Local analysis of fast magnetic reconnection

In the limit of non-ideality becoming arbitrarily small, $\epsilon_{ni}\rightarrow 0$, a pair of magnetic field lines must pass within a very small distance of each other, $\Delta_{d}\propto\epsilon_{ni}$, to lose their distinguishability and reconnect.

Nevertheless, when a pair of field lines have become indistinguishable anywhere along their trajectories, they have also become indistinguishable everywhere along their trajectories—no matter how great their separation may become. In other words, magnetic reconnection occurs over a large spatial scale when the separation between a pair of magnetic field lines undergoes sufficiently large changes along the length of the lines, $\Delta_{max}/\Delta_{min}$, the ratio of the maximum to the minimum separation.

Three distance scales characterize reconnection: The first is the distance scale $\Delta_{d}$ at which field lines become indistinguishable due to direct non-ideal effects. The second is the distance scale $a_{r}$ over which reconnection must occur to have a significant effect on the evolution and to be of practical importance. The third is the distance scale $a_{u}$ of spatial variations in $\vec{u}_{\bot}$.

Plasma resistivity causes magnetic field lines to interdiffuse, become indistinguishable, and reconnect over the scale $\Delta_{d}=\sqrt{(\eta/\mu_{0})\tau_{\eta}}$ during the time $\tau_{\eta}$. The shortest timescale $\tau_{\eta}$ defines the shortest distance over which lines can remain distinguishable, and that timescale is $\tau_{\eta}=\Delta_{d}/\tilde{u}_{\bot}$. Consequently, resistivity alone gives $\Delta_{d}\sim\eta/(\mu_{0}\tilde{u}_{\bot})$. Other non-ideal effects, including the grid scale in numerical simulations, contribute to determining the minimum scale $\Delta_{d}$ over which pairs of magnetic field lines can be distinguished.

Naively, one would expect reconnection to be irrelevant when the distance required for the distinguishability of the lines, $\Delta_{d}$, is tiny compared to the spatial scale for reconnection to be significant, $a_{r}$, but this is clearly false if the ratio of separations between pairs of magnetic field lines, $\Delta_{max}/\Delta_{min}$, becomes as large as $a_{r}/\Delta_{d}$ with $a_{r}/\Delta_{d}\sim 1/\epsilon_{ni}$.

The spatial scale $a_{u}$ of variations in $\vec{u}_{\bot}$ is important because the maximum separation $\Delta_{max}$ can increase exponentially in time for field lines with a minimum separation $\Delta_{min}=\Delta_{d}$ only when $\Delta_{max}\lesssim a_{u}$. When the scale of a chaotic flow $a_{u}$ is large compared to $\Delta_{max}$, an initially circular tube of magnetic flux distorts into an ellipse with a large diameter $\Delta_{max}$ that increases exponentially with time and a small diameter $\Delta_{min}$ that decreases exponentially with time. However, when $\Delta_{max}$ becomes large compared to the scale of the chaotic flow $a_{u}$, the evermore elongated tubes of magnetic flux develop folds, and the maximum separation increases only diffusively, which in simple cases means as $\sqrt{t}$.

The natural scale for fast magnetic reconnection across the magnetic field lines is $a_{u}$. The importance of fast magnetic reconnection is determined by the relative size of $a_{u}$ and $a_{r}$. Natural flows may have many spatial scales, and the scales that are comparable to or longer than $a_{r}$ are of greatest practical importance. Extremely short scales, as in turbulent flows, are only important for rapid reconnection when the flow speed of the turbulent eddies is far larger than the speed of the longer scale parts of the flow Boozer:rec-phys .

The 1984 paper of Aref Aref:1984 showed that neighboring streamlines of flows, such as $\vec{u}_{\bot}$, intrinsically separate exponentially in time, which implies the natural behavior of $\ln(\Delta_{max}/\Delta_{min})\sim t/\tau_{ev}$, where $\tau_{ev}$ is the timescale of the ideal evolution. In other words, the natural timescale for the onset of a large scale magnetic reconnection is $\sim\tau_{ev}\ln(a_{r}/\Delta_{d})$. In systems of interest, $\ln(a_{r}/\Delta_{d})\sim\ln(1/\epsilon_{ni})$ is generally of order ten, but certainly less than a hundred.

Aref’s work explained the ubiquity of temperature relaxation in a room in tens of minutes rather than the few weeks that would be required by diffusive transport. The flow that causes this relaxation is constrained by an energy argument to be divergence free, which requires the region of relaxation have at least two non-trivial spatial dimensions Boozer:rec-phys . For magnetic reconnection, an analogous energy argument requires the region of relaxation have at least three non-trivial spatial dimensions Boozer:rec-phys . This essentially follows from the requirement that the flow involve at least two directions with the magnetic field pointing in a third.

The current density required to achieve $\Delta_{max}/\Delta_{min}\sim a_{r}/\Delta_{d}$ scales Boozer:B-line.sep as $\ln(a_{r}/\Delta_{d})$ and lies in numerous thin but broad ribbons throughout the region of magnetic reconnection Boozer:rec-phys . This was illustrated for a smooth large-scale flow of the field lines $\vec{u}_{\bot}$ by Boozer and Elder Rec-example .

The traditional theory of magnetic reconnection, which developed out of two-dimensional models, ignores exponentially large variations in separation between the magnetic field lines in neighboring pairs. In two dimensions, an exponential increase in $\Delta_{max}/\Delta_{min}$ for pairs of field lines throughout a volume is impossible Boozer:rec-phys . The maximum current density required in traditional theory scales as $1/\epsilon_{ni}$ rather than as the $\ln(1/\epsilon_{ni})$ scaling that arises when exponentiation is taken into account. A feeling for the ratio of $1/\epsilon_{ni}$ to $\ln(1/\epsilon_{ni})$ in the solar corona is gained by noting that this ratio is comparable to the ratio of the age of the universe to the life expectancy of a person.

Traditional reconnection theory explains fast magnetic reconnection by assuming that, even though the non-ideality, $\epsilon_{ni}$ is extremely small, the actual reconnection takes place in a thin layer in which

This was the solution given in 1988 by Schindler, Hesse, and Birn Schindler:1988 . In their model, $\mathcal{E}$ is given by a field-line average of $\eta j_{||}$. The thickness of the layer in which the current must be concentrated, $\Delta_{\eta}$, is related to the reconnection scale $a_{r}$ by

$R_{m}$ is called the magnetic Reynolds number and can be greater than $10^{10}$ in coronal problems. The current density in the thin layer is approximately $R_{m}$ times larger than the characteristic current density, which is given by the scale $a_{r}$.

Traditional reconnection theory Hesse-Cassak2020 , including plasmoid theory plasmoid , Figure 1, has not focused on how such an intense current can be produced in a timely manner, but rather how it can be maintained assuming its presence as an initial condition.

As discussed in Section VI, one can find cases in which the current density will become singular on a surface in the limit as time goes to infinity, but the required time can be too long to explain fast reconnection. The required time for a current to concentrate in a thin layer about a surface depends on how long is required for a shear Alfvén wave propagating along magnetic field lines to cover the region enclosed by this layer Rec-example ; Boozer:surf2022 .

What is the range of applicability of traditional reconnection theory? In systems with only two non-trivial spatial coordinates, there is no real alternative. But, the applicability of traditional reconnection theory to three-dimensional systems is far from obvious when the magnetic Reynolds number $R_{m}$ is large. The intrinsic reconnection described in Section II.1 would be expected to arise and fundamentally change the magnetic evolution when the current density is smaller by a factor $R_{m}/\ln{R_{m}}\approx 4\times 10^{8}$ when $R_{m}=10^{10}$.

Understanding fast reconnection as an intrinsic implication of Faraday’s law is a sharp shift in paradigm from traditional reconnection theory. The concept of a paradigm shift in science was introduced by Thomas Kuhn Kuhn in 1962, one of the most influential works of history and philosophy written in the 20th century Britannica:Kuhn .

The shift in paradigm between traditional reconnection theory and the intrinsic implications of Faraday’s law produces changes (1) in questions, (2) in concepts, and (3) in standards of validation. These are the standard changes of any paradigm shift.

1. Changes in questions

a. How can an arbitrary magnetic field naturally and sufficiently rapidly evolve into a state of fast magnetic reconnection? This question is neither a focus nor convincingly addressed in traditional reconnection theory.

b. What are the implications of different properties along the direction perpendicular to the page in Figure 1, the $y$ direction? The $y$ direction could have perfect symmetry from plus to minus infinity or have perfect symmetry in a periodic angle. The $y$ direction could also be bounded by a perfectly conducting or by an insulating wall. This question is neither a focus nor convincingly addressed in traditional reconnection theory.

c. Over what regions can particles be accelerated? Appendix A shows that when a magnetic field is evolving—even evolving ideally—the maximum parallel kinetic energy of collisionless charged particles naturally increases exponentially with time.

This follows from Hamiltonians of the $H(p,q,t)$ form generically having chaotic trajectories, which means neighboring pairs of trajectories $p(t),q(t)$ separate exponentially in time. The simplest Hamiltonian for charged particle motion along a magnetic field line is $H_{p}(p_{||},\ell,t)=(p_{||}^{2}/2m)+\mu B(\ell,t)$ is of the $H(p,q,t)$ form when the magnetic field is evolving.

The acceleration region is spread through a large volume in intrinsic reconnection unlike in the forms of traditional reconnection theory in which the primary acceleration mechanism is taken to be the non-ideal electric field $\mathcal{E}\vec{\nabla}\ell$, which is extremely large but only in thin sheets.

Although the dominant acceleration mechanism was traditionally thought to be from the non-ideal part of the electric field, Dahlin, Drake, and Swisdak noted in 2016 that the parallel electric field is not an efficient mechanism for particle acceleration. In the next year, they discussed an alternative mechanism Drake:2017 . Boozer gave the analogous theory of particle acceleration in 2019 for intrinsic reconnection Boozer:acc . The derivations given in these three papers are subtle and complicated unlike the demonstration given in Appendix A.

2. Changes in concepts

a. The primary conceptual failure in traditional reconnection theory is the neglect of the dominant effect in making Faraday’s law exponentially sensitive to non-ideal effects. This is the characteristic exponential increase in the ratio between the maximum and minimum separation of neighboring pairs of magnetic field lines.

b. Because the dominant effect is neglected, traditional reconnection theory has focused on the maintenance of the extremely large electric field thought to be required by Maxwell’s equations. In three-dimensional reconnection, the electric field need not be large and the region in which reconnection actually takes place need not be thin—a fundamental conceptual change.

c. Traditional reconnection theory fails to distinguish the plasma and the magnetic-field-line velocities. This distinction, which has been known since the 1958 paper of Newcomb Newcomb , greatly simplifies Faraday’s law, and clarifies the relative importance of different plasma effects to reconnection.

3. Changes in standards of validation

a. A valid theory of magnetic reconnection should be implied by mathematics coupled with Maxwell’s equations. Despite the traditional belief, there is no implication that a large electric field is required to produce a fast reconnection when there is a non-trivial dependence on all three spatial coordinates. What is implied by mathematics and Maxwell’s equations is that the magnetic-field-line velocity of an ideal evolution will generally make a magnetic evolution exponentially sensitive to non-ideal effects Boozer:rec-phys .

b. The observation of current sheets is often assumed to validate traditional reconnection theory. However, the existence of thin but wide ribbons of current is characteristic of a near ideal evolution in a system with three spatial coordinates Boozer:rec-phys ; Rec-example . This is true even when the $\vec{u}_{\bot}$ that defines the ideal evolution is a simple non-turbulent flow Rec-example . The maximum current density along the magnetic field, $j_{||}$, is exponentially larger in traditional theory in comparison to what is expected from Maxwell’s equations in a generic evolution. An important observational question is whether there are actual measurements of the extreme values of $j_{||}$ that traditional theory predicts.

c. As discussed in Appendix A, particle acceleration can occur even when the non-ideal part of the electric field $\mathcal{E}$ is zero. Consequently, particle acceleration does not demonstrate the existence of a large $\mathcal{E}$. Studies of intrinsic reconnection are needed in order to determine its observational implications on particle acceleration.

As discussed by Thomas Kuhn, paradigm changes are always met with great resistance. The practitioners of the old paradigm naturally cling to a method of thought that has long been accepted as a basis of truth upon which careers, publications, and grants can be based. Nevertheless, truth in science is not demonstrated by history or a majority vote of scientists.

Two other theories of magnetic reconnection should be discussed: (1) plasma turbulence and (2) theories associated with Eric Priest.

Positions near an arbitrarily chosen magnetic field line, $\vec{x}_{0}(s)$, which means $d\vec{x}_{0}/ds\equiv\hat{b}(\vec{x}_{0})$, can be described using the intrinsic coordinates of Courant and Snyder Courant:1958 ;

The unit vector $\hat{b}$ is defined as $\hat{b}\equiv\vec{B}/B$, and $s$ has the interpretation as the distance along the line $\vec{x}_{0}(s)$.

A smooth curve $\vec{x}_{0}(s)$, which is parameterized by the distance along the curve $s$, can be used to define three orthonormal unit vectors: the tangent unit vector $\hat{b}_{0}(s)\equiv\hat{b}(\vec{x}_{0})$, the curvature unit vector $\hat{\kappa}_{0}$, and the torsion unit vector $\hat{\tau}_{0}$. The definitions are

The signs of the curvature $\kappa_{0}$ and the torsion $\tau_{0}$ are chosen so Equation (12) has the standard sign. These relations, which were first derived by Jean Frédéric Frenet in 1847 and independently by Joseph Alfred Serret in 1851, are easily obtained.

By their definition, orthogonal unit vectors satisfy $\hat{e}_{i}\cdot\hat{e}_{j}=\delta_{ij}$, so $(d\hat{e}_{i}/ds)\cdot\hat{e}_{j}+\hat{e}_{i}\cdot(d\hat{e}_{j}/ds)=0$. Differentiating $\hat{b}_{0}\cdot\hat{b}_{0}=1$, one finds $d\hat{b}_{0}/ds$ must have the form $d\hat{b}_{0}/ds=\kappa_{0}\hat{\kappa}_{0}$, where $\hat{\kappa}_{0}$ must be orthogonal to $\hat{b}_{0}$. Differentiating $\hat{\kappa}_{0}\cdot\hat{b}_{0}=0$, one finds the form $d\hat{\kappa}_{0}/ds=-(\kappa_{0}\hat{b}_{0}+\tau_{0}\hat{\tau}_{0})$. Finally differentiating $\hat{\tau}_{0}\cdot\hat{b}_{0}=0$ and $\hat{\kappa}_{0}\cdot\hat{\tau}_{0}=0$, one finds that $d\hat{\tau}_{0}/ds=\tau_{0}\hat{\kappa}_{0}$.

The derivatives of the position vector $\vec{x}_{I}(\rho,\alpha,s)$ with respect to the three coordinates are called the tangent vectors. Using the relations obeyed by the Frenet unit vectors:

The coordinate Jacobian is

The appendix to Reference Boozer:RMP gives a two-page derivation of the theory of general coordinates. This includes the dual relations,

Consequently, the tangent vectors of intrinsic coordinates can be written in terms of the gradients of the intrinsic coordinates as

This section gives the derivation of the exact Hamiltonian $\tilde{H}(\tilde{\psi},\alpha,s)=\tilde{\psi}h(\alpha,s)$ for the magnetic field lines that are in the neighborhood of an arbitrarily chosen line, called the central line, which means in the limit as the distance from the line $\rho$ goes to zero. This Hamiltonian is determined by four functions of the distance $s$ along the central magnetic field line.

Any field that is intrinsically divergence free, such as the magnetic field, can be written Boozer:B-H in the form

This follows from writing the vector potential in the general form

where $(\rho,\alpha,s)$ are the coordinates of the intrinsic coordinate system. Any vector in three space can be described by three functions of the coordinates. In Equation (26) these are $\tilde{\psi}(\rho,\alpha,s)$, $\tilde{H}(\rho,\alpha,s)$, and $g(\rho,\alpha,s)$.

Equation (25) implies Hamiltonian equations for the magnetic field lines:

When the functional dependence of $\tilde{H}$ is chosen to be $(\tilde{\psi},\alpha,s)$ rather than the coordinates $(\rho,\alpha,s)$ of the intrinsic coordinate system, $\tilde{H}$ is the Hamiltonian for the magnetic field lines. The coordinates $(\tilde{\psi},\alpha,s)$ are the canonical coordinates of $\tilde{H}$.

The magnetic flux enclosed by a circle of radius $\rho$ is given by $\mathcal{J}_{I}\vec{B}\cdot\vec{\nabla}s$ integrated over $\alpha$ from zero to $2\pi$ and over $\rho$. But, $\mathcal{J}_{I}\vec{B}\cdot\vec{\nabla}s=(\partial\tilde{\psi}/\partial\rho)/2\pi$, so $\oint\tilde{\psi}(\rho,\alpha)d\alpha/2\pi$ is that flux. To the lowest non-trivial order,

where $B_{0}(s)$ is the field strength along the line $\vec{x}_{0}(s)$.

$\tilde{H}$ to lowest non-trivial order in $\rho$ will be derived in the coordinates $(\rho,\alpha,s)$ using the expression for the current density parallel to the central magnetic field line. This requires writing $\vec{B}$ in the covariant form

which is be calculated in the limit $\rho\rightarrow 0$.

A derivative with respect to $s$, which means along the magnetic field, is intrinsically smaller in problems of interest compared to the derivatives with respect to $\rho$ and $\alpha$, which mean derivatives across the field. When magnetic field lines exponentiate apart, derivatives across the field lines are exponentially larger than derivatives along the lines.

The differential equation for $\tilde{H}$ is then

Writing $\tilde{H}=\pi B_{0}\rho^{2}h(\alpha,s)$, the differential equation for $\tilde{H}$ implies

The torsion $\tau_{0}(s)$ measures the departure of the central line from lying in a plane.

The position vector of the Courant and Snyder intrinsic coordinate system, Equation (7), can be rewritten using the canonical coordinates of the Hamiltonian $\tilde{H}(\tilde{\psi},\alpha,s)$ of Equation (40),

In time dependent problems, it is important to have the position vector $\vec{x}_{Ic}$ given as a function of the canonical coordinates and time. When the magnetic field is time dependent, each of the functions that appears in $\vec{x}_{Ic}(\tilde{\psi},\alpha,s,t)$ depends on time. These functions are $B_{0}(s,t)$, $\hat{\kappa}_{0}(s,t)$, $\hat{\tau}_{0}(s,t)$, and $\vec{x}_{0}(s,t)$.

The evolution of a magnetic field is determined by the electric field, which can be obtained from the time-derivative of the vector potential. Equation (26) for the vector potential in intrinsic coordinates implies

The derivation of Equation (42) is non-trivial but is given in the appendix to Reference Boozer:RMP . The electric field equals $-(\partial\vec{A}/\partial t)_{\vec{x}}$ minus the gradient of a potential:

When the evolution is ideal, the electric field is also given by

where $\vec{u}_{\bot}$ is the velocity of the magnetic field lines through space. An ideal evolution of a magnetic field is uniquely described by $\vec{u}_{\bot}$ since $\partial\vec{B}/\partial t=\vec{\nabla}\times(\vec{u}_{\bot}\times\vec{B})$.

The interpretation of Equations (44) and (45) is clarified by a related canonical representation of the magnetic field, Appendix B, that is not limited to the vicinity of a given magnetic field line.

The freedom of canonical transformations is given by $\phi$. When Equations (44) and (45) are used to eliminate the electric field from the analysis:

Dotting this equation with $\vec{B}$,

Crossing with $\vec{\nabla}s$,

Although $\vec{u}_{\bot}$ is defined by an ideal evolution, the choice of the velocity of the canonical coordinates $\vec{u}_{c}$ has freedom, the freedom of the canonical transformations given by $\phi$. The choice $\vec{u}_{c}=\partial\vec{x}_{Ic}/\partial t$ fixes $\phi$ and through Equation (47) the evolution of the Hamiltonian.

The $\hat{b}_{0}$ component of the vorticity of the magnetic field line flow will be shown to have an important relationship to $K_{0}=\mu_{0}j_{||}/B_{0}$.

The vorticity of the field line flow about the central line is calculated using Equation (48) for $\vec{u}_{c}-\vec{u}_{\bot}$:

since to lowest order in $\rho$, $\vec{\nabla}s=\hat{b}_{0}(s)$ and $\hat{b}_{0}\cdot\vec{\nabla}s=1$. The derivatives along the magnetic field, $\partial/\partial s$ are small compared to derivatives across the magnetic field—in fact exponentially smaller when the magnetic field is chaotic. Therefore, to lowest order in $\rho$,

Equation (34) for $\nabla_{\bot}^{2}\tilde{H}$, Equation (47) for $(\partial H/\partial t)_{c}$, and Equation (51) for $\Omega$ imply

Equation (52) is a generalization of the expression $\partial\Omega/\partial s=\partial K_{0}/\partial t$, Equation (4) of Reference Boozer:j-relax on the Alfvénic relaxation of variation in $j_{||}/B$ along magnetic field lines and Equation (26) of Reference Rec-example on how simple non-turbulent footpoint motions in the solar corona can give complicated ribbons of current. The primary difference is the inclusion of torsion $\tau_{0}$. As explained in Section V, including $\tau_{0}$ generally gives only a small correction.

In the solar corona, the evolution of magnetic field lines is driven by the motion of the footpoints at their two interceptions with a surface outside of the photosphere. As with any natural flow, this flow is in general chaotic even when it is a smooth and simple function of position. By the definition of a chaotic flow, neighboring streamlines separate exponentially in time until the separation is comparable to the longest spatial scales of the flow Rec-example .

When for simplicity the footpoint flow is assumed to be zero at one interception of the field lines and non-zero at the other, the boundary conditions imply the ratio of separations of neighboring magnetic field lines $\Delta_{max}/\Delta_{min}$ must be greater or equal to the separation of neighboring streamlines of the flowing footpoints. Consequently, $\ln(\Delta_{max}/\Delta_{min})\gtrsim t/\tau_{ev}$, where the ideal evolution time $\tau_{ev}$ is defined by the flowing footpoints.

The existence of a spatial scale $\Delta_{d}$ below which magnetic field lines cannot be distinguished implies a fast magnetic reconnection over the spatial scale of the footpoint flow $a_{u}$ on the timescale $\sim\tau_{ev}\ln(a_{u}/\Delta_{d})$. What is subtle is the force-free current density in the plasma $K=\mu_{0}j_{||}/B$. Boozer and Elder Rec-example showed this current characteristically lies in thin but wide ribbons along the magnetic field lines with the maximum current density scaling as $\ln(a_{u}/\Delta_{d})$, as predicted in Reference Boozer:rec-phys .

What is new is that in Boozer and Elder Rec-example the force-free current $K$ was calculated using $\partial\Omega/\partial s=\partial K/\partial t$. The more accurate and rigorous derivation of this paper gives Equation (52). As discussed below, the difference in these two equations produces no qualitative difference in the results.

In the solar corona, the timescale of the footpoint flow is generally far longer than the time for shear Alfvén waves to propagate from one field line interception to the other. Force balance then implies $dK_{0}/ds=0$.

The constraint $\vec{\nabla}\cdot\vec{j}=0$ is mathematically equivalent to

where $\vec{f}_{L}\equiv\vec{j}\times\vec{B}$ is the Lorentz force. When $dK_{0}/ds$ is large, the plasma inertia $\rho d\vec{v}/dt$ must balance the Lorentz force. The $\hat{b}_{0}$ component of the curl of the force balance equation gives

is the Alfvén speed Boozer:j-relax ; Rec-example . Consequently, $\partial K_{0}/\partial s=0$ when the evolution of $\Omega$ is slow compared to the time for a shear Alfvén wave to propagate all along a magnetic field line.

When an ideal evolution is driven by a perfectly-conducting flowing-wall, which moves with a divergence-free velocity $\vec{v}_{w}$ tangential to the wall, a field line strikes the wall at $\vec{x}_{0}(t)$, which is given by $d\vec{x}_{0}/dt=\vec{v}_{w}(\vec{x}_{0},t)$. The vorticity is

where $\vec{v}_{w}=\hat{n}\times\vec{\nabla}f_{w}$, with $\hat{n}$ the unit normal to the wall. If the wall is flat, the vorticity at the particular point is $\vec{\Omega}_{w}=\hat{n}(\nabla^{2}f_{w})_{\vec{x}_{0}(t)}$. The boundary condition on the vorticity about the magnetic field line that is attached to the point $\vec{x}_{0}(t)$ is $\Omega=(\hat{b}\cdot\hat{n}\nabla^{2}f_{w})_{\vec{x}_{0}}$.

As discussed in Appendix C.1, intrinsic coordinates have a subtlety when the curvature and torsion are produced by a curl-free perturbation in which $\vec{B}/B_{0}=\hat{z}+\epsilon\big{(}-\hat{x}\sin kz+\hat{y}\cos kz\big{)}$. Even when $\epsilon\rightarrow 0$, the curvature unit vector $\hat{\kappa}_{0}$, rotates by $2\pi$ as $z$ changes by $2\pi/k$. This is a coordinate system oddity, not a physical effect of the relative twist of the two interceptions of the magnetic field lines, and can be removed by subtracting the $\epsilon\rightarrow 0$ torsion from that given by the perturbation. The physically relevant torsion is then $\tau_{0}\approx-\epsilon^{2}/2$.

The motion of the footpoints that drive the solar corona is sufficiently slow compared to the Alfvén speed that $K_{0}\equiv\mu_{0}j_{||}/B$ must be independent of $s$, but $K_{0}$ can depend on time. Equation (52) for the evolution of $K_{0}$ then implies that the vorticity $\Omega$ associated with the two boundary interceptions, at $s_{1}$ and $s_{2}$, of a field line give