Modeling the Magnetosphere Dr. Ramon Lopez

The Magnetosphere  When the solar wind encounters a magnetized body, it is slowed and deflected  The resulting cavity in the solar wind controlled by the body’s magnetic field is called a magnetosphere

Simulations show how this happens  The Earth’s undisturbed field is basically a dipole  When the solar wind flows in from the left, the Earth’s field is deformed

Magnetospheric “Geography”

Currents flow along the magnetic field

Field Aligned currents light up the sky. We call it the Aurora

Other planets have aurora too!

What is the right physics? 1) Electricity and Magnetism - the Maxwell Equations 2) Fluid Flow equations 3) Relativity 4) Both 1 and 2 5) Both 2 and 3 6) 1, 2, and 3 Given the problem of modeling the flow of the solar wind past the Earth’s magnetic field, what is the proper physics we need to include?

Ideal MHD Equations Non Conservative Formulation  No strict numerical conservation of energy and momentum  Various numerical issues Errors in propagating strong shocksErrors in propagating strong shocks Errors in RH ConditionsErrors in RH Conditions Incorrect shock speedsIncorrect shock speeds

Ideal MHD Equations Full Conservative Formulation  Strict numerical conservation of mass, momentum and energy  Numerical difficulties in regions where p<<B 2 negative pressures possible because p becomes difference of two large numbers

Ideal MHD Equations Gas Conservative Formulation  Strict numerical conservation of mass, momentum and plasma energy no strict conservation of total energyno strict conservation of total energy  No difficulties in regions where p<<B 2  Could use ‘  switch’ to combine with full conservative scheme

Time out to think  MHD equations can be cast in different forms. We chose a particular form to solve numerically because 1) they require less computer time1) they require less computer time 2) they will not allow numerically unphysical results2) they will not allow numerically unphysical results 3) they are easier to solve mathematically3) they are easier to solve mathematically 4) they are smaller set of equations4) they are smaller set of equations

Computation Grids  Simulation boundaries should be in supermagnetosonic flow regimes 18 Re from Earth on Sunward side18 Re from Earth on Sunward side 200 Re in tailward direction200 Re in tailward direction 50 Re in transverse directions50 Re in transverse directions  A variety of grid types exist with varying degrees of complexity Uniformed CartesianUniformed Cartesian Stretched CartesianStretched Cartesian Nested CartesianNested Cartesian Regular NoncartesianRegular Noncartesian Irregular NoncartesianIrregular Noncartesian

 Stretched Cartesian Grid Low programming overheadLow programming overhead Low computing overheadLow computing overhead No memory overheadNo memory overhead Easy parallelizationEasy parallelization Somewhat adaptableSomewhat adaptable  Example from Raeder UCLA MHD Model  Uniformed Cartiesian Grid Low programming overhead Low computing overhead No memory overhead Easy parallelization Not very adaptable

 Regular Noncartesian Medium programming overheadMedium programming overhead Low memory overheadLow memory overhead small computing overheadsmall computing overhead parallelizes like regular cartesian gridparallelizes like regular cartesian grid somewhat adaptablesomewhat adaptable  Example from LFM  Nested Cartesian Medium/High programming overhead Medium/High memory overhead small computational overhead difficult to parallelize very (self) adaptable  Example from BATS-R-US

Boundary Conditions  Upstream Fixed or time dependent values for 8 plasma parametersFixed or time dependent values for 8 plasma parameters  Can be idealized for derived from solar wind observations Problem with B XProblem with B X  Need to know 3D structure of solar wind because  Implies B X =B N cannot change if solar parameters are independent of Y and Z  Find n direction with no variation and then sweep these fronts across front boundary

Boundary Conditions II  All other sides Free flow conditions for plasma and transverse components of BFree flow conditions for plasma and transverse components of B normal component of B flows from  B=0normal component of B flows from  B=0  Inner Boundary Condition MI Coupling moduleMI Coupling module Hard wall boundary condition for normal component of velocity and densityHard wall boundary condition for normal component of velocity and density

Magnetosphere-Ionosphere Coupling  Inner boundary of MHD domain is placed between 2-4 R E from the Earth High Alfven speeds in this region would impose strong limitations on global step size High Alfven speeds in this region would impose strong limitations on global step size Physical reasonable since MHD not the correct description of the physics occuring within this region Physical reasonable since MHD not the correct description of the physics occuring within this region Covers the high latitude region of the ionosphere (45  -90  ) Covers the high latitude region of the ionosphere (45  -90  )  Parameters in MHD region are mapped along static dipole field lines into the ionosphere  Field aligned currents (FACs) and precipitation parameters are used to solve for ionospheric potential which is mapped back to inner boundary as boundary condition for flow

Ionosphere Model 2D Electrostatic Model –  (  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

Auroral Precipitation Model  Emperical relationships are used to convert MHD parameters into a characteristic energy and flux of the precipitating electrons Initial flux and energyInitial flux and energy Parallel Potential drops (Knight relationship)Parallel Potential drops (Knight relationship) Effects of geomagnetic fieldEffects of geomagnetic field Hall and Pederson Conductance from electron precp (Hardy)Hall and Pederson Conductance from electron precp (Hardy)

Time Out to Think  LFM contains how many separate ‘computational’ models and ‘physical’ domains 1) 2 models and 1 domain1) 2 models and 1 domain 2) 3 models and 2 domains2) 3 models and 2 domains 3) 1 model and 2 domain3) 1 model and 2 domain 4) 2 models and 2 domains4) 2 models and 2 domains

MHD Magnetosphere Simulation  The Lyon-Fedder-Mobary (LFM) code is a fully 3-D MHD simulation run with real solar wind input  Magnetosphere modeled via ideal MHD equations within 30 to -300R E (x) and 100 R E (y,z)  Upstream and side BCs -> Solar wind data  Downstream BC -> Supersonic outflow  Inner BC -> 2-D Ionospheric simulation  Reconnection occurs due to numerical effects

Visualizing the results  In order to understand a a global sense what the simulation produced, you have to visualize the results  We use package called OpenDX, which is an open-source version of IBM DataExplorer  Another package, SPDX, provides modules for reading the simulation HDF files  Images are rendered frame by fame, then strung together as movies

Methods of Model Validation  Computation of theoretical problems with known analytic answers Provides a ground truth that code is workingProvides a ground truth that code is working Very limited number of MHD problemsVery limited number of MHD problems  Direct comparison with observations Limited number of spacecraft observationsLimited number of spacecraft observations  Check general characteristics with superposed epoch studies Include comparison with indirect observationsInclude comparison with indirect observations Use metrics to quantitatively asses validityUse metrics to quantitatively asses validity

The March 9, 1995 Substorm  Clean, isolated, multiple onset substorm  Simulated AL and CANOPUS AL agree quite well  Other simulated features agree well with the observations

LFM simulation results are similar to AIME results

Where does the energy come from?  Magnetic reconnection is the process by which magnetic energy is converted into plasma energy  Reconnection plays a basic role in the dynamics of space plasmas

IMF coupling to the magnetosphere  Magnetic merging on the dayside between the northward geomagnetic field (1) and southward IMF (2) allows solar wind energy to enter the magnetosphere (3,4)  Reconnection on the nightside (6) releases energy stored as magnetic flux in the tail lobes (5)