Flux tube

In magnetohydrodynamics, Alfvén's theorem states that the magnetic flux through a surface, such as the surface of a flux tube, moving along with a perfectly conducting fluid is conserved. In other words, the magnetic field is constrained to move with the fluid or is "frozen-in" to the fluid.

This can be shown mathematically for a flux tube using the induction equation of a perfectly conducting fluid

$${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}={\boldsymbol {\nabla }}\times (\mathbf {v} \times \mathbf {B} )}$$

where ${\displaystyle \mathbf {B} }$ is the magnetic field and ${\displaystyle \mathbf {v} }$ is the velocity field of the fluid. The change in magnetic flux over time through any open surface of the flux tube ${\displaystyle \mathbf {S} }$ enclosed by ${\displaystyle C}$ with a differential line element ${\displaystyle d\mathbf {l} }$ can be written as

$${\displaystyle {\frac {d\Phi _{B}}{dt}}=\int _{S}{\partial \mathbf {B} \over \partial t}\cdot d\mathbf {S} +\oint _{C}\mathbf {B} \cdot \mathbf {v} \times d\mathbf {l} }$$

.

Using the induction equation gives

$${\displaystyle {\frac {d\Phi _{B}}{dt}}=\int _{S}{\boldsymbol {\nabla }}\times (\mathbf {v} \times \mathbf {B} )\cdot d\mathbf {S} +\oint _{C}\mathbf {B} \cdot \mathbf {v} \times d\mathbf {l} }$$

which can be rewritten using Stokes' theorem and an elementary vector identity on the first and second term, respectively, to give[6]

$${\displaystyle \int _{S}\mathbf {B} \cdot d\mathbf {S} ={\text{const}}.}$$

In ideal magnetohydrodynamics, if a cylindrical flux tube of length ${\displaystyle L_{0}}$ is compressed while the length of tube stays the same, the magnetic field and the density of the tube increase with the same proportionality. If a flux tube with a configuration of a magnetic field of ${\displaystyle B_{0}}$ and a plasma density of ${\displaystyle \rho _{0}}$ confined to the tube is compressed by a scalar value defined as ${\displaystyle \lambda }$, the new magnetic field and density are given by:[4]

$${\displaystyle B={\frac {B_{0}}{\lambda ^{2}}}}$$

$${\displaystyle \rho ={\frac {\rho _{0}}{\lambda ^{2}}}}$$

If ${\displaystyle \lambda <1}$, known as transverse compression, ${\displaystyle B}$ and ${\displaystyle \rho }$ increase and are scaled the same while transverse expansion decreases ${\displaystyle B}$ and ${\displaystyle \rho }$ by the same value and proportion where ${\displaystyle B/\rho }$ is constant.[4]

Extending the length of the flux tube by ${\displaystyle \lambda ^{*}}$ gives a new length of ${\displaystyle L=\lambda ^{*}L_{0}}$ while the density of the tube remains the same, ${\displaystyle \rho _{0}}$, which then results in the magnetic field strength increasing by ${\displaystyle \lambda ^{*}B_{0}}$. Reducing the length of the tubes results in a decrease of the magnetic field's strength.[4]

In magnetohydrostatic equilibrium, the following condition is met for the equation of motion of the plasma confined to the flux tube:[4]

$${\displaystyle 0=-\nabla p+j\times B-\rho g}$$

where

• ${\displaystyle p}$ is the plasma pressure
• ${\displaystyle j}$ is the current density of the plasma
• ${\displaystyle \rho g}$ is the gravitational force

With the magnetohydrostatic equilibrium condition met, a cylindrical flux tube's plasma pressure of ${\displaystyle p(R)}$ is given by the following relation written in cylindrical coordinates with ${\displaystyle R}$ as the distance from the axis radially:[4]

$${\displaystyle 0={\frac {dp}{dR}}+{\frac {d}{dR}}\left({\frac {B_{\phi }^{2}+B_{z}^{2}}{2\mu }}\right)+{\frac {B_{\phi }^{2}}{\mu R}}}$$

The second term in the above equation gives the magnetic pressure force while the third term represents the magnetic tension force.[4] The field line's twist around the axis from one end of the tube of length ${\displaystyle L}$ to the other end is given by:[4]

$${\displaystyle \Phi (R)={\frac {LB_{\phi }(R)}{RB_{z}(R)}}}$$

Diagram of coronal loops that consist of plasma confined to magnetic flux tubes.

Examples of solar flux tubes include sunspots and intense magnetic tubes in the photosphere and the field around the solar prominence and coronal loops in the corona.[4]

Sunspots occur when small flux tubes combine into a large flux tube that breaks the surface of the photosphere.[1] The large flux tube of the sunspot has a field intensity of around 3 kG with a diameter of typically 4000 km.[1] There are extreme cases of when the large flux tubes have diameters of ${\displaystyle 6\times 10^{4}}$ km with a field strength of 3 kG.[1] Sunspots can continue to grow as long as there is a constant supply of new flux from small flux tubes on the surface of the Sun.[1] The magnetic field within the flux tube can be compressed by decreasing the gas pressure inside and therefore the internal temperature of the tube while maintaining a constant pressure outside.[1]

Intense magnetic tubes are isolated flux tubes that have diameters of 100 to 300 km with an overall field strength of 1 to 2 kG and a flux of around ${\displaystyle 3\times 10^{9}}$Wb.[4] These flux tubes are concentrated strong magnetic fields that are found between solar granules.[7] The magnetic field causes the plasma pressure in the flux tube to decrease, known as the plasma density depletion region.[7] If there is a significant difference in the temperatures in the flux tube and the surroundings, there is a decrease in plasma pressure as well as a decrease in the plasma density causing some of the magnetic field to escape the plasma.[7]

Plasma that is trapped within magnetic flux tubes that are attached to the photosphere, referred to as footpoints, create a loop-like structure known as a coronal loop.[8] The plasma inside the loop has a higher temperature than the surroundings causing the pressure and density of the plasma to increase.[8] These coronal loops get their characteristic high luminosity and ranges of shapes from the behavior of the magnetic flux tube.[8] These flux tubes confine plasma and are characterized as isolated. The confined magnetic field strength varies from 0.1 to 10 G with diameters ranging from 200 to 300 km.[8][9]

The result of emerging twisted flux tubes from the interior of the Sun cause twisted magnetic structures in the corona, which then lead to solar prominences.[10] Solar prominences are modeled using twisted magnetic flux tubes known as flux ropes.[11]

Graphic of the magnetosphere of Jupiter with a flux tube connecting Jupiter and Io shown in yellow.

Magnetized planets have an area above their ionospheres which traps energetic particles and plasma along magnetic fields, referred to as magnetospheres.[12] The extension of the magnetosphere away from the sun known as a magnetotail is modeled as magnetic flux tubes.[12] Mars and Venus both have strong magnetic fields resulting in flux tubes from the solar wind gathering at high altitudes of the ionosphere on the sun side of the planets and causing the flux tubes to distort along the magnetic field lines creating flux ropes.[12] Particles from the solar wind magnetic field lines can transfer to the magnetic field lines of a planet's magnetosphere through the processes of magnetic reconnection that occurs when a flux tube from the solar wind and a flux tube from the magnetosphere in opposite field directions get close to one another.[12]

Flux tubes that occur from magnetic reconnection forms into a dipole-like configuration around the planet where plasma flow occurs.[12] An example of this case is the flux tube between Jupiter and its moon Io approximately 450 km in diameter at the points closest to Jupiter.[13]