Aurora & Precipitation
The aurora is the visible manifestation of magnetosphere-ionosphere coupling: energetic particles accelerated along magnetic field lines precipitate into the upper atmosphere, exciting atomic and molecular emissions. This chapter derives the key physics from Birkeland currents through the Knight relation to ionospheric electrodynamics.
3.1 Field-Aligned Currents (Birkeland Currents)
Magnetosphere-ionosphere coupling is mediated by field-aligned currents (FACs) that flow along magnetic field lines. These were predicted by Birkeland (1908) and confirmed by satellite observations (Iijima & Potemra, 1976). Two main systems exist:
- Region 1: Flow on the poleward boundary of the auroral oval (downward on dusk, upward on dawn)
- Region 2: Flow on the equatorward boundary (opposite polarity to Region 1)
Derivation: Current Continuity at the Ionosphere
Step 1. In steady state, current must be divergence-free: ∇ · J = 0. Separating parallel and perpendicular components:
Step 2. Integrating through the ionospheric height-integrated layer:
where Σ is the height-integrated conductivity tensor and E is the ionospheric electric field.
Step 3. Expanding in terms of Pedersen (ΣP) and Hall (ΣH) conductivities:
The first term represents current divergence from the electric field pattern; the second and third arise from conductivity gradients (day-night asymmetry, auroral enhancement).
Vasyliunas Equation
The magnetospheric source of Region 1 currents comes from pressure gradients in the equatorial magnetosphere. Vasyliunas (1970) showed:
where V = ∫ ds/B is the flux tube volume. This shows that FACs arise wherever pressure gradients are not aligned with flux-tube-volume gradients — i.e., at the inner edge of the plasma sheet where ∇p is radial but ∇V has an azimuthal component.
Total Region 1 current during quiet times: ~1–2 MA per hemisphere; during storms: ~5–10 MA.
3.2 The Knight Relation
Upward field-aligned currents require a supply of downgoing electrons from the magnetosphere. When the required current density exceeds the thermal flux, a parallel electric field must develop to accelerate electrons. The Knight (1973) relation gives the current-voltage characteristic of the auroral acceleration region.
Derivation: Knight Current-Voltage Relation
Step 1. Consider electrons with a Maxwellian distribution at the magnetospheric source (subscript m). By Liouville's theorem, the distribution function is conserved along particle trajectories:
Step 2. At the ionosphere (subscript i), the field is stronger: Bi ≫ Bm. A particle reaching the ionosphere must satisfy the mirror condition. With a parallel potential drop Φ∥, the energy conservation gives:
Step 3. Conservation of the magnetic moment μ = mev⊥²/2B:
Step 4. The current density at the ionosphere is obtained by integrating the downward flux over all electrons that reach the ionosphere (accounting for the acceleration by Φ∥ and the expanded loss cone):
Step 5. Evaluating the integral (transforming to magnetospheric velocity space and integrating over the filled loss cone, now widened by Φ∥):
Limiting cases:
- No acceleration (Φ∥ = 0): j∥ = enevth/(2√π) × (Bi/Bm − 1) — the thermal precipitation flux, typically ~1 μA/m²
- Strong acceleration (eΦ∥ ≫ kBTe):
This linear j–Φ relation defines the Knight conductance K ≈ 10−9 S/m². For j∥ ≈ 10 μA/m² (bright aurora), the required potential is Φ∥ ≈ 1–10 kV. These accelerated electrons produce the "inverted-V" energy spectra observed by satellites.
3.3 Auroral Emission Physics
Precipitating electrons lose energy primarily through ionization, excitation, and heating of the neutral atmosphere. The altitude of maximum energy deposition depends on the electron energy.
Energy Deposition Profile
An electron beam with characteristic energy E0 deposits most of its energy at the altitude where the column mass equals the stopping range:
where E0 is in keV. This gives peak emission altitudes of:
- E0 ≈ 1 keV → ~250 km (thermosphere, red aurora)
- E0 ≈ 10 keV → ~110 km (E-region, green aurora)
- E0 ≈ 100 keV → ~90 km (D-region, blue/purple aurora)
Principal auroral lines:
| Wavelength | Species | Transition | Altitude |
|---|---|---|---|
| 557.7 nm (green) | O(¹S → ¹D) | Forbidden | ~110 km |
| 630.0 nm (red) | O(¹D → ³P) | Forbidden | ~250 km |
| 427.8 nm (blue) | N₂⁺ (1NG) | Allowed | ~100 km |
| 391.4 nm (violet) | N₂⁺ (1NG) | Allowed | ~100 km |
| 844.6 nm (IR) | O(³P → ³S°) | Allowed | ~200 km |
The green line (557.7 nm) dominates because the O(¹S) state has a radiative lifetime of 0.74 s, short enough to emit before collisional quenching below ~95 km. The red line (630.0 nm) has τ ≈ 110 s, so it is quenched below ~200 km and only appears in high-altitude, soft-electron aurora.
Robinson Conductivity Formulas
The ionization produced by precipitating electrons enhances the ionospheric conductivity. Robinson et al. (1987) provided empirical formulas:
where E0 is the mean energy in keV and ΦE is the energy flux in mW/m². For a bright aurora (E0 = 10 keV, ΦE = 10 mW/m²): ΣP ≈ 11 S, ΣH ≈ 35 S.
3.4 Ionospheric Electrodynamics
The ionosphere is a weakly ionized plasma where the conductivity is anisotropic due to the magnetic field. The generalized Ohm's law in the ionosphere takes the form:
Derivation: Pedersen Conductivity
Step 1. In a magnetized, weakly ionized plasma, the ion equation of motion is:
where νin is the ion-neutral collision frequency.
Step 2. In steady state (dv/dt = 0), for E⊥ perpendicular to B (taking B = Bẑ):
where Ωi = eB/mi is the ion cyclotron frequency.
Step 3. The Pedersen current is the component of J parallel to E⊥. For electrons (νen ≪ Ωe) and ions:
The Pedersen conductivity peaks near 125 km altitude where νin ≈ Ωi. Below this altitude, ions are collision-dominated; above, both species are magnetized and drift together.
Hall Conductivity
The Hall current flows perpendicular to both E and B:
The Hall conductivity peaks near 110 km altitude. It drives the auroral electrojet — an intense east-west current in the E-region ionosphere that produces ground magnetic perturbations.
Joule heating: The dissipation of electromagnetic energy in the ionosphere is:
where E' = E + vn × B is the electric field in the neutral wind frame. During active auroral conditions, the total Joule heating power can reach 100–500 GW, comparable to the total auroral particle precipitation power.
3.5 Auroral Substorm Dynamics
During a magnetospheric substorm, the aurora undergoes dramatic changes:
Growth phase: The auroral oval moves equatorward as open flux accumulates. Quiet arcs in the midnight sector gradually brighten.
Onset: A sudden brightening of the most equatorward arc near magnetic midnight signals substorm onset. This is followed by a rapid poleward expansion (the "auroral breakup").
Expansion: The aurora expands poleward, westward (traveling surge), and eastward (omega bands). The substorm current wedge forms:
Recovery: The aurora retreats poleward and the oval contracts. Diffuse aurora persists as energized ring current particles scatter into the loss cone.
The total power dissipated during an auroral substorm:
partitioned roughly as 30% particle precipitation, 40% Joule heating, and 30% ring current injection.