an intuitive feeling, but it is worth pointing out that the square of $\text{d}$ is zero or, in words, an exact form (a form that is the exterior derivative of another one) is closed (its exterior derivative vanishes). Under certain conditions, to which we shall come back, the converse is true.

2.2 Lie advection: an extension of the Lagrangian (material) derivative

In this section we present some standard mathematical concepts needed to introduce our theorem on generalised Cauchy invariants, stated in the next section. For this, we need to generalise the fluid mechanics concept of Lagrangian invariant, which applies to a scalar quantity that does not change along fluid-particle trajectories. The generalisation is called Lie-advection invariance (alternative terminologies found in the literature are ‘Lie-transport’ and ‘Lie-dragging’).

First we introduce the pullback and pushforward operations, which arise naturally when applying a change of variable, here, between Lagrangian and Eulerian coordinates at a fixed time $t$ (later, we shall let this dynamical time vary). The Lagrangian variable (initial position of the fluid particle), denoted by $a$ , is on a manifold $M$ (called here for concreteness Lagrangian), while the Eulerian variable (current position of the fluid particle), denoted by $x$ , is on a manifold $N$ (called here Eulerian). The sets $M$ and $N$ may or may not coincide. The Lagrangian map linking $a\in M$ to $x\in N$ is defined as follows

The change of variable $a\rightarrow x=\unicode[STIX]{x1D711}(a)$ induces two operations that connect objects (such as functions, vectors, forms and tensors), defined on $M$ to corresponding ones, defined on $N$ . They are the pushforward operator, which sends objects defined on $M$ to ones defined on $N$ and its inverse, the pullback operator. To define these transformations precisely, it is convenient to consider successively the cases where these operators act on real-valued functions (scalars), then on vectors, then on 1-forms, and finally on more involved objects such as $p$ -forms, obtainable from the former ones by linear combinations of tensor products (see § B.2).

For the case of scalars, namely elements of ${\mathcal{F}}(M)$ , the set of real-valued smooth functions defined on $M$ , the pullback is simply a change of variable from Eulerian to Lagrangian variables and the pushforward is the converse. Specifically, the pushforward of a function $f:M\rightarrow \mathbb{R}$ on $M$ is $\unicode[STIX]{x1D711}_{\ast }f=f\circ \unicode[STIX]{x1D711}^{-1}$ , where the symbol $\circ$ denotes the usual composition of maps. Conversely, the pullback of a function $f:N\rightarrow \mathbb{R}$ on $N$ is $\unicode[STIX]{x1D711}^{\ast }f=f\circ \unicode[STIX]{x1D711}$ .

Now, we turn to vector fields, denoted ${\mathcal{T}}_{0}^{1}(M)$ on $M$ , a subset of the tangent bundle $TM$ . (Why we use the notation ${\mathcal{T}}_{0}^{1}(M)$ will become clear later.) At this point we cannot just make a change of variable, because the Lagrangian and the Eulerian vectors take values in different tangent spaces. But we can reinterpret tangent vectors to a manifold in terms of differentials of scalar functions defined on that manifold. To implement this, it is useful to consider a vector field $X(a)$ on $M$ as the generator of a suitable flow on $M$ . For this, we need an auxiliary time variable, denoted $s$ , to parametrise a family of smooth maps $\unicode[STIX]{x1D6FE}_{s}:M\rightarrow M$ . Observe that the time $s$ is not related to the dynamical time $t$ , which so far is held fixed. The maps satisfy the following equations

The pushforward of the vector field $Y\in {\mathcal{T}}_{0}^{1}(N)$ (also called the differential of the map $\unicode[STIX]{x1D711}$ or the tangent map) is now defined locally at the point $a\in M$ , as the linear map $\unicode[STIX]{x1D711}_{\ast }:=T_{a}\unicode[STIX]{x1D711}:TM_{a}\rightarrow TN_{\unicode[STIX]{x1D711}(a)}$ , obtained by simply identifying the resulting vector with the tangent vector to the mapped curve. This is illustrated in figure 1. Translated into equations it means that

where $T$ denotes the tangent map and is given locally by the Jacobian matrix $J_{\unicode[STIX]{x1D711}}=J(\unicode[STIX]{x1D711})=\unicode[STIX]{x2202}\unicode[STIX]{x1D711}/\unicode[STIX]{x2202}a$ . Recalling that $a^{i}$ denotes local coordinates on $M$ and $x^{i}$ local coordinates on $N$ , in terms of these local coordinates, this formula is expressed equivalently as

To define the inverse operation, the pullback denoted $\unicode[STIX]{x1D711}^{\ast }$ , we just interchange $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D711}^{-1}$ . Thus we have $\unicode[STIX]{x1D711}_{\ast }=(\unicode[STIX]{x1D711}^{-1})^{\ast }=(\unicode[STIX]{x1D711}^{\ast })^{-1}$ and $\unicode[STIX]{x1D711}^{\ast }=(\unicode[STIX]{x1D711}^{-1})_{\ast }=(\unicode[STIX]{x1D711}_{\ast })^{-1}$ . It thus follows that the pullback of a vector field $Y\in {\mathcal{T}}_{0}^{1}(N)$ on $N$ is

Therefore we find that $\unicode[STIX]{x1D711}^{\ast }Y=(\unicode[STIX]{x1D711}^{\ast }Y)^{i}(a)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}a^{i})=Y^{i}(x)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x^{i})$ and $\unicode[STIX]{x1D711}_{\ast }X=(\unicode[STIX]{x1D711}_{\ast }X)^{i}(x)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x^{i})=X^{i}(a)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}a^{i})$ . Notice that $\unicode[STIX]{x1D711}$ must be a diffeomorphism (one-to-one smooth map) in order for the pullback and pushforward operations to make sense; the only exception to this is the pullback of functions (and covariant tensors, see § B.4), since the inverse map is then not needed. Thus vector fields can only be pulled back and pushed forward by diffeomorphisms.

Figure 1. The curve $\unicode[STIX]{x1D6FE}_{s}$ is the integral curve of a given vector field $X$ , while the curve $\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D6FE}_{s}$ is the integral curve of $\unicode[STIX]{x1D711}_{\ast }X$ . The pullback transformation $\unicode[STIX]{x1D711}^{\ast }$ is a change of variables from Eulerian to Lagrangian coordinates, while the pushforward transformation $\unicode[STIX]{x1D711}_{\ast }$ is a change of variables from Lagrangian to Eulerian coordinates.

We can extend pullback and pushforward operations to linear forms on vector fields, that is 1-forms or covectors. The set of such 1-forms fields on $M$ is denoted by ${\mathcal{T}}_{1}^{0}(M)\subset T^{\ast }M$ (see §§ B.2 and B.7). In order to define the pullback of a 1-form $\unicode[STIX]{x1D6FC}\in {\mathcal{T}}_{1}^{0}(N)$ , we introduce the linear map $\unicode[STIX]{x1D711}^{\ast }:T^{\ast }N_{\unicode[STIX]{x1D711}(a)}\rightarrow T^{\ast }M_{a}$ , defined by

where the duality bracket $\langle \cdot ,\cdot \rangle$ is the natural pairing between the spaces $TM_{a}$ and $T^{\ast }M_{a}$ or between the spaces $TN_{\unicode[STIX]{x1D711}(a)}$ and $T^{\ast }N_{\unicode[STIX]{x1D711}(a)}$ . The pushforward of a 1-form $\unicode[STIX]{x1D6FD}\in {\mathcal{T}}_{1}^{0}(M)$ , is defined by changing $\unicode[STIX]{x1D711}$ to $\unicode[STIX]{x1D711}^{-1}$ , i.e. $\unicode[STIX]{x1D711}_{\ast }:=(\unicode[STIX]{x1D711}^{-1})^{\ast }$ . In terms of local coordinates we have

Therefore we find that $\unicode[STIX]{x1D711}^{\ast }\unicode[STIX]{x1D6FC}:=(\unicode[STIX]{x1D711}^{\ast }\unicode[STIX]{x1D6FC})_{i}(a)\,\text{d}a^{i}=\unicode[STIX]{x1D6FC}_{i}(x)\,\text{d}x^{i}$ and $\unicode[STIX]{x1D711}_{\ast }\unicode[STIX]{x1D6FD}:=(\unicode[STIX]{x1D711}_{\ast }\unicode[STIX]{x1D6FD})_{i}(x)\,\text{d}x^{i}=\unicode[STIX]{x1D6FD}_{i}(a)\,\text{d}a^{i}$ .

Pullback and pushforward operations are easily generalised to tensor fields $\unicode[STIX]{x0398}\in {\mathcal{T}}_{p}^{q}(M)$ , where ${\mathcal{T}}_{p}^{q}(M)$ denotes the set of $p$ -covariant and $q$ -contravariant tensor fields on  $M$ . Such generalisations follow naturally since a $p$ -covariant and $q$ -contravariant tensor can be written as linear combinations of tensor products of $p$ 1-forms and $q$ vectors (see § B.2).

In order to define the Lie derivative, we bring in the dynamical time  $t$ . For this, we specialise to the case where the Lagrangian and the Eulerian points are on the same manifold (with $N=M$ ) and we consider a time-dependent vector field $v_{t}$ , the velocity field, taken in ${\mathcal{T}}_{0}^{1}(M)$ for all $t\geqslant 0$ . This velocity field is prescribed a priori and we do not have to specify which dynamical equation it satisfies. We define a time-dependent Lagrangian map $\unicode[STIX]{x1D711}_{t}$ in the usual fluid-mechanical sense as mapping the initial position of a fluid particle, following the flow, to its position at time $t$ , namely as the solution of the ordinary differential equation (ODE)

From this equation, we also define a 2-time Lagrangian map $\unicode[STIX]{x1D711}_{t,s}$ with $t\geqslant 0$ and $s\geqslant 0$ as the map from the position of a fluid particle at time $s$ to its position at time  $t$ . Allowing the time to run backwards, we do not impose $t>s$ . Obviously, we have $\unicode[STIX]{x1D711}_{t,0}=\unicode[STIX]{x1D711}_{t}$ . Furthermore, we obviously have the group composition rule

In this dynamical setting, the pullback and pushforward operations consist roughly in following a given tensor field, while taking into account the geometrical deformation of the tensor basis, along the Lagrangian flow. This will naturally lead to considering a derivative with respect to the Lagrangian flow, called the Lie derivative. The Lie derivative of a structure (for instance a vector, a 1-form or a tensor field) with respect to the time-dependent vector field $v_{t}$ measures the instantaneous rate of geometrical variation of the structure (tensor basis) as it is transported and deformed by the Lagrangian flow $\unicode[STIX]{x1D711}_{t}$ generated by  $v_{t}$ .

Specifically, we first define the Lie derivative acting on a time-independent tensor field $\unicode[STIX]{x0398}\in {\mathcal{T}}_{p}^{q}(M)$ . To the Lagrangian map $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F},t}$ we associate its pullback $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F},t}^{\ast }$ , constructed just as earlier in this section, when the dynamical time was held fixed. The Lie derivative with respect to $v_{t}$ is defined by

Now, we turn to a time-dependent tensor field $\unicode[STIX]{x0398}_{t}\in {\mathcal{T}}_{p}^{q}(M)$ and we derive the Lie-derivative theorem $\forall t\geqslant 0$ . For this, we calculate the time derivative of $\unicode[STIX]{x1D711}_{t,s}^{\ast }\unicode[STIX]{x0398}_{t}$ , using the product rule for derivatives and obtain

Then, using the group composition rule (2.9), this equation becomes

Using a property for the pullback of map composition (see § B.4), namely $(\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D713})^{\ast }=\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D711}^{\ast }$ , we obtain

Finally, using the definition of the Lie derivative (2.10), this equation leads to the following formula, known as the Lie-derivative theorem:

In this paper, a central role will be played by tensor fields that are Lie-advection invariant (in short Lie invariant). A Lie-advection invariant tensor field $\unicode[STIX]{x0398}_{t}$ is such that its Lagrangian pullback, i.e. its pullback to time $t=0$ , is equal to the initial tensor field, that is

In fluid mechanics terms, one then states that the tensor field $\unicode[STIX]{x0398}_{t}$ is frozen into the flow  $\unicode[STIX]{x1D711}_{t}$ . From the Lie-derivative theorem (2.14), we immediately find that this is equivalent to having the tensor field $\unicode[STIX]{x0398}_{t}$ satisfying the equation

which is called the Lie-advection equation. A tensor field $\unicode[STIX]{x0398}_{t}$ satisfying the Lie-advection equation (2.16) is said to be Lie-advected by the flow of  $v_{t}$ .

It is easily checked that when $\unicode[STIX]{x0398}_{t}$ is a scalar field (denoted $\unicode[STIX]{x1D703}_{t}$ ) and when the manifold reduces to an Euclidean space, equation (2.16) becomes just

where $\unicode[STIX]{x2202}_{i}$ is the Eulerian derivative. Hence, in the scalar case, Lie-advection invariance of $\unicode[STIX]{x1D703}_{t}$ is the same as stating that $\unicode[STIX]{x1D703}_{t}$ is a Lagrangian (material) invariant in the usual fluid-mechanical sense. The advantage of the Lie-advection invariance formulation for higher-order objects is that, e.g., in 3D the vorticity, when considered as a 2-form, is then also Lie-advection invariant, as noticed for the first time (in 19th century language) by Helmholtz (Reference von Helmholtz1858).

2.3 Generalised Cauchy invariants

In this section we state a general theorem about Lie-advection invariance using differential geometry tools. The result is a natural generalisation of Cauchy invariants that arises when we consider, in an Euclidean space $\mathbb{R}^{d}$ or on a $d$ -dimensional Riemannian manifold $(M,g)$ , a Lie-advected $p$ -form with a crucial additional constraint of exactness (or some generalisation). We recall that a $d$ -dimensional Riemannian manifold $(M,g)$ is a differentiable manifold $M$ of dimension $d$ , together with a 2-covariant tensor field, the metric tensor $g$ , which associates to any point $a\in M$ a 2-covariant tensor ${\mathcal{T}}_{2}^{0}(M)$ (see §§ B.2 and B.3). The metric tensor $g$ allows one both to define a metric on $M$ for measuring distances between two points on $M$ , and to define a suitable scalar product for vectors lying in a tangent space (see § B.3).

The main new result of the present section will be to show that, to each exact Lie-advected $p$ -form, corresponds a generalised Cauchy invariant. This is of course a result with applications beyond hydrodynamics, but it is not just a rewriting of Lie-advection invariance in Lagrangian coordinates: the Cauchy invariants formulation requires an additional condition other than Lie-advection invariance. The method of proving this is quite general but, of course, also applies to Euler flow in the ordinary flat 3D space. In that case, we already have the original proof of Cauchy (Reference Cauchy1815), which juggles with Eulerian and Lagrangian coordinates and thus has a flavour of pullback–pushforward argument. In addition, we have all the relatively recent derivations using Noether’s theorem in conjunction with a variational formulation of the Euler equations and the relabelling invariance (see, e.g., Salmon Reference Salmon1988). What we now present constitutes in a sense a third approach, rooted in differential geometry and allowing generalisation to a variety of hydrodynamical and MHD problems, discussed in §§ 2.5, 3 and 4.

Let $\unicode[STIX]{x1D6FA}\subset M$ be a bounded region of the $d$ -dimensional Riemannian manifold $(M,g)$ . We remind the reader that a $p$ -form $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x039B}^{p}(\unicode[STIX]{x1D6FA})$ is exact if it is the exterior derivative of a $(p-1)$ -form $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x039B}^{p-1}(\unicode[STIX]{x1D6FA})$ , that is

where d denotes the exterior derivative (see § B.8). We recall that a family of $p$ -forms $\unicode[STIX]{x1D6FE}_{t}\in \unicode[STIX]{x039B}^{p}(\unicode[STIX]{x1D6FA})$ , $t>0$ , are Lie-advected by the flow of $v_{t}$ if they satisfy the Lie-advection equation

Here the vector field $v_{t}$ is the generator of the Lagrangian flow $\unicode[STIX]{x1D711}_{t}$ defined by (2.8).

Here, $x=\unicode[STIX]{x1D711}_{t}$ denotes Eulerian coordinates and $\unicode[STIX]{x1D6FF}_{j_{1}\ldots j_{p}}^{i_{1}\ldots i_{p}}$ the generalised Kronecker symbol (see § B.6). Note that, henceforth, in connection with Cauchy invariants, we use the singular for ‘equation’, since in modern writing a vector or a tensor is considered a single object.

2.4 Alternative formulations and extensions of Theorem 1

Hereafter we discuss alternative representations of Theorem 1, which are local, such as the generalised Cauchy formula, or global, such as the integral formulation of the Cauchy invariants equations. We also give extensions of Theorem 1 for some non-exact differential forms.

2.5 Broad applicability of Theorem 1

Our key result, namely Theorem 1, may be viewed as a new fundamental result in linear transport theory, giving an alternative Lagrangian formulation of Lie advection for a large class of differential forms. (This was pointed out to us by Peter Constantin who made us realise that this new result in linear transport theory may be of independent interest.) Indeed there is no need to have a self-consistent coupling between the transporter (vector fields  $v$ ) and the transported (differential forms  $\unicode[STIX]{x1D6FE}$ ) to obtain generalised Cauchy invariants equations. For the first time, it is here shown that Cauchy invariants equations exist for non-self-consistent linear transport. It must be pointed out that, when the Cauchy invariants were rediscovered in the 20th century, most of the time it was by making use of Noether’s theorem in the case of self-consistent nonlinear equations (Frisch & Villone Reference Frisch and Villone2014). Although Noether’s theorem is usually not available for linear transport equations, our key result shows that such generalised Cauchy invariants still do exist in linear transport theory. Consequently, our result is applicable to a large class of fluid dynamical equations that rely on Lie advection. Hereafter, we give some important examples. Some more material, dealing specifically with helicity problems in fluids and MHD, will be presented in § 4.

3.1 Covariant formulation of the vorticity equation

In this section the vorticity will be considered as a 2-form $\unicode[STIX]{x1D714}$ . We start with the incompressible Euler equations on a $d$ -dimensional Riemannian manifold $(M,g)$ . Written in terms of the velocity vector field $v$ and of the scalar pressure field $p$ , they read

Here the symbol $\unicode[STIX]{x1D735}_{k}$ denotes the covariant derivative, which can be seen as a generalisation to curved spaces of the classical partial derivative $\unicode[STIX]{x2202}_{k}$ of Euclidean spaces (for a more detailed definition, see § B.10). The geometric interpretation of the incompressible Euler equations is recalled in § A.1, while their simplest derivation is obtained from a variational formulation (least-action principle), as explained in § A.2.

The Euler equations and incompressibility condition, written in the contravariant formulation (3.1), can be rewritten in the covariant formulation, i.e. in term of 1-form fields instead of vector fields. Let $v^{\flat }$ be the 1-form field obtained from the vector field by the index lowering operator $(\cdot )^{\flat }:{\mathcal{T}}_{0}^{1}(\unicode[STIX]{x1D6FA})\rightarrow {\mathcal{T}}_{1}^{0}(\unicode[STIX]{x1D6FA})$ ; that is, we set $v^{\flat }=(v^{i}\unicode[STIX]{x2202}_{i})^{\flat }=(v^{\flat })_{i}\,\text{d}x^{i}=\text{g}_{ij}v^{\,j}\,\text{d}x^{i}$ . Using the preservation of the metric of the Riemann–Levi-Civita connection, namely $\unicode[STIX]{x1D735}_{k}\text{g}_{ij}=0$ , we easily find

and

In compact form, equation (3.2) can be written as

Now we rewrite (3.4) as

an equation which differs from a Lie-advection condition for $v^{\flat }$ by just an additional exact differential (which will disappear upon application of yet another exterior differential). To obtain (3.5) we use the Cartan formula $\unicode[STIX]{x00A3}_{v}v^{\flat }=\text{d}\text{i}_{v}v^{\flat }+\text{i}_{v}\,\text{d}v^{\flat }$ for the Lie derivative (§ B.8) and a rewrite of the right-hand side of the Cartan formula, precisely $\text{d}\text{i}_{v}v^{\flat }+\text{i}_{v}\,\text{d}v^{\flat }=(\unicode[STIX]{x1D735}_{v}v)^{\flat }+(1/2)\text{d}(v,v)_{g}$ . This is established in § A.5. In these equations $\text{i}_{v}:\unicode[STIX]{x039B}^{p}(\unicode[STIX]{x1D6FA})\rightarrow \unicode[STIX]{x039B}^{p-1}(\unicode[STIX]{x1D6FA})$ is the interior (or inner) product with the vector $v$ , which acts as an integration; as to $(\cdot ,\cdot )_{g}$ , it denotes the Riemannian scalar product for vector fields in ${\mathcal{T}}_{0}^{1}(\unicode[STIX]{x1D6FA})$ , defined by $(v,w)_{g}=\text{g}_{ij}v^{i}w^{\,j}$ (see § B.3).

From a fluid-mechanical point of view, specialising to the Euclidean case, it is of interest to rewrite the Euler equations (3.5) in standard vector notation as

where $\times$ denotes the vector product and $\unicode[STIX]{x1D735}$ is the standard gradient operator in Euclidean coordinate. This has some similarity to what is known as Lamb’s form of the incompressible Euler equations, in which $\unicode[STIX]{x1D74E}\times \boldsymbol{v}$ also appears. It would not be advisable to simplify (3.6) to Lamb’s form by combining the two terms involving a gradient of the local kinetic energy, because the second and third term on the left-hand side of (3.6) are both needed to obtain a Lie derivative and all the nice consequences.

Indeed, we can now define the vorticity 2-form as the exterior derivative of the infinitesimal velocity circulation 1-form $v^{\flat }$ , that is

Taking the exterior derivative of the covariant formulation (3.5) of the Euler equations, and using the commutation relation $[\text{d},\unicode[STIX]{x00A3}_{v}]=0$ , we obtain

This establishes that the vorticity 2-form is Lie-advected, a result essentially known since Helmholtz (Reference von Helmholtz1858). In terms of the 1-form $v^{\flat }$ , the incompressibility condition $\unicode[STIX]{x1D735}_{i}v^{i}=0$ reads $\text{d}^{\star }v^{\flat }=0$ (see § B.9). Using the Hodge theorem (see § B.13), we obtain the Biot–Savart law $v=(\text{d}^{\star }\unicode[STIX]{x0394}_{H}^{-1}\unicode[STIX]{x1D714})^{\sharp }$ , which self-consistently expresses the velocity vector field $v$ in terms of the vorticity 2-form $\unicode[STIX]{x1D714}$ . Indeed, using the incompressibility condition $\text{d}^{\star }v^{\flat }=0$ , we have $v=(\text{d}^{\star }\unicode[STIX]{x0394}_{H}^{-1}\unicode[STIX]{x1D714})^{\sharp }=(\unicode[STIX]{x0394}_{H}^{-1}\text{d}^{\star }\,\text{d}v^{\flat })^{\sharp }=(\unicode[STIX]{x0394}_{H}^{-1}(\text{d}^{\star }\text{d}+\text{d}\text{d}^{\star })v^{\flat })^{\sharp }=(v^{\flat })^{\sharp }=v$ .

Finally from (3.8), using the lesser known commutation relation $[\sharp ^{d-p}\,\star ,\unicode[STIX]{x00A3}_{v}]=0$ (with $p=2$ ), whose proof is given in § A.6, we obtain that the vorticity vector is also Lie-advected. Here, by vorticity vector, we understand the $(d-2)$ -vector $\unicode[STIX]{x1D74E}:=(\star \unicode[STIX]{x1D714})^{\sharp ^{d-2}}$ (in other words a $(d-2)$ -contravariant tensor). Namely, we have

3.2 Cauchy invariants equation and Cauchy formula

We are now ready to extend to Riemannian manifolds of any dimension the Cauchy invariants equation and the Cauchy formula. We begin by observing that all assumptions of Theorem 1 are now satisfied: the vorticity 2-form $\unicode[STIX]{x1D714}=\text{d}v^{\flat }$ is exact and is Lie-advected. There from follows Corollary 2 for which we also give a direct simplified proof.

We turn now to a corollary that clarifies the relationship between the Cauchy invariants equation and the Cauchy vorticity formula, which are actually Hodge dual of each other. We refer the reader to § B.9 for detailed definition of the Hodge duality operator $\star :\unicode[STIX]{x039B}^{p}(\unicode[STIX]{x1D6FA})\rightarrow \unicode[STIX]{x039B}^{d-p}(\unicode[STIX]{x1D6FA})$ , which implements the already mentioned Hodge duality. Indeed, applying the Hodge dual operator to (3.11), we obtain the following.

In dimensions $d>3$ the Cauchy vorticity formula is no more an equality in terms of 1-forms (or vectors by the lowering–raising duality) but an equality in terms of $(d-2)$ -forms (or $(d-2)$ -contravariant tensors by the lowering–raising duality). Thus for $d>3$ , there still exists a Cauchy-type formula for the vorticity, but given in general by (3.20).

Specialising further, we then consider the flat 3D case and obtain the relations actually written by Cauchy (Reference Cauchy1815) in modern vector notation (Cauchy, of course, wrote them component by component).

4.1 Local helicity in ideal hydrodynamics

Here we assume that $\unicode[STIX]{x1D6FA}$ is of dimension three ( $d=3$ ). Let us recall the covariant Euler equations (3.5), written in terms of the velocity circulation 1-form $v^{\flat }$ :

Here the 0-form $\unicode[STIX]{x1D705}$ , is given by

in the incompressible case and the barotropic compressible case, respectively. Let us introduce the 0-form $\ell$ defined by the following equation

Equation (4.6) can be integrated along the flow $\unicode[STIX]{x1D711}_{t}$ generated by the velocity vector field $v$ , since (4.6) is equivalent to

Integrating (4.6) in time, we obtain

with the initial condition $\ell (0,a)=\ell _{0}(a)$ . The function $\ell$ appears for the first time in the work of Weber (Reference Weber1868) and might be called the Weber function. Let us introduce, $u$ , the modified velocity circulation 1-form defined by

From the definition (4.9), using (4.4)–(4.6), the 1-form $u$ satisfies

and is thus Lie-advected. The 1-form $u$ appears for the first time in Clebsch (Reference Clebsch1859), where it takes the form $u=m\,\text{d}\unicode[STIX]{x1D713}$ . Here, $m$ and $\unicode[STIX]{x1D713}$ are two material invariants (Lie-advected 0-forms), now called the Clebsch variables; $u$ might thus be called the Clebsch 1-form and the associated vector the Clebsch velocity. Of course, the vorticity 2-form $\unicode[STIX]{x1D714}=\text{d}u=\text{d}v^{\flat }$ , still satisfies the Lie-advection equation

From (4.10)–(4.11), we deduce that the local helicity 3-form $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x039B}^{3}(\unicode[STIX]{x1D6FA})$ , which is defined by

satisfies

This is a result of Oseledets (Reference Oseledets1988, where helicity is called spirality). Taking the Hodge dual of (4.13) and using the properties of the Lie derivative (see § B.5) and of the Hodge dual operator (see § B.9), we observe that the scalar local helicity $\star \unicode[STIX]{x1D70E}$ also satisfies a Lie-advection equation; thus it is also a local conserved quantity, as shown by Kuzmin (Reference Kuzmin1983) in the 3D flat space. Given that $\unicode[STIX]{x1D70E}$ is a 3-form in a three-dimensional space, we obviously have $\text{d}\unicode[STIX]{x1D70E}=0$ , and thus $\unicode[STIX]{x1D70E}$ is closed on $\unicode[STIX]{x1D6FA}$ . The situation is different for $d>3$ , because the 4-form $\text{d}v^{\flat }\wedge \text{d}v^{\flat }$ no longer vanishes. Indeed, the wedge product is not commutative in general (see § B.7); hence, the wedge product $\unicode[STIX]{x1D6FC}\wedge \unicode[STIX]{x1D6FC}$ is identically zero only if the degree of the differential form $\unicode[STIX]{x1D6FC}$ is odd (as is the case for the cross-product of two identical vectors). Hence, $\unicode[STIX]{x1D70E}$ is not closed; nevertheless, the helicity 3-form $\unicode[STIX]{x1D70E}$ is still a local invariant since (4.13) remains valid on Riemannian manifolds of any dimension.

Thus local helicity, as a Lie-advection-invariant 3-form, actually exists in any dimension $d\geqslant 3$ , although it cannot in general be associated (by Hodge duality) to a material-invariant scalar.

Returning to the three-dimensional case, we now suppose that the Betti number $b_{3}=0$ (see Remark 1 and § B.13). This guarantees that the closed 3-form $\unicode[STIX]{x1D70E}$ is exact; that is, there exists a 2-form $\unicode[STIX]{x03C0}\in \unicode[STIX]{x039B}^{2}(\unicode[STIX]{x1D6FA})$ such that

From (4.13)–(4.14), and using Theorem 1, we obtain the following Cauchy invariants equation

In principle $\unicode[STIX]{x1D70E}_{0}=u_{0}\wedge \unicode[STIX]{x1D714}_{0}=v_{0}^{\flat }\wedge \text{d}v_{0}^{\flat }-\text{d}\ell _{0}\wedge \text{d}v_{0}^{\flat }$ , but if we choose the initial condition $\ell _{0}=0$ , we obtain $\unicode[STIX]{x1D70E}_{0}=v_{0}^{\flat }\wedge \text{d}v_{0}^{\flat }$ . As stated in Corollary 1, equation (4.15) can actually be inverted to obtain the 2-form  $\unicode[STIX]{x03C0}$ . In the present case, this is particularly simple: from (4.15), using the inverse Lagrangian map, one obtains componentwise

Taking the divergence of this equation and inverting a Laplacian, one formally obtains

where $\unicode[STIX]{x0394}_{a}^{-1}$ denotes the formal inverse of the Laplacian operator $\unicode[STIX]{x0394}_{a}=\sum _{i=1}^{3}\unicode[STIX]{x2202}_{a^{i}}^{2}$ in Cartesian coordinates, and $\unicode[STIX]{x1D6FF}^{kl}=1$ if $k=l$ and zero otherwise.

4.2 Local helicities in ideal MHD

4.3 Other high-order local invariants in hydrodynamics

Here we consider a $d$ -dimensional Riemannian manifolds $(M,g)$ , with $d$ an odd natural integer and $\unicode[STIX]{x1D6FA}$ a bounded region of  $M$ . Again, we consider the velocity circulation 1-form $u$ , which is defined by (4.9). Using the 1-form $u$ , we define the $d$ -form $J\in \unicode[STIX]{x039B}^{d}(\unicode[STIX]{x1D6FA})$ (Serre Reference Serre1984; Gama & Frisch Reference Gama, Frisch, Proctor, Matthews and Rucklidge1993) by

where $(\wedge \,\text{d}u)^{(d-1)/2}$ stands for $(d-1)/2$ times the exterior product of the 2-form $\text{d}u$ . It was proven by Gama & Frisch (Reference Gama, Frisch, Proctor, Matthews and Rucklidge1993) that $J$ is Lie-advected by the velocity field  $v$ . Indeed, first the 1-form $u$ satisfies the Lie-advection equation (4.10). Second, taking the exterior derivative of equation (4.10) the 2-form $\text{d}u$ satisfies the same Lie-advection equation (4.10), because Lie derivative and exterior derivative commute. Therefore we obtain

Since $J\in \unicode[STIX]{x039B}^{d}(\unicode[STIX]{x1D6FA})$ , we obviously have $\text{d}J=0$ , and thus $J$ is closed on $\unicode[STIX]{x1D6FA}$ . We now assume again that the Betti number $b_{p}=0$ (see Remark 1 and § B.13). This guarantees that the closed form $J$ is exact; that is, there exists a $(d-1)$ -form $I$ such that $J=\text{d}I$ . From exactness of the $d$ -form $J$ and (4.46)–(4.47), using Theorem 1, we then obtain our last Cauchy invariants equation

In principle $J_{0}=u_{0}\wedge (\wedge \,\text{d}v_{0}^{\flat })^{(d-1)/2}=v_{0}^{\flat }\wedge (\wedge \,\text{d}v_{0}^{\flat })^{(d-1)/2}-\text{d}\ell _{0}\wedge (\wedge \,\text{d}v_{0}^{\flat })^{(d-1)/2}$ , but if we choose a gauge such that $\ell _{0}=0$ , we obtain $J_{0}=v_{0}^{\flat }\wedge (\wedge \,\text{d}v_{0}^{\flat })^{(d-1)/2}$ . By Corollary 1, the $(d-1)$ -form $I$ can be written as