MHD modeling of the kink double-gradient branch of the ballooning - PowerPoint PPT Presentation

MHD modeling of the kink “double-gradient” branch of the ballooning instability in the magnetotail Korovinskiy 1 D., Divin 2 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; 2. 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 Graz, Austria; 9. Departement Wiskunde, Katholieke Universiteit Leuven, Belgium; 10. PDC Center for High Performance Computing, KTH Royal Institute of Technology, Sweden; 11. Graz University of Technology, Graz, Austria. Astronum-2013

Introduction: Flapping oscillations 10- 30 R E Kink mode Sergeev et al. (2006), Ann. Geophys., Golovchanskaya et al. (2006), J. 24, 2015–2024. Geophys. Res., 111, A11216. = − = − λ = − 100 200 , 30 70 / , 2 5 R T s V km s g E Sergeev et al. (2003), Geophys. Res. Lett. 30, 1327; Runov et al. (2005), Ann. Geophys. 23, 1391; Petrukovich et al. (2006), Ann. Geophys. 24, 1695.

Introduction: Equilibrium In equilibrium state ∂ ∂ 1 P B = z B ∂ π ∂ x 4 z x Displacement along the Z axis yields the restoring force ∂ ∂   1 B B = − δ x z   F z π ∂ ∂ z   4 z x = 0 z Equation of motion of the plasma element ∂ δ 2 z = − ω δ 2 , z ∂ f 2 t ∂ ∂ 1 B B ω = A plasma element at the center 2 x z πρ ∂ ∂ f 4 z x of the current sheet (CS) = 0 z

Introduction: (in)stability ω > 2 0 f Minimum of the total 2 Δ pressure in the center of the CS, Stable situation, L Oscillations ω < 2 0 f Maximum of the total pressure in the center Features of the configuration: of the CS, ν = ∆ << Unstable situation, 1 ∂ ∂ L   B B ε = <<  z x  1 Wave growth ν ∂ ∂ ~ 0.1   x z = 0 z ε << ν ε ~ 0.01

Introduction: Analytical solution of Erkaev et al., Ann. Geophys., 27, 417, 2009 System of ideal MHD equations Simplifying assumptions 1 ( d V ρ + ∇ = ⋅∇ ) , B B P πρ 4  incompressibility dt ρ d B d = ⋅∇ = ( ) , 0,  B = [ B x (z), 0, B z (x) ] B V dt dt ∇⋅ = ∇⋅ = 0, 0. V B  perturbations are slow waves propagating in Y direction Normalization *2 B  perturbations depend on Y ρ ∆ = * * * , , &L, , B P π 4 and Z coordinates only, not on X * B = = ∆ * , V t V A A πρ * 4

Introduction: Analytical solution • Linearize the ideal MHD system • background configuration = = + ε εν 2 2 tanh( ), B z B a bx • Neglect small terms , ~ x z ω − exp[ ( )] • Substitute Fourier harmonics of perturbations ~ i t ky • Derive a second order ordinary differential equation for the ( ) amplitude of v z perturbation: + ω ω − = 2 2 2 2 2 1 0 d v dz k v z z f ω • Obtain two modes of solution for ( ) k Even function v z (z) – Odd function v z (z) – kink-like mode of the solution sausage-like mode of the solution ω ∆ k ∆ k ω = f ω = ω s ∆ + ∆ + ∆ + k f 2 ( ) 3 2 1 k k k

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

Two different magnetic configurations “Double-gradient” branch “Ballooning” branch 1. 2. Curvature Radius Wave Length System Size

The “Ballooning” instability When plasma pressure decreases too sharply on R, plasma Qualitatively: 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: The analytical dispersion relation for the small-scale, ⋅ ≠ oblique-propagating ( ) disturbances. 0 k B k → ∞ The limiting ( ) value of the “ballooning” growth rate: For our y particular case:   − β + κβ ∂ κ 2 2 1 2 ln(p) 2 V ω = − 2 A   B = ( B x , 0, B z ) + κβ ∂ b 2  2  R x R c c k = (0, k y , 0) κ – Polytropic index, β – Plasma parameter, p – Plasma pressure, V A – Alfvenic velocity.

DG and Ballooning growth rates BALLOONING BRANCH DOUBLE-GRADIENT BRANCH Mazur et al. (2012) [ Geomagnetism Erkaev et al. (2007) and Aeronomy, 52, 603–612 ] [Phys. Rev. Lett., 99, 235003] → ∞ π − π k ~ 2 2 k L R c   − β + κβ ∂ κ ∂ ∂ 2 2 1 2 ln(p) 2 V 1 B B ω = − ω = 2 2 A   x z + κβ ∂ b πρ ∂ ∂ f 2  2  R x R 4 z x c c The unstable ballooning − ε branch exists when 2 1 β ≤ The common physical , ω b 2 < 0. κ + ε 1 nature of these two 2 For equilibrium state β = branches (the Ampere ∂ ∂ B B this condition requires: κ ε = ∂ z x . force against the ∂ x z pressure gradient) is ω 2 ω = seen clearly in one 2 f b particular case : 2

Generally: Ballooning Double- Ballooning ω ? gradient segment segment k β increases Long- Short- wave- wave- length β = κ length 2 band band 0

Aim Analytical solution of Erkaev et al. [Phys. Rev. Lett., 99, 235003, 2007] has Advantages: Disadvantages:  Match observational data  Simplicity of the equations: on flapping oscillations quasy-1-D problem is solved  Simplicity of the configuration [Erkaev et al., 2007; Forsyth et al., Ann. Geophys., 27, 2457 – Isn’t it excessively simple? 2474, 2009]  Simplicity, clearness Aim: Numerical examination of the double-gradient instability in the frame of linearized 2D / fully 3D ideal MHD to confirm / amend / disprove the Erkaev model.

2D simulations: Equations Normalization: Δ , B* = B(0,z max ) , ρ * = ρ (0,0) , t* = Δ /V A , V A = B*/(4 π ρ * ) 1/2 , p* = B* 2 /(4 π ). = ρ = ρ Linearization: ( , , , ), ( , , , ), U V B E U V B E 0 0 0 0 0 1 1 1 1 1 = + = ρ + ρ + 2 2 . U U U 0.5( ) E e V B 0 1 = δ 1 ( , , ; ) ( , , )exp( ) Perturbations: U x z t y U x z t iky ∂ δ ∂ ∂ ( ) Linearized system U F F + + = x z . S for the amplitudes: ∂ ∂ ∂ t x z [ ] Solving this system = δ { , , } ( , ), ( , , ); . F F S f U x z U x z t k 0 x z for several fixed k ω Korovinskiy et al. (2011), Adv. Space Res., 48, 1531–1536. ( ) k we obtain

2D simulations: Method γ = γ − γ = Im[ ω ]. Assume, * ( ) A h h Calculated value True value Scheme damping Mesh step → γ → × = × Grid [ ] 0.1 0.025 h h Lax-Friedrichs scheme x z h one-step method → γ → × = × Grid [ ] 0.05 0.0125 h h I -order accuracy 2 h x z The Richardson 1 γ = γ − γ II -order * 2 2 h h extrapolation: accuracy δ = Seed Courant top : 0 BC U bottom perturbation number ∂ δ ∂ = left : 0 BC U x δ = − C = 2 exp( ) V z 0.1 right z 1 Richardson, Phil. Trans. Royal Soc. Lond., A 210, 307 – 357, 1911.

2D simulations: Growth rate The sample solution for some fixed wave number  δ = δ γ ( , , ) ( , )exp( ) U x z t U x z t δ − δ ln ( ) ln ( ) U t U t γ = 2 1 − t t 2 1 γ = ω Im[ ]

Erkaev’s background: Dispersion curve 20%

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

The Pritchett solution 1 : Profiles The Pritchett approximate solution of the Grad-Shafranov equation for the magnetic potential A Reverse grad B z (normalized units),   cosh[ ( ) ] F x z = Earth ln   A 0 y   ( ) F x 1 exp( 2   ρ = − + ) 1 A   0 y 2 1 Pritchett and Coroniti, JGR,115, A06301, doi:10.1029/2009JA014752, 2010

Magnetic configuration and ω f Small R c Large R c γ = Im[ ω f ] γ max = 0.127 Stable Unstable region region

The configuration features < λ < < > , R L R L R L c c c ∀ < 5 ~10 k L ω b is real Stable The DG- ballooning part favourable mode is of segment of stabilized the the CS CS

Disp. curves: DGI-favorable segment + 25% 5% 2% ρ = max 1.24 ρ min ρ = 1 1.05 ρ 2

ρ = 1 ρ 0 (z) 0 matters − ρ = 2 cosh (0.4z) 0 Erkaev et al., Ann. Geophys., 27, 417–425, 2009.

Disp. curves: stable segment + 5%

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