Corrugation instability and bifurcation of slow viscous MHD shocks in a duct. Kevin Zumbrun

Transcription

1 Corrugation instability and bifurcation of slow viscous MHD shocks in a duct Kevin Collaborators: Blake Barker Rafael Monteiro Department of Mathematics Indiana University Sponsored by NSF Grants no. DMS and DMS Séminaire EDP/Physique Math.: Bordeaux, May 2015

2 I. Introduction Numerical/analytical study of instability/bifurcation of slow MHD shocks in a duct. Inviscid instability (in contrast to fast, gas-type MHD shocks) known for some time, numerically ([Fillipova,...]) and analytically [Blokhin et al]. Corrugation instabilities seen in astrophysics, familiar but not well-understood. Following our general program, study viscous problem, deduce rigorous nonlinear bifurcation results (natural extension of several previous lines of study). To some extent, informational somewhat exotic MHD scenario and results (hopefully with useful simplification!) But, also substantial new results.

3 Shock propagation in a duct (John Lee, detonation) 182 Unstable Detonations: Experimental Observations 0 Time, µs cm Distance, cm Figure Simultaneous soot record and laser schlieren photographs of a detonation wave in a rectangular channel (Oppenheim, 1985). Figure Schlieren photographs of multiheaded detonations in a thin channel at (a) P0 = 13 kpa and (b) P0 = 8kPa(courtesyofM.Radulescu).

4 Plasma dynamics (MHD): magnetically/inertially confined fusion, geosciences, astrophysics Figure: NASA picture of our sun.

5 Issues Dealing with constraints/involutions (div. free in this case). Multi-d Evans computation (first data for viscous problem!). Large- H asymtotics and rigorous verification of instability. Viscous vs. inviscid instability (*).

6 II. Equations and shock solutions ([Dafermos, p. 71]) ]Maxwell s equations: µh t= curl(e) E t=curl(h) J, Constraints: divh = 0, µdive = r. H, E 2 R 3 magnetic and electric Fields, J current, r charge. Reduction: Taking E t! 0, havej = curl(h), Lorentz force: J µh = µh curl(h), u 2 R 3 fluid velocity. Generalized Ohm s Law: E = µh u, ( mass e mass i.)

7 Magnetohydrodynamics (MHD) Coupling with compressible gas dynamics, MHD equations in R 3 : t+div( u) = 0 ( u) t+div( u u)+h curl(h)+rp = u+( + )rdiv(u) H t+r (H u) = H (E+ 1 2 H2 ) t +div((e+p)u+h (u H)) = div( u)+apple T + (H (r H)), where u 2 R n :fluidvelocity, >0: density; T: temperature, p = p(, T ): pressure, H 2 R n : magnetic field; : viscous stress tensor; E = E(, T ): energy., 0: viscosity coe cients. apple, 0: heat conductivity and electrical resistivity, respectively.

8 Planar shock layers, parallel case (H k u) Planar shock layer (WLOG stationary): U(x, t) =Ū(x 1 ), U =(, u, H, T ). FACT: For gas-dynamical shock (H 0), u 2 = u 3 constant, WLOG 0. In parallel case H (H 1, 0, 0) k u, equations decouple, H 1 constant by div-free condition. Thus, standard gas-dynamical shock, constant imposed magnetic field.

9 Type of the shock Shocks are categorized as Lax, undercompressive, overcompressive, depending on number of characteristics entering the shock. Lax shocks are further characterized by their characteristic field: the unique family entering on both sides. Depending on H rel. gas-dynamical parameters, Ū is a fast (i.e., extreme, or gas-dynamical type) Lax shock for 0 apple H appleh. intermediate overcompressive type for H apple H appleh,and slow Lax type for H apple H (in particular, H!1).

10 III. Evans/Lopatinski conditions and constraints Conservative form: u t + F(u) =0, F(u) := X j f j (u) xj, (1) zero-speed planar background shock in x 1 direction: u(x, t) =ū(x 1 )=u ± for x 1? 0, (2) satisfying Rankine-Hugoniot conditions: [f 1 ]=0.

11 Lopatinski problem Recall from, e.g., [Z-Serre], the normal modes equations after shifting to a frame (z, x 2,...,x d ) with discontinuity at z = 0: v + A 1 ±v z + X j6=1 ik j A j ± v =0, x? 0, (3) (interior equation) and Y ( [u]+ X j6=1 ik j [f j (u)]) [Av] = 0 (4) (linearized jump conditions), where [h] denotes jump across z = 0. Here, v is the Laplace Fourier transform of the normal mode, Y 2 C the Laplace Fourier transform of the front location, and a spectral parameter indicating growth e t of the normal mode e t+p j6=1 ik j x j v(z), with < >0 corresponding to instability. Here, A j ± := df j (u ± ).

12 Problem: indeterminacy of equations, how to impose constraint? Can add any multiple of div(h) to the equations and obtain an equivalent system modulo constraint. Which is best? And, what to do with constraint? Standard solution was to add udiv(h) toh equation to obtain symmetrizable (so hyperbolic) but nonconservative system. Works, but introduces technical di culties in Lopatinski study. (Does show that constrained system is hyperbolic...!) Constraint issue was handled in ad hoc way, and only for inviscid case.

13 Resolution: the -model The inviscid MHD equations may be rewritten as t + div( u) =0, ( u) t + div( u u H H) +rq =0, H t r (u H) + div(h)e 1 =0, (E + H 2 /2) t + div((e + p)u + H (u H)) = 0, and the viscous MHD equations by t + div( u) =0 ( u) t + div( u u H H) +rq = µ u +(µ + )rdivu H t r (u H) + div(h)e 1 = H (E + H 2 /2) t + div((e + p)u + H (u H)) = div( u) +apple T + div(h (r H)), along with the divergence free condition div(h) = 0, = constant 6= 0: conservative, hyperbolic, noncharacteristic.

14 Involutions and constraints General situation: U t = F(U) := X j F j (U) xj =0, constraint U = 0, first-order linear constant coe cient. KEY PROPERTY ([Dafermos]): M linear first-order constant coe noncharacteristic. F(U) =M U, L = M, cient, hyperbolic, ) constraint is preserved by time t ( U) =M U, U t=0 0, ) U 0fort 0 by uniqueness/well-posedness.

15 Involutions, cont d In our case, U = div(h), MU x1 H 1,transport, 6= 0... Proposition Under the above assumptions, hyperbolicity with constraint is equivalent to hyperbolicity without constraint. Likewise a shock is Lax type+ noncharacteristic with constraint i without. Proof. (L(k) µ)r = 0 implies(m(k) µ) = 0, with = (k)r (N(k) denoting Fourier symbol of N), giving result by properties of M, L ker. M const.-coe. implies M-char s independent of ±1, so don t a ect the type of the shock. Hence, unconstrained shock falls into standard framework, Lopatinski condition well-defined.

16 Fundamental Theorem for constraints Proposition Under the assumptions above, for < >0, normalmodes e t+p j6=2 ik j x j v(x 1 ) of the unconstrained linearized shock problem, v piecewise smooth, satisfy := v 0. Thatis, unstablenormal modes automatically satisfy the constraint. Proof. Direct computations using relations implied by the involution properties yield ( + M) =0, x? 0, [M 1 ]=0, x =0, i.e., is a weak solution of the e-value problem for M with e-value (contradicting < >0) or else 0. (NOT EASY!) (5)

17 Conclusion Consequence: Stability for unconstrained system is equivalent to stability with constraint (no issue!). Remark: Seemingly related to Alinhac s good unknown; however not reflected in proof. Generalization of the remarkable property shown by Blokhin to hold for his special nonconservative mixed formulation of MHD. (Applies also, for example, to viscoelasticity.) The same conclusion follows (with trivial argument) in the viscous case (see also [Pogan-Yao-Z2015]). SIMPLE, BUT EXTREMELY USEFUL: demystifies previous analyses, allows o -the-shelf treatment of stability.

18 IV. Numerical Evans computation in multi-d Apart from constraint issues, substantial new di from 1 to multi-d Evans computation: culties in going Complexity/number of equations/parameters!: not machine, but human (error-free coding) limits. Solution: Maple/Mathematica front ends to fill in code (e.g., compute 7 7 Jacobians of flux fns.), general STABLAB platform for generalized Kawashima class systems. (NOTE: -model allows use of previously tested code...) Unexpected issues with Evans function asymptotics, related to Eulerian vs. Lagrangian coordinates, made computation practically impossible. Only recently resolved, one of first 2 multi-d numerical Evans computations (both in progress)...

19 Lagrangian coordinates in 1d The isentropic Navier Stokes equations in Eulerian coordinates in one space dimension are t +( u) x =0, ( u) t +( u 2 + p( )) x = µu xx. By the change of independent variables y(x, t) = dx dt Z x x (t) (z, t) dz, = u(x (t), t), x (0) = 0,

20 Lagrangian coordinates, cont d or, (x, t) = (x, t), and setting (y(x, t), t) = 1 (x,t), w(y(x, t), t) =u(x, t), we recover the standard Lagrangian formulation: t w y =0, w t + P( ) y = µ wy See [Courant & Friedrichs] or [Serre] for further discussion. Constant-coe cient in hyperbolic ( ) mode,) better largeasymptotics. IN MULTI-D: perform linearized version of (??) with similar benefit (since background shock 1d). CRUCIAL! y.

21 Pseudo-Lagrangian versus Lagrangian coordinates Evans function asymptotic behavior: e Cp verse e C (a) (b) Figure: Plot of the Evans function for isentropic gas in 1d ( =5/3, u 1+ =0.3) computed on a semicircular contour of radius R = 10 using the polar coordinate method and the adjoint formulation on the right. (a) Plot of the Evans function in Eulerian coordinates (blue), Lagrangian coordinates (black), and pseudo-lagrangian coordinates (red). (b) A zoom in of Figure (a).

22 V. Results: Multi-d Evans/Lopatinski computations First planar multi-d study, with [Humpherys-Lyng-Z]. Evans function ODE set up with symbolic code Planar multi-d Evans function now supported in STABLAB Excellent agreement between Evans and Lop. in low-freq. limit.

23 Isentropic MHD inviscid stability diagram (Lopatinski) Figure: Plot of inviscid stability diagram for to stability and red to instability. = 5/3. Black corresponds

24 MHD: Numerical verification that root is simple λ h 1 Figure: Plot of the roots of the Lopatinski determinant against h 1 for =1and =5/3, u 1+ =0.9.

25 Roots of the Evans function for isentropic MHD 4 x λ(ξ) ξ Figure: Plot of ( ) against where D( ( ), )=0forh 1 = 3, =5/3, u 1+ =0.6, µ = = apple =0.1, = 0.067, = 1.

26 Large- H asymptotics of unstable root (inviscid) Figure: Lopatinski root. Red points (FT prediction); green points - analytical prediction; black points - numerical prediction (both in BMZ)

27 2d Hopf bifurcation in a channel (in our case, stationary) =( ) =( ) MHD Lopatinski Lopatinski MHD <( ) <( ) =( ) Lopatinski MHD <( )

28 Eigenfunction for isentropic MHD superimposed on wave Re(u1) x x Figure: Plot of the real part of the u 1 component of the Eigenfunction superimposed on background wave solution to approximate the nonlinear solution for h 1 = 3, =5/3, u 1+ =0.6, = µ = apple =0.1, =

29 Summary and open questions Summary Beyond stability to physical behavior Use mathematical model to provide data First multi-d Evans function computations! Already interesting phenomena (not in 1d) and expect more. Non-planar char. propagation mode in slow fields, seen in astrophysics. Quantitave improvement (10-20%!) in asymptotics for instability (vs. Edelman-.., Freistühler-Trakhinin, etc.). Key open questions: Nonlinear bifurcation for physical viscosity? Are there systems where viscous instability precedes inviscid instability as shock amplitude is increased from 0 (stable)?

30 References Blake Barker, Rafael A. Monteiro, and Kevin. Anumerical study of transverse steady bifurcation of viscous mhd shocks in a channel. In preparation. Heinrich Freistühler and Yuri Trakhinin. On the viscous and inviscid stability of magnetohydrodynamic shock waves. Phys. D, 237(23): , John H.S. LeeThe detonation phenomenon; Rafael A. Monteiro Transverse steady bifurcation of viscous shock solutions of a system of parabolic conservation laws in a strip. J. Di erential Equations, 257(6): , J.M. Stone and M. Edelman. The corrugation instability in slow mhd shocks. The Astrophysical Journal, 454:182,1995.; K. and D. Serre. Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J., 48(3): , 1999