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HomePosts tagged 'leaks'
— 'Global consequences' of lethal radiation leak at Fukushima; TEPCO botches recent measurement inside reactor; "insane" levels detected last year, exceed Chernobyl
February 14, 2018 February 14, 2018 admin leaks, radiation, TEPCO
From Sputnik News
Lethal levels of radiation have been observed inside Japan's damaged Fukushima nuclear power plant. And they are arguably way higher than you suspect.
According to Tokyo Electric Power Company (Tepco), radiation levels of eight Sieverts per hour (Sv/h) have been discovered within the Fukushima nuclear power plant, which was destroyed after a massive earthquake and a tsunami in March 2011.
Japanese Regulator Advocates Releasing Toxic Water Into Sea at Fukushima
Tepco, the company that operated the plant and is now tasked with decommissioning it, reported the discovery after making observations in a reactor containment vessel last month.
Eight Sv/h of radiation, if absorbed at once, mean certain death, even with quick treatment. One Sv/h is likely to cause sickness and 5.5 Sv/h will result in a high chance of developing cancer.
While 8 Sv/h is deadly, outside of Fukushima's Reactor Number 2 foundations of a much higher level of 42 Sv/h was detected.
A strange occurrence, and experts are still arguing what caused the discrepancy. One possible explanation is that cooling water washed radioactive material off debris, taking it somewhere else.
But here's a truly terrifying catch: according to the report, Tepco highly doubts the new readings, because, as was discovered later, a cover was not removed from the robot-mounted measurement device at the time of the inspection, NHK World reports.
Exactly one year ago, Sputnik reported that Tepco engineers discovered absolutely insane levels of radiation of about 530 Sv/h within the reactor. Such levels of radiation would kill a human within seconds. By comparison, the Chernobyl reactor reads 34 Sv/h radiation level, enough to kill a human after 20 minutes of exposure.
Japan Begins Disposal of Radioactive Waste From Fukushima Disaster
The levels of radiation within Fukushima reactor number 2 were so high that Tepco's toughest robot, designed to withstand 1000 Sv/h of radiation, had to be pulled out, as it started glitching due to high radiation levels. Nuclear experts called the radiation levels "unimaginable" at the time.
On November 2017, the New York Times and other news outlets reported a much smaller figure of 70 Sv/h of radiation, more or less on par with a 74 Sv/h reading gathered before an anomalous 530 Sv/h spike.
While that radiation dosimeter cover negligence prevents precise calculations, the actual picture inside Unit 2 is thought to be much worse.
Japanese state broadcaster NHK World quoted experts saying that if the cleaning of the stricken power plant is not properly addressed, it will result in major leak of radioactivity with "global" consequences.
Richard Black, director of the Energy and Climate Intelligence Unit, says that while the readings are not reliable, they still "demonstrate that, seven years after the disaster, cleaning up the Fukushima site remains a massive challenge — and one that we're going to be reading about for decades, never mind years."
Mycle Schneider, independent energy consultant and lead author of the World Nuclear Industry Status Report, criticized Tepco, saying the power company has "no clue" what it is doing.
"I find it symptomatic of the past seven years, in that they don't know what they're doing, Tepco, these energy companies, haven't a clue what they're doing, so to me it's been going wrong from the beginning. It's a disaster of unseen proportions."
In observing the poor maintenance of plant radiation leaks, Schneider also pointed out that the company stores nuclear waste at the site in an inappropriate way.
"This is an area of the planet that gets hit by tornadoes and all kinds of heavy weather patterns, which is a problem. When you have waste stored above ground in inappropriate ways, it can get washed out and you can get contamination all over the place."
https://sputniknews.com/environment/201802041061337934-global-consequences-fukushima-leak/
Posted under Fair Use Rules.
— Lethal radiation outside Fukushima reactor foundations- 42 sieverts/hour; expert says contaminated water going into ocean would cause global disaster, TEPCO "hasn't a clue what it's doing"
February 14, 2018 February 14, 2018 admin Chernobyl, contamination, global, leaks, Pacific Ocean, reactors
"It needs to be clear that this problem is not gone, this is not just a local problem. It's a very major thing." — Mycle Schneider
From ENE News:
The Independent, Feb 2, 2018 (emphasis added): Fukushima nuclear disaster: Lethal levels of radiation detected in leak… Expert warns of 'global' consequences unless the plant is treated properly… [Tepco] found eight sieverts per hour of radiation, while 42 [sieverts] were also detected outside its foundations… It came as Tepco said the problem of contaminated water pooled around the plants three reactors that is seeping into the ground has caused a major headache in its efforts to decommission the plant… Mycle Schneider, an independent energy consultant and lead author of the World Nuclear Industry Status Report, said that Tepco "hasn't a clue what it is doing" in its job to decommission the plant. He added that the contaminated water that is leaking at the site could end up in the ocean if the ongoing treatment project fails and cause a "global" disaster, he told The Independent… "I find it symptomatic of the past seven years, in that they don't know what they're doing, Tepco, these energy companies haven't a clue what they're doing, so to me it's been going wrong from the beginning. It's a disaster of unseen proportions." Mr Schneider added that the radiation leaks coupled with the waste from the plant stored in an "inappropriate" way in tanks could have global consequences… "This can get problematic anytime, if it contaminates the ocean there is no local contamination, the ocean is global, so anything that goes into the ocean goes to everyone." He added: "It needs to be clear that this problem is not gone, this is not just a local problem. It's a very major thing."
NHK, Feb 1, 2018: High radiation detected at Fukushima plant… A remote-controlled inspection of the Unit 2 reactor containment vessel last month detected a maximum of 8 sieverts per hour of radiation… [Tepco] said the radiation reading was taken near what appeared to be fuel debris, the term used to describe a mixture of molten fuel and broken interior parts… radiation levels remain so high that they present a major challenge to decommissioning work. During the probe, 42 sieverts per hour of radiation was also detected outside the foundations of the reactor. But officials said they have doubts about the accuracy of the reading because a cover had not been removed from the measuring instrument at the time. They added that they don't know why radiation levels were lower near the suspected fuel debris than around the foundations. They gave a number of possible reasons, such as that cooling water may have washed radioactive materials off the debris…
RT, Feb 2, 2018: An inspection of the Fukushima nuclear plant has detected extremely highamounts of radiation, says operator TEPCO… Experts can't explain why radiation levels in fuel debris were lower than outside the reactor's foundations…
Sputnik, Feb 4, 2018: 'Global Consequences' of Lethal Radiation Leak at Destroyed Japan Nuclear Plant… While 8 Sv/h is deadly, outside of Fukushima's Reactor Number 2 foundations… a much higher level of 42 Sv/h was detected. A strange occurrence, and experts are still arguing what caused the discrepancy. One possible explanation is that cooling water washed radioactive material off debris, taking it somewhere else. But here's a truly terrifying catch: according to the report, Tepco highly doubts the new readings, because, as was discovered later, a cover was not removed from the robot-mounted measurement device at the time of the inspection, NHK World reports… While that radiation dosimeter cover negligence prevents precise calculations, the actual picture inside Unit 2 is thought to be much worse…
Watch NHK's broadcast here
Leak worsens in massive Hanford tank holding nuclear waste
April 18, 2016 admin Hanford, leaks, tanks
From King5 TV
VIDEO on website
Susannah Frame, KING
(Photo: KING)
A leak in a massive nuclear waste storage tank at the Hanford Site has expanded significantly, KING 5 learned this weekend.
After leak detector alarms sounded early Sunday morning, crews at Hanford lowered a camera into the two-foot-wide space between the tank's inner and outer walls. They discovered 8.4 inches of radioactive and chemically toxic waste has seeped into the annulus.
The U.S. Department of Energy released a statement Monday calling the leak an "anticipated" outcome of an ongoing effort to empty the tank in question. The Washington state Department of Ecology said, "There is no indication of waste leaking into the environment or risk to the public at this time."
But one former tank farm worker said the leak should be considered a major problem.
"This is catastrophic. This is probably the biggest event to ever happen in tank farm history. The double shell tanks were supposed to be the saviors of all saviors (to hold waste safely from people and the environment)," said former Hanford worker Mike Geffre.
Geffre is the worker who first discovered that the tank, known as AY-102, was failing in 2011. In a 2013 series, "Hanford's Dirty Secrets," the KING 5 Investigators exposed that the government contractor in charge of the tanks, Washington River Protection Solutions (WRPS), ignored Geffre's findings for nearly a year. The company finally admitted the problem in 2012.
Until now, the leak found by Geffre was very slow. The liquid would almost immediately dry up, leaving a salt-like substance on the floor of the two-foot space between the tank's walls, called the annulus.
Approximately three weeks ago, work began to pump out the contents of AY-102, which has the capacity to hold one million gallons of the deadly waste. The state of Washington has been pressuring the federal government, which owns Hanford, to pump out AY-102 for three-and-a-half years because of the cracking and slow leaking discovered by Geffre in 2011. Sources told KING the disturbance caused by the pumping must have exacerbated the leak: essentially blowing a hole in the aging tank allowing the material to leak more quickly into the outer shell.
Tank AY-102 is one of 28 double-shell tanks at Hanford (there are 177 underground tanks total) holding nuclear byproducts from nearly four decades of plutonium production on the Hanford Nuclear Site, located near Richland. Initially the plutonium was used to fuel the bombed dropped on Nagasaki, Japan, in World War II.
Plutonium production continued throughout the Cold War. Since 1989 the work at Hanford has focused solely on cleanup – the most difficult being getting rid of the liquid waste left behind that threatens the health of people, wildlife and the environment, including the nearby Columbia River.
The new leak poses problems on several fronts. The outer shell of AY-102 does not have the exhaust or filtration system needed to keep the dangerous gases created by the waste in check. Workers have been ordered to wear full respiratory safety gear in the area, but the risk remains.
"The hazards to workers just went up by a factor of 10," said Geffre.
In addition, the breakdown calls into question the viability of three other double-shell tanks at Hanford that have the exact design of AY-102.
"The primary tanks weren't designed to stage waste like this for so many years," said a current worker. "There's always the question, 'Are the outer shells compromised'"?
The accumulation of waste in the outer shell also means the deadliest substance on earth is that much closer to the ground surrounding the tank. And currently there is no viable plan in place to take care of it.
"It makes me sad that they didn't believe me that there was a problem in 2011," said Geffre. "I wish they would have listened to me and reacted faster. Maybe none of this would be happening now. It's an example of a culture at Hanford of 'We don't have problems here. We're doing just fine.' Which is a total lie," said Geffre.
Full statement from Dept. of Energy:
The U.S. Department of Energy (DOE) and Hanford tank farms contractor Washington River Protection Solutions (WRPS) are evaluating recently identified, anticipated changes in the amount of waste between the inner and outer shells (the annulus) of Hanford double-shell tank AY-102.
Since March 3, DOE and WRPS have been retrieving waste from Tank AY-102, and to date approximately 95 percent of the material has been retrieved. Early on the morning of April 17, crews identified an increase in the waste level in the tank's annulus. Early on the morning of April 18, a slight decrease in the height of the waste in the annulus was detected. Out of an abundance of caution, DOE and WRPS are in the process of evaluating the tank's condition.
DOE and WRPS are committed to ensuring the safety of the Hanford workforce, the public and the environment.
WA Dept. of Ecology statement:
RICHLAND – An alarm was activated on Sunday, April 17, at the Hanford Nuclear Reservation while crews working for the U.S. Department of Energy were pumping waste out of the double-shell tank AY-102.
There is no indication of waste leaking into the environment or risk to the public at this time.
The Department of Energy notified the Washington Department of Ecology that the leak detector alarm went off.
The alarm indicates an increase in waste seeping from the primary tank into the space between the primary and secondary tank, known as the annulus.
Crews have been actively removing waste from AY-102 since March because mixed radioactive and chemical waste had previously leaked into the secondary containment area. Approximately 20,000 gallons of waste remains from the original 800,000 gallons in the tank.
According to the Department of Energy, the removal work is currently on hold while engineers evaluate the situation and prepare a plan to recover the material that leaked between the two walls of the tank.
This morning, an Ecology Nuclear Waste Program engineer assessed the situation with the Department of Energy waste retrieval engineers to assure that contingency response plans are being followed.
Additional leaking into the annulus was a known possibility during pumping and is addressed in the Department of Energy's contingency plan that was submitted to Ecology as part of a Settlement Agreement. That plan delineates actions for Energy to take.
Ecology continues to monitor the situation as spelled out in the settlement agreement that directs waste retrieval.
There are 28 double-shell tanks at the Hanford site.
Copyright 2016 KING
Posted under Fair Use Rules
http://www.king5.com/news/local/investigations/catastrophic-event-at-hanford-prompts-emergency-response/140990679
Leaking beachfront nuclear reactor near Miami threatening Florida everglades
March 10, 2016 admin algae blooms, Biscayne Bay, Florida, Florida Power & Light, FPL, leaks, Miami, Tritium, Turkey Point
Posted on Zero Hedge
Submitted by Claire Bernish via TheAntiMedia.org,
According to a study released by Miami-Dade County Mayor Carlos Gimenez on Monday, the waters of Biscayne Bay measured 215 times the level of radioactive tritium as is found in normal ocean water.
Tritium is a radioactive isotope traceable to nuclear plant cooling tower operations. In this case, the leak appears to be emanating from the aging canals in the Turkey Point Nuclear Generating Station located nearby.
"This is one of several things we were very worried about," said South Miami Mayor and biological sciences professor, Philip Stoddard, as the Miami New Times reported. "You would have to work hard to find a worse place to put a nuclear plant, right between two national parks and subject to hurricanes and storm surge."
Biscayne Bay harbors one of the largest coral reefs on the planet and is situated near the Everglades. Hot, salty water from the canals appears to be flowing back into both national parks, which has caused concern among environmentalists and others from the time Turkey Point planned to expand its reactors in 2013.
"They argued the canals were a closed system, but that's not how water works in South Florida," Stoddard remarked.
"How much damage is that cooling canal system causing the bay is a question to be answered," Everglades Law Center Attorney Julie Dick told the Miami Herald prior to reviewing the report. "There are a lot more unknowns than knowns and it just shows how much more attention we need to be paying to that cooling canal system."
Tritium, a hydrogen isotope, is considered a precursor indication of leaks from nuclear plants, as it 'travels' or spreads faster than, and often precedes, other radioactive agents.
"While the tritium levels far fall below levels experts consider dangerous, the telltale tracer provided the critical link that high levels of ammonia and phosphorus in sections of bay bottom — pollution that is more damaging to marine life — likely came from the canals," the Herald explained. Samples for the county monitoring study were gathered during December and January — and the tritium levels seem to show Florida Power & Light in violation of both local water laws and federal operating permits.
FPL, which operates Turkey Point, will likely receive another violation due to the leak — the county issued a citation in October for tainted groundwater — to force FPL to bring the plant into compliance, the Herald reported Tuesday.
After news of the report made headlines, critics, including environmentalists, nearby rock miners, and Rep. Jose Javier Rodriguez, came forward in full force calling for the Environmental Protection Agency to intervene in the matter.
"For years our state regulators have failed to take seriously the threat to our public safety, to our drinking water, and to our environment posed by FP&L's actions at Turkey Point," Rodriguez asserted, according to the New Times."Evidence revealed this week of radioactive material in Biscayne Bay is the last straw and I join those calling on the U.S. EPA to step in and do what our state regulators have so far refused to do — protect the public."
The leak also serves as possible confirmation for environmentalists who have suspected radioactive leaks from Turkey Point as the cause of algae blooms appearing in the bay for years.
"Biscayne Bay has not traditionally had algae blooms," explained executive director for Miami Waterkeeper Rachel Silverstein, reported the Herald. "That's from pollution. From sewers, septic tanks and now we know, cooling canals."
Indeed, though FPL claims it continues to protect the health of the bay, as the Herald noted, Turkey Point has created issues for the waterways since the facility began producing more energy three years ago. "When you look at the big picture," FPL environmental director, Matt Raffenberg, insisted, the canals "are not impacting Biscayne Bay."
At a meeting on Tuesday, county commissioners discussed the imperative need to bring FPL and Turkey point into compliance with the law.
"We've had stop gap measures we've approved," Gimenez said. "So far they've not proved to be the solution."
Referencing the last time FPL was forced to implement changes following a lawsuit in the 1970s, he added, "It's time we enter the 21st century."
FPL's continued problems with Turkey Point might have finally crossed the legal line by violating the federal Clean Water Act.
"There's a certain validation to critics in seeing this result in the study," Stoddard said. "But more important, it's now crossed the threshold of federal law here."
http://www.zerohedge.com/news/2016-03-10/leaking-beachfront-nuclear-reactor-near-miami-threatening-florida-everglades
"Leak First, Fix Later: Uncontrolled and Unmonitored Radioactive Releases at Nuclear Power Plants" — report
February 11, 2016 admin Beyond Nuclear, leaks, NRC, radioactive, radioactive hydrogen, releases, strontium, Tritium
LEAK FIRST, FIX LATER
Uncontrolled and Unmonitored Radioactive Releases from Nuclear Power Plants
A Beyond Nuclear Report
By Paul Gunter, Director, Reactor Oversight Project
Revised Edition: March 2015
"Leak First, Fix Later" was first published in April 2010. Now nearly five years later, Beyond Nuclear takes another look at the problem of aging and deteriorating piping systems carrying
radioactive liquids that still run under every nuclear power plant.
Nuclear power plants have an extensive network of piping systems dozens of which transport liquids that contain radioactive isotopes including tritium — a radioactive form of hydrogen — and long-lived strontium-90. These piping systems are not adequately inspected or maintained due to their inaccessibility.
U.S. reactors continue to experience leaks and spills of radioactive material into groundwater the unmonitored pathways from unknown and unanticipated sources.
Now, five years after our initial 2010 report, Beyond Nuclear has determined that the NRC has failed to mandate any corrective action programs that focus on inspection and maintenance programs aimed at groundwater protection by preventing ongoing radioactive leaks and contamination of water resources.
Full report at: http://static1.1.sqspcdn.com/static/f/356082/26211376/1431107993237/LeakFirst_ReportLater_BeyondNuclear_March2015.pdf?token=z1pOj4O3mtw9GUIJX27aU%2FNIDIU%3D


Swarovski has long since had a partnership with Vienna Opera Ball wherein they create the delightful tiaras worn by the debutantes at the ball. This year the Swiss crystal company enlisted Italian designers Dolce and Gabbana to design the tiaras- and the results were spectacular.
The designers drew inspiration from Mozart's opera Le nozze di Figaro, specifically the words: "Giovani lieti – fiori spargete!" ("Happy maids, scatter flowers!"). This phrase was interpreted into the floral design of the tiaras created for the 2018 Vienna Opera Ball. The design uses 702 Swarovski crystals and "Xirius Chatons" in colors like Crystal and Golden Shadow in addition to six rose-colored enamel flowers. The flawless designs are absolutely fit for a princess.
"Two opera ball dreams have come true for me this year: firstly, that thanks to the outstanding commitment of Nadja Swarovski and her team, we have once again managed to secure one of the world's most prestigious design houses for the Swarovski-Tiara design for the Vienna Opera Ball 2018: Dolce & Gabbana. Moreover, Dolce & Gabbana, whose style I personally love, have not only made my dream of a connection with opera come true, but exceeded my every expectation. The designers selected a passage from my suggested Mozart opera Le nozze di Figaro, which was then used as the inspiration for the tiara design. It is the ultimate crowning of the overall concept 'Alles Oper!' – I'm really happy!" said Maria Großbauer, Organizer of the Vienna Opera Ball.


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Hand Tool School #37: The Beginner Workbench
Made from construction lumber and just a few tools
By Shannon Rogers - July 6, 2017
"What is the best beginner workbench?" is a loaded question, and one I get all the time.
I usually tell beginning woodworkers to hold off on building a bench right away because they don't know what they don't know yet and a workbench isn't actually needed to get started. For example, I've built a few things in the garage of my in-laws' place in Maine using a sawhorse and a rickety table. But the time has come to build a proper workbench in that space.
This workbench is designed to be possibly the first bench that a new woodworker would build. Though it could also end up being the workbench you use for many years as it is rock solid and highly functional. It is built using just construction lumber, seven boards to be exact, and just four tools.
The beginner's workbench should be simple to build, yet highly customizable for the future. It should be a rock solid, blank chassis that doesn't require a lot of lumber or tools to build.
Its a straightforward built that introduces the new woodworker to some key concepts while not sweating the details that could make this project drag on for months and months. While hardly a new design, I think the approach I took to it could enable the brand new woodworker with no tools and no bench to actually get started building, and come out of this experience a better woodworker.
The full build is coming up soon. Semester 5 at The Hand Tool School will be a back to basics course designed to speak to the brand new woodworker who has no tools and has possibly never picked up a saw or plane. It will consist of a few introductory lessons and then many applied lessons in the course of building three essential projects for their shop.
It's in production now and scheduled to be released this Fall. Watch this space or subscribe to my email list to be the first to hear about it when it becomes available.

redpajama_set_name: RedPajamaCommonCrawl


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Q: Tokudb file space low error I have my mongo data files in a temporary directory.
I start the server by giving :
    ./mongod --dbpath /nlu/users/ig/mongodb
It was working fine and i could write into the db using a python driver.  I get this error today.
Please help on how to solve.
Also how to set tokudb_fs_reserve_percent?
Thanks!
bash-3.2$ ./mongod --dbpath /nlu/users/ig/tokudata
Mon Jul 15 18:22:28  nssize is a deprecated parameter
Mon Jul 15 18:22:28 [initandlisten] TokuMX starting : pid=9880 port=27017               dbpath=/nlu/users/indrani_gorti/tokudata 64-bit host=unv-ig
Mon Jul 15 18:22:28 [initandlisten] TokuMX mongod server v1.0.1-mongodb-2.2.4, using TokuKV rev 8f8d303
Mon Jul 15 18:22:28 [initandlisten] git version: 101a2f8
Mon Jul 15 18:22:28 [initandlisten] build info: Linux lex1.tokutek.com 2.6.18-    308.8.2.el5 #1 SMP Tue Jun 12 09:58:12 EDT 2012 x86_64 BOOST_LIB_VERSION=1_49
Mon Jul 15 18:22:28 [initandlisten] options: { dbpath: "/nlu/users/indrani_gorti/tokudata" }
Mon Jul 15 18:22:29 [initandlisten] [tokumx] startup
Mon Jul 15 18:22:29 [initandlisten] [tokumx] locktree max memory set to 418906316 bytes.
Mon Jul 15 18:22:32 2013 Tokudb file system space is low
Mon Jul 15 18:22:32 [initandlisten] ERROR: listen(): bind() failed errno:98 Address already in use for socket: 0.0.0.0:27017
Mon Jul 15 18:22:32 [initandlisten] ERROR:   addr already in use
Mon Jul 15 18:22:32 [initandlisten] now exiting
Mon Jul 15 18:22:32 dbexit:
Mon Jul 15 18:22:32 [initandlisten] shutdown: going to close listening sockets...
Mon Jul 15 18:22:32 [initandlisten] shutdown: going to flush diaglog...
Mon Jul 15 18:22:32 [initandlisten] shutdown: going to close sockets...
Mon Jul 15 18:22:32 [initandlisten] shutdown: going to close databases...
Mon Jul 15 18:22:32 [initandlisten] shutdown: going to shutdown TokuKV...
Mon Jul 15 18:22:32 [initandlisten] [tokumx] shutdown
Mon Jul 15 18:22:32 [TTLMonitor] ERROR: Client::shutdown not called: TTLMonitor
Mon Jul 15 18:22:32 [websvr] ERROR: listen(): bind() failed errno:98 Address already in use for socket: 0.0.0.0:28017
Mon Jul 15 18:22:32 [websvr] ERROR:   addr already in use
Mon Jul 15 18:22:33 [initandlisten] shutdown: removing fs lock...
Mon Jul 15 18:22:33 dbexit: really exiting now

A: The fatal error here is this:
Mon Jul 15 18:22:32 [initandlisten] ERROR: listen(): bind() failed errno:98 Address already in use for socket: 0.0.0.0:27017
Mon Jul 15 18:22:32 [initandlisten] ERROR:   addr already in use

This indicates that you already have another mongod running on port 27017.  You need to either use the other mongod process that's running, shut it down before starting this one, or use a different port for this mongod process.
In TokuMX, the setting you're looking for is called --fsRedzone, rather than tokudb_fs_reserve_percent.

redpajama_set_name: RedPajamaStackExchange


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text:
\section{Introduction}

The phenomena described by numerical simulations are usually highly
simplified compared to the physical reality. However simplified, these
phenomena are sometimes complicated enough to challenge our ability to
understand them. The flow of gas onto a gravitating accretor moving
supersonically is a classic astrophysical problem (Hoyle \& Lyttleton
1939, Bondi \& Hoyle 1944) which is relevant in many astrophysical
contexts such as wind fed X-ray binaries, supermassive black holes,
star formation, and also galaxies in a cluster (see the recent review
by Edgar 2004).
Early numerical simulations in 2D (Matsuda et al.  1987, Fryxell \& Taam
1988) revealed that this flow is unstable. 3D simulations (Matsuda
et al.  1991, Ruffert \& Arnett 1994) confirmed the unstable character of
the flow, displaying however a weaker variability than in 2D.
The instability is not understood even in the simple case of an ideal
uniform gas.
Is this instability physical, or numerical? Is it the same instability
mechanism in 2D and 3D? How can this unstable behaviour be extrapolated
to small accretor sizes, which are relevant for wind fed X-ray binaries
but out of reach of numerical simulations?
Some authors recently doubted that this instability is physical
(Pogorelov, Ohsugi \& Matsuda 2000). The most recent relativistic
simulations by Font \& Ibanez (1998a, 1998b) and Font, Ibanez \&
Papadopoulos (1999) also showed stable flows.
What about the published instability mechanisms which have been
proposed over the years? What is left of the longitudinal and
transverse  instabilities of the accretion line (Cowie 1977, Soker
1990, 1991), the "relatively simple" mechanism based on the shock
opening angle by Livio et al.  (1991), the "vortex shedding in the Von
Karman manner" (Koide et al.  1991, Matsuda et al.  1991, 1992)?
The present uncertain situation reveals how unconvincing the proposed
mechanisms were. Some were inconclusive, lacked quantitative criteria,
and others might also have been incompletely understood. The present
work aims at clarifying the different instability mechanisms and
confront them with existing numerical simulations, paying particular
attention to possible numerical artefacts.\\
The general trends based on the existing simulations, and the proposed
mechanisms are recalled in Sect.~2. The longitudinal and transverse
instabilities of the accretion line are revisited in Sect.~3 and
Sect.~4. The advective-acoustic instability is adapted to the BHL flow
in Sect. ~5.
This enables a new look at the simulations in Sect.~6, in an attempt to
reconcile them.
The basis of new simulations, free of numerical artefacts, testing
these ideas, is described in Sect.~7. For the sake of the clarity of
the paper, the main text contains only the most important equations,
which summarize the analytical arguments proven in appendices A to H.

\section{An overview of existing simulations of BHL accretion and
proposed instability mechanisms}

\subsection{Numerical simulations are numerous}

\begin{table*}
\caption[]{Overview of the published numerical simulations of BHL
accretion using a polytropic equation of state and a totally absorbing
accretor: plane accretion in 2D, 3D axisymmetric accretion and full 3D
accretion. The first column contains an abreviated reference which can
be found in the bibliography.}
\label{tableoverview}
\centering
\begin{tabular}{ccccccc}
\hline\hline
          ref.           & grid   &     Mach    &   accretor  &    index
    &    transverse   & stability \\
                       &             &   number  &   $r_*/r_{\rm A}$
&$\gamma$ & gradients& \\
\hline\\
3D& & & & & & \\
\hline\\
R99     &    Cart.  &  1.4 - 10 &   0.02 - 1  &   1.01, ${4/3}$,
${5/3}$ &  den & no \\
R97     &    Cart.  &  0.6 - 10 &   0.02 - 1  &   ${4/3}$, ${5/3}$    &
    vel & no \\
R96       &  Cart.    & 0.6 - 10  &  0.02 - 1   &  ${1.01}$   & -  &
no     \\
R95       &  Cart.    & 0.6 - 10  &  0.02 - 1   &  ${4/3}$   & -  &
no    \\
RA95 &  Cart.    &  3      &   0.01 - 10  &  ${5/3}$    & vel &
no \\
RA94 &  Cart.    &  3      &   0.01 - 10  &  ${5/3}$    & - &       no
\\
R94       &  Cart.    & 0.6 - 10  & 0.02 - 10    & ${5/3}$     & -  &
no    \\
IMS93  &   cyl.  & 3      &     0.125    &   isothermal   &   den/vel
& no\\
MIS92 &  Cart.     & 3                  &   0.1   &  $1.005$ - ${5/3}$
   & - &       no           \\
MSS91  & Cart.   &  3          & 0.06 - 0.25    & ${5/3}$
& - &   no  \\
SMA89 & curv. & 1.4& 0.1 & ${5/3}$& vel& quasi \\
LSK86       & Cart.   &   3, 16   &  0.15   &   ${7/6}$ - ${5/3}$    &
den & quasi  \\
SLK86        & Cart.   &   2, 4   &  0.15   &   $1$    & den & yes  \\
\hline\\
3D axisym.& & & & & & \\
\hline\\
POM00 & polar   &    3 - 10  &  0.05    &    1.01, 1.4, ${5/3}$   & - &
        yes   \\
FI98a &   polar &   0.6 - 10  &   0.1 - 2.4 &     1.1, ${4/3}$, ${5/3}$
   & - & yes   \\
KMS91      &  polar      &  1.4 - 10  & 0.005 - 0.015  & ${5/3}$
& - &     no  \\
MSS89   &   polar &  1.4    &   0.01 - 0.05  &     ${5/3}$      & - &
no       \\
SMA89 & curv. & 1.4& 0.1 & ${5/3}$&  - &yes \\
PSS89   & polar & 0.6 - 5   &  0.125  & 1.1, ${4/3}$, ${5/3}$, 2   & -
&  yes \\
FTM87 & polar & 1.4 -  4 & 0.016 - 0.13& ${5/ 3}$ &- & no\\
SMT85&  polar & 0.6 - 5 &0.1 &1.1, ${4/3}$, ${5/3}$& -&yes  \\
H79 &  polar & 0.6 - 3.6&0.01 & ${4/3}$& -& - \\
H71 &  polar &0.6 - 2.4 &0.01&${5/3}$ & -& -\\
\hline\\
2D planar& & & & & & \\
\hline\\
POM00 & polar     &    3 - 10  &  0.05    &    1.4, ${5/3}$   &
-/den/vel&        yes  \\
POM00 & polar     &    3 - 10  & 0.05     &    1.01   & -&
no/yes   \\
POM00 & polar     &    4  &   0.05  &    ${4/3}$   & den&        no
\\
FIP99  & polar   &   5      &  0.25  &    ${4/3}$, ${5/3}$, 2         &
- &    yes   \\
FI98b &  polar & 3 - 10 &  0.25    &      1.1, ${4/3}$, ${5/3}$     & -
& yes  \\
SMA98   & polar    &     1 - 16   & 0.005 - 0.05 & isothermal     & - &
   no  \\
BLT97 &   polar &  4    &    0.001, 0.005   &   ${4/3}$    & -/den/vel
&     no \\
ZWN95 & Cart.& 3 &0.03 - 0.13 & ${5/3}$ & - & yes \\
ZWN95 & Cart.& 4 &0.03 - 0.13 & ${4/3}$ & den & no \\
BA94 & SPH & 3&0.04 - 0.13 &1.1, 1.3, 1.5 & - & no  \\
IMS93  &   Cart.  &   3      &     0.13   &    isothermal   &  den/vel
& no \\
MIS92 &  Cart.    & 1.4 - 10    &     0.04 - 0.3    &   1.005 - ${5/3}$
  & - &        no          \\
MSS91 &  Cart.  &   3 - 5    &  0.06 - 0.25  &  1.2, ${5/3}$    & - &
no     \\
SMA89 & curv. & 3& 0.1& 1.5, 2& vel&no  \\
TF89 &  polar &    4     &    0.037   &     ${4/3}$       &  vel  &no
\\
FT88 &  polar &    4     &    0.037   &     ${4/3}$       &  den  &no
\\
MIS87  & curv.  &  1 - 5   &   0.03 - 0.6  &    ${4/3}$, ${5/3}$    &
binary & no \\
ABM87        & SPH   &   3    & 0.13, 0.15   &   1.5    & vel, den &
yes\\
\hline
\end{tabular}
\end{table*}

Numerical simulations of the BHL problem started with the work of Hunt
(1971). The instability first appeared in Matsuda et al.  (1987), Fryxell
\& Taam (1988). The many subsequent simulations are listed in
Table~\ref{tableoverview}. Simulations involving more complicated
ingredients such as realistic heating and cooling (e.g.  Blondin et al.
1990, Taam, Fu \& Fryxell 1991) are not included for the sake of
simplicity. The simulations of  Table~\ref{tableoverview} are divided
into three groups corresponding to plane accretion, axisymmetric
accretion and full 3D accretion.
In each of these groups, the listing in chronological order follows the
progress in computing speed and numerical techniques over the last 30
years. This progress enabled the simulation of smaller and smaller
accretors, improving the first attempts by a factor 10. Denoting by
$r_{\rm A}\equiv 2GM/v_\infty^2$ the accretion radius of an accretor of
mass $M$ and velocity $v_\infty$, the most recent simulations reach
$r_*/r_{\rm A}=0.005$ in axisymmetric flows (KMS91), $r_*/r_{\rm
A}=0.001$ in planar accretion (BLT97), and $r_*/r_{\rm A}=0.01$ in 3D
accretion (RA94). Global trends can be summarized as follows:
\par-The shock is always attached to the accretor in simulations of 2D
planar flow. By contrast, 3D simulations revealed a detached bow shock,
ahead of the accretor, if the accretor is small enough and
$\gamma\ge4/3$. A calculation in Appendix~A suggests that the shock
should be detached in planar flows with $\gamma\sim3$.
\par-Although the strength of the instability varies from one code to
another, the instability is found in numerical simulations using any of
the coordinate systems, Cartesian, polar, cylindrical or special
curvilinear, even with SPH. If numerical, the phenomenon is not
specific to a particular grid or a specific method.
\par-Simulations of plane accretion exhibit the most unstable
behaviour. The shock moves sideways in a flip-flop manner (MIS87). The
instability is strongest for small accretors, possibly for intermediate
Mach numbers (${\cal M}=3$ according to MIS87 and POM00).
\par-The presence of velocity or density gradients in the transverse
direction of the flow was considered in the earliest unstable
simulations (MIS87, FT88, SMA89), but MSS91 realized that this
ingredient is not crucial for instability. This conclusion was
challenged by ZWN95 who confirmed the flip-flop instability when the
accretor is a square single cell (as in MSS91), but found a stable flow
when the accretor is spatially resolved and modelized as a polygon.
 From their point of view, the instability is related to the detachement
of the shock.
\par -Stable planar accretion flows were also found by FI98b and FIP99
who considered a rather large ratio $r_*/r_{\rm A}=0.25$ and POM00
whose method is further discussed below.
\par-Axisymmetric flows are generally stable, with few exceptions.
Among them, SMT85, FTM87 and MSS89 showed vortex shedding when the
accretor is a hard, non absorbing sphere : in this case the problem is
similar to the classical flow around a  sphere, modified by gravity. An
important exception to the stability of axisymmetric flow is KMS91, who
also found vortex shedding for an absorbing accretor, if ${\cal M}\ge2.4$. It
can be noted that the accretor size they considered is the smallest
ever used in axisymmetric simulations.
\par-The instability of full 3D accretion is never as strong as the
flip-flop observed for planar flows, but seems present in all published
simulations, at least for small enough accretors.
Even the earliest simulations of LSK86 and SMA89 showed some persistant
oscillations. The instability seems to be more violent if the shock is
detached ($\gamma=4/3$ and $5/3$).

\subsection{Several mechanisms were proposed}

The question of the stability of BHL accretion could in principle be
solved by performing a perturbation analysis on a stationary solution
such as obtained by POM00. This procedure would be very heavy,
and has never been achieved. Using various simplifications, six
different physical instability mechanisms have been proposed so far. In
chronological order:
\par (i) longitudinal instability of the accretion line (Cowie 1977),
\par (ii) transverse instability of the accretion line (Soker 1990,
1991),
\par (iii) shock opening angle (Livio et al.  1991),
\par (iv) vortex shedding in the Von Karman manner (Koide et al.  1991,
Matsuda et al.  1991, 1992),
\par (v)  local Rayleigh-Taylor (RT) and Kelvin-Helmhotz (KH)
instabilities (Foglizzo \& Ruffert 1999),
\par (vi) advective-acoustic cycle (Foglizzo \& Tagger 2000, Foglizzo
2001, 2002).\\
Some of the proposed mechanisms (iii, iv, v) are interesting ideas,
which need to be developed, but which are not conclusive at present:
\par -The calculation of Livio et al.  (1991) is not a stability analysis
of the shock surface, but rather an attempt to express in equations the
idea that the pressure should decrease along the shock surface.
Since no growth rate or typical timescale is computed, extrapolating on
the possibility that this is responsible for the violent and chaotic
behaviour observed in 2D simulations is not convincing.
\par -Vortex shedding, by the interaction of the incoming gas with the
"atmosphere" captured by the accretor, was demonstrated by SMT85 and
MSS89 in simulations where the accretor is non-absorbing.
The fate of this instability in BHL accretion, where the gas is
absorbed at supersonic velocity, is rather speculative.
\par -The local analysis of FR99 investigates two natural causes of
instability (RT and KH), and concluded that these are not
quantitatively convincing without a feedback mechanism.\\

Among the cited mechanisms, the only conclusive stability analysis are
those of Cowie (1977) and Soker (1990, 1991) in the approximation of
the accretion line model. This simplification is known to be valid only
when the shock opening angle is very narrow, i.e.  at very high Mach
number. Even then, it misses the possible interaction between a bow
shock and the accretor (as described by the advective-acoustic cycle).
However simplified, these instability mechanisms are physical. They
should be understood well enough to predict what is left of them beyond
the accretion line model.

\section{A new look at the longitudinal instability of the accretion
line}

\subsection{Physical cause of the longitudinal instability}

The mechanism of the longitudinal instability of the accretion line was
briefly explained by Cowie (1977) as being due to the effect of
accreted momentum on density perturbations. Soker (1990) challenged
this explanation by assessing that this instability mechanism is
independent of accretion and is a mere consequence of the acceleration
of the flow. A closer look at the equations,  in Appendix~B, shows the
weakness of this argument. Let us consider more generally the following
dynamical system:
\begin{eqnarray}
{\partial \rho\over \partial t}+{\partial \rho v\over \partial r}&=&H(r),\label{dzH}\\
{\partial v\over \partial t}+v{\partial v\over \partial r}&=&F(r,v,\rho),\label{dyF}
\end{eqnarray}
where $H,F$ are regular functions describing the mass input and the
force per unit mass. The particular case of the accretion line model is
described in Table~\ref{tableacline} and Appendix~A, where velocities
are in units of $v_\infty$, distances $r$ are in units of the accretion
radius $r_{\rm A}$ , and the line density $\rho$ is normalized using
the mass accretion rate (the normalization of distances and densities
used by Cowie (1977) are different by a factor $2$).
\begin{table}
\caption[]{Accretion line models in 2D and 3D. }
\label{tableacline}
\begin{tabular}{ccc}
                  & 2D & 3D\\
\hline
\\
$H$     &    $\displaystyle{r^{-{1\over2}}}$ &  $1$\\
\\
$F$     &    $\displaystyle{{1-v\over r^{1\over2}\rho}-{1\over 2r^2}}$
&
$\displaystyle{{1-v\over \rho}-{1\over 2r^2}}$\\
\\
\hline
\end{tabular}
\end{table}
A linearization of Eqs.~(\ref{dzH}-\ref{dyF}) gives the differential
equation satisfied by a perturbation of the mass flux $h\equiv
\rho_0\delta v+v_0\delta\rho$ (Appendix~B).
In what follows, the subscript for unperturbed quantities $v_0,\rho_0$
is omitted.
The solution is written at high frequency $\omega$ using the WKB
approximation  :
\begin{eqnarray}
h\sim\left({\rho\over v {\partial F\over\partial\rho}}\right)^{1\over4}
\exp\left\lbrack\int\left( v{\partial F\over\partial v}-\rho{\partial
F\over\partial\rho}\right){{\rm d} r\over 2v^2}
\right\rbrack\nonumber\\
\exp\left\lbrack i\omega\int{{\rm d} r\over v}
\pm {1-i\over 2^{1\over2}}\omega^{1\over 2}\int \left({\rho\over
v^3}{\partial F\over\partial \rho}\right)^{1\over2}
{\rm d} r\right\rbrack.\label{wkbh}
\end{eqnarray}
The flow is thus unstable at high frequency if the force per unit mass
$F$, acting on the accretion line, depends on density. The instability
does not depend on the accretion of mass, as stressed by Soker (1990),
in the sense that it does not depend on the function $H$. Nevertheless
it does depend on the accreted momentum through the function $F$. In
this sense, accretion plays a crucial role in this instability, as
initially sketched by Cowie (1977). Contrary to the conclusions of
Soker (1990), acceleration within the accretion line is not crucial for
this instability (see a counter example in Appendix~B).
In the 3D accretion line model, Eq.~(\ref{wkbh}) becomes:
\begin{eqnarray}
h(r)\propto {r^{1\over2}\over\log^{1\over4}r}\exp\left\lbrack i\omega r
\pm{1+i\over 3}(2\omega)^{1\over2}\log^{3\over2}
r\right\rbrack,\nonumber\\
{\rm for}\;\;r\gg\alpha,\label{ampliout3}\\
h(r)\propto r^{5\over8}\exp
\left\lbrack-{2\over3}i\omega r^{3\over 2}
\pm{1-i\over 5}\left({8\omega\over\alpha}\right)^{1\over2}
r^{5\over4}\right\rbrack,\nonumber\\
{\rm for}\;\;{1\over\omega^{2\over 5}}\ll r\ll\alpha,\label{ampliin3}
\end{eqnarray}
where $\alpha$ is the distance of the stagnation point. The amplitude
of high frequency perturbations increases far from the accretor
($r\gg\alpha$).
Perturbations in the accreting part of the flow ($r<\alpha$) are also
unstable at high frequency $\omega$ down to a point $r\propto
\omega^{-{2\over5}}$ where the amplication ceases and the WKB
approximation breaks down. In a numerical simulation, the treatment of
high frequency perturbations is limited by the numerical resolution.
According to Eqs.~(\ref{ampliout3}-\ref{ampliin3}), the better the
resolution, the stronger the instability on both sides of the
stagnation point. Formulae computed in Appendix~B for 2D flows show
slight differences which are not significant on the scale of a few
accretion radii.

\subsection{The longitudinal instability modified by pressure forces:
analogy with radiation driven winds \label{radwind}}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{2201fig1.eps}
\caption[]{Schematic view of the accretion line model. Stream lines are
drawn as solid lines with arrows. When pressure is taken into account
(Sect.~\ref{radwind}), density perturbations in the accretion line
propagate as acoustic waves (wavy lines with arrows).}
\label{figaccline}
\end{center}
\end{figure}

The region of the stagnation point is necessarily subsonic, and
longitudinal pressure forces cannot be neglected there. Pressure forces
in the stationary accretion line were considered by Wolfson (1977),
Yabushita (1978a) and Horedt (2000). The simplest formulation
corresponds to the isothermal hypothesis, in which the dynamical
equations (\ref{dzH}-\ref{dyF}) are changed into:
\begin{eqnarray}
{\partial \rho\over \partial t}+{\partial \rho v\over \partial r}&=&H(r),\\
{\partial v\over \partial t}+v{\partial v\over \partial r}&=&F(r,\rho,v)-{c^2\over \rho}{\partial
\rho\over\partial r},
\end{eqnarray}
where $c$ is the isothermal sound speed.
Again, a single differential equation is obtained for the radial
structure of the linear perturbation $h$ of the mass flux
(Eq.~(\ref{singlediffpress})),
describing the propagation of acoustic waves modified by the external
forces $F$. The solution is approximated at high frequency through a
WKB analysis, away from the sonic points ($v=\pm c$):
\begin{eqnarray}
h\sim\rho^{1\over2}
\exp\int {{\rm d} r\over v\pm c}\left\lbrack i\omega+{1\over2}\left(
{\partial F\over\partial v}
\pm{\rho\over c}{\partial F\over\partial \rho} \right)\right\rbrack,\nonumber\\
{\rm for}\;\;\omega\gg {\rho v\over c^2}{\partial
F\over\partial\rho}.\label{freqwkb}
\end{eqnarray}
This stability analysis resembles that of radiation driven winds,
studied by Mestel, Moore \& Perry (1976) and Mathews (1976).
Considering a uniform gravitational acceleration $g$ and a linearized
radiative force ($F\equiv A\rho-g$, $H\equiv 0$), they showed that
acoustic waves propagating outwards are amplified by radiation.
In addition to the density effect $\partial F/\partial\rho$, the velocity effect
$\partial F/\partial v$ is formally well known in line-driven winds (see e.g.
Carlberg 1980, and more recent reviews by Owocki 1994, Feldmeier \&
Owocki 1998). $\partial F/\partial v <0$ in the accretion line model, implying that
this velocity effect is always stabilizing, independently of the
direction of propagation. This could have been anticipated directly
from the Euler equation, since a positive perturbation of velocity
results in a decreased external force.
By contrast, the effect of the density dependence (term $\partial F/\partial\rho$)
is opposite for outgoing and ingoing waves:
\begin{eqnarray}
{\partial F\over\partial v}\pm{\rho\over c}{\partial F\over\partial \rho}
&=&-{v\over r-\alpha}\left(1\pm {v_\infty-v\over c}\right)\;\;{\rm
in\;3D},\label{accline}\\
&=&-{v\over r^{1\over 2}(r^{1\over 2}-\alpha)}\left(1\pm
{v_\infty-v\over c}\right)\;\;{\rm in\;2D}.
\label{accline2D}
\end{eqnarray}
The density effect $\partial F/\partial\rho<0$ is thus stabilizing for waves
propagating outwards and destabilizing for waves propagating inwards.
This can be understood as follows: with $\partial F/\partial\rho<0$, a positive
density perturbation is associated with a decreased external force.
According to the Euler equation, this decreased force has a damping
effect on the positive velocity perturbation associated with a wave
propagating outwards, whereas it amplifies the negative velocity
perturbation associated with a wave propagating inwards. In contrast
with the instability without pressure found by Cowie (1977), this
possible amplification of ingoing acoustic waves is not oscillatory.
Altogether, the density and velocity effects damp outgoing acoustic
waves.
According to Eqs.~(\ref{freqwkb}) and (\ref{accline}-\ref{accline2D}),
acoustic waves propagating inwards may be amplified only if
\begin{eqnarray}
c+v<v_\infty.
\end{eqnarray}
This condition cannot be fulfilled far from the accretor since $v\sim
v_\infty$ for $r\gg\alpha$. From this we conclude that the only
possible amplification of high frequency acoustic waves is restricted
to ingoing waves in a region of finite size. The size of this region is
independent of frequency. The amplification factor ${\cal A}$, deduced
from the WKB analysis,
is also independent of the perturbation as long as its frequency is
high enough to satisfy Eq.~(\ref{freqwkb}). It can be estimated as
follows in 3D:
\begin{eqnarray}
{\cal A}&\sim& \exp\int {v_\infty-v-c\over 2c}{v\over r-\alpha}{{\rm d}
r\over v-c}
\;\;{\rm for}\;\;\omega\gg{v^2\over c^2}{v_\infty-v\over
r-\alpha}.\label{ampliA}\\
&\le&{\rm e}^{{\cal M}_{\infty}}
\end{eqnarray}
This contrasts with the instability found by Cowie (1977) which could
be arbitrarily fast at high frequency
(Eqs.~\ref{ampliout3}-\ref{ampliin3}).

\subsection{The longitudinal instability beyond the accretion line
model}

The longitudinal instability of the accretion line proves that if the
Mach number is high enough, the amplification of ingoing acoustic waves
should be visible in numerical simulations.
Such a transient amplification, however, cannot be considered to be a
convincing mechanism to explain the instability observed in numerical
simulations for moderate Mach numbers ${\cal M}_\infty\sim 3-5$.
A true instability would require a feedback loop, in order to build an
acoustic cycle. Ingoing acoustic waves may be partially reflected
outwards near the accretor. Outgoing waves, however, are likely to
escape to infinity rather than be reflected inwards again. Moreover,
their amplitude is damped by the effect of the accreted momentum. In
conclusion, the longitudinal instability of the accretion line does not
explain the instability of BHL accretion.

\section{A new look at the transverse instability of the accretion line}

\subsection{High frequency approximation of the transverse instability}

Soker (1990) extended the stability analysis of Cowie (1977) to the
case of transverse perturbations in 2D planar flows.
The position of the accretion line is described in polar coordinates
$\theta=\Theta(r)$. The angle $\Psi$ between the tangent to the accretion
line and the symmetry axis, and the transverse velocity $v_\theta$ are
related to $\Theta$ as follows:
  \begin{eqnarray}
\Psi=\Theta+r{\partial \Theta\over\partial r},\\
v_\theta = r\left({\partial \over\partial t}+v{\partial\over\partial r}\right)\Theta.
\end{eqnarray}
As remarked by Soker (1990), the transverse instability is decoupled
from the longitudinal one. The angle $\Theta$ of the accretion line
satisfies a second order differential equation, which is approximated
at high frequency in Appendix~D using a WKB analysis:
\begin{eqnarray}
\Theta&\propto& {1\over r^{5\over4}}\exp \left\lbrack i\omega r
  \pm 2^{1\over2}(1+i)\omega^{1\over 2}r^{1\over2}\right\rbrack\;\;{\rm
for}\;\;r\gg\alpha,\label{gsup}\\
\Theta&\propto& {1\over r^{7\over8}}\exp\left\lbrack i\omega\int{{\rm d} r \over
v}
\pm {2(1-i)\over5\alpha^{1\over2}}\omega^{1\over 2}
r^{5\over4}\right\rbrack
\;\;{\rm for}\;\;r\ll\alpha.\label{ginf}
\end{eqnarray}
According to Eq.~(\ref{gsup}), the larger the distance from the
accretor, the larger the amplification of perturbations. As for the
longitudinal instability, the amplification in the region of accretion
ceases close to the accretor at $r\propto \omega^{-2/ 5}$, and thus
depends on the numerical resolution.
These high frequency estimates could be used to compare the radial
profiles of the transverse and longitudinal instabilities of the
accretion line in the linear regime in 2D flows: the differences
between Eqs.~(\ref{gsup}-\ref{ginf}) and
Eqs.~(\ref{ampliout2}-\ref{ampliin2}) are not significant on the scale
of few accretion radii, and cannot be responsible for the dominant
longitudinal instability observed in the non linear simulations of
Soker (1991).

\subsection{Comparison of the transverse instability with the
instability of a flag}

More generally, a system satisfying a transverse equation of the form
\begin{eqnarray}
\left({\partial \over\partial t}+v{\partial\over\partial
r}\right)v_{\theta}=A\Theta+B\Psi+Cv_{\theta},\label{geneflop}
\end{eqnarray}
with arbitrary coefficients $A,B,C$, is unstable at high frequency. The
differential equation (\ref{diflag}) satisfied by $\Theta$ is written in
Appendix~D. If $v\ne0$, the WKB approximation at high frequency is as
follows:
\begin{eqnarray}
\Theta\propto \left({v\over r^2B}\right)^{1\over4}
\exp\int\left\lbrack{i\omega \over v}
+{C\over2v}+{B\over2v^2}\pm\left({i\omega B\over v^3}\right)^{1\over 2}
\right\rbrack{\rm d} r,\nonumber\\
{\rm for}\;\;\omega\gg {v\over r},C,{Av\over rB}, {\partial\log\over\partial
r}{B\over v^2}.\label{geneflop2}
\end{eqnarray}
This formulation outlines the role of the restoring force $B \Psi$ in
driving the high frequency instability. $B\equiv -1/(r^{1\over2}\rho)$
in the 2D accretion line model. This mechanism is reminiscent of the
instability of a flag as described by Argentina et al. (2004), where
the hydrodynamical force acting on the flag is also proportional to the
inclination $\Psi$.
The equation describing the transverse motion of a flag with infinite
flexibility corresponds to the same Eq.~(\ref{geneflop}), with $v\equiv
0$, $A\equiv0$, $B\equiv -\alpha U_0$ and $C\equiv\alpha$, where $U_0$
is the wind velocity and the coefficient $\alpha>0$ characterizes the
aerodynamic force acting on the flag. The solution of
Eq.~(\ref{geneflop}) when $v\equiv 0$ is unstable at high frequency  if
$B<0$:
\begin{eqnarray}
\Theta\propto {1\over r} \exp\int \left\lbrack
i\omega {C\over B}-{\omega^2\over B}-{A\over rB}\right\rbrack{\rm d} r.
\end{eqnarray}
The presence of finite flexibility ("flexural rigidity") in a realistic
flag material would set an upper bound to unstable frequencies. The
comparison with the instability of a flag suggests the possible
existence of a transverse instability of the subsonic region of the
stagnation point ($v\sim 0$).

\subsection{The transverse instability beyond the accretion line model}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{2201fig2.eps}
\caption[]{Transverse instability growing in the vicinity of the
stagnation point. The shock (thick line), the sonic surface (dashed
line) and the stream lines (thin lines with arrows) are drawn. The thin
dotted line delineates the accreted gas. The transverse motion of the
shock on one side is transmitted to the other side through acoustic
waves (wavy lines with arrows).
}
\label{figflag}
\end{center}
\end{figure}

\subsubsection{Transient growth of the transverse instability in a
shock cone}

The accretion line model relies on the asumption that the half angle
$\theta_0$ of the shock cone is small compared to other distances.
Identifying the shock cone with the Mach cone,
$\theta_0\sim1/{\cal M}_\infty$ is indeed small if the incident flow is
highly supersonic.
With a longitudinal wavelength comparable to $\sim 2\pi v/\omega$, the
transverse instability of the accretion line model is valid only for
\begin{eqnarray}
{2\pi v\over\omega}\gg{2 r\over{\cal M}_{\infty}}.
\end{eqnarray}
The maximum exponential amplification ${\cal A}$ deduced from
Eqs.~(\ref{gsup}-\ref{ginf}) is thus bounded by:
\begin{eqnarray}
{\cal A}\le \exp (2\pi{\cal M}_{\infty})^{1\over2}.
\end{eqnarray}
Taking into account the finite width of the shock cone sets an upper
bound $\propto {\cal M}_\infty$ to the frequency of the most unstable
perturbations in a plane 2D flow. An additional limitation to the
accretion line model comes from the acoustic time across the shock
cone, which is a lower bound to the growth time of the instability.
This also favours the subsonic region surrounding the stagnation point
rather than the supersonic regions away from it.
The instability as described by Soker (1990) is a local mechanism,
taking place in the advected flow. A global mode as observed in the 2D
plane simulations of the flip-flop instability requires an acoustic
feedback inside the subsonic region.
This instability could be described as a purely acoustic cycle between
the opposite sides of the shock cone as in Fig.~\ref{figflag}.

\subsubsection{Towards a global transverse instability in BHL accretion
?}

The amplification of transverse perturbations should be visible if the
Mach number is high enough. Whether this is sufficient to explain the
instability observed in 2D flows at low Mach number is not clear since
this instability is transient.
When pressure is taken into account, the region of the stagnation point
seems to be a privileged place for the growth of a global mode
involving transverse displacement and acoustic propagation. \\
The fate of the transverse instability in 3D BHL accretion is rather
uncertain, as remarked by Soker (1991). Obviously transverse motions of
the accretion line are forbidden in 3D, since incoming symmetrical
trajectories which do not intersect the displaced accretion line meet
along the symmetry axis and generate a new accretion line. This does
not exclude a possible unstable oscillation of the shock cone in a
transverse direction, such that the symmetry axis stays inside the
accretion shock. According to Eqs.~(\ref{geneflop}-\ref{geneflop2}),
the existence of a transverse restoring force proportional to the
inclination angle is enough to generate a high frequency instability.
Beyond geometrical factors, this force is still present in the 3D
geometry. This instability mechanism could thus be present for high
enough Mach numbers, although it has never been observed clearly in 3D
numerical simulations. A correct description of the stability with
respect to transverse oscillations would require taking into account
pressure effects and the 3D dynamical deformation of the shock cone,
which is much more difficult than the accretion line formalism.

\section{The advective-acoustic mechanism in BHL accretion\label{adac}}

\subsection{Schematic formulation of a global cycle}

The advective-acoustic instability deals with the cycle of advected
perturbations (entropy, vorticity) coupled to acoustic waves, between
the shock and the sonic surfaces. The coupling at the shock is a local
process associated to the conservation laws through the shock. By
contrast, the coupling due to the inhomogeneity of the subsonic flow
occurs all the way from the shock to the sonic surface. F01 showed in a
simple radial geometry that this acoustic feedback is described by an
integral over the subsonic region, dominated by the region close to the
sonic point, where the temperature is highest. This allows us to
decompose the advective-acoustic cycle in four steps as follows:
\par(1) advection of an entropy/vorticity perturbation from the shock
to the sonic point,
\par(2) excitation of an acoustic feedback due to the inhomogeneity of
the flow,
\par(3) propagation of this acoustic feedback towards the shock,
\par(4) excitation of a new entropy/vorticity perturbation on the shock
surface.\\
Each of these steps $j=1$ to $4$ is characterized by an efficiency
${\cal Q}_j$ measuring the amplification of perturbations. This
decomposition is motivated by the existence of invariants, allowing a
direct calculation of ${\cal Q}_1$ based on the conservation of
entropy, or ${\cal Q}_3$ based on the conservation of acoustic energy.
The stability of the global cycle depends on the product ${\cal Q}$:
\begin{eqnarray}
{\cal Q}\equiv{\cal Q}_1\times{\cal Q}_2\times{\cal Q}_3\times{\cal Q}_4
\end{eqnarray}
The linear growth rate of the advective-acoustic cycle, measured by the
imaginary part of the eigenfrequency $\omega\equiv
(\omega_r,\omega_i)$, can then be approximated by
\begin{eqnarray}
\omega_i\sim{1\over\tau}\log|{\cal Q}|,
\end{eqnarray}
where $\tau$ is the duration of the advective-acoustic cycle, generally
dominated by the advection time. Note that this schematized approach
neglects the purely acoustic cycle, which can influence the stability
threshold (FT00, F02).

\subsection{Efficiencies ${\cal Q}_i$ based on radial accretion}

In a radial flow with $\gamma>1$, we choose to measure, in the
entropic-acoustic cycle, the amplification of the perturbation $f$ of
the Bernoulli constant :
\begin{eqnarray}
f\equiv v\delta v+{2\over\gamma-1}c\delta c.
\end{eqnarray}
Let us denote by $f^\pm$ its values for pressure perturbations $\delta
p^\pm$ corresponding to an energy flux $F^\pm$ propagating with $(+)$
or against ($-$) the stream, and $f^S$ for entropy perturbations
$\delta S$. At high frequency, for low degree waves $l=0,1,2$:
\begin{eqnarray}
F^\pm&\propto&{1\over{\cal M} c^2}|f^\pm|^2,\label{acflux}\\
f^\pm&\sim& (1\pm{\cal M})c^2{\delta p^{\pm}\over\gamma p} ,\label{fpm}\\
f_S&\sim& {c^2\over\gamma}\delta S.\label{fS}
\end{eqnarray}
Non radial entropy perturbations $\delta S$ are associated to vorticity
perturbations $\delta w$ in shocked spherical accretion through
Eqs.~(\ref{wr}) to (\ref{wp}),
so that the amplification of the entropy $\delta S\rightarrow\delta
p^-\rightarrow\delta S'$ also measures the simultaneous amplification
of vorticity $\delta w\rightarrow\delta p^-\rightarrow \delta w'$. In
the isothermal limit ($\gamma\rightarrow 1$), vorticity is more
appropriate than entropy to describe the advective-acoustic cycle,
which becomes a vortical-acoustic cycle (F02) .

The steps (1) and (3) of advection and propagation in the spherical
accretion flow studied by F01, F02 are deduced from Eqs.~(\ref{acflux})
and (\ref{fS}) and the
conservations of entropy $\delta S$ and acoustic energy $F^-$:
\begin{eqnarray}
{\cal Q}_1&\equiv& {f_{\rm son}^S\over f_{\rm sh}^S}\sim {c_{\rm
son}^2\over c_{\rm sh}^2}
\;\; {\rm for}\;\;\omega_r\sim {c_{\rm son}\over r_{\rm
son}},\label{Q1}\\
{\cal Q}_3&\equiv& {f_{\rm sh}^-\over f_{\rm son}^-}\sim {\cal M}_{\rm
sh}^{1\over 2}{c_{\rm sh}\over c_{\rm son}},
\end{eqnarray}
where the subscripts "son" and "sh" refer to the sonic point and the
shock respectively.
Although the acoustic energy is conserved, the amplitude of outgoing
waves $f^-$ decreases (${\cal Q}_3<1$) due to the geometric dilution in
a diverging flow.\\
The advection of an entropy perturbation in a hot region may greatly
increase the thermal energy carried by this perturbation, by a factor
comparable to the temperature ratio (Eq.~(\ref{Q1})). As sketched by
FT00, the difference of energy is carried away by acoustic waves,
propagating both upward and downward. Even the waves propagating
downward may be
refracted upward at low enough frequency (F01). The ratio ${f_{\rm
son}^-/ f_{\rm son}^S}$ is thus of order unity:
\begin{eqnarray}
{\cal Q}_{2}\equiv {f_{\rm son}^-\over f_{\rm son}^S}\sim 1.
\end{eqnarray}
The advective-acoustic coupling ${\cal Q}_4$ at the shock is deduced
from a local analysis of a perturbed shock, performed in Appendix~F:
\begin{eqnarray}
{\cal Q}_{4}\equiv {f_{\rm sh}^S\over f_{\rm sh}^-}\propto{1-{\cal M}_{\rm
sh}\over{\cal M}_{\rm sh}}.\label{Q4}
\end{eqnarray}
This equation is identical to Eqs.~(16) and (C.11) in FT00, obtained
for radial perturbations. ${\cal Q}_4$ may reach large values in nearly
isothermal flows with strong shocks (${\cal M}_{\rm sh}\sim 1/{\cal M}_1$), in
which case the dominant advected perturbations are non radial (F02):
this is the basis of the vortical-acoustic cycle. The factor
$(1-{\cal M}_{\rm sh})$ in Eq.~(\ref{Q4}) is responsible for the damping of
the advective-acoustic coupling for weak shocks. \\
Altogether,
\begin{eqnarray}
{\cal Q}\propto {c_{\rm son}\over c_{\rm sh}}{1-{\cal M}_{\rm sh}\over{\cal M}_{\rm
sh}^{1\over2}}.
\label{qglobal}
\end{eqnarray}
This approximate formula illustrates the two regimes of efficient
advective-acoustic coupling identified by FT00, F01, F02:
\par (i) Strong temperature gradients are responsible for an efficient
triggering of acoustic waves from advected entropy perturbations, which
is the basis of the entropic-acoustic cycle (FT00, F01).
\par (ii) Strong entropy/vorticity perturbations can be produced at the
shock if the post shock Mach number ${\cal M}_{\rm sh}$ is small.\\
These two sources of amplification can be used as guidelines for
anticipating the properties of advective-acoustic cycles in BHL
accretion flows. However, the dependence of ${\cal Q}$ on the frequency
and the degree $l$ of the perturbation requires further calculations.
In particular, F01 showed that  the acoustic feedback is strongest for
non radial modes $l=1$. Note also that Eq.~(\ref{qglobal}) is singular
if $c_{\rm son}\to \infty$ as is the case for 3D accretion with
$\gamma=5/3$, which deserves a more careful analysis (Appendix~G).

\subsection{From radial to BHL accretion\label{radialBHL}}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{2201fg3a.eps}
\includegraphics[width=\columnwidth]{2201fg3b.eps}
\caption[]{Schematic view of a global mode if the shock is detached or
attached. Drawn lines have the same meaning as in Fig.~\ref{figflag}.
Perturbations of entropy/vorticity (circular arrows) follow the flow
lines from the shock to the accretor, producing an acoustic feedback
which propagates to the shock. A fraction of the acoustic energy may
leak outside of the accretion cylinder. }
\label{figmode}
\end{center}
\end{figure}

\subsubsection{Extrapolation to detached shocks}

The locus of the advective-acoustic cycle is different if the shock is
attached or detached (Fig.~\ref{figmode}). A global mode between a bow
shock and the accretor resembles the advective-acoustic cycle in a
shocked radial flow. In this case the efficiency of the
advective-acoustic cycle can be estimated from the study of radial
accretion.
By contrast, if the shock is attached to the accretor, most of the flow
is accreted from behind in a supersonic manner. The value of ${\cal
Q}_2$ in this particular geometry cannot be directlly extrapolated from
its value in a radial flow.

\subsubsection{Geometrical factors}

In BHL accretion, the shape of the shock surface is not only
non-spherical but open to infinity.
The value of ${\cal Q}_3$ might be reduced by a geometrical factor
$\sim 2$ compared to spherical accretion because a significant fraction
of acoustic waves may propagate away from the region of accretion and
leave the cycle (Fig.~\ref{figmode}).\\
Conversely, the value of ${\cal Q}_1$ should be increased by the
amplification of vorticity perturbations through the local KH and RT
mechanisms (FR99), due to vorticity and entropy gradients in the
post-shock flow. \\
The efficiencies ${\cal Q}_i$ estimated in spherical geometry should
thus be considered, at best, as very rough approximations of those in
the BHL flow.

\subsection{Influence of the parameters $\gamma$, ${\cal M}_\infty$ and $r_*$}

According to FT00, F01, the stability of the entropic-acoustic cycle
depends essentially on the
temperature increase between the shock and the sonic point. The most
unstable cycle involves high frequency acoustic waves, those able to
explore the hottest parts of the flow but still be refracted out, with
a wavelength slightly larger than the smallest size of the sonic
surface. Using the Bernoulli equation, the temperature on a point of
the sonic surface is directly related to its distance $r_{\rm son}$ to
the accretor:
\begin{eqnarray}
c_{\rm son}^2= {\gamma-1\over\gamma+1}\left\lbrack
{2GM\over r_{\rm son}}+c_\infty^2\left(
{\cal M}_\infty^2+{2\over\gamma-1}\right)\right\rbrack.
\end{eqnarray}
The closer the sonic surface to the accretor, the higher the
temperature, and the more efficient the entropic-acoustic cycle.
The calculation of Appendix~A indicates that the sonic surface is
always attached to the accretor if $\gamma=\gamma_{\rm max}$, with
\begin{eqnarray}
\gamma_{\rm max}&\equiv&3\;\;{\rm in\;2D},\\
\gamma_{\rm max}&\equiv&{5\over 3}\;\;{\rm in\;3D},
\end{eqnarray}
even if the shock is detached. Fig.~17 of R94 shows the shape of the
attached sonic surface for 3D accretion with $\gamma=5/3$. Together
with Eq.~(\ref{qglobal}), the strength $\cal Q$ of the instability
should increase when $\gamma$ approaches $\gamma_{\rm max}$ and $r_*$
decreases. The strength of the instability should be asymptotically
independent of the incident Mach number for strong shocks. Conversely,
the instability should be suppressed if the shock is weak (${\cal M}_{\rm
sh}\sim 1$). \\
According to Appendix~A, a critical index $\gamma_{\rm
crit}<\gamma_{\rm max}$ must exist, below which the shock is always
attached to the accretor, whatever its size. Numerical simulations with
$\gamma=4/3$ (R95) suggest that $\gamma_{\rm crit}<4/3$ in 3D flows.
The entropic-acoustic cycle is expected to be an efficient instability
mechanism in the range $[\gamma_{\rm crit},\gamma_{\rm max}]$, as long
as the distance of the sonic surface is small enough. \\
Nearly isothermal flows ($\gamma\sim 1$) could be unstable through the
vortical-acoustic cycle, fed by ${\cal Q}_4\gg1$ for strong shocks.
However, the effect of the acoustic feedback in the particular geometry
of an attached shock is uncertain. Neither the vortical-acoustic
mechanism nor the extrapolated transverse instability manage to explain
why isothermal BHL accretion seems so much more unstable in planar
flows (SMA98) than in 3D simulations (R96).

\section{Can the different numerical simulations be reconciled?}

\subsection{Some simulations are stable\label{stable}}

The stability of the 2D planar flow simulated by FI98b, FI99, with a
relativistic accretor contrasts with the many unstable simulations
performed in Newtonian gravity.
This is by no mean a relativistic stabilization of BHL accretion
through relativistic effect,
as recognized by FI98b.  Indeed,  they restricted their studies to
$v_\infty/c_{\rm light}=0.5$, which corresponds to a rather big
Schwarzschild radius in units of the accretion radius: $r_{\rm
Sch}/r_{\rm A}=0.25$. These flows would have been stable even in the
Newtonian limit.\\
According to ZWN95, the flow simulated by MSS91 becomes stable if the
square accretor is replaced by a polygon. Although the shape of the
accretor may influence the instability threshold,
the existence of strongly unstable simulations in polar coordinates
(BLT97 and SMA98) shows that 2D accretion can be unstable even if the
accretor is perfectly spherical. The smaller accretor size and higher
resolution used by BLT97 and SMA98, compared to ZWN95 (see
Table~\ref{tableoverview}), may be a hint in favour of an
advective-acoustic mechanism.
A direct comparison, however, is hampered by the fact that the
adiabatic indices are different in these three simulations. The
influence of the shape of the accretor, demonstrated by ZWN95, speaks
against the transverse acoustic instability. These rather indirect
arguments, if not conclusive, show at least that it is possible to
reconcile existing simulations of 2D plane accretion in the framework
of the advective-acoustic mechanism.

The stable 3D simulations can also be analyzed in the framework of the
entropic-acoustic cycle. This cycle is stabilized by a weak shock, and
destabilized by a small accretor size.
\par -Table~\ref{tableoverview} indicates that most axisymmetric
simulations of an absorbing accretor are stable (SMT85, PSS89, SMA89),
with the exception of KMS91 which shows an instability when $r_*/r_{\rm
A}\le0.05$ and ${\cal M}\ge2.4$. The existence of a threshold for the
accretor size and the minimum Mach number fits perfectly with the
entropic-acoustic mechanism.

\par -In the axisymmetric simulations of FI98a with $v_\infty/c_{\rm
light}=0.5$, the
Schwarzschild radius is too big to allow for a detached shock.
According to Newtonian 3D simulations, the shock distance scales like a
fraction (e.g.  typically $0.2-0.4$ in R94) of the accretion radius. A
slower black hole would have a detached shock, and the
entropic-acoustic instability could develop naturally if the
temperature gradient is sufficient. The shock gets detached in the
simulation of FI98a for $\gamma=5/3$, $v_\infty/c=0.15$, but the shock
is then too weak (${\cal M}_\infty=1.5$) to be destabilized. Indeed, the
Newtonian simulations of Ruffert with ${\cal M}_\infty=1.4$ were also stable.
A decisive test could be made by simulating an axisymmetric flow with
$\gamma=5/3$, ${\cal M}_\infty=3$, and $v_\infty/c \le 0.1$.

\par -The apparent stability observed in the simulations of POM00, in
particular for $\gamma=5/3$, cannot be explained by simple
considerations about the accretor size and shock strength. This result
may be attributed to the particular numerical method of local time step
used by the authors. In Sect. 3 of POM00, the authors make the
following statement:
  "It is important to note from the very begining that we seek steady
state solution and generally do not perform time-accurate calculations
(local time step inside each cell for the sake of computational
efficiency)". This method might not be adequate to propagate high
frequency acoustic waves across the subsonic region.

\subsection{Numerical artefacts in the simulations\label{artefacts}}

Numerical issues are numerous. Besides the damping effect of numerical
viscosity (SPH and Eulerian codes were compared by BA94) and the
possible axis effect in axisymmetric simulations (FTM87), more subtle
effects can be understood through the advective-acoustic cycle.
This instability mechanism is physical, but may be artificially
triggered or damped by numerical effects
such as the carbuncle phenomenon at the shock, the boundary condition
at the surface of the accretor and the grid size in between.
The accuracy of both the advection of entropy/vorticity perturbations
and the propagation of acoustic waves, between the shock and the sonic
surface, is crucial for advective-acoustic instabilities.

\subsubsection{Carbuncle phenomenon at the shock}

POM00 drew attention to possible numerical instabilities at the shock
in the BHL flow. In the region where the shock is parallel to the grid,
the carbuncle instability (e.g.  Robinet et al.  2000) may favour the
generation of vorticity and entropy perturbations, which in turn can
feed an advective-acoustic cycle. Conversely, one should carefully
check the effect of any numerical procedure designed to damp the
carbuncle instability at the shock, since it may also damp the coupling
between advected and acoustic perturbations.

\subsubsection{Grid resolution between the shock and the accretor}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{2201fig4.eps}
\caption[]{Numerical damping of a sinusoidal perturbation of entropy
(full line) or vorticity (dotted line) advected over one wavelength, in
a direction parallel to the grid, as a function of the number of grid
points per wavelength.
The flow is homogeneous with $\gamma=5/3$, and moves uniformly at Mach
$0.2$.
The dashed line corresponds to the damping of an acoustic wave
propagated over one wavelength. In the three cases, the wavevector of
the perturbation is parallel to the flow.
}
\label{figgrid}
\end{center}
\end{figure}
Vorticity is not usually computed with as much accuracy as momentum or
energy in classical numerical schemes such as those used for BHL
simulations. The artificial generation of vorticity at the interface of
nested grids could feed a vortical-acoustic cycle.\\
Conversely, insufficient numerical resolution is responsible for an
artificial damping of vorticity and entropy waves. As an example,
Fig.~\ref{figgrid} shows a measure of the damping of the amplitude of
entropy and vorticity perturbations advected over one wavelength, in a
direction parallel to the grid, as a function of the number of grid
cells per wavelength. This test is performed on a Cartesian grid in 2D
using the same PPM technique as in Ruffert (1994).
The correct advection of entropy and vorticity in this numerical
simulation requires as much as 10-15 grid cells per wavelength.
Perturbations with a wavelength shorter than 10 grid cells are
significantly damped over one wavelength. The same test performed on
acoustic waves propagating over one wavelength shows a smaller damping:
5-10 grid cells are enough.
This is a strong constraint for the correct calculation of an
advective-acoustic cycle at high frequency which involves perturbations
with a wavelength comparable to the accretor size, when the sonic
surface is attached to it. The grid size should thus be at least 15-20
times smaller than the accretor size in order to properly describe the
advective-acoustic coupling in the inner regions of the flow. This
example illustrates the fact that the advective-acoustic instability
can be impeded in a numerical simulation with insufficient resolution.
Determining how these numbers depend on the numerical technique is
beyond the scope of the present paper.

\subsubsection{Boundary condition on the accretor}

The size of the accretor is known to play an important role in the
strength of the instability. One reason for this is that the sonic
surface ahead of the accretor is generally attached to the accretor in
numerical simulations. Boundary conditions at the surface of the
accretor are therefore even more crucial, since they are in contact
with a subsonic flow. The advection of entropy and vorticity
perturbations through the boundary condition should be considered
carefully in order to avoid an artificial acoustic feedback from the
accretor.
For example, imposing a transverse velocity equal to zero in the
perturbed Bondi flow would generate a spurious feedback $f^-$ from any
non spherical entropy/vorticity perturbations $f^S$, and artificially
excite a vortical-acoustic cycle. A calculation in Appendix~H of this
condition at the boundary of a uniform, parallel flow shows that
\begin{eqnarray}
\left({f^-\over f^S}\right)_{\rm boundary}=-1\;\;{\rm if}\;\;l\ne0.
\end{eqnarray}
The artificial coupling due to inadequate boundary conditions can be as
strong as the physical coupling ${\cal Q}_2$ expected from the
temperature gradients in the flow.

\subsection{Physical or numerical instability?}

Many of the numerical artefacts discussed in Sect.~\ref{artefacts} were
not considered by the authors of the existing simulations. Could new
simulations of BHL accretion, corrected from numerical artefacts, be
stable? The physical arguments of Sect.~\ref{adac} prove that the flow
must be unstable, at least in the case $\gamma\sim 5/3$,
${\cal M}_\infty\ge3$, for a small enough accretor: in this case, the
efficiency ${\cal Q}$ of the entropic-acoustic cycle diverges when the
accretor size $r_*\to 0$. Even the leak of acoustic energy cannot
diminish the efficiency ${\cal Q}$ by more than a finite geometrical
factor $2-3$, at most. The argument is weaker for $\gamma<5/3$, because
${\cal Q}$ depends strongly on the unknown shape of the sonic surface.

\section{Future numerical tests of the instability mechanism}

Future simulations should consider carefully the numerical artefacts
listed in Sect.~\ref{artefacts}. The boundary condition at the surface
of the accretor should be designed to absorb entropy and vorticity
perturbations silently: this can be tested in a uniform flow.
An alternative would be to find a set of parameters such that the
accretor is fully embedded inside the sonic surface. Since $\gamma=5/3$
and $\gamma\sim1$ are ruled out by FR97 in 3D, an intermediate choice
could be $\gamma=4/3$ with ${\cal M}_\infty=3$, and a small enough accretor.
The numerical issue of the absorbing boundary condition can also be
solved naturally with general relativity, for any value of $\gamma$,
since the flow is bound to be supersonic on the horizon of the black
hole (FI98a,b).

The stable simulations analyzed in Sect.~\ref{stable} call for new
simulations which would directly test the efficiency of the
advective-acoustic mechanism in accretion flows where the shock is
detached:
\par-all the axisymmetric, Newtonian simulations with $\gamma=5/3$
should be unstable if ${\cal M}_\infty\ge 3$ and a small enough accretor
($r_*/r_{\rm A}\sim 0.005$ seems to be enough according to KMS91). In
particular, new axisymmetric, relativistic simulations similiar to
FI98a should become unstable when considering a slower accretor with a
moderate shock, such as $v_\infty/c_{\rm light}=0.1$, $\gamma=5/3$,
${\cal M}_\infty=3$. The axisymmetric simulations of POM00 should also be
unstable if their numerical technique allows for the advective-acoustic
cycle.
\par-2D planar simulation should be unstable through the
entropic-acoustic cycle if the shock is detached: comparing a
simulation with $\gamma\sim 3$ (see Appendix~A) and the classic
flip-flop obtained for $\gamma\le 5/3$ could help to understand the
respective influences of the entropic-acoustic cycle and the purely
acoustic transverse instability.

In an unstable simulation, the advective-acoustic mechanism could be
tested directly by measuring the incoming entropy/vorticity
perturbations and the outgoing acoustic flux in the subsonic region of
the flow, as measured by Blondin et al.  (2004) in his simulations of
spherical accretion. An indirect way of testing the instability
mechanism is to measure the effect of the various physical parameters.
The linear growth rate of the entropic-acoustic cycle should increase
when the accretor size is decreased, decrease for a weak shock and
saturate for strong shocks.
The present understanding of the 3D advective-acoustic instability when
$\gamma\sim5/3$ implies that its strength for an accretor size $r_*\sim
10^{-5}r_{\rm A}$ may be significantly under-estimated by the existing
simulations ($r_*/r_{\rm A}>10^{-2}$). Tracing the power spectrum of
the mass accretion rate (e.g.  R95), as a function of the numerical
resolution in the range accessible to computers, could give a hint on
the extrapolation to smaller accretor sizes.

In order to better understand the instability of isothermal flows, it
would be interesting to try
to discriminate between a transverse instability (purely acoustic) and
a vortical-acoustic cycle.
The accretor size and boundary condition should play a more important
role in a vortical-acoustic cycle than in a transverse instability.

\section{Conclusion}

The physical mechanisms underlying the longitudinal and transverse
instabilities of the accretion line have been clarified. The
destabilizing factor for the longitudinal instability is the density
dependence of the external force per unit mass. The transverse
instability is ruled by the restoring force proportional to the local
inclination of the accretion line. The analogy with the instability of
a flag has been outlined. \\
The overstable amplification of longitudinal high frequency
perturbations of the accretion line found by Cowie (1977) is greatly
affected by pressure forces. Density perturbations are propagated as
acoustic waves. Those propagating outwards are damped, whereas those
propagating inwards are transiently amplified. The analogy with the
instability of radiation driven winds has been drawn. \\
The transverse instability of the accretion line is limited if the
finite width of the shock cone is taken into account. A feedback
process is necessary to explain the global flip-flop instability
observed in planar 2D simulations. It seems possible that the mechanism
of the transverse instability, modified by the propagation of acoustic
waves within the subsonic region of the flow, leads to a global
unstable mode. However, the influence of the shape of the accretor,
demonstrated by ZWN95, may be a hint against this explanation of the
instability. \\
For the first time, the advective-acoustic instabilitiy has been
analyzed in the context of BHL accretion. This analysis is based on the
extrapolation of the properties established in spherically symmetric
flows (F01, F02). For this reason, the relevance of the
advective-acoustic instabilities in BHL accretion is convincingly
demonstrated only when the shock is detached from the accretor. The
difference of geometry precludes an accurate prediction of the
instability threshold in BHL accretion, especially since it is very
sensitive to the size and shape of the sonic surface.
Nevertheless, the analysis is predictive enough to assess that the
advective-acoustic mechanism
\par - must be stable in the limit of a weak shock,
\par - must be unstable for 3D accretion flows with $\gamma\sim 5/3$, a
reasonable shock strength ${\cal M}_\infty\ge 3$ and a small enough accretor
size.\\
Numerical artefacts have also been discussed, specifically the grid
size and the boundary conditions. We comment on how these artefacts
must be circumvented to
produce reliable numerical simulations of BHL accretion. \\
Surprisingly, it seems that all the existing numerical simulations can
be reconciled in the framework of the advective-acoustic instability,
with no striking contradiction, even when the shock is attached to the
accretor. Several numerical tests of these ideas have been proposed.\\

Besides new numerical simulations, future analytic work ought to
describe in more detail both the transverse instability modified by
pressure forces, and the efficiency of the advective-acoustic cycle
when the shock is attached. This could help understand why isothermal
accretion is so much more unstable in 2D than in 3D.

\begin{acknowledgements}
The authors are grateful to Egide and the British Council for their
exchange programme. The anonymous referee is thanked for his
constructive comments.
\end{acknowledgements}

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