MHD and frozen-in flux¶
On large scales and slow timescales we can treat the plasma as a single conducting fluid. This is magnetohydrodynamics (MHD).
Ohm's law and the induction equation¶
For a fluid moving with velocity \(\mathbf{v}\) and electrical conductivity \(\sigma\), the simplest Ohm's law is
Combine this with Faraday's law \(\partial_t \mathbf{B} = -\nabla\times\mathbf{E}\) and Ampère's law \(\mu_0\mathbf{J} = \nabla\times\mathbf{B}\) (displacement current neglected) to get the induction equation:
Here \(\eta\) is the magnetic diffusivity (units m\(^2\)/s).
Magnetic Reynolds number¶
Compare the two terms using a length scale \(L\) and velocity \(U\):
When the velocity is the Alfvén speed,
this ratio is called the Lundquist number:
Alfvén's frozen-in theorem¶
When \(R_m \gg 1\), diffusion can be dropped and
One can show that the magnetic flux through any surface moving with the fluid is then constant. Field lines behave as if they are frozen into the plasma: two fluid elements on the same field line stay on the same field line forever.
How frozen is it?
In the solar corona, \(L\sim10^7\) m, \(v_A\sim10^6\) m/s and \(\eta\sim1\) m\(^2\)/s, so
The diffusion time \(\tau_D = L^2/\eta \sim 10^{14}\) s is millions of years. In ideal MHD, the topology of the coronal field can never change.
Yet solar flares release stored magnetic energy in minutes. Something must break the frozen-in condition. That something is magnetic reconnection.
Practice problems¶
Try each problem before opening its solution. Click an option to check a concept question;
type a number and press Check for a numerical one (forms like 6.2e7 or 6.2x10^7 both work).
Concept questions¶
Problem 1
What does \(R_m\gg1\) imply?
- The magnetic field diffuses through the plasma quickly
- Field lines move with the plasma (frozen in)
- The plasma is collisionless
- The magnetic field is weak
\(R_m\) compares advection to diffusion. When it is large, diffusion is negligible and the field is carried with the flow.
Problem 2
If the system size \(L\) doubles, how does the magnetic diffusion time \(\tau_D = L^2/\eta\) change?
- Doubles
- Quadruples
- Halves
- Unchanged
\(\tau_D\propto L^2\). Large systems diffuse very slowly, which is why astrophysical fields are so well frozen in.
Problem 3
Which speed appears in the Lundquist number \(S = Lv/\eta\)?
- The flow speed
- The Alfvén speed
- The sound speed
- The speed of light
\(S\) is the magnetic Reynolds number evaluated with the Alfvén speed \(v_A = B/\sqrt{\mu_0\rho}\), the natural speed of magnetic disturbances.
Show that…¶
Problem 4: the induction equation
Starting from \(\mathbf{E} + \mathbf{v}\times\mathbf{B} = \mathbf{J}/\sigma\), Faraday's law and Ampère's law (with uniform \(\eta\)), derive \(\partial_t\mathbf{B} = \nabla\times(\mathbf{v}\times\mathbf{B}) + \eta\nabla^2\mathbf{B}\).
Hint
Take the curl of Ohm's law and use \(\nabla\times(\nabla\times\mathbf{B}) = \nabla(\nabla\cdot\mathbf{B}) - \nabla^2\mathbf{B}\).
Proof
\(\mathbf{E} = -\mathbf{v}\times\mathbf{B} + \eta\mu_0\mathbf{J} = -\mathbf{v}\times\mathbf{B} + \eta\nabla\times\mathbf{B}\). Then \(\partial_t\mathbf{B} = -\nabla\times\mathbf{E} = \nabla\times(\mathbf{v}\times\mathbf{B}) - \eta\nabla\times(\nabla\times\mathbf{B})\), and with \(\nabla\cdot\mathbf{B} = 0\) the last term is \(+\eta\nabla^2\mathbf{B}\).
Problem 5: the ratio of timescales
Show that the diffusion time divided by the Alfvén crossing time equals the Lundquist number: \(\tau_D/\tau_A = S\).
Hint
\(\tau_D = L^2/\eta\) and \(\tau_A = L/v_A\).
Proof
\(\tau_D/\tau_A = (L^2/\eta)/(L/v_A) = Lv_A/\eta = S\).
Numbers¶
Problem 6
What is the Alfvén speed in the corona for \(B = 0.01\) T and a proton density \(n = 10^{15}\) m\(^{-3}\)? Give it in km/s.
Solution
\(\rho = nm_p = 1.67\times10^{-12}\) kg/m³, \(v_A = B/\sqrt{\mu_0\rho} = 6.9\times10^6\) m/s.
Problem 7
How long would it take a coronal field with \(L = 10^7\) m and \(\eta = 1\) m²/s to diffuse away? Give it in years.
Solution
\(\tau_D = L^2/\eta = 10^{14}\) s \(\approx 3.2\times10^6\) years.
Problem 8
A laboratory plasma has \(L = 0.1\) m, \(v_A = 5\times10^4\) m/s and \(\eta = 5\) m²/s. What is its Lundquist number?
Solution
\(S = Lv_A/\eta = 0.1\times5\times10^4/5 = 1000\).
Problem 9
The Spitzer magnetic diffusivity is roughly \(\eta\approx10^9\,T^{-3/2}\) m²/s (with \(T\) in K). What is \(\eta\) in the chromosphere, where \(T = 10^4\) K?
Solution
\(\eta\approx10^9/(10^4)^{3/2} = 10^9/10^6 = 1000\) m²/s: a thousand times more diffusive than the corona.