Chandrasekhar number

Chandrasekhar number is a fundamental dimensionless number in magnetohydrodynamics (MHD) that quantifies the relative strength of the Lorentz force to the viscous force in a conducting fluid. Named in honor of the astrophysicist Subrahmanyan Chandrasekhar, it is a crucial parameter in analyzing the stability of magnetized fluid layers and the onset of convection under the influence of a magnetic field. It plays a role analogous to the Rayleigh number for thermal convection but incorporates magnetic effects, determining thresholds for phenomena like the Chandrasekhar–Kendall state and the suppression of instability in stellar and planetary interiors.

Definition and mathematical expression

The Chandrasekhar number is typically denoted by $Q$ or $\backslash mathrm\{Ch\}$. For a fluid layer of characteristic thickness $d$, with kinematic viscosity $\backslash nu$, magnetic diffusivity $\backslash eta$, and permeated by a uniform vertical magnetic field of strength $B\_0$, it is defined as $Q\; =\; \backslash frac\{B\_0^2\; d^2\}\{\backslash mu\_0\; \backslash rho\; \backslash nu\; \backslash eta\}$. Here, $\backslash mu\_0$ is the vacuum permeability and $\backslash rho$ is the fluid density. This formulation arises directly from the non-dimensionalization of the Navier–Stokes equations coupled with the induction equation from Maxwell's equations. In some contexts, particularly when comparing magnetic to thermal effects, it is expressed using the Alfvén speed $v\_A\; =\; B\_0\; /\; \backslash sqrt\{\backslash mu\_0\; \backslash rho\}$, giving $Q\; =\; \backslash frac\{v\_A^2\; d^2\}\{\backslash nu\; \backslash eta\}$.

Physical interpretation

Physically, the Chandrasekhar number represents the square of the ratio of the Alfvén wave travel time across the layer to the geometric mean of the viscous and magnetic diffusion times. A high value indicates that the Lorentz force, which arises from the interaction of the induced electric currents with the ambient magnetic field, dominates over dissipative viscous forces. This magnetic dominance can strongly suppress fluid motions, stabilize otherwise unstable stratified layers, and organize flow into structured patterns like magnetoconvection rolls. In astrophysics, large values are characteristic of highly magnetized environments such as the solar tachocline, neutron star atmospheres, and the interiors of gas giants like Jupiter.

Derivation and scaling

The derivation proceeds by non-dimensionalizing the governing equations of incompressible MHD: the momentum equation and the magnetic induction equation. Scaling length by $d$, time by the magnetic diffusion time $d^2/\backslash eta$, velocity by $\backslash eta/d$, and magnetic field by the imposed field $B\_0$, the Lorentz force term becomes proportional to $Q$ multiplied by the curl of the induced field. The parameter emerges naturally as the coefficient quantifying the strength of this magnetic term relative to the viscous term, which is scaled by the magnetic Prandtl number. This scaling analysis, foundational to the work of Subrahmanyan Chandrasekhar in his treatise Hydrodynamic and Hydromagnetic Stability, reveals that the critical condition for the onset of convection is governed by a functional relationship between the Chandrasekhar number and the Rayleigh number.

Applications in magnetohydrodynamics

A primary application is in determining the critical stability threshold for the onset of magnetoconvection. For a layer heated from below, the required Rayleigh number to initiate convection increases with increasing Chandrasekhar number, demonstrating the stabilizing effect of the magnetic field. It is central to studies of the Sun's convection zone and the generation of sunspots via magnetic suppression of turbulence. In geophysics, it helps model the dynamics of Earth's outer core and the generation of the geomagnetic field through the dynamo theory. Furthermore, it appears in analyses of pinch instabilities in plasma physics and the design of magnetic confinement devices like the tokamak.

Relation to other dimensionless numbers

The Chandrasekhar number is intimately related to several other key dimensionless groups in fluid dynamics and MHD. It is the magnetic analogue of the Rayleigh number $\backslash mathrm\{Ra\}$, with both governing convective instability but under different forcing mechanisms. It can be expressed as the product $Q\; =\; \backslash mathrm\{Ha\}^2\; \backslash cdot\; \backslash mathrm\{Pm\}$, where $\backslash mathrm\{Ha\}$ is the Hartmann number (ratio of Lorentz to viscous forces in steady flow) and $\backslash mathrm\{Pm\}\; =\; \backslash nu\; /\; \backslash eta$ is the magnetic Prandtl number. In systems with rotation, it interacts with the Taylor number to determine magneto-rotational instabilities. Its inverse relationship with the magnetic Reynolds number $\backslash mathrm\{Rm\}$ highlights the competition between advection and diffusion of the magnetic field in systems like the Milky Way's interstellar medium.

Category:Dimensionless numbers Category:Magnetohydrodynamics Category:Fluid dynamics Category:Astrophysics