Download Magnetospheric Simulations

Lecture 14
Two Simulation Codes:
BATSRUS and Global GCM
BATSRUS (Block-adaptive tree solar wind
Roe-type upwind scheme) (Powell et al.,J.
Comp. Phys., 154, 284,1999)
• Conservation of mass
• Faraday’s Law


    u   0
t


B
   E
t
• Total time rate of change of magnetic flux across a given
surface S bounded by δS

 
 
d  
B 
B  dS  
 dS   B  u  dl     Bu  dS

S
s
S
S
dt
t
'
d  
  B  dS   E  dl
S
dt S
BATSRUS
Equations



 
 
B
   uB  Bu  u  B
t
• Although B is divergenceless it is kept in order to have all

of the equations in normalized
divergence form


U
 F
t

T

S
• Conservation of Momentum

  


 u 
    u u  pI   j  B
t
 



 
  


  u 
B

B
B
B
1
   B  B
I 
     u u   p 


t
20 
 0 
0


The second equation was derived from the first by using

 1
Ampere’s Law ( j    B ) and vector identities.
0
BATSRUS Equations
• Conservation of Energy



e
B B
1  
u  
    e  p 
u  B   B

t
20 
0


   
p
u u B  B
e


g 1
2
20
 
g is the ratio of specific heats.
BATSRUS Ohm’s Law

 
E  u  B
• Uses an ideal Ohm’s Law with η=0.
• Achieves numerical stability by eliminating numerical
noise.
BATSRUS Adaptive Grid
• The grid is adaptive and Cartesian arranged using a tree
data structure.
• It consists of grid blocks each of which corresponds to a
node of the tree.
• The root of the tree is a coarse grid covering the entire
region to be simulated.
• In region where refinement occurs each block is divided
into eight octants – cube is halved.
• Each of the new blocks can be refined in the same way.
Adaption Continued
• The decision on whether to refine the grid (or reduce the
grid spacing) is made by comparison of local flow
quantities with threshold values.

c   u V

r  u V
t    B V
• Local values of compressibility, rotationality and current
density are used where V is the cell volume.
The Adaptive Grid
The grid refinement for an actual simulation
Local Time Stepping
• Time-accurate mode – each time step advanced
according to the Courant condition.
• Iterative local time stepping mode – each cell takes
different time steps.
– Converges to a steady state solution through unphysical
intermediate states.
– Upstream boundary conditions are held constant and a steady
state magnetospheric configuration is derived for those upstream
conditions and the dipole and corotation tilt.
– No history to the solution – wrong for the magnetosphere.
– For directly driven part of magnetospheric response method
works.
BATSRUS Ionosphere
• The Hall and Pedersen conductances are determined
from the F10.7 flux.
• Particle precipitation induced conductance was included.
The conductance was determined from the local fieldaligned current.
• Ionospheric conductances and field-aligned currents
derived from the assimilative mapping of ionospheric
electrondynamics technique (AMIE). (Ridley et al., J.
Geophys. Res., 106, A12, 30,067)
  0e
 AJ
• Ran the AMIE code 8500 times to get patterns of
conductance, binned the results according to location
and current density and fit the results.
BATSRUS Ionospheric Conductance and
Field Aligned Currents
• Simulation does not reproduce
region 2 field-aligned currents.
• Ionospheric boundary extends
into the region where the R2
currents are generated.
• R2 currents are thought to result
from pressure gradients in the
inner magnetosphere. Need
even better resolution.
• Contribution from gradient and
curvature drift missing.
BATSRUS Coordinate System
• Calculations are carried out in Geocentric Solar
Magnetospheric (GSM) Coordinates.
– X-axis from the Earth to the Sun
– Y-axis is defined to be perpendicular to the Earth’s magnetic
dipole so that the X-Z plane contains the dipole axis.
– Z-axis is chosen to be in the same sense as the northern
magnetic dipole.
– The difference between the GSM and GSE is a rotation about
the X-axis.
– The rotation has the form below where θ is a function of time of
day and time of year
0
1

 0 cos 
 0 sin 

0 

 sin  
cos  
Field Lines and Pressure in Noon-Midnight
Meridian Plane
3-D Field Lines for Northward IMF
OpenGCM
• Originally developed by Jimmy Raeder at UCLA.
• CCMC has a version he developed at the University of
New Hampshire.
• It is very similar to the current code used by Mostafa ElAlaoui at UCLA.
Open-GCM Magnetohydrodynamic Equations
•
Macroscopic plasma properties are governed by basic conservation laws for mass,
momentum and energy in a fluid.
Mass
Momentum
Energy*
Faraday’s Law
Gauss’ Law
Ohm’s Law
Ampere’s Law


   (  v )
t

  

 v
   (  v v  p I ) + j  B
t
  
e
1
p
2
   [(e  p ) v ] + j  E
e v 
t
2
g 1


B
   E
t

B  0


 
E   v  B   j    j2
 1

j
B
0
Open-GCM Boundary Conditions
• The MHD equations are solved as an
initial value problem.
•Solar wind parameters enter the
upstream edge of the simulation and
interact with a fields and plasmas in the
simulation box.
•Boundary conditions at the
downstream edge, the north and south
edges and the east and west edges are
set to approximate infinity.
–Downstream- y  x  0 where
y represents the parameters in the
MHD equations.
– The bow shock frequently passes
through the side and top and
bottom boundaries. Here setting
the derivative approximately
parallel to the shock to zero works
well.
Open-GCM Ionospheric Conductance Model
• Solar EUV ionization
– Empirical model- [Moen and Brekke, 1993]
• Diffuse auroral precipitation
– Strong pitch angle scattering at the inner boundary of the
simulation-[Kennel and Petschek, 1966]1
FE  ne kTe 2me 2
E0  kTe
Te  CTMHD
– Electron precipitation associated with upward field-aligned
currents (Knight, [1972] relationship but empirical models).
FE   j
E0  e
e 2 ne
 
min( 0, j )
2me kTe
• Conductance [Robinson et al, 1987]



 P  ne 40 E0 16  E02 FE1 2
 H  0.45 E05 8  P
The Open-GCM Electric Field


 
E  v  B   J
where v is bulk velocity, B the magnetic field, J the
current density, and η is the resistivity.
2



J
• The resistivity
where  is a small constant. The
resistivity is turned on when the current density exceeds
a threshold. This threshold has been calibrated so it only
appears in strong current sheets. This avoids spurious
dissipation.
Open-GCM Coordinate System
• The Open-GCM calculations are in the Geocentric Solar
Ecliptic Coordinate System (GSE).
• In GSE the x-axis points toward the Sun
• The Y-axis is in the ecliptic plane pointing toward dusk
(i.e. opposite to planetary motion).
• The z-axis is parallel to the ecliptic pole.
• Relative to an inertial system this system has a yearly
rotation.
Including the IMF Bx component in the OpenGCM Model
• Special care must be taken
 when including the IMF BX in
the calculations or   B  0 .
• Two approaches are used.
– Some use a method developed by John Lyon and colleagues.
They assume that the BX component is given by a linear
combination of the other two field components
Bx t   Bx ,0  cBy t   dBz t 
where c and d are determined by least squares fitting to the
observed solar wind time series.
– Mostafa El-Alaoui developed a method thereby he carries out a
minimum variance analysis and rotates the simulation domain
into the minimum variance direction. The magnetic field is kept
constant along the minimum variance direction thereby assuring
that   B  0.
• Both approaches only approximate BX.
Southward and Away IMF
(BATSRUS Code)
Northward and Earthward IMF
MHD Models and the Ring Current
• The ring current involves physics not included in MHD.
For instance particles move around the Earth due to
gradient and curvature drift motion.
• Additional models must be used to include the ring
current in the MHD models.
– Empirically based ring current models can be added to the MHD
models (e.g. the Fok ring current model).
– Theoretical models can be added such as the Rice Convection
Model which include the particle motion.