2.1. The stationary cylindrical principal pair

We work in cylindrical coordinates $x^{a}=(t,r,\phi,\chi )$ with the flat metric

(2.1) \begin{align} {\rm d}s^{2} = -{\rm d}t^{2} +{\rm d}r^{2} +r^{2}{\rm d}\phi ^{2} +{\rm d}\chi ^{2}, \end{align}

and orientation $\varOmega :={\rm d}t\wedge {\rm d}r\wedge {\rm d}\phi \wedge {\rm d}\chi \gt 0.$ The configuration is stationary and cylindrically symmetric, and all profile functions depend only on $r$ . Both the electric and magnetic fields will be tangent to the cylindrical surfaces.

The general mathematics of metric-conjugate rank-two sectors and Bianchi-closed principal completion is developed by Kim (Reference Kim2026a ). We now specify directly the finite stationary principal pair used throughout the paper. Introduce the one-form legs

(2.2) \begin{align} \vartheta _G &:= {\rm d}r+C(r)\,{\rm d}t, & \beta _G &:= B(r)\,{\rm d}\phi +A(r)\,{\rm d}\chi, \end{align}

(2.3) \begin{align} \vartheta _H &:= {\rm d}t+C(r)\,{\rm d}r, & \beta _H &:= B(r)\,{\rm d}\chi -r^{2}A(r)\,{\rm d}\phi . \end{align}

The two rank-two sectors are defined by

(2.4) \begin{eqnarray} G := \vartheta _G\wedge \beta _G = ({\rm d}r+C\,{\rm d}t) \wedge (B\,{\rm d}\phi +A\,{\rm d}\chi), \end{eqnarray}

(2.5) \begin{eqnarray} H := h(r)\,\vartheta _H\wedge \beta _H = h(r) ({\rm d}t+C\,{\rm d}r) \wedge (B\,{\rm d}\chi -r^{2}A\,{\rm d}\phi). \end{eqnarray}

No profile in (2.4) and (2.5) is assumed to be small.

The functions have a direct role in the cylindrical pair. The ratio $A/B$ determines the helical direction carried by the first sector, $C$ describes a boost-like mixing of its radial and temporal legs, and $h$ gives the amplitude of the metric-conjugate sector. The quantity $B(r)$ is a reconstruction variable and should not be confused with the magnitude of the observable total magnetic field.

On any interval where $B\neq 0$ , we introduce

(2.6) \begin{align} a(r):=\frac {A(r)}{B(r)}. \end{align}

The relation of $a(r)$ to the observable magnetic pitches is derived in § 3.

The two pairs of legs are mutually metric orthogonal: $ g^{-1}(\vartheta _G,\vartheta _H)=0$ and $g^{-1}(\beta _G,\beta _H)=0.$ Their norms are $g^{-1}(\vartheta _G,\vartheta _G)=1-C^{2}$ , $ g^{-1}(\vartheta _H,\vartheta _H)=-(1-C^{2})$ , $ g^{-1}(\beta _G,\beta _G)= D/r^{2}$ and $ g^{-1}(\beta _H,\beta _H)= D$ , where

(2.7) \begin{align} D(r):=B^{2}(r)+r^{2}A^{2}(r). \end{align}

A regular Lorentzian splitting into one spacelike and one timelike principal plane therefore requires

(2.8) \begin{align} \fbox {${ C^{2}\lt 1, \qquad D\gt 0. }$} \end{align}

The first condition keeps the radial–temporal mixing subluminal. The second excludes the trivial limit in which the first sector vanishes.

With the orientation chosen here, the Hodge dual of $G$ is $ \star G = ({1}/{r}) \vartheta _H\wedge \beta _H.$ Consequently,

(2.9) \begin{align} \fbox {${ H=hr\,\star G. }$} \end{align}

Thus, $G$ and $H$ occupy metric-conjugate principal planes, with relative principal amplitude $ \rho (r)=hr.$

The total electromagnetic field is $ F:=G+H.$ Since both sectors are simple, $G\wedge G=H\wedge H=0$ , while

(2.10) \begin{align} F\wedge F = 2G\wedge H = 2h(1-C^{2}){\rm D}\varOmega . \end{align}

Subject to (2.8), the total field is therefore rank four wherever $h\neq 0$ . The parallel electric field of the completed configuration is produced by the coupling between the two rank-two sectors.

2.2. Maxwell closure and the stationary locking relation

The principal-pair structure does not by itself guarantee that $F$ is an admissible electromagnetic field. The homogeneous Maxwell equations require (1.7). For the pair defined here, the exterior derivatives are

(2.11) \begin{align} {\rm d}G ={}& -(BC)'\, {\rm d}t\wedge {\rm d}r\wedge {\rm d}\phi -(AC)'\, {\rm d}t\wedge {\rm d}r\wedge {\rm d}\chi, \nonumber \\[3pt] {\rm d}H ={}& (hr^{2}A)'\, {\rm d}t\wedge {\rm d}r\wedge {\rm d}\phi -(h\!B)'\, {\rm d}t\wedge {\rm d}r\wedge {\rm d}\chi . \end{align}

Hence, ${\rm d}F=0$ is equivalent to the two radial equations $ (hr^{2}A)'=(BC)'$ and $(h\!B)'=-(AC)'.$ After one radial integration, $ hr^{2}A-BC=k_{1}$ and $ h\!B+AC=k_{2},$ where $k_{1}$ and $k_{2}$ are constants. Using $A=aB$ , these equations form a pointwise algebraic system for the two combinations $ \eta :=h\!B$ and $\varLambda :=BC.$ Solving it gives the central stationary locking relation.

Proposition 1 (Stationary principal-sector locking). Let $G$ and $H$ be given by ( 2.4 ) and ( 2.5 ), with all profiles depending only on $r$ . Then, $F=G+H$ satisfies ${\rm d}F=0$ if and only if

(2.12) \begin{align} \fbox {${ h\!B = \dfrac {k_{2}+k_{1}a} {1+r^{2}a^{2}}, \qquad BC = \dfrac {k_{2}r^{2}a-k_{1}} {1+r^{2}a^{2}}, }$} \end{align}

for $B \neq 0$ . Thus, the principal-sector amplitude $h\!B$ and the radial-mixing combination $BC$ are fixed pointwise once the pitch profile $a(r)$ , the radial profile $B(r)$ and the two drive constants are specified.

The electric interpretation of these constants is given in § 3.2: $k_{1}$ fixes the azimuthal electromotive period, whereas $k_{2}$ is the axial electric drive before compactification. Thus, within the present stationary cylindrical ansatz, Maxwell closure fixes $h\!B$ and $BC$ pointwise from $a$ , $B$ and the two drive constants.

Accordingly, the stationary locking relation can be stated as follows: once the magnetic pitch profile and the two electric drives are specified, Maxwell closure fixes the amplitude of the conjugate principal sector and the radial–temporal mixing at every radius. The locking is algebraic rather than differential after the two Maxwell equations have been integrated.

This result is specific to the stationary, radially dependent cylindrical pair. It is not a general relation for arbitrary flux ropes, nor is it the general kinematic locking relation derived by Kim (Reference Kim2026a ). The latter follows from metric conjugacy before imposing a particular field equation. Equation (2.12) is instead the additional profile locking imposed by the homogeneous Maxwell equations in the present cylindrical geometry. Electric charge and current are determined separately from the inhomogeneous Maxwell equation in § 3.

If $k_{1}=k_{2}=0$ , (2.12) gives $h\!B=BC=0.$ For $B\neq 0$ , this implies $h=C=0$ and the field reduces to a single stationary rank-two magnetic sector. A stationary rank-four solution within the present ansatz therefore requires a non-vanishing electric drive. Non-zero $k_{1}$ or $k_{2}$ is necessary, although the local non-degeneracy condition $h\neq 0$ must still be checked throughout the chosen radial domain.

2.3. Radial pitch shear and sector anholonomy

The general relation between principal-sector integrability, anholonomy and Bianchi closure is established by Kim (Reference Kim2026a ). In the present cylindrical pair, the corresponding diagnostics reduce to radial pitch-shear expressions that can be read directly from the profiles.

Using (2.11), one obtains

(2.13) \begin{align} \beta _G\wedge {\rm d}G &= -CB^{2}a'\,\varOmega, \qquad \beta _H\wedge {\rm d}H = -h\!B^{2} (r^{2}a)'\varOmega . \end{align}

We therefore define the anholonomies

(2.14) \begin{align} \fbox {${ q_G(r) := -CB^{2}a', \qquad q_H(r) := -h\!B^{2} (r^{2}a)'. }$} \end{align}

These quantities measure the failure of the two principal planes to remain integrable as their helical directions vary across the annulus. The pitches are derived in (3.9) later. Thus, $q_G$ is controlled by the radial shear of $a$ or, equivalently, of the first-sector pitch, while $q_H$ is controlled by the radial shear of the conjugate pitch $r^{2}a$ .

Pitch shear alone is not sufficient to produce non-zero anholonomy. For the $G$ sector, it must be accompanied by non-zero radial–temporal mixing $C$ , whereas for the $H$ sector, the conjugate amplitude $h$ must be non-zero. Assuming $B\neq 0$ ,

(2.15) \begin{align} q_G\neq 0 &\quad \Longleftrightarrow \quad C\neq 0 \quad \text{and}\quad a'\neq 0, \end{align}

(2.16) \begin{align} q_H\neq 0 &\quad \Longleftrightarrow \quad h\neq 0 \quad \text{and}\quad (r^{2}a)'\neq 0. \end{align}

This qualification is important: a magnetic pitch may vary radially while the corresponding rank-two sector remains integrable in a degenerate limit such as $C=0$ .

Combining (2.14) with the stationary locking relation, both sectors are strictly anholonomic wherever $ a'\neq 0$ , $ (r^{2}a)'\neq 0$ , $k_{2}r^{2}a-k_{1}\neq 0$ and $ k_{2}+k_{1}a\neq 0.$ The Lorentzian signature condition (2.8) additionally requires $|{k_{2}r^{2}a-k_{1}}/{B(1+r^{2}a^{2})}| \lt 1.$ Limits in which one of the two anholonomy coefficients vanishes remain admissible Maxwell fields; the corresponding sector simply becomes integrable. An explicit regular configuration for which both coefficients remain non-zero throughout a finite annulus is constructed in § 3.5 and Appendix A.

3.1. Observable fields and the sector contributions

3.1.1. The total fields

We use the static observer $u=\partial _{t}$ and the orthonormal spatial basis $\hat {\boldsymbol e}_{r}=\partial _{r}$ , $\hat {\boldsymbol e}_{\phi }=r^{-1}\partial _{\phi }$ , $\hat {\boldsymbol e}_{\chi }=\partial _{\chi }$ . Imposing the locking relation (2.12), the total electric and magnetic fields of the completed configuration are

(3.1) \begin{align} \fbox {${ \boldsymbol E = -\dfrac {k_{1}}{r}\, \hat {\boldsymbol e}_{\phi } + k_{2}\, \hat {\boldsymbol e}_{\chi }, \qquad \boldsymbol B = - B(a+ \kappa)\,\hat {\boldsymbol e}_{\phi } +\dfrac {B}{r} (1-r^2 a \kappa)\,\hat {\boldsymbol e}_{\chi }, }$} \end{align}

where

(3.2) \begin{align} \kappa (r):=h(r)C(r) \end{align}

is the magnetic participation of the conjugate sector, discussed later. Both fields are tangent to the cylinders $r=\mathrm{const.}$ , which is what allows the entire decomposition to be drawn in a single plane, as in figure 1(b). The electric field is remarkably simple: it is fixed entirely by the two constants of Proposition1 and carries no radial component.

For later use, we name the two observable magnetic components

(3.3) \begin{align} \mathcal B_{\phi } :=-B (a + \kappa), \qquad \mathcal B_{\chi } :=\frac {B}r(1 - r^2 a \kappa), \end{align}

in terms of which the completed two-form takes the compact form

(3.4) \begin{align} \fbox {${ F = -k_1\,{\rm d}t\wedge {\rm d}\phi + k_2\,{\rm d}t\wedge {\rm d}\chi + r\mathcal B_{\chi }(r)\, {\rm d}r\wedge {\rm d}\phi - \mathcal B_{\phi }(r)\, {\rm d}r\wedge {\rm d}\chi . }$} \end{align}

Written this way, ${\rm d}F=0$ is manifest: every coefficient depends on $r$ alone and every term already contains ${\rm d}r$ or is constant. Equation (3.4) is the form used in the periodic-identification analysis of § 4.

3.1.2. How the two sectors contribute

The observable fields (3.1) are the sum of contributions from the two principal sectors. From (2.4),

(3.5) \begin{align} \boldsymbol E_{G} = \frac {BC}{r}(\hat {\boldsymbol e}_{\phi } + \textit{ra}\, \hat {\boldsymbol e}_{\chi }), \qquad \boldsymbol B_{G} =\frac { B}{r}( -\textit{ra}\, \hat {\boldsymbol e}_{\phi } + \hat {\boldsymbol e}_{\chi }), \end{align}

and from (2.5),

(3.6) \begin{align} \boldsymbol E_{H} = h\!B (-\textit{ra}\, \hat {\boldsymbol e}_{\phi } + \, \hat {\boldsymbol e}_{\chi }), \qquad \boldsymbol B_{H} = -\kappa B( \hat {\boldsymbol e}_{\phi } + \textit{ra}\,\hat {\boldsymbol e}_{\chi }). \end{align}

These are decomposition-selected contributions; only their sum is the physical Maxwell field.

The magnetic participation $\kappa =hC$ controls how much of the observable magnetic field the conjugate sector carries. When $\kappa =0$ , the sector $H$ need not vanish, but it is purely electric with respect to the static observer; the branch with two non-vanishing magnetic helices is characterised by $\kappa \neq 0$ .

Each sector is separately rank two, $ \boldsymbol E_{G}\boldsymbol{\cdot }\boldsymbol B_{G}=0$ , $\boldsymbol E_{H}\boldsymbol{\cdot }\boldsymbol B_{H}=0$ , and the two exchange their electric and magnetic directions,

(3.7) \begin{align} \boldsymbol E_{H} = hr\,\boldsymbol B_{G}, \qquad \boldsymbol B_{H} = -hr\,\boldsymbol E_{G} \quad \rightarrow \quad \fbox {${\ \boldsymbol E_{H} \parallel \boldsymbol B_{G}, \qquad \boldsymbol B_{H} \parallel \boldsymbol E_{G},\ }$} \end{align}

which is the spatial form of the Hodge-dual relation (2.9), which is schematically plotted in figure 1.

3.1.3. The two helical directions

Define the pitch of any vector field tangent to the cylinders by

(3.8) \begin{align} p_{V}(r) := r\frac {V_{\hat \chi }}{V_{\hat \phi }} . \end{align}

For the two principal magnetic directions,

(3.9) \begin{align} p_{G} = r\frac {B_{G\hat \chi }}{B_{G\hat \phi }} = -\frac {B}{A} = -\frac {1}{a}, \qquad p_{H} = r\frac {B_{H\hat \chi }}{B_{H\hat \phi }} = \frac {r^{2}A}{B} = r^{2}a, \end{align}

whose product is fixed entirely by the cylindrical metric,

(3.10) \begin{align} \fbox {${ p_{G}(r)\,p_{H}(r)=-r^{2}. }$} \end{align}

As illustrated in figure 1(b), the two sector magnetic fields are perpendicular within the tangent plane of each cylinder. Their helices therefore wind oppositely and satisfy $p_Gp_H=-r^2$ , with a shallow helix paired with a steep one. This metric relation is independent of Maxwell locking, which instead enforces ${\rm d}G+{\rm d}H=0$ .

3.1.4. Rank four from two rank-two pieces

Although $G$ and $H$ are each rank two and satisfy $\boldsymbol E_I\boldsymbol{\cdot }\boldsymbol B_I=0$ , their sum is generically rank four. The mixed invariant,

(3.11) \begin{align} \boldsymbol E\boldsymbol{\cdot }\boldsymbol B = \frac {h(1-C^{2})}{r} (B^{2}+r^{2}A^{2}) = hr\big( |\boldsymbol B_{G}|^{2}-|\boldsymbol E_{G}|^{2} \big), \end{align}

is generically non-zero. The second form makes the mechanism transparent: since the sector-diagonal products $\boldsymbol E_I\boldsymbol{\cdot }\boldsymbol B_I$ vanish, the only surviving terms are the cross-sector pairings, $\boldsymbol E_{G}\boldsymbol{\cdot }\boldsymbol B_{H}+\boldsymbol E_{H}\boldsymbol{\cdot }\boldsymbol B_{G}$ . The parallel electric field of a non-ideal rope is therefore generated entirely by the coupling between the two orthogonal contributions, and vanishes if either one is absent. On a region satisfying (2.8), the completed field is non-degenerate precisely when $h\neq 0$ , that is, when $k_{2}+k_{1}a\neq 0$ in terms of the locking data (2.12).

3.1.5. The electric current

The homogeneous Maxwell equation ${\rm d}F=0$ imposes the stationary locking relation between the two principal sectors, while the inhomogeneous equation determines the physical electric current. From ${\rm d}\star F=J_{\textrm {e}}$ ,

(3.12) \begin{align} \fbox {${ \boldsymbol j_{\mathrm e} = -\mathcal B_{\chi }'\, \hat {\boldsymbol e}_{\phi } + \dfrac {1}{r} (r\mathcal B_{\phi })' \hat {\boldsymbol e}_{\chi }, }$} \end{align}

so radial gradients of the observable magnetic components are supported by azimuthal and axial currents, generically helical. The completed configuration is thus a current-carrying electromagnetic rope, not a vacuum field; imposing the vacuum condition $d\star F=0$ in addition would leave only the uniform axial and cylindrical $1/r$ branches. This is also why the phrase ‘source free’ is avoided throughout: the completion is closed, ${\rm d}F=0$ , but it is emphatically not current free.

Finally, a current-carrying rank-four completion can be neither force-free nor ideal, which sharply separates the present configurations from the force-free electrodynamics (Uchida Reference Uchida1997; Gralla & Jacobson Reference Gralla and Jacobson2014; Gralla & Iqbal Reference Gralla and Iqbal2019). Since $F\wedge F\neq 0$ , the matrix $F_{\mu \nu }$ is invertible, so

(3.13) \begin{align} F_{\mu \nu }J^{\nu }=0 \quad \Longrightarrow \quad J^{\mu }=0, \end{align}

and for the same reason, the ideal-MHD condition $F_{\mu \nu }u^{\nu }=0$ admits no non-zero timelike solution. The generic two-sector rope must therefore be interpreted as a non-ideal, matter-supported and externally driven structure.

3.2. Global support and boundary data

The two constants of the locking relation encode externally maintained electric drives. Because $\phi$ is already periodic on the annulus, $k_{1}$ fixes the azimuthal electromotive period. Before axial compactification, $k_{2}$ is instead a uniform axial electric drive, which may be maintained by end electrodes. Once the identification $\chi \sim \chi +L_{\chi }$ is imposed in § 4, it also defines the axial loop electromotive period. Thus, on the periodic annulus and with $E=-k_{1}\,{\rm d}\phi +k_{2}\,{\rm d}\chi$ ,

(3.14) \begin{align} \oint _{S^1_{\phi }} E=-2\pi k_{1}, \qquad \oint _{S^1_{\chi }} E=L_{\chi }k_{2}. \end{align}

Through (2.12), these electric-drive data fix the complementary amplitude $h\!B$ and the radial mixing $BC$ pointwise. The non-degenerate two-helix branch is therefore globally supported by non-zero electric drives, which must be maintained by boundary conditions or external driving.

This has an immediate consequence for the admissible domain. Regularity on the cylindrical axis removes the azimuthal $1/r$ electric component, $k_{1}=0$ , while localisation at radial infinity removes the uniform axial component, $k_{2}=0$ . Imposing both, for $B\neq 0$ , reduces the locking relations to $h=C=0$ . A non-degenerate stationary completion must therefore live on a finite or annular radial domain and be maintained by non-trivial electric drives; it cannot be a globally regular, isolated, self-contained object.

Although $\boldsymbol E\boldsymbol{\cdot }\boldsymbol B\neq 0$ on this branch, this does not by itself imply helicity decay. The corresponding global balance, on the periodic annulus and for this stationary solution, is derived in § 5.7.

3.3. Plasma support and stationary force–energy balance

The construction so far fixes the electromagnetic field and the Ampère current it carries. It does not specify the material medium. This subsection records the force and energy balance that a stationary plasma realisation must satisfy, and it rests on assumptions that are listed here rather than left implicit.

3.3.1. Assumptions of this subsection

(i) The state is stationary. (ii) Bulk inertia is negligible, so the momentum balance reduces to a static force balance. (iii) The plasma pressure is isotropic, $p=p(r)$ . (iv) The domain is annular, excluding the cylindrical axis. The purpose is not to select a unique material model, but to identify the matter support that the completed field requires; more general realisations are discussed at the end of the subsection.

On the annular domain, the electric field (3.1) is divergence free, $\rho _{\mathrm e}=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol E=0$ , and the current density (3.12) gives a purely radial electromagnetic force density:

(3.15) \begin{align} (\rho _{\mathrm e}\boldsymbol E + \boldsymbol j_{\mathrm e}\times \boldsymbol B)_{r} &= -\mathcal B_{\chi }\mathcal B_{\chi }' - \frac {\mathcal B_{\phi }}{r} (r\mathcal B_{\phi })' \nonumber \\[3pt] &= -\frac {1}{2} \frac {d}{{\rm d}r} \big( \mathcal B_{\phi }^{2} + \mathcal B_{\chi }^{2} \big) - \frac {\mathcal B_{\phi }^{2}}{r}. \end{align}

The first term is the radial magnetic-pressure force and the second the curvature force associated with the azimuthal component – the familiar screw-pinch combination.

Under assumptions (i)–(iv), the radial momentum balance $0=-\boldsymbol{\nabla }p+\rho _{\mathrm e}\boldsymbol E +\boldsymbol j_{\mathrm e}\times \boldsymbol B$ reduces to

(3.16) \begin{align} \frac {{\rm d}p}{{\rm d}r} = -\frac {1}{2} \frac {\rm d}{{\rm d}r} \big( \mathcal B_{\phi }^{2} + \mathcal B_{\chi }^{2} \big) - \frac {\mathcal B_{\phi }^{2}}{r}, \end{align}

so that the pressure profile is fixed, up to a boundary value, by

(3.17) \begin{align} \begin{split} p(r) ={}& p(r_{0}) - \frac {1}{2} \big[ \mathcal B_{\phi }^{2}(r) + \mathcal B_{\chi }^{2}(r) - \mathcal B_{\phi }^{2}(r_{0}) - \mathcal B_{\chi }^{2}(r_{0}) \big] - \int _{r_{0}}^{r} \frac {\mathcal B_{\phi }^{2}(s)}{s}\,{\rm d}s . \end{split} \end{align}

An admissible isotropic support requires this pressure to remain finite and non-negative throughout the chosen radial domain. Equation (3.16) supplies one admissible support model, not a unique plasma closure: realisations with bulk inertia, pressure anisotropy, viscous stresses or a solid mechanical support satisfy a different balance, and the corresponding terms must then be added.

3.3.3. Energy balance

The radial Poynting flux of the configuration is

(3.18) \begin{align} S_{r} = \frac {C(B^{2}+r^{2}A^{2})}{r^{2}} (1+h^{2}r^{2}), \end{align}

which is non-zero whenever $C\neq 0$ and, hence, on the whole two-helix branch $\kappa =hC\neq 0$ . On the non-trivial $h\neq 0$ branch, the zero-flux limit $C=0$ forces $\boldsymbol B_{H}=0$ , ${\rm d}G=-{\rm d}H=0$ and $r^{2}a=\text{constant}$ , so the structure degenerates to a pair of closed sectors. Because the field is stationary, Poynting’s theorem reads

(3.19) \begin{align} \fbox {${ \dfrac {1}{r} \dfrac {d}{{\rm d}r} (rS_{r}) = -\boldsymbol j_{\mathrm e}\boldsymbol{\cdot }\boldsymbol E = - \dfrac {k_{1}}{r}\mathcal B_{\chi }' - \dfrac {k_{2}}{r} (r\mathcal B_{\phi })'. }$} \end{align}

It should be stated explicitly that stationary does not mean energetically isolated. Equation (3.19) says that the radial variation of the Poynting flux is balanced by electromagnetic work done on the current-carrying medium; a strictly stationary state therefore requires that power input to be removed again by matter-energy transport, radiation, heat conduction or work at the radial boundaries. The same two constants $k_{1}$ and $k_{2}$ that fix the locking also control the global energy supply to the rope. The electromagnetic geometry fixes the required current, radial force, parallel non-ideal response and power transfer, while their material realisation depends on the chosen constitutive model.

3.4. Driven Ohmic realisation and force support

The same Maxwell field admits a simple Ohmic benchmark. Its purpose is narrow: it shows that the geometrically reconstructed field can be carried by a familiar helical current system. The scalar Ohm law fixes the current, magnetic profile and Joule dissipation, but does not by itself specify how the radial Lorentz force is supported. Two alternative force-support choices will be distinguished herein: mechanical stress in a resistive conductor and isotropic pressure balance in a plasma. The explicit constant- $\sigma$ example of § 3.5 adopts the latter force support. The two interpretations share the same $(\boldsymbol E,\boldsymbol B,\boldsymbol j_{e})$ and differ only in the material stress that balances the radial force.

3.4.1. Assumptions of this subsection

The conducting medium is at rest, obeys a scalar Ohm law with $\sigma (r)\gt 0$ and is driven by an external circuit; the Ohmic heat is removed so that the material state is stationary. For the conductor interpretation, the Lorentz force is carried by mechanical support rather than fluid pressure. Stationary fields generated by helical conductors and homogenised helical current sheets are standard in magnet theory, and their profiles, Lorentz forces and structural support have been studied analytically and numerically (Cain Reference Cain1983; Tominaka, Okamura & Katayama Reference Tominaka, Okamura and Katayama2001; Quéval & Gottkehaskamp Reference Quéval and Gottkehaskamp2015).

With $\boldsymbol j_{e}=\sigma (r)\boldsymbol E$ , the total field (3.1) gives

(3.20) \begin{align} j_{\phi }= -\frac {\sigma k_{1}}{r}, \qquad j_{\chi }= \sigma k_{2}, \end{align}

and the bulk charge density vanishes, $\rho _{e}=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol E=0$ , although surface charges and the external circuit are generally required at the conductor boundaries. The two constants $ k_1$ and $k_2$ have a direct circuit interpretation as done in § 3.2.

Comparing (3.20) with (3.12) gives $ \mathcal B_{\chi }'=\sigma (r)k_{1}/r$ and $ \bigl (r\mathcal B_{\phi }\bigr )'=\sigma (r)k_{2}r,$ so a prescribed conductivity yields an explicit magnetic profile. For constant $\sigma$ ,

(3.21) \begin{align} \mathcal B_{\chi }(r) = \mathcal B_{\chi 0} + \sigma k_{1} \ln \left (\frac {r}{r_{0}}\right ), \qquad \mathcal B_{\phi }(r) = \frac {\sigma k_{2}}{2}\,r + \frac {I_{0}}{r}, \end{align}

where $\mathcal B_{\chi 0}$ is a background axial field and $I_{0}/r$ represents a current enclosed inside the inner boundary. The logarithmic term is regular on the annulus $r_{\mathrm{in}}\leqslant r\leqslant r_{\mathrm{out}}$ and shows again why an axis-excluding geometry is natural when $k_{1}\neq 0$ .

3.4.2. Inverting the locking map

These magnetic profiles can be matched to the principal-sector variables through the stationary locking relations. At each fixed radius, define $\mathcal K_{1}:=hr^{2}A-BC$ and $\mathcal K_{2}:=h\!B+AC$ ; closure imposes $\mathcal K_{1}=k_{1}$ and $\mathcal K_{2}=k_{2}$ . The pointwise algebraic map $(A,B,C,h)\mapsto (\mathcal B_{\phi },\mathcal B_{\chi },\mathcal K_{1},\mathcal K_{2})$ has Jacobian

(3.22) \begin{align} \det \frac {\partial ( \mathcal B_{\phi }, \mathcal B_{\chi }, \mathcal K_{1}, \mathcal K_{2} )} {\partial (A,B,C,h)} = \frac { (1-C^{2}) (B^{2}+r^{2}A^{2}) (1+r^{2}h^{2})} {r}, \end{align}

which is non-zero on every regular Lorentzian branch, that is, wherever (2.8) holds. By the implicit-function theorem, the geometric data $(A,B,C,h)$ can then be reconstructed, locally uniquely, from the observable data $(\mathcal B_{\phi },\mathcal B_{\chi },k_{1},k_{2})$ . This is the sense in which the principal decomposition is not merely a rewriting: the two electric-drive data and the two magnetic components determine the entire principal-plane geometry.

3.4.3. Forces and power

The electromagnetic force density is purely radial,

(3.23) \begin{align} f_{r}^{\mathrm{EM}} &= (\boldsymbol j_{e}\times \boldsymbol B)_{r} = -\sigma \left ( \frac {k_{1}}{r}\mathcal B_{\chi } + k_{2}\mathcal B_{\phi } \right ). \end{align}

In the conductor realisation, this force is carried by the mechanical stress tensor of the conductor and its support, $\boldsymbol{\nabla} _{j}\tau _{\mathrm{mat}}^{\,ji}+f_{\mathrm{EM}}^{\,i}=0$ . In the plasma realisation, the same force may instead be balanced by an isotropic pressure gradient through (3.16); this is the choice made in the explicit solution later. More general realisations may involve anisotropic pressure or other material stresses.

Combining Joule heating with Poynting’s theorem gives

(3.24) \begin{align} \frac {1}{r}\frac {\rm d}{{\rm d}r}(rS_{r}) = -\,\boldsymbol j_{e}\boldsymbol{\cdot }\boldsymbol E = -\,\sigma (r) \left ( \frac {k_{1}^{2}}{r^{2}} + k_{2}^{2} \right ) \lt 0, \end{align}

which is strictly negative for $\sigma \gt 0$ : the Ohmic medium is a local sink of electromagnetic energy, the external circuit supplies the power and heat removal maintains the stationary state. The field is externally driven and requires both a force-support mechanism and heat removal; it is not an isolated soliton.

3.5. An explicit driven stationary solution

To show that the reconstruction conditions are simultaneously realisable, rather than merely formal pointwise constraints, we solve them explicitly for a constant-conductivity configuration on the annulus $1\leqslant r\leqslant 2$ . Here, the Ohm law fixes the magnetic profile and dissipation, while the radial Lorentz force is assigned to isotropic pressure balance rather than mechanical conductor support. For $\sigma =0.2$ , $k_{1}=0.1$ , $k_{2}=0.2$ , $\mathcal B_{\chi 0}=1$ and $I_{0}=0.3$ , the magnetic components (3.21) are regular and the algebraic inverse of § 3.4 selects a smooth branch throughout the annulus. On that branch, ${\max}C^{2}\simeq 0.0211\lt 1$ , so the Lorentzian signature condition holds everywhere, while both pitch-shear diagnostics (2.14) remain strictly non-zero. With the boundary condition $p(2)=0$ , the radial force balance (3.16) yields a monotonically decreasing pressure with $0\leqslant p(r)\leqslant p(1)\simeq 0.0235$ .

Thus, the closure locking of two individually sheared principal sectors is realised in an explicit, regular, Lorentzian configuration supported by a non-negative isotropic pressure. The full reconstruction and the numerical checks are given in Appendix A and summarised in figure 2.

What this example does and does not establish deserves a sentence of its own. It proves the non-emptiness of a smooth regular branch on a finite annulus: the constraints of Proposition1, the signature bound (2.8), the two shear conditions and non-negative pressure support can all hold at once. It is not a global existence result and nothing here shows that an arbitrary prescribed magnetic profile admits a regular principal completion.

Figure 2.

Constant- $\sigma$ Ohmic profile with isotropic-pressure support: the regular constant- $\sigma$ completion on $1\leqslant r/r_{0}\leqslant 2$ , with $\sigma =0.2$ , $k_{1}=0.1$ and $k_{2}=0.2$ . (a) The prescribed observable magnetic components $\mathcal B_{\phi }$ and $\mathcal B_{\chi }$ – this is the field a magnetic diagnostic would measure. (b) The principal-plane variables $a$ , $C$ and $h$ reconstructed from it by inverting the locking map: the pitch $a$ , the frame mixing $C$ and the amplitude $h$ of the complementary contribution. (c) The two pitch-shear (anholonomy) coefficients $-CB^{2}a'$ and $-h\!B^{2}(r^{2}a)'$ . Both are strictly non-zero across the annulus, so each sector’s winding direction shears with radius even though the total field is a Maxwell field. (d) The isotropic pressure profile with $p(2)=0$ ; it is non-negative throughout, so the configuration can be held by a non-negative isotropic pressure profile.

Figure 2. Long description

Four plots show the regular constant-conductivity reconstruction on 1 less than or equal to r over r zero less than or equal to 2, for conductivity 0.2 and drive constants k 1 equals 0.1 and k 2 equals 0.2. Panel (a) shows the prescribed observable magnetic components: B chi increases slightly from 1 to about 1.014, while B phi decreases from 0.32 to 0.19. Panel (b) shows reconstructed principal variables: pitch a increases from minus 0.295 to minus 0.086, mixing C increases from minus 0.145 to minus 0.081, and complementary amplitude h decreases from 0.156 to 0.091. Panel (c) shows two positive pitch-shear coefficients. q G decreases from about 0.084 to 0.023, while q H rises from about 0.0034 to 0.027 and crosses q G near the outer edge. Their non-zero values indicate radial shear of both sector winding directions. Panel (d) shows isotropic pressure decreasing smoothly from about 0.0235 at the inner radius to zero at the outer radius and remaining non-negative throughout.

4.1. What the periodic identification does and what it costs

Physically, $\chi \sim \chi +L_\chi$ is the periodic-cylinder or large-aspect-ratio surrogate for toroidal closure: a long plasma column whose two ends are joined or, equivalently, a single periodic cell of an axially repeating system. Periodic domains underlie the classical relaxation and helicity constructions (Woltjer Reference Woltjer1958; Taylor Reference Taylor1974, Reference Taylor1986; Bevir & Gray Reference Bevir and Gray1980; Berger Reference Berger1997; Xiao, Prior & Yeates Reference Xiao, Prior and Yeates2025). Related resistive relaxations of braided, line-tied fields demonstrate how strongly the resulting topology depends on the boundary condition (Yeates, Hornig & Wilmot-Smith Reference Yeates, Hornig and Wilmot-Smith2010; Pontin et al. Reference Pontin, Wilmot-Smith, Hornig and Galsgaard2011). What it does not do is include toroidal curvature: there is no major radius, no $1/R$ field variation and no Shafranov shift in the present model. It captures the topology of toroidal closure without its geometry.

The difference this makes is best stated as a comparison, illustrated in figure 3.

1. (i) Before the identification – on an open or line-tied cylinder – each surface of fixed $t$ and $r$ is a cylinder. Its first homology is generated by the single class $[S^{1}_{\phi }]$ . A field line that is not closed simply runs out of the domain and terminates on a boundary, and any helicity statement must be made relative to a reference field with boundary transport terms accounted for (Berger & Field Reference Berger and Field1984; Finn & Antonsen Reference Finn and Antonsen1985).

2. (ii) After the identification, each fixed-radius surface is a genuine two-torus $T^{2}_{r}=S^{1}_{\phi }\times S^{1}_{\chi }$ . Three things become available at once: a second non-contractible cycle, hence, a rank-two first homology and a winding lattice $H_{1}(T^{2}_{r},\mathbb Z)\simeq \mathbb Z^{2}$ ; two harmonic periods of the vector potential, which is what forces the corrected helicity of § 5; and closed field lines at resonant radii, which carry integral homology classes.

The one-forms ${\rm d}\phi$ and ${\rm d}\chi$ are globally defined closed one-forms on $T^{2}_{r}$ , although the angular functions $\phi$ and $\chi$ themselves need not be globally single valued. The completed field is the one obtained in § 3, (3.1) and (3.4).

Figure 3.

The topology of this section is introduced by a boundary condition, not by the local decomposition. (a) Two axial boundary conditions (schematic alternatives) that do not produce the topology of this section. (b) Imposing $\chi \sim \chi +L_{\chi }$ closes the axial direction. (c) After the identification, each fixed-radius surface is a two-torus with two independent cycles.

Figure 3. Long description

The three panels compare the topology before and after the additional periodic axial boundary condition. Panel (a) shows the open-end and line-tied-end cases as schematic alternatives on a cylinder. In either case, a fixed-radius surface is a cylinder and has only the azimuthal cycle S phi. Panel (b) shows the identification chi equivalent to chi plus L chi, represented by a curved double arrow joining the axial ends. Red and blue radial strips represent the two flux surfaces Sigma r phi and Sigma r chi, with fluxes Phi r phi and Phi r chi. Panel (c) shows the resulting fixed-radius surface as a torus. Its independent azimuthal cycle S phi and axial cycle S chi generate first homology equal to Z squared, and the vector potential consequently has two harmonic periods. Thus the two-torus topology used for winding and generalized helicity comes from the periodic boundary condition rather than from the local field decomposition.

4.2. Vanishing flux through fixed-radius tori and radial-strip fluxes

The pullback of $F$ to a fixed- $t$ , fixed- $r$ torus vanishes, $F|_{T^{2}_{r}}=0$ , because the completed field has no ${\rm d}\phi \wedge {\rm d}\chi$ component. The magnetic fluxes through the two radial strips $ \varSigma _{r\phi }= [r_{1},r_{2}]\times S^{1}_{\phi }$ and $ \varSigma _{r\chi }=[r_{1},r_{2}]\times S^{1}_{\chi }$ are

(4.1) \begin{align} \fbox {${ \varPhi _{r\phi } = \displaystyle\int_{\varSigma _{r\phi }}F = 2\pi \displaystyle\int_{r_{1}}^{r_{2}} r\mathcal B_{\chi }(r)\,{\rm d}r, \qquad \varPhi _{r\chi } = \displaystyle\int_{\varSigma _{r\chi }}F = -L_{\chi } \displaystyle\int_{r_{1}}^{r_{2}} \mathcal B_{\phi }(r)\,{\rm d}r. }$} \end{align}

The sign in the second expression corresponds to the orientation ${\rm d}r\wedge {\rm d}\chi$ . These strips have boundaries at $r_{1}$ and $r_{2}$ , so their fluxes are fixed by boundary data and are not topologically quantised. In plasma terms, $\varPhi _{r\phi }$ is the axial (‘toroidal’) flux through the annular cross-section and $\varPhi _{r\chi }$ is the azimuthal (‘poloidal’) flux through a radial cut.

4.3. Helical winding of the total magnetic field

We treat the physically observable total field first and, only afterwards, the winding assigned to its two principal contributions; the two must not be conflated.

Because the total magnetic field has no radial component, its field lines remain on surfaces of fixed $r$ . For a tangent vector field $\boldsymbol V=V_{\hat \phi }\hat {\boldsymbol e}_{\phi }+V_{\hat \chi }\hat {\boldsymbol e}_{\chi }$ on $T^{2}_{r}$ , with dimensional pitch (3.8), it is convenient to use the dimensionless normalised pitch $\nu _{V}(r):=(2\pi /L_{\chi })\,p_{V}(r)$ . A field line closes after $m$ turns around $S^{1}_{\phi }$ and $n$ turns around $S^{1}_{\chi }$ when

(4.2) \begin{align} \Delta \phi =2\pi m, \quad \Delta \chi =nL_{\chi }, \quad \rightarrow \quad p_{V}(r) = \frac {n}{m}\frac {L_{\chi }}{2\pi }, \quad \nu _{V}(r) = \frac {n}{m}\in \mathbb Q, \end{align}

and for a primitive closed curve, one may take $\gcd (m,n)=1$ . The closed field line then represents the homology class (Prior & MacTaggart Reference Prior and MacTaggart2020) $[\gamma ]=m[S^{1}_{\phi }]+n[S^{1}_{\chi }]\in H_{1}(T^{2}_{r},\mathbb Z)$ .

For the total magnetic field (3.1), the pitch is

(4.3) \begin{align} p_{B}(r) = r\frac {\mathcal B_{\chi }}{\mathcal B_{\phi }} = -\frac {1-r^{2}a \kappa }{ a+ \kappa }, \end{align}

so a physical field line closes at a radius $r=r_{\ast }$ when

(4.4) \begin{align} \fbox {${ -\dfrac {2\pi }{L_{\chi }} \dfrac {1 - r_{\ast }^{2}a(r_{\ast }) \kappa (r_\ast ) }{ a(r_{\ast }) + \kappa (r_{\ast }) } = \dfrac {n}{m}. }$} \end{align}

These are the rational, or resonant, magnetic surfaces of the completed rope, and they are the same objects familiar from tokamak and pinch theory.

It is worth being explicit about what (4.4) does and does not do. The pitch $p_{B}(r)$ varies continuously with radius, and the radial strip fluxes (4.1) vary continuously with the boundary data. The integers $(m,n)$ do not quantise either of them. What the condition does is select radii: it picks out those surfaces on which a continuously varying helical field line happens to close and, therefore, defines an integral homology class. Requiring a prescribed rational winding of the total field restricts the allowed continuous loop-voltage data through $\kappa$ , but selects no discrete values of the electric-field amplitude. It is a resonance condition, not a flux-quantisation condition.

The special cases $\mathcal B_{\chi }=0$ and $\mathcal B_{\phi }=0$ describe purely azimuthal and purely axial magnetic lines, and may be regarded as the limiting winding classes $(m,n)=(1,0)$ and $(m,n)=(0,1)$ .

4.4. Winding assigned to the two principal contributions

The winding of the total field just described is a property of the measurable rope, and is defined whenever $\mathcal B_{\phi }$ and $\mathcal B_{\chi }$ are; in particular, it is defined when $\kappa =0$ . The winding of the conjugate contribution, by contrast, is a property of the decomposition and exists only on the two-helix branch $\kappa \neq 0$ . It is a finer classification of the same physical field, not an additional field.

The total magnetic field is the sum of the two sector contributions, $\boldsymbol B=\boldsymbol B_{G}+\boldsymbol B_{H}$ , with components given in (3.5) and (3.6). Where $A\neq 0$ , $B\neq 0$ and $\kappa \neq 0$ , both sector pitches are finite and obey (3.10). Their normalised pitches are

(4.5) \begin{align} \nu _{B_{G}}(r) = -\frac {2\pi }{L_{\chi }} \frac {1}{a(r)}, \qquad \nu _{B_{H}}(r) = \frac {2\pi }{L_{\chi }}r^{2}a(r), \end{align}

and a sector line closes when

(4.6) \begin{align} \nu _{B_{G}}(r) = \frac {n_{G}}{m_{G}}, \quad \nu _{B_{H}}(r) = \frac {n_{H}}{m_{H}}, \qquad m_{G}, n_{G}, m_{H}, n_{H} \in \mathbb Z, \end{align}

the corresponding closed curves representing

(4.7) \begin{align} [\gamma _{G}] = m_{G}\big[S^{1}_{\phi }\big] + n_{G}\big[S^{1}_{\chi }\big], \qquad [\gamma _{H}] = m_{H}\big[S^{1}_{\phi }\big] + n_{H}\big[S^{1}_{\chi }\big]. \end{align}

The two sector fields need not close at the same radius; simultaneous closure requires both normalised pitches to be rational. Using (3.10), their product is

(4.8) \begin{align} \nu _{B_{G}}(r)\, \nu _{B_{H}}(r) = -\left ( \frac {2\pi r}{L_{\chi }} \right )^{2}, \end{align}

so simultaneous rational winding imposes a further condition on the dimensionless radius $2\pi r/L_{\chi }$ . Whenever both sector magnetic fields are non-zero, their pitches have opposite signs and obey the orthogonality relation (3.10), while their non-closures cancel according to ${\rm d}G+{\rm d}H=0$ . These two facts are logically independent: the first is geometry of the cylinder, the second is Maxwell’s equations.

5.1. Why the ordinary helicity integral is ambiguous here

Let $\mathscr B:=F|_{{\rm d}t=0}$ denote the spatial magnetic two-form on a constant-time slice and let $\mathscr A$ be a spatial potential one-form satisfying $\mathscr B={\rm d}\mathscr A$ . On a simply connected domain, the magnetic helicity $\int \mathscr A\wedge \mathscr B$ is gauge invariant when the magnetic field is tangent to the boundary, because a gauge transformation $\mathscr A\to \mathscr A+{\rm d}\lambda$ with single-valued $\lambda$ leaves the integral unchanged. The domain $V$ is not simply connected and this is where the problem arises. On $V$ , one may add to $\mathscr A$ a closed but not exact one-form,

(5.1) \begin{align} \mathscr A \longrightarrow \mathscr A + {\rm d}\lambda + c_{\phi }\,{\rm d}\phi + c_{\chi }\,{\rm d}\chi, \end{align}

with constants $c_{\phi },c_{\chi }$ . Such a shift leaves the magnetic field $\mathscr B={\rm d}\mathscr A$ completely unchanged – it is invisible to any measurement of $\boldsymbol B$ – yet it changes $\int _{V}\mathscr A\wedge \mathscr B$ . The raw Chern–Simons integral therefore depends on an arbitrary choice, namely the two loop periods of the vector potential around the non-contractible cycles, and it is not by itself a property of the magnetic field. Physically, the time derivatives of the two potential periods are fixed by the loop electromotive forces (3.14), so this ambiguity cannot be dismissed as a gauge artefact of no consequence in a driven system.

The standard remedy is to add correction terms that cancel the ambiguity, giving the generalised magnetic helicity for multiply connected domains (Bevir & Gray Reference Bevir and Gray1980; MacTaggart & Valli Reference MacTaggart and Valli2019). We set this up now. For the completed field, the magnetic two-form introduced here takes the explicit form

(5.2) \begin{align} \mathscr B := F\big |_{{\rm d}t=0} = X(r)\,{\rm d}r\wedge {\rm d}\phi + Y(r)\,{\rm d}r\wedge {\rm d}\chi, \end{align}

with $X(r):=r\mathcal B_{\chi }=B(1-r^{2}a\kappa )$ and $Y(r):=-\mathcal B_{\phi }=B(a+\kappa )$ . A compatible spatial potential may be chosen in the angular form $ \mathscr A = \psi _{\phi }(r)\,{\rm d}\phi + \psi _{\chi }(r)\,{\rm d}\chi,$ with $\psi _{\phi }'=X$ and $ \psi _{\chi }'=Y,$

the total magnetic fluxes through the two radial strips are

(5.3) \begin{align} \varPhi _{r\phi } := 2\pi \int _{r_{1}}^{r_{2}} X(r)\,{\rm d}r, \qquad \varPhi _{r\chi } := L_{\chi } \int _{r_{1}}^{r_{2}} Y(r)\,{\rm d}r, \end{align}

and the two loop periods of the potential on the inner boundary are

(5.4) \begin{align} P_{\phi } := \oint _{S^{1}_{\phi }(r_{1})}\mathscr A = 2\pi \psi _{\phi }(r_{1}), \qquad P_{\chi } := \oint _{S^{1}_{\chi }(r_{1})}\mathscr A = L_{\chi }\psi _{\chi }(r_{1}). \end{align}

For $V=[r_{1},r_{2}]\times T^{2}$ , the two-cycle specialisation of the generalised Bevir–Gray helicity is

(5.5) \begin{align} \fbox {${ H_{M} := H_0 + P_{\phi }\varPhi _{r\chi }-P_{\chi }\varPhi _{r\phi }; \qquad H_{0} := \displaystyle\int _{V} \mathscr A\wedge \mathscr B. }$} \end{align}

The first term is the usual Chern–Simons integral; the remaining two remove its dependence on the loop periods. We refer to $H_{M}$ simply as the generalised magnetic helicity.

The invariance is immediate. For the exact part of (5.1),

(5.6) \begin{align} \Delta _{\lambda }H_{0} = \int _{V}{\rm d}\lambda \wedge \mathscr B =\int _{\partial V}\lambda \,\mathscr B - \int _{V}\lambda \,{\rm d}\mathscr B = 0, \end{align}

since ${\rm d}\mathscr B=0$ and the pullback of $\mathscr B$ to each radial boundary torus vanishes – every term in (5.2) contains ${\rm d}r$ . The periods are unchanged because $\lambda$ is single valued. For the harmonic part, $\Delta _{\mathrm{harm}}H_{0}=-2\pi c_{\phi }\varPhi _{r\chi }+L_{\chi }c_{\chi }\varPhi _{r\phi }$ , while $\Delta P_{\phi }=2\pi c_{\phi }$ and $\Delta P_{\chi }=L_{\chi }c_{\chi }$ , so that $\Delta (P_{\phi }\varPhi _{r\chi }-P_{\chi }\varPhi _{r\phi }) =2\pi c_{\phi }\varPhi _{r\chi }-L_{\chi }c_{\chi }\varPhi _{r\phi }$ cancels it exactly. Hence, $ \Delta H_{M}=0,$ and $H_{M}$ is the appropriate helicity functional for this geometry.

5.2. Flux-plane area representation

Choose the representative for which the potential periods vanish at the inner boundary, $\psi _{\phi }(r_{1})=\psi _{\chi }(r_{1})=0$ ; since $H_{M}$ is invariant under changes of harmonic representative, this does not affect its value. Define the accumulated radial-strip flux vector

(5.7) \begin{align} \boldsymbol{\varPhi }(r) := ( \varPhi _{r\phi }(r), \varPhi _{r\chi }(r)) = \left ( 2\pi \int _{r_{1}}^{r}X(s)\,{\rm d}s, L_{\chi }\int _{r_{1}}^{r}Y(s)\,{\rm d}s \right ). \end{align}

Using ${\rm d}\boldsymbol{\varPhi }=(2\pi X,\,L_{\chi }Y)\,{\rm d}r$ , the generalised helicity becomes

(5.8) \begin{align} \fbox {${ H_M = \displaystyle\int (\varPhi _{r\chi }\,{\rm d}\varPhi _{r\phi } - \varPhi _{r\phi }\,{\rm d}\varPhi _{r\chi }) = -\displaystyle\int \omega (\boldsymbol{\varPhi },{\rm d}\boldsymbol{\varPhi }), }$} \end{align}

with $ \omega (a,b) := a_{\phi }b_{\chi } - a_{\chi }b_{\phi } .$ This is the cylindrical specialisation of the familiar flux-coordinate representation of helicity in toroidal domains (Boozer Reference Boozer1986; MacTaggart & Valli Reference MacTaggart and Valli2019).

The two axes of the flux plane have a direct physical meaning and it is worth naming it before the picture is used. The horizontal coordinate $\varPhi _{r\phi }(r)$ is the axial magnetic flux accumulated between the inner boundary and radius $r$ ; the vertical coordinate $\varPhi _{r\chi }(r)$ is the azimuthal flux accumulated over the same interval. The direction of $\boldsymbol{\varPhi }(r)$ therefore records the relative axial and azimuthal magnetic-flux content accumulated out to that radius – in effect, records the cumulative axial-to-azimuthal flux composition – and the turning of that direction as $r$ increases is what the helicity measures.

As illustrated in figure 4(a), $\boldsymbol{\varPhi }(r)$ traces a curve in this plane as $r$ increases, and $H_{M}$ is minus twice the oriented area enclosed by the curve and the straight segment joining its endpoint to the origin. A straight flux trajectory gives $H_{M}=0$ : if the accumulated flux always points in the same direction, there is no helicity. The helicity vanishes when the flux direction is radially constant; when that direction varies, the flux amplitudes determine the weight of each contribution.

Figure 4.

Flux-plane representation of the generalised magnetic helicity. The axes are the accumulated axial and azimuthal fluxes. (a) For an extended rope, $H_M$ is minus twice the oriented area enclosed by $\boldsymbol{\varPhi }(r)$ and its closing chord, (5.8). (b) In the formal localised-shell limit, two sector flux increments form a triangle whose area gives $2H_{GH}=-\operatorname {sgn}(r_H-r_G)I_{GH}\varphi _G\varphi _H$ . This distributional limit illustrates the intersection pairing, but is not asserted to be a smooth equilibrium.

Figure 4. Long description

Both panels use a flux plane with horizontal coordinate Phi r phi and vertical coordinate Phi r chi. Panel (a) shows an extended rope. As radius increases from r 1 to r 2, the accumulated flux vector traces a curved blue path from the origin to an endpoint. A dotted straight chord returns from the endpoint to the origin, enclosing a shaded oriented area with the curve. The generalized magnetic helicity H M is minus twice this oriented area; a straight radial flux trajectory would enclose no area and have zero helicity. Panel (b) shows the formal radially localized limit with a G-sector shell at smaller radius than an H-sector shell. A blue increment Delta Phi G runs from the origin to an upper-left point, followed by a red dashed increment Delta Phi H to an endpoint on the vertical axis. These increments and the closing dotted segment form a shaded triangle. Its oriented area gives the mutual contribution 2 H G H, including the sign set by radial ordering and the intersection pairing I G H. The localized construction is distributional and illustrates the pairing; it is not asserted to be a smooth equilibrium.

5.3. Sector decomposition and radial ordering

The magnetic two-forms carried by the two principal sectors, (3.5) and (3.6), are

(5.9) \begin{align} \mathscr B_{I} = X_I\,{\rm d}r\wedge {\rm d}\phi + Y_I\,{\rm d}r\wedge {\rm d}\chi, \end{align}

with

(5.10) \begin{align} ( X_{G}, Y_G) =(B, A), \qquad (X_{H}, Y_H)= \kappa (-r^{2}A, B), \end{align}

and $\mathscr B=\mathscr B_{G}+\mathscr B_{H}$ . The corresponding running flux vectors $\boldsymbol{\varPhi }_{I}(r)=(\varPhi ^{I}_{r\phi }(r),\varPhi ^{I}_{r\chi }(r))$ , $I=G,H$ , satisfy $\boldsymbol{\varPhi }=\boldsymbol{\varPhi }_{G}+\boldsymbol{\varPhi }_{H}$ and the local increment

(5.11) \begin{align} {\rm d}\varPhi ^{I}_{r\phi } = 2\pi X_{I}\,{\rm d}r, \qquad {\rm d}\varPhi ^{I}_{r\chi } = L_{\chi }Y_{I}\,{\rm d}r \end{align}

is dual to the winding direction. Magnetic helicity is widely used as a global measure of magnetic-field organisation and, in suitably closed systems, as a constraint on magnetic relaxation (Woltjer Reference Woltjer1958; Taylor Reference Taylor1974, Reference Taylor1986). Its total value alone, however, does not identify which radial feature of a flux rope carries that helicity. The principal decomposition provides a profile-level diagnostic of this internal organisation: as shown later, the sector-diagonal terms measure flux-weighted radial pitch shear within each principal magnetic direction, whereas the cross-term measures the radially ordered coupling between the two directions. Expanding (5.8) gives an algebraic self–mutual split, analogous to the standard self-mutual decompositions of magnetic helicity (Moffatt Reference Moffatt1969; Cantarella Reference Cantarella2000; Candelaresi et al. Reference Candelaresi, Hornig, MacTaggart and Simitev2021),

(5.12) \begin{align} \fbox {${ H_{M} = H_{GG} + H_{HH} + 2H_{GH}, }$} \end{align}

with

(5.13) \begin{align} H_{II} := - \int _{r_{1}}^{r_{2}} \omega (\boldsymbol{\varPhi }_{I}, {\rm d}\boldsymbol{\varPhi }_{I}), \quad 2H_{GH} := - \int _{r_{1}}^{r_{2}} \big [ \omega ( \boldsymbol{\varPhi }_{G}, {\rm d}\boldsymbol{\varPhi }_{H} ) + \omega ( \boldsymbol{\varPhi }_{H}, {\rm d}\boldsymbol{\varPhi }_{G} ) \big ]. \end{align}

The three terms are not independent observables. Only their sum $H_M$ is the generalised helicity of the physical magnetic field; $H_{GG}$ , $H_{HH}$ and $H_{GH}$ are contributions selected by the principal decomposition and are no more separately measurable than the sector fields themselves. Once the principal variables have been reconstructed from the observable magnetic profile and the applied electric drives, however, the split identifies whether the total helicity arises predominantly from radial pitch shear in either principal direction or from their mutual radial ordering. It can therefore distinguish ropes having similar total helicity, but different internal radial organisation. This is a geometrical diagnostic of the helicity budget, not by itself a criterion for stability, relaxation or transport.

Writing $\boldsymbol{\varPhi }(r)=\int _{r_{1}}^{r}{\rm d}\boldsymbol{\varPhi }(s)$ and substituting into (5.8) gives

(5.14) \begin{align} H_{M} = - \int _{r_{1}\lt s\lt r\lt r_{2}} \omega \left ( {\rm d}\boldsymbol{\varPhi }(s), {\rm d}\boldsymbol{\varPhi }(r) \right ) = -\frac {1}{2} \int _{r_{1}}^{r_{2}} \int _{r_{1}}^{r_{2}} \operatorname {sgn}(r-s)\, \omega \left ( {\rm d}\boldsymbol{\varPhi }(s), {\rm d}\boldsymbol{\varPhi }(r) \right ). \end{align}

Both $\omega ({\rm d}\boldsymbol{\varPhi }(s),{\rm d}\boldsymbol{\varPhi }(r))$ and $\operatorname {sgn}(r-s)$ are antisymmetric under $r\leftrightarrow s$ , so their product is symmetric; this is how an antisymmetric flux-plane pairing produces a symmetric mutual-helicity contribution. The helicity therefore measures not only the relative directions of the magnetic flux increments, but also their radial ordering – which flux family sits inside which.

5.4. Sector self-helicity as integrated radial pitch shear

The sector-diagonal contributions have a direct interpretation in terms of the radial shear of the principal planes – the same shear that measures the local anholonomy in § 2.3. With the above-mentioned radial ordering convention,

(5.15) \begin{align} H_{II} = -\int _{r_1\lt s\lt r\lt r_2} \omega \!\left ( {\rm d}\boldsymbol{\varPhi }_I(s), {\rm d}\boldsymbol{\varPhi }_I(r) \right ). \end{align}

Introduce $ b_G(r) \equiv a(r)$ , $ b_H(r) \equiv r^2a(r)$ , $ W_G(r) \equiv B(r)$ and $ W_H(r) \equiv \kappa (r)B(r),$ where $\kappa =hC$ ; the $b_{I}$ are sector shear variables and the $W_{I}$ are the corresponding flux weights. The infinitesimal flux vectors are $ {\rm d}\boldsymbol{\varPhi }_G(r) = W_G(r) \bigl ( 2\pi, L_\chi b_G(r) \bigr )\,{\rm d}r$ and $ {\rm d}\boldsymbol{\varPhi }_H(r) = W_H(r) (-2\pi b_H(r), L_\chi)\,{\rm d}r,$ so that the self-helicity becomes

(5.16) \begin{align} H_{II} = -2\pi L_\chi \int _{r_1\lt s\lt r\lt r_2} W_I(s)W_I(r) [b_I(r)-b_I(s)]\,{\rm d}s\,{\rm d}r . \end{align}

Using $ b_I(r)-b_I(s) = \int _s^r b_I'(u)\,{\rm d}u$ and exchanging the order of integration gives the equivalent single-integral representation

(5.17) \begin{align} \fbox {${ H_{II} = -2\pi L_\chi \displaystyle\int _{r_1}^{r_2} b_I'(u) \left [ \displaystyle\int _{r_1}^{u}W_I(s)\,{\rm d}s \right ] \left [ \displaystyle\int _{u}^{r_2}W_I(r)\,{\rm d}r \right ]{\rm d}u . }$} \end{align}

Thus, $H_{II}$ is a flux-weighted integral of the radial pitch shear $b_I'(r)$ . If $W_I(r)$ has a fixed sign throughout the domain, then $ b_I'(r) \gtrless 0\ \Longrightarrow \ H_{II} \lessgtr 0,$ and a radially constant pitch, $b_I'(r)=0$ , gives $H_{II}=0$ .

The same shear appears in the local anholonomy diagnostic (2.14). Setting $C_G(r)=C(r)$ and $C_H(r)=h(r)$ , (5.17) may be written as

(5.18) \begin{align} H_{II} = 2\pi L_\chi \int _{r_1}^{r_2} \frac {q_I(u)}{C_I(u)B^2(u)} \left [ \int _{r_1}^{u}W_I(s)\,ds \right ] \left [ \int _{u}^{r_2}W_I(r)\,{\rm d}r \right ]{\rm d}u . \end{align}

Equation (5.18) establishes the direct relation between local principal-sector shear and global sector self-helicity, for $C_I\neq 0$ . On the regular two-helix branch, the same pitch controls both $q_I$ and $H_{II}$ .

In plasma language, the sector self-helicity measures how much the winding direction of that sector’s flux turns between the inner and the outer boundary, weighted by how much flux sits at each radius. It is not the framed self-linking of an individual thin flux tube: no framing is involved and no thin tube is being linked with itself. Non-zero self-helicity requires shear somewhere, but a non-monotonic shear profile can produce cancellations, so vanishing total self-helicity does not imply vanishing local shear.

5.4.2. The constant- $\sigma$ solution

For the driven stationary solution of § 3.5 and Appendix A, take $L_\chi =2\pi$ . The reconstructed pitch satisfies $ a(1)=-0.295227$ , $ a(2)=-0.086086$ and $ a'(r)\gt 0,$ whereas $ r^2a(r)\big |_{r=1} = -0.295227$ , $r^2a(r)\big |_{r=2} = -0.344345$ and $ \bigl (r^2a(r)\bigr )'\lt 0 .$ Numerical evaluation of the ordered helicity integrals gives

(5.19) \begin{align} H_{GG} = -2.6093644, \qquad H_{HH} = 1.09171\times 10^{-4}, \end{align}

whose signs agree with the directions of the corresponding radial shears (see equation just below (5.17)). The smallness of $H_{HH}$ follows from the magnetic participation factor, $ -0.02263 \lesssim \kappa (r) \lesssim -0.00738,$ since $H_{HH}$ is weighted quadratically by $\kappa (s)\kappa (r)$ : the conjugate sector carries very little of the observable magnetic field in this solution. The mutual contribution is $2H_{GH}=-0.2018921$ , so that

(5.20) \begin{align} H_M = H_{GG}+H_{HH}+2H_{GH} = -2.8111473. \end{align}

The helicity of this configuration is thus dominated by the accumulated radial pitch shear of the $G$ sector. More specifically, the negative value of $H_M$ arises mainly from the radial variation of the dominant principal magnetic direction, while the weakly participating $H$ sector and the mutual term provide only subleading contributions. The total helicity should therefore not be interpreted here primarily as mutual winding between two helical flux families; it mainly records the internal radial reorientation of the $G$ -sector flux. This identifies the $G$ -sector pitch profile as the dominant geometrical carrier of helicity in the reconstructed solution.

5.5. Winding lattice and dual flux directions

We now connect the continuous helicity to the integer winding classes introduced in § 4.4. Suppose a sector magnetic line closes on a resonant torus $T^{2}_{r_{0}}$ , with homology class $ [\gamma _{I}] = m_{I}[S^{1}_{\phi }] + n_{I}[S^{1}_{\chi }],$ for $ I=G,H,$ and winding vector $ w_{I} := (m_{I},n_{I}) \in H_{1}(T^{2},\mathbb Z) \simeq \mathbb Z^{2}.$ The product structure $[r_{1},r_{2}]\times T^{2}$ gives a canonical radial identification of the homology groups of the nested tori and $I_{GH}$ denotes the intersection pairing after this transport to a common reference torus. With ${\rm d}\phi \wedge {\rm d}\chi \gt 0$ and $[S^{1}_{\phi }]\boldsymbol{\cdot }[S^{1}_{\chi }]=+1$ , the torus intersection pairing (Hatcher Reference Hatcher2002) is

(5.21) \begin{align} I_{GH} := [\gamma _{G}] \boldsymbol{\cdot }[\gamma _{H}] = \omega (w_{G},w_{H}) = m_{G}n_{H} - n_{G}m_{H}, \end{align}

with $I_{HG}=-I_{GH}$ . It counts, with sign, how many times the two closed helical curves cross on the torus.

The flux increment associated with a winding class is not parallel to the winding vector itself. From

(5.22) \begin{align} \frac {2\pi }{L_{\chi }} r\frac {\mathcal B_{\chi }}{\mathcal B_{\phi }} = -\frac {{\rm d}\varPhi _{r\phi }}{{\rm d}\varPhi _{r\chi }}, \end{align}

the resonance condition $(2\pi /L_{\chi })\,r\mathcal B_{\chi }/\mathcal B_{\phi }=n/m$ implies ${\rm d}\boldsymbol{\varPhi }\propto (-n,m)$ . For a radially localised sector with winding vector $w_I=(m_I,n_I)$ , we may therefore write $\Delta \boldsymbol{\varPhi }_{I}=\varphi _{I}(-n_I,m_I)$ , with $\varphi _I$ a signed flux amplitude, and it follows directly that

(5.23) \begin{align} \omega \left ( \Delta \boldsymbol{\varPhi }_{G}, \Delta \boldsymbol{\varPhi }_{H} \right ) = \varphi _{G}\varphi _{H} \left [ (-n_G)m_H - m_G(-n_H) \right ] = \varphi _{G}\varphi _{H}I_{GH}. \end{align}

The antisymmetric pairing of flux increments thus reproduces the intersection pairing of the corresponding winding classes; the direction $(-n,m)$ is the flux direction dual to the winding class $(m,n)$ . The same canonical antisymmetric form therefore appears in two representations,

(5.24) \begin{equation} \fbox {${ \begin{aligned} \omega \left ( \boldsymbol{\varPhi }, {\rm d}\boldsymbol{\varPhi } \right ) &: \quad \text{continuous flux-plane pairing}, \\ \omega (w_{G},w_{H}) &: \quad \text{integer winding-lattice pairing}. \end{aligned} }$} \end{equation}

5.6. Formal localised-shell and single-flux-family limits

We now take a formal distributional limit of the helicity decomposition. It is a kinematic limit of the helicity functional, taken to expose the pairing in its simplest form; it is not a solution of the locking relation and no claim is made that a smooth principal-locked configuration reduces to it.

Suppose the two sector fluxes are concentrated near distinct resonant radii $r_{G}$ and $r_{H}$ :

(5.25) \begin{align} {\rm d}\boldsymbol{\varPhi }_{I} = \Delta \boldsymbol{\varPhi }_{I}\, \delta (r-r_I)\,{\rm d}r, \qquad \Delta \boldsymbol{\varPhi }_{I} = \varphi _I(-n_I,m_I), \qquad I=G,H. \end{align}

Each sector then contributes a single flux increment, so its self-helicity vanishes, $H_{GG}=H_{HH}=0$ , and the flux-plane trajectory reduces to the two-step path of figure 4(b). Using the radially ordered form together with $\omega (\Delta \boldsymbol{\varPhi }_G,\Delta \boldsymbol{\varPhi }_H)=\varphi _G\varphi _H I_{GH}$ , one obtains, for either radial ordering,

(5.26) \begin{align} \fbox {${ H_M = 2H_{GH} = - \operatorname {sgn}(r_H-r_G)\, I_{GH}\, \varphi _G\varphi _H. }$} \end{align}

Both $I_{GH}$ and $\operatorname {sgn}(r_H-r_G)$ change sign under $G\leftrightarrow H$ , so their product is symmetric, as a mutual-helicity contribution must be. The intersection number supplies the discrete geometric coefficient; the signed fluxes and their radial ordering supply its continuous weight. If $r_G=r_H$ , the ordering must be specified by a radial push-off or an equivalent framing prescription. The resulting quantity is the cross-contribution selected by the principal decomposition, and it should be identified with a three-dimensional Gauss linking number only after an ambient embedding and a framing prescription have been fixed.

To repeat the caveat, since (5.26) is the most quotable formula in this section: the delta-shell model is a kinematic limit of the helicity functional and its smooth realisability within the principal-locked family is not established here.

For a single flux family localised on one resonant surface, ${\rm d}\boldsymbol{\varPhi }_{I}=\Delta \boldsymbol{\varPhi }_{I}\,\delta (r-r_{I})\,{\rm d}r$ , one has $\omega (\Delta \boldsymbol{\varPhi }_{I},\Delta \boldsymbol{\varPhi }_{I})=0$ and, hence, $H_{II}=0$ . More generally, if the flux direction is constant throughout the rope, ${\rm d}\boldsymbol{\varPhi }(r)=f(r)\,\boldsymbol{v}\,{\rm d}r$ for a fixed vector $\boldsymbol{v}$ , the flux-plane trajectory is a straight line and $H_{M}=0$ .

The generalised magnetic helicity defined here therefore measures radial shear of the winding direction together with the ordered coupling between distinct flux families. It is not the framed self-linking helicity of an individual thin tube, which would have the form $H_{\mathrm{self}}=\operatorname {SL}\,\varphi ^{2}$ , with a self-linking number $\operatorname {SL}$ that requires an additional framing (Moffatt & Ricca Reference Moffatt and Ricca1992) and is not fixed by the winding class $(m,n)$ alone.

5.7. Conservation on the stationary periodic annulus

Write the total field (3.4) as $F={\rm d}t\wedge E+\mathscr B$ , with $E=-k_{1}\,{\rm d}\phi +k_{2}\,{\rm d}\chi$ closed, $\textrm{d}E=0$ . A space–time potential may be chosen as $\mathcal A = \mathscr A_{0} + tE,$ so that ${\rm d}\mathcal A=\mathscr B+{\rm d}t\wedge E=F$ . The magnetic two-form is stationary, $\partial _{t}\mathscr B=0$ , but the loop part of the spatial potential evolves, $\partial _{t}\mathscr A=E$ , so the inner-boundary periods satisfy

(5.27) \begin{align} \dot P_{\phi } = \oint _{S^{1}_{\phi }}E = -2\pi k_{1}, \qquad \dot P_{\chi } = \oint _{S^{1}_{\chi }}E = L_{\chi }k_{2}. \end{align}

The magnetic strip fluxes are stationary, $\dot \varPhi _{r\phi }=\dot \varPhi _{r\chi }=0$ . The raw Chern–Simons integral therefore is not conserved: ${{\rm d}H_{0}}/{{\rm d}t} = \int _{V} E\wedge \mathscr B = 2\pi k_{1}\varPhi _{r\chi } + L_{\chi }k_{2}\varPhi _{r\phi } \neq 0 .$ At the same time, ${\rm d}/{{\rm d}t}(P_{\phi }\varPhi _{r\chi }-P_{\chi }\varPhi _{r\phi }) =-2\pi k_{1}\varPhi _{r\chi }-L_{\chi }k_{2}\varPhi _{r\phi }$ and the two contributions cancel exactly, so that

(5.28) \begin{align} \frac {{\rm d}H_{M}}{{\rm d}t}=0. \end{align}

Thus, $H_M$ , but not the raw Chern–Simons integral, is time independent on the stationary periodic annulus: the bulk $\boldsymbol E\boldsymbol{\cdot }\boldsymbol B$ contribution is cancelled by the loop-emf terms. This domain-specific balance is not a general helicity-conservation theorem and does not carry over unchanged to open or line-tied cylinders. It is a balance between a bulk term and a boundary term that happens to be exact for this stationary solution on this domain, and it is the explicit realisation, in the two-cycle annular geometry, of the standard helicity balance between bulk $\boldsymbol E\boldsymbol{\cdot }\boldsymbol B$ and loop-voltage or boundary contributions (Jensen & Chu Reference Jensen and Chu1984; Boozer Reference Boozer1986; Boozer Reference Boozer2024). In an open or line-tied cylinder, the corresponding contribution appears instead as helicity transport through the end boundaries and the same conservation statement does not follow automatically: one must then work with a relative helicity and account for the boundary flux explicitly (Berger & Field Reference Berger and Field1984; Finn & Antonsen Reference Finn and Antonsen1985).

Finally, $H_{M}$ is a continuous conserved functional, $H_{M}\in \mathbb R$ . It is not in general quantised, because the radial strip fluxes $\varphi _{I}$ are continuous. The intersection pairing $I_{GH}=m_{G}n_{H}-n_{G}m_{H}\in \mathbb Z$ is integer valued, and in the formal localised limit, the two are related by (5.26); but the helicity remains flux weighted and therefore continuous, unless a separate physical flux-quantisation condition is imposed.