Lundquist_number

plasmapy.formulary.dimensionless.Lundquist_number(
L: Annotated[Quantity, Unit('m')],
B: Annotated[Quantity, Unit('T')],
density: Annotated[Quantity, Unit('1 / m3')],
sigma: Annotated[Quantity, Unit('S / m')],
ion: str | int | integer | Particle | CustomParticle | Quantity | None = None,
z_mean: float | None = None,
) → Annotated[Quantity, Unit(dimensionless)][source]

Compute the ratio of the Alfén wave crossing timescale to the magnetic diffusion timescale.

The Lundquist number \(S\) is a dimensionless quantity that compares the Alfvén wave crossing timescale to the magnetic diffusion timescale in a conducting medium. It is given by

\[S = \frac{L V_A}{η}\]

where L is the length scale, \(V_A = B / \sqrt{μ_0 ρ}\) is the Alfvén speed, \(B\) is the magnetic field, \(ρ\) is the mass density, \(μ_0\) is the permeability of free space, \(η = 1 / (μ_0 σ)\) is the magnetic diffusivity, and \(σ\) is the electrical conductivity.

Parameters:

  • L (Quantity) – The length scale of the plasma.

  • B (Quantity) – The magnetic field magnitude in units convertible to tesla.

  • density (Quantity) – Either the ion number density \(n_i\) in units convertible to m-3 or the total mass density \(ρ\) in units convertible to kg m-3.

  • sigma (Quantity) – The conductivity of the plasma.

  • ion (Particle, optional) – Representation of the ion species (e.g., 'p+' for protons, 'D+' for deuterium, 'He-4 +1' for singly ionized helium-4, etc.). If no charge state information is provided, then the ions are assumed to be singly ionized. If the density is an ion number density, then this parameter is required in order to convert to mass density.

  • z_mean (float, optional) – The average ionization state (arithmetic mean) of the ion composing the plasma. This is used in calculating the mass density \(ρ = n_i (m_i + Z_{mean} m_e)\). z_mean is ignored if density is passed as a mass density and overrides any charge state info provided by ion.

Returns:

S – The Lundquist number.

Return type:

Quantity

Raises:

  • RelativityError – If the Alfvén velocity is greater than or equal to the speed of light.

  • TypeError – If B and/or density are not of type Quantity, or convertible to one.

  • TypeError – If ion is not of type or convertible to Particle.

  • TypeError – If z_mean is not of type int or float.

  • UnitTypeError – If the magnetic field B does not have units equivalent to tesla.

  • UnitTypeError – If the density does not have units equivalent to a number density or mass density.

  • ValueError – If density is negative.

Warns:

  • RelativityWarning – If the Alfvén velocity exceeds 5% of the speed of light.

  • UnitsWarning – If units are not provided, SI units are assumed.

Notes

For calculating the Alfvén speed Alfven_speed is used and for calculating the Lundquist number Mag_Reynolds is used.

The Lundquist number is an important quantity in the study of magnetic reconnection. For example, reconnection rates in both the Sweet-Parker and Petschek models of magnetic reconnection can be expressed in terms of the Lundquist number. In the Sweet-Parker model, a current sheet with half-width \(L\), conductivity \(σ\), magnetic diffusivity \(η = 1 / (μ_0 σ)\), and Alfvén speed \(V_A\) at the inflow has a Lundquist number of \(S = L V_A / η\). The dimensionless reconnection rate \(R\), i.e., the ratio of the inflow to outflow speeds, can then be expressed as \(R \sim 1 / \sqrt{S}\). Similarly, the maximum reconnection rate in the Petschek model can be expressed as approximately \(π / (8 \ln S)\) [Priest and Forbes, 2000].

Examples

>>> import astropy.units as u
>>> from astropy.constants.si import m_p, m_e
>>> L = 10**8 * u.m
>>> B = 10**2 * u.G
>>> n = 10**19 * u.m**-3
>>> rho = n * (m_p + m_e)
>>> sigma = 10**-7 * u.S / u.m
>>> Lundquist_number(L, B, rho, sigma)
<Quantity 0.866538...>
>>> Lundquist_number(L, B, n, sigma, ion="p+")
<Quantity 0.866538...>
>>> Lundquist_number(L, B, n, sigma, ion="He +2")
<Quantity 0.434819...>
>>> Lundquist_number(L, B, n, sigma, ion="He", z_mean=1.8)
<Quantity 0.434819...>
>>> sigma = 10**-2 * u.S / u.m
>>> Lundquist_number(L, B, n, sigma, ion="He", z_mean=1.8)
<Quantity 43481.96672...>