Interplanetary Magnetic Field

Let us now investigate how the solar wind and the interplanetary magnetic field affect one another.

The hot coronal plasma making up the solar wind possesses an extremely high electrical conductivity. In such a plasma, we expect the concept of “frozen-in” magnetic field-lines, discussed in Section 8.3, to be applicable. The continuous flow of coronal material into interplanetary space must, therefore, result in the transport of the solar magnetic field into the interplanetary region. If the Sun did not rotate then the resulting magnetic configuration would be very simple. The radial coronal expansion considered previously (with the neglect of any magnetic forces) would produce magnetic field-lines extending radially outward from the Sun.

Of course, the Sun does rotate, with a (latitude dependent) period of about 25 days.^8.1 Because the solar photosphere is an excellent electrical conductor, the magnetic field at the base of the corona is frozen into the rotating frame of reference of the Sun. A magnetic field-line starting from a given location on the surface of the Sun is drawn out along the path followed by the element of the solar wind emanating from that location. As before, let us suppose that the coronal expansion is purely radial in a stationary frame of reference. Consider a spherical coordinate system $(r,\,\theta,\,\phi)$ that co-rotates with the Sun. Of course, the symmetry axis of the coordinate system is assumed to coincide with the axis of the Sun's rotation. In the rotating coordinate system, the velocity components of the solar wind are written

$$u_r = u,$$ (8.78)

$$u_\theta =0,$$ (8.79)

$$u_\phi = - {\mit\Omega}\,r\,\sin\theta,$$ (8.80)

where ${\mit\Omega}= 2.7\times 10^{-6}\,{\rm rad\,sec}^{-1}$ is the angular velocity of solar rotation (Yoder 1995). The azimuthal velocity $u_\phi$ is entirely due to the transformation to the rotating frame of reference. The stream-lines of the flow satisfy the differential equation (Richardson 2019)

$$\frac{1}{r\,\sin\theta}\frac{dr}{d\phi} \simeq \frac{u_r}{u_\phi} = -\frac{u}{ {\mit\Omega}\,r\,\sin\theta}$$ (8.81)

at constant $\theta$. The stream-lines are also magnetic field-lines, so Equation (8.81) can be regarded as the differential equation of a magnetic field-line. For radii, $r$, greater than several times the critical radius, $r_c$, the solar wind solution (8.77) predicts that $u(r)$ is almost constant. (See Figure 8.3.) Thus, for $r\gg r_c$, it is reasonable to write $u(r) = u_s$, where $u_s$ is a constant. Equation (8.81) can then be integrated to give the equation of a magnetic field-line,

$$r-r_0 = -\frac{u_s}{{\mit\Omega}}\,(\phi-\phi_0),$$ (8.82)

where the field-line is assumed to pass through the point $(r_0,\,\theta,\,\phi_0)$. Maxwell's equation $\nabla\cdot{\bf B}=0$, plus the assumption of a spherically symmetric magnetic field, easily yield the following expressions for the components of the interplanetary magnetic field:

$$B_r(r,\theta,\phi) = B(r_0,\theta,\phi_0)\left(\frac{r_0}{r}\right)^2,$$ (8.83)

$$B_\theta(r,\theta,\phi) =0,$$ (8.84)

$$B_\phi(r,\theta,\phi) = - B(r_0,\theta,\phi_0)\,\frac{{\mit\Omega}\,r_0}{u_s} \,\frac{r_0}{r}\,\sin\theta.$$ (8.85)

Figure 8.4 illustrates the interplanetary magnetic field close to the ecliptic plane (i.e., $\theta=\pi/2$). The magnetic field-lines of the Sun are drawn into Archimedean spirals by the solar rotation. Transformation to a stationary frame of reference gives the same magnetic field configuration, with the addition of an electric field

$${\bf E} = - {\bf u} \times {\bf B} = u_s\,B_\phi\,{\bf e}_\theta$$ (8.86)

The latter field arises because the radial plasma flow is no longer parallel to magnetic field-lines in the stationary frame.

Figure 8.4: The interplanetary magnetic field in the ecliptic plane.

The interplanetary magnetic field at 1 AU is observed to lie in the ecliptic plane, and is directed at an angle of approximately $45^\circ$ from the radial direction to the Sun (Priest 1984). This is in basic agreement with the spiral configuration predicted previously.

The previous analysis is premised on the assumption that the interplanetary magnetic field is too weak to affect the coronal outflow, and is, therefore, passively convected by the solar wind. In fact, this is only the case if the interplanetary magnetic energy density, $B^{2}/(2\,\mu_0)$, is much less that the kinetic energy density, $\rho \,u^2/2$, of the solar wind. Rearrangement yields the condition

$$u > V_A,$$ (8.87)

where $V_A$ is the Alfvén speed. It turns out that $u\sim 10\,V_A$ at 1 AU. On the other hand, $u\ll V_A$ close to the base of the corona. In fact, the solar wind becomes super-Alfvénic at a radius, denoted $r_A$, which is typically $50\,R_\odot$, or $1/4$ of an astronomical unit (Priest 1984). We conclude that the previous analysis is only valid well outside the Alfvén radius: that is, in the region $r\gg r_A$.

Well inside the Alfvén radius (i.e., in the region $r\ll r_A$), the solar wind is too weak to modify the structure of the solar magnetic field. In fact, in this region, we expect the solar magnetic field to force the solar wind to co-rotate with the Sun. Observe that flux-freezing is a two-way-street: if the energy density of the flow greatly exceeds that of the magnetic field then the magnetic field is passively convected by the flow, but if the energy density of the magnetic field greatly exceeds that of the flow then the flow is forced to conform to the magnetic field.

The previous discussion leads us to the following, rather crude, picture of the interaction of the solar wind and the interplanetary magnetic field. We expect the interplanetary magnetic field to be the undistorted continuation of the Sun's magnetic field for $r<r_A$. On the other hand, we expect the interplanetary field to be dragged out into a spiral pattern for $r>r_A$. Furthermore, we expect the Sun's magnetic field to impart a non-zero azimuthal velocity $u_\phi(r)$ to the solar wind. In the ecliptic plane ( $\theta=\pi/2$), we infer that

$$u_\phi = {\mit\Omega}\,r$$ (8.88)

for $r<r_A$, and

$$u_\phi = {\mit\Omega}\,r_A\left(\frac{r_A}{r}\right)$$ (8.89)

for $r>r_A$. This corresponds to co-rotation with the Sun inside the Alfvén radius, and outflow at constant angular velocity outside the Alfvén radius. We, therefore, expect the solar wind at 1 AU to possess a small azimuthal velocity component. This is indeed the case. In fact, the direction of the solar wind at 1 AU deviates from purely radial outflow by about $1.5^\circ$ (Priest 1984).