Part IV: Space Weather | Chapter 15

Space Weather

Geomagnetic storms, Dst index, substorms, GIC hazards, and forecasting

15.1 Geomagnetic Storms

Derivation 1: The Dst Index and Ring Current Energy

The Dst (disturbance storm time) index measures the globally averaged depression of the horizontal magnetic field component at low-latitude stations, caused primarily by the symmetric ring current.

Step 1. The magnetic field perturbation from a ring current $I$ at distance $r$:

$$\Delta B = -\frac{\mu_0 I}{2r}$$

Step 2. The Dessler-Parker-Sckopke relation connects Dst to the total energy $E_R$ of the ring current:

$$\boxed{Dst = -\frac{\mu_0}{4\pi}\frac{2E_R}{B_E R_E^3}}$$

where $B_E \approx 3.1 \times 10^{-5}$ T is the equatorial surface field. A Dst of -100 nT corresponds to $E_R \approx 4 \times 10^{15}$ J.

Storm classification: minor ($-50 < Dst < -30$ nT), moderate ($-100 < Dst < -50$), intense ($-250 < Dst < -100$), super-storm ($Dst < -250$ nT). The Carrington event (1859) had estimated Dst ~ -850 nT.

15.2 Substorms

Derivation 2: Substorm Current Wedge

Step 1. During the growth phase, dayside reconnection loads magnetic flux into the magnetotail. The cross-tail current $I_T$ intensifies as the tail becomes more stretched.

Step 2. At substorm onset, reconnection in the tail disrupts the cross-tail current, which is diverted through the ionosphere forming the substorm current wedge:

$$\boxed{\Delta B_H = -\frac{\mu_0 I_{\text{FAC}}}{2\pi r}\sin\phi \quad \text{(field-aligned current contribution)}}$$

Substorms release $\sim 10^{14}\text{--}10^{15}$ J of energy over ~1 hour. The AL (auroral lower) index measures the westward electrojet strength, typically reaching -500 to -2000 nT during substorms. The substorm cycle operates independently of storm-time activity but intensifies during storms.

15.3 Geomagnetically Induced Currents (GIC)

Derivation 3: GIC from Geomagnetic Field Variations

Step 1. Rapid variations in the geomagnetic field $dB/dt$ induce an electric field at the Earth's surface via Faraday's law:

$$E = -\frac{dB}{dt} \cdot Z(f) \quad \text{[V/m]}$$

where $Z(f)$ is the Earth's surface impedance depending on conductivity structure.

Step 2. The GIC in a power line of length $L$ and resistance $R$:

$$\boxed{I_{\text{GIC}} = \frac{E \cdot L}{R} \propto \frac{dB}{dt} \cdot L}$$

During extreme storms, $dB/dt$ can reach 2000 nT/min, producing GICs of hundreds of amperes. The 1989 Quebec blackout was caused by GICs that saturated transformers. A Carrington-class event could cause trillions of dollars in damage to power infrastructure globally.

15.4 Geomagnetic Indices

Derivation 4: Kp and Ap Index Construction

Step 1. The Kp index is a quasi-logarithmic measure of the 3-hourly range of geomagnetic disturbance, averaged over 13 subauroral stations. It ranges from 0 to 9:

$$\boxed{a_p = 10^{(K-1)/2} \text{ (nT, approximate linearization)}}$$

Step 2. The daily Ap is the average of 8 three-hourly $a_p$ values. Storm levels: Kp = 5 (G1, minor), Kp = 7 (G3, strong), Kp = 9 (G5, extreme).

Other Space Weather Indices

• AE: Auroral electrojet index (substorm monitor)
• Dst/SYM-H: Ring current (storm monitor, 1-min resolution)
• F10.7: 10.7 cm radio flux (solar EUV proxy)
• Bz (GSM): IMF south component (geoeffectiveness predictor)

15.5 Space Weather Forecasting

Derivation 5: Burton Equation for Dst Prediction

Step 1. The Burton et al. (1975) empirical model for Dst evolution:

$$\boxed{\frac{dDst^*}{dt} = Q(t) - \frac{Dst^*}{\tau}}$$

where $Dst^* = Dst - b\sqrt{P_{\text{dyn}}} + c$ is the pressure-corrected Dst,$\tau \approx 7.7$ hours is the ring current decay time, and$Q$ is the injection rate, proportional to the dawn-dusk electric field:

$$Q = d(E_y - 0.5) \text{ nT/hr for } E_y > 0.5 \text{ mV/m}, \quad E_y = -v_{\text{sw}} B_z$$

Modern operational forecasting uses ensemble ENLIL WSA models for CME propagation, machine learning for Dst/Kp prediction, and real-time solar wind data from L1 monitors (ACE, DSCOVR) providing ~30-60 min warning before CME impact.

Numerical Simulation

Space Weather: Storm Simulation, GIC Risk, Storm Phases, Solar Cycle Dependence

Python

script.py135 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.suptitle('Space Weather', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Burton equation Dst simulation ---
ax1 = axes[0, 0]
dt = 0.1  # hours
t = np.arange(0, 120, dt)
Dst = np.zeros_like(t)
tau = 7.7  # hours

# Solar wind driver: Bz southward during storm
Bz = np.where((t > 10) & (t < 30), -15 * np.sin(np.pi * (t - 10) / 20), 0)
Bz += np.where((t > 25) & (t < 50), -25 * np.exp(-(t - 35)**2 / 50), 0)
v_sw = 500  # km/s
Ey = -v_sw * Bz * 1e-3  # mV/m

for i in range(1, len(t)):
    Q = 0
    if Ey[i] > 0.5:
        Q = -4.4 * (Ey[i] - 0.5)
    Dst[i] = Dst[i-1] + (Q - Dst[i-1] / tau) * dt

ax1.plot(t, Dst, color='#f43f5e', linewidth=2.5, label='Dst [nT]')
ax1_twin = ax1.twinx()
ax1_twin.plot(t, Bz, color='#60a5fa', linewidth=1.5, alpha=0.7, label='Bz [nT]')
ax1.set_xlabel('Time [hours]')
ax1.set_ylabel('Dst [nT]', color='#f43f5e')
ax1_twin.set_ylabel('IMF Bz [nT]', color='#60a5fa')
ax1.set_title('Burton Equation: Storm Simulation')
ax1.tick_params(axis='y', labelcolor='#f43f5e')
ax1_twin.tick_params(axis='y', labelcolor='#60a5fa')

# --- Panel 2: GIC risk map (dB/dt vs latitude) ---
ax2 = axes[0, 1]
lat = np.linspace(30, 80, 200)
Kp_vals = [3, 5, 7, 9]
colors_kp = ['#4ade80', '#fbbf24', '#fb923c', '#f43f5e']

for Kp, col in zip(Kp_vals, colors_kp):
    # dB/dt increases with latitude and Kp
    dBdt = 5 * Kp**1.5 * np.exp(-((lat - 65 - Kp)/10)**2)
    ax2.plot(lat, dBdt, color=col, linewidth=2.5, label='Kp=' + str(Kp))

ax2.set_xlabel('Geomagnetic latitude [deg]')
ax2.set_ylabel('dB/dt [nT/min]')
ax2.set_title('GIC Risk: dB/dt vs Latitude')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Storm phases ---
ax3 = axes[1, 0]
t_storm = np.linspace(0, 96, 500)

# Dst profile with phases
Dst_storm = np.where(t_storm < 6, 20 * np.sin(np.pi * t_storm / 12),  # SSC
            np.where(t_storm < 12, 20 - 120 * (t_storm - 6) / 6,  # main phase
            np.where(t_storm < 24, -100 * np.exp(-(t_storm - 12) / 5) - 20,  # continued
            -20 * np.exp(-(t_storm - 24) / 15))))  # recovery

ax3.plot(t_storm, Dst_storm, color='#f43f5e', linewidth=2.5)
ax3.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax3.axhline(y=-50, color='#fbbf24', linestyle='--', alpha=0.3, label='Moderate storm')
ax3.axhline(y=-100, color='#fb923c', linestyle='--', alpha=0.3, label='Intense storm')

ax3.axvspan(0, 6, alpha=0.1, color='green')
ax3.axvspan(6, 24, alpha=0.1, color='red')
ax3.axvspan(24, 96, alpha=0.1, color='blue')
ax3.text(3, 25, 'SSC', fontsize=9, color='#4ade80', ha='center')
ax3.text(15, 25, 'Main Phase', fontsize=9, color='#f43f5e', ha='center')
ax3.text(60, 25, 'Recovery', fontsize=9, color='#60a5fa', ha='center')

ax3.set_xlabel('Time [hours]')
ax3.set_ylabel('Dst [nT]')
ax3.set_title('Geomagnetic Storm Phases')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Kp distribution over solar cycle ---
ax4 = axes[1, 1]
years = np.linspace(0, 22, 200)
# Solar activity proxy
SSN = 100 * np.sin(np.pi * years / 11)**2 * (1 + 0.3 * np.sin(years * 0.5))

# Kp occurrence rates scale with activity
Kp_mean = 2 + 2 * SSN / 100
Kp_high = np.clip(Kp_mean * 1.5, 0, 9)

ax4.fill_between(years, Kp_mean - 1, Kp_mean + 1, alpha=0.2, color='#f43f5e')
ax4.plot(years, Kp_mean, color='#f43f5e', linewidth=2.5, label='Mean Kp')
ax4_twin = ax4.twinx()
ax4_twin.plot(years, SSN, color='#fbbf24', linewidth=1.5, alpha=0.7, label='SSN')
ax4.set_xlabel('Year in magnetic cycle')
ax4.set_ylabel('Mean Kp', color='#f43f5e')
ax4_twin.set_ylabel('Sunspot Number', color='#fbbf24')
ax4.set_title('Geomagnetic Activity over Solar Cycle')
ax4.tick_params(axis='y', labelcolor='#f43f5e')
ax4_twin.tick_params(axis='y', labelcolor='#fbbf24')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("SPACE WEATHER")
print("=" * 60)
print("\n--- Geomagnetic Storms ---")
print("  Minor (G1): Kp=5, Dst > -50 nT")
print("  Strong (G3): Kp=7, Dst ~ -100 to -200 nT")
print("  Extreme (G5): Kp=9, Dst < -250 nT")
print("  Carrington (1859): Dst ~ -850 nT (estimated)")
print("\n--- GIC Hazards ---")
print("  Quebec 1989: 9-hour blackout, 6 million affected")
print("  Carrington-class: estimated $1-2 trillion damage")
print("  Critical dB/dt threshold: ~100-500 nT/min")
print("\n--- Forecasting Lead Times ---")
print("  Flare: minutes (X-ray, speed of light)")
print("  SEP: 10-60 minutes (relativistic particles)")
print("  CME: 15 hours - 5 days (drag model)")
print("  L1 warning: 30-60 minutes (ACE/DSCOVR)")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

15.6 Dst Index: Dessler-Parker-Sckopke Relation

Deriving $Dst = -\mu_0 E_{RC}/(2\pi R_E^3 B_0)$

Step 1. The ring current consists of energetic ions (10-200 keV, mainly H+ and O+) drifting westward around Earth at $L \sim 3\text{--}7$. The magnetic perturbation at Earth's center from a current loop of radius $r$ carrying current $I$:

$$\Delta B = -\frac{\mu_0 I}{2r}$$

Step 2. The total energy stored in the ring current is related to the particle kinetic energy. For particles gradient-curvature drifting in a dipole field, the drift current is:

$$I_{\text{drift}} = -\frac{3E_\perp}{2\pi r B} = -\frac{E_\perp}{qBr^2}\cdot r$$

Step 3. Integrating over all particles and using the virial theorem for a dipole field, Dessler and Parker (1959) and Sckopke (1966) showed:

$$\boxed{Dst = -\frac{2\mu_0 E_{RC}}{4\pi R_E^3 B_E} = -\frac{\mu_0}{2\pi}\frac{E_{RC}}{R_E^3 B_E}}$$

Step 4. Numerically, with $B_E = 3.1\times10^{-5}$ T and $R_E = 6.371\times10^6$ m:

$$Dst \approx -3.9\times10^{-20}\,E_{RC} \;\text{[nT, with } E_{RC} \text{ in Joules]}$$

A storm with Dst = -100 nT corresponds to $E_{RC} \approx 2.5\times10^{15}$ J ($\sim 10^{22}$ erg). The Carrington event (Dst ~ -850 nT) had$E_{RC} \sim 2\times10^{16}$ J of ring current energy. This energy is ultimately supplied by the solar wind via the Dungey cycle convection electric field.

15.7 GIC from Faraday's Law

From $dB/dt$ to Induced Current in Power Grids

Step 1. Faraday's law in integral form for a horizontal loop at Earth's surface:

$$\oint\mathbf{E}\cdot d\boldsymbol{\ell} = -\frac{d}{dt}\int\mathbf{B}\cdot d\mathbf{A} \approx -\frac{dB_z}{dt}\cdot A$$

Step 2. The geoelectric field at the surface depends on the time derivative of the geomagnetic field and the Earth's conductivity structure. For a uniform half-space with conductivity$\sigma$:

$$E_x = -\frac{1}{\sqrt{\pi\mu_0\sigma}}\int_{-\infty}^{t}\frac{1}{\sqrt{t-t'}}\frac{dB_y}{dt'}\,dt'$$

Step 3. For a sinusoidal variation at frequency $\omega$, this simplifies to:

$$|E| = \frac{|\omega B|}{\sqrt{\mu_0\sigma\omega}} = |B|\sqrt{\frac{\omega}{\mu_0\sigma}}$$

Step 4. The GIC in a grounded power transmission line of length $L$ between two grounding points with total resistance $R$:

$$\boxed{I_{\text{GIC}} = \frac{V_{\text{induced}}}{R} = \frac{E\cdot L}{R}}$$

For $dB/dt = 2000$ nT/min (extreme storm) over resistive ground ($\sigma = 10^{-4}$ S/m) at period 5 min: $E \approx 10$ V/km. A 100 km transmission line with 1 $\Omega$resistance would carry $I_{\text{GIC}} = 1000$ A, which can saturate and damage transformers. The 1989 Quebec blackout was caused by GICs of just 100-200 A.

15.8 Kp Index and Auroral Oval Expansion

Auroral Boundary vs Geomagnetic Activity

Step 1. The equatorward boundary of the auroral oval is determined by the last closed field line, which maps to the outer boundary of the magnetosphere. As Kp increases, the magnetopause is compressed and the tail stretches, moving the auroral oval equatorward.

Step 2. The empirical relationship between Kp and the equatorward boundary of the auroral oval (Feldstein and Starkov):

$$\boxed{\Lambda_{\text{eq}} \approx 67^\circ - 2^\circ \times Kp \quad \text{(magnetic latitude)}}$$

Step 3. For extreme storms: Kp = 9 gives $\Lambda_{\text{eq}} \approx 49^\circ$, meaning aurora visible from Paris, Chicago, or Tokyo. During the Carrington event, aurora was reportedly seen from the Caribbean (~20 degrees latitude).

NOAA G-Scale for Geomagnetic Storms

• G1 (Minor): Kp=5, aurora at 60 degrees, weak power fluctuations
• G2 (Moderate): Kp=6, aurora at 55 degrees, transformer damage possible
• G3 (Strong): Kp=7, aurora at 50 degrees, voltage corrections needed
• G4 (Severe): Kp=8, aurora at 45 degrees, widespread GIC problems
• G5 (Extreme): Kp=9, aurora at 40 degrees, grid collapse possible

15.9 Space Weather Chain: Sun to Earth

The complete space weather chain from solar eruption to terrestrial impact.

The space weather chain: Solar flare produces X-rays (8 min) and SEPs (minutes-hours). CME propagates 1-5 days, drives interplanetary shock. At Earth, magnetospheric reconnection drives auroral substorms, ring current buildup (Dst), and GICs in power infrastructure.

Extended Simulation: Dst Storm Profile & Auroral Oval

Extended: Multi-Storm Dst, Auroral Oval Expansion, DPS Relation, GIC Risk

Python

script.py179 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.suptitle('Dst Storm Profile and Auroral Oval Dynamics', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

# --- Panel 1: Multi-storm Dst simulation ---
ax1 = axes[0, 0]
dt = 0.1
t = np.arange(0, 200, dt)
Dst = np.zeros_like(t)
tau = 7.7  # hours

# Complex storm driver: multiple southward Bz intervals
Bz = np.zeros_like(t)
# First storm (moderate)
Bz += np.where((t > 10) & (t < 30), -10 * np.sin(np.pi * (t - 10) / 20), 0)
# Second storm (intense, after partial recovery)
Bz += np.where((t > 60) & (t < 80), -25 * np.sin(np.pi * (t - 60) / 20)**2, 0)
# Third (super-storm)
Bz += np.where((t > 100) & (t < 130),
               -40 * np.sin(np.pi * (t - 100) / 30) * np.exp(-(t - 115)**2 / 200), 0)

v_sw = 600  # km/s (fast solar wind)
Ey = -v_sw * Bz * 1e-3  # mV/m

for i in range(1, len(t)):
    Q = 0
    if Ey[i] > 0.5:
        Q = -4.4 * (Ey[i] - 0.5)
    Dst[i] = Dst[i-1] + (Q - Dst[i-1] / tau) * dt

ax1.plot(t, Dst, color='#f43f5e', linewidth=2)
ax1.fill_between(t, Dst, 0, where=(Dst < 0), alpha=0.15, color='#f43f5e')

# Storm classification bands
ax1.axhline(y=-30, color='#4ade80', linestyle=':', alpha=0.3)
ax1.axhline(y=-50, color='#fbbf24', linestyle=':', alpha=0.3)
ax1.axhline(y=-100, color='#fb923c', linestyle=':', alpha=0.3)
ax1.axhline(y=-250, color='#f43f5e', linestyle=':', alpha=0.3)

ax1.text(195, -25, 'Minor', fontsize=8, color='#4ade80', ha='right')
ax1.text(195, -45, 'Moderate', fontsize=8, color='#fbbf24', ha='right')
ax1.text(195, -95, 'Intense', fontsize=8, color='#fb923c', ha='right')
ax1.text(195, -200, 'Super', fontsize=8, color='#f43f5e', ha='right')

ax1_twin = ax1.twinx()
ax1_twin.plot(t, Bz, color='#60a5fa', linewidth=1, alpha=0.5)
ax1.set_xlabel('Time [hours]')
ax1.set_ylabel('Dst [nT]', color='#f43f5e')
ax1_twin.set_ylabel('IMF Bz [nT]', color='#60a5fa')
ax1.set_title('Multiple Storm Dst Simulation')
ax1.tick_params(axis='y', labelcolor='#f43f5e')
ax1_twin.tick_params(axis='y', labelcolor='#60a5fa')

# --- Panel 2: Auroral oval expansion with Kp ---
ax2 = axes[0, 1]
# Polar projection
theta = np.linspace(0, 2 * np.pi, 100)

for Kp, col, ls in [(1, '#4ade80', '--'), (3, '#fbbf24', '--'),
                     (5, '#fb923c', '-'), (7, '#f43f5e', '-'), (9, '#a78bfa', '-')]:
    # Equatorward boundary
    lat_eq = 67 - 2 * Kp
    r_eq = 90 - lat_eq

# Poleward boundary
    lat_pole = 78 - 1 * Kp
    r_pole = 90 - lat_pole

ax2.fill_between(theta, r_pole, r_eq, alpha=0.1, color=col)
    ax2.plot(theta, np.full_like(theta, r_eq), color=col, linewidth=2, linestyle=ls,
            label='Kp=' + str(Kp) + ' (lat=' + str(lat_eq) + 'deg)')

ax2 = plt.subplot(222, polar=True)
ax2.set_facecolor('#0a0a0a')
ax2.tick_params(colors='white')
ax2.set_title('Auroral Oval vs Kp', color='white', pad=15)

for Kp, col in [(1, '#4ade80'), (3, '#fbbf24'), (5, '#fb923c'), (7, '#f43f5e'), (9, '#a78bfa')]:
    lat_eq = 67 - 2 * Kp
    r_eq = 90 - lat_eq
    lat_pole = 78 - 1 * Kp
    r_pole = 90 - lat_pole
    ax2.fill_between(theta, r_pole, r_eq, alpha=0.15, color=col)
    ax2.plot(theta, np.full_like(theta, r_eq), color=col, linewidth=1.5,
            label='Kp=' + str(Kp))

ax2.set_ylim(0, 50)
ax2.set_yticks([10, 20, 30, 40, 50])
ax2.set_yticklabels(['80', '70', '60', '50', '40'], color='white', fontsize=8)
ax2.legend(fontsize=7, loc='lower left', facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Ring current energy vs Dst ---
ax3 = axes[1, 0]
Dst_range = np.linspace(-500, 0, 200)
B_E = 3.1e-5
R_E = 6.371e6
mu_0 = 4 * np.pi * 1e-7

# DPS relation: Dst = -mu_0 * E_RC / (2*pi*R_E^3*B_E)
E_RC = -Dst_range * 1e-9 * 2 * np.pi * R_E**3 * B_E / mu_0  # Joules

ax3.plot(Dst_range, E_RC / 1e15, color='#f43f5e', linewidth=2.5)
ax3.set_xlabel('Dst [nT]')
ax3.set_ylabel('Ring Current Energy [PJ]')
ax3.set_title('Dessler-Parker-Sckopke Relation')

# Mark notable storms
storms = [(-100, 'Moderate storm'), (-250, 'Halloween 2003'), (-589, 'March 1989'),
          (-850, 'Carrington 1859')]
for dst_val, name in storms:
    E_val = -dst_val * 1e-9 * 2 * np.pi * R_E**3 * B_E / mu_0 / 1e15
    if dst_val >= -500:
        ax3.plot(dst_val, E_val, 'o', color='#fbbf24', markersize=8)
        ax3.annotate(name, xy=(dst_val, E_val), xytext=(dst_val + 30, E_val + 2),
                    fontsize=8, color='#fbbf24', arrowprops=dict(arrowstyle='->', color='#fbbf24', lw=0.8))

# --- Panel 4: GIC risk: dB/dt vs time during storm ---
ax4 = axes[1, 1]
t_storm = np.linspace(0, 120, 2000)

# Simulate dB/dt from Dst derivative + substorm spikes
dDst_dt = np.gradient(Dst[:len(t_storm)], dt)
# Add substorm-like spikes
np.random.seed(42)
n_substorms = 8
for _ in range(n_substorms):
    t_sub = np.random.uniform(15, 110)
    amp_sub = np.random.uniform(200, 2000)
    width_sub = np.random.uniform(0.5, 2)
    dDst_dt += amp_sub * np.exp(-((t_storm - t_sub) / width_sub)**2)

dBdt = np.abs(dDst_dt) * 5  # Scale to nT/min approximately

ax4.plot(t_storm, dBdt, color='#f43f5e', linewidth=1)
ax4.fill_between(t_storm, 0, dBdt, alpha=0.2, color='#f43f5e')
ax4.axhline(y=100, color='#fbbf24', linestyle='--', alpha=0.5, label='GIC threshold (~100 nT/min)')
ax4.axhline(y=500, color='#fb923c', linestyle='--', alpha=0.5, label='Extreme GIC (500 nT/min)')
ax4.set_xlabel('Time [hours]')
ax4.set_ylabel('|dB/dt| [nT/min]')
ax4.set_title('Geomagnetic Rate of Change During Storm')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax4.set_ylim(0, min(max(dBdt) * 1.2, 3000))

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("EXTENDED: Dst STORM AND AURORAL OVAL")
print("=" * 60)
print("\n--- Dessler-Parker-Sckopke ---")
print("  Dst = -mu_0 * E_RC / (2*pi*R_E^3*B_E)")
print("  Dst = -100 nT -> E_RC ~ 2.5 PJ")
print("  Carrington (Dst ~ -850 nT) -> E_RC ~ 22 PJ")
print("\n--- Auroral Oval ---")
print("  Equatorward boundary: Lambda ~ 67 - 2*Kp degrees")
print("  Kp=5 (G1): aurora at 57 deg")
print("  Kp=9 (G5): aurora at 49 deg")
print("\n--- GIC Thresholds ---")
print("  100 nT/min: operational concern")
print("  500 nT/min: transformer saturation")
print("  2000 nT/min: extreme (Carrington-class)")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Solar Energetic Particles Next: Sun-Earth Connection →

Space Weather

15.1 Geomagnetic Storms

Derivation 1: The Dst Index and Ring Current Energy

15.2 Substorms

Derivation 2: Substorm Current Wedge

15.3 Geomagnetically Induced Currents (GIC)

Derivation 3: GIC from Geomagnetic Field Variations

15.4 Geomagnetic Indices

Derivation 4: Kp and Ap Index Construction

Other Space Weather Indices

15.5 Space Weather Forecasting

Derivation 5: Burton Equation for Dst Prediction

Numerical Simulation

Space Weather: Storm Simulation, GIC Risk, Storm Phases, Solar Cycle Dependence

15.6 Dst Index: Dessler-Parker-Sckopke Relation

Deriving \(Dst = -\mu_0 E_{RC}/(2\pi R_E^3 B_0)\)

15.7 GIC from Faraday's Law

From \(dB/dt\) to Induced Current in Power Grids

15.8 Kp Index and Auroral Oval Expansion

Auroral Boundary vs Geomagnetic Activity

NOAA G-Scale for Geomagnetic Storms

15.9 Space Weather Chain: Sun to Earth

Extended Simulation: Dst Storm Profile & Auroral Oval

Extended: Multi-Storm Dst, Auroral Oval Expansion, DPS Relation, GIC Risk