Magnetic helicity

The helicity ${\displaystyle H^{\mathbf {f} }}$ of a smooth vector field ${\displaystyle \mathbf {f} }$ defined on a domain in 3D space is the standard measure of the extent to which the field lines wrap and coil around one another.[14][2] It is defined as the volume integral of the scalar product of ${\displaystyle \mathbf {f} }$ and its curl ${\displaystyle \nabla \times {\mathbf {f} }}$:

${\displaystyle H^{\mathbf {f} }=\int {\mathbf {f} }\cdot {\nabla \times {\mathbf {f} }}\,d^{3}{\mathbf {r} }}$,

where ${\displaystyle d^{3}{\mathbf {r} }}$ is the differential volume element for the volume integral, the integration taking place over the whole considered domain.

As to magnetic helicity ${\displaystyle H^{\mathbf {M} }}$, it is the helicity of the magnetic vector potential ${\displaystyle {\mathbf {A} }}$, such that ${\displaystyle {\mathbf {B} }=\nabla \times {\mathbf {A} }}$ is the magnetic field:[9]

${\displaystyle H^{\mathbf {M} }=\int {\mathbf {A} }\cdot {\mathbf {B} }\,d^{3}{\mathbf {r} }}$.

Magnetic helicity has units of Wb² (webers squared) in SI units and Mx² (maxwells squared) in Gaussian Units.[15]

Magnetic helicity should not be confused with the helicity of the magnetic field ${\displaystyle H^{\mathbf {J} }=\int {\mathbf {B} }\cdot {\mathbf {J} }\,d^{3}{\mathbf {r} }}$, with ${\displaystyle {\mathbf {J} }=\nabla \times {\mathbf {B} }}$ the current. This quantity is called the "current helicity".[16] Contrary to magnetic helicity, current helicity is not an ideal invariant (it is not conserved even when the electrical resistivity is zero).

Since the magnetic vector potential is not gauge invariant, the magnetic helicity is also not gauge invariant in general. As a consquence, one cannot measure directly the magnetic helicity of a physical system. In certain conditions and under certain assumptions, one can however measure the current helicity of a system and from it, when further conditions are fulfilled and under further assumptions, deduce the magnetic helicity.[17]

The name "helicity" relies on the fact 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 ${\displaystyle \textstyle H^{K}=\int \mathbf {v} \cdot {\boldsymbol {\omega }}\neq 0}$. When ${\displaystyle \textstyle H^{K}>0}$, the helix is right-handed and when ${\displaystyle \textstyle H^{K}<0}$ it is left-handed. This behaviour is very similar for magnetic field lines.

Magnetic helicity is a generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field.[6] As with many quantities in electromagnetism, magnetic helicity (which describes magnetic field lines) is closely related to fluid mechanical helicity (which describes fluid flow lines) and their dynamics are interlinked.[5][18]

If magnetic field lines follow the strands of a twisted rope, this configuration would have nonzero magnetic helicity; left-handed ropes would have negative values and right-handed ropes would have positive values.

If the magnetic field is turbulent and weakly inhomogeneous, a magnetic helicity density and its associated flux can be defined in terms of the density of field line linkages.[16]

In the late 1950s, Lodewijk Woltjer and Walter M. Elsässer discovered independently the ideal invariance of magnetic helicity,[4][3] that is, its conservation in case of a zero resistivity. Woltjer's proof, valid for a closed system, is repeated in the following:

In ideal MHD, the magnetic field and magnetic vector potential time evolution are governed by:

${\displaystyle \partial _{t}{\mathbf {B} }=\nabla \times ({\mathbf {v} }\times {\mathbf {B} }),}$ ${\displaystyle \partial _{t}{\mathbf {A} }={\mathbf {v} }\times {\mathbf {B} }+\nabla (\Phi ),}$

where the second equation is obtained by "uncurling" the first one and ${\displaystyle \nabla (\Phi )}$ is a scalar potential given by the gauge condition (see the paragraph about gauge consideration). Choosing the gauge so that the scalar potential vanishes (${\displaystyle \nabla (\Phi )}$=0), the magnetic helicity time evolution is given by:

${\displaystyle \partial _{t}H^{\mathbf {M} }=\int (\partial _{t}{\mathbf {A} })\cdot {\mathbf {B} }+{\mathbf {A} }\cdot \partial _{t}{\mathbf {B} }d^{3}{\mathbf {r} }=\int ({\mathbf {v} }\times {\mathbf {B} })\cdot {\mathbf {B} }d^{3}{\mathbf {r} }+\int {\mathbf {A} }\cdot (\nabla \times \partial _{t}{\mathbf {A} })d^{3}{\mathbf {r} }}$.

The first integral is zero since ${\displaystyle {\mathbf {B} }}$ is orthogonal to the cross-product ${\displaystyle {\mathbf {v} }\times {\mathbf {B} }}$. The second integral can be integrated by parts, giving:

${\displaystyle \partial _{t}H^{\mathbf {M} }=\int _{V}\nabla \times {\mathbf {A} }\cdot (\partial _{t}{\mathbf {A} })d^{3}{\mathbf {r} }+\int _{S}{\mathbf {A} }\times (\partial _{t}{\mathbf {A} })d^{2}{\mathbf {r} }=0.}$

The first integral is done over the whole volume and is zero because ${\displaystyle \nabla \times {\mathbf {A} }={\mathbf {B} }\perp \partial _{t}{\mathbf {A} }}$ as written above. The second integral corresponds to the surface integral over ${\displaystyle S}$, the boundaries of the closed system. It is zero because motions inside the closed system cannot affect the vector potential outside, so that at the boundary surface ${\displaystyle \partial _{t}A=0}$, since the magnetic vector potential is a continuous function.

In all situations where magnetic helicity is gauge invariant (see paragraph below), magnetic helicity is hence ideally conserved without the need of the specific gauge choice ${\displaystyle \nabla (\Phi )=0}$ .

Magnetic helicity remains conserved in a good approximation even with a small but finite resistivity, in which case magnetic reconnection dissipates energy.[6][9]

Magnetic helicity is subject to an inverse transfer in Fourier space. This possibility has first been proposed by Uriel Frisch and collaborators[5] and has been verified through many numerical experiments.[19][20][21][22][23][24] This confirms that through the inverse transfer of magnetic helicity, larger and larger magnetic structures are consecutively formed from small scale fluctuations.

An argument for this inverse transfer taken from[5] is repeated here, which is based on the so-called "realizability condition" on 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" corresponds to an application of Cauchy-Schwarz inequality, which yields:

${\displaystyle |{\hat {H}}_{\mathbf {k} }^{M}|\leq {\frac {2E_{\mathbf {k} }^{M}}{|{\mathbf {k} }|}}}$,

with ${\displaystyle E_{\mathbf {k} }^{M}={\frac {1}{2}}{\hat {\mathbf {B} }}_{\mathbf {k} }^{*}\cdot {\hat {\mathbf {B} }}_{\mathbf {k} }}$ the magnetic energy spectrum. To obtain this inequality, the fact that ${\displaystyle |{\hat {\mathbf {B} }}_{\mathbf {k} }|=|{\mathbf {k} }||{\hat {\mathbf {A} }}_{\mathbf {k} }^{\perp }|}$ (with ${\displaystyle {\hat {\mathbf {A} }}_{\mathbf {k} }^{\perp }}$ the solenoidal part of the Fourier transformed magnetic vector potential, orthogonal to the wavevector in Fourier space) has been used, since ${\displaystyle {\hat {\mathbf {B} }}_{\mathbf {k} }=i{\mathbf {k} }\times {\hat {\mathbf {A} }}_{\mathbf {k} }}$. The factor 2 is not present in the paper[5] since the magnetic helicity is defined there alternatively as ${\displaystyle {\frac {1}{2}}\int {\mathbf {A} }\cdot {\mathbf {B} }d^{3}{\mathbf {r} }}$.

One can then imagine an initial situation with no velocity field and a magnetic field only present at two wavevectors ${\displaystyle \mathbf {p} }$ and ${\displaystyle \mathbf {q} }$. We assume a fully helical magnetic field, which means that it saturates the realizability condition: ${\displaystyle |{\hat {H}}_{\mathbf {p} }^{M}|={\frac {2E_{\mathbf {p} }^{M}}{|{\mathbf {p} }|}}}$ and ${\displaystyle |{\hat {H}}_{\mathbf {q} }^{M}|={\frac {2E_{\mathbf {q} }^{M}}{|{\mathbf {q} }|}}}$. Assuming that all the energy and magnetic helicity transfers are done to another wavevector ${\displaystyle \mathbf {k} }$, the conservation of magnetic helicity on the one hand and of the total energy ${\displaystyle E^{T}=E^{M}+E^{K}}$ (the sum of (m)agnetic and (k)inetic energy) on the other hand gives:

${\displaystyle H_{\mathbf {k} }^{M}=H_{\mathbf {p} }^{M}+H_{\mathbf {q} }^{M},}$

${\displaystyle E_{\mathbf {k} }^{T}=E_{\mathbf {p} }^{T}+E_{\mathbf {q} }^{T}=E_{\mathbf {p} }^{M}+E_{\mathbf {q} }^{M}.}$

The second equality for the energy comes from the fact that we consider an initial state with no kinetic energy. Then we have necessarily ${\displaystyle |\mathbf {k} |\leq \max(|\mathbf {p} |,|\mathbf {q} |)}$. Indeed, if we would have ${\displaystyle |\mathbf {k} |>\max(|\mathbf {p} |,|\mathbf {q} |)}$, then:

${\displaystyle H_{\mathbf {k} }^{M}=H_{\mathbf {p} }^{M}+H_{\mathbf {q} }^{M}={\frac {2E_{\mathbf {p} }^{M}}{|\mathbf {p} |}}+{\frac {2E_{\mathbf {q} }^{M}}{|\mathbf {q} |}}>{\frac {2(E_{\mathbf {p} }^{M}+E_{\mathbf {q} }^{M})}{|\mathbf {k} |}}={\frac {2E_{\mathbf {k} }^{T}}{|\mathbf {k} |}}\geq {\frac {2E_{\mathbf {k} }^{M}}{|\mathbf {k} |}},}$

which would break the realizability condition. This means that ${\displaystyle |\mathbf {k} |\leq \max(|\mathbf {p} |,|\mathbf {q} |)}$. In particular, for ${\displaystyle |{\mathbf {p} }|=|{\mathbf {q} }|}$, the magnetic helicity is transferred to a smaller wavevector, which means to larger scales.

Magnetic helicity is a gauge-dependent quantity, because ${\displaystyle \mathbf {A} }$ can be redefined by adding a gradient to it (gauge choosing). However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity contained in the whole domain is gauge invariant,[16] that is, independent of the gauge choice. A gauge-invariant relative helicity has been defined for volumes with non-zero magnetic flux on their boundary surfaces.[6]

• Woltjer's theorem

• A. A. Pevtsov's Helicity Page
• Mitch Berger's Publications Page