1 The Inner Magnetosphere Nathaniel Stickley George Mason University

2 Overview Particle populations –Radiation belts, plasmasphere, ring current Particle injection and energization –Diffusion, wave-particle interaction Electric fields and drift paths –Shielding, co-rotational electric field, Alfvén layer DPS relation –Derivation, discussion Modeling –Rice Convection Model (RCM)

3 Particle populations Radiation belts (Van Allen, 1958) Inner belt Located at L ≈ Primarily cosmic ray albedo protons of high energy (>10MeV) Very stable Outer belt Located at L≈3-9 Primarily high energy electrons with energy up to 10MeV Population is unstable (particles are not trapped as efficiently)

4 Particle populations Radiation belts Electron “slot” region Located at L ≈ 2.2 Apparently due to increased wave-particle interactions There is no corresponding slot for ions

5 Plasmasphere Cool particles (~1eV-1keV) High particle density (~10 3 cm -3 ) Extends to L=3-6 Distinct from radiation belts but shares same region of space Primarily ExB drifting particles (because of low temperature) Particle populations

6 Ring current Mostly indistinguishable from trapped radiation belts Gradient and curvature drift Composed of mostly keV ions Typically in the range L=3-6 O + is dominant Ion in terms of abundance H + begins to dominate > few keV Total energy density dominated by O + and H + Energy midpoint: 85keV

7 Particle injection and energization How are particles injected into the inner magnetosphere? Cosmic rays Ionosphere injection Substorm and storm particle injections Diffusion (adiabatic invariants do not strictly hold).

8 Brief theoretical aside (1) Sample derivation of the 1st adiabatic invariant: Definition of adiabatic invariant: p and q are canonical momentum and coordinates respectively. Integration is performed over one cycle If system changes slowly during each cycle, the action S is a constant. We could use this definition to show that μ is an adiabatic invariant, but we will use a less direct approach in order to illustrate a point.

9 Brief theoretical aside (2) Starting with Faraday’s law: and the equation of motion: where Taking the scalar product of this with

10 Brief theoretical aside (3) Left hand side is rate of change of KE: Variation in KE is: Key point: if the field changes slowly, this is the same as

11 Brief theoretical aside (4) Re-write using Stokes’ theorem: Now we use Faraday’s law: for ionsfor electrons

12 Brief theoretical aside (5) Factoring out Thus: Since

13 Particle injection and energization How particles in the inner magnetosphere become energized : Electric fields Wave-particle interaction Whistlers (review) Play whistler audio in Adobe Audition:

14 Sun Dawn Dusk f Orientation Reminder Convection Electric Field

15 In the absence of the convection electric field, the plasma sheet appears as illustrated: What happens when the convection electric field is included? Polarization field

16 Polarization field Positive charge collects on duskward edge of plasma sheet while negative charge collects on the dawnward edge. Resulting electric field is called the “polarization electric field”. The inner magnetosphere is thus shielded from convection electric field. Other features: Over-shielding Partial ring current Region 2 Birkeland current

17 Overview of current systems

18 Co-rotation Electric field (review) Small due to shielding Dominates for high- energy particles Dominates for low-energy particles

19 The Alfvén Layer

20 Recall the shape of the plasmasphere Compare with the shape of low-temperature Alfvén layer. The Alfvén Layer

21 Hot ions vs. Hot electrons The Alfvén Layer

22 Variability in position: The Alfvén Layer

23 Dessler-Parker-Schopke (DPS) Relation Derived originally in 1959 by Dessler and Parker Relates the total energy in the ring current to the magnetic field perturbation at the center of the Earth

24 Derivation of DPS Relation (1) at equatorial plane

25 Derivation of DPS Relation (2)

26 Derivation of DPS Relation (3)

27 Derivation of DPS Relation (4)

28 Derivation of DPS Relation (5)

29 Problems with the DPS relation False assumptions: Ring current is circular and concentric with Earth Ring current is azimuthally symmetric (no partial ring) Magnetic field is purely dipolar Field perturbation due to ring current is not important –Nonlinear “feedback” is not accounted for. Earth is non-conducting Assumes ring current is confined to the equatorial plane. –However Schopke proved that the relation holds for arbitrary pitch angle

30 Experimental: DPS typically estimates Dst index to within 20% (the relation does not pretend to include affects from other current systems, so this is rather impressive) Computational: Liemohn (2003) used computational model of ring current to test DPS relation 1.Calculate realistic particle distributions 2.Calculate pressures from particle distributions (include non-zero pressure outside of volume of integration) 3.Calculate currents using pressure information 4.Calculate ΔB(0) from currents using Biot-Savart DPS systematically over-estimates ΔB(0) for isotropic pressure distribution. Testing the DPS relation

31 Modeling the Inner Magnetosphere Goals –Calculate-particle distributions / drifts –Self-consistently calculate electric fields –Self-consistently calculate magnetic fields –Couple inputs and outputs to global MHD and ionosphere models What physics should be included in such a model?

32 Rice Convection Model Multi-fluid model (typically 100 fluids) Self-consistently computes electric fields Isotropic particle distribution Calculates adiabatic drifts Requires specified magnetic field Input: Magnetic field model Polar cap potential distribution Initial plasma density Plasma boundary conditions Some possible outputs: velocity distribution particle fluxes potential distribution (and therefore electric fields) Ionospheric precipitation

33 Fundamentals Rice Convection Model ds 1/B

34 Fundamentals Algorithm: Iterate through eq(2) and eq(3), updating velocities with eq(1). Rice Convection Model

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36 Coupling the models RCM requires as boundary conditions: Ionosphere conditions (conductivities, etc) Magnetic field Outer plasma conditions Solution: Couple RCM with MHD model for outer magnetosphere Couple RCM with ionosphere model. Self-consistently compute magnetic field with MHD model. There are difficulties in actually implementing this.