PPMLR-MHD Scheme for
Global Simulation of the SMI System
Chi Wang and Xiaocheng Guo
State Key Laboratory of Space Weather, NSSC, CAS, Beijing
Youqiu Hu
University of Science and Technology of China, Hefei
2012.04.20
OUTLINES
1. Construction of the PPMLR-MHD Scheme
2. A Summary of Main Simulation Results
3. Prospects for the Future
1. Construction of the PPM-MHD Scheme
 Modelling of the SMI System (same as most models)
Z

Y
Inner Boundary
(r = 3 Re)
SW
Earth
Sun
X
Global MHD for the solar windmagnetosphere (SM) system
 Electrostatical model for the
ionosphere
 M-I coupling
(1) The magnetospehere feeds the
ionosphere with FAC
(2) The ionosphere maps its electric
field (EF) to the inner boundary
of the magnetosphere
 Existing 3D Global MHD Simulation Codes
Code
Affiliation
Reference
Lax-Wendroff
Nagoya Uni.
Ogino et al. (1992)
FV-TVD
openGGCM
Kyushu Uni.
UCLA, UNH
Tanaka (1994)
Raeder (1995)
GIMICS-3
Finnish Meteo. Inst.
Janhunen (1996)
LFM
Dartmouth College
Lyon et al. (1998)
BATS-R-US
Michigan Uni.
Powell et al. (1999)
PPMLR-MHD
USTC, CSSAR
Hu et al. (2005,2007)
 Difference Schemes of Conservational Type:
Godunov Scheme
 Conserved quantities Ui are defined at the zone centers as
volume averages, and their fluxes Fi1/2(Ui1/2) are defined at
the zone interfaces as surface averages
 Ui1/2 are obtaned by an accurate or approximate Riemann
solver, in which interpolations made among discret data
determine the spatial accuracy of the scheme.
1. Godonov Scheme: step function, accuracy of 1st order
2. MUSCL Scheme: piecewise linear, accuracy of 2nd order
3. PPM Scheme: piecewise parabolic, accuracy of 3rd order
 A Distinguishing Feature of the PPM Scheme
 Schemes of the first-order accuracy:
with strong numerical dissipation and poor resolution of MHD
discontinuities and small-scale structures.
 Schemes of the second-order accuracy:
with weak numerical dissipation but strong dispersion, and the latter
demands additional artificial dissipation to stabilize the schemes,
leading to a poor spatial resolution of the scheme, too.
 Schemes of the third-order accuracy like PPM Scheme:
with weak numerical dissipation and weak dispersion, so the resultant
scheme can operate stably with little or without additional artificial
dissipation, leading to a higher spatial resolution of MHD
discontinuities (~2 meshes) and small-scale structures.
 Construction of the PPMLR－MHD Scheme
(1) To start from the MHD equations of conservational type in
d

t
Lagrangian coordinates V  1 ,
  v  ,
r  r0  0 vdt, (1)

dt t
dU 1
   F  S  Sx  S y  Sz ,
(2)
dt 
where U denotes the conserved quantities，F the fluxes, and S the source terms.
(2) To separate them into 3 one-dimensional equations:
dU 1 Fx
dU 1 Fy
dU 1 Fz

 Sx ,

 Sy ,

 S z , Lagrangian step(3)
dt  x
dt  y
dt  z
U
U
U
U
U
U
Remapping step (4)
 vx
 0,
 vy
 0,
 vz
 0,
dt
x
dt
y
dt
z
t
t
t
x  x0  0 vx dt , y  y0  0 v y dt , z  z0  0 vz dt , (for coordinate mapping) (5)
and to sweep along 3 coordinates alternatively: first x, y, z, followed by z, y, x.
Construction of the PPMLR-MHD scheme (cont.)
(3) To execute Lagrangian step and to remap to Eulerian grid

Taking x-direction as an example： dt  F

F
1
1
xi  , jk
xi  , jk
 Lagnrangian step:
(1)
(0)
U ijk
 U ijk


2
2
ijk  xi  1  xi  1 

2
2
  dtS
xijk
,
(5)

where F xi1/2,jk  F(Ui1/2,jk) and Ui1/2,jk is defined at the zone interface,
evaluated by an approximate Riemann Solver in terms of PPM-TVD (Total
Variation Diminshing) interpolation.
 Remapping to the Eulerian grid (Convection step)
v xijk t (1)
( 2)
(1)
U ijk  U ijk 
(U i 1 / 2, jk  U i(11)/ 2, jk ),
xi
where U (1) are determined by PPM-TVD interpolation, too.
i 1 / 2, jk
A similar treatment is made along y- and z-directions.
(6)
Construction of the PPMLR-MHD scheme (cont.)
(4) Artificial Dissipation
Taking x-direction as an example, the Euler grid interface coordinates
xj+1/2 are shifted globally by
x j 1 / 2  x j 1 / 2  K ,
where 0  K < 1, and “+” for odd steps, “” for even steps, and  is the
minimum spatial step length. It stands for a convection followed by a
reverse one, resulting in an artificial dissipation, whose strength depends
on the magnitude of K .
 For ordinary schemes of 2nd accuracy, K =1 for stabilizing the scheme
 For PPM scheme, taking K = 0.1 is enough, so that the numerical
dissipation is about tenth of that for ordinary schemes.
 One may reduce K below 0.1 so as to further decrease the dissipation
For instance, we took K = 0 for due north IMF cases to reproduce K-H
waves at magnetopause and standing shocks in the magnetosheath, and
K = 0.0001 for due south IMF cases to reproduce standing shocks in the
magnetosheath.
Sweeping in
proper order
along 3 axes
 Flow Chart of the PPMLR-MHD Scheme
Magnetospheric MHD
X-sweeping
Lagrangian step
Y-sweeping
Ionospheric Electrostatics
Mapping
FAC to
ionosphere
Z-sweeping
Remapping to the
Fixed Eulerian grid
Solving for
ionospheric
potential
Mapping EF
to magnetosphere
OUTLINES
1. Construction of the PPMLR-MHD Scheme
2. A Summary of Main Simulation Results
3. Prospects for the Future
2. A Summary of Main Simulation Results
I. Response of Magnetic Field at the
Synchronous Orbit to IP Shocks (Wang et al., 2000)
Simulation of a negative response of the magnetospheric
magnetic field at night side to IP shocks
Positive response
Negative response
Physical Explanation
Flows caused by IP shocks
near the flanks
Develop toward the x axis
Diverge at the x axis
earthward flow formation
Decrease of Bz
II. Large-Scale Electric Current System in the
Magnetosphere and Ionosphere (Tang et al., 2009)
 For the case of IMF BZ
= 0, a classical -shaped
structure is well
reproduced by simulations
 For cases with south IMF
BZ , part of the cross-tail
current is closed via the
bow shock, forming a
double -shaped structure
X = -15Re
IMF Bz = -5 nT
( Tang et al. JGR, 2009)
III. Intensification of the Cowling Electrojet via
Magnetosphere-Ionosphere Coupling (Tang et al., 2011)
Given a south turning of the IMF, the Perdersen and Hall conductances
are increased artificially from 5S to 25S and 60S, respectively, in 10
minutes after the NEXL is formed.
The enhancement of ionospehric conductances leads to
intensification of the Cowling electrojet in the ionosphere
IV. K-H Waves at the Magnetopause under Due
North IMF Cases (Guo et al., 2010)
 Previous studies of
K-H waves were limited
to local analyses
 The PPMLR-MHD
code reproduces the
global structure of the
K-H waves across the
magnetopause
 Contours of Vx in the
equatorial plane, showing
vortex structures of the K-H
waves at the magnetopause
OUTLINES
1. Construction of the PPMLR-MHD Scheme
2. A Summary of Main Simulation Results
3. Prospects for the Future
3. Prospects for the Future
 Global simulations aimed at substorms by diagnozing
 The role of M-I coupling in the formation of electrojets, NEXL, and
dipolarization of the magnetospheric magnetic field
 The propagation and reflection of Alfvén waves between the magnetotail and
the ionosphere
 Extending these diagonoses from quasi-steady to time-dependent regime
 Conbine simulations with ground-based geomagnetic field
and ionospheric observations in order to closely coordinate
with the Meridional Chain Program
 Equivalent current system in the ionosphere
 Auroral patterns and their temporal evolution in the ionosphere
 Magnetic field and particle observations at the synchronous orbit
 High spatial resolution simulations of middle- and small-scale
structures of the magnetopshere, such as K-H waves, FTE events, BBF
events, magnetotail current sheet, and their fine configuration,
mechanism of formation, and temporal evolution.
Thank You！