# Flux tube

A flux tube is a generally tube-like (cylindrical) region of space containing a magnetic field, such that the field at the side surfaces is parallel to those surfaces. Both the cross-sectional area of the tube and the field contained may vary along the length of the tube, but the magnetic flux is always constant.

Flux tubes can be easily created with the use of a superconductor. A superconductor is a material that when cooled to a certain temperature, conducts electricity with no power loss. Superconductors are also known to repel magnetic fields. Placing a thin superconductor over a magnet will result in flux tubes passing through the small cracks of it.

As used in astrophysics, a flux tube generally has a larger magnetic field and other properties that differ from the surrounding space. They are commonly found around stars, including the Sun, which has many flux tubes of around 300 km diameter. Sunspots are also associated with larger flux tubes of 2500 km diameter. Some planets also have flux tubes. A well-known example is the flux tube between Jupiter and its moon Io.

## Formulation

The studies of evolution of magnetic fields in a stably-stratified, compressible atmosphere, subject to a driven parallel shear flow. The problem is examined in a Cartesian domain with aspect ratio xm : ym : 1, with z increasing downwards. In this geometry, the x direction is considered to be toroidal and the y and z directions poloidal. It is assumed as that the domain contains a perfect gas and that the specific heats cp, cv, dynamic viscosity µ, thermal conductivity K, magnetic diffusivity η and gravitational acceleration are finite and constant. The evolution of the velocity u = (u, v, w), magnetic field B = (Bx, By, Bz), density ρ, temperature T, and pressure p is then described by the compressible MHD equations (not reproduced here for brevity; see CBC for details). At the upper and lower boundaries, It is imposed in stress-free, impenetrable velocity conditions, and require that the vertical gradient of the horizontal components of the magnetic field vanish. It is imposed in a constant temperature on the upper surface and a fixed heat flux through the lower one. It is assumed that periodic boundary conditions in the horizontal directions.

A forcing function F is included in the momentum equations, chosen to drive, in the absence of magnetic effects, a steady, stable target velocity profile U0. The chosen profile contains shear in both the y and z directions: U0 = (U0, 0, 0) with U0(y, z) = P(z) cos(2πy/ym). Here, P(z) is a polynomial function of z chosen so that the velocity is non-zero between two horizontal levels, z0 = 0.4 and z1 = 0.95, with maximum amplitude Um at zv = (z0 + z1)/2 = 0.675, joined smoothly to the surrounding quiescent layers. Since the nonlinear advective terms for this profile are identically zero, setting F = (F, 0, 0) with F = −PrCk∇2U0 with no magnetic fields present, forces an initial condition u = 0 to evolve to U0. Provided that Um is not too large, this velocity profile is stable to hydrodynamic perturbations.

The atmosphere is initially in polytropic hydrostatic balance with polytropic index m, and is threaded by a uniform, weak, horizontal, poloidal (y-directed) magnetic field of strength B0.

Under standard nondimensionalisation of the equations, a number of parameters arise that govern the problem: Ck is the thermal dissipation parameter (related to the thermal conductivity, K), Pr = µcp/K is the Prandtl number, ζ = ηcpρ0/K, and the Chandrasekhar number Q = B2 0 d2/ (µ0µη), with µ0 the magnetic permeability, measures the strength B0 of this background magnetic field (although a related parameter appears explicitly in the equations, α = PrζQC2k). The other parameters are fixed: γ = cp/cv = 5/3, xm = ym = 0.5, θ = 2, m = 1.6. These parameters, together with the Pr and Ck chosen, ensure that the Rayleigh number is large and negative, and thus that the system is stable to convective motions.

## Equations

Reynolds number
```Re ≡Uf ymρ/ σCk
```
Magnetic Reynolds number
```Rm ≡Uf ym / ζCk
```
Peclet number
```Pe ≡Uf ymρ/ γCk
```