The Bow Shock and the Magnetosheath

Chapter 6 The Bow Shock and the Magnetosheath The solar wind plasma travels sally at speeds which are faster than any flid plasma wave relative to the magnetosphere. Therefore a standing shock wave forms arond the magnetosphere jst as in front of an aircraft traveling at spersonic speeds. The bow shock is the shock in front of the magnetosphere and the magnetosheath is the shocked solar wind plasma. Therefore it is not directly the solar wind plasma which constittes the bondary of the magnetosphere bt the strongly heated and compressed plasma behind the bow shock. The region is rich in varios wave phenomena, bondaries and shocks are often treated as discontinities. 6.1 Solar Wind Solar coronal otflow: static atmosphere In a steady state with radial otflow the solar wind mst satisfy continity, momentm, and energy eqations d r nv ) dr = 0 6.1) v dv 6.) d dr dr ) p n γ with the gravitational constant G and the solar mass M. = GM 1 dp r m i n dr Exercise: Derive the above eqation from the set of MHD eqations = 0 6.3) The first soltions of these eqations have assmed a static plasma Chapman) yielding from 6.) and 6.3) GM r 1 d dr dp m i n dr ) p n γ = 0 6.4) = 0 6.5) 64

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 65 The energy eqation can be written as ) d p = m dr n γ γ d dr ) d p dr n γ ) c s n γ 1 = 1 dp n γ dr γ p dn n γ+1 dr = 0 = γ 1 γ m c s i dn n γ dr + 1 γ m 1 dc s i n γ 1 dr = 0 with c s = γp/m i n = γk B T/m i which yields γ 1 n dp dr dn dr = m i c dn s dr = 1 dc s c s dr 6.6) 6.7) or 1 dp n dr = and sbstittion into the force balance eqation yields m i dc s γ 1 dr 6.8) 1 dc s γ 1 dr = GM r where c s = γp/m i n = γk B T/m i with the soltion T T 0 = γ 1 GM 1 1 ) γ k r0 r 6.9) This represents the general soltion for gravitational bond atmospheres if the radial velocity can be neglected. The temperatre decreases at the so-called adiabatic lapse rate with height. The soltion for the pressre p and n can be obtained from 6.3) sing the ideal gas law p = nk B T to relate pressre and temperatre. In the special case of constant temperatre the force balance eqation can be directly integrated yielding Parker s steady state solar wind eqation Using the continity eqation in the energy eqation 6.6) Sbstittion in the force balance eqation yields GMm 1 p = p 0 exp kt r 1 )) R S r + 1 dn n dr + 1 dv v dr = 0 r c s + c s dv v dr = 1 dp m i n dr 1 dv dr = Φ r + r c s + c s dv v dr 6.10)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 66 or r v c s) 1 dv v dr = c s Φ Normalization to solar radii R s with Φ 0 = GM/R S and R = r/r S yields 1 R v c ) 1 dv s v dr = c s Φ 0 R 6.11) An alternative form in terms of the sonic Mach nmber can be obtained by sbstittion of v = c sms. ) Φ0 R + K R M s 1 dm s = 1 + γ 1 ) [ M Ms s 4K + 3γ 5 ] Φ 0 6.1) dr γ 1 R where K is an integration constant. There is no known analytical soltion to these eqations bt it is possible to draw several qalitative conclsions: For M s = 1 the lhs of 6.11) vanishes sch that the rhs mst also be 0. This sonic point is located at a distance R c = Φ 0 c s To have the sonic point above the solar srface implies R c > 1 or Discssion for Λ = c s Φ 0 /R 3 c s < Φ 0 Mach Nmber 1 4 1 1 3 Critical Radii Figre 6.1: The for types of the Parker soltion. a) Λ < 0 : i. v > c s => dv/dr < 0 ii. v < c s => dv/dr > 0 and v -> c s for dv/dr -> ± b) Λ > 0 : i. v > c s => dv/dr > 0 ii. v < c s => dv/dr < 0 c) Λ > 0 : Singlarity for v = c s or dv/dr = 0

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 67 Properties of the Solar Wind Typical velocities of the solar wind range between 300 km/s and 1400 km/s with a typical vale of abot 500 km/s. Sorce region of the solar wind are the so-called coronal holes, which are region where the magnetic field of the sn stretches ot into interplanetary space, i.e., is not closed in a loop back to the sn. In the solar corona the temperatre is abot 1.6 10 6 K and density abot 5 10 17 cm 3. Velocities of the solar wind: between 300 km/s and 1400 km/s with a typical vale of abot 500 km/s. Density: decreases from abot 10 4 cm 3 at 0.1 AU to abot 5cm 3 at 1 AU. Temperatre: decreases from abot 10 6 K at 0.1 AU to abot 10 5 K at 1 AU corresponding to approximately 10 ev Thermal electron velocity 1500 km/s and thermal ion velocity 35 km/s Magnetic field: abot 5 nt Alfvén speed: v A = B sw / µ 0 m i n 40 km/s Fast mode speed: c f = Plasma β > 1 often B + γp) /µ 0 m i n 60 km/s B Sector Bondary Rotating Sn radially expanding plasma Earth Figre 6.: Illstration of spiral and sector strctre of the solar wind. Ths the solar wind is mch faster than the sond, Alfvén, and fast mode wave speeds in the frame of the Earths. This implies the existence of a shock in front of the magnetosphere - the bow shock - across which the solar wind plasma is decelerated to sb-fast velocities which then interact with the Earth. Cased by the sn s rotation the magnetic field has a spiral shape Parker spiral configration) and is sally closely aligned with the ecliptic plane.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 68 6. MHD Discontinities and Shocks The typical solar wind is mch faster than the fastest MHD mode speed c f = B + γp) /µ 0 m i n. Similar to spersonic flow past an aircraft the solar wind develops a shock in front of the magnetosphere across which the solar wind is decelerated to velocities below the fast mode speed. In a flid approximation this shock can be represented as a discontinity across which mass, momentm, and energy has to be conserved. The jmp conditions on large flid) scales can be derived from the MHD eqations. 6..1 Rankine Hgoniot Conditions and MHD Discontinities To derive the jmp conditions across a flid bondary it is assmed that this bondary is infinitesimally thin and that the system is in a stationary state. The assmption of zero width is eqivalent to assming a one-dimensional bondary. Assming a property which is conserved f t = f 6.13) these assmption imply f n = const where n is the velocity normal to the discontinity. Another way to express this reslt is [f n ] f d nd f n = 0 6.14) where the indexes d and indicate the downstream and pstream regions. It is easy to show that [ab] = 1 a [b] + 1 [a] b where In other words the flx of f is constant across the thin bondary. To apply this method it is convenient to write the basic eqation the set of MHD eqations) in a conservative form. This is already the case for the continity eqations. The momentm eqation can easily be broght into conservative form and similarly the total energy density thermal, blk flow, and magnetic energy) can be expressed in conservative from implying energy conservation. The complete set of eqations which need to be solved are ρ t ρ t w tot = ρ 6.15) = [ ) ρ + p + B 1 1 ] BB µ 0 µ 0 6.16) [ 1 = t ρ + γp γ 1 + 1 ) B B B η ] j B µ 0 µ 0 µ 0 6.17) B t = B ηj) 6.18) 0 = B 6.19) with the total energy density w tot = 1 ρ + p γ 1 + 1 B 6.0) µ 0 Exercise: Derive the conservative form of the momentm eqation. Exercise: Derive the conservative form of the total energy density eqation.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 69 Exercise: Why is the electric field energy not considered in the energy eqation. B = 0 is sally an initial condition. Why is it inclded in the set of eqation to derive the jmp conditions. Using the above set of MHD eqations any steady state has to satisfy [ ρ + p + B µ 0 ) ρ = 0 6.1) ] 1 1 µ 0 BB [ 1 ρ + γp γ 1 + 1 ) B B B η j B µ 0 µ 0 µ 0 ] = 0 6.) = 0 6.3) B ηj) = 0 6.4) B = 0 6.5) To obtain the so-called Rankine Hgoniot conditions We now assme a one-dimensional bondary with zero width and assme ideal MHD η = 0): Properties: n n [ρ] = 0 6.6) [ ] n [ρ] + n p + B 1 n [BB] = 0 6.7) µ 0 µ 0 [ 1 γp + γ 1) ρ + 1 ) ] µ 0 ρ B ρ 1 n [ B) B] = 0 6.8) µ 0 n [ B] = 0 6.9) n [B] = 0 6.30) Variables: ρ,, p, B; No variables: 8; No of eqations 8 [B n ] = 0 => B n = const [ρ n ] = 0 => ρ n = f n = const Classification: [ n ] = 0 => MHD discontinity [ n ] 0 => shock Applying the trivial properties f n [ n ] + [ p + B t µ 0 ] = 0 6.31) f n [ 1 + f n [ t ] 1 µ 0 B n [B t ] = 0 6.3) B n [ t ] [ n B t ] = 0 6.33) γp γ 1) ρ + 1 ] µ 0 ρ B t 1 B n [ B] = 0 6.34) µ 0

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 70 Discontinities: [ n ] = 0 Contact Discontinity n ρ Τ B Tangential Discontinity ρ 1 Τ 1 v n =0 p B t n p 1 Rotational Discontinity B B t1 n B t =-B t1 v n =v An B t1 Figre 6.3: MHD discontinities. a) Contact Discontinity: n = 0, B n 0 => [ t ] = 0, [B t ] = 0, [p] = 0 => [ρ] 0 and [T ] 0. b) Tangential Discontinity: n = 0, B n = 0 => [ p + B t µ 0 ] = 0 c) Rotational Discontinity: n = const => f n [ t ] 1 µ 0 B n [B t ] = 0 B n [ t ] n [B t ] = 0 [ρ] = 0, [ p + B t µ 0 ] = 0 } n B n µ 0 ρ ) [B t ] = 0 => n = An and [ t ] = ± An B n [B t ] = ± [ At ] with v At = B t / µ 0 ρ tangential direction defined [B t ] with t1 = At1 => t = v At v At1 + t1 = v At energy eqation [ γp γ 1) + 1 ] Bt = 0 µ 0 => [p] = 0 and B t1 = B t or trivial soltion B td = B t ) 6.. Hydrodynamic Shocks Formation: steepening of large amplitde waves -> compression

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 71 p t=t 1 p t=t c s >c s0 c s0 x Figre 6.4: Illstration of wave steepening. x fast motion of an obstacle in a gas or liqid e.g., aircraft, space shttle, low altitde satellites) waves propagating into a rarefied medim Eqations: n d d = n 6.35) p d + mn d d = p + mn 6.36) ) ) 1 mn d γ 1 d + γ 1) p d d = mn γ + γ 1) p 6.37) Introdce: Mach nmber M = /c s with c s = γp/mn Relations: n d n = d γ + 1) M + γ 1) M 6.38) = n n d 6.39) p d = γm γ 1) p γ + 1 6.40) Entropy irreversibility): s = c V ln p ρ γ 6.41) with c V = 1 k Bm γ 1 = 3k B m sch that s = sm ) ds dm = 4γ γ 1) M 1) c v M [γm γ 1)] [ + γ 1) M ] 6.4) sch that ds = 0 dm for M = 1 ds > 0 dm for M > 1 Since s d = s for M = 1 => s d > s for M > 1 => Entropy mst increase for shock! Shock properties:

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 7 pressre and density increase velocity decreases entropy increases Mach nmber M d < 1 Strong shocks M 1 with γ = 5/3: n d = γ + 1 n γ 1 = 4 6.43) d = γ 1 γ + 1 = 1 4 6.44) p d = γm p γ + 1 6.45) Md = γ 1 = 1 γ 5 6.46) 6..3 MHD Shocks Coplanarity: [ n ] 0, and B n 0 => B n µ 0 f [B t] = [ n B t ] Assme B t = B t e y sch that the z component becomes B n µ 0 f B zd = nd B zd 6.47) with the soltion nd = B n or B µ 0 f zd = 0. Since in general nd B n µ 0 except for Alfvén waves) the f general soltion implies that the tangential fields on the two sides of the shock are aligned B t B td. 6.48) It follows that [ t ] [B t ]. Ths a Galilei transformation can always render the velocities parallel to the magnetic fields. Basic Eqations for MHD Shocks: Assmptions: B t = B y e y, t = y e y, and x in the normal n direction. ρ d nd = ρ n 6.49) p d + 1 Byd + ρ d nd = p + 1 By + ρ n 6.50) µ 0 µ 0

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 73 ρ d nd yd 1 µ 0 B n B yd = ρ n y 1 µ 0 B n B y 6.51) 1 ρ d d + γp d γ 1) + B d µ 0 nd B yd yd B n = n B y y B n 6.5) ) nd B n µ 0 d B d = 1 ρ + γp ) γ 1) + B n B n B 6.53) µ 0 µ 0 Parallel Shock For this case the magnetic field is parallel to the shock normal. The velocity is aligned with the magnetic field and one can always transform into a frame in which y = 0. Exercise: Demonstrate that the parallel shock redces to the pre hydrodynamic shock. Perpendiclar Shock B n = 0. This is a special case we need to discss becase it is not contained in the general soltion. Here the magnetic field is exactly perpendiclar to the shock. Parallel Shock Perpendiclar Shock ρ p ρ d p d B B d d d ρ p ρ d p d B The basic eqations redce to Straightforward relations: B d n n Figre 6.5: Illstration of parallel and perpendiclar shocks. ρ d nd = ρ n 6.54) p d + 1 Byd + ρ d nd = p + 1 By + ρ n 6.55) µ 0 µ 0 ρ d nd yd = ρ n y 6.56) nd B yd = n B y 6.57) 1 ρ d d + γp ) d γ 1) + B yd 1 nd = µ 0 ρ + γp ) γ 1) + B y n 6.58) µ 0 y = 0 6.59) X = ρ d ρ 6.60) nd n = 1 X 6.61) B yd B y = X 6.6)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 74 In addition we se M = c s 6.63) β = p th = µ 0p = p B B γ γp c s = ρ c s A 6.64) 6.65) A = B µ0 ρ 6.66) From 6.55) p d = 1 + 1 B ) y 1 B yd + ρ n 1 ρ d ) nd p µ 0 p By p ρ n = 1 + β 1 1 X ) + γm 1 X 1) and from 6.58) p d p = n + γ 1 nd γ = X + γ 1 ρ p n ρ d ) nd nd ρ n M X X 1) + γ 1 γ B y + γ 1 γ µ 0 p ) X X β ) n B yd nd By Eqating the two expressions for p d /p and mltiplying with X and dividing by X 1 X + γ 1 M X + 1) γ 1 γ Rearranging this expression X is the positive root of β X + β 1 X 1 + X) γm = 0 fx) = γ) X + [ β + γ 1) βm + ] γx γ γ + 1) βm = 0 6.67) Note: Only one positive root! fx) 1 X Figre 6.6: fx) for the shock eqation. i) Shock mst be compressive, i.e., X 1 implies f1) < 0. f1) = γ) + [ β + γ 1) βm + ] γ γ γ + 1) βm = M [βγ γ 1) βγ γ + 1)] + γ + γ + βγ) = βγm + + βγ) < 0

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 75 => M 1 + γβ = c s + A c s 6.68) or c s + A ii) Limit M 1 or sch that fx) = γ 1) βm γx γ γ + 1) βm = 0 X = γ + 1 γ 1 = 4 ρ d ρ = 4 6.69) nd n = 1 4 6.70) B yd B y = 4 6.71) p d p 5 4 M 6.7) General Shock Soltion Using a frame of reference with E z = x B y + y B x = 0 sch that y = n B y B n 6.73) Sbstittion in 6.49) - 6.53) with nd = n /X, X = ρ d /ρ, And = An/X, and An = An yields ρ d nd 1 µ 0 B n ) B yd = ρ n 1 µ 0 B n ) B y or B yd = n An ρ B y nd = n An X And ρ d n X An B yd = A X 6.74) B y X A The last reslt ses the property that the magnetic field and velocity are parallel i.e., n / = B n /B, and the same for An ). This also implies from Ohm s law y / B y n ) = 1/B n = const yd y = B yd B y nd n = A X A 6.75)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 76 The last two terms in the energy eqation B yd + Bn ) nd B n B d B d = yd + B ) n nd B n µ 0 µ 0 µ 0 µ 0 nd B yd B n + nd B n ) = 0 Sch that p d + 1 Byd + ρ d nd = p + 1 By + ρ n 6.76) µ 0 µ 0 1 ρ d d + γp ) d 1 nd = γ 1) ρ + γp ) n 6.77) γ 1) remain to be solved. The energy eqation yields Finally p d = γ 1 p γ = γ 1 = X + γ 1 X c s ρ d d + n 1 + γ 1 ρ ) p nd γ p ρ d d + X 1 + γ 1 ) ρ c s c s ) 1 d p d + 1 µ 0 B yd + ρ d nd = p + 1 µ 0 B y + ρ n p d = 1 + γ n 1 ρ d ) ) nd + 1β 1 B yd p c s ρ n By = 1 + γm ) 1 X 1 + 1 1 X ) A β X A Combining the pressre eqations yields X [ 1 + γ 1 1 c s ) ] d = 1 + γm ) 1 X 1 + 1 1 X ) A β X A with yd from 6.75). Introdcing the angle θ between the incident magnetic field and the shock normal n, X is the soltion of [ XA) Xc s + 1 ] cos θ {X γ 1) γ + 1)} + 1 A X sin θ [ γ + X γ)) X A γ + 1) X γ 1)) ] = 0 6.78)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 77 Relations: Properties: 6.78) has 3 soltions ρ d ρ = X 6.79) nd = 1 6.80) n X yd = A 6.81) y X A B yd = A X 6.8) B y X A p d = X + γ 1 X ) 1 d 6.83) p c s Intermediate Alfvén) wave for X = 1 and = A X 1 slow and intermediate waves => shocks for X > 1 Slow Shock Fast Shock B B d n θ d B d θ n B d Figre 6.7: Illstration of slow and fast shocks. Slow and fast shocks: Compressive with X > 1 => p d > p Sign of B y conserved for A => B yd < B y and yd < y - slow shock X A => B yd > B y and yd > y - fast shock limit B n -> 0 fast: B yd /B y -> X perpendiclar shock slow: -> TD

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 78 Two special cases of slow and fast shocks are switch-on and switch-off shocks a) Switch-off slow) shock ) => B yd = 0 tangential magnetic field is switched off! Since B it follows that n = = A 6.84) B n µ0 ρ = An 6.85) => switch-off shock propagates at Alfvén speed. With the above relations the shock eqation becomes [ 1 X) X c s + 1 ] A cos θ {X γ 1) γ + 1)} + 1 X sin θ [ γ + X γ) X γ + 1) + X γ 1) ] = 0 One can factorize this expression by noting that X = 1 is a soltion [ 1 X) X c s + 1 ] A cos θ {X γ 1) γ + 1)} + 1 X sin θ X 1) γx X γ)) = 0 or X 1) [ X c s + cos θ {X γ 1) γ + 1)} A ] + X sin θ γ 1) X γ) = 0 Re-arranging yields the eqation which has one positive soltion Properties: i. c s/ A > 1/: c s + γ 1 A ) X X [ 1, 1 + c s/ A + γ 1) 1] X increases) for θ [0, π/] ii. c s/ A < 1/: c s + γ 1 + cos θ )) X + γ + 1) cos θ = 0 6.86) A X [ γ + 1) / c s/ A + γ 1), 1 + c s/ A + γ 1) 1] X decreases) for θ [0, π/] Switch-on fast) Shock For propagation along the magnetic field we assme B y = 0 6.87) θ = 0 6.88)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 79 Switch-off Shock Switch-on Shock n θ = 0 B d B d B d n θ B d Figre 6.8: Illstration of switch-off and switch-on shocks. Using this assmption yields the shock eqation [ XA) Xc s + 1 ] {X γ 1) γ + 1)} = 0 6.89) The slow shock soltion is given by [..] = 0 and is a prely hydrodynamic shock. The fast shock soltion is given by X = A Eliminating p d from 6.50) and 6.53) yields B yd as B yd B n = X 1) [ γ + 1) γ 1) X γµ 0p B ] 6.90) Since the rhs mst be positive the compression ratio mst satisfy 1 < X < γ + 1 c s/ A γ 1 < γ + 1 γ 1 6.91) This implies that the field ratio is largest for c s A and reqires c s/ A < γ + 1) /. Finally the general soltion also contains the intermediate wave as a soltion with X = 1 Smmary of shock properties: ρ d > ρ p d > p s d > s sign B yd = sign B y Slow shocks: A

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 80 B yd < B y B d < B yd < y special case: switch-off shock n = ± An B yd = 0 Strongest compression for c s/ A 1, maximm X = 4 Fast Shocks X A B yd > B y B d > B yd > y compression increases with Mach nmber special case: switch-on shock B yd = 0 Strongest compression for c s/ A 1, maximm X = 4 6.3 Properties of the Bow Shock and the Magnetosheath 6.3.1 Foreshocks and dehoffmann-teller Frame With the MHD plasma approximation one can analyze basic shock strctre and determine downstream conditions as a fnction of the pstream solar wind properties. We have arged that the reason for the formation of the bow shock is the sper fast solar wind speed and that no information can travel pstream from a fast shock. However, considering a kinetic plasma this is not anymore tre. In particlar any Maxwellian distribtion will not only contain thermal particle bt also particle althogh few which have mch higher energies than implied by the thermal motion. To stdy the dynamics of particles in the vicinity of a fast shock it is instrctive to se the dehoffmann- Teller frame, i.e., a frame of reference in which the shock is at rest and the electric field is 0. We have sed this frame already in deriving the general shock eqation. Usally it is simple to identify the pstream velocity normal to the shock plane n. Moving with this velocity the pstream electric field is 0. However, from or derivation we know that the downstream electric field is nonzero. To transform into the dehoffmann-teller frame in which the pstream and

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 81 x Electron Foreshock B Solar Wind Ion Foreshock ht ' θ Magnetopase Bow Shock B y M'sheath Figre 6.9: Illstration of foreshocks and the dehoffmann-teller frame. the downstream electric field is zero and the flow is aligned with the magnetic field on both sides we need a transformation by the dehoffmann-teller velocity v ht. From Figre 6.9) we see that v ht = n tan θ 6.9) where θ is the angle between the shock normal and the pstream magnetic field. As θ approaches 90 the dehoffmann-teller velocity increases rapidly and approaches relativistic vales. For θ = 90 there is no dehoffmann-teller frame. A general expression for v ht is obtained as follows. For the transformation the electric field E = B = 0 sch that = n v ht v ht B = n B Taking the cross prodct with the shock normal nit vector one obtains n v ht B ) = v ht n B or v ht = n n B ) n B 6.93) In the dehoffmann-teller frame particles have only the gyro motion and the motion parallel to the magnetic field sch that particle motion is mch easier to stdy. Particles can otrn the shock if there velocity is sfficiently large. In the dehoffmann-teller frame the velocity needed for particle to escape from the shock is. From Figre 6.9) we see that the marginal escape velocity esc = n / cos θ. Ths particles moving faster than this velocity can otrn the shock and poplate the pstream region. The velocity of these particle is a combination of their pstream velocity and the dehoffmann-teller velocity or E B drift in the pstream region). Since particle with higher velocities move ahead of slower escaping particle an observer wold expect to see higher energy particles first or frther from the shock.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 8 At the Earth the IMF is often in a Parker spiral configration Figre 6.) implying that it has an angle of in the average 45 with the Sn-Earth line. The last field line which toches the bow shock has a shock angle of θ = 90. Field lines frther downstream have decreasing shock angles. Particles originating from the vicinity of the first field line are the most energetic becase only those can escape p stream. The pstream region which is filled with the escaping electrons is called the electron foreshock. Ions appear to escape only if θ 70 sch that they fill a region - the ion foreshock - downstream of the electron foreshock region. The foreshock regions are highly trblent. The stream of energetic particles against the incoming solar wind is nstable with respect to many instabilities of the two-stream type. Ths the foreshock regions are rich in many different plasma waves excited by the energetic particles. 6.3. Shock Strctre and Heating The typical solar wind is a high-mach nmber stream with M 8 sch that the bow shock is a fast magnetosonic shock. This also implies that density and magnetic field jmp by almost a factor of 4 at the sbsolar the location closest to the sn) location of the bow shock. This distance from Earth of this location is approximated by R bs = 1 + 1.1 n ) sw R mp 6.94) n msh where R mp is the stand-off distance of the magnetopase the location where the obstacle magnetosphere begins), n sw is the solar wind nmber density, and n msh is the nmber density in the sbsolar region in the magnetosheath downstream of the bow shock). Away from the sbsolar point the shock is crved Figre 6.9). Becase of the crvatre the solar wind flow is not anymore exactly normal to the shock and the normal component of the solar wind velocity is given by n = n bs sw = sw cos ϕ 6.95) where ϕ is the angle between the shock normal and the Sn-Earth line GSM x direction). For a sbsolar Mach nmber of 8 the Mach nmber is redced to 1 for ϕ 80. Ths the bow shock is finite in size limited to normal directions with ϕ < 80 and the shock strctre changes from the sbsolar region to the periphery of the bow shock becase the Mach nmber and the magnetic field and orientation relative to the shock changes. The changing Mach nmber has additional implications. High-Mach nmber shocks are called spercritical with M > M c and have a different strctre than sb-critical M < M c shocks. The critical Mach nmber is sally defined as the Mach nmber for which the downstream velocity is eqal to the downstream sond speed. This yields a critical Mach nmber of M c.7. For practical prposes observations indicate a critical Mach nmber < for the bow shock. The shock strctre is also different for qasi-perpendiclar almost perpendiclar) and qasi-parallel shocks. Particles cannot travel far into the pstream region for perpendiclar shocks becase the gyro motion brings them back into the shock. Typically perpendiclar shocks show a shock foot where the magnetic field gradally increases in front of the main shock. Behind the main ramp the shock shows an overshoot with field vales slightly larger than the asymptotic downstream vales.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 83 Qasperpendiclar Foot Ramp Overshoot Qasperparallel Figre 6.10: Typical magnetic shock profiles. Qasi-parallel shocks allow a more efficient the escape of particles. Small variations of the pstream magnetic field orientation are amplified by the shock and generate considerable trblence. In addition - as mentioned - foreshock particles contribte to trblent fields sch that both the pstream and the downstream region show oscillations in the plasma and magnetic field properties. Shock Crrents and Ion Acceleration A shock leads by definition to an irreversible to heating and compression of the plasma. It therefore reqires the presence of irreversible physics sch as viscosity or resistivity. The plasma at the bow shock is highly collisionless bt the viscos-like or resistive-like processes can occr de the interactions of particles with the trblent wave fields. However, resistivity and viscosity are insfficient to explain shock strctre and typical particle properties sch as preferential ion heating. For instance, resistivity wold preferentially heat electron. The dissipation in spercritical) collisionless shock is largely controlled by the actal ion dynamics. The change in the tangential field [B t ] corresponds to a crrent in the bow shock j sh = [B t] µ 0 l sh 6.96) where l sh is the width of the shock. This crrent is eqivalent to the increase in the tangential field behind the shock. Becase of the larger gyroradii ions can penetrate deep into the compressed field than electrons reslting in a thin layer with an electric field pointing toward the sn. Electrons are accelerated by this electric field into the shock while some ions are can be reflected by this electric field. The electric field is given by ɛ 0 E s = e n is + n es ) l e where l e is the width of the electric field layer and n is and n es are ion and electron densities in the layer. Assming the nmber of reflected ions as n ir the nmber of ions in the shock is n is = n es n ir and the electric field is ) E s = en esl e ɛ 0 1 n ir n es 6.97)

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 84 E sw sw j f B sw l f l e E s j e l sh j sh msh B msh Figre 6.11: Ion motion at a perpendiclar shock. and all ions with energies less than eφ = ee s l e will be reflected back into the solar wind. In the E B the reflected ions are accelerated to abot twice the solar wind velocity. These reflected ions carry the crrent observed in the foot region. In addition, the electric field layer cases the electrons to E B drift while the large gyro-radii for the ions preclde this drift sch that the electron carry a Hall) crrent in this region. Note, however, that the interpretation of the crrent strctre based on single particle dynamics and drifts shold be considered with care. While the basic shock strctre is reasonably represented crrent based on single particle drifts may not be the correct representation becase of diamagnetic or in general collective) effects which are not inclded in the single particle dynamics. For instance, the foot crrent as derived from the particle drift and acceleration is opposite to the crrent actally needed for the magnetic field increase at the foot of the shock. 6.3.3 Magnetosheath Flow and Strctre The magnetosheath strctre depends strongly on the properties of the bow shock and ths on the pstream solar wind conditions. However, the overall shock location and the flow in the magnetosheath is to lowest order relatively well determined by prely gas dynamic models. Bt the magnetosheath is a very trblent medim and there are many sorce for the trblent natre of the magnetosheath. The magnetosheath is en entirely open system with a large inflx of energy from the solar wind. This is the basic case for the presence of the trblence in the magnetosheath. Ths the trblence is mainly the expression of the many ways that the plasma at the bow shock and in the magnetosheath) dissipates the energy which is carried into the system by the solar wind. Varios aspect of this trblence are kinetic and two-flid electrostatic and electromagnetic ) plasma waves close to bow shock cased downstream plasma conditions: Whistler Lower-Hybrid Ion-Acostic Non-MHD waves in the magnetosheath.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 85 Mirror mode cased by pressre anisotropy. Ion-cyclotron waves driven by electrical crrent driven. Large scale magnetic field flctations associated with switch-on qasi-parallel) shocks. MHD wave generation in the bow shock by changes in the pstream solar wind conditions. Fast mode wave boncing between the bow shock and the magnetopase. Z Bowshock Bowshock Solar Wind Z X X Solar Wind M'sheath M'sheath Qasi-Perpendiclar Shock Qasi-Parallel Shock Figre 6.1: Illstration of qasi-perpendiclar and qasi-parallel bow shock sitations. Qasi-Perpendiclar Shock A largely perpendiclar shock leads to a more prononced pressre anisotropy with larger perpendiclar pressre. The pressre anisotropy can be frther enhanced as plasma travels from the bow shock toward the magnetopase becase field aligned flow cools the parallel pressre component This has particlar importance for the presence of mirror waves in the magnetosheath. Firehose and Mirror Waves: Low freqency ω ω gi and long wavelength kr gi 1 limit kinetic waves. Withot blk motion the tensor component ɛ s = 0 and with k = k e + k e the dispersion relation splits into k c + k c ω ω ) k c ɛ ω s1 = 0 ) ɛ s1 ɛ s0 + ɛ s5 ɛ s3 = 0 The first term redces to [ ω = k v A 1 1 ) ] βs β s s

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 86 This is the kinetic dispersion relation for the so-called fire hose instability. The instability operates when the rhs becomes negative ) βs β s > Assming ɛ s3 1 in the low freqency limit the dispersion relation becomes k c s + k c ω ω or with ζ s = ω/kv ths the dielectric yy component is ɛ yy = ωps ωgs k c ω ) ɛ s1 ɛ s0 ) = 0 β s β s + β s k k + β ) s Z ζ s ) β s In the very low freqency limit and small phase velocities, ω/kv thi ω/kv A 1 one can expand the plasma dispersion fnction for large argments which yields ) π β i i β i ω kv ths = 1 + s β s β s β s ) [ + k 1 + 1 k ) ] βs β s s δv Satellite Path B n Growth: Figre 6.13: Illstration of the mirror mode. Only the ion component contribtes to the imaginary part and the freqency is prely imaginary. If the imaginary freqency is negative the mode is prely damped and if it is positive the mode is prely growing. Let s consider two cases: For strongly parallel propagation k k : In this case growth occrs if the second term on the rhs becomes negative or which is again the firehose condition. β s > + s s β s B

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 87 For strongly perpendiclar propagation k k : Growth occrs for and the growth rate is γ m = s β s β s > 1 + s β s [ ) ] β i βs β π βi s 1 1 kv thi s β s Electrons and ion anisotropy contribte eqally to instability Growth rate is proportional to ion thermal speed. Growth is favored for higher plasma beta β s = µ 0p s B = γ c s A Ion-Cyclotron Waves Dielectric fnction for electrostatic plasma waves for propagation with a component parallel to the magnetic field: ɛω k) = 1 ω [ psλ l η s ) Z ζ s k vths s.l ) lω ] gs Zζ s.l ) k v ths with l= ζ i.l = ω lω gi k v thi ζ e.l = ω lω ge k v d k v the Here a niform plasma drift v d parallel to the magnetic field is assmed for the electron. For prely parallel propagation the dispersion relation also contains ion acostic wave for T e T i. For ion cyclotron waves we assme k k and T e T i with the soltion ] Λ 1 η i ) ω ω gi [1 + 1 + T i /T e G with G = Λ 1 + 1 Λ 0 ) /η i, and η e 1, ζ e.l 1 and ζ i.l 1 are sed. Since Λ 1 η i ) < 1, and G < 1 the correction term is sally smaller than 0.5 sch that the freqency is close to the ion gyro freqency. In the limit η i -> 0 the freqency is ω ω gi 1 + k c ia/ω gi ) = ωgi 1 + ) with c ia = k B T e /m i Onset conditions: Solving for the drift velocity for the nstable wave sing the above dispersion relation one can obtain the threshold condition critical drift velocity) for the growth of ion-cyclotron wave as v dc = 1 + 1 ) { [ ln v thi ηi ) mi T e m e T i ) ]} 1/ Λ 1 ηi ) where η i corresponds to the vale of k where the drift velocity maximizes.

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 88 1.0 Unstable vdc/vthe l=1 0.1 Stable l= 1 10 T e /T i Figre 6.14: Stability of ion-cyclotron waves. Qasi-parallel shock We know already that a qasi-parallel shock generates more trblence both in the pstream and in the down stream region. The shock for θ 0 is a switch-on fast shock. Therefore the downstream magnetic field direction can vary strongly and depends on the history of the plasma and on the small changes in the pstream magnetic field orientation. Ths pstream changes are extremely amplified and small changes can entirely change the magnetosheath magnetic field. Northward verss sothward IMF Switching the IMF from a northward to a sothward direction shold not have any inflence on the magnetosheath flow and strctre if this strctre is entirely controlled by the bow shock. Albeit the magnetosheath strctre is different for north- and sothward fields. A characteristic property of a northward IMF is a region of depleted density and thermal pressre in front of the sbsolar magnetopase. This region is absent dring periods of strongly sothward IMF. V B V B Magnetosphere Streamlines V V Figre 6.15: Illstration of two- and three-dimensional stagnation flow. The sketch on the left shows a view from the sn onto the dayside magnetopase circle). The sketch on the right illstrates the flow in a plane determined by the magnetic field orientation and the stagnation flow. The reason for this difference is magnetic reconnection which we will discss later) for the case of sothward IMF and a reslting difference in the transport of plasma and vs. the transport of

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 89 magnetic flx. The magnetosheath flow and therefore the transport of plasma is three-dimensional, i.e., it occrs in the dawn and dsk directions as well as in the north and soth directions as illstrated in Figre 6.15. However, the magnetic flx extends in the north-soth direction and for northward IMF it can only be removed by dawn and dskward flow. Ths plasma can be transported more efficiently away from the sbsolar region becase the removing flow has an additional degree of freedom compared to the flow which can remove the magnetic field. This may be better illstrated in two dimensions where we assme the Earth to be a cylinder and the magnetosphere a two-dimensional dipole with infinite extend in the GSM y direction. In this sitation plasma can still flow arond this two-dimensional magnetosphere. However, magnetic flx is stck in front of the cylinder and cannot be removed if the IMF is northward. Ths the magnetic flx piles p in front of this magnetosphere. For sothward IMF reconnection at the dayside magnetopase can disconnect a magnetic field line and remove it from the dayside. The fact that plasma is more easily removed along the magnetic field for northward IMF leads to an increase in the magnetic flx and magnetic field strength in front of the magnetopase. Pressre balance then reqires that the thermal pressre in this region has to decrease giving rise to the socalled plasma depletion layer. The bow shock as a filter for solar wind pertrbations Ths far we have focsed on the stationary strctre of the bow shock for given solar wind conditions. However, the solar wind is a relatively nsteady medim. The typical correlation time for solar wind conditions is of the order of 10 to 0 mintes, i.e, solar wind conditions typically change on this time scale. Any pertrbation present in the solar wind is transmitted throgh the bow shock. These pertrbations clearly contribte significantly to the trblent nonlinear waves in the magnetosheath. p t=t 1 t=t t=t 3 t=t 4 BS MP -x Figre 6.16: Illstration of fast mode boncing between the bow shock and the magnetopase. To illstrate the basic aspects of this wave interaction consider a one-dimensional system along the Sn-Earth line. Let s assme that a fast mode rectanglar wave plse travels in the solar wind

CHAPTER 6. THE BOW SHOCK AND THE MAGNETOSHEATH 90 toward the Earth. At the bow shock this plse cannot be reflected sch that all energy, momentm, mass, and magnetic flx which is transported by the plse mst be transmitted into the downstream region. Dring the interaction the plasma condition jst pstream of the shock are modified by the wave. Ths shock properties like the compression, tangential field, velocity change. In particlar, also the dehoffmann-teller frame will change implying that the shock location moves p- or downstream depending on the properties of the wave. The downstream properties are changed becase of the intermittent different shock properties. In general the pertrbation created downstream cannot be represented by a single rectanglar fast wave. Ths the transmitted mass, momentm, energy, and flx transport reqires the presence of several MHD waves one o which will be a now modified rectanglar fast wave. The fast wave propagates faster downstream than Alfvén and slow waves sch that it otrns the other waves. Since the fast mode wave is the only large scale flid wave with a significant grop velocity perpendiclar to the magnetic field, the fast wave is the only wave that can reach the magnetopase. At the magnetopase some of the wave energy is transmitted into the magnetosphere, however most of it is reflected again as fast wave becase of the steep density gradient at the magnetopase. Dring the reflection/transmission other wave modes are generated to maintain mass, momentm, energy, and magnetic flx conservation. The transmitted waves will ndergo frther modifications and reflections in the magnetosphere. The reflected fast wave is traveling back pstream throgh the magnetosheath to the bow shock location where it is reflected again with the side effects to move the location of the bow shock again and to generate other MHD wave. In the entire process the amplitde of the fast wave is decreasing becase of copling to other waves, transmission into the magnetosphere, transport in the stagnation flow away from the sbsolar region, and becase of crvatre three-dimensionality which is not considered in this simplified example.