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\begin{document}

\title{Unusual Properties of Convection and Dynamos in Rotating Spherical Shells} 

% First author block:
\author{F. H. Busse}
\email{busse@uni-bayreuth.de}
% \homepage{An author's web page; optional}
\affiliation{Institute of Physics, University of Bayreuth, D-95440  Bayreuth, Germany}
% You may list several affiliation, using separate commands for each:
\begin{abstract}
Convection in rotating spherical fluid shells is the basic dynamical
process in the description of heat transports in planets and in
stars. Since the respective fluids are often electrically conducting,
the occurrence of convection is frequently associated with the
generation of magnetic fields. The interaction between velocity and
magnetic fields gives rise to rich dynamical structures in dependence
on the numerous parameters of the problem. 
\end{abstract}

% Insert suggested PACS numbers (up to 4) in braces.
% The PACS (Physics and Astronomy Classification Scheme) 
% can be accessed on the web at http://www.aip.org/pacs/\pacs{47.10.A-,47.15.ki,47.20.Bp,47.20.Qr,47.32.-y,97.10.Gz,97.10.Kc,97.25.Za}

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\section{\label{intro}Introduction}

The interest in the problem of convection in rotating spheres has been
stimulated by its astrophysical applications and by the search for the
origin of geomagnetism in the Earth's liquid outer core. The
exploration of the solar system and the discovery of the magnetic
fields of other planets has amplified this interest and in the past
decades a large number of papers dealing with convection in fluid
shells and its dynamo action have appeared. Instead of reviewing all
of this work we wish just wish to draw the attention to a number of
surprising features that have appeared in the course of this work and
which are  of more general interest in the field of fluid dynamics. In
order to make this article more readily accessible to a general
resdership the use of mathematical equations and formulas is
minimized. For more complete introductions to the subject we refer to
the book  \cite{DS} and the review article \cite{Bu}. 


\section{Thermal Rossby waves}
\begin{figure}[t] 
\begin{center}
\includegraphics[width=12cm]{fig-01.pdf} 
\caption{Sketches for the dynamics of Rossby waves}
\end{center} 
\end{figure}

A basic theorem of the dynamics of rotating fluids is the Proudman-Taylor theorem which states that steady small amplitude motions of a barotropic rotating fluid do not vary in the direction of the axis of rotation when viscous effects can be neglected. "Small amplitude" means in this connection that the vorticity of the motion  is negligible in comparison to the rotation rate of the system. The Proudman-Taylor condition is a consequence of the complete balance between Coriolis force and pressure gradient.  This balance is  also called geostrophic balance since it holds in good approximation for the large scale motions in the Earth's atmosphere.

Two-dimensional fluid motions can not often be accommodated in physical reality and in a spherical shell in particular two-dimensional motions are prevented by the boundary conditions. Fluid flows are thus forced to become time dependent. In the simplest cases the motions assume the form of nearly two-dimensional propagating Rossby waves. As indicated in figure 1, Rossby waves can be understood on the basis of the conservation of angular momentum. When a column of fluid (aligned with the axis of rotation) moves into a shallower place it becomes compressed and - because of the conservation of mass - its moment of inertia increases.
To conserve angular momentum its rotation relative to an inertial frame of reference must decrease. Relative to the rotating system it thus acquires anticyclonic vorticity. The opposite process happens when the column moves to a deeper place where it gets stretched in the direction of the axis of rotation and acquires cyclonic vorticity.

In the annular fluid layer of figure 1 the depth decreases with increasing distance from the axis. A sinusoidal displacement of the initially static fluid columns leads to a flow structure in the form of vortices which tend to move the columns to new positions as indicated by the dashed line in the lower plot of the figure, i.e. the initial sinusoidal displacement propagates as a wave in the prograde direction. A retrograde propagation relative to the sense of rotation will be obtained  when the depth of the annular layer increases with distance from the axis. 

\begin{figure}[t] 
\begin{center}
\includegraphics[width=8cm]{fig-02.pdf} 
\caption{Geometrical configuration of the rotating annulus.}
\end{center} 
\end{figure}

Rossby waves Like water waves Rossby waves decay, of course, when there is no force sustaining them against viscous dissipation. The possibility for such a sustenance exists in a thermally unstably stratified system where thermal Rossby waves may be generated. Growing disturbances are obtained in an annular configuration as sketched in figure 2 when a temperature difference, $T_2- T_1$, and a gravity force are applied in the $x$-direction. In the experimental realization of the problem \cite{76} the centrifugal force $\Omega^2r_0$ is used as gravity and the temperature gradient must point outward in order to create the unstable density stratification. For geophysical applications one may think of the opposite directions for gravity and temperature gradient, but the mathematical problem remains the same in both cases. 

\begin{figure}[t] 
\begin{center}
\includegraphics[width=5cm]{fig-03.pdf} 
\caption{Convection columns in a rotating spherical fluid shell for $\tau=10^4, R=3.8\cdot 10^5, P=1$. Dark and light surfaces correspond to a constant positive and negative value of the radial velocity.}
\end{center} 
\end{figure}
The annulus configuration of figure 2 may be regarded as an annular
section of a rotating spherical shell. Indeed, the theory developed
for the rotating annulus can be applied in good approximation to the  
case of convection in a self-gravitating rotating fluid sphere heated
from within \cite{70a}. 
As a standard model a spherical fluid shell of thickness $d$ rotating
with a constant angularvelocity $\Omega$ is often assumed for which a
static state exists when thetemperature depends only on the distance
from the center. Here we shall assume the particular temperature
distribution $T_S = T_0 -\beta d^2 r^2 /2$ which corresponds to a
homogeneously heated sphere. $rd$ isthe length of the position vector,
$\vec{r}$, with respect to the center of thesphere and the gravity
field is assumed in the form $\vec g = - d \gamma \vec r$. Usingthe
length $d$, the time $d^2 / \nu$ and  the temperature $\nu^2 /\gamma
\alpha d^4$ dimensionless equations can be formulated in which the
Rayleigh number $R$, the Coriolis number $\tau$ and the Prandt lnumber
$P$,
\begin{equation}R = \frac{\alpha \gamma \beta d^6}{\nu \kappa}
  ,\enspace \tau = \frac{2\Omega d^2}{\nu} , \enspace P =
  \frac{\nu}{\kappa} ,  
\end{equation}
appear as dimensionless parameters. Here $\nu$ denotes
the kinematic viscosity of the fluid, $\kappa$ its thermal
diffusivity, $\varrho$ its density and $\alpha$ is the coefficient of
thermal expansion. Since the Boussinesq approximation is assumed, the
velocity field $\vec u$ is solenoidal and the general representation
in terms of poloidal and toroidal components can be used,
\begin{equation}
%\begin{gather}
\vec u = \nabla \times ( \nabla v
                \times \vec r) + \nabla w \times\vec r \enspace,  
%\end{gather}
\end{equation}such that the radial velocity depends on
$v$ alone and is given by $u_r=L_2v\equiv \partial^2ru/\partial r^2 -
r\nabla^2v$. A typical picture of thermal Rossby waves in a rotating
spherical fluid shell is shown in figure 3. Here as well as in the
following the ratio 0.4 between inner and outer radius of the shell
has been used. Because of the symmetry of the progradely propagating
velocity field with respect to the equatorial plane it is sufficient
to plot streamlines in this plane, given by $r \partial v/\partial r=
const.$, in order to characterize the convection flow as has been
done in figures 4, 5 and 6. Even in the case of turbulent convection
the part of the velocity field that is antisymmetric with respect to
the equatorial plane is rather small. 

\begin{figure}%[t]
\begin{center}
\includegraphics[width=5cm,angle=90]{fig-04.pdf} 
\end{center}
\caption{Time periodic vacillations of convection at $Ra= 2.8 \times
  10^5$ (left side) and $Ra=3 \times 10^5$ (right side) for $\tau =
  10^4$, $Pr=1$ The streamlines, $r \partial u_p / \partial \varphi =$
  const. are shown in one quarter of the equatorial plane. The four
  quarters are equidistant in time (with $\Delta t =0.015$ $(\Delta t
  =  0.024)$ in the left (right) case in the clockwise sense such
  that  approximately a full period is covered by the circles.}
\label{f8}
\end{figure}
%\begin{figure}[t] 
%\begin{center}
%\includegraphics[height=5cm,angle=90]{fig8.pdf} 
%\caption{Convection  streamlines, $r \partial v / \partial \varphi=$ const., in the
%  equatorial plane of a rotating spherical fluid shell for $\tau=3\cdot10^4, R=3\cdot 10^5, P=0.1$.}
%\end{center} 
%\end{figure}
As the Rayleigh number $R$ increases beyond its critical value $R_c$
for the onset of convection in the form of thermal Rossby waves, a
sequence of bifurcations can be observed similar to those found in
other problems of convection. First, typical oscillations in amplitude
are observed as shown in figure 4, then another bifurcation may add
low wavenumber modulations in the azimuthal direction as shown in
figure 5. Finally, a chaotic state of convection is obtained.  
\begin{figure}%[t] 
\begin{center}
\includegraphics[width=18cm]{fig-05.pdf} 
\caption{Modulated shape vacillations of convection for $Ra=2.9 \times
  10^5$, $\tau=10^4$, $Pr=1$. The plots show streamlines,
  $r \frac{\partial u_p}{\partial \phi}=const.$, in the equatorial
  plane and are equidistant in time with $\Delta t =0.04$ so that
  approximately a full period is covered.} 
\end{center} 
\end{figure}
\begin{figure}%[t] 
\begin{center}
\includegraphics[width=16cm,height=12cm,angle=-90]{fig-06new.pdf} 
\caption{Time series of energy densities of convection for $P = 0.5$,
  $\tau= 1.5\times10^4$ and $R = 3\times10^5$ ,  
$3.2 \times 10^5$, $3.45 \times 10^5$, $5 \times 10^5$, $7 \times
10^5$ , $10^6$ , (from bottom to top). 
Solid, dotted and dashed lines indicate $\overline{E}_t$ , $\check{E}_t$ , and $\check{E}_p$ ,
respectively. The critical Rayleigh number for onset  $R_c = 215142$.}
%\includegraphics[width=16cm,height=12cm]{fig-06.pdf}
%\caption{Time series of energy densities $E_t^m$ (thick solid line)
%and $E_t^f$ (thin solid line), the  and of the Nusselt number $Nu$
%(dotted line, right ordinate) are plotted for $\tau=3\cdot10^4,
%R=3\cdot 10^5, P=0.1$.} 
\end{center} 
\end{figure}

\section{Chaotic Convection in Rotating Spherical Shells}

The sequence of transitions is also evident in the time dependence of average quantities such as the convective heat transport plotted in figure 6. The latter is described by the Nusselt number which is defined as ratio between the average gradients of the temperature at the inner boundary of the shell in the presence of convection and in its absence. Also of interest are the components of the kinetic energy density which are defined by
\begin{subequations}
\begin{align}
\label{7}
E_p^m = \frac{1}{2} \langle \mid \nabla \times ( \nabla \bar v \times \vec r 
) \mid^2 \rangle , \quad & E_t^m = \frac{1}{2} \langle \mid \nabla \bar w_t \times
\vec r \mid^2 \rangle \\
\label{8}
E_p^f = \frac{1}{2} \langle \mid \nabla \times ( \nabla \check v_p \times \vec r) 
\mid^2 \rangle , \quad&  E_t^f = \frac{1}{2} \langle \mid \nabla \check w_t \times
\vec r \mid^2 \rangle
\end{align}
\end{subequations}
where the angular brackets indicate the average over the fluid shell
and where $\bar v_p$ refers to the azimuthally averaged 
component of $v_p$ and $\check v_p$ is given by $\check v_p = v_p -
\bar v_p $.  
$E_t^m$ describes the energy density of the differential rotation
which increases substantially with increasing $R$ as is evident from
figure 6. This increase is caused by the strong Reynolds stress of the
convection eddies resulting from their inclination with respect to
radial direction which is evident in  figures 3 through 5. The shear
of the differential rotation, however, tends to inhibit convection in
that it shears off the convection eddies. This is a consequence of the
nearly two-dimensional nature of the dynamics in a rotating system: In
a non-rotating system the convection rolls would simply align
themselves with the direction of the shear and the heat transport
would thus remain unchanged. In the rotating sphere a precarious
balance results realized in the form localized convection. As shown in
figure 7 convection occurs only in a restricted azimuthal section of
the spherical shell where its amplitude is sufficiently strong to
overcome the inhibiting influence of the shear. The axisymmetric
differential rotation continues to be driven by the localized
convection. The advection by the differential rotation of the  thermal
boundary layers which have expanded in the non-convecting region of
the shell actually strengthens the localized convection in that its
available buoyancy is replenished. 

Instead of a spatial localization the localization of convection in
time offers another possibility for the precarious balance as
demonstrated in figures 8 and 9. Here convection exist only for a
short period while the differential rotation is sufficiently weak. As
the amplitude of convection grows the differential rotation grows even
more strongly as the Reynolds stress increases with the square of the
amplitude. Soon the shearing action becomes strong enough to cut off
convection. Now a viscous diffusion time must pass before the
differential rotation has decayed sufficiently such that convection
may start growing again. It is remarkable to see how the chaotic
system exhibits its nearly periodic relaxation oscillations as shown
in figure 8. 

\begin{figure} 
\begin{center}
\includegraphics[height=9cm]{fig-07.pdf} 
\caption{Localized convection for $Ra=7 \times 10^5, \tau =1.5 \times 10^4, Pr=0.5$
The streamlines, $r \partial u_p / \partial \varphi =$ const. (first row) and the
isotherms, $\Theta =$ const. (second row), are shown in the equatorial plane for
equidistant times (from left to right) with $\Delta t = 0.03$.}
%\includegraphics[height=9cm]{fig-07new.pdf}
%\caption{Localized convection for $P = 1$, $\tau= 10^4$. In this case,
%convection is driven by imposed temperature difference $\Delta T$ between inner
%and outer boundary rather than by internally distributed heat
%sources. The Rayleigh number is defined by $R = \frac{\alpha \gamma
%  \Delta T d^4}{\nu \kappa} $ and has the value $R = 1.4 \times  10^6$. 
%The streamlines, $r \partial u_p / \partial \varphi =$ const. (first row) and the
%isotherms, $\Theta =$ const. (second row), are shown in the
%equatorial plane for equidistant times (from left to right) with
%$\delta t = 0.02$.}
\end{center} 
\end{figure}
\begin{figure} 
\begin{center}
\includegraphics[height=16cm,angle=-90]{fig-08new.pdf} 
\caption{Relaxation oscillations of chaotic convection  in the case
  $\tau=10^4, R=6.5\cdot 10^5, P=0.5$. The energy densities $E_t^m$
  (solid line), $E_t^f$ (dotted line), $E_p^f$ (dashed line) and the
  Nusselt number (dot-dashed, right ordinate) are shown as function of time.} 
%\includegraphics[height=14cm,angle=-90]{fig-08.pdf} 
%\caption{Relaxation oscillations of chaotic convection  in the case
%$\tau=10^4, R=4\cdot 10^5, P=0.25$. The energy densities $E_t^m$
%(solid line), $E_t^f$ (dotted line), and the Nusselt number
%(long-dashed line, right ordinate) are shown as function of time.} 
\end{center} 
\end{figure}
\begin{figure} 
\begin{center}
%\includegraphics[width=18cm]{fig-09.pdf} 
%\caption{Sequence of plots starting at $t=0.12015$ and equidistant in
%  time $(\Delta t=0.016$) for 
%  the same case as in Fig.~\ref{f14}.   Lines of constant
%  $\bar{u_{\varphi}}$ and mean temperature perturbation,
%  $\bar{\Theta}=$ const. in the meridional plane, are shown in  the left 
%  and right halves, respectively, of the first row. The second 
%  row shows streamlines, $r \partial u_p / \partial \varphi =$
%  const., in the equatorial plane.}
\includegraphics[width=18cm]{fig-09new.pdf} 
\caption{Sequence of plots starting at $t=2.31143$ and equidistant in
  time $(\Delta t=0.01$) for 
  the same case as in Fig.~\ref{f14}.   Lines of constant
  $\bar{u_{\varphi}}$ and mean temperature perturbation,
  $\bar{\Theta}=$ const. in the meridional plane, are shown in  the left 
  and right halves, respectively, of the first row. The second 
  row shows streamlines, $r \partial u_p / \partial \varphi =$
  const., in the equatorial plane.}
\end{center} 
\end{figure}
\begin{figure} 
\begin{center}
\includegraphics[height=14cm,angle=-90]{fig-10new.pdf} 
\caption{Transition from a dynamo state to a state of chaotic
  relaxation oscillations for $\tau=3\cdot10^4, R=29\cdot 10^5,
  Pm=0.4$. The energy densities $\overline{E}_t$ (thin solid line),
  $\check{E}_t$ (dotted line), $\check{E}_p$ (dashed line), the total
  magnetic energy density multiplied by a factor 8 (thick solid line)
  and the Nusselt number  $Nu$ (dash-dotted line) are shown as function of
  time.}  
\end{center} 
\end{figure}

The convective heat transport in the case of localized convection as
well as in the case of the relaxation oscillations is, of course, much
reduced relative to a case without strong differential rotation. Here
the magnetic field enters in an important way. By putting the brakes
on the differential rotation through its Lorentz force the magnetic
field permits a much higher heat transport than would be possible in
an electrically-insulating fluid. This is the basic reason that
rapidly rotating stars and planets with convecting cores exhibit
magnetic fields. A demonstration of this effect is seen in figure 10
where by chance the convection driven dynamo was just marginal such
that it could not recover after a downward fluctuation of the magnetic
field. Hence the relaxation oscillations took over from the dynamo
state with their much reduced average heat flux.  
\begin{figure} 
\begin{center}
\includegraphics[height=14cm,angle=-90]{fig-11.pdf} 
\caption[]
{ Time series of two different chaotic attractors are
  shown - a MD (left column (a,b)) and a
FD dynamo (right column (c,d)) both in the case $R=3.5\times10^6$,
$\tau=3\times10^4$,
  $P=0.75$ and  $P_m=0.75$.
The top two panels (a,c) show magnetic energy densities. and the
bottom two panels (b,d) show kinetic energy densities in the presence
The component $\overline{X}_p$ is shown by thick solid black line,
while
$\overline{X}_t$, $\widetilde{X}_p$, and $\widetilde{X}_t$ are indicated
by squares, triangles and circles respectively, and
$X$ stands for either $M$ or $E$.
}
\end{center} 
\end{figure}
\begin{figure} 
\begin{center}
\includegraphics[height=18cm,angle=-90]{fig-12.pdf} 
\caption{ The upper row shows the hysteresis effect in
  the ratio  $E/M$ at $\tau=3\times10^4$
  (a) as a function of the Prandtl number in the case
  of $R=3.5\times10^6$, $P/P_m=0.5$; (b) as a function of
  the ratio  $P/P_m$ in the case of $R=3.5\times10^6$,
  $P=0.75$ and (c)  as a function of the Rayleigh number in the case
  $P=0.75$, $P_m=1.5$. Full full and empty
  circles indicate FD and   MD dynamos, respectively.
  The critical value of $R$ for the onset of
  thermal convection for the cases shown in (c) is
  $R_c=659145$. A
  transition from FD to MD dynamos as $P/P_m$ is decreased in
  (b) is expected, but is not indicated owing to lack of data.
  The lower row shows the value of the Nusselt number at
  $r=r_i$ for the same dynamo cases. Values for non-magnetic
  convection are
  indicated by squares for comparison. }
\end{center} 
\end{figure}



\section{Two distinct turbulent dynamos at identical parameter values}

Convection driven dynamos in rotating spherical fluid shells are often
subcritical as is already indicated in figure 10 where at slightly
higher Rayleigh number convection with a strong magnetic field
persists, while the dynamo will decay when the magnetic field is
artificially reduced to, say, a quarter of its averaged energy. There
thus exists the possibility of the existence of a convection driven
dynamo state and of a non-magnetic convection state at identical
values of the external parameters $R, \tau, P, Pm$ where the latter
parameter denotes the ratio between kinematic viscosity and magnetic
diffusivity, $Pm =\nu/\lambda$.  The magnetic diffusivity itself is
defined as the inverse of the electrical conductivity times the
magnetic permeability, $\lambda=\sigma^{-1}\mu^{-1}$.  

More surprising is the fact that two different turbulent dynamo states
can exist at identical values of the external parameters as has been
shown in \cite{SB} and is  demonstrated by the two examples shown in
figure 11. Here the representation of the magnetic flux density $\vec
B$ in terms of poloidal and toroidal components, 
\begin{equation}
%\begin{gather}
\vec B = \nabla \times ( \nabla h \times \vec r) + \nabla g \times\vec r \enspace, 
%\end{gather}
\end{equation}
has been used and the magnetic energy densities have been defined in
analogy to expressions (3), 
\begin{subequations}
\begin{align}
\label{7}
M_p^m = \frac{1}{2} \langle \mid \nabla \times ( \nabla \bar h \times \vec r 
) \mid^2 \rangle , \quad & M_t^m = \frac{1}{2} \langle \mid \nabla \bar g_t \times
\vec r \mid^2 \rangle \\
\label{8}
M_p^f = \frac{1}{2} \langle \mid \nabla \times ( \nabla \check h_p \times \vec r) 
\mid^2 \rangle , \quad&  M_t^f = \frac{1}{2} \langle \mid \nabla \check g_t \times
\vec r \mid^2 \rangle.
\end{align}
\end{subequations}
The two turbulent dynamo states differ strongly in their magnetic
energies and their kinetic energies as is apparent from figure
11. While a strong mean poloidal magnetic field acts as an efficient
brake on the differential rotation measured by $E_t^m$, it also
inhibits convection. The alternative dynamo is characterized by a
relatively weak mean magnetic field and dominant fluctuating
components. Here the kinetic energy densities of convection are
larger, but the differential rotation is still much weaker than in the
non-magnetic case. As a result both types of dynamos provide nearly
the same convective heat transport as measured by the Nusselt number
$Nu$.  
This evident from figure 12 where it is demonstrated that the two
separate types of dynamos exist over an extended region of the
parameter space.   



\section{Concluding Remarks}

The existence of two distinct turbulent states is a rare phenomenon,
although examples exist in hydrodynamics, see, for instance, \cite{Ra}
\cite{Mu} . In magnetohydrodynamics the magnetic field offers new
degrees of freedom which allow the co-existence of more than a single
turbulent state. Initial conditions determine which of the competing
states is actually realized.   

The possibility of bistability could be of interest for the
interpretation of planetary and stellar magnetism. At least this
possibility should remind colleagues involved in numerical simulations
of convection driven dynamos that solutions quite different from those
they have obtained may exist for the same set of parameters.  

% If you have appendix consisting of several sections, uncomment 
% the following command. Use \appendix* if there is only one section 
% in the appendix
%\appendix
%\section{}

\begin{acknowledgments}
The help of Dr. R. Simitev in creating the figures is gratefully acknowledged.
\end{acknowledgments}

% Uncomment the following line if you use BibTeX to produce 
% the reference section
% \bibliography{EE250sample}

\begin{thebibliography}{1}
\bibitem{DS}
E. Dormy, A.M. Soward, eds., Mathematical Aspects of Natural Dynamos, CRC Press, Taylor \& Francis Group,  2007
\bibitem{76}
F.~H. Busse, C.~R. Carrigan, Laboratory simulation of thermal
convection in rotating planets and stars, {\it SCIENCE}, {\bf 191},
81-83 (1976)
\bibitem{70a}
F.~H. Busse, Thermal instabilities in rapidly rotating systems {\it
  J. Fluid Mech.}, 
{\bf44}, 441-460 (1970)
\bibitem{86}
F.~H. Busse, Asymptotic theory of convection in a rotating, cylindrical annulus. {\it
  J. Fluid Mech.}, 
{\bf173}, 545-556 (1986)
\bibitem{Bu}
F.~H. Busse, Convective flows in rapidly rotating spheres and their dynamo action, {\it
  Phys. Fluids}, 
{\bf14}, 1301-1314 (2002)
\bibitem{SB}
R. D. Simitev and F.~H. Busse, Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells, {\it
Euro Phys. Lett.}, in press 
{\bf14}, 1301-1314 (2002)
\bibitem{Ra}
 F. Ravelet, L. Mari\'e, A. Chiffaudel and F. Daviaud, Multistability and Memory Effect in a Highly Turbulent Flow: Experimental Evidence for a Global bifurcation, {\it Phys.~Rev.~Lett.} {\bf93},
 164501 (2004).
\bibitem{Mu}
N. Mujica and D. P. Lathrop, Bistability and hystreresis in a highly turbulent swirling flow, {\it Physica A}{\bf49}, 162-166 (2005)
 
\end{thebibliography}

\end{document}
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