MHD modeling of the kink double-gradient branch of the ballooning instability in the magnetotail

Transcription

1 MHD modeling of the kink double-gradient branch of the ballooning instability in the magnetotail Korovinskiy 1 D., Divin A., Ivanova 3 V., Erkaev 4,5 N., Semenov 6 V., Ivanov 7 I., Biernat 1,8 H., Lapenta 9 G., Markidis 10 S., Zellinger 11 M. 1. Space Research Institute, Austrian Academy of Sciences, Austria;. Swedish Institute of Space Physics, Sweden; 3. Orel State Technical University, Russia; 4. Institute of Computational Modelling, Siberian Branch of the RAS, Russia; 5. Siberian Federal University, Russia; 6. State University of St. Petersburg, Russia; 7. Theoretical Physics Division, Petersburg Nuclear Physics Institute, Russia; 8. Institute of Physics, University of Gra, Austria; 9. Departement Wiskunde, Katholieke Universiteit Leuven, Belgium; 10. PDC Center for High Performance Computing, KTH Royal Institute of Technology, Sweden; 11. Gra University of Technology, Gra, Austria. Astronum-013

2 Introduction: Flapping oscillations R E Kink mode Sergeev et al. (006), Ann. Geophys., 4, Golovchanskaya et al. (006), J. Geophys. Res., 111, A1116. T = s, V = km / s, λ = 5 R g Sergeev et al. (003), Geophys. Res. Lett. 30, 137; Runov et al. (005), Ann. Geophys. 3, 1391; Petrukovich et al. (006), Ann. Geophys. 4, E

3 Introduction: Equilibrium 1 4 x P B B x π = x B B F x δ π = = In equilibrium state Displacement along the Z axis yields the restoring force Equation of motion of the plasma element, f t δ ωδ = x f B B x ω πρ = = A plasma element at the center of the current sheet (CS)

4 Introduction: (in)stability Δ L ω > f 0 Minimum of the total pressure in the center of the CS, Stable situation, Oscillations Features of the configuration: ν = L << 1 B Bx ε = << 1 ν ~ 0.1 x = 0 ε ~ 0.01 ε << ν ω < f 0 Maximum of the total pressure in the center of the CS, Unstable situation, Wave growth

5 Introduction: Analytical solution of Erkaev et al., Ann. Geophys., 7, 417, 009 System of ideal MHD equations dv 1 ρ + P = ( B ) B, dt 4πρ db dρ = ( B ) V, = 0, dt dt V = 0, B= 0. B Normaliation * B, ρ, &L, P =, 4π B = = 4πρ * * * * * VA, t V * A Simplifying assumptions incompressibility B = [ B x (), 0, B (x) ] perturbations are slow waves propagating in Y direction perturbations depend on Y and Z coordinates only, not on X

6 Introduction: Analytical solution Linearie the ideal MHD system Neglect small terms ~ ε, εν Substitute Fourier harmonics of perturbations Obtain two modes of solution for ω( k) background configuration B = tanh( ), B = a + bx Derive a second order ordinary differential equation for the amplitude of v perturbation: x ~ ( ) d v d + k v ω ω 1 = 0 f exp[ i( ωt ky)] Even function v () kink-like mode of the solution ω k = ω f k k + 1 Odd function v () sausage-like mode of the solution ω s = ω k f ( ) 3 k + k +

7 Introduction: Analytical solution Kink mode is faster Dispersion curves of the double-gradient oscillations (Im[ω] =0) / instability (Re[ω]=0).

8 Two different magnetic configurations Ballooning branch Double-gradient branch 1.. Curvature Radius Wave Length System Sie

9 The Ballooning instability Qualitatively: When plasma pressure decreases too sharply on R, plasma becomes unstable to the ballooning mode, which represents a locally swelling blobs. Some analogue to the Rayleigh-Taylor instability, where the curvature of the magnetic field replaces the gravitational force. Mathematically: Consider a system of coupled equations for poloidal Alfvenic and SMS modes in a curvilinear magnetic field. Result: For our particular case: B = (B x, 0, B ) k = (0, k y, 0) The analytical dispersion relation for the small-scale, oblique-propagating ( k B 0) disturbances. The limiting ( ω k y ) value of the ballooning growth rate: V β + κβ κ A 1 ln(p) b = + κβ Rc x Rc κ Polytropic index, β Plasma parameter, p Plasma pressure, V A Alfvenic velocity.

10 DG and Ballooning growth rates BALLOONING BRANCH Maur et al. (01) [Geomagnetism and Aeronomy, 5, ] DOUBLE-GRADIENT BRANCH Erkaev et al. (007) [Phys. Rev. Lett., 99, 35003] k ω V β + κβ κ A 1 ln(p) b = + κβ Rc x Rc k ~π L π R ω f = 1 B 4πρ x c B x The unstable ballooning branch exists when ω b < 0. For equilibrium state this condition requires: 1 ε β, κ 1+ ε B B ε = x x. The common physical nature of these two branches (the Ampere force against the pressure gradient) is seen clearly in one particular case : β = κ ω f ωb =

11 Generally: Ballooning ω Doublegradient segment? Ballooning segment k Longwavelength band β = κ Shortwavelength band β increases 0

12 Aim Analytical solution of Erkaev et al. [Phys. Rev. Lett., 99, 35003, 007] has Advantages: Disadvantages: Match observational data on flapping oscillations [Erkaev et al., 007; Forsyth et al., Ann. Geophys., 7, , 009] Simplicity, clearness Aim: Simplicity of the equations: quasy-1-d problem is solved Simplicity of the configuration Isn t it excessively simple? Numerical examination of the double-gradient instability in the frame of linearied D / fully 3D ideal MHD to confirm / amend / disprove the Erkaev model.

13 D simulations: Equations Normaliation: Δ, B* = B(0, max ), ρ* = ρ(0,0), t* = Δ/V A, V A = B*/(4πρ*) 1/, p* = B* /(4π). Lineariation: U = ( ρ, V, B, E ), U = ( ρ, V, B, E ), = E = ρe+ 0.5( ρv + B ) U U U Perturbations: U (,, ; ) (,, )exp( ) 1 xt y = δ U xt iky Linearied system for the amplitudes: ( δ U) Fx F + + = t x S. [ δ ] { F, F, S} f U (, ), U(,, );. x = 0 x xt k Korovinskiy et al. (011), Adv. Space Res., 48, Solving this system for several fixed k we obtain ω( k)

14 D simulations: Method γ = Im[ω]. Assume, γ = γ * Ah ( ) Calculated value True value Scheme damping Mesh step h Grid [ h h ] = x Grid [ h h ] = x Lax-Friedrichs scheme one-step method I-order accuracy γ h γ h The Richardson 1 extrapolation: γ γ γ * = h h II-order accuracy BC top bottom : δ U = 0 Seed perturbation left BCright : δ U x = 0 δ V = exp( ) 1 Richardson, Phil. Trans. Royal Soc. Lond., A 10, , Courant number C = 0.1

15 D simulations: Growth rate The sample solution for some fixed wave number δu( xt,, ) = δu ( x, )exp( γt) γ = γ = ln δu( t ) ln δu( t ) Im[ ω] 1 t t 1

16 Erkaev s background: Dispersion curve 0%

17 Dispersion curves for different p(x,) 5%

18 The Pritchett solution 1 : Profiles Earth Reverse grad B The Pritchett approximate solution of the Grad-Shafranov equation for the magnetic potential A (normalied units), cosh[ Fx ( ) ] A0 y = ln F( x) 1 ρ = exp( A0 y ) Pritchett and Coroniti, JGR,115, A06301, doi:10.109/009ja01475, 010

19 Magnetic configuration and ω f Small R c Large R c Stable region Unstable region γ = Im[ω f ] γ max = 0.17

20 The configuration features Rc < λ < L, Rc < L Rc k < 5 L ~ 10 > L Stable part of the CS The DGfavourable segment of the CS ω b is real ballooning mode is stabilied

21 Disp. curves: DGI-favorable segment + 5% 5% % ρ ρ max min = 1.4 ρ 1 ρ = 1.05

22 ρ 0 = 1 ρ 0 () matters ρ 0 = cosh (0.4 ) Erkaev et al., Ann. Geophys., 7, , 009.

23 Disp. curves: stable segment + 5%

24 Disp. Curves: Large-R c region Looks more or less DG-like Transient region?

25 3D MHD: background relaxation 1 ρ + ( ρv) = 0, -dimensional t friction MHD ρv simulation is + ( V ρv B B) + P = αρv, performed to t minimie the B + ( V B B V) = 0, net force t p j B e + ( ) Ve + VP B B V = αρv, t p ρv B e = + +, cos( πt40 ), t 0, κ 1 α( t) = 0.1, t 0, B P = p+. 0, t > Hesse&Birn (1993), JGR, 98, , doi:10.109/9ja0905

26 Initial (green) and relaxed (black) background configurations

27 Relaxation efficiency Fx & F Total Net Force F = x f f k k i j i, j k [ j B p] = (x, ), k. k x, blue =, red

28 L L L = x y N N N = x y BC in relaxation phase (D in XZ plane) fix the magnetic flux entering domain Run parameters n ρ, B, p = 0, { } B t = 0, V = 0. In the main phase the same BC are applied at Z-boundaries, and the Earthward X-boundary n τ Free BC are imposed at the tailword X-boundary and Y-boundaries n ρ, BV,, p = 0 { } The instability is seeded with a mode m y = kick of V velocity: ( ) ( ) y δ V = f( x)sin k y exp, ( ) ( ) f( x) = 0.5 tanh x Lx 4 tanh x 3Lx 4. k = π m L = y y y

29 3D simulation: Seed perturbation δv (x,y) =0, t=0

30 V perturbation modes m y = {1,,3,4} integrated over all and x [3.75, 11.5] At t =130 the non-linear evolution starts

31 ρ, t = 117, x-slices

32 Estimation of γ: Method Use Fourier transform to find the amplitude of the mode of the plasma density variation in Y direction, A (x,) Calculate A at different times t 1, t Calculate growth rate: γ = 1 A( t) ln t t A ( t ) 1 1

33 Growth rate by 3D simulation = 0, t = 86 Instability develops in the stable part of the CS also ω = ωf k k % 1 The inclination angle: *10 4 growth rate is uniform on X γ A analytical prediction γ 3D numerically obtained values

34 Fully 3D MHD sim. DGI does not develop in the regions of too large R c. DGI can develop in the domains with mixed uniform / tailward-growing B. Conclusions 1 D linearied MHD sim. The uniform / Earthwardgrowing-B regions produce strong stabiliing effect. The growth rate is close to the analytical estimation averaged over the domain. The growth rate is close to the maximal analytical estimation in the domain. 1 Korovinskiy et al. (013), J. Geophys. Res., 118,