Shocks in Heliophysics

Dana Longcope LWS Heliophysics Summer School & Montana State University Shocks in Heliophysics

Vasyliunas, v.1 ch.10 Shocks in Heliophysics

Tsuneta and Naito 1998

Forbes, T.G., and Malherbe 1986 (v.2 ch. 6) Shocks in Heliophysics

shock

Gold 1962

Cane, Reames, and von Rosenvinge 1988 (v. 2 ch. 7) Shocks in Heliophysics

Forbes 2000 (v. 2, ch. 6) Shocks in Heliophysics

Crooker, Gosling, et al. 1999 (v3, ch. 8) Shocks in Heliophysics

Opher v2, ch. 7

Shocks in the air Shocks are…

• Collision between 2 different plasmas

• A hydrodynamic surprise Shocks in Hydrodynamics Viscosity, exchanges ∂ρ mass = −∇⋅ [ρv] thermal & kinec energy ∂t ∂(ρv)   momentum = −∇⋅ [ρvv + pI +σ ] ∂t € ⎛ ⎞ ⎡ ⎤ ∂ 1 2 p 1 2 γ  energy ⎜ 2 ρv + ⎟ = −∇⋅ ⎢ 2 ρv v + pv +σ ⋅ v⎥ ∂t ⎝ γ −1⎠ ⎣ γ −1 ⎦ € kinec thermal

€ ⎛ ⎞ ∂vi ∂v j 2 viscous stress tensor σij = µ⎜ + − 3 δij∇⋅ v⎟ ⎝ ∂x j ∂xi ⎠

€ ∂ρ mass = −∇⋅ [ρv] ∂t ⎛ v ⎞ ∂(ρv)   ∂vi ∂ j 2 momentum σij = µ⎜ + − 3 δij∇⋅ v⎟ = −∇⋅ [ρvv + pI +σ ] ⎜ ∂x ∂x ⎟ ∂t ⎝ j i ⎠ € ⎛ ⎞ ⎡ ⎤ ∂ 1 2 p 1 2 γ  energy ⎜ 2 ρv + ⎟ = −∇⋅ ⎢ 2 ρv v + pv +σ ⋅ v⎥ ∂t ⎝ γ −1⎠ ⎣ γ −1 ⎦ € €

vs € collision vi

vi ∂ρ mass = −∇⋅ [ρv] ∂t ⎛ v ⎞ ∂(ρv)   ∂vi ∂ j 2 momentum σij = µ⎜ + − 3 δij∇⋅ v⎟ = −∇⋅ [ρvv + pI +σ ] ⎜ ∂x ∂x ⎟ ∂t ⎝ j i ⎠ € ⎛ ⎞ ⎡ ⎤ ∂ 1 2 p 1 2 γ  energy ⎜ 2 ρv + ⎟ = −∇⋅ ⎢ 2 ρv v + pv +σ ⋅ v⎥ ∂t ⎝ γ −1⎠ ⎣ γ −1 ⎦ € € pre-shock (1) post-shock (2) Assume € vi -vs -vs 1. Soluon is staonary in v1 v2 shock frame

v 1 v2 2. Regions 1 & 2 are uniform: transform to ref. frame co-moving ρ1 v1 p1 ρ2 v2 p2 w/ shock - shock frame 0 = −∇⋅ [ρv] jump 0 = [[ρvn ]]

  [[ f ]] = f 2 − f1 0 = −∇⋅ ρvv + pI +σ ˆ [ ] 0 = [[ρvn v + pn]] € ⎡ ⎤ € 1 2 γ  1 2 γ 0 = −∇⋅ ⎢ 2 ρv v + pv +σ ⋅ v⎥ 0 = [[ ρv v + pv ]] € 1 2 n γ −1 n ⎣ γ − ⎦ € nˆ € Jump condions pre-shock (1) post-shock (2) (Rankine-Hugoniot) € € € v1 v2

2. Regions 1 & 2 are v 1 v2 uniform: ⎛ ⎞ ∂vi ∂v j 2 σij = µ⎜ + − 3 δij∇⋅ v⎟ = 0 ρ1 ρ2 ⎝ ∂x j ∂xi ⎠

€ 0 = [[ρvn ]] ρvn = const. ˆ 0 = [[ρvn v + pnˆ ]] ρvn [[v]]+[[p]]n = 0 1 2 γ v [[ 1 v 2 γ (p/ )]] 0 0 = [[ 2 ρvnv + γ −1 pvn ]] ρ n 2 + γ −1 ρ = € € € € Case 1: ρvn = 0 - no flow across interface €  p2=p1 pressure balance € 0 = [[ρvn ]] ρvn = const. ˆ 0 = [[ρvn v + pnˆ ]] ρvn [[v]]+[[p]]n = 0 1 2 γ v [[ 1 v 2 γ (p/ )]] 0 0 = [[ 2 ρvnv + γ −1 pvn ]] ρ n 2 + γ −1 ρ = € € € € Case 1: ρvn = 0 - no flow across interface €  p2=p1 pressure balance €

v2 Contact HOT COLD disconnuity v1 p p 1 2 related to linear

T1 T2 waves w/ ω=0

entropy mode: [[ρ]]≠0 ρ1 ρ2 shear mode: [[vt]] ≠0 0 = [[ρvn ]] Case 2: ρvn = const.≠ 0 ρv [[v ]] = 0 0 = [[ρvn v + pnˆ ]] n ⊥

1 2 γ ⇒ v⊥,1 = v⊥,2 = v⊥ 0 = [[ 2 ρvnv + γ −1 pvn ]] € € transform away € (ρv 2 + p)2 γ −1 (γM 2 +1€)2 = = F(M) 2( v) 1 v 3 γ pv γ 2 M 2 ( 1)M€2 2 € ρ ( 2 ρ + γ −1 ) [ γ − + ]

v M = Mach € γp/ρ number

F(M1) = F(M2 ) €

−1 M2 = F (F(M1)) €

€ pre-shock (1) post-shock (2) −1 M2 = F (F(M1)) super-sonic 2 sub-sonic (γ −1)M + 2 M = 1 2 2γM 2 −γ +1 v1 v2 € 1 c v1 s,2 2 v2 (γ −1)M1 + 2 v2 = 2 cs,1 € v1 (γ +1)M1 ρ (γ +1)M 2 ρ1 ρ2 2 1 = 2 ρ1 (γ −1)M1 + 2 € p 2 2 p 2γM − γ +1 2 = 1 p1 p γ +1 € 1 ⎛ ⎞ ⎛ ⎞ p2 ρ2 s1 s2 − s1 = ln⎜ ⎟ −γ ln⎜ ⎟ s ⎝ p1 ⎠ ⎝ ρ1 ⎠ 2 €

€ Hypersonic limit

M1 >> 1 γ = 5/3

2 (γ −1)M1 + 2 (γ −1) 1 M2 = 2 M2 → = 2γM1 −γ +1 2γ 5

2 v (γ −1)M + 2 v2 (γ −1) 1 2 = 1 → = 2 v (γ +1) 4 € v1 (γ +1)M1€ 1 € ρ (γ +1)M 2 ρ (γ +1) 2 = 1 2 → = 4 ρ (γ −1)M 2 + 2 ρ (γ −1) 1 1 1 € € € 2 p 2γ 2 p2 2γM1 − γ +1 2 → M 3 1 2 = € 1 p2 → 2 × 2 ρ1v1 p γ +1 p1 γ +1 € 1 €

€ € € Entropy’s tale

s2 ≥ s1  pre-shock must be supersonic  fluid goes from super- to sub-sonic  fluid slows down crossing shock  fluid compresses crossing shock

viscosity = 0  s2 = s1 ⎛ ⎞ ⎡ ⎤ ∂ 1 2 p 1 2 γ  ⎜ 2 ρv + ⎟ = −∇⋅ ⎢ 2 ρv v + pv +σ ⋅ v⎥ t 1 1 ∂ ⎝ γ − ⎠ ⎣ γ − ⎦ viscosity does not appear in jump condions (conservaon laws) but it is necessary for transion from 1 to 2 € Q: what goes wrong if we set µ = 0 ? Q: is that diff. from taking µ  0 ? vs rarefacon

vi

vi

v -v i s -vs co-moving frame v1 v2

sasfies conservaon v1 v 2 but violates 2nd law TD

What happens instead? ρ2 ρ1 cs rarefacon

vi

vi

vi

v t i

cs

vi A nearby shock M=1 asin(1/M∞) Mach cone

M∞

M∞=8 γ = 5/3

M=5.2

49o

8.0

M∞=8.0 Spreiter, Summers & Alksne 1966 Farther afield…

shock

contact disconnuity

shock

Opher v2, ch. 7 Hundhausen 1973 (v.3, ch.8) Q: which way is for each? Both move outward ( Forward shock 2 shocks: with slower wind creates High speed stream colliding Reverse shock € n ˆ v s >0) Shocks in MHD ∂ρ = −∇⋅ [ρv] ∂t

∂(ρv) 1 1 2   = −∇⋅ ρvv - 4π BB + (p + 8π B )I +σ ∂t [ ] € ∂ 1 2 1 1 2 1 2 γ 1  ( 2 ρv + γ −1 p + 8π B ) = −∇⋅ [ 2 ρv v + γ −1 pv − 4π (v × B) × B +σ ⋅ v] ∂ t € kinec thermal magnec

€ ∂B = −∇ × [v × B] ∂t

0 = ∇⋅ B

€

€ 0 = −∇⋅ [ρv]   0 = −∇⋅ ρvv - 1 BB + p + 1 B2 I +σ [ 4π ( 8π ) ]  € 0 = −∇⋅ 1 ρv 2v + γ pv − 1 (v × B) × B +σ ⋅ v [ 2 γ −1 4π ] € 0 = −∇ × [v × B]

€ 0 = ∇⋅ B

€

€ v2

B1 B2

v1 0 = −∇⋅ [ρv] ρvn = const. B const. 0 = ∇⋅ B n =   0 = −∇⋅ ρvv - 1 BB + p + 1 B2 I +σ € [ 4π ( €8π ) ]

€ € 2 ρvn [[v]] − Bn[[B]]/4π +[[p + B /8π]]nˆ = 0 € ρvn [[v⊥]] = Bn[[B⊥]]/4π €

v2 € B1 B2

v1 0 = −∇⋅ [ρv] ρvn = const. B const. 0 = ∇⋅ B n =   0 = −∇⋅ ρvv - 1 BB + p + 1 B2 I +σ € [ 4π ( €8π ) ] € € ρvn [[v⊥]] = Bn[[B⊥]]/4π 0 = Bn[[B]]

€ Case 1: ρvn = 0

Case 1a: Bn = 0 € Case 2: Bn = 0 €

v2 Tangenal Disconnuity

B1 B2 Case 1b: [[ B ]] = 0

v1 Contact Disconnuity Case 1a: Tangenal Disc. 0 = −∇⋅ [ρv] ρvn = const.

vn = 0, Bn = 0 0 = ∇⋅ B Bn = const.

  0 = −∇⋅ ρvv - 1 BB + p + 1 B€2 I +σ € 2 [ 4π ( 8π ) ] [[p + B /8π]]nˆ = 0 € €

J € €

An equilibrium current B1  v2 sheet separang uniform B  fields w/ different angles. v 1 B2 • total pressure balance  • possible shear

2 B1 /8π p2 p 2 1 B2 /8π Case 1b: Contact Disc. 0 = −∇⋅ [ρv] ρvn = const.

vn = 0, [[ B ]] = 0 0 = ∇⋅ B Bn = const.   0 = −∇⋅ ρvv - 1 BB + p + 1 B2 I +σ ˆ [ 4π ( 8π € ) ] € [[p]]n = 0 € € 0 = −∇ × [v × B] [[v⊥]] = 0 € € B € HOT COLD €

p p2 1 related to linear

T1 T2 waves w/ ω=0

entropy mode: [[ρ]]≠0 ρ1 ρ2 shear mode: [[vt]] ≠0 Case 2: perpendicular shock ρvn [[v⊥]] = Bn[[B⊥]]/4π

Bn = 0, vn ≠ 0 ⇒ v⊥,1 = v⊥,2 = v⊥ transform away   0 = −∇⋅ ρvv - 1 BB + p + 1 B2 I +σ 2 2 [ 4π ( 8π ) ]€ [[ρv + p + B /8π]] = 0

1 2 γ 1 € 0 = −∇⋅ ρv v + pv − (v × B) × B +σ ⋅ v [ 2 γ −1 4π ] 1 2 γ 1 2 € [[ 2 ρvnv + 1 pvn + 4 B vn ]] = 0 € γ − π

€ 0 = −∇ × [v × B] [[Bvn ]] = 0 v v 1 €2 € HD shock w/ 2 ``fluids’’ B1 B 2 €• plasma: γ=5/3 • mag. field: γ=2 p B1 B2 2 p1 Gabriel 1976

hot Bn = 0 Bn ≠ 0 vn = 0 Tangenal Contact disconnuity disconnuity cold vn ≠ 0 Perpendicular General shock MHD shock Case 3: General MHD shock ρvn [[v⊥]] = Bn[[B⊥]]/4π Bn ≠ 0, vn ≠ 0 ⇒ v⊥,1 ≠ v⊥,2

ρvn = const.≠ 0 0 = −∇⋅ [ρv] cannot transform away v⊥ € 0 = ∇⋅ B Bn = const.≠ 0   can transform to 1 1 2 € deHoffman-Teller frame 0 = −∇⋅ ρvv - 4π BB + (p + 8π B )I +σ [ € ] €  vn,1 1 2 γ 1 v €= v − B € 0 = −∇⋅ 2 ρv v + γ −1 pv − 4π (v × B) × B +σ ⋅ v dHT ⊥,1 ⊥,1 [ € ] Bn,1 0 = −∇ × [v × B] € In dH-T frame: • v || B € 1 1 B2 € • E1 = v1 X B1 = 0 v1 € • E 2 = v2 X B2 = 0 B v2 1 • v2 || B2 • flow is parallel to field lines on both sides Shock characterized by 3 dimensionless Jump condions in pre-shock parameters: de Hoffman-Teller frame

v1 v1n ρvn [[v⊥]] − Bn[[B⊥]]/4π = 0 M = = 1A B / 4πρ B / 4πρ 2 2 1 1 1n 1 [[ρvn + p + B⊥ /8π]] = 0 ⎛ B ⎞ 1 2 γ −1 ⊥1 θ1 = tan ⎜ ⎟ [[ 2 ρvnv + γ −1 pvn ]] = 0 € ⎝ Bn1 ⎠ € p1 € β1 = 2 B1 /8π B2 € v1 € Use 3 jump. conds B v2 β β1 1 2 to solve for € θ2 downstream parameters θ1 MA2 θ2 β2 2 2 ρvn [[v⊥]] − Bn[[B⊥]]/4π = 0 [[(MA −1)tan θ]] = 0

2 2 eliminate θ2 in terms of θ1, MA1 and MA2 etc. unl… € €

B2 v1 B v2 β β1 1 2

θ2

θ1 NB: 2 2 0 • ρ2/ρ1 = MA1 / MA2 θ1 =30 β1=0.05 • region M A2 > MA1 (ρ2 < ρ1) disallowed by 2nd law thermo.

• M A2 = MA1 reps. 3 linear waves of MHD • each shock: flow faster (slower) than lin.

wave in region 1 (2) 3 MHD shocks

fast: MA1 > MA2 > 1 intermediate: MA1 > 1 > MA2 slow: 1 > MA1 > MA2 3 MHD shocks

fast: MA1 > MA2 > 1 intermediate: MA1 > 1 > MA2 slow: 1 > MA1 > MA2

2 2 [[(MA −1)tan θ]] = 0 2 B⊥2 tanθ2 MA1 −1 = = 2 B⊥1 tanθ1 MA 2 −1 € fast intermediate slow B2 B B2 B1 B1 2 B € 1

θ 2 θ 2 θ θ1 θ1 2 θ1

θ2 > θ1> 0 B1 < B2 θ1 > 0 > θ2 θ1 > θ2> 0 B1 > B2 B1 B1 FMW

SMW SAW θ1 θ1 quasi-parallel quasi-perpendicular o, o, θ1=20 β1=0.2 θ1=70 β1=0.2 B1 B1 FMW

SMW SAW θ1 θ1 quasi-parallel quasi-perpendicular o, o, θ1=20 β1=2.0 θ1=70 β1=2.0 Hypersonic limit

Magnec  4 pressure: only x16

Plasma pressure, no limit H-sphere: θ =84o, β =15 Fast Shocks 1 1 Observed Voyager I @ r=1.6 AU

(vesc = 34 km/s) T R 282 km/s 321  N 305 km/s

-3 7.6 cm-3 19 cm

M = 8.0/cos(θ ) A1 1 Vinas & Scudder 1986 ISEE-1 crosses bow shock

MA1 = 8.1/cos(θ1) o, θ1=74 β1=6



Vinas & Scudder 1986 Forward shock

Voyager 2 at r=2.17 AU

o, θ1=23 β1=1.7, MA1=2.4 Scudder et al. 1984

upstream downstream Gosling et al. 1994

=3.5 AU r

@ Ulysses 2 (1?) TDs 2 FM shocks Solar Fast Shocks

Type II Liu et al. 2009

n =107 cm-3 e Band spling reveals both 6 -3 ne=10 cm ne1 & ne2 Type III ne2/ne1=1.59

MA1~ 1.26-1.47 Solar Fast Shocks

Type II Liu et al. 2009

ground

7 -3 ne=10 cm 6 -3 ne=3 x 10 cm Band spling reveals both n =106 cm-3 n & n e Space (STEREO) e1 e2 Type III ne2/ne1=1.59

MA1~ 1.26-1.47 Slow shocks: reconnection

SS’ Lin & Lee 1999

p Model reconnection

Tsuneta and Naito 1998

Forbes, T.G., and Malherbe 1986 (v.2 ch. 6) Walthour et al. 1994

 magnetopause: SS from reconnecon ISEE-1 at the B Intermediate Shocks z  By Reconnecon between skewed fields  Lin & Lee 1999

Petschek & Thorne 1967 Entropy’s Tale … the sequel

• A shock converts kinetic to thermal energy • How? What is the µ-physics? – Collisions (i.e. viscosity) – high enough density – Otherwise…(story not finished yet) • Instabilities, chaotic particle motion, … • Related to acceleration of non-thermal particles • Easier if particles gyrate near shock: quasi-perp. thermalizes easier than quasi-parallel Summary