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\begin{document}
\Large


\centerline{\bf HIGH ENERGY ASTROPHYSICS} 


%\medskip 
%Understanding Conflicting Results in Mean Field Dynamo Suppression Studies
\bigskip


\medskip
\bigskip

\centerline{--------------------------------------------------------------}

\bigskip

\ni {\bf  High energy processes, ionized plasmas, ionizing photons,
x-rays, gamma-rays, compact objects, strong magnetic fields,
strong gravity, early universe cosmology}

\bigskip

\ni {\bf \ \ $\bullet$  They are observed astrophysical entities:}

{Where do they come from?}  

\bigskip
\bigskip

\ni {\bf \ \ $\bullet$  Intermediary between gravity
and radiation:}  
\bigskip

{\bf Sites}: Stars, stellar coronae, winds, Galactic 
ISM, AGN, 

Clusters, Gamma-
Ray Bursts, microquasars, accretion 
disks...

{\bf Processes:} jets, particle acceleration/reconnection, 

angular momentum
transport

\bigskip
\bigskip




\ni {\bf \ \ $\bullet$  Field amplification:} problem of
MHD/plasma  turbulence
\bigskip

\ni {\bf \ \ $\bullet$  Note: Plasma vs. MHD field amplification} 

\ni {\bf \ \ $\bullet$  Note: Seed field generation vs. field amplification}


\bigskip
\bigskip






\vfill
\eject



\bigskip


\ni {\bf THE GALAXY AS AN EXAMPLE MAGNET}


\bigskip
\bigskip
\bigskip

\ni {\bf Synchrotron polarization and Faraday rotation reveal:} 

\bigskip
\bigskip

\ni {\bf Small Scale Field:}  

\bigskip

\ni { \ \ $\bullet$ random,  $5\mu$G;  dominating scale $\sim$ few 50-100pc}


\bigskip

\ni {\ \ \ $\bullet$ equipartition with random kinetic motions}



\bigskip
\bigskip
\bigskip

\ni {\bf Large Scale Field:}

\bigskip


\ni  { \ \ $\bullet$ toroidal outside 200pc}

\bigskip

\ni  { \ \ $\bullet$ poloidal inside 200pc;  $10^{-3}$G }

\bigskip

\ni{ \ \  $\bullet$ few $\mu$G;  scale  $\gsim$ few kpc, reversals }

\bigskip

\ni{ \ \  $\bullet$ no field reversal of planar component
across disk midplane  }

\bigskip

\ni { \ \  $\bullet$ {Cosmological, Protogalactic, or In Situ origins?}

\bigskip



\vfill 
\eject


\centerline{\bf MHD, IN SITU AMPLIFICATION: SMALL SCALE FIELDS}

\bigskip

\bigskip







\ni {\bf ``Turbulent Amplification'' $\equiv$ ``Small Scale Dynamo''}


\medskip
\ni{ \bf \ \ $\bullet$  MHD turbulence}:

1. Randomly forced fluid motions produce kinetic 
energy 

spectrum  from non-linear interactions.  

2. Kinetic energy develops $E_k ^{ -5/3}$ Kolmogorov direct cascade.

3. Flux freezing means seed magnetic field responds $\rightarrow$ MHD 

turbulence, with spectra for magnetic and kinetic energies.


\bigskip




\bigskip
\ni{\ \ \ $\bullet$ B grows on and below scale of input forcing.
Fastest growth at 

small scales, but saturation value largest at input scale.}

%\ni{ \ \ $\bullet$ ``SUB-INPUT SCALE'' Inverse Cascade

\bigskip
\ni{\ \ \ $\bullet$ B-energy near equipartition with v-energy}


\bigskip

\ni{ \ \ $\bullet$ In Galaxy: 
supernova driven forcing, input scale 50-100pc} 
\bigskip

\ni{ \ \ $\bullet$  growth rate = (turnover time for largest eddies)$^{-1}$
$\sim 10^{-7}$ yr 

for Galaxy} 

\bigskip

\ni{ \ \ $\bullet$  Think ``random walk field line stretching''}
\bigskip




\ni{ \ \ $\bullet$ ``Not too controversial'' but:
  1. mag. ``Prandtl number'' important and 2.  Hawley, Gammie, Balbus 1996 
puzzle.






\bigskip



%\ni {\bf $\bullet$ Growth: shear $\rightarrow$ linear; turbulence $\rightarrow%$ exponential}
%\medskip




\vfill 
\eject

\centerline{\bf MHD, IN SITU AMPLIFICATION: LARGE SCALE FIELDS}
\bigskip

\bigskip

\ni 1. {\bf  Simple Shear}=``$\Omega$'' effect$\rightarrow$ linear growth  


\bigskip

\ni 2.  {\bf  Mean Field Dynamo (MFD)} $\rightarrow$ exponential growth


\bigskip

\ni{ \  \ $\bullet$ 
$\bf B$ grows on scales $>$ forcing scales}

\bigskip


\ni{\ \ \ $\bullet$  ``$\alpha$-helicity'' required for growth: forcing 
with 
finite $\lb{\bf v}\cdot\curl {\bf v}\rb$


\bigskip

\ni{ \ \ \ $\bullet$  involves turbulence + rotation + density gradient} 

\bigskip

\ni{ \ \ \ $\bullet$  also 
requires ``turbulent diffusion coefficient $\beta$'' 
to explain obs.}

\bigskip

\ni{ \ \ \ $\bullet$  can account for fast changes in large scale {\bf B}}

\bigskip

\ni{ \ \ \ $\bullet$  growth rate longer than for ``small scale dynamo''} 

\bigskip

\ni{\ \ \ \ $\bullet$  spatial mean $\sim$ ensemble mean for 2 scaled systems} 
\bigskip


\ni{ \ \ \ $\bullet$  controversial: are $\alpha$ and $\beta$ 
prematurely quenched? }

\bigskip
\bigskip

\ni{\bf MFD can also be viewed as a simple framework for understanding the inverse cascade of magnetic helicity in turbulent flows}.

\bigskip

%{\bf A Comment on accretion disks and coronae}




\vfill 
\eject 

{\bf SPECTRAL VIEW OF IN SITU AMPLIFICATION}

\bigskip


{$\bullet$ Ultimately 
want to understand the shape of this curve 
}

{$\bullet$ Note the distinction between regions $k < k_{input}$ and
$k > k_{input}$.
}








\vfill
\eject
{\bf DIRECT VS INVERSE CASCADE}

\bigskip

\ni{ \ \ \ $\bullet$}
Two important conservation laws for MHD turbulence in systems

with infinite or periodic boundaries:
\bigskip

1.  Energy conservation.

2. ``Magnetic helicity'' conservation.
\bigskip

The latter measures the net ``twist'' or net ``linkage'' of

magnetic field lines, a ``topological invariant'' for a highly

conducting  plasma.




\bigskip
\bigskip
\bigskip



\ni{ \ \  $\bullet$} In Steady state 3-D, these can be shown to imply that:

1. Kinetic energy cascades primarily DOWN 

(to larger $k$): {\bf direct cascade}
\medskip

2. Magnetic energy goes down {\bf direct cascade}, 
but before saturation is reached one sees
{\bf sub-input inverse cascade}.

3. Magnetic helicity cascades UP (to smaller $k$) than input: 
{\bf super-input inverse  cascade}, and it is non-local.











\vfill
\eject




\centerline{\bf SOLAR DYNAMO}
\bigskip
\bigskip

\vfill
\eject


\centerline{\bf MFD BASIC EQUATIONS AND BACKREATION}
\bigskip
\bigskip

%$$
%{\bfB}=\bbB+\bfb; \ {\bbV}=\bbV+\bfv; \ 
%$$

%$$
%c\bbE \simeq 
%-\bbV\ts\bbB-\lb\bfv\ts\bfb\rb
%$$

Write ${\bf V}= \bfv +\bbV$  and ${\bf B}= \bfb +\bbB$, then 

$$
\partial_t{\bbB}=
\curl(\alpha^{(0)}\bbB)
-\curl (\beta^{(0)}\curl\bbB)
%\curl\lb\bfv\ts\bfb\rb+\curl(\bbV\ts\bbB)
%\curl(\alpha \curl\bbB)-\curl(\beta\curl\bbB)
+\curl(\bbV \ts \bbB)
$$


\bigskip

$\alpha^{(0)}
=-\tau_c [\lb \bfv
\cdot\curl\bfv\rb^{(0)}
-\lb\bfb\cdot\curl\bfb\rb^{(0)}]
/3$


$
\beta^{(0)}=\tau_c[\lb\bfv\cdot\bfv\rb^{(0)}
+\lb\bfb\cdot\bfb\rb^{(0)}]/3
$

\bigskip


\ni $\bullet$  First order theory (the $^{(0)}$) implies
ignoring  nonlinear backreaction but $|\bfb|^2>> |\bbB|^2$ very quickly.

  
\ni $\bullet$ ``Backreaction''  for even a weak $\bB$ and large $b$
could shut down diffusive motions required for field growth.



\ni $\bullet$  
Does backreaction to non-linear order in $\bbB$ 
quench dynamo coefficients catastrophically?


\ni $\bullet$  
Revival of this concern in the 90s, but concern
has been around since the 50s.(Cowling, Batchelor, Schluter, Biermann..).


\ni $\bullet$  
Numerical simulations help in the revivial, but can be confusing
too...










\vfill
\eject


%\centerline{\bf BACK-REACTION PROBLEM}
%\bigskip
%\bigskip

\vfill
\eject


\ni {\bf PREVIOUS ARGUMENTS FOR SUPPRESSION}

\ni $\bullet$ { Since} $|\alpha| \le |k\beta|$ {$\beta$ suppression
$\rightarrow \alpha$ suppression}.


\bigskip




\ni $\bullet$ {Equipartition suppression, still gives ``fast dynamo''}
$$\alpha,\beta \simeq {\alpha_{kin},\beta_{kin}
\over  (1 + {\bbB^2/\bfb^2})}$$
%\ \  
%or \ \ {\alpha_{kin},\beta_{kin}\over  (1 + {\bfb^2/\bfv^2})}$$


\ni $\bullet$ 
{Catastrophic suppression ``slow dynamo'': would kill MFD in astrophysics} 
$$\alpha,\beta \simeq {\alpha_{kin},\beta_{kin}\over  (1 + R_M{\bbB^2/\bfb^2})}$$



\ni {from interpreting 2-D Zeldovich (1957)} 
%limit 
%$b^2 \le R_M {\bbB}^2$, 
(fixed $\bbB$, no bdry terms)




\ni \ \ \ \ \ $\bullet$ 
%$\lb A^2 \rb$ diffuses as
diffusion occurs with time $\tau_D \sim  \delta^2(t)/ 2\eta$.

\ni \ \ \ \ \ $\bullet$  $\delta(t)$ marks scale of maximum magnetic energy.

\ni \ \ \ \ \ $\bullet$ Kinematic regime: $d \delta /dt < 0$, $\tau_D$ 
decreases as $b$ energy grows.  

\ni \ \ \ \ \ $\bullet$ Kinematic regime ends when 
$b^2  = R_M\bbB^2$, here $\tau_D$ is shortest.

\ni \ \ \ \ $\bullet$ ``Backreaction'' regime begins, then $d \delta/dt > 0$, 
and $\tau_D$ increases. 





%\ni {\bf Interpretation:} field cascade inhibited for 
%$\bbB^2\ge {\bfb}^2/R_M$, locking motions into oscillations rather than diffus%ion. 

\bigskip

\ni {\bf BUT}:

\ni 3-D $\ne$ 2-D:  

{\bf 1}. Extra terms in 3-D that act as effective diffusion and do not

depend on resisitivity.

{\bf 2}. Interchange motions in 3-D, and intermittency. 




%\ni {\bf 1}. Z57 is 2-D limit on $\bfb^2$.   


%\ni {\bf 2}. Thus Z-relation irrelevant for 3-D. Irrelevant for 2-D when 
%$\bfv^2 << R_M\bbB^2$, or when bdry terms included.


%\ni {\bf 3}. In 3-D should just mean the end of the kinematic
%limit, not clear implications in nonlinear 2-D let alone 3-D regime. 


%\ni {\bf 4}. In 3-D simulations,  equipartition reached 
%on small scales, but then saturates building up to progressively
%larger scales, never quite reaching exact equipartition on the 
%largest scale.  


%\ni $\bullet$ {\bf Let's consider existing numerical simulations}












\vfill
\eject

\centerline{\bf SOME 3-D NUMERICAL SUPPRESSION STUDIES}
\bigskip


\bigskip


\ni $\bullet$ 3-D simulations of molecular clouds (Stone \& Ostriker)
may suggest diffusion is not inhibited
in contrast to 2-D of Cattaneo (1994).


\ni {\bf However}:


\ni $\bullet$ 3-D simulations of $\alpha$ 
in periodic box 
with uniform field show catastrophic $\alpha$ suppression
(Cattaneo \& Hughes 1996)





\bigskip

\ni $\bullet$ recent 3-D periodic box simulations (Brandenburg 2000)
shows initially ``fast'' dynamo action but then ``slow''

\bigskip
\bigskip




\ni {\bf Do these prove that  real dynamos are suppressed?}:  
  

\ni {\bf Claim: No---boundary conditions} (Blackman \& Field 2000) 





\bigskip
\bigskip
\bigskip

\ni (Note symbiosis between simulations and analytic studies.)

{${Lim}_{computers \rightarrow perfect}$ Simulation $\simeq$ Experiement $\ne$ Theory}
















 












\vfill
\eject










\ni {\bf IMPORTANT ROLE OF BOUNDARY CONDITIONS}

\medskip


\ni {\bf 1. For Quenching Studies}

\ni{ $\bullet$ Real astrophysical dynamo vs. quenching studies}

\bigskip

%\ni {\bf $\bullet$ MFD equation:}  $\bf B= b+{\bbB}$; \  ${\bf V=v}$

\ni{ $\bullet$ Can actually constrain dynamo coefficient $\alpha$
for 3-D using the magnetic helicity conservation law.

The result is (Blackman \& Field 2000)
%=-\eta\lb\bfj\cdot\bfb\rb-\lb{\bf e}\cdot\bfb\rb$$
% $$\lb{\bf E}\cdot{\bf B}\rb-\eta{\bar {\bf J}}\cdot
%{\bar {\bf B}}=

%$$\lb\bfv\ts\bfb\rb\cdot\bbB 
%=-\eta\lb\bfj\cdot\bfb\rb/{ - \lb\bfe\cdot\bfb\rb$$

$$\alpha \lsim \ {\alpha^{(0)}  \over  R_M  {(\bbB/\bfb})^{2}}
-{\partial_t\lb {\bf a}\cdot {\bf b}\rb\over 2c \bbB^2} + 
\nabla \cdot \lb{\bf Q} \rb
,$$



\bigskip

\ni {$\bullet$} for stationary turbulence in periodic averaging
region, last terms on right vanish!



\ni {$\bullet$ $\rightarrow$ \bf ``quenching'' from boundary conditions}


\ni {$\bullet$ this is not the dynamical quenching people were worried
about but an artifact of ``inappropriate'' boundary conditions}

\ni {$\bullet$ it can largely explain the results of the simulations which
currently show quenching.} (and some of the associated fist-fighting).



\ni {\bf 2. For Flux Generation}





{$\rightarrow$ \bf B.C. and BOUNADRY DIFFUSION IMPORTANT}

\vfill
\eject



\ni {\bf ROLE OF MAGNETIC HELICITY }

\medskip

\ni{ $\bullet$ B-energy growth at $k < k_{input}$ related
to ``inverse cascade'' of B-helicity $(H\equiv \int{\bf A\cdot B}d^3x)$}. 
(Frisch et al 75; Pouquet et al 76)
\medskip

\ni{$\bullet$ $E_k \gsim k H_k$}
\medskip

\ni{$\bullet$ If large scale B-helicity grows,  
small scale B-helicity must dissipate or leave system .

\bigskip

\ni{\bf Conservation of magnetic helicity}

$$
0 \simeq \partial_t H_{U}=
-2c\int_{U} {\bf E}\cdot{\bf B}d^3x +\int_{U} {\bf Q}\cdot d{\bf S}
\label{r4aab1}
$$

\ni {For periodic, or infinite boundaries {surface terms} $\rightarrow 0$}.

%$$
%=-2c\eta \int_{U=U_{in}} 
%{\bf J}\cdot{\bf B}d^3{\bf x},
%$$



\bigskip

\ni {\bf For real systems:}   
$$
0\simeq \partial_t H_{U}
= \partial_t{H}_{R,U_{in}}+\partial_t{H}_{R,U_{out}}
%=-2c\int_{U>U_{in}} {\bf E}\cdot{\bf B}d^3x
\label{r4aab1}
$$

%$$   \partial_t \int{\bbA}\cdot \bbB d^3x=
%-2c \int_{SYS} {\overline {\bf E}}\cdot {\bbB}d^3x$$
%\smallskip
%$$
$$
\partial_t {{H}}_{R,U_{in}}=
-2c\int_{U_{in}} {\bf E}\cdot{\bf B} d^3x
+2c\int_{S_{in}}
%({\bf A}_p\ts 
%{\bf E})
{\bf Q}\cdot d{\bf S}
\label{r4aad}
$$

%$$
%\partial_t{{H}}_{R,in}(\bfb)=
%-2c\int_{in}{\bfe\cdot\bfb} d^3x+2c\int_{S_{in}}({\bf a}_p\ts 
%{\bfe})\cdot d{\bf S}.
%\label{r4aae}
%$$
Surface term DOES NOT vanish.

${\bf \bullet}$ 
If  $H_{R,U_{in}}$ is steady, then $\alpha$ effect depends on magnitude 

of surface terms.

${\bf \bullet}$ Surface terms can be large. $\rightarrow$ implications for simulations.

${\bf \bullet}$ Note also Ji 1999ab



\vfill
\eject

{\bf SCHEMATIC OF RELEVANT SCALES}

\vfill
\eject

{\bf IMPLICATIONS FOR CORONAE}

\bigskip
\bigskip

\ni {$\bullet$ Boundary flow can allow steady-state field growth}

\bigskip

\ni{$\bullet$ $E_k \gsim k H_k$}: flow of helicity to corona $\rightarrow$ 
energy flow to surface}

\bigskip

\ni {$\bullet$  Can estimate energy flow to surface for steady dynamo}

$$dE/dt \gsim \alpha^2 {\bbB}^2 R^2$$ 

\bigskip

\ni (Steady means on time scales longer than eddy turnover time)

\bigskip

\ni {$\bullet$  Exponential field growth
inside a rotator should imply less coronal variability than linear growth}

\bigskip

\bigskip

{\bf SPECIFIC ASTROPHYSICAL SETTINGS}

\bigskip

\ni {$\bullet$  Predicts rate of energy supply into
Galactic corona}   $\ge 10^{40}$erg/s.

\bigskip


\ni {$\bullet$  Consistent with helicity supply to Solar Corona,
$\ge 10^{27}\rm erg/s$.


\bigskip

\ni {$\bullet$  Consistent with inferred X-ray 
emission from AGN accretion disks $(\ge 10^{43}{\rm erg/s})$, 


\bigskip




\bigskip




\vfill
\eject

{\bf PICTURES OF SUN}



\vfill
\eject
\centerline{\bf ``EXPLAINING'' QUENCHING RESULTS}

\bigskip
\ni {\bf Numerical}
\bigskip


\ni{$\bullet$ Periodic b.c. leave ambiguity of dynamical
vs. boundary induced quenching in existing 3-D simulations.}

%\ni{$\bullet$ More quenching studies 
%with non-periodic boundary conditions and scale separation recommended.}

\ni{$\bullet$ Everyone is  ``correct'' for problems solved, but 
implications for real systems need to be considered 
carefully.}

\ni{$\bullet$  Currently there are no 
dynamo simulations which invoke astrophysically realistic boundary 
physics.}


\bigskip
\ni {\bf  Analytical }
\bigskip

\ni{ $\bullet$ 
Analytic studies sugg. strong quenching have either:}

\ni \ \ \ { (1) been 2-D}   


\ni \ \ \ { (2) ignored  boundary terms}

\ni \ \ \ { (3) assumed that kinematic regime violation = strong quenching

\ni \ \ \ { (4) or not distinguished ``isotropic, homogeneous'' 
part}

\ni { \ \ \ \ \ \ \ of turbulence from ``anisotropic'' part:}

\vfill
\eject

{\bf SOME OPEN ISSUES FOR MHD DYNAMOS}

\bigskip
\bigskip


\ni $\bullet$ {Role of boundary diffusion, buoyancy winds for real systems;
are open boundaries enough?} 
(e.g. Rafikov \& Kulsrud 2000)



\ni $\bullet$ {Effect of magnetic Prandtl number (mag. diffusivity/viscosity)} 


\ni $\bullet$ {relation between Balbus-Hawley Instabiltiy (MRI) and dynamos} 

\bigskip


\ni $\bullet$ {Simulation approaches}:  
Can a periodic box be used?


\bigskip

{\bf BROADER QUESTIONS}

\bigskip

\ni $\bullet$ When are dynamos needed? e.g. can fields be otherwise 
accreted rather than generated in situ?  UNDERSTANDING TURBULENT
DIFFUSION/BOUNDARIES ARE KEY

* dynamo is required for sun clearly the field was not accreted.


* in Galaxies, more obervations needed-- but even if primordial

\ \ or 
protogalactic field, dynamo still likely required 

\ \ IF TURBULENT DIFFUSION OPERATES.


\ni $\bullet$
specific applications: e.g. planetary nebulae (Blackman et al. 2000ab),
jets in AGN, young NS, general relativistic flows


\ni $\bullet$ Other relevant field amplification besides
MHD Dynamos? 

* seed fields

* gamma-ray bursts (Weibel Instability)



\vfill
\eject

\ni {\bf Pseudo-scalar Term in Analytic Disk Theory $\rightarrow$ 
Vorticity Growth ?}


\bigskip

%{\bf Vorticity Growth}

\bigskip
\bigskip
\ni $\bullet$ {Shakura-Sunyaev viscosity: 
simply the turbulent diffusion coefficient for the mean velocity.
It is a correlation of fluctuating quantities in  mean field formalism.}

\ni $\bullet$ {But in this approach, there also the analgous 
pseudo-scalar to the dynamo $\alpha$.}

\ni $\bullet$ { Mean vorticity equation looks like this:}

$\partial_t{\overline {\bf \omega}}=\curl \gamma {\overline {\bf \omega}}
-\curl\beta\curl{\overline {\bf \omega}}+{\overline {\bf \omega}}\cdot\nabla
{\overline {\bf v}}-{\overline {\bf v}}\cdot\nabla{\overline {\bf \omega}}$

\ni$\bullet$ { $\beta$ is usual viscosity, but $\gamma \propto \lb{\bf v}'\cdot \curl {\bf v}'\rb^{(0)}$ is a pseudo-scalar helicity. 
(see also Frisch et al 1987).

\ni$\bullet$ { Important: ${\gamma}$ 
term appears when one vertically averages over 1/2 of disk thickness, not full thickness}

\ni$\bullet$ { Analagous to mean magnetic field theory. Also, 
both equations are actually coupled.}

\ni$\bullet$ { Consequences of this term not fully explored}

\ni$\bullet$ { Can lead to vorticity growth.
Simplest treatment $\rightarrow$ scale and  survival time
ok for planet formation...}

\vfill
\eject

{\bf Relevance to Planet Formation(?)}

\bigskip

\ni$\bullet$ {Planets may form
from settling dust grains in primordial
stellar nebulae, Edgeworth 1949, Safranov 1969, Goldreich \& Ward 1973;
But turbulence is a problem.}

\ni$\bullet$ { Cm-size and smaller probably from sticking, 
above m-size from grav. instab.}

\ni$\bullet$ { But what about cm to m-sized growth}?

\ni$\bullet$ { Barge \& Sommeria 95; Tanga et al 96;
Godon \& Livio 1999,2000; Chavanis 2000)
dust particles settle in cores of (anti-cyc) vortices}

\ni$\bullet$ { B\&S: trap time to form planetesimal of $10 ^{-5}
M_{E}$ is about
10-100 orbits (100-1000yr) for vortex of size $D=H/10$ at 5 AU.

\ni$\bullet$ 
Tanga et al. found 10-15$M_e$ could form in vortex of order a 200-500 hundred
orbit times for $D\sim H$.

\ni$\bullet$ {{\bf Formation  of long lived vortices} in these disks 
is less understood.}

\ni$\bullet$ { Simplest growth found from prev. discussion  
shows dominant gr. scale $\sim H$. Survives $\gsim$ few $\ts\ 1/\alpha_{ss}$} orbits 

\ni$\bullet$ { Will  saturate such that  
${\overline \omega}H\sim  v_{t}$ $\rightarrow\ {\overline \omega}\sim |0.1\Omega_k|$..

%\ni $\bullet$ simplest growth found from prev. discussion  
%shows dominant gr. scale $\sim H$ and
%survives minimum few $\ts\ 1/\alpha_{ss}$} orbits 

\ni$\bullet$ {Note alternative to core accretion by Boss 2000}


\ni$\bullet$ { Other consequences of vortices?}




\bigskip









\vfill
\eject






\centerline{\bf GAMMA-RAY BURST FIELD GENERATION PROBLEM}


\bigskip


\ni{$\bullet$}
Gamma-ray burst fireball model (e.g. M\'esz\'aros and Rees): $10^{51-54}$ erg
released in small volume without much baryon load.

\ni{$\bullet$}
Baryon accelerated to relativistic velocities $\rightarrow$
shocks, energy transfer to electrons (not Fermi acceleration though).

\ni{$\bullet$}
Inverse compton and synchrotron cooling in internal (GRB) 
and/or external (afterglow) shocks. 
Magnetic fields up to $100$G needed to explain observations of GRB/afterglow.

\ni{$\bullet$} Not trivial to  
amplify the field 8 orders of magnitude on $t_{dyn}$.


\ni{$\bullet$} 
Must occur below gyro-radius scale $\rightarrow$ relativistic plasma process:
small scale turbulence, or beam instabilities (e.g. Weibel
instability e.g. Medvedev \& Loeb 1999).



\ni{$\bullet$} 
{\bf Possible Alternative:}  

* Poynting flux dominated rotator
(e.g. Usov 1994, Smolsky \& 

\ \ Usov 1996;
Blackman \& Yi 1996, M\'esz\'aros \& Rees 1997;  

\ \ Lyutikov \& Blackman 2001).


* $B\sim 10^{15-16}(r_o/r)$ outside lightcone. Can supply 
required field.  


* Large amplitude e-m waves accelerate  particles (``internal'').

* ``External'' afterglow: interplay between 
 magnetized piston  and 

\ \ ambient medium.






\vfill
\eject





\end{document}










{\bf In situ $B-$Field Growth in Turbulent Disks}
\bigskip

\ni $\bullet${Turbulent and shear are sources of  B-energy}

\ni $\bullet$
{Two types of concurrent growth:}


\ni {\bf (1)  ``Turbulent amplification/input scale dynamo''

{$\bullet$} {\bf Exponential B-growth on scale as input turbulence.}

$\bullet$ {\bf Random walk/field line stretching. Growth time $t_{ed}$}

$\bullet$ {\bf Simulations all show 
(near) equipartition $\lb v_A^2\rb \sim \lb v_t^2\rb$.


$\bullet$ {\bf Not controversial}

\ni {\bf (2)  ``Mean field dynamo/Inverse Cascade Field Growth''}

$\bullet$ {\bf Field grows on scale $>$ turbulent input scale}.


 $\bullet$ {\bf Combines turbulence, rotation, \& density gradient}

 $\bullet$ {\bf Needs ``pseudo-scalar helicity,'' and  `` turb. diffus.''}

 $\bullet$ {\bf Controversial (backreaction), but bdry. conds key.}

 $\bullet$ {\bf Jets: dynamo vs. flux dragging? BH vs. NS, GRB}


\ni $\bullet$ {\bf Field growth $\rightarrow$ 
buoyant field supply to corona, spatial intermittency important,
(Miller \& Stone 98)}

\ni $\bullet$ {\bf Steady exponential field growth in disk
${\rightarrow}$ predominantly steady corona} 

\ni $\bullet$ {\bf Ironically, B-H inst. limits max. field growth value}





\vfill 
\eject









\bigskip
\bigskip

\ni $\bullet$ {\bf In turb disk, each quantity has mean and fluct. part, e.g. ${\bf B} ={\overline {\bf B}}+{\bf b}'$


\ni $\bullet$ {\bf Mean field dynamo results from mean induction eqn}



\ni $\bullet$ {\bf Same really true for usual acc. disk equations. 
Must formally average equations: 
usual momentum etc. equations are for mean quantities.}

\ni $\bullet$ {\bf Turbulent Shakura-Sunyaev viscosity 
$\ne$ mere replacement of mol. viscosity. Its a correlation of fluctuations:}

e.g. $\nu ={\lb {\bf v}'(t)\cdot \int {\bf v'}(t')dt'\rb}^{(0)}\simeq v_t^2t_{ed}
= \alpha_{ss}c_s H$


\ni $\bullet${ ``Intrinsic'' 
variability expected since mean quantities, smooth over turbulence}

\ni $\bullet$
${\Delta L_{\nu}/L_{\nu}} \simeq |R\partial_R({\rm Ln}L_\nu)|
{{\De R}/ R}\sim \Psi 
l_{ed}/[N_{\phi}N_z(1 + t_{obs}/t_{ed})]^{1/2}$

\medskip
$\sim \Psi \alpha_{ss} (H/R)^{3/2}(1 + \Omega t_{obs})^{-1/2}$


\ni $\bullet$ {\bf For $H/R \sim 1$ and $\alpha_{ss}\sim 1$,
large variability expected unless $t_{obs} >> 1 /\Omega$.

\ni $\bullet$ {\bf Possible issue for thick disk models, e.g. ADAFs,
as applied to large ellipticals, where $t_{obs}\sim \Omega^{-1}$. (NGC 4649, NGC 4472, NGC 4636, M87..DiMatteo et al. 99)}


\vfill
\eject

\ni {\bf ``Missing'' Term in Analytic Theory and Vorticity Growth}


\bigskip

%{\bf Vorticity Growth}

\bigskip
\bigskip
\ni $\bullet$ {\bf Recall Shakura-Sunyaev viscosity, from
correlation of fluctuating quantities}

\ni $\bullet$ {\bf Not the only transport coefficient: also a pseudo-scalar(!)}

\ni $\bullet$ {\bf Mean vorticity equation looks like this:}

$\partial_t{\overline {\bf \omega}}=\curl \gamma {\overline {\bf \omega}}
-\curl\beta\curl{\overline {\bf \omega}}+{\overline {\bf \omega}}\cdot\nabla
{\overline {\bf v}}-{\overline {\bf v}}\cdot\nabla{\overline {\bf \omega}}$

\ni$\bullet$ {\bf $\beta$ is usual viscosity, but $\gamma \propto \lb{\bf v}'\cdot \curl {\bf v}'\rb^{(0)}$ is a pseudo-scalar helicity. Not usually
considered}.

\ni$\bullet$ {\bf Important: ${\gamma}$ 
term appears when one vertically averages over 1/2 of disk thickness, not full thickness}

\ni$\bullet$ {\bf Analagous to mean B-field theory. 
(Equations are actually coupled).}

\ni$\bullet$ {\bf Consequences of this term not fully explored}

\ni$\bullet$ {\bf Can lead to vorticity growth.
Simplest treatment $\rightarrow$ scale and  survival time
ok for planet formation...}

\vfill
\eject

{\bf Relevance to Planet Formation(?)}

\bigskip

\ni$\bullet$ {Planets form from accumulation of dust grains in primordial
stellar nebulae}

%\ni$\bullet$ { probably cm-size grains can form from sticking,
ice formation, charge attraction,  
%km-size from grav. instab.}

\ni$\bullet$ { What about cm to m-sized growth}?

\ni$\bullet$ { Barge \& Sommeria 95; Tanga et al 96;



Godon \& Livio (1999,2000), Chavanis (2000)
dust particles settle in cores of (anti-cyc) vortices}

\ni$\bullet$ { B\&S: trap time to form planetesimal is about
10-100 orbits (100-1000yr) for vortex of size $D=H/6$ at 5 AU.

\ni$\bullet$ 
Tanga et al. found 10$M_e$ could form in vortex of order a few hundred
orbit times for $D\sim H$.

\ni$\bullet$ { Studies of vortex survival once formed in disk 
but analytic picture of generation less studied}

\ni$\bullet$ { Simplest growth found from prev. discussion  
shows dominant gr. scale $\sim H$. Survies $\gsim$ few $\ts\ 1/\alpha_{ss}$} orbits 

\ni$\bullet$ { Will  saturate such that  
${\overline \omega}H\sim  v_{t}$ $\rightarrow\ {\overline \omega}\sim |0.1\Omega_k|$..

%\ni $\bullet$ simplest growth found from prev. discussion  
%shows dominant gr. scale $\sim H$ and
%survives minimum few $\ts\ 1/\alpha_{ss}$} orbits 


\ni$\bullet$ {\bf Other consequences ?}


\vfill
\eject



{\bf Motivations for Studying Fe Lines from Non-Flat Disks}

\bigskip
\bigskip


\ni $\bullet$ {\bf Data are still noisy}


\ni $\bullet$ {\bf Nishiura et al 98. 
find BLR not coplanar with Fe line interpretation in e.g. NGC4151}


\ni $\bullet$ {\bf Nandra et al (97) may show preponderance at $i=30^o$,
(concave)?

\ni $\bullet$ {\bf Excessive reprocessed fraction of USNLS (Brandt) (concave, warp blobs?)}

\ni $\bullet$ {\bf Variability (non axisymmetry, warp?)

\ni $\bullet$ {\bf Some Sey II show steeper red wing than blue
(Turner et al 97) (shadowing?)}

\ni $\bullet$ {\bf Theoretical warp models (Terquem and Bartout 93,96;
Pringle 97; Maloney et al 96.)}

\ni $\bullet$ {\bf Warps on larger scales NGC 4258, Miyoshi et al 95;
NGC 660 (Arnaboldi \& Galleta 92), T-Tauri disks.

\ni $\bullet$ {\bf Precession of X-ray binaries (e.g. Wijers \& Pringle 99), 
SS433}


\ni $\bullet$ {\bf ADAFs, winds, thick disks, blobby disks?}






\vfill
\eject

\end{document}



The role magnetic fields in accretion disks is perhaps
most prominently recognized as being an agent which can 
supply a source of angular momentum transport.
The need for turbulence in accretion disks, motivated 
by short time scales in lifetimes of CVs disks.

Not as clear for the AGN disks. I dont really know of direct evidence
that tells us AGN require large Alphas.

It does this through the Balbus-Hawley instability:
Balbus Hawley instability, a local instability however, not only story.
There may be global modes, rossby waves, driven by entropy gradients.


Explain Balbus-Hawley instability.

Once the turbulence is generated, the $\lb b^2\rb$
energy rapidly approaches equipartition on the small scale.
The magnetic energy slightly dominantes because it is being 
stretched a bit from shear, but the eddy turnover time
is of order a rotation time as well.  

Classic theory of random field amplification-Random walk

Note that the Balbus hawley instabilty  suppresses actually 
limits field growth to what it could be if only linear shear.

Balbus Hawley is a local instability.


{\bf Given  MHD turbulent accretion disk what other impliciations?}


\ni {\bf 1. variability}


-Note that in understanding a disk with a turbulent field one
must use MEAN FIELD theory.  That is, one must take the equations
and average them over the turbulence.  

-e.g. axisymmetry etc. only hold in a ``mean'' sense.

-this is important because it means that the equations do not follow
from a simple replacement of the molecular viscosity with the turbulent
one.  One should break the dynamical equations up into mean and fluctuating
compoentents and then the effective turbulent viscosity is the result
of correlations between turbulent quantities.

-this has a number of implications.   First, it means that there
is an intrinsic variablitiy built into accretion disk theory.
That is, the larger the scale of the fluctuations, the less accurate
the mean field theory is for a fixed time.  One must 


there are different was to estimate this but for thin disks, it means
that 

formula for luminosity variability


presumed thick disk systems should have a larger variability 



\ni {\bf 2. Accretion disk theory is complement of mean field dynamo theory,
vorticity growth}


The way of averaging the equations and splitting into fluctuatating
components etc. is just the approach used in mean field dynamo theory.

In fact kinematic dynamo theory is the complement of analytic accretion disk
theory:  MFD theory averages the induction eqaution and
considers evolution of mean field, while 

averaging only over half the height leads to a second transport
coefficient, in addition to diffusion coefficient leads to kinetic helicity
coefficient in the vorticity equation

vorticity growth of interest for planets


\ni {\bf 3.  Mean Field dynamo in an accretion disk as well}

Should find generation of large scale fields as well as 
small scale fields in global accretion disk with vertical density
gradient, and pseudo scalar.

mean field growth of interest for coronae, jets

how do fields of the required type form for jets?

for example in various systems in which jets are observed,
not clear how the fields are formed.   if one only
considers accretion, field may diffuse rather than accrete.

jets and mean velocity as well can be fluctuating
component as well as mean component driving jets

Role of GR effects on field generation.  BH cannot
carry currents, and so their fields must be accreted from exterior.
anytime one studies accretion of field one must also include
the "alpha" helicity term.

flux production from dynamos, flux freezing of mean field is violated


do thick disks favor jets more than thin disks? 

helicity---inverse cascade 

magnetic helicity inverse cascade. total magnetic helicity
is conserved.


\ni {\bf 4. Coronae and Winds/Particle acceleration}

Both mean field and small scale field
dynamos are observed in accretion disks.
While the random field stuff is agreed up, the
small scale stuff is 

Flux tubes general rule in high beta turbulent plasmas

General rule : filling fraction of magnetic field in turbulent
medium $= P_{mag}/P_{part}$


affects particle acceleration



Magnetic fields are buoyant, will rise to surface.

Here, the physics of the solar corona again highlights what
needs to be understood. 

flux tubes general rule would be intersting to understand
the kind of activity expected in accretion disk coronae from magnetic
fields.  has to do with understanding how magentic field
is generated in corona


Interestingly, there seems to be  steady coroan rather than
pulsy. suggests exponential growth of field since the dissiaption
is catastrophic, with occasional larger bursts.  winds

In any case, the daunting task of understanding acc disk corona
is emphasized by the solar corona which we still dont understand


it is believed that the energy dissipation by magnetic fields
generates x-ray flares something like what is seen on the sun.

hard to fully interpret line emission

x-ray emission 

solar flares much like accretion disk flares

The generation of turbulence and the use of turbulence
leads one to consider mean field theory.
shakura and sunyaev is infact a mean field theory.


\ni {\bf 5. Corona + Disk-->Iron Line probe of accretion engine}

Iron lines, profiles




\ni {\bf 6. Conclusions/open problems:}

buoyant rise of flux tubes

backreaction of magnetic field

in mean field formalism with turbulence

mean field dynamo

vorticity growth

jets how do they form, relation to formation of
large scale fields...

do conditions determine jets...

%\end{document}


%\ni {\bf CASE 1:  System = universal scale: no boundary terms:}
%$$\partial_t \lb{\bf A}\cdot \bfB \rb=
%-2c \lb{\bf E}\cdot {\bf B}\rb=
%-2c {\bbE}\cdot {\bbB}-2c \lb{\bfe}\cdot {\bfb}\rb\simeq 0$$
%% c \eta \lb {\bf J}\cdot{\bf B}\rb
%$\partial_t ({\bbA}\cdot \bbB) =-2c {\overline {\bf E}}\cdot {\bbB}
%\\ ; \ \ \partial_t \lb{\bfa}\cdot \bfb \rb=-2c \lb {\bfe}\cdot {\bfb}\rb$$

\bigskip





need

berger and field
lyutikov blackman
solar dynamo



jet formation, cannot accrete flux

coronal variability

solar flare issues

a word about gamma-ray bursts

accretion disk issues buoyancy

open problems in solar dynamo

dynamos in other stars.

\ni {\bf 2-D}

\ni $\bullet$ 2-D diffusion sims., periodic box:
Scalar diffusion is catastrophically restricted for small $\bbB$
(Cattaneo 1994). Motions locked up.

\ni $\bullet$ does not measure $\bbB$ diffusion (fixed in periodic box). 

%\ni $\bullet$ nqpinewot quite following the Z57 trail: Lagrangian
%spectrum locks up, but magnetic energy and is not peaked on small scales.

\ni $\bullet$ Alfv\'en crossing time of the box $\le$ diffusion time 

\ni $\bullet$ Additional terms and processes present in 3-D.



\ni $\bullet$ early 
3-D simulations of accretion disks  can be interpreted to exhibitrapid MFD action (Brandenburg \& Donner 1997) but Magnetic Reynolds numbers
not high enough.


\ni $\bullet$  3-D simulations (Brandenburg 2000)
shows dynamo action but ``resistively limited'' i.e. rate to final
state proceeds at resistive rates.







\ni $\bullet$ 
Accretion disk simulations of Balbus,Hawley, Stone et al., 
have periodic boundary conditions and have been generally local.
Show large scale fields generated but may just be from linear
shear.



\centerline{\bf Dynamos and Jets in Planetary Nebuale}

\bigskip
\ni {\bf Numerical}
\bigskip


\ni{$\bullet$ Periodic b.c. leave ambiguity of dynamical
vs. boundary induced quenching in existing 3-D simulations.}

%\ni{$\bullet$ More quenching studies 
%with non-periodic boundary conditions and scale separation recommended.}

\ni{$\bullet$ Everyone is  ``correct'' for problems solved, but 
implications for real systems need to be considered 
carefully.}

\ni{$\bullet$  Currently there is no evidence of suppression
in 3-D simulations which invoke open boundary conditions and scale
separation.  There is a dearth of simulations in the astrophysically
relevant regimes.}


\bigskip
\ni {\bf  Analytical }
\bigskip

\ni{ $\bullet$ 
Analytic studies sugg. strong quenching have either:}

\ni \ \ \ { (1) been 2-D}   


\ni \ \ \ { (2) ignored  boundary terms}

\ni \ \ \ { (3) assumed that kinematic regime violation = strong quenching

\ni \ \ \ { (4) or not distinguished ``isotropic, homogeneous'' 
part}

\ni { \ \ \ \ \ \ \ of turbulence from ``anisotropic'' part:}

$$\lb \bfv \ts \bfb \rb =\alpha\bbB\ \propto\  
(\lb {\bfb}\cdot\curl\bfb\rb -\lb {\bfv}\cdot\curl\bfv\rb)\bbB$$

{$\bf v,b$  on left $\ne  \bf v,b$ on right for expansion in $\bbB$} 


\vfill
\eject


 
