# MHD Physics of Ground Magnetic Perturbations

Current Systems, Biot-Savart Decomposition, and What Ground Magnetometers Actually See
Bruce Stephenson & Argus · May 2026

# 1. Earth's Magnetic Envelope

Earth's internal dynamo generates a roughly dipolar magnetic field that extends into space. The solar wind — a supersonic plasma flowing at 400–800 km/s from the Sun — compresses this field on the dayside and stretches it into a long tail on the nightside, creating the magnetosphere: a bullet-shaped cavity in the solar wind about 10 Earth radii (R_E ≈ 6371 km) wide on the dayside and extending >100 R_E tailward.

The magnetosphere is not empty space — it's filled with hot, tenuous plasma organized by the magnetic field. The plasma and field are coupled through the MHD equations: the plasma can't cross field lines (except through reconnection), and the magnetic field can't change without moving the plasma (except through resistivity). This coupling is what makes the system both complex and tractable — complex because everything is connected, tractable because the MHD equations provide a complete description.

Figure 1. Schematic magnetosphere (not to scale). The solar wind compresses the dayside and stretches the nightside into a long tail. Key current systems flow at the magnetopause, in the ring current, through the cross-tail, and along field lines connecting to the ionosphere.

# 2. The Current Systems

Electric currents flow everywhere in the magnetosphere-ionosphere system. Each current system has a distinct physical origin, spatial scale, and ground signature. Understanding these is essential for interpreting what ground magnetometers measure.

### Chapman-Ferraro Currents (Magnetopause)

Location: Magnetopause surface, ~10 R_E on dayside

Physical origin: Pressure balance. Solar wind dynamic pressure compresses the geomagnetic field; the boundary condition requires a surface current flowing dawn-to-dusk on the dayside to shield the Earth's field from the solar wind. Named after Chapman and Ferraro who predicted them in 1930.

Ground signature: Increases the horizontal field at the equator (adds to the main field). Magnitude: ~20 nT during quiet, ~50 nT during compression events (sudden commencements). Spatially very smooth — uniform over thousands of km. Low Moran's I contribution because variation between stations is minimal.

Typical δB at ground: 10–50 nT, very smooth

### Ring Current

Location: Equatorial plane, 3–8 R_E, flowing westward

Physical origin: Energetic ions (10–100 keV, primarily H⁺ and O⁺) trapped in the dipolar magnetic field undergo gradient and curvature drift, creating a net westward current. During storms, fresh injection from the plasma sheet intensifies this current.

Ground signature: Depresses the horizontal field globally — this is what the Dst index measures. A Dst of −100 nT means the ring current is producing a 100 nT southward perturbation at the equator. Spatially very smooth because the current is at 3–8 R_E distance: all ground stations see essentially the same perturbation.

Why it matters for Moran's I: The ring current produces a common-mode perturbation — it shifts all stations similarly. Because Moran's I measures deviation from the mean, a uniform shift doesn't contribute to spatial pattern. The ring current tells you how intense the storm is, but it doesn't create spatial structure on the ground.

Typical δB at ground: 50–400+ nT during storms, nearly uniform

### Cross-Tail Current

Location: Plasma sheet, 10–30 R_E tailward, flowing dawn-to-dusk

Physical origin: Pressure gradient in the plasma sheet drives a current sheet that maintains the stretched tail field configuration. During substorms, part of this current is disrupted ("current wedge") and diverted along field lines to the ionosphere.

Ground signature: Small at the ground due to large distance. But during substorms, the current disruption drives the substorm current wedge — see FAC below.

Typical δB at ground: Small directly; large indirectly via FAC

### Field-Aligned Currents (FAC) — Region 1 and Region 2

Location: Along geomagnetic field lines, connecting magnetosphere to ionosphere

Physical origin: Magnetospheric convection (driven by reconnection) maps an electric field into the ionosphere. The resulting potential difference drives currents that close through the ionosphere. Iijima & Potemra (1976) identified the two-ring pattern:

- Region 1 (R1): Higher latitude (~70–75°), flows INTO the ionosphere on the dawnside, OUT on the duskside. Driven by the boundary between open and closed field lines.
- Region 2 (R2): Lower latitude (~65–70°), opposite polarity to R1. Driven by the inner magnetosphere pressure gradient (ring current shielding).

Ground signature: FAC are the primary source of spatially organized ground magnetic perturbations at high latitudes. The R1/R2 system creates a large-scale (~5000 km diameter) pattern of perturbations that is coherent across many stations.

THIS IS THE KEY CURRENT FOR SPATIAL COHERENCE.

Typical δB at ground: 100–1000+ nT during storms, spatially organized

### Ionospheric Hall Currents

Location: Ionospheric E-region, ~100–130 km altitude

Physical origin: When FAC close through the ionosphere, the perpendicular electric field drives horizontal currents. Hall currents flow perpendicular to both E and B — in the ionosphere, this means perpendicular to the applied electric field. The Hall current is the dominant contributor to the auroral electrojets (the intense east-west current bands at auroral latitudes).

Ground signature: Creates strong north-south (N-component) perturbations directly below. But because Hall currents respond to the local electric field and conductance, their spatial pattern is more fragmented than the FAC pattern that drives them. They partially cancel the FAC ground signal.

Typical δB at ground: 200–2000+ nT during substorms, locally intense

### Ionospheric Pedersen Currents

Location: Ionospheric E and F-region, 110–300+ km

Physical origin: Pedersen currents flow parallel to the applied electric field (perpendicular to B) — the "resistive" component. In the ionosphere, they flow in the direction of the convection electric field and close the FAC circuit through the ionosphere.

Ground signature: Relatively weak at the ground because Pedersen currents flow radially (from FAC entry point outward) and tend to cancel when projected to the ground. The main contribution is through Joule heating (J·E), which modifies TEC by changing ionospheric density and composition.

Typical δB at ground: 20–100 nT, spatially diffuse

# 3. Biot-Savart: From Currents to δB

The Biot-Savart law relates a current distribution J(r) to the magnetic field perturbation at any point r₀:

δB(r₀) = (μ₀/4π) ∫ J(r) × (r₀ − r) / |r₀ − r|³ dV

This integral has three key properties that matter for our analysis:

1. Superposition: Each current source contributes independently. The total δB is the sum of contributions from every current in the system. This is what makes the SWMF decomposition possible — you can integrate over just the FAC current density, or just the Hall current density, and get their individual contributions.
2. Distance weighting: The 1/|r|² dependence means nearby currents dominate. A 100 nT perturbation from Hall currents at 110 km altitude is comparable to a 100 nT perturbation from FAC at 100–10,000 km. Hall currents are close; FAC span a huge range of distances.
3. Geometry sensitivity: The cross product J × r̂ means the direction matters. A current flowing east-west produces north-south δB at the ground (and vice versa). A radial current (like the vertical segment of FAC near the ground) produces a circular pattern of horizontal δB.

Why decomposition is non-trivial: The Biot-Savart integral is over all space, so the "FAC contribution" includes not just the FAC themselves but their distant magnetic field projected to the ground. A field-aligned current at 5 R_E altitude contributes to ground δB through this long-range integral. The SWMF evaluates this integral separately for each current source using the full 3D current distribution from the MHD solution.

# 4. The Current Closure Circuit

Currents don't start and end — they form closed circuits (∇·J = 0). The M-I current closure circuit is:

Magnetospheric driving → FAC (along field lines) → Ionospheric Pedersen + Hall → FAC return → Magnetosphere

In more detail:

1. Solar wind reconnection drives magnetospheric convection, creating a dawn-dusk electric field mapped along field lines to the ionosphere.
2. The electric field drives FAC along field lines — downward on one side of the convection cell, upward on the other. Region 1 flows at ~70° latitude; Region 2 at ~65°.
3. In the ionosphere, the FAC must close horizontally. Pedersen currents flow along E, Hall currents flow along E×B. The total horizontal current connects the downward and upward FAC channels.
4. The upward FAC returns to the magnetosphere, completing the circuit.

A ground magnetometer sitting below this circuit sees the superposition of contributions from ALL segments of the circuit — FAC overhead, Hall currents at 110 km, Pedersen currents at 150 km, and the distant magnetospheric currents that drive the whole system. This is why the total ground δB is such a complicated quantity to interpret.

# 5. Why Hall Currents Cancel FAC at the Ground

This is the central puzzle our pipeline illuminates: why does the total ground δB show less spatial coherence than the FAC component alone?

The answer lies in geometry. Consider a single FAC sheet — an infinite line of current flowing vertically downward into the ionosphere at some latitude. By Biot-Savart, this current produces a circular pattern of horizontal δB at the ground, centered on the footpoint. If there are two FAC sheets (R1 and R2, with opposite polarity), they create a large-scale organized pattern with a characteristic scale equal to the R1-R2 separation (~500–1000 km).

Now consider what happens when this FAC closes through the ionosphere. The Hall current flows east-west in the E-region, directly below the FAC footprint. By Biot-Savart, this east-west current produces a north-south δB at the ground. But the Hall current flows in a direction that partially opposes the ground projection of the FAC itself.

The cancellation mechanism: The FAC (at varying altitude) produces a broad, smooth ground δB pattern. The Hall current (at ~110 km, very close to the ground) produces a strong, local δB that opposes part of the FAC signal. Because the Hall current is close, its amplitude is large relative to its spatial extent — it creates local perturbations that fragment the large-scale FAC pattern.

In our SWMF results, the numbers are striking:

- FAC alone: Moran's I ≈ +0.15 (coherent large-scale pattern)
- Hall alone: Moran's I ≈ 0.0 (no large-scale pattern)
- FAC + Hall + others: Moran's I ≈ 0.0 (cancellation!)

The Hall current doesn't just add noise — it specifically destroys the large-scale structure that FAC creates. This is because the Hall current is driven by the FAC closure and therefore flows in a spatial pattern that is anti-correlated with the FAC ground signature at large scales.

# 6. The SWMF Model: What It Computes

The Space Weather Modeling Framework (SWMF) couples multiple physics models:

### GM: Global Magnetosphere (BATS-R-US)

BATS-R-US solves the ideal MHD equations on an adaptive mesh grid covering the magnetosphere (−250 R_E to +32 R_E in GSM coordinates). It computes the full 3D magnetic field, plasma velocity, density, and pressure at each grid point. From the MHD solution, the current density J = ∇×B/μ₀ is computed everywhere in the volume.

### IE: Ionosphere Electrodynamics (Ridley)

The Ridley model solves a 2D Poisson equation for the ionospheric potential Φ on a spherical shell at ~110 km altitude. Inputs: FAC pattern from BATS-R-US (mapped along field lines to the ionosphere), ionospheric conductance model (solar illumination + auroral precipitation). Outputs: electric field E = −∇Φ, Hall and Pedersen current densities.

### Virtual Magnetometers

At specified ground locations, SWMF evaluates the Biot-Savart integral for each current source separately:

Component | Biot-Savart domain | Columns
--- | --- | ---
MHD | All magnetospheric currents (GM volume) | N, E, D
FAC | Field-aligned currents (at the IE boundary) | N, E, D
Hall | Hall currents (IE sheet) | N, E, D
Pedersen | Pedersen currents (IE sheet) | N, E, D
Total | Sum of all four | N, E, D

N = geographic north, E = geographic east, D = vertical (downward positive). Each timestep produces 15 values per station: 5 sources × 3 components.

# 7. Reading the 15-Column Decomposition

When you look at SWMF .mag output, the 15 columns are ordered:

total_N, total_E, total_D, mhd_N, mhd_E, mhd_D, fac_N, fac_E, fac_D, hall_N, hall_E, hall_D, ped_N, ped_E, ped_D

The total should equal the sum of the four components: total = mhd + fac + hall + ped. (In practice, there can be small numerical differences due to interpolation between the GM and IE grids.)

### What to look for

- MHD columns: Usually small compared to FAC and Hall. The magnetospheric currents are far away. Largest during sudden commencements (magnetopause compression) and for low-latitude stations (ring current).
- FAC columns: Large, spatially organized. The N and E components are comparably sized (FAC creates both horizontal components through the Biot-Savart cross product). D is usually smaller.
- Hall columns: Large in N (the east-west electrojet creates north-south δB), smaller in E. Often opposite sign to FAC_N at the same station — this is the cancellation.
- Pedersen columns: Usually small. Pedersen currents flow radially from the FAC footpoint and their ground δB tends to cancel by symmetry.

Caution: the decomposition is model-dependent. The split between FAC, Hall, and Pedersen depends on how the model defines the ionospheric boundary, how conductance is computed, and how the Biot-Savart integral handles the interface between GM and IE. Different models (e.g., GAMERA, LFM, OpenGGCM) will give different decompositions for the same event, though the total should be similar if the models are well-calibrated.

# 8. What the Model Gets Right and Wrong

### What SWMF/BATS-R-US gets right

- Large-scale current systems: The R1/R2 FAC pattern, magnetopause currents, and ring current are well-reproduced. The model captures the basic convection pattern and its dependence on IMF Bz.
- Decomposition insight: Even if individual values are off by 30–50%, the relative importance of FAC vs Hall vs MHD is meaningful. The cancellation pattern is robust.
- Temporal evolution: Storm onset, main phase, and recovery are qualitatively correct. Substorm dynamics are captured (though timing is often wrong).

### What it gets wrong

- Spatial resolution: BATS-R-US uses adaptive mesh but even with refinement, the ionospheric mapping is limited. Small-scale FAC structure (<100 km) is not resolved.
- Ionospheric conductance: The Ridley model uses empirical conductance (solar EUV + Hardy auroral model). Real conductance has much more structure (discrete arcs, diffuse vs discrete precipitation). This affects the Hall/Pedersen current distribution.
- Numerical diffusion: Ideal MHD with numerical resistivity smears out current sheets and reduces sharp gradients. This may reduce the actual FAC spatial structure below what nature produces.
- Missing physics: No kinetic effects, no ring current chemistry (O⁺ injection), no radiation belt dynamics, no induction effects in the ground. Substorm onset timing is notoriously difficult for MHD models.

Bottom line: Trust the decomposition for qualitative insights (which current system dominates spatial structure?), but be cautious about quantitative comparisons with observations. The FAC-Hall cancellation is a robust prediction across models; the exact cancellation ratio is model-dependent.

# 9. Induction Effects: The Ground Matters

One important effect that neither our pipeline nor SWMF currently accounts for is electromagnetic induction in the Earth. Time-varying magnetic fields from ionospheric currents induce electric currents in the conducting ground (oceans, sedimentary basins, ore bodies), which produce their own δB that adds to the measured signal.

### How big is this?

Induction effects can contribute 20–40% of the measured horizontal δB, depending on local ground conductivity. Coastal stations (near conducting ocean) see stronger induction than inland stations. This creates systematic spatial variation that is NOT due to the ionospheric source — it's due to geology.

### Impact on Moran's I

Induction effects could either increase or decrease spatial autocorrelation:

- Increase: If nearby stations have similar ground conductivity (likely in Scandinavia or Canadian Shield), induction amplifies the signal uniformly, adding coherent spatial pattern.
- Decrease: If nearby stations have very different conductivity (one coastal, one inland), induction creates local differences that reduce coherence.

The Z-component is most sensitive to induction (vertical δB from induced currents is large and conductivity-dependent). Our finding that Z has the lowest Moran's I (mean = −0.02) is consistent with geologically-driven local variations masking any ionospheric spatial pattern in the vertical component.

# 10. Connecting MHD to Spatial Coherence

Now we can connect the MHD physics to what our pipeline measures:

Physical Quantity | Creates Spatial Coherence? | Scale | Ground Effect
--- | --- | --- | ---
Ring current | No (common mode) | Global | Uniform δB, subtracted by mean
Chapman-Ferraro | No (very smooth) | Global | Small, uniform
FAC (R1/R2) | Yes — primary source | ~5000 km | Large, organized pattern
Hall currents | No (local, cancels FAC) | ~1000 km | Fragments FAC pattern
Pedersen | No (weak, symmetric) | ~500 km | Small contribution
Substorm current wedge | Yes (during substorms) | ~2000 km | Transient organized pattern

The ground magnetometer sees the sum of all contributions. The ring current adds a large common-mode offset (this is Dst). FAC creates the spatial pattern. Hall currents partially cancel it. The result is a much weaker spatial coherence signal than the driving FAC structure would predict.

The fundamental insight: Ground magnetometers are excellent amplitude detectors (Dst captures storm intensity with high fidelity) but poor spatial structure detectors because the closure currents that enable FAC to flow also destroy the FAC's ground signature through cancellation. Understanding this requires the model decomposition — which is exactly what SWMF provides and our pipeline analyzes.

## Implications for Observation Strategy

Given this physics, how should we optimize spatial coherence detection?

1. Use the E-component preferentially: It's less contaminated by Hall currents (Finding 7).
2. Focus on auroral latitudes: Spatial structure is concentrated at 60–90° (Finding 8).
3. Require N ≥ 25 stations: Below this, sampling noise dominates (Finding 3).
4. Use 2–4 hour time windows: Coherence oscillates on this timescale, so snapshot analysis may miss it (Finding 9).
5. Consider AMPERE (direct FAC observation): Bypass the cancellation problem entirely by measuring FAC directly from low-orbit magnetometers.

MHD physics tutorial · May 2026 · See also: Baseline Pipeline Tutorial, Extended Analysis Tutorial