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wd)1**, WANG Shuai-Chuang(王RM)1, DENG Ai-Dong("O东)1, GU Zhuo-Wei(谷R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- EB − EB0 = Ek0; (1) erate a strong magnetic field, especially at a scale of 2 2 over 1000 T. where EB0 = B0 휋ro=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- FB = PB = B =2휇0, clear and needs further investigation. The theory of r Z t an equivalent circuit has been commonly used to an- A(F ) = E − E = − 2휋rF dr: (2) alyze and design this kind of generator.[7] However, B B B0 B r0 *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] (Br2 = On the other hand, we know that the magnetic en- 2 B0ro, ro 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] r vaporized EB0 rt = ro : (3) 2 EB0 + Ek0 B =2휇0 = cvT: (12) The initial kinetic energy is related to the final im- p ploding velocity v of liner The characteristic field B is defined as Bc = 2휇0/훽, a which means that the nonlinear effect may play an 1 2 important role if the magnetic field is over this value Ek0 = Mva; (4) 2 2 (43 T for Fe) (the same reference as above). If (B=Bc) where M is the liner mass per unit length. To obtain is larger than 1, we have this velocity, we use rout as the initial charge external 휎 휎 = 0 ≈ 휎 (B =B)2: (13) radius and rin is the initial charge inner radius (the 2 0 c 1 + (B=Bc) charge’s initial thickness rout − rin). The distribution of gas imploding velocity v(r) within the detonation Therefore, the modified skin depth that includes the products is assumed to be nonlinear diffusion is rout − r p v(r) = va : (5) 훿d = 휏/휇0휎0(B=Bc); (14) rout − rin 2 2 where B usually does not exceed the maximum value, If C = 휋(rout − rin)휌 is the explosive mass unit length and Wk is the detonation product kinetic energy per which corresponds to the total vaporized point. For unit length, energy conservation of the liner-explosive example, B 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 CE = Mv2 + W ; (6) expressions as follows: 2 a k where E is an energy density, and a cylindrical charge Rt = rt + 훿c; and liner geometry have been assumed. By definition, if nonlinear diffusion is not considered W is k Rt = rt + 훿c + 훿d; Z rout 1 2 if nonlinear diffusion is considered; (15) Wk = 휌 · 2휋r · v dr r 2 in then the final compressed magnetic field is 1 2h 2rin i = Cva 1 + : (7) 2 12 rout + rin (︁ ro )︁ B = B0 : (16) Then, we obtain the final imploding liner velocity Rt −1=2 p hM 1(︁ 2rin )︁]︁ Now we apply the theory to the experiment. Recently, va = 2E + 1 + : (8) C 6 rout + rin a series of experiments related to strong magnetic field p generation by the explosive driven implosion compres- The value 2E 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 p of the experiment while Fig. 2 is a simplified analy- 2E = vD=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 p 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 rt = 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 + 훽cvT 100702-2 CHIN.