Meridional flow velocities on solar-like stars with known activity cycles
Name\corref{cor1
Dilyara Baklanova\corref{cor
Sergei Plachinda
%% Text of abstract The direct measurements of the meridional flow velocities on stars are impossible today. To evaluate the meridional flow velocities on solar-like stars with stable activity periods, we supposed that during the stellar Hale cycle the matter on surfaces of stars passes the meridional way equivalent to $2\pi R_\star$. We present here the dependence of the mean meridional flow velocity on Rossby number, which is an effective parameter of the stellar magnetic dynamo.
baklanova.tex
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\title{Meridional flow velocities on solar-like stars with known activity cycles}%\tnoteref{footnote1}}
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\author{Dilyara Baklanova\corref{cor}}%\fnref{footnote2}}
\address{Crimean Astrophysical Observatory, Nauchny 298409, Crimea}
\cortext[cor]{Corresponding author}
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\ead{dilyara@crao.crimea.ua}

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\author{Sergei Plachinda}%\fnref{footnote3}}
\address{Crimean Astrophysical Observatory, Nauchny 298409, Crimea}
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\begin{abstract}
%% Text of abstract
The direct measurements of the meridional flow velocities on stars are impossible today. 
To evaluate the meridional flow velocities on solar-like stars with stable activity periods, we supposed that during the stellar Hale cycle the matter on surfaces of stars passes the meridional way equivalent to $2\pi R_\star$. We present here the dependence of the mean meridional flow velocity on Rossby number, which is an effective parameter of the stellar magnetic dynamo.
\end{abstract}

\begin{keyword}
%first keyword \sep second keyword \sep more keywords
stars: solar-like; stars: activity; stars: Hale cycle; stars: meridional flow velocity
% keywords here, in the form: keyword \sep keyword
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\end{keyword}

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%% Main text
\section{Introduction}

Today the physics of the large-scale flows on the Sun and stars are intensively simulated by scientists to determine the detailed mechanisms of activity cycles. Recent results and bibliography are presented, for example, in the papers by \citet{Upton2014}, \citet{Zhao2013}, \citet{Guerrero2013}, \citet{Moss2012}, \citet{Kitchatinov2013}, \citet{Kitchatinov2012}.

The empirical dependence of the mean meridional flow speed on the number of the 22-year Hale cycle \citep{Plachinda2011} was obtained under the assumption that during the Hale cycle the total length of the reverse track of the poloidal dipole polarity is equivalent to the circumference of the Sun as the magnetic dipole moment does not vanish and migrates between the poles \citep{Livshits2006, Moss2012}.

The mean velocity $\left \langle v \right\rangle = 6.29$~m~s$^{-1}$, which gives $P_{Hale} = 22$~years for the Sun, corresponds to the 7.3 years activity period for the solar-like star 61~Cyg~A and is well agreed with the observations \citep{Plachinda2011}.
Therefore we supposed that the magnetic flux transported by meridional flows on the surfaces of solar-like stars with stable activity period also passes the way equivalent to $2\pi R_\star$ during the own Hale cycle.
We use this approach to draw the dependence of the mean meridional flow velocity on the Rossby number.
In other words, in this paper we use the empirical data to see whether the duration of activity cycle on convective stars depends on the effective parameter of dynamo processes.
\sloppy

\section{Parameters of stars}
 
We selected stars with well-known activity periods. In the Table~\ref{tab:param1} and Table~\ref{tab:param3} we summarized the physical parameters of the stars that we have got from literature. 
 
The names of the stars are listed in the first columns of the Tables. Columns 2-4 in the Table~\ref{tab:param1} contain magnitude $V$, spectral type and color index $B-V$. The rotation periods of the stars and the references are given in the 5th and 6th columns, the duration of activity cycles and the references are given in the 7th and 8th columns.

The masses of the stars and the references are listed in the 9th and 10th columns in the Table~\ref{tab:param1}, columns 11 and 12 represent the radii of the stars and the references, columns 13-14 give the logarithm of gravity and the references. Finally, columns 15-16 show  the effective temperature and the references.

The 2nd column in the Table~\ref{tab:param3} shows the parameter $\left\langle R'_{HK} \right\rangle$ defined as the ratio of the chromospheric emission in the cores of the CaII H and K lines to the total bolometric emission of the star, and the 3rd column contains the references.
The logarithm of the convective turnover time, the Rossby number obtained from the dependence of $\tau_c$ from $B-V$ and the mean meridional flow velocity are listed in the last three columns in the Table~\ref{tab:param3}.

The convective turnover time $\tau_c$ and the Rossby number were calculated using the methods described in the Section~\ref{sec:Rhk}.
The mean meridional flow velocity has been calculated using the equation $2\pi R_\star / \left\langle v \right\rangle = P_{Hale}$ \citep{Plachinda2011}, where $R_\star$ is the radius of a star, $P_{Hale}$ is the stellar magnetic activity period, $P_{Hale} = 2P_{cyc}$, where $P_{cyc}$ is a star-spot activity period.
The mean activity period of the Sun calculated by averaging of sunspot numbers for all years of observations from 1755 to 2008 is equals to $P_{cyc \odot} = 11$ years.
\sloppy

\renewcommand{\tabcolsep}{1.1mm}   %% increase table column spacing
\begin{table*}[ht]
\caption{Parameters of stars}%\\ % title of Table 
\footnotesize
\label{tab:param1}
\begin{minipage}{\textwidth}
\begin{tabular}{l|crD{.}{.}{1.3}|D{.}{.}{2.4}D{.}{.}{1}|D{.}{.}{2.2}D{.}{.}{1}|D{.}{.}{1.3}D{.}{.}{1}|D{.}{.}{1.3}D{.}{.}{1}|D{.}{.}{2}D{.}{.}{1}|cD{.}{.}{1}}
\hline\hline
\multicolumn{1}{c|}{Name}	&	V	&	\multicolumn{1}{c}{Sp}	&	\multicolumn{1}{c|}{$B-V$}	&	\multicolumn{1}{c}{$P_{rot}$, days}	&	\multicolumn{1}{c|}{Ref$^a$}	&	\multicolumn{1}{c}{$P_{cyc}$, yr}	&	\multicolumn{1}{c|}{Ref$^b$}	&	\multicolumn{1}{c}{$M_\star$, $M_\odot$}	&	\multicolumn{1}{c|}{Ref$^c$}	&	\multicolumn{1}{c}{$R_\star$, $R_\odot$}	&	\multicolumn{1}{c|}{Ref$^d$}	&	\multicolumn{1}{c}{$\log g$}	&	\multicolumn{1}{c|}{Ref$^e$}	&	$T_{eff}$, K	&	\multicolumn{1}{c}{Ref$^f$}	\\
\hline
Sun	&		&	G2~V	&	0.65	&	25.4	&	5	&	11.0	&	-	&	1.00	&		&	1.00	&		&	4.44	&		&	5780	&		\\
BE~Cet	&	6.39	&	G2~V	&	0.659	&	7.78	&	7	&	6.7	&	2	&	1	&	6	&	1.04	&	6	&	4.4	&	6	&	5790	&	6	\\
54~Psc	&	5.87	&	K0~V	&	0.85	&	48.0	&	5	&	13.8	&	1	&	0.76	&	12	&	0.94	&	12	&	4.51	&	21	&	5250	&	28	\\
HD4628	&	5.75	&	K2~V	&	0.88	&	38.5	&	7	&	8.37	&	1	&	0.77	&	6	&	0.69	&	6	&	4.64	&	6	&	5004	&	6	\\
107~Psc	&	5.24	&	K1~V	&	0.84	&	35.2	&	7	&	9.6	&	1	&	0.816	&	21	&	0.82	&	21	&	4.54	&	21	&	5098	&	11	\\
HD16160	&	5.82	&	K3~V	&	0.98	&	48.0	&	7	&	13.2	&	1	&	0.809	&	21	&	0.76	&	21	&	4.62	&	21	&	5262	&	29	\\
$\kappa^1$~Cet	&	4.80	&	G5~V	&	0.68	&	9.214	&	2	&	5.9	&	2	&	1.02	&	6	&	0.877	&	6	&	4.5	&	16	&	5630	&	16	\\
40o$^2$~Eri	&	4.43	&	K1~V	&	0.82	&	43.0	&	8	&	10.1	&	1	&	0.81	&	21	&	0.82	&	21	&	4.31	&	16	&	5090	&	16	\\
HD32147	&	6.22	&	K3~V	&	1.06	&	47.4	&	9	&	11.1	&	1	&	0.838	&	21	&	0.78	&	21	&	4.4	&	18	&	4945	&	18	\\
HD78366	&	5.93	&	F9~V	&	0.585	&	9.67	&	7	&	12.2	&	1	&	1.13	&	21	&	1.075	&	23	&	4.46	&	21	&	5938	&	23	\\
HD81809	&	5.38	&	G2~V	&	0.64	&	18.0	&	13	&	8.17	&	1	&	1.33	&	14	&	2.24	&	14	&	3.86	&	14	&	5888	&	14	\\
DX Leo	&	7.01	&	K0~V	&	0.78	&	5.377	&	10	&	3.21	&	2	&	0.93	&	11	&	0.84	&	11	&	4.4	&	18	&	5121	&	11	\\
CF~UMa	&	6.45	&	K1~V	&	0.75	&	31.0	&	8	&	7.3	&	1	&	0.661	&	21	&	0.681	&	17	&	4.63	&	21	&	4759	&	17	\\
$\beta$~Com	&	4.26	&	F9.5~V	&	0.57	&	12.35	&	7	&	16.6	&	1	&	1.17	&	11	&	1.1	&	17	&	4.4	&	16	&	5960	&	16	\\
HD115404	&	6.66	&	K2~V	&	0.93	&	18.47	&	11	&	12.4	&	1	&	0.86	&	11	&	0.77	&	11	&	4.3	&	25	&	4852	&	11	\\
18~Sco	&	5.49	&	G2~V	&	0.652	&	23.7	&	9	&	7.1	&	3	&	1.01	&	24	&	1.03	&	24	&	4.4	&	27	&	5433	&	17	\\
V2133~Oph	&	5.75	&	K2~V	&	0.827	&	21.07	&	7	&	17.4	&	1	&	0.91	&	15	&	0.84	&	15	&	4.5	&	18	&	5924	&	18	\\
V2292~Oph	&	6.64	&	G7~V	&	0.76	&	11.43	&	11	&	10.9	&	1	&	0.97	&	11	&	0.87	&	9	&	4.56	&	21	&	5266	&	11	\\
V2215 Oph	&	6.34	&	K5~V	&	1.16	&	18.0	&	11	&	21.0	&	1	&	0.72	&	11	&	0.63	&	11	&	4.67	&	9	&	4319	&	11	\\
HD160346	&	6.52	&	K3~V	&	0.96	&	36.4	&	11	&	7.0	&	1	&	0.86	&	11	&	0.77	&	11	&	4.3	&	25	&	4862	&	11	\\
HD166620	&	6.40	&	K2~V	&	0.87	&	42.4	&	7	&	15.8	&	1	&	0.89	&	11	&	0.791	&	21	&	4.0	&	18	&	5035	&	18	\\
61~Cyg~A	&	5.21	&	K5~V	&	1.18	&	35.37	&	7	&	7.3	&	1	&	0.69	&	20	&	0.665	&	20	&	4.63	&	6	&	4400	&	20	\\
61~Cyg~B	&	6.03	&	K7~V	&	1.37	&	37.84	&	7	&	11.7	&	1	&	0.605	&	20	&	0.595	&	20	&	4.71	&	21	&	4040	&	20	\\
HN~Peg	&	5.94	&	G0~V	&	0.587	&	4.86	&	7	&	5.5	&	2	&	1.1	&	21	&	1.041	&	21	&	4.48	&	21	&	5967	&	23	\\
94~Aqr~A	&	5.20	&	G8.5~IV	&	0.79	&	42.0	&	8	&	21.0	&	1	&	1.04	&	14	&	1.99	&	14	&	3.86	&	14	&	5370	&	14	\\
94~Aqr~B	&	8.88	&	K2~V	&	0.88	&	43.0	&	8	&	10.0	&	1	&	0.96	&	15	&	0.93	&	15	&	4.54	&	15	&	5136	&	15	\\
BY~Dra	&	8.07	&	K6~V	&	1.2	&	3.83	&	4	&	13.7	&	4	&	0.58	&	22	&	0.71	&	22	&	4.65	&	18	&	4622	&	18	\\
V833~Tau	&	8.42	&	K5~V	&	1.19	&	1.7936	&	4	&	6.4	&	4	&	0.93	&	26	&	0.77	&	22	&	4.5	&	22	&	4450	&	22	\\
\hline
\end{tabular}
\end{minipage}
\par
   \vspace{0.05\skip\footins}
$^a${References for rotation periods;}
$^b$References for periods of the activity cycles;
$^c$References for stellar masses;
$^d$References for radii of stars;
$^e$References for $\log g$;
$^f$References for effective temperatures $T_{eff}$
\\
(1)~\citet{Baliunas1995};
(2)~\citet{Messina2002};
(3)~\citet{Hall2007};
(4)~\citet{Olah2000};
(5)~\citet{Noyes1984};
(6)~\citet{Cranmer2011};
(7)~\citet{Donahue1996};
(8)~\citet{Baliunas1996};
(9)~\citet{Cincunegui2007};
(10)~\citet{Messina1999};
(11)~\citet{Wright2011};
(12)~\citet{Santos2004};
(13)~\citet{Isaacson2010};
(14)~\citet{AllendePrieto1999};
(15)~\citet{Fuhrmann2008};
(16)~\citet{Chmielewski2000};
(17)~\citet{Boyajian2012a};
(18)~\citet{Mishenina2012a};
(20)~\citet{Kervella2008};
(21)~\citet{Takeda2007};
(22)~\citet{Eker2008};
(23)~\citet{Masana2006};
(24)~\citet{Lammer2012};
(25)~\citet{Soubiran2010};
(26)~\citet{Roser2011};
(27)~\citet{Mishenina2003};
(28)~\citet{Hillen2012};
(29)~\citet{VanBelle2009}
\end{table*}				

\begin{table}[t]
\caption{Parameters of stars}
\footnotesize
\label{tab:param3}
\begin{minipage}{\textwidth}
\begin{tabular}{l|D{.}{.}{2.3}D{.}{.}{1}|D{.}{.}{1.2}D{.}{.}{2.3}D{.}{.}{2.2}}
\hline\hline
\multicolumn{1}{c|}{Name}	&	\multicolumn{1}{c}{$\log \left\langle R'_{HK} \right\rangle$}	&	\multicolumn{1}{c|}{Ref$^g$}	&	\multicolumn{1}{c}{$\log \tau_c$}	&	\multicolumn{1}{c}{Ro}	&	\multicolumn{1}{c}{$v$, m s$^{-1}$}	\\
\hline
Sun	&	-4.937	&	5	&	1.08	&	2.080	&	6.92	\\
BE~Cet	&	-4.441	&	5	&	1.10	&	0.616	&	7.91	\\
54~Psc	&	-4.960	&	5	&	1.32	&	2.297	&	4.72	\\
HD4628	&	-4.852	&	5	&	1.29	&	1.988	&	5.71	\\
107~Psc	&	-4.874	&	5	&	1.31	&	1.722	&	5.84	\\
HD16160	&	-4.847	&	5	&	1.34	&	2.179	&	4.46	\\
$\kappa^1$~Cet	&	-4.45	&	16	&	1.14	&	0.672	&	10.29	\\
40o$^2$~Eri	&	-4.944	&	5	&	1.30	&	2.145	&	5.55	\\
HD32147	&	-4.94	&	5	&	1.37	&	2.013	&	5.05	\\
HD78366	&	-4.631	&	5	&	0.62	&	2.327	&	6.10	\\
HD81809	&	-4.907	&	5	&	1.06	&	3.596	&	18.98	\\
DX Leo	&	-4.08	&	18	&	1.27	&	0.290	&	18.12	\\
CF~UMa	&	-4.930	&	5	&	1.24	&	1.772	&	6.46	\\
$\beta$~Com	&	-4.756	&	5	&	0.88	&	1.636	&	4.61	\\
HD115404	&	-4.467	&	5	&	1.35	&	0.840	&	4.58	\\
18~Sco	&	-4.950	&	19	&	1.08	&	1.983	&	11.31	\\
V2133~Oph	&	-4.541	&	5	&	1.31	&	0.544	&	3.34	\\
V2292~Oph	&	-4.438	&	5	&	1.25	&	0.643	&	5.53	\\
V2215 Oph	&	-4.627	&	5	&	1.38	&	0.750	&	2.08	\\
HD160346	&	-4.787	&	5	&	1.36	&	1.589	&	7.62	\\
HD166620	&	-4.910	&	5	&	1.33	&	1.974	&	3.72	\\
61~Cyg~A	&	-4.800	&	5	&	1.38	&	1.473	&	6.31	\\
61~Cyg~B	&	-4.909	&	5	&	1.42	&	1.461	&	4.02	\\
HN~Peg	&	-4.424	&	5	&	0.92	&	0.580	&	13.11	\\
94~Aqr~A	&	-4.999	&	5	&	1.28	&	2.209	&	6.56	\\
94~Aqr~B	&	-4.902	&	5	&	1.34	&	1.952	&	6.44	\\
BY~Dra	&	-4.01	&	18	&	1.39	&	0.156	&	3.59	\\
V833~Tau	&	-	&	-	&	1.39	&	0.076	&	8.33	\\
\hline
\end{tabular}
\end{minipage}
\par
   \vspace{0.05\skip\footins}
$^g$References for $\log R'_{HK}$\\
(5)~\citet{Noyes1984};
(16)~\citet{Chmielewski2000};\par
(18)~\citet{Mishenina2012a};
(19)~\citet{Raghavan2010}
\end{table}				


\section{The dependence of the Rossby number on the mean chromospheric emission ratio \texorpdfstring{$\left\langle R'_{HK} \right\rangle$}{Rhk}} \label{sec:Rhk}

\begin{figure}[h!]%[htbp] 
	\begin{center}
				\includegraphics[width=1\linewidth]{baklanova_fig1.eps}
				\caption{$\log \left\langle R'_{{HK}} \right\rangle$ versus the logarithm of the Rossby number $\log (P_{{obs}}/ \tau_c)$, where $\tau_c = f(B-V)$. The symbol $\odot$ identifies the place of the Sun. The dashed line is the function from \citet{Noyes1984}. The solid line represents the linear fit to all data except one point marked by the cross.} 
				\label{fig:rossby-b-v-Rhk}
	\end{center}
\end{figure}

\begin{figure}[h]%
		\centering
%	\begin{center}
				\includegraphics[width=1\linewidth]{baklanova_fig2.eps}
				\caption{$\log \left\langle R'_{{HK}} \right\rangle$ versus the logarithm of the Rossby number $\log (P_{{obs}}/ \tau_c)$, where $\tau_c = f(M_\star)$. The symbol $\odot$ points out the place of the Sun. The solid line represents the linear fit to all data points.}
				\label{fig:rossby-mass-Rhk}
%	\end{center}
\end{figure}

The Rossby number, $Ro = P_{rot}/\tau_c$, is the ratio of the stellar rotation period $P_{rot}$ to the convective turnover time $\tau_c$.
To find the convective turnover time \citet[eq.~4]{Noyes1984} used the empirical dependence of $\tau_c$ on color index $B-V$, and \citet{Wright2011} chose the empirical dependence of $\tau_c$ on stellar masses.

The dependence of $\log \left\langle R'_{{HK}} \right\rangle$ on $\log Ro$ for $\tau_c = f(B-V)$ according to \citet{Noyes1984} is shown in Figure~\ref{fig:rossby-b-v-Rhk} by dashed curve.
The dependence of $\log \left\langle R'_{{HK}} \right\rangle$ on $\log Ro$, where $\tau_c = f(M_\star)$, is plotted in Figure~\ref{fig:rossby-mass-Rhk}.

The relation between $\log \left\langle R'_{{HK}} \right\rangle$ and $\log Ro$ is more accurate in the case of using of the dependence of $\tau_c$ on color index $B-V$.
The significance level of the difference between the scatterings is more than 99.99\%. Therefore we used the relation $\tau_c = f(B-V)$ to evaluate the Rossby number. 

We have supplemented the list of stars of \citet{Noyes1984} using, in particular, the stars with higher level of the chromospheric activity.
The relation between $\log \left\langle R'_{{HK}} \right\rangle$ and $\log Ro$ in this case may be presented as $\log R'_{{HK}} = -4.63 - 0.83 \log Ro$ (solid line in Figure~\ref{fig:rossby-b-v-Rhk}).
\sloppy

\section{The dependence of the mean meridional flow velocity on the Rossby number and the activity cycle period}

The Figure~\ref{fig:rossby-v} shows that the mean meridional flow velocities $\left\langle v \right\rangle$ for solar-type stars located near $5.4\pm1.5$~m\,s$^{-1}$ that is in good agreement with the mean value of the meridional flow velocity of the Sun ($6.29$~m\,s$^{-1}$, \citep{Plachinda2011}) obtained in the same manner.
We could suggest that the mean meridional flow velocity does not depend on the Rossby number. 
The only five stars out of 28 show higher values which significantly (more than 3$\sigma$) deviates from the mean value of the meridional flow velocity.
So, we can suppose that in the case of 80\% stars with the stable activity period the meridional flow determines the duration of the Hale's cycle.

\begin{figure}[h!]%[htbp]
		\centering
		\includegraphics[width=1\linewidth]{baklanova_fig3.eps}
		\caption{Mean meridional flow velocity versus Rossby number. The dotted line is a fit to all data excluding 5 points (stars symbols) which lie out of 10~m\,s$^{-1}$.}
		\label{fig:rossby-v}
	\end{figure}

\begin{figure}[h!]%[htbp]
		\centering
		\includegraphics[width=1\linewidth]{baklanova_fig4.eps}
		\caption{$P_{cyc}$ versus mean meridional flow velocity $\left\langle v \right\rangle$.}
		\label{fig:v-Pcyc}
	\end{figure}

On the other hand, Figure~\ref{fig:v-Pcyc} demonstrates that the observed cycle period shows the exponential decay when the velocity of the flow increases.
The relation between $P_{cyc}$ and $\left\langle v \right\rangle$ may be presented as $P_{cyc} = 5.74 + 35 \exp \left(- \left( \left\langle v \right\rangle - 2.1\right)/1.66\right)$ (dashed line in Figure~\ref{fig:v-Pcyc}).

The dynamo models show qualitatively similar but much slower decay \citep[see][Figure 8]{Bonanno2002}.
Therefore, we can not eliminate another way to interpret the detected velocity as the phase velocity of the dynamo wave drift \citep[e.g.,][]{Kitchatinov2002a}.

\section{Acknowledgement}	
We thank both anonymous referees and V.~Butkovskaya for their constructive comments and useful suggestions.

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