February 2001 GAMM, Zuerich, Switzerland 1

Overcompressive shocks Overcompressive shocks in 3D MHD bow shock in 3D MHD bow shock

flowsflows

H. De SterckH. De Sterck

Centre for Plasma Astrophysics, K.U.Leuven, Centre for Plasma Astrophysics, K.U.Leuven, BelgiumBelgium

Department of Computer Science, University Department of Computer Science, University of Colorado at Boulder, USAof Colorado at Boulder, USA

andand

February 2001 GAMM, Zuerich, Switzerland 2

MagnetohydrodynamicsMagnetohydrodynamics

0

)()2/p(e

)2/(ρ

ρ

e

ρ

ρ

BBvvBB

vBBv

BBIBBvv

v

B

v pt

Nonlinear hyperbolic conservation Nonlinear hyperbolic conservation lawlawDescribes plasma as fluidDescribes plasma as fluidApplies to laboratory and Applies to laboratory and astrophysical plasmasastrophysical plasmas

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MagnetohydrodynamicsMagnetohydrodynamicsThree wave modes:Three wave modes:

(fast, Alfven, slow)(fast, Alfven, slow)

MHD is non-strictly hyperbolic MHD is non-strictly hyperbolic (wave speeds may coincide)(wave speeds may coincide)

MHD flux function rotationally MHD flux function rotationally

invariantinvariant

sAf ccc

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Magnetohydrodynamic Magnetohydrodynamic shocksshocks

three types of shocks: (coplanar)three types of shocks: (coplanar)(fast (fast 1-21-2, intermediate , intermediate 1-3,1-4,2-3,2-41-3,1-4,2-3,2-4, , slow slow 3-43-4))

(thick=shock, arrowed=magnetic field line)(thick=shock, arrowed=magnetic field line)

intermediate shocks are intermediate shocks are overcompressive! (overcompressive! (1-3, 1-4, 2-41-3, 1-4, 2-4))

up to 4 fixed points in RH relations up to 4 fixed points in RH relations 1,2,3,41,2,3,4

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Stability of overcompressive Stability of overcompressive shocksshocks

Theory of hyperbolic systems: Theory of hyperbolic systems: Overcompressive shocksOvercompressive shocks

• are completely are completely unstableunstable in in dissipation-freedissipation-free case (non- case (non-evolutionary)evolutionary)

• can have can have stablestable viscous profilesviscous profiles when when dissipationdissipation is added is added

•for for rotationally invariant systemsrotationally invariant systems, the stability is , the stability is conditionalconditional: : only stable against perturbations that have small non-coplanar only stable against perturbations that have small non-coplanar componentcomponent

• for for rotationally invariant systemsrotationally invariant systems, the stability is , the stability is not not uniformuniform: stability disappears for vanishing dissipation: stability disappears for vanishing dissipation

regular undercompressive regular undercompressive overcompressiveovercompressive

1-3, 1-4, 2-4 1-3, 1-4, 2-4 intermediate MHD intermediate MHD shocks are shocks are overcompressiveovercompressive

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Overcompressive MHD Overcompressive MHD shocksshocks

1D, 1D, simulationsimulation: overcompressive MHD : overcompressive MHD shocks obtained (e.g. Wu 1987, …)shocks obtained (e.g. Wu 1987, …)but: only when co-planarity imposedbut: only when co-planarity imposed

1D, 1D, theorytheory: (conditionally) stable : (conditionally) stable viscous profiles exist for small viscous profiles exist for small dissipation (e.g. Freistuehler 1991, …)dissipation (e.g. Freistuehler 1991, …)

What remained unknown:What remained unknown:

What had been learned:What had been learned:

Do overcompressive MHD shocks occur in Do overcompressive MHD shocks occur in 3D?3D?Are they common or exceptional?Are they common or exceptional?Do they occur in real physical plasmas?Do they occur in real physical plasmas?Some answers:Some answers: 3D MHD simulations of bow 3D MHD simulations of bow

shocksshocks

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Bow shock in neutral gas Bow shock in neutral gas (~Euler)(~Euler)

•Experiment: Experiment: supersonic flow of supersonic flow of air over sphereair over sphere

•Regular bow shock Regular bow shock shape, single front, shape, single front, acoustic shockacoustic shock

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Bow shock in MHD plasmaBow 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 simulation: MHD plasma flow over plasma flow over perfectly perfectly conducting sphere, conducting sphere, 3D3D

•complex bow complex bow shock shape, shock shape, double frontdouble front

•All types of All types of shocks present: shocks present: fast, slow, fast, slow, intermediateintermediate

•Overcompressive Overcompressive shocks present!shocks present!

(density contours, magnetic field lines)

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When does the new topology When does the new topology (with intermediate shocks) (with intermediate shocks)

arise?arise?

Pressure-Pressure-dominateddominated

Magnetically dominated Magnetically dominated regimeregime

(low B strength)(low B strength) (high B strength)(high B strength)(also(also ))

When upstream B is When upstream B is largelarge

(density contours, cut in symmetry plane)

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Why does the new topology Why does the new topology arise?arise?

Three Three MHD MHD shock shock typestypes

= switch-on shocks (d) occur= switch-on shocks (d) occur

Regular bow shock (a) Regular bow shock (a) is then impossible, is then impossible, new to-pology (b) can new to-pology (b) can be explain-ed from RH be explain-ed from RH relationsrelations

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Why does the new topology Why does the new topology arise?arise?

MHD Rankine-MHD Rankine-Hugoniot Hugoniot relationsrelations

•At point At point BB: 1-3 switch-on shock as : 1-3 switch-on shock as in (a)in (a)

•Between points Between points BB and and DD: 1-3 : 1-3 overcompressiveovercompressive shock as in (b) shock as in (b)

•At point At point DD: 1-3=4 overcompressive : 1-3=4 overcompressive shock as in (c) shock as in (c)

•Beyond point Beyond point DD: splits into two : splits into two shocks, leading shock 1-2 as in (d)shocks, leading shock 1-2 as in (d)

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Conclusion: 3D MHD simulations Conclusion: 3D MHD simulations show...show...

they must also occur in real physical they must also occur in real physical plasmas!plasmas!

overcompressive overcompressive MHD shocks occur in MHD shocks occur in 3D3D

they are commonthey are commonAny bow shock flow Any bow shock flow with strong upstream with strong upstream BBrestrictive co-planar restrictive co-planar setup not requiredsetup not required

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Physical plasma with overcompressive Physical plasma with overcompressive shocks?shocks?

Planetary bow shocks, terrestrial bow shockPlanetary bow shocks, terrestrial bow shock

• supersonic solar wind supersonic solar wind forms shock in front of forms shock in front of planetsplanets

• new topology with new topology with overcompressive shocks overcompressive shocks not directly observed yet not directly observed yet at earth (possibly at earth (possibly indirectly: Song et al, JGR, indirectly: Song et al, JGR, 1992)1992)

• CLUSTERII (recently CLUSTERII (recently launched satellite) may launched satellite) may see overcompressive see overcompressive shocks in earth’s bow shocks in earth’s bow shock soonshock soon

•Kivelson et al, Science, Kivelson et al, Science, 1991: observation of 1991: observation of overcompressive shock overcompressive shock segment in Venus’ bow segment in Venus’ bow shockshock