Three-dimensional simulations of nonlinear MHD wave propagation in the magnetosphere

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Three-dimensional simulations of nonlinear MHD wave propagation in the magnetosphere. Kyung- Im Kim¹, Dong-Hun Lee¹, Khan- Hyuk Kim ¹ , and Kihong Kim 2 1 School of Space Research, Kyung Hee Univ . 2 Division of Energy Systems Research, Ajou Univ.

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### Three-dimensional simulations of nonlinear MHD wave propagation in the magnetosphere

Kyung-Im Kim¹, Dong-Hun Lee¹, Khan-Hyuk Kim¹,

and Kihong Kim2

1School of Space Research, Kyung Hee Univ.

2Division of Energy Systems Research, AjouUniv.

- Introduction: Effects of nonlinearity in a time-dependent system - Observations: e.g., Russell et al. (GRL, 2009) STEREO observations of shock formation in the solar wind - Theory: What kinds of exact nonlinear MHD solutions available? - Numerical approach 1. Theory vs. Numerical test2. check the profiles from ACE (Venus Express) to the Earth (STEREO)3. Apply to the 3-D homogeneous magnetosphere

- Conclusion

Introduction

B

A

• Propagation of nonlinear MHD waves is studied in the interplanetary space.
• As realistic variations in the solar wind become often nonlinear,  it is important to investigate time-dependent behaviors of the solar wind fluctuations.
Introduction
• If the disturbance is linear in a uniform space, f_A = f_B:
• If the disturbance is linear in a nonuniform space, only
• the effects of refraction/reflection are to change f_A & f_B:

A

A

B

B

Introduction
• If the disturbance is nonlinear, f_A and f_B become differentiated even in a uniform space:
• The disturbance in the SW rest frame can evolve in a time-dependent manner, which is far from the steady-state.
• cf) Rankine-Hugoniot relations

A

A

B

B

Introduction
• For instance, the eqof motion for adiabatic MHD:

: Linear MHD waves

: Nonlinear MHD waves

Observations

Venus Express

0.72 AU

STEREO

1 AU

Russell et al., GRL, 2009

Theory

The assumption of simple waves (similarity flow) is often used to obtain a solution.

Exact solution for the nonlinear MHD wave is available

if it is a one-dimensional uniform system.

Theory

Sources

Simple waves

Shock waves

* Simple waves

or

Theoretical solutions

* Exact solution for the MHD wave [Lee & Kim, JGR, 2000]

Ex) piston-like motion

V (Km/s)

V (Km/s)

V (Km/s)

V (Km/s)

Time (s)

Time (s)

Numerical test vs. Exact solutions

V (Km/s)

V (Km/s)

V (Km/s)

V (Km/s)

Numerical solutions are corresponding to the exact solutions before the shock formation!

Numerical test vs. Exact solutions

V (Km/s)

X (Re)

Numerical solutions = Analytic solutions

Numerical effect test

Nx = 1000

Nx = 5000

Solution

Simulation

V (Km/s)

V (Km/s)

Time (s)

Time (s)

Numerical effect test

B (nT)

Nx=2000

Time (Hr)

B (nT)

Nx=20000

Time (Hr)

Numerical Model (1. Small scale : 300Re)

cf) Lx to ACE ~ 234 Re

* Impulse

V (Km/s)

* Simulation Parameters

- nx = 5000 , Lx = 300 Re

- Total time = 5000 s

- Sound speed = 50 Km/s

- Alfven speed = 49 Km/s

- Plasma density = 10

- Magnetic field Bz = 8 nT

t/t0

Shock formation (1. Small scale : 300Re)

* Shock formations for different background magnetic fields (3nT, 5nT, 8nT) and impulse timescales ( 𝛕0 = 300s, 500s, 700s)

B0 = 3nT

B0 = 5nT

B0 = 8nT

B (nT)

Time (s)

Time (s)

Time (s)

Considered Solar Wind flow speed : 400 Km/s

: Timescale of the source motion

53Re

48Re

45Re

246Re

256Re

256Re

Shock formation (1. Small scale : 300Re)

* Shock formations for different amplitudes of impulse

45Re

256Re

B (nT)

v = v0

v = 2v0

v = 3v0

Time (s)

Time (s)

Time (s)

Considered Solar Wind flow speed : 400 Km/s

: Timescale of the source motion

Numerical Model (2. Large scale : 0.28 AU)

cf) ~ 6560 Re

* Impulse

* Simulation Parameters

- nx = 20000

- Total time = 105,000 s (~29.16 hr)

- Lx = 7000 Re (~0.28 AU)

- Sound speed = 60 Km/s

- Alfven speed = 66 Km/s

- Plasma density = 7

- Magnetic field Bz = 9 nT

V (Km/s)

t/𝛕0

Vx

Simulation Box (2. Large scale : 0.28 AU)

Simulation Box

z

I

II

x

Solar Wind (Vsw = 400 Km/s)

0.72 AU

1 AU

Sun

Venus Express

STEREO A

Shock formation (2. Large scale : 0.28 AU)

I (at 0.72 AU)

2 nT

B (nT)

Time (Hr)

II (at 1 AU)

2 nT

B (nT)

Time (Hr)

Considered Solar Wind flow speed : 400 Km/s, 𝛕0 : Timescale of the source motion

Shock formation (2. Large scale : 0.28 AU)

I (at 0.72 AU)

B (nT)

Time (Hr)

II (at 1 AU)

B (nT)

Time (Hr)

Considered Solar Wind flow speed : 400 Km/s, 𝛕0 : Timescale of the source motion

Shock formation (2. Large scale : 0.28 AU)

I (at 0.72 AU)

B (nT)

Time (Hr)

II (at 1 AU)

B (nT)

Time (Hr)

Considered Solar Wind flow speed : 400 Km/s, 𝛕0 : Timescale of the source motion

Shock formation (2. Large scale : 0.28 AU)

I (at 0.72 AU)

B (nT)

Time (Hr)

II (at 1 AU)

B (nT)

Time (Hr)

Considered Solar Wind flow speed : 400 Km/s, 𝛕0 : Timescale of the source motion

3D model (Homogeneous Magnetosphere)

* Simulation Parameters

- Total time = 140 s

- Sound speed = 60 Km/s

- Alfven speed = 600 Km/s

- Plasma density = 1

- Magnetic field Bz = 600 nT

V (Km/s)

t/t0

Z (20 Re)

Nz=512

B0

Y (10 Re)

Ny=64

X (40Re)

Nx=512

Conclusion
• Propagation of nonlinear MHD waves is studied in the interplanetary space. We examined how these fluctuations are changed by steepening process and/or shock formation.
• The simulation results should be first validated by the exact analytical solution (Lee and Kim, 2000), which showed excellent correspondence between theory and simulation.
• The profiles tend to significantly evolve at the different locations, which strongly depends on the given parameters.
• The observations in the IP space (e.g., ACE) cannot directly deliver the SW condition around the earth unless it belongs to the linear case.