Magnetic Helicity Reversal in the Corona at Small Plasma Beta

BBP solved the continuity equation, the equation of motion, the induction equation, and an energy equation, which includes the necessary energy sinks to get realistic and self-consistent coronal heating and cooling terms:

$$\begin{eqnarray}&&\displaystyle \frac{D\,\mathrm{ln}\,\rho }{{Dt}}=-{\boldsymbol{\nabla }}\cdot {\boldsymbol{u}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\rho \displaystyle \frac{D{\boldsymbol{u}}}{{Dt}}=-{\boldsymbol{\nabla }}P+\rho {\boldsymbol{g}}+{\boldsymbol{J}}\times {\boldsymbol{B}}+{\boldsymbol{\nabla }}\cdot (2\nu \rho {\mathsf{S}}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\rho T\,\displaystyle \frac{{Ds}}{{Dt}}=-{\boldsymbol{\nabla }}\cdot {\boldsymbol{F}}-{\rho }^{2}{\rm{\Lambda }}(T)+{\mu }_{0}\eta {{\boldsymbol{J}}}^{2}+2\rho \nu {{\mathsf{S}}}^{2},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{A}}}{\partial t}={\boldsymbol{u}}\times {\boldsymbol{B}}-{\mu }_{0}\eta {\boldsymbol{J}},\end{eqnarray} \tag{ 4 }$$

where $P=({ \mathcal R }/\mu )\rho T$ is the gas pressure, ${ \mathcal R }$ is the universal gas constant, $\mu =0.67$ is the mean atomic mass, T is the temperature, ${F}_{i}=-\rho {c}_{P}{\chi }_{{ij}}{{\rm{\nabla }}}_{j}T$ is the conductive heat flux, $s={c}_{V}\mathrm{ln}P-{c}_{P}\mathrm{ln}\rho +{s}_{0}$ is the specific entropy, s₀ is a constant, ${c}_{P}$ and ${c}_{V}$ are the specific heats at constant pressure and constant volume, respectively, ${\chi }_{{ij}}={\chi }_{0}{\delta }_{{ij}}+{\chi }_{\mathrm{Spitz}}{\hat{B}}_{i}{\hat{B}}_{j}$ is the thermal diffusivity, $\hat{{\boldsymbol{B}}}$ is the unit vector of the magnetic field, $\nu$ is the kinematic viscosity, $\eta$ is the magnetic diffusivity, ${\chi }_{\mathrm{Spitz}}$ is the Spitzer field-aligned heat conductivity, ${\chi }_{0}$ is an isotropic contribution, ${\boldsymbol{g}}$ is the gravitational acceleration, and ${\mathsf{S}}$ is the traceless rate-of-strain tensor with the components ${{\mathsf{S}}}_{{ij}}=\tfrac{1}{2}({u}_{i,j}+{u}_{j,i})-\tfrac{1}{3}{\delta }_{{ij}}{u}_{k,k}$. Here, we solve for ${\boldsymbol{A}}$ because then ${\boldsymbol{B}}$ is automatically divergence free. Instead of solving for s, we use the logarithmic temperature $\mathrm{ln}T$, which is directly related to s and $\mathrm{ln}\rho$. Using the logarithmic density $\mathrm{ln}\rho$, we are able to capture many orders of magnitude in the density stratification that our model atmosphere covers.

For the radiative cooling function $\rho {\rm{\Lambda }}(T)$, we use a realistic tabulation of Λ provided by Cook et al. (1989). In that work, the important emission peak from highly ionized iron lines is included, which efficiently cools the model corona. The characteristic half-time of this cooling is below 30 minutes.

During the first 35 minutes of physical time, however, this loss term is turned off together with the heat conduction along the field. This is to prevent excessive cooling of the corona during the initial phase in which the granular motions cause magnetic disturbances that still need to propagate from the photosphere into the corona. The simulation is then continued for another about 35 minutes after all physical terms in the equations are turned on. Further details about the switching on can be found in Bourdin et al. (2014) and Bourdin (2014). We use here data from a fully developed state at 63 minutes physical time.

Unlike ideal models of MHD, where η = ν = 0 and dissipation is modeled by highly nonlinear diffusion operators that cannot easily be stated in concise form and effective Reynolds or Lundquist numbers are difficult to specify, we use in our model constant values of ν, η, and also χ₀. The value of the magnetic diffusivity $\eta ={10}^{10}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$ ($={10}^{14}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$) is about eight orders of magnitude bigger than that estimated for the solar corona. This choice is required for numerical stability and for having a grid Reynolds number near unity. On the other hand, we use a realistic value for the viscosity, $\nu ={10}^{10}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$, which results in a Prandtl number of unity because η = ν. However, as pointed out by Rempel (2017), the relative importance of Ohmic and viscous heating changes toward the latter when realistically large values of the magnetic Prandtl numbers are taken into account; see Brandenburg (2014) for the relation between the dissipation ratio and the magnetic Prandtl number. The isotropic heat conduction is set to ${\chi }_{0}=5\times {10}^{8}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$ ($=5\times {10}^{12}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$). We use a realistic coronal value of ${\kappa }_{\mathrm{Spitz}}=1.8\cdot {10}^{-10}\,{T}^{5/2}/\mathrm{ln}\,{{\rm{\Lambda }}}_{C}$ with $\mathrm{ln}\,{{\rm{\Lambda }}}_{C}=20$ being the Coulomb logarithm. We obtain the field-aligned Spitzer conductivity as

$$\begin{eqnarray}&&{\chi }_{\mathrm{Spitz}}=\displaystyle \frac{1}{{c}_{P}}{\kappa }_{\mathrm{Spitz}}{T}^{5/2}/\rho .\end{eqnarray} \tag{ 5 }$$

The model is periodic in the horizontal directions and employs a potential-field extrapolation on the top boundary. To formulate the potential-field boundary condition, we define the Fourier-transformed magnetic vector potential as

$$\begin{eqnarray}&&\tilde{{\boldsymbol{A}}}({k}_{x},{k}_{y},z,t)=\int {\boldsymbol{A}}(x,y,z,t){e}^{{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}}{d}^{2}{\boldsymbol{r}},\end{eqnarray} \tag{ 6 }$$

where ${\boldsymbol{k}}=({k}_{x},{k}_{y})$ and ${\boldsymbol{r}}=(x,y)$. On the lower and upper z boundaries, we thus have

$$\begin{eqnarray}&&\displaystyle \frac{\partial \tilde{{\boldsymbol{A}}}}{\partial z}=-| {\boldsymbol{k}}| \tilde{{\boldsymbol{A}}}({k}_{x},{k}_{y},{z}_{* },t),\end{eqnarray} \tag{ 7 }$$

where z_* denotes the locations of the boundaries. Apart from this, the top boundary is closed for any plasma flows and is thermally insulating.

At the bottom boundary, realistic atmospheric temperature and density are imposed. We also adopt Equation (7) at the bottom boundary, but keep the minus sign on the right-hand side. This corresponds to an inverted potential-field extrapolation, whereby contrasts in the magnetic field are increased, instead of smearing them out like for the top boundary. This mimics the effect of flux tubes becoming narrower when entering below the photosphere. Because this increase of contrast would quickly lead to artifacts, like wiggles in the B_z component, this increase of contrast is limited to about one-third of the pressure scale height some 100 km below the photosphere so that artifacts in B_z are avoided. With this method, we obtain the ghost zones for all three components of ${\boldsymbol{A}}$ just beneath the lower photospheric boundary.

In general, the local magnetic helicity density

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{M}}}={\boldsymbol{A}}\cdot {\boldsymbol{B}}\end{eqnarray} \tag{ 8 }$$

depends on the gauge of the vector potential ${\boldsymbol{A}}$. On the lower and upper boundaries of the simulation domain, our magnetic field is driven toward a potential state. The resulting ${\boldsymbol{A}}$ at these boundaries is in the Weyl gauge. However, the gauge could in principle still drift because we have no boundary restrictions other than periodicity along $x$ and $y$. Therefore, we must check if such a gauge drift occurs and if it significantly changes our simulation results.

We use the relative helicity from Equation (5) of Finn & Antonsen (1985), similar to the formulation of Berger & Field (1984), to obtain a gauge-independent helicity as

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}(z)={\iiint }_{z}^{\infty }\displaystyle \frac{1}{2}({\boldsymbol{A}}+{{\boldsymbol{A}}}_{\mathrm{pot}})\cdot ({\boldsymbol{B}}-{{\boldsymbol{B}}}_{\mathrm{pot}}){dz}\,{dy}\,{dx}.\end{eqnarray} \tag{ 9 }$$

${{\boldsymbol{A}}}_{\mathrm{pot}}$ and ${{\boldsymbol{B}}}_{\mathrm{pot}}$ are the nonhelical potential fields extrapolated from the known state of ${B}_{z}(z)$ at the height $z$. On the upper boundary $z={{L}}_{z}$, our magnetic field is already almost potential and very close to the nonhelical extrapolation ${{\boldsymbol{B}}}_{\mathrm{pot}}({{L}}_{z})$. The magnetic helicity therefore vanishes toward the top of the domain and we may omit the volume from ${{L}}_{z}$ to $\infty$ in the integrals of ${H}_{\mathrm{rel}}$. Our factor of one-half compensates for the addition of the two similar quantities ${\boldsymbol{A}}$ and ${{\boldsymbol{A}}}_{\mathrm{pot}}$ in Equation (9), which allows the relative magnetic helicity to be quantitatively similar to the magnetic helicity ${{ \mathcal H }}_{{\rm{M}}}$. Because the components of ${\boldsymbol{B}}$ and ${{\boldsymbol{B}}}_{\mathrm{pot}}$ are normal to all nonperiodic boundaries (here, the top and bottom of the simulation domain), they are either identical by construction or their differences are negligible. Equation (9) gives us therefore a gauge-independent relative helicity (Berger & Field 1984; Finn & Antonsen 1985).

To compare a vertical profile of ${{ \mathcal H }}_{{\rm{M}}}$ and ${{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}$, we compute the horizontal averages of both quantities. To get $\langle {{ \mathcal H }}_{{\rm{M}}}(z){\rangle }_{{xy}}$ of the magnetic helicity density ${{ \mathcal H }}_{{\rm{M}}}(z)$, we simply average over horizontal slices from the height $z$ and with a thickness of ${\rm{\Delta }}z$ equal to our vertical grid spacing. For the profile of ${{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}(z)$, we simply subtract the volume integrals above the heights $z$ and $z+{\rm{\Delta }}z$. Each integral uses different potential fields, $A,{B}_{\mathrm{pot},{\rm{z}}}$ and $A,{B}_{\mathrm{pot},{\rm{z}}+{\rm{\Delta }}{\rm{z}}}$, which we extrapolate from the known states ${B}_{z}(z)$ and ${B}_{z}(z+{\rm{\Delta }}z)$, respectively. The average relative magnetic helicity density contained in this ${xy}$ layer is then just the difference

$$\begin{eqnarray}&&\langle {{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}{\rangle }_{{xy}}(z)=\displaystyle \frac{{{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}(z)-{{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}(z+{\rm{\Delta }}z)}{{V}_{{xy}{\rm{\Delta }}z}},\end{eqnarray} \tag{ 10 }$$

which we normalize to the volume of the layer ${V}_{{xy}{\rm{\Delta }}z}$ to be comparable to $\langle {{ \mathcal H }}_{{\rm{M}}}(z){\rangle }_{{xy}}$; see Figure 1.

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We find that the horizontal averages of our magnetic helicity density ${{ \mathcal H }}_{{\rm{M}}}$ and the relative helicity density ${{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}$ are very similar. Both magnetic helicities do show sign reversals that are located roughly at the same height, like the maxima and minima; see the dotted and red lines in Figure 1. The vertical profile of the current helicity, ${{ \mathcal H }}_{{\rm{C}}}={\mu }_{0}{\boldsymbol{j}}\cdot {\boldsymbol{B}}$, shows a qualitatively similar trend also with a sign reversal in the corona, albeit higher up; see the blue dashed line. We show that our magnetic helicity density is therefore not significantly influenced by the gauge drift along the periodic directions. Therefore, we may continue to use the gauge-dependent magnetic helicity ${{ \mathcal H }}_{{\rm{M}}}$ as a good proxy of the gauge-independent relative helicity ${{ \mathcal H }}_{{\rm{M}},\mathrm{rel}}$.

Through most of this work, we show both current and magnetic helicity. In particular, we consider two-dimensional current and magnetic helicity spectra defined as

$$\begin{eqnarray}&&{H}_{{\rm{C}}}(k)=\tfrac{1}{2}\displaystyle \sum _{{k}_{-}\lt | {\bf{k}}| \leqslant {k}_{+}}(\tilde{{\boldsymbol{J}}}\cdot {\tilde{{\boldsymbol{B}}}}^{* }+{\tilde{{\boldsymbol{J}}}}^{* }\cdot \tilde{{\boldsymbol{B}}}),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{H}_{{\rm{M}}}(k)=\tfrac{1}{2}\displaystyle \sum _{{k}_{-}\lt | {\bf{k}}| \leqslant {k}_{+}}(\tilde{{\boldsymbol{A}}}\cdot {\tilde{{\boldsymbol{B}}}}^{* }+{\tilde{{\boldsymbol{A}}}}^{* }\cdot \tilde{{\boldsymbol{B}}}),\end{eqnarray} \tag{ 12 }$$

where ${k}_{\pm }=k\pm \delta k/2$ and $\delta k=2\pi /L$ with $L=235\,\mathrm{Mm}$ being the size of the magnetograms and tildes denote, again, Fourier transformation. Under horizontally isotropic conditions, we have

$$\begin{eqnarray}&&{\mu }_{0}{H}_{{\rm{C}}}(k)={k}^{2}{H}_{{\rm{M}}}(k),\end{eqnarray} \tag{ 13 }$$

i.e., the current helicity spectrum is directly related to the magnetic helicity spectrum, but weighted with a k² factor, so high wavenumber contributions in ${H}_{{\rm{M}}}(k)$ get enhanced.

It is sometimes convenient to define the magnetic and current helicity densities as

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{M}}}={\boldsymbol{A}}\cdot {\boldsymbol{B}},\quad {{ \mathcal H }}_{{\rm{C}}}={\boldsymbol{J}}\cdot {\boldsymbol{B}}.\end{eqnarray} \tag{ 14 }$$

Note, in particular, that at each value of z, we have

$$\begin{eqnarray}&&\int {H}_{{\rm{M}}}\,{dk}=\langle {{ \mathcal H }}_{{\rm{M}}}{\rangle }_{{xy}},\quad \int {H}_{{\rm{C}}}\,{dk}=\langle {{ \mathcal H }}_{{\rm{C}}}{\rangle }_{{xy}}.\end{eqnarray} \tag{ 15 }$$

In the following, however, we often retain the more explicit notation in terms of ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ and ${\boldsymbol{J}}\cdot {\boldsymbol{B}}$.

To set the stage, we show in Figure 2 vertical profiles of $\langle {H}_{{\rm{M}}}{\rangle }_{{xy}}$, $\langle {H}_{{\rm{M}}}{\rangle }_{{xy}}/\langle {{\boldsymbol{B}}}^{2}{\rangle }_{{xy}}$, and the plasma beta, $2{\mu }_{0}\langle P{\rangle }_{{xy}}/\langle {{\boldsymbol{B}}}^{2}{\rangle }_{{xy}}$ (cf. Bourdin 2017). For comparison, we also plot the corresponding profiles for averages over the AR core and the complementary quiet-Sun (QS) area. For the plasma beta, we also show minimum and maximum values (dotted). In the lower part, for $z\lesssim 5\,\mathrm{Mm}$, $\langle {\boldsymbol{A}}\cdot {\boldsymbol{B}}{\rangle }_{{xy}}$ is positive, while for $z\gtrsim 5\,\mathrm{Mm}$ it is negative. In fact, for the full field of view (FOV), and in a small z interval very close to the lower boundary, the sign of $\langle {\boldsymbol{A}}\cdot {\boldsymbol{B}}{\rangle }_{{xy}}$ changes once again. We return to this aspect again later.

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The main focus of this paper is instead the sign reversal of $\langle {\boldsymbol{A}}\cdot {\boldsymbol{B}}{\rangle }_{{xy}}$ at $z\approx 5\,\mathrm{Mm}$. This happens at a height where the magnetic field begins to become almost force free; see Figure 2(a). To assess this quantitatively, we plot in Figure 3 vertical profiles of the characteristic nondimensional wavenumbers ${\kappa }_{{\boldsymbol{J}}\cdot {\boldsymbol{B}}}$ and ${\kappa }_{{\boldsymbol{J}}\times {\boldsymbol{B}}}$ defined through (Warnecke & Brandenburg 2010)

$$\begin{eqnarray}&&{\kappa }_{{\boldsymbol{J}}\cdot {\boldsymbol{B}}}^{2}\equiv \displaystyle \frac{\langle {({\boldsymbol{J}}\cdot {\boldsymbol{B}})}^{2}{\rangle }_{{xy}}}{\langle {{\boldsymbol{J}}}^{2}{{\boldsymbol{B}}}^{2}{\rangle }_{{xy}}},\quad {\kappa }_{{\boldsymbol{J}}\times {\boldsymbol{B}}}^{2}\equiv \displaystyle \frac{\langle {({\boldsymbol{J}}\times {\boldsymbol{B}})}^{2}{\rangle }_{{xy}}}{\langle {{\boldsymbol{J}}}^{2}{{\boldsymbol{B}}}^{2}{\rangle }_{{xy}}}.\end{eqnarray} \tag{ 16 }$$

Note that ${\kappa }_{{\boldsymbol{J}}\cdot {\boldsymbol{B}}}^{2}+{\kappa }_{{\boldsymbol{J}}\times {\boldsymbol{B}}}^{2}=1$, so the two are complementary in the sense that when ${\kappa }_{{\boldsymbol{J}}\times {\boldsymbol{B}}}\to 0$, we have ${\kappa }_{{\boldsymbol{J}}\cdot {\boldsymbol{B}}}\to 1$, and vice versa. Looking at Figure 3, we see that in $5\,\mathrm{Mm}\lesssim z\lesssim 50\,\mathrm{Mm}$, the magnetic field is indeed nearly force free and ${\kappa }_{{\boldsymbol{J}}\cdot {\boldsymbol{B}}}$ reaches values close to unity, the largest possible value. Consequently, ${\kappa }_{{\boldsymbol{J}}\times {\boldsymbol{B}}}$ is very small in this range for AR and QS.

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In Figure 4, we show a visualization of ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$. We see that the magnetic helicity density changes in a horizontal plane at approximately $5\,\mathrm{Mm}$. We also show the magnetic helicity density in several yz planes through two particularly prominent magnetic field lines labeled as CL1 and CL2, where CL1 is a EUV-emissive loop in the core of the AR. While one of the two field lines passes through regions where ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ is positive (red) throughout, the other field line traverses yz planes in which ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ is positive near the apex of the line (denoted by CL2) and negative in the xy plane through $5\,\mathrm{Mm}$ (denoted by CL1). This shows that at least one magnetic helicity reversal is possible right in the middle of a field line or loop.

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The loops SL1–3 connect from one of the main polarities to the periphery of the AR. SL1–3 show strongly asymmetric heating and EUV emissivity. We find that the coronal heating is particularly strong on that side, where SL1–3 are rooted in strong negative magnetic helicity (blue in Figure 4). The other end of these side loops connects to low-helicity areas and there we also see less heating and EUV emissivity. Note that under the assumption of horizontal isotropy leading to Equation (13), both the current helicity and magnetic helicity spectra are related. In particular, since the spectral magnetic helicity reflects the large-scale properties of current helicity, it corresponds to the integral over all large-scale patches of the current helicity density.

We also note that the most strongly heated core-loop CL1 is rooted in two strong positive helicity regions (red in Figure 4), and at the same time, we find a negative helicity (blue) near the loop apex. While we do not want to claim a direct relation of magnetic helicity and the coronal Ohmic dissipation of currents that heats our model loops, we need to point out that local injection of helicity is a way of transporting magnetic energy to the corona and induce currents there. Nonetheless, this could only demonstate that the volumetric heating and the EUV emissivity is of course strongly modulated by density variations. In particular, when the density is low, the heating per particle is high. Therefore, one would not see a clear one-to-one correlation of helicity and the coronal heating or EUV emission.

The study of magnetic helicity spectra has revealed important insights about the nature of the turbulent dynamo; see Brandenburg & Subramanian (2005) for a review. Owing to magnetic helicity conservation, the α effect in mean-field electrodynamics (Moffatt 1978; Krause & Rädler 1980) can only produce positive and negative magnetic helicities to equal amounts, but at different length scales (Seehafer 1996; Ji 1999). This leads to a bihelical magnetic field (Blackman & Brandenburg 2003; Yousef & Brandenburg 2003) with one sign at the scale of the energy carrying eddies (referred to as "small scale") and another sign at the scale of the domain (referred to as "large scale"). In the solar wind, the spectrum is also found to be bihelical, but the signs at both small and large scales are reversed (Brandenburg et al. 2011). In the MHD model, the photospheric structures are "small scale" and smear out when reaching higher atmospheric layers. Coronal loops then define the "large scale" structures. The basic question is now whether this apparent swap in sign at small and large scales happens abruptly at one particular height and across all scales, or gradually through an effective shift of the spectrum in wavenumber, as perhaps suggested by the idea of an inverse cascade behavior, where the height in the domain plays the role of time in a decaying MHD simulation; see Christensson et al. (2001) for an example in the cosmological context.

The result is shown in Figure 5 where we compare visualizations of both ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ and ${\boldsymbol{J}}\cdot {\boldsymbol{B}}$ in six horizontal planes with the corresponding spectra ${k}^{2}{H}_{{\rm{M}}}(k)$ and ${H}_{{\rm{C}}}(k)$ obtained in the same six planes. Note first of all that the two spectra look similar in shape and magnitude at corresponding heights, suggesting that the relation between them, as given in Equation (13) for isotropic turbulence, is reasonably well obeyed. The spectra vary over more than 10 orders of magnitude, falling steeply with wavenumber, with its largest values corresponding to the smallest few wavenumbers that dominate the overall sign of the total integrated magnetic and current helicities. In the first three slices up to $z\approx 5\,\mathrm{Mm}$, the dominant signs of ${k}^{2}{H}_{{\rm{M}}}(k)$ and ${H}_{{\rm{C}}}(k)$ are negative for $k\gt 3\,{\mathrm{Mm}}^{-1}$ and positive for $k\lt 3\,{\mathrm{Mm}}^{-1}$. Above this layer, the sign of ${k}^{2}{H}_{{\rm{M}}}(k)$ reverses abruptly in the sense that it is now negative (positive) for k smaller (larger) than $3\,{\mathrm{Mm}}^{-1}$. However, the sign of ${H}_{{\rm{C}}}(k)$ varies more gradually with height, showing a similar reversal only at $z\approx 13\,\mathrm{Mm};$ see also Figure 7. Interestingly enough, at a fixed height, both below and above the transition layer at $z\approx 5\,\mathrm{Mm}$ where the sign reversal of the magnetic helicity occurs, the spectrum ${k}^{2}{H}_{{\rm{M}}}(k)$ changes its sign in k space at roughly the same value of k, namely at $k\approx 3\,{\mathrm{Mm}}^{-1}$. This supports the notion that this phenomenon is related to a change in the relative dominance of structures of opposite sign of ${\boldsymbol{J}}\cdot {\boldsymbol{B}}$, as discussed above in Section 3.2, and is not due to a shift in k, which would be more reminiscent of an inverse cascade-type behavior.

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We reiterate that ${H}_{{\rm{M}}}(k)$ is gauge independent. It is therefore important to emphasize that the magnetic and current helicity reversals are also seen in specific wavenumber intervals (e.g., for k larger or smaller than $3\,{\mathrm{Mm}}^{-1}$). Moreover, the reversals occur at the same height as those in ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$. This supports the notion that the sign change in ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ is not compromised by its gauge dependence; see also Figure 1 and Section 2.4. Furthermore, the upper boundary condition Equation (7) always tends to relax ${\boldsymbol{A}}$ back to zero, as any contrasts get smeared out by the potential-field extrapolation. The A_x and A_y components are set through the lower boundary condition to match the observed B_z component. Hence, any drift in ${\boldsymbol{A}}$ will be suppressed.

Within the lower corona, in the range $5\,\mathrm{Mm}\lesssim z\lesssim 50\,\mathrm{Mm}$, the plasma beta is around 10⁻² or less; see Figure 2(c). The magnetic field here is nearly fully force free; see Figure 3. In this range, the angle

$$\begin{eqnarray}&&\sphericalangle ({\boldsymbol{J}},{\boldsymbol{B}})=\arccos ({\boldsymbol{J}}\cdot {\boldsymbol{B}}/(| {\boldsymbol{J}}| \,| {\boldsymbol{B}}| ))\end{eqnarray} \tag{ 17 }$$

between ${\boldsymbol{J}}$ and ${\boldsymbol{B}}$ is on average very small. Closer to the surface, for $z\lt 5\,\mathrm{Mm}$, larger values can be found, but even then the angles are hardly much larger than $\pm 2^\circ ;$ see Figure 6. Only above the AR can larger angles of up to $\pm 8^\circ$ be found.

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Very near the surface, we have seen in Figure 2 for the full FOV an additional reversal in magnetic helicity very close to the surface. Looking at a similar plot of current helicity, we see that this secondary reversal is now more pronounced and includes even the AR. In current helicity, the secondary reversal is seen at $z\approx 2\,\mathrm{Mm}$. Furthermore, the primary reversal occurs higher up at about $13\,\mathrm{Mm}$. The reason for this secondary reversal becomes more plausible when looking at the horizontal distribution of ${\boldsymbol{J}}\cdot {\boldsymbol{B}}$ in Figure 7, which shows that there are always nearly equally many and nearly equally large patches of both helicities. Thus, the dominance of one sign over the other depends on small changes in the relative strengths of structures with positive and negative contributions to ${\boldsymbol{J}}\cdot {\boldsymbol{B}}$. The second reversal in current helicity is obviously a real phenomenon in the present simulations, but it is unclear whether it is also a generic phenomenon of stratified and magnetized atmospheres in general. Furthermore, in magnetic helicity, it was only seen in the full FOV and not above the AR. Comparing the maps of magnetic and current helicities shown in Figure 5, we see that ${{ \mathcal H }}_{C}$ is more noisy, and therefore the additional reversal does not appear to be a systematic feature. More important to note is that the magnetic helicity associated with the AR is positive near the surface, exactly as would be expected for the southern hemisphere based on an α effect-driven turbulent dynamo.

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Looking once more at Figure 5, it becomes clear that the negative sign of magnetic helicity in the uppermost layers can be associated with a single structure that persists in all the horizontal maps of ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ between 3 and $22\,\mathrm{Mm}$. This persistent helicity patch is located in the AR core near the legs of the loop CL1 that are indicated by black circles in Figure 5. Structures of opposite sign tend to be associated with the periphery of the core of the AR.

As discussed above, under the assumption of isotropy, the magnetic and current helicity spectra are related to each other through Equation (13). It was already clear from Figure 5 that this assumption holds reasonably well. The purpose of this section is to analyze this in more detail. Therefore, we show in Figure 8 scatter plots of ${\mu }_{0}{H}_{{\rm{C}}}(k)$ versus ${k}^{2}{H}_{{\rm{M}}}(k)$ for the same six height as in Figure 5. It turns out that most of the data points lie on the diagonal, which covers about eight orders of magnitude. Some of the data points, however, have mutually opposite signs, which correspond to an anticorrelation. Thus, Equation (13) holds primarily for the moduli of ${H}_{{\rm{C}}}(k)$ and ${H}_{{\rm{M}}}(k)$.

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The fact that some of the data points have the opposite sign was already evident when examining the colors in Figure 5. Below $z=5\,\mathrm{Mm}$, about 80% of the points have the expected sign, but at higher levels, the number of exceptions increases. For large values of $| {H}_{{\rm{M}}}(k)|$, and especially for $z\gt 5\,\mathrm{Mm}$, there is a noticeable number of data points below the diagonal, i.e., $| {H}_{{\rm{C}}}(k)|$ is somewhat smaller than expected.

It is clear from Figure 5 that ${H}_{{\rm{M}}}(k)$ shows fewer sign reversals with k than ${H}_{{\rm{C}}}(k)$ and shows a more systematic behavior in that sense. One would therefore be tempted to trust the magnetic helicity spectra more than the current helicity spectra. However, two other considerations come to mind. First, both spectra are intrinsically noisy and one can expect meaningful results only after some degree of averaging. This could be accomplished by averaging the spectra over broader wavenumber bins. Second, Equation (13) is only valid under the assumption of isotropy. Again, this statement only applies in the statistical sense, i.e., after sufficient averaging. This is particularly evident in the present case where there is only one AR with its resulting coronal structure. In view of these caveats, one must say that the agreement found in Figure 8 is actually rather remarkable.