Lundquist Number Calculator

The Lundquist number, S, is a dimensionless number that compares the timescale for resistive magnetic diffusion to the timescale for Alfvén-wave propagation across a plasma system. It is the specific case of the magnetic Reynolds number where the characteristic velocity is fixed to the Alfvén speed, the natural velocity scale for magnetized plasma dynamics.

S = μ₀σLv_A, where μ₀ is the vacuum permeability, σ is the plasma's electrical conductivity, L is a characteristic length scale, and v_A is the Alfvén speed computed from the local magnetic field and mass density.

They share the same mathematical form, Rm = μ₀σLV, but the magnetic Reynolds number allows V to be any characteristic flow velocity relevant to the problem, while the Lundquist number specifically fixes V to the Alfvén speed v_A. This makes S the natural choice whenever Alfvén-wave-scale dynamics, especially magnetic reconnection, are the process of interest.

Classical resistive (Sweet-Parker) magnetic reconnection theory predicts a reconnection rate that scales as S^(-1/2), and since astrophysical Lundquist numbers are enormous (often 10¹⁰ or more), this predicts reconnection far too slow to explain observed solar flares and magnetospheric substorms, a long-standing puzzle that motivated theories of faster, plasmoid-mediated reconnection at high S.

Like the magnetic Reynolds number, S scales directly with the characteristic length L, and astrophysical systems (the solar corona, stellar interiors, accretion disks) span enormous length scales, pushing S to values far beyond anything achievable in a laboratory, even with modest plasma conductivity.

A high Lundquist number means the magnetic field is very strongly frozen into the plasma on the global scale (Alfvén-wave-timescale advection dominates over resistive diffusion), yet this same high-S condition is what permits current sheets to become thin and unstable enough to trigger fast reconnection in localized regions, an apparent paradox central to modern reconnection theory.

Yes, indirectly, through the Alfvén speed, which depends on the mass density via v_A = B/√(μ₀ρ). Higher density lowers the Alfvén speed and therefore lowers the Lundquist number for the same conductivity, length, and field strength.

It is typically estimated from independently measured or inferred plasma parameters, magnetic field strength, density (via the Alfvén speed), electrical conductivity (often via the Spitzer resistivity formula), and a characteristic system size, then combined using the Lundquist number formula rather than measured directly.

No, laboratory reconnection experiments typically achieve Lundquist numbers many orders of magnitude smaller than solar or astrophysical values, due to the much smaller achievable length scales, which is a key challenge in experimentally validating fast-reconnection theories intended to explain solar and astrophysical observations.

It uses the same formula as the dedicated <a href="/science/plasma-physics/alfven-wave-speed-calculator/">Alfvén Wave Speed Calculator</a>, v_A = B/√(μ₀ρ) with ρ = n·mᵢ, computed internally from the magnetic field, ion density, and ion mass you enter.