2. One-dimensional relaxation in a pressureless plasma

Suppose that a low-resistivity, low-density plasma is contained between two perfectly conducting planes $x=\pm {\rm\pi}L$ . Suppose that, at time $t=0$ , the plasma has uniform density ${\it\rho}_{0}$ , and is at rest and permeated by a two-component magnetic field,

where $B_{0}$ is a representative value of $|\boldsymbol{B}(x,0)|$ . The $\boldsymbol{B}$ -lines are rectilinear and lie in planes $x=\text{const.}$ , the magnitude and direction of $\boldsymbol{B}$ being dependent on $x$ . The associated current is

and the Lorentz force is

where $p_{M0}(x)=\boldsymbol{B}^{2}/2$ is the magnetic pressure distribution. The electric field is $\boldsymbol{E}={\it\eta}\,\boldsymbol{j}$ , where ${\it\eta}$ is the magnetic resistivity of the plasma, assumed small; and since $E_{y}$ and $E_{z}$ must be zero at the perfectly conducting boundaries, it follows that

conditions that we shall assume to persist for all $t>0$ .

We further assume that the density of the plasma is so low that its pressure is negligible compared with the magnetic pressure (the ‘low- ${\it\beta}$ ’ approximation). For $t>0$ , the plasma flows in response to the magnetic pressure with velocity

The electric field $\boldsymbol{E}(x,t)$ in the plasma is then given by Ohm’s law,

and the magnetic field $\boldsymbol{B}(x,t)$ with components $[0,b_{y}(x,t),b_{z}(x,t)]$ evolves for $t>0$ according to the induction equation

with components

We note that, together with the boundary conditions (2.4) and (2.5), these equations imply that

i.e. the flux vector $(0,{\it\Phi}_{y},{\it\Phi}_{z})$ is constant, being trapped between the perfectly conducting boundaries. We may choose the axes $Oyz$ so that ${\it\Phi}_{y}=0$ . Flux conservation then implies that, whatever the initial field $\boldsymbol{b}_{0}$ may be, the long-term asymptotic state is just the uniform field

Following Bajer & Moffatt (Reference Bajer and Moffatt2013), we adopt a viscous model for the relaxation process, with an effective viscosity ${\it\mu}$ ; then $u$ and ${\it\rho}$ evolve according to the one-dimensional equations

where $p_{M}=B_{0}^{2}(b_{y}^{2}+b_{z}^{2})/2$ . We note that, for this geometry, (2.7) and (2.11b ) may be combined to give

where $\text{D}/\text{D}t$ is the Lagrangian derivative following a material element of fluid.

As explained in the introduction, a primary aim of this paper is to explore the extent to which the Taylor conjecture is valid for the model under consideration. That is, is the global helicity the only relevant invariant during relaxation to a minimum energy state? And does the field relax to a ‘Taylor state’ in which the Beltrami condition (1.1) is satisfied? For this purpose we need to introduce a vector potential $\boldsymbol{A}=[0,a_{y}(x,t),a_{z}(x,t)]$ with

It is evident that any vector potential of this kind satisfies $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{A}=0$ . We may further pin down the gauge of $\boldsymbol{A}$ so that $a_{y}(0,t)=a_{z}(0,t)=0$ . Since from (2.6), $\boldsymbol{E}$ has no $x$ component, we may assume that $\boldsymbol{E}=-\partial \boldsymbol{A}/\partial t$ , so that $\boldsymbol{A}$ then satisfies the equation

With $\boldsymbol{u}\times \boldsymbol{B}=(0,-ub_{z},ub_{y})$ , we then have

or equivalently

The helicity density is given by

and it is evident from (2.12) and (2.16) that, when ${\it\eta}=0$ ,

i.e.  ${\mathcal{H}}/{\it\rho}$ is a Lagrangian invariant. Thus, a fortiori, the ‘global’ helicity per unit area of the $(y,z)$ plane, given byFootnote ²

is invariant when ${\it\eta}=0$ . When ${\it\eta}\neq 0$ , standard manipulations give

The vector potential $\boldsymbol{A}$ is of course generally not unique: under the gauge transformation $\boldsymbol{A}\rightarrow \boldsymbol{A}^{\ast }=\boldsymbol{A}+\boldsymbol{{\rm\nabla}}{\it\phi}$ , the helicity density becomes ${\mathcal{H}}^{\ast }(x,t)={\mathcal{H}}+\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\phi}$ . If we insist that $\boldsymbol{A}$ as well as $\boldsymbol{B}$ be one-dimensional, then $\boldsymbol{{\rm\nabla}}{\it\phi}=(\partial {\it\psi}/\partial x,c_{1},c_{2})$ , where ${\it\psi}(x,t)$ is arbitrary and $c_{1}$ and $c_{2}$ are at most functions of time. Hence

Here $c_{1}$ and $c_{2}$ , with dimensions of magnetic flux, are actually fluxes that may be imposed (in the $z$ and $y$ directions, respectively!) external to the slab $|x|<{\rm\pi}L$ in which the plasma is located. We shall suppose that these external fluxes are zero for all $t$ , i.e.  $c_{1}=c_{2}=0$ . They cannot therefore contribute to gauge transformation. With this understanding, the helicity density, like the total helicity, is in fact gauge-invariant.

It will be convenient to use non-dimensional variables. Defining

equations (2.11) and (2.8) become (dropping the hats)

where the dimensionless parameters ${\it\epsilon}$ and ${\it\kappa}$ (both assumed small) are given by

Here, ${\it\epsilon}$ is like a Reynolds number based on the velocity scale $B_{0}^{2}L/{\it\mu}$ , and is small by virtue of the smallness of ${\it\rho}_{0}$ ; and ${\it\kappa}$ is small by virtue of the smallness of ${\it\eta}$ . For this combination of reasons, we must suppose that the magnetic Prandtl number ${\it\mu}/{\it\rho}_{0}{\it\eta}=({\it\epsilon}{\it\kappa})^{-1}$ is large, not unreasonable in a very low-density, high-conductivity plasma.

3. Two-stage process

We may ensure that the initial helicity is non-zero by adopting initial conditions of the form

The coefficients ${\it\beta}$ and ${\it\gamma}$ are arbitrary; the only important thing is that they are unequal, and, for the sake of illustration, we shall adopt values ${\it\beta}=2,~{\it\gamma}=3$ . The variable wavenumber $m(x)$ must be chosen in such a way that the conditions $b_{y0}^{\prime }(\pm {\rm\pi})=b_{z0}^{\prime }(\pm {\rm\pi})=0$ are satisfied. A simple choice is

where $k$ is constant. The field (3.1) provides an initial non-uniform magnetic pressure distribution,

It is the gradient of this magnetic pressure that drives the subsequent flow. Note that $p_{M0}(x)$ nowhere vanishes, i.e. there are no null points of $\boldsymbol{B}$ . In this respect, the situation differs from that considered by Bajer & Moffatt (Reference Bajer and Moffatt2013), wherein the null points played a crucial role.

Obviously $b_{y0}(-x)=-b_{y0}(x)$ and $b_{z0}(-x)=b_{z0}(x)$ , and, with the further initial conditions

and boundary conditions

these symmetries are respected for $t>0$ , as are the symmetries

With the choice $k=0.1$ (again arbitrary), the initial field components $b_{y0}(x)$ and $b_{z0}(x)$ are as shown in figure 1. The corresponding fluxes are then

and these values are conserved. Adopting the suitably small parameter values ${\it\epsilon}=0.001$ and ${\it\kappa}=0.001$ , (2.23) and (2.24) may be integrated numericallyFootnote ³ . Figure 2 shows the manner in which the initial magnetic energy distribution $M(x,t)~(=p_{M}(x,t))$ relaxes to a near-uniform (i.e. nearly force-free) state. Figure 2(a) shows that, by time $t=1$ , the energy density (and so the magnetic pressure) is nearly uniform in $x$ , and remains so for later times. It does not however become exactly uniform, as indicated by figure 2(b), which shows $M(x,50)$ expanded in the vertical by a factor of about $100$ . Figure 3 shows the relatively rapid initial decay of the mean magnetic energy $M(t)$ up to about $t=0.2$ (with $M(0)=3.396$ ), and the subsequent relatively slow decay driven by magnetic resistivity (which forces a continuing weak $x$ dependence of magnetic energy density and pressure).

Figure 1.

Initial field components in the interval $(-{\rm\pi},{\rm\pi})$ , defined by (3.1) and (3.2), with $k=0.1$ .

Figure 2.

(a) Magnetic energy $M(x,t)~(=p_{M}(x,t))$ for $t=0,0.2,1$ and 50. (b) Expanded view of $M(x,t)$ at time $t=50$ ; the ordinate here runs just from 1.86 to 1.89.

Figure 3.

Relaxation of the mean magnetic energy $M(t)$ for $0<t<1$ ; $M(0)=3.396$ . Note the initial rapid relaxation followed by relatively slow decay for $t\gtrsim 0.2$ .

As previously noted, the lines of force are rectilinear and lie in planes $x=\text{const}$ . At time $t$ , as $x$ increases from $-{\rm\pi}$ to ${\rm\pi}$ , the line of force through the point $(x,0,0)$ sweeps out a ruled surface $S[t]$ with parametric representation $[x,sb_{y}(x,t),sb_{z}(x,t)]$ . This surface is shown in figure 4 for the range $\{-{\rm\pi}<x<{\rm\pi},-1<s<1\}$ at the initial instant $t=0$ and at time $t=50$ when the field is well relaxed; the helical character of the field structure is very evident from this representation.

Figure 4.

Ruled surface $S\;[t]$ containing the $\boldsymbol{B}$ -lines through the points $(x,0,0)$ for $-{\rm\pi}<x<{\rm\pi}$ : (a) non-equilibrium initial field; (b) nearly force-free relaxed field at time $t=50$ .

Our interest is primarily focused on the behaviour of the helicity during the relaxation process. The helicity density is shown in figure 5(a) for times $t=0,~0.5$ and $10$ . Up to about time $t=0.5$ , there is merely a redistribution of lines of force, with conservation of mean helicity ${\mathcal{H}}(t)={\mathcal{H}}_{V}(t)/2{\rm\pi}$ (actually about a 1 % decrease due to diffusive decay). For greater values of $t$ , this diffusive decay continues (figure 5 b), with no sign of any change of behaviour as $t$ increases through values $O(1)$ . This is entirely consistent with conservation of global helicity in the non-diffusive limit ${\it\kappa}=0$ . However, the non-uniformity of ${\mathcal{H}}(x,t)$ remains marked throughout, as expected given that, as shown above, ${\mathcal{H}}(x,t)/{\it\rho}$ is a Lagrangian invariant when ${\it\kappa}=0$ . The contrasting behaviour of $M(t)/M(0)$ (figure 3) and ${\mathcal{H}}(t)/{\mathcal{H}}(0)$ (figure 5 b) is to be noted.

Figure 5.

(a) Helicity density ${\mathcal{H}}(x,t)$ at times $t=0$ (solid), $0.5$ (dashed) and $10$ (dotted). (b) Mean helicity ${\mathcal{H}}(t)~(={\mathcal{H}}_{V}(t)/2{\rm\pi})$ for $0<t<1$ ; ${\mathcal{H}}(0)=2.367$ . The ordinate runs from 0.95 to 1.005, as in figure 3. There is a redistribution of helicity density but mean helicity is almost constant.

Figure 6.

The $x$ dependence of $j_{y}/b_{y}$ (solid) and $j_{z}/b_{z}$ (dashed) at times $t=0$ , 0.1 and 1. (The vertical segments on these curves appear at points where the ratios are not well defined other than by continuity.) By time $t=1$ , the two curves are almost indistinguishable, as implied by (3.8).

We may view this differently in terms of an ${\it\alpha}$ -effect. After the early stage of relaxation when the magnetic pressure has become approximately uniform, the field is nearly force-free, so that at leading order $\boldsymbol{j}={\it\alpha}(x,t)\boldsymbol{B}$ , i.e.

Figure 6 shows curves for $j_{y}/b_{y}$ and $j_{z}/b_{z}$ for $t=0,~0.1$ and $1$ . There is indeed a convergence of the two curves as $t$ increases, and by time $t=1$ they are almost indistinguishable. From this point on, the single curve evidently gives an excellent approximation for the function ${\it\alpha}(x,t)$ . This is obviously non-uniform in $x$ , providing a clear signal that the Taylor state (1.1) is not attained, and therefore that global helicity is certainly not the only relevant invariant during the initial rapid relaxation process.

During this initial phase, there is a transfer of energy from the magnetic field to the velocity field, i.e. to kinetic energy $K(t)$ defined by

Figure 7(a) shows this kinetic energy for $0<t<0.5$ . It rises initially very rapidly to a maximum, $u$ being subsequently determined in quasi-static manner. The kinetic energy $K(t)$ falls to a low value ( $K\lesssim 10^{-3}$ for $t>0.5$ ), a further indication of the nearly force-free state. Figure 7(b) shows $K(t)$ for $0<t<0.01$ , showing how this kinetic energy grows initially on an $O({\it\epsilon})$ time scale.

Figure 7.

Kinetic energy $K(t)$ in (3.9): (a) for $0<t<0.5$ , $K(t)$ rises rapidly from zero to a maximum of 0.393 at $t=0.004$ , and then decays monotonically; (b) for $0<t<0.01$ , showing the initial rise on an $O({\it\epsilon})$ time scale.

We may note that, if ${\it\epsilon}=0$ , then the development is truly quasi-static as far as the velocity is concerned. In this situation, a single integration of (2.23) using the boundary conditions (3.5) gives

where $b^{2}=b_{y}^{2}+b_{y}^{2}$ , and the angular brackets indicate an average over the range $(-{\rm\pi},{\rm\pi})$ . This gives an initial kinetic energy (with ${\it\rho}_{0}=1$ )

which may be evaluated numerically using (3.3). The result is $K(0)=0.416$ , marginally greater than the maximum of $K(t)$ shown in figure 7 (actually, with ${\it\epsilon}=0.001$ ,  $K_{max}(t)=0.393$ at $t=0.004$ ). This is consistent with the establishment of quasi-static evolution in a time $t=O({\it\epsilon})$ .

Note that if we return to dimensional variables via (2.22), the dimensional magnetic and kinetic energies are

so that

and this ratio is certainly small. Nevertheless, the initial drop in $M(t)$ evident in figure 2(c) is due to conversion $M(t)\rightarrow K(t)$ and immediate dissipation by viscosity.

Finally, we may consider the evolution of the density field ${\it\rho}(x,t)$ , bearing in mind that, in the zero-resistivity limit ( ${\it\kappa}=0$ ), $b_{y}/{\it\rho}$ and $b_{z}/{\it\rho}$ are both Lagrangian invariants so that evolution of $b_{y}$ and $b_{z}$ is necessarily accompanied by corresponding evolution of ${\it\rho}$ . This correspondence is of course broken when ${\it\kappa}\neq 0$ . Figure 8 shows the density field at times $t=0,~1$ and $50$ . During the early relaxation phase, the density field, initially uniform, is significantly perturbed, as expected. The evolution continues during the subsequent slow diffusive phase, due to the cumulative effect of the non-vanishing velocity field (as similarly found by Bajer & Moffatt (Reference Bajer and Moffatt2013)), and persistent convergence towards the four minima of magnetic pressure (see figure 2 a) causes these maxima of density to become increasingly pronounced while the magnetic field slowly decays.

Figure 8.

Density field ${\it\rho}(x,t)$ at times $t=0$ (solid), $t=1$ (dashed) and $t=50$ (dotted). Note the development of increasingly sharp maxima as $t$ increases.

In a fully three-dimensional situation, $\boldsymbol{B}$ and ${\it\rho}$ are still strongly coupled in the non-diffusive limit through the equation $\text{D}(\boldsymbol{B}/{\it\rho})/\text{D}t=(\boldsymbol{B}/{\it\rho})\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}$ , so a similar development of density variations is to be expected, although perhaps not with such strong peaks. Previous computations have mainly focused on the evolution of the magnetic field, ignoring the associated evolution of the density field that must clearly be significant in any low- ${\it\beta}$ plasma.

Figure 9.

Plots of $\langle {\it\alpha}^{2}\rangle$ and $\langle {\it\alpha}\rangle$ as functions of time for $0<t<30$ .

4. Energy and helicity decay during the slow diffusion stage

During the slow diffusion stage, although $b_{y}$ and $b_{z}$ are still strongly non-uniform in $x$ , $b^{2}=b_{y}^{2}+b_{z}^{2}$ is approximately uniform, and $\boldsymbol{j}\approx {\it\alpha}(x,t)\boldsymbol{B}$ . The exact equation for $b^{2}$ may be readily found from (2.24) in the form

Here, $\partial u/\partial x$ is now given by the quasi-static relation (3.10), and it is the first term on the right that is responsible for keeping $b^{2}$ approximately uniform. (Note that $b^{2}$ would not remain uniform under purely diffusive decay, without the all-important smoothing effect of magnetic pressure.) Here, using $\boldsymbol{j}={\it\alpha}(x,t)\boldsymbol{B}$ , the diffusive term becomes, at leading order,

Now, we may average (4.1) over the range $(-{\rm\pi},{\rm\pi})$ using $u(\pm {\rm\pi},t)=0$ , giving, again at leading order using the approximate uniformity of $b^{2}$ ,

where $b_{0}^{2}$ is the value of $\langle b^{2}\rangle$ at some nominal instant $t=t_{d}\sim 1$ by which time the slow diffusive stage may be deemed to be established. The function $\langle {\it\alpha}^{2}\rangle (t)=\langle (\,\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{b}/b^{2})^{2}\rangle$ is shown in figure 9. It varies by only $\pm 1\,\%$ from the value 5.7 over the (diffusive) range $t_{d}<t<30$ , so that (4.3) indicates mild exponential decrease of energy over this range:

The decay of mean helicity ${\mathcal{H}}(t)~(=(2{\rm\pi})^{-1}{\mathcal{H}}_{V}(t))$ may be calculated to the same modest degree of accuracy. We have

exploiting again the near-uniformity of $b^{2}$ . The function $\langle {\it\alpha}\rangle$ is also shown in figure 9; it is remarkably constant near the value 1.98, over the whole range $0<t<30$ . Adopting this value, and noting that ${\mathcal{H}}(t)$ must tend to zero as $t\rightarrow \infty$ , (4.5) integrates to give

at least for $t_{d}\lesssim t\lesssim 30$ . What is important here is that both energy and helicity decrease at the same rate, the characteristic decay time for both being $t_{d}\sim (2{\it\kappa}\langle {\it\alpha}^{2}\rangle )^{-1}$ . Clearly, the ratio ${\mathcal{H}}(t)/\langle b^{2}\rangle$ remains constant throughout the diffusive stage.

We note however that this behaviour cannot continue for ever, since ultimately $b_{y}(x,t)$ must decay to zero, whereas $b_{z}(x,t)$ must level off at the level ${\it\Phi}_{z}/2{\rm\pi}\approx 0.5491$ determined by the conserved flux. This levelling off occurs when $M(t)$ has fallen to such a low level that the nonlinear convection terms of (2.24) become negligible compared with the diffusion terms. For the parameter values chosen here, this occurs when $t\sim 250$ , well beyond the range shown in figures 3 and 5(b). For $t\gtrsim 250$ , the field gradient is so weak that the continuing decay is due to ohmic diffusion alone.

5. Lagrangian invariance of $\langle {\it\alpha}\rangle$

The near-constancy of $\langle {\it\alpha}\rangle$ evident in figure 9 contrasts with the behaviour of $\langle {\it\alpha}^{2}\rangle$ and calls for explanation. First, whether the force-free equation $\boldsymbol{j}={\it\alpha}(x,t)\boldsymbol{b}$ is satisfied or not, we may define ${\it\alpha}(x,t)$ (a measure of the rate of rotation of the field vector with increasing $x$ , as shown in figure 4) by the equation

Then we have the following theorem.

6. An example with concentrated initial field near the boundaries

We now briefly consider what happens if the initial field is more concentrated near the boundaries $x=\pm {\rm\pi}$ . We adopt initial conditions (again somewhat arbitrarily)

These field components are shown in figure 10(a). Again, in this case, by symmetry ${\it\Phi}_{y}=0$ , and the conserved flux in the $z$ direction is ${\it\Phi}_{z}=9.0881$ . The initial magnetic pressure distribution is shown by the solid curve of figure 10(b). Note how this distribution is quite concentrated near the boundaries; there are two minima at $x=\pm 1.5$ , and a very flat weak maximum (not visible in this plot) at $x=0$ . The pressure maxima near the boundaries push in towards the centre, as evidenced by the dashed curve (for time $t=0.01$ ). By time $t=0.2$ (dotted curve), the magnetic pressure is close to its ultimate uniform value (indicated by the thin red line). Figure 10(c,d) shows the ruled surface representation of the field; just as in figure 4, the field lines are the intersections of these surfaces with planes $x=\text{const}$ .

Figure 10.

(a) Initial field components $b_{y0}$ (solid) and $b_{z0}$ (dashed), as defined by (6.1). (b) Associated magnetic pressure distribution at times $t=0$ (solid), 0.01 (dashed), 0.2 (dotted) and 0.6 (faint). (c) Three-dimensional representation of field at $t=0$ , as in figure 4. (d) Same as (c) at $t=5$ . (e) Density ${\it\rho}(x,t)$ and (f) helicity density ${\mathcal{H}}(x,t)$ at times $t=0$ (solid), 0.2 (dashed) and 3 (dotted). (g) Mean magnetic energy $M(t)$ and mean helicity ${\mathcal{H}}(t)$ , both relative to their initial values, for $0<t<0.6$ . (h) Plots of $10\langle {\it\alpha}(t)\rangle$ and $\langle {\it\alpha}^{2}(t)\rangle$ for $0<t<5$ .

Figure 10(e,f) shows the density and helicity fields at times $t=0$ (solid), $t=0.2$ (dashed) and $t=3$ (dotted). Peaks of density form near $x=\pm 0.4$ , and there is also a build-up of density at the boundaries due to the local magnetic pressure gradient. As regards the helicity density, there is a corresponding redistribution (as implied by (2.18)), but it remains distinctly non-uniform after the initial relaxation is complete. This initial relaxation is evident in the lower curve of figure 10(g), which shows that the mean magnetic energy has in fact relaxed by time $t\approx 0.2$ . Here, $M(0)=11.943$ and we may note that the asymptotic mean magnetic energy as $t\rightarrow \infty$ is $M(\infty )=1.0461$ , so that $M(\infty )/M(0)=0.0876$ . The upper curve of figure 10(g) indicates the very nearly constant value of mean helicity right through the initial relaxation phase and well beyond. This behaviour is much the same as found in the previous sections (figures 3 and 5 b).

Finally, figure 10(h) shows the time dependence of $10\langle {\it\alpha}\rangle$ and $\langle {\it\alpha}^{2}\rangle$ , where ${\it\alpha}$ is again defined as $\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{b}/b^{2}$ . Here, the anomalous behaviour of $\langle {\it\alpha}\rangle$ for $t\lesssim 0.1$ must be attributed to diffusion near the boundaries where $\text{d}^{2}b_{y0}/\text{d}x^{2}$ is largeFootnote ⁴ . This curvature actually has a maximum value of approximately $496$ at $x={\rm\pi}$ , so that diffusion is immediately (and permanently) significant in this neighbourhood, and similarly near $x=-{\rm\pi}$ .

All in all, although the initial condition (6.1) is very different in this example from that in § 3 (3.1), the gross features of the behaviour are very similar: magnetic pressure drives a flow that tends to make this pressure uniform, thus establishing a nearly force-free state, energy being dissipated indirectly by viscosity; helicity density is redistributed, the mean helicity being approximately constant; and density peaks develop near minima of magnetic pressure, a process that persists during the slow diffusive stage.