Part III: Solar Magnetic Activity | Chapter 11

Sunspots & Active Regions

Sunspot structure, Evershed flow, Joy's law, Hale's law, and the solar cycle

11.1 Sunspot Structure

Derivation 1: Umbral-Penumbral Magnetic Field Geometry

A mature sunspot consists of a dark umbra (T ~ 3800 K) surrounded by a lighter penumbra (T ~ 5200 K). The magnetic field is vertical in the umbra and becomes increasingly inclined in the penumbra.

Step 1. The magnetohydrostatic force balance:

$$-\nabla P + \rho\mathbf{g} + \frac{1}{\mu_0}(\nabla \times \mathbf{B}) \times \mathbf{B} = 0$$

Step 2. The Lorentz force can be decomposed into magnetic pressure and tension:

$$\frac{(\nabla \times \mathbf{B}) \times \mathbf{B}}{\mu_0} = -\nabla\left(\frac{B^2}{2\mu_0}\right) + \frac{(\mathbf{B} \cdot \nabla)\mathbf{B}}{\mu_0}$$

Step 3. In a simple thin flux tube model, lateral pressure balance gives:

$$\boxed{P_{\text{ext}}(z) = P_{\text{int}}(z) + \frac{B^2(z)}{2\mu_0}}$$

Since $P_{\text{int}} < P_{\text{ext}}$, the internal temperature must be lower at the same geometric height. The Wilson depression (~400 km) is the geometric sinking of the sunspot photosphere due to reduced gas pressure. Umbral dots (bright features ~300 km across) indicate residual convection even in strong fields.

11.2 Evershed Flow

Derivation 2: Siphon Flow Model

The Evershed effect is a systematic radial outflow of ~2 km/s in the penumbra, discovered by Evershed in 1909. The siphon flow model explains it as a pressure-driven flow along arched flux tubes.

Step 1. Along a thin magnetic flux tube, Bernoulli's equation gives:

$$\frac{v^2}{2} + \frac{\gamma}{\gamma - 1}\frac{P}{\rho} + gz + \frac{B^2}{2\mu_0\rho} = \text{const}$$

Step 2. If the external pressure differs between the two footpoints (different field strengths), there is a pressure difference driving the flow:

$$\boxed{v \approx \sqrt{\frac{2\Delta P}{\rho}} = \sqrt{\frac{B_1^2 - B_2^2}{\mu_0\rho}} \approx 1\text{--}3 \text{ km/s}}$$

The inverse Evershed flow (inflow) observed in the chromosphere above sunspots may represent the return flow in a different atmospheric layer.

11.3 Joy's Law and Hale's Law

Derivation 3: Tilt Angle from Coriolis Force

Joy's law states that bipolar active regions are tilted with respect to the east-west direction, with the leading polarity closer to the equator. The tilt angle increases with latitude.

Step 1. As a rising flux tube (omega loop) expands in the convection zone, the Coriolis force acts on the diverging flows:

$$\mathbf{f}_{\text{Cor}} = -2\boldsymbol{\Omega} \times \mathbf{v}$$

Step 2. The component of the Coriolis force perpendicular to the tube axis produces a twist proportional to the sine of the latitude:

$$\boxed{\gamma_{\text{tilt}} \approx \frac{2\Omega \tau_{\text{rise}} d}{a} \sin\lambda \approx (0.5\text{--}0.7) \sin\lambda}$$

where $\tau_{\text{rise}}$ is the rise time, $d$ is the footpoint separation, and $a$ is the initial tube radius. Observations give$\gamma \approx 0.5\sin\lambda$ (about 5-7 degrees at typical active latitudes).

Hale's polarity law (1919): The leading polarity (closer to the equator) in each hemisphere has the same magnetic sign throughout a given cycle, and the pattern reverses every 11 years—hence the full magnetic cycle is 22 years.

11.4 The Solar Cycle and Butterfly Diagram

Derivation 4: Sporer's Law of Equatorward Migration

Step 1. At cycle onset, sunspots appear at latitudes ~30-35 degrees. As the cycle progresses, the emergence latitude migrates equatorward (Sporer's law). This creates the famous butterfly diagram when plotting sunspot latitude versus time.

Step 2. The migration can be understood from the Parker-Yoshimura dynamo wave propagation:

$$\boxed{\mathbf{v}_{\text{wave}} = -\hat{r} \times \nabla(\alpha \nabla\Omega)}$$

Equatorward propagation requires $\alpha \partial\Omega/\partial r < 0$ in the northern hemisphere, which is satisfied when the $\alpha$-effect is positive in the north and the radial shear is negative (as at the base of the convection zone).

11.5 Solar Cycle Statistics

Derivation 5: Waldmeier Effect

Step 1. The Waldmeier effect is an anticorrelation between the rise time and amplitude of solar cycles: stronger cycles rise faster.

$$\tau_{\text{rise}} \propto R_{\max}^{-\beta}, \quad \beta \approx 0.4\text{--}0.5$$

Step 2. This can be understood in BL dynamo models: a stronger poloidal seed field produces more toroidal flux, which emerges as more sunspots, and the nonlinear quenching saturates the growth faster.

Solar Cycle Statistics

• Average period: 11.04 years (activity cycle), 22 years (magnetic cycle)
• Rise time: 3-5 years; decline time: 6-8 years
• Sunspot number range: 50 (weak) to 250 (strong)
• Grand minima (Maunder minimum, 1645-1715): very few sunspots for ~70 years

Numerical Simulation

Sunspots: Butterfly Diagram, Solar Cycle, Joy's Law, Magnetic Structure

Python

script.py137 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('Sunspots and the Solar Cycle', 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: Synthetic butterfly diagram ---
ax1 = axes[0, 0]
np.random.seed(42)
n_cycles = 4
t_all = []
lat_all = []
for cyc in range(n_cycles):
    t_start = cyc * 11
    n_spots = 3000
    t_spots = t_start + np.random.exponential(4, n_spots)
    t_spots = t_spots[t_spots < t_start + 11]
    phase = (t_spots - t_start) / 11.0
    lat_max = 35 - 25 * phase
    lat_spots = lat_max * (0.5 + 0.5 * np.random.randn(len(t_spots)) * 0.3)
    lat_spots = np.clip(lat_spots, 0, 40)
    sign = np.random.choice([-1, 1], len(t_spots))
    t_all.extend(t_spots)
    lat_all.extend(sign * lat_spots)

ax1.scatter(t_all, lat_all, s=0.5, c='#f43f5e', alpha=0.3)
ax1.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax1.set_xlabel('Time [years]')
ax1.set_ylabel('Latitude [deg]')
ax1.set_title('Synthetic Butterfly Diagram')
ax1.set_ylim(-45, 45)

# --- Panel 2: Sunspot number time series ---
ax2 = axes[0, 1]
t = np.linspace(0, 44, 1000)
# Asymmetric cycle shape (faster rise, slower decline)
SSN = np.zeros_like(t)
for cyc in range(4):
    t_start = cyc * 11
    phase = np.clip((t - t_start) / 11.0, 0, 1)
    amplitude = 80 + 60 * np.sin(cyc * 1.5)
    # Asymmetric profile
    rise = np.where(phase < 0.35, amplitude * np.sin(np.pi * phase / 0.7)**2, 0)
    decline = np.where((phase >= 0.35) & (phase <= 1.0),
                       amplitude * np.exp(-3 * (phase - 0.35)), 0)
    SSN += rise + decline

SSN += 5 * np.random.randn(len(t))
SSN = np.clip(SSN, 0, None)

ax2.plot(t, SSN, color='#f43f5e', linewidth=1.5)
ax2.fill_between(t, 0, SSN, alpha=0.2, color='#f43f5e')
ax2.set_xlabel('Time [years]')
ax2.set_ylabel('Sunspot Number')
ax2.set_title('Synthetic Sunspot Number')

# --- Panel 3: Joy's law - tilt angle vs latitude ---
ax3 = axes[1, 0]
lat_joy = np.linspace(0, 40, 200)
tilt_theory = 0.5 * np.sin(np.radians(lat_joy)) * 180 / np.pi

# Simulated scatter
np.random.seed(123)
lat_data = np.random.uniform(2, 38, 200)
tilt_data = 0.5 * np.sin(np.radians(lat_data)) * 180 / np.pi + np.random.normal(0, 5, 200)

ax3.scatter(lat_data, tilt_data, s=10, c='#fb923c', alpha=0.4, label='Simulated data')
ax3.plot(lat_joy, tilt_theory, color='#f43f5e', linewidth=2.5, label='Joy law: gamma ~ 0.5 sin(lambda)')
ax3.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax3.set_xlabel('Latitude [deg]')
ax3.set_ylabel('Tilt angle [deg]')
ax3.set_title("Joy's Law")
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Sunspot magnetic structure ---
ax4 = axes[1, 1]
r_spot = np.linspace(0, 1.5, 200)  # normalized to penumbral radius

# Field strength profile
B_umbra = 0.3  # T
B_profile = B_umbra * np.exp(-r_spot**2 / 0.8)
B_profile = np.where(r_spot < 0.5, B_umbra, B_profile)

# Inclination
incl = np.where(r_spot < 0.5, 0, 70 * (r_spot - 0.5) / 0.5)
incl = np.clip(incl, 0, 80)

ax4_twin = ax4.twinx()
ax4.plot(r_spot, B_profile * 1000, color='#f43f5e', linewidth=2.5, label='|B| [mT]')
ax4_twin.plot(r_spot, incl, color='#60a5fa', linewidth=2.5, label='Inclination [deg]')

ax4.axvspan(0, 0.5, alpha=0.1, color='gray', label='Umbra')
ax4.axvspan(0.5, 1.0, alpha=0.05, color='orange')
ax4.axvline(x=0.5, color='white', linestyle=':', alpha=0.3)
ax4.axvline(x=1.0, color='white', linestyle=':', alpha=0.3)
ax4.set_xlabel('r / r_penumbra')
ax4.set_ylabel('|B| [mT]', color='#f43f5e')
ax4_twin.set_ylabel('Inclination [deg]', color='#60a5fa')
ax4.set_title('Sunspot Magnetic Structure')
ax4.tick_params(axis='y', labelcolor='#f43f5e')
ax4_twin.tick_params(axis='y', labelcolor='#60a5fa')

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("SUNSPOTS AND THE SOLAR CYCLE")
print("=" * 60)
print("\n--- Sunspot Properties ---")
print("  Umbral T: ~3800 K, B: 200-400 mT")
print("  Penumbral T: ~5200 K, B: 100-200 mT")
print("  Evershed flow: ~2 km/s radial outflow")
print("  Wilson depression: ~400 km")
print("\n--- Solar Cycle Laws ---")
print("  Hale: Leading polarity same sign in each hemisphere, reverses each cycle")
print("  Joy: Tilt angle ~ 0.5 * sin(latitude)")
print("  Sporer: Spots migrate equatorward during cycle")
print("  Waldmeier: Stronger cycles rise faster")
print("\n--- Cycle Numbers ---")
print("  Period: 11.04 yr (Schwabe), 22 yr (Hale)")
print("  We are in Cycle 25 (began Dec 2019)")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

11.6 Detailed Sunspot Cooling from Magnetic Pressure

Temperature Reduction: $\Delta T/T \approx B^2/(8\pi P)$

Step 1. Total pressure balance across the sunspot boundary:

$$P_{\text{gas,ext}} = P_{\text{gas,int}} + \frac{B^2}{2\mu_0}$$

Step 2. Using the ideal gas law $P = \rho k_B T / (\mu m_H)$ and assuming the density is approximately the same (a rough first estimate):

$$\frac{\rho k_B T_{\text{ext}}}{\mu m_H} = \frac{\rho k_B T_{\text{int}}}{\mu m_H} + \frac{B^2}{2\mu_0}$$

Step 3. The fractional temperature deficit:

$$\frac{T_{\text{ext}} - T_{\text{int}}}{T_{\text{ext}}} = \frac{B^2/(2\mu_0)}{P_{\text{ext}}} = \frac{P_{\text{mag}}}{P_{\text{ext}}}$$

Step 4. In CGS notation this is often written as:

$$\boxed{\frac{\Delta T}{T} \approx \frac{B^2}{8\pi P_{\text{ext}}} \equiv \beta^{-1}}$$

Step 5. For the photosphere, $P_{\text{ext}} \approx 1.4\times10^4$ Pa and$B = 0.3$ T:

$$\frac{\Delta T}{T} \approx \frac{(0.3)^2}{2\times4\pi\times10^{-7}\times1.4\times10^4} \approx \frac{3.6\times10^4}{1.4\times10^4} \approx 2.5$$

This is an overestimate because the density also adjusts (the sunspot is partially evacuated). A more careful treatment accounting for hydrostatic stratification gives$\Delta T \approx 2000$ K, reducing $T$ from 5800 K to ~3800 K in the umbra, consistent with observations of sunspot umbral brightness ($\sim 20\%$ of the quiet Sun).

11.7 Joy's Law from Coriolis Force on Flux Tubes

Tilt Angle from Angular Momentum Conservation

Step 1. Consider a thin flux tube rising from the base of the convection zone. As the tube expands, the two footpoints separate in longitude. The Coriolis force acts on this diverging flow.

Step 2. The Coriolis acceleration perpendicular to the tube axis (in the east-west direction):

$$a_{\text{Cor}} = 2\Omega_\odot v_r \sin\lambda$$

where $v_r$ is the radial rise velocity and $\lambda$ is the latitude.

Step 3. Over the rise time $\tau_{\text{rise}}$, the angular displacement of each footpoint:

$$\delta\phi \sim \frac{1}{2}a_{\text{Cor}}\tau_{\text{rise}}^2 / d = \Omega_\odot v_r \tau_{\text{rise}}^2 \sin\lambda / d$$

where $d$ is the footpoint separation.

Step 4. The tilt angle $\gamma \approx \delta\phi$ (small angle), and using$\tau_{\text{rise}} \sim H/v_r$ where $H$ is the convection zone depth:

$$\boxed{\gamma \approx \frac{\Omega_\odot H^2}{v_r d}\sin\lambda \approx A\sin\lambda}$$

Observations give $A \approx 0.5$ (i.e., $\gamma\approx 0.5\sin\lambda$ radians$\approx 30\sin\lambda$ degrees). At typical active region latitude $\lambda=15^\circ$, the tilt is about $7^\circ$. This latitude-dependent tilt is critical for the Babcock-Leighton dynamo: it provides the systematic asymmetry needed to convert toroidal to poloidal flux.

11.8 Butterfly Diagram Schematic

The butterfly diagram shows the latitude of sunspot emergence over time, revealing the 11-year equatorward migration pattern described by Sporer's law.

Schematic butterfly diagram showing equatorward migration of sunspot emergence zones over two 11-year cycles. Colors indicate magnetic polarity (Hale's law): leading polarity reverses every cycle.

Extended Simulation: Solar Cycle, Maunder Minimum, & Butterfly Diagram

Extended: Maunder Minimum, Waldmeier Effect, Butterfly Diagram, Magnetic Flux Cycle

Python

script.py153 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('Solar Cycle, Maunder Minimum, and Butterfly Diagram', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

# --- Panel 1: Multi-century SSN with Maunder Minimum ---
ax1 = axes[0, 0]
years = np.linspace(1600, 1800, 2000)
SSN = np.zeros_like(years)

for yr in years:
    idx = int((yr - 1600) / 200 * 2000)
    if idx >= len(SSN):
        idx = len(SSN) - 1
    cycle_phase = ((yr - 1610) % 11) / 11.0
    # Base amplitude varies with grand cycle
    if 1645 <= yr <= 1715:
        # Maunder Minimum: very few spots
        amp = 5 + 3 * np.sin(2 * np.pi * cycle_phase)
    elif yr < 1645:
        amp = 60 + 30 * np.sin(2 * np.pi * (yr - 1610) / 80)
    else:
        amp = 80 + 40 * np.sin(2 * np.pi * (yr - 1715) / 90)
    # Asymmetric cycle shape
    if cycle_phase < 0.35:
        shape = np.sin(np.pi * cycle_phase / 0.7)**2
    else:
        shape = np.exp(-3 * (cycle_phase - 0.35))
    SSN[idx] = max(0, amp * shape + np.random.normal(0, 3))

ax1.plot(years, SSN, color='#f43f5e', linewidth=0.8)
ax1.fill_between(years, 0, SSN, alpha=0.2, color='#f43f5e')
ax1.axvspan(1645, 1715, alpha=0.15, color='#60a5fa')
ax1.text(1680, max(SSN)*0.85, 'Maunder\nMinimum', fontsize=11, color='#60a5fa', ha='center', fontweight='bold')
ax1.set_xlabel('Year')
ax1.set_ylabel('Sunspot Number')
ax1.set_title('Solar Activity 1600-1800')

# --- Panel 2: Waldmeier Effect ---
ax2 = axes[0, 1]
np.random.seed(99)
# Generate synthetic cycles with Waldmeier effect
n_cycles = 25
amplitudes = 50 + 100 * np.random.rand(n_cycles)
rise_times = 5.5 - 0.015 * amplitudes + np.random.normal(0, 0.5, n_cycles)
rise_times = np.clip(rise_times, 2, 6)

ax2.scatter(amplitudes, rise_times, s=60, c='#f43f5e', alpha=0.7, edgecolors='white', linewidths=0.5)

# Fit line
z = np.polyfit(amplitudes, rise_times, 1)
amp_fit = np.linspace(40, 180, 100)
ax2.plot(amp_fit, np.polyval(z, amp_fit), color='#fbbf24', linewidth=2, linestyle='--',
        label='Linear fit (Waldmeier effect)')
ax2.set_xlabel('Cycle amplitude (SSN_max)')
ax2.set_ylabel('Rise time [years]')
ax2.set_title('Waldmeier Effect')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Detailed butterfly diagram ---
ax3 = axes[1, 0]
np.random.seed(42)
t_years = np.linspace(0, 44, 4000)
lat_grid = np.linspace(-40, 40, 200)
T_grid, L_grid = np.meshgrid(t_years, lat_grid)

B_butterfly = np.zeros_like(T_grid)
for cyc in range(4):
    t0 = cyc * 11
    phase = np.clip((T_grid - t0) / 11.0, 0, 1)
    polarity = 1 if cyc % 2 == 0 else -1

# Activity envelope
    activity = np.where(phase < 0.35,
                       np.sin(np.pi * phase / 0.7)**2,
                       np.exp(-3 * (phase - 0.35)))

# Latitude of activity band
    lat_center = (30 - 25 * phase)
    lat_width = 5 + 3 * (1 - phase)

# Northern hemisphere
    north = activity * np.exp(-(L_grid - lat_center)**2 / (2 * lat_width**2))
    # Southern hemisphere
    south = activity * np.exp(-(L_grid + lat_center)**2 / (2 * lat_width**2))

B_butterfly += polarity * north - polarity * south

im3 = ax3.pcolormesh(T_grid, L_grid, B_butterfly, cmap='RdBu_r', shading='auto', vmin=-1, vmax=1)
ax3.set_xlabel('Time [years]')
ax3.set_ylabel('Latitude [deg]')
ax3.set_title('Detailed Butterfly Diagram')
plt.colorbar(im3, ax=ax3, label='Magnetic polarity', shrink=0.8)

# --- Panel 4: Sunspot number and magnetic flux ---
ax4 = axes[1, 1]
t = np.linspace(0, 44, 1000)
SSN_synth = np.zeros_like(t)
for cyc in range(4):
    t0 = cyc * 11
    phase = np.clip((t - t0) / 11.0, 0, 1)
    amp = 80 + 60 * np.sin(cyc * 1.5 + 0.3)
    rise_part = np.where(phase < 0.35, amp * np.sin(np.pi * phase / 0.7)**2, 0)
    fall_part = np.where((phase >= 0.35) & (phase <= 1),
                        amp * np.exp(-3 * (phase - 0.35)), 0)
    SSN_synth += rise_part + fall_part

# Total magnetic flux (from SSN, roughly)
flux = 2e22 + 3e22 * SSN_synth / 150  # Mx

ax4.plot(t, SSN_synth, color='#f43f5e', linewidth=2, label='Sunspot Number')
ax4_twin = ax4.twinx()
ax4_twin.plot(t, flux, color='#60a5fa', linewidth=2, label='Total Magnetic Flux')
ax4.set_xlabel('Time [years]')
ax4.set_ylabel('Sunspot Number', color='#f43f5e')
ax4_twin.set_ylabel('Total Flux [Mx]', color='#60a5fa')
ax4.set_title('SSN and Total Magnetic Flux')
ax4.tick_params(axis='y', labelcolor='#f43f5e')
ax4_twin.tick_params(axis='y', labelcolor='#60a5fa')

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: SOLAR CYCLE AND MAUNDER MINIMUM")
print("=" * 60)
print("\n--- Sunspot Cooling ---")
print("  Delta_T/T ~ B^2/(2*mu_0*P_ext) = beta^-1")
print("  For B=0.3 T: P_mag ~ 3.6e4 Pa ~ P_photosphere")
print("  T_umbra ~ 3800 K (vs 5800 K quiet Sun)")
print("\n--- Maunder Minimum (1645-1715) ---")
print("  ~70 years of very low activity")
print("  Corresponded to Little Ice Age in Europe")
print("  Caused by stochastic fluctuations in dynamo")
print("\n--- Waldmeier Effect ---")
print("  Strong cycles rise faster: tau_rise ~ R_max^{-0.45}")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Solar Dynamo Next: Solar Flares →

Sunspots & Active Regions

11.1 Sunspot Structure

Derivation 1: Umbral-Penumbral Magnetic Field Geometry

11.2 Evershed Flow

Derivation 2: Siphon Flow Model

11.3 Joy's Law and Hale's Law

Derivation 3: Tilt Angle from Coriolis Force

11.4 The Solar Cycle and Butterfly Diagram

Derivation 4: Sporer's Law of Equatorward Migration

11.5 Solar Cycle Statistics

Derivation 5: Waldmeier Effect

Solar Cycle Statistics

Numerical Simulation

Sunspots: Butterfly Diagram, Solar Cycle, Joy's Law, Magnetic Structure

11.6 Detailed Sunspot Cooling from Magnetic Pressure

Temperature Reduction: \(\Delta T/T \approx B^2/(8\pi P)\)

11.7 Joy's Law from Coriolis Force on Flux Tubes

Tilt Angle from Angular Momentum Conservation

11.8 Butterfly Diagram Schematic

Extended Simulation: Solar Cycle, Maunder Minimum, & Butterfly Diagram

Extended: Maunder Minimum, Waldmeier Effect, Butterfly Diagram, Magnetic Flux Cycle