Cylindrical_Liner_MRT_Instability_Coupling_Paper_2015

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ViewOnlineExportCitationRESEARCH ARTICLE| MARCH 24 2015Coupling of sausage, kink, and magneto-Rayleigh-Taylorinstabilities in a cylindrical liner M. R. Weis; P. Zhang; Y. Y. Lau; P. F. Schmit; K. J. Peterson; M. Hess; R. M. Gilgenbach Phys. Plasmas 22, 032706 (2015)https://doi.org/10.1063/1.4915520 CHORUSArticles You May Be Interested InLinear analytical model for magneto-Rayleigh-Taylor and sausage instabilities in a cylindrical linerPhys. Plasmas (February 2023)Seeded and unseeded helical modes in magnetized, non-imploding cylindrical liner-plasmasPhys. Plasmas (October 2016)Technique for fabrication of ultrathin foils in cylindrical geometry for liner-plasma implosion experimentswith sub-megaampere currentsRev. Sci. Instrum. (November 2015)

PHYSICS OF PLASMAS 22, 032706 (2015)

(Received 20 January 2015; accepted 5 March 2015; published online 24 March 2015)

Coupling of sausage, kink, and magneto-Rayleigh-Taylor instabilities in a cylindrical liner

M. R. Weis,1 P. Zhang,1 Y. Y. Lau,1,a) P. F. Schmit,2 K. J. Peterson,2 M. Hess,2 and R. M. Gilgenbach1 1Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109-2104, USA 2Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

This paper analyzes the coupling of magneto-Rayleigh-Taylor (MRT), sausage, and kink modes in an imploding cylindrical liner, using ideal MHD. A uniform axial magnetic field of arbitrary value is included in each region: liner, its interior, and its exterior. The dispersion relation is solved exactly, for arbitrary radial acceleration (-g), axial wavenumber (k), azimuthal mode number (m), liner aspect ratio, and equilibrium quantities in each region. For small k, a positive g (inward radial acceleration in the lab frame) tends to stabilize the sausage mode, but destabilize the kink mode. For large k, a positive g destabilizes both the kink and sausage mode. Using the 1D-HYDRA simu- lation results for an equilibrium model that includes a pre-existing axial magnetic field and a pre- heated fuel, we identify several stages of MRT-sausage-kink mode evolution. We find that the m ¼ 1 kink-MRT mode has a higher growth rate at the initial stage and stagnation stage of the im- plosion, and that the m ¼ 0 sausage-MRT mode dominates at the main part of implosion. This anal- ysis also sheds light on a puzzling feature in Harris’ classic paper of MRT [E. G. Harris, Phys. Fluids 5, 1057 (1962)]. An attempt is made to interpret the persistence of the observed helical struc- tures [Awe et al., Phys. Rev. Lett. 111, 235005 (2013)] in terms of non-axisymmetric eigenmode. VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4915520]

When a strong axial current is present, the dominant instabilities on a cylindrical plasma column are the sausage and kink mode, with azimuthal mode numbers, m ¼ 0 and m ¼ 1, respectively. If the plasma column is in the form of a cylindrical liner of outer radius R and thickness D, the sau- sage and kink mode are still the dominant modes if it is assumed that there is a sufficient internal pressure in the cen- tral region of the liner to prevent any radial acceleration of the liner. However, if this internal pressure is weak, there would be inward acceleration of the liner, and the outer inter- face of the liner would be subjected to the magneto- Rayleigh-Taylor instability (MRT).1-5 If, on the other hand, the internal pressure in the central region is very high, there is a radially outward acceleration and the inner interface of the cylindrical liner would be subjected to MRT. In the rest frame of the interface, the effective gravity, g, is equal to the negative of the radial acceleration. Thus, g > 0 (g < 0) corre- sponds to implosion (stagnation or explosion), and the con- ventional above correspond to g ¼ 0. It is clear that the nature of sausage, kink, MRT, and the coupling among them depends much on the cylindrical geometry, on the magnitude and sign of g, on the aspect ratio, R/D, and on the dominant magnetic field. In this paper, we use the ideal MHD model and present a gen- eral linear stability analysis including the m ¼ 0 MRT- sausage mode and the m ¼ 1 MRT-kink mode, for arbitrary

The coupling of the m ¼ 0 MRT and sausage mode, and the coupling of the m ¼ 1 MRT and kink mode, has received only scant attention in the past.4-7 They have become an im- portant issue in the recent magnetized liner inertial fusion (MagLIF) experiments8-14 on the Z-machine at Sandia National Laboratories. While there have been extensive stud- ies of MRT on the MagLIF liner without an axial magnetic field,9-11 many of these MRT theories were based on a slab geometry which is incapable of describing the conventional sausage mode and kink mode.2-4,15-18 Without an axial mag- netic field, MRT structure is typically oriented along nearly horizontal planes perpendicular to the z-axis with limited or no pitch.12,13 However, with the inclusion of an axial mag- netic field, helical structures were found with significant in- clination during the implosion phase.12,13 In the fully integrated MagLIF experiments (with axial magnetic field and a preheated fuel in the central region inside the liner), possible kink-like perturbations of the plasma column were reported at stagnation.14 These non-axisymmetric MHD activities are yet to be explained.

To keep the problem analytically tractable, we use ideal MHD and apply a linear stability analysis on a sharp bound- ary model (Fig. 1). For the linear stability analysis, we assume (a) that the liner has a uniform and constant density, the density elsewhere is practically zero in comparison, (b) that there is a constant, uniform axial magnetic field in each region, (c) that the azimuthal magnetic field, generated by

value (and sign) of g, for arbitrary aspect ratio R/D (>1), and for general values of the axial magnetic field outside, inside, and within the liner.

a)Corresponding author: [email protected]

1070-664X/2015/22(3)/032706/7/$30.00

I. INTRODUCTION

VC 2015 AIP Publishing LLC

kink mode

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described

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sausage

and

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II. EQUILIBRIUM AND STABILITY

FIG. 1. MHD model for an imploding cylindrical liner. Uniform axial mag- netic field is included in each region: liner (a < r < R), its interior (r < a), and its exterior (r > R), as B02, B03, and B01, respectively.

and azimuthal magnetic fields. The results of the stability analysis for general m and k are presented. Numerical results are presented only for the m ¼ 0 and m ¼ 1 modes.

The model under study is shown in Fig. 1. It consists of three regions, I, II, and III. In each region, we assume that the ﬂuid is perfectly conducting. We solve the ideal MHD equations: qð@=@t þ v (cid:4) rÞv ¼ (cid:2)rp þ J (cid:5) B þ qgr, @q=@t þr (cid:4) ðqvÞ ¼ 0, @B=@t ¼ r (cid:5) ðv (cid:5) BÞ, and r (cid:5) B ¼ l0J in the cylindrical coordinates. Here, q is the mass density, v is the ﬂuid velocity, p is the ﬂuid pressure which is assumed to be isotropic, J is the current density, B is the magnetic field, g is the gravity which is positive when acceleration is radi- ally inward [Fig. 1], r is the unit radial vector, and l0 is the free space permeability.

the axial current (a surface current), exists only in the exte- rior region (r > R), (d) that R and D are constants, and (e) that the effective gravity, g, is uniform and constant. These assumptions may be justified if we pretend that the liner is inward acceleration subjected to an instantaneous, initial (¼ (cid:2)g), so that the MRT mechanism is switched on, but without creating any motion of the liner so that R, D, and the liner density are essentially constants. Such a simplified con- ceptual model enables a close examination of the relation between a purely MRT mode (one without any internal pres- sure to reduce the acceleration), and a purely sausage and kink mode (one with g ¼ 0). In so doing, we are able to resolve a little-noted puzzle in Harris’ classic paper which shows that there is a finite MRT growth rate, c ¼ ﬃﬃﬃﬃﬃﬃﬃﬃg=Rp , for a thin shell even when k ¼ m ¼ 0, where R is the radius of the thin shell and k is the axial wavenumber of the imploding thin liner.4 This is a surprising result because the MRT

In experiments, the axial current has a finite rise time. Quantities such as the outer radius R, and g will then evolve in time. In fact, the effective gravity g even changes sign from implosion to stagnation. To obtain some rough understanding of the relative importance of MRT, sausage, and kink, on a liner like MagLIF, we use the HYDRA simulation code19 to examine the temporal evolution of a cylindrical liner with a realistic current rise profile reaching (cid:3)20 MA in 150 ns, a pre- seeded axial magnetic field of 10 T, and a preheated fuel ((cid:3)250 eV), similar to an ideal version of the fully integrated MagLIF experiments by Gomez et al.14 From 1D HYDRA simulations, we extract the instantaneous “equilibrium” pa- rameters for R, D, g, Bh; B03, etc., and apply these profiles to the linear stability theory of the sharp boundary model. This analysis, reported below, reveals several stages of evolution of sausage, kink, and MRT, from implosion to stagnation. Figure 5 below illustrates the relative importance of the m ¼ 0 and m ¼ 1 modes at the different stages.

where D ¼ R - a is the liner thickness, pI is the equilibrium ﬂuid pressure in region I at the outer liner surface, and pIII is the equilibrium ﬂuid pressure in region III at the inner liner surface. The acceleration, which equals (cid:2)g, may therefore be driven by an arbitrary mix of ﬂuid pressure (pIII; pIÞ or magnetic pressure (B01; B03; BhÞ, as long as the above equi- librium conditions are satisfied. Note that if the internal pres- sure pIII is dominant among all pressures, as in the stagnation stage, then g is negative (acceleration is radially outward) in Eq. (2). Hereafter, we will call the case g ¼ 0 pure kink mode and pure sausage mode, i.e., the total pres- sure exactly balances when there is no acceleration in the laboratory frame. On the other hand, if one square bracket in

where B0 ¼ jB0j is the magnitude of the equilibrium magnetic field B0 and B0h(r) is the azimuthal component of B0. We assume that the effective gravity, g, is a constant, though fully incompressible ﬂow would require a non-uniform g to account for cylindrical convergence and mass conservation consistently. This may be justified if we use 1D HYDRA simulations to pro- vide these equilibrium profiles, which fully take into account compressibility of the ﬂuid as it implodes. Also, the thickness of the shell remains on the order of 100s of micrometers, which is relatively thin so that the effect of non-uniform g remains small. Integrate Eq. (1) across region II to yield

In equilibrium, the magnetic field in the three regions, I (r > R), II (a < r < R), and III (r < a) is assumed to be, respec- tively, B01 ¼ zB01 þ hBhR/r, B02 ¼ zB02, and B03 ¼ zB03, where B01, B02, B03, and Bh are constants. We further assume that the mass density in each region is also constant with the liner density being the dominant, i.e., q01 (cid:6) q02, and q03 (cid:6) q02. For a scenario such as MagLIF, region I is a vacuum field region, region II is the liner region, and region III is the fuel region. The equilibrium pressure profile p0(r) is adjusted so that it satisfies the equilibrium condition,20 for all r,

growth rate is expected to be c ¼ , and for m and k equal to zero, we would have c ¼ 0. The nature of this finite growth rate of Harris and its relation to the m ¼ 0 sau- sage mode will be discussed.

We shall first consider the equilibrium model (Fig. 1), in which the internal pressure in each region is assumed to be adjusted so that g may be assigned an arbitrary constant value (zero, positive, or negative), for a given set of axial

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z Þ1=2g ðk2 h þ k2

01 þ B2 B2 h 2l0

gq02D ¼ pI þ

B2 0h rð Þ rl0

B2 0 rð Þ 2l0

¼ q0 rð Þg;

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(cid:2) pIII þ

2l0

p0 rð Þ þ

@ @r

(2)

(1)

”

;

Þ :

(8)

(3)

c ¼

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3k2aRÞ;

B ¼ b1 þ b2;

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Ax4 (cid:2) Bx2 þ C ¼ 0;

A ¼ ðX1X2 þ 1Þ=ðX2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B2 h ; l0q02DR

pure sausage mode kR (cid:6) 1; R=D (cid:8) 1 ð

Equation (8) may be derived from Eq. (3) in the asymptotic limits shown, after setting g ¼ 0.

Eq. (2) is much larger than the other square bracket, so that jgj is maximized, we call the unstable mode pure MRT or pure RT mode. In general, jgj is between zero and its maximum value; and the resulting instability is somewhere between a pure sausage (or kink) mode and a pure MRT mode.

is plotted by the solid lines in Fig. 2(a) for various aspect ratios, from a thin shell (R/D ¼ 10) to an almost solid cylin- der (R/D ¼ 1.0101). Note that for a thin liner, R/D (cid:8) 1, the pure sausage mode growth rate, c ¼ (cid:2)Im(x), approaches the asymptotic limit for small kR,

For the pure MRT mode, there is no internal pressure, pIII ¼ 0, in Eq. (7). There is then a maximum inward acceler- ation, with a maximum g ¼ gmax ¼ B2 h=2l0q02D according to Eq. (7). The normalized growth rate for this pure MRT mode with m ¼ 0 is given by the dashed curves in Fig. 2(a). Asymptotically, one may show from Eq. (3) that, for a thin liner, R/D (cid:8) 1, and small kR,

We next consider a small signal perturbation of the form, u1ðrÞeixtþimh(cid:2)ikz, on the equilibrium of the 3-region ge- ometry shown in Fig. 1 and assume these perturbations to be incompressible (r (cid:4) v ¼ 0). For the above sharp boundary model, the linearized MHD equations for each region may be distilled into a second order ODE for the perturbed displace- ment of the plasma, the solution of which requires two bound- ary conditions, plus constraint on the perturbed magnetic field: (1) A perfect conductor (ideal MHD) requires that the mag- netic field component normal to the liner surface to be identi- cally zero, which links the perturbation magnetic field and displacement and (2) The continuity of total pressure across each interface. This leads to a dispersion relation of the form,

Equation (9) is identical to Eq. (61) of Harris4 in the limit k ¼ 0, m ¼ 0, thus confirming Harris’ finite MRT growth rate in this limit for a collapsing thin shell. Note from Eqs. (8) and (9) that the pure MRT mode has a growth rate lower than the pure sausage mode in the long axial wavelength limit. Since the value of g ranges between g ¼ 0 and g ¼ gmax, we con- clude that inward acceleration (g > 0) tends to stabilize the long wavelength sausage mode, and this is even more appa- ratios, R/D ¼ 2, and rent lower aspect R/D ¼ 1.0101. This is also true when B0z/Bh ¼ 0.1, as shown in Fig. 2(c). However, if the axial magnetic field is increased to B0z/Bh ¼ 1 (Fig. 2(e)), the pure sausage mode is stable for all R/D, and the inward acceleration destabilizes the pure sau- sage mode for the R/D ¼ 10 case (and only for this case among the values of R/D shown in Fig. 2(e)). Note that the inward acceleration (g > 0) tends to destabilize the short wavelength m ¼ 0,1 modes, when kR > 1. This result is impor- tant when we consider the experiments of Sinars et al.9,10 and simulations and experiments of McBride et al.11 where such modes are observed. They show that while pre-seeded m ¼ 0 (k > 0) modes persist, retaining azimuthal symmetry, unmodi- fied liners show a rapid departure from azimuthal symmetry. One explanation with our theory is while the growth rate for m ¼ 0 (k > 0) remains the largest, m > 0 and k > 0 modes have comparable enough growth rates that, unless preseeded with m ¼ 0, directly compete and destroy the m ¼ 0 symmetry. In fact, to accurately simulate a typical unseeded liner, fully 3D modes (m,k > 0) must be allowed (2D simulations are insuffi- cient), though even this can be challenging11 and is discussed brieﬂy in the conclusion.

jmj, X2 ¼ ^I jmjK0 jmj, In Eqs. (4)-(6), X1 ¼ Kjmj

jmj, Ijmj ¼IjmjðjkjaÞ, Kjmj ¼KjmjðjkjaÞ, ^I jmj X3 ¼ ^I jmj (cid:2) ^K jmjK0 ¼IjmjðjkjRÞ, ^K jmj ¼KjmjðjkjRÞ, b1 ¼2k2V2 02A, b2 ¼ (cid:2)½ðX1=X3Þ , V03 ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jkjg0 þðX2=X3Þjkjg(cid:7), V02 ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p B2 03=l0q02 02=l0q02 ,

jmjjkjRÞð(cid:2)mBh þ kRB01Þ2(cid:7), h þ ð ^K jmj= ^K g0 ¼ g þ ð1=l0q02RÞ ½B2 Im and Km are, respectively, the modified Bessel function of order m of the first and second kind, and a prime denotes dif- ferentiation of these modified Bessel functions with respect to their arguments. Of the four eigenvalues of x in the dispersion relation (3), we shall henceforth consider only the most unsta- ble mode with the largest negative imaginary part of x. This is the exponentially growing mode, the second root is expo- nentially decaying and the 3rd and 4th are purely oscillatory. The numerical values of the growth rate, obtained from Eq. (3), are presented below. They have all been validated with an independent, alternative approach that directly solved the governing differential equation including q03>0.

The difference between a pure sausage mode, and a pure MRT mode, both with m ¼ 0, is shown in Fig. 2(a), where we have set the axial magnetic field equal to zero every- where. We further assume that pI ¼ 0, and Eq. (2) then reads,

For the kink mode (m ¼ 1), an inward acceleration (g > 0) tends to destabilize the kink mode for long axial wavelength (kR(cid:6) 1), regardless of the value of B0z/B0h ¼ 0, 0.1 or 1, as shown Figs. 2(b), 2(d), and 2(f). The curves with

For the pure sausage mode, we set pIII ¼ B2 h=2l0, so that g ¼ 0. The normalized growth rate of this pure sausage mode

III. m 5 0 AND m 5 1 MODE WITH g 5 0 AND NONZERO g

pure MRT mode kR (cid:6) 1; R=D (cid:8) 1 ð

02X2=X3 þ jkjg0Þ þ k2gg0:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B2 h ; 2l0q02DR

jmj (cid:2) ^K jmjI0

jmjÞð(cid:2)k2V2

^I jmj (cid:2) Ijmj

in Fig. 2(a)

03ðIjmj=I0

C ¼ k2V2

02A þ b2Þ

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02ðk2V2

gq02D ¼

B2 h 2l0

þ k2V2

(cid:2) pIII:

jmjI0

c ¼

(7)

(4)

(5)

(6)

(9)

for

Þ :

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g ¼ 0 in Figs. 2(d) and 2(f) show disappearance of m ¼ 1 instability for sufficiently large kR. This only means that the Kruskal-Shafranov criterion for kink mode stabilization is satisfied for sufficiently large kR.20 Note that the inward acceleration (g > 0) tends to destabilize the short wavelength m ¼ 1 mode, when kR > 1.

The pure sausage and pure kink mode assumes PI ¼ PIII so that g ¼ 0. Their normalized growth rates are shown by the solid lines in Fig. 3(a) for m ¼ 0, and in Fig. 3(b) for m ¼ 1. The pure MRT case assumes PI ¼ 0 in Eq. (10), and the nor- malized growth rates are shown by the dotted lines in Fig. 3,

FIG. 2. The normalized growth rate calculated from Eq. (3) for (a), (c), (e), the m ¼ 0 mode, and (b), (d), (f), the m ¼ 1 mode, with B0z/Bh ¼ 0, 0.1, and 1, and R/ D ¼ 1.0101 (almost a solid cylinder), 2, and 10, where B0z ¼ B01 ¼ B02 ¼ B03. Here, g ¼ 0 corresponds to the pure sausage mode (m ¼ 0), or the pure kink mode (m ¼ 1), and g ¼ gmax > 0 corresponds to the pure MRT mode for an imploding liner.

To gain some understanding of the coupling between MRT, sausage and kink mode at the stagnation stage, we now assume that B01 ¼ B02 is so small compared with B03

that we may now set B01 ¼ B02 ¼ 0. We further assume B03/ Bh ¼ 1. The equilibrium condition, Eq. (2), then reads,

gq02D ¼ pI (cid:2) pIII:

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(10)

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the HYDRA 1D simulations using the procedure outlined elsewhere.18 Note from Fig. 4 that during the entire interval of 148 ns, q01 ¼ 0, q03 (cid:6) q02 and that g < 0 only within the last 7 ns. The instantaneous instability growth rates for the m ¼ 0 and the m ¼ 1 mode are compared in Fig. 5(a) as a function of time, for various axial wavelengths k ¼ 2p/k. We interpret Fig. 5(a) as follows, focusing on the k ¼ 1 mm case. For k ¼ 1 mm, Fig. 5(a) shows that there are five (5) dif- ferent stages of MRT-sausage-kink growth for the evolving B01, B02, B03, Bh, g, R, a, q02, and q03 shown in Fig. 4. (i) Initially, for the first 20 ns, the azimuthal magnetic field is small compared with the pre-seeded axial magnetic field of 10 T, both the m ¼ 0 and m ¼ 1 modes are stable. (ii) As the azimuthal magnetic field increases, but is still less than the axial magnetic field, the kink mode (m ¼ 1) becomes unsta- ble but the sausage mode (m ¼ 0) remains stable. This stage is very similar to a tokamak with a safety factor q < 1, but will quickly pass. (iii) As the azimuthal magnetic field is increased further, from about 25 to 55 ns, both the kink and

We next calculate the instantaneous instability growth rate for the m ¼ 0 and the m ¼ 1 modes according to Eq. (3), using the data from 1D HYDRA to obtain the instantaneous equilibrium profiles. The m ¼ (cid:2)1 mode could also be unsta- ble but in general has a smaller growth rate than m ¼ 0, 1 so we focus on the more dangerous growth rate. Figure 4 shows the evolution of B01, B02, B03, Bh, g, R, a, q02, and q03, for a liner geometry and current profile similar to the fully inte- grated MagLIF experiments of Gomez et al. (with pre- magnetization of a 10 T axial magnetic field, and a preheated fuel14). These results are expected to be equally applicable to the experiments of Awe et al.,12,13 except for the decelera- tion phase since the experiments by Gomez et al. included a laser pre-heat. This allows stagnation at smaller convergence ratio, a key feature of MagLIF. Figure 4 was extracted from

where we set pIII ¼ (cid:2)gmaxq02D ¼ B2 h=l0. From Fig. 3, we see that the radially outward acceleration (g < 0) destabilizes only the thinnest liner for the sausage mode, while all thick- nesses are destabilized for the kink mode. Overall, the m ¼ 1 MRT mode has a higher growth rate during this deceleration phase with a highly compressed axial field. This may have implications for MagLIF as we will discuss below.

FIG. 3. The normalized growth rate calculated from Eq. (3) for (a), the m ¼ 0 mode, and (b), the m ¼ 1 mode, with B01 ¼ B02 ¼ 0, and B03/Bh ¼ 1, for R/D ¼ 1.0101 (almost solid cylinder), 2, and 10. Here, g ¼ 0 corresponds to the pure sausage mode (m ¼ 0), or the pure kink mode (m ¼ 1), and g ¼ (cid:2)jgmaxj < 0 corresponds to the pure MRT mode for an exploding liner near its stagnation.

FIG. 4. (a) Evolution of magnetic fields and average liner acceleration, a ¼ (cid:2)g, from 1D HYDRA simulations. (b) Liner trajectory and evolution of t ¼ 0: the fuel and liner density from 1D HYDRA simulations. At B01 ¼ B02 ¼ B03 ¼ 10 T, Bh ¼ 0 T.

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are unstable, but the m ¼ 0 mode becomes dominant. At this point g is large, such that MRT is the dominant driver of instability decreasing the e-folding time substantially. This situation remains for the major part of the implosion, all the way until the fuel region is heated up to such a high pressure that the sign of radial acceleration reverses, and the stagna- tion stage begins. (v) During the deceleration stage (stagna- tion) in the last 7 ns of the simulation, both the m ¼ 0 and m ¼ 1 modes are unstable, but the m ¼ 1, kink-like mode has a higher total amplitude gain than the m ¼ 0 mode during this final stage. Figure 5(b) shows that the kink mode’s am- plitude gain is about three times that of the sausage mode during this final stage if k ¼ 1 mm. Thus, if helical structure has fed-through the liner and seeded the inner surface, this could generate substantial helical growth within the fuel/ liner inner surface during the deceleration phase. In fact, from Fig. 3(b), the pure deceleration-MRT m ¼ 1 mode exhib- its a growth rate with complete lack of dependence on wave- length except for an upper wavenumber cutoff kmax (short wavelength). This would suggest that the dominant kink-like perturbation near stagnation will correspond to the most strongly-seeded kink-like liner deformation during run-in.

sausage mode become unstable, but the kink mode is domi- nant. This is not the case if there is no axial magnetic field. Thus, the axial magnetic field gives a preference to the growth of the kink mode if helical perturbations are present. One might wonder if the subdominance of the m ¼ 0 mode in these early stages has anything to do with the appearance of the helical structures in Awe’s experiments.12,13 Early on, MRT is not important because there is little inward accelera- tion of the liner (g is small). Over the first 60 ns, the maxi- mum number of e-folds is on the order of 1 for axial wavelengths around 100 lm, while wavelengths on the order of 1 mm have undergone around 0.1 e-folds. Because we use a sharp boundary model, the number of e-folds also depends on the density used. Certainly, there will be some ablation and it is difficult to tell whether the early time growth occurs in the ablated plasma or the bulk, above we have used the near-solid density which may underestimate growth. It is also possible these modes collude with electrothermal insta- bility which tends to occur for short wavelengths (<200 lm) at these early times.21,22 (iv) As the azimuthal magnetic field further increases much beyond the axial magnetic field (from the current rise after 55 ns), both the m ¼ 0 and m ¼ 1 mode

This paper concentrates on the cylindrical effects of the stability of a current-carrying liner of various aspect ratios, from a thin liner to a solid cylinder. We focused mainly on the m ¼ 0 and m ¼ 1 modes and on the effect of radial accel- eration on the liner stability. When the radial acceleration is negligible, as is the case during the initial axial current rise, the sausage mode is dominant without an axial magnetic field, but the kink mode is dominant with a pre-seeded axial magnetic field. During the main part of the implosion, the m ¼ 0 mode grows faster than the m ¼ 1 mode. At the stagna- tion stage where the radial acceleration changes sign, the m ¼ 1 MRT-kink mode grows faster than the m ¼ 0 MRT- sausage mode for shorter wavelengths ((cid:3)1 mm) when there is a pre-seeded axial magnetic field.

Finally, we note that the instability growth rates pre- sented above are also in excellent agreement with the bench- mark MRT data in Sinars et al.9,10 Experiments showing the growth of seeded m ¼ 0, kz ¼ 400 lm modes were presented which also compared very favorably with planar growth rates10,18 since kR was large. The m ¼ 0 modes are fairly straightforward to verify via 2D simulations and also in experiments, however 3D perturbations are much more difficult to investigate. As such, growth rates with m 6¼ 0 and k 6¼ 0 could hopefully be used to benchmark 3D simulation codes, of which there are very few tests.

The intricate interplay between the m ¼ 0 and m ¼ 1 modes, which also depends on the magnitude and sign of g, makes the interpretation of the helical structure observed by Awe et al.12,13 and the apparent kink-like activities in Gomez et al.14 difficult. Most questions on them remain unanswered. Among them include the sharpness of helical structures, the underlying reasons for the observed mode numbers (m, k), the role of initial seeding, the origin and maintenance of the helical structures, and their relation (if

FIG. 5. (a) Relative dominance of sausage and kink modes for a MagLIF like implosion. g > 0 up until final 7 ns where it changes sign. Observed ex- perimental axial wavelengths are on the order of 1 mm. (b) Magnification of the last 7 ns, comparing the amplitude gain of the sausage and kink mode as a function of wavelength. Stronger axial fields allow the kink mode to domi- nate over shorter wavelengths.

IV. CONCLUDING REMARKS

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Phys. Plasmas 22, 032706 (2015)

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any) to the kink-like mode that seems to have shown up at the final stage in the liner experiment of Gomez et al.14 Our analysis does show that if helical perturbations are present on the liner surface, the axial magnetic field opens a window for the kink to grow instead of the sausage mode.

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Despite the uncertainties, one puzzle on the experiment of Awe et al.,12,13 namely, why the helical structures opened up (instead of very tightly wound up as the azimuthal mag- netic field increases; see Fig. 4(a)) may be explained in terms of an eigenmode with a specific m and k. The axial wave- number, k, is assumed to be constant. The frames in the experiment of Awe et al.,12,13 occur over a narrow time win- dow so this seems adequate. The pitch angle of the helix, /, of the eigenmode is given by / ¼ tan(cid:2)1ðm=kRÞ (cid:9) m=kR, relating the values of /(t2) and /(t1) at different times t2 and t1: /ðt2Þ ¼ /ðt1Þ (cid:5) Rðt1Þ=Rðt2Þ. For the Z-Machine Shot 2480, the measured (mean) values are /ðt1Þ ¼ 16:48, aðt1Þ ¼ 870 lm, aðt2Þ ¼ 365 lm. We then have, Rðt1Þ ¼ aðt1Þ þD ¼ 870 lm þ 465 lm ¼ 1335 lm, and Rðt2Þ ¼ 365 lm þ465 lm ¼ 830 lm, where we have assumed that the liner thickness D ¼ 465 lm remains unchanged through- out (see Fig. 4(b) and Awe et al.12,13). The predicted helix pitch angle at t2 is then /ðt2Þ ¼ 16:48 (cid:5) 1335=830 ¼ 26:48, which is quite close to the observed mean value of /ðt2Þ ¼ 25:68. Using this technique, the predicted value for /ðt2Þ in Shot 2481 is within the experimental uncertainties of the measured value also. This interpretation of the persist- ence of helical structures was motivated by the density wave theory that also used eigenmodes to explain the persistence of spiral structures in disk galaxies despite strong differential rotation.23 Reproduction of the experimentally observed heli- cal structures from simulations, without artificial seeding, has proved very challenging. 3D MHD simulations have required initial seeding of a helix to reproduce the observed helical structure; it has not simply arisen out of white noise.13 Unfortunately, this seeding has also produced helical structure when no axial magnetic field is present which is not in line with experimental results. The sharpness of the heli- ces and very specific mode numbers (in m and k) that were observed present the biggest challenge in comparing (unseeded) 3D simulations and experiments.

This work was supported by DoE Award Nos. DE- SC0002590 and DE-SC0012328. Matt Weis was supported by the Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

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13T. J. Awe, C. A. Jennings, R. D. McBride, M. E. Cuneo, D. C. Lamppa, M. R. Martin, D. C. Rovang, D. B. Sinars, S. A. Slutz, A. C. Owen, K. Tomlinson, M. R. Gomez, S. B. Hansen, M. C. Herrmann, M. C. Jones, J. L. McKenney, G. K. Robertson, G. A. Rochau, M. E. Savage, D. G. Schroen, and W. A. Stygar, Phys. Plasmas 21, 056303 (2014).

12T. J. Awe, R. D. McBride, C. A. Jennings, D. C. Lamppa, M. R. Martin, D. C. Rovang, S. A. Slutz, M. E. Cuneo, A. C. Owen, D. B. Sinars, K. Tomlinson, M. R. Gomez, S. B. Hansen, M. C. Herrmann, J. L. McKenney, C. Nakhleh, G. K. Robertson, G. A. Rochau, M. E. Savage, D. G. Schroen, and W. A. Stygar, Phys. Rev. Lett. 111, 235005 (2013).

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