SPHERICAL SHOCK WAVES IN MAGNETO-GAS-DYNAMICS

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 15 Number 1 Spring 27 SPHERICAL SHOCK WAVES IN MAGNETO-GAS-DYNAMICS RAJAN ARORA ABSTRACT. A Spherically symmetric model describing a blast wave in the solar wind caused by the explosive energy release of a solar flare is considered. The solution of this model is obtained using method of multiple time scale. We derived the transport equations for the amplitudes of resonantly interacting high frequency spherically symmetric waves propagating into perfectly conducting gas pervaded by a magnetic field. Also the evolutionary behavior of nonresonant wave modes culminating into shock waves is studied. 1 Introduction The propagation of shock waves under the influence of strong magnetic field constitutes a problem of great interest to researchers in many branches of science especially in studies of coronal heating problem which relates to the question of why the temperature of Sun s corona is millions of kelvins higher than that of its surface. A solar flare accompanied by enhanced coronal temperatures gives rise to the ejection of plasma at a velocity 5-15 km/s into interplanetry space and to a shock wave propagating outward from the sun reaching the orbit of the earth one or two days later. The work of Summers [1] Sharma et al. [2] Kumar et al. [3] Hunter and Ali [4] Tsiklauri [5] Gunderson [6] and Whitham [7] is worth mentioning in this context. We use Asymptotic method for the analysis of weakly nonlinear hyperbolic waves. We mention the work of Moodie et al. [8] He and Moodie [9 12] Sharma and Arora [13] Arora and Sharma [14] in the study of nonlinear hyperbolic waves. Choquet-Bruhat [15] proposed a method to discuss shockless solutions of hyperbolic systems which depend upon a single phase function. Germain [16] has given the general discussion of single phase progressive waves. Hunter and Keller [17] established a general nonresonant multi-wave mode theory which has led to several interesting generalizations by Majda and Rosales [18] and Keywords: Weakly nonlinear hyperbolic waves resonance flare produced shock ideal gas magnetic field. Copyright c Applied Mathematics Institute University of Alberta. 11

2 12 RAJAN ARORA Hunter et al [19] to include resonantly interacting multi-wave mode features. We use the resonantly interacting multi-wave theory to examine small a mplitude high frequency asymptotic waves for one-dimensional unsteady spherical symmetric model incorporating a self-consistent magnetic field describing a blast wave in the solar wind caused by the explosive energy release of a solar flare where at the leading order many waves coexist and interact with one another resonantly and obtain evolution equations which describe the resonant wave interactions inherent in the system. 2 Basic equations The basic equations governing the one-dimensional spherically symmetric motion of an invicid and perfectly conducting gas in a magnetic field can be written as Summers [1] 1 u t + u u x + 1 ρ ρ t + ρ u x + u ρ x + 2ρu x = p x + h h + h2 x ρ x = u x + 2u = x p t + u p x + ρa2 h t + u h x + h u x + hu x = where u is the fluid velocity p is the pressure ρ is the density h is the magnetic field strength a = γp/ρ 1/2 is the speed of sound with γ as the adiabatic exponent t is the time and x is the radial distance. Also all the variables in system 1 are dimensionless. The equation of state for an ideal gas is 2 p = ρr T where R is the gas constant and T is the temperature. Equations 1 may be written in matrix form as 3 U t + A U x + B = where U is the column vector defined by 2ρu 4 U = ρ u p h tr and B = x h2 ρx 2uγp x hu tr x

3 SPHERICAL SHOCK WAVES 13 A is the 4x4 matrix having the components A ij the nonzero ones are as follows 5 A 11 = A 22 = A 33 = A 44 = u A 12 = ρ A 23 = 1 ρ A24 = h ρ A32 = ρ a 2 A 42 = h. System 3 being strictly hyperbolic admits four families of characteristics among them two represent waves propagating in ±x directions with the speed u ± c where c = a 2 + b 2 1/2 represents the magnetoacoustic speed with b = h 2 /ρ 1/2 as the Alfvén speed. The remaining two families form a set of double characteristics representing particle paths propagating with velocity u. We consider waves propagating into a background state U = ρ x p h x tr where p is constant and ρ and h are functions of x. The characteristic speeds at U = U are given by λ 1 = λ 2 = λ 3 = c and λ 4 = c. The subscript refers to evaluation at U = U and is synonymous with a state of equilibrium. 3 Interaction of high frequency waves We denote the left and right eigenvectors of A associated with the eigenvalue λ i by L i and R i. These eigenvectors satisfy the normalization condition L i R j = δ ij 1 i j 4 where δ ij represents the Kronecker s delta. These eigenvectors are given as 6 L 1 = 1 1 c 2 h tr c 2 R 1 = 1 a 2 a2 h L 2 = a 2 1 R2 = 1 1 tr h L 3 = L 4 = ρ 2c ρ 2c 1 2c 2 h 2c 2 R 3 = 1 c tr a 2 ρ b2. h 1 h 2c 2 R 4 = 1 c tr a 2 ρ b2. h 2c 2 We look for asymptotic solution for 3 as of the form 7 U U + U 1 x t θ + 2 U 2 x t θ + O 3 where U 1 is a smooth bounded function of its arguments and U 2 is bounded in x t in a certain bounded region of interest having at most

4 14 RAJAN ARORA sub-linear growth in θ as θ ±. Here θ = θ 1 θ 2 θ 3 θ 4 represents the fast variables characterized by the functions φ i as θ i = φ i / where φ i 1 i 4 is the phase of the i-th wave associated with the characteristic speed λ i. Now we use 7 in 3 expand A and B in Taylor s series in powers of about U = U replace the partial derivatives / X X being either x or t by X φ i i=1 X coefficients of and 1 in the resulting expansions to obtain 8 9 O : O 1 : i=1 i=1 I φ i t + A φ i U1 = x θ i I φ i t + A φ i U2 x θ i θ i and equate to zero the = U 1 U 1 A t x U 1. B φ i x U U 1 1. A θ i where I is the 4x4 unit matrix and is the gradient operator with respect to the dependent variable U. Now since the phase functions φ i 1 i 4 satisfy the eikonal equation 1 Det i=1 I φ i t + A φ i = x we choose the simplest phase function of this equation namely 11 φ i x t = x λ i t 1 i 4. It follows from 5 that for each phase φ i U 1 / θ i is parallel to the right eigen vector R i of A and thus 12 U 1 = σ i x t θ i R i i=1 where σ i = L i U 1 is a scalar function called the wave amplitude that depends only on the i-th fast variable θ i. We assume that σ i x t θ i has zero mean value with respect to the fast variable θ i that is 1 13 lim T 2T T T σ i x t θ i dθ i =.

5 SPHERICAL SHOCK WAVES 15 We then use 12 in 9 and solve for U 2. To begin with we write U 2 = m j R j ; j=1 substitute this value in 9 and premultiply the resulting equation by L i to obtain the system of decoupled inhomogeneous first order partial differential equations: 14 λ i λ j m i θ j j=1 = σ i t λ σ i i x Li U 1 B j=1 L i U 1 A U 1 θ j 1 i 4. The characteristic ODEs for the ith equation in 14 are given by 15 θj = λ i λ j for j i θi = ṁ i = H i where H i x t θ 1 θ 2 θ 3 θ 4 = σ i t λ σ i i x Li U 1 B j=1 L i U 1 A U 1 θ j. We asymptotically average 14 along the characteristics and appeal to the sub-linearity of U 2 in θ which ensures that the expression 7 does not contain secular terms. The constancy of θ i along the characteristics and the vanishing asymptotic mean value of ṁ i along the characteristics implies that the wave amplitudes σ i 1 i 4 satisfy the following system of coupled integro-differential equations 16 σ i t + λ σ i i x + a iσ i + Γ i σ i ii σ i + Γ i jk θ i i j k lim T 1 T σ j θ i + λ i λ j s σ k θ i + λ i λ k s ds = 2T T

6 16 RAJAN ARORA where σ k = σ k / θ k and the coefficients a i and Γ i jk are given by 17 a i = L i R i B Γ i jk = Li R j A R k. The interaction coefficients Γ i jk which are asymmetric in j and k denote the strength of coupling between the j-th and k-th wave modes j k that can generate an i-th wave i j k. The coefficients Γ i ii which refer to the nonlinear self-interaction are nonzero for genuinely nonlinear waves and zero for linearly degenerate waves. It is also interesting to note that if all the coupling coefficients Γ i jk i j k are zero or the integral in 16 vanishes the waves do not resonate and 16 reduces to a system of uncoupled Burgers equations. The coefficients a i Γ i ii and Γ i jk given by 16 provide a picture of the nonlinear interaction process present in the system under consideration and can be easily determined in the following form; the nonzero ones being: a 3 = c x a 4 = c x Γ 1 23 = Γ1 24 = 1 γ ρ c 18 Γ 1 34 = Γ1 43 = γ 1 a2 c ρ Γ 2 13 = Γ2 43 = Γ2 14 = Γ2 34 = γ 1a2 Γ 3 14 = Γ4 13 = c2 + a2 1 γ = β 1 2ρ c Γ 3 24 = Γ 4 23 = 1 γ 2ρ c = β 2. c ρ

7 SPHERICAL SHOCK WAVES 17 The resonance equations 16 can now be written as 19 σ 1 = t σ 3 t + c σ 2 t = σ 3 x + c x σ σ Γσ 3 θ 3 1 lim P 2P P P K x t θ 3 + φ 2 σ 4 x t φ dφ = σ 4 t c σ 4 x c x σ σ 4 4 Γσ lim P 2P P P K θ 4 x t θ 4 + φ 2 σ 3 x t φdφ = where Γ = Γ 3 33 = Γ 4 44 = [2b 2 + γ + 1a 2 ]/2c ρ and the kernel K is defined as 2 K x t θ + φ = β 1 σ 1 x t θ + φ + β 2 σ 2 x t θ + φ 2 2 θ θ 2 2 with β 1 and β 2 being the same as in 18. Let the initial value of σ j be σ j t= = σ j x θ j. Hence gives σ 1 x t θ 1 = σ 1 x θ 1 and σ 2 x t θ 2 = σ 2 x θ 2 and subsequently the system 19 reduces to a pair of equations for the wave fields σ 3 and σ 4 coupled through the linear integral operator involving the kernel 21 Kx t θ = β 1 2 σ1 x θ + β 2 θ 1 2 σ 2 θ 2 x θ. If the initial data σj x θ are 2π periodic function of the phase variable θ then the pair of resonant asymptotic equations in system 19 becomes 22 σ 3 t + c σ 3 x + c 1 2π π π x σ σ Γσ 3 θ 3 x t θ 3 + φ 2 K σ 4 t c σ 4 x c + 1 2π π π where K is given by 21. x σ σ 4 4 Γσ 4 θ 4 x t θ 4 + φ 2 K σ 4 x t φdφ = σ 3 x t φdφ =

8 18 RAJAN ARORA 4 Nonlinear geometrical acoustics solution The asymptotic solution 7 of hyperbolic system 3 satisfying small amplitude oscillating initial data 23 Ux = U + U 1 x x/ + O 2 is nonresonant if U1 x x/ are smooth functions with a compact support [18]. The expansion 7 is uniformly valid to the leading order till shock waves have formed in the solution. The characteristic equations are 24 dθ j dx = Γσ j c and dt dx = e j c j = 3 4 where e j = { +1 if j = 3 1 if j = 4. In terms of the characteristic equations the decoupled equations 22 2 and 22 3 can be written as 25 dσ j dx = σ j j = 3 4 x which yields on integration 26 σ j = σ j s j ξ j sj x j = 3 4 along the rays s j = x e j c t = constant where the function σ j j = 3 4 is obtained from the initial condition 23 and the fast variable ξ j parametrizes the set of characteristic curves Thus we obtain from ξ j = θ j e j Γ σ j s j ξ j I j t j = 3 4 where 28 I j t = t 1 + e 1 j c t d t s j

9 SPHERICAL SHOCK WAVES 19 The solution of 3 satisfying 23 where U1 x x/t has compact support is obtained as ρx t = ρ + σ1 x x + x c tσ3 x s 3 ξ 3 + x + c tσ 4 s 4 ξ 4 + O 2 ux t = c x c tσ3 xρ s 3 ξ 3 x + c tσ4 s 4 ξ 4 + O 2 px t = p + a 2 σ1 x x + σ2 x x 29 + a2 x c tσ3 x s 3 ξ 3 + x + c tσ4 s 4 ξ 4 + O 2 hx t = h a2 σ1 x x 1 σ2 x x h h + b2 xh x c tσ 3s 3 ξ 3 + x + c t σ 4 s 4 ξ 4 + O 2 where the fast variables ξ j 1 j 4 are given in 27 and the initial values for σ i 1 i 4 are obtained from the solution 29 specified at t = as 3 σ 1 σ 2 = ρ 1 x x 1 x x c 2 x x = a 2 ρ 1 x x + p 1 σ 3 x ξ 3 = ρ 2c 1 + σ4 x ξ ρ 4 = 2c u 1 x ξ 3 2c p 1 x x + h h 1 x x x x p 1x ξ 3 + h h 1x ξ 3 u 1 x ξ 4 2c 2 p 1x ξ 4 + h h 1x ξ 4.

10 2 RAJAN ARORA This is the complete solution of 3 and 23; any multivalued overlap in this solution is resolved by introducing shocks into the solution. 5 Shock waves Following [17] it can be shown that the shock location θj s satisfies the relation 31 dθj s dt = 1 2 Λj jj σ j + σ + j j = 3 4 which is the shock speed in θ j - t plane. Here σ j and σ + j respectively are the the values of σ j just ahead and behind the shock. We have σ j = for the undisturbed region ahead of the shock. Now we use 26 and drop the superscripts on θj s and σ+ j to obtain 32 dθ j dt = Γ 2 e j σj s sj j ξ j. x Now using 32 and 27 we obtain the following equation which determines the shock path parametrically 33 θ j = ξ j 2 σ j ξj σ j t d t. If σ j then the shock forms right at the origin. 6 Conclusion A Spherically symmetric model describing a blast wave in the solar wind caused by the explosive energy release of a solar flare is considered. The model incorporates a self-consistent magnetic field. The method of multiple scales is used to obtain small amplitude high frequency asymptotic solution to the above model where it is assumed that the electrical conductivity is infinite. The theory of weakly nonlinear geometrical acoustics is used to analyze situations where many waves coexist and interact with one another resonantly. The basic idea underlying the procedure is to separate the rapidly varying part of the solution from the slowly varying part; this is accomplished by averaging the solution with respect to the fast variables. We derived the transport equations for the wave amplitudes along the rays of the governing system; these transport equations constitute a system of inviscid Burger s equations with quadratic nonlinearity coupled through a linear integral operators with a known kernel. The coefficients

11 SPHERICAL SHOCK WAVES 21 appearing in the transport equations provide a measure of coupling between the various modes and set a qualitative picture of the interaction process involved therein. It is observed that the wave fields associated with the particle paths do not interact with each other; however they do interact with an acoustic wave field to produce resonant contribution towards the other acoustic field. The acoustic wave fields may or may not interact but in either case their net contribution which is directed towards the entropy field is always zero. In our analysis the governing system of Euler equations reduces to a pair of resonant asymptotic equations for the acoustic wave fields. For a nonresonant multi wave mode case proposed by Hunter and Keller [17] the reduced system of transport equations gets decoupled with vanishing integral average terms and the occurrences of shocks in the acoustic wave fields are analyzed. It is found that in a contracting piston motion having spherical symmetry a shock is always formed before the formation of a focus no matter how small be the initial wave amplitude. REFERENCES 1. D. Summers An idealised model of a magnetohydrodynamic spherical blast wave applied to a flare produced shock in the solar wind Astron. & Astrophys V. D. Sharma L. P. Singh and R. Ram The progressive wave approach analyzing the decay of a sawtooth profile in magnatogasdynamics Phys. Fluids A. Kumar V. D. Sharma and B. D. Pandey Nonlinear waves in gases Acta Astronautica J. K. Hunter and G. Ali Wave interactions in magnetohydrodynamics Wave Motion D. Tsiklauri T. D. Arber and V. M. Nakariakov A weakly nonlinear Alfvenic pulse in a transversely inhomogeneous medium Astronomy and Astrophysics R. M. Gunderson Linearized analysis of one-dimensional magneto-hydrodynamic flows Springer Tracts in Natural Philosophy G. B. Whitham Linear and Nonlinear Waves Wiley New York T. B. Moodie Y. He and D. W. Barclay avefront expansions for nonlinear hyperbolic waves Wave Motion Y. He and T. B. Moodie Two-wave interactions for weakly nonlinear hyperbolic waves Stud. Appl. Math Y. He and T. B. Moodie Shock wave tracking for nonlinear hyperbolic systems exhibiting local linear degeneracy Stud. Appl. Math Y. He and T. B. Moodie Solvability and nonlinear geometrical optics for systems of conservation laws having spatially dependent flux functions Can. Appl. Math. Quart

12 22 RAJAN ARORA 12. Y. He and T. B. Moodie Geometrical optics and post shock behaviour for nonlinear conservation laws Applicable Analysis V. D. Sharma and Rajan Arora Similarity solutions for strong shocks in an ideal gas Stud. Appl. Math Rajan Arora and V. D. Sharma Convergence of strong shock in a Van der Waals gas SIAM J. Applied Mathematics V. Choquet-Bruhat Ondes asymptotique et approchees pour systemes d equations aux derivees partielles nonlineaires J. Math. Pures Appl P. Germain Progressive waves 14th Prandtl Memorial Lecture Jarbuch der DGLR J. K. Hunter and J. Keller Weakly nonlinear high frequency waves Comm. Pure Appl. Math A. Majda and R. Rosales Resonantly interacting weakly nonlinear hyperbolic waves Stud. Appl. Math J. K. Hunter A. Majda and R. Rosales Resonantly interacting weakly nonlinear hyperbolic waves II. Several space variables Stud. Appl. Math Department of Paper Technology IIT Roorkee Saharanpur Campus Saharanpur-2471 UP India address: rajanfpt@iitr.ernet.in rajan a1@yahoo.com