MAGNETOHYDRODYNAMICS USING PATH OR STREAM FUNCTIONS

Let ${\boldsymbol{l}}$ be an infinitesimal vector adjoining nearby fluid elements that lie on the same magnetic field line (henceforth: length element). The magnetic field can be shown to evolve in proportion to $\rho {\boldsymbol{l}}$ (e.g., Landau & Lifshitz 1960). We derive this relation in an Eulerian picture in Section 2.1.1, and in a Lagrangian framework in Section 2.1.2.

2.1.1. Eulerian Picture

The ideal MHD equations,

$$\begin{eqnarray}&&{\rm{Gauss'}}\;{\rm{law,}}\quad \quad {\boldsymbol{\nabla }}\cdot {\boldsymbol{B}}=0;\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\rm{continuity,}}\quad \quad \displaystyle \frac{\partial \rho }{\partial t}+{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{v}})=0;\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\rm{convection,}}\quad \quad \displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\boldsymbol{\nabla }}\times ({\boldsymbol{v}}\times {\boldsymbol{B}});\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{and}}\;{\rm{Ohm\_\_MPSAPOS\_\_s}}\;{\rm{law,}}\quad {\boldsymbol{E}}=-\displaystyle \frac{{\boldsymbol{v}}}{c}\times {\boldsymbol{B}},\end{eqnarray} \tag{ 5 }$$

can be combined (e.g., Landau & Lifshitz 1960) to give the Helmholtz equation

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}(\displaystyle \frac{{\boldsymbol{B}}}{\rho })=(\displaystyle \frac{{\boldsymbol{B}}}{\rho }\cdot {\boldsymbol{\nabla }}){\boldsymbol{v}}.\end{eqnarray} \tag{ 6 }$$

Here, ${\boldsymbol{E}}$ is the electric field, and c is the speed of light.

As ${\boldsymbol{l}}$ is fixed to the flow, it satisfies the equation

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{l}}}{{dt}}=({\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}){\boldsymbol{v}}.\end{eqnarray} \tag{ 7 }$$

Comparing Equations (6) and (7) indicates that as $({\boldsymbol{B}}/\rho )$ and ${\boldsymbol{l}}$ evolve, their ratio remains constant, such that

$$\begin{eqnarray}&&{\boldsymbol{B}}=\tilde{B}\displaystyle \frac{\rho {\boldsymbol{l}}}{\tilde{\rho }\tilde{l}}.\end{eqnarray} \tag{ 8 }$$

This equation holds for both steady and time-dependent flows, and even across surfaces of discontinuity. Using Equation (8) to evolve ${\boldsymbol{B}}$ from $\tilde{H}$ guarantees that the MHD equations are satisfied everywhere, assuming continuity and that Gauss' law holds on $\tilde{H}$; see Appendix A.

In our non-relativistic, ideal MHD limit, ${\boldsymbol{E}}$ is small, and may be estimated from Equation (5) once ${\boldsymbol{B}}$ is known.

2.1.2. Lagrangian Picture

For the subsequent analysis, it is useful to visualize the magnetic field amplitude as proportional to the density of field lines, i.e.inversely proportional to the distance ψ between them. Therefore, ${\boldsymbol{B}}$ in the direction $\hat{{\boldsymbol{l}}}$ is inversely proportional to the perpendicular (to $\hat{{\boldsymbol{l}}}$) area element ${\mathcal{A}}$ spanned by two such distances, ${{\boldsymbol{\psi }}}_{1}$ and ${{\boldsymbol{\psi }}}_{2}$. Hence,

$$\begin{eqnarray}&&{\boldsymbol{B}}=\tilde{B}\displaystyle \frac{\tilde{{\mathcal{A}}}}{{\mathcal{A}}}\hat{{\boldsymbol{l}}},\end{eqnarray} \tag{ 9 }$$

where ${\mathcal{A}}=\hat{{\boldsymbol{l}}}\cdot ({{\boldsymbol{\psi }}}_{1}\times {{\boldsymbol{\psi }}}_{2})$ is assumed infinitesimally small. As the mass $\rho {\mathcal{A}}l$ of the fluid element in the parallelogram spanned by $\{{\boldsymbol{l}},{{\boldsymbol{\psi }}}_{1},{{\boldsymbol{\psi }}}_{2}\}$ is constant, $\tilde{{\mathcal{A}}}/{\mathcal{A}}=\rho l/(\tilde{\rho }\tilde{l})$, and we recover Equation (8).

2.2.1. General Analysis

The flow can be fully determined by specifying two stream functions, ${\lambda }_{\kappa }$ where $\kappa \in \{1,2\}$ (henceforth), and an additional scalar field φ (Giese 1951). Two surfaces, defined by constant values of the stream functions, intersect along a streamline, so

$$\begin{eqnarray}&&{\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}{\lambda }_{\kappa }=0.\end{eqnarray} \tag{ 10 }$$

This equation, along with the time-independent continuity equation

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{v}})=0,\end{eqnarray} \tag{ 11 }$$

are identically satisfied by the λ gauge (Yih 1957)

$$\begin{eqnarray}\displaystyle \frac{\rho {\boldsymbol{v}}}{\breve{\rho }\breve{v}}=({\boldsymbol{\nabla }}{\lambda }_{1}\times {\boldsymbol{\nabla }}{\lambda }_{2})=| \begin{array}{ccc}{\hat{{\boldsymbol{x}}}}_{1} & {\hat{{\boldsymbol{x}}}}_{2} & {\hat{{\boldsymbol{x}}}}_{3}\\ \displaystyle \frac{\partial {\lambda }_{1}}{\partial {x}_{1}} & \displaystyle \frac{\partial {\lambda }_{1}}{\partial {x}_{2}} & \displaystyle \frac{\partial {\lambda }_{1}}{\partial {x}_{3}}\\ \displaystyle \frac{\partial {\lambda }_{2}}{\partial {x}_{1}} & \displaystyle \frac{\partial {\lambda }_{2}}{\partial {x}_{2}} & \displaystyle \frac{\partial {\lambda }_{2}}{\partial {x}_{3}}\end{array}| ,\end{eqnarray} \tag{ 12 }$$

where the determinant, as written without $\mathrm{Lam}\acute{{\rm{e}}}$ coefficients, is valid only in Cartesian coordinates. We use stream functions (and later, path functions) as coordinates. For convenience, we take them in units of length by introducing the overall normalizations denoted by breve (henceforth; e.g., $\rho /\breve{\rho }$ is a dimensionless mass density). Equation (12), which is known as the Clebsch representation for the mass flux (e.g., Stern 1967), indicates that $\rho {\boldsymbol{v}}$ cannot be chosen as the independent φ, and that swapping ${\lambda }_{1}$ and ${\lambda }_{2}$ would reverse the flow.

In order to parameterize the position of fluid elements in the volume spanned by the flow, we must introduce, in addition to ${\lambda }_{1}$ and ${\lambda }_{2}$, a third variable $q({\boldsymbol{x}})$, not necessarily equal to φ. The surfaces defined by ${\lambda }_{1}$, ${\lambda }_{2}$ and q may never overlap; namely, the vectors

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{1}={\boldsymbol{\nabla }}{\lambda }_{1};\quad {{\boldsymbol{A}}}_{2}={\boldsymbol{\nabla }}{\lambda }_{2};\quad {{\boldsymbol{A}}}_{3}=(\displaystyle \frac{v}{{v}_{q}}){\boldsymbol{\nabla }}q\end{eqnarray} \tag{ 13 }$$

cannot be coplanar anywhere in the flow (except at stagnation points, where the analysis breaks down; see below). Here, ${v}_{q}\equiv {dq}/{dt}={\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}q$ is the rate of change in $q$ along the flow (in units of $q/t$). The normalization factor $(v/{v}_{q})$ is included to keep the $\{{{\boldsymbol{A}}}_{j}\}$ basis dimensionless. The freedom in the choice of $q$ will be utilized in Section 2.2.2.

It is useful to introduce the reciprocal basis ${{\boldsymbol{a}}}_{j}={{\boldsymbol{A}}}_{j}^{-1}$, defined (e.g., Kishan 2007) by ${{\boldsymbol{a}}}_{i}\cdot {{\boldsymbol{A}}}_{j}={\delta }_{{ij}}$, where ${\delta }_{{ij}}$ is Kronecker's delta. Namely,

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{1}\equiv {(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{1}})}_{q}{\rm{;}}\;{{\boldsymbol{a}}}_{2}\equiv {(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{2}})}_{q}{\rm{;}}\;{{\boldsymbol{a}}}_{3}\equiv \displaystyle \frac{{{\boldsymbol{v}}}_{q}}{{\boldsymbol{v}}}{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial q})}_{\vec{{\lambda }}}=\hat{{\boldsymbol{v}}}.\end{eqnarray} \tag{ 14 }$$

Here, $\vec{{\lambda }}\equiv \{{\lambda }_{1},{\lambda }_{2}\}$, and it is understood (henceforth) that derivatives with respect to ${\lambda }_{1}({\lambda }_{2})$ are taken at a fixed ${\lambda }_{2}({\lambda }_{1})$. Like $\{{{\boldsymbol{A}}}_{j}\}$, the reciprocal basis too is dimensionless, and is nowhere coplanar. Note that the ${{\boldsymbol{a}}}_{j}$ are not necessarily perpendicular to one another, and that the $q$-dependent ${{\boldsymbol{a}}}_{\mathrm{1,2}}$ are not necessarily unit vectors.

An advantage of the $\{{{\boldsymbol{a}}}_{j}\}$ basis is that the $\kappa =\{1,2\}$ components of any vector ${\boldsymbol{V}}$ in this basis are independent of $q$, as

$$\begin{eqnarray}&&{\boldsymbol{V}}={V}_{\kappa }{{\boldsymbol{a}}}_{\kappa }+{V}_{3}{{\boldsymbol{a}}}_{3}=[\begin{array}{c}{V}_{1}\\ {V}_{2}\\ {V}_{3}\end{array}]=[\begin{array}{c}{\boldsymbol{V}}\cdot {{\boldsymbol{A}}}_{1}\\ {\boldsymbol{V}}\cdot {{\boldsymbol{A}}}_{2}\\ {\boldsymbol{V}}\cdot {{\boldsymbol{A}}}_{3}\end{array}]=[\begin{array}{c}{\boldsymbol{V}}\cdot {\boldsymbol{\nabla }}{\lambda }_{1}\\ {\boldsymbol{V}}\cdot {\boldsymbol{\nabla }}{\lambda }_{2}\\ \displaystyle \frac{v}{{v}_{q}}{\boldsymbol{V}}\cdot {\boldsymbol{\nabla }}q\end{array}].\end{eqnarray} \tag{ 15 }$$

Moreover, for ${\boldsymbol{l}}$, the $\kappa$ components ${l}_{\kappa }$ are conserved along streamlines, as we show below. We use (henceforth) square brackets to denote vectors in the reciprocal basis, which in general is neither orthogonal nor normalized, and reserve round brackets for vectors in orthonormal (in particular, cylindrical, except in Sections 4.3 and 5.1) bases.

Now, consider two infinitesimally close fluid elements, which at some instant are at the locations ${\boldsymbol{x}}({\lambda }_{\kappa },q)$ and

$$\begin{eqnarray}{\boldsymbol{x}}+{\boldsymbol{l}} & = & {\boldsymbol{x}}({\lambda }_{\kappa }+{l}_{\kappa },q+{\delta }_{q})\\ & = & {\boldsymbol{x}}({\lambda }_{\kappa },q)+{l}_{\kappa }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{\kappa }})}_{q}+{\delta }_{q}{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial q})}_{\vec{{\lambda }}}+O({l}^{2}),\end{eqnarray} \tag{ 16 }$$

where we generalized the Einstein summation rule also for stream function indices $\kappa =\{1,2\}$ (and, later on, for path function indices $\alpha =\{1,2,3\}$). As the $\vec{{\lambda }}$ of each of the two fluid elements is conserved along the flow, the intervals ${l}_{\kappa }={\tilde{l}}_{\kappa }$ are constant along the streamline.

Henceforth, we work to first order in the infinitesimal l, such that

$$\begin{eqnarray}&&{\boldsymbol{l}}={l}_{\kappa }{{\boldsymbol{a}}}_{\kappa }+{l}_{3}{{\boldsymbol{a}}}_{3}={\tilde{l}}_{\kappa }{{\boldsymbol{a}}}_{\kappa }+{l}_{3}{{\boldsymbol{a}}}_{3}=[\begin{array}{c}{\tilde{l}}_{1}\\ {\tilde{l}}_{2}\\ {l}_{3}\end{array}],\end{eqnarray} \tag{ 17 }$$

where ${l}_{3}=v{\delta }_{q}/{v}_{q}$. As ${l}_{\kappa }={\tilde{l}}_{\kappa }$, the $\kappa$ components of ${\boldsymbol{l}}$ in the reciprocal basis are indeed seen to be constant along the streamline. In contrast, ${l}_{3}$ is in general not conserved. The evolution of ${\boldsymbol{l}}$ and its decomposition in the reciprocal basis are illustrated in Figure 1.

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In order to evaluate ${l}_{3}$, consider the travel time ${\rm{\Delta }}t$ of a fluid element from $\tilde{H}$ to the position ${\boldsymbol{x}}$,

$$\begin{eqnarray}&&{\rm{\Delta }}t({\boldsymbol{x}})={\displaystyle \int }_{\tilde{q}({\boldsymbol{x}})}^{q({\boldsymbol{x}})}\displaystyle \frac{{dq}^{\prime} }{{v}_{q}}.\end{eqnarray} \tag{ 18 }$$

During the same time, the ${\boldsymbol{x}}+{\boldsymbol{l}}$ fluid element travels from $\tilde{q}+{\tilde{\delta }}_{q}$ (not necessarily on $\tilde{H}$) to $q+{\delta }_{q}$, so

$$\begin{eqnarray}&&{\displaystyle \int }_{\tilde{q}}^{q}\displaystyle \frac{{dq}^{\prime} }{{v}_{q}({\lambda }_{\kappa },q^{\prime} )}={\rm{\Delta }}t={\displaystyle \int }_{\tilde{q}+{\tilde{\delta }}_{q}}^{q+{\delta }_{q}}\displaystyle \frac{{dq}^{\prime} }{{v}_{q}({\lambda }_{\kappa }+{\tilde{l}}_{\kappa },q^{\prime} )}.\end{eqnarray} \tag{ 19 }$$

Expanding to first order in l indicates that

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{q}}{{v}_{q}}=\displaystyle \frac{{\tilde{\delta }}_{q}}{{\tilde{v}}_{q}}-{\tilde{l}}_{\kappa }{(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }})}_{q,\tilde{q}},\end{eqnarray} \tag{ 20 }$$

where derivatives of ${\rm{\Delta }}t={\rm{\Delta }}t(\vec{{\lambda }},q,\tilde{q})$ with respect to ${\lambda }_{1}$ or ${\lambda }_{2}$ are taken (henceforth) with the other three arguments of ${\rm{\Delta }}t$ fixed. The component ${l}_{3}$ can now be related to the Lagrangian constants through

$$\begin{eqnarray}&&{l}_{3}=v\displaystyle \frac{{\delta }_{q}}{{v}_{q}}={\tilde{l}}_{3}\displaystyle \frac{v}{\tilde{v}}-v{\tilde{l}}_{\kappa }\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }}.\end{eqnarray} \tag{ 21 }$$

If a surface of constant $q$ overlaps with one of the constant ${\lambda }_{\kappa }$ surfaces, ${v}_{q}$ vanishes and Equation (20) is rendered invalid; a piecewise analysis may still be possible. The analysis breaks down at stagnation points, where ${\rm{\Delta }}t$ (and so also ${\boldsymbol{l}}$ and ${\boldsymbol{B}}$) diverges.

Finally, the length element ${\boldsymbol{l}}$ is now given, throughout the flow connected to $\tilde{H}$, by

$$\begin{eqnarray}&&{\boldsymbol{l}}={\tilde{l}}_{\kappa }{{\boldsymbol{a}}}_{\kappa }+\displaystyle \frac{{\boldsymbol{v}}}{\tilde{v}}({\tilde{l}}_{3}-{\tilde{l}}_{\kappa }\tilde{v}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }})=[\begin{array}{c}{\tilde{l}}_{1}\\ {\tilde{l}}_{2}\\ \displaystyle \frac{v}{\tilde{v}}{\tilde{l}}_{3}\end{array}]-[\begin{array}{c}0\\ 0\\ v\end{array}]{\tilde{l}}_{\kappa }\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }}.\end{eqnarray} \tag{ 22 }$$

Note that in addition to ${{\boldsymbol{a}}}_{\kappa }$, the ${\rm{\Delta }}t$ derivative also depends on $q$. It can be written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }}\equiv {(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }})}_{q,\tilde{q}}={\displaystyle \int }_{\tilde{q}}^{q}{(\displaystyle \frac{\partial {v}_{q}^{-1}}{\partial {\lambda }_{\kappa }})}_{{q}^{\prime }}{dq}^{\prime} .\end{eqnarray} \tag{ 23 }$$

The Lagrangian constants are fixed at $\tilde{{\boldsymbol{x}}}$, where the streamline meets $\tilde{H}$. The point $\tilde{{\boldsymbol{x}}}({\boldsymbol{x}})$ can be computed by inverting $\tilde{H}(\tilde{{\boldsymbol{x}}})$, as $\vec{{\lambda }}({\boldsymbol{x}})=\vec{{\lambda }}(\tilde{{\boldsymbol{x}}})$ is given, and remains constant along the streamline. The components ${\tilde{l}}_{j}$ are found by projecting $\tilde{{\boldsymbol{l}}}$ onto the reciprocal basis as in Equation (15),

$$\begin{eqnarray}&&{\tilde{l}}_{\kappa }={\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{\lambda }_{\kappa }={({\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{\lambda }_{\kappa })}_{\tilde{H}}\quad {\rm{and}}\quad {\tilde{l}}_{3}={(\displaystyle \frac{v}{{v}_{q}}\;{\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}q)}_{\tilde{H}}.\end{eqnarray} \tag{ 24 }$$

Equations (8) and (22) yield the magnetic field, as

$$\begin{eqnarray}&&\displaystyle \frac{\tilde{\rho }}{\rho }{\boldsymbol{B}}=\tilde{B}\displaystyle \frac{{\boldsymbol{l}}}{\tilde{l}}=[\begin{array}{c}{\tilde{B}}_{1}\\ {\tilde{B}}_{2}\\ \displaystyle \frac{v}{\tilde{v}}{\tilde{B}}_{3}\end{array}]-[\begin{array}{c}0\\ 0\\ v\end{array}]{\tilde{B}}_{\kappa }\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }},\end{eqnarray} \tag{ 25 }$$

where the Lagrangian constants ${\tilde{B}}_{j}=(\tilde{B}/\tilde{l}){\tilde{l}}_{j}$ are fixed by the projection Equation (15) on $\tilde{H}$,

$$\begin{eqnarray}&&{\tilde{B}}_{\kappa }={({\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}{\lambda }_{\kappa })}_{\tilde{H}}\quad {\rm{and}}\quad {\tilde{B}}_{3}={(\displaystyle \frac{v}{{v}_{q}}\;{\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}q)}_{\tilde{H}}.\end{eqnarray} \tag{ 26 }$$

Next, consider the electric field ${\boldsymbol{E}}$. The vector product $({\boldsymbol{v}}\times {\boldsymbol{B}})$ in Equation (5) eliminates the ${{\boldsymbol{a}}}_{3}$ terms in Equation (25), leaving

$$\begin{eqnarray}&&{\boldsymbol{E}}=\displaystyle \frac{\breve{\rho }\breve{v}}{\tilde{\rho }c}({\tilde{B}}_{2}{\boldsymbol{\nabla }}{\lambda }_{1}-{\tilde{B}}_{1}{\boldsymbol{\nabla }}{\lambda }_{2}).\end{eqnarray} \tag{ 27 }$$

Therefore, ${\boldsymbol{E}}$ is entirely determined by the stream functions, independent of φ, once ${\boldsymbol{B}}$ and ρ are given on $\tilde{H}$. Notice that deriving the electric field directly from the stream (and below, from the path) functions is computationally much simpler than first computing the magnetic field, which typically involves integration, and then using Ohm's law to find ${\boldsymbol{E}}$.

2.2.2. Choice of a Third Variable $q({\boldsymbol{x}})$

There are various choices for the third variable, $q$, depending on the specific problem at hand. A useful choice is the travel time ${\rm{\Delta }}t$, which in some cases is easily estimated or computed. For $q={\rm{\Delta }}t$, which is the specific case illustrated in Figure 1, Equations (22) and (25) reduce to

$$\begin{eqnarray}&&{\boldsymbol{l}}={\tilde{l}}_{\kappa }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{\kappa }})}_{{\rm{\Delta }}t}+{\tilde{l}}_{3}\displaystyle \frac{{\boldsymbol{v}}}{\tilde{v}}=[\begin{array}{c}{\tilde{l}}_{1}\\ {\tilde{l}}_{2}\\ \displaystyle \frac{v}{\tilde{v}}{\tilde{l}}_{3}\end{array}],\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}&&{\boldsymbol{B}}=[{\tilde{B}}_{\kappa }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{\kappa }})}_{{\rm{\Delta }}t}+{\tilde{B}}_{3}\displaystyle \frac{{\boldsymbol{v}}}{\tilde{v}}]\displaystyle \frac{\rho }{\tilde{\rho }}=[\begin{array}{c}{\tilde{B}}_{1}\\ {\tilde{B}}_{2}\\ \displaystyle \frac{v}{\tilde{v}}{\tilde{B}}_{3}\end{array}]\displaystyle \frac{\rho }{\tilde{\rho }}.\end{eqnarray} \tag{ 29 }$$

Notice that Equation (28) could have been directly inferred from the reciprocal basis projection Equation (15), by noting that ${\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}({\rm{\Delta }}t)={\rm{\Delta }}t({\boldsymbol{x}}+{\boldsymbol{l}})-{\rm{\Delta }}t({\boldsymbol{x}})+O({l}^{2})$ is constant (to first order) along a stream line, as this is the difference between the travel times of two nearby fluid elements.

In the case of an irrotational flow, a potential Φ can be defined such that ${\boldsymbol{v}}=\breve{v}{\boldsymbol{\nabla }}{\rm{\Phi }}$, where the normalization factor assigns Φ with units of length. Here we may choose $q={\rm{\Phi }}$, and read ${\boldsymbol{l}}$ and ${\boldsymbol{B}}$ off Equations (22) and (25), respectively, with the time derivative term (see Equation (23))

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }}={(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial {\lambda }_{\kappa }})}_{{\rm{\Phi }},\tilde{{\rm{\Phi }}}}=\breve{v}{\displaystyle \int }_{\tilde{{\rm{\Phi }}}}^{{\rm{\Phi }}}d{{\rm{\Phi }}}^{\prime }{(\displaystyle \frac{\partial {v}^{-2}}{\partial {\lambda }_{\kappa }})}_{{{\rm{\Phi }}}^{\prime }}.\end{eqnarray} \tag{ 30 }$$

Equipotential surfaces are perpendicular to the streamlines, so here ${{\boldsymbol{a}}}_{1}$ and ${{\boldsymbol{a}}}_{2}$ are conveniently perpendicular to ${{\boldsymbol{a}}}_{3}$. In simple, e.g., axisymmetric, flows where ${{\boldsymbol{a}}}_{1}$ and ${{\boldsymbol{a}}}_{2}$ are perpendicular to each other, the reciprocal basis thus becomes orthogonal.

Among the infinite other possible choices of $q$, noteworthy are the natural coordinates of the flow, for example the axial coordinate z in simple axisymmetric flows, and the length $s({\boldsymbol{x}})$ of the streamline from $\tilde{H}$ to ${\boldsymbol{x}}$. In general, the simplest variable which is monotonic along the flow is advantageous.

A time-dependent flow can be fully specified by three path functions, ${{\rm{\Lambda }}}_{\alpha =\mathrm{1,2,3}}$, when particle diffusion can be neglected (Yih 1957). Three spacetime manifolds, defined by some constant values of these path functions, intersect along a pathline. Therefore, along the flow

$$\begin{eqnarray}&&\displaystyle \frac{d{{\rm{\Lambda }}}_{\alpha }({\boldsymbol{x}},t)}{{dt}}=(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}){{\rm{\Lambda }}}_{\alpha }({\boldsymbol{x}},t)=0,\end{eqnarray} \tag{ 31 }$$

which determines ${\boldsymbol{v}}$ through

$$\begin{eqnarray}&&{\boldsymbol{v}}=-\displaystyle \frac{{\epsilon }_{{ijk}}({\partial }_{t}{{\rm{\Lambda }}}_{i})({\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{j}\times {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{k})}{2({\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{1}\times {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{2})\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{3}},\end{eqnarray} \tag{ 32 }$$

where ${\epsilon }_{{ijk}}$ is the Levi-Civita symbol. The continuity Equation (3) is then identically satisfied by the Λ gauge (Yih 1957)

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{\breve{\rho }}=-({\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{1}\times {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{2})\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{3}.\end{eqnarray} \tag{ 33 }$$

Equations (32) and (33) can be written as (Yih 1957)

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{\breve{\rho }}(\begin{array}{c}1\\ {\boldsymbol{v}}\end{array})=\displaystyle \frac{\rho }{\breve{\rho }}({\bf{1}}+{\boldsymbol{v}})=| \begin{array}{cccc}{\hat{{\boldsymbol{x}}}}_{1} & {\hat{{\boldsymbol{x}}}}_{2} & {\hat{{\boldsymbol{x}}}}_{3} & {\bf{1}}\\ \displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {x}_{1}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {x}_{2}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {x}_{3}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial t}\\ \displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {x}_{1}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {x}_{2}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {x}_{3}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial t}\\ \displaystyle \frac{\partial {{\rm{\Lambda }}}_{3}}{\partial {x}_{1}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{3}}{\partial {x}_{2}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{3}}{\partial {x}_{3}} & \displaystyle \frac{\partial {{\rm{\Lambda }}}_{3}}{\partial t}\end{array}| ,\end{eqnarray} \tag{ 34 }$$

where ${\bf{1}}$ is a unit vector in some non-spatial dimension, and the determinant, as written without $\mathrm{Lam}\acute{e}$ coefficients, holds only in Cartesian coordinates. Equivalently (Van Roessel & Hui 1991), if ${v}^{\mu }\equiv (1,{\boldsymbol{v}})$,

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{\breve{\rho }}\;{v}^{\alpha }={-\epsilon }^{\alpha \beta \gamma \delta }\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}\partial {{\rm{\Lambda }}}_{2}\partial {{\rm{\Lambda }}}_{3}}{\partial {x}^{\beta }\partial {x}^{\gamma }\partial {x}^{\delta }},\end{eqnarray} \tag{ 35 }$$

where $\{\alpha ,\beta ,\gamma ,\delta \}$ are four-indices, ${x}^{0}\equiv t$, and ${\epsilon }^{\alpha \beta \gamma \delta }$ is the four-dimensional Levi-Civita symbol.

To parameterize spacetime using the path functions ${{\rm{\Lambda }}}_{1}$, ${{\rm{\Lambda }}}_{2}$, and ${{\rm{\Lambda }}}_{3}$, we normalize them in units of length, and introduce a fourth coordinate, $Q({\boldsymbol{x}},t)$. The fourth coordinate can be chosen arbitrarily, as long as the equivalue hypersurfaces of the four parameters never overlap.

Define ${\boldsymbol{l}}$ as the distance between two simultaneous, infinitesimally close spacetime events,

$$\begin{eqnarray}&&{x}^{\mu }({{\rm{\Lambda }}}_{\alpha },Q)=(\begin{array}{c}t\\ {\boldsymbol{x}}\end{array})\end{eqnarray} \tag{ 36 }$$

and

$$\begin{eqnarray}&&(\begin{array}{c}t\\ {\boldsymbol{x}}+{\boldsymbol{l}}\end{array}) & = & {x}^{\mu }+{{dx}}^{\mu }={x}^{\mu }({{\rm{\Lambda }}}_{\alpha }+{L}_{\alpha },Q+{\delta }_{Q})\\ & = & {x}^{\mu }+{L}_{\alpha }\displaystyle \frac{\partial {x}^{\mu }}{\partial {{\rm{\Lambda }}}_{\alpha }}+{\delta }_{Q}\displaystyle \frac{\partial {x}^{\mu }}{\partial Q}+O({l}^{2}){\rm{.}}\end{eqnarray} \tag{ 37 }$$

As ${x}^{\mu }={x}^{\mu }({{\rm{\Lambda }}}_{\alpha },Q)$, derivatives with respect to $Q$ or one of the ${{\rm{\Lambda }}}_{\alpha }$ are understood as taken with the other three arguments fixed. Here, the infinitesimal ${L}_{\alpha }={\tilde{L}}_{\alpha }$ are constant along the pathline, because they are the differences between the conserved path function values. In contrast, ${\delta }_{Q}$ varies in general along the pathline.

Working to first order in l, the spacetime length element is now given by

$$\begin{eqnarray}&&(\begin{array}{c}0\\ {\boldsymbol{l}}\end{array})={\tilde{L}}_{\alpha }(\begin{array}{c}\partial t/\partial {{\rm{\Lambda }}}_{\alpha }\\ \partial {\boldsymbol{x}}/\partial {{\rm{\Lambda }}}_{\alpha }\end{array})+\displaystyle \frac{{\delta }_{Q}}{{v}_{Q}}(\begin{array}{c}1\\ {\boldsymbol{v}}\end{array}),\end{eqnarray} \tag{ 38 }$$

where we defined ${v}_{Q}\equiv {dQ}/{dt}$ as the rate of change in $Q$ along the flow (in units of $Q/t$). The vanishing temporal component requires

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{Q}}{{v}_{Q}}=-{\tilde{L}}_{\alpha }\displaystyle \frac{\partial t}{\partial {{\rm{\Lambda }}}_{\alpha }},\end{eqnarray} \tag{ 39 }$$

so the spatial component gives

$$\begin{eqnarray}&&{\boldsymbol{l}}={\tilde{L}}_{\alpha }(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{\alpha }}-{\boldsymbol{v}}\displaystyle \frac{\partial t}{\partial {{\rm{\Lambda }}}_{\alpha }}).\end{eqnarray} \tag{ 40 }$$

Taking the scalar product of this result with ${\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\alpha }$, and using Equation (31) and the chain rule, implies that the constant ${\tilde{L}}_{\alpha }$ satisfy

$$\begin{eqnarray}&&{\tilde{L}}_{\alpha }={\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\alpha }={({\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\alpha })}_{\tilde{H}},\end{eqnarray} \tag{ 41 }$$

and are independent of $Q$. They may thus be determined at any position where ${\boldsymbol{l}}$ is known, in particular on $\tilde{H}$.

Finally, combining Equations (8) and (40) yields

$$\begin{eqnarray}&&{\boldsymbol{B}}={\tilde{B}}_{\alpha }\displaystyle \frac{\rho }{\tilde{\rho }}(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{\alpha }}-{\boldsymbol{v}}\displaystyle \frac{\partial t}{\partial {{\rm{\Lambda }}}_{\alpha }}),\end{eqnarray} \tag{ 42 }$$

where the Lagrangian constants ${\tilde{B}}_{\alpha }\equiv (\tilde{B}/\tilde{l}){\tilde{L}}_{\alpha }$ can be determined using

$$\begin{eqnarray}&&{\tilde{B}}_{\alpha }=\displaystyle \frac{\tilde{\rho }}{\rho }{\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\alpha }={({\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\alpha })}_{\tilde{H}}.\end{eqnarray} \tag{ 43 }$$

As in the time-independent case, here too there is considerable freedom in choosing $Q$. A useful choice is the temporal coordinate, $Q=t$, for which Equation (40) reduces to

$$\begin{eqnarray}&&{\boldsymbol{l}}={\tilde{L}}_{\alpha }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{\alpha }})}_{t}.\end{eqnarray} \tag{ 44 }$$

The time-dependent vectors ${(\partial {\boldsymbol{x}}/\partial {{\rm{\Lambda }}}_{\alpha })}_{t}$ can now be identified as the generalizations of the reciprocal basis vectors found for steady flows; see Equations (14) and (17). Equation (42) now gives the magnetic field,

$$\begin{eqnarray}&&{\boldsymbol{B}}={\tilde{B}}_{\alpha }\displaystyle \frac{\rho }{\tilde{\rho }}{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{\alpha }})}_{t}.\end{eqnarray} \tag{ 45 }$$

Other choices of $Q$ may be advantageous in certain circumstances.

The electric field in a time-dependent flow becomes, according to Equations (5) and (42),

$$\begin{eqnarray}&&{\boldsymbol{E}}={\epsilon }_{{ijk}}\displaystyle \frac{\breve{\rho }{\tilde{B}}_{i}}{\tilde{\rho }c}(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{j}}{\partial t}){\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{k}.\end{eqnarray} \tag{ 46 }$$

To ascertain that Sections 2.2 and 2.3 are mutually consistent, we consider an arbitrary steady flow, and regard it as time-dependent. If the steady flow is characterized by the stream functions ${\lambda }_{1}$ and ${\lambda }_{2}$, we can choose two path functions as time independent,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{1}={\lambda }_{1}\quad {\rm{and}}\quad {{\rm{\Lambda }}}_{2}={\lambda }_{2}.\end{eqnarray} \tag{ 47 }$$

Solving Equations (32) and (33) with the aid of Equation (12) and the definitions of Section 2.2, we find that the third path function must be

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{3}=\breve{v}(t-{\rm{\Delta }}t).\end{eqnarray} \tag{ 48 }$$

We can now use the time-dependent results of Section 2.3, and attempt to reproduce the steady flow results of Section 2.2.

For simplicity, we choose Q = t, such that the solution Equation (44) for ${\boldsymbol{l}}$ in the time dependent regime gives

$$\begin{eqnarray}{\boldsymbol{l}} & = & {\tilde{L}}_{\kappa }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{\kappa }})}_{{{\rm{\Lambda }}}_{3},t}+{\tilde{L}}_{3}{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {{\rm{\Lambda }}}_{3}})}_{{{\rm{\Lambda }}}_{\kappa },t}\\ & = & {\tilde{L}}_{\kappa }{(\displaystyle \frac{\partial {\boldsymbol{x}}}{\partial {\lambda }_{\kappa }})}_{{\rm{\Delta }}t,t}+(-{\tilde{L}}_{3}\displaystyle \frac{\tilde{v}}{\breve{v}})\displaystyle \frac{{\boldsymbol{v}}}{\tilde{v}}{\rm{.}}\end{eqnarray} \tag{ 49 }$$

This is formally identical to Equation (28), derived for a steady flow under the choice $q={\rm{\Delta }}t$, provided that one may consistently identify ${\tilde{l}}_{\kappa }={\tilde{L}}_{\kappa }$ and ${\tilde{l}}_{3}=-{\tilde{L}}_{3}(\tilde{v}/\breve{v})$. Indeed, Equation (41) yields ${\tilde{L}}_{\kappa }={\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{\kappa }=$ ${\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{\lambda }_{\kappa }={\tilde{l}}_{\kappa }$, and, on $\tilde{H}$, ${\tilde{L}}_{3}={\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}{{\rm{\Lambda }}}_{3}$ = $-\breve{v}{\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}({\rm{\Delta }}t)$ = $-(\breve{v}/\tilde{v})$ $(v/{v}_{q}){\boldsymbol{l}}\cdot {\boldsymbol{\nabla }}q=-(\breve{v}/\tilde{v}){\tilde{l}}_{3}$; consistent with the definitions of Equation (24).

In conclusion, the time-dependent analysis with Q = t yields the time-independent analysis with $q={\rm{\Delta }}t$, when applied to a steady flow. This shows that the two formalisms, involving path functions and stream functions, are consistent with each other.

With the above ${{\rm{\Lambda }}}_{\alpha }$, the time-dependent electric field in Equation (46) directly reduces to the time-independent ${\boldsymbol{E}}$ in Equation (27), as $\partial {{\rm{\Lambda }}}_{j}/\partial t=\breve{v}{\delta }_{3,j}$.

Consider a steady, axisymmetric flow. Using cylindrical coordinates $\{\varrho ,\phi ,z\}$, such flows are characterized by ${v}_{\phi }=0$, and by vanishing ϕ derivatives of the flow parameters. Therefore, constant ϕ surfaces are perpendicular to the flow, and we may choose ${\lambda }_{1}\equiv \breve{R}\phi$, where $\breve{R}$ is some arbitrary length scale of the problem. Since the velocity field ${\boldsymbol{v}}$ is ϕ-independent, we may choose ${\lambda }_{2}\equiv \lambda (\varrho ,z)$. For these stream functions, Equation (12) yields

$$\begin{eqnarray}&&{\boldsymbol{v}}=({v}_{\varrho },0,{v}_{z})=\displaystyle \frac{\breve{\mu }}{\varrho \rho }(\displaystyle \frac{\partial \lambda }{\partial z},0,-\displaystyle \frac{\partial \lambda }{\partial \varrho }),\end{eqnarray} \tag{ 50 }$$

where ${\boldsymbol{\mu }}\equiv \rho \varrho {\boldsymbol{v}}$ is the mass flux in a ring of radius $\varrho$ (and has units of viscosity). Here, the normalizations $\breve{R}$ and $\breve{\mu }\equiv \breve{\rho }\;\breve{v}\;\breve{R}$ arise from our use of stream functions as coordinates with units of length. Round bracket vectors in this section pertain to cylindrical coordinates.

In addition to ${\lambda }_{1}$ and ${\lambda }_{2}$, we must now choose a third variable q to parameterize space. The choice $q=z$ is advantageous for simple flow regions in which v_z does not change sign. For our choice of stream functions and q, the reciprocal basis Equation (14) becomes

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{1}=\displaystyle \frac{\varrho }{\breve{R}}\hat{{\boldsymbol{\phi }}}{\rm{;}}\qquad {{\boldsymbol{a}}}_{2}=-\displaystyle \frac{\breve{\mu }}{{\mu }_{z}}\hat{{\boldsymbol{\varrho }}}{\rm{;}}\qquad {{\boldsymbol{a}}}_{3}=\hat{{\boldsymbol{v}}}.\end{eqnarray} \tag{ 51 }$$

A given $\tilde{{\boldsymbol{B}}}=({\tilde{B}}_{\varrho },{\tilde{B}}_{\phi },{\tilde{B}}_{z})$ may now be decomposed as ${\tilde{B}}_{i}{\tilde{{\boldsymbol{a}}}}_{i}$, where (see Equation (26))

$$\begin{eqnarray}&&\tilde{{\boldsymbol{B}}}=[\begin{array}{c}{\tilde{B}}_{1}\\ {\tilde{B}}_{2}\\ {\tilde{B}}_{3}\end{array}]=[\begin{array}{c}{\tilde{B}}_{\phi }\displaystyle \frac{\breve{R}}{\tilde{\varrho }}\\ ({\tilde{\mu }}_{\varrho }{\tilde{B}}_{z}-{\tilde{\mu }}_{z}{\tilde{B}}_{\varrho })/\breve{\mu }\\ {\tilde{B}}_{z}\displaystyle \frac{\tilde{v}}{{\tilde{v}}_{z}}\end{array}].\end{eqnarray} \tag{ 52 }$$

Recall that in our notations, vectors in square brackets (as in Equation (52)) are in the reciprocal basis.

Equation (25) then yields

$$\begin{eqnarray}\displaystyle \frac{\tilde{\rho }}{\rho }{\boldsymbol{B}} & = & \displaystyle \frac{\varrho }{\breve{R}}{\tilde{B}}_{1}\hat{{\boldsymbol{\phi }}}-\displaystyle \frac{\breve{\mu }}{{\mu }_{z}}{\tilde{B}}_{2}\hat{{\boldsymbol{\varrho }}}+\displaystyle \frac{{\boldsymbol{v}}}{\tilde{v}}({\tilde{B}}_{3}-{\tilde{B}}_{2}\tilde{v}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda })\\ & = & (\begin{array}{c}\displaystyle \frac{{\tilde{\mu }}_{z}}{{\mu }_{z}}{\tilde{B}}_{\varrho }\\ \displaystyle \frac{\varrho }{\tilde{\varrho }}{\tilde{B}}_{\phi }\\ \displaystyle \frac{{v}_{z}}{{\tilde{v}}_{z}}{\tilde{B}}_{z}\end{array})+(\begin{array}{c}\displaystyle \frac{{v}_{\varrho }}{{\tilde{v}}_{z}}-\displaystyle \frac{{\tilde{\mu }}_{\varrho }}{{\mu }_{z}}\\ 0\\ 0\end{array}){\tilde{B}}_{z}-(\begin{array}{c}{v}_{\varrho }\\ 0\\ {v}_{z}\end{array}){\tilde{B}}_{2}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }{\rm{.}}\end{eqnarray} \tag{ 53 }$$

The only computation necessary is an integral for the time derivative term,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }={[\displaystyle \frac{\partial {\rm{\Delta }}t(\varrho ,z)}{\partial \lambda }]}_{\phi ,z,\tilde{z}}={\displaystyle \int }_{\tilde{z}}^{z}{(\displaystyle \frac{\partial {v}_{z}^{-1}}{\partial \lambda })}_{\phi ,z^{\prime} }{dz}^{\prime} .\end{eqnarray} \tag{ 54 }$$

The electric field is found from Equations (27), (50), and (52), and given simply by

$$\begin{eqnarray}&&{\boldsymbol{E}}=\displaystyle \frac{\tilde{v}}{c}(\begin{array}{c}\displaystyle \frac{{\mu }_{z}}{\tilde{\mu }}{\tilde{B}}_{\phi }\\ \displaystyle \frac{\tilde{\varrho }\breve{\mu }}{\varrho \tilde{\mu }}{\tilde{B}}_{2}\\ -\displaystyle \frac{{\mu }_{\varrho }}{\tilde{\mu }}{\tilde{B}}_{\phi }\end{array})=\displaystyle \frac{\tilde{v}}{\tilde{\mu }c}(\begin{array}{c}{\mu }_{z}{\tilde{B}}_{\phi }\\ \displaystyle \frac{\tilde{\varrho }}{\varrho }\;({\tilde{\mu }}_{\varrho }{\tilde{B}}_{z}-{\tilde{\mu }}_{z}{\tilde{B}}_{\varrho })\\ -{\mu }_{\varrho }{\tilde{B}}_{\phi }\end{array}).\end{eqnarray} \tag{ 55 }$$

Next, we focus on the vicinity of the axis of symmetry, $\varrho =0$, along which ${v}_{\varrho }=0$. Taking the incident flow in the ($-\hat{{\boldsymbol{z}}}$) direction, we denote the incoming velocity as $u(\rho \simeq 0,z)\equiv -{v}_{z}$, and assume it is finite. Then, Equation (50) indicates that in the vicinity of the axis, $\lambda =(\rho u/2\breve{\mu }){\varrho }^{2}+O({\varrho }^{3})$. Along a streamline, λ is constant, therefore so is $\rho u{\varrho }^{2}$; this is a direct expression of mass conservation. In the vicinity of the axis, we may thus write $\varrho /\tilde{\varrho }={(\tilde{\rho }\tilde{u}/\rho u)}^{1/2}$ along a streamline. This holds even in the presence of a shock, which conserves the normal, in this case axial, mass flux $j=\rho u$.

Equation (53) now becomes

$$\begin{eqnarray}&&{\boldsymbol{B}}=(\begin{array}{c}{(\displaystyle \frac{\rho /u}{\tilde{\rho }/\tilde{u}})}^{\frac{1}{2}}{\tilde{B}}_{\varrho }\\ {(\displaystyle \frac{\rho /u}{\tilde{\rho }/\tilde{u}})}^{\frac{1}{2}}{\tilde{B}}_{\phi }\\ \displaystyle \frac{j}{\tilde{j}}{\tilde{B}}_{z}\end{array})+C(\begin{array}{c}0\\ 0\\ {\tilde{B}}_{\varrho }\end{array}),\end{eqnarray} \tag{ 56 }$$

where

$$\begin{eqnarray}&&C(z)\equiv -\displaystyle \frac{j{\tilde{\mu }}_{z}}{\tilde{\rho }\breve{\mu }}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }=\displaystyle \frac{j\tilde{u}}{\sqrt{\tilde{j}}}{\displaystyle \int }_{\tilde{z}}^{z}\displaystyle \frac{{u}_{1}{{dz}}^{\prime }}{{u}_{0}^{2}\sqrt{j}},\end{eqnarray} \tag{ 57 }$$

and we expanded $u={u}_{0}(z)+{u}_{1}(z)\rho +O({\rho }^{2})$.

For a potential flow, ${u}_{1}=0$, and so C = 0. Such a flow arises, for example, if a homogeneous incident flow remains subsonic or mildly supersonic (e.g., Landau & Lifshitz 1959).

It is typically possible to derive $(\rho /\tilde{\rho })$ as a function of u, such that ${\boldsymbol{B}}={\boldsymbol{B}}(u)$ along the axis. For example, consider a steady, inviscid flow in a polytropic gas of an adiabatic index γ, with dynamically insignificant magnetic field. Such a flow is governed, in addition to continuity, by Euler's equation,

$$\begin{eqnarray}&&({\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}){\boldsymbol{v}}=-\displaystyle \frac{{\boldsymbol{\nabla }}P}{\rho }=-\displaystyle \frac{{c}_{s}^{2}\;{\boldsymbol{\nabla }}\rho }{\rho },\end{eqnarray} \tag{ 58 }$$

and Bernoulli's equation,

$$\begin{eqnarray}&&\displaystyle \frac{{u}^{2}}{2}+\displaystyle \frac{{c}_{s}^{2}}{\gamma -1}=\displaystyle \frac{{\bar{c}}_{s}^{2}}{\gamma -1}={\rm{constant.}}\end{eqnarray} \tag{ 59 }$$

Here, P is the fluid pressure, ${c}_{s}={(\gamma P/\rho )}^{1/2}$ is the (local) speed of sound, and a bar denotes (henceforth) a putative stagnation point where v = 0, whether or not such a point lies along a given streamline.

The axial mass density solution is ${\rho }^{2}\propto {({S}^{2}-{M}_{0}^{2})}^{{S}^{2}}$, where $S\equiv {[2/(\gamma -1)]}^{1/2}$, and ${M}_{0}\equiv u/{\bar{c}}_{s}$ is the axial Mach number with respect to the stagnation sound speed. Hence, away from shocks,

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{\tilde{\rho }}={(\displaystyle \frac{{S}^{2}-{M}_{0}^{2}}{{S}^{2}-{\tilde{M}}_{0}^{2}})}^{1/(\gamma -1)}\end{eqnarray} \tag{ 60 }$$

can be used in Equation (56) to find ${\boldsymbol{B}}$ once u is determined. When a shock is present, the Lagrangian constants here are taken downstream; if $\tilde{H}$ is located upstream, the shock compression ratio ${({\tilde{M}}_{0}/W)}^{2}$ should be incorporated. Here, $W\equiv {[2/(\gamma +1)]}^{1/2}$ and S are the weak and strong shock limits of M₀; see Keshet & Naor (2014).

Consider the inviscid, incompressible, steady flow around a solid sphere. We assume that the incident flow is homogeneous, unidirectional, and subsonic, so the entire flow is irrotational. Here, the mass density $\rho =\tilde{\rho }$ is constant, the radius of the sphere is taken (henceforth) as R = 1, and the constant $\tilde{{\boldsymbol{v}}}=-\tilde{u}\hat{{\boldsymbol{z}}}$ is taken on $\tilde{H}$ far upstream, $\tilde{z}\to \infty$. It is thus natural to choose $\breve{\rho }=\tilde{\rho }$, $\breve{R}=1$ and $\breve{v}=\tilde{u}$, so $\breve{\mu }=\tilde{\rho }\tilde{u}$.

As ${\boldsymbol{\nabla }}\cdot {\boldsymbol{v}}=0$ and ${\boldsymbol{\nabla }}\times {\boldsymbol{v}}=0$, the flow satisfies the Poisson equation ${{\boldsymbol{\nabla }}}^{2}{\rm{\Phi }}=0$ for the velocity potential Φ, defined by ${\boldsymbol{v}}=\tilde{u}{\boldsymbol{\nabla }}{\rm{\Phi }}$. The solution for boundary conditions of a vanishing normal velocity across the sphere is

$$\begin{eqnarray}&&{\rm{\Phi }}=-z(1+\displaystyle \frac{1}{2{r}^{3}}).\end{eqnarray} \tag{ 61 }$$

Here we primarily use cylindrical coordinates, but occasionally allude to spherical coordinates $\{r,\theta ,\phi \}$, where ${r}^{2}={\varrho }^{2}+{z}^{2}$. Thus,

$$\begin{eqnarray}&&{v}_{z}=\tilde{u}(-1-\displaystyle \frac{1}{2{r}^{3}}+\displaystyle \frac{3{z}^{2}}{2{r}^{5}})\quad {\rm{and}}\quad {v}_{\varrho }=\displaystyle \frac{3\tilde{u}\varrho z}{2{r}^{5}}.\end{eqnarray} \tag{ 62 }$$

The stream functions can be chosen as ${\lambda }_{1}=\phi$ and, according to Equation (50),

$$\begin{eqnarray}&&\lambda \equiv {\lambda }_{2}=\displaystyle \frac{{\varrho }^{2}}{2}(1-\displaystyle \frac{1}{{r}^{3}}),\end{eqnarray} \tag{ 63 }$$

such that $\tilde{\varrho }={(2\lambda )}^{1/2}$.

Solving Euler's Equation (58) gives the pressure,

$$\begin{eqnarray}&&P=\tilde{P}+\tilde{\rho }{\tilde{u}}^{2}\displaystyle \frac{4{r}^{3}-5+3(4{r}^{3}-1)\mathrm{cos}2\theta }{16{r}^{6}},\end{eqnarray} \tag{ 64 }$$

where $\mathrm{cos}2\theta =({z}^{2}-{\varrho }^{2})/{r}^{2}$. Note that an incident pressure smaller than $(5/8)\tilde{\rho }{\tilde{u}}^{2}$ would lead to a nonphysical $P\lt 0$ around z = 0, as the plasma would be too cold to remain incompressible.

Consider the evolution of a magnetic field, initially given by $\tilde{{\boldsymbol{B}}}=({\tilde{B}}_{\varrho },{\tilde{B}}_{\phi },{\tilde{B}}_{z})$, for different choices of q. In this subsection, round-bracket vectors are in cylindrical coordinates.

4.1.1. The Natural Choice $q=z$

For the choice $q=z$, which is natural because z monotonically decreases for all fluid elements, the magnetic field is given by (see Equation (53))

$$\begin{eqnarray}&&{\boldsymbol{B}}={\tilde{B}}_{\phi }\displaystyle \frac{\varrho }{\tilde{\varrho }}\hat{{\boldsymbol{\phi }}}-{\tilde{B}}_{\varrho }\displaystyle \frac{\tilde{\varrho }\tilde{u}}{\varrho {v}_{z}}\hat{{\boldsymbol{\varrho }}}-\displaystyle \frac{{\boldsymbol{v}}}{\tilde{u}}({\tilde{B}}_{z}+{\tilde{B}}_{\varrho }\tilde{\varrho }\tilde{u}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }),\end{eqnarray} \tag{ 65 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }={(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda })}_{\phi ,z,\tilde{z}}=-\displaystyle \frac{12}{\tilde{u}}{\displaystyle \int }_{\infty }^{z}\displaystyle \frac{r{^{\prime} }^{8}(r{^{\prime} }^{2}-5z{^{\prime} }^{2}){dz}^{\prime} }{{(r{^{\prime} }^{2}+2r{^{\prime} }^{5}-3z{^{\prime} }^{2})}^{3}}.\end{eqnarray} \tag{ 66 }$$

The integral is taken at a constant λ, so $r^{\prime} (\lambda ,z^{\prime} )$ is found from Equation (63).

Figure 2 shows a constant ϕ slice through the flow, depicting the pressure (grayscale) and the magnetic field components parallel (colorscaled arrows) or initially perpendicular (colorscaled curves) to the velocity. For illustrative purposes, in the figure we take $\tilde{P}=\tilde{\rho }{\tilde{u}}^{2}$.

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The electric field here is a special case of Equation (55), namely

$$\begin{eqnarray}&&{\boldsymbol{E}}=\displaystyle \frac{\varrho }{\tilde{\varrho }c}(\begin{array}{c}{v}_{z}{\tilde{B}}_{\phi }\\ \displaystyle \frac{2\lambda }{{\varrho }^{2}}\;\tilde{u}{\tilde{B}}_{\varrho }\\ -{v}_{\varrho }{\tilde{B}}_{\phi }\end{array}).\end{eqnarray} \tag{ 67 }$$

4.1.2. The Problematic Choice $q=r$

Another seemingly possible choice is $q=r$. According to Equation (25), the magnetic field may then be written as

$$\begin{eqnarray}&&{\boldsymbol{B}}={\tilde{B}}_{\phi }\displaystyle \frac{\varrho }{\tilde{\varrho }}\hat{{\boldsymbol{\phi }}}+{\tilde{B}}_{\varrho }\displaystyle \frac{\varrho r}{\tilde{\varrho }z}\hat{{\boldsymbol{\theta }}}-\displaystyle \frac{{\boldsymbol{v}}}{\tilde{u}}({\tilde{B}}_{z}+{\tilde{B}}_{\varrho }\tilde{\varrho }\tilde{u}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }),\end{eqnarray} \tag{ 68 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }={(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda })}_{\phi ,r,\tilde{r}}={\displaystyle \int }_{\infty }^{r}\displaystyle \frac{-r{^{\prime} }^{4}{dr}^{\prime} /\tilde{u}}{{(r{^{\prime} }^{3}-1)}^{2}{(1-\displaystyle \frac{2\lambda r^{\prime} }{r{^{\prime} }^{3}-1})}^{\frac{3}{2}}}.\end{eqnarray} \tag{ 69 }$$

However, this result, obtained previously (Dursi & Pfrommer 2008; Romanelli et al. 2014) by solving the MHD equations, is strictly valid only in the half space $\theta \lt \pi /2$. At the surface $\theta =\pi /2$, the denominator of the integrand diverges, and the integration cannot be continued. This is to be expected, as surfaces of constant λ and surfaces of constant r overlap at $\theta =\pi /2$, as ${\boldsymbol{\nabla }}\lambda$ and ${\boldsymbol{\nabla }}r$ become parallel. The problem can be circumvented by computing the converging ${\rm{\Delta }}t$, rather than its diverging derivative.

The integral in Equation (69) can be solved analytically as a power series in λ; see Appendix B.

4.1.3. The Orthogonal Choice $q={\rm{\Phi }}$

Another optional choice for $q$ is the velocity potential Φ. Here, the reciprocal basis becomes

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{1}=\varrho \;\hat{{\boldsymbol{\phi }}}{\rm{;}}\quad {{\boldsymbol{a}}}_{2}=\displaystyle \frac{\tilde{u}}{\varrho v}(\hat{{\boldsymbol{v}}}\times \hat{{\boldsymbol{\phi }}}){\rm{;}}\quad {{\boldsymbol{a}}}_{3}=\hat{{\boldsymbol{v}}}.\end{eqnarray} \tag{ 70 }$$

These vectors are perpendicular to one another, and therefore may be useful as an orthogonal (albeit not orthonormal) basis. The magnetic field here is given by

$$\begin{eqnarray}&&{\boldsymbol{B}}=[\begin{array}{c}{\tilde{\varrho }}^{-1}{\tilde{B}}_{\phi }\\ \tilde{\varrho }{\tilde{B}}_{\varrho }\\ -\displaystyle \frac{v}{\tilde{u}}{\tilde{B}}_{z}\end{array}]-[\begin{array}{c}0\\ 0\\ \tilde{\varrho }v\end{array}]{\tilde{B}}_{\varrho }\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda },\end{eqnarray} \tag{ 71 }$$

where

$$\begin{eqnarray}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda } & = & {(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda })}_{\phi ,{\rm{\Phi }},\tilde{{\rm{\Phi }}}}\\ & = & \displaystyle \frac{48}{\tilde{u}}{\displaystyle \int }_{-\infty }^{{\rm{\Phi }}}\displaystyle \frac{z{^{\prime} }^{4}+4r{^{\prime} }^{8}{\rm{\Phi }}{^{\prime} }^{2}+10z{^{\prime} }^{4}r{^{\prime} }^{3}(1-2r{^{\prime} }^{3})}{{r}^{-11}{z}^{-4}{[4r{^{\prime} }^{8}{\rm{\Phi }}{^{\prime} }^{2}+3z{^{\prime} }^{4}(1-4r{^{\prime} }^{3})]}^{3}}d{\rm{\Phi }}^{\prime} {\rm{.}}\end{eqnarray} \tag{ 72 }$$

The parameters $r^{\prime} (\lambda ,{{\rm{\Phi }}}^{\prime })$ and $z^{\prime} (\lambda ,{{\rm{\Phi }}}^{\prime })$ are found from Equations (61) and (63).

Consider the Stokes, incompressible, steady flow around a solid sphere. As in Section 4.1, $\rho =\tilde{\rho }$ is constant, the radius of the sphere is taken as R = 1, we assume that the incident flow $\tilde{{\boldsymbol{v}}}\equiv -\tilde{u}\hat{{\boldsymbol{z}}}$, taken on $\tilde{H}$ far upstream ($\tilde{z}\to \infty$), is homogeneous and unidirectional, and use a mixture of cylindrical and spherical coordinates. It is again natural to choose $\breve{\rho }=\tilde{\rho }$, $\breve{R}=1$ and $\breve{v}=\tilde{u}$, so $\breve{\mu }=\tilde{\rho }\tilde{u}$.

Stokes flows are characterized by viscous forces much greater than the inertial forces, so the flow is governed by the approximate Navier–Stokes equation

$$\begin{eqnarray}&&0=-{\boldsymbol{\nabla }}(\displaystyle \frac{P}{\tilde{\rho }})+\nu {{\boldsymbol{\nabla }}}^{2}{\boldsymbol{v}},\end{eqnarray} \tag{ 73 }$$

where ν is the kinematic viscosity, assumed constant. Taking the curl of this equation gives

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\times ({{\boldsymbol{\nabla }}}^{2}{\boldsymbol{v}})=0.\end{eqnarray} \tag{ 74 }$$

The solution, for boundary conditions of a hard sphere, is

$$\begin{eqnarray}&&{v}_{z}=-\tilde{u}(1-\displaystyle \frac{1}{r})[1+\displaystyle \frac{({r}^{2}-3{z}^{2})(r+1)}{4{r}^{4}}],\end{eqnarray} \tag{ 75 }$$

and

$$\begin{eqnarray}&&{v}_{\varrho }=3\tilde{u}(1-\displaystyle \frac{1}{{r}^{2}})\displaystyle \frac{\varrho z}{4{r}^{3}}.\end{eqnarray} \tag{ 76 }$$

Solving Equation (73) gives the pressure distribution,

$$\begin{eqnarray}&&P=\tilde{P}+\tilde{\rho }\tilde{u}\nu \displaystyle \frac{3z}{2{r}^{3}}.\end{eqnarray} \tag{ 77 }$$

Here too, the incident plasma cannot be too cold, as an incident pressure $\tilde{P}\lt (3/2)\tilde{\rho }{\tilde{u}}^{2}/\mathrm{Re}$, where $\mathrm{Re}=\tilde{u}R/\nu$ is the incident Reynolds number, would lead to a nonphysical $P\lt 0$ around the $z\lt 0$ axis.

The stream functions can be chosen as ${\lambda }_{1}=\phi$ and, according to Equation (50),

$$\begin{eqnarray}&&\lambda \equiv {\lambda }_{2}=\displaystyle \frac{{\varrho }^{2}}{2}{(1-\displaystyle \frac{1}{r})}^{2}(1+\displaystyle \frac{1}{2r}).\end{eqnarray} \tag{ 78 }$$

As in the case of a potential flow, here too $\tilde{\varrho }={(2\lambda )}^{1/2}$, and we proceed similarly with the natural choice $q=z$. The magnetic field is given by

$$\begin{eqnarray}&&{\boldsymbol{B}}={\tilde{B}}_{\phi }\displaystyle \frac{\varrho }{\tilde{\varrho }}\hat{{\boldsymbol{\phi }}}-{\tilde{B}}_{\varrho }\displaystyle \frac{\tilde{\varrho }\tilde{u}}{\varrho {v}_{z}}\hat{{\boldsymbol{\varrho }}}-\displaystyle \frac{{\boldsymbol{v}}}{\tilde{u}}({\tilde{B}}_{z}+{\tilde{B}}_{\varrho }\tilde{\varrho }\tilde{u}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }),\end{eqnarray} \tag{ 79 }$$

which is formally equivalent to Equation (65). Here,

$$\begin{eqnarray}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda } & = & {(\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda })}_{\phi ,z,\tilde{z}}\\ & = & \displaystyle \frac{48}{\tilde{u}}{\displaystyle \int }_{\infty }^{z}\displaystyle \frac{r{^{\prime} }^{6}-5z{^{\prime} }^{2}r{^{\prime} }^{2}+r{^{\prime} }^{4}(1+3z{^{\prime} }^{2})}{{(1-\displaystyle \frac{1}{r{^{\prime} }^{2}})}^{3}{(r{^{\prime} }^{2}-3z{^{\prime} }^{2}+\displaystyle \frac{4r{^{\prime} }^{4}}{1+r^{\prime} })}^{3}}\;{dz}^{\prime} ,\end{eqnarray} \tag{ 80 }$$

where $r^{\prime} (\lambda ,z^{\prime} )$ is found from Equation (78).

Figure 3 shows the pressure and magnetization in such a flow, with the same notations used in Figure 2. For illustrative purposes, in the figure $\tilde{P}=2\tilde{\rho }{\tilde{u}}^{2}/\mathrm{Re}$.

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Here too, the electric field is a special case of Equation (55), and is given by Equation (67).

As a simple illustration of magnetization in the time-dependent regime, consider the case of an arbitrary 1D flow. Classical examples include the plasma in a cylindrical pipe, disturbed by a moving piston, or a 1D rarefaction wave. If discontinuities are present in the flow, the following applies separately to each region bounded by surfaces of discontinuity or $\tilde{H}$.

Without loss of generality, take the flow in the $\hat{{\boldsymbol{x}}}$ direction, such that ${\boldsymbol{v}}=v\hat{{\boldsymbol{x}}}$ and the perpendicular velocity components vanish. A simple choice of path functions is then

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{2}=y\quad {\rm{and}}\quad {{\rm{\Lambda }}}_{3}=z,\end{eqnarray} \tag{ 81 }$$

as both are constant along the flow. Equations (32) and (33) therefore constrain ${{\rm{\Lambda }}}_{1}$, as

$$\begin{eqnarray}&&{(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial x})}_{y,z,t}=-\displaystyle \frac{\rho }{\breve{\rho }}\quad {\rm{and}}\quad {(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial t})}_{x,y,z}=\displaystyle \frac{\rho v}{\breve{\rho }}.\end{eqnarray} \tag{ 82 }$$

The solution is the path function family

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{1}(x,t)=-\displaystyle \frac{1}{\breve{\rho }}{\displaystyle \int }^{x}\rho (x^{\prime} ,t){dx}^{\prime} +{\mathcal{C}},\end{eqnarray} \tag{ 83 }$$

where we used the continuity equation, and ${\mathcal{C}}$ is an insignificant constant.

For a given $\tilde{{\boldsymbol{l}}}=({\tilde{l}}_{x},{\tilde{l}}_{y},{\tilde{l}}_{z})$, Equation (41) yields the Lagrangian constants

$$\begin{eqnarray}&&{\tilde{L}}_{1}=-\displaystyle \frac{\tilde{\rho }}{\breve{\rho }}{\tilde{l}}_{x}{\rm{;}}\quad {\tilde{L}}_{2}={\tilde{l}}_{y}{\rm{;}}\quad {\tilde{L}}_{3}={\tilde{l}}_{z},\end{eqnarray} \tag{ 84 }$$

where here and in the rest of Section 4.3, round brackets are in Cartesian coordinates. Equivalently, for $\tilde{{\boldsymbol{B}}}=({\tilde{B}}_{x},{\tilde{B}}_{y},{\tilde{B}}_{z})$, Equation (43) gives

$$\begin{eqnarray}&&{\tilde{B}}_{1}=-\displaystyle \frac{\tilde{\rho }}{\breve{\rho }}{\tilde{B}}_{x}{\rm{;}}\quad {\tilde{B}}_{2}={\tilde{B}}_{y}{\rm{;}}\quad {\tilde{B}}_{3}={\tilde{B}}_{z}.\end{eqnarray} \tag{ 85 }$$

Finally, Equations (44) and (45) give, respectively

$$\begin{eqnarray}&&{\boldsymbol{l}}=(\displaystyle \frac{\tilde{\rho }}{\rho }{\tilde{l}}_{x},{\tilde{l}}_{y},{\tilde{l}}_{z}),\end{eqnarray} \tag{ 86 }$$

and

$$\begin{eqnarray}&&{\boldsymbol{B}}=({\tilde{B}}_{x},\displaystyle \frac{\rho }{\tilde{\rho }}{\tilde{B}}_{y},\displaystyle \frac{\rho }{\tilde{\rho }}{\tilde{B}}_{z}).\end{eqnarray} \tag{ 87 }$$

Equations (86) and (87) can be seen directly from Lagrangian considerations. Since the gas flows only in the x direction, the perpendicular (to the flow) distance between fluid elements is constant. Perpendicular length elements $\{{l}_{y},{l}_{z}\}$ and the parallel magnetic field B_x, are thus conserved. The parallel distance between nearby fluid elements is simply ${l}_{x}\propto {\rho }^{-1}$, so the perpendicular fields follow $\{{B}_{y},{B}_{z}\}\propto \rho$.

Given a flow and an initial magnetic configuration ${\tilde{{\boldsymbol{B}}}}_{\mathrm{in}}$ on $\tilde{H}$, the electromagnetic field solution in Equations (25), (27), (29), (42), (45), or (46), depends only on knowing the Lagrangian constant $\tilde{{\boldsymbol{B}}}$ corresponding to every spacetime location. For an initially uniform magnetic field, $\tilde{{\boldsymbol{B}}}$ is everywhere constant, and the result is straightforward even for time-dependent flows. For a spacetime-dependent initial magnetic field, this mapping can become complicated, especially for time-dependent flows.

Consider however the case of a steady flow, with the choice $q={\rm{\Delta }}t$. Here, even if ${\tilde{{\boldsymbol{B}}}}_{\mathrm{in}}({\boldsymbol{x}},t)$ varies in space and time, we may directly determine $\tilde{{\boldsymbol{B}}}$ using

$$\begin{eqnarray}\tilde{{\boldsymbol{B}}}({\boldsymbol{x}};t) & = & \tilde{{\boldsymbol{B}}}({\lambda }_{1},{\lambda }_{2},{\rm{\Delta }}t;t)=\tilde{{\boldsymbol{B}}}({\lambda }_{1},{\lambda }_{2},0;\tilde{t}\equiv t-{\rm{\Delta }}t)\\ & = & {\tilde{{\boldsymbol{B}}}}_{\mathrm{in}}({\lambda }_{1},{\lambda }_{2},0;\tilde{t}),\end{eqnarray} \tag{ 88 }$$

where $\tilde{t}$ is the time at which the fluid element, which presently lies at $({\boldsymbol{x}},t)$, was on $\tilde{H}$. Equation (88) applies for any choice of q, if Equation (18) is used to compute the travel time. For example, in cylindrical coordinates with $\tilde{H}$ at $z\to \infty$, we may compute it by

$$\begin{eqnarray}&&{\rm{\Delta }}t={\displaystyle \int }_{\infty }^{z}\displaystyle \frac{{dz}^{\prime} }{{v}_{z}}.\end{eqnarray} \tag{ 89 }$$

An arbitrary unperturbed $\tilde{{\boldsymbol{B}}}({\boldsymbol{x}},t)$ can be Fourier decomposed into components of wavenumber k and angular frequency ω. As a simple example, assume a flow in the $(-z)$ direction, and consider a Fourier component of the form

$$\begin{eqnarray}&&\tilde{{\boldsymbol{B}}}={B}_{0}(\mathrm{cos}(\omega \tilde{t}),0,\mathrm{cos}(k\tilde{\varrho })),\end{eqnarray} \tag{ 90 }$$

where B₀ is a constant, and round-bracket vectors in this subsection are in Cartesian coordinates. Notice that no spatial component of this $\tilde{{\boldsymbol{B}}}$ contributes to Gauss' law, and that a t-dependence on $\tilde{H}$ induces a z-dependence in its vicinity, i.e.Equation (90) becomes

$$\begin{eqnarray}&&\tilde{{\boldsymbol{B}}}={B}_{0}(\mathrm{cos}[\omega (t+z/\tilde{v})],0,\mathrm{cos}(k\varrho ))\end{eqnarray} \tag{ 91 }$$

in the region where the flow is approximately uniform.

Figure 4 presents the evolution of the Fourier component Equation (90), in an incompressible potential flow around a sphere (see Section 4.1). As Figures 4 and 5 show, substructure in the far upstream magnetic field is non-trivially imprinted on the downstream flow. In situ measurements of ${\boldsymbol{B}}$ downward of an object such as a planet would thus be sensitive to both the substructure and details of the flow.

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As an example, consider measurements of the inverse polarity reversal layer (henceforth IPRL) induced by a planetary body (Romanelli et al. 2014). The IPRL is defined as the layer in which the magnetic field component in the direction of the initial flow changes its sign, implying in our case that B_z = 0. According to Equation (53), for $q=z$, the IPRL is described for a general flow by the simple formula

$$\begin{eqnarray}&&{\tilde{B}}_{z}={\tilde{B}}_{2}\tilde{v}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }=\displaystyle \frac{\tilde{v}({\tilde{\mu }}_{\varrho }{\tilde{B}}_{z}-{\tilde{\mu }}_{z}{\tilde{B}}_{\varrho })}{\breve{\mu }}\displaystyle \frac{\partial {\rm{\Delta }}t}{\partial \lambda }.\end{eqnarray} \tag{ 92 }$$

Consider the IPRL for a magnetic field with initial oscillatory substructure. For an initial magnetic inclination ${\theta }_{0}$, let

$$\begin{eqnarray}&&\tilde{{\boldsymbol{B}}}={B}_{0}(\mathrm{sin}{\theta }_{0},0,\mathrm{cos}{\theta }_{0})+\hat{{\boldsymbol{x}}}\zeta {B}_{0}\mathrm{sin}({\theta }_{0})\mathrm{cos}\omega \tilde{t},\end{eqnarray} \tag{ 93 }$$

where $\zeta \lt 1$ is a small number; this $\tilde{{\boldsymbol{B}}}$ trivially satisfies Gauss' law near $\tilde{H}$. A snapshot of the resulting magnetic configuration is shown in Figure 5, where for illustrative purposes we adopt the incompressible, steady potential flow of Section 4.1. As the figure shows, the far upstream magnetic substructure with wavelength $2\pi \tilde{u}/\omega$ is directly imprinted on the IPRL.

Such a model may describe the magnetization not only around planets and moons, but also around stellar systems moving through the ISM, around clumps and bubbles moving through the IGM, etc.

Under our assumptions, as the flow approaches the stagnation point in front of an object, the magnetic field initially perpendicular to the flow diverges, as seen in Figures 2–5. The divergence here, around the object, and in the wake behind it, may be regarded as due to the accumulation of infinitely many field lines, continuously stretched around the object, eventually wrapped closely around it and stretching to infinity along the wake.

These nonphysical divergencies signal the breakdown of our assumptions in the above regions. In particular, the magnetic field here can no longer be regarded as dynamically insignificant, and eventually resistivity can no longer be neglected (e.g., Chacko & Hassam 1997; Lyutikov 2006).

It is useful to consider the extent of the highly magnetized layer forming in front of an object in an arbitrary axisymmetric flow. Such layers form as the magnetized solar wind drapes around a planetary magnetopause, as the magnetized ISM drapes around a bubble or a stellar heliopause, and as the magnetized IGM drapes around a large scale-clump or bubble. The regions surrounding such objects could involve plasma depletion layers with dynamically significant magnetic fields, and lead to plasma instabilities, magnetic reconnection, and turbulence (e.g., Dibraccio et al. 2013; Sulaiman et al. 2014). In the following, we neglect the small modifications to the flow far from the object due to the presence of a highly magnetized region.

Define the magnetic pressure fraction

$$\begin{eqnarray}&&{\epsilon }_{B}\equiv {\beta }_{p}^{-1}=\displaystyle \frac{{B}^{2}/8\pi }{{P}_{{\rm{th}}}},\end{eqnarray} \tag{ 94 }$$

where ${\beta }_{p}$ is the beta of the plasma, and ${P}_{{\rm{th}}}$ is the thermal pressure. The fractional magnetic amplification may then be defined as

$$\begin{eqnarray}&&\displaystyle \frac{{\epsilon }_{B}}{{\tilde{\epsilon }}_{B}}={(\displaystyle \frac{B}{\tilde{B}})}^{2}{(\displaystyle \frac{{P}_{{\rm{th}}}}{{\tilde{P}}_{{\rm{th}}}})}^{-1}.\end{eqnarray} \tag{ 95 }$$

Figure 6 shows the amplifications of magnetic pressure $({B}^{2}/{\tilde{B}}^{2})$ and pressure fraction $({\epsilon }_{B}/{\tilde{\epsilon }}_{B})$, as a function of the distance $\delta \equiv r-1$ in front of the unit sphere, for various steady flows: an incompressible potential flow, a Stokes incompressible flow, and inviscid compressible flows in the subsonic and mildly supersonic regimes. As the above flows are approximately (for the mildly supersonic case), or precisely, irrotational along the axis, Equations (56) and (57) become

$$\begin{eqnarray}&&B{(z)}^{2}=\displaystyle \frac{\rho /u}{\tilde{\rho }/\tilde{u}}{\tilde{B}}_{\perp }^{2}+\displaystyle \frac{{j}^{2}}{{\tilde{j}}^{2}}{\tilde{B}}_{\parallel }^{2},\end{eqnarray} \tag{ 96 }$$

where ${\tilde{B}}_{\perp }$ and ${\tilde{B}}_{\parallel }$ are the axial magnetic field components initially perpendicular and parallel to the flow. The figure focuses on the diverging, perpendicular component.

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As Figure 6 shows, viscosity induces magnetic amplification farther from the body. Compressibility has a similar, but somewhat weaker, effect on magnetization, but the magnetic pressure fraction is not strongly affected. The large difference in ${\epsilon }_{B}$ between the incompressible and compressible inviscid flows arises from the different pressure profiles corresponding to the different equations of state.

Close to stagnation, the flow is adiabatic, ρ and ${P}_{{\rm{th}}}$ approach constant values, and ${\boldsymbol{B}}$ becomes primarily transverse, so

$$\begin{eqnarray}&&{\epsilon }_{B}\propto \displaystyle \frac{{B}_{\perp }^{2}}{{P}_{{\rm{th}}}}\propto {B}_{\perp }^{2}\propto \displaystyle \frac{{\rho }^{2}}{j}\propto {u}^{-1},\end{eqnarray} \tag{ 97 }$$

which scales as ${\delta }^{-2}$ for the Stokes flow, and as ${\delta }^{-1}$ for the incompressible potential flow; the latter approximately characterizes the stagnation region even for compressible flows, except in the highly supersonic regime.

Magnetic amplification beyond ${\epsilon }_{B}\simeq 1$, say, is not self-consistent, as our passive magnetization assumption fails. Instead, one expects a highly magnetized, self-regulating layer to form; its width δ depends on the initial magnetization level ${\tilde{\epsilon }}_{B}$. For simplicity, consider an initially hot plasma, such that ${\tilde{P}}_{{\rm{th}}}\gg \tilde{\rho }{\tilde{u}}^{2}$ for an inviscid flow, or ${\tilde{P}}_{{\rm{th}}}\gg \tilde{\rho }{\tilde{u}}^{2}/\mathrm{Re}$ for the Stokes flow. In this regime ${P}_{{\rm{th}}}\simeq {\tilde{P}}_{{\rm{th}}}$, so according to Equation (97)

$$\begin{eqnarray}\delta \sim \{\begin{array}{ll}{\tilde{\epsilon }}_{B} & {\rm{for}}\;{\rm{negligible}}\;{\rm{viscosity;}}\\ {\tilde{\epsilon }}_{B}^{1/2} & {\rm{for}}\;{\rm{strong}}\;{\rm{viscosity.}}\end{array}\end{eqnarray} \tag{ 98 }$$

For the compressible flows shown in Figure 6, we use the approximate solutions of Keshet & Naor (2014, using $\gamma =5/3$ and $\beta =0.52$). In the supersonic regime, $\tilde{H}$ is assumed to be upstream, so the jumps in ${B}_{\perp }$ and ${P}_{{\rm{th}}}$ across the shock are taken into account. Here, the flow remains nonmagnetized until reaching the shock, at the so called standoff distance $\delta ={\rm{\Delta }}$; for high ${\tilde{M}}_{0}$, the magnetized layer slightly shrinks as Δ diminishes.

The electric field in ideal MHD was shown to be a simple function of the stream functions (see Equations (27), (46), (55)), so it is easy to infer ${\boldsymbol{E}}$ directly in various astronomical circumstances. As an example, in what follows we consider the electric field outside a heliosphere (of some star) or magnetosphere (of some stellar object). We approximate the bow shock as weak (or absent; for the possibility that there is no bow shock in front of the solar system, see McComas et al. 2012), and model the discontinuity (heliopause or magnetopause) using a Rankine half-body. These assumptions may be marginally invalid for the magnetopause of Earth, but roughly apply for the Sun's heliosphere, and may be valid for other objects such as the induced magnetospheres of Venus and Mars.

The Rankine half-body model was applied to the Sun's heliopause by Parker (1961). Models for Earth's magnetopause are more sophisticated (e.g., Fairfield 1971; Tsyganenko 1989; Shue et al. 1997; Chao et al. 2002; Lin et al. 2010; Lu et al. 2011; Tsyganenko 2013, and references threin), and better calibrated by satellite observations. However, no present analytic model captures the full complexity of the magnetopause structure, including the time dependence of the geodipole tilt direction and the stellar wind variability (Olson 1969; Zhou & Russell 1997), so for simplicity we resort to Equation (99).

The normalized potential of the Rankine half-body is

$$\begin{eqnarray}&&{\rm{\Phi }}=-(z+\displaystyle \frac{{L}_{m}^{2}}{r}),\end{eqnarray} \tag{ 99 }$$

where mixed coordinates are used, as in Section 4.1, and L_m is the distance between the object (star, planet, or moon) and the nose of the discontinuity. This simplified potential flow is axisymmetric and incompressible. It approximates the flow ahead of the discontinuity as homogeneous and unidirectional, $\tilde{{\boldsymbol{v}}}=-\tilde{u}\hat{{\boldsymbol{z}}}$, which is strictly valid only if the bow shock is weak or absent. The resulting discontinuity profile is approximately paraboloid at the nose, and quickly flattens to a $\varrho =2{L}_{m}$ cylinder far downstream.

The velocity field derived from Equation (99) is

$$\begin{eqnarray}&&{\boldsymbol{v}}\equiv (\begin{array}{c}{v}_{\varrho }\\ {v}_{\phi }\\ {v}_{z}\end{array})=\tilde{u}(\begin{array}{c}\varrho {L}_{m}^{2}/{r}^{3}\\ 0\\ -1+{{zL}}_{m}^{2}/{r}^{3}\end{array}).\end{eqnarray} \tag{ 100 }$$

The stream functions may thus be chosen as ${\lambda }_{1}=\breve{R}\phi$ and

$$\begin{eqnarray}&&{\lambda }_{2}=\displaystyle \frac{1}{2\breve{R}}[{\varrho }^{2}+2{L}_{m}^{2}(\displaystyle \frac{z}{r}-1)]=\displaystyle \frac{{\tilde{\varrho }}^{2}}{2\breve{R}},\end{eqnarray} \tag{ 101 }$$

where, as above, the length scale $\breve{R}$ is arbitrary. The thermal pressure is found from Bernoulli's Equation (59)

$$\begin{eqnarray}&&{P}_{{\rm{th}}}={\tilde{P}}_{{\rm{ram}}}(\tilde{\chi }+\displaystyle \frac{{L}_{m}^{2}z}{{r}^{3}}-\displaystyle \frac{{L}_{m}^{4}}{2{r}^{4}}),\end{eqnarray} \tag{ 102 }$$

where ${P}_{{\rm{ram}}}\equiv \rho {v}^{2}$ is the ram pressure, and $\chi \equiv {P}_{{\rm{th}}}/{P}_{{\rm{ram}}}\geqslant 3/2$ is necessary to guarantee a positive ${P}_{{\rm{th}}}$ throughout the flow.

The electric field outside the discontinuity is given, according to the axisymmetric flow Equation (55), by the simple analytic expression

$$\begin{eqnarray}&&{\boldsymbol{E}}=\displaystyle \frac{\varrho }{\tilde{\varrho }c}(\begin{array}{c}{v}_{z}{\tilde{B}}_{\phi }\\ \displaystyle \frac{{\tilde{\varrho }}^{2}}{{\varrho }^{2}}\;\tilde{u}{\tilde{B}}_{\varrho }\\ -{v}_{\varrho }{\tilde{B}}_{\phi }\end{array}).\end{eqnarray} \tag{ 103 }$$

The magnetic field can be written in terms of elliptic functions (Röken et al. 2014). Close to the discontinuity $\tilde{\varrho }$ vanishes, so Equation (53) yields

$$\begin{eqnarray}&&{\boldsymbol{B}}=\displaystyle \frac{{L}_{m}}{\tilde{\varrho }}(\begin{array}{c}-{\tilde{B}}_{\varrho }{v}_{\varrho }/\tilde{u}\\ {\tilde{B}}_{\phi }\varrho /{L}_{m}\\ -{\tilde{B}}_{\varrho }{v}_{z}/\tilde{u}\end{array})-\displaystyle \frac{{\tilde{B}}_{z}+\displaystyle \frac{3}{8}{\tilde{B}}_{\varrho }}{\tilde{u}}(\begin{array}{c}{v}_{\varrho }+O(\tilde{\varrho })\\ 0\\ {v}_{z}+O(\tilde{\varrho })\end{array}).\end{eqnarray} \tag{ 104 }$$

Near the discontinuity surface, where $\tilde{\varrho }$ is small, ${\boldsymbol{E}}$ is dominated by its $\varrho$ and z components, and ${\boldsymbol{B}}$ is dominated by the first term in Equation (104). In the cylindrical regime, ${\boldsymbol{v}}\to (-\tilde{u}\hat{{\boldsymbol{z}}})$ and ϱ→2L_m, so ${E}_{\phi }\ll {E}_{\varrho }\ll {E}_{z}$ and ${B}_{\varrho }\ll {B}_{z}\simeq ({\tilde{B}}_{\varrho }/2{\tilde{B}}_{\phi }){B}_{\phi }$.

The magnetic field diverges near the discontinuity. Hence, as in Section 5.2, we identify the region where ${\epsilon }_{B}\gt 1$ as a highly-magnetized, self-limiting layer. The outer edge of the magnetized layer, defined, say, by ${\epsilon }_{B}=1$, can now be used to estimate the electromagnetic fields in the discontinuity region Ω, namely the heliopause or the magnetopause.

The width δ of the highly magnetized layer varies slightly along the discontinuity,

$$\begin{eqnarray}&&\delta ={(\displaystyle \frac{2}{5+3\mathrm{cos}\theta })}^{1/2}\;{{f}{L}}_{m},\end{eqnarray} \tag{ 105 }$$

where $\theta$ is the polar angle, taken with respect to the z axis with the object at the origin, and we defined

$$\begin{eqnarray}&&f\equiv \displaystyle \frac{{\tilde{\epsilon }}_{B}}{1+\displaystyle \frac{1+4\mathrm{cos}\theta +3\mathrm{cos}2\theta }{16\tilde{\chi }}}.\end{eqnarray} \tag{ 106 }$$

The electromagnetic fields on Ω are thus given by

$$\begin{eqnarray}{{\boldsymbol{E}}}_{{\rm{\Omega }}} & = & \displaystyle \frac{-{\tilde{B}}_{\phi }\tilde{u}}{c}\sqrt{\displaystyle \frac{f}{2+f}}(\begin{array}{c}1+\displaystyle \frac{3-2\mathrm{cos}\theta -\mathrm{cos}2\theta }{f(2+f)}\\ -{\tilde{B}}_{\varrho }/{\tilde{B}}_{\phi }\\ \displaystyle \frac{4{\mathrm{cos}}^{2}(\theta /2)\mathrm{sin}\theta }{f(2+f)}\end{array})\\ & = & -\displaystyle \frac{2{\tilde{B}}_{\phi }\tilde{u}\;{\mathrm{sin}}^{2}(\theta /2)}{c\sqrt{f}}(\begin{array}{c}1+\displaystyle \frac{\mathrm{cos}\theta }{2}\\ 0\\ \displaystyle \frac{{\mathrm{cos}}^{3}(\theta /2)}{\mathrm{sin}(\theta /2)}\end{array})+O(\sqrt{{\tilde{\epsilon }}_{B}}),\end{eqnarray} \tag{ 107 }$$

and

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{\rm{\Omega }}}=\displaystyle \frac{{\tilde{B}}_{\varrho }}{\sqrt{f}}(\begin{array}{c}-{\mathrm{cos}}^{3}(\theta /2)\\ -{\tilde{B}}_{\phi }/{\tilde{B}}_{\varrho }\\ (1+\displaystyle \frac{\mathrm{cos}\theta }{2})\mathrm{sin}(\theta /2)\end{array})+O(\sqrt{{\tilde{\epsilon }}_{B}}).\end{eqnarray} \tag{ 108 }$$

As an example, consider the magnetopause of Earth. Satellite measurements (the 1999 Wind spacecraft trajectory; Wang et al. 2003) suggest that halfway between the bow shock and the plasma depletion layer, $\tilde{u}\simeq 300\;{\rm{km}}\;{{\rm{s}}}^{-1}$, $\tilde{B}\simeq 0.2\;{\rm{mG}}$, $\tilde{{\boldsymbol{B}}}\simeq (0.07,0.17,-0.05)\;{\rm{mG}}$,

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{P}}_{B}}{{\tilde{P}}_{{\rm{ram}}}}=\displaystyle \frac{1}{2{\tilde{M}}_{{\rm{A}}}^{2}}\simeq 0.1,\end{eqnarray} \tag{ 109 }$$

and

$$\begin{eqnarray}&&\tilde{\chi }=\displaystyle \frac{{\tilde{P}}_{{\rm{th}}}}{{\tilde{P}}_{{\rm{ram}}}}=\displaystyle \frac{1}{\gamma {\tilde{M}}^{2}}\simeq 0.4,\end{eqnarray} \tag{ 110 }$$

where ${P}_{B}\equiv {B}^{2}/(8\pi )$ is the magnetic pressure, and ${M}_{{\rm{A}}}\equiv v/{({B}^{2}/4\pi \rho )}^{1/2}$ is the $\mathrm{Alfv}\acute{{\rm{e}}}{\rm{n}}$ Mach number. Hence the plasma is warm, and mildly magnetized with ${\tilde{\epsilon }}_{B}\simeq 0.25$.

With these $\tilde{H}$ parameters, the passive evolution of the fields in the Rankine half-body model Equations (105)–(108) yields (at $\theta \sim 80^\circ$; Wang et al. 2003) $\delta \simeq 1.8{R}_{\oplus }$, ${B}_{{\rm{\Omega }}}\simeq 0.35\;{\rm{mG}}$, and ${E}_{{\rm{\Omega }}}\simeq 10\;\mathrm{mV}\;{{\rm{m}}}^{-1}$. These should be compared with the depletion layer thickness ${\delta }_{{\rm{dep}}}\simeq 2.2{R}_{\oplus }$ (Wang et al. 2003) and with the magnetopause fields. Wind measured ${B}_{{\rm{\Omega }}}=0.35\;{\rm{mG}}$ (Wang et al. 2003), but not the electric field; the CLUSTER spacecraft measured ${E}_{{\rm{\Omega }}}\simeq 10\;\mathrm{mV}\;{{\rm{m}}}^{-1}$ (but three years layer, at $\theta \sim 70^\circ$; Panov et al. 2006). The good agreement between measurements and model is largely coincidental, as the latter is crude, pertains only to the draped component, and fails to account for the complexity of the magnetosphere, the variability of the solar wind, and physical effects such as magnetic reconnection.