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

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 propagation in the magnetosphere: 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
Introduction propagation in the magnetosphere

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.


Introduction1
Introduction propagation in the magnetosphere

  • 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


Introduction2
Introduction propagation in the magnetosphere

  • 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


Introduction3
Introduction propagation in the magnetosphere

  • For instance, the eqof motion for adiabatic MHD:

: Linear MHD waves

: Steady-state cf) R-H relations

: Nonlinear MHD waves


Observations
Observations propagation in the magnetosphere

Venus Express

0.72 AU

STEREO

1 AU

Russell et al., GRL, 2009


Theory
Theory propagation in the magnetosphere

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.


Theory1
Theory propagation in the magnetosphere

Sources

Simple waves

Shock waves

* Simple waves

or


Theoretical solutions
Theoretical solutions propagation in the magnetosphere

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


Ex piston like motion
Ex) piston-like motion propagation in the magnetosphere

V (Km/s)

V (Km/s)

V (Km/s)

V (Km/s)

Time (s)

Time (s)


Numerical test vs exact solutions
Numerical test propagation in the magnetospherevs. 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 solutions1
Numerical test propagation in the magnetospherevs. Exact solutions

V (Km/s)

X (Re)

Numerical solutions = Analytic solutions


Numerical effect test
Numerical effect test propagation in the magnetosphere

Nx = 1000

Nx = 5000

Solution

Simulation

V (Km/s)

V (Km/s)

Time (s)

Time (s)


Numerical effect test1
Numerical effect test propagation in the magnetosphere

B (nT)

Nx=2000

Time (Hr)

B (nT)

Nx=20000

Time (Hr)


Numerical model 1 small scale 300re
Numerical Model (1. Small scale : 300Re) propagation in the magnetosphere

* 1-D Adiabatic MHD eqs

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 formation (1. Small scale : 300Re) propagation in the magnetosphere

* 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 300re1
Shock formation (1. Small scale : 300Re) propagation in the magnetosphere

* 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
Numerical Model (2. Large scale : 0.28 AU) propagation in the magnetosphere

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 (2. Large scale : 0.28 AU) propagation in the magnetosphere

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
Shock formation (2. Large scale : 0.28 AU) propagation in the magnetosphere

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 au1
Shock formation (2. Large scale : 0.28 AU) propagation in the magnetosphere

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 au2
Shock formation (2. Large scale : 0.28 AU) propagation in the magnetosphere

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 au3
Shock formation (2. Large scale : 0.28 AU) propagation in the magnetosphere

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
3D model (Homogeneous Magnetosphere) propagation in the 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


3d model homogeneous case
3D model (Homogeneous case) propagation in the magnetosphere


3d model homogeneous case1
3D model (Homogeneous case) propagation in the magnetosphere


Conclusion
Conclusion propagation in the magnetosphere

  • 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.


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