Example of exponentially enhanced magnetic reconnection driven by a spatially-bounded and laminar ideal flow

Magnetic reconnection has an interesting early history Cargill:2015 and was defined in 1956 by Parker and Krook Parker-Krook:1956 as the “severing and reconnection of lines of force.”

The ideal evolution of magnetic fields is the opposing concept to magnetic reconnection. Newcomb showed in 1958 that magnetic field lines move with a velocity $\vec{u}_{\bot}$ and do not break, if and only if the field obeys the ideal evolution equation Newcomb

$$\frac{\partial\vec{B}}{\partial t}=\vec{\nabla}\times(\vec{u}_{\bot}\times\vec{B}).$$

(1)

The exact equation for the evolution of $\vec{B}$ is Faraday’s Law, $\partial\vec{B}/\partial t=-\vec{\nabla}\times\vec{E}$. The ideal evolution of a magnetic field that is embedded in a plasma is broken Boozer:ideal-ev when the electric field parallel to the magnetic field, $\vec{E}_{||}$, cannot be balanced by the parallel component of the gradient of a well-behaved potential, $\vec{\nabla}\Phi$. The components $\vec{E}_{\bot}+\vec{\nabla}_{\bot}\Phi$, which are perpendicular to $\vec{B}$, are associated with the velocity $\vec{u}_{\bot}$ of the magnetic field lines, $\vec{u}_{\bot}=(\vec{E}_{\bot}+\vec{\nabla}_{\bot}\Phi)\times\vec{B}/B^{2}$.

Two effects that cause deviations from an ideal evolution are resistivity along the magnetic field $\eta$, which contributes to $E_{||}$ as $\eta\vec{j}_{||}$, and electron inertia, which contributes Spitzer to $E_{||}$ as $(c/\omega_{pe})^{2}\partial j_{||}/\partial t$. The electron skin depth, $c/\omega_{pe}$, is the speed of light divided by the electron plasma frequency.

The Oxford English Dictionary defines a paradox as “a strongly counter-intuitive (statement), which investigation…may nevertheless prove to be well-founded or true.” Both the resistivity and the electron skin depth are so small in many plasmas of practical interest that it is paradoxical that magnetic reconnection could be of any relevance. Yet it is. Approximately 1500 papers have been written on magnetic reconnection, which demonstrate the importance of the topic to the understanding of both laboratory and naturally occurring plasmas. The speed and prevalence of magnetic reconnection are so great that they must be derivable from the properties of Equation (1) for the ideal evolution of a magnetic field Boozer:ideal-ev .

To define the reconnection paradox, let $a$ be a characteristic spatial scale over which $u_{\bot}$ varies across the magnetic field lines; the time scale for an ideal evolution is

$$\tau_{ev}\equiv\frac{a}{u_{\bot}},$$

(2)

Resistivity spatially interdiffuses magnetic field lines, which implies that lines that approach each other closer than the distance $\Delta_{d}=\sqrt{(\eta/\mu_{0})\tau_{\eta}}$ remain distinguishable only for a time $t$ less than $\tau_{\eta}$. The ratio of the resistive time scale of the overall system to the evolution time scale $\tau_{ev}$ is the magnetic Reynolds number

$$R_{m}\equiv\left(\frac{a^{2}}{\eta/\mu_{0}}\right)\left(\frac{u_{\bot}}{a}\right)=\frac{\mu_{0}u_{\bot}}{\eta}a.$$

(3)

Magnetic field lines that come closer than $\Delta_{d}=a/R_{m}$ lose their distinguishability, where $R_{m}\sim 10^{12}$ in the solar corona.

The electron inertia causes evolving magnetic field lines that approach each other closer than $\Delta_{d}=c/\omega_{pe}$, the electron skin depth, to become indistinguishable, Appendix C of Boozer:null-X . In the solar corona $a/(c/\omega_{pe})\sim 10^{9}$.

What is paradoxical is that magnetic field lines are observed to reconnect and become indistinguishable over a region of width $a$ across the magnetic field lines on a time scale only an order of magnitude or so longer than the characteristic ideal-evolution time of the magnetic field, $\tau_{ev}$. Effects such as the resistivity would be expected to cause a loss of distinguishability of magnetic field lines in the solar corona only on a time scale of order $10^{12}$ times longer than $\tau_{ev}$. This paradox was clearly described in 1988 by Schindler, Hesse, and Birn Schindler:1988 .

Schindler et al Schindler:1988 sought to resolve the paradox of reconnection occurring over a region of far greater width than $\Delta_{d}$. In their resolution, the current density would become and remain extremely large, $j\sim B_{rec}/(\mu_{0}\Delta_{d})$, in a layer of width $\Delta_{d}$, where $B_{rec}$ is the part of the magnetic field that is reconnecting. This assumption has formed the basis of most of the reconnection literature from even before Schindler et al developed their reconnection theory. Nonetheless, it is difficult to understand how an ideal evolution would result in such a strong current. Most of the reconnection literature has not dealt with that issue but has focused instead on how such a large current density could be maintained if it were initially present. A foundational publication for recent work on the maintenance issue, with more than two hundred citations, is the 2010 paper by Uzdenksy, Louriero, and Schekochihin Loureiro:2010 on plasmoids.

The paradoxical speed of magnetic reconnection when non-ideal effects are extremely small has analogues with other phenomena, such as thermal equilibration in air. The analogy between magnetic reconnection and thermal equilibration in a room was demonstrated Boozer:rec-phys by Boozer in 2021. In both reconnection and thermal equilibration, the time scale of the relaxation of the ideal constraint is the ideal evolution time $\tau_{ev}$ times a term that is logarithmically dependent on the ratio of the non-ideal time scale divided by the $\tau_{ev}$. This explains why a radiator can heat a room in tens of minutes rather than the several weeks that would be expected from thermal diffusion alone. Many may find the subtle mathematics used in Boozer:rec-phys difficult to follow. Here a simple model of magnetic field evolution driven by footpoint motion, as in the solar corona, is used to illustrate the general principles that were explained in the companion paper Boozer:rec-phys .

The equation for temperature equilibration in air moving with a divergence-free velocity $\vec{v}(\vec{x},t)$ is $\partial T/\partial t+\vec{v}\cdot\vec{\nabla}T=\vec{\nabla}\cdot(D\vec{\nabla}T)$, which is the standard form for the advection-diffusion equation. In many problems of practical importance, the Péclet number, $va/D$, is many orders of magnitude greater than unity. The diffusion coefficient for magnetic field lines is $\eta/\mu_{0}$, so the magnetic Reynolds number would be better called the magnetic Péclet number.

The classic paper on the advection-diffusion equation was written in 1984 by Hassan Aref Aref;1984 and has had over two thousand citations. This paper showed that a laminar flow can equilibrate $T$ on the evolution time $a/v$ times a logarithm of the Péclet number as $va/D\rightarrow\infty$. Before Aref’s paper, it had been assumed a turbulent $\vec{v}$ was required to obtain fast equilibration. Section I.B in an article Aref:2017 published in the Reviews of Modern Physics discusses the merits of the use of turbulent versus laminar flows to speed the mixing of fluids. What that article did not discuss is that for a given maximum flow speed $v$ a laminar flow can give a faster mixing over a large region than a turbulent flow. A demonstration starts with the last sentence on p. 3 of Boozer:rec-phys . The laminar part of a flow, which is defined as an average over the small spatial scales of $\vec{v}(\vec{x},t)$, generally dominates the advective part of the advection-diffusion equation. A flow $\vec{v}(\vec{x},t)$, no matter how complicated and rapid the dependence on $\vec{x}$ and $t$, cannot directly produce diffusive mixing but can exponentially enhance the speed with which diffusion can act over the spatial scale covered by the advection of the fluid.

The essential element in an enhanced relaxation by a laminar flow is that the flow be chaotic. Using standard terminology, a flow is deterministic but chaotic when neighboring streamlines have a separation that increases exponentially with time. Articles on the mathematics of deterministic chaos and topological mixing can easily be found on the web, but their importance to this paper is only that such effects are common. As noted in Boozer:rec-phys , although the condition that the flow be chaotic may sound restrictive, it is a non-chaotic natural flow that is essentially impossible to realize. No special effort is required to achieve enhanced mixing by stirring. Every cook knows that stirring enhances the mixing of fluids—no particular pattern of stirring or detailed computations are required.

The terms chaotic and stochastic are sometimes considered synonyms in descriptions of flows or magnetic fields. Here a distinction is made that is consistent with a distinction made in the mathematical literature. Chaos, or more precisely deterministic chaos, when applied to a flow within a bounded region of space, means the streamlines of the flow are predictable but exponentiate apart throughout a finite fraction of that space, the region in which the flow is chaotic. In contrast, a stochastic motion is not deterministic and has a random component on all time and spatial scales. Stochastic flows were used by L. F. Richardson in a paper Richardson:1926 published in 1926 to explain the enhancement of diffusion produced by atmospheric flows. His assumption was that that the velocity of air $\Delta\vec{x}/\Delta t$ has no well defined limit as $\Delta t$ goes to zero. Weak magnetic field line stochasticity makes field lines indeterminant in a way that is similar to the the effect that Richardson’s assumption has on streamlines of air. The enhancement of reconnection by weak stochasticity of the magnetic field lines was studied by A. Lazarian and E. T. Vishniac Lazarian:1999 in 1999. Here we consider only deterministic flows, which means velocities are well defined. In an ideal magnetic evolution, Equation (1), deterministic flows give deterministic magnetic field lines.

The evolution equation for a magnetic field is also of the advection-diffusion type but more subtly so than the equation for thermal equilibration. The companion paper Boozer:rec-phys derives two salient differences: (1) A flow-enhanced equilbration of the temperature requires a dependence of $\vec{v}$ on at least two spatial coordinates to be energetically feasible, but flow-enhanced magnetic reconnection requires a dependence of $\vec{u}_{\bot}$ on all three spatial coordinates for energetic feasibility. (2) In thermal relaxation, $\big{|}\vec{\nabla}T\big{|}$ increases exponentially with time but what one might think would be the analogous quantity for the magnetic field, $\vec{j}=\vec{\nabla}\times\vec{B}/\mu_{0}$ does not. In the model developed in this paper, the increase in the current density is limited to being proportional to time. Related limits on the current density are known for resonant perturbations to toroidal plasmas Hahm-Kulsrud and for currents flowing near magnetic field lines that intercept a magnetic null Elder-Boozer .

When the ideal flow velocity of a magnetic field $\vec{u}_{\bot}$ is chaotic, the ratio $\Delta_{max}/\Delta_{min}$, the maximum to the minimum separation between two field lines at any particular time, tends to increase exponentially during the evolution when the closest approach of the lines is small, $\Delta_{min}/a\rightarrow 0$. Large scale reconnection occurs for magnetic field lines that satisfy $\Delta_{min}\lesssim\Delta_{d}$ and $\Delta_{max}\approx a$. As will be seen, a simple laminar flow can be followed as $\Delta_{max}/\Delta_{min}$ increases by more than nine orders of magnitude.

The model that will be used to illustrate features of magnetic reconnection is developed in Section II. In this model, the magnetic field evolution is driven in a bounded plasma in the same way as footpoint motion drives the magnetic field in the solar corona.

Section III will show that when the footpoint velocity is sufficiently small compared to the Alfvén speed in the plasma that important features can be determined without solving the difficult problem of determining $\vec{B}(\vec{x},t)$ throughout the plasma. These features include the exponentiation in time of the separation between neighboring magnetic field lines and the current density $j_{||}$ along each magnetic field line, or more precisely $K\equiv\mu_{0}j_{||}/B$. A bound on the magnitude of $K$ is obtained, which increases linearly in time. For these features, only the determination of the streamlines of a time-dependent flow in two dimensions is required.

The simplicity of the treatment given in Section III depends on the evolution being slow compared to the Alfvén transit time. This would be violated if the plasma became kink unstable. Section IV shows that the flow example introduced in Section II ensures kink stability.

Section V shows that the rate of production of the magnetic flux associated with a particular magnetic field line by the footpoint motion is the magnetic Reynolds number $R_{m}$ times larger than its rate of destruction by resistivity. The sign of the flux differs from field line to field line in kink-stable evolutions, and the rate of flux production and destruction can be brought into balance by reconnection.

Section VI shows that even if the plasma becomes kink unstable, the magnetic helicity contained in the bounded plasma will increase without limit when $R_{m}>>1$ unless the footpoint motion obeys a constraint. If the plasma were unbounded, an ever increasing helicity would presumptively result in a plasma eruption.

Section VII derives the power that must be supplied to maintain the specified footpoint motion. For a steady-state flow, the required power increases linearly with $K$. Section VIII discusses the results and their implications.

Features of magnetic reconnection will be illustrated using a simplified model of magnetic loops in the solar corona. The evolution of coronal loops is driven by the motion of their foopoints by photospheric flows.

The description of the model has three parts: (1) Section II.1 defines a spatially bounded perfectly-conducting cylinder that encloses the loop as well as the flow $\vec{v}_{t}$ in the top surface of the cylinder, which represents the photospheric motion. (2) Section II.2 explains the specific form chosen for $\vec{v}_{t}$. (3) Section II.3 explains how the chaotic properties of $\vec{v}_{t}$ are quantified.

The numerical results for the streamlines of the flow $\vec{v}_{t}$ are given in Section II.4.

The magnetic field evolution in the model described in Section II becomes remarkably simple when the height of the cylinder is far greater than its radius, $L/a\rightarrow\infty$ and the time scale of ideal evolution is very long compared the the Alfvén transit time. For simplicity the plasma pressure is assumed to be zero. The derivation is essentially a simplified version of the derivation of Reduced MHD Kadomtsev ; Strauss .

This section consists of four subsections. The first two subsections, Section III.1 and III.2, derive two differential equations that relate the Lagrangian derivatives of time and distance along the magnetic field lines $\ell$ of the distribution of parallel current $K\equiv\mu_{0}j_{||}/B$ and the vorticity $\Omega\equiv\hat{z}\cdot\vec{\nabla}\times\vec{u}_{\bot}$ of the magnetic field line flow. The third subsection, Section III.3, discusses the implications of these two equations and is the most important part of the paper. The fourth subsection, Section III.4, gives a heuristic argument that the magnitude of the current distribution $K$ should typically scale as the strength of the exponential separation of the magnetic field lines, their Frobenius norm. This relation is shown to hold accurately for the ensemble of many field lines but not for each line. Indeed, the correlation between large values of $K$ and large values of the Frobenius norm is found to be remarkably weak.

The current flowing along the magnetic field lines causes the lines to twist through an angle $\Theta=KL/2$ from one end of the cylinder to the other. When the twist has a smooth variation with radius, Hood and Priest Hood:kink1979 found the magnetic field becomes unstable to an ideal kink when $\Theta$ is greater than a critical value, which in their calculations lay in the range $2\pi$ to $6\pi$. Studies of the onset of reconnection in the model of Figure 1 are much simpler when ideal kink instabilities are not an issue.

The largest $KL$ values in Figure 2c correspond to $\Theta\approx 8\pi$, but the current $K$ has an extremely complicated spatial distribution, not only in magnitude but also in sign; spatial averages are far smaller than the maximum value. As will be discussed, the anisotropy of the derivatives of $K$ across the magnetic field lines and the smallness of the spatial averages of $K$ makes the system highly stable to kinks.

It is not required that $K$, or equivalently the twist $\Theta$, have small spatial averages when the flow is chaotic. The spatially averaged twist $\Theta$ can be made arbitrarily large by choosing $c_{0}$ to be large and $\omega_{0}$ to be either zero or small in Equation (6) for $\tilde{h}$. The choice $c_{0}=0$ was made to show that a large average field line twist is not needed to obtain chaos.

Even when $c_{0}=0$, the spatial average of $K$ over small regions can be non-zero. This is illustrated by Figure 2c and by the first row of Figure 5.

When the stream function is chosen so the flow is chaotic but with a large spatially-averaged $K$, the resulting magnetic field will generally evolve not only into a kinked but also into an eruptive state. As shown in Section VI, the evolution properties of magnetic helicity imply the spatial and temporal average of $h_{t}$ must be zero for a non-eruptive steady-state solution for the magnetic field when $R_{m}>>1$—no matter how spatially concentrated the current may become. Consequently, non-eruptive chaotic models tend to have spatially complicated distributions of $\Theta$ in which $\Theta$ has both signs and a near-zero spatial average as in the $\vec{v}_{t}$ example used to construct Figure 2.

As discussed in Section V.B.1 of Boozer:RMP , the stability of force-free equilibria can be determined using the perturbed equilibrium equation $\vec{\nabla}\times\delta\vec{B}=(\mu_{0}\delta j_{||}/B)\vec{B}$, where $\delta\vec{B}=\vec{\nabla}\times(\delta A_{||}\hat{z})$. The perturbed parallel current is determined by the constancy of $K\equiv\mu_{0}j_{||}/B$ along magnetic field lines, which in linear order in the perturbation implies $\vec{B}\cdot\vec{\nabla}\delta K+\delta\vec{B}\cdot\vec{\nabla}K=0$. Stability is determined by whether it takes positive or negative energy to drive a perturbation that obeys the equations

	$\displaystyle\nabla_{\bot}^{2}\delta A_{||}$	$\displaystyle=$	$\displaystyle-\delta KB_{0};$		(46)
	$\displaystyle B_{0}\left(\frac{\partial\delta K}{\partial\ell}\right)_{x_{0}y_{0}}$	$\displaystyle=$	$\displaystyle\hat{z}\cdot\left(\vec{\nabla}_{\bot}A_{||}\times\vec{\nabla}_{\bot}K\right).\hskip 14.45377pt$		(47)

The system is at marginal stability when $\delta K$ is just strong enough to produce a solution $\delta A_{||}$ that fits within the perfectly conducting cylindrical walls. The implication is that when Equation (46) is multiplied by $\delta A_{||}$, then at marginal stability

$\displaystyle\int\left\{\left(\vec{\nabla}_{\bot}\delta A_{||}\right)^{2}-\delta K\delta A_{||}B_{0}\right\}d^{3}x=0.$

(48)

When $\delta K$ has only a rapid spatial variation, as it does when $K$ does, then $\delta A_{||}$ must also have a rapid variation to avoid a self-cancelation of the destabilizing term $\delta K\delta A_{||}$ in Equation (48). Equation (47) for $\delta K$ involves two spatial derivatives across $\vec{B}$ and one might think they could be large and balance the stabilizing effect of the two spatial derivatives in $\left(\vec{\nabla}_{\bot}\delta A_{||}\right)^{2}$, but that is not the case. The two spatial derivatives across $\vec{B}$ in the equation for $\delta K$ are orthogonal and the spatial derivatives of $K$ in the two directions across the magnetic field lines tend to be of exponentially different magnitudes, so the large term in $\vec{\nabla}_{\bot}K$ forces a large spatial derivative in $\delta A_{||}$, which quadratically enhances $\left(\vec{\nabla}_{\bot}\delta A_{||}\right)^{2}$ but only linearly enhances $\delta K$.

The change in the magnetic flux associated with a particular magnetic field line $\psi(x_{0},y_{0},t)$ is the integral from one perfectly conducting surface to the other, $\partial\psi/\partial t=-\int\vec{E}\cdot d\vec{\ell}$. When the cylindrical conductor is stationary and the plasma is resistive, the flux decays as $\partial\psi/\partial t=-\int\eta\vec{j}\cdot d\vec{\ell}$.

As was shown in the derivation of Equation (19) for $\partial\vec{A}/dt$, the effective inductive electric field along the magnetic field is $-\vec{B}\cdot(\partial A/\partial t)=\vec{B}\cdot(B_{0}\vec{\nabla}h)$, which gives a change in the flux, $\partial\psi/\partial t=-\int\vec{E}\cdot d\vec{\ell}$, or

$$\frac{\partial\psi}{\partial t}=-B_{0}h_{t}(x_{0},y_{0},t)$$

(49)

since the at the top of the cylinder $h=h_{t}$.

The appearance of $B_{0}h_{t}$ in the electric-field integral can be understood using the expression $\vec{E}=-\vec{v}_{t}\times\vec{B}_{0}$ for the electric field in the flowing conductor when observed from a stationary frame of reference. The velocity is $\vec{v}_{t}=\hat{z}\times\vec{\nabla}h_{t}$ and $\vec{B}_{0}=B_{0}\hat{z}$, so $\vec{E}=-B_{0}\vec{\nabla}_{\bot}h_{t}$. The electromotive force from the intersection point of the field line to a stationary point, where $h_{t}=0$, is $B_{0}h_{t}$.

The rate at which plasma resistivity destroys magnetic flux is $\mathcal{E}_{\eta}=\int\eta j_{||}d\ell$. Since $K=\mu_{0}j_{||}/B$ is constant along a magnetic field line, $\mathcal{E}_{\eta}=B_{0}(\eta/\mu_{0})K(x_{0},y_{0},t)L$. Equation (41) implies $\partial\mathcal{E}_{\eta}/\partial t=-B_{0}(\eta/\mu_{0})\Omega_{t}(x_{0},y_{0},t)$. Since $\Omega_{t}=\nabla^{2}h_{t}$, the ratio of flux creation to flux destruction is

$$\Big{|}\frac{2\partial h_{t}/\partial t}{(\eta/\mu_{0})\nabla^{2}h_{t}}\Big{|}\approx R_{m}\sim 10^{12}$$

(50)

in the solar corona.

As will be shown, an argument based on magnetic helicity implies that a long-term relevant solution to the problem outlined in Figure 1 requires the long-term spatial and temporal average of $h_{t}$ to be zero. When the average of $h_{t}$ is zero over a chaotic region, the interchange of penetration points implies the poloidal magnetic flux associated with a field line $x_{0},y_{0}$ fluctuates but has no systematic increase.

Chaotic streamlines can cause two field lines that penetrate the bottom of the cylinder at two distinct points $x_{0},y_{0}$ and $x^{\prime}_{0},y^{\prime}_{0}$ to interchange their penetration points through the top plane due to exponentially small non-ideal effects.

Equation (60) for the evolution of the magnetic helicity limits the degree to which a magnetic field driven as in Figure 1 can be simplified by magnetic field lines exchanging connections even if the current density were to obtain arbitrarily high local values by being concentrated in thin sheets. As has been shown, the maximum current density increases only linearly in time, Equation (43), and does not reach the enhancement by a factor of order $1/R_{m}$, which would be required for the loop voltage to balance the poloidal flux creation, before reconnection has already occurred. But, even if it did the rate of helicity dissipation would not be significantly enhanced, Equation (60). Magnetic turbulence can reduce the magnetic energy, but not the helicity Taylor:1974 ; Berger:1984 as $R_{m}\rightarrow\infty$.

Equation (58) for the rate of helicity increase implies that unless the stream function integrated over each chaotic region, $\int h_{t}da_{t}$, has a zero time average, the magnetic helicity can increase without limit. In the model of this paper, the perfectly conducting cylindrical boundary conditions will keep the system confined no matter how strong or contorted the magnetic field may become. But, in a natural system, such as the solar corona, a drive $h_{t}$ that does not have a zero long-term average will presumably cause the eruption of a magnetic flux tube.

The derivation of the helicity evolution equation starts with the definition of the magnetic helicity enclosed by the cylinder,

$\displaystyle\mathcal{K}$

$\displaystyle\equiv$

$\displaystyle\int\vec{B}\cdot\vec{A}d^{3}x.$

(51)

Equations (17) and (18) together with $\vec{\nabla}\cdot\vec{x}_{\bot}=2$ imply $\vec{B}\cdot\vec{A}=B_{0}^{2}(-2H+\vec{\nabla}_{\bot}\cdot(H\vec{x}_{\bot})$. The helicity is then

$$\mathcal{K}=-2B_{0}^{2}\int Hd^{3}x.$$

(52)

The time derivative of the helicity is calculated using

	$\displaystyle\frac{\partial\vec{B}\cdot\vec{A}}{\partial t}$	$\displaystyle=$	$\displaystyle\vec{B}\cdot\frac{\partial\vec{A}}{\partial t}+\vec{A}\cdot\vec{\nabla}\times\frac{\partial\vec{A}}{\partial t}$		(53)
		$\displaystyle=$	$\displaystyle 2\vec{B}\cdot\frac{\partial\vec{A}}{\partial t}-\vec{\nabla}\cdot\left(\vec{A}\times\frac{\partial\vec{A}}{\partial t}\right)\mbox{ and }$
	$\displaystyle\vec{A}\times\frac{\partial\vec{A}}{\partial t}$	$\displaystyle=$	$\displaystyle-B_{0}^{2}\frac{\vec{x}_{\bot}}{2}\frac{\partial H}{\partial t},\mbox{ so }$		(54)
	$\displaystyle\frac{\partial\vec{B}\cdot\vec{A}}{\partial t}$	$\displaystyle=$	$\displaystyle 2\vec{B}\cdot\frac{\partial\vec{A}}{\partial t}+\vec{\nabla}\cdot\left(B_{0}^{2}\frac{\vec{x}_{\bot}}{2}\frac{\partial H}{\partial t}\right)$		(55)

The side of the cylinder is a rigid perfect conductor, so $\partial H/\partial t=0$ within its sides. Consequently,

	$\displaystyle\frac{d\mathcal{K}}{dt}$	$\displaystyle=$	$\displaystyle 2\int\vec{B}\cdot\frac{\partial\vec{A}}{\partial t}d^{3}x$		(56)
		$\displaystyle=$	$\displaystyle-2B_{0}\int\vec{\nabla}\cdot(h\vec{B})d^{3}x$		(57)
		$\displaystyle=$	$\displaystyle-2B_{0}^{2}\int h_{t}da_{t},$		(58)

using Equation (19) with $da_{t}=rdrd\theta$ the area integral over the top of the cylinder. Since the magnetic field lines are tied to the flow of the perfectly conducting top, $h=h_{t}$ within the top surface.