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., Divin2 A., Ivanova 3 V., Erkaev 4,5 N., Semenov 6 V., Ivanov7 I., Biernat 1,8 H., Lapenta9 G., Markidis10 S., Zellinger11 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

100 200 , 30 70 / , 2 5 R

g E

T s V km s λ = − = − = −

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. Golovchanskaya et al. (2006), J.

• Geophys. Res., 111, A11216.

Sergeev et al. (2006), Ann. Geophys., 24, 2015–2024.

E

10- 30 R

Kink mode

Introduction: Equilibrium

1 4

z x

P B B z x π ∂ ∂ = ∂ ∂

1 4

x z z z

B B F z z x δ π

=

∂ ∂   = −   ∂ ∂  

In equilibrium state Displacement along the Z axis yields the restoring force Equation of motion of the plasma element

2 2 2

,

f

z z t δ ω δ ∂ = − ∂

2

1 4

x z f z

B B z x ω πρ

=

∂ ∂ = ∂ ∂

A plasma element at the center

• f the current sheet (CS)

Introduction: (in)stability

2 f

ω >

Minimum of the total pressure in the center

• f the CS,

Stable situation, Oscillations

2 f

ω <

Maximum of the total pressure in the center

• f the CS,

Unstable situation, Wave growth

2Δ

L

1 ~ 0.01

x z z

B B x z ε ε

=

∂ ∂   = <<   ∂ ∂   1 ~ 0.1 L ν ν = ∆ << ε ν <<

Features of the configuration:

Introduction: Analytical solution of

Erkaev et al., Ann. Geophys., 27, 417, 2009

System of ideal MHD equations

1 ( ) , 4 ( ) , 0, 0, 0. d P dt d d dt dt ρ πρ ρ + ∇ = ⋅∇ = ⋅∇ = ∇⋅ = ∇⋅ = V B B B B V V B

Normalization

*2 * * * * * *

, , &L, , 4 , 4

A A

B B P B V t V ρ π πρ ∆ = = = ∆

Simplifying assumptions

• incompressibility
• B = [ Bx(z), 0, Bz(x) ]
• perturbations are slow waves

propagating in Y direction

• perturbations depend on Y

and Z coordinates only, not on X

Introduction: Analytical solution

1

k f

k k ω ω ∆ = ∆ +

Even function vz(z) – kink-like mode of the solution Odd function vz(z) – sausage-like mode of the solution

2

( ) 3 2

f s

k k k ω ω ∆ = ∆ + ∆ +

• Obtain two modes of solution for ( )

k ω

( )

2 2 2 2 2

1

z z f

d v dz k v ω ω + − =

• Linearize the ideal MHD system
• Neglect small terms
• Substitute Fourier harmonics of perturbations
• Derive a second order ordinary differential equation for the

amplitude of vz perturbation:

2 2

, ε εν ~ exp[ ( )] i t ky ω − ~

tanh( ),

x z

B z B a bx = = +

• background configuration

Introduction: Analytical solution

Dispersion curves of the double-gradient

• scillations (Im[ω] =0) / instability (Re[ω]=0).

Kink mode is faster

Two different magnetic configurations

Curvature Radius System Size Wave Length 1. 2.

“Ballooning” branch “Double-gradient” branch

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:

The analytical dispersion relation for the small-scale,

• blique-propagating ( ) disturbances.

⋅ ≠ k B

For our particular case: B = (Bx, 0, Bz) k = (0, ky, 0)

2 2

2 1 2 ln(p) 2 2 2

A b c c

V R x R β κβ κ ω κβ   − + ∂ = −   + ∂  

The limiting ( ) value of the “ballooning” growth rate:

y

k → ∞

κ – Polytropic index, β – Plasma parameter, p – Plasma pressure, V

A – Alfvenic velocity.

DG and Ballooning growth rates

BALLOONING BRANCH

Mazur et al. (2012) [Geomagnetism and Aeronomy, 52, 603–612]

DOUBLE-GRADIENT BRANCH

Erkaev et al. (2007) [Phys. Rev. Lett., 99, 235003]

2 2

2 1 2 ln(p) 2 2 2

A b c c

k V R x R β κβ κ ω κβ → ∞   − + ∂ = −   + ∂  

2

~ 2 2 1 4

c x z f

k L R B B z x π π ω πρ − ∂ ∂ = ∂ ∂

The unstable ballooning branch exists when ωb

2 < 0.

For equilibrium state this condition requires:

2 1 , 1 .

x z

B B x z ε β κ ε ε − ≤ + ∂ ∂ = ∂ ∂

The common physical nature of these two branches (the Ampere force against the pressure gradient) is seen clearly in one particular case :

2 2

2 2

f b

β κ ω ω = =

Generally: Ballooning

Double- gradient segment Ballooning segment

?

k ω

2 β κ =

Long- wave- length band Short- wave- length band β increases

Aim

Isn’t it excessively simple? 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. Analytical solution

• f Erkaev et al. [Phys. Rev. Lett., 99, 235003, 2007] has

Advantages:

• Match observational data
• n flapping oscillations

[Erkaev et al., 2007; Forsyth et al., Ann. Geophys., 27, 2457 – 2474, 2009]

• Simplicity, clearness

Disadvantages:

• Simplicity of the equations:

quasy-1-D problem is solved

• Simplicity of the configuration

2D simulations: Equations

2 2

0.5( ) E e V B ρ ρ = + +

Normalization: Δ, B* = B(0,zmax), ρ* = ρ(0,0), t* = Δ/VA, VA = B*/(4πρ*)1/2, p* = B*2 /(4π). Linearization: Perturbations: Linearized system for the amplitudes:

1 1 1 1 1 1

( , , , ), ( , , , ), . E E ρ ρ = = = + U V B U V B U U U

1( , , ; )

( , , )exp( ) x z t y x z t iky δ = U U

( ) .

x z

t x z δ ∂ ∂ ∂ + + = ∂ ∂ ∂ F F U S

[ ]

{ , , } ( , ), ( , , ); .

x z

x z x z t k δ = F F S f U U

Korovinskiy et al. (2011), Adv. Space Res., 48, 1531–1536.

Solving this system for several fixed k we obtain

( ) k ω

2D simulations: Method

γ = Im[ω]. Assume,

* ( )

h

A h γ γ = −

Calculated value True value Scheme damping Mesh step

1Richardson, Phil. Trans. Royal Soc. Lond., A 210, 307 – 357, 1911.

: :

top bottom left right

x δ δ = ∂ ∂ = BC U BC U

2

exp( )

z

V z δ = −

Seed perturbation Courant number

0.1 C =

h

γ →

2 h

γ →

Lax-Friedrichs scheme

• ne-step method

I-order accuracy Grid [

] 0.1 0.025

x z

h h × = ×

Grid [

] 0.05 0.0125

x z

h h × = ×

→ →

2

* 2

h h

γ γ γ = −

The Richardson1 extrapolation: II-order accuracy

2D simulations: Growth rate

The sample solution for some fixed wave number

( , , ) ( , )exp( ) x z t x z t δ δ γ = U U 

2 1 2 1

ln ( ) ln ( ) Im[ ] t t t t δ δ γ γ ω − = − = U U

Erkaev’s background: Dispersion curve

20%

Dispersion curves for different p(x,z)

5%

The Pritchett solution1: Profiles

The Pritchett approximate solution

• f the Grad-Shafranov

equation for the magnetic potential A (normalized units),

cosh[ ( ) ] ln ( ) 1 exp( 2 ) 1 2

y y

F x z A F x A ρ   =       = − +  

1Pritchett and Coroniti, JGR,115, A06301, doi:10.1029/2009JA014752, 2010

Reverse grad Bz

Earth

Magnetic configuration and ωf

Stable region Unstable region γ = Im[ωf] γmax = 0.127 Small Rc Large Rc

The configuration features

, 5

c

R L k λ < < ∀ < ~10

c

R L L <

c

R L >

Stable part

• f

the CS ωb is real ballooning mode is stabilized The DG- favourable segment of the CS

• Disp. curves: DGI-favorable segment +

25% 5% 2%

max min

1.24 ρ ρ =

1 2

1.05 ρ ρ =

ρ0(z) matters

Erkaev et al.,

• Ann. Geophys.,

27, 417–425, 2009.

1 ρ =

2

cosh (0.4z) ρ

−

=

• Disp. curves: stable segment +

5%

• Disp. Curves: Large-Rc region

Looks more

• r less

DG-like Transient region?

3D MHD: background relaxation1

1Hesse&Birn (1993), JGR, 98, 3973–3982, doi:10.1029/92JA02905

( ) ( ) ( ) ( )

2 2 2 2

0, , 0, , , 1 2 2 . 2 t P t t e e P V t p V B e B P p ρ ρ ρ ρ αρ αρ ρ κ ∂ + ∇⋅ = ∂ ∂ + ∇⋅ ⊗ − ⊗ + ∇ = − ∂ ∂ + ∇⋅ ⊗ − ⊗ = ∂ ∂ + ∇⋅ + − ⊗ ⋅ = − ∂ = + + − = + V V V V B B V B V B B V V V B B V

( )

0.1 10cos 40 , 20, ( ) 0.1, 20, 0, 80. t t t t t π α  + ≤  = ≥   > 

2-dimensional friction MHD simulation is performed to minimize the net force

p ∇ − × j B

Initial (green) and relaxed (black) background configurations

Relaxation efficiency

[ ]

,

(x ,z ), .

k k i j i j k k

F x z f f p = ∆ ∆ = × −∇

∑

j B , , x k z  =  

blue red Total Net Force

x

F

z

F

&

Run parameters

{ }

, , 0, 0, 0.

n

p B t

τ

ρ ∂ ∂ = ∂ ∂ = = n B V

{ }

, , , p ρ ∂ ∂ = n B V

BC in relaxation phase (2D in XZ plane) fix the magnetic flux entering domain In the main phase the same BC are applied at Z-boundaries, and the Earthward X-boundary Free BC are imposed at the tailword X-boundary and Y-boundaries The instability is seeded with a mode my = 2 kick of Vz velocity:

15 7.5 7.5 384 192 192

x y z x y z

L L L N N N × × = × × × × = × ×

( ) ( )

( ) ( )

2

0.003 ( )sin exp 2 , ( ) 0.5 tanh 4 tanh 3 4 .

z y x x

V f x k y z f x x L x L δ = −   = − − −  

2 1.675

y y y

k m L π = =

3D simulation: Seed perturbation

δVz(x,y)|z=0, t=0

|Vz| perturbation modes my = {1,2,3,4}

integrated over all z and x ∈ [3.75, 11.25]

At t =130 the non-linear evolution starts

ρ, t = 117, x-slices

Estimation of γ: Method

• Use Fourier transform to find the amplitude of the mode

2 of the plasma density variation in Y direction, A2(x,z)

• Calculate A2 at different times t1, t2
• Calculate growth rate:

2 2 2 1 2 1

1 ( ) ln ( ) A t t t A t γ = −

Growth rate by 3D simulation |z = 0, t = 86

1 The inclination angle: –2*10–4 – growth rate is uniform on X 35% γA – analytical prediction γ 3D – numerically obtained values

1

f

k k ω ω = +

Instability develops in the stable part of the CS also 1

Conclusions1

• DGI does not develop in the

regions of too large Rc.

• DGI can develop in the

domains with mixed uniform / tailward-growing Bz.

• The uniform / Earthward-

growing-Bz regions produce strong stabilizing effect.

• The growth rate is close to

the analytical estimation averaged over the domain.

Fully 3D MHD sim. 2D linearized MHD sim.

• 
• 
• 
• The growth rate is close to

the maximal analytical estimation in the domain.

1Korovinskiy et al. (2013), J. Geophys. Res., 118, 1146 – 1158.