Magnetosphere-Ionosphere coupling in global MHD simulations and its improvement

Hiroyuki Nakata

The Korea-Japan Space Weather Modeling workshop2008/8/12, KASI, Daejeon, Korea

Graduate School of Engineering, Chiba University, Japan

Outline

•Process of MI coupling in global MHD simulations

•The effect of MI coupling on the variation of the magnetosphere

•Our studies for the improvements of MI coupling

•Future works

Recent global MHD simulation codes

Code TVD Order Mesh Cleaning B∇ Ionosphere

LFM(CISM) Y 8 Adapted Y Y

Raeder

(OpenGGCM) Y 4 Stretched Y Y

BATSRUS Y 2 AMR ? Y

Ogino N 2 Cartesian Y N

Tanaka Y 2 Spherical? Y Y

Janhunen

(GUMICS)

? 1 AMR ? Y

From Lyon [2005, Proceeding of ISSS]

MRC, Washington (Winglee), …

Fedder and Lyon [1987]

• The first global MHD simulation with MI coupling.

• In this paper, the relationship between the cross polar cap potential and the FAC (Region 1) is examined.

• IMF Bz = -5 nT, the uniform ionospheric conductivity

• The ionospheric conductivity controls the amount of open mangetic flux.

=0.1 S

=1.25 S

=2. 5 S

=5 S

Separation of the ionosphere and inner boundary of the magnetosphere

• Since the Alfven speed ( ) in the inner magnetosphere is very high, the inner boundary of the magnetosphere is located at ~3RE.

• Therefore, the magnetosphere and the ionosphere have to be connected by some method which constitute a realistic physical model for MI coupling.

Inner boundary

Ionosphere

MI coupling in global MHD simulations

A) Determine FAC maps in the inner boundary of the magnetosphere

B) Map FAC onto the ionosphere along the field line and derive the potential map from

using the conductivity distribution determined by models or calculations.

C) Map the potential distribution back to the inner boundary. The electric field determined by this potential contributes to the variation of plasma velocity via ExB drift.

A) j|| C)

Inner boundary

B) In the ionosphere

Conductivity (1)• Ionospheric conductivity is very important parameter to

determine the electric potential exactly.• In recent global MHD simulations, the effect on the conductivities

are considered as solar EUV, diffusive, and discrete precipitation. • As for the effect of solar EUV, Moen and Brekke [1993] have

shown that the effect of the solar radio flux on the conductivity (F10.7 :solar radio flux, :solar zenith angle)

The effect of MI coupling in the variations of the magnetosphere• Some papers show the variations of the

magnetosphere due to the effects of MI coupling.▫Ridley et al. [2004], Merkin et al. [2004]

The relationship between the cross polar cap potential and the ionospheric conductivity is examined.

▫Nakata et al. [2003] Electric potentials for 5 phases of substorms are

determined using Ogino code.

Ridley et al. [2004]

• The relationship between the ionospheric conductivity and the global state of the magnetosphere is examined.

S1Σ S100Σ

Relationship between conductivity and Potential/FAC

• As the ionospheric conductivity increase, the FACs increases but the electric potential decrease -> consistent with the Hill model

Ridley et al. [2004](Michigan code)

Merkin et al. [2004](LFM code)

j||

Φ Φ

J||

Position of Magnetopause

• Using the magnetogram inversion method (aka KRM method), Kamide et al. [1996] have shown the variation of the ionospheric electric potential for each phases of substorm.

• The polar cap potentials are calculated using FAC determined by the global MHD simulation developed by Ogino [1986] and the average conductivity map used in Kamide et al.[1996]

• The polar cap potential, , is derived by

where Σ is Height-integrated conductivity tensor from Kamide et al.[1996], j|| is Field-aligned current from the global MHD simulation.

Nakata et al. [2003]

Ij sin( || Σ

Average patterns of the polar cap potentials derived from the KRM method

～ 5/cc

Vsw ～ 500km/s

Bz=-5nT ～ 5nT

• These patterns are derived by superposing a number of substorms normalized by AL [Kamide et al., 1996]. Substorms are observed for March 17-19,1978

Potential variations during the substorms [Nakata et al., 2003]

Problems in MI coupling

•Mapping only FAC• Inner boundary = 3 RE

•Mapping along the field lines▫Informations propagate instantaneously to the

ionosphere▫The lower latitude region is not affected.▫Energy conservation does not hold

•Thin ionosphere▫The real ionosphere has 3-dimensional structures.

Self-consistent MI coupling

• Energy conservation does not hold because only the FACs are mapped on the ionosphere and only the electric potential is mapped back to the inner boundary of the magnetosphere.

• In this study, a self-consistent M-I coupling algorithm proposed by Yoshikawa and Nakata [2004] which treats M-I coupling as a wave-reflection is adopted into the global MHD simulation developed by Tanaka [1995]. In this presentation, a substorm determined by the global MHD simulation using this algorithm is compared with one determined by the original MHD simulation.

Modified MI Coupling Algorithm

• Exact values of the field-aligned current (FAC) and the electric potential are separated as

Here, are the exact values of FAC and the potential, recpectively. are those determined by MHD scheme. are the additional perturbed components produced by the MI coupling.

• In the inner boundary of the magnetosphere, therefore, the following equation is satisfied:

Modified MI Coupling Algorithm

• The parameters determined by MHD schemes show temporal variations due to the perturbations propagating from the magnetosphere whether MI coupling is included or not. Thus, are expressed as

where, are the exact values at the previous step and 　　　　　　　　

are the perturbations propagating from the magnetosphere to the inner boundary. From equations (1) and (4), therefore, the exact values are separated as follows:

Modified MI Coupling Algorithm

• As the exact values satisfy equation (1),

• From these equations, we have

This equation shows that the perturbations propagating from the magnetosphere and the additional components due to MI coupling are balanced. Therefore, the process of MI coupling is treated as reflection of waves.

Modified M-I Coupling Algorithm

• As these components are associated with the Shear Alfven waves, therefore, we have

• Substituting Eq. (8) into (3) , the following equation is obtained:

In the present study, this equation is adopted as M-I coupling equation. Giving , is determined.

Since the magnetosphere receives ,

are replaced by Then the MHD parameters at the next step are

calculated by the MHD scheme.

Simulation parameters

• Using the MHD scheme with the present M-I coupling algorithm, a substorm is simulated, which is produced by southward turning of northward IMF. The solar wind parameters used in the present study are as follows;▫ IMF (By, Bz) = (2.5 nT, 4.3 nT) → (0 nT, -4.3 nT)▫Solar wind speed = 400 km/s▫Solar wind density = 5/cc▫Radius of inner boundary = 3.5 Re

• To examine the effect of the present algorithm, we compare the potentials determined by the present scheme with original TVD scheme developed by Tanaka [1995].

Variations of cross polar cap potentials

TIME (Min.)

T=40 minPote

nti

al (k

V)

Conductivities at T=40 min

Σ ｘｘ Alfven conductivity

Potentials at T=40 min

Original(93.7 kV)

New(87.5 kV)

Original – New(4.9 kV)

FACs at T=40 min

Discussion

•Why the cross polarcap potential in the steady state determined by the present algorithm differs from that determined by the original MHD scheme ?▫Their steady states may differ because the

difference of the potentials accumulate as the substorm develops. In the present results, the difference of the cross polarcap potential increase with the value of the potential and the difference corresponds to the ratio of .

3-D ionospheric potential distribution

• The ionosphere is treated as a thin layer in these global MHD simulation although the real ionosphere has a three-dimensional structure.

• To examine 3-dimensional distributions of the ionospheric potential and the current system in the ionosphere, a solver of the 3-dimensional distribution of the ionospheric electric potential is adopted in M-I coupling process of the global MHD simulation code developed by Tanaka [1995, JGR].

Model of the ionosphere

• In order to avoid a singular point, spherical coordinates whose polar axis is inclined 90 degrees is used.

• The electric potential is determined from field-aligned currents mapped from the inner boundary of the magnetosphere and the anisotropic inhomogeneous conductivity distribution.

• The horizontal potential maps determined by the original global simulation scheme are applied to the lower boundary. At the side surface boundaries ( boundary), the gradient of the potential normal to the surface is set to zero.

Upper boundary : 400 km Lower boundary : 80 km - 40 < < 40 [deg]

Basic equation

• Assuming the electric field is expressed by the potential,, the ionospheric current, J, is determined by

where is conductivity tensor. From the volume integral of the divergence of this equation, we have

• Calculating the sparse matrix equation, AX=B (A : the sparse matrix of the coefficients determined by the conductivity distribution, X : the values of the potential in each grid, B: the source term determined by field-aligned currents), which derived by the discretization of this equation, the electric potential is determined.

• In solving the matrix equation, BiCGStab method is used.

• The original simulation scheme calculates the height-integrated ionospheric conductivities considering the effects of solar EUV, diffuse auroral precipitation, and upward field-aligned current.

• In the present calculation, the distributions of height-integrated conductivities are used as the horizontal distribution. The vertical profiles of the conductivities are shown in the right panel.

Ionospheric conductivity

Height-integrated Hall conductivity at T=35 min.

Vertical profiles of the conductivities at the north pole in January 2000 determined by IRI and MSIS models

||

P

H

MHD simulation parameters• The original global MHD simulation code is based on total

variation diminishing (TVD) scheme developed by Tanaka [1994, 1995]. The simulation scheme includes M-I coupling with the 2-D ionosphere.

• In the present study, the variation of M-I system due to southward turning from steady northward IMF is simulated.

• The following parameters are used.▫ IMF (By, Bz) = (2.5 nT, 4.3 nT) → (0 nT, -4.3 nT)▫ Solar wind speed = 400 km/s▫ Solar wind density = 5/cc▫ Radius of inner boundary = 3 Re

The pressure distribution in the magnetosphere after southward turning of IMF (T = 35 min.)

Variation of cross polar cap potential• After southward turning of IMF, the field-aligned currents develop and

the magnitudes of polar-cap potentials increase. In this presentation, the parameters at T=3 and 35 minutes are examined.

T = 3 min. T = 35 min.

Vertical profiles of the parameters at T=35 min.

JPJH J||

||

P

H

Pedersen and Hall currents at T=35 min.

• Upper panel : Pedersen current

• Lower panel : Hall current

• Left column : 105 km

• Right column : 131 km

• Hall and Pedersen conductivities reach their maximum at 105 and 131 km, respectively.

JP

JH

131 km105 km

• The parallel current mainly connects to the Pedersen current. Due to the inhomogeneity of the Hall conductivity, the divergence of the Hall current enhances especially in the lower ionosphere of the dayside and nightside.

Divergences of ionospheric currents at 131 km

T = 3 min. T = 35 min.

Yugo ( 融合 ; Fusion, Integration) Project

• Especially in America, the many models are connected each other and the solar-terrestrial environment from the solar surface to the upper atmosphere of the earth can be solved. (CISM, OpenGGCM..)

• Some Japanese (especially young?) scientist unify the models of magnetosphere, ionosphere, neutral atmosphere.

• The simulation code for the entire model is easily constructed by the models for each regions are stored in web page.

• Ultimate purpose of this project is the correct calculation of the magnetic storms.

Web pages for automated connecting tools

Summary• The effect of MI coupling on the variation of the

magnetosphere is important and inevitable to determine the conditions of the magnetosphere.

• Although the present process of MI coupling is sophisticated, they have some problems which should be overcome.

• Recently, Japanese scientists collaborate for constructing the unified model from solar wind to neutral atmosphere.

• In addition, improvements for the models and the coupling processes are important. We have to progress the improvements.