Flow oscillations in radial expansion of an inhomogeneous plasma layer

Physics Letters A 375 (2011) 2629–2636

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Physics Letters A www.elsevier.com/locate/pla

Flow oscillations in radial expansion of an inhomogeneous plasma layer A.R. Karimov a , M.Y. Yu b,c,∗ , L. Stenflo d,e a

Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412, Russia Institute for Fusion Theory and Simulation, Department of Physics, Zhejiang University, 310027 Hangzhou, China c Institut für Theoretische Physik I, Ruhr-Universität Bochum, D-44780 Bochum, Germany d Department of Physics, Linköping University, SE-58183 Linköping, Sweden e Department of Physics, Umeå University, SE-90187 Umeå, Sweden b

a r t i c l e

i n f o

Article history: Received 26 March 2011 Received in revised form 20 May 2011 Accepted 25 May 2011 Available online 27 May 2011 Communicated by F. Porcelli

a b s t r a c t The cylindrically symmetric radial evolution of an inhomogeneous plasma layer expanding into vacuum is investigated nonperturbatively by first determining the spatial structure of the plasma flow structure. The evolution is then governed by a set of ordinary differential equations. The effect of the plasma inhomogeneity on the nonlinear coupling among the electron and ion flow components and oscillations is investigated. © 2011 Elsevier B.V. All rights reserved.

Keywords: Plasma-layer expansion Nonperturbative analysis Nonneutrality

1. Introduction Expansion of plasma into vacuum remains a topic of great interest in physics because of its importance in areas such as laserplasma interaction, astrophysical and space phenomena, intense detonations, etc. [1–14]. Many of the existing works are concerned with the long-time, usually self-similar, stage of the expansion process. The corresponding asymptotic solutions generally do not contain information about the initial state and behavior of the expanding process. Since the latter is usually highly nonlinear during the initial stage, it is often difficult to identify the initial state leading to a particular asymptotic solution, and if the latter is attainable from a certain initial state and/or unique. Moreover, often one is also interested in the short-time behavior of the expansion and the sensitivity of the expansion dynamics to the initial conditions, especially when the system can be far from equilibrium [5,6,10]. In the present Letter, we consider electrostatic expansion of an inhomogeneous plasma layer into vacuum in cylindrical geometry by solving the electron and ion fluid equations together with the Poisson equation. Plasma expansion, together with possible electron and ion oscillations and their nonlinear coupling, are investigated nonperturbatively. The problem is treated by first constructing a spatial structure of the motion of the electron and ion fluids. The

*

Corresponding author at: Institute for Fusion Theory and Simulation, Department of Physics, Zhejiang University, 310027 Hangzhou, China. E-mail addresses: [email protected] (A.R. Karimov), [email protected] (M.Y. Yu), lennart.stenfl[email protected] (L. Stenflo). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.05.053

time evolution of the system is then reduced to a set of ordinary differential equations. The approach is somewhat similar to that of separation of variables for solving linear partial differential equations [15], except that here the spatial dependence is predetermined by trial and error [16–23]. This implies that the number of problems that can be handled by this approach is somewhat restricted. Nevertheless, as shown here, new physically significant results can be obtained. 2. Formulation We shall first obtain a basis set of solutions by considering cylindrically symmetric radial expansion of an initially inhomogeneous nondissipative plasma consisting of cold electrons and ions. We also assume that there is no magnetic field, and that at t = 0 the plasma occupies a finite volume, where its density is given by

n0s (r ) = N 0s + D 0s /r ,

for a0  r  b0 ,

(1)

where the subscripts s = e and i denote electron and ion quantities, respectively. The constants a0 , b0 , N 0s , and D 0s define the initial density profile of the plasma shell layer. 1 2 1/ 2 It is convenient to use a0 , N 0e , and ω− , pe = (me /4π N 0e e ) where e and me are the charge and mass of the electrons, as the scalings for the length, density, and time of the expansion process. Since there is no background nor self-consistent magnetic field, charge continuity can be expressed as,

∂t E + 4π e (ni v i − ne v e ) = 0,

(2)

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where E is the self-consistent radial electric field, −e is the charge of the electron, ne , ni , and v e , v i are the densities and radial velocities of the electron and ion fluids, respectively. The ions are singly charged. To avoid possible singularity at the axis, we require that the initial velocities v 0e and v 0i be much smaller than v c , where

v c = ω pe a0 ,

(3)

so that the governing equations in dimensionless form are

Eqs. (9)–(13) and (16) can be considered as the base for investigating the fully nonlinear evolution of the expansion of a cylindrical plasma layer. It is instructive to first consider the expansion of a neutral fluid, which can be realized by setting E r = 0, N e (t ) = N i (t ) ≡ N (t ), and μe = μi = 0, so that D e (t ) = D i (t ) ≡ D (t ). Setting A 0 (t ) = A 0e (t ) = A 0i (t ) and A 1 (t ) = A 1e (t ) = A 1i (t ), we can obtain from Eqs. (9)–(13) and (16)

A 0 (t ) = A 0 (0)θ(t ),

∂t (rns ) + ∂r (rns v s ) = 0,

(4)

A 1 (t ) = A 1 (0)θ(t ),

∂t v s + v s ∂r v s = μs E ,

(5)

N (t ) = N (0)θ 2 (t ),

∂r (r E ) = r (ni − ne ),

(6)

D (t ) = D 2 θ(t ) + D 1 θ 2 (t ),

where μe = −1 and μi = me /mi . The velocities v s have been normalized by a0 ω pe and the electric field E by 4π eN 0e a0 . We look for exact solutions of Eqs. (4)–(6) that satisfy the initial condition (1). Accordingly, we set

v s = A 0s (t ) + A 1s (t )r ,

ns = N s (t ) + D s (t )/r ,

(7)

which together with Eq. (4) yield the relation

dt D s + A 1s D s + A 0s N s + (dt N s + 2N s A 1s )r = 0,

(8)

which is valid for any r. Thus the coefficient of each power of r should be equated to zero, or

dt D s + A 1s D s + A 0s N s = 0,

(9)

dt N s + 2N s A 1s = 0.

(10)

From Eqs. (6) and (7), we can obtain

E r = ε0 + ε1 r = D i − D e + ( N i − N e )r /2.

(11)

Substitution of (11) and (7) into (5) leads to

dt A 0s + A 1s A 0s = μs ( D i − D e ), dt A 1s +

A 21s

(12)

= μs ( N i − N e )/2.

(13)

Eqs. (9), (10), (12), and (13) describe the fully nonlinear evolution of the plasma layer that occupied the region a0  r  b0 at t = 0. It if of interest to see how does these initially given boundaries evolve. If there is no particle source or sink in the region of interest, during the expansion the total particle numbers

bs N s = 2π

ns r dr ,

s = e, i

(14)

as

must be conserved, or ∂t Ns = 0. Here as (t ) and b s (t ) are the boundaries of the expanding ion and electron layers and are determined as follows. From Eq. (4) and (14) we obtain









as dt as − v s (t , as ) − b s dt b s − v s (t , b s ) = 0,

(15)

which should be satisfied for arbitrary as (t ) and b s (t ). Thus, we have

dt as = v s (t , as ) and dt b s = v s (t , b s ),

(16)

which defines as (t ) and b s (t ). Inserting the velocity Ansatz (7) into (16), we obtain

dt as = A 0s + as A 1s ,

and dt b s = A 0s + b s A 1s ,

(17)

so that the dynamics of the expansion, including that of the electron- and ion-layer boundaries ae,i (t ) and be,i (t ), are fully determined.

(18)

where θ(t ) = 1/[1 + A 1 (0)t ], D 1 = A 0 (0) N (0)/ A 1 (0), and D 2 = D (0) − A 0 (0) N (0)/ A 1 (0). 3. Expansion of neutral layer In Fig. 1 the expansion dynamics of the neutral fluid layer satisfying the condition (3) is shown. In this and the following figures, the velocity parameters A 0 (t ) and A 1 (t ) have been renormalized by A 0 (0). It is worth noting that if A 0 (0) < 0 or/and A 1 (0) < 0, the expansion can be singular. These cases can be unphysical within the scope of the present model. We now relax the neutrality condition. For ease of comparison and search for physical solutions we shall consider plasma expansion with initial conditions close to that of the basis flow discussed above. For the same reason we shall set μi = 10−5 in the following cases. For the case where the plasma flow has exactly the same initial conditions as the neutral expansion case above, i.e., A 0e = A 0i (0) = A 0 (0), A 1e (0) = A 1i (0) = A 1 (0), and N e (0) = N i (0) = D e (0) = D i (0) = N 0 (0), the numerical solutions of the governing equations show the same expansion behavior as in Fig. 1 for the neutral fluid. This can be expected since in the absence of external forces an initially perfectly neutral plasma layer will remain neutral during the expansion. However, any small deviation from the ideal initial state discussed above can lead to different plasma behavior. To see this, we impose at t = 0 a small perturbation in the electron velocity. The ions remain unperturbed. The results are shown in Figs. 2 and 3. One can see that during the expansion large-amplitude electron velocity oscillations occur without any perturbation in the electron or ion density. Note that the oscillations are initiated only in A 0e (curves 1 in Figs. 2 and 3). This result can be attributed to the fact that the temporal dependence of the densities given by Eqs. (9) and (10) depends only on time integrals of A 0s and A 1s , so that oscillations in the latter are smoothed out [20]. That is, here a temporal nonlocal effect is involved. We can also see that the large-amplitude oscillations in A 0e appear because of strong coupling between A 0e and A 1e via the term A 0e A 1e in Eq. (12). On the other hand, the variation of A 1e is only determined by the electric field, as given by Eq. (13). Such a behavior is made possible by the inhomogeneity, which allows the existence of a velocity field like (7). We now examine the effect of a small initial electron density perturbation, or more precisely, charge separation. Figs. 4 and 5 illustrate what typically happens to the plasma expansion when initially there is a slight deviation from the perfect neutrality condition via a small difference between N e (0) and N i (0) and between D e (0) and D i (0). Curve 1 in Fig. 4 shows that a small perturbation in N 0e can have a large effect on the plasma expansion. In particular, highly nonlinear oscillations in A 0e appear. On the other hand, curve 1 in Fig. 5 shows that an initial perturbation in D 0e leads only to high-frequency plasma oscillations in A 0e , but does not affect the overall behavior on the slow expansion time scale.

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Fig. 1. The evolution of A 0 / A 0 (0) (curve 1), A 1 / A 0 (0) (curve 3), N / N (0) (curve 5), D / D (0) (curve 7) for a neutral-fluid (μe = μi = 0) layer under the initial conditions A 0 (0) = 10−3 , A 1 (0) = 9 × 10−4 , and N (0) = D (0) = 1. The figure is also valid for an expanding plasma (μe = −1 and μi = 10−5 ) layer with the initial conditions A 0e,i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , and N e (0) = N i (0) = D e (0) = D i (0) = 1. That is, the electron and ion fluids have exactly the same initial conditions as that of the neutral fluid above. For this case, the curves 1, 3, 5, 7 represent A 0e,i / A 0e (0), A 1e,i / A 0e (0), N e,i / N e,i (0), and D e,i / D e,i (0), respectively. The odd numbers used for the curves here are consistent with the numbering of the curves in the following figures.

Fig. 2. The evolution of a plasma layer for the initial conditions A 0e (0) = 1.1 × 10−3 , A 0i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , and N e (0) = N i (0) = D e (0) = D i (0) = 1. That is, the initial conditions are the same as that in Fig. 1, except that now A 0e (0) and A 0i (0) are slightly different. Here and in the following figures, the curves 1–8 are for A 0e / A 0e (0), A 0i / A 0e (0), A 1e / A 0e (0), A 1i / A 0e (0), N e / N e (0), N i / N i (0), D e / De (0), and D i / D i (0), respectively.

The behavior of the other parameters remain similar to that for the case of perfectly neutral expansion shown in Fig. 1. As mentioned, the back and front boundaries ae,i (t ) and be,i (t ) of the expanding plasma layer are fully determined by the other flow parameters. As an example, in Fig. 6 we show the evolution

of the plasma-layer boundaries for the case of Fig. 4. One can see that the layer gradually becomes broadened, as to be expected. Since it is difficult to see the effect of each flow component in the exact evolution equations (9), (10), (12), and (13), it is instructive to simplify the governing equations by considering the

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Fig. 3. The evolution of a plasma layer for the initial conditions A 0e (0) = A 0i (0) = 10−3 , A 1e (0) = 9.1 × 10−4 , A 1i = 9 × 10−4 , and N e (0) = N i (0) = D e (0) = D i (0) = 1. That is, only A 1e and A 1i are initially slightly different.

Fig. 4. The evolution of a plasma layer for the initial conditions A 0e (0) = A 0i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , N e (0) = 1.00001, N i (0) = 1, and D e (0) = D i (0) = 1. That is, only N e and N i are initially slightly different. 1 initial stage of the expansion. For t  ω− pe , the expansion process satisfies N i ∼ N (t ), D i ∼ D (t ), A 0i ∼ A 0 (t ) and A 1i ∼ A 1 (t ). For | N i (0) − N e (0)|  1, Eq. (13) can be reduced to dt A 1e + A 21e ∼ 0, i.e., A 1e ∼ A 1 (t ). From (10) it then follows N e = N (t ). As long as A 1 (t )  1, one can replace (9) by dt D e + N A 0e = 0, and rewrite (12) for the electrons as

dt A 0e + A 1 A 0e − D e + D = 0.

(19)

Combining these equations we get





dt2 A 0e + A 1 dt A 0e + N − A 21 A 0e + dt D = 0,

(20)

where A 1 (t ), N (t ), and D (t ) are given by (18). Eq. (20) describes a driven oscillator, whose characteristics are determined by the time-dependent coefficients A 1 (t ) and N (t ) as well as the source dt D (t ), which are from the other flow com-

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Fig. 5. The evolution of a plasma layer for A 0e (0) = A 0i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , N e (0) = N i (0) = 1, D e (0) = 1.0001 and D i (0) = 1. That is, only D e and D i are initially slightly different.

Fig. 6. The evolution of the plasma-layer boundaries for the case in Fig. 4.

ponents. That is, all the flow components can affect the oscillations in A 0e even at the very beginning of the evolution, although the reverse may not hold. Thus, small changes in the initial condition can essentially influence the evolution of the profile and oscillation frequency of A 0e , as can be observed in Figs. 2–5.

4. Expansion of nonneutral layer We now consider the expansion of a non-neutral plasma layer [24,25]. Accordingly, we set A 0e (0) = A 0i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , N e (0) = 0.499, N i (0) = 0.5, and D e (0) = D i (0) = 1. That is, N e and N i differ initially by 0.2%.

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Fig. 7. The evolution of a nonneutral plasma layer for A 0e (0) = A 0i (0) = 10−3 , A 1e (0) = A 1i = 9 × 10−4 , N e (0) = 0.499, N i (0) = 0.5, and D e (0) = D i (0) = 1. Here, N e and N i are quite different at t = 0.

Fig. 8. Same as Fig. 7, but for t < 10. Here one can see that A 0e and A 0i evolve much faster than the other flow quantities. One can see in Fig. 7 that the oscillation amplitude of A 0e can become very large, but that of A 0i remains small.

In Fig. 7, the parameters of the expanding nonneutral layer are shown. One can see that although the initial density difference is extremely small, the evolution appears to be quite different from that of the neutral layers shown in the preceding cases. In particular, the oscillation in electron velocity component A 0e dwarfs the

profiles of the other flow parameters. Actually, both A 0e and A 1e oscillate, as can be seen in Fig. 8 for the initial stage (t < 10) of the expansion, but the oscillation in A 1e remains much smaller during the expansion. The behavior of the other parameters are similar to that of the neutral layer.

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Fig. 9. The evolution of the plasma-layer boundaries for the nonneutral-layer case of Figs. 7 and 8. The electron boundaries oscillate strongly in phase, so that the expanding electron layer behaves like a thick oscillating band. Both front and back boundaries of the ion layer also oscillate, but the oscillation is at much smaller amplitudes and is thus not observable in the figure.

It is also of interest to point out that A 0e can become negative 1 during the expansion. That is, on the ω− pe time scale the electron flow can change directions because of the large-amplitude oscillations. However, this does not affect the overall expansion, except at the layer boundaries. The evolution of the electron and ion front (be,i (t )) and back (ae,i (t )) boundaries of the nonneutral layer are shown in Fig. 9. One can see that as the expanding layer becomes slowly broadened, both the back and front boundaries of the electron layer oscillate. In fact, the boundaries of the ion layer also oscillate, but at much lower amplitudes, and are thus not noticeable in the figure. 5. Discussion and conclusion The dissipationless hydrodynamic description of the expansion of a plasma layer given by Eqs. (4)–(6) is valid provided that the instantaneous scale length L inhom (t ) = be (t ) − ae (t ) remains much larger than the collision mean free path λei = V T e /νei , √ where V T e = T e /me , T e , and νei are the electron thermal speed, electron temperature, and electron–ion collision frequency, respectively. Since for the cases considered the layer thickness and the inhomogeneity scale length always increase during the expansion, the sufficient condition for the validity of the cold fluid approach can be written as

3me2 V T4e 4(2π )1/2 e 4 N i0 Λ

 L inhom (0),

(21)

where Λ is the Coulomb logarithm and L inhom (0) = b0 − a0 . The cold plasma assumption also requires that the thermal force due the pressure gradient in the momentum equations be negligible. Assuming that the electron pressure is given by P e = ne T e , one can see that the pressure force term (∼ O ( V T2e / L inhom )) can be neglected compared to the convection term (∼ O (ω2pe a0 )) if

V T e  vc,

(22)

for the plasma layer. This is also consistent with the requirement in Section 2 for the initial electron and ion velocities. In this Letter we have considered the nonlinear evolution of a non-rotating cylindrically symmetric radially expanding inhomogeneous plasma layer. The effect of radial inhomogeneity on the linear and nonlinear coupling between the excited electron and ion flows and oscillations is isolated by predetermining a specific spatial flow structure. The time evolution is then described by a set of ordinary differential equations for the density components N s (t ) and D s (t ) and velocity components A 0s (t ) and A 1s (t ), which are then solved numerically. As examples, we have obtained a class of solutions (under various initial conditions) for the expansion of an almost neutral plasma layer with a specific plasma flow structure. It is shown that, despite the strong coupling among the different components, for the initial conditions considered oscillations appear mainly in the velocity component A 0e . This behavior is similar to that for a fully nonlinear homogeneous rotating plasma flow, where, depending on the initial conditions, natural oscillations tend to appear mainly in certain degrees of freedom and the energy tends to be transferred into that of rotation [20,21]. Our investigation is based on finding exact solutions of the electron and ion fluid equations. A simple but nevertheless physically meaningful case of the fully nonlinear hydrodynamic expansion of an inhomogeneous plasma layer has been studied. The results here are relevant to the highly nonlinear plasma expansions occurring in the interaction of intense lasers with plasmas, clusters, and solids [5,8,10,26,27], strong ionizing detonations [4,11], as well as energetic cosmic phenomena [4,12,14] and terrestrial and space weather [28]. More elaborate and realistic problems still await investigation. Of particular interest would be to consider the effect of plasma inhomogeneity on the expansion process a collisionless plasma layer, as described by the Vlasov–Poisson or other systems, which can more properly represent the later stages of the expansion when the plasma becomes highly rarefied and can involve other novel physics features [1,2,27,29–32].

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Acknowledgements M.Y.Y. was supported by the National Natural Science Foundation of China (10835003), the National Hi-Tech Inertial Confinement Fusion Committee of China, the National Basic Research Program of China (2008CB717806), and the ITER Program in China (2009GB105005). References [1] A.V. Gurevich, L.V. Pariiskaya, Reviews of Plasma Physics, vol. 10, Consultants Bureau, New York, 1986, and the references therein. [2] C. Sack, H. Schamel, Phys. Rep. 156 (1987) 311. [3] A.V. Gurevich, R.Z. Sagdeev, S.I. Anisimov, Yu.V. Medvedev, Sov. Sci. Rev. A Phys. 13 (1989) 1. [4] Ya.B. Zel’dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover, Mineola, NY, 2002. [5] P. Mora, Phys. Rev. Lett. 90 (2003) 185002. [6] H. Schamel, Phys. Rep. 392 (2004) 279. [7] R. Engeln, S. Mazouffre, P. Vankan, I. Bakker, D.C. Schram, Plasma Sources Sci. Technol. 11 (2002) A100. [8] S. Atzeni, J. Meyer-ter-Vehn, The Physics of Inertial Fusion, Oxford University Press, Oxford, 2004. [9] E.A. Cummings, J.E. Daily, D.S. Durfee, S.D. Bergeson, Phys. Plasmas 12 (2005) 123501.

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