Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 118 124 Invariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations in 3 + 1) Dimensions A.M. GRUNDLAND and P. PICARD Centre de Recherches Mathématiques, Université demontréal, C.P. 6128, Succ. Centre-ville, Montréal, QC) H3C 3J7, Canada E-mail: grundlan@crm.umontreal.ca Département de Physique, Université demontréal, C.P. 6128, Succ. Centre-ville, Montréal, QC) H3C 3J7, Canada E-mail: picard@crm.umontreal.ca The symmetry reduction method is used to obtain invariant and conditionally invariant solutions of magnetohydrodynamic equations in 3+1) dimensions. Detail description of the procedure for constructing such solutions is presented. These solutions represent simple and double Riemann waves. 1 Introduction We consider the magnetohydrodynamic MHD) equations in 3+1) dimensions which can be written in the matrix evolutionary form u t + A 1 u)u x + A 2 u)u y + A 3 u)u z =0, H =0, 1) where the 8 by 8 matrix functions A 1, A 2 and A 3 have the form A 1 = A 2 = u 0 ρ 0 0 0 0 0 0 u κp 0 0 0 0 0 0 1/ρ u 0 0 0 H 2 /ρ H 3 /ρ 0 0 0 u 0 0 H 1 /ρ 0 0 0 0 0 u 0 0 H 1 /ρ 0 0 0 0 0 u 0 0 0 0 H 2 H 1 0 0 u 0 0 0 H 3 0 H 1 0 0 u v 0 ρ 0 0 0 0 0 0 v κp 0 0 0 0 0 0 0 v 0 0 H 2 /ρ 0 0 0 1/ρ 0 v 0 H 1 /ρ 0 H 3 /ρ 0 0 0 0 v 0 0 H 2 /ρ 0 0 H 2 H 1 0 v 0 0 0 0 0 0 0 0 v 0 0 0 0 H 3 H 2 0 0 v, 2),
Invariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations 119 and A 3 = w 0 0 0 ρ 0 0 0 0 w 0 0 κp 0 0 0 0 0 w 0 0 H 3 /ρ 0 0 0 0 0 w 0 0 H 3 /ρ 0 0 1/ρ 0 0 w H 1 /ρ H 2 /ρ 0 0 0 H 3 0 H 1 w 0 0 0 0 0 H 3 H 2 0 w 0 0 0 0 0 0 0 0 w. 3) Here ρ and p represent the density and the pressure of the fluid, respectively; κ is an adiabatic exponent; v =u, v, w)and H =H 1,H 2,H 3 ) represent the velocity of the fluid and the magnetic field, respectively. This system describes an ideal nonstationary isentropic flow of a compressible conductive fluid with negligible dissipative effect, and the electrical conductivity of the fluid is assumed to be infinitely large) placed in magnetic fluid H. The MHD system of equations 1) is composed of nine equations involving eight unknown functions u = ρ, p, v, H) R 8. The aim of this paper is to analyse two special cases of point symmetry of MHD equations 1) that describe the nonlinear propagation and superposition of waves. The methodological approach accepted here is based on the Lie group methods of infinitesimal transformations [1, 2]. For a comprehensive review of this subject see references [3 5]. Examples of invariant and conditionally invariant solutions of system 1) are presented, and some physical and mathematical consequences are discussed. 2 Conditionally invariant solutions and multiples waves Consider a properly determined homogeneous quasilinear hyperbolic system of first order partial differential equations in p independent variable x = x 1,...,x p )andq dependent variables u=u 1,...,u q ) of the form A i u)u i =0, i =1,...,p, 4) where A 1,...,A p are q q matrix functions of u. A vector λ is said to be a wave vector of the system 4) if there exists a non zero function λ u) = λ 1 u),...,λ p u)) such that kerλ i A i ) 0. The function r ofxanduissaidtobeariemann invariant associated to the wave vector λ if the following relation holds rx, u) = λ i u)x i. 5) Now let us fix 1 k p linearly independent wave vectors λ 1,...,λ k with corresponding Riemann invariants r 1 x, u) = λ 1 i u)x i,...,r k x, u) = λ k i u)x i. 6) The equation u=f r 1 x, u),...,r k x, u) ) 7) defines a unique function ux) on a neighborhood of x = 0, for any function f : R k R q,and the Jacobian matrix is given by u α x i = φ 1 x) ) l j λj i ux) )f α r l rx, u) ), l,j =1,...,k, i=1,...,p, α=1,...,q, 8)
120 A.M. Grundland and P. Picard where the k k matrix φ takes the form ) l φ x) j = δl j rl f α ) r x, u). 9) u α r j Note that the rank of ux) is at most equal to k. If the vector-valued function ξ a u) = ξ 1 au),...,ξ p au) ) T, a =1,...,p k 10) satisfies the orthogonality condition λ s i ξ i a =0, s =1,...,k, a=1,...,p k 11) then we get ξa i u α x) ) u α x i =0. 12) Therefore u 1 x),...,u q x) are invariants of the vector field ξa i ) ux), a =1,...,p k 13) xi on R p. Hence the graph of a solution Γ f is invariant under the vector field X a = ξau) i, a =1,...,p k 14) xi on R p R q. Conversely, if ux) is a q-component function defined on a neighborhood of x = 0 such that the graph Γ f is invariant under all vector fields 14) with the property 11), then ux) is the solution of 7) for some function f, because r 1,...,r k,u 1,...,u q constitute a complete set of invariants of the Abelian algebra of such vector fields 14). This geometrically characterizes the solutions of the equations of the form 7). If ux) is the solution of 7) then A i ux) ) u x i x) = φ 1 x) ) l ) j λj i ux) A i ux) ) f )) r x, ux) =0. 15) r l Now we choose an appropriate coordinate system on space R p R q in order to rectify the vector fields 14). We assume that the k by k matrix Λ= λ i j), 1 i, j k 16) is invertible. The independent vector fields X k+1 = x k+1 k Λ 1 ) j i,j=1 i λi k+1 x j,..., X p = x p k Λ 1 ) j i,j=1 i λi p x j 17) have the required form 14) with the orthogonality property 7). If we change the independent and dependent variables as follows x 1 = r 1 x, u),..., x k = r k x, u), x k+1 = x k+1,..., x p = r p, u 1 = u 1,..., u q = u q 18) then the vector fields 17) take rectified form X k+1 = x k+1,..., X p = x p. 19)
Invariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations 121 The graph of the solution Γ f invariant under X κ+1,...,x p is defined by equations of the form ux) = f x 1,...,x k), 20) where f : R k R q is an arbitrary function. This implies that expression 20) is the general solution of the invariance conditions u k+1 =0,..., u p =0. 21) The system 4) is described in the coordinates x, u) by A i x, u, ux ) ui =0, 22) where, A 1 = Dr1 Dx i Ai,..., A k = Drk Dx i Ai, A k+1 = A k+1,..., A p = A p. 23) Now, we have Dr i Dx i uk+1,...,u p =0 = Φ 1) i j λj i, where Φ 1) i j = δi j ri u α uα j, i,j =1,...,k. 24) Adding the invariance conditions 21) to the system 22) we obtain the reduced system of the form k p i,j=1 i=1 which must then be solved. Φ 1 ) i j λj i Ai u i =0, u k+1 =0,..., u p =0, 25) 3 Double Alfvén-entropic wave Using a version of the conditional symmetry method presented above we construct an example of a double Alfvén-entropic wave. According to the proposed method we consider the case where two wave vectors λ 1 and λ 2 corresponding to the Alfvén and entropic wave vectors are given by λ 1 = δ A λ 1 v), λ 1), λ 2 = λ 2 v), λ 2), δ A = ε H λ 1 ) ρ, ε = ±1, 26) λ 1 =cosϕ, sin ϕ, 0), λ 2 =cosθ, sin θ, 0). 27) We consider additionally two vectors ξ 1, ξ 2 which are orthogonal to the given vectors λ 1 and λ 2. We look for the graph of the double wave solution which is invariant under these vector fields X 1 =sinθ ϕ) t [δ A sin θ + u sinϕ θ)] x +[δ A cos θ + v sinθ ϕ)] y, X 2 = z. 28) Note that the vector fields X 1 and X 2 form an Abelian distribution on the space M R 4 R 8, i.e. [X 1,X 2 ] = 0. This requirement constitutes the necessary and sufficient conditions for the existence of a local curvilinear system of coordinates on the integral surface spanned by the fields
122 A.M. Grundland and P. Picard X 1 and X 2 in the space M. Hence the unknown functions can be written in the parametric form u = fr, s) where r and s are the Riemann invariants of the vector fields X 1 and X 2 r = λ 2 x λ 2 v), s = λ 1 x+ δ A λ 1 v) ) t. 29) Substituting the unknown functions in term of r and s into the MHD equations 1) we obtain the following solution ρ = ρr), pr)+ 1 2 H2 r) =p 0, v= ε H ρr) + wr) e 3, H = Hr)[cos ϕs) e 1 +sinϕs) e 2 ], p 0 R, ε = ±1, 30) where ρ, p, w, H and ϕ are arbitrary functions of their arguments r = y cos ϕ x sin ϕ, s = x cos ϕ + y sin ϕ, 31) and e i, i =1, 2, 3 denote the constant unit vectors along x, y, z axes of R 3. Note that the rank of the solution determined by equations 30) and 31) is equal two. The Alfvén-entropic double wave describes a stationary incompressible fluid flow. We have an equilibrium between hydrodynamic and magnetic pressures resulting from the entropic wave. We notice that the density ρ and the pressure p are unchanged by the Alfvén wave. They do however depend on the Riemann invariant r, which is related to the entropic wave. 4 Spherically invariant solution We intend now to discuss the construction of nonstationary spherically symmetric solution of the MHD equations 1). For the purpose of this investigation we limit ourselves to the reduction obtained from the subalgebra g 1 =span { J 1,J 2,J 3 }, 32) where the infinitesimal generators are of the form J 1 = z y y z + w v v w + H 3 H2 H 2 H3, J 2 = x z z x + u w w u + H 1 H3 H 3 H1, 33) J 3 = y x x y + v u u v + H 2 H1 H 1 H2. Note that for the subalgebra g 1 the transversality condition is not satisfied rank {Ω 1 } < rank {Ω 2 }, 34) where matrices Ω 1 and Ω 2 are of the form 0 z y 0 z y 0 w v 0 H 3 H 2 Ω 1 = z 0 x, Ω 2 = z 0 x w 0 u H 3 0 H 1, 35) y x 0 y x 0 v u 0 H 2 H 1 0 and in principle the classical symmetry reduction method cannot be applied. However, there exists a certain domain D given by u j = x j ft, x, y, z), H j = x j gt, x, y, z), j =1, 2, 3 36)
Invariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations 123 for which the rank condition is fulfilled, i.e., rank {Ω 1 } D = rank {Ω 2 } D. 37) In this case the subgroup G 1 acts regularly and transversally and invariant solutions can be constructed [2]. The corresponding invariants of subalgebra g 1 are t, s = x 2 + y 2 + z 2, Φ 1 = u 2 + v 2 + w 2, Φ 2 = xu + yv + zw, Φ 3 = H1 2 + H2 2 + H2 3, Φ4 = xh 1 + yh 2 + zh 3, Φ 5 = ρ, Φ 6 = p. 38) Note that the condition ) Φ i rank u j =7, i =1,...,6, j =1,...,8 39) implies that the dimension of the graph of the solution Γ f is not preserved by the action of the symmetry group G 1. Therefore, we cannot obtain invariant solutions from this set of invariants by the classical symmetry reduction method. This is due to the fact that the defect structure δ of the solution with respect the the group G 1 is equal to one, because the rank of the characteristic matrix Q associated to the subalgebra g 1 is equal to one δ = rank Q x,u 1))) =1. 40) The graph of the solution sweeps out an orbit of dimension dim G 1 Γ f )=p + δ = 5. equations determining the orbit of the graph of the solution are given by The V xu) zw ρ = Rt, s), p = At, s), v =, y w = zv xu)+εy [U 2 u 2 ]y 2 + z 2 ) xu V ) 2 y 2 + z 2, H 2 = X xh 1) zh 3 y 2 + z 2, H 3 = zy xh 1)+εy [X 2 H 2 1 ]y2 + z 2 ) xh 1 Y ) 2 y 2 + z 2, ε = ±1, 41) where u and H 1 are arbitrary functions of t, x, y, z); Φ 2 = V t, s), Φ 3 = Xt, s), Φ 4 = Y t, s), Φ 5 = Rt, s) andφ 6 = At, s) are arbitrary functions of their arguments. Assuming that conditions 36) and p = A 0 ρ κ hold and substituting 41) into the MHD equations 1) we find the reduced system R s f t + R + sf s + 3f + s f s f + s f ) s ) R =0, f + κ A 0 R Rκ 2) =0, 42) s s 3g + s g s =0, A 0 R. Using the method of separation of variables we can solve system 42). We obtain the following invariant solution ρ = ρ o [ s 2 2α p = A o ρ κ, d 2 α dt 2 s2 α 2 ) ] dα 2 1/κ 1) + C 1 α 3κ 1), dt u j = xj dα α dt, H j = α s 3 xj, 43)
124 A.M. Grundland and P. Picard where ) κ 1 1/κ 1) [ ρ 0 =, αt) =C 2 exp κa 0 ηt ) dt ], C 1,C 2 R, j =1, 2, 3. 44) The function η satisfies the ordinary differential equation d 2 η dη 1 + 3κ)η dt2 dt +3κ 1)η3 = 0 45) which can be solved as follows. First, we introduce the change of variable ζη) = dη dt, 46) so that the equation 45) can be rewritten as ζ dζ dη 1 + 3κ)ηζ +3κ 1)η3 =0. 47) Using the second change of variable ζη) =η 2 χτ), where τ =lnη, 48) we solve 47) to finally obtain χdχ τ = 2χ 2 1 + 3κ)χ +3κ 1) + τ 0, τ 0 R. 49) The solution 43) represents a nonstationary compressible irrotational fluid flow for which, according Kelvin s Theorem, the circulation around a fluid element is preserved. Note that F m v = 0 which indicates that the interaction effect between the hydrodynamic and magnetic expressions in the MHD equations can be decoupled. [1] Grundland A.M. and Picard P., On conditionally invariant solutions of magnetohydrodynamic equations. Multiple waves, J. Nonlinear Math. Phys., 2004, V.11, N 1, 47 74. [2] Grundland A.M., Tempesta P. and Winternitz P., Weak transversality and partially invariant solutions, J. Math. Phys., 2003, V.44, N 6, 2704 2722. [3] Ovsiannikov L.V., Group analysis of differential equations, New York, Academic, 1982. [4] Olver P.J. and Vorobev E.M., Nonclassical and conditional symmetries, in CRC Handbook of Lie Group Analysis, Editor N.H. Ibragimov, Boca Raton, Fl., CRC Press, 1996, Vol. 3, Chapt XI, 291 328. [5] Fushchych W.I., Conditional symmetry of equations of mathematical physics, Ukrain. Math. J., 1991, V.43, 1456 1470.