Part III: Solar Magnetic Activity | Chapter 10

The Solar Dynamo

Magnetic induction, magnetic Reynolds number, alpha-omega dynamo, and mean-field theory

10.1 The Magnetic Induction Equation

Derivation 1: Induction Equation from Maxwell and Ohm

The evolution of the magnetic field in a conducting fluid is governed by the induction equation, derived from Faraday's law and Ohm's law.

Step 1. Faraday's law: $\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t$. Ohm's law for a moving conductor: $\mathbf{J} = \sigma(\mathbf{E} + \mathbf{v} \times \mathbf{B})$. Ampere's law (MHD limit): $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$.

Step 2. Eliminating $\mathbf{E}$ and $\mathbf{J}$:

$$\mathbf{E} = \frac{\mathbf{J}}{\sigma} - \mathbf{v} \times \mathbf{B} = \frac{\nabla \times \mathbf{B}}{\mu_0 \sigma} - \mathbf{v} \times \mathbf{B}$$

Step 3. Substituting into Faraday's law and defining $\eta = 1/(\mu_0\sigma)$:

$$\boxed{\frac{\partial \mathbf{B}}{\partial t} = \underbrace{\nabla \times (\mathbf{v} \times \mathbf{B})}_{\text{induction}} + \underbrace{\eta \nabla^2 \mathbf{B}}_{\text{diffusion}}}$$

The first term represents flux freezing (field is carried with the plasma) and the second represents resistive diffusion (field decays on a timescale $\tau_d = L^2/\eta$).

10.2 Magnetic Reynolds Number

Derivation 2: $R_m$ and Flux Freezing

Step 1. The ratio of the induction to diffusion terms defines the magnetic Reynolds number:

$$\boxed{R_m = \frac{|\nabla \times (\mathbf{v} \times \mathbf{B})|}{|\eta \nabla^2 \mathbf{B}|} \sim \frac{v L}{\eta}}$$

Step 2. For the solar convection zone: $v \sim 100$ m/s,$L \sim 2 \times 10^8$ m, $\eta \sim 1$ m$^2$/s (Spitzer value), giving:

$$R_m \sim 10^{10} \gg 1$$

Step 3. When $R_m \gg 1$, Alfven's theorem holds: magnetic flux through any surface moving with the fluid is conserved. The field lines are "frozen" to the plasma. The Ohmic diffusion time for the Sun is:

$$\tau_d = \frac{R_\odot^2}{\eta} \sim 10^{10} \text{ years}$$

Since $\tau_d$ greatly exceeds the solar age, resistive decay alone cannot explain the 22-year magnetic cycle. A dynamo mechanism is needed to regenerate the field.

10.3 The $\alpha$-$\Omega$ Dynamo

Derivation 3: Toroidal and Poloidal Field Generation

The solar dynamo converts between poloidal (dipolar) and toroidal (azimuthal) field components through two processes.

Step 1: The $\Omega$-effect. Differential rotation stretches poloidal field lines into toroidal field. Starting from a poloidal field $B_r$and differential rotation $\Omega(r, \theta)$:

$$\frac{\partial B_\phi}{\partial t} = r\sin\theta \, (\mathbf{B}_p \cdot \nabla)\Omega + \ldots$$

At the tachocline where $d\Omega/dr$ is large, this process is very efficient.

Step 2: The $\alpha$-effect. Helical turbulence (cyclonic convection influenced by the Coriolis force) twists toroidal field to regenerate poloidal field:

$$\mathcal{E} = \langle \mathbf{v}' \times \mathbf{B}' \rangle = \alpha \langle\mathbf{B}\rangle - \beta \nabla \times \langle\mathbf{B}\rangle$$

Step 3. The $\alpha$ coefficient from first-order smoothing theory:

$$\boxed{\alpha \approx -\frac{1}{3}\tau_c \langle \mathbf{v}' \cdot (\nabla \times \mathbf{v}')\rangle = -\frac{1}{3}\tau_c \langle \text{kinetic helicity}\rangle}$$

The $\alpha$-effect is antisymmetric about the equator (positive in the north, negative in the south), which naturally produces the observed equatorial antisymmetry of the solar magnetic field.

10.4 Babcock-Leighton Mechanism

Derivation 4: Surface Flux Transport

Step 1. In the Babcock-Leighton (BL) framework, the poloidal field is regenerated by the decay of tilted bipolar active regions (Joy's law). The surface flux transport equation:

$$\boxed{\frac{\partial B_r}{\partial t} = -\Omega(\theta)\frac{\partial B_r}{\partial\phi} - \frac{1}{R\sin\theta}\frac{\partial}{\partial\theta}(v_\theta B_r \sin\theta) + \frac{\eta_h}{R^2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial B_r}{\partial\theta}\right)\right] + S(\theta,\phi,t)}$$

Step 2. Here $v_\theta$ is the meridional flow (~15 m/s poleward at the surface),$\eta_h \sim 600$ km$^2$/s is the horizontal turbulent diffusivity, and $S$ is the source term from active region emergence.

The BL mechanism has emerged as the leading paradigm because it directly connects the observed surface flux evolution to the polar field that serves as the seed for the next cycle. The strength of the polar field at cycle minimum is the best predictor of the amplitude of the following cycle.

10.5 Mean-Field Electrodynamics

Derivation 5: Dynamo Waves and the Cycle Period

Step 1. The mean-field induction equation for the axisymmetric components in an $\alpha\Omega$ dynamo:

$$\frac{\partial A}{\partial t} = \alpha B_\phi + \eta_T \left(\nabla^2 - \frac{1}{s^2}\right) A$$$$\frac{\partial B_\phi}{\partial t} = s(\mathbf{B}_p \cdot \nabla)\Omega + \eta_T \left(\nabla^2 - \frac{1}{s^2}\right) B_\phi$$

where $A$ is the poloidal vector potential and $s = r\sin\theta$.

Step 2. These equations support dynamo waves that propagate according to the Parker-Yoshimura sign rule:

$$\boxed{\text{Propagation direction} = -\alpha \frac{\partial\Omega}{\partial r} \hat{e}_\theta}$$

Step 3. The dynamo number determines whether oscillatory solutions exist:

$$D = \frac{\alpha_0 \Delta\Omega L^3}{\eta_T^2}$$

When $|D| > D_c \approx 1\text{--}10$, the dynamo is supercritical and oscillates. The cycle period $P \sim L^2/\eta_T$ is set by the turbulent diffusion time. For $\eta_T \sim 10^{8}\text{--}10^{9}$ m$^2$/s and $L \sim R_\odot/3$, this gives $P \sim 10\text{--}30$ years, consistent with the observed 22-year magnetic cycle.

Numerical Simulation

Solar Dynamo: Differential Rotation, Butterfly Diagram, Alpha-Omega Profiles

Python

script.py117 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 Dynamo Theory', 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: Omega effect - differential rotation ---
ax1 = axes[0, 0]
theta_lat = np.linspace(-90, 90, 200)
theta_rad = np.radians(theta_lat)

# Differential rotation profile (sidereal, nHz)
Omega = 460 - 65 * np.sin(theta_rad)**2 - 55 * np.sin(theta_rad)**4

ax1.plot(theta_lat, Omega, color='#f43f5e', linewidth=2.5)
ax1.axhline(y=430, color='#fbbf24', linestyle='--', alpha=0.5, label='Radiative interior')
ax1.set_xlabel('Latitude [deg]')
ax1.set_ylabel('Rotation rate [nHz]')
ax1.set_title('Differential Rotation (Omega-effect)')
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: 1D dynamo wave ---
ax2 = axes[0, 1]
x = np.linspace(0, 1, 100)  # latitude-like coordinate
t = np.linspace(0, 4, 200)  # time in units of P_cyc
X, T = np.meshgrid(x, t)

# Propagating dynamo wave (butterfly diagram pattern)
k = 2 * np.pi * 3  # wavenumber
omega = 2 * np.pi  # frequency
B_toroidal = np.sin(k * X - omega * T) * np.exp(-2 * (X - 0.3 - 0.15 * T % 1)**2 / 0.1)

im2 = ax2.pcolormesh(T, 90 * (1 - 2*X), B_toroidal, cmap='RdBu_r', shading='auto', vmin=-1, vmax=1)
ax2.set_xlabel('Time [cycles]')
ax2.set_ylabel('Latitude [deg]')
ax2.set_title('Dynamo Wave (Butterfly Diagram)')
plt.colorbar(im2, ax=ax2, label='B_phi (normalized)', shrink=0.8)

# --- Panel 3: Alpha and Omega profiles ---
ax3 = axes[1, 0]
r = np.linspace(0.5, 1.0, 200)

# Alpha profile (antisymmetric, concentrated near surface)
alpha_profile = np.sin(np.pi * (r - 0.7) / 0.3) * np.exp(-((r - 0.85) / 0.1)**2)
alpha_profile = np.where((r > 0.7) & (r < 1.0), alpha_profile, 0)

# dOmega/dr profile (concentrated at tachocline)
dOmega_dr = -10 * np.exp(-((r - 0.693) / 0.02)**2)

ax3.plot(r, alpha_profile / np.max(np.abs(alpha_profile)), color='#f43f5e', linewidth=2.5,
        label='alpha(r) (normalized)')
ax3.plot(r, dOmega_dr / np.min(dOmega_dr), color='#60a5fa', linewidth=2.5,
        label='dOmega/dr (normalized)')
ax3.axvline(x=0.693, color='#fbbf24', linestyle=':', alpha=0.5, label='Tachocline')
ax3.axvline(x=0.713, color='#4ade80', linestyle=':', alpha=0.5, label='CZ base')
ax3.set_xlabel('r / R_sun')
ax3.set_ylabel('Normalized amplitude')
ax3.set_title('Alpha and Omega Profiles')
ax3.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Dynamo number and growth rate ---
ax4 = axes[1, 1]
D = np.linspace(-1000, 1000, 500)

# Growth rate (schematic): real part of eigenvalue
# For alpha-omega dynamo, critical D_c ~ 10
sigma_real = np.where(np.abs(D) > 10,
    0.1 * np.log(np.abs(D) / 10),
    -0.1 * (1 - np.abs(D) / 10))
sigma_imag = np.where(np.abs(D) > 10,
    2 * np.pi / 22.0 * np.ones_like(D),
    0)

ax4.plot(D, sigma_real, color='#f43f5e', linewidth=2.5, label='Growth rate')
ax4.plot(D, sigma_imag, color='#60a5fa', linewidth=2, linestyle='--', label='Oscillation freq')
ax4.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax4.axvline(x=10, color='#fbbf24', linestyle=':', alpha=0.5, label='D_c')
ax4.axvline(x=-10, color='#fbbf24', linestyle=':', alpha=0.5)
ax4.set_xlabel('Dynamo number D')
ax4.set_ylabel('Growth rate / Frequency')
ax4.set_title('Dynamo Eigenvalue vs D')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

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("SOLAR DYNAMO")
print("=" * 60)
print("\n--- Key Parameters ---")
print("  Magnetic Reynolds number R_m ~ 10^10")
print("  Ohmic diffusion time: ~10^10 years")
print("  Turbulent diffusion: eta_T ~ 10^8-10^9 m^2/s")
print("  Cycle period: 22 years (magnetic), 11 years (activity)")
print("\n--- Dynamo Components ---")
print("  Omega-effect: differential rotation at tachocline")
print("  Alpha-effect: helical turbulence / Babcock-Leighton")
print("  Meridional flow: ~15 m/s poleward at surface")
print("\n--- Parker-Yoshimura Rule ---")
print("  Equatorward migration requires alpha * dOmega/dr < 0")
print("  Consistent with alpha > 0 in N hemisphere, dOmega/dr < 0 at CZ base")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

10.6 Full Induction Equation Derivation

From Maxwell + Ohm to $\partial\mathbf{B}/\partial t$

We derive every step explicitly from the fundamental equations.

Step 1. Start with Faraday's law:

$$\frac{\partial\mathbf{B}}{\partial t} = -\nabla\times\mathbf{E}$$

Step 2. Ohm's law in a moving conductor with resistivity $\eta_r = 1/\sigma$:

$$\mathbf{E} = -\mathbf{v}\times\mathbf{B} + \frac{\mathbf{J}}{\sigma} = -\mathbf{v}\times\mathbf{B} + \eta\mu_0\mathbf{J}$$

Step 3. Ampere's law (neglecting displacement current in MHD):$\mathbf{J} = \nabla\times\mathbf{B}/\mu_0$. Substitute:

$$\mathbf{E} = -\mathbf{v}\times\mathbf{B} + \eta(\nabla\times\mathbf{B})$$

Step 4. Insert into Faraday's law:

$$\frac{\partial\mathbf{B}}{\partial t} = -\nabla\times\left[-\mathbf{v}\times\mathbf{B} + \eta(\nabla\times\mathbf{B})\right]$$

Step 5. For uniform $\eta$, use the vector identity$\nabla\times(\nabla\times\mathbf{B}) = \nabla(\nabla\cdot\mathbf{B})-\nabla^2\mathbf{B} = -\nabla^2\mathbf{B}$:

$$\boxed{\frac{\partial\mathbf{B}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{B}) + \eta\nabla^2\mathbf{B}}$$

Step 6: Frozen-in flux for $R_m\gg 1$. When diffusion is negligible:

$$\frac{\partial\mathbf{B}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{B})$$

Expanding with the BAC-CAB rule and using $\nabla\cdot\mathbf{B}=0$:

$$\boxed{\frac{d}{dt}\left(\frac{\mathbf{B}}{\rho}\right) = \left(\frac{\mathbf{B}}{\rho}\cdot\nabla\right)\mathbf{v}}$$

This is Alfven's frozen-in flux theorem: $\mathbf{B}/\rho$ evolves like a material line element, meaning the magnetic flux through any surface comoving with the fluid is conserved. For the solar convection zone with $R_m\sim 10^{10}$, flux freezing is an excellent approximation except at thin current sheets where reconnection occurs.

10.7 Mean-Field Dynamo Equations

Derivation of $\partial\bar{\mathbf{B}}/\partial t = \nabla\times(\bar{\mathbf{v}}\times\bar{\mathbf{B}}+\alpha\bar{\mathbf{B}}) + \eta_T\nabla^2\bar{\mathbf{B}}$

Step 1. Decompose all quantities into mean and fluctuating parts:$\mathbf{B} = \bar{\mathbf{B}} + \mathbf{b}$,$\mathbf{v} = \bar{\mathbf{v}} + \mathbf{v}'$.

Step 2. Average the induction equation. The key term is the mean electromotive force (EMF):

$$\boldsymbol{\mathcal{E}} = \overline{\mathbf{v}'\times\mathbf{b}}$$

Step 3. Under the first-order smoothing approximation (FOSA), expand $\boldsymbol{\mathcal{E}}$in terms of $\bar{\mathbf{B}}$ and its derivatives:

$$\mathcal{E}_i = \alpha_{ij}\bar{B}_j + \beta_{ijk}\frac{\partial\bar{B}_j}{\partial x_k} + \ldots$$

Step 4. For isotropic, homogeneous turbulence, the tensors simplify:

$$\alpha_{ij} = \alpha\delta_{ij}, \quad \alpha = -\frac{1}{3}\tau_c\overline{\mathbf{v}'\cdot(\nabla\times\mathbf{v}')}$$

$$\beta_{ijk} = -\beta\epsilon_{ijk}, \quad \beta = \frac{1}{3}\tau_c\overline{v'^2} = \eta_T$$

Step 5. The complete mean-field induction equation:

$$\boxed{\frac{\partial\bar{\mathbf{B}}}{\partial t} = \nabla\times(\bar{\mathbf{v}}\times\bar{\mathbf{B}} + \alpha\bar{\mathbf{B}}) + \eta_T\nabla^2\bar{\mathbf{B}}}$$

The $\alpha$ term is proportional to the kinetic helicity of the turbulence, which is non-zero when rotation (Coriolis force) breaks the mirror symmetry. The turbulent diffusivity$\eta_T \sim \frac{1}{3}v'l \sim 10^8\text{--}10^9$ m$^2$/s vastly exceeds the molecular value $\eta\sim 1$ m$^2$/s, reducing the effective diffusion time from $10^{10}$ years to $\sim 10$ years.

10.8 Babcock-Leighton Mechanism

The diagram below illustrates the toroidal-to-poloidal field conversion through the emergence and decay of tilted bipolar active regions.

The Babcock-Leighton mechanism: (1) Toroidal field at the tachocline is wound by differential rotation. (2) Buoyant flux tubes rise as omega-loops. (3) They emerge as tilted bipolar regions (Joy's law). (4) Trailing polarity diffuses poleward, reversing the polar field and creating new poloidal flux.

Extended Simulation: Dynamo Wave & Butterfly Diagram

Extended: Dynamo Wave Solution, Butterfly Diagram, Rm Regimes, Mean-Field Coefficients

Python

script.py156 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('Dynamo Wave Solution and Butterfly Diagram', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

# --- Panel 1: 1D alpha-omega dynamo wave ---
ax1 = axes[0, 0]
# Solve the dispersion relation for dynamo waves
# For plane waves: B ~ exp(i(kx - omega*t))
# omega = (alpha*k*dOmega/dr)^{1/2} * (1+i)/sqrt(2)
# This gives propagating, growing/decaying waves

x = np.linspace(0, 1, 300)  # normalized latitude (0=equator, 1=pole)
t = np.linspace(0, 4, 400)  # time in units of cycle period

X, T = np.meshgrid(x, t)

# Equatorward-propagating dynamo wave
k = 6 * np.pi
omega = 2 * np.pi  # one cycle per unit time
# Gaussian envelope migrating equatorward
lat_center = 0.6 - 0.4 * (T % 1.0)
width = 0.15

B_wave = np.sin(k * X - omega * T) * np.exp(-(X - lat_center)**2 / (2 * width**2))

# Convert to latitude in degrees
lat_deg = 90 * (1 - 2 * X)  # equator at center

im1 = ax1.pcolormesh(T, lat_deg, B_wave, cmap='RdBu_r', shading='auto', vmin=-1, vmax=1)
ax1.set_xlabel('Time [cycle periods]')
ax1.set_ylabel('Latitude [deg]')
ax1.set_title('Alpha-Omega Dynamo Wave')
plt.colorbar(im1, ax=ax1, label='B_phi', shrink=0.8)

# --- Panel 2: Improved butterfly diagram ---
ax2 = axes[0, 1]
np.random.seed(42)
t_all = []
lat_all = []
color_all = []

for cyc in range(6):
    t_start = cyc * 11
    polarity = 1 if cyc % 2 == 0 else -1
    # Number of spots proportional to cycle amplitude
    amp = 120 + 60 * np.sin(cyc * 1.2 + 0.5)
    n_spots = int(amp * 25)

# Phase within cycle
    t_spots = t_start + np.abs(np.random.normal(4, 2.5, n_spots))
    t_spots = t_spots[t_spots < t_start + 11]
    phase = (t_spots - t_start) / 11.0

# Latitude: starts at ~30 deg, migrates to ~5 deg
    lat_mean = 30 - 25 * phase
    lat_scatter = 3 + 2 * (1 - phase)
    lat_spots = np.abs(np.random.normal(lat_mean, lat_scatter))
    lat_spots = np.clip(lat_spots, 1, 40)

# Both hemispheres
    for sign in [1, -1]:
        t_all.extend(t_spots)
        lat_all.extend(sign * lat_spots)
        color_all.extend([polarity * sign] * len(t_spots))

scatter = ax2.scatter(t_all, lat_all, s=0.3, c=color_all, cmap='RdBu_r', vmin=-1.5, vmax=1.5, alpha=0.4)
ax2.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax2.set_xlabel('Time [years]')
ax2.set_ylabel('Latitude [deg]')
ax2.set_title('Synthetic Butterfly Diagram (6 Cycles)')
ax2.set_ylim(-45, 45)

# --- Panel 3: Magnetic Reynolds number regimes ---
ax3 = axes[1, 0]
Rm = np.logspace(-2, 14, 500)
# Characteristic timescales normalized
tau_diff = Rm  # L^2/eta proportional to Rm
tau_adv = np.ones_like(Rm)  # L/v = 1

ax3.loglog(Rm, tau_diff, color='#60a5fa', linewidth=2.5, label='Diffusion time (L^2/eta)')
ax3.loglog(Rm, tau_adv, color='#f43f5e', linewidth=2.5, label='Advection time (L/v)')
ax3.axvline(x=1, color='#fbbf24', linestyle='--', linewidth=2, alpha=0.7, label='Rm = 1')
ax3.fill_betweenx([1e-3, 1e15], 1e-2, 1, alpha=0.1, color='#60a5fa')
ax3.fill_betweenx([1e-3, 1e15], 1, 1e14, alpha=0.1, color='#f43f5e')
ax3.text(0.05, 1e6, 'Diffusion\ndominates', fontsize=10, color='#60a5fa', ha='center')
ax3.text(1e7, 1e6, 'Advection\ndominates\n(frozen-in)', fontsize=10, color='#f43f5e', ha='center')

# Mark solar values
ax3.axvline(x=1e10, color='#4ade80', linestyle=':', alpha=0.5)
ax3.text(1e10, 1e-2, 'Sun CZ', fontsize=9, color='#4ade80', ha='center')
ax3.set_xlabel('Magnetic Reynolds Number Rm')
ax3.set_ylabel('Timescale (normalized)')
ax3.set_title('Rm Regimes: Diffusion vs Advection')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax3.set_ylim(1e-3, 1e15)

# --- Panel 4: Turbulent EMF alpha and beta ---
ax4 = axes[1, 1]
theta_lat = np.linspace(-90, 90, 300)
theta_rad = np.radians(theta_lat)

# Alpha proportional to cos(theta) * sin^2(theta) (antisymmetric)
alpha_profile = np.cos(theta_rad) * np.sin(theta_rad)**2
alpha_profile = alpha_profile / np.max(np.abs(alpha_profile))

# Turbulent diffusivity (symmetric, peaks at mid-latitudes)
eta_T_profile = np.sin(theta_rad)**2
eta_T_profile = eta_T_profile / np.max(eta_T_profile)

ax4.plot(theta_lat, alpha_profile, color='#f43f5e', linewidth=2.5, label='alpha-effect (antisymmetric)')
ax4.plot(theta_lat, eta_T_profile, color='#60a5fa', linewidth=2.5, label='eta_T (turbulent diffusivity)')
ax4.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax4.axvline(x=0, color='white', linestyle=':', alpha=0.3)
ax4.fill_between(theta_lat, 0, alpha_profile, where=(alpha_profile > 0), alpha=0.15, color='#f43f5e')
ax4.fill_between(theta_lat, 0, alpha_profile, where=(alpha_profile < 0), alpha=0.15, color='#60a5fa')
ax4.set_xlabel('Latitude [deg]')
ax4.set_ylabel('Normalized amplitude')
ax4.set_title('Mean-Field Coefficients vs Latitude')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax4.text(45, 0.5, 'alpha > 0\n(N hemisphere)', fontsize=9, color='#f43f5e', ha='center')
ax4.text(-45, -0.5, 'alpha < 0\n(S hemisphere)', fontsize=9, color='#60a5fa', ha='center')

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: DYNAMO WAVE AND MEAN-FIELD THEORY")
print("=" * 60)
print("\n--- Induction Equation ---")
print("  dB/dt = curl(v x B) + eta * laplacian(B)")
print("  Frozen-in: d(B/rho)/dt = (B/rho . grad)v")
print("\n--- Mean-Field Parameters ---")
print("  alpha ~ -(1/3) * tau_c * <v . curl(v)>")
print("  eta_T ~ (1/3) * v_rms * l_corr ~ 10^8-10^9 m^2/s")
print("  Effective diffusion time: ~10 years (vs 10^10 yr Ohmic)")
print("\n--- Dynamo Wave ---")
print("  Propagation: equatorward when alpha * dOmega/dr < 0")
print("  Period set by: P ~ L^2 / eta_T ~ 10-30 years")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Magnetic Fields Next: Sunspots & Active Regions →

The Solar Dynamo

10.1 The Magnetic Induction Equation

Derivation 1: Induction Equation from Maxwell and Ohm

10.2 Magnetic Reynolds Number

Derivation 2: \(R_m\) and Flux Freezing

10.3 The \(\alpha\)-\(\Omega\) Dynamo

Derivation 3: Toroidal and Poloidal Field Generation

10.4 Babcock-Leighton Mechanism

Derivation 4: Surface Flux Transport

10.5 Mean-Field Electrodynamics

Derivation 5: Dynamo Waves and the Cycle Period

Numerical Simulation

Solar Dynamo: Differential Rotation, Butterfly Diagram, Alpha-Omega Profiles

10.6 Full Induction Equation Derivation

From Maxwell + Ohm to \(\partial\mathbf{B}/\partial t\)

10.7 Mean-Field Dynamo Equations

Derivation of \(\partial\bar{\mathbf{B}}/\partial t = \nabla\times(\bar{\mathbf{v}}\times\bar{\mathbf{B}}+\alpha\bar{\mathbf{B}}) + \eta_T\nabla^2\bar{\mathbf{B}}\)

10.8 Babcock-Leighton Mechanism

Extended Simulation: Dynamo Wave & Butterfly Diagram

Extended: Dynamo Wave Solution, Butterfly Diagram, Rm Regimes, Mean-Field Coefficients