February 2001GAMM, Zuerich, Switzerland1 Overcompressive shocks in 3D MHD bow shock flows H. De Sterck Centre for Plasma Astrophysics, K.U.Leuven, Belgium Department of Computer Science, University of Colorado at Boulder, USA and

February 2001GAMM, Zuerich, Switzerland2 Magnetohydrodynamics  Nonlinear hyperbolic conservation law  Describes plasma as fluid  Applies to laboratory and astrophysical plasmas

February 2001GAMM, Zuerich, Switzerland3 Magnetohydrodynamics  Three wave modes: (fast, Alfven, slow) (fast, Alfven, slow)  MHD is non-strictly hyperbolic (wave speeds may coincide)  MHD flux function rotationally invariant

February 2001GAMM, Zuerich, Switzerland4 Magnetohydrodynamic shocks  three types of shocks: (coplanar) (fast 1-2, intermediate 1-3,1-4,2-3,2-4, slow 3-4 ) (thick=shock, arrowed=magnetic field line)  intermediate shocks are overcompressive! ( 1-3, 1-4, 2-4 )  up to 4 fixed points in RH relations 1,2,3,4

February 2001GAMM, Zuerich, Switzerland5 Stability of overcompressive shocks Theory of hyperbolic systems: Overcompressive shocks are completely unstable in dissipation-free case (non-evolutionary) are completely unstable in dissipation-free case (non-evolutionary) can have stable viscous profiles when dissipation is added can have stable viscous profiles when dissipation is added for rotationally invariant systems, the stability is conditional: only stable against perturbations that have small non-coplanar componentfor rotationally invariant systems, the stability is conditional: only stable against perturbations that have small non-coplanar component for rotationally invariant systems, the stability is not uniform: stability disappears for vanishing dissipation for rotationally invariant systems, the stability is not uniform: stability disappears for vanishing dissipation regular undercompressive overcompressive regular undercompressive overcompressive 1-3, 1-4, 2-4 intermediate MHD shocks are overcompressive

February 2001GAMM, Zuerich, Switzerland6 Overcompressive MHD shocks  1D, simulation: overcompressive MHD shocks obtained (e.g. Wu 1987, …) but: only when co-planarity imposed  1D, theory: (conditionally) stable viscous profiles exist for small dissipation (e.g. Freistuehler 1991, …) What remained unknown: What had been learned:  Do overcompressive MHD shocks occur in 3D?  Are they common or exceptional?  Do they occur in real physical plasmas? Some answers: 3D MHD simulations of bow shocks

February 2001GAMM, Zuerich, Switzerland7 Bow shock in neutral gas (~Euler) Experiment: supersonic flow of air over sphereExperiment: supersonic flow of air over sphere Regular bow shock shape, single front, acoustic shockRegular bow shock shape, single front, acoustic shock

February 2001GAMM, Zuerich, Switzerland8 Bow shock in MHD plasma (H. De Sterck and S. Poedts. Intermediate shocks in 3D MHD bow shock flows with multiple interacting shock fronts. Physical Rev. Lett., 84, 5524, 2000.) simulation: MHD plasma flow over perfectly conducting sphere, 3Dsimulation: MHD plasma flow over perfectly conducting sphere, 3D complex bow shock shape, double frontcomplex bow shock shape, double front All types of shocks present: fast, slow, intermediateAll types of shocks present: fast, slow, intermediate Overcompressive shocks present!Overcompressive shocks present! (density contours, magnetic field lines)

February 2001GAMM, Zuerich, Switzerland9 When does the new topology (with intermediate shocks) arise? Pressure-dominated Magnetically dominated regime (low B strength) (high B strength) (also) When upstream B is large (density contours, cut in symmetry plane)

February 2001GAMM, Zuerich, Switzerland10 Why does the new topology arise? Three MHD shock types = switch-on shocks (d) occur Regular bow shock (a) is then impossible, new to- pology (b) can be explain- ed from RH relations

February 2001GAMM, Zuerich, Switzerland11 Why does the new topology arise? MHD Rankine- Hugoniot relations At point B: 1-3 switch-on shock as in (a)At point B: 1-3 switch-on shock as in (a) Between points B and D: 1-3 overcompressive shock as in (b)Between points B and D: 1-3 overcompressive shock as in (b) At point D: 1-3=4 overcompressive shock as in (c)At point D: 1-3=4 overcompressive shock as in (c) Beyond point D: splits into two shocks, leading shock 1-2 as in (d)Beyond point D: splits into two shocks, leading shock 1-2 as in (d)

February 2001GAMM, Zuerich, Switzerland12 Conclusion: 3D MHD simulations show...  they must also occur in real physical plasmas!  overcompressive MHD shocks occur in 3D  they are common  Any bow shock flow with strong upstream B  restrictive co-planar setup not required

February 2001GAMM, Zuerich, Switzerland13 Physical plasma with overcompressive shocks? n Planetary bow shocks, terrestrial bow shock supersonic solar wind forms shock in front of planets supersonic solar wind forms shock in front of planets new topology with overcompressive shocks not directly observed yet at earth (possibly indirectly: Song et al, JGR, 1992) new topology with overcompressive shocks not directly observed yet at earth (possibly indirectly: Song et al, JGR, 1992) CLUSTERII (recently launched satellite) may see overcompressive shocks in earth’s bow shock soon CLUSTERII (recently launched satellite) may see overcompressive shocks in earth’s bow shock soon Kivelson et al, Science, 1991: observation of overcompressive shock segment in Venus’ bow shockKivelson et al, Science, 1991: observation of overcompressive shock segment in Venus’ bow shock