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\title{Magnetic helicity proxy from linear polarization in anisotropic turbulence}
\author{}
\date{\today,~ $ $Revision: 1.22 $ $}
\begin{document}
\maketitle

\section{Introduction}

Helicity characterizes the swirl of a flow or a magnetic field.
Examples include the cyclonic and anticyclonic events on weathermaps,
which are systematically different in the northern and southern hemispheres.
Similar differences are also seen on the solar surface, where not only
the flow vectors show swirl, but also the magnetic field vectors.
Both play an important role in the solar dynamo and are believed to be
responsible for the generation of the solar large-scale magnetic field
\citep{Mof78,KR80,BS05}.

To determine the solar magnetic helicity, one first needs to determine
the magnetic field $\BB$.
This is done by measuring all four Stokes parameters to compute $\BB$
at the visible surface.
Historically, the first evidence for a helical magnetic field came from
estimates of the mean current helicity density, $\bra{\JJ\cdot\BB}$,
where $\JJ$ is the current density and $\mu_0$ is the vacuum permeability.
Under isotropic conditions, using a local Cartesian coordinate system,
one can assume $\bra{\JJ\cdot\BB}\approx\bra{J_z B_z}/3$, where
$J_z=\partial_x B_y-\partial_y B_x$ is the vertical component of
the current density, which involves only horizontal derivatives.
\cite{See90} and \cite{PCM95} computed instead the ratio $\alpha=J_z/B_z$,
which they found to be negative in the northern hemisphere and positive
in the southern.
However, that was also large scatter, so the hemispheric sign rule is
only obeyed on average.
Another possibility is to estimate the magnetic helicity
$\int\AAA\cdot\BB\,\dd V$ within a semi-open volume $V$ above the surface.
Here, $\AAA$ is the magnetic vector potential with $\BB=\nab\times\AAA$.
The magnetic helicity changes in time, depending on the amount of magnetic
helicity flux that enters the volume from below.
This is usually done by performing a force-free field extrapolation
to obtain the magnetic field in the volume above the surface
\citep[e.g.,][]{VKK05,PNDB06,PVDD15}.
Alternatively, one computes the magnetic helicity density at the solar
surface subject to certain assumptions \citep{ZBS14,ZBS16,BPS17}.

A general difficulty with all these approaches lies in the fact that
the horizontal magnetic field components are only determined up to the
$\pi$ ambiguity.
In other words, one only measures horizontal magnetic field vectors
without arrow heads.
The full horizontal field direction can be reconstructed by comparing
with that expected from a potential magnetic field extrapolation, but it
is unclear to what extent this introduces errors and how those affect, for
example, the scale dependence of the magnetic helicity that was determined
in some of the aforementioned approaches \citep{ZBS14,ZBS16,BPS17}.

To address the question about the limitations resulting from the $\pi$
ambiguity, we compare with another potential proxy of magnetic helicity
that is independent of the $\pi$ ambiguity.
To this end, we decompose the linear polarization into the E and B
polarizations that are routinely used in the cosmological context
\citep{SZ97,Kamion97}.
These two parts of the polarization are derived from those two components
($Q$ and $U$) of the Stokes vector that characterize linear polarization.
The E and B polarizations characterize parity-even and parity-odd
contributions.
The correlation between E and B polarizations is believed to be a proxy
of the helicity of the underlying magnetic field \cite{KMLK14}.
However, as we will show in this paper, this seems to be only true
under certain anisotropic conditions, for example in the proximity of
the surface and under certain additional assumptions.
%Nevertheless, in spite of these limitations, this technique may prove
%useful for obtaining information about the solar magnetic helicity that
%is directly based on the $Q$ and $U$ polarizations.

Observations of E and B polarizations have been obtained at various
frequencies using the {\em Planck} satellite \citep{Adam16}.
Much of this emission is now believed to come from the galactic foreground.
Although the EB cross-correlation has not yet been determined, we do know
that the EE correlation is about twice as large as the BB correlation.
This is unexpected.

%A relatively simple proxy of magnetic helicity that has been used for
%many years \citep{See90,PCM95} is the normal component
%of the current helicity, $J_\| B_\|$, where
%$J_\|=(\nab_\perp\times\BB_\perp)_\|$ is the horizontal curl.
%Again, its calculation involves the horizontal magnetic field
%components, which are subject to the $\pi$ ambiguity.
The purpose of this work is to illustrate the relation between these
proxies and the E and B correlations that are derived from just the
horizontal field vectors without knowledge of which of the two horizontal
directions the vector points into (i.e., the $\pi$ ambiguity).
We begin with a simple example of a vector field that is helical and
anisotropic.

\section{E and B polarization}

\subsection{Formalism}

We consider magnetic field vectors, $\BB=(B_x, B_y, B_z)$, in
two-dimensional $xy$ cross-sections.
We compute the observable complex polarization (neglecting a $\pi/2$
rotation between electric and magnetic fields),
\EQ
P(x,y)=(B_x+\ii B_y)^2
\EN
in the $xy$ plane, ignoring for now the fact that electric and magnetic
fields are at right angles to each other.
Next, we compute the complex EB-polarization in Fourier space,
\begin{equation}
\tilde{Q}(k_x,k_y)=(\hat{k}_x-\ii \hat{k}_y)^2\tilde{P}(k_x,k_y),
\end{equation}
where $\hat{k}_x$ and $\hat{k}_y$ are the $x$ and $y$ components of the
planar unit vector $\hatkk=\kk/k$, $k=(k_x^2+k_y^2)^{1/2}$ is the
length of $\kk$, and tildes indicate Fourier transformation.
We transform $\tilde{Q}$ back into real space,
\EQ
Q(x,y)=E(x,y)+\ii B(x,y),
\EN
where $E(x,y)$ and $B(x,y)$ are the E and B polarizations.
%Those are then transformed separately again into Fourier space
%and correlation spectra are determined as
This $B(x,y)$ is not to be confused with the modulus of $\BB$,
which will not be used in this paper.

We consider both one-point correlations of the form 
$\bra{XY}_{xy}$, where $X$ and $Y$ stand for $E$, $B$, or $B_z$,
as well as spectra
\begin{equation}
C_{XY}(k)=\int\tilde{X}\tilde{Y}^*\,k\,\dd\phi_k,
\end{equation}
where $\phi_k$ is the azimuthal angle in Fourier space, and
$\tilde{X}$ and $\tilde{Y}$ are Fourier transformed fields.
Note that $\bra{XY}_{xy}=\int C_{XY}(k)\,\dd k$.

\subsection{Simple test cases}

We consider gradient- and curl-type vector fields,
\begin{equation}
F_i(x,y)=\partial_i f(x,y),\quad
G_i=\epsilon_{ij}\partial_j g,
\end{equation}
using
\begin{equation}
f=f_0\cos kx\cos ky,\quad
g=g_0\cos kx\cos ky,
\end{equation}
Assuming $k=1$ in a domain $-\pi<(x,y)<\pi$, we have
\begin{equation}
\FF=-f_0\pmatrix{ \sin x\cos y\cr \cos x\sin y},\quad
\GG= g_0\pmatrix{-\cos x\sin y\cr+\sin x\cos y}.
\end{equation}
In \Fig{p} we show examples for {\em different combinations}
of the coefficients $(f_0,g_0)$.
The polarization vectors correspond to $\BB$ vectors without an arrow.

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{p}
\end{center}\caption[]{
(a) and (b): pure E polarization for cases
$(f_0,g_0)=(1,0)$ and $(0,1)$, respectively.
(c) and (d): pure B polarization for cases
$(f_0,g_0)=(1,\pm1)$, for upper and lower
signs, respectively.
}\label{p}\end{figure}

Pure E polarization occurs whenever either $f_0$ or $g_0$ vanish, whereas
pure B polarization occurs whenever $|f_0|=|g_0|$.
%Thus, it is not true that, as one may understand from \cite{Dur08},
Thus, it is not true that, as one may have thought,
that $f_0\neq0$ implies pure E polarization and
that $g_0\neq0$ implies pure B polarization.

\subsection{Convection}

Unlike isotropic turbulence, convection has boundaries
and, in the presence of rotation, this leads to swirl in the
spreading outflows and the converging inflows.
As a simple model, the following vector field would have
this property:
\begin{equation}
\BB=\nab\times\phi\zzz+s\nab\times\nab\times\phi\zzz
\label{RobFlow1}
\end{equation}
with $\phi(x,y,z)=z\cos x\cos y$, so
\begin{equation}
\BB=\pmatrix{
-z\cos x\sin y-\sin x\cos y\cr
+z\sin x\cos y-\cos x\sin y\cr
2zs\cos x\cos y}.
\label{RobFlow2}
\end{equation}
Here, $s\neq0$ determines the sign of magnetic helicity.
We compute E and B polarizations, $E(x,y,z)$ and $B(x,y,z)$
and compare also with $B_z(x,y,z)$.
It turns out that the parameter $z$ determines the sign of the EB
polarization.
In particular, the sign of EB is independent of the sign of $s$,
{\em so there is no correspondence between magnetic helicity and
the sign of $s$}.

Note that $\nab_\perp\cdot\BB_\perp=-2\cos x\cos y$ and
$J_\|=2z\cos x\cos y$, so $J_\|\nab_\perp\cdot\BB_\perp=-z(2\cos x\cos y)^2$,
but $J_\|B_\|=s(2z\cos x\cos y)^2$, so the sign of helicity is determined
by the sign of $s$, but not by $z$.
This is shown in \Fig{prob}, where we see that B (second column),
EB (third column), B, and E$B_z$ change sign.
The planar average of EB is negative (positive) for $z>0$ ($z<0$),
while the planar averages of B and E$B_z$ vanish.

\section{Results}

\subsection{Isotropic turbulence}

We use 3-D simulation data of isotropic MHD turbulence
and consider separately all $xy$ planes.
We consider decaying MHD turbulence with and without
magnetic helicity in cases where the magnetic field is either
stronger or weaker than the kinetic energy.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\columnwidth]{ppEBmodes_p144a}
\includegraphics[width=\columnwidth]{ppkEBmodes_p144a}
\end{center}\caption[]{
Isotropic turbulence with {\em positive} kinetic helicity.
Top left: $\bra{EB}_{xy}$ (black line) with positive (negative)
values indicated by red (blue) symbols, $\bra{EE}_{xy}$ (blue line),
and $\bra{BB}_{xy}$ (red line), versus $z$ for Run~\url{p144a}.
Bottom left: $\bra{EB_z}_{xy}$ (blue) and $\bra{BB_z}_{xy}$ (red)
The dotted lines denote the respective mean values.
\blue{Note that $\bra{EE}_{xy}$ is slightly larger than $\bra{BB}_{xy}$.}
Top right: $C_{EB}(k)$ (black), $C_{EE}(k)$ (blue),
and $C_{BB}(k)$ (red).
The short straight line is proportional to $k^{-5/3}$.
Bottom right: $C_{EB_z}(k)$ (blue), $C_{EE_z}(k)$ (red).
}\label{ppEBmodes_p144a}\end{figure*}

Our simulations of decaying hydromagnetic turbulence are
magnetically dominated, i.e., the magnetic energy exceeds
the kinetic energy by typically a factor of ten.
In those cases, we find that the EE correlation exceeds the BB correlation
by a factor of about 1.6, which is less than what is observed in the
foreground polarization detected with {\em Planck} \citep{Adam16}.
but more than what is theoretically expect \citep{CHK17}, and may
be explicable with Alfv\'en waves \citep{KLP17}.

Correlations between $B$ and either $E$ or $B_z$
are found to be compatible with zero in all cases; see
\Figs{ppEBmodes_p144a}{ppEBmodes_m144a}, i.e., there is {\em no
observational signature of magnetic helicity from B polarization}
or EB correlation.
The detectability of magnetic helicity from the EB correlations
is theoretically expected from the CMB of sufficient quality
\citep{KMLK14}.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\columnwidth]{ppEBmodes_m144a}
\includegraphics[width=\columnwidth]{ppkEBmodes_m144a}
\end{center}\caption[]{
Like \Fig{ppEBmodes_p144a}, but for isotropic turbulence with
{\em negative} kinetic helicity.
\blue{Again, $\bra{EE}_{xy}$ is slightly larger than $\bra{BB}_{xy}$.}
}\label{ppEBmodes_m144a}\end{figure*}

Images of E and B, as computed from helical turbulence, are shown
in \Figs{ppebpol}{ppebpol_576b} appear to be different.
However, this distinction is subjective.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.8\textwidth]{ppebpol}
\end{center}\caption[]{
E and B polarizations from $\Pm=100$ run of BK17
in an arbitrarily chosen plane.
The two images look different, but the difference is hard to quantify.
}\label{ppebpol}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.8\textwidth]{ppebpol_576b}
\end{center}\caption[]{
E and B polarizations from $\Pm=1$ run 576b of \cite{BPS17}
in an arbitrarily chosen plane.
Again, the two images look different, but it is unclear whether
this difference is significant.
}\label{ppebpol_576b}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{prob}
\end{center}\caption[]{
Roberts field with $z=\pm1$ (upper/lower row).
From left to right: $E$, $B$, $EB$, $EB_z$, $BB_z$,
for $z=1$ (upper row) and $z=-1$ (lower row).
\blue{Only $EB$ has a definite sign (negative in the upper
row and positive in the lower row).}
}\label{prob}\end{figure*}

\subsection{Convection}

We perform simulations with gravity $\grav$ and angular velocity $\OO$
in a layer $z_{\rm bot}\leq z\leq z_{\rm top}$, where
$z_{\rm top}-z_{\rm bot}\equiv d$ is the thickness of the layer.

We adopt a polytropic stratification with background temperature
$T=-gz/\cp$, so $T=0$ at $z=0$.
We fix $d$ and choose $|z_{\rm top}|/d$ to set the degree of stratification.
The smaller $|z_{\rm top}|/d$, the stronger is the stratification, i.e.,
the stronger is the temperature contrast.
In the following we choose $|z_{\rm top}|/d=0.1$, so the temperature
changes by a factor of 10; see \cite{HTM84} for a similar setup.
We choose $\grav\cdot\OO$ to be either positive or negative.
A vertical magnetic field is imposed and tangled by this velocity field.
The results are shown in
\Figs{ppEBmodes_Bm144x144x48a}{ppEBmodes_B144x144x48a}
and summarized in \Tab{Tconv}.

\begin{table}[htb]\caption{
Result for convection, as shown in
\Figs{ppEBmodes_Bm144x144x48a}{ppEBmodes_B144x144x48a}.
}\vspace{12pt}\centerline{\begin{tabular}{c|cc}
$\grav\cdot\OO$ & $\bra{EB}$ & $\bra{BB_z}$ \\ 
\hline
$+$ & $-$ & $+$ \\
$-$ & $+$ & $-$ \\
\label{Tconv}\end{tabular}}\end{table}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.8\columnwidth]{ppEBmodes_Bm144x144x48a}
\includegraphics[width=.8\columnwidth]{ppkEBmodes_Bm144x144x48a}
\end{center}\caption[]{
Convection with $\grav\cdot\OO>0$.
Top left: $\bra{EB}_{xy}$ (black), $\bra{EE}_{xy}$ (blue),
and $\bra{BB}_{xy}$ (red), versus $z$ for Run~\url{p144a}
with {\em positive} magnetic helicity.
Bottom left: $\bra{EB_z}_{xy}$ (blue) and $\bra{BB_z}_{xy}$ (red)
for the same run.
The dotted lines denote the respective mean values.
\blue{Note that $\bra{EE}_{xy}$ and $\bra{BB}_{xy}$ have the same amplitudes
and now $\bra{EB}_{xy}<0$.}
\url{ppEBmodes_Bm144x144x48a}
Top right: $C_{EB}(k)$ (black), $C_{EE}(k)$ (blue),
and $C_{BB}(k)$ (red) for Run~\url{p144a}
with {\em positive} magnetic helicity.
Bottom right: $C_{EB_z}(k)$ (blue), $C_{EE_z}(k)$ (red),
for the same run.
\url{ppkEBmodes_Bm144x144x48a}
\blue{Note that $\bra{EE}_{xy}$ and $\bra{BB}_{xy}$ have the same amplitudes
and now $\bra{EB}_{xy}>0$.}
}\label{ppEBmodes_Bm144x144x48a}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.8\columnwidth]{ppEBmodes_B144x144x48a}
\includegraphics[width=.8\columnwidth]{ppkEBmodes_B144x144x48a}
\end{center}\caption[]{
Convection with $\grav\cdot\OO<0$.
Top left: $\bra{EB}_{xy}$ (black), $\bra{EE}_{xy}$ (blue),
and $\bra{BB}_{xy}$ (red), versus $z$ for Run~\url{p144a}
with {\em positive} magnetic helicity.
Bottom left: $\bra{EB_z}_{xy}$ (blue) and $\bra{BB_z}_{xy}$ (red)
for the same run.
The dotted lines denote the respective mean values.
\url{ppEBmodes_B144x144x48a}
Top right: $C_{EB}(k)$ (black), $C_{EE}(k)$ (blue),
and $C_{BB}(k)$ (red) for Run~\url{p144a}
with {\em positive} magnetic helicity.
Bottom right: $C_{EB_z}(k)$ (blue), $C_{EE_z}(k)$ (red),
for the same run.
\url{ppkEBmodes_B144x144x48a}
}\label{ppEBmodes_B144x144x48a}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pppEBmodes_Bm144x144x48a}
\end{center}\caption[]{
\url{pppEBmodes_Bm144x144x48a}
}\label{pppEBmodes_Bm144x144x48a}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pppEBmodes_B144x144x48a}
\end{center}\caption[]{
\url{pppEBmodes_B144x144x48a}
}\label{pppEBmodes_B144x144x48a}\end{figure*}

Again, EB has a definite sign and is negative (positive) for
$\grav\cdot\OO>0$ ($\grav\cdot\OO<0$).
Here, unlike ur simple model given by \Eqs{RobFlow1}{RobFlow2},
there is a net sign in B$B_z$ resulting from a dominance of the deeper
layers relative to the midlayer.

In rotating convection on the northern hemisphere, we have $\grav\cdot\OO<0$.
Near the upper surface, a downdraft ($u_z<0$) will suffer a counter-clockwise
spin ($\omega_z>0$), so $\omega_z u_z<0$, corresponding to negative
kinetic helicity.
This applies to the center point of the top row of \Fig{prob}.
Likewise, an updraft ($u_z<0$) will suffer a clockwise
spin ($\omega_z<0$), so again $\omega_z u_z<0$, i.e., the
kinetic helicity unchanged and its sign equal to that of $\grav\cdot\OO$.
Since the polarization vectors have no vector tip, this will therefore
applies to the {\em same} center point of the top row of \Fig{prob}.
Therefore $EB$ is negative for $\grav\cdot\OO<0$ and
positive for $\grav\cdot\OO>0$.
In this case, $EB$ does reflect the sign of kinetic helicity.
Obviously, the vertical velocity is different up- and downdrafts,
so the products $EB_z$ and $BB_z$ have different signs in
up- and downdrafts, so their average vanishes.
Thus, only $EB$, but neither $EB_z$ or $BB_z$ have a chance
in being useful for characterizing the sign of the helicity.

\section{Solar EB polarization}

%\subsection{Spectral line shapes}

We ignore Stokes $I$ and $V$ and only look at $Q$ and $U$
at the frequency pixel 81; see \Fig{pv2}
The $E$ and $B$ polarizations are computed from those data;
see \Fig{pEBmodes81}.
The spectra are currently mixed over north and south;
see \Fig{pEBmodes81p}.
The dynamical range is rather small.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pv2}
\end{center}\caption[]{
{\em Upper row}: Components of the Stokes vector $I$, $V$, $Q$, and $U$
at frequency pixel 85 (for $I$ and $V$) and 81 (for $Q$ and $U$).
{\em Lower row}: spectral line profiles through the two points marked
in the upper row in red and blue, respectively.
}\label{pv2}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pEBmodes81}
\end{center}\caption[]{
Solar $E$, $B$, and $EB$ plots for 2015-03-11.
\url{pEBmodes81}
}\label{pEBmodes81}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pEBmodes81p}
\end{center}\caption[]{
Spectra of solar $EE$, $BB$, and $EB$ polarization for 2015-03-11.
}\label{pEBmodes81p}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{p150416_AR12325}
\end{center}\caption[]{
Solar $E$, $B$, and $EB$ plots for AR12325 on 2015-04-16.
}\label{p150416_AR12325}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{p150416_AR12321}
\end{center}\caption[]{
Solar $E$, $B$, and $EB$ plots for AR12321 on 2015-04-16.
This is the active region that is seen in the disk center
at $\phi=540\degr$ and shown in the lower row of \Fig{pspec_vs_t}.
}\label{p150416_AR12321}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pspec_vs_t1}
\includegraphics[width=\textwidth]{pspec_vs_t2}
\end{center}\caption[]{
Left: time dependence of $EB$, but mapped in terms of degrees
of Carrington rotation, just like in \cite{BPS17}.
Red is for a patch of $375\Mm\times375\Mm$ in the northern hemisphere
and blue is for a similar patch in the southern hemisphere.
Right: spectra for the two patches at the time that is marked
with a vertical dashed-dotted line in the left panel.
The upper and lower rows are for two different times.
}\label{pspec_vs_t}\end{figure*}

To have a chance in finding a definite sign, one should separate the
signs in the northern and southern hemisphere by using the two-scale
method of \cite{BPS17}.

\appendix
There is an interesting analogy with another quantity that is sometimes
used in solar physics to estimate {\em kinetic} helicity from the surface
velocity field.
It employs the horizontal divergence, $\nab_\perp\cdot\uu_\perp$ and the
normal component of the curl, $\omega_\|\equiv(\nab_\perp\times\uu_\perp)_\|$.
Here, $\uu_\perp=(u_x,u_y,0)$ is the horizontal velocity in a Cartesian
coordinate system where $u_z\equiv u_\|$ is the vertical velocity.
The horizontal divergence and $\omega_\|$ are derived from the
horizontal components $\uu_\perp$ of the velocity vector.
Again, the correlation of $\nab_\perp\cdot\uu_\perp$ and $\omega_\|$
is only a proxy, but, unlike the case of polarization measurements, the
$\pi$ ambiguity does here not exist.
\cite{RBP99} showed that it is different from zero in a rotating
stratified layer and has the same sign as the kinetic helicity.

Unlike the velocity field at the solar surface or the magnetic field in
the cosmological context, the normal component of the magnetic field,
$B_\|$, can usually be determined quite accurately from the circular
polarization.
(For the cosmic microwave background radiation, which is produced by
Thomson scheduling, the circular polarization vanishes.)

%r e f
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\end{thebibliography}

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