3  Kinematics

3.1 Faraday’s Law

To determine how magnetic fields evolve in time, we need another fundamental law from electromagnetism. This concerns the magnetic flux through a closed curve \(\gamma\), which is defined as \[ \Phi_\gamma = \int_S\Bb\cdot\dS, \] where \(S\) is a spanning surface. According to Faraday’s Law (of induction), the flux through any closed material curve \(\gamma_t\) obeys \[ \boxed{ \ddt{\Phi_{\gamma_t}} = -\oint_{\gamma_t}\big(\Eb + \ub\times\Bb\big)\cdot\dl.} \tag{3.1}\] Here \(\Eb\) is the electric field in a fixed frame of reference, and \(\Eb+\ub\times\Bb\) is the electric field in the frame of reference moving with the fluid. The right-hand side is called the electromotive force induced in \(\gamma_t\) by the changing magnetic field.

Sketch of Faraday's Law.

Figure 3.1: Definition sketch for Faraday’s Law.

To illustrate why the electric field in a moving frame is \(\Eb+\ub\times\Bb\), consider the equations \[ \nabla\cdot\Bb=0, \quad \nabla\times\Bb=\mu_0\Jb, \quad \ddy{\Bb}{t} = -\nabla\times\Eb. \] Under a Galilean transformation \(\xb'=\xb-\Ub t\), \(t'=t\), for a constant vector \(\Ub\), you can show that these equations remain invariant if \(\Bb'=\Bb\) and \(\Eb'=\Eb + \Ub\times\Bb\), but not if \(\Eb'=\Eb\).

Sketch of how turbines induce electric current.

Figure 3.2: Faraday’s Law is the principle by which most of our electricity is generated: a turbine turns a conducting loop in a fixed magnetic field, and the electromotive force induces a current in the loop.

The S.I. units of electric field, \(\Eb\), are Volts per meter, \(\mathrm{V}\,\mathrm{m}^{-1}\). We will see shortly that \(\Eb\) can effectively be eliminated from the MHD equations provided that \(\ub\) is non-relativistic.

Faraday himself recognised that salt water could conduct electricity, and famously tried to measure the current induced by the flow of the River Thames under Waterloo Bridge, through the Earth’s magnetic field. (The idea was sound but the effect too weak for him to measure.)

Again, in any region where \(\Eb\), \(\ub\) and \(\Bb\) are differentiable, we can derive a differential form of Equation 3.1. To rewrite the left-hand side of Equation 3.1, we use the transport formula (F23) for the rate of change of an integral over a material surface, which gives \[ \ddt{\Phi_{\gamma_t}} = \ddt{}\int_{S_t}\Bb\cdot\dS=\int_{S_t}\left(\ddy{\Bb}{t} - \nabla\times(\ub\times\Bb)+\ub\nabla\cdot\Bb\right)\cdot\dS, \] where \(S_t\) spans \(\gamma_t\). Noting that \(\nabla\cdot\Bb=0\), we then apply Stokes Theorem to the right-hand side of Equation 3.1 and obtain \[\begin{align} \int_{S_t}\left(\ddy{\Bb}{t} - \nabla\times(\ub\times\Bb)\right)\cdot\dS &= -\int_{S_t}\Big(\nabla\times\Eb + \nabla\times(\ub\times\Bb)\Big)\cdot\dS\\ \iff\quad \int_{S_t}\ddy{\Bb}{t}\cdot\dS &= -\int_{S_t}\nabla\times\Eb\cdot\dS. \end{align}\] Since this holds for arbitrary \(\gamma_t\) within our region, we must have \[ \frac{\partial\Bb}{\partial t} = -\nabla\times\Eb. \tag{3.2}\]

Equation 3.2 is famous as one of Maxwell’s equations for electromagnetism.

If \(\ub\) were not smooth, then the curve \(\gamma_t\) in Faraday’s Law could bifurcate into separate disconnected pieces.

3.2 Ohm’s Law

To complete the MHD description we need another equation relating \(\Eb\) and \(\Bb\). This takes the form of Ohm’s Law. It is not a fundamental law of nature, but rather an approximation that holds in non-relativistic fluids.

cf. the equation of state in fluid mechanics.

Ohm’s Law says that the current density in our conducting fluid is proportional to the electric field in the frame moving with the fluid, so \[ \boxed{\Jb = \sigma\big(\Eb + \ub\times\Bb\big).} \tag{3.3}\] The scalar \(\sigma\) is the electrical conductivity of the fluid, which we will assume to be constant in this course.

It is actually possible to derive Ohm’s Law from statistical mechanics of the underlying charged particles, under suitable assumptions. But this is outside the scope of this course. Sometimes these assumptions are relaxed and additional terms included in Ohm’s Law, for example the Hall term.

3.3 The induction equation

If we substitute Ohm’s Law Equation 3.3 into the differential form Equation 3.2 of Faraday’s Law, we can eliminate \(\Eb\) to obtain \[\begin{align} \frac{\partial\Bb}{\partial t} &= -\nabla\times\left(\frac{1}{\sigma}\Jb - \ub\times\Bb\right)\\ &= -\frac{1}{\mu_0\sigma}\nabla\times\nabla\times\Bb + \nabla\times\big(\ub\times\Bb\big)\\ &= -\frac{1}{\mu_0\sigma}\Big[\nabla(\nabla\cdot\Bb) - \Delta\Bb\Big] + \nabla\times\big(\ub\times\Bb\big), \end{align}\] where we used (F6) on the Formula Sheet and assumed \(\sigma\) to be constant (for simplicity). Using the solenoidal condition Equation 2.2, we obtain the important induction equation \[ \boxed{\frac{\partial\Bb}{\partial t} = \nabla\times\big(\ub\times\Bb\big) + \eta\Delta\Bb,} \tag{3.4}\] where \(\displaystyle \eta = \frac{1}{\mu_0\sigma}\) is called the magnetic diffusivity (units \(\mathrm{m}^2\mathrm{s}^{-1}\)).

Warning: many authors use \(\eta\) to denote resistivity, with \(\eta=\sigma^{-1}\).

Notice that Equation 3.4 is mathematically equivalent to the vorticity equation for a viscous fluid, if \(\Bb\leftrightarrow\omb\). There is however the important difference that \(\omb=\nabla\times\ub\), whereas \(\Bb\neq\nabla\times\ub\).

We can compare the sizes of the two terms on the right-hand side of Equation 3.4 by supposing that the solution varies on a characteristic lengthscale \(L_0\), and taking characteristic values \[ |\ub| \sim u_0, \quad t\sim\frac{L_0}{u_0}, \quad |\Bb|\sim B_0. \] Then \[ \begin{cases} \displaystyle\big|\nabla\times\big(\ub\times\Bb\big)\big| \approx \frac{u_0 B_0}{L_0}\\ \displaystyle\big|\eta\Delta\Bb\big| \approx \frac{\eta B_0}{L_0^2} \end{cases} \implies \frac{\big|\nabla\times\big(\ub\times\Bb\big)\big|}{\big|\eta\Delta\Bb\big|} \approx \frac{u_0 L_0}{\eta} = \mathrm{Rm}, \] where the dimensionless number \(\mathrm{Rm}\) is called the magnetic Reynolds number. High \(\textrm{Rm}\) means less resistive/diffusive.

This is analogous to the Reynolds number \(\displaystyle\Rey=\frac{L_0u_0}{\nu}\) in viscous fluids, which measures the relative importance of viscosity. Like viscosity, resistivity is more important on smaller scales and in slower flows, at least for a given conductivity \(\sigma\).

You can show that defining the dimensionless variables \(\displaystyle \ub_1 = \frac{1}{u_0}\ub\), \(\displaystyle \xb' = \frac{1}{L_0}\xb\), \(\displaystyle t' = \frac{L_0}{u_0}t\), \(\displaystyle \Bb' = \frac{1}{B_0}\Bb\) would reduce Equation 3.4 to the form \[ \ddy{\Bb'}{t'} = \nabla'\times\big(\ub_1\times\Bb'\big) + \frac{1}{\textrm{Rm}}\Delta'\Bb'. \]

3.4 The ideal limit

The infinite-conductivity limit \(\textrm{Rm}\to\infty\), where \(\eta= 0\), is called ideal MHD, in which case \[\frac{\partial\Bb}{\partial t} = \nabla\times\big(\ub\times\Bb\big). \tag{3.5}\]

For a given steady flow \(\ub(\xb)\), the evolution of \(\Bb\) is entirely determined by this equation.

Later we will see how \(\ub\) is, in general, also dependent on \(\Bb\), so that the full dynamical MHD equations form a coupled system.

Consider the effect of an imposed velocity field \(\ub\) on a magnetic field under Equation 3.5. If \(\ub=\Ub\) is a uniform flow, then \[\begin{align} \nabla\times\big(\Ub\times\Bb\big) &= (\Bb\cdot\nabla)\Ub - (\Ub\cdot\nabla)\Bb + \Ub\nabla\cdot\Bb - \Bb\nabla\cdot\Ub \quad \textrm{[by formula sheet (F6)]}\\ &= - (\Ub\cdot\nabla)\Bb \quad \textrm{[using the solenoidal condition and fact that $\Ub$ is constant]}, \end{align}\] so the ideal induction equation reduces to pure advection, \[ \ddy{\Bb}{t} = -(\Ub\cdot\nabla)\Bb \qquad \iff \DDt{\Bb} = 0. \tag{3.6}\]

Here Equation 3.6 gives \[ \ddy{\Bb}{t} = -U_0\ddy{\Bb}{x} = -U_0\evy \qquad \implies \Bb = -y\evx + (x-U_0t)\evy. \] So for \(U_0>0\) the original magnetic field lines are just moved steadily in the direction of \(\ub\).

However, when \(\ub\) depends on \(\xb\), the behaviour can be more complicated. In particular, it is possible for flows to amplify magnetic field, at different rates depending on the flow.

Recall that \(\displaystyle\mathrm{erf}(y)= \frac{2}{\sqrt{\pi}}\int_0^y\mathrm{e}^{-t^2}\,\mathrm{d}t\).

Here we have \(\ub\times\Bb = -u_xB_z\evy + u_xB_y\evz\), so \[ \nabla\times\big(\ub\times\Bb\big) = \left(\ddy{}{y}(u_xB_y) + u_x\ddy{B_z}{z}\right)\evx - u_x\ddy{B_y}{x}\evy - u_x\ddy{B_z}{x}\evz. \] Now at \(t=0\), we have \(B_z=0\) and \(B_y=B_0\) (constant), so the only non-zero component will be the \(x\)-component. It follows that \(B_z=0\) and \(B_y=B_0\) for all time, so the ideal induction equation Equation 3.5 reduces to \[ \ddy{B_x}{t} = \ddy{}{y}(u_xB_y) = B_0\ddy{u_x}{y} = \frac{2B_0}{\sqrt{\pi}}\mathrm{e}^{-y^2}. \] Integrating gives \[ \Bb = \frac{2B_0t}{\sqrt{\pi}}\mathrm{e}^{-y^2}\evx + B_0\evy, \] so this shear flow layer generates a linear growth rate in \(B_x\).

The streamlines of this flow are given by \[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-y}{x} \quad \implies xy = \textrm{constant}. \] Thus we expect \(\Bb\) to be carried towards the \(y\)-axis. Notice that \[ \nabla\times\big(\ub\times B_y(x,0)\evy\big) = -\nabla\times\big(x B_y(x,0)\evz\big) = \ddy{}{x}\big(x B_y(x,t)\big)\evy, \] so \(\Bb\) will remain of the form \(\Bb(\xb,t)=B_y(x,t)\evy\).

For such a field, the ideal induction equation Equation 3.5 reduces to \[ \ddy{B_y}{t} - x\ddy{B_y}{x}= B_y. \tag{3.7}\] This type of PDE can be solved by the method of characteristics. The idea is to consider the characteristic curves \(x(t)\) that satisfy \[ \ddt{x} = -x, \] where the right-hand side is the coefficient of \(\displaystyle\ddy{B_y}{x}\) in Equation 3.7. Along such a curve, Equation 3.7 reduces to an ODE, \[ \ddt{B_y} = B_y \qquad \implies B_y(x,t) = B_y(x_0,0)\mathrm{e}^t = (1-x_0^2)\mathrm{e}^t, \] where \(x_0\) is the initial position of the characteristic through \(x\). We obtain this initial position by solving the characteristic equation to find \[ x(t) = x_0\mathrm{e}^{-t} \qquad \implies x_0 = x\mathrm{e}^t, \] so \[ B_y(x,t) = (1 - x^2\mathrm{e}^{2t})\mathrm{e}^t = \mathrm{e}^t - x^2\mathrm{e}^{3t}. \] So the field lines with \(B_y>0\) pile up at \(x=0\). This time the amplification is exponential, even though the flow is incompressible.

3.5 Topology conservation

A fundamental property of ideal MHD, whether or not \(\ub\) is prescribed kinematically or determined dynamically, is the conservation of magnetic topology.

This is analogous to the conservation of vortex topology in an ideal (inviscid) fluid.

Theorem 3.1 (Alfvén) In the limit of infinite conductivity, \(\sigma\to\infty\), then \[ \ddt{\Phi_{\gamma_t}} = 0 \quad \textrm{for any closed material curve $\gamma_t$}. \]

Proof. Substituting Ohm’s Law Equation 3.3 into the integral form of Faraday’s Law Equation 3.1 gives \[ \ddt{\Phi_{\gamma_t}} = -\oint_{\gamma_t} \frac{1}{\sigma}\Jb\cdot\dl. \] Taking \(\sigma\to\infty\) gives the result.

This is known as flux freezing, as the magnetic field is frozen-in to the fluid. For suitably smooth fields, Theorem Theorem 3.1 is equivalent to the ideal induction equation Equation 3.5. To see this, note that the Transport Theorem (F23) for surface integrals shows, for any material curve \(\gamma_t\), that \[ \ddt{\Phi_{\gamma_t}} = \ddt{}\int_{S_t}\Bb\cdot\dS = \int_{S_t}\left(\ddy{\Bb}{t} - \nabla\times\big(\ub\times\Bb\big) + \ub\nabla\cdot\Bb\right)\cdot\dS \] \[= \int_{S_t}\left(\ddy{\Bb}{t} - \nabla\times\big(\ub\times\Bb\big)\right)\cdot\dS \quad \textrm{[using the solenoidal condition]}. \tag{3.8}\] So the integral vanishes for any \(\gamma_t\) in some region of space if and only if Equation 3.5 holds throughout that region.

Hannes Alfvén was a Swedish physicist who won the Nobel Prize for Physics in 1970 for his pioneering work on MHD, published in the 1940s.

By Stokes’ Theorem, we can write \(\Phi_{\gamma_t}=\oint_{\gamma_t}\Ab\cdot\dl\). So Alfvén’s Theorem is similar to Kelvin’s Theorem for an ideal barotropic fluid, whereby \[ \displaystyle\ddt{}\oint_{\gamma_t}\ub\cdot\dl = 0, \] and it is the vorticity \(\omb\) rather than \(\Bb\) that is frozen-in to the fluid. The analogy is not total, however, since in the Alfvén case, \(\Ab\) is not the velocity field transporting \(\gamma_t\).

A typical star such as the Sun has radius \(r_*\sim 10^6\,\mathrm{km}\) and magnetic field \(B_*\sim 10^{-2}\,\mathrm{T}\). If its mass is in a certain range, it can collapse at the end of its lifetime to a neutron star with radius \(r_{\textrm{N}}\sim 10\,\textrm{km}\) and \(B_{\textrm{N}}\sim 10^8\,\mathrm{T}\). Why is \(B_{\textrm{N}}\) so much higher than \(B_*\)? Consider the magnetic flux through the star’s equator.

Initially, we have \(\displaystyle \int_{S_*}\Bb\cdot\dS \sim B_*\pi r_*^2 = 10^{-2}\pi(10^6)^2 \sim 10^{10}\,\mathrm{km}^2\,\mathrm{T}.\)

After collapse, we have \(\displaystyle \int_{S_{\textrm{N}}}\Bb\cdot\dS \sim B_{\textrm{N}}\pi r_{\textrm{N}}^2 = 100B_{\textrm{N}}.\)

Assuming an infinitely-conducting evolution, the two fluxes must be equal so \(\displaystyle B_{\textrm{N}} \sim \frac{10^{10}}{100} = 10^8\,\mathrm{T}.\)

A similar argument explains why stars, which form due to collapse of interplanetary magnetic clouds, have much stronger magnetic fields than the background galactic magnetic field.

Another important consequence of Equation 3.5 is field line conservation.

Corollary 3.1 In an infinitely conducting fluid, magnetic field lines are material lines.

Proof. We will derive the general condition for field line conservation, then show that it follows from Equation 3.5. Let \(\gamma_t\) be a material curve, given by \(\xb(s; t)\) where \(s\) is arclength along the curve. Suppose that \(\gamma_t\) coincides with a magnetic field line at \(t=0\), so that initially its tangent vector is parallel to \(\Bb\), \[ \ddy{\xb}{s}\times\Bb = \bfzero \quad \iff\ \quad \ddy{\xb}{s} = \lambda_0\Bb \quad \textrm{for some scalar $\lambda_0(\xb)$.} \tag{3.9}\]

Now consider the time evolution \[\begin{align} \DDt{}\left( \ddy{\xb}{s}\times\Bb \right) &= \DDt{}\left(\ddy{\xb}{s}\right)\times\Bb + \ddy{\xb}{s}\times\DDt{\Bb}\\ &= \ddy{}{s}\left(\DDt{\xb}\right)\times\Bb + \ddy{\xb}{s}\times\DDt{\Bb}\\ &= \ddy{\ub}{s}\times\Bb + \ddy{\xb}{s}\times\DDt{\Bb} \quad \textrm{[since $\xb(s,t)$ is a material curve]}\\ &= \left[\left(\ddy{\xb}{s}\cdot\nabla\right)\ub\right]\times\Bb + \ddy{\xb}{s}\times\DDt{\Bb} \quad \textrm{[by the chain rule]}. \end{align}\] At \(t=0\), we can substitute Equation 3.9 to find \[\begin{align} \left.\DDt{}\left( \ddy{\xb}{s}\times\Bb \right)\right|_{t=0} &= \Big[\lambda_0(\Bb\cdot\nabla)\ub\Big]\times\Bb + \lambda_0\Bb\times\DDt{\Bb}\\ &= \lambda_0\Bb\times\left[\DDt{\Bb} - (\Bb\cdot\nabla)\ub\right]\\ &= \lambda_0\Bb\times\left[\ddy{\Bb}{t} + (\ub\cdot\nabla)\Bb - (\Bb\cdot\nabla)\ub\right]\\ &= \lambda_0\Bb\times\left[\ddy{\Bb}{t} - \nabla\times\big(\ub\times\Bb\big)\right] \quad \textrm{[by formula (F6)]}. \end{align}\] It follows that \(\gamma_t\) will remain a magnetic field line for all times provided that \[ \ddy{\Bb}{t} - \nabla\times\big(\ub\times\Bb\big) = \lambda\Bb \quad \textrm{for some $\lambda(\xb,t)\in\mathbb{R}$}. \tag{3.10}\] Clearly this holds for the ideal MHD induction equation Equation 3.5, which corresponds to \(\lambda\equiv 0\).

Notice more generally from Equation 3.8 and Equation 3.10 that flux conservation implies line conservation, but not the converse.

Since magnetic field lines are material lines, they cannot pass through each other so their linkage and topology must remain invariant. As for vorticity in inviscid fluids, there is an integral invariant of Equation 3.5 associated with the linkage between magnetic field lines, called magnetic helicity \(H\).

Theorem 3.2 Let \(\Ab\) be any vector potential for \(\Bb\) in a volume \(V\). If \(\nb\cdot\Bb=0\) and \(\nb\cdot\ub=0\) on the boundary \(\partial V\), then under ideal MHD, \[ \ddt{H}=0\quad \textrm{where} \quad H=\int_{V}\Ab\cdot\Bb\dV. \]

Loop around a magnetic field line.

Figure 3.3: Why is \(H\) related to field line linkage? This was studied in more detail for kinetic helicity \(\ub\cdot\omb\) in Fluid Mechanics III, but it ultimately boils down to Stokes’ Theorem: since \(\Bb=\nabla\times\Ab\), a magnetic field line must have some component of \(\Ab\) “around it”, so if \(\Ab\cdot\Bb\neq 0\) then \(\Bb\) must be “wrapping around itself”.

Proof. Since the volume is fixed we have \[ \ddt{H} = \ddt{}\int_V\Ab\cdot\Bb\dV \] \[ = \int_V\ddy{}{t}(\Ab\cdot\Bb) \dV \] \[ = \int_V\left(\ddy{\Ab}{t}\cdot\Bb + \Ab\cdot\ddy{\Bb}{t}\right)\dV. \tag{3.11}\] It is neatest to work with Faraday’s Law Equation 3.2, bearing in mind that the ideal Ohm’s Law fixes \(\Eb=-\ub\times\Bb\). Uncurling Equation 3.2 gives \[ \ddy{\Ab}{t} = -\Eb + \nabla\phi \] for some arbitrary scalar function \(\phi(\xb,t)\) that depends on the particular choice of \(\Ab\). Then \[\begin{align} \ddt{H} &= \int_V\Big(-\Eb\cdot\Bb +\nabla\phi\cdot\Bb - \Ab\cdot\nabla\times\Eb \Big)\dV\\ &= \int_V\Big(\nabla\phi\cdot\Bb - \Ab\cdot\nabla\times\Eb \Big) \quad\textrm{[by ideal Ohm's Law]}\\ &= \int_V\Big(\nabla\cdot(\phi\Bb) - \Ab\cdot\nabla\times\Eb \Big)\dV\quad \textrm{[using $\nabla\cdot\Bb=0$]}\\ &= \int_V\Big(\nabla\cdot(\phi\Bb + \Ab\times\Eb) - \Eb\cdot\nabla\times\Ab \Big)\dV \quad \textrm{[by formula (F5)]}\\ &= \int_V\Big(\nabla\cdot(\phi\Bb + \Ab\times\Eb) - \Eb\cdot\Bb \Big)\dV\\ &= \int_V\nabla\cdot(\phi\Bb + \Ab\times\Eb)\dV \quad \textrm{[by ideal Ohm's Law again]}\\ &= \oint_{\partial V}(\phi\Bb + \Ab\times\Eb)\cdot\dS\\ &= \oint_{\partial V}\big(\phi\Bb - \Ab\times(\ub\times\Bb)\big)\cdot\dS\\ &= \oint_{\partial V}\big(\phi\Bb - (\Ab\cdot\Bb)\ub + (\Ab\cdot\ub)\Bb\big)\cdot\dS. \end{align}\] This integral vanishes thanks to the boundary conditions \(\nb\cdot\Bb=\nb\cdot\ub=0\) on \(\partial V\).

Under the conditions of Theorem Theorem 3.2, one can also show that the value of \(H\) is independent of the choice of \(\Ab\). [See problem sheet.]

Observation of a solar filament eruption.

Figure 3.4: There are spectacular consequences of magnetic helicity conservation in solar physics, for example solar filaments. These form when magnetic helicity supplied from the Sun’s interior into its atmosphere becomes concentrated along narrow filament channels. Once they become too twisted, they erupt, as in the following photograph (in extreme-ultraviolet) from NASA Solar Dynamics Observatory. These filament eruptions lead to coronal mass ejections. When the resulting interplanetary magnetic clouds are detected by near-Earth spacecraft, they are found to have twisted structures because they still retain their conserved magnetic helicity.

3.6 The diffusive limit

For small \(\textrm{Rm}\), or large \(\Delta\Bb\), the resistive term in the induction equation Equation 3.4 cannot be ignored. Here we consider the case where this term dominates, so the induction equation reduces (for constant \(\eta\)) to the diffusion equation \[ \ddy{\Bb}{t} = \eta\Delta\Bb. \tag{3.12}\]

Firstly, observe that steady states of this equation satisfy \(\Delta\Bb=\bfzero\), or equivalently \(\nabla\times\nabla\times\Bb=\bfzero\). So, in particular, potential fields with \(\nabla\times\Bb=0\) will remain unchanged under pure diffusion.

In fact, we will see later (when considering energy) that potential fields \(\nabla\times\Bb\) are the only equilibria that can actually be reached in practice by a diffusive evolution.

An important physical effect of diffusion is to smooth out so-called current sheets – thin layers where \(\Jb=\mu_0^{-1}\nabla\times\Bb\) is large.

Taking the curl of Equation 3.12 shows that \(\Jb\) satisfies the same equation, and taking the ansatz \(\Jb(\xb,t) = J(x,t)\evz\) gives the scalar diffusion equation \[ \ddy{J}{t} = \eta\ddy{^2J}{x^2}. \tag{3.13}\] One way to solve this initial value problem is to use the Fourier transform, defined as \[ \widetilde{J}(k,t) = \frac{1}{2\pi}\int_{-\infty}^\infty J(x,t)\mathrm{e}^{-ikx}\,\mathrm{d}x, \] for which the inverse is \[ J(x,t) = \int_{-\infty}^\infty \widetilde{J}(k,t)\mathrm{e}^{ikx}\,\mathrm{d}k. \] If we multiply Equation 3.13 by \((2\pi)^{-1}\mathrm{e}^{-ikx}\) and integrate, we obtain \[\begin{align} \frac{1}{2\pi}\int_{-\infty}^\infty \ddy{J}{t}\mathrm{e}^{-ikx}\,\mathrm{d}x &= \frac{\eta}{2\pi}\int_{-\infty}^\infty \ddy{^2J}{x^2}\mathrm{e}^{-ikx}\,\mathrm{d}x\\ \iff \quad \ddy{\widetilde{J}}{t} &= \frac{\eta}{2\pi}\int_{-\infty}^\infty\ddy{}{x}\left(\ddy{J}{x}\mathrm{e}^{-ikx}\right)\,\mathrm{d}x - \frac{\eta}{2\pi}\int_{-\infty}^\infty\ddy{J}{x}\ddy{}{x}\big(\mathrm{e}^{-ikx}\big)\,\mathrm{d}x\\ \iff \quad \ddy{\widetilde{J}}{t} &= ik\frac{\eta}{2\pi} \int_{-\infty}^\infty \ddy{J}{x}\mathrm{e}^{-ikx}\,\mathrm{d}x \quad \textrm{[assuming $\ddy{J}{x}\to 0$ as $|x|\to\infty$]}\\ \iff \quad \ddy{\widetilde{J}}{t} &= ik\frac{\eta}{2\pi}\int_{-\infty}^\infty\ddy{}{x}\big(J\mathrm{e}^{-ikx}\big)\,\mathrm{d}x + (ik)^2\frac{\eta}{2\pi}\int_{-\infty}^\infty J \mathrm{e}^{-ikx}\,\mathrm{d}x\\ \iff \ddy{\widetilde{J}}{t} &= - k^2\eta\widetilde{J} \quad \textrm{[assuming $J\to 0$ as $|x|\to\infty$]}. \end{align}\] Solving this equation for each \(k\) separately gives \(\widetilde{J}(k,t) = \widetilde{J}(k,0)\mathrm{e}^{-k^2\eta t}\), and so the solution is \[ J(x,t) = \int_{-\infty}^\infty \widetilde{J}(k,0)\mathrm{e}^{ikx -k^2\eta t}\,\mathrm{d}k. \] From our initial condition, we calculate \(\displaystyle\widetilde{J}(k,0) = \frac{1}{2\pi}\int_{-\infty}^\infty \delta(x)\mathrm{e}^{-ikx}\,\mathrm{d}x = \frac{1}{2\pi}\). Finally, we can evaluate the integral in \(k\) by completing the square so as to change it to a standard gaussian integral: \[ J(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty \mathrm{e}^{-\eta t\left( k - \frac{ix}{2\eta t}\right)^2 - \frac{x^2}{4\eta t}}\,\mathrm{d}k \] \[ = \frac{1}{2\pi\sqrt{\eta t}}\mathrm{e}^{-\frac{x^2}{4\eta t}}\int_{-\infty}^\infty\mathrm{e}^{-u^2}\,\mathrm{d}u = \frac{1}{\sqrt{4\pi\eta t}}\mathrm{e}^{-\frac{x^2}{4\eta t}}. \] So the initial current sheet with infinitesimal width spreads out and reduces its amplitude over time. The corresponding magnetic field is given by \(\Bb=B(x,t)\evy\) with \(J_z = \partial B_y/\partial x\) giving \[ B(x,t) - B_0 = \int_0^x J(w,t)\,\mathrm{d}w = \frac{1}{\sqrt{4\pi\eta t}}\int_0^x\mathrm{e}^{-\frac{w^2}{4\eta t}}\,\mathrm{d}w = \frac{1}{\sqrt{\pi}}\int_0^{\frac{x}{\sqrt{4\eta t}}}\mathrm{e}^{-u^2}\,\mathrm{d}u \] \[ = \frac12\mathrm{erf}\left(\frac{x}{\sqrt{4\eta t}}\right). \]

We could also have derived this solution using a similarity ansatz (cf. Fluid Mechanics III).

In general, diffusion causes non-potential magnetic fields to decay. To estimate the decay timescale \(T_\eta\), suppose a magnetic field has characteristic strength \(|\Bb|\sim B_0\) and lengthscale \(L_0\). Then Equation 3.12 gives \[ \begin{cases} \displaystyle\left|\ddy{\Bb}{t}\right| \approx \frac{B_0}{T_\eta}\\ \displaystyle\left|\eta\Delta\Bb\right|\sim\frac{\eta B_0}{L_0^2} \end{cases} \implies T_\eta = \frac{L_0^2}{\eta}. \tag{3.14}\] So magnetic structures with small lengthscale decay faster, and the decay is faster when \(\eta\) is larger.

3.7 Kinematic dynamos

A (fluid) dynamo is a system where an initial “seed” magnetic field is amplified by fluid motions. Here we will consider only the kinematic dynamo problem, where we ask whether a given \(\ub\) will generate an increasing \(\Bb\).

The full dynamical problem, where \(\ub\) and \(\Bb\) are coupled, is beyond the scope of this course (and very much the subject of current research).

Fluid dynamos are required in order to explain the observed magnetic fields of many astrophysical bodies.

Planets are observed with magnetic fields of \(10^{-8}\,\mathrm{T}-4\times 10^{-4}\,\mathrm{T}\) (including the Earth with \(\sim 0.3\times 10^{-4}\,\mathrm{T}\)). Stars are observed with magnetic fields of \(10^{-4}\,\mathrm{T}-10\,\mathrm{T}\) (including the Sun with \(\sim 10^{-1}\,\mathrm{T}\)).

Dynamos may be required for several reasons:

  1. Permanent magnets have magnetic fields (ferromagnetism, where small internal dipoles align and lead to a net external \(\Bb\)). But iron, for example loses magnetism above a critical temperature \(\sim 1000\,\mathrm{K}\), and this is exceeded inside planets and stars.
  2. Fossil (primordial) magnetic fields are possible. But for Earth, the lengthscale is \(L_0\sim 10^3\,\mathrm{km}\) and \(\eta_{\textrm{iron}}\) gives a resistive decay time from Equation 3.14 of \(T_\eta\sim 20\,000\,\mathrm{years}\). This is much less than the age of the Earth \(\sim 10^9\,\mathrm{years}\). (For the Sun, \(T_\eta\sim 10^{12}\,\mathrm{years}\) so a fossil field would not be ruled out by this argument.)
  3. In many stars, observations show cycles of varying magnetic activity on a short period (for the Sun, the activity cycle period is about 22 years), which contradicts the possibility of a (very) slowly decaying fossil field.

There are some well-known mathematical results showing that \(\ub\) and \(\Bb\) must be sufficiently “complicated” in order to permit dynamo action.

Theorem 3.3 (Cowling’s Antidynamo Theorem) A magnetic field \(\Bb(x,y,t)\) that is invariant in \(z\) and vanishes at infinity (in the \(x\), \(y\) directions) cannot be maintained by an incompressible velocity field \(\ub(x,y,t)\) that is also invariant in \(z\).

Proof. Since \(\Bb\) is invariant in \(z\), we may write \[ \Bb = \Bb_h + B(x,y,t)\evz \quad \textrm{where} \quad \Bb_h= \nabla\times(A(x,y,t)\evz), \tag{3.15}\] and similarly \[ \ub = \ub_h + u_z(x,y,t)\evz \quad \textrm{where} \quad \ub_h= \nabla\times(\psi(x,y,t)\evz). \tag{3.16}\] Observe that this implies \(\nabla\cdot\ub=0\). Substituting Equation 3.15 and Equation 3.16 into the induction equation gives \[ \ddy{\Bb_h}{t} + \ddy{B}{t}\evz = \nabla\times\Big(\ub_h\times\Bb_h + \ub_h\times(B\evz) + u_z\evz\times\Bb_h \Big) + \eta\Delta\Bb_h + \eta\Delta(B\evz). \tag{3.17}\] Now \[\begin{align} \ub_h\times\Bb_h &= \ub_h\times(\nabla A\times\evz)\\ &= \evz\cdot\ub_h\nabla A - (\ub_h\cdot\nabla A)\evz\\ &= - (\ub_h\cdot\nabla A)\evz. \end{align}\] and formula (F6) gives \[\begin{align} \nabla\times\big(\ub_h\times(B\evz)\big) &= (B\evz\cdot\nabla)\ub_h - (\ub_h\cdot\nabla)B\evz + \ub_h\nabla\cdot(B\evz) - B\evz\nabla\cdot\ub_h\\ &= - (\ub_h\cdot\nabla)B\evz \end{align}\] and \[\begin{align} \nabla\times(u_z\evz\times\Bb_h) &= (\Bb_h\cdot\nabla)u_z\evz - (u_z\evz\cdot\nabla)\Bb_h + u_z\evz\nabla\cdot\Bb_h - \Bb_h\nabla\cdot(u_z\evz)\\ &= (\Bb_h\cdot\nabla)u_z\evz. \end{align}\] Thus the horizontal component of Equation 3.17 gives \[\begin{align} &\ddy{\Bb_h}{t} = -\nabla\times\Big((\ub_h\cdot\nabla A)\evz\Big) + \eta\Delta\Bb_h\\ &\iff \ddy{A}{t} = -(\ub_h\cdot\nabla A) + \eta\Delta A + \nabla\phi, \end{align}\] for some arbitrary scalar function \(\phi(x,y,z,t)\) which we may set to zero without loss of generality. Considering also the \(z\)-component of Equation 3.17 thus gives \[ \ddy{A}{t} = -(\ub_h\cdot\nabla) A + \eta\Delta A, \tag{3.18}\] \[ \ddy{B}{t} = -(\ub_h\cdot\nabla) B + (\Bb_h\cdot\nabla)u_z + \eta\Delta B. \tag{3.19}\] Notice that both equations contain advection and diffusion terms, but only the \(B\) equation contains a “source term”, \((\Bb_h\cdot\nabla)u_z\), that could amplify the magnetic field.

To prove that no dynamo is possible, multiply Equation 3.18 by \(A\) and integrate over all space to find \[\begin{align} &\int_VA\ddy{A}{t}\dV = \int_V\Big(-A(\ub_h\cdot\nabla)A + \eta A\Delta A\Big)\dV\\ &\iff\quad \ddt{}\int_V\frac{A^2}{2}\dV = \int_V\Big( -\nabla\cdot(\tfrac12 A^2\ub_h) + \eta\nabla\cdot(A\nabla A) - \eta|\nabla A|^2\Big)\dV\\ &\iff\quad \ddt{}\int_V\frac{A^2}{2}\dV = \oint_{\partial V}\Big(-\tfrac12A^2\ub_h + \eta A\nabla A\Big)\cdot\dS - \eta\int_V|\nabla A|^2\dV. \end{align}\] Now if \(\Bb\) vanishes at infinity then \(A\) must tend to a constant so that the boundary term vanishes as \(\partial V\to\infty\) (using \(\nabla\cdot\ub_h=0\)). We conclude that \(A^2\) continually decays until \(A\) is constant.

In that case, \(\Bb_h=\bfzero\), so there is no source term in Equation 3.19, and we can repeat the same argument to show that \(B\) decays to zero.

Cowling’s original axisymmetric version of this result [see problem sheet] was influential, and for some time the possibility that a dynamo could explain the magnetic fields in astrophysical bodies was in doubt. However, we now know that kinematic dynamos are possible in principle – explicit examples are too complex to consider here, one of the simplest being the Ponomarenko dynamo where a non-axisymmetric \(\Bb\) is generated by an axisymmetric \(\ub\).

It is possible to circumvent Cowling’s Theorem and generate a magnetic field that is axisymmetric on a large scale, but with small-scale non-axisymmetric fluctuations. This is the idea behind mean field dynamo theory. The idea is to split \(\Bb\) and \(\ub\) into mean and fluctuating parts, \[ \Bb(\xb,t) = {\Bb}_0(\xb,t) + \Bb_1(\xb,t), \qquad \ub(\xb,t) = {\ub}_0(\xb,t) + \ub_1(\xb,t), \tag{3.20}\] and choose an averaging process that obeys the Reynolds averaging rules, namely:

  1. It is linear: \(\overline{\ab + \bb} = \overline{\ab} + \overline{\bb}\) and \(\overline{c\ab} = c\overline{\ab}\) for \(c\in\mathbb{R}\).
  2. Multiple averages are equivalent: \(\overline{\overline{\ab}} = \overline{\ab}\).
  3. Averages behave like constants under further averaging: \(\overline{\overline{\ab}\times\bb}=\overline{\ab}\times\overline{\bb}\).

Defining \(\Bb_0=\overline{\Bb}\) and \(\ub_0=\overline{\ub}\), it follows from Equation 3.20 that \(\overline{\Bb_1} = \overline{\ub_1}=\bfzero\).

The averaging rules are named after Osborne Reynolds because they originate in a similar idea obeyed to fluid turbulence. We don’t need to worry here about exactly how the average is defined, but you could, for example, take \(\overline{\ab}(\xb) = |V|^{-1}\int_V\ab\dV\) for some volume \(V\) around \(\xb\) that is larger than the fluctuating scale but smaller than the global scale.

Now substitute Equation 3.20 into the induction equation Equation 3.4 to obtain \[ \ddy{}{t}({\Bb}_0 + \Bb_1) = \nabla\times\Big[({\ub}_0 + \ub_1)\times({\Bb}_0 + \Bb_1)\Big] + \eta\Delta({\Bb}_0 + \Bb_1). \] To obtain an equation for \({\Bb}_0\), we now average both sides of this equation. Assuming that averaging commutes with differentiation, we have \[ \ddy{}{t}(\Bb_0 + \cancel{\overline{\Bb_1}}) = \nabla\times\Big[\ub_0\times\Bb_0 + \overline{\ub_0\times\Bb_1} + \overline{\ub_1\times\Bb_0} + \overline{\ub_1\times\Bb_1} \Big] + \eta\Delta(\Bb_0 + \cancel{\overline{\Bb_1}}) \] \[ \iff\quad \ddy{\Bb_0}{t} = \nabla\times\big(\ub_0\times\Bb_0\big) + \nabla\times\big(\overline{\ub_1\times\Bb_1}\big) + \eta\Delta\Bb_0 \quad \textrm{[using the Reynolds rules]} \tag{3.21}\] This is the mean field induction equation. Notice that it differs from the usual induction equation by the term involving \(\boldsymbol\epsilon \equiv \overline{\ub_1\times\Bb_1}\), known as the turbulent electromotive force (e.m.f.). In principle, this means that small scale flows can interact with small-scale magnetic fluctuations to generate a contribution to the large-scale mean field.

Now consider a magnetic field \(\Bb(x,y,t)\) and incompressible flow \(\ub(x,y,t)\) that are invariant in \(z\), as in Cowling’s Theorem. Writing \(\Bb_0 = \Bb_h + B\evz\) and \(\ub_0 = \ub_h + u_z\evz\), with \(\Bb_h=\nabla\times(A\evz)\), the same derivation as in the proof of Theorem 3.3 leads to the equations \[ \ddy{A}{t} = -(\ub_h\cdot\nabla)A \textcolor{red}{+ \epsilon_z} + \eta\Delta A, \] \[ \ddy{B}{t} = -(\ub_h\cdot\nabla)B + (\Bb_h\cdot\nabla)u_z \textcolor{red}{+ \evz\cdot\nabla\times\boldsymbol\epsilon} + \eta\Delta B. \tag{3.22}\] Now there is a possible source term for \(A\), so Cowling’s Theorem can, in principle, be circumvented by the fluctuations. In general, the form of \(\boldsymbol{\epsilon}\) will be very complicated because it depends on the details of the small-scale evolution.

E. N. Parker famously showed that the basic features of the Sun’s dynamo can be explained using mean field theory, if we assume that \(\boldsymbol\epsilon = \alpha\Bb_0\), called the alpha effect. For simplicity, we model the Sun in Cartesian coordinates (!) with \(x\) corresponding to the \(r\)-direction, \(y\) to the \(\theta\)-direction and \(z\) to the \(\phi\)-direction:

We assume a radial shear flow \(\ub_0 = U'x\evz\) inside the Sun (so \(\ub_h=\bfzero\)), and further that the term \(\evz\cdot\nabla\times(\alpha\Bb_0)\) in Equation 3.22 can be neglected compared to the shear term, so that \[\begin{align} \ddy{A}{t} &= \alpha B + \eta\Delta A,\\ \ddy{B}{t} &= U'\ddy{A}{y} + \eta\Delta B. \end{align}\] These equations are linear with constant coefficients, so we will look for propagating wave like solutions \[ A(y,t) = \widetilde{A}\mathrm{e}^{iky + s t}, \qquad B(y,t) = \widetilde{B}\mathrm{e}^{iky + s t} \] where \(\widetilde{A}\) and \(\widetilde{B}\) are constants. Substituting into the equations gives \[\begin{align} \begin{cases} s A &= \alpha B - \eta k^2 A\\ s B &= ikU'A - \eta k^2 B \end{cases} \quad \iff \quad \begin{bmatrix} s + \eta k^2 & -\alpha\\ -ikU' & s + \eta k^2 \end{bmatrix} \begin{bmatrix} A\\B \end{bmatrix} = \begin{bmatrix} 0\\0 \end{bmatrix}. \end{align}\] A non-zero solution for \(A\) and \(B\) requires the determinant to vanish, \[ (s + \eta k^2)^2 = ik\alpha U' \quad \iff \quad s = -\eta k^2 \pm \sqrt{-ik\alpha U'} = -\eta k^2 \pm \frac{i+1}{\sqrt{2}}\sqrt{k\alpha U'}, \] where we used \(\displaystyle\sqrt{i} = \pm\frac{i+1}{\sqrt{2}}\). Now there are two cases to consider (assuming \(k>0\)):

Case 1: \(\alpha U' > 0\). In this case, \(\Re(s) = \pm\sqrt{\frac{k\alpha U'}{2}} - \eta k^2\). Dynamo action requires \(\Re(s) \geq 0\), which is possible only for the positive root, provided that \(k\) satisfies the condition \[ \sqrt{\frac{k\alpha U'}{2}} \geq \eta k^2 \quad \iff \quad \frac{k\alpha U'}{2} \geq \eta^2 k^4 \quad \iff \quad k^3 \leq \frac{\alpha U'}{2\eta^2}. \] The marginally sustained magnetic field when \(\Re(s)=0\) will be a wave of the form \[ A(y,t) = \widetilde{A}\mathrm{e}^{iky + i\sqrt{k\alpha U'/2}t} = \widetilde{A}\mathrm{e}^{ik(y + \eta kt)}, \quad \textrm{and similarly for $B$,} \] which travels in the negative \(y\)-direction (poleward).

Case 2: \(\alpha U' < 0\). In this case, \(\Re(s) = \mp\sqrt{\frac{k|\alpha U'|}{2}} - \eta k^2\). So now dynamo action requires the other root and the condition \[ k^3 \leq \frac{|\alpha U'|}{2\eta^2}. \] Because we took the second root, the marginally sustained field is now \[ A(y,t) = \widetilde{A}\mathrm{e}^{ik(y - \eta k t)}, \] so the wave travels in the positive \(y\)-direction (equatorward).

The observed sunspot butterfly diagram.

Figure 3.5: If \(\alpha U'<0\) in the Northern hemisphere, Parker’s model predicts equatorward propagation of the dynamo wave. This is consistent with the equatorward propagation of sunspots over the solar cycle – the famous butterfly diagram. Sunspots are believed to track the latitude of strong magnetic fields in the solar interior. This plot of observed sunspot data is taken from Muñoz-Jaramillo et al. https://doi.org/10.1038/s41550-018-0638-2

To (roughly) motivate the assumption that \(\boldsymbol\epsilon\) is linearly proportional to \(\Bb_0\), look again at equation Equation 3.21. If \(\Bb_1=\bfzero\) at \(t=0\), then the linearity of the equation shows that \(\boldsymbol\epsilon\) and \(\Bb_0\) are linearly related. Setting \(\boldsymbol\epsilon = \alpha\Bb_0\) is the simplest such relation.