Magnetospheric Charged Fluid Transport

1
Magnetospheric Charged Fluid Transport

• Robert Sheldon
• University of Alabama in Huntsville
• September 22, 1999

2
History of Diffusive Transport
Thanks Barry, Dick Haje

• Early 1960s, radiation belt theory was
  developed, first analytically, then with fluid
  PDEs.
• By 1970s, ring current theory followed rad
  belts. Analytic theory was not able to describe
  the data. Fluid models dominated.
• 1980s, saw increasing use of particle tracing
  and Monte-Carlo solutions.
• 90s developed hybrid models e.g. w/MHD

3
Whats wrong with this picture?

• In the 30 years since these early models,
  computers have increased in speed by 215,
  according to Moores law.
• This should have produced models that are perhaps
  32,000 times better, but the deviations (model
  - data)/data have barely improved a factor 2.
  Err100 is still considered good agreement.
• Why has the convergence been so slow?

4
Degrees of Freedom

• Phase space is 6-Dimensional. This is the maximum
  of degrees of freedom.
• Adiabatic invariants involve an average over one
  of these dimensions.
• Time is NOT a degree of freedom. For one thing,
  diffusion in all 6 phase space dimensions is
  possible. Time does not diffuse. In PDEs it
  introduces a parabolic dimension, d/dt d2/dq2.
  One can always add a homogeneous solution, 0
  d2/dq2

5
An Aside on MHD

• MHD includes 2 more vector fields with the phase
  space density This 12-D field gets simplified
  down to 5-D using coupling equations.
  Unfortunately, the phase space is averaged into
  moments, and therefore not applicable for inner
  magnetosphere.
• We assume both E and B model and keep as much
  information in phase space.

6
Non-MHD Inner Msphere bltlt1 6-D 5-D
4-D 3D

• Particle tracing
• Vx,Vy,Vz, X,Y,Z
• m, J, L, Fm,FJ,FL
• m,J, L,Fm, FJ,FL,t
• Vx,Vy,Vz, X,Y,Z,t

• Gyro-kinetic
• m, J, L, FJ, FL
• m, J, L, FJ, FL,t

• Radial diffusion
• B-L space
• m, J, L
• m, J, L, t
• Salammbo

• Guiding center
• m,J,L,FL
• m,U,B,K
• m,U,B,K,t

Dynamic, time-dependent models
7
1-D Diffusion Modelling

• Analytic Solution
• Errorgt0
• 6-Dimensional No effect.

• Fluid PDE
• df/dt D d2f/d2x
• Error gt dx
• 6-Dimensional
• (X/dx)1/6

• Particle tracing Monte-Carlo
• Error gt dx/X dN1/2
• 6-Dimensional (X/dx)1/6(N1/12)

215 gt(215)1/12 2.37!
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Lessons from Numerics

• 1) Use as low a dimension as possible. As simple
  as possible, but no simpler.
• 2) Use as analytic a technique as possible.
• Example Radiation Belt Models.
• 3-D, averaging over Fm,FJ,FL
• symmetry in B, allows L radial distance,
• thus diffusion in L is separable from convection
• Extension to Ring Current Belt Models?
• NOT SYMMETRIC! Drift shell splitting, ...
• NOT SEPARABLE!

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2-Diffusive Transport

• If coordinates are not separable, gtnew physics
• Vortices, convection
• Anomalous, or enhanced diffusion, migration
• Convection diffusion confusion

• If coordinates are separable, then D D1 D2
  gt diagonally dominant, easily generalized from
  1-D.
• DLL, Daa, Dmm gt radial diffusion models (Schulz
  74)

Is there a coordinate system which is separable?
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Yes! UBK (Whipple, 78)

• Average over Fm,FJ gt 1st 2nd adiabatic
  invariants. Then the total energy is
• H0 K.E. P.E.
• H0 mBm(x,y) q U(x,y)
• dH0/dt 0 mdBm/dt q dU/dt
• dU/dBm -m/q
• Convection conserves energy, so constant energy
  trajectories convection. In UB space these are
  straight lines. Coordinates SEPARATE! (Sheldon
  Gaffey, 95)

11
Schematic UBK
U
dawn
e-
i
Magnetotail
D
(2)
(1)
Earth
dusk
(3)
Bm(K)
12
2 ways to diffusion convection
Hamiltonian Lagrangian

• (Stern 78, Northrop 63)
• Time is implicit
• Energy explicit
• Lie groups perturbation theory, high-order
  accuracy is built-in. (Design of the SSC magnets
  used some 15 orders to achieve 106 orbits.
  Dragts book.)

• (Chen 70, Smith 74)
• Time is explicit
• Energy implicit
• ODE solvers, Runge-Kutta 4/5, high-order accuracy
  hard to achieve (Gears Implicit methods,
  Adams-Bashford-Moultons predictor-corrector,
  Bulirsch-Stoers extrapolation)

13
Hamiltonian Diffusion

• HH0 H1 H2
• H mBm qU mdBm/dt q dU/dt
• ltHgt 0 (conservation of energy)
• ltH2gt / 0 gt D a m(dBm/dt)2 q (dU/dt)2
• Diffusion coefficient is easily derived (cf.
  Falthammer 65,68, Lyons 75)
• Asymptotic series converge rapidly, error is
  minimized. (cf. Dragt 9x and Lie algebra)
• Explains the L4 dependence seen in the data.

(Sheldon, 97)
14
Uncontrolled Exuberance

• We now have the tools to analyze thermal to
  radiation belt plasmas (0ltEltMeV) with radial
  diffusion models
• Diffusion coefficient D(U,B,K,m)D(x,y,K,m)
• Convection coefficient v(U,B,K,m)
• Coulomb drag coefficient C(U,B,K,m)
• Charge exchange rate X(U,B,K,m)
• df/dt d/dq(Dqq df/dq) vq df/dq C df/dm Xf

15
Conclusions

• UBK is a 4-D method, the simplest system able to
  describe magnetospheric convection
• UBK then is ideal for inner mag plasma at 0
  lt E lt MeV conserving the first 2 invariants
• UBK straightens trajectories, helping intuition
• UBK has separable coordinates for diffusion and
  convection avoiding confusion.
• UBK permits a Hamiltonian perturbation expansion
  gt trivial calculation of diffusion
• Applications Jupiter. Heliosphere. Cusp!

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Examples

• Separatrix between plasmasphere bananasphere
  explains Williams Frank 84,88. mystery peak.
  (Sheldon 94)
• This peak explains fast diffusive transport
  (Sheldon 91) and RC storm injection (Sheldon 99)
• Separatrix between plasmasphere and open drift
  explains LANL signatures (Sylvestre, Korth) and
  determines E-field (McIlwain).
• Matching particle features between ISTP
  spacecraft determines E-field (Whipple 98)