Magnetic helicity

In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field.^[1][2]

Magnetic helicity
Unit system	Wb2 (SI), Mx2 (Gaussian)
Unit of	Plasma physics, Magnetohydrodynamics
Symbol	HM
Dimension	[M L4 T−2 I−2]
Conserved	Yes (in ideal MHD)

Magnetic helicity is used to analyze systems with very low resistivity, including many astrophysical environments. When resistivity is low, magnetic helicity is approximately conserved over long timescales. Magnetic helicity dynamics are important in studies of solar flares and coronal mass ejections.^[3] It is relevant in the dynamics of the solar wind.^[4] Its approximate conservation is significant in dynamo processes. It also plays a role in fusion research, including reversed field pinch experiments.^{[5][6][7][8][9]}

When a magnetic field contains magnetic helicity, it can drive the formation of large-scale structures from small-scale ones.^[10] This process is referred to as inverse transfer in Fourier space. In three dimensions, magnetic helicity supports growth toward larger scales. In contrast, many three-dimensional flows in ordinary fluid mechanics are turbulent and exhibit a direct cascade in which large-scale vortices break up into smaller ones that dissipate through viscous effects. By a parallel but inverted process, small helical magnetic structures with nonzero magnetic helicity combine to form large-scale magnetic fields. This behavior is observed in the dynamics of the heliospheric current sheet,^[11] a large magnetic structure in the Solar System.

History

The concept of helicity emerged in the mid-20th century within fluid dynamics, where British fluid dynamicist H. K. Moffatt connected the knottedness of vortex lines to a conserved integral he termed helicity.^[12] In magnetohydrodynamics, Dutch-American astrophysicist Lodewijk Woltjer proved that magnetic helicity is an ideal invariant and characterized minimum energy states at fixed helicity. German-American geophysicist Walter M. Elsasser's dynamo work provided an early theoretical foundation for such invariants in cosmic magnetism.^[13][14]

During the 1970s and 1980s, the concept was further developed through advances in turbulence theory, laboratory plasma experiments, and topology. Uriel Frisch and collaborators predicted an inverse transfer of magnetic helicity toward larger scales, which was later confirmed numerically and interpreted as a pathway to self-organization in magnetized turbulence.^[15] American plasma physicist J. B. Taylor introduced relaxation theory for confined plasmas, arguing that low resistivity allows rapid relaxation to a force-free state that preserves helicity. He emphasized that during relaxation "only total magnetic helicity survives."^[16][17] On the topological front, American mathematician Mitchell A. Berger and American astrophysicist George B. Field introduced relative magnetic helicity to extend the invariant to volumes with magnetic flux crossing their boundaries. American plasma physicists John M. Finn and Thomas M. Antonsen Jr. provided an equivalent gauge-invariant expression, describing a "general gauge invariant definition."^[18][19]

From the 1990s onward, magnetic helicity became an important observational and diagnostic tool in solar physics and space physics. German solar physicist Norbert Seehafer reported that current helicity in active regions is "predominantly negative in the northern" and "positive in the southern hemisphere," establishing an empirical hemispheric rule that motivated extensive follow-up research.^[20] American solar physicists Alexei A. Pevtsov, Richard C. Canfield, and Thomas R. Metcalf mapped helicity patterns in active regions and demonstrated its latitudinal variation, helping to connect photospheric measurements to coronal dynamics and ejections.^[21][22] Analyses of the solar wind and heliosphere used helicity to interpret large-scale magnetic structure and transport.^[23]

Scientists have debated how best to define and measure helicity in realistic, open systems and how to interpret local proxies. Relative magnetic helicity is now the standard approach for volumes with flux crossing the boundary, while current helicity and other proxies are used when full three-dimensional measurements are unavailable.^[18][19][24] Ongoing discussions address gauge issues and whether a meaningful local helicity density can be defined in weakly inhomogeneous turbulence, leading to proposed gauge-invariant local measures and improved numerical diagnostics.^[25] In dynamo theory, magnetic helicity conservation constrains the growth of large-scale fields. Research on helicity fluxes and open boundaries suggests that such fluxes can relax these constraints, a perspective developed in astrophysical dynamo modeling.^[26][27][28]

Mathematical definition

Generally, the helicity ${\displaystyle H^{\mathbf {f} }}$ of a smooth vector field ${\displaystyle \mathbf {f} }$ confined to a volume ${\displaystyle V}$ is a measure of the extent to which field lines wrap and coil around one another.^[29][2] It is defined as the volume integral over ${\displaystyle V}$ of the scalar product of ${\displaystyle \mathbf {f} }$ and its curl, ${\displaystyle \nabla \times {\mathbf {f} }}$: $${\displaystyle H^{\mathbf {f} }=\int _{V}{\mathbf {f} }\cdot \left(\nabla \times {\mathbf {f} }\right)\ dV.}$$

Magnetic helicity

Magnetic helicity ${\displaystyle H^{\mathbf {M} }}$ is the helicity of a magnetic vector potential ${\displaystyle {\mathbf {A} }}$ where ${\displaystyle \nabla \times {\mathbf {A} }={\mathbf {B} }}$ is the associated magnetic field confined to a volume ${\displaystyle V}$. Magnetic helicity can then be expressed as^[5] $${\displaystyle H^{\mathbf {M} }=\int _{V}{\mathbf {A} }\cdot {\mathbf {B} }\ dV.}$$

Since the magnetic vector potential is not gauge invariant, the magnetic helicity is also gauge dependent in general. As a consequence, the magnetic helicity of a physical system cannot be measured directly. Under certain conditions, one can measure the current helicity of a system and, when further conditions are fulfilled, deduce the magnetic helicity.^[30]

Magnetic helicity has units of magnetic flux squared: Wb² (webers squared) in SI units and Mx² (maxwells squared) in Gaussian units.^[31]

Current helicity

The current helicity, or helicity ${\displaystyle H^{\mathbf {J} }}$ of the magnetic field ${\displaystyle \mathbf {B} }$ confined to a volume ${\displaystyle V}$, can be expressed as $${\displaystyle H^{\mathbf {J} }=\int _{V}{\mathbf {B} }\cdot {\mathbf {J} }\ dV}$$ where ${\displaystyle {\mathbf {J} }=\nabla \times {\mathbf {B} }}$ is the current density.^[32] Unlike magnetic helicity, current helicity is not an ideal invariant. It is not conserved even when the electrical resistivity is zero.

Gauge considerations

Magnetic helicity is a gauge-dependent quantity, because ${\displaystyle \mathbf {A} }$ can be redefined by adding a gradient to it, a change of gauge. However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity contained in the whole domain is gauge invariant,^[32] that is, independent of the gauge choice. A gauge-invariant relative helicity has been defined for volumes with nonzero magnetic flux on their boundary surfaces.^[11]

Topological interpretation

The term helicity reflects that the trajectory of a fluid particle in a fluid with velocity ${\displaystyle {\boldsymbol {v}}}$ and vorticity ${\displaystyle {\boldsymbol {\omega }}=\nabla \times {\boldsymbol {v}}}$ forms a helix in regions where the kinetic helicity ${\displaystyle \textstyle H^{K}=\int \mathbf {v} \cdot {\boldsymbol {\omega }}dV\neq 0}$. When ${\displaystyle \textstyle H^{K}>0}$, the resulting helix is right-handed. When ${\displaystyle \textstyle H^{K}<0}$ it is left-handed. This behavior is closely analogous to that of magnetic field lines.

Regions where magnetic helicity is not zero can also contain other sorts of magnetic structures, such as helical magnetic field lines. Magnetic helicity is a continuous generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field.^[11] Where linking numbers describe how many times curves are interlinked, magnetic helicity describes how many magnetic field lines are interlinked.^[5]

Examples of curves with varying values of writhe and twist. Magnetic helicity measures the sum of these two quantities for magnetic field lines. The sum is conserved under all transformations where curves are not cut or joined.

Magnetic helicity is proportional to the sum of the topological quantities twist and writhe for magnetic field lines. The twist is the rotation of the flux tube around its axis, and writhe is the rotation of the flux tube axis itself. Topological transformations can change twist and writhe individually, but conserve their sum. As magnetic flux tubes, collections of closed magnetic field line loops, tend to avoid crossing in magnetohydrodynamic fluids, magnetic helicity is well conserved.

Magnetic helicity is closely related to fluid mechanical helicity, the corresponding quantity for fluid flow lines, and their dynamics are interlinked.^[10][33]

Properties

Ideal quadratic invariance

In the late 1950s, Lodewijk Woltjer and Walter M. Elsässer discovered independently the ideal invariance of magnetic helicity,^[34][35] that is, its conservation when resistivity is zero. The following outlines Woltjer's proof for a closed system.

In ideal magnetohydrodynamics, the time evolution of a magnetic field and magnetic vector potential can be expressed using the induction equation as $${\displaystyle {\frac {\partial {\mathbf {B} }}{\partial t}}=\nabla \times ({\mathbf {v} }\times {\mathbf {B} }),\quad {\frac {\partial {\mathbf {A} }}{\partial t}}={\mathbf {v} }\times {\mathbf {B} }+\nabla \Phi ,}$$ respectively, where ${\displaystyle \nabla \Phi }$ is a scalar potential given by the gauge condition, see Gauge considerations. Choosing the gauge so that the scalar potential vanishes, ${\displaystyle \nabla \Phi =\mathbf {0} }$, the time evolution of magnetic helicity in a volume ${\displaystyle V}$ is given by: $${\displaystyle {\begin{aligned}{\frac {\partial H^{\mathbf {M} }}{\partial t}}&=\int _{V}\left({\frac {\partial {\mathbf {A} }}{\partial t}}\cdot {\mathbf {B} }+{\mathbf {A} }\cdot {\frac {\partial {\mathbf {B} }}{\partial t}}\right)dV\\&=\int _{V}({\mathbf {v} }\times {\mathbf {B} })\cdot {\mathbf {B} }\ dV+\int _{V}{\mathbf {A} }\cdot \left(\nabla \times {\frac {\partial {\mathbf {A} }}{\partial t}}\right)dV.\end{aligned}}}$$ The dot product in the integrand of the first term is zero since ${\displaystyle {\mathbf {B} }}$ is orthogonal to the cross product ${\displaystyle {\mathbf {v} }\times {\mathbf {B} }}$. The second term can be integrated by parts to give $${\displaystyle {\frac {\partial H^{\mathbf {M} }}{\partial t}}=\int _{V}\left(\nabla \times {\mathbf {A} }\right)\cdot {\frac {\partial {\mathbf {A} }}{\partial t}}\ dV+\int _{\partial V}\left({\mathbf {A} }\times {\frac {\partial {\mathbf {A} }}{\partial t}}\right)\cdot d\mathbf {S} }$$ where the second term is a surface integral over the boundary surface ${\displaystyle \partial V}$ of the closed system. The dot product in the integrand of the first term is zero because ${\displaystyle \nabla \times {\mathbf {A} }={\mathbf {B} }}$ is orthogonal to ${\displaystyle \partial {\mathbf {A} }/\partial t.}$ The second term also vanishes because motions inside the closed system do not affect the vector potential outside, so that at the boundary surface ${\displaystyle \partial {\mathbf {A} }/\partial t=\mathbf {0} }$ since the magnetic vector potential is a continuous function. Therefore, $${\displaystyle {\frac {\partial H^{\mathbf {M} }}{\partial t}}=0,}$$ and magnetic helicity is ideally conserved. In all situations where magnetic helicity is gauge invariant, magnetic helicity is ideally conserved without the need for the specific gauge choice ${\displaystyle \nabla \Phi =\mathbf {0} .}$

Magnetic helicity remains conserved to a good approximation even with small but finite resistivity. In that case magnetic reconnection dissipates energy.^[11][5]

Inverse transfer

Small-scale helical structures tend to form larger magnetic structures. This is called inverse transfer in Fourier space, as opposed to the direct energy cascade in three-dimensional turbulent hydrodynamical flows. The possibility of such an inverse transfer was first proposed by Uriel Frisch and collaborators^[10] and has been verified through many numerical experiments.^{[36][37][38][39][40][41]} As a consequence, the presence of magnetic helicity is a candidate explanation for the existence and sustainment of large-scale magnetic structures in the Universe.

The following argument for inverse transfer follows Frisch et al.^[10] It is based on the "realizability condition" for the magnetic helicity Fourier spectrum ${\displaystyle {\hat {H}}_{\mathbf {k} }^{M}={\hat {\mathbf {A} }}_{\mathbf {k} }^{*}\cdot {\hat {\mathbf {B} }}_{\mathbf {k} }}$ where ${\displaystyle {\hat {\mathbf {B} }}_{\mathbf {k} }}$ is the Fourier coefficient at the wavevector ${\displaystyle {\mathbf {k} }}$ of the magnetic field ${\displaystyle {\mathbf {B} }}$, and similarly for ${\displaystyle {\hat {\mathbf {A} }}}$, the star denoting the complex conjugate. The realizability condition is an application of the Cauchy–Schwarz inequality and yields $${\displaystyle \left|{\hat {H}}_{\mathbf {k} }^{M}\right|\leq {\frac {2E_{\mathbf {k} }^{M}}{|{\mathbf {k} }|}},}$$ with ${\textstyle E_{\mathbf {k} }^{M}={\frac {1}{2}}{\hat {\mathbf {B} }}_{\mathbf {k} }^{*}\cdot {\hat {\mathbf {B} }}_{\mathbf {k} }}$ the magnetic energy spectrum. To obtain this inequality, use the relation ${\displaystyle |{\hat {\mathbf {B} }}_{\mathbf {k} }|=|{\mathbf {k} }||{\hat {\mathbf {A} }}_{\mathbf {k} }^{\perp }|}$, with ${\displaystyle {\hat {\mathbf {A} }}_{\mathbf {k} }^{\perp }}$ the solenoidal part of the Fourier transformed magnetic vector potential orthogonal to the wavevector, since ${\displaystyle {\hat {\mathbf {B} }}_{\mathbf {k} }=i{\mathbf {k} }\times {\hat {\mathbf {A} }}_{\mathbf {k} }}$. The factor 2 is not present in Frisch et al.^[10] because magnetic helicity is defined there as ${\textstyle {\frac {1}{2}}\int _{V}{\mathbf {A} }\cdot {\mathbf {B} }\ dV}$.

Consider an initial state with no velocity field and a magnetic field present only at two wavevectors ${\displaystyle \mathbf {p} }$ and ${\displaystyle \mathbf {q} }$. Assume a fully helical magnetic field that saturates the realizability condition, ${\displaystyle \left|{\hat {H}}_{\mathbf {p} }^{M}\right|={\frac {2E_{\mathbf {p} }^{M}}{|{\mathbf {p} }|}}}$ and ${\displaystyle \left|{\hat {H}}_{\mathbf {q} }^{M}\right|={\frac {2E_{\mathbf {q} }^{M}}{|{\mathbf {q} }|}}}$. If all the energy and magnetic helicity transfer to another wavevector ${\displaystyle \mathbf {k} }$, conservation of magnetic helicity and of the total energy ${\displaystyle E^{T}=E^{M}+E^{K}}$, the sum of magnetic and kinetic energy, gives $${\displaystyle H_{\mathbf {k} }^{M}=H_{\mathbf {p} }^{M}+H_{\mathbf {q} }^{M},}$$ $${\displaystyle E_{\mathbf {k} }^{T}=E_{\mathbf {p} }^{T}+E_{\mathbf {q} }^{T}=E_{\mathbf {p} }^{M}+E_{\mathbf {q} }^{M}.}$$

Because the initial state has no kinetic energy, it follows that ${\displaystyle |\mathbf {k} |\leq \max(|\mathbf {p} |,|\mathbf {q} |)}$. If instead ${\displaystyle |\mathbf {k} |>\max(|\mathbf {p} |,|\mathbf {q} |)}$, then $${\displaystyle H_{\mathbf {k} }^{M}=H_{\mathbf {p} }^{M}+H_{\mathbf {q} }^{M}={\frac {2E_{\mathbf {p} }^{M}}{|\mathbf {p} |}}+{\frac {2E_{\mathbf {q} }^{M}}{|\mathbf {q} |}}>{\frac {2\left(E_{\mathbf {p} }^{M}+E_{\mathbf {q} }^{M}\right)}{|\mathbf {k} |}}={\frac {2E_{\mathbf {k} }^{T}}{|\mathbf {k} |}}\geq {\frac {2E_{\mathbf {k} }^{M}}{|\mathbf {k} |}},}$$ which would violate the realizability condition. Therefore ${\displaystyle |\mathbf {k} |\leq \max(|\mathbf {p} |,|\mathbf {q} |)}$. In particular, for ${\displaystyle |{\mathbf {p} }|=|{\mathbf {q} }|}$, the magnetic helicity is transferred to a smaller wavevector, which corresponds to larger spatial scales.

See also

• Woltjer's theorem