1. Third Invariant Kinetic Codes: 4D phase space Robert Sheldon June 29, 2005 National Space Science & Technology Center

2. MHD vs Kinetic Codes MHD assumes cold plasma, isotropic, single-fluid, small B-gradients, etc. If any process that violates MHD begins to dominate the dynamics then MHD becomes a poor approx. Kinetic codes usually assume particles in fixed B- fields, and are not self-consistent with B. The inner magnetosphere has hot particles, in strong gradients. Ring currents form that are not MHD, and they dominate the dynamics as well as distort B. But full kinetic codes, e.g. PIC codes that track 1000’s of particles, are terribly expensive & slow.

3. Two approaches: Top-down, Bottom-up Complicate the MHD: –Add pressure tensors, Hall effects, etc. –Ad hoc, with no guarantee that all the “important” physics is captured. Simplify the Kinetic code –Use symmetries of problem to integrate over ignorable coordinates –Not self-consistent E & B. Iterative refinement is possible.

4. Why not do it all? Problem 1 Why not trace 1,000,000’s of particles and get all the physics? (PIC codes of the world unite!) –10 20 particles are needed to get a cubic Re of just solar wind plasma. So even if we can trace 10 6 particles, we are sampling only 10 -14 of the particles. It’s a set of measure zero. – This means that we have to select a chosen few to trace, and we can easily miss an important subset. Here’s an example of an important phenomena not seen in such particle tracing codes. (Either forward-time or backward- time propagated!)

5. Hot Plasma Subsets Energy dependence and grad-B drift can lead to brand-new physics, not in the MHD. Williams, Frank & Shelley peak seen in ISEE-3 data (35 keV protons). Local acceleration or transport?

6. Problem2: Monte Carlo Integration What do we want to predict? S/C fluences. What does particle tracing give us? Single particle transport, f(x,y,z,x’,y’,z’) How do we get fluences? Integrate over energy, pitchangle, MLT, radius [gyrophase, bounce phase] How many particles to do this integration? –Monte Carlo methods: sample parameter space evenly –Two drawbacks of Monte Carlo (Press, p. 299): Smoothness—sample spacing defined by smallest feature Dimensions—error in solution goes as N -1/2*D  10%=10 12 !

7. Phase Space Compression If we can reduce the number of dimensions, then Monte Carlo integration starts to look reasonable. 6D  4D = 10 8 <<10 12 for 10% accuracy. Adiabatic invariants are equivalently integrals over some canonical dimension. –1 st =magnetic moment =  GYRO –2 nd =J (“pitchangle”) =  BOUNCE –3 rd =L-shell =  DRIFT Thus radial diffusion is 6 - 3 = 3-D calculation. Gyrokinetic = 6 - 1 = 5-D calculation.

8. 8 Non-MHD Inner M’sphere  6-D 5-D 4-D 3D Particle tracing Vx,Vy,Vz, X,Y,Z , J, L,    J  L ,J, L,     J  L,t Vx,Vy,Vz, X,Y,Z,t Radial diffusion B-L space , J, L , J, L, t Salammbo Gyro-kinetic , J, L,  J   L , J, L,  J   L,t Guiding center ,J,L,  L ,U,B,K ,U,B,K,t Dynamic, time-dependent models

9. What kind of Transport Code? 1.Einstein: “As simple as possible but no simpler” Lowest dimensionality possible by considering time- and spatial- scales required for solution. 2. Hamilton-Jacobi theory for set of action-angle variables with best convergence properties. 3.Coupled PDE with appropriate diffusion coefficients, preferably diagonalized 4.(Flowfield separation of diffusion from convection to allow operator splitting.)

10. 3D Canonical Coordinate Diffusion ( Roederer 1970 ) Basis set 1 bounce-reson 2 longitud. avg. 3 J~0 only Distribution Function RadialAsymm Pitch- angle Symm Pitch- angle Energy Loss ,J,M ƒ (X, ,J,M,t) ,J,M 1 J,M 1 J,M L,K,Mƒ (X,L,K,M,t) LL,K,M 1 K,M 1 M L, B m,Tƒ (X,L,B m,t) L,TL, B m BmBm T L,  T ƒ (L, ,T,t) L,T L,  2  T L,  ƒ (L, , ,t) L L,  3 33  Diffusion

11. The 3 rd Invariant: Problem 3 In 1960’s, inner radiation belts were the issue. –Offset-tilted dipole worked really well. No external B perturbations  azimuthally symmetric magnetic model –High energy particles ignore E-field. No external E perturbations  azimuthally symmetric E potentials Therefore 3 rd invariant=L-shell  3D phase space. However it has problems: –outer zone rad belts NOT azimuthally symmetric B-field –Ring current does NOT ignore E-field potentials. Therefore L-shell (or L*) isn’t a good coordinate.

12. 12 If coordinates are not separable, ==>new physics: –Vortices, convection –Anomalous, or enhanced diffusion, “migration” Convection + diffusion = confusion Diffusive Transport: Problem 4 If coordinates are separable, then D = D1 * D2 ==> diagonally dominant, easily generalized from 1-D. D LL, D  D  ==> radial diffusion models (Schulz 74) Is there a coordinate system which is separable?

13. What is better than L? 3 rd invariant doesn’t vanish due to distortions of B and E, rather, we need a generalized 3 rd invariant that captures the real existing symmetries. In conservative force fields (no friction) such as E & B fields, the total energy is conserved. (The first invariant is really only about kinetic energy), so H = KE + PE =  B m + q  Since a trajectory conserves energy, dH/dt = 0 Iso-H contours are trajectories through the magnetosphere. (x,y)  (B m,  ) is convenient. Lagrange  Hamilton

14. Lagrange vs Hamilton So we can simplify our equations if we use adiabatic (=constant energy) invariants. From classical mechanics, we can recast the equations of motion either as (x,v) or (E,t). Numerically, Lagrangian solvers include time explicitly x = f(t), and energy is implicit, whereas Hamiltonian solvers include energy explicitly, time implicit x = f(E). Adiabats beg for Hamilton. Accelerator designers find Hamiltonian methods converge faster to higher accuracy. (MaryLIE)

15. 4-D UBK Hamiltonian We use two, energy conserving, 4D coordinate systems: – ƒ 1 (M,K,U,B m,n,t): “radial” diffusion + convection – ƒ 2 (T, ,X,Y,t): Coulomb collisions + pitchangle diffusion –Where “n” separates quadrupole & dipole trapping Operator splitting enables us to carry out radial transport on ƒ 1, and pitchangle+energy loss on ƒ 2. The Quadrupole trap is located at a specific MLT as well as high-latitude. Thus we need to keep the 3 rd invariant phase  4-D

16. PDE Diffusion Equation df k /dt= ∂f k /∂t+ C∂f/∂y+ D z ∂ 2 f k /∂z 2 + ∂/∂x i (D ij ∂f k /∂x j ) –i,j= index for (M,K) or (E,  ); k = quad or dipole trap –1 st term is the source + loss terms –2 nd term is convective velocity in (U,B m )-space –3 rd term is diffusion in (U,B m )-space –4 th term is remaining diffusion in (E,a)-space Mixed parabolic & elliptic PDE solved with standard finite element techniques. Operator splitting, mapping between ƒ 1  ƒ 2 (Fast, efficient, not numerically diffusive).

17. Conclusions Just as going from j  f = j/p 2 was necessary to understand trapped plasma, locate sources, solve for diffusion etc, so it is also necessary to find a better 3 rd invariant. We propose to go from L (or even L*)  H to organize the PSD and for once, eliminate the spurious diffusion caused by bad coordinates. The cost is high, going from 3D  4D, but the benefits are also high:

18. More Conclusions The UBK transform has the unique property of solving several requirements at once: –Separates convection from diffusion (MLT-dependence) –Permits Hamilton-Jacobi solution to equations of motion –Treats both Quadrupole & Dipole Regions (hi-latitudes) –Permits rapid calculation of diffusion coefficients –Can use operator splitting to diagonalize Diffusion tensor Cannot handle time- or spatial- scales that violate 1 st invariant. E.g. rare shock acceleration events (1991). It should indicate whether a hi-latitude source exists.

19. PDE vs ODE Diffusion BUT many processes are not purely adiabatic, they DIFFUSE. How is this handled? –Lagrangian ODE tracing can be done for many different magnetospheric conditions, and averaged over that as well. Equivalent to adding a 7 th additional time dimension. –Hamiltonian adiabatic methods have already averaged over some temporal scales implicitly. Non-adiabatic behavior introduces 2 nd order cross terms (diffusion) into the system, equivalent to going from ODE  PDE with the same number of dimensions. Now one uses PDE not MonteCarlo integration. Time is implicit, so careful attention must be spent splitting the diffusion timescale from the convection timescale, or else the PDE solves the wrong problem. Choice of appropriate coordinates is CRITICAL for PDE.