ISSN: 0256-307X 中国物理快报 Chinese Physics Letters

Volume 31 Number 10 October 2014 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y

CHIN. PHYS. LETT. Vol. 31, No. 10 (2014) 100702 Mechanism and Simulation of Generating Pulsed Strong Magnetic Field *

YANG Xian-Jun(杨显俊)1**, WANG Shuai-Chuang(王帅创)1, -Dong(邓爱东)1, GU Zhuo-Wei(谷卓伟)2, LUO Hao(罗浩)2 1Institute of Applied Physics and Computational Mathematics, Beijing 100094 2Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900

(Received 13 March 2014) A strong magnetic field (over 1000 T) was recently experimentally produced at the Academy of Engineering Physics in China. The theoretical methods, which include a simple model and MHD code, are discussed to investigate the physical mechanism and dynamics of generating the strong magnetic field. The analysis and simulation results show that nonlinear magnetic diffusion contributes less as compared to the linear magnetic diffusion. This indicates that the compressible hydrodynamic effect and solid imploding compression mayhavea large influence on strong magnetic field generation.

PACS: 07.55.Db, 47.40.Rs, 89.30.Jj DOI: 10.1088/0256-307X/31/10/100702

Magneto-inertial fusion (MIF) has become a hot this model is too simple to figure out the real phys- research topic in recent years.[1] Its remarkable char- ical picture of the dynamical process. The magneto- acter, the strong magnetic field, which is hundreds of hydrodynamic theoretical method is a more applicable Tesla or larger usually, plays a very important role solution in this study and is a suitable tool in the de- in the process of ignition for magnetized plasma.[2] scription of the generation process.[8] However, there In addition to MIF, the generation of a strong mag- is still an undergoing argument existing in this method netic field in the laboratory expands new frontiers in with regard to the contribution of the nonlinear mag- plasma and beam physics, astro- and solar-physics, netic diffusion. material science, and atomic and molecular physics.[3] In this Letter, a simple mechanism of the strong There are usually three typical methods to generate magnetic field generation process is analyzed, which such a strong magnetic field. One method is to use includes the explosive driven liner model and the sim- the stationary assembled capacitors. However, with ple energy conservation model from first principles. this method, it is difficult to generate a strong enough The MHD method is established to describe the whole magnetic field over hundreds of Tesla. Laser driven process. The physical results are simulated to com- compression is another method commonly utilized for pare with the experiment data conducted at CAEP. generating a very strong magnetic field. For example, The minimum turning point radius and possible max- an over kilo Tesla magnetic field has been achieved imum strength of a strong magnetic field from the pro- at the OMEGA laser facility via this method.[4] How- cess of explosive flux compression can be estimated via ever, its sub-millimeter geometry characteristic makes using these theoretical methods. it difficult to deal with the large scale geometry re- To simplify the generating process of a strong quirement. The explosive driven flux compression as magnetic field from the explosive imploding compres- an efficient method is widely applied to generate the sion method, the three-phase model is assumed firstly, strong magnetic field since it can resolve the limited which includes an energy conservation principle, linear scale geometry problem with low cost compared to the diffusion, and nonlinear diffusion. At the turnaround other two methods. In 1965, the first magnetic field point, the initial energy (kinetic energy plus embed- with 400 T strength was produced by slip explosion.[5] ded magnetic energy) should be totally transformed According to records, even a magnetic field with to the amplified magnetic field if there is no energy up to 2500 T strength was occasionally generated by loss, using the explosive driven flux compression,[6] while currently, it is still a great challenge for China to gen- 퐸퐵 − 퐸퐵0 = 퐸푘0, (1) erate a strong magnetic field, especially at a scale of 2 2 over 1000 T. where 퐸퐵0 = 퐵0 휋푟o/2휇0 is the initial magnetic en- On the other hand, although great progress has ergy per length. On the left-hand side, the increment been achieved to experimentally generate the mag- magnetic energy should be equal to the work carried out by explosives to overcome the magnetic pressure netic field with the explosive driven flux compression 2 method, the specific generation mechanism is still un- 퐹퐵 = 푃퐵 = 퐵 /2휇0, clear and needs further investigation. The theory of 푟 ∫︁ t an equivalent circuit has been commonly used to an- 퐴(퐹 ) = 퐸 − 퐸 = − 2휋푟퐹 푑푟. (2) alyze and design this kind of generator.[7] However, 퐵 퐵 퐵0 퐵 푟0

*Supported by the National Natural Science Foundation of China under Grant No 11175028. **Corresponding author. Email: [email protected]; [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd 100702-1 CHIN. PHYS. LETT. Vol. 31, No. 10 (2014) 100702

If the flux is assumed to be conservative[10] (퐵푟2 = On the other hand, we know that the magnetic en- 2 퐵0푟o, 푟o is the initial liner inner radius), we have the ergy per length is approximately equal to the ther- turnaround radius without energy loss mal energy per length before the solid goes on to be [10] √︂ vaporized 퐸퐵0 푟t = 푟o . (3) 2 퐸퐵0 + 퐸푘0 퐵 /2휇0 = 푐v푇. (12) The initial kinetic energy is related to the final im- √︀ ploding velocity 푣 of liner The characteristic field 퐵 is defined as 퐵c = 2휇0/훽, a which means that the nonlinear effect may play an 1 2 important role if the magnetic field is over this value 퐸푘0 = 푀푣a, (4) 2 2 (43 T for Fe) (the same reference as above). If (퐵/퐵c) where 푀 is the liner mass per unit length. To obtain is larger than 1, we have this velocity, we use 푟out as the initial charge external 휎 휎 = 0 ≈ 휎 (퐵 /퐵)2. (13) radius and 푟in is the initial charge inner radius (the 2 0 c 1 + (퐵/퐵c) charge’s initial thickness 푟out − 푟in). The distribution of gas imploding velocity 푣(푟) within the detonation Therefore, the modified skin depth that includes the products is assumed to be nonlinear diffusion is

푟out − 푟 √︀ 푣(푟) = 푣a . (5) 훿d = 휏/휇0휎0(퐵/퐵c), (14) 푟out − 푟in 2 2 where 퐵 usually does not exceed the maximum value, If 퐶 = 휋(푟out − 푟in)휌 is the explosive mass unit length and 푊k is the detonation product kinetic energy per which corresponds to the total vaporized point. For unit length, energy conservation of the liner-explosive example, 퐵 takes the maximum value of 203 T for Cu system is now before it goes on to be totally vaporized.[11] Therefore, the radius of the turning point should have different 1 퐶퐸 = 푀푣2 + 푊 , (6) expressions as follows: 2 a k where 퐸 is an energy density, and a cylindrical charge 푅t = 푟t + 훿c, and liner geometry have been assumed. By definition, if nonlinear diffusion is not considered 푊 is k 푅t = 푟t + 훿c + 훿d, ∫︁ 푟out 1 2 if nonlinear diffusion is considered, (15) 푊k = 휌 · 2휋푟 · 푣 푑푟 푟 2 in then the final compressed magnetic field is 1 2[︁ 2푟in ]︁ = 퐶푣a 1 + . (7) 2 12 푟out + 푟in (︁ 푟o )︁ 퐵 = 퐵0 . (16) Then, we obtain the final imploding liner velocity 푅t

−1/2 √ [︁푀 1(︁ 2푟in )︁]︁ Now we apply the theory to the experiment. Recently, 푣a = 2퐸 + 1 + . (8) 퐶 6 푟out + 푟in a series of experiments related to strong magnetic field √ generation by the explosive driven implosion compres- The value 2퐸 can be estimated by the explosive det- sion has been conducted at the Chinese Academic En- onation velocity gineering Physics (CAEP). Figure 1 is a configuration √ of the experiment while Fig. 2 is a simplified analy- 2퐸 = 푣D/3, (9) sis model, which corresponds to the simplified center and it is about 0.8 cm per unit microsecond. profile on the left hand. First of all, the coils pro- In fact, some kind of energy loss should exist dur- duce the initial embedded magnetic field in the center ing the process of transferring explosive chemical en- of the liner, then a synchronous detonating explosive ergy to magnetic energy. To simplify, the skin depth drives the liner implosion to compress the flux in the from linear magnetic diffusion can be written as inner of the liner. The explosive is a mixture of RDX and TNT and its detonation velocity is approximately √︀ 훿c = 휏/휇0휎0, (10) 0.8 cm per microsecond. The radii are 10.5 cm and 5 cm, respectively, and the explosive length is 6.5 cm. where 휏 is the current or magnetic field’s character The liner is made of steel, which has the inner radius rise time, which is roughly a microsecond in this case. 4.85 cm with the thickness of 0.15 cm while its length is The nonlinear diffusion is mainly related to anon- 21.5 cm. From this geometry, the turning point radius solid process, which means that electric conductivity and skin depth can be estimated to be 푟t = 0.267 cm, [10] is no longer a constant, 훿c = 0.026 cm, which results in the final maximum 휎0 strong magnetic field about 1365 T for initial field 5T, 휎 = . (11) which is quite close to the experimental value 1450 T 1 + 훽푐v푇 100702-2 CHIN. PHYS. LETT. Vol. 31, No. 10 (2014) 100702 in the following. This means that the linear diffusion cannot be discussed by using our 1D model. Second, may play an important role in generating the strong with a Lagrangian grid, we cannot simulate the non- field. On the other hand, if the additional nonlinear linear stage of any instability that may develop, due contribution 훿d=0.125 cm is included, the final maxi- to the strong grid deformation. This problem can be mum of the strong magnetic field should decrease from avoided by using an Eulerian grid; however, the Eu- 1, 370 T to 673 T. lerian option is not good at handling strong gradients and moving between boundaries. c d e (a)

a

b (b)

Fig. 3. (a) The evaluation of the inner radius of the liner: (a) in the experiment, (b) by the MHD model. Fig. 1. The configuration of the experimental layout: (a) synchronous detonating device; (b) explosive; (c) coil of

Experimenta l

initial magnetic field; (d) liner; and (e) fixed device. 1 0 0

This wo rk

1

8 0

6 0

4 0 Diam eter (m m )

2 0

0 2 0 5 1 0 1 5

3 Time (ms)

Fig. 2. The simplified simulating model: (1) explosive; Fig. 4. The evaluation of the inner radius of the liner: (2) liner; (3) magnetic field. the black line is the experimental data and the red line is the simulation data.

A one-dimensional (푟–푧) magneto hydrodynamic

(a ) computer code with a single-fluid, single-temperature,

2

ax symmetric, explicit finite-difference with artificial T /s) 9 viscosity, Lagrangian mesh, and elastic-plastic model 0 has been developed here by us to simulate the pro- (10 2

(b) cess of generating a strong field, which includes con- T) 3

1

servation of mass, equation of motion, conservation of energy and the magnetic induction. (10

0 ) 2

The following initial conditions are used. The ini- (c)

3 .6 m tial embedded magnetic field is assumed to be parallel S

T to the axis and keep constant as well as the tempera- 3 .4 - 2 ture. The initial velocity is zero everywhere. 3 .2 (10 The boundary conditions are specified for velocity, 0 5 1 0 1 5 2 0 temperature, and magnetic field in the following way. Time (ms) The radial velocity component at the explosive wall is Fig. 5. Upper: the diversity rate of magnetic intensity; taken to be zero and at the outer liner velocity it is middle: the magnetic intensity; lower: the magnetic flux dependent on the properties of the explosive and the changed with time by the MHD model. process of implosion. The temperature boundary con- The dynamic process of explosive driven liner im- dition is either an insulating wall with 휕푇/휕푟 = 0 or plosion compression has been scanned by the high a perfectly conducting wall with 푇 =constant. It is as- speed rotating mirror streak camera,[12] which is sumed that the magnetic flux is conserved at the cross shown in Fig.3(a), at CAEP recently, while Fig. 3(b) section, which includes the linear skin depth. Other shows the calculation result from our MHD model. parameters can be seen from the above text. The experimental data and simulation result are in Two major limitations of the present study must agreement with what was roughly anticipated from be noted here. First, the 2D effect, such as RMT, these two figures, especially as both of them do not 100702-3 CHIN. PHYS. LETT. Vol. 31, No. 10 (2014) 100702 meet together finally, due to the counteractive force experiment and simple model at the turning point. from the compressed magnetic flux. A more precise This suggests that the MHD model needs more proper comparison is given in Fig.4. The black curve is the magnetic diffusion parameters in further work. experimental result while the red one is from the the- In conclusion, the physical mechanism and dy- oretical calculation. Both of them are quite consistent namic process of generating a strong magnetic field with each other except for the turning point of the ex- are investigated by building a suit of describing meth- perimental curve, which is due to our current technical ods from a simple model to the MHD model, which limitation of data acquisition. is used to analyze recent experiments conducted by our academe. The simple model is based on the first physical principle, which can give the minimum ra- 1 5 0 0 dius and maximum magnetic field that are more close 2 0 1 3 0 3 2 8

2 0 1 3 0 6 2 8 to the experimental results, as listed in Table 1. The MHD model can describe the dynamic process of gen- 1 0 0 0 erating a strong magnetic field in detail while its mag-

T netic diffusion parameters still need to be improved. The analysis and simulations show that the hydrody- 5 0 0 namic effect and liner magnetic diffusion effect may dominate the generation process of a strong magnetic field, which can be supported by being close toex- 0 perimental data from the simple model and the high 5 7 6 5 7 8 5 8 0 5 8 2 5 8 4 5 8 6 5 8 8 5 9 0 5 9 2 5 9 4 speed rotating mirror streak camera’s photos from the

Time (ms) MHD model. Fig. 6. Data from two experiments of generating mag- netic field. Table 1. The comparisons among the simple model, experi- ment and the MHD model. References

Simple model Experiment MHD model [1] Perkins L J, Logan B G, Zimmerman G B and Werner C J Minimum 2.85 2.93 2.56 2013 LLNL-JRNL-628792 radius (mm) (extrapolated) [2] Lindemuth I and Kirkpatrick R 1983 Nucl. Fusion 23 263 Maximum [3] Herlach F and Miura N 2003 High Magnetic Field: Science 1370 1450 1800 field (T) and Technology (Singapore: World Scientific) [4] Gotchev O V et al 2009 Phys. Rev. Lett. 103 215004 A series of calculation results, such as the magnetic [5] Herlach Fand Knoepfel H 1965 Rev. Sci. Instrum. 36 1088 intensity, its diversity rate, and magnetic flux changed [6] Hawke R S, Duerre D E et al 1972 J. Appl. Phys. 43 2734 with time, can be given by MHD simulation. The [7] Goforth J H, Oona H, Herrera D H and Torres D T 2010 13th International Conference on Mega Gauss Magnetic curve of magnetic intensity is presented in the mid- Field Generation and Related Topics (Suzhou, China 6–10 dle of Fig.5 and shows its maximum value ∼1800 T. July 2010) p 102 Compared with the theoretical result, the maximum [8] Flower C A, Freeman B L et al 1994 LA-UR-93-3459 magnetic field was recorded during the last middle of [9] Neuber A A 2005 Explosively Driven Pulsed Power (Berlin: Springer-Verlag Heidelberg) 2013 at CAEP about 1450 T from two typical experi- [10] Knoepfel H 1970 Pulsed High Magnetic Fields (Amsterdam: ments in Fig. 6, which means that the highest record North-Holland Publishing Company) of generating the experimental strong field has been [11] Turchi P J 2012 14th International Conference on Mega achieved in China. Although our MHD model can de- gauss Magnetic Field Generation and Related Topics (Maui Hawaii USA 14–19 October 2012) p 26 scribe the dynamic process very well, it gives a slightly [12] Gu Z W, Luo H et al 2013 Acta Phys. Sin. 62 17071 (in higher value of the maximum magnetic field than the Chinese)

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