## Abstract

We consider the implosion of a hollow cylinder of ideal gas with non-zero electrical resistivity. It is shown that there exist self-similar solutions that collapse in a finite time for a range of power-law dependences of the resistivity on the plasma temperature, *η*∼*T*^{δ}. In contrast to the earlier work with zero resistivity, all field variables are finite up to the instant of collapse and the compression is homogeneous. A solution in closed form is found for constant diffusivity, *δ*=0. It is shown that *δ*→0 is a singular limit with the density and sound speed adjusting over a layer of thickness |*δ*|^{1/2} at the inner boundary.

## 1. Introduction

The confinement of a cylindrical column of plasma by means of an azimuthal magnetic field was originally studied with a view to fusion applications. However, this static configuration, known as the ‘*Z*-pinch’, was found to be too unstable for the prolonged confinement necessary for sustained fusion. Interest in *Z*-pinch devices has recently been rekindled as collapsing hollow cylinders are found to emit powerful bursts of X-rays (Ryutov *et al*. 2000).

In an earlier paper, we examined the radial collapse of hollow cylinders of an ideal gas (Lock & Mestel 2008). It was shown that in the absence of resistivity, the magnetogasdynamic (MGD) equations permitted self-similar solutions. However, these solutions are afflicted with singular behaviour on the boundaries. It was speculated that this might be due to the absence of dissipative terms in the governing equations. In this paper, we therefore include non-zero electrical resistance in the problem, which alters the magnetic induction and energy equations. It will be shown that annular self-similar solutions are still possible, even when the resistivity has a power-law dependence on the plasma temperature. Furthermore, these solutions are finite in all field variables up to collapse.

As the geometry is cylindrically symmetric, the density, *ρ*; the velocity, *u*; the sound speed, *c*; the magnetic field, *h*; and the specific internal energy, *e*, are assumed to depend only on the cylindrical radius *r* and time *t*. The flow is wholly radial and the magnetic field is azimuthal. Thus, *Z*-dependent instabilities are not considered here.

Bud'ko & Liberman (1989) studied the problem of cylindrically symmetric solutions of the resistive MGD equations, but restricted their analysis to a Spitzer (2006) behaviour (*T*^{−3/2}, where *T* is temperature) for the plasma resistivity. They considered subsonic motion only, neglecting the inertial terms in the momentum equation. In this paper, we include inertia in our analysis and allow the resistivity to have a more general power-law dependence on temperature, *η*∼*T*^{δ}. We are particularly interested in hollow annular self-similar solutions in , satisfying free-surface boundary conditions at *r*=*r*_{in} and *r*_{out}.

The pressure, *p*, of an ideal gas is given by *p*=*ρc*^{2}/*γ*, where *γ* is the ratio of specific heats. The governing equations for cylindrical MGD flow of an ideal gas, including resistivity, are(1.1)(1.2)(1.3)and(1.4)where *η*(*r*, *t*) is the magnetic diffusivity. With the inclusion of Joule heating in (1.3), the flow is no longer adiabatic.

For an ideal gas, the temperature *T*∝*e*∝*c*^{2} and so (1.3) can be given as(1.5)The magnetic Reynolds number is essentially the ratio of the first of the terms in this equation to the second(1.6)where *L* is an appropriate length scale and *U*_{0} is a representative fluid speed. Ideal MGDs clearly correspond to the case of infinite .

We seek compressing solutions to these equations in the annulusIn §2, we define self-similar imploding solutions and develop the resulting similarity ordinary differential equations (ODEs). In §3, we discuss some properties of these ODEs and their implications for the possibility of annular solutions. Unlike in the ideal case (Lock & Mestel 2008), it is possible to construct similarity solutions with homogeneous compression (*u*∼*r*/*t*). Furthermore, all field variables are finite at the boundaries. We find annular solutions for constant resistivity (*δ*=0) in closed form and numerical solutions for more general *δ*. It is shown that the limit *δ*→0 is singular, with the density and sound speed adjusting across a layer of thickness *O*(*δ*^{1/2}) at the inner boundary.

## 2. Self-similar form of the non-ideal equations

We investigate self-similar imploding solutions to equations (1.1)–(1.5) in terms of the similarity variable(2.1)for some positive parameter *ν*. Time is negative and the entire domain collapses onto the axis *r*=0 at *t*=0. In terms of *X*, the solution domain is *X*_{in}<*X*<*X*_{out}, whereIn terms of a second parameter, *λ*, the ideal MGD similarity ansatz used for perfectly conducting gas is (ter Vehn & Schalk 1982; Lock & Mestel 2008)(2.2)Introduction of a finite diffusivity, *η*(*r*, *t*), constrains the similarity ansatz. The magnetic diffusivity of a fluid is usually a function of temperature, generally decreasing with increasing temperature. For instance, in a classical plasma, theory indicates that *η*∼*T*^{−3/2} (Spitzer 2006), but we consider a more widely applicable power law. Since *T*∝*c*^{2}, we assume that(2.3)where *D* is a prescribed constant and *δ* is a parameter. Physically, we expect *δ*≤0 so that the resistance decreases with temperature. The *O*(1)-factor *ν*^{−2δ} and some constant reference temperature are included in *D*. The similarity ansatz (2.2) then requires(2.4)so that 1<*ν*≤2. In the case *δ*=0, we have the standard constant diffusivity variable *r*/(−*t*)^{1/2}, while as *δ*→−∞, *η*→0 and we have the advective behaviour *r*/(−*t*). Substitution of (2.2) into the governing equations leads to a system of ODEs for the similarity variables,(2.5)(2.6)(2.7)and(2.8)A prime refers to differentiation with respect to *X*. The boxed terms here are those that follow from the introduction of a finite resistivity. They disappear when *D*=0, and these equations reduce to the ideal MGD similarity equations of Liberman & Velikovich (1986) and Lock & Mestel (2008).

If the physics dictates the value of *δ*, e.g. constant resistivity, the constraint (2.4) reduces the number of similarity parameters in the problem to one. However, an additional constant *D* has been introduced into the problem, which is related to the magnetic Reynolds number(2.9)The similarity equations for non-ideal MGDs differ in three important respects from the equivalent system for ideal MGDs (Liberman & Velikovich 1986; Lock & Mestel 2008). Firstly, in the ideal case, the ODE system can be made autonomous by taking ln *X* as the independent variable. This cannot be achieved here since it would require *δ*=1 and *ν*=0. Secondly, the dimension of the ideal ODE system could be reduced by one via the introduction of the dependent variable , related to the square of the Alfvén speed. Finally, the second derivative in the diffusive term in the magnetic evolution (2.8) increases the order of the system by one. The system described by equations (2.5)–(2.8) is equivalent to a system of five first-order ODEs, whereas in the ideal case the system consisted of three first-order equations (Liberman & Velikovich 1986; Lock & Mestel 2008).

In order to solve the system of ODEs numerically, the system must be manipulated to provide an explicit ODE for *U*′. Eliminating *G*′ and *V*′ from (2.5) using (2.6) and (2.7) gives(2.10)The system has a singular curve *V*=(*U*−1)^{2}, which separates regions in causal contact.

## 3. Annular similarity solutions

Imploding solutions of the above equations in a finite annulus must satisfy a kinematic condition at each boundary, *X*=*X*_{in} and *X*_{out}. Requiring the material derivative of *X* to vanish gives the simple constraints . In the perfectly conducting case, this requires at least one of the dependent variables to be infinite on the boundaries (Lock & Mestel 2008). It is interesting to consider whether the introduction of a finite diffusivity into the equations changes this. Assume that at *X*=*X*_{b} when *U*(*X*_{b})=1, the other variables take *finite* values. Then, equation (2.6) indicates that *U*′(*X*_{b}) must satisfy the condition(3.1)This implies that the sign of *U*′ for a solution that is finite in all the dependent variables as *U*→1 is fixed by the similarity parameters *λ* and *ν* and is independent of *X*_{b}. It follows that a non-constant (*U*≢1) solution for which *U*=1 at both boundaries can be finite in all variables at most one of these endpoints, since such a solution requires *U*′ to have opposite signs at each boundary in contradiction to (3.1). However, (3.1) suggests the interesting possibility that if(3.2)a solution for which *U*≡1 identically throughout the domain is possible. Such a solution has uniform compression. In gasdynamics (*h*=0), a solution with this property can be constructed, but in ideal MGDs (*h*≠0,*η*=0) it is prohibited (Kidder 1976; Lock & Mestel 2008).

If *U*≡1, the system simplifies somewhat, and (2.5) becomes(3.3)while (2.7) indicates(3.4)and (2.8) simplifies to(3.5)It should be remembered that *δ* is related to *ν* by (2.4), and thus to *λ* by (3.2). For physical reasons, we are only interested in *δ*≤0 which, by (2.4), implies that 1<*ν*≤2. In fact the constraints on *ν* are even tighter. Equation (3.4) requires that *ν*>*γ* if *G*(*X*) is to be positive and so(3.6)defines the allowable range of values for *ν*. The case of constant diffusivity *δ*=0 corresponds to *ν*=2 and *λ*=−3. We first consider possible solutions for this case.

It should be noted, however, that *δ*→0 is a singular limit. When *δ*≠0, equation (2.4) relates *H*″, *H*′, *H*, *V* and *V*′. Eliminating *G* from (3.3) and (3.4) leads to another equation of this type, and eliminating *V* from these two equations would lead to a third-order ODE in *H*(*X*), requiring a total of three boundary conditions, perhaps the values of *H*(*X*_{in}), *H*(*X*_{out}) and *G*(*X*_{in}). By contrast, when *δ*=0, (3.5) gives rise to a second-order ODE in *H*(*X*), while *G* and *V* then follow uniquely from (3.3) and (3.4). When there is no current inside the annulus so that *H*(*X*_{in})=0, we find that necessarily *G*(*X*_{in})=0 as *δ*→0, and we therefore anticipate a thin layer near the inner boundary in that limit.

### (a) Constant diffusivity (*δ*=0)

When *δ*=0, equation (3.5) simplifies to(3.7)which is the modified Bessel equation of order one. If there is no net current inside the annulus, then we must impose *H*(*X*_{in})=0 and so the magnetic field profile takes the form(3.8)where *I*_{1} and *K*_{1} are modified Bessel functions of order one, and . *G*(*X*) and *V*(*X*) can then be found from (3.4) and (3.3). From (3.4) with *γ*=5/3, the self-similar pressure *P*(*X*)=*G*(*X*)*V*(*X*) can be calculated(3.9)so, from (3.8),(3.10)This demonstrates that *P*(*X*) is finite at the inner boundary. *G*(*X*) then follows from (3.3):(3.11)Substituting (3.8) and (3.10) into (3.11) gives finally(3.12)This shows that *G*(*X*_{in})=0 and consequently, from (3.10), that *V*(*X*) is singular at *X*_{in}, since *P*(*X*_{in}) is finite. This solution is depicted in figure 1.

It is particularly interesting to consider the *D*→0 limit of this system of equations. In this case, a skin-layer structure develops at the outer boundary. Superficially, the *D*→0 limit of the system should correspond to ideal MGDs. However, a non-trivial (*ν*≠1) constant *U*≡1 solution of the ideal MGD system is not possible and so this solution does not naturally tend to a continuous solution of the ideal MGD equations as *D*→0 (Lock & Mestel 2008). As *D*→0, terms such as *DH*′^{2} and *DH*″ remain important in thin regions and so equations (3.3)–(3.5) do not simply reduce to the ideal limit unless discontinuities are permitted.

### (b) *U*≡1 solutions with *δ*≠0

When *δ*≠0, the system described by (3.3)–(3.5) can only be solved numerically. To do this, the equations must be massaged into a form amenable to computation. Eliminating *G* from (3.3) and (3.4) leads to an ODE for *V*(*X*) and *H*(*X*), which is first order in *V* and second order in *H*. Then, *H*″ can be eliminated using (3.5), which we reproduce here for the sake of clarity(3.13)giving the following equation for *V*:(3.14)The problem is now easy to represent as three first-order equations in the variables *H*, *H*′ and *V* and requires three boundary conditions, for example, on *H*(*X*_{in}), *H*(*X*_{out}) and *V*(*X*_{in}). The system is defined by the two physical parameters *δ* and *D* with *ν* and *λ* given by (2.4) and (3.2).

We might expect that in the limit of *δ*→0_{−} numerical solutions would, for appropriate initial (boundary) conditions, tend to the analytic solution developed in §3*a*. However, this is a singular limit, as the system with *δ*=0 is only second order. When *H*(*X*_{in})=0, the equations for *δ*=0 require *G*(*X*_{in})=0. We therefore expect a thin region near the inner boundary, across which *G* will change rapidly. This region could be investigated using asymptotic matching techniques, but it is simpler to proceed directly from (3.14).

As *δ*→0, we have *ν*→2 and *H* and *H*′ are given by (3.8). Away from *X*=*X*_{in}, the derivative *V*′ can be neglected and *V* can be found. Near *X*=*X*_{in}, however, *H* approaches zero and *V*∼(*X*−*X*_{in})^{−1}. In this region, writing *H*=*Ax*, where *x*=*X*−*X*_{in}, (3.14) becomes(3.15)which has the solution(3.16)where(3.17)For 1≫*y*≫|*δ*|^{1/2}, the boundary condition loses significance and the integral can be approximated, giving(3.18)The density can be recovered from (3.4), which essentially states that the pressure is constant across the layer(3.19)so that(3.20)

We now compare numerical solutions of the full *U*≡1 system with the asymptotic results. We take *X*_{in}=1, *X*_{out}=5, *H*(*X*_{in})=0 and *H*(*X*_{out})=1. In figure 2, we show *V* for solutions of the full system in which *V*_{in}, *H*(*X*_{out}) and *D* are held constant for varying *δ*. Also shown in this figure are solutions of (3.14) with *H* taken from the *δ*=0 solution described earlier. As |*δ*| decreases, these curves converge because *H* is insensitive to the value of *δ* when |*δ*| is small. This figure also demonstrates the expected thinning of the small *x* layer as |*δ*| decreases.

For *δ*=−0.01 and *D*=1, figure 3 compares *V* for the full problem with the asymptotic form (3.16), which assumes *H*=*Ax*. As expected, for small *x* these agree well, but they diverge as *x* increases because *H*=*Ax* ceases to be valid. Also shown is the *δ*=0 solution, which is singular at *x*=0, and the approximation to (3.16) given by (3.18). These agree with the full and asymptotic solutions, respectively, at large *x*.

In figure 4, *V*_{in} is varied for fixed values of *δ* and *D*. It is clear that the inner boundary condition has little effect outside the layer where the solution for *δ*=0 is recovered. These results indicate that the system given by (3.13) and (3.14) does reproduce the *δ*=0 behaviour analysed in §3*a*.

Figure 5 shows numerical solutions for the arbitrary value *ν*=1.83 with *V*_{in}=1 for a range of values of the diffusivity constant *D*. The effect of reducing *D* is that *H*(*X*) develops a boundary layer at the outer boundary, as was noted in §3*a* for the *δ*=0 case. However, it should be remembered from (2.3) that the magnetic diffusivity *η*∝*D*/*V*^{|δ|} and hence that as *D*→0, *η* need not be small in regions where *V*≪1 if *δ*<0. Once more, a singular region near *X*=*X*_{in} may develop to match with *V*_{in} as in figure 5.

This also demonstrates that if *δ*<0, it is possible to find solutions of the system for which all of the dependent variables remain finite in the domain of interest.

Ultimately, we are interested in the physical flow predicted by the self-similar solutions developed here. An example of the physical flow determined by these *U*≡1 solutions is depicted in figure 6. This example is based on the solution in figure 5 with *D*=1 and *ν*=1.83, so that and *λ*=−2.83.

The current *I*(*X*, *t*) inside the annulus [*X*_{in}, *X*] is(3.21)The first thing to note is that because *H*(*X*) is bounded, the total current remains finite up to collapse at *t*=0. This was not the case for annular self-similar solutions of the ideal MGD equations, because in those solutions *H* was singular at the outer boundary (Lock & Mestel 2008). Recalling that the physical constraint *δ*<0 requires *ν*>1, we see that as *t*→0, the current becomes infinite if *λ*<−2. If we restrict attention to solutions with all variables regular at the endpoints, then we must have *U*≡1 and the constraint (3.6) implies and(3.22)This is consistent with the behaviour depicted for *h*(*r*, *t*) in the solution shown in figure 6, where the total current increases approximately as *I*∼(−*t*)^{−0.454} as the collapse is approached.

### (c) The relationship to the *U*≡1 gasdynamic solution

A hollow imploding solution, the cylindrical analogue of Kidder's spherical solution (Kidder 1976; ter Vehn & Schalk 1982) of the self-similar gasdynamic equations, was discussed in Lock & Mestel (2008). This solution also had the property that *U*≡1 throughout its domain of definition. The gasdynamic equations are a limiting case of the MGD equations as *h*→0, so we might expect there to be a connection between the two cases.

The *U*≡1 gasdynamic solution requires *ν*=*γ* and *λ*=−(1+*γ*). These values satisfy the constraint (3.2), *λ*+*ν*+1=0, required for self-similarity in resistive MGDs. A further constraint is that *ν*>*γ*, from (3.6). It is apparent that we can approach the gasdynamic solutions if we choose , where *H*_{0} is a characteristic scale of *H*.

It is tempting to consider the limit *D*→0, when the field becomes confined to a surface skin layer, and might be expected to provide a surface magnetic pressure to drive the gasdynamic flow. Such self-similar solutions do not, however, approach the gasdynamic solutions. This is because the Joule heating term *D*(*H*′)^{2} in (3.4) does not tend to zero inside the skin layer as *D*→0. Thus, no constant value of *ν* can satisfy (3.4) both inside and outside the skin layer. It may be possible for this limit to approach a self-similar solution outside the skin layer only, but this is not of independent interest.

## 4. Summary and concluding remarks

In this paper, we have shown that a temperature-dependent electrical conductivity does not inhibit self-similar solutions of the MGD equations. In particular, collapsing solutions in a cylindrical annulus have been found, a geometry that models imploding *Z*-pinch devices in use as powerful X-ray sources. All field variables can be finite on the boundaries and a uniformly compressing flow is possible, unlike in the ideal case (Lock & Mestel 2008).

The assumption that the electrical resistivity has a simple power-law dependence on the local temperature is physically reasonable, since we expect the resistivity of a fluid (plasma) to be a function of temperature, which, for an ideal gas, is proportional to sound speed squared. The well-known Spitzer law for the resistivity of a fully ionized plasma indicates that *η*∼*T*^{−3/2} (Spitzer 2006), for example. This power, *δ*, determines the similarity parameter, *ν*, while *λ* is still free. In contrast to the gasdynamic and ideal MGD equations, the resulting system of ODEs cannot be cast in an autonomous form.

We are interested in similarity solutions that satisfy the kinematic boundary condition (*U*=1) at least twice, which physically corresponds to an annular imploding shell. The ideal MGD solutions allowed this possibility only when at each boundary some of the dependent variables were singular.

Regular behaviour at both boundaries can be obtained only for the solutions with *U*=1 throughout the annular domain. This is reminiscent of the *U*≡1 solution of the gasdynamic equations discussed by Lock & Mestel (2008), which is the cylindrical analogue of Kidder's (1976) solution. This possibility leads to a reduced system of equations and a further constraint on the similarity parameter (*λ*+*ν*+1=0). Thus, the *U*≡1 structure determines the parameters *ν* and *λ* in terms of the diffusivity exponent *δ*.

For constant diffusivity (*δ*=0), the solution has a simple closed form in terms of Bessel functions. If there is no current inside the annulus, so that the magnetic field vanishes on the inner boundary, then the density will also be found to vanish there, which renders the sound speed singular. At the outer boundary, the solution remains finite. With *δ*<0, this difficulty disappears and self-similar solutions can be constructed with all variables finite. However, then the equations can only be solved numerically. It has been shown that *δ*→0 is a singular limit, with a layer of thickness |*δ*|^{1/2} appearing near the inner boundary. When *γ*=5/3, the possible range of *δ* is −1/4<*δ*≤0 (from (2.4)). It is interesting to note that *δ*=−3/2, as is the case for the Spitzer theory, is not in this range. However, for a gas with *γ*<5/4, a self-similar solution with *δ*=−3/2 and all variables finite is possible.

We have not considered the stability of the imploding flows. These are likely to be less serious than for the static configuration aimed at plasma confinement. Neither have we considered the possibility of shock formation in this paper. These are discussed by Lock (2008) and were not found to change the dynamics qualitatively.

## Acknowledgments

© British Crown Copyright 2008/MOD.

## Footnotes

- Received February 6, 2008.
- Accepted April 16, 2008.

- © 2008 The Royal Society