3-D Simulations of Ergospheric Disk Driven Poynting Jets
Brian Punsly
4014 Emerald Street No.116, Torrance CA, USA 90503 and International Center for
Relativistic Astrophysics, I.C.R.A.,University of Rome La Sapienza, I-00185 Roma, Italy
brian.m.punsly@L-3com.com or brian.punsly@gte.net
ABSTRACT
This Letter reports on 3-dimensional simulations of Kerr black hole magne-
tospheres that obey the general relativistic equations of perfect magnetohydro-
dynamics (MHD). In particular, we study powerful Poynting flux dominated jets
that are driven from dense gas in the equatorial plane in the ergosphere. The
physics of which has been previously studied in the simplified limit of an ergop-
sheric disk. For high spin black holes, a/M > 0.95, the ergospheric disk is promi-
nent in the 3-D simulations and is responsible for greatly enhanced Poynting flux
emission. Any large scale poloidal magnetic flux that is trapped in the equatorial
region leads to an enormous release of electromagnetic energy that dwarfs the jet
energy produced by magnetic flux threading the event horizon. The implication
is that magnetic flux threading the equatorial plane of the ergosphere is a likely
prerequisite for the central engine of powerful FRII quasars.
Subject headings: Black hole physics - magnetohydrodynamics -galaxies: jets—
galaxies: active — accretion disks
1. Introduction
Recent studies of luminous radio quasars indicate that the power of the radio jet can
exceed the bolometric luminosity associated with the accretion flow thermal emission (Punsly
2006b, 2007). This has proven to be quite challenging for current 3-D numerical simulations
of MHD black hole magnetospheres. Based on table 4 of Hawley and Krolik (2006) and the
related discussion of Punsly (2006b, 2007), the most promising 3-D simulations for achieving
this level of efficiency are those of the highest spin, a/M ≈ 1 (where the black hole mass, M ,
and the angular momentum per unit mass, a, are in geometrized units). More generally, such
high spins have been inferred in some black hole systems based on observational constraints
(McClintock et al 2006). Thus, there is tremendous astronomical relevance to these highest
http://arxiv.org/abs/0704.0816v1
– 2 –
spin configurations, in particular the physical origin of the relativistic Poynting jet. The first
generation of long term 3-D simulations produced one Poynting flux powerhouse, the a/M =
0.995 simulation, KDE (De Villiers et al 2003, 2005a; Hirose et al 2004; Krolik et al 2005).
The source of most of the Poynting flux was clearly shown to be outside the event horizon
in KDE (Punsly 2006a). However, without access to the original data, the details of the
physical mechanism could not be ascertained. A second generation of 3-D simulations were
developed in Hawley and Krolik (2006), the highest spin case was KDJ, a/M = 0.99, with
by far the most powerful Poynting jet within the new family of simulations; three times the
Poynting flux (in units of the accretion rate of mass energy) of the next closest simulation
KDH, a/M = 0.95. The last three data dumps, at simulation times, t = 9840 M, t = 9920 M
and t = 10000 M, were generously made available to this author. The late time behavior of
the simulations is established after t = 2000 M (when the large transients due to the funnel
formation have died off) making these data dumps of particular interest for studying the
Poynting jet (Hawley and Krolik 2006). This paper studies the origin of the Poynting jet
at these late times.
The analysis of the data from the KDJ simulation clearly indicates that the Poynting
flux in the outgoing jet is dominated by large flares. Typically, one expects the turbulence
in the field variables to mask the dynamics of Poynting flux creation in an individual time
slice of one of the 3-D simulations (Punsly 2006a). Surprisingly, the flares are of such a large
magnitude that they clearly standout above the background field fluctuations as evidenced
by figure 1. The flares are created in the equatorial accretion flow deep in the egosphere
between the inner calculational boundary at r=1.203 M and r= 1.6 M (the event horizon is at
r= 1.141 M). Powerful beams of Poynting flux emerge perpendicular to the equatorial plane
in the ergospheric flares and much of the energy flux is diverted outward along approximately
radial trajectories that are closely aligned with the poloidal magnetic field direction in the jet
(see figure 1). The situation is unsteady, whenever some vertical magnetic flux is captured
in the accretion flow it tends to be asymetrically distributed and concentrated in either
the northern or southern hemisphere. This hemisphere then receives a huge injection of
electromagnetic energy on time scales ∼ 60M .
The source of Poynting flux in KDJ resembles a nonstationary version of the ergospheric
disk (see Punsly and Coroniti (1990) and chapter 8 of Punsly (2001) for a review). The
ergospheric disk is modeled in the limit of negligible accretion and it is the most direct
manifestation of gravitohydromagnetics (GHM) Punsly (2001). A GHM dynamo arises
when the magnetic field impedes the inflow of gas in the ergosphere, i.e., vertical flux in an
equatorial accretion flow. The strong gravitational force will impart stress to the magnetic
field in an effort to move the plasma through the obstructing flux. In particular, the metric
induced frame dragging force will twist up the field azimuthally. These stresses are coupled
– 3 –
into the accretion vortex around a black hole by large scale magnetic flux, and propagate
outward as a relativistic Poynting jet. The more obstinate the obstruction, the more powerful
the jet. There are two defining characteristics that distinguish the GHM dynamo from a
Blandford-Znajek (B-Z) process, Blandford and Znajek (1977), on field lines that thread the
ergopshere:
1. The B-Z process is electrodynamic so there is no source within the ergosphere, it
appears as if the energy flux is emerging from the horizon. In the GHM mechanism,
the source of Poynting flux is in the ergospheric equatorial accretion flow.
2. In a B-Z process in a magnetosphere shaped by the accretion vortex, the field line
angular velocity is, ΩF ≈ ΩH/2 (where ΩH is the angular velocity of the horizon)
near the pole and decreases with latitude to ≈ ΩH/5 near the equatorial plane of
the inner ergosphere (Phinney 1983). In GHM, since the magnetic flux is anchored
by the inertia of the accretion flow in the inner ergosphere, frame dragging enforces
dφ/dt ≈ ΩH . One therefore has the condition, ΩF ≈ ΩH .
In order to understand the physical origin of the Poynting flux, these two issues are studied
below.
2. The KDJ Simulation
The simulation is performed in the Kerr metric (that of a rotating, uncharged black
hole), gµν . Calculations are carried out in Boyer-Lindquist (B-L) coordinates (r, θ, φ, t). The
reader should refer to Hawley and Krolik (2006) for details of the simulation. We only
give a brief overview. The initial state is a torus of gas in equilibrium that is threaded
by concentric loops of weak magnetic flux that foliate the surfaces of constant pressure.
The magnetic loops are twisted azimuthally by the differentially rotating gas. This creates
significant magnetic stress that removes angular momentum from the gas, initiating a strong
inflow that is permeated by magneto-rotational instabilities (MRI). The end result is that
after t = a few hundred M, accreted poloidal magnetic flux gets trapped in the accretion
vortex or funnel (with an opening angle of ∼ 60◦ at the horizon tapering to ∼ 35◦ at
r > 20M). This region is the black hole magnetosphere and it supports a Poynting jet. The
surrounding accretion flow is very turbulent.
In order to understand the source of the strong flares of radial Poynting flux, one
needs to merely consider the conservation of global, redshifted, or equivalently the B-L
coordinate evaluated energy flux (Thorne et al 1986). In general, the divergence of the
– 4 –
Fig. 1.— The source of Poynting flux. The left hand column is Sθ and the right hand column
is Sr in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920
M and t= 10000 M. The relative units (based on code variables) are in a color bar to right
of each plot for comparison of magnitudes between the six plots. The contours on the Sθ
plots are of the density, scaled from the peak value within the frame at relative levels 0.5
and 0.1. The contours on the Sr plots are of Sθ scaled from the peak within the frame at
relative levels 0.67 and 0.33. The inside of the inner calculational boundary (r=1.203 M)
is black. The calculational boundary near the poles is at 8.1◦ and 171.9◦. Notice that any
contribution from an electrodynamic effect associated with the horizon appears minimal.
The white contour is the stationary limit surface. There is no data clipping, so plot values
that exceed the limits of the color bar appear white.
– 5 –
Fig. 2.— The central engine. The left hand column is Bθ and the right hand column is ΩF
in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920 M and
t= 10000 M. The relative units (based on code variables) are in a color bar to right of each
plot for comparison of magnitudes between the plots. The calculational boundaries are the
same as figure 1. The contours on the Bθ plots are of the density, scaled from the peak value
within the frame at relative levels 0.5 and 0.1. There is no data clipping, so plot values that
exceed the limits of the color bar appear white.
– 6 –
time component of the stress-energy tensor in a coordinate system can be expanded as,
T νt ;ν = (1/
−g)[∂(
−g T νt )/∂(x
ν)] + Γ
µ , where Γ
t β is the connection coefficient and
g = −(r2 + a2 cos2 θ)2 sin2 θ is the determinant of the metric. However, the Kerr metric has
a Killing vector (the metric is time stationary) dual to the B-L time coordinate. Thus, there
is a conservation law associated with the time component of the divergence of the stress-
energy tensor. Consequently, if one expands out the inhomogeneous connection coefficient
term in the expression above, it will equate to zero. The conservation of energy evaluated
in B-L coordinates reduces to, ∂(
−g T νt )/∂(x
ν) = 0, where the four-momentum −T νt has
two components: one from the fluid, −(T νt )fluid, and one from the electromagnetic field,
−(T νt )EM. The reduction to a homogeneous equation with only partial derivatives is the
reason why the global conservation of energy can be expressed in integral form in (3.70) of
Thorne et al (1986). It follows that the poloidal components of the redshifted Poynting flux
are Sθ = −
−g (T θt )EM and S
r = −
−g (T rt )EM. We can use these simple expressions to
understand the primary source of the Poynting jet in KDJ. Figure 1 is a plot of Sθ (on the
left) and Sr (on the right) in KDJ at the last three time steps of data collection. Each frame
is the average over azimuth of each time step. This greatly reduces the fluctuations as the
accretion vortex is a cauldron of strong MHD waves. The individual φ = constant slices show
the same dominant behavior, however it is embedded in large MHD fluctuations. On the left
hand column of figure 1, density contours have been superimposed on the images to indicate
the location of the equatorial accretion flow. The density is evaluated in B-L coordinates
with contours at 0.5 and 0.1 of the peak value within r < 2.5M . Notice that in all three left
hand frames, Sθ is created primarily in regions of very high accretion flow density. In all
three of the right hand frames of figure 1, there is an enhanced Sr that emanates from the
ergosphere (defined by the interior of the stationary limit, rs = M +
M2 − a2 cos2 θ, note
that there are 40 grid points between r = 1.203M and rs at θ = π/2). This radial energy
beam diminishes precipitously just outside the horizon, near the equatorial plane in all three
time steps. The region in which Sr diminishes is adjacent to a region of strong Sθ that orig-
inates in the inertially dominated accretion flow in the inner ergosphere, 1.2M < r < 1.6M
(this region is resolved by 28 radial grid zones). In fact, if one looks at the conservation of
energy equation, the term ∂(Sθ)/∂θ is sufficiently large to be the source of ∂(Sr)/∂r at the
base of the radial beam in all three frames. This does not preclude the transfer of energy
to and from the plasma. It merely states that the magnitude is sufficient to source Sr. In
general, the hydrodynamic energy flux is negligible in the funnel. In order to illustrate this,
contours of Sθ are superimposed on the color plots of Sr. The contour levels are chosen to
be 2/3 and 1/3 of the maximum value of Sθ emerging from the dense equatorial accretion
flow. One clearly sees Sθ switching off where Sr switches on. We conclude that a vertical
Poynting flux created in the equatorial accretion flow is the source of the strong beams of
Sr. This establishes condition 1 of the Introduction.
– 7 –
The left column of figure 2 contains plots of the magnetic field component, Bθ ≡ Frφ, at
the three time steps. At every location in which Sθ is strong in figure 1, there is a pronounced
enhancement in Bθ in figure 2. Recall that the sign of Sθ is not determined by the sign of Bθ.
These intense flux patches penetrate the inertially dominated equatorial accretion flow in all
three frames. The density contours indicate that the regions of enhanced vertical field greatly
disrupt the equatorial inflow. As noted in the introduction, a GHM interaction is likely to
occur when the magnetic field impedes the inflow in the ergosphere. The regions of large
Bθ are compact compared to the global field configuration of the jet, only ∼ 1.0M − 2.0M
long. Considering the turbulent, differentially rotating plasma in which they are embedded,
these are most likely highly enhanced regions of twisted magnetic loops created by the MRI.
The strength of Bθ at the base of the flares is comparable to, or exceeds the radial magnetic
field strength. The situation is clearly very unsteady and vertical flux is constantly shifting
from hemisphere to hemisphere. The time slice t = 10000 M, although primarily a southern
hemisphere event, also has a significant contribution in the northern hemisphere (see the
blue fan-like plume of vertical Poynting flux in figure 1). The GHM interaction is provided
by the vertical flux that links the equatorial plasma to the relatively slowly rotating plasma
of the magnetosphere within the accretion vortex. The vertical flux transmits huge torsional
stresses from the accretion flow to the magnetosphere.
Further corroboration of this interpretation can be found by looking at the values of ΩF
in the vicinity of the Sr flares. In a non-axisymmetric, non-time stationary flow, there is still
a well defined notion of ΩF : the rate at which a frame of reference at fixed r and θ would
have to rotate so that the poloidal component of the electric field, E⊥, that is orthogonal
to the poloidal magnetic field, BP , vanishes. This was first derived in Punsly (1991) (see
the extended discussion in Punsly (2001) for the various physical interpretations), and has
recently been written out in B-L coordinates in Hawley and Krolik (2006) in terms of the
plasma three-velocity, vi and the Faraday tensor as
ΩF = v
φ − Fθr
rFφθ + gθθv
(Fφθ)2grr + (Frφ)2gθθ
. (2-1)
This expression was studied in the context of the simulation KDH, a/M = 0.95, in Hawley and Krolik
(2006). They found that a long term time and azimuth average yielded ΩF ≈ 1/3ΩH and
there was no enhancement at high latitudes as was anticipated by Phinney (1983). The t
= 10000 M time slice of KDH was generously provided to this author. At t = 10000 M,
there are no strong flares emerging from the equatorial accretion flow. Inside the funnel at
r < 10M , at t=10000 M, 0 < ΩF < 0.5ΩH .
The right hand column of figure 2 is ΩF plotted at three different time steps for KDJ.
By comparison to figure 1, notice that each flare in Sr is enveloped by a region of enhanced
– 8 –
ΩF , typically 0.7ΩH < ΩF < 1.2ΩH . The regions of the funnel outside the ergosphere are
devoid of large flares in Sr and typically have 0 < ΩF < 0.5ΩH , similar to what is seen in
KDH.. Unlike KDH, there are huge enhancements in ΩF at lower latitudes in the funnel.
It seems reasonable to associate this large difference in the peak values of ΩF in KDJ and
KDH (at t= 10000 M) with the spatially and temporally coincident flares in Sr that occur
in KDJ. Furthermore, this greatly enhanced value of ΩF indicates a different physical origin
for ΩF in the flares than for the remainder of the funnel or in KDH at t = 10000 M. The
most straightforward interpretation is that it is a direct consequence of the fact that the
flares originate on magnetic flux that is locked into approximate corotation with the dense
accreting equatorial plasma (i.e., the inertially dominated equatorial plasma anchors the
magnetic flux). In the inner ergosphere, frame dragging enforces 0.7ΩH < dφ/dt < 1.0ΩH
on the accretion flow. This establishes condition 2 of the Introduction.
3. Discussion
In this Letter we showed that in the last three data dumps of the 3-D MHD numerical
simulation, KDJ, the dominant source of Poynting flux originated near the equatorial plane
deep in the ergopshere. The phenomenon is unsteady and is triggered by large scale vertical
flux that is anchored in the inertially dominated equatorial accretion flow. The situation
typifies the ergospheric disk in virtually every aspect, even though there is an intense accre-
tion flow. There is one exception, unlike the ergospheric disk, the anchoring plasma rarely
achieves the global negative energy condition that is defined by the four-velocity, −Ut < 0,
because of the flood of incoming positive energy plasma from the accretion flow. The plasma
attains −Ut < 0 only near the base of the strongest flares seen in the φ = constant slices.
The switch-on of a powerful beam of Sr outside the horizon at r ≈ 1.3M in the a/M =
0.995 simulation, KDE, of Krolik et al (2005) was demonstrated in Punsly (2006a). It
seems likely the the source of Sr in KDE is Sθ from an ergopsheric disk. The ergospheric
disk appears to switch on at a/M > 0.95 as evidenced by the factor of 3 weaker Poynting
flux in KDH. Furthermore, if the funnel opening angle at the horizon in KDH at t= 10000
M is typical within ±5◦ then figure 5 and table 4 of Hawley and Krolik (2006) indicate
that only 35% to 40% of the funnel Poynting flux at large distances is created outside the
horizon during the course of the simulation. A plausible reason is given by the plots of Bθ
in figure 2. The vertical magnetic flux at the equatorial plane is located at r < 1.55M . The
power in the ergospheric disk jet ∼ [Bθ(SA)(ΩH)]2, where SA is the proper surface area
of the equatorial plane threaded by vertical magnetic flux (Semenov et al 2004; Punsly
2001). The proper surface area in the ergospheric equatorial plane increases dramatically
– 9 –
at high spin, diverging at a = M . For example, between the inner calculational boundary
and 1.55 M the surface area is only significant for a/M > 0.95 and grows quickly with a/M ,
exceeding twice the surface area of the horizon for a/M = 0.99. Thus, if Bθ in the inner
ergosphere were independent of spin to first order, then a strong ergospheric disk jet would
switch-on in the 3-D simulations at a/M > 0.95. Note that if the inner boundary were truly
the event horizon instead of the inner calculational boundary then this argument would
indicate that the ergospheric disk would likely be very powerful even at a/M = 0.95 and the
switch-on would occur at a/M ≈ 0.9. The implication is that a significant amount of large
scale magnetic flux threading the equatorial plane of the ergopshere (which implies a large
black hole spin based on geometrical considerations) catalyzes the formation of the most
powerful Poynting jets around black holes. Thus, we are now considering initial conditions
in simulations that are conducive to producing significant vertical flux in the equatorial plane
of the ergosphere.
It should be noted that 2-D simulations from a similar initial state of torii threaded by
magnetic loops have been studied in McKinney and Gammie (2004). However, the magnetic
flux evolution can be much different in this setting as discussed in Punsly (2006a) and poloidal
flux configurations conducive to GHM could be highly suppressed. In summary, there are
no interchange instabilities, so flux tubes cannot pass by each other or move around each
other in the extra degree of freedom provided by the azimuth. Thus, there is a tendency
for flux tubes to get pushed into the hole by the accretion flow. This is in contrast to the
formation of the ergospheric disk in Punsly and Coroniti (1990) in which buoyant flux tubes
are created by reconnection at the inner edge of the ergospheric disk and recycle back out
into the outer ergosphere by interchange instabilities. Ideally, a full 3-D simulation with a
detailed treatment of resistive MHD reconnection is preferred for studying the relevant GHM
physics.
I would like to thank Jean-Pierre DeVilliers for sharing his deep understanding of the
numerical code and these simulations. I was also very fortunate that Julian Krolik and John
Hawley were willing to share their data in the best spirit of science.
REFERENCES
Blandford, R. and Znajek, R. 1977, MNRAS. 179, 433
De Villiers, J-P., Hawley, J., Krolik, 2003, ApJ 599 1238
De Villiers, J-P., Hawley, J., Krolik, J.,Hirose, S. 2005, ApJ 620 878
– 10 –
De Villiers, J-P., Staff, J., Ouyed, R.. 2005, astro-ph 0502225
Hawley, J., Krolik, K. 2006, ApJ 641 103
Hirose, S., Krolik, K., De Villiers, J., Hawley, J. 2004, ApJ 606, 1083
Krolik, K., Hawley, J., Hirose, S. 2005, ApJ 622, 1008
McKinney, J. and Gammie, C. 2004, ApJ 611 977
McClintock, J.E. et al 2006, ApJ 652, 518
Phinney, E.S. 1983, PhD Dissertation University of Cambridge.
Punsly, B., Coroniti, F.V. 1990, ApJ 354 583
Punsly, B. 1991, ApJ 372 424
Punsly, B. 2001, Black Hole Gravitohydromagnetics (Springer-Verlag, New York)
Punsly, B. 2006, MNRAS 366 29
Punsly, B. 2006, ApJL 651 L17
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Semenov, V., Dyadechkin, S. and Punsly, B. 2004, Science 305978
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University Press, New Haven)
This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0502225
	Introduction
	The KDJ Simulation
	Discussion

  This Letter reports on 3-dimensional simulations of Kerr black hole
magnetospheres that obey the general relativistic equations of perfect
magnetohydrodynamics (MHD). In particular, we study powerful Poynting flux
dominated jets that are driven from dense gas in the equatorial plane in the
ergosphere. The physics of which has been previously studied in the simplified
limit of an ergopsheric disk. For high spin black holes, $a/M > 0.95$, the
ergospheric disk is prominent in the 3-D simulations and is responsible for
greatly enhanced Poynting flux emission. Any large scale poloidal magnetic flux
that is trapped in the equatorial region leads to an enormous release of
electromagnetic energy that dwarfs the jet energy produced by magnetic flux
threading the event horizon. The implication is that magnetic flux threading
the equatorial plane of the ergosphere is a likely prerequisite for the central
engine of powerful FRII quasars.

Introduction
Recent studies of luminous radio quasars indicate that the power of the radio jet can
exceed the bolometric luminosity associated with the accretion flow thermal emission (Punsly
2006b, 2007). This has proven to be quite challenging for current 3-D numerical simulations
of MHD black hole magnetospheres. Based on table 4 of Hawley and Krolik (2006) and the
related discussion of Punsly (2006b, 2007), the most promising 3-D simulations for achieving
this level of efficiency are those of the highest spin, a/M ≈ 1 (where the black hole mass, M ,
and the angular momentum per unit mass, a, are in geometrized units). More generally, such
high spins have been inferred in some black hole systems based on observational constraints
(McClintock et al 2006). Thus, there is tremendous astronomical relevance to these highest
http://arxiv.org/abs/0704.0816v1
– 2 –
spin configurations, in particular the physical origin of the relativistic Poynting jet. The first
generation of long term 3-D simulations produced one Poynting flux powerhouse, the a/M =
0.995 simulation, KDE (De Villiers et al 2003, 2005a; Hirose et al 2004; Krolik et al 2005).
The source of most of the Poynting flux was clearly shown to be outside the event horizon
in KDE (Punsly 2006a). However, without access to the original data, the details of the
physical mechanism could not be ascertained. A second generation of 3-D simulations were
developed in Hawley and Krolik (2006), the highest spin case was KDJ, a/M = 0.99, with
by far the most powerful Poynting jet within the new family of simulations; three times the
Poynting flux (in units of the accretion rate of mass energy) of the next closest simulation
KDH, a/M = 0.95. The last three data dumps, at simulation times, t = 9840 M, t = 9920 M
and t = 10000 M, were generously made available to this author. The late time behavior of
the simulations is established after t = 2000 M (when the large transients due to the funnel
formation have died off) making these data dumps of particular interest for studying the
Poynting jet (Hawley and Krolik 2006). This paper studies the origin of the Poynting jet
at these late times.
The analysis of the data from the KDJ simulation clearly indicates that the Poynting
flux in the outgoing jet is dominated by large flares. Typically, one expects the turbulence
in the field variables to mask the dynamics of Poynting flux creation in an individual time
slice of one of the 3-D simulations (Punsly 2006a). Surprisingly, the flares are of such a large
magnitude that they clearly standout above the background field fluctuations as evidenced
by figure 1. The flares are created in the equatorial accretion flow deep in the egosphere
between the inner calculational boundary at r=1.203 M and r= 1.6 M (the event horizon is at
r= 1.141 M). Powerful beams of Poynting flux emerge perpendicular to the equatorial plane
in the ergospheric flares and much of the energy flux is diverted outward along approximately
radial trajectories that are closely aligned with the poloidal magnetic field direction in the jet
(see figure 1). The situation is unsteady, whenever some vertical magnetic flux is captured
in the accretion flow it tends to be asymetrically distributed and concentrated in either
the northern or southern hemisphere. This hemisphere then receives a huge injection of
electromagnetic energy on time scales ∼ 60M .
The source of Poynting flux in KDJ resembles a nonstationary version of the ergospheric
disk (see Punsly and Coroniti (1990) and chapter 8 of Punsly (2001) for a review). The
ergospheric disk is modeled in the limit of negligible accretion and it is the most direct
manifestation of gravitohydromagnetics (GHM) Punsly (2001). A GHM dynamo arises
when the magnetic field impedes the inflow of gas in the ergosphere, i.e., vertical flux in an
equatorial accretion flow. The strong gravitational force will impart stress to the magnetic
field in an effort to move the plasma through the obstructing flux. In particular, the metric
induced frame dragging force will twist up the field azimuthally. These stresses are coupled
– 3 –
into the accretion vortex around a black hole by large scale magnetic flux, and propagate
outward as a relativistic Poynting jet. The more obstinate the obstruction, the more powerful
the jet. There are two defining characteristics that distinguish the GHM dynamo from a
Blandford-Znajek (B-Z) process, Blandford and Znajek (1977), on field lines that thread the
ergopshere:
1. The B-Z process is electrodynamic so there is no source within the ergosphere, it
appears as if the energy flux is emerging from the horizon. In the GHM mechanism,
the source of Poynting flux is in the ergospheric equatorial accretion flow.
2. In a B-Z process in a magnetosphere shaped by the accretion vortex, the field line
angular velocity is, ΩF ≈ ΩH/2 (where ΩH is the angular velocity of the horizon)
near the pole and decreases with latitude to ≈ ΩH/5 near the equatorial plane of
the inner ergosphere (Phinney 1983). In GHM, since the magnetic flux is anchored
by the inertia of the accretion flow in the inner ergosphere, frame dragging enforces
dφ/dt ≈ ΩH . One therefore has the condition, ΩF ≈ ΩH .
In order to understand the physical origin of the Poynting flux, these two issues are studied
below.
2. The KDJ Simulation
The simulation is performed in the Kerr metric (that of a rotating, uncharged black
hole), gµν . Calculations are carried out in Boyer-Lindquist (B-L) coordinates (r, θ, φ, t). The
reader should refer to Hawley and Krolik (2006) for details of the simulation. We only
give a brief overview. The initial state is a torus of gas in equilibrium that is threaded
by concentric loops of weak magnetic flux that foliate the surfaces of constant pressure.
The magnetic loops are twisted azimuthally by the differentially rotating gas. This creates
significant magnetic stress that removes angular momentum from the gas, initiating a strong
inflow that is permeated by magneto-rotational instabilities (MRI). The end result is that
after t = a few hundred M, accreted poloidal magnetic flux gets trapped in the accretion
vortex or funnel (with an opening angle of ∼ 60◦ at the horizon tapering to ∼ 35◦ at
r > 20M). This region is the black hole magnetosphere and it supports a Poynting jet. The
surrounding accretion flow is very turbulent.
In order to understand the source of the strong flares of radial Poynting flux, one
needs to merely consider the conservation of global, redshifted, or equivalently the B-L
coordinate evaluated energy flux (Thorne et al 1986). In general, the divergence of the
– 4 –
Fig. 1.— The source of Poynting flux. The left hand column is Sθ and the right hand column
is Sr in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920
M and t= 10000 M. The relative units (based on code variables) are in a color bar to right
of each plot for comparison of magnitudes between the six plots. The contours on the Sθ
plots are of the density, scaled from the peak value within the frame at relative levels 0.5
and 0.1. The contours on the Sr plots are of Sθ scaled from the peak within the frame at
relative levels 0.67 and 0.33. The inside of the inner calculational boundary (r=1.203 M)
is black. The calculational boundary near the poles is at 8.1◦ and 171.9◦. Notice that any
contribution from an electrodynamic effect associated with the horizon appears minimal.
The white contour is the stationary limit surface. There is no data clipping, so plot values
that exceed the limits of the color bar appear white.
– 5 –
Fig. 2.— The central engine. The left hand column is Bθ and the right hand column is ΩF
in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920 M and
t= 10000 M. The relative units (based on code variables) are in a color bar to right of each
plot for comparison of magnitudes between the plots. The calculational boundaries are the
same as figure 1. The contours on the Bθ plots are of the density, scaled from the peak value
within the frame at relative levels 0.5 and 0.1. There is no data clipping, so plot values that
exceed the limits of the color bar appear white.
– 6 –
time component of the stress-energy tensor in a coordinate system can be expanded as,
T νt ;ν = (1/
−g)[∂(
−g T νt )/∂(x
ν)] + Γ
µ , where Γ
t β is the connection coefficient and
g = −(r2 + a2 cos2 θ)2 sin2 θ is the determinant of the metric. However, the Kerr metric has
a Killing vector (the metric is time stationary) dual to the B-L time coordinate. Thus, there
is a conservation law associated with the time component of the divergence of the stress-
energy tensor. Consequently, if one expands out the inhomogeneous connection coefficient
term in the expression above, it will equate to zero. The conservation of energy evaluated
in B-L coordinates reduces to, ∂(
−g T νt )/∂(x
ν) = 0, where the four-momentum −T νt has
two components: one from the fluid, −(T νt )fluid, and one from the electromagnetic field,
−(T νt )EM. The reduction to a homogeneous equation with only partial derivatives is the
reason why the global conservation of energy can be expressed in integral form in (3.70) of
Thorne et al (1986). It follows that the poloidal components of the redshifted Poynting flux
are Sθ = −
−g (T θt )EM and S
r = −
−g (T rt )EM. We can use these simple expressions to
understand the primary source of the Poynting jet in KDJ. Figure 1 is a plot of Sθ (on the
left) and Sr (on the right) in KDJ at the last three time steps of data collection. Each frame
is the average over azimuth of each time step. This greatly reduces the fluctuations as the
accretion vortex is a cauldron of strong MHD waves. The individual φ = constant slices show
the same dominant behavior, however it is embedded in large MHD fluctuations. On the left
hand column of figure 1, density contours have been superimposed on the images to indicate
the location of the equatorial accretion flow. The density is evaluated in B-L coordinates
with contours at 0.5 and 0.1 of the peak value within r < 2.5M . Notice that in all three left
hand frames, Sθ is created primarily in regions of very high accretion flow density. In all
three of the right hand frames of figure 1, there is an enhanced Sr that emanates from the
ergosphere (defined by the interior of the stationary limit, rs = M +
M2 − a2 cos2 θ, note
that there are 40 grid points between r = 1.203M and rs at θ = π/2). This radial energy
beam diminishes precipitously just outside the horizon, near the equatorial plane in all three
time steps. The region in which Sr diminishes is adjacent to a region of strong Sθ that orig-
inates in the inertially dominated accretion flow in the inner ergosphere, 1.2M < r < 1.6M
(this region is resolved by 28 radial grid zones). In fact, if one looks at the conservation of
energy equation, the term ∂(Sθ)/∂θ is sufficiently large to be the source of ∂(Sr)/∂r at the
base of the radial beam in all three frames. This does not preclude the transfer of energy
to and from the plasma. It merely states that the magnitude is sufficient to source Sr. In
general, the hydrodynamic energy flux is negligible in the funnel. In order to illustrate this,
contours of Sθ are superimposed on the color plots of Sr. The contour levels are chosen to
be 2/3 and 1/3 of the maximum value of Sθ emerging from the dense equatorial accretion
flow. One clearly sees Sθ switching off where Sr switches on. We conclude that a vertical
Poynting flux created in the equatorial accretion flow is the source of the strong beams of
Sr. This establishes condition 1 of the Introduction.
– 7 –
The left column of figure 2 contains plots of the magnetic field component, Bθ ≡ Frφ, at
the three time steps. At every location in which Sθ is strong in figure 1, there is a pronounced
enhancement in Bθ in figure 2. Recall that the sign of Sθ is not determined by the sign of Bθ.
These intense flux patches penetrate the inertially dominated equatorial accretion flow in all
three frames. The density contours indicate that the regions of enhanced vertical field greatly
disrupt the equatorial inflow. As noted in the introduction, a GHM interaction is likely to
occur when the magnetic field impedes the inflow in the ergosphere. The regions of large
Bθ are compact compared to the global field configuration of the jet, only ∼ 1.0M − 2.0M
long. Considering the turbulent, differentially rotating plasma in which they are embedded,
these are most likely highly enhanced regions of twisted magnetic loops created by the MRI.
The strength of Bθ at the base of the flares is comparable to, or exceeds the radial magnetic
field strength. The situation is clearly very unsteady and vertical flux is constantly shifting
from hemisphere to hemisphere. The time slice t = 10000 M, although primarily a southern
hemisphere event, also has a significant contribution in the northern hemisphere (see the
blue fan-like plume of vertical Poynting flux in figure 1). The GHM interaction is provided
by the vertical flux that links the equatorial plasma to the relatively slowly rotating plasma
of the magnetosphere within the accretion vortex. The vertical flux transmits huge torsional
stresses from the accretion flow to the magnetosphere.
Further corroboration of this interpretation can be found by looking at the values of ΩF
in the vicinity of the Sr flares. In a non-axisymmetric, non-time stationary flow, there is still
a well defined notion of ΩF : the rate at which a frame of reference at fixed r and θ would
have to rotate so that the poloidal component of the electric field, E⊥, that is orthogonal
to the poloidal magnetic field, BP , vanishes. This was first derived in Punsly (1991) (see
the extended discussion in Punsly (2001) for the various physical interpretations), and has
recently been written out in B-L coordinates in Hawley and Krolik (2006) in terms of the
plasma three-velocity, vi and the Faraday tensor as
ΩF = v
φ − Fθr
rFφθ + gθθv
(Fφθ)2grr + (Frφ)2gθθ
. (2-1)
This expression was studied in the context of the simulation KDH, a/M = 0.95, in Hawley and Krolik
(2006). They found that a long term time and azimuth average yielded ΩF ≈ 1/3ΩH and
there was no enhancement at high latitudes as was anticipated by Phinney (1983). The t
= 10000 M time slice of KDH was generously provided to this author. At t = 10000 M,
there are no strong flares emerging from the equatorial accretion flow. Inside the funnel at
r < 10M , at t=10000 M, 0 < ΩF < 0.5ΩH .
The right hand column of figure 2 is ΩF plotted at three different time steps for KDJ.
By comparison to figure 1, notice that each flare in Sr is enveloped by a region of enhanced
– 8 –
ΩF , typically 0.7ΩH < ΩF < 1.2ΩH . The regions of the funnel outside the ergosphere are
devoid of large flares in Sr and typically have 0 < ΩF < 0.5ΩH , similar to what is seen in
KDH.. Unlike KDH, there are huge enhancements in ΩF at lower latitudes in the funnel.
It seems reasonable to associate this large difference in the peak values of ΩF in KDJ and
KDH (at t= 10000 M) with the spatially and temporally coincident flares in Sr that occur
in KDJ. Furthermore, this greatly enhanced value of ΩF indicates a different physical origin
for ΩF in the flares than for the remainder of the funnel or in KDH at t = 10000 M. The
most straightforward interpretation is that it is a direct consequence of the fact that the
flares originate on magnetic flux that is locked into approximate corotation with the dense
accreting equatorial plasma (i.e., the inertially dominated equatorial plasma anchors the
magnetic flux). In the inner ergosphere, frame dragging enforces 0.7ΩH < dφ/dt < 1.0ΩH
on the accretion flow. This establishes condition 2 of the Introduction.
3. Discussion
In this Letter we showed that in the last three data dumps of the 3-D MHD numerical
simulation, KDJ, the dominant source of Poynting flux originated near the equatorial plane
deep in the ergopshere. The phenomenon is unsteady and is triggered by large scale vertical
flux that is anchored in the inertially dominated equatorial accretion flow. The situation
typifies the ergospheric disk in virtually every aspect, even though there is an intense accre-
tion flow. There is one exception, unlike the ergospheric disk, the anchoring plasma rarely
achieves the global negative energy condition that is defined by the four-velocity, −Ut < 0,
because of the flood of incoming positive energy plasma from the accretion flow. The plasma
attains −Ut < 0 only near the base of the strongest flares seen in the φ = constant slices.
The switch-on of a powerful beam of Sr outside the horizon at r ≈ 1.3M in the a/M =
0.995 simulation, KDE, of Krolik et al (2005) was demonstrated in Punsly (2006a). It
seems likely the the source of Sr in KDE is Sθ from an ergopsheric disk. The ergospheric
disk appears to switch on at a/M > 0.95 as evidenced by the factor of 3 weaker Poynting
flux in KDH. Furthermore, if the funnel opening angle at the horizon in KDH at t= 10000
M is typical within ±5◦ then figure 5 and table 4 of Hawley and Krolik (2006) indicate
that only 35% to 40% of the funnel Poynting flux at large distances is created outside the
horizon during the course of the simulation. A plausible reason is given by the plots of Bθ
in figure 2. The vertical magnetic flux at the equatorial plane is located at r < 1.55M . The
power in the ergospheric disk jet ∼ [Bθ(SA)(ΩH)]2, where SA is the proper surface area
of the equatorial plane threaded by vertical magnetic flux (Semenov et al 2004; Punsly
2001). The proper surface area in the ergospheric equatorial plane increases dramatically
– 9 –
at high spin, diverging at a = M . For example, between the inner calculational boundary
and 1.55 M the surface area is only significant for a/M > 0.95 and grows quickly with a/M ,
exceeding twice the surface area of the horizon for a/M = 0.99. Thus, if Bθ in the inner
ergosphere were independent of spin to first order, then a strong ergospheric disk jet would
switch-on in the 3-D simulations at a/M > 0.95. Note that if the inner boundary were truly
the event horizon instead of the inner calculational boundary then this argument would
indicate that the ergospheric disk would likely be very powerful even at a/M = 0.95 and the
switch-on would occur at a/M ≈ 0.9. The implication is that a significant amount of large
scale magnetic flux threading the equatorial plane of the ergopshere (which implies a large
black hole spin based on geometrical considerations) catalyzes the formation of the most
powerful Poynting jets around black holes. Thus, we are now considering initial conditions
in simulations that are conducive to producing significant vertical flux in the equatorial plane
of the ergosphere.
It should be noted that 2-D simulations from a similar initial state of torii threaded by
magnetic loops have been studied in McKinney and Gammie (2004). However, the magnetic
flux evolution can be much different in this setting as discussed in Punsly (2006a) and poloidal
flux configurations conducive to GHM could be highly suppressed. In summary, there are
no interchange instabilities, so flux tubes cannot pass by each other or move around each
other in the extra degree of freedom provided by the azimuth. Thus, there is a tendency
for flux tubes to get pushed into the hole by the accretion flow. This is in contrast to the
formation of the ergospheric disk in Punsly and Coroniti (1990) in which buoyant flux tubes
are created by reconnection at the inner edge of the ergospheric disk and recycle back out
into the outer ergosphere by interchange instabilities. Ideally, a full 3-D simulation with a
detailed treatment of resistive MHD reconnection is preferred for studying the relevant GHM
physics.
I would like to thank Jean-Pierre DeVilliers for sharing his deep understanding of the
numerical code and these simulations. I was also very fortunate that Julian Krolik and John
Hawley were willing to share their data in the best spirit of science.
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This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0502225
	Introduction
	The KDJ Simulation
	Discussion
