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Sandbox Physics
From magnetic stress to reconnection

L1 · Magnetic reconnection and solar-flare signals

When magnetic flux follows the fluid

Sandbox Physics ·

Flux freezing conserves magnetic flux through a loop that moves with a smooth ideal fluid. The loop can bend and stretch. To identify the conserved quantity, follow the moving loop and keep its boundary tied to the same fluid elements.

Choose the moving object before defining the flux

Take a small closed loop made of labelled fluid elements. Every point on the loop moves with the local bulk-fluid velocity. Give a surface spanning the loop a positive normal direction; the right-hand rule then fixes the positive direction around its edge.

Magnetic flux ΦB\Phi_B is the sum of the normal magnetic field over that surface. Its SI unit is the weber. A field parallel to the surface contributes no flux. For a perpendicular uniform field, flux is field strength times area.

The material loop is a way of following the gas. A fixed loop in the laboratory is a different measuring object. Magnetic flux through it can change simply because a magnetized fluid moves through it.

ΦB(t)=∫S(t)B⋅dS\Phi_B(t)=\int_{S(t)}\mathbf B\cdot d\mathbf S
The surface moves with its fluid boundary; the oriented area element supplies the sign.

Sources: Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)

Motion adds a term to Faraday’s law

For a fixed boundary, Faraday’s law relates changing flux to the circulation of the electric field. A moving edge also sweeps across magnetic field. Adding the narrow strips swept out in a short time gives the motional term. Both effects use the same orientation.

Write the electric field as E\mathbf E in volts per metre, velocity as v\mathbf v in metres per second, and field as B\mathbf B in tesla. The cross product picks the direction perpendicular to velocity and field. Its units match the electric field.

In ideal MHD, the electric field seen by the local bulk fluid vanishes in this nonrelativistic description. The integrand below is then zero, so the material loop carries constant flux. This conclusion assumes smooth fields and a well-defined fluid motion; it is the starting limit, not a statement that every microscopic electron follows a drawn line.

dΦBdt=−∮∂S(t)(E+v×B)⋅dℓ\frac{d\Phi_B}{dt}=-\oint_{\partial S(t)}(\mathbf E+\mathbf v\times\mathbf B)\cdot d\boldsymbol\ell
The negative sign follows the chosen surface/edge orientation. A material loop samples the local fluid velocity along its whole edge.
E+v×B=0⟹dΦBdt=0\mathbf E+\mathbf v\times\mathbf B=0\quad\Longrightarrow\quad\frac{d\Phi_B}{dt}=0
Smooth ideal evolution freezes flux through a moving material loop.

Sources: Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R) · Burch et al. (2016), Electron-scale measurements of magnetic reconnection in space, Science 352, aaf2939, Figures 3–5

Resistance and diffusion have different units

For a uniform scalar electrical resistivity ηe\eta_{\rm e} in ohm-metres, the resistive-fluid closure makes the nonideal field proportional to current density J\mathbf J. Combining this closure with low-frequency Ampère’s law and Faraday’s law gives magnetic diffusion.

The magnetic diffusivity ηm\eta_{\rm m} has units of square metres per second. Dividing electrical resistivity by vacuum permeability converts one quantity into the other. They cannot be exchanged numerically without the conversion.

As a hypothetical example, electrical resistivity of two millionths of an ohm-metre gives magnetic diffusivity about 1.59 square metres per second. A 10-metre variation then has a characteristic diffusion time about 63 seconds. This is a scale estimate for uniform diffusion, not a fitted solar reconnection time.

E+v×B=ηeJ,ηm=ηeμ0\mathbf E+\mathbf v\times\mathbf B=\eta_{\rm e}\mathbf J,\qquad\eta_{\rm m}=\frac{\eta_{\rm e}}{\mu_0}
Electrical resistivity times current density gives electric field. Magnetic diffusivity enters the induction equation.
∂tB=∇×(v×B)+ηm∇2B,τdiff∼L2ηm\partial_t\mathbf B=\nabla\times(\mathbf v\times\mathbf B)+\eta_{\rm m}\nabla^2\mathbf B,\qquad\tau_{\rm diff}\sim\frac{L^2}{\eta_{\rm m}}
This form uses constant scalar diffusivity and zero magnetic divergence. The Laplacian smooths field variation on the chosen length scale.

Sources: Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)

A magnetic contour is a useful two-dimensional description

In a plane with no variation along the third coordinate, introduce a magnetic potential ψ\psi. Its derivatives give a divergence-free in-plane magnetic field. The dot product of this field with the potential gradient vanishes, so constant-potential contours run along the field.

The potential has magnetic-field-times-length units. A difference between two contours gives signed magnetic flux per unit invariant depth, with the sign set by the surface orientation. It describes field geometry; a fluid element instead moves with the velocity.

The scalar induction equation follows the same ideal-versus-diffusive distinction. In the ideal limit, a material fluid element keeps its potential value. A resistive term permits it to change. The numerical chapter will test that local statement at magnetic nulls, where the field itself vanishes.

B=(∂yψ,−∂xψ,0),B⋅∇ψ=0\mathbf B=(\partial_y\psi,-\partial_x\psi,0),\qquad\mathbf B\cdot\nabla\psi=0
The coordinate directions are right-handed, with the third direction invariant.
DψDt≡∂tψ+v⋅∇ψ=ηm∇2ψ\frac{D\psi}{Dt}\equiv\partial_t\psi+\mathbf v\cdot\nabla\psi=\eta_{\rm m}\nabla^2\psi
The material derivative combines change at a fixed point with change caused by fluid transport.

Sources: Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)

A conserved flux can coexist with a changing shape

Consider a compressible ideal-fluid example whose loop stays perpendicular to a uniform field. If the area halves while field strength doubles, the flux remains unchanged. The field strength was not the conserved scalar; the oriented surface integral was.

The interactive model takes a narrower, incompressible two-dimensional slice. Its four cases let us compare ideal and resistive induction while using the same coordinates and initial potential. The gold crosshair is a measuring probe, and moving it changes which part of the solution we inspect.

ΦB′=B′A′=(2B)(A/2)=BA\Phi_B'=B'A'=(2B)(A/2)=BA
This area-change example permits compression; the moving-loop flux relation applies to compressible as well as incompressible ideal flows.

Sources: Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)

Work through the question

A magnetized fluid carries flux through a fixed window. Must the flux through the window stay constant in ideal MHD?

No. The window is fixed, so it is not the material loop in the flux-freezing statement. Different fluid elements can enter and leave it.

Why does dividing electrical resistivity by vacuum permeability produce a diffusivity?

Their ratio has units m2 s−1\mathrm{m^2\,s^{-1}}. It multiplies the spatial second derivative in an equation for a time derivative. This also gives the diffusion-time scale length squared divided by diffusivity.

Does a moving magnetic contour identify a particular electron?

No. It describes the field geometry. Flux freezing is a smooth bulk-fluid limit, while individual particle motion and electron-scale departures require additional equations.

Compare the ideal and resistive null flux

  1. Select Ideal flow and inspect the signed null-flux curve over its supported short interval.
  2. Compare with the resistive default, keeping the two different tested endpoints visible.
  3. Move the probe without changing the case. Explain how the moved probe samples another contour within the same numerical state.

Primary sources

  1. Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)
  2. Burch et al. (2016), Electron-scale measurements of magnetic reconnection in space, Science 352, aaf2939, Figures 3–5
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