mrt-instability-cylindrical-liner

 View Online  Export Citation RESEARCH ARTICLE| MARCH 24 2015 Coupling of sausage, kink, and magneto-Rayleigh-Taylor instabilities 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  CHORUS Articles You May Be Interested In Linear analytical model for magneto-Rayleigh–Taylor and sausage instabilities in a cylindrical liner Phys. Plasmas (February 2023) Seeded and unseeded helical modes in magnetized, non-imploding cylindrical liner-plasmas Phys. Plasmas (October 2016) Technique for fabrication of ultrathin foils in cylindrical geometry for liner-plasma implosion experiments with sub-megaampere currents Rev. Sci. Instrum. (November 2015) 01 September 2025 20:25:40 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.Gilgenbach 1 1 Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109-2104, USA 2 Sandia National Laboratories, Albuquerque, New Mexico 87185, USA (Received 20 January 2015; accepted 5 March 2015; published online 24 March 2015) 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 smallk, a positiveg(inward radial acceleration in the lab frame) tends to stabilize the sausage mode, but destabilize the kink mode. For largek, a positivegdestabilizes 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 m¼1 kink-MRT mode has a higher growth rate at the initial stage and stagnation stage of the im- plosion, and that them¼0 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. Fluids5, 1057 (1962)]. An attempt is made to interpret the persistence of the observed helical struc- tures [Aweet al., Phys. Rev. Lett.111, 235005 (2013)] in terms of non-axisymmetric eigenmode. V C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4915520] I. INTRODUCTION 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,m¼0 and m¼1, respectively. If the plasma column is in the form of a cylindrical liner of outer radius Rand 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 sausage and kink mode described above correspond tog¼0. 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 ofg,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 them¼0 MRT- sausage mode and them¼1 MRT-kink mode, for arbitrary value (and sign) ofg, for arbitrary aspect ratio R/D(>1), and for general values of the axial magnetic field outside, inside, and within the liner. The coupling of them¼0 MRT and sausage mode, and the coupling of them¼1 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 (Mag LIF) experiments 8–14 on the Z-machine at Sandia National Laboratories. While there have been extensive stud- ies of MRT on the Mag LIF 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 thez-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 Mag LIF 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 a) Corresponding author: [email protected] 1070-664X/2015/22(3)/032706/7/$30.00 V C 2015 AIP Publishing LLC22, 032706-1 PHYSICS OF PLASMAS22, 032706 (2015) 01 September 2025 20:25:40 the axial current (a surface current), exists only in the exte- rior region (r>R), (d) that Rand Dare constants, and (e) that the effective gravity,g, is uniform and constant. These assumptions may be justified if we pretend that the liner is subjected to an instantaneous, initial inward acceleration (¼g), 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 withg¼0). 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,c¼ ffiffiffiffiffiffiffiffi g=R p , for a thin shell even whenk¼m¼0, where Ris the radius of the thin shell andkis the axial wavenumber of the imploding thin liner. 4 This is a surprising result because the MRT growth rate is expected to bec¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk 2 h þk 2 z Þ 1=2 g q , and form andkequal to zero, we would havec¼0. The nature of this finite growth rate of Harris and its relation to them¼0 sau- sage mode will be discussed. In experiments, the axial current has a finite rise time. Quantities such as the outer radius R,andgwill then evolve in time. In fact, the effective gravitygeven changes sign from implosion to stagnation. To obtain some rough understanding of the relative importance of MRT, sausage, and kink, on a liner like Mag LIF, we use the HYDRA simulation code 19 to examine the temporal evolution of a cylindrical liner with a realistic current rise profile reaching20 MA in 150 ns, a pre- seeded axial magnetic field of 10 T, and a preheated fuel (250 e V), similar to an ideal version of the fully integrated Mag LIF experiments by Gomezet al. 14 From 1D HYDRA simulations, we extract the instantaneous “equilibrium” pa- rameters for R,D,g,B h ;B 03 , 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. Figure5below illustrates the relative importance of them¼0 andm¼1 modes at the different stages. We shall first consider the equilibrium model (Fig.1), in which the internal pressure in each region is assumed to be adjusted so thatgmay be assigned an arbitrary constant value (zero, positive, or negative), for a given set of axial and azimuthal magnetic fields. The results of the stability analysis for generalmandkare presented. Numerical results are presented only for them¼0 andm¼1 modes. II. EQUILIBRIUM AND STABILITY 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@=@tþvrÞv¼rpþJBþqgr,@q=@t þrðqvÞ¼0,@B=@t¼rðvBÞ, andrB¼l 0 Jin the cylindrical coordinates. Here,qis the mass density,vis the fluid velocity,pis the fluid pressure which is assumed to be isotropic,Jis the current density,Bis the magnetic field, gis the gravity which is positive when acceleration is radi- ally inward [Fig.1],ris the unit radial vector, andl 0 is the free space permeability. 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,B 01 ¼z B 01 þh B h R/r,B 02 ¼z B 02 ,and B 03 ¼z B 03 , where B 01 ,B 02 ,B 03 ,and B h are constants. We further assume that the mass density in each region is also constant with the liner density being the dominant, i.e.,q 01 q 02 ,andq 03 q 02 . For a scenario such as Mag LIF, region I is a vacuum field region, region II is the liner region, and region III is the fuel region. The equilibrium pressure profilep 0 (r) is adjusted so that it satisfies the equilibrium condition, 20 for allr, @ @r p 0 r ðÞ þ B 2 0 r ðÞ 2l 0 ! þ B 2 0h r ðÞ rl 0 ¼q 0 r ðÞ g;(1) where B 0 ¼j B 0 jis the magnitude of the equilibrium magnetic field B 0 and B 0h (r) is the azimuthal component of B 0 .We assume that the effective gravity,g, is a constant, though fully incompressible flow would require a non-uniformgto account for cylindrical convergence andmass 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-uniformgremains small. Integrate Eq.(1)across region II to yield gq 02 D¼p I þ B 2 01 þB 2 h 2l 0 "# p III þ B 2 03 2l 0 "# ;(2) where D¼R–ais the liner thickness,p I is the equilibrium fluid pressure in region I at the outer liner surface, andp III is the equilibrium fluid pressure in region III at the inner liner surface. The acceleration, which equalsg, may therefore be driven by an arbitrary mix of fluid pressure (p III ;p I Þor magnetic pressure (B 01 ;B 03 ;B h Þ, as long as the above equi- librium conditions are satisfied. Note that if the internal pres- surep III is dominant among all pressures, as in the stagnation stage, thengis negative (acceleration is radially outward) in Eq.(2). Hereafter, we will call the caseg¼0 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 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 B 02 ,B 03 , and B 01 , respectively. 032706-2 Weisetal. Phys. Plasmas22, 032706 (2015) 01 September 2025 20:25:40 Eq.(2)is much larger than the other square bracket, so thatjgj is maximized, we call the unstable mode pure MRT or pure RT mode. In general,jgjis between zero and its maximum value; and the resulting instability is somewhere between a pure sausage (or kink) mode and a pure MRT mode. We next consider a small signal perturbation of the form,u 1 ðrÞe ixtþimhikz , on the equilibrium of the 3-region ge- ometry shown in Fig.1and assume these perturbations to be incompressible (rv¼0). 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 onthe 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, Ax 4 Bx 2 þC¼0;(3) A¼ðX 1 X 2 þ1Þ=ðX 2 3 k 2 a RÞ;(4) B¼b 1 þb 2 ;(5) C¼k 2 V 2 02 ðk 2 V 2 02 Aþb 2 Þ þk 2 V 2 03 ðI jmj =I 0 jmj Þðk 2 V 2 02 X 2 =X 3 þjkjg 0 Þþk 2 gg 0 :(6) In Eqs.(4)–(6),X 1 ¼K jmj ^ I 0 jmj I jmj ^ K 0 jmj ,X 2 ¼ ^ I jmj K 0 jmj  ^ K jmj I 0 jmj , X 3 ¼ ^ I 0 jmj K 0 jmj  ^ K 0 jmj I 0 jmj ,I jmj ¼I jmj ðjkjaÞ,K jmj ¼K jmj ðjkjaÞ, ^ I jmj ¼I jmj ðjkj RÞ, ^ K jmj ¼K jmj ðjkj RÞ,b 1 ¼2k 2 V 2 02 A,b 2 ¼½ðX 1 =X 3 Þ jkjg 0 þðX 2 =X 3 Þjkjg,V 02 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 02 =l 0 q 02 p ,V 03 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 03 =l 0 q 02 p , g 0 ¼gþð1=l 0 q 02 RÞ½B 2 h þð ^ K jmj = ^ K 0 jmj jkj RÞðm B h þk RB 01 Þ 2 , I m and K m are, respectively, the modified Bessel function of ordermof 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 ofxin the dispersion relation(3), we shall henceforth consider only the most unsta- ble mode with the largest negative imaginary part ofx. 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 includingq 03 >0. III. m50ANDm51 MODE WITH g50 AND NONZERO g The difference between a pure sausage mode, and a pure MRT mode, both withm¼0, is shown in Fig.2(a), where we have set the axial magnetic field equal to zero every- where. We further assume thatp I ¼0, and Eq.(2)then reads, gq 02 D¼ B 2 h 2l 0 p III :(7) For the pure sausage mode, we setp III ¼B 2 h =2l 0 , so that g¼0. The normalized growth rate of this pure sausage mode is plotted by the solid lines in Fig.2(a)for various aspect ratios, from a thin shell (R/D¼10) to an almost solid cylin- der (R/D¼1.0101). Note that for a thin liner,R/D1, the pure sausage mode growth rate,c¼Im(x), approaches the asymptotic limit for smallk R, c¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 h l 0 q 02 DR s ;pure sausage modek R1;R=D1 ðÞ : (8) Equation(8)may be derived from Eq.(3)in the asymptotic limits shown, after settingg¼0. For the pure MRT mode, there is no internal pressure, p III ¼0, in Eq.(7). There is then a maximum inward acceler- ation, with a maximumg¼g max ¼B 2 h =2l 0 q 02 Daccording to Eq.(7). The normalized growth rate for this pure MRT mode withm¼0 is given by the dashed curves in Fig.2(a). Asymptotically, one may show from Eq.(3)that, for a thin liner,R/D1, and smallk R, c¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 h 2l 0 q 02 DR s ;pure MRT modek R1;R=D1 ðÞ : (9) Equation(9)is identical to Eq. (61) of Harris 4 in the limit k¼0,m¼0, 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 ofgranges betweeng¼0andg¼g max ,wecon- clude that inward acceleration (g>0) tends to stabilize the long wavelength sausage mode, and this is even more appa- rent in Fig.2(a)for lower aspect ratios,R/D¼2, and R/D¼1.0101. This is also true when B 0z /B h ¼0.1, as shown in Fig.2(c). However, if the axial magnetic field is increased to B 0z /B h ¼1(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/D¼10 case (and only for this case among the values of R/Dshown in Fig.2(e)). Note that the inward acceleration (g>0) tends to destabilize the short wavelengthm¼0,1 modes, whenk R>1. This result is impor- tant when we consider the experiments of Sinarset al. 9,10 and simulations and experiments of Mc Brideet al. 11 where such modes are observed. They show that while pre-seededm¼0 (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 m¼0(k>0) remains the largest,m>0andk>0 modes have comparable enough growth rates that, unless preseeded with m¼0, directly compete and destroy them¼0 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 challenging 11 and is discussed briefly in the conclusion. For the kink mode (m¼1), an inward acceleration (g>0) tends to destabilize the kink mode for long axial wavelength (k R1), regardless of the value of B 0z /B 0h ¼0, 0.1 or 1, as shown Figs.2(b),2(d), and2(f). The curves with 032706-3 Weisetal. Phys. Plasmas22, 032706 (2015) 01 September 2025 20:25:40 g¼0 in Figs.2(d)and2(f)show disappearance ofm¼1 instability for sufficiently largek R. This only means that the Kruskal-Shafranov criterion for kink mode stabilization is satisfied for sufficiently largek R. 20 Note that the inward acceleration (g>0) tends to destabilize the short wavelength m¼1 mode, whenk R>1. To gain some understanding of the coupling between MRT, sausage and kink mode at the stagnation stage, we now assume that B 01 ¼B 02 is so small compared with B 03 that we may now set B 01 ¼B 02 ¼0. We further assume B 03 / B h ¼1. The equilibrium condition, Eq.(2), then reads, gq 02 D¼p I p III :(10) The pure sausage and pure kink mode assumes P I ¼P III so thatg¼0. Their normalized growth rates are shown by the solid lines in Fig.3(a)form¼0, and in Fig.3(b)form¼1. The pure MRT case assumes P I ¼0 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), them¼0 mode, and (b), (d), (f), the m¼1 mode, with B 0z /B h ¼0, 0.1, and 1, and R/ D¼1.0101 (almost a solid cylinder), 2, and 10, where B 0z ¼B 01 ¼B 02 ¼B 03 . Here, g¼0 corresponds to the pure sausage mode (m¼0), or the pure kink mode (m¼1), andg¼g max >0 corresponds to the pure MRT mode for an imploding liner. 032706-4 Weisetal. Phys. Plasmas22, 032706 (2015) 01 September 2025 20:25:40 where we setp III ¼g max q 02 D¼B 2 h =l 0 . 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, them¼1 MRT mode has a higher growth rate during this deceleration phase with a highly compressed axial field. This may have implications for Mag LIF as we will discuss below. We next calculate the instantaneous instability growth rate for them¼0 and them¼1 modes according to Eq.(3), using the data from 1D HYDRA to obtain the instantaneous equilibrium profiles. Them¼1 mode could also be unsta- ble but in general has a smaller growth rate thanm¼0, 1 so we focus on the more dangerous growth rate. Figure4shows the evolution of B 01 ,B 02 ,B 03 ,B h ,g,R,a,q 02 , andq 03 , for a liner geometry and current profile similar to the fully inte- grated Mag LIF experiments of Gomezet al.(with pre- magnetization of a 10 T axial magnetic field, and a preheated fuel 14 ). These results are expected to be equally applicable to the experiments of Aweet al., 12,13 except for the decelera- tion phase since the experiments by Gomezet al.included a laser pre-heat. This allows stagnation at smaller convergence ratio, a key feature of Mag LIF. Figure4was extracted from the HYDRA 1D simulations using the procedure outlined elsewhere. 18 Note from Fig.4that during the entire interval of 148 ns,q 01 ¼0,q 03 q 02 and thatg<0 only within the last 7 ns. The instantaneous instability growth rates for the m¼0 and them¼1 mode are compared in Fig.5(a)as a function of time, for various axial wavelengthsk¼2p/k. We interpret Fig.5(a)as follows, focusing on thek¼1 mm case. Fork¼1 mm, Fig.5(a)shows that there are five (5) dif- ferent stages of MRT-sausage-kink growth for the evolving B 01 ,B 02 ,B 03 ,B h ,g,R,a,q 02 , andq 03 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 them¼0 andm¼1 modes are stable. (ii) As the azimuthal magnetic field increases, but is still less than the axial magnetic field, the kink mode (m¼1) becomes unsta- ble but the sausage mode (m¼0) remains stable. This stage is very similar to a tokamak with a safety factorq<1, but will quickly pass. (iii) As the azimuthal magnetic field is increased further, from about 25 to 55 ns, both the kink and FIG. 3. The normalized growth rate calculated from Eq.(3)for (a), the m¼0 mode, and (b), them¼1 mode, with B 01 ¼B 02 ¼0, and B 03 /B h ¼1, for R/D¼1.0101 (almost solid cylinder), 2, and 10. Here,g¼0 corresponds to the pure sausage mode (m¼0), or the pure kink mode (m¼1), and g¼jg max j<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, a¼g, from 1D HYDRA simulations. (b) Liner trajectory and evolution of the fuel and liner density from 1D HYDRA simulations. Att¼0: B 01 ¼B 02 ¼B 03 ¼10 T,B h ¼0T. 032706-5 Weisetal. Phys. Plasmas22, 032706 (2015) 01 September 2025 20:25:40 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 them¼0 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 (gis small). Over the first 60 ns, the maxi- mum number of e-folds is on the order of 1 for axial wavelengths around 100lm, 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 (<200lm) 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 them¼0 andm¼1mode are unstable, but them¼0 mode becomes dominant. At this pointgis 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 them¼0 and m¼1 modes are unstable, but them¼1, kink-like mode has a higher total amplitude gain than them¼0 mode during this final stage. Figure5(b)shows that the kink mode’s am- plitude gain is about three times that of the sausage mode during this final stage ifk¼1 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-MRTm¼1modeexhib- itsagrowthratewithcompletelackofdependenceonwave- length except for an upper wavenumber cutoffk max (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. Finally, we note that the instability growth rates pre- sented above are also in excellent agreement with the bench- mark MRT data in Sinarset al. 9,10 Experiments showing the growth of seededm¼0,k z ¼400lm modes were presented which also compared very favorably with planar growth rates 10,18 sincek Rwas large. Them¼0 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 withm6¼0 and k6¼0 could hopefully be used to benchmark 3D simulation codes, of which there are very few tests. IV. CONCLUDING REMARKS 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 them¼0 andm¼1 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 m¼0 mode grows faster than them¼1 mode. At the stagna- tion stage where the radial acceleration changes sign, the m¼1 MRT-kink mode grows faster than them¼0 MRT- sausage mode for shorter wavelengths (1 mm) when there is a pre-seeded axial magnetic field. The intricate interplay between them¼0 andm¼1 modes, which also depends on the magnitude and sign ofg, makes the interpretation of the helical structure observed by Aweet al. 12,13 and the apparent kink-like activities in Gomezet 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 Mag LIF 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. 032706-6 Weisetal. Phys. Plasmas22, 032706 (2015) 01 September 2025 20:25:40 any) to the kink-like mode that seems to have shown up at the final stage in the liner experiment of Gomezet 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. Despite the uncertainties, one puzzle on the experiment of Aweet al., 12,13 namely, why the helical structuresopened up (instead of very tightly wound up as the azimuthal mag- netic field increases; see Fig.4(a)) may be explained in terms of aneigenmodewith a specificmandk. The axial wave- number,k, is assumed to be constant. The frames in the experiment of Aweet 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/¼tan 1 ðm=k RÞ m=k R, relating the values of/(t2) and/(t1) at different times t2 and t1:/ðt2Þ¼/ðt1ÞRðt1Þ=Rðt2Þ. For the Z-Machine Shot 2480, the measured (mean) values are/ðt1Þ¼16:48, aðt1Þ¼870lm,aðt2Þ¼365lm. We then have,Rðt1Þ¼ aðt1ÞþD¼870lmþ465lm¼1335lm, and Rðt2Þ¼ 365lmþ465lm¼830 lm, where we have assumed that the liner thickness D¼465lm remains unchanged through- out (see Fig.4(b)and Aweet al. 12,13 ). The predicted helix pitch angle at t2 is then/ðt2Þ¼16:481335=830¼26:48, which is quite close to the observed mean value of /ðt2Þ¼25:68. Using this technique, the predicted value for /ðt2Þin 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. 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