1. Magnetospheric Modeling
M. Wiltberger and E. J. Rigler
NCAR/HAO

2. Outline What is the magnetosphere? How do we model it?
Possible areas where statistics can help us

3. Solar Origins Solar Flares - abrupt release of energy
localized solar region
mainly radiation (UV, X- rays,γ-rays)
occur near complex sunspot configurations
Coronal Mass Ejections (CMEs)
Releases of massive amounts of solar material
Usually with higher speeds and greater magnetic fields than surrounding solar wind
Usually cause shocks in solar wind
Solar Wind
Steady ionized gas outflow with average velocity 400 km/s
Magnetic field direction variable
Exact properties depend upon solar origins

4. Earth's Magnetosphere
The magnetosphere is region near the Earth where it's magnetic field forms a protective bubble which impedes the transfer of energy and momentum from the solar wind plasma
A variety of different phenomenon
Substorms
impulsive energy release over hours
Storms
globally enhanced activity over days
Radiation belts
trapped particles which are omnipresent

5. Magnetospheric Currents
Dayside and R1 Currents
Neutral Sheet Current
Ring and R2 Currents
From the COMET Aurora Web Site
Because the magnetopause boundary separates a region of relatively strong magnetic field (the magnetosphere) from a region of relatively weak magnetic field (the magnetosheath), the boundary must carry a surface current to maintain force balance across the boundary.
The magnetopuase current encircling the two magnetotail lobes closes through the middle of the tail in a neutral sheet current that seperates the oppositely directed magnetic fields above and below.
Part of the dayside magnetopause current closes on the magnetopause and part closes via the Region 1 field-aligned currents in the ionosphere.
The current encircles the Earth and is not uniform, with the stronger current flowing on the night side of Earth. The ring current closes partially within the magnetosphere and partially through the ionosphere via the region 2 field aligned currents.
Magnetopause current systems are created by the force balance between the Earth’s dipole and the incoming solar wind

6. Ionospheric Currents
Region 1
Pedersen currents flow parallel to the electric field and perpendicular to the magnetic field.
Hall currents flow perpendicular to both the electric and magnetic fields
If the ionospheric conductivity where uniform were uniform, then the closing would accomplished solely by Pedsersen
Region 2
FAC from the magnetosphere close though Pedersen and Hall Currents in the ionosphere

7. LFM Magnetospheric Model
Uses the ideal MHD equations to model the interaction between the solar wind, magnetosphere, and ionosphere
Computational domain
30 RE < x < -300 RE & ±100RE for YZ
Inner radius at 2 RE
Calculates
full MHD state vector everywhere within computational domain
Requires
Solar wind MHD state vector along outer boundary
Empirical model for determining energy flux of precipitating electrons
Cross polar cap potential pattern in high latitude region which is used to determine boundary condition on flow

8. Computational Grid of the LFM
Distorted spherical mesh
Places optimal resolution in regions of a priori interest
Logically rectangular nature allows for easy code development
Yee type grid
Magnetic fluxes on faces
Electric fields on edges
Guarantees B = 0

9. Ionosphere Model 2D Electrostatic Model Conductivity Models
(P+H)=J||
=0 at low latitude boundary of ionosphere
Conductivity Models
Solar EUV ionization
Creates day/night and winter/summer asymmetries
Auroral Precipitation
Empirical determination of energetic electron precipitation

10. Calculation of Particle Fluxes
Empirical relationships are used to convert MHD parameters into an average energy and flux of the precipitating electrons
Initial flux and energy (Fedder et al., [1995])
Parallel Potential drops (Knight [1972], Chiu [1981])
Effects of geomagnetic field (Orens and Fedder [1978])

11. Determining Energy Flux
According to Lummerzheim [1997], it is possible using the UVI instrument on POLAR to determine both the characteristic energy and energy flux
Energy flux is proportional to emission rates in the LBH bands
Characteristic energy determines altitude of emission so it is determined by monitoring brightness of features that decay away and those that persist

12. Estimating Optimal Parameters
LFM electron flux/energy estimates are projected onto irregular grid corresponding to Polar UVI observations:
1-D “state vector” constructed from all available observations; time is simply treated as a third coordinate, in addition to MLT and ALAT;
Levenberg-Marquardt (nonlinear least-squares) algorithm adjusts model parameters a, b, and R to minimize the sum of the squared prediction errors:

14. Can advanced statistics help us?
Are there better parameter estimation algorithms then Levenberg-Marquart?
What can statistical comparisons with observations tell us about the bias present in our models?
What are the best measures to monitor model improvement over time?

15. Some Research Problems
1. How can uncertain model parameters be optimized to provide the best agreement, on the average, with observations?
2. How can model variability about the average, including information about scale sizes of this variability, best be compared with variability in observations to determine agreement or disagreement?
3. How can we improve the interpolation/extrapolation of observations of model input parameters in space and time to get complete specification of the boundary conditions?
4. In developing parameterizations of sub-grid phenomena, such as the transport of momentum and the creation of turbulence by breaking gravity waves, what is a good measure of intermittency, and how can its effects be parameterized?
5. How can relatively rare and sparse observations of extreme events like large magnetic storms be used to characterize upper-atmospheric behavior and test simulations for such events?
6. What can statistical comparisons tell us about underlying biases in our models?
7. What are the best measures to monitor model improvement over time?

17. MI Coupling Eqs
As described in Kelley [1989] The fundamental equation for MI coupling is obtain by breaking the ionospheric current into parallel and perpendicular components and requiring continuity
Assuming no current flows out the bottom of the ionosphere we get
Further assuming the electric field is uniform with height we get
And finally using and electrostatic approximation in the MI coupling region we obtain
Start by adding that a significant portion electric field is mapped down to the high latitude ionosphere where it becomes the driver for the convection of ionospheric plasma.
Also stress that in the Magnetospheric simulation the computed electric field is used as a boundary condition.

18. Conductances from Particle Flux
Spiro et al. [1982] used Atmospheric Explorer observations to determine a set of empirical relationships between the average electron energy and the electron energy flux
Robinson et al. [1987] revised the relationships using Hilat data and careful consideration of Maxwellian used to determine the average energy
Recall the height integrated conductivity = conductance
Hardy et al. [1987] reports a version of the Robinson et al. with slight typographical error