**
**

**
Results
**

In this section the results from a range of three dimensional
simulations are presented. The simulations are listed in the summary
Table which gives a model designation, the grid resolution used, the
gravitational potential, the initial torus angular momentum
distribution *q*, the initial magnetic field topology, and the time to
which the simulation was run.

Model | Grid | Potential | q |
Initial Field | Endtime |
---|---|---|---|---|---|

CK1 | 90 × 80 × 24 | Cylindrical Newton | 1.5 | B_{z} ~ sin(R) |
188 |

CK2 | 90 × 80 × 24 | Cylindrical Newton | 1.5 | Toroidal | 285 |

CT1 | 90 × 80 × 24 | Cylindrical Newton | 2 | Vertical | 102 |

CT2 | 90 × 80 × 24 | Cylindrical Newton | 2 | Toroidal | 403 |

CT3 | 128 × 64 × 32 | Cylindrical Pseudo-Newton | 2 | Vertical | 420 |

GT1 | 128 × 64 × 128 | Global Pseudo-Newton | 2 | Poloidal loops | 780 |

GT2 | 120 × 64 × 90 | Global Pseudo-Newton | 2 | Poloidal loops | 727 |

GT3 | 128 × 128 × 128 | Global Pseudo-Newton | 2 | Toroidal | 727 |

GT4 | 128 × 128 × 128 | Global Pseudo-Newton | 1.68 | Poloidal loops | 1283 |

*
*

*
Cylindrical Disks
*

The first set of simulations make use of a cylindrical
gravitational potential *~ 1/R* (the cylindrical disk limit).
This is a significant simplification, as it permits the use of periodic
boundary conditions in the vertical direction. Further, with this
potential there is only one important vertical lengthscale, namely that
of the MRI; this reduces the number of *z* grid zones required.
Finally, in the cylindrical limit one can study the evolution of models
with vertical fields without the stringent Courant limits due to the
high Alfven speeds associated with strong fields passing through the
low density region above the disk.

The first two cylindrical simulations assume a Keplerian initial
disk. Keplerian simulations provide an immediate comparison with the
local shearing box results which also, for the most part, assumed
Keplerian flow. Consider first the run labeled CK1 in the summary
Table; this is a cylindrical computational domain *(R, phi,z)*
running from 1 to 4 in *R*, 0 to 1 in *z*, and 0 to *
pi/2* in *phi*. The grid has 90 × 80 × 24 zones. An
outflow boundary condition is used at both the outer and inner radial
boundaries, and periodic boundary conditions are used for *phi*
and *z*. The initial disk has a constant mass density from
*r=1.5* to the outer boundary. The adiabatic sound speed is
constant and equal to 5% of the orbital velocity at the inner edge of
the disk. The initial magnetic field is vertical and proportional to
sin*(R)/R* between *R=1.5* and 3.5, with a maximum strength
corresponding to *beta=400*. The strength of the field, and hence
the Alfven speed *v _{A}*, was chosen so that the
characteristic wavelength of the MRI was

As the evolution proceeds, the MRI sets in, field energy is
amplified, and soon the characteristic radial streaming structures
(referred to as the channel solution) of the vertical field instability
appear, much as they do in the local shearing box models. In the
present simulation, these structures develop first at the inner part of
the disk where the rotation frequency is the highest. The amplitude of
the MRI becomes nonlinear by 3 orbits at the center of the grid
*(P _{orb} = 24.8*), and filaments of strong magnetic field
are carried inward and outward by fluid elements well out of Keplerian
balance. These reach the outer part of the disk even before the local
MRI in that region becomes fully nonlinear. Thus there are two
immediate global effects not seen in local simulations: linear growth
rates that vary strongly with radius (

In local simulations with initial vertical fields that vary
sinusoidally with radius, the initial phase of the instability promptly
breaks down into MHD turbulence. This happens in the global simulation
as well. After the onset of turbulence, the disk displays many tightly
wrapped (i.e., large radial wavenumber) trailing spiral features with
low azimuthal wavenumber, *m*. The magnetic field energy
saturates at about *beta=4*, with the toroidal component
dominant: *B _{R}^{2}/B_{phi}^{2}
=* 0.075 and

The MHD turbulence produces rapid angular momentum transport and
mass accretion, with *alpha*peaking at 0.21 at *t=*90, and
declining beyond this point, dropping to 0.06 at the end of the
simulation. The Maxwell stress displays a high correlation with the
total magnetic pressure. After an early peak at 0.68,
*alpha _{m}* declines with time; at the end of the
simulation

The mass flux varies both in space and time. After 8 orbits of
evolution, over half of the initial disk mass has been lost through the
boundaries, particularly the outer boundary. Since the initial
Keplerian disk abuts the outer boundary, and the mass increases with
radius for a constant initial *rho*, it is relatively easy for a
considerable amount of the mass to leave the grid, driven in part by a
pressure gradient created by a rarefaction wave.

This simulation is similar to that of Armitage (1998) except that
it uses a smaller domain in the angular direction, and a constant
initial density in the disk. Qualitatively, the two runs exhibit
similar behavior, and both are consistent with results from local
simulations. One difference noted by Armitage, is that *alpha ~ *
0.1 in the global simulation; this is larger than the value of *alpha
~ * 0.01 seen in local simulations with zero net vertical field.
The larger *alpha* is, however, consistent with what is seen in
local simulations with net vertical field. The global simulations have
much larger radial and azimuthal wavelengths available to the MRI.
Further, the radial scale over which the vertical field sums to zero in
the global simulation is much larger than in the local shearing box.
Here *B _{z}* initially varies between

The next simulation, labeled CK2, begins with the same Keplerian disk,
but with an initial toroidal field of constant strength, *beta=4*,
lying between *r=*2 and 3.5. Generally speaking, toroidal field
simulations are more challenging than vertical field simulations.
Although the presence of the linear instability is independent of the
orientation of the background magnetic field, the fastest growing
toroidal field modes are associated with large vertical and radial
wavenumbers. Further, the critical azimuthal wavenumber is large,
unless the magnetic energy is comparable to the rotation energy. Here
*m _{crit} = 2 pi R/lambda_{MRI}* ~ 40.
Since there are 80
azimuthal grid zones over

With a toroidal initial field, the disk evolves at a slower rate
that CK1. The instability grows over the first few orbits,
with the fastest rate of growth associated with the innermost radius.
The growing modes of the instability have the same appearance as seen
in the local simulations; high *m* structures appear first, building to
lower *m* with time. The vertical and radial structure also features
high wavenumbers. The rapid growth phase of the instability ceases
after about 8 orbits at the grid center. At this
point the field exhibits a low *m*, tightly wrapped structure, with
rapid variations in *R* and *z*. This behavior is consistent with the
linear analysis (Balbus & Hawley 1992). Turbulence and accretion
begin, and by 12 orbits over half of the disk mass has been lost from
the grid (much of it through the outer boundary); what remains is piled
up near *R=*1.7 and is slowly accreting. The magnetic field is
subthermal and dominated by the toroidal component. The component
magnetic energies at late time are quite similar to those seen in the
vertical field run CK1.

Angular momentum transport is again mainly due to the Maxwell stress.
Compared with CK1, the toroidal field model has lower overall stress.
This is partly because the vertical field model has a very vigorous
initial phase associated with the saturation of the linear
instability. At the end of both CK1 and CK2, *alpha _{m} =*
0.3-0.4. The mass accretion rate, magnetic field strength, and

These Keplerian disk models suffer from significant mass loss through
the outer boundary. In contrast, tori can be completely and
self-consistently contained initially within the computational domain. We next
turn to cylindrical models of constant angular momentum (*q=2*) tori,
beginning with a torus with *R _{in} = *2.0 and

Model CT1 begins with a vertical field with constant *beta =* 100 from
*r=*2.1 to 3.1. This gives *lambda _{MRI} = *0.25
at the pressure
maximum. As the evolution proceeds, the magnetorotational instability
rapidly develops, again with the characteristic channel. Early on, the
beginnings of the Papaloizou-Pringle principal mode can also be seen as

The magnetic field is amplified to an overall average value of
*beta=*2. Toroidal field dominates:
*B _{r}^{2}/B_{phi}^{2} = *0.1, and

Next take the same torus and apply an initial toroidal field with
*beta=*100 (Model CT2). With this strength field, the critical
azimuthal wavenumber at the pressure maximum is *m _{crit}=*
63, so the
fastest growing modes are underresolved on this grid. As with the
Keplerian simulation CK2, the initial toroidal field model becomes
turbulent at a later time compared with an initial vertical field
model. The total poloidal magnetic field amplification is also
considerably smaller than seen in the vertical field model. Despite
this, the qualitative outcome of the instability for the torus is
largely the same. The slower onset of the MRI allows the principal
mode of the Papaloizou-Pringle instability to appear, but soon the
transport of angular momentum brings this to a halt. The disk spreads
outward with the bulk of the mass slowly moving inward. The overall
angular momentum distribution changes from constant to Keplerian from
the inner boundary to

At late time the magnetic energy has risen to *beta=*15, with the
toroidal field dominant by a large factor:
*B _{r}^{2}/B_{phi}^{2}= *0.04.
The ratio of the vertical to radial field energy is 0.16. The value of

Finally, consider a constant angular momentum torus, CT3, with an
initial field constructed by setting the azimuthal component of the
vector potential equal to the density in the torus, *A _{phi} =
rho(R)*. The resulting field is normalized to an average

The MRI grows rapidly in the inner regions of the disk, again with the
characteristic radial channel appearance. Accretion through the inner
boundary begins at about *t=*100. The magnetic energy rises to
peak at *beta =* 8 at *t=*150. The magnetic energy grows more slowly
after that point; additional small peaks are observed due to the growth
of the MRI in the outer parts of the disk. Mass loss through the outer
boundary begins at *t=*200, after which the initial linear growth
phase is over and the disk is fully turbulent. Between *t=*200 and the
end of the simulation, *beta* decreases from 10 to 6. As always, the
toroidal field exceeds the radial field, with
*B _{r}^{2} / B_{phi}^{2} =* 0.16.
The radial field is, in turn, greater than the vertical field,

The overall Maxwell stress in the MHD turbulence reaches a peak at
*t=*150, drops off, and then slowly climbs again. At late time
*alpha _{m} = *0.5, and the total stress parameter

By the end of the simulation (*t=*420) over one quarter of the
disk mass has been lost, and 70% of it has gone in past the marginally
stable orbit. An examination of the various vertically- and
azimuthally-averaged velocities at the end of the run (Fig. 2) reveals
that the inflow velocity *v _{r}* accelerates rapidly
inside of

FIGURE 2:Vertically- and azimuthally-averaged velocities as a function of radius in simulation CT3 at time t=420 (4.8 orbits at the initial pressure maximum). The curves trace the toroidal speedv, the adiabatic sound speed_{phi}c, the toroidal and poloidal Alfven speeds_{s}vand_{A phi}v, and the radial speed_{Ap}v. The vertical dashed line indicates the location of the marginally stable orbit in the pseudo-Newtonian potential. The short-dashed line corresponds to the Keplerian velocity._{r}

*
*

* Axisymmetry: Simulations in the (R,z) plane
*

The cylindrical disk limit represents a useful way of simplifying the
full global problem. Another potentially useful simplification is the
axisymmetric limit. In this series of simulations, the torus evolution
problem is considered in the axisymmetric *(R,z)* plane. These
simulations use a pseudo-Newtonian potential, and begin with a
pressure-supported torus that is fully contained on the grid. The
initial magnetic field is chosen to satisfy two requirements: it must
have a poloidal component to allow for the development of the MRI, and,
as a practical matter, it should be contained completely within the
torus to avoid the Courant limitations caused by high Alfven speeds
due to strong fields in low density regions. A suitable initial
configuration consists of magnetic field loops lying along equidensity
surfaces in the torus. This initial setup will develop strong toroidal
field due to shearing of the initial radial field. However, experience
has shown that strong toroidal fields are the outcome of all initial
field choices, so this should not represent too great an idealization.
All of the two-dimensional simulations considered here will also be run
in three dimensions.

The first simulation has a radial grid running from *R=1.5* to
an outer boundary at *R=11.5* and a vertical grid running from
*z=-5* to 5. The grid resolution is 128 × 128. A
constant-*l* torus is placed on the grid with an inner boundary at
the marginally stable orbit, *R _{in}=*3 and a pressure
maximum at

Density grayscale plots from this simulation are presented in Figure
3. The initial period of evolution is dominated by the growth of
toroidal magnetic field due to shear. As the magnetic pressure
increases, the torus expands, particularly at the inner edge. At
first, low density material is driven outward perpendicular to the
torus surface; subsequently, it flows radially out and around the
torus. Because of the initial reflection symmetry across the equator,
the toroidal field changes sign there and a strong current sheet
forms. This current sheet proves to be unstable, and oscillates around
the equator. This is an important part of the evolution, and indicates
that it is necessary to simulate the full *(R,z)* plane rather than
adopt the equator as an explicit boundary.

FIGURE 3:Plots of log density in a two-dimensional axisymmetric simulation of a constant angular momentum torus containing poloidal field loops. Each image is labeled by time. At t=50 the torus has expanded due to shear amplification of the toroidal field. At $t=180$ the poloidal field MRI has set in. A period of turbulence follows (t=250, 350) which dies out by the end of the simulation (t=850, 17 orbits at the initial pressure maxiumum).

As the toroidal field pressure increases, it drives inflow through
the marginally stable orbit; this initial accretion flow peaks at
*t=*50 then begins to decline. In the meantime, the poloidal
field MRI grows within the torus, and begins to manifest itself visibly
in the typical form of radial channels by *t=*180 (3.6 orbits).
There follows a period of violent readjustment within the disk,
featuring strong mass inflow punctuated by episodic accretion events.
This phase lasts until *t~* 350 (7 orbits), beyond which the
poloidal magnetic energy declines, and with it the level of turbulence
in the disk, and the accretion rate. At the end of the simulation
(*t=*850, 17 orbits) about 60% of the initial disk mass has been
lost. Most of this mass is lost by *t=*500; after this time the
inflow accretion rate is very small.

Thus there are three distinct phases to the two-dimensional torus
evolution: expansion due to the shear-amplified toroidal magnetic
pressure, strong nonlinear evolution of the poloidal field MRI, and
finally a more quiescent turbulent state with declining poloidal
magnetic field strengths. Angular momentum transport occurs in all
three phases at different rates. In the first phase there is a growing
Maxwell stress from the amplified *B _{phi}* field, mainly
in the inner region of the disk where the orbital frequency is
highest. The Reynolds stress is negligible. The initially constant
angular momentum distribution is unchanged within the disk except in
this inner region. With the onset of the MRI, angular momentum
transport occurs everywhere, and the specific angular momentum begins
to increase with radius. During the middle of this second phase,
angular momentum transport peaks, with

The next model is a torus located initially at a larger radius,
which increases the amount of time that it can evolve prior to reaching
the marginally stable orbit. The grid runs from *R=*1.5 to
*R=*13.5 and from -4.5 to 4.5 in *z*. The grid resolution is
120 × 90. The constant-*l* torus has an inner edge at
*R _{in}=*4.5 and a pressure maximum at

As before, this torus undergoes three phases of evolution: toroidal
field amplification, development of the nonlinear poloidal MRI, and
subsequent turbulence. The MRI saturates around *t=*300 (3.4 orbits)
with total magnetic energy *beta =* 2. The phase of strong MRI
turbulence is over by *t=*450 (5.1 orbits) when the average *beta*
rises to 6; fairly steady accretion follows for the remainder of the
simulation which runs to time *t=*830 (9.5 orbits).

In the third simulation, the initial torus has an angular momentum
distribution closer to Keplerian, specifically *q=1.68*. The torus
inner boundary is at 4 and its pressure maximum at *R _{kep}=*10
(orbital period =179). The computational domain runs from

Although it began with a much different initial angular momentum
distribution, the evolution is quite similar to the two previous
cases. In the early stage, toroidal field is amplified by shear. The
poloidal MRI soon comes into play, producing the characteristic radial
streams. The total magnetic energy peaks at about *t=*800 (4.5
orbits). After this the poloidal field energy declines steadily, as
does the Maxwell stress and the mass accretion through the inner radial
boundary. Toroidal field dominates, with the ratio
*B _{r}^{2}/B_{phi}^{2} =*
0.017. Beyond

Angular momentum transport begins almost immediately. Initially it is
confined to the inner regions of the torus where shear is strongest,
and dominated by the Maxwell stress associated with the growing
toroidal field. However, the growth of the MRI produces stress
throughout the torus. By *t=*1000 the averaged *alpha* value is
between 0.1 and 0.2. At the end of the simulation at *t=*2840,
*alpha* ranges between 0.01 and 0.04 within the torus. The ratio of
the Maxwell stress to the magnetic pressure, *alpha _{m}*, begins small,
rises to a value of 0.1 during the initial saturation of the MRI, then
steadily declines to 0.03 by the end of the simulation as poloidal
field is preferentially destroyed. Since this torus began with an
angular momentum distribution similar to the end state of the initially

*
*

*
Three Dimensional Tori: Full Global Simulations
*

Now we turn to the evolution of fully three dimensional tori using
a pseudo-Newtonian potential. The simulations described in this
section are three-dimensional versions of the tori considered above in
the axisymmetric limit. The first, model GT1, is the constant angular
momentum torus with *r _{kep}*=4.7 and

Animation:Click on the image to see an MPEG
animation of run GT1 compared with its 2D axisymmetric counterpart.

The evolution of this torus is illustrated with a series of
grayscale plots in log density (Fig. 4) of vertical *(R,z)* and
equatorial *(R, phi)* slices. As with the axisymmetric torus
(Fig.~3), the initial phase of evolution is controlled by the shear
amplification of the toroidal field. However, in three-dimensions this
toroidal field is itself unstable to the MRI. The characteristic
large-wavenumber structures associated with the instability quickly
appear in the inner regions of the disk (by *t=*80). This creates
turbulence, and momentarily increases the mass accretion rate over that
seen in the axisymmetric simulation (Fig.~5). By *t=*200 the disk
contains tightly wrapped, low *m* trailing spiral waves. The
presence of this nonaxisymmetric structure affects the development of
the poloidal field instability, eliminating the organized radial
streaming flows seen in axisymmetry. The flow is more turbulent and
the mass accretion rate is steadier, without the large impulsive spikes
in accretion rate associated with the radial streaming in two
dimensions. This overall turbulence declines after saturation of the
MRI, but continues through the end of the run. This is the stage when
the accretion rate in the two-dimensional simulation drops to very
small levels. While in three dimensions one might hope that the disk
could achieve a nearly steady state mass accretion rate, there is, of
course, only a finite amount of disk mass available to accrete. At the
end of the run 85% of the initial total mass has been lost from the
grid, most of it through the innermost stable orbit. The amount of
mass ejected off the outer boundary is about 30% of that accreted
through the inner boundary.

FIGURE 4:Plots of log density in simulation GT1, initially a constant angular momentum torus containing poloidal field loops. Each image consists of a side view and an equatorial view, and is labeled by time. At t=80 the torus has expanded due to shear amplification of the toroidal field which has become visibly unstable in the inner regions. Full turbulence sets in by t=200 (4 orbits at the initial pressure maximum) and continues for the remainder of the simulation. The total disk mass drops steadily due to accretion. These images should be compared with the axisymmetric model in Figure 3.

FIGURE 5:Mass flux through the inner grid radiusRas a function of time for global models (solid lines) and their axisymmetric counterparts where applicable (dashed lines). In each case, the mass accretion rate is normalized by the initial mass of the torus. In models GT1, GT2 and GT4, the initial accretion is driven mainly by the growth of the toroidal field due to shear. Subsequently the poloidal MRI drives strong accretion. The coherent channel solution in axisymmetry produces particularly strong accretion events. At late times, accretion in the axisymmetric models dies down while in the global models accretion continues at a reduced, but significant, level. Note that despite their quite different initial field strengths and topologies, models GT2 and GT3 have similar accretion rates at late times._{min}

Although the total volume-averaged magnetic energy drops with time
beyond *t=*400, most of this decline is due to loss from accretion.
Average *beta* after *t=*200 is ~4 and remains fairly
constant thereafter. The magnetic energy is dominated by the toroidal
component; the poloidal field *beta* value has an average value of
~30, and also remains nearly constant after *t=*200. In the
axisymmetric simulation the poloidal *beta* value attains a value of
15 at *t=*280, but rises steadily to *beta=*75 by *t=*800,
indicating a poloidal magnetic energy loss rate exceeding that due to
accretion alone.

Angular momentum transport results in rapid restructuring of the disk.
The angular momentum distribution in the inner half of the torus
steepens to a nearly Keplerian value by *t=*50 (Fig.~6). The angular
momentum in the outer part of the torus increases over the remainder of
the simulation. At the end time the angular velocity *Omega ~
R ^{-q}* has a slope near

FIGURE 6:Mass distribution as a function of radius (top), and averaged angular momentum as a function of radius (bottom) at several different times in model GT1. The angular momentum curves are labeled by time; the corresponding mass curves run from top to bottom with increasing time. The orbital time at the initial pressure maximum is 50. The long dashed line corresponds to the Keplerian value in the pseudo-Newtonian potential. The short dashed line is the angular momentum distribution at time t=420 in the axisymmetric version of this torus.

Figure 6 shows that the specific angular momentum continues to
decline even inside the marginally stable orbit. This indicates that
even here there remains a significant net stress. In fact, inside the
marginally stable orbit *alpha* rapidly increases because the gas
pressure drops while the Maxwell stress remains roughly constant. The
presence of this stress means there is no maximum in the epicyclic
frequency, *appa ^{2} = 2 Omega/r dl/dr*. In the
pseudo-Newtonian potential (and, of course in the relativistic
potential that the pseudo-Newtonian potential was designed to imitate),
the Keplerian value of

In some respects, the two- and three-dimensional simulations are
similar. They both have shear amplification of toroidal field, they
both evolve due to the resulting increase in toroidal magnetic
pressure, and they both develop turbulence due to the poloidal field
MRI. Both rapidly evolve from constant to nearly Keplerian specific
angular momentum distributions (Fig.~6). The three dimensional
simulation, however, permits the development of the nonaxisymmetric MRI
which increases and sustains turbulence and mass accretion. In two
dimensions the organized poloidal field channel solution produces an
impulsive accretion rate greater than that seen in three dimensions
during the initial saturation of the poloidal MRI. But axisymmetry
causes the two-dimensional poloidal field to decline, and with it the
Maxwell stress. The contrast between *alpha _{m}* in the
three- and the two-dimensional simulations is instructive (Fig.~7). In
three dimensions

FIGURE 7:Time evolution of volume-integrated value of the magnetic alpha value in the global torus models (solid lines) and the axisymmetric version (dashed lines) where appropriate. Individual graphs are labeled by global torus model number. In three dimensions, poloidal field amplitudes are maintained relative to the toroidal field, and the Maxwell stress remains appreciable compared to the total magnetic pressure. In two dimensions the poloidal fields die out and the stress drops.

Although two-dimensional simulations cannot capture these essential
features of global evolution, they do have one clear advantage: they
are considerably easier to compute. Two dimensional simulations are
useful for searching a wide range of initial conditions in support of
the more challenging three-dimensional models. To test the idea of
using an evolved two-dimensional simulation as an initial condition,
consider next a constant-*l* torus with an initial inner edge at
*r _{in}=* 4.5 and a pressure maximum at

While this initialization procedure reduces the total computational
time required, it doesn't allow the toroidal field instability the
opportunity to grow during the initial phase. This means that the
strong coherent poloidal field instability develops as it does in two
dimensions. Nevertheless, significant nonaxisymmetric structure
appears by *t=*400, and the overall turbulence is increased over
that seen in axisymmetry. Accretion inflow at the inner boundary is
about twice what it is in two dimensions after this point in time.
After *t=*400 the average magnetic field strength is
*beta=*4.7, and the poloidal field strength *beta _{p} =
*24. At late time

This run can also be compared with the cylindrical run CT3. Figure 8 is a plot of the radial dependence of the azimuthally and vertically averaged speeds. These curves are very similar to those in Figure 2 from CT3. GT3 shows a stronger initial field amplification phase and a stronger MRI saturation (created by initializing from the axisymmetric run). However the magnetic field amplitudes and stress levels are comparable at late times in both simulations. These similarities indicate that the cylindrical limit provides a useful approximation for investigating the nonlinear evolution of MHD turbulence near the midplane of a disk.

FIGURE 8:Vertically- and azimuthally-averaged velocities as a function of radius at time t=727 (8.3 orbits at the initial pressure maximum) in simulation GT2. The curves trace the toroidal speedv, the adiabatic sound speed_{phi}c, the toroidal and poloidal Alfv\'en speeds_{s}vand_{Aphi}v, and the radial speed_{Ap}v. The vertical dashed line indicates the location of the marginally stable orbit in the pseudo-Newtonian potential. The short-dashed line corresponds to the Keplerian velocity._{r}

To summarize, GT2 shows that fully three dimensional turbulence can be rapidly produced and sustained in a simulation initialized from the output of an axisymmetric simulation. The late-time properties of such a simulation, essentially one with complicated nonlinear initial perturbations, are quite similar to those obtained from a simulation evolved from a formal equilibrium and linear perturbations. The two-dimensional simulations, therefore, serve as just another type of initial condition, albeit more complicated than usually adopted.

One characteristic of all these simulations is that at late times
the magnetic field is predominantly toroidal. The next simulation
considers a torus that begins with only a toroidal field. Model GT3
is of a *q=2* torus with an inner boundary at 4.5 and a pressure
maximum at *R _{kep}=*6.5 (orbital period =88). The
initial toroidal field has

No attempt is made to keep the initial torus in pressure balance;
the initial magnetic field simply adds to the equilibrium hydrodynamic
pressure. As a consequence, the torus undergoes an axisymmetric
readjustment due to this additional magnetic pressure. The toroidal
field MRI develops rapidly at the inner edge of the disk where
*Omega* is largest, before spreading throughout the disk. The
poloidal field energy grows steadily with time until about *t=*250
when it reaches a value of *beta _{p}= 34*. After

Animation:Click on the image to see an MPEG
animation of run GT3.

FIGURE 9:Plots of log density in simulation GT3, a torus withq=2 andbeta=4 toroidal field initially. Each frame consists of a side and an equatorial view, and is labeled by time. At t=120 the MRI has appeared at the inner edge of the torus. Turbulence is soon established throughout the torus.

Models GT2 and GT3 began with the same hydrodynamic torus; they
differ in their initial magnetic fields, in the size of the *phi*
domain, and in resolution. Despite these differences, the late time
evolution in these two runs is very similar. They have comparable mass
accretion rates through *R _{min}* (Fig. 5). Both have

FIGURE 10:Time evolution of the vertically- and azimuthally-averaged mass (top) and specific angular momentum (bottom) as a function of radius in simulation GT3. The different lines correspond to different times throughout the evolution: $t=0$, 117, 209, 298, 382, 470, 550, 637, 727. The orbital period at the initial torus pressure maximum is 88. The long dashed line in the angular moment plot corresponds to the Keplerian value. The short dashed line is the curve from the final time in model GT1.

Figure 11 shows the radial run of density *Sigma*, velocity
* v _{R}*, and mass flux at time

FIGURE 11:From top to bottom, the vertically averaged density, radial velocity, and mass flux as a function of radius in model GT3 at t=720 (8.2 orbits at the initial pressure maximum). The dashed line in the radial velocity plot is the orbital velocity.

FIGURE 12:Vertically averaged gas pressureP, toroidal magnetic field pressure, radial field pressure, and Maxwell stress as a function of radius at t=720 in model GT3. The pressures are normalized to the initial pressure maximum value.

The final global simulation, GT4, is the *q=1.68* torus with
an inner boundary at 4 and a pressure maximum at
*R _{kep}=*10. This angular momentum distribution yields a
torus that extends over a large radial distance without becoming too
thick in the vertical direction. The computational domain runs from

Animation:Click on the image to see an MPEG
animation of run GT4.

Density plots from GT4 are presented in Figure 13. At the beginning,
the toroidal field grows by shearing out the radial field, but as it
does so, the toroidal MRI sets in. This soon leads to turbulence. The
poloidal MRI develops rapidly in the inner regions of the disk and
subsequently spreads throughout. The inner edge of the disk moves
slowly inward until it reaches the marginally stable orbit at
*t=*145 after which gas plunges inward. Initially the inflowing
fluid is confined largely to the equatorial plane. As time passes,
however, this region fills with gas and becomes thicker.

FIGURE 13:Plots of log density in simulation GT4, initially a torus withq=1.68 and poloidal field loops. Each frame consists of a side and an equatorial view, and is labeled by time. The orbital period at the initial pressure maximum is 179. At t=320 the torus has expanded due to shear amplification of the toroidal field which has, in turn, become unstable withm=4dominating in the equatorial plane. Beyond this time the poloidal field MRI begins and turbulence follows.

As with the previous global simulations, the mass accretion rate is
larger in three dimensions compared to two. Near the end of the
simulation, the accretion rate at the inner radial boundary is
approximately 2.5 times that seen in axisymmetry. The value of
*alpha _{m}* is rising with time indicating that the
poloidal field strength is still increasing at the end of the
simulation.

Figure 14 shows the radial mass and angular momentum distributions at
the initial and final times in GT4. Also shown are curves from two
times in the equivalent axisymmetric calculation. The average angular
velocity parameter *q* decreases by a very small amount over the course
of the evolution. The torus mass, on the other hand, has been substantially
redistributed.

FIGURE 14:Vertically and azimuthally averaged mass (top) and specific angular momentum (bottom) as a function of radius for the initial and final time, t=1283 in model GT4. The slope has steepened very slightly over the course of the evolution. The long dashed line in the angular momentum plot corresponds to the Keplerian value. The short dashed lines are from times t=1559 and t=2840 in the axisymmetric simulation of this torus.