Quantum effects on Rayleigh Taylor instability in magnetized plasma

Transcription

1 Quantum effects on Rayleigh Taylor instability in magnetized plasma Jintao Cao, 1 Haijun Ren, 1 Zhengwei Wu, 1,,a and Paul K. Chu 1 CAS Key Laboratory of Basic Plasma Physics, University of Science and Technology of China, Hefei 3006, People s Republic of China Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong PHYSICS OF PLASMAS 15, Received 16 October 007; accepted 14 December 007; published online 4 January 008 The effects of the quantum mechanism and magnetic field on Rayleigh Taylor RT instability in an ideal incompressible plasma are investigated. The explicit expression of the linear growth rate is obtained in the presence of fixed boundary conditions. It is shown that the magnetic field has a stabilizing effect on RT instability similar to the behavior in classical plasmas and RT instability is affected significantly by quantum effects. Quantum effects are also shown to suppress RT instability with the appropriate physical quantities. Some astrophysical parameters are discussed as an example to investigate the new effects. 008 American Institute of Physics. DOI: / I. INTRODUCTION The Rayleigh Taylor RT instability occurs at the plane interface between two fluids of different densities whenever the heavy is accelerated by the light ones or superimposed onto the latter fluid. 1 4 Slight perturbation at the interface leads to tangential acceleration and is amplified by material flowing downward under the influence of this force. It is magnified into a set of interpenetrating structures as the heavy materials moves downward and the light ones flows upward. This results in fundamental hydrodynamic instability that plays a critical role in the implosion of inertial confinement fusion ICF capsules. Previous studies on RT instability based on the sharp boundary model SBM show that the small perturbations accumulate linearly in terms of e t in a simple case where there is no shear flow or ablation. 1,4 6 Here, = AT kg is the linear growth rate of RT instability, A T = 1 / + 1 is the Atwood number and 1 are the densities of the heavy and the light fluids, respectively, k=/ is the perturbation wave number, and g is the gravitation, i.e., acceleration. At the ablation front, the classical growth rate is modified as described, for example, by Takabe. 7 The SBM is effectively used to describe RT instability in an ablating plasma while other density distribution conditions such as exponential density distributions are also frequently used to study for some other physical processes. The linear growth rate of RT instability has been obtained by Goldston 8 under fixed boundary conditions. The equilibrium flow and magnetic field effects on RT instability have also been studied using exponential density distributions. 9 Studies on RT instability show that this instability is troublesome because it obstructs the realization of ICF. 10,11 In astrophysics, this instability is related to the stellar structure and evolution Hence, it is important to understand the physical mechanisms that can affect such instability, especially its suppression as well as the details of this instability. Such knowledge will aid our understanding of the origin of white dwarfs and type-ia supernovas. As a new emerging area in plasma physics, quantum plasmas have received a great deal of attention see Refs , and references therein. It is well know that quantum effects become important in the behavior of the charged plasma particles when the de Broglie wavelength of the charge carriers becomes equal to or greater than the dimension of the quantum plasma system. 19 Quantum plasmas can be composed of electrons, ions, positrons, holes and/or grains. Two well-known models are used to study quantum plasmas systems. The first one is the Wigner model which describes the statistical behavior of plasmas based on the Wigner Poisson system. The other is the Hartree model which describes the hydrodynamic behavior of plasmas based on the Schrödinger Poisson system. 19,0 The quantum hydrodynamic QHD model 1 describing the transport of charge, momentum and energy in plasmas has been introduced to semiconductor physics and astrophysics. 3 5 Quantum effects also appear in ultrasmall electronic devices 6 and strong laser plasmas. 7,8 The quantum magnetohydrodynamic QMHD model has been obtained by Haas 9 with the help of QHD with magnetic fields based on the Wigner Maxwell equations. 30 Recently, many investigations in quantum plasmas see Refs. 17, 18, and 31 4 have been extensively studied with quantum effects corrections. A range of classical subjects such as waves and stream instability were studied again in quantum plasmas systems. The present work is based on an ideal incompressible magnetized plasma described by the QMHD model. 9 Here, we study the RT instability by ignoring shear flow or ablation effects and investigate the effects introduced by the quantum mechanism and magnetic field. The second-order ordinary differential equations ODE are gained to describe the velocity perturbation with quantum effects corrections and magnetic field. We solve the ODE by using the fixed boundary condition and obtain an explicit analytical expression of the growth rate of RT instability. The effect of magnetic field on RT instability is discussed and proved to be coincident with the previous result. 4,9,43 Then quantum effects are disa Author to whom correspondence should be addressed. Electronic mail: wzhengwei@student.cityu.edu.hk X/008/151/01110/6/$ , American Institute of Physics

2 Cao et al. Phys. Plasmas 15, cussed and proved to restrain RT instability. We give some parameters to measure the importance of quantum effects on RT instability. One dense astrophysical case 13,14 is used as an example to explain our results. The manuscript is organized as follows: In Sec. II, we describe the QMHD model for ideal incompressible plasmas. The growth rate of RT instability is derived in Sec. III. Discussions on the effects of magnetic field and quantum mechanism on RT instability are presented in Sec. IV, and Sec. V contains a brief summary. II. MATHEMATICAL MODEL In this piece of work, an ideal incompressible quantum plasma comprising electrons and single charged ions is considered. We assume that the plasma is immersed in a magnetic field B 0. The ideal QMHD equations are 9 d + u =0, dt 1 du dt = p + g + 1 B B + m e m i E = B t, E = ub + em e,, where is the mass density and p is the thermal pressure. m e is the mass of electrons and m i is the mass of ions. e is the magnitude of the electrons charge, d/dt=/t+u is the convective derivative. E, B, and g are the electric field, magnetic field, and gravitational acceleration, respectively. The last -dependent term in the right-hand side of Eq., which is called Bohm potential term, is due to quantum corrections in the density fluctuation. The classical magnetohydrodynamic MHD results can be recovered in the limit of =0. Since the fluid is assumed to be incompressible, we have u =0. Every quantity representing u, B,, p is supposed to have the following form: = + 1, where is the unperturbed value and 1 is a small perturbation, 1 zexp it + ikx. Here, k is the wave number and is the complex oscillating frequency. The perturbed velocity is assumed to take the form u 1 =u x1 ê x +u y1 ê y +u z1 ê z, where ê x ê y,ê z is the unit vector along the xy,z direction. Each scalar component of the perturbed velocity is proportion to exp it+ikx. The density and pressure gradients are both set opposite to the gravitational acceleration which is assumed to be along the direction of the positive z axis. Therefore, it is reasonable to assume that the inhomogeneity occurs only in the z direction. Besides, equilibrium flow is neglected in this study and hence, the equilibrium profiles can be expressed in the following form: = z and p 0 = p 0 z, 8 B 0 = zê x + B y0 zê y, u 0 = E 0 = Furthermore, dissipation effects arising from resistance and viscosity are ignored in our study for simplicity. III. ANALYTICAL DEVELOPMENT The first-order linearized equations derived from Eqs. 1 5 are 1 t u 1 t B 1 t + u 1 =0, 11 = 1 g p 1 + B 0 B B 0 B 1 + B 1 B 0 + H 1, 1 = B 0 u 1 u 1 B 0, 13 u 1 =0, 14 where the last term in the right-hand side of Eq. 1 is the first order perturbation of Bohm potential term, with the exact expression being H 1 = m e m i A 1, where A 1 = The incompressible condition and divergence-free property of magnetic field, B=0, are employed. Equation 13 describes the motion of perturbed magnetic field. As the perturbed quantity is proportional to expikx it, we obtain the perturbed magnetic field as

3 Quantum effects on Rayleigh Taylor instability Phys. Plasmas 15, B 1 = k u 1 + u z1 db 0 i. 16 * + k B d ũ z1 + d * + k A dũ z1 By using thus formula, we obtain the tension provided by the magnetic field on the right-hand side of Eq. 1 as B 0 B 1 + B 1 B 0 = ik u The incompressible condition from Eq. 14 now becomes iku x1 + u z1 z =0, and the continual Eq. 11 can be expressed as 18 = k * g + C d ũ z1, where we have denoted A = B = 1 d 4m e m i d 0 d, 1 4m e m i d 0, 3 i 1 + d u z1 =0. 19 and Inserting Eq. 17 and the perturbed mass density 1 derived from the formula above into Eq. 1, we obtain k u 1 = ip 1 + B 0 B 1 + d u z1g + ih 1. 0 Now we carry out the following manipulation ê y 0, through which the calculation can be greatly simplified by eliminating the perturbed thermal pressure and magnetic pressure terms in Eq. 0 without loss of generality of the results. Consequently, we have ê y k u 1 = ê y d z1g u + i ê y A 1. m e m i 1 By substituting A 1 from Eq. 15 and perturbed velocity u 1 and mass density 1 from above into Eq. 1, we obtain the second-order ODE for the dispersion relation in quantum magnetized plasmas, d z1 d ũ with k G = = k k + k d d 4m e m i 1 d g + G dũ z1 d z1 ũ 4m e m i k d + d 1 d. An early result 43 is recovered if we ignore quantum effects by setting =0 in Eq.. Suppose * z = k B x0 /, the above equation can be written as C = k d 4m e m i. Quantum effects are represented by the three parameters. According to Eq. 3, the growth rates of RT instability under different boundary conditions can be obtained with quantum effects corrections. To solve the second order differential equation 3, it is necessary to define the mass density distribution z. In that literature, it is generally assumed that has an exponential dependence on z. In this work, we follow this way and suppose that the mass density distribution satisfies z= f exp z/l D, where f is the mass density at z=0. From Eq., we are allowed to suppose that the magnetic field z has the similar z-dependent distribution as z = B f exp z/l D, 4 where B f is the magnetic field component along the x direction at z=0. Inserting and into Eq. 3, we obtain k v af q d ũ z1 1 k v L af q dũ z1 D = k k v af + 1 g L q L D ũ z1, D 5 where q =k /L D 4m e m i represents quantum effects and v af =B f / f 1/ is Alfvén speed. As we have used the exponential density distribution, correspondingly, the fixed boundary condition ũ z1 0=ũ z1 h=0 should be adopted to solve the equation above. That means ũ z1 = ũ 0 sin nz h expz. Substituting the above formula into Eq. 5, we gain

4 Cao et al. Phys. Plasmas 15, = 1 L D, k v af k L D q k L D n h n h k g L D =0. 6 By inserting the value of into Eq. 6, we obtain the frequency, = 4gh k L D h +4h k L D +4L D n + q + k v af. 7 This is the dispersion relation of RT instability in quantum magnetized plasmas. When quantum effects corrections are eliminated by setting q equal to zero, the previous results are recovered. 9 IV. DISCUSSIONS The effects of the quantum mechanism and equilibrium magnetic field on RT instability can be discussed by using Eq. 7 obtained in Sec. III. That equation is the explicit analytical expression of the growth rate, imaginary part of, with quantum effects corrections and magnetic field effect. For mathematical convenience, we introduce some dimensionless variables to illuminate this problem clearly, H = /4m e m i pe L 4 D, v * af =v af /L D pe, g * =g/l D pe, K * =k L D, and h * =h/l D with pe = f e /m e 1/, which is the plasma frequency. These parameters describe the importance of quantum effects, equilibrium magnetic field, gravitational acceleration, wave number, and plasma shell thickness, respectively. Hence, we gain the normalized dispersion as follows: * = = 4h * K * g * pe 4h * K * + h * +4n + K* H + K * v * af. 8 Thus we obtain = * = 4h * K * g * 4h * K * + h * +4n K* H K * v * af, 9 where is the square of normalized growth rate of RT instability. In order to see how quantum effects corrections and magnetic field affect the growth rate of RT instability more explicitly, we plot the relationship between the normalized growth rate of RT instability and the square of the normalized wave number K * in Figs The effects of the magnetic field on RT instability are first discussed. Equation 9 indicates that there is /v * af = K *, which implies that the growth rate decreases monotonically as Alfvén speed v * af increases. In other words, Rayleigh Taylor instability can be suppressed by the magnetic field, as confirmed by Fig. 1. These stabilizing effects are provided by FIG. 1. Dependence of the value of on the square of the normalized wave number for quantum case H=0.5 while the value of v * af varies. Other parameters used in our plotting are: g * =10, n=1, and h * =1. the Lorentz force. 4 Our result shows great consistency with previous results such as in Refs. 4, 9, and 43. Next, we focus our attention on the quantum effects on RT instability. From Eq. 9, we can obtain /H = K *. Similarly, it is concluded that the growth rate is a monotonic function that decreases as quantum effects are enhanced. It is known that quantum effects are introduced by mass density fluctuation. The Bohm potential term provides a force in the same direction as the density gradient, which is in the negative z direction against the gravity. Obviously, the quantum mechanism has the same stabilizing effect on RT instability as the magnetic field. This is confirmed in Fig.. If the last term on the right-hand side of Eq. 9 is large enough, the instability will be totally restrained and not occur. It means that if the normalized wave number becomes FIG.. Dependence of the value of on the square of normalized wave number for both classic case H=0 and quantum cases H=0.1,0.,0.5. The value of v * af is fixed as 0.1. All other parameters are the same as in Fig. 1.

5 Quantum effects on Rayleigh Taylor instability Phys. Plasmas 15, FIG. 3. Dependence of the value of on the square of the normalized wave number for classic case H=0 and quantum case H=0.5 with different values of v * af. All other parameters are the same as in Fig. 1. large enough, the parameter H becomes large enough and/or equilibrium magnetic field is strong enough, the perturbation becomes stable. Therefore, we define K c1 as the critical wave number at which =0. If K * K c1, and perturbation is suppressed and becomes a steady wave. We derive the expression of K c1 as K c1 = g * H * + v 1 af 4 n h *. 30 This formula implies that K c1 decreases as H * and/or v af increases. The threshold becomes smaller and easier to be reached as quantum effects and/or the effect of the magnetic field is enhanced. Accordingly, the RT instability can be quenched easily, as be confirmed in Figs. 1 and. Byneglecting the quantum effects term and magnetic field terms in Eq. 9, does not increase indefinitely with k. It approaches a definite limit g *. K c1 has a limited value, implying that RT instability will occur only in the long wave region. The instability appearing in the short wave region is easily restrained by the magnetic field and quantum effects. Finally, the dependence of growth rate on the normalized wave number is discussed. Figure 3 shows that the growth rate increases monotonically as wave number K * increases under classical conditions. However, in the quantum case illustrated in these three figures, the relationship becomes more complicated. When K * is small, the growth rate increases as K * increases. On the other hand, the growth rate decreases as K * increases when K * is large enough. In fact, it is easy to find out from Eq. 9 that K * = 1/4+n /h * g * K * +1/4+n /h * H v * af, 31 which implies that the growth rate increases monotonically as K * increases when 0K * K c, where K c =1/4 +n /h * g * /H +v * af 1/ 1/4+n /h *. Therefore, growth rate reaches its maximum value max =g * +1/4 +n /h * H +v * af g * 1/4+n /h * H +v * af 1/ at K * =K c. When K * K c, the growth rate starts to decrease as K * increases. These are confirmed in Figs. 1 3, especially Fig. 3. In order to discuss physical problems more explicitly, we define the ratio of the second term to the first term in the right-hand side of Eq. 9 as. The quantum effects corrections can be neglected safely if is much less than 1. We insert the following parameters into our numerical calculation: m e = kg, m i =1m p m p = kg is the mass of proton, and = J s. Setting h * =1 and n=1, we discuss two physical conditions. For the k L D 1 case, from Eq. 9 we gain k gl D. Another instance is k L D O1 or k L D 1, then one gets gl D 3. The typical value of the effective gravitational acceleration g is 10 6 ms in white dwarfs and some supernovas, and there are some flames with the density-scale length L D about m which are involved in RT instability In these circumstances, k. If the wave number k is greater than 10 6 m 1, that is, the very short wavelength perturbations, quantum effects should be take into consideration to discuss RT instability. Following the same parameters, we obtain Therefore, quantum effects are remarkable for RT instability due to the flames of white dwarfs. In Eq. 30, we notice that the RT instability can be completely suppressed by quantum effects and magnetic field. To discuss quantum effects on the normalized cut-off wave number K c1 in Eq. 30, we can neglect magnetic field by setting v * af equal to zero. Using the same physical parameters as above, we obtain k c1 = 4m em i gl D 1 4L n D h L gl D 10.1 D L, D where k c1 =K c1 /L D is the cut-off wave number. Then we obtain k c m 1. This means that the short wavelength perturbations for the RT instability due to the flames of white dwarfs will be calmed by quantum effects. We define the ratio of H to v * af in Eq. 30 as B f L D, where a typical mass density f =10 5 kg m 3 in ICF capsules is adopted. Considering the self-generated magnetic fields, 44 we get 1 and the effects of magnetic field on RT instability are more important compared to quantum effects in ICF experiments. V. CONCLUSION In this work, based on the quantum magnetohydrodynamic model, we investigate the RT instability in an ideal incompressible quantum magnetized plasma and obtain the second-order ODE with equilibrium magnetic field and

6 Cao et al. Phys. Plasmas 15, quantum effect corrections. Using the exponential density distribution and fixed boundary conditions, we derive the analytical expression of the growth rate of RT instability. The effects due to quantum mechanism and magnetic field on RT instability are discussed. Our results show that the equilibrium magnetic field in quantum plasmas has the same effect on RT instability as in classical plasmas without quantum effect corrections. On the other hand, our results indicate that the RT instability is affected significantly by quantum effects. Quantum mechanical effects are shown to suppress the RT instability in proper circumstances. We use an example in the celestial bodies to discuss the importance of quantum effects on RT instability and find that the very short wavelength RT instability is calmed. This is a new feature in RT instability study in astrophysics, space physics, and dusty plasmas. Furthermore, it is shown that the self-generated magnetic field plays a more significant role in ICF experiments than quantum effects. ACKNOWLEDGMENTS This work was supported by the Chinese National Natural Science Foundation under Grant No and City University of Hong Kong Direct Allocation Grant No Lord Rayleigh, Scientific Papers Cambridge University Press, Cambridge, 1900, Vol. II, p. 00. G. I. Taylor, Proc. R. Soc. London, Ser. A 01, S. Bodner, Phys. Rev. Lett. 33, S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability Dover, New York, J. G. Wouchuk and A. R. Piriz, Phys. Plasmas, A. R. Piriz, J. Sanz, and L. F. Ibanz, Phys. Plasmas 4, H. Takabe, L. Montierth, and R. L. Morse, Phys. Fluids 6, R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics Institute of Physics, London, 1997, p Z. Wu, W. Zhang, D. Li, and W. Yang, Chin. Phys. Lett. 1, J. Sanz, Phys. Rev. Lett. 73, J. D. Lindl, Phys. Plasmas, W. H. Cabot and A. W. Cook, Nat. Phys., F. X. Timmes and S. E. Woosley, Astrophys. J. 396, S. Blinnikov and E. Sorokina, Astrophys. Space Sci. 90, D. Shaikh and P. K. Shukla, Phys. Rev. Lett. 99, J. Lundin, J. Zamanian, M. Marklund, and G. Brodin, Phys. Plasmas 14, A. Bret, Phys. Plasmas 14, S. Ali, W. M. Moslem, P. K. Shukla, and R. Schlickeiserd, Phys. Plasmas 14, G. Manfredi, Fields Inst. Commun. 46, G. Manfredi and F. Haas, Phys. Rev. B 64, E. Madelung, Z. Phys. 40, C. Gardner, SIAM J. Appl. Math. 54, M. Marklund, G. Brodin, and L. Stenflo, Phys. Rev. Lett. 91, R. Bingham, J. T. Mendonca, and P. K. Shukla, Plasma Phys. Controlled Fusion 46, R M. Marklund, P. K. Shukla, L. Stenflo, G. Brodin, and M. Servin, Plasma Phys. Controlled Fusion 47, L5005; M. Marklund and P. K. Shukla, Rev. Mod. Phys. 78, P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations Springer, Vienna, D. Kremp, Th. Bornath, M. Bonitz, and M. Schlanges, Phys. Rev. E 60, M. Marklund, D. D. Tskhakaya, and P. K. Shukla, Europhys. Lett. 7, F. Haas, Phys. Plasmas 1, T. B. Materdey and C. E. Seyler, Int. J. Mod. Phys. B 17, N. Suh, M. R. Feix, and P. Bertrand, J. Comput. Phys. 94, G. Manfredi and M. Feix, Phys. Rev. E 53, B. Shokri and A. A. Rukhae, Phys. Plasmas 6, B. Shokri and S. M. Khorashady, Pramana 61, A. Luque, H. Schamel, and R. Fedele, Phys. Lett. A 34, L. G. Garcia, F. Haas, L. P. L. de Oliveira, and J. Goedert, Phys. Plasmas 1, F. Haas, G. Manfredi, and M. Feix, Phys. Rev. E 6, D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, S. Ali and P. K. Shukla, Eur. Phys. J. D 41, B. Shokri and A. A. Rukhae, Phys. Plasmas 6, F. Haas, L. G. Garcia, J. Goedert, and G. Manfredi, Phys. Plasmas 10, P. K. Shukla and S. Ali, Phys. Plasmas 13, W. Zhang, Z. Wu, and D. Li, Phys. Plasmas 1, J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11,