Ohmic Elsasser Number in MHD

• The Ohmic Elsasser Number is a dimensionless parameter that quantifies the balance between magnetic Lorentz forces and rotational effects under Ohmic diffusion in MHD systems.
• It is defined using the Alfvén speed, rotation rate, and Ohmic diffusivity to assess magnetic field–fluid coupling in astrophysical disks and laboratory flows.
• Applications include delineating MRI-active and dead zones in protoplanetary disks as well as characterizing boundary-layer regimes in Bödewadt–Hartmann flows.

The Ohmic Elsasser number is a dimensionless parameter fundamental to magnetohydrodynamics (MHD) in rotating, weakly ionized astrophysical and laboratory systems. It quantifies the relative strength of magnetic forces (specifically, Lorentz force) to rotational (Coriolis) forces when Ohmic diffusion is present. The number is central to the theory of protoplanetary disks, the analysis of magnetorotational instability (MRI), and classic boundary-layer problems such as the Bödewadt–Hartmann layer.

1. Mathematical Definition

The Ohmic Elsasser number, customarily denoted $\Lambda_O$ (or $A$ in some engineering literature), is defined as

$\Lambda_O = \frac{v_A^2}{\Omega\,\eta_O}$

where $v_A = B/\sqrt{4\pi\rho}$ is the Alfvén speed for magnetic field strength $B$ and mass density $\rho$; $\Omega$ is the local rotation rate (e.g., Keplerian angular frequency in disks); $\eta_O$ is the Ohmic (magnetic) diffusivity. An alternative SI-unit form is

$A = \frac{B^2}{\rho\,\mu\,\eta\,\Omega}$

with $\mu$ the magnetic permeability and $\eta$ the magnetic diffusivity (Davidson et al., 2020).

In the context of astrophysical disks, $\Lambda_O$ appears in the induction equation as the ratio of the inductive term to the Ohmic diffusive term. It measures the efficacy of magnetic field–fluid coupling in the presence of resistive losses (Nolan et al., 2023, Bai, 2011).

2. Physical Interpretation

$\Lambda_O$ compares the rate of magnetic field–induced acceleration (via the Lorentz force) to the intrinsic timescale set by system rotation and Ohmic dissipation. If $\Lambda_O \gg 1$, Lorentz forces can effectively influence the neutral fluid—magnetic fields are "frozen in" and ideal-MHD-like behavior prevails. Conversely, $\Lambda_O \ll 1$ implies Ohmic dissipation dominates, leading to rapid field slippage relative to the fluid and substantial suppression of MHD phenomena such as the MRI.

In the boundary-layer setting (e.g., Bödewadt–Hartmann flows), $A$ serves as the parameter distinguishing regimes dominated by Coriolis forces (thick, oscillatory layers) from those dominated by Lorentz force (thin, monotonic Hartmann layers). The magnetic diffusivity $\eta$ in the denominator acts to diminish induced currents and therefore the Lorentz force, so increased $\eta$ reduces $\Lambda_O$ and weakens magnetic coupling (Davidson et al., 2020, Nolan et al., 2023).

3. Computation via Conductivity and Chemistry

In fully self-consistent models, $\eta_O$ is not a fixed parameter but is computed from a detailed microphysical treatment:

• The Ohmic conductivity is

$$\sigma_O = \frac{e c\, n(\mathrm{H}_2)}{B} \sum_i Z_i\, x_i\, \beta_{i,\mathrm{H}_2}$$

where $n(\mathrm{H}_2)$ is the molecular hydrogen number density, $Z_i$ the charge of species $i$, $x_i$ its abundance, and $\beta_{i,\mathrm{H}_2}$ the Hall parameter for collisions with H$_2$.

• Ohmic diffusivity:

$\eta_O = \frac{c^2}{4\pi\,\sigma_O}$

Abundances of charged species (electrons, ions, charged grains, PAHs) are obtained from chemical equilibrium networks as a function of density, temperature, and the total ionization rate (stellar X-rays, cosmic rays, and radionuclides) [(Nolan et al., 2023); (Bai, 2011)].

The process ensures $\Lambda_O$ (and thus disk physics) is inherently tied to microphysical parameters, including grain size distributions and chemical state.

4. Application to Protoplanetary Disks and MRI

In protoplanetary disks, the spatial structure of $\Lambda_O$ varies radially and vertically and is critically shaped by local chemistry and dust population. The system is typically divided into MRI-"active" zones ($\Lambda_O \gtrsim 1$) and MRI-"dead" zones ($\Lambda_O \ll 1$), where the MRI cannot operate due to excessive Ohmic damping.

Key findings include:

• Small grains (e.g., standard ISM or truncated MRN) yield low electron fractions, high $\eta_O$, and hence $\Lambda_O \ll 1$ near the midplane—fields decouple and no large-scale rings form.
• Large grains (eMRN) result in higher electron fractions and $\Lambda_O \sim 1$ over 10–30 AU, enabling stable ring and gap structures.
• The inclusion of polyaromatic hydrocarbons (PAHs) modifies ambipolar diffusion but Ohmic terms dominate in suppressing $\Lambda_{\text{eff}}$ (an effective Elsasser number including both Ohmic and ambipolar terms)—hence, even with PAHs, rings are suppressed if $\eta_O$ is large (Nolan et al., 2023).

A quantitative summary of disk regimes: | Grain Model | $\Lambda_{\text{eff}}$ Value | MRI/Ring Outcome | |--------------------|-----------------------------|-----------------------------------| | MRN, trMRN | $10^{-4}$–$10^{-3}$ | Magnetic decoupling, no rings | | eMRN | $\sim 1$ | Stable, periodic rings and gaps | | MRN-PAH, trMRN-PAH | drops to $\sim 10^{-2}$ | Rings suppressed with $\eta_O$ |

5. Thresholds and Criteria for Instability

MRI can develop and turbulence or wind-driven substructures (e.g., rings and gaps) can persist only if

$\Lambda_O \gtrsim 1$

Linear analyses, nonlinear simulations, and empirical correlations all confirm this threshold. For $\Lambda_O < 1$, Ohmic resistivity damps MRI entirely; for $\Lambda_O > 1$, MRI can become vigorous, but values much larger than unity may lead to non-axisymmetric, unsteady dynamics rather than orderly structures. When both Ohmic and ambipolar effects are considered, the relevant effective Elsasser number $$\Lambda_{\rm eff} = v_A^2/[\Omega (\eta_O + \eta_A)]$$ must satisfy the same criterion (Nolan et al., 2023, Bai, 2011).

The criticality of $\Lambda_O \sim 1$ is not only found in disk studies but also underpins the transition from Coriolis- to Lorentz-dominated regimes in laboratory and geophysical MHD (Davidson et al., 2020).

6. Elsasser Number in Boundary-Layer Magnetohydrodynamics

In the Bödewadt–Hartmann boundary layer, the Ohmic Elsasser number $A$ governs both the boundary-layer thickness and dynamical evolution:

• For $A \ll 1$, Coriolis forces dominate, and the boundary is thick and oscillatory (Ekman/Bödewadt regime).
• For $A \gg 1$, Lorentz forces dominate, giving a thin, monotonic Hartmann layer.
• For $A \sim 1$, the system transitions smoothly between these extremes, and the decay of core rotation (spin-down) rapidly shifts from algebraic (Coriolis regime) to exponential (Hartmann regime).

These regimes and transitions are quantitatively described by the composite parameter $R = [A+\sqrt{1+A^2}]^{1/2}$ and follow detailed analytical and numerical solutions, with precise agreement with experimental results for $A$ from $0.01$ to $10$ (Davidson et al., 2020).

7. Relationship to Other Non-Ideal MHD Parameters

The Ohmic Elsasser number is one of a family of Elsasser-type dimensionless numbers in non-ideal MHD. Others include:

• The Hall Elsasser number $\chi = v_A^2/(\eta_H\,\Omega)$.
• The ambipolar Elsasser number $$Am = v_A^2/(\eta_A\,\Omega) \approx \gamma_i \rho_i/\Omega$$.

Astrophysical disk models require not only $\Lambda_O \geq 1$ but also that the magnetic field strength does not exceed thresholds set by the plasma $\beta = P_{\rm gas}/P_{\rm mag}$ and ambipolar diffusion parameters. The lower boundary of the MRI-active layer is set by $\Lambda_O = 1$, while the upper (vertical) extent is sensitive to $Am$ and Hall physics (Bai, 2011).

8. Broader Significance and Regimes of Application

The Ohmic Elsasser number governs the dynamical and morphological character of MHD flows in both astrophysical disks and rotating laboratory systems. In protoplanetary disks, variations in $\Lambda_O$ (as mapped in radius, height, and grain population) underpin the existence of dead zones, the structure of turbulent and laminar layers, and the conditions for ring and gap formation. In laboratory rotating flows, $\Lambda_O$ ($A$) prescribes the transition from Coriolis-dominated to Lorentz-dominated dynamics, controlling boundary-layer thickness and the decay laws of rotating cores (Nolan et al., 2023, Bai, 2011, Davidson et al., 2020).

Values of $\Lambda_O$ are highly sensitive to local chemical and microphysical environments, especially dust properties and ionization sources, making it a direct diagnostic of MHD coupling in diverse natural and experimental settings.