2.1 Separating field strength and topological information

To disentangle the field strength (flux) and topological information associated with the helicity input, we can also consider the net winding input

(2.7) $$\begin{eqnarray}\frac{\text{d}L}{\text{d}t}=-\frac{1}{2\unicode[STIX]{x03C0}}\int _{P}\int _{P}\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a}_{1})\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a}_{2})\frac{\text{d}\unicode[STIX]{x1D703}_{12}(\boldsymbol{a}_{1},\boldsymbol{a}_{2},t)}{\text{d}t}\,\text{d}x_{1}\,\text{d}y_{1}\,\text{d}x_{2}\,\text{d}y_{2},\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{z}(\boldsymbol{x})=\left\{\begin{array}{@{}cc@{}}1 & \text{ if }B_{z}(\boldsymbol{x})>0,\\ -1 & \text{ if }B_{z}(\boldsymbol{x})<0,\\ 0 & \text{ if }B_{z}(\boldsymbol{x})=0,\end{array}\right.\end{eqnarray}$$

(e.g. Prior & Yeates Reference Prior and Yeates2014). The rate $\text{d}L/\text{d}t$ measures the input of field line entanglement through the photosphere and ignores the extra flux information provided by the helicity. A similar suggestion made in the literature is to divide the helicity input by the square of the flux (e.g. Yamamoto et al. Reference Yamamoto, Kusano, Maeshiro, Yokoyama and Sakurai2005; LaBonte et al. Reference LaBonte, Georgoulis and Rust2007; Vemareddy & Démoulin Reference Vemareddy and Démoulin2017). We propose that $\text{d}L/\text{d}t$ is more meaningful in this context as it is based on a quantity, the net winding, which is also an ideal invariant (Prior & Yeates Reference Prior and Yeates2014). Indeed, it was shown recently to be more topologically meaningful than helicity as it can be used, under certain circumstances, to classify field topologies and hence precisely measure changing field connectivity (Prior & Yeates Reference Prior and Yeates2018).

2.2 Evolving photosphere

The standard practice in both theoretical and observational studies of helicity is to treat the lower (photospheric) boundary as fixed. In almost all studies, the photosphere is a plane (e.g. Pariat et al. Reference Pariat, Démoulin and Berger2005, Reference Pariat, Nindos, Démoulin and Berger2006; Jeong & Chae Reference Jeong and Chae2007; Sturrock et al. Reference Sturrock, Hood, Archontis and McNeill2015; Vemareddy Reference Vemareddy2015; Vemareddy & Démoulin Reference Vemareddy and Démoulin2017), although calculations in spherical geometry have also been performed (Berger Reference Berger1985; MacTaggart et al. Reference MacTaggart, Gregory, Neukirch and Donati2016; Moratis et al. Reference Moratis, Pariat, Savcheva and Valori2018). In our calculations, which we will describe shortly, our fixed photospheric plane $P$ will represent the fixed height $z=0$ in a Cartesian domain.

In addition to this approach, we also consider the effect of an evolving photosphere. Including this effect means that the lower boundary is no longer fixed but is a surface that deforms in space and time. In our calculations, we will define the photospheric boundary as the surface where the plasma density $\unicode[STIX]{x1D70C}$ has the non-dimensional value of 1. In order to evaluate how a changing photosphere will affect the helicity and winding fluxes, we use the following procedure. First, we calculate the varying surface on which $\unicode[STIX]{x1D70C}=1$ , i.e. the set

(2.9) $$\begin{eqnarray}\left\{P_{v}(\boldsymbol{a},t)\,|\,\boldsymbol{a}\in P,\,\unicode[STIX]{x1D70C}(P_{v})=1\right\}.\end{eqnarray}$$

$P$ will always represent the horizontal plane at $z=0$ in our model (more details will be given later). At $t=0$ in the simulations, $P_{v}\equiv P$ . As indicated, $P_{v}$ is a function of the coordinates $(x,y)$ of $P$ .

We then evaluate $\text{d}\boldsymbol{a}/\text{d}t$ at $P_{v}(\boldsymbol{a},t)$ rather than at $P(\boldsymbol{a})$ . We can use this quantity to calculate an effective rotational derivative $\text{d}\unicode[STIX]{x1D703}_{12}/\text{d}t$ using (2.3). To account for the surface geometry, we replace the area element $\text{d}x_{1}\text{d}y_{1}$ with $J(\boldsymbol{a}_{1})\text{d}x_{1}\text{d}y_{1}$ , where $J$ the Jacobian of the surface $P_{v}$ at the coordinates of $\boldsymbol{a}_{1}$ (and similarly for $\boldsymbol{a}_{2}$ ). For example, the calculation (2.7) will become

(2.10) $$\begin{eqnarray}\displaystyle \frac{\text{d}L}{\text{d}t} & = & \displaystyle -\frac{1}{2\unicode[STIX]{x03C0}}\int _{P}\int _{P}\bigg[\unicode[STIX]{x1D70E}_{z}(P_{v}(\boldsymbol{a}_{1},t))\unicode[STIX]{x1D70E}_{z}(P_{v}(\boldsymbol{a}_{2},t))\frac{\text{d}\unicode[STIX]{x1D703}_{12}(P_{v}(\boldsymbol{a}_{1},t),P_{v}(\boldsymbol{a}_{2},t))}{\text{d}t}\nonumber\\ \displaystyle & & \displaystyle \,\times J(P_{v}(\boldsymbol{a}_{1},t))J(P_{v}(\boldsymbol{a}_{2},t))\bigg]\text{d}x_{1}\,\text{d}y_{1}\,\text{d}x_{2}\,\text{d}y_{2}.\end{eqnarray}$$

Equation (2.10) does not represent the exact winding flux through a spatially non-uniform surface. We could, for example, have made use of the differential form definition of relative helicity given in Chapter 3 of Arnold & Khesin (Reference Arnold and Khesin1999) in order to calculate the actual flux. However, we argue that what the above procedure represents is more akin to the effective projection of information onto the plane $P$ which occurs in the creation of magnetogram and vector magnetogram data (e.g. Démoulin & Pariat Reference Démoulin and Pariat2009; Scherrer et al. Reference Scherrer, Schou, Bush, Kosovichev, Bogart, Hoeksema, Liu, Duvall, Zhao and Schrijver2012). These magnetogram data are obtained from line-of-sight information, Zeeman splitting and normal to line-of-sight linear polarization information. These data are effectively projected onto a plane $P$ . Since we are trying to gauge the consequences of an effect which is already uncertain (the interpretation of optical information) we feel the above approach is sensible first step. To the best of our knowledge, this is the first theoretical helicity study which has attempted to take account of a moving photosphere.

3.1 Initial background atmosphere and velocity perturbation

The idealized initial equilibrium atmosphere is given by prescribing the temperature profile

(3.11) $$\begin{eqnarray}T(z)=\left\{\begin{array}{@{}cc@{}}1-{\displaystyle \frac{\unicode[STIX]{x1D6FE}-1}{\unicode[STIX]{x1D6FE}}}z, & z<0,\\ 1, & 0\leqslant z\leqslant 10,\\ 150^{[(z-10)/10]}, & 10<z<20,\\ 150, & z\geqslant 20.\end{array}\right.\end{eqnarray}$$

Starting from the top of (3.11), the sections represent the solar interior, the photosphere/chromosphere, the transition region and the corona. The above model temperature profile is standard in many works on flux emergence (e.g. Murray et al. Reference Murray, Hood, Moreno-Insertis, Galsgaard and Archontis2006; Fan Reference Fan2009; Sturrock et al. Reference Sturrock, Hood, Archontis and McNeill2015). The above temperature profile is also used in Prior & MacTaggart (Reference Prior and MacTaggart2016).

The other state variables, pressure and density, are found by solving the hydrostatic equation in conjunction with the ideal equation of state

(3.12) $$\begin{eqnarray}\frac{\text{d}p}{\text{d}z}=-\unicode[STIX]{x1D70C}g,\quad p=\unicode[STIX]{x1D70C}T.\end{eqnarray}$$

To study emergence, we must place a particular form for the magnetic field in the solar interior and apply a perturbation to allow it to emerge. In the following simulations, we will apply an initial velocity perturbation of the form

(3.13) $$\begin{eqnarray}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{z}}=u_{0}\exp \left(-\frac{x^{2}}{x_{0}^{2}}\right)\exp \left(-\frac{y^{2}}{y_{0}^{2}}\right)\exp \left(-\frac{(z+\unicode[STIX]{x1D716}_{0}-R)^{2}}{z_{0}^{2}}\right)\sin \left(\frac{t}{t_{0}}\unicode[STIX]{x03C0}\right),\end{eqnarray}$$

where we set the constants $u_{0}=0.05$ , $x_{0}=5$ , $y_{0}=3$ , $z_{0}=5$ , $\unicode[STIX]{x1D716}_{0}=2.5$ and $t_{0}=6$ ; $R$ is the major radius of the tube and is described below. After $t=6$ the perturbation is switched off.

We will consider two magnetic field models. The first model is a twisted toroidal tube that has been used in many other studies (e.g. MacTaggart & Hood Reference MacTaggart and Hood2009; MacTaggart & Haynes Reference MacTaggart and Haynes2014; MacTaggart et al. Reference MacTaggart, Guglielmino, Haynes, Simitev and Zuccarello2015; Sturrock et al. Reference Sturrock, Hood, Archontis and McNeill2015). The second model is a ‘mixed helicity’ field which has a complex topology but zero total helicity and will serve as a model for mixed helicity emergence. We now discuss the construction of such fields.

3.2 How to construct mixed helicity fields

The following represents a specific case of the general mathematical form of magnetic flux ropes with general field line topology, introduced in Prior & Yeates (Reference Prior and Yeates2016) and first applied to flux emergence in Prior & MacTaggart (Reference Prior and MacTaggart2016). We assume the flux rope has a toroidal shape with an axis curve $\boldsymbol{r}(s)$ parameterized by its arclength $s$ as

(3.14) $$\begin{eqnarray}\boldsymbol{r}(s)=(-R\cos (s/R),0,R\sin (s/R)+z_{0}),\quad s\in [0,\unicode[STIX]{x03C0}R],\end{eqnarray}$$

where $R$ is the major radius of the torus and $z_{0}$ is the height of the rope’s anchoring footpoints at the base of the computational domain. This expression can be used to define a moving orthonormal frame $\{\boldsymbol{T},\boldsymbol{d}_{1},\boldsymbol{d}_{2}\}$ for $\boldsymbol{r}(s)$ , which can, in turn, be used to define a tubular coordinate system through the map $\boldsymbol{f}(s,x_{1},x_{2})$ , where

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{f}(s,x_{1},x_{2})=\boldsymbol{r}(s)+x_{1}\boldsymbol{d}_{1}+x_{2}\boldsymbol{d}_{2}, & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{d}_{1}=\left(\cos (s/R),0,-\sin (s/R)\right),\quad \boldsymbol{d}_{2}=(0,1,0). & \displaystyle\end{eqnarray}$$

From this mapping, we can find a set of basis vector fields $\{\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}s},\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{1}},\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{2}}\}$ which can be used to define general vector fields in this toroidal coordinate system as

(3.17) $$\begin{eqnarray}\boldsymbol{B}=B_{s}\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}s}+B_{1}\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{1}}+B_{2}\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{2}}.\end{eqnarray}$$

For such fields to be divergence free the following condition must hold

(3.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}JB_{s}}{\unicode[STIX]{x2202}s}+\frac{\unicode[STIX]{x2202}JB_{1}}{\unicode[STIX]{x2202}x_{1}}+\frac{\unicode[STIX]{x2202}JB_{2}}{\unicode[STIX]{x2202}x_{2}}=0,\end{eqnarray}$$

where $J=(R-x_{1})/R$ is the Jacobian of the map $\boldsymbol{f}$ . It follows that a magnetic field defined in a cylinder can be transferred to a torus simply by dividing the components by $J$ . In particular, we take the mixed helicity (braided) field used in numerous studies (e.g. Wilmot-Smith, Pontin & Hornig Reference Wilmot-Smith, Pontin and Hornig2010; Wilmot-Smith et al. Reference Wilmot-Smith, Pontin, Yeates and Hornig2011; Pontin & Hornig Reference Pontin and Hornig2015), which is composed of a series of overlapping counter-twists. The magnetic field has no helicity (the average total entanglement) but subsets of its field lines are braided. In our toroidal coordinate system this field is composed of exponential twists $\boldsymbol{B}_{t}(b_{0},k,a,l,x_{1c},x_{2c},s_{c})$ given by

(3.19) $$\begin{eqnarray}\displaystyle \hspace{-12.0pt} & \displaystyle \boldsymbol{B}_{t}(b_{0},k,a,l,x_{1c},x_{2c},s_{c})=\frac{2b_{0}k}{aJ}\text{exp}\left(-\frac{(x_{1}-x_{1c})^{2}+(x_{2}-x_{2c})^{2}}{a^{2}}-\frac{(s-s_{c})^{2}}{l^{2}}\right)\boldsymbol{R},\qquad & \displaystyle\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle \hspace{-10.79993pt} & \displaystyle \boldsymbol{R}=-(x_{2}-x_{2c})\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{1}}+(x_{1}-x_{1c})\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}x_{2}}, & \displaystyle\end{eqnarray}$$

where the parameter $b_{0}$ determines the strength of the field, $a$ the horizontal width of the twist zones, $l$ their vertical extent and $k$ the handedness of the twist ( $k=1$ is right handed). The centre of rotation is $(x_{1c},x_{2c},s_{c})$ . The mixed helicity field is then defined as a superposition of $n$ pairs of positive and negative twists and an axial background field

(3.21) $$\begin{eqnarray}\displaystyle \boldsymbol{B}_{br}(b_{0},a,l,d,R,n) & = & \displaystyle \mathop{\sum }_{i=1}^{n}\left[\boldsymbol{B}_{t}(b_{0},1,a,l,0,-d,s_{d}i)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\boldsymbol{B}_{t}(b_{0},-1,a,l,0,d,s_{d}(i+1))\right]+\frac{b_{0}}{J}\frac{\unicode[STIX]{x2202}\boldsymbol{f}}{\unicode[STIX]{x2202}s},\end{eqnarray}$$

where $s_{d}=\unicode[STIX]{x03C0}R/(2n+1)$ and $d$ is the axial offset of the two opposing twists (see figure 1). The final component is the axial field (see figure 1 b). Finally, to express (3.21) in the ambient Cartesian coordinate system $(x,y,z)$ , we use the following maps between the tubular coordinates $(s,x_{1},x_{2})$ and the Cartesian coordinates

(3.22) $$\begin{eqnarray}s=\arctan (z,x)+\frac{\unicode[STIX]{x03C0}}{2},\quad x_{1}=(x+R\cos s)\cos s-(z-R\sin s)\sin s,\quad x_{2}=y,\end{eqnarray}$$

where the branch cut for the arctan function is at $\unicode[STIX]{x03C0}$ . For a magnetic field confined to a tube of minor radius $r$ , the field $\boldsymbol{B}_{b}(x,y,z)$ takes the following form

(3.23) $$\begin{eqnarray}\boldsymbol{B}_{b}(x,y,z)=\left\{\begin{array}{@{}cc@{}}\boldsymbol{B}_{br}(b_{0},a,l,d,R,n) & \text{ if }x_{1}^{2}(x,y,z)+x_{2}^{2}(x,y,z)\leqslant r,\\ 0 & \text{ if }x_{1}^{2}(x,y,z)+x_{2}^{2}(x,y,z)>r.\end{array}\right.\end{eqnarray}$$

For completeness, we can define a uniformly twisted field $\boldsymbol{B}_{t}$ as

(3.24) $$\begin{eqnarray}\boldsymbol{B}_{tw}(x,y,z)=\left\{\begin{array}{@{}cc@{}}{\displaystyle \frac{\unicode[STIX]{x1D719}}{J}}\boldsymbol{R} & \text{ if }x_{1}^{2}(x,y,z)+x_{2}^{2}(x,y,z)\leqslant r,\\ 0 & \text{ if }x_{1}^{2}(x,y,z)+x_{2}^{2}(x,y,z)>r.\end{array}\right.\end{eqnarray}$$

with $\unicode[STIX]{x1D719}$ determining the rate of rotation of the field and $\boldsymbol{R}$ given by (3.20). The actual twisted model that we will use in the simulations is that in (3.24) but weighted with an exponential term (see MacTaggart & Hood Reference MacTaggart and Hood2009). In these studies, we set the tube parameters to be $z_{0}=-30$ , $R=17.5$ and $r=2.5$ , so that the flux rope is anchored at the bottom boundary and its maximum initial height is $-10$ . The mixed helicity parameters are $a=\sqrt{0.04}$ , $l=0.04\unicode[STIX]{x03C0}R$ , $d=2.5/3$ , $b_{0}=5$ and we consider $n=2$ .

Figure 1.

An example of the mixed helicity field we consider. (a) A field with $n=2$ , i.e. two right-handed twist elements (red) and two left handed (blue). (b) Shows the background axial field which is added to (a) in equation (3.21).

Finally, we remark that the more general field specification in Prior & Yeates (Reference Prior and Yeates2016) allows for arbitrary tube shapes $\boldsymbol{r}(s)$ (as well as varying tube radius). The frame vectors $\boldsymbol{d}_{1}(s)$ and $\boldsymbol{d}_{2}(s)$ are defined by parallel transport, which requires an arclength integration. Thus, in general, the relationship between the tube coordinates $(s,x_{1},x_{2})$ and $(x,y,z)$ will require numerical integration.

4.1 Field line helicity input rate

For completeness, we restate (2.6),

(4.1) $$\begin{eqnarray}\frac{\text{d}{\mathcal{H}}}{\text{d}t}(\boldsymbol{a}_{0})=-\frac{1}{2\unicode[STIX]{x03C0}}B_{z}(\boldsymbol{a}_{0})\int _{P}B_{z}(\boldsymbol{a})\frac{\text{d}\unicode[STIX]{x1D703}(\boldsymbol{a}_{0},\boldsymbol{a})}{\text{d}t}\,\text{d}x\,\text{d}y,\end{eqnarray}$$

which represents the contribution to the magnetic helicity input rate $\text{d}H/\text{d}t$ from a single point $\boldsymbol{a}_{0}\in P$ . For the moving photosphere calculation we evaluate the field at the points of the surface $P_{v}(\boldsymbol{a},t)$ and account for the surface Jacobian, i.e. $\text{d}x\,\text{d}y\rightarrow J\text{d}x\,\text{d}y$ . We label this quantity $\text{d}{\mathcal{H}}_{v}/\text{d}t$ .

4.1.1 Net helicity input

The net helicity flux is given by

(4.2) $$\begin{eqnarray}\frac{\text{d}H}{\text{d}t}=\int _{P}\frac{\text{d}{\mathcal{H}}}{\text{d}t}(\boldsymbol{a}_{0})\,\text{d}x\,\text{d}y,\end{eqnarray}$$

where the integration is taken over all $\boldsymbol{a}_{0}\in P$ . The total helicity input over a period $[t_{0},t]$ through $P$ is then given by

(4.3) $$\begin{eqnarray}H(t)=\int _{t_{0}}^{t}\frac{\text{d}H}{\text{d}t^{\prime }}\,\text{d}t^{\prime }.\end{eqnarray}$$

In this study we choose $t_{0}$ to be the time at which the emerging magnetic field first reaches $z=0$ . The equivalent moving photosphere calculations will be labelled $\text{d}H_{v}/\text{d}t$ and $H_{v}(t)$ . We point out that whilst the calculation concerns a moving surface, the quantity $\text{d}H/\text{d}s$ is calculated on a fixed domain $P$ by projection (see 2.10). This applies to all calculations which consider the moving domain.

4.2 Winding input rate

The field line winding input is

(4.4) $$\begin{eqnarray}\frac{\text{d}{\mathcal{L}}}{\text{d}t}(\boldsymbol{a}_{0})=-\frac{1}{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a}_{0})\int _{P}\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a})\frac{\text{d}\unicode[STIX]{x1D703}(\boldsymbol{a}_{0},\boldsymbol{a})}{\text{d}t}\,\text{d}x\,\text{d}y,\end{eqnarray}$$

and the integral of this over $P$ gives $\text{d}L/\text{d}t$ . The net winding input over a time period $[t_{0},t]$ is

(4.5) $$\begin{eqnarray}L(t)=\int _{t_{0}}^{t}\frac{\text{d}L}{\text{d}t^{\prime }}\,\text{d}t^{\prime }.\end{eqnarray}$$

The moving photosphere calculations will be labelled with a $v$ subscript as for the helicity inputs, with analogous changes to the calculations.

4.3 Weak field corrections

In our simulations, there is no magnetic field outside the emerging regions. Therefore, there is a thin current sheet around the emerging field where the field strength rapidly decreases to zero. In this region the, field line topology is incoherent and not likely to be resolved numerically (typical field strengths in this layer are measured to be ${<}0.01\,\%$ of the peak field strength). Such regions make a negligible contribution to the helicity, due to the field strength weighting, but can have a significant effect on the winding calculations. To remove such potentially spurious information we will, in some calculations, modify the definition of the indicator function $\unicode[STIX]{x1D70E}_{z}$ to be

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{z}(\boldsymbol{x})=\left\{\begin{array}{@{}cc@{}}1 & \text{if }B_{z}(\boldsymbol{x})>0\text{ and }|\boldsymbol{B}|>\unicode[STIX]{x1D716},\\ -1 & \text{if }B_{z}(\boldsymbol{x})<0\text{ and }|\boldsymbol{B}|>\unicode[STIX]{x1D716},\\ 0 & \text{if }B_{z}(\boldsymbol{x})=0\text{ or }|\boldsymbol{B}|\leqslant \unicode[STIX]{x1D716}.\end{array}\right.\end{eqnarray}$$

Where we use this definition in the article, we will be explicit and specify the value of $\unicode[STIX]{x1D716}$ . Otherwise it is to be assumed that $\unicode[STIX]{x1D70E}_{z}$ has its usual definition (2.8).

4.4 Winding and helicity ratios

Both the helicity and winding densities can be positive and negative corresponding to right- and left-handed field entanglement. For mixed helicity flux ropes there can be significant entanglement, locally, but little overall average. This is true of the mixed helicity fields $\boldsymbol{B}_{b}$ which have zero total helicity. To quantify whether the helicity input for a given simulation is biased to one sign or is mixed, we calculate the following ratios

(4.7) $$\begin{eqnarray}\displaystyle H_{t}^{r}=\frac{\text{d}H}{\text{d}t}\Big/\frac{\text{d}H^{abs}}{\text{d}t},\quad \frac{\text{d}H^{abs}}{\text{d}t}=\frac{1}{2\unicode[STIX]{x03C0}}\int _{P}\int _{P}\left|B_{z}(\boldsymbol{a}_{1})B_{z}(\boldsymbol{a}_{2})\frac{\text{d}\unicode[STIX]{x1D703}_{12}(\boldsymbol{a}_{1},\boldsymbol{a}_{2},t)}{\text{d}t}\right|\text{d}x_{1}\,\text{d}y_{1}\,\text{d}x_{2}\,\text{d}y_{2}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}L_{t}^{r}=\frac{\text{d}L}{\text{d}t}\Big/\frac{\text{d}L^{abs}}{\text{d}t},\quad \frac{\text{d}L^{abs}}{\text{d}t}=\frac{1}{2\unicode[STIX]{x03C0}}\int _{P}\int _{P}\left|\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a}_{1})\unicode[STIX]{x1D70E}_{z}(\boldsymbol{a}_{2})\frac{\text{d}\unicode[STIX]{x1D703}_{12}(\boldsymbol{a}_{1},\boldsymbol{a}_{2},t)}{\text{d}t}\right|\text{d}x_{1}\,\text{d}y_{1}\,\text{d}x_{2}\,\text{d}y_{2}.\end{eqnarray}$$

4.5 Weighted velocity flux ${\mathcal{V}}_{z}$

The plasma flow across the photosphere will transpire to be an important quantity in what follows. We calculate the net rate of plasma flow through the photosphere weighted by the absolute plasma flow rate,

(4.9) $$\begin{eqnarray}{\mathcal{V}}_{z}=\frac{\int _{P}u_{z}\,\text{d}x\,\text{d}y}{\int _{P}|u_{z}|\text{d}x\,\text{d}y}.\end{eqnarray}$$

5.1 Twisted flux emergence

First, we analyse the helicity input of a twisted field with a non-dimensional twist of $\unicode[STIX]{x1D719}=-0.4$ and a negative (left-handed) chirality (the other parameters are as stated in § 3.2). This value of twist is a common choice for flux emergence simulations (e.g. Hood et al. Reference Hood, Archontis and MacTaggart2012, and references therein) and represents a level of twist slightly below what would be required for the onset of the kink instability.

5.1.1 Physical characteristics of emergence

Figure 2.

Illustrative distributions characterizing the emergence of the twisted flux rope into the Sun’s atmosphere. (a,d,g) are contour plots of the current density, showing the buoyancy instability-triggered rise and expansion into the corona. Also indicated is the plane $z=0$ . (b,e,h) represent subsets of the field lines at these times. (c,f,i) are the corresponding magnetograms with a clear bipole structure.

Figure 3.

Photospheric helicity and winding input for the emergence of the twisted field. (a) The helicity input rate $\text{d}H/\text{d}t$ . It is negative on average and has a consistent oscillation. The vertical lines indicate the times $t=67$ –72 at which the field’s helicity distribution is analysed in what follows. (b) The net helicity input $H(t)$ which is always negative and increases in magnitude over time. (c) The winding input rate $\text{d}L/\text{d}t$ and (d) the total winding input $L(t)$ .

A description of the emergence of twisted flux tubes has been described in detail in many other studies (e.g. MacTaggart & Hood Reference MacTaggart and Hood2009; Hood et al. Reference Hood, Archontis and MacTaggart2012) so we only highlight aspects critical to the subsequent helicity input analysis. Illustrative figures are shown in figure 2 at times $t=35,55$ and $71$ . The magnetic field reaches the photosphere where it remains until the plasma $\unicode[STIX]{x1D6FD}$ drops to order unity and the field becomes subject to the magnetic buoyancy instability, subsequently expanding into the corona (Hood et al. Reference Hood, Archontis and MacTaggart2012). The initial toroidal flux tube leads to a bipole emergence which is clearly visible in the magnetograms.

Figure 4.

Time series of the ratios (a) $H_{t}^{r}$ and (b) $L_{t}^{r}$ for the twisted field emergence.

5.1.2 Helicity and winding input time series

The temporal evolutions of both $\text{d}H/\text{d}t$ and $H$ are shown in figures 3(a) and 3(b) respectively. The tracking of the helicity input begins when the magnetic field reaches the photospheric boundary which, in this simulation, is almost coincident with the onset of the magnetic buoyancy instability ( $t=18$ ). With the initial expansion of the field into the corona there is an input of negative helicity (figure 3 a), as is to be expected for this negative helicity flux rope (MacTaggart & Hood Reference MacTaggart and Hood2009). Later, the helicity input settles into an oscillatory pattern with the input rate, occasionally, becoming net positive. The net helicity input $H$ , shown in figure 3(b), exhibits, on average, an almost linear input increase of negative helicity (in line with the results of Sturrock et al. (Reference Sturrock, Hood, Archontis and McNeill2015) accounting for the opposing sign of twist with a smaller oscillation about this trend). The time series $\text{d}L/\text{d}t$ and $L$ , shown in figure 3(c,d), are qualitatively similar to their helicity counterparts so we do not focus on them in what immediately follows. In contrast to these results, the oscillations in the $\text{d}H/\text{d}t$ time series calculated in Sturrock et al. (Reference Sturrock, Hood, Archontis and McNeill2015) are relatively small (though similarly coherent) and do not cause the change in sign we find here. We will expand more on this comparison later. Finally, the ratios $H_{t}^{r}$ and $L_{t}^{r}$ , whose time series are shown in figure 4, indicate, as expected, a preference towards one sign of helicity input, i.e. the quantity is often close to 1 in magnitude so that most of the winding/helicity density is coherent in sign (the sign, of course, oscillates from positive to negative). An important question to address is what is causing the oscillations and, further, why are they so significant in comparison to previous studies?

Figure 5.

Helicity input rate density distributions $\text{d}{\mathcal{H}}/\text{d}t$ of the emerged region during the period $t\in [67,72]$ over which the sign of $\text{d}H/\text{d}t$ varies form positive to negative (net). The plots shown in (a–f) correspond to times $t=67$ – $72$ indicated by the set of vertical lines on figure 3(a). (a) The density is dominantly positive around the PIL. In (b–f), the helicity input at the PIL changes from dominantly positive to negative. The helicity input in the two flux poles becomes dominantly negative over the cycle.

Figure 6.

Magnetic and velocity field distributions in the photosphere at $t=67$ . (a) The transverse field $\boldsymbol{B}_{\Vert }$ superimposed on a scalar plot of its magnitude (stronger red colouring implies a stronger field). The twist at the positive pole is left-handed and the twist at the negative pole is right-handed. (b) The transverse velocity field $\boldsymbol{u}_{\Vert }$ superimposed on a scalar plot of its magnitude (stronger blue colouring implies a stronger field). Both vortices have right-handed rotation.

Figure 7.

Distributions of the components of the helicity producing field $\boldsymbol{w}$ , at times $t=67$ and $72$ . Panels (a,b) are the distributions of $\sqrt{w_{x}(\boldsymbol{a}_{0})}$ and $\sqrt{w_{y}(\boldsymbol{a}_{0})}$ respectively at $t=67$ . Panels (c,d) are the same distributions but at $t=72$ . For both $w_{x}(\boldsymbol{a}_{0})$ and $w_{y}(\boldsymbol{a}_{0})$ , the sign of the distribution either side of the PIL reverses.

We now focus on the time period $t\in [67,72]$ , indicated in figure 3(a) by a set of vertical lines. There is a variation between positive and negative input rates over this period. There is no particular reason why we choose to present the results of this period over the rest of the input cycle (except the rise stage). Indeed we checked that the following analysis would have led to similar conclusions for the behaviour of the system over the period $t\in [40,76]$ .

Distributions of $\text{d}{\mathcal{H}}/\text{d}t$ , during the considered time interval, are shown in figure 5. At $t=67$ (a) and $t=72$ (f) there are respectively strong positive and then negative field line helicity input densities around the polarity inversion line (PIL). For the intermediate times (b–d), the field line helicity density in this region shows a gradual variation from positive to negative. Patches of strong negative density at the two poles of flux distribution also develop over the period. There are fluid vortices, in the in-plane velocity field, which are centred on these poles. It is shown in figure 6 that at the magnetic field’s positive pole, the vortex opposes the direction of magnetic twist, whilst at the negative pole the signs of the vortical and magnetic twists agree. We might speculate that some kind of relative balance between the twisted field input and fluid rotation might lead to the oscillation in helicity input around the inversion line. However, we now show this oscillation occurs instead due to the cyclic submergence and re-emergence of the flux, that is, the movement of part of the magnetic field below and above the photospheric plane. In what follows, submergence does not necessarily indicate that the field passes deep beneath the photosphere. As will be demonstrated, any part of the field that makes contact with the photospheric boundary, and perhaps passing only slightly beneath it, will register a signal in the magnetograms and in the helicity and winding inputs. Therefore, in this work, submergence is any previously emerged field moving down from the atmosphere or otherwise making contact with the photosphere from above.

5.2 Helicity sign change around the PIL

Figure 8.

Plots of the winding input rate over the period $t=67$ to $71$ for (a) the unrestricted calculation $\text{d}L/\text{d}t$ and (b) the capped version $\text{d}L^{c}/\text{d}t$ for which the threshold is $c=2$ .

The distributions of the $x$ and $y$ -components of $\boldsymbol{w}=\text{d}\boldsymbol{a}/\text{d}t$ at $t=67$ and $t=72$ are shown in figure 7. The most significant helicity producing velocities arise in the neighbourhood of the PIL (the plots are of $\sqrt{w}_{x}$ and $\sqrt{w}_{y}$ are shown for clarity, thus relatively exaggerating the magnitudes in weaker regions). There is a clear switch in sign of both the $w_{x}$ and $w_{y}$ densities on both sides of the PIL over this cycle. To test if it is the temporal change in the sign of the field line velocity field $\boldsymbol{w}$ close to the PIL which determines the change in sign of the helicity input, we consider a modified vector field $\boldsymbol{w}^{c}$ as follows

(5.1) $$\begin{eqnarray}\boldsymbol{w}^{c}=\left\{\begin{array}{@{}cc@{}}\boldsymbol{w} & \text{ if }\left|\boldsymbol{w}\right|\leqslant c,\\ 0 & \text{ if }\left|\boldsymbol{w}\right|>c,\end{array}\right.\end{eqnarray}$$

where $c$ is some chosen threshold. We choose $c=2$ here, which represents a cutoff speed much smaller than typical field line speeds at the PIL ( ${\approx}10$ ). As long as the cutoff is small enough, the following results are robust. The main behaviour of the following results was also obtained for $c=1$ and $c=5$ .

In essence, we cut out the higher helicity producing velocities which reside primarily in the region of the PIL and not at the centres of the magnetic footpoints. We also calculate the modified field line winding input rate $\text{d}L^{c}/\text{d}t$ using $\boldsymbol{w}^{c}$ . We see in figure 8(a) that the full winding input rate $\text{d}L/\text{d}t$ changes sign from positive to negative over the time period $t\in [67,72]$ . By contrast, for the restricted input $\text{d}L^{c}/\text{d}t$ (b), the sign of winding input is always negative over this period and two orders of magnitude smaller. Therefore, ignoring the winding input due to the field line motions $\boldsymbol{w}$ around the PIL leads to a time series which does not have a positive input rate over this period (the story is the same for the helicity input rate). It was confirmed that this is true throughout the simulation (for the time period when the oscillations in $\text{d}H/\text{d}t$ occur).

Figure 9.

Distributions of the vertical velocity $u_{z}(\boldsymbol{a}_{0})$ at times (a) $t=67$ and (b) $72$ . (c) A plot of the net velocity flux ${\mathcal{V}}_{z}$ through $P$ . (d) Plots of $\text{d}H/\text{d}t$ and ${\mathcal{V}}_{z}$ scaled to have values between $0$ and $1$ .

The cause of the helicity sign change is found to be related to a switch in the sign of $u_{z}$ in the region around the PIL. To establish this, we first evaluate the relative magnitudes of the terms $\boldsymbol{u}_{\Vert }$ and $\boldsymbol{B}_{\Vert }u_{z}/B_{z}$ . The submergence/emergence term $\boldsymbol{B}_{\Vert }u_{z}/B_{z}$ is typically found to be an order of magnitude higher in the regions where $\boldsymbol{w}$ is (relatively) large. The $u_{z}$ distributions at the start ( $t=67$ ) and end ( $t=72$ ) of the cycle are shown in figure 9(a,b). Initially, there is a net negative velocity with most of the flow surrounding the PIL. Then at the end of the period the velocity is net positive with most of the flow being at the PIL. A time series plot of the net velocity flux ${\mathcal{V}}_{z}$ is shown in figure 9(c). A scaled comparison of the maxima and minima of $\text{d}H/\text{d}t$ and ${\mathcal{V}}_{z}$ is shown in figure 9(d) emphasizing that it is motion across the photosphere that is dominating the helicity in this phase of emergence.

5.2.1 Flux rope centre at the photosphere

Figure 10.

Arrows indicating the magnetic field in the plane orthogonal to the PIL at $(0,0,0)$ at $t=67$ . This is superimposed on a plot of the current magnitude in the same plane. The core of the flux rope is clearly visible and is centred at the PIL.

The fact that the oscillations in helicity (and winding) are sufficient to lead to a sign of input which opposes the field’s chirality is a result of the fact that the bulk of the field’s initially twisted flux rope remains trapped at the photosphere, see figure 10. The plasma $\unicode[STIX]{x1D6FD}$ $(=2p/|\boldsymbol{B}|^{2})$ has a value of approximately 5 at the photosphere in this simulation. This implies that the magnetic field is not dynamically dominant (as in the low- $\unicode[STIX]{x1D6FD}$ corona) and so can moved by the surrounding plasma. A combination of upward motion, from emergence, and downward motion, from draining plasma (e.g. Hood et al. Reference Hood, Archontis and MacTaggart2012) causes the flux rope centre (axis) to oscillate about the $z=0$ plane. In this topologically simple field (relative to mixed helicity model that we will study shortly) the bulk of the topological information is concentrated at the flux rope centre. Therefore, if the centre crosses the photospheric plane, the response in the helicity and winding rates is large. As mentioned before, it is transport across the photospheric plane and not (un)twisting motions on the plane that causes the largest changes in helicity and winding.

If the plasma $\unicode[STIX]{x1D6FD}$ were smaller, it would be expected that the oscillations would be less pronounced. In Sturrock et al. (Reference Sturrock, Hood, Archontis and McNeill2015), who perform a very similar simulation but with a flux rope of almost double the initial field strength to the one we consider, they still find oscillations in $\text{d}H/\text{d}t$ but much less pronounced than those found here (their oscillations do not change the sign of the helicity input rate). Two factors are important in contributing to this change in behaviour. The first is that the axes of flux tubes with higher field strengths can emerge further in the atmosphere compared to those with weaker field strengths (MacTaggart & Hood Reference MacTaggart and Hood2009). This makes the flux rope centre further away from the $z=0$ plane and so there is less flux crossing the plane in any partial submergence event. Secondly, the plasma $\unicode[STIX]{x1D6FD}$ is smaller and the magnetic field is less susceptible to surrounding plasma motions.

5.3 Tracking the moving photosphere

Figure 11.

Photospheric helicity and winding inputs for the emergence of the twisted field which account for the tracking of the changing photosphere geometry. (a) The helicity input rate $\text{d}H_{v}/\text{d}t$ . (b) The net helicity input $H_{v}(t)$ . (c) The winding input rate $\text{d}L_{v}/\text{d}t$ . The vertical lines are at times $t=30,31$ when the field structure is analysed further in what follows. (d) The total winding input $L_{v}(t)$ .

Figure 12.

Vertical velocity distributions at $t=28$ . (a) The velocity density $u_{z}(P_{v}(x,y))J$ at the moving photosphere line $\unicode[STIX]{x1D70C}=1$ . (b) The difference between the varying photosphere velocity density $u_{z}(P_{v}(x,y))J$ and the $z=0$ velocity density $u_{z}(x,y)$ .

The velocity flux oscillations indicated in figure 9 imply that the surface $\unicode[STIX]{x1D70C}=1$ will be varying in space and time. As discussed in the introduction, we can track this variation and calculate modified quantities which account for this motion. The time series of the adjusted quantities $\text{d}H_{v}/\text{d}t$ , $H_{v}$ , $\text{d}L_{v}/\text{d}t$ and $L_{v}$ are shown in figure 11. On a qualitative level, the helicity time series (panels (a,b) respectively) are effectively the same as the $z=0$ calculations shown in figure 3. The magnitudes, however, of the various peaks of the input rate $\text{d}H_{v}/\text{d}t$ are roughly four times larger than those of $\text{d}H/\text{d}t$ . This magnification is due to increased variations in velocities at the $\unicode[STIX]{x1D70C}=1$ surface. As an example, figure 12 displays the varying photosphere velocity distribution $Ju_{z}(P_{v}(x,y))$ in (a) and the difference $Ju_{z}(P_{v}(x,y))-u_{z}(x,y)$ between the moving surface and static surface velocity distributions in (b). In (a), the strongest velocities are, as before, at the PIL. In (b), major differences are at the main footpoints. We note that the differences can be greater in magnitude than a typical velocity $Ju_{z}(P_{v}(x,y))$ . Therefore, it is not simply a case of larger velocity magnitudes at the $\unicode[STIX]{x1D70C}=1$ surface compared to the $z=0$ plane, rather a difference in the distribution of flows on these surfaces.

Figure 13.

A slice of the vector field in the plane orthogonal to the PIL at $t=30$ , just prior to the spike in the time series $\text{d}L_{v}/\text{d}t$ shown in figure 11(c). Also shown as a green curve is the intersection of the surface $P_{v}$ and this plane. The background distribution is the out of plane component of the current density.

Figure 14.

Magnified representations of a slice of the vector field in the plane orthogonal to the PIL at $t=30$ and $t=31$ , prior to and at the spike in the time series $\text{d}L_{v}/\text{d}t$ shown in figure 11(d). Shown as a green curve is the intersection of the surface $P_{v}$ and this plane. The background distribution is the out of plane component of the current density. (a) shows ( $t=30$ ) a magnification of the region of figure 13 which contains the curve $\unicode[STIX]{x1D70C}=1$ . (b) shows ( $t=31$ ) the same distribution as in (a) but at the time of the spike. The $\unicode[STIX]{x1D70C}=1$ curve has dropped vertically by a value of approximately $0.5$ .

The winding time series, panels (c,d) of figure 9, show significant qualitative differences compared to the $z=0$ input calculations shown in panels (c,d) of figure 3. There is one significant period of negative input rate $\text{d}L_{v}/\text{d}t$ at $t=31$ . This is near the end of the period when the central part of the field (containing the original tube axis) emerges at the photosphere (see figure 13). The spike coincides with a relatively sharp (negative) change in the height of the $\unicode[STIX]{x1D70C}=1$ surface from $t=30$ (figure 14 a) to $t=31$ (figure 14 b). This change results in a significant change in the negative helicity input as the flux rope core moves further beyond the photospheric boundary. It was confirmed that this jump in the moving photospheric surface was atypically large from $t=30$ to $t=31$ by comparison to typical changes in its morphology over single time step of size $1$ . After this, the input rate $\text{d}L_{v}/\text{d}t$ falls to a relatively small rate. It is interesting to note that this occurs due to a relatively fast change in the geometry of the $\unicode[STIX]{x1D70C}=1$ surface, rather than a sudden change in field topology, indicating that this may be an important factor in interpreting observational inputs of field topology. The oscillations shown in the helicity input rate $\text{d}H_{v}/\text{d}t$ are still present but, as shown in figure 11(d), these have a small effect on the net helicity injected into the photosphere, by comparison to the sharp negative input around $t=30$ .

It is intriguing that the (moving photosphere) winding input more clearly represents the transition between the emergence of the field’s twisted core to the photosphere (the $\unicode[STIX]{x1D70C}=1$ surface) followed by the lack of changing topological input due to the core getting stuck at this surface. The significantly reduced (at least on a relative scale) size of the oscillations in comparison to the helicity input results from the lack of field strength weighting in the winding input. The inclusion of field strength in the helicity input rate magnifies the oscillatory signal due to the central core of the field being immediately surrounded by strong field – slight dips under the photospheric boundary produce high signals because of the strong field strengths. Another way to view this result is that the winding input treats the oscillation at the photosphere as not especially interesting at later times as it is not really indicating the input of new topological information into the atmosphere.

5.4 Mixed helicity emergence

We now consider a mixed helicity field $\boldsymbol{B}_{b}$ , specified by (3.23) with $n=2$ . The emergence of mixed helicity (braided) magnetic fields and their associated dynamics was first described in Prior & MacTaggart (Reference Prior and MacTaggart2016).

5.4.1 Physical characteristics of the emergence

Figure 15.

Illustrative distributions characterizing the emergence of a mixed helicity flux rope with $n=2$ twist pairs. (a,d,g) are magnetograms. (a) depicts ( $t=27$ ) a bipole structure associated with the initial emergence phase. In (d) ( $t=38$ ), this distribution has separated slightly and there also appear to be some weak thin horizontal structures appearing at the centre of the domain. In (g) ( $t=62$ ), these weak additional structures have developed into pairs of bipoles whose polarity oppose that of the larger initial bipole. (b,e,h) are current contours ( $|\boldsymbol{j}|=0.01$ ) which indicate the field’s expansion. Also indicated is the plane $z=0$ . (c,f,i) are representative field lines.

Various illustrative visualizations of the emerging field’s evolution are shown in figure 15. As for the twisted tube, the emerging field gets trapped below the photosphere before becoming subject to the magnetic buoyancy instability and rising into the coronal region. The magnetograms (a,d,g) show that, in addition to a larger bipole structure developing, there is also an additional formation of smaller bipole structures in between these larger poles with orientations in opposition to that of the large bipole. In (b,e,h), contours of the current density structure show the field’s expansion, which is restricted along the PIL of the main bipole. This behaviour is in contrast to the single structure found in the twisted case (figure 2) which shows a more uniform expansion. The field lines, shown in  (c,f,i) indicate a pair of intertwined field line arcades linked by a centrally dipped section (which partially submerges, as we will see later). There is no significant twisting in the field lines. It was established in Prior & MacTaggart (Reference Prior and MacTaggart2016) that the dips in the current contours coincide with plasma draining and are the nonlinear manifestations of a Rayleigh–Taylor instability.

Figure 16.

Photospheric helicity and winding inputs for the emergence of the mixed helicity field. (a) The helicity input rate $\text{d}H/\text{d}t$ . The input is both positive and negative. Also shown are lines at $t=25,42$ and $48$ . (b) The net helicity input $H(t)$ . (c) The winding rate $\text{d}L/\text{d}t$ . (d) The total winding input $L(t)$ , the sign of this input is generally opposite to that of the helicity.

5.4.2 Helicity and winding input: static photosphere

Plots of the time series of the quantities $\text{d}H/\text{d}t,\,H,\,\text{d}L/\text{d}t$ and $L$ are shown in figure 16. The helicity rate $\text{d}H/\text{d}t$ (a) is both positive and negative and shows oscillatory behaviour (less temporally regular than for the twisted field). The net helicity input $H(t)$ (b) switches from negative to net positive. The winding input rate $\text{d}L/\text{d}t$ (c) shows much more rapid variations. The net input $L$ (d) has a pattern qualitatively similar to the net helicity but with opposing sign. The ratios $H_{t}^{r}$ and $L_{t}^{r}$ are found to be typically 1–2 % (figure 17), indicating the topological input is not far off neutral, as to be expected.

The distributions $\text{d}{\mathcal{H}}/\text{d}t$ and $\text{d}{\mathcal{L}}/\text{d}t$ are shown in figure 18. The snapshots are chosen to display critical characteristics of these distributions which are present at various times of the emergence process.

There are always four distinct regions of substantial positive and negative helicity (and winding) and these occur in pairs. These regions are generally of similar size and magnitude, which explains why the net helicity (and winding) input is much smaller than its absolute total. In practice, it is difficult to detect the imbalances in the maps of $\text{d}{\mathcal{H}}/\text{d}t$ that lead to the time series variations shown in figure 16(a). By contrast, the winding distributions $\text{d}{\mathcal{L}}/\text{d}t$ capture more features related to emergence and submergence (we will return to this point later). We ignore the distribution values around the edge of the magnetic domain as the field is very weak there. The field line structure in the boundary current sheet is incoherent and, thus, any apparent winding input is not meaningful.

Figure 17.

Time series of the ratios (a) $H_{t}^{r}$ and (b) $L_{t}^{r}$ for the $n=2$ mixed helicity field emergence.

Figure 18.

Distributions of $\text{d}{\mathcal{H}}/\text{d}t$ and $\text{d}{\mathcal{L}}/\text{d}t$ which indicate the various characteristic stages of the distribution evolutions (indicated with vertical lines on figure 16). (a) $\text{d}{\mathcal{H}}/\text{d}t$ and (b) $\text{d}{\mathcal{L}}/\text{d}t$ at $t=25$ . In both cases there are two significant regions of positive input and two with negative input, a quadrupolar distribution. In (b) the straight PIL is visible along the $y$ axis. (c) $\text{d}{\mathcal{H}}/\text{d}t$ and (d) $\text{d}{\mathcal{L}}/\text{d}t$ at $t=42$ . In (c) the four regions from (a) remain but are surrounded by thin strips of helicity input of opposing sign. In (d), by contrast, the sign of the four regions have swapped compared to (b) and there are additional sub regions of positive and negative input centred on the PIL. (e) $\text{d}{\mathcal{H}}/\text{d}t$ and (f) $\text{d}{\mathcal{L}}/\text{d}t$ at $t=48$ . The helicity input (e) still has four dominant domains but they have swapped sign from the distributions in (a,c). In (f) the flux region seen in the latter magnetograms (figure 15) appears as a distorted PIL and there are strong concentrations of helicity either side of it.

The PIL is clear in all $\text{d}{\mathcal{L}}/\text{d}t$ distributions and its shape varies significantly as various new helicity structures appear (note that all additional structures appear at the PIL).

5.5 Helicity and winding input: moving photosphere

Figure 19.

Helicity and winding inputs for the emergence of the mixed helicity field, allowing for the varying photospheric geometry. (a) The helicity input rate $\text{d}H_{v}/\text{d}t$ . (b) The net helicity input $H_{v}(t)$ . (c) The winding rate $\text{d}L_{v}/\text{d}t$ . The vertical lines mark times $t=45,46,47$ at which the field is analysed in detail in the text. (d) The total winding input $L_{v}(t)$ .

Time series of $\text{d}H_{v}/\text{d}t$ , $H_{v}$ , $\text{d}L_{v}/\text{d}t$ and $L_{v}$ , which account for the varying photospheric geometry, are shown in figure 19. In order to prevent the winding measure being affected by any weak (unresolved) field, we utilize the modified definition of the function $\unicode[STIX]{x1D70E}_{z}$ (4.6) with $\unicode[STIX]{x1D716}=0.0001$ for the $\text{d}L_{v}/\text{d}t$ and $L_{v}$ calculations. As was the case for the twisted field, the only significant change in the helicity time series compared to those shown in figure 16(a,b) is in regard to the magnitudes. This difference could, again, be traced to differences in the vertical velocity distribution. The time series of the winding quantities $\text{d}L_{v}/\text{d}t$ and $L_{v}$ , shown in figure 19(c,d), differ significantly from those shown in figure 16(c,d). We find that this is, in part, due to the varying photosphere $P_{v}$ and partly due to ignoring the contributions from significantly weak regions of the field. As discussed previously, the winding is more sensitive to large input from topologically complex field lines of very weak field strength. Hence, the calculation of the winding requires some care.

The sign of the net input $L_{v}$ is largely dictated by a significant spike in the input $\text{d}L_{v}/\text{d}t$ at $t=47$ . After this, there are two large oscillations which are more balanced. The net inputs $H_{v}$ and $L_{v}$ now agree in terms of the net sign of their input over the simulation (as opposed to the quantities $H$ and $L$ , emphasizing the sensitivity of $L$ or $L_{v}$ ).

5.6 Field submergence and deformation

As mentioned above, a clearer transition is revealed by the winding compared to the helicity. The key question to answer here is, what is causing this transition? Some indications can be found in the distributions of $\text{d}{\mathcal{L}}_{v}/\text{d}t$ shown in figure 20. These distributions are at the times indicated by the parallel lines in figure 19(c). Figure 20(a) displays $\text{d}{\mathcal{L}}_{v}/\text{d}t$ when $\text{d}L_{v}/\text{d}t$ is changing from positive to negative. A small region of negative $\text{d}{\mathcal{L}}_{v}/\text{d}t$ is visible at the centre (0,0). The region grows in (b), which corresponds to when $\text{d}L_{v}/\text{d}t$ takes its largest negative value. In (c), which corresponds to the large positive spike in $\text{d}L_{v}/\text{d}t$ and positive jump in $L_{v}$ , the distribution changes to a clear antisymmetrical pattern. There is no longer the isolated patch of negative $\text{d}{\mathcal{L}}_{v}/\text{d}t$ at the centre and the PIL is highly deformed.

Figure 20.

Distributions of the winding input density $\text{d}{\mathcal{L}}_{v}/\text{d}t$ covering the period of the time series shown in figure 19(d) at which there is a strong spike in input. (a) At $t=45$ , before the spike. (b) At $t=46$ . (c) At $t=47$ at the spike’s peak.

Figure 21.

A contour plot of density $\unicode[STIX]{x1D70C}=0.2$ , slightly above the $\unicode[STIX]{x1D70C}=1$ photospheric level at $t=62$ . There is a an $s$ -shaped peak in the density profile at the centre of the domain which is parallel to the $y$ -direction and has been established to correspond to a pooling of dense plasma in this location. This shape matches the morphology of line observed in the winding input densities $\text{d}L/\text{d}t$ , shown in figure 18(f), and $\text{d}L_{v}/\text{d}t$ , shown in figure 20(c).

Figure 22.

Slices in the $y$ – $z$ plane, at the PIL at (a) $t=27$ and (b) $t=42$ , revealing different phases of emergence. The slices display distributions of the out of plane component of the current density. Also shown, as a green line, the $\unicode[STIX]{x1D70C}=1$ surface of the moving photosphere. (a) ( $t=27$ ) There is a spiral structure, indicative of a locally twisted field just off the centre of the photospheric domain. It is part way through emerging through the $\unicode[STIX]{x1D70C}=1$ photosphere line. (b) ( $t=42$ ) The previous twist has split in two due to the deformation of the current structure in the atmosphere.

Figure 23.

Slices in the $y$ – $z$ plane at the PIL at (a) $t=46$ and (b) $t=47$ . The slices display distributions of the out of plane component of the current density. Also shown, as a green line, the $\unicode[STIX]{x1D70C}=1$ surface of the moving photosphere.

Since the emerging field has very little twist, it cannot easily support dense plasma in the atmosphere. This manifests itself in the atmosphere as the buckling of the central part of the emerging field, where dense plasma drains and restricts the magnetic field in the lower atmosphere. This process is visualized in figure 15 and the pooling of dense plasma which forms can be seen in figure 21. The patch of negative $\text{d}{\mathcal{L}}_{v}/\text{d}t$ , described above, indicates the beginning of motion leading to submergence, that is, the movement of helicity carrying field from the atmosphere down to the photospheric boundary. We now characterize various stages of the emergence up to and including this submergence event.

Figure 22(a) displays the emergence of the mixed helicity field at the (relatively) early time of $t=27$ . The slice at $x=0$ displays the $x$ -component of current density, the projection of magnetic field arrows on the plane and the position of the moving photosphere. What is shown is a twisted structure beginning to emerge into the atmosphere. Comparison with figure 16 shows that the helicity and winding inputs are still relatively weak. Figure 22(b) displays a similar slice but for the much later time of $t=42$ . Comparison with figure 16 shows that this corresponds to the helicity approaching a local (positive) maximum and the winding near its most negative value. A current structure exits in the atmosphere in figure 22(b) that was not present in (a). This current developed due to the buckling of the field due to dense plasma drainage, as described previously. This nonlinear deformation of the field has resulted in two ‘twist units’, as opposed to the single unit displayed in (a). Where the two twist units meet, at $y=0$ and just above the photosphere, there is a small patch of negative current. This patch corresponds to the patch of negative winding displayed in figure 20(a,b), which reveals this patch growing at the later times of $t=45$ and $t=46$ .

As mentioned above, figure 20(c) reveals that the patch of negative winding found in figure 20(a,b) disappears. Instead, the winding distribution reveals a highly deformed PIL and winding magnitudes double that of the previous time step. This behaviour is caused by submergence. Consider figure 23. This figure shows $y$ – $z$ slices (at $x=0$ ) of the $x$ -component of the current density and the photospheric boundary at times (a) $t=46$ and (b) $t=47$ , corresponding to the times of the winding distributions in figure 20(b,c). In figure 23(a) the structure of positive current, created by dipped field in the atmosphere, is just glancing the photospheric boundary. In figure 23(b), this structure has passed slightly beneath the photospheric boundary. Despite the appearance of figure 23(a,b) being only marginally different, the submergence of a very small part of the field can have a large impact on the winding distributions (as well as the magnetograms). As submergence continues, the changing current structure leads to a large spike in the winding input (figure 19 c,d).

It is interesting to note that the original emerging field (such as the twisted unit in figure 22 a) did not lead to a spike in the winding input. It was only after the emerged field above the photosphere changed and this new structure was submerged that an event was detected in the time series.

Further confirmation that submergence dominates these helicity and winding time series is found, as we did for the twisted case, by examining the velocity flux. Time series of the velocity flux and the $\text{d}H/\text{d}t$ are compared in figure 24(a). After $t=40$ , ${\mathcal{V}}_{z}<0$ for the period corresponding to the draining and submergence described above. We also note that there is a correlation between oscillations in the helicity input time series and the velocity flux (figure 24 b).

Figure 24.

Plots of the velocity flux ${\mathcal{V}}_{z}$ and its correlation to the helicity input rate. (a) The temporal variation of ${\mathcal{V}}_{z}$ . (b) A comparison of the temporal variation of the quantities ${\mathcal{V}}_{z}$ and the helicity input rate $\text{d}H/\text{d}t$ (both scaled to lie in the range $[0,1]$ ).