Visualizing the Earth's dipolar magnetic field

(2 comments)

The magnetic field due to a magnetic dipole moment, $\boldsymbol{m}$ at a point $\boldsymbol{r}$ relative to it may be written $$ \boldsymbol{B}(\boldsymbol{r}) = \frac{\mu_0}{4\pi r^3}[3\boldsymbol{\hat{r}(\boldsymbol{\hat{r}} \cdot \boldsymbol{m}) - \boldsymbol{m}}], $$ where $\mu_0$ is the vacuum permeability. In geomagnetism, it is usual to write the radial and angular components of $\boldsymbol{B}$ as: $$ \begin{align*} B_r & = -2B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\cos\theta, \\ B_\theta & = -B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\sin\theta, \\ B_\phi &= 0, \end{align*} $$ where $\theta$ is polar (colatitude) angle (relative to the magnetic North pole), $\phi$ is the azimuthal angle (longitude), and $R_\mathrm{E}$ is the Earth's radius, about 6370 km. See below for a derivation of these formulae.

With this definition, $B_0$ denotes the magnitude of the mean value of the field at the magnetic equator on the Earth's surface, about $31.2\; \mathrm{\mu T}$. With these definitions, we can plot the dipole component of the Earth's magnetic field as a Matplotlib streamplot. We first construct a meshgrid of $(x,y)$ coordinates and convert them into polar coordinates with the relations: $$ \begin{align*} r &= \sqrt{x^2 + y^2},\\ \theta &= \mathrm{atan2}(y/x), \end{align*} $$ where the two-argument arctangent function, $\mathrm{atan2}$, is implemented in NumPy as arctan2. We can then use the formulae above to calculate $B_r$ and $B_\theta$ and convert back to Cartesian coordinates to plot the streamplot. The $\theta$ coordinate is offset by $\alpha = 9.6^\circ$ to account for the current tilt of the Earth's magnetic dipole with respect to its rotational axis.

Earth's dipolar magnetic field

import sys
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle

# Mean magnitude of the Earth's magnetic field at the equator in T
B0 = 3.12e-5
# Radius of Earth, Mm (10^6 m: mega-metres!)
RE = 6.370
# Deviation of magnetic pole from axis
alpha = np.radians(9.6)

def B(r, theta):
    """Return the magnetic field vector at (r, theta)."""
    fac = B0 * (RE / r)**3
    return -2 * fac * np.cos(theta + alpha), -fac * np.sin(theta + alpha)

# Grid of x, y points on a Cartesian grid
nx, ny = 64, 64
XMAX, YMAX = 40, 40
x = np.linspace(-XMAX, XMAX, nx)
y = np.linspace(-YMAX, YMAX, ny)
X, Y = np.meshgrid(x, y)
r, theta = np.hypot(X, Y), np.arctan2(Y, X)

# Magnetic field vector, B = (Ex, Ey), as separate components
Br, Btheta = B(r, theta)
# Transform to Cartesian coordinates: NB make North point up, not to the right.
c, s = np.cos(np.pi/2 + theta), np.sin(np.pi/2 + theta)
Bx = -Btheta * s + Br * c
By = Btheta * c + Br * s

fig, ax = plt.subplots()

# Plot the streamlines with an appropriate colormap and arrow style
color = 2 * np.log(np.hypot(Bx, By))
ax.streamplot(x, y, Bx, By, color=color, linewidth=1, cmap=plt.cm.inferno,
              density=2, arrowstyle='->', arrowsize=1.5)

# Add a filled circle for the Earth; make sure it's on top of the streamlines.
ax.add_patch(Circle((0,0), RE, color='b', zorder=100))

ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_xlim(-XMAX, XMAX)
ax.set_ylim(-YMAX, YMAX)
ax.set_aspect('equal')
plt.show()

Derivation

For now, consider the dipole to lie along the $y$-axis: $\boldsymbol{m} = m\boldsymbol{\hat{\jmath}}$ and define $\boldsymbol{B_0}$ to be the field perpendicular to $\boldsymbol{m}$ at a distance $R_\mathrm{E}$ from it (i.e. on the Earth's magnetic equator): $$ \boldsymbol{B_0} = -\frac{\mu_0m}{4\pi R_\mathrm{E}^3} \boldsymbol{\hat{\jmath}} = B_0\boldsymbol{\hat{\jmath}}. $$ Then, $$ \boldsymbol{B}(\boldsymbol{r}) = -B_0 \left(\frac{R_\mathrm{E}}{r}\right)^3 [3\cos\theta \boldsymbol{\hat{r}} - \boldsymbol{\hat{\jmath}}]. $$ The radial component of the magnetic field is therefore $$ B_r = \boldsymbol{B}\cdot\boldsymbol{\hat{r}} = -B_0\left(\frac{R_\mathrm{E}}{r}\right)^3[3\cos\theta - \cos\theta] = -2B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\cos\theta, $$ Its angular component is perpendicular to this: $$ B_\theta = \boldsymbol{B}\cdot\boldsymbol{\hat{\theta}} = - B_0 \left(\frac{R_\mathrm{E}}{r}\right)^3 [-\boldsymbol{\hat{\jmath}} \cdot \boldsymbol{\hat{\theta}}] = -B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\sin\theta. $$ Finally, $B_\phi = 0$: the field is symmetric about the axis of the dipole.

Further Reading

Current rating: 5

Comments

Comments are pre-moderated. Please be patient and your comment will appear soon.

Zion_Samurai 1 week, 5 days ago

As I see it, as you may allow. I think that the magnetic field of the Earth is reversed, and some of the beginning and tail end are not connected to the sphere. Is there any chance could you please improve that?

Link | Reply
Currently unrated

christian 1 week, 5 days ago

Thank you for your comment: there's a great article about the north magnetic pole ("dip pole") vs the north geomagnetic pole ("dipole") here: https://www.scientificamerican.com/article/the-earth-has-more-than-one-north-pole/

The streamlines are plotted according to Matplotlib's own algorithm and where they converge some tend to get dropped. I think the overall impression is clear enough, though.

Link | Reply
Currently unrated

New Comment

required

required (not published)

optional

required