The Lundquist Number

The ratio of resistive diffusion time to Alfven transit time, controlling the rate of magnetic reconnection.

Definition

The Lundquist number is S = mu0 L v_A / eta, where L is a characteristic length, v_A the Alfven speed, and eta the magnetic diffusivity (resistivity divided by mu0). Equivalently it is the ratio of the resistive diffusion time L^2/eta to the Alfven crossing time L/v_A. It is the magnetic Reynolds number evaluated with the Alfven speed as the velocity scale.

What It Measures

A large Lundquist number means the field is nearly frozen to the plasma over dynamical timescales, with resistive slippage confined to thin current layers. Fusion plasmas have enormous S, often above ten to the sixth or higher, because the Spitzer resistivity of a hot plasma is tiny. This makes ideal MHD an excellent description almost everywhere, with resistivity mattering only in narrow layers.

Reconnection Scaling

Classical Sweet-Parker reconnection through a current sheet proceeds at a rate scaling as S^{-1/2}, which at fusion-relevant S is far too slow to explain observed fast events like sawtooth crashes. Above a critical S of roughly ten thousand the thin current sheet becomes unstable to the plasmoid instability, breaking into chains of magnetic islands and restoring a fast, nearly S-independent reconnection rate. Hall physics does the same at small scales.

Relevance

The Lundquist number sets which reconnection regime governs tearing modes, sawteeth, and disruptions in a device. For the Hyperion breeder concept, the high plasma temperature implies very high S, placing reconnection in the plasmoid-mediated fast regime in extended-MHD modeling. Such simulations inform disruption and stability planning for a machine not yet built.