ISSN 20700482, Mathematical Models and Computer Simulations, 2011, Vol. 3, No. 6, pp. 732–743. © Pleiades Publishing, Ltd., 2011. Original Russian Text © N.A. Zavyalova, A.I. Lobanov, 2011, published in Matematicheskoe Modelirovanie, 2011, Vol. 23, No. 4, pp. 103–119.
Numerical Calculations of the Dynamics of Liners Formed by Copper Vapors’ N. A. Zavyalova and A. I. Lobanov Moscow Institute of Physics and Technology (State University), Moscow, Russia email:
[email protected];
[email protected] Received July 20, 2010
Abstract—The paper reports the calculations of the dynamics of fast copper liners. The implosion time is of the order of 80 ns. The problem is considered in a twodimensional (axisymmetrical) state ment. For simulation, a system of equations of electron magnetic hydrodynamics (EMH) is solved, using splitting into physical processes. At the first stage, the ideal MHD with a frozenin magnetic field is considered. An implicit variation differential scheme is constructed for the solution of the equations of ideal twotemperature magnetic hydrodynamics. The dynamics of the liner splitting is calculated at different values of the Hall parameters. With an increased value of the Hall parameters, the character of the magnetic field’s penetration into plasma changes radically and the field’s “tongues” penetrating along one of the electrodes are formed. Keywords: electron magnetic hydrodynamics, implicit difference scheme, method of splitting into physical processes, radiation losses. DOI: 10.1134/S2070048211060111
INTRODUCTION In plasma dynamics, numerical calculations are very important; sometimes they are the only way to obtain detailed information about the liner plasma. Analytical solutions in nonlinear nonstationary prob lems of plasma physics are, as a rule, impossible. Thus, the numerical modeling of plasmadynamic pro cesses has become a traditional method of investigation, along with a natural experiment. The authors of [1] present the results of an experimental and numerical investigation into the effect of the preliminary electrical explosion of the cylindrical bunched liner made from wolfram on the generation of Xray radiation in Zpinch geometry. In [2], the results are described of experimental and theoretical studies, as well as calculations, of plasma threedimensional dynamics resulting from the current implo sion of the taper wire assemblies (liners) in the mode of a Zpinch discharge at discharge currents of up to 3 MA on an Angara51 plant. The authors of [3] provide the results of the numerical MHD2Dmodelling of the Xpinch compres sion in rz and xy geometries with a load of the homogeneous (dense plasma) and heterogeneous (kern crown) types. The processes of the formation of a minidiode, a narrow neck, and Xray flash of the hot point, as well as the effect of the crown on the dynamics of an Xpinch dense plasma compression, are investigated. In the calculation of the evolution by the shockcapturing method, taper necks are used, while for the detailed description of the flash parabolic necks are employed. The MHD accumulation pro cesses of the oblique shock waves and of the development of the plasma cord instability are analyzed. The processes forming the cascade of the neck compressions are found to have a quasicyclic character. The state of plasma in the hot area of the flash directly before the neck break and the possibility of the gener ation of accelerated beams are investigated. In [4], a statement of the problem is given and the numerical model for the calculation of MHD fluxes in chambers of the plasma focus. For the solution of the motion equation, a difference system implicit rel ative to the magnetic field and the procedures for its numerical solution are proposed. This scheme describes the plasma flux with highly different densities of different domains and, thus, allows for the pres ence of a vacuum domain behind the currentplasma envelope in the plasma focus. The allowance for this domain proves salient for numerical simulation of the process of focusing and calculating the effect of the acceleration neutrons. The results of comparisons of the calculations with the experimental data charac terize the plasma dynamics in two types of chambers. In order to estimate the contribution of the acceler ation mechanism to the total neutron yield, a model of accelerated ions is suggested. In the calculations 732
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using this model, the difference for three different kinds of the geometry of plasma focus chambers with current values of 0.5– 1.5 MA, the value of the calculated neutron yield does not differ from the experimental result by more than twofold.
The authors of [5] provide operations demonstrating the inap plicability of the classical magnetic hydrodynamics in respect of the highcurrent fast Z pinches. For such objects a substantial 0 R r role is played by the Hall effect. It is possible to simplify the two liquid MHD by constructing a model of electron magnetic hydrodynamics (EMH). A characteristic feature of the EMH Fig. 1. Scheme of the discharge modes is the magnetic field being frozen in the electron fluxes. It chamber. is accepted that such a flux can be completely characterized by the electron current velocity [6]. This model is an object for the oretical studies. In [7], the transfer of spatial heterogeneity of the magnetic field in plasma is theoretically studied in the EMH approximation. Recently, the Z pinches have been used as a powerful source of Xray radiation with a narrow spectrum and high efficiency. About 30%–40% of the inserted energy enters into the flux of the radiation. The authors of [1, 8] consider the compression dynamics, taking into account radiation losses. Publication [1] describes the results of numerical modeling by the onedimensional radiation MHDcode. The model demonstrates good agreement of the radiation parameters generated by the Z pinch in experiments both with the preliminary electron flash of the liner wires and without it. The authors of [8] describe the results of experiments on measuring the emission of the hard X ray and neutrons by 5 scintillation detectors, two mounted in the axial direction and one mounted in the radial direction. PROBLEM STATEMENT The Stand300 facility is an impulse highcurrent generator. It is used for experiments with different types of loads such as plasma breakers, z pinches, and wire assemblies. An example of the load can be pinch formed by copper vapors blown out from the nozzle. The physical model of the Stand300 discharge chamber is presented in Figure 1. A system of two disc electrodes is connected to the current generator. The sizes of the discharge chamber are a radius of R =1 cm and clearance between electrodes Z = 1 cm. Two current impulses are generated in the circuit. In the preimpulse action, an electric breakdown occurs; the material found between the electrodes is evaporated and ionized. Then, the second, i.e., the basic, current impulse is fed. The model does not include the equation of the external electric circuit; use is made of interpolation of the experimental time dependence of the fullload current I (t ) = I 0 sin ( πt t 0 ), where t is the current time, and t0 is the time of the impulse increase, t0 = 150 ns. The time of transient processes is short; thus, in the model they can be neglected. We believe that the homogeneous plasma is found after switching the current in the external circuit (the feed of the second impulse) between the electrodes. The pinch is formed by the copper vapors. The starting distribution of density was assumed to be Gaussian
(
)
⎛ r − rmax 2 ⎞ ρ ( r ) = ρ0 + ρ1 exp ⎜ − ⎟, δ ⎝ ⎠ where rmax = 0.8 cm, δ was varied, and ρ0 and ρ1 were chosen from the condition of the specified line den sity. The starting temperature was set the same for electrons and ions and was Te = Ti = 0.01 keV. During the operation of the facility the electrodes were cooled by liquid nitrogen, and their model temperature is considered equal to zero. MATHEMATICAL MODEL Model Equations The mathematical model of the fast zpinch dynamics includes the equations of EMH written in a dimensionless form [5, 9]: MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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dρ + ρ div v = 0, dt
dv = − 1 ⋅ gradP + [rot B × B] , ρ dt 4π ρ
ρd ( z eff ε e + J ( z eff )) = −Pe ⋅ div v e + div ( κ e grad Te ) + Q − Qei − div S, A dt ρ dεi = −Pi div v + div ( κ i ⋅ grad Ti ) + Qei , A dt ∂B = − rot E, j = 1 rot B, E = − v × B + j − A gradP − R , ( ) [ e ] e ∂t 4π σ ρz
R = −0.71
ρz ρz [B × gradTe] , grad ||Te − 1.5 B A Aωe τ e
Q=
2
j + ( j, R ) , σ
v e = v − j A , ε e = AeTe, Pe = A pρzTe, ε i = AeTi , Pi = A pρTi , ρz 1 div S = − 4ε1σ СБT 4 − U , S = − l grad U , κ e = 3.16z eff ρ Te τ e , κ i = 3.9ρTi τi . l 3 The following has been taken as the base scale in putting these equations in a dimensionless form: the linear dimension L = 1 cm, time t = 100 ns, and temperature T = 1 keV. The twodimensional problem statement in the cylindrical (r, z) geometry was considered. The standard denotation of most of the vari ables in the equations is employed. Indices i and e correspond to electrons and ions, zeff is the effective charge of the nucleus, J(zeff) are ionization losses, ωeτe is the Hall parameter, κe and κi are electron and ion heat conductivity, À is the number of nucleons in the nucleus, and the constants included in the condition equations are Ae = 14.38 and Ap = 9.68 × 10−2. For inclusion of the twoliquid effects, the electron velocity ve is included. U is the unbalanced intensity of radiation, σ is the plasma conductivity, and σSB is the Ste fan−Boltzmann constant. For the length of the free path of photons, the dependence l = l 0Te2ρ −2.was used. For the calculation of ionization losses, scaling [11] was used:
(
)
3 2 ⎛ z eff z eff z eff ⎞ 0.131 J ( z eff ) = + + ⎜ ⎟, 2/3 2 6 ⎠ 0.85 + 0.15Z ⎝ 6 is the temperature dependence of the average ion charge z eff = 9Te1/3 and Z is the charge number. For the electronion relaxation, the relation
Qei = 8.48 × 10 −1
3 ρ 2 z eff Te − Ti A 3 Te3/2
was used. The expression for the frequency of electronion collisions is needed for estimation of the coefficients of heat and electric conductivity, and for estimation of the Hall parameter. For the time of collisions we have [9]. 3/2 3 τe = T , λ = ln Λ, Λ is the Coulomb logarithm. 2 ( e) 4 2πniλz eff For conductivity formulas [10] 3ne 3/2 3 σ = neτe = T = (Te )3/2 2 ( e) 4 2πniλz eff 4 2πλz eff were used. We estimate the Hall parameter according to
ρ eB0 3.5 × 10 4 Te , Λ ~ 10. , ni = 2 Am p mec Λ 10 z eff ni Then, for the characteristic values of the parameters of electron temperature, magnetic [10] field strength and plasma density, as well as the degree of the ionization corresponding to the Stand300 facility, we obtain A = 63.5, ρ = 0.3 microgram/cm3, and the value of the magnetic [10] field strength is estimated by the value of the total current in the circuit B0 = 0.2 × I0/r. The current increases from 0 to 5 mA, the pinch from the starting size of 1 cm is compressed tenfold in radius. Thus, in the conditions of this prob lem, the Hall parameter ranges from 0 to 25. 3/2
ωe τ e =
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Numerical Method The experience in solving the problems of twotemperature hydrodynamics shows that the ion pressure is a more conservative value and for small times it changes considerably less than the electron [11]. Thus, the physical nature of the problem dictates the choice of the following scheme of splitting. The layer bylayer transition goes according to the scheme of splitting into physical processes [13]. Within this approach, the solution of the system of equations is reduced to a sequence of problems, which separately do not approxi mate the starting system, but for the solution of particular subproblems, it is possible to suggest effective numer ical algorithms. The totality of the splitting stages approximates the starting problem. At the first stage, the ideal magnetic hydrodynamics was considered, i.e., the motion of plasma without taking into account dissipative effects and the radiation. The implicit difference scheme of this stage was set up on the basis of variation method [14]. It is known that for systems of the hyperbolic type, the calcu lation by implicit schemes yields adequate results with the Courant number less than unity. However, the use of the implicit scheme makes it possible to remove substantial constraints on the steps by time in the interaction of the shock waves. In this case, the condition of the Courant stability for the explicit schemes yields the value of the time step tending to zero. Similar implicit methods have been earlier implemented for the problems of the onetemperature MHD [15, 16]. A characteristic feature in the solution of the twotemperature electron hydrodynamic problem is splitting into ion and electron pressures in solving a system of equations for the pressure at the upper time layer.The Lagrangian in the MHD models [14] is
L=
∫∫ V
2 ⎛V2 B ⎞ ρdV . − ε − ε − e i ⎜ ⎟ 8πρ ⎠ ⎝2 t1
The Hamiltonian action functional will then be S = ∫ L (t ) dt . On the trajectories, its variation δS = 0, t0
therefore, t1
∫
t1
δ L (t ) dt =
t0
2 ⎛ B δ B + B δρ ⎞ρ dVdt = 0. v δ v − δε − δε − e i ⎜ ⎟ 4πρ 8πρ 2 ⎠ ⎝ V
∫∫
t0
In the rated operating conditions, a movable grid with quadrangular cells was introduced. The scalar values are related to the centers of cells, the vector values are related to nodes, and the flows are related to the centers of the appropriate facets. We use an implicit approximation, here, and, henceforth, the values at the upper layer are denoted by “caps.” In motion equations, the following denotations are introduced: u is the radial velocity, v is the axial velocity, and uˆ and vˆ are radial and axial velocities at the upper layer. The volumes of the grid cells are denoted as Ωml and their mass as mml. The discrete analogue of the action functional is presented as t n +τ
2 ⎛ u2 + v 2 B ⎞ ml mml ⎜ − ( ε e + ε i ) ml − ml ⎟ dt. ⎜ 2 8πρ ml ⎟ ⎝ ⎠ t n ml 2 2 Here, 〈u + v 〉ml is the cellaveraged density of the kinetic energy
S =
∫∑
(1)
t n +τ
2 ⎛ u2 + v 2 B ⎞ ml u +v S = mml ⎜ − ( ε e + ε i ) ml − ml ⎟ dt. ml ⎜ 2 8πρ ml ⎟ ⎝ ⎠ t n ml The condition of the discrete analogue minimum of the action functional is the equality to zero of all partial derivatives (1) by the velocities of the nodes uml, vml. As the equation of connections in variation of the discrete analogue of the action functional, the mass conservation law δ(ρdΩ) = δ(dm) = 0 is used. After differentiation, we have
1
2
2
∑(
)
=1 um2 +q,l + p + v m2 +q,l + p . 4 p,q =0
t n +τ
t n +τ
∫
1u m + m ml ( ml m −1l + mml −1 + mm −1l −1 ) dt − 4
∫
1v m + m ml ( ml m −1l + mml −1 + mm −1l −1 ) dt − 4
tn t n +τ
tn
∫∑
∫
ρ ml ( ε i + ε e )
∂Ω ml ∂rml B 2 ∂Ω ml + dt = 0, ∂rml ∂uml 8π ∂rml
∫
ρ ml ( ε i + ε e )
∂Ω ml ∂z ml B 2 ∂Ω ml dt = 0. + ∂z ml ∂v ml 8π ∂z ml
tn t n +τ
tn
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We introduce the node mass M kl = 0.25 × ( mkl + mk −1l + mkl −1 + mk −1l −1 ) , and we write the projections of the equation of the nodes of the grid motion on the axis of the system of coordinates:
uˆml − uml = 1 τ M ml
4
⎛
∑ ⎜⎝ Pˆ
k
k =1 4
+
Bk2 ⎞ ∂Ω k , ⎟ 8π ⎠ ∂rml
(2)
⎛ ˆ Bk2 ⎞ ∂Ω k vˆ ml − v ml (3) = 1 , ⎜ Pk + ⎟ τ M ml k =1 ⎝ 8π ⎠ ∂z ml In order to exclude pressure from the motion equations, use is made of the equation of ion energy ρ dεi = − pi div v, A dt which, at the first stage of splitting, does not include heat conductivity and dissipating terms associated with the energy exchange of electrons and ions as a result of collisions. In order to write the difference ana logue of the divergence operator, the differential approximation [14] was used. The discrete analogue of the equation for ion energy is
∑
ρ εˆ i − ε i = − pˆi 1 Ω ml A τ
4
ˆ ml ˆ ml ⎞ ⎛ ∂Ω ∂Ω uˆk + vˆ k ⎟. ˆ ˆ ∂ ∂ r z ⎠ k k k =1
∑ ⎜⎝
Pressure depends on temperature pi = A pρTi A , so Ti = pi A A pρ , and the internal energy is expressed through temperature by the correlation ε i = AeTi , therefore, ε i = Ae A A p pi ρ. Substituting into the energy equation the value of the internal ion energy at the upper layer expressed through pressure, we obtain difference equations: 4 ˆ ˆ ˆ ml ⎛⎜ Ae ⎞⎟ pˆi − mml ε i + τpˆi ⎛ ∂Ω ml uˆk + ∂Ω ml vˆ k ⎞ = 0. Ω ⎜ ⎟ A ∂rk ∂z k ⎠ ⎝ Ap ⎠ k =1 ⎝ From them, it is possible to exclude uˆml and vˆml , taking advantage of motion equations (2) and (3). Let us consider now the equation of electron energy
∑
4 ˆ ml ˆ ml ⎞ ⎛ ∂Ω ∂Ω ρ zˆeff εˆ e + J ( zˆeff ) − z eff ε e − J ( z eff ) 1 ˆ u vˆ k ⎟. = − pˆe + k A τ Ω ml k =1 ⎜⎝ ∂rk ∂z k ⎠ In the same way, we exclude the electron internal energy at the upper layer. Then,
∑
4 ˆ ml ⎛ Ae ⎞ ˆ ml ˆ ml ⎞ ˆ ml ˆ ml ⎛ ∂Ω m m Ω ∂Ω 1 ˆ ˆ ˆ p J z z J z p uˆk + + − ε + + τ vˆ k ⎟ = 0. ( eff ) ( eff e ( eff )) e ⎜ ⎟ e ⎜ A ⎝ Ap ⎠ A A Ω ml k =1 ⎝ ∂rk ∂z k ⎠ The obtained difference scheme is nonmonotonic. In order to decrease the amplitude of oscillations on dis continuous solutions, use was made of the artificial viscosity η, w, the weight coefficient [17]. In the calcula tions, its value was assumed to be 0.6–0.7. Besides, from the difference equation for electron energy, we exclude the electron motion at the upper layer by substituting for it the difference of total and ion pressures. Eventually, we obtain a system of difference equations for total and ion pressure: ˆ ml ⎛ Ae ⎞ ˆ ml ˆ m m Ω J ( zˆeff ) − ml ( z eff ε e + J ( z eff )) + ⎜ ⎟ ( pˆ − pˆi + wη) + A ⎝ Ap ⎠ A A 4 ˆ ml ˆ ml ⎞ ⎛ ∂Ω ∂Ω τ( pˆ − pˆi + wη) 1 uˆk + vˆ k ⎟ = 0, ⎜ Ω ml k =1 ⎝ ∂ rk ∂z k ⎠
∑
∑
4 ˆ ml ⎛ Ae ⎞ ˆ ml ˆ ml ⎞ ˆ ml ⎛ ∂Ω m Ω ∂Ω 1 uˆk + ε i + τ( pˆi + (1 − w)η) vˆ k ⎟ = 0. ⎜ ⎟ ( pˆi + (1 − w)η) + ⎜ A ⎝ Ap ⎠ A Ω ml k =1 ⎝ ∂rk ∂z k ⎠ The template for pressure is given in Fig. 2. The magnetic field stress vector in the considered problems has only one component by the azimuthal angle B = (0, Bϕ,0). Then, the equation of the magnetic field freezingin into the plasma material is dB = −B div v. dt
∑
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This equation leads to the integral B ml S ml = Bˆml Sˆml . The freezingin integral written for each cell closes the system of difference equations. Having determined the ion motion, the total pressure and the magnetic field stress at the upper layer can be determined and the velocities can be found by using (2), (3). The new coordinates of the grid nodes are obtained through numerical integration of motion equations. The thermodynamic parameters are computed by use of the equation of condition.
i, j + 1
For solution of a nonlinear difference equation, New ton’s method is used. The system of linear equations for 2' 1' increments of the argument arising in Newton’s method is solved by a method of subsequent upper relaxation with programs of computation library of numerical analysis by use of programs from the Scientific Research Computing Center of Moscow State University (SRCC MSU) [188]. Then, having fixed the found value of the ion pressure on Fig. 2. Template for pressure including nine cells. the upper layer and using the linearization by Newton, the value of the total pressure and magnetic field strength (magnetic induction) is found without accounting for dissipative effects (the final electrical conductivity of the medium). For reconstruction of the computation grid, the variation method [20] was used. If, after completion of the first stage of the computation (MHDpredictor), the value of the step by time reaches the mini mum acceptable value, the procedure of movable grid realignment was performed with the subsequent conservative recalculation of values for the new grid.The quadrangular cell was divided diagonally into two triangular ones. If the area of at least one of such cells became negative, then the quadrangular cell was thought of as nonconvex and the grid was rearranged. This procedure was also used after the set number of steps by time. The algorithm of grid realignment is based fully on the optimization method [19]. After the grid’s realignment, the conservative recalculation of all values was performed for the new calculation grid. At the next stages of splitting, the dissipative processes were included: electron and ion heat conduc tivity, diffusion of the magnetic field, and plasma radiation. i, j
3'
BOUNDARY CONDITIONS Magnetic Field It was assumed that current exits from one electrode and enters another normally to the surface, with the current along the electrode being absent and ∂Bϕ ∂n = 0, where n is the direction of the external nor mal. This approximation is not consistent with EMH; it is true only if ω e τ e Ⰶ 1. On the border of the plasma with the vacuum, the magnetic field strength is associated with the total current in the external cir cuit, Bϕ = 0.2I R . This expression is written in dimensionless variables. Thermodynamic Values and Velocity In order to achieve the homogeneity of the difference scheme, fictitious cells having a common facet with the boundary ones were introduced. Their form and way of setting the values in them depended on the specific species of the boundary condition. On electrodes, a condition of leakproof ( V, n) = 0 , ∂P ∂n = 0 was stated. The fictitious cells were made as the mirror mapping of nodes relative to the electrode surface; the values of density and pressure were determined by deflection. On the border with the vacuum, the gas kinetic pressure is P = 0 . The areas were surrounded by ficti tious cells of zero volume; the density and pressure in them were assumed to be zero. Near the exceptional point r → 0, only limited solutions are of interest. From the nonseparability Bϕ, we obtain condition B ϕ r = 0 = 0 . The velocity on the symmetry axis is v r = 0. The heat flow also equals zero MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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ZAVYALOVA, LOBANOV 1:2:4 0.2 0.15 0.1 0.05
T 0.25 0.20 0.15 0.10 0.05 0
0
0.1 0.2
0.3
0.050 0.150.10 0.4 0.5 0.250.20 0 .3 z 0 0.6 r 0 0.70.8 0 .35 0.45 .40 0.9 1.0 0.50
2:1:5 0 –1 –2 –3
B 0.5 0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5 –4.0
0
0.05
0.10
0.15 r 0.20
0.25
0.30 0.9 0.8 0.35 1.0
0 0.1 0.2 0.3 0.50.4 0.7 0.6 z
Fig. 3. At the top, the distribution of the electron temperature for a time of 60 ns. A temperature profile is seen which is typical of the heat wave propagating about the cold background. At the bottom, the distribution of the magnetic field at a time of 65 ns, ωeτe = 5. The beginning of the tongue formation near the electrode.
κ ∂T = 0 and 1 ∂ r κ ∂T is limited, i.e., 1 κ ∂ T + ∂ κ ∂T is limited. Then, we obtain the requirement ∂r r ∂r ∂r r ∂r ∂r ∂r α ∂ T κ ~ r , where α > 1 . From this, we obtain the condition for derivative temperature ∂ T = 0. ∂r ∂ r r =0 On the pinch’s axis, the condition for symmetry is stipulated for the system of differential equations. How ever, the system of differential equations with r → 0 loses its approximation. Thus, in order to maintain the approximation, the condition on the axis was changed to conditions on the hard wall situated at 0.01 from the symmetry axis. Radiation It is thought that there is no radiation incident from the outside. Then, the intensity of radiation is asso ciated with the flow by the relation U + 1 (S,n) = 0 , where U is radiation intensity. 2
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T 16 14 12 10 8 6 4 2 0
0
739
0.1
0.2
0 0.3 0.4 0.5 0.6 z
0.70.8
0.9 0.30 1.0
0.15 0.20 r 0.25
0.10
0.05
1:2:3 60 50 40 30 20 10
ro 70 60 50 40 30 20 10 0
1.0 0.9 0.8 0.7 0.6 z 0.50.4 0.3 0.2 0.1 0
0
0.02
0.04
0.06
0.08
0.10
0.12
r
Fig. 4. At the top, the distribution of the ion temperature at a time of 76 ns. The formation of hot and cold bands are seen along the electrodes. At the bottom, the distribution of density is 80 ns. In the density distribution, a mapped shock wave is seen.
CALCULATION RESULTS Dynamics of Plasma The test calculations on a coarse grid of cells measuring 20 × 20 are given since the buildup time is inconsistent with the duration of the impulse on the Stand300 generator. At the starting stage of the process, when the magnetic pressure in the system is less than the gas kinetic pressure, the plasma expands into the vacuum. Within about 10 ns, the process of expansion stops and the plasma is slowed by the magnetic pressure. Due to the release of Joule heat, the plasma starts to be heated and a crown is formed. After 20 ns, the plasma contracts intensively. A shock wave is formed that spreads to the center of the area. Closer to 30 ns, the plasma density in the proximity of the stressfree boundary increases, because of the “snow plough” effect. The process develops as a onedimensional one. The tem perature gradients caused by the electrodes’ cooling and the electron effects do not much influence the liner’s dynamics. The magnetic field is displaced into the skinlayer near the free boundary. These areas correspond to the high values of the magnetic field rotor. As a result of Joule heating, heat release occurs here. Ions are MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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ZAVYALOVA, LOBANOV 1:2:4 1.0 0.8 0.6 0.4
T 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
0 0.02
T 120 100 80 60 40 20 0
0.04 r 0.06
1.0 0.8 0.9 0.7 0.6 0.08 0.4 0.5 z 0.10 0.3 0.2 0.12 0 0.1 1:2:7 100 80 60 40 20
0 0.02
0.04 0.06 r 0.08
0.10 0.12 0 0.1
1.0 0.8 0.9 0.7 0.6 0.4 0.5 z 0.2 0.3
Fig. 5. At the top, the distribution of the electron temperature after pinch accumulation on the axis, 80 ns. At the bottom, the distribution of the ion temperature at this time.
heated behind the shock wave due to adiabatic compression. At times of 40–60 ns, the value of the plasma’s flowing velocity is about constant 2 (2 × 107 cm/s). The value of the ion temperature behind the shock wave is about 500 eV, and the value of the electron is about 180–250 eV. With about 60 ns, the first nonlinear effect is observed in the calculation. When the temperature behind the shock wave exceeds 250 eV, the propagation of electron temperature occurs as a heat wave. Such solu tions are intrinsic in the quasilinear equations of the parabolic type. The profile of temperature has a form typical of these solutions (Fig. 3). The velocity of the heat wave exceeds the velocity of the shock wave. Toward 70 ns, the heat wave reaches the center of the area, heating the plasma material up to 250–300 eV and increasing the degree of ionization. As the pressure difference on the shock wave decreases, the veloc ity of its propagation by the liner’s material also drops. Because of the electrodes’ cooling, the temperature minima are formed near them. It is in this time period that the twodimensional effects begin to show themselves considerably. Immediately after the shock wave has been focused on the pinch’s axis, the local areas of increased electron and ion temperatures are formed. With the parameters of density which are used in this calcula tion, the moment of accumulation of the shock wave is about 78 ns. The distribution of density after the shock wave mapping is given in Fig. 4. The Hall parameter for this calculations is ωeτe = 0.01. By the time MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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0.5 0.4 z
10 5 0
B 14 12 10 8 6 4 2 0 –2
0
0.1 0.2 0.3 0.4 0.5 0.6 z 0.7 0.8 0.9 1.0 0
0.16 0.14 0.12 0.10 0.08 r 0.06 0.04 0.02
Fig. 6. At the top, the distribution of the magnetic field strength at the value of the Hall parameter ωe τe = 5 for a time of 79.5 ns. At the bottom, the distribution of the magnetic field strength at the change of polarity of electrodes and at the value of the Hall parameter ωe τe = 0.01 for a time of 82 ns. At a higher value of the Hall parameters, the tongues of the magnetic field’s penetration, are seen with strong penetration along the anode (with r = 1). With a decrease in the Hall parameter, the tongue about the anode is not quite distinct.
of the accumulation, the pinch is compressed about tenfold by radius.Fig. 5 gives the distribution of tem peratures after shock wave focusing. Our attention is engaged by the high (about 100 keV) value of the ion temperature on the axis. Such a high increase is the consequence of the differential schemes of entropy effects typical for this class. Radiation Dynamics In order to understand the pinch’s accumulation processes, it is important to estimate the balance of energy and losses on radiation. Even in the diffusion onegroup approximation, it is possible to investigate the effect of the compression dynamics on the radiation process.Figure 7 shows the dependence of the full flow of radiation through the free boundary of the liner on time and the degree of the liner’s compression. It is seen that the flow of energy greatly increases after an approximately tenfold compression. The inset to Fig. 7 shows separately the flow dependence on time at the moment of focusing of the shock wave. Before accumulation, a part of the liner’s energy is lost due to highlighting. The focusing moment has the corresponding minimum of radiation. After the shock wave is mapped from the pinch axis, the energy losses on radiation sharply increase.Figure 8 shows the dynamics of energy losses on radi MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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ZAVYALOVA, LOBANOV R/R0 Moment of the shock wave arrival at the axis
160 1.00 140 120 0.75 100 – 80 0.50 60 40 0.25
Moment of the shock wave arrival at the axis
10 8 –6 4 2 0
Integral radiation flow through the boundary of the calculation domain R/R0
0.74 0.76 0.78 0.80 T, ns
20 0
0.1
0.2
0.3
0.4 0.5 T, ns
0.6
0.7
0.8
0.9
Fig. 7. Energy radiation flow through the free boundary and degree of liner compression.
20 18 16 14 Heat wave start
ln(S)
12 10 8
Formation of the snow plough
6
Moment of the shock wave arrival at the axis
4 2 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T, ns
Fig. 8. Integral radiation flow through the boundary.
ation depending on time (in the logarithmic scale). At first, the energy losses sharply increase at the begin ning of the motion to the center (25 ns). The next increase of the energy yield occurs in the formation of the heat wave. In its propagation, it increases the temperature of electrons when fillingup, which involves increased losses of energy on radiation. Finally, the next change of the energy yield derivative is caused by the focusing on the shock wave axis (77 ns). On the basis of the obtained numerical data, it is possible to conclude that the processes of radiation are of a rather complicated character. It is impossible to definitely state that the maximum energy losses corresponds roughly to the liner’s achievement of the maximum degree of compression. In the dynamics of radiation, there are local maxima preceding the moment of the focusing on the axis. It is rather that a more detailed analysis of the processes of energy losses on radiation should be based on the application of more accurate mathematical models. CONCLUSIONS Our numerical calculations showed the following regularities in liner compression. With the value of the electron temperature of 250 eV, a heat wave is formed. Its propagation velocity exceeds the velocity of MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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the shock wave. The heat wave reaches the center of the area, thus heating the plasma material up to 250 300 eV, and increasing the average ion charge. The losses on radiation slightly increase.Because of the Hall effect, the magnetic field does not penetrate into the plasma material uniformly. The influence of electron effects becomes even more noticeable at high times: magnetic field tongues are formed.On the basis of the numerical data, it is possible to conclude that a greater part of energy is lost by the pinch on radiation after the shock wave is focused; moreover, the pinch continues being compressed. ACKNOWLEDGMENTS The work was supported by the Russian Foundation for Basic Research, grants 070100381a, 1001 00751a and by the Federal Targeted Program “Research and Academic Personnel” for 2009–2013, GK P954. REFERENCES 1. Repin, P. B., Selemir, V. D., et al, “Investigation of the Influence of Preliminary Explosion of the Wire Liner on Generation of XRay Radiation in the Zpinch Geometry,” Fizika plazmy 35 (1) 48–55 (2008). 2. Grabovskii, Ye. V., Aleksandrov, V. V., et al., “Use of Tapered Wire Assemblies for Simulation of ThreeDimen sional Effects of MHDcompression,” Fizika plazmy 34 10 (2008). 3. G.V. Ivanenkov, G. V., Stepnevsi, and Gus’kov, S. Yu., “MHDprocesses of Cascade Development of the Neck and Hot Point Flashes in the X pinch,” Fizika plazmy 34 (8) 675–694 (2008). 4. Garanin, S. F. and Mamyshev, V. P., “TwoDimensional MHD Simulation of Operation of the Plasma Focus with Account of the Accelerating Mechanism of Neutrons Generation,” Fizika plasmy 34 (8) 695–706 (2008). 5. Imshennik, V. S. and Bobrova, N. A., Dynamics of the Collision Plazma (Energoatomizdat, Moscow, 1997) [in Russian]. 6. Kingsep, A. S., Chukbar, K. V., and Yan’kov, V. V., “Electron Magnetic Hydrodynamics,” Issues of the Theory of Plasma (Atomizdat, Moscow, 1987), Number 16, pp. 243–291. 7. Tolstik, D. A., “Features in the Magnetic Field Diffusion in NnHomogeneous Plasma in EMHApproxima tion,” Fizika plazmy 34 (6) 556–562 (2008). 8. Kubesh, P., Korolev, V. D., et al., “Generation of Neutrons in Compression of the Wire Liner on Deuterated Fibre,” Fizika plazmy 34 (1) 57–65 (2008). 9. Morozov, A. I., Introduction into NonLinear Plasmodynamics (Fismatlit, Moscow, 2006) [in Russian]. 10. Braginskii, S. I., Issues of Theory of Plasma (Atomizdat, Moscow, 1969), Number 1 [in Russian]. 11. Post, D. E., Jensen, R. V., Tarter, C. V., et al., “SteadyState Radiative Cooling Rates for LowDensity High Temperature Plasmas,” PPPL1352 ( Princeton Univ., 1977). 12. Kingsep, A. S., Kovalenko, I. V., Lobanov, A. I., et al., “Simulation of Fast Plasma Flow Disengager in the Mode of electron Magnetic Hydrodynamics,” Mat. Mod. 16 (10) 93–106 (2004). 13. Yanenko, N. N. and Kovenia, V. M., Method of Splitting in Problems of Gas Dynamics (Nauka, Novosibirsk, 1981) [in Russian]. 14. Samarskii, A. A., Koldoba, A. V., Poveshchenko Yu. A., et al. Difference Schemes on Irregular Grids (Kriterii, Minsk, 1996) [in Russian]. 15. Gasilov, V. A., Krukovskii, A. Yu., and Otochin A. A. (Software Package for Calculation of TwoDimensional AxisSymmetrical Flows of Radiation Gas Dynamics) Preprint, 1990. 16. Gasilov, V. A. and Golovisnin V. M. (The Use of Newton Method for Solution of Difference Equations of Hydrodynamics) Preprint No. 100. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 1978) 17. A.A. Samarskii, A. A. And Popov, Yu. P., Difference Methods for Solution of the Problems of Gas Dynamics (Nauka, Moscow, 1980) [in Russian]. 18. Library of Numerical Analysis of the SRCC MSU. http://www.srcc.msu.su/num_anal/lib_na/cat/cat5.htm 19. Knupp, P. Margolin, and L. Shashkov M., “Reference Jacobian OptimizationBased Rezone Strategies for Arbitrary Lagrangian Eulerian Methods,” Journal of Computational Physics, No. 176, 93128 (2002). 20. Charakhch’yan, A. ”A Variational Form of the Winslow Grid Generator”, No. 136, 385–398 (1997). MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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