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    {\small\rm\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm  suppl.\hfill  H. Men \& J. L. Han: Generation of the
    Toroidal Field in the Galactic Halo  %%�����ߣ���Ŀ���ƣ�
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\zihao{5}\songti

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    Vol.44~ Suppl.
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    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf
Generation of the Toroidal Field in the Galactic Halo   %%�����⣩
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\vs\vs{\normalsize Hui Men\hs\&\hs J. L. Han}  %%���������ߣ�

\vs
 (\footnotesize  \nao)
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
We demonstrate that the antisymmetric toroidal magnetic field can
be generated from the dipole field through velocity shear, and
noticed that the effect of the radial shear is opposite to and
greater than the effect of the vertical shear.

\end{quotation}

%%%% ����
%\para{\large\bf{1}~~ Introduction}

\vs\vs The large scale global magnetic field structure of our
Galaxy has been obtained from the rotation measures(RMs) of
pulsars and extragalactic radio sources. In the Galactic halo and
near the Galactic Center, there are toroidal fields with opposite
directions below and above the Galactic plane and poloidal fields
of dipole form (Han 2002). This fields have been believed to be
produced by A0 dynamo (Han et al. 1997). We investigate here
whether these fields can be generated by velocity shear.

Galaxies have extended ionized diffuse layer. The velocity
shear exists in our Galaxy and nearby galaxies. In NGC 891, the
halo gas appears to rotate 25 to 100\,km\,s$^{-1}$ more slowly
than the gas in the plane (Swaters et al. 1997). Exterior to $R
>$10\,kpc, a strip averaged between 2.8$>z>$1.4\,kpc shows a
rotation speed that is $\cong$ 25\,km s$^{-1}$ lower than the
value in the plane. Interior to $R < $7\,kpc, the difference gets
greater, reaching 100\,km\,s$^{-1}$ at $R = $ 5\,kpc.  The velocity
shear has been detected in NGC 5775 (T\"ullmann et al. 2000). The
gas velocity strongly depend on $z$. It reaches the highest of
1822$\pm$8\,km\,s$^{-1}$ at $z$ around 0\,kpc, but decrease slowly
towards larger $z$, finally reach to the systermic velocity of
1681\,km\,$^{-1}$ at $z=9$\,kpc. In our Galaxy, the neutral
hydrogen has a component with a scaleheight of $\sim$500\,pc (Dickey \&
Lockman 1990). Observations of the velocities at high $z$ of our
Galaxy are obviously difficult. Benjamin \& Danly (1997) discussed
the volocity stratification of high velocity clouds. On the other
hand, we noticed the velocity shear exists in all gravitational
potential models, such as that of Paczynski (1990) which consists
of a contribution of disk, halo and bulge. The radial acceleration
and the velocity of the Galaxy can be derived from the model.
Fig.1 shows the rotation velocity decreases with increasing
heights.
\begin{figure}[h]
\center {\includegraphics [bb=0 182 408 455,scale=0.6]{fig1.eps}}

   %\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize

      Fig.\,1~~ The rotation velocities as a function of $z$ and $R$
calculated using the potential model of Paczynski (1990)}

    \end{figure}
\begin{figure}[h]
\center {\includegraphics [bb=85 256 477 504,scale=0.6]{fig2.eps}}

   %\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize

      Fig.\,2~~ The toroidal field gets stronger with time due to the
vertical velocity shear}

    \end{figure}

To explain the the toroidal field found in our Galaxy, we now show whether
the toroidal field can be generated through velocity shear defined by the
potential model of Paczynski. As we know, the magnetic lines are frozen in
plasma. We suppose that the initial magnetic field in our Galaxy is of
dipolar form. When there is vertical velocity shear, azimuthal stretching
of vertical componet of field lines will also produce the azimuthal toroidal
field, as shown in Fig.~2. The strength of toroidal fields grows with time
in the form of

 $$
  B_{\varphi}(R,z)=B_{z}(R,z)\frac{\partial v_{rot}(R,z)}{\partial
 z}t\,,
 $$
where the $B_{z}$ is the vertical component of dipole field, $v_{rot}$ is the
function of rotation velocity, $B_\varphi$ is the toroidal field.

On the one hand, when the Galaxy rotates differentially, so that there is
radial velocity shear. The radial component of field lines will be winded
up, producing the toroidal field. The toroidal field thus generated should
be
$$
 B_{\varphi}(R,z)=B_{R}(R,z)R\frac{\partial \omega(R,z)}{\partial
 R}t\,,
$$
where the $B_{R} $ is the radial component of dipole field, and $\omega$ is
the function of angle velocity.

Both kinds of velocity shear can produce the toroidal fields.
Observationally, these field can be revealed by rotation measures of radio
sources.  Rotation measure depends on the magnetic field and electron
density, given by
$$
RM=0.820\int_0^{\rm dist}n_{e}(x,y,z)\cdot\vec{B}(x,y,z)\cdot {\rm
d}\vec{l}(x,y,z){\rm rad~m^{-2}}\,,
$$
To demonstrate the RM distribution of the produced toroidal fields,
we use the model for galactic electron density in a smooth and axisymmetric
pattern (Comez et al. 2001)
$$
 n_{e}(R,z)=n_{0}\frac{f(R/R_{0})}{f(R_{\odot}/R_{0})}f(\frac{z}{z_{0}})
 +n_{1}\frac{f(R/R_{1})}{f(R_{\odot}/R_{1})}f(\frac{z}{z_{1}})\,,
$$
where $ f(x)=\mathrm{sech}^{2}x $.

\begin{figure}[h]
\center {\includegraphics [bb=0 238 408 439,scale=0.5]{fig3.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \vspace{2mm}\footnotesize

      Fig.\,3~~ The simulated RM distribution by using the toroidal fields
        produced by vertical velocity (t = 2000\,Myr) shear plus the dipole
        field. The electron density model is from  Comez et al. (2001)}

    \end{figure}
\begin{figure}[h]
\center {\includegraphics [bb=0 238 408 439,scale=0.5]{fig4.eps}}

   %\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize

      Fig.\,4~~ The same to Fig.~3, but for the radial shear at t = 1\,Myr}

    \end{figure}
\begin{figure}[tbph]
\center {\includegraphics [bb=0 239 408 439,scale=0.5]{fig5.eps}}

   %\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize

      Fig.\,5~~ The observed RM sky from Han, Manchester \& Qiao (1999)}

    \end{figure}

The simulate RM distribution are shown in Fig.~3 and Fig.~4 for the vertical
and radial velocity shear, respectively. We found that both shears produce
the antisymmetric rotation measure sky. However, it is interesting to notice
that the directions of toroidal fields produced by the vertical and radial
velocity shears are contrary to each other. As effects of both kinds of shears
should exist in our Galaxy and they should cancel each other somehow
finally. Although the effect of radial shear appears much stronger than
that of vertical shear, the observed rotation measure sky (Fig.~5) is
very similar to Fig.~3. Two possible explanations: First,
the assumed dipole field should have an opposite direction comparing to
the real dipole in our Galaxy, so that toroidal field directions produced
by the radial shear consist with observed ones and the toroidal field
produced by the vertical shear does not dominate but cancle partially
the field from radial shear. Secondly, the dipole
field assumed is fine, and the toroidal field by vertical velocity shear
dominates. The radial shear only works at very low $z$ and so produce the
field does not affect the rotation measure sky much.

\vn{\bf ACKNOWLEDGEMENTS}~~ We thank Katia
Ferriere for helpful comments. This work is supported by
the National Natural Science
Foundation of China (19903003 and 10025313), the National
Key Basic Research Science Foundation of China (G19990754)
and the partner group of MPIfR at NAOC.

\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \nref Benjamin R.A., Danly L., 1997, ApJ, 1997, 481, 764
\nref Comez G.C., Benjamin R.A., Cox D.P., AJ, 2001, 122, 908
\nref Dickey J.M., Lockman F.J., ARA\&A, 1990, 28, 215 \nref Han
J.L., in: Astrophysical Polarized Backgrounds, held 9-12 October,
2001 in Bologna Italy. Edited by Stefano Cecchini, Stefano
Cortiglioni, Robert Sault, and Carla Sbarra. AIP Conference
Proceedings, Vol. 609. Melville, NY: American Institute of
Physics, 2002., p.96 \nref Han J.L., Manchester R.N., Berkhuijsen
E.M., Beck R., A\&A, 322, 1997, 98 \nref Han J.L., Manchester
R.N., Qiao G.J., MNRAS, 1999, 306, 371 \nref Paczynski B., ApJ,
1990, 348, 485 \nref Swaters R.A., Sancisi R., van der Hulst J.M.,
ApJ, 1997, 491, 140 \nref T\"ullmann R., Dettmar R.J., Soida M.,
et al., A\&A, 2000, 364, L36

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