Lecture 14Lecture 14

Two Simulation Codes: BATSRUS and Global

GCM

BATSRUS (Block-adaptive tree solar BATSRUS (Block-adaptive tree solar wind Roe-type upwindwind Roe-type upwind schemescheme) ) (Powell (Powell

et al.,J. Comp. Phys., 154, 284,1999)et al.,J. Comp. Phys., 154, 284,1999)• Conservation of mass

• Faraday’s Law

• Total time rate of change of magnetic flux across a given surface S bounded by δS

0

ut

Et

B

SduBdluBSdt

BSdB

dt

dSSsS

dlESdBdt

dS

'

S

BATSRUS Equations

• Conservation of Momentum

The second equation was derived from the first by using Ampere’s Law ( ) and vector identities.

BBBB

IBB

puut

u

BjIpuut

u

�

�

000

1

2

Bj

0

1

BuuBBut

B

• Although B is divergenceless it is kept in order to have all of the equations in normalized divergence form

SFt

U T

BATSRUS Equations

• Conservation of Energy

is the ratio of specific heats.

0

00

221

1

2

BBuupe

BBuuBB

pet

e

BATSRUS Ohm’s Law

• Uses an ideal Ohm’s Law with η=0.• Achieves numerical stability by eliminating numerical

noise.

BuE

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.

• Local values of compressibility, rotationality and current density are used where V is the cell volume.

VB

Vu

Vu

t

r

c

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 field-aligned 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)

• Ran the AMIE code 8500 times to get patterns of conductance, binned the results according to location and current density and fit the results.

AJe

0

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

cossin0

sincos0

001

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 El-

Alaoui at UCLA.

Open-GCM Magnetohydrodynamic Equations

• Macroscopic plasma properties are governed by basic conservation laws for mass, momentum and energy in a fluid.

Bj

jjBvE

B

EB

Ejv

BjIvvv

v

�

0

2

2

1

0

t

1v

2

1+])[(

t

+)(t

)(t

pepe

e

p

Mass

Momentum

Energy*

Faraday’s Law

Gauss’ Law

Ohm’s Law

Ampere’s Law

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- where 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.

0 x

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]

– Electron precipitation associated with upward field-aligned currents (Knight, [1972] relationship but empirical models).

• Conductance [Robinson et al, 1987]

21

2 eeeE mkTnF ekTE 0

),0min(2

2

0

jkTm

ne

eE

jF

ee

e

E

PH

EeP

E

FEEn85

0

21200

45.0

1640

MHDe CTT

The Open-GCM Electric Field

where v is bulk velocity, B the magnetic field, J the current density, and η is the resistivity.

• 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.

JBvE

2J

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 Open-GCM Model

• Special care must be taken when including the IMF BX in the calculations or .

• 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

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 .

• Both approaches only approximate BX.

0 B

tdBtcBBtB zyxx 0,

0 B

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.