Flux tube

A flux tube is a generally tube-like (cylindrical) region of space containing a magnetic field, B, such that the cylindrical sides of the tube are everywhere parallel to the magnetic field lines. Since no magnetic flux passes through the sides of the tube, the flux through any cross section of the tube is equal, and the flux entering the tube at one end is equal to the flux leaving the tube at the other. Both the cross-sectional area of the tube and the magnetic field strength may vary along the length of the tube, but the magnetic flux inside is always constant.'If upon any surface which cuts the lines of fluid motion we draw a closed curve, and if from every point of this curve we draw lines of motion, these lines of motion will generate a tubular surface which we may call a tube of fluid motion.' F = ∫ S B ( t ) → ⋅ d S {displaystyle F=int _{S}{vec {B(t)}}cdot dS} R m = U L η {displaystyle R_{m}={frac {UL}{eta }}} ∂ B → ∂ t = ∇ → × ( v → × B → ) {displaystyle {partial {vec {B}} over partial t}={vec { abla }} imes ({vec {v}} imes {vec {B}})} d d t Φ ( S , t ) = d d t ∫ S B ( t ) → ⋅ d S = ∫ S ( ∂ B → ∂ t − ∇ × ( v → × B → ) ) ⋅ d S = 0 {displaystyle {frac {d}{dt}}Phi (S,t)={frac {d}{dt}}int _{S}{vec {B(t)}}cdot dS=int _{S}left({frac {partial {vec {B}}}{partial t}}- abla imes ({vec {v}} imes {vec {B}}) ight)cdot dS=0} B = B 0 λ 2 {displaystyle B={frac {B_{0}}{lambda ^{2}}}} ρ = ρ 0 λ 2 {displaystyle ho ={frac { ho _{0}}{lambda ^{2}}}} 0 = − ∇ p + j × B − ρ g {displaystyle 0=- abla p+j imes B- ho g} 0 = d p d R + d d R ( B ϕ 2 + B z 2 2 μ ) + B ϕ 2 μ R {displaystyle 0={frac {dp}{dR}}+{frac {d}{dR}}left({frac {B_{phi }^{2}+B_{z}^{2}}{2mu }} ight)+{frac {B_{phi ^{2}}}{mu R}}} A flux tube is a generally tube-like (cylindrical) region of space containing a magnetic field, B, such that the cylindrical sides of the tube are everywhere parallel to the magnetic field lines. Since no magnetic flux passes through the sides of the tube, the flux through any cross section of the tube is equal, and the flux entering the tube at one end is equal to the flux leaving the tube at the other. Both the cross-sectional area of the tube and the magnetic field strength may vary along the length of the tube, but the magnetic flux inside is always constant. 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. In 1861, James Clerk Maxwell gave rise to the concept of a flux tube inspired by Michael Faraday's work in electrical and magnetic behavior in his paper titled 'On Physical Lines of Force'. Maxwell described flux tubes as: The flux tube's strength, F, is defined to be the magnetic flux through a surface, S, with F constant throughout the flux tube as a result of Maxwell's Equation: ∇ ⋅ B = 0 {displaystyle abla cdot B=0} . Under the condition that the cross-sectional area, A, of the flux tube is small, F can be approximated as F ≈ B A {displaystyle Fapprox BA} . If the area is decreased then the magnetic field must increase in order to satisfy the condition of constant F. From the condition of perfect conductivity in ideal Ohm's Law, ( σ ⟶ ∞ {displaystyle sigma longrightarrow infty } ), in ideal magnetohydrodynamics, the change in magnetic flux, Φ {displaystyle Phi } , is zero in a flux tube, known as Alfvén's Theorem of flux conservation. With flux conservation, the topology of the flux tube does not change. This effect arises when there is a high Magnetic Reynolds number, Rm >> 1, where induction dominates and diffusion is neglected, allowing for the magnetic field to follow the flow of the plasma resulting in 'frozen-in' flux.