text:
September 24th 2019 | Central London
Richard Vize
Director, Public Policy Media
Richard Vize has been a commentator on public policy and public services reform for more than 25 years. He is a columnist for Guardian Society, contributes to the British Medical Journal and works with a range of organisations across health and local government.
Richard is a former Editor of both Health Service Journal and Local Government Chronicle, was Managing Director Media at Dods Parliamentary Communications and Head of Communications at regulator Ofsted. He and his teams have won a number of awards.
He is a trustee of the whistleblowing charity Protect. Richard's passions are photography, independent travel and political history.
Twitter: @RichardVize
Dr Phil Moore
Chair, Mental Health Commissioners Network at NHSCC, and Deputy Chair (Clinical) at NHS Kingston CCG
Phil Moore is a long-standing and enthusiastic GP committed to transforming the NHS through the opportunities afforded by the current system. He has held public roles in the NHS and beyond for many years including in Kingston, London-wide in mental health and primary care, and nationally as board member of NHS Clinical Commissioners and member of various national advisory groups on mental health. He is a trustee and chair of a variety of charities, and a Visiting Fellow in Healthcare Management at the University of Surrey. He has a strong educational background having been a GP Trainer and Honorary Teaching Fellow at Imperial College. He frequently speaks and chairs at national conferences.
Twitter: @DittonPhil
Dr Ed Beveridge
Consultant Psychiatrist, CNWL NHS Foundation Trust
MA (Hons Cantab) MB BCh (Oxon) MRCPsych
Ed trained in medicine at Cambridge and Oxford. He undertook specialist training in London. He has worked as a consultant general adult psychiatrist in London since 2008, currently with CNWL NHS Foundation Trust.
His is passionate about improving physical health of patients with severe mental illness (SMI) and worked with NIHR CLAHRC NW London designing interventions to improve physical health screening. He is Clinical Lead for Physical Health in CNWL Mental Health Services, co-chair of the Pan-London Physical Health Leads' Network and leads an SMI Physical Health improvement collaborative with Healthy London Partnership. An advocate for patient and public involvement (PPI) in health, he has worked on several coproduction ventures and published on the 4Pi standards for PPI in quality improvement.
Other interests include psychodynamic approaches to SMI, and psychosomatic disorders (in association with the Institut Psychosomatique Pierre Marty, Paris, where he is currently completing training as a psychosomatic therapist).
Twitter: @dredbeveridge
Rhiannon England
Clinical Lead, Mental Health, City and Hackney CCG
Details to appear shortly
Catherine Gamble
RCN mental health professional lead and Head of Nursing Practice, Education and Research
Catherine's role involves providing evidence for policy makers, commissioners and providers, whilst ensuring the RCN influences and supports the mental health nursing workforce to face new challenges in many specialist and diverse areas of nursing practice, research, education and management. Working across the UK, the role has a wide sphere of influence, covering a range of activities – namely supporting members, promoting professional networks and sharing good practice. Her NHS Trust role at South West London and St Georges Mental Health NHS Trust supports this activity, as she is responsible for driving clinical innovation and evaluating nursing workforce development initiatives. Catherine is an Eileen Skellern laureate, Royal College of Nursing Mental Health Nurse of the Year award holder, Royal College of Psychiatrists Team winner and a 2019 RCNI Nurse Leader finalist. Catherine has broad experience in coproduction and practice development work. Her clinical expertise lies in working with families and those with psychosis; she has held various NHS roles in acute and community psychiatric services, has published extensively and is an elected member of the NHS Confederation Mental Health Network board. Catherine works for the RCN three days/week.
Twitter: @minifeet2
Dr Julie Hankin
Executive Medical Director, Nottinghamshire Healthcare NHS Foundation Trust
Julie is Executive Medical Director for the Trust. She is a General Adult Consultant psychiatrist by background and has worked in clinical leadership for the last 15 years.
Locally she is SRO for Mental Health in the Nottingham and Nottinghamshire Integrated Care System.
From 2012 to 2014, she was National Professional Advisor for Mental Health for the Care Quality Commission. Currently she is the Medical Director representative on the Mental Health Network Board at the NHS Confederation and is chair of the Mental Health Medical Directors Forum.
She is Honorary Professor of Healthcare Leadership at Nottingham University Business School and sits as a panel member for the HSDR researcher led stream within the NIHR.
Twitter: @julie_hankin
Brinda Paramothayan
GP CLINICAL LEAD CHILDREN'S HEALTH AND CAMHS, RICHMOND CCG
Brinda is a GP working in Teddington, Richmond Borough.
She has always been interested in Young People's health and completed a diploma in child health during her GP training.
Over the past few years she has worked as the GP Lead for CAMHS for Richmond CCG and more recently as the SWL STP GP Lead for CAMHS, working closely with both Commissioners and Providers.
As a working GP, she recognises the distress caused by poor mental health to YP and their families and the importance of early and easy access to effective treatment.
With 2 teenagers of her own Brinda is well aware of the rising rates of anxiety and self harm across the UK and is particularly interested in how we can build resilience at an early age by working with families and schools.
Sarah Holloway
Head of Mental Health, NHS England and NHS Improvement
Sarah is a Head of Mental Health at NHS England and NHS Improvement, leading work across Crisis and acute services, Community services for people with SMI and support for people with Dementia services.
Sarah has been passionate about improving mental health services throughout her career and has worked on the NHS England mental health programme for four years. Prior to joining NHS England, Sarah worked in the Department of Health and Cabinet Office and before that at Mind, leading on research and impact measurement for the organisation and its network of local services.
Sarah Chaired small charity, Disability Law Service, for five years, working towards stability and sustainability for the charity that ensures access to justice for people with disabilities in the areas of community care, employment discrimination and welfare and benefits.
Twitter: @sarahofholloway
If you have wish to register onto the conference or have any questions about registration please contact:
customer.services@westminster-briefing.com
More from Dods
©2019 Dods Group Plc.


text:
<?php

namespace Oro\Bundle\SegmentBundle\Query;

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{
    /** @var QueryBuilderInterface[] */
    private $queryBuilders = [];

    /**
     * Registers a query builder for a given segment type.
     *
     * @param string                $type
     * @param QueryBuilderInterface $queryBuilder
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    public function addQueryBuilder($type, QueryBuilderInterface $queryBuilder)
    {
        $this->queryBuilders[$type] = $queryBuilder;
    }

    /**
     * Returns a data transformer for a given data type.
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    public function getQueryBuilder($type)
    {
        return isset($this->queryBuilders[$type]) ? $this->queryBuilders[$type] : null;
    }
}

redpajama_set_name: RedPajamaGithub


Living Colorfully: Outlets Are Very Dangerous Places.
Dom has already made a rule that I am not allowed to go grocery shopping hungry, and I definitely can't go to Whole Foods unattended.
I think the same rule is about to be made for outlet malls.
I was very good at staying controlled in the beginning...but it's all fun and games until you spot J. Crew. This precious tee might have jumped into my bag by itself immediately upon arrival.
Hopefully, I can style it half as cute as she does! Rumor has it from Hallie that there will be an outlet mall really close to me down in Florida. I've never had one close and I have a feeling it will be a very, very dangerous!
I had to drive past that outlet mall everyday on my way to work at my last job. Holy moly- talk about temptation!
Unfortunately I live super close to the Philly premium outlets & not too far from Lancaster... It is deadly! Outlets may be the death of me!
Yes there will be an outlet mall very close to you! I might have to join you when I visit!
Haha "It's all fun and games until you spot J. Crew." is actually one of the most relatable things I've ever heard :) You're not alone with an outlet addiction!
I saw this shirt the other day at J.Crew and loved it, kind of sad I didn't get it!


If you've been wondering how many states had significant issues with test participation last spring, there's an answer from the U.S. Department of Education.
Twelve states have received letters from the Education Department in recent months asking them to address lower-than-required participation rates on state exams for groups of students or districts, or statewide.
The states to receive these letters are: California, Colorado, Connecticut, Delaware, Idaho, Maine, New York, North Carolina, Oregon, Rhode Island, Washington, and Wisconsin. (Click the individual link to see each state's letter from the department.) Some of the copies of the letters are undated, but several were sent in either early November or early December. UPDATE: Illinois also received a letter from the department dated Dec. 23 stating that at least one district had not met the participation rate requirement.
The Education Deparment's letters ask states to specify how they will address their local or state participation rates for the 2015-16 school year. The department includes possible examples of how states could act, such as lowering a school or district's rating on state accountability systems, and counting non-participating students as not proficient for accountability purposes.
On Tuesday, the department also released guidance for states in which it said that under the Every Student Succeeds Act, states need plans to address situations in which participation rates that dip below the required 95 percent.
The testing opt-out movement became a prominent story in education this year, particularly in some of the states listed above, like New York. Reasons for skepticism and opposition to the test included backlash to the Common Core State Standards, concerns about the amount of testing, and teacher evaluations that factored in test scores.
So what were some of the various statewide and other test-participation issues encountered by each state?
California reported that it did not assess at least 95 percent of its students, or individual student subgroups, on both the English/language arts and math exams. UPDATE: The California Department of Education has told me that in fact, with respect to statewide participation on exams, California did in fact clear the 95 percent threshold; the state reported that 97 percent of students took the Smarter Balanced exams. Some districts in California did not meet the 95 percent threshold, an issue the Education Department letter also specifies might have been the issue with respect to participation on the state exams.
Colorado missed the 95-percent requirement for all students, and participation rates in grades 7 through high school were particularly low, ranging from 50 percent to 89 percent in E/LA and 60 percent to 89 percent in math.
Connecticut did not assess at least 95 percent of all its students.
Delaware did not assess at least 95 percent of its English-language learners and high school students.
Maine did not assess at least 95 percent of all its students on both exams.
North Carolina missed participation rate targets for English-language learners and students with disabiliites.
Oregon did not assess at least 95 percent of all students, and 21 districts did not assess at least 95 percent of all their students.
Rhode Island did not assess at least 95 percent of all students, and "at least" 34 districts did not assess 95 percent of all their students.
Washington state did not assess at least 95 percent of all students.
In each letter, the department stresses the importance of a "high-quality, annual Statewide assessment system" to provide better information to local leaders and educators.
The department also specified that if a state fails to address the issue, the department might withhold a portion of the state's Title I funds, and that other state funds from Title II, Title III, and Title VI, among others, could be put at risk.


text:
Not the right job for you?
Head of Spanish, Essex
The King John SchoolEssex
Start date: Required September 2019/January 2020
Salary: Main/Upper Pay Range (salary dependent on experience) TLR 2b - £4,204 per annum
Contract term: Permanent
Suitable for NQTs: No
Head of Spanish
Main/Upper Pay Range (salary dependent on experience)
TLR 2b - £4,204 per annum
Required September 2019/January 2020
We are seeking an outstanding and dedicated head of Spanish
The King John School in Benfleet is a heavily oversubscribed 11-18 mixed comprehensive of over 2000 students, of which approximately 420 are in the sixth form. The school is exceptionally proud of the achievements of our students and the ethos of the school. Our results are consistently strong and over 76% of our students achieved a standard pass in English and mathematics at GCSE (Grade 4 or above) in 2018 and we are recognised by the Department for Education for our proactive promotion of the English Baccalaureate within our curriculum.
Our A-level results have been exemplary, with over 50% of year 13 students gaining A* - B grades in all of their subjects. We are in the top 15% of the country for A-level outcomes compared to 2817 schools and colleges. The school enjoys an excellent reputation, having received a letter from the Director of Education in September 2017 commending the school on the 'fantastic' results that year.
The King John School offers a high quality educational experience through a broad curriculum and a caring, friendly environment, and sets the highest expectations for all. Through high quality teaching and learning and exceptional wide ranging enrichment opportunities the school seeks to cultivate students' motivation, confidence and intelligence. The school strives to be at the forefront of school improvement and innovation and invests heavily in the creation of a learning community and to this end the extension to the sixth form, humanities building and state of the art assembly hall have been constructed to accommodate the growing number of students wishing to join us.
Our school is a dynamic learning environment where all students are encouraged to aspire towards their very best. We are growing rapidly and our reputation in the local area has attracted an increasing number of external students to the sixth form who wish to continue their studies with us. We challenge our students to achieve ambitious goals and to never settle for second best.
Our school has good staff retention and is wholly committed to continued professional development. We offer free healthcare through the Benenden Healthcare scheme which all employees are entitled to join. If you feel you could contribute to this ethos and enhance it further, please contact us.
We can offer you:
A dedicated hardworking team of staff
Enthusiastic, friendly and talented students
The opportunity to develop within a highly effective multi academy trust
Have a proven track record of achieving high standards for students
Previous experience as head of department would be an advantage
Be rigorous, determined, creative and inspirational in your style
Commit to working collaboratively with colleagues and students
Utilise a variety of teaching methods and ensure that your lessons are engaging at all levels
For further information please contact Tanya Bonner, Director of HR: tanyab@thekjs.com or alternatively to download an applicant form please visit www.thekjs.essex.sch.uk/about/recruitment/
Completed applications should be submitted via email to Tanya Bonner.
We reserve the right to close this vacancy early if there is a good level of response. Therefore, we would recommend that you submit your application as early as possible.
Closing date for applications: 30 June 2019 (midnight)
CVs alone or CVs submitted via agencies will not be accepted. Applications received after the closing date will not be considered.
Zenith Multi Academy Trust is committed to safeguarding the welfare of young people and expects all staff and volunteers to share this commitment. This post will be subject to full employment checks, including an enhanced disclosure from the Disclosure and Barring Service.
The King John School is a Zenith Multi Academy Trust school as part of Zenith Multi Academy Trust
Share this role via email
More about The King John School
Type: Mainstream School
Funding status: State - Academy
Age Range: 11 - 18 years
Denomination: Non-denominational
Address: Shipwrights Drive, Thundersley, Benfleet, Essex, SS7 1RQ, United Kingdom. View in Google Maps
Website: http://www.thekjs.essex.sch.uk
The King John School is a heavily oversubscribed 11-18 mixed comprehensive of 2000 students, of which 450 are in the sixth form. We are one of the highest performing schools in the country. The school enjoys an excellent reputation, and has bee...
Learn more about this school
Contact The King John School
Thanks for your interest in this role.
Simply fill in the form below to ask any questions, request a call, or schedule a school visit.
Questions, requests or comments:
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redpajama_set_name: RedPajamaCommonCrawl


text:
J.Lo and A-Rod's birthday week continues with another birthday bash with his daughters and one big cake
"American Idol" producers are looking for two more judges, and they have some favorites in mind
Alex Carrigan, July 26, 2017 3:51 pm
97-Year-Old WWII Veteran Is Still a Flamethrower Expert
Emily Mack, June 8, 2021 11:41 am
Frank Abagnale Jr: The Man Behind 'Catch Me If You Can'
Haley Van Horn, June 7, 2021 4:56 pm
Claire Danes Credits a One-Night Stand with Someone Else in Helping Her Marry Hugh Dancy
Photo by Kevork Djansezian/Getty Images
HOLLYWOOD, CALIFORNIA - APRIL 07: American Idol past winners and finalists perform together onstage during FOX's "American Idol" Finale For The Farewell Season at Dolby Theatre on April 7, 2016 in Hollywood, California. at Dolby Theatre on April 7, 2016 in Hollywood, California. (Photo by Kevork Djansezian/Getty Images)
The search for the remaining cast roles on "American Idol" is underway, with tons of speculation as to who will join host Ryan Seacrest and judge Katy Perry on the show. According to TMZ, there are some particular singers to whom "Idol" producers may soon be making offers.
An "Idol" source told TMZ that R&B singer Lionel Richie is "still a favorite" for one of the judge spots. A few weeks ago, Richie had a meeting with "Idol" producers as a possible "nice judge." Sources told TMZ that the producers thought Richie would be "a great fit for the show" and that they like his "vibe."
RELATED: After former "American Idol" contestant got into a physical altercation at a Chicago bar, a bartender has been arrested.
Another front-runner for the judging table is pop singer Charlie Puth. Puth is a young singer known for tunes like "Marvin Gaye" with Meghan Trainor and, most notably, "See You Again" with Wiz Khalifa from the film "Furious 7."
TMZ reports that country singer Luke Bryan is also being considered for a judging spot. They also report that series executives will meet with ABC this week to "finalize their choices and make offers."
About the author: Alex Carrigan
Alex Carrigan is a copy editor for Rare. He is originally from Newport News, Virginia and graduated from Virginia Commonwealth University in 2014. He has a B.S. in Mass Communications: Print/Online Journalism and volunteers as the Communications and PR Manager for the Cambridge Writers' Workshop. He longs to travel, write, ...Read more


AMC takes us behind the scenes on this past week's episode of The Walking Dead (Episode 412, Still) the 100% Daryl Dixon and Beth Greene episode.
Emily Kinney (Beth) and Norman Reedus (Daryl) discuss the storyline from the beginning where they are both sort of just surviving to finding the country club and finally the abandoned moonshiner's cabin. The show's producers explain what they were going for with the episode and the relationship between the two characters.


Mr Knox has over 35 years of experience in the upstream oil and gas sector. He has worked extensively throughout Australasia, South East Asia and North America with several entities. He was formerly CFO of Cue Energy Resources Limited, a position he had held for 22 years. His prior role was CFO at Cultus Petroleum Limited. Mr Knox has significant experience in financial and commercial activities, involving acquisitions, M&A and strategy.
He is a chartered accountant (CA ANZ), public accountant (CPA Australia) and a fellow of the Australian Institute of Company Directors (FAICD). He holds a Bachelor of Commerce degree from the University of Western Australia.
Mr Carey has 20+ years as a manager and director in a range of listed companies in the resources, Oil & Gas and technology sectors. Mr. Carey has extensive experience in the resource sector providing management, capital raising and corporate advisory services to a range of listed mining and energy companies. He has worked for mining and energy companies in Russia, Brazil, Australia and England..
Mr Le Page is currently a Director & Corporate Adviser of RM Corporate Finance specialising in resources. He is actively involved in a range of corporate initiatives from mergers and acquisitions, initial public offerings to valuations, consulting and corporate advisory roles.
Mr Le Page was Head of Research at Morgan Stockbroking Limited (Perth) prior to joining Tolhurst Noall as a Corporate Advisor in July of 1998. As Head of Research, Mr Le Page was responsible for the supervision of all Industrial and Resources Research. As a Resources Analyst, Mr Le Page published detailed research on various mineral exploration and mining companies listed on the Australian Securities Exchange. The majority of this research involved valuations of both exploration and production assets.
Prior to entering the stockbroking industry, he spent 10 years as an exploration and mining geologist in Australia, Canada and the United States. His experience spans gold and base metal exploration and mining geology, and he has acted as a consultant to private and public companies. This professional experience included the production of both technical and valuation reports for resource companies.
Mr Le Page holds a Bachelor of Arts, a Bachelor of Science and a Masters' Degree in Business Administration from the University of Adelaide, a Bachelor of Applied Science (Hons) from the Curtin University of Technology and a Graduate Diploma in Applied Finance and Investment from the Securities Institute of Australia.
Mr Wing is a Certified Practising Accountant and specialises in the public company environment. He practised in the audit and corporate divisions of a chartered accounting firm before providing corporate/accounting consultanting and company secretary services to public companies.
His experience includes accounting, administration, compliance, company secretarial, re-structuring and recapitalising companies on ASX. Mr Wing currently provides these services to several public companies on the ASX. He has served on a number of public company boards and related committees.


text:
Nordic Research Councils for the Humanities and the Social Sciences (NOS-HS)
NOS-HS provides research funding for studies within the fields of Humanities and Social Sciences. The principal recipients of funding are universities, university colleges and research institutes.
Mail Facebook Linkedin Twitter Apply for funding
Opportunity category
Nordic Council of Ministers for Education and Research (MR-U)
External organization
Joint Committee for Nordic Research Councils for the Humanities and the Social Sciences (NOS-HS)
Helle Glen Petersen
helpet@norden.org
Nordic Culture Point: Volt, culture and language programme for children and young people
Volt is a culture and language programme for children and young people. It aims to make young people interested in each other's arts, culture and language. Funding can be applied for by p...
NORDITA - Scientific Programmes
NORDITA - Nordic Institute for Theoretical Physics, annually awards funding for a number of scientific programmes. A Nordita Scientific programme is an extended workshop where a limited n...
Grants from the Foundation for Danish-Icelandic Co-operation
The Foundation for Danish-Icelandic Co-operation aims to promote understanding and co-operation between Denmark and Iceland. In particular, it has a duty to promote the Danish language in...
The Foundation for Danish-Swedish Co-operation
The Foundation for Danish-Swedish Co-operation provides funding for work and study visits to Denmark and Sweden, scientific work of interest to both countries, training and exchange progr...
Scandinavian Research Council for Criminology: Research and travel grants
The Scandinavian Research Council for Criminology provides support for research projects, working party meetings, travel, study tours and translation.


text:
\section{Introduction}
\label{S:Intro}

According to our current structure formation scenarios, the cold dark
matter (CDM) models, structures in the Universe form via gravitational instability.
When a mass overdensity reaches a critical threshold, the expansion of the
density fluctuation eventually stops and the region collapses.
On large scales (\simgreat$\,$Mpc), the cooling time is much longer than the
gravitational dynamical time, and the super--sonic collapse of gas in overdense regions
into sheets, filaments and halos is halted by large--scale accretion shocks.
These shocks in the low-density intergalactic gas convert the kinetic energy of
the collapsing gas into thermal energy, and are responsible for heating the gas.
Accretion shocks are also important for generating high energy photons
and cosmic rays (CR) via diffusive first order Fermi acceleration of charged particles
(e.g., \citealt{Miniet01a,Miniet01b,GaBl03,Mini03,Ryu_et03,Kanget07,Pfroet07,Skilet08}
and references therein).

Semi-analytic solutions exist only for the one-dimensional accretion problem.
\cite {FillGold1984} derived self-similar solutions for collisionless matter perturbations
in an expanding Einstein--de Sitter universe with planar, cylindrical and spherical geometry.
Assuming spherical geometry, self-similar solutions for positive density perturbations of
collisionless and collisional matter in an Einstein--de Sitter universe (ignoring radiation transfer,
heat conduction and magnetic fields) were derived by Bertschinger (1985).
Semi-analytic models for large-scale structure shocks produced by spherical density
perturbations were derived by \cite{FuLo04}.
Barkana (2004) studied the physical properties of the infalling gas
using a spherical model that started with different initial density distributions.
Three--dimensional (3D) cosmological N--body and hydrodynamical simulations, based on CDM models,
confirmed the existence of large-scale accretion shocks at about the virial radius, \ensuremath{R\rmsub{vir}},
often referred to as ``virial shocks'', around individual collapsed and
bound structures, i.e. massive clusters of galaxies
(e.g., Evrard 1990; Bryan \& Norman 1998; Keshet et al. 2003).

Other studies based on cosmological simulations have focused on
the global properties of structure formation shocks on very large scales
\citep{Miniet00,Ryu_et03,Pfroet06,Kanget07, Skilet08}.
For example, shocks generated by structure formation were tracked and studied in detail
by Miniati et al. (2000) using Eulerian cosmological simulations.
They have found that the spatial structure of shocks is very complex,
multiply connected, and that shocks plunge deep into clusters of galaxies along overdense filaments.
Shocks can be classified based on their location and/or on the physical mechanism which drives them.
Based on their location, shocks can be classified as external and internal shocks \citep{Miniet00}.
Large-scale, external shocks form where pristine, previously un-shocked, gas is falling onto sheets,
filaments, and halos, due to gravitational instability. External shocks are accretion shocks surrounding voids.
Internal shocks form in gas that has already been shock-heated,
and can be further divided, based on the driving physical mechanism \citep{Ryu_et03},
into the following categories:

\begin{itemize}
  \item
    Internal accretion shocks, due to coherent continued infall of diffuse uncollapsed gas, following an external shock
  \item
    Merger shocks, which occur when bound or collapsing objects
                  (such as halos, filaments, and sheets) collide
  \item    Flow shocks, which are generated within collapsed objects by internal, in general non--radial, bulk flows
   (which are not necessary gravitationally driven)
\end{itemize}

The global energetics of external and internal cosmological shocks, focusing on their role
in accelerating nonthermal, cosmic-ray (CR) particles, were studied in detail by \cite{Ryu_et03}
using Eulerian simulations and by \cite{Pfroet06}
using the smoothed particle hydrodynamics (SPH) code {\sc GADGET-2}\ \citep{Spri05}.
Using self-similar solutions, Abadi, Bower\& Navarro (2000) assessed the reliability of
SPH cosmological numerical simulations.
They concluded that SPH simulations obtain the physical quantities describing
the collapse reliably within the virial radius. However, due to the low-density in the
pre-shock region, the smoothing length becomes large, and the pre-shock region
is smoothed out by about 2\ensuremath{R\rmsub{vir}}.
The most recent SPH simulations perform much better in the low-density regions due to
their higher resolution.
Shock acceleration and $\gamma$-ray production were also studied by \cite{Mini02},
\cite{Keshet03}, and \cite{Kanget07}.
\cite{Skilet08} studied cosmological shocks and shock acceleration using the Eulerian code {\sc ENZO}\
\citep{BryaNorm98,OSheet04}.
Skillman et al. found that merger and internal flow shocks have Mach numbers less than about 5, and
accretion shocks into clusters of galaxies have large Mach numbers, between about 20 and 300
(they did not distinguish between internal and external shocks).
Our work here is similar, but there are two major differences:
(i) rather than using a large simulation box to compute global statistics, we utilize
a discrete sample of well--resolved individual clusters, and (ii) we focus on the
detectability of accretion shocks near the virial radius through the Sunyaev-Zel'dovich (SZ) effect
(Sunyaev \& Zel'dovich 1980; for recent reviews see Rephaeli 1995; Birkinshaw 1999;
Carlstrom, Holder \& Reese 2002).

Observational confirmation of large--scale shocks would be an important test for
our structure formation scenarios.
However, in general, it is difficult to detect such shocks, since the intra-cluster gas
density is very low near, and beyond, the virial radius.
Evidence for large--scale gas infall has been claimed in the Lyman $\alpha$
spectra of some high-redshift quasars
(see, e.g., Dijkstra, Haiman \& Spaans 2006 and references therein),
and internal shocks are thought to be responsible for radio relics observed at smaller radii in
nearby galaxy clusters (Ensslin et al. 1998; Ensslin 2002).
However, direct observational confirmation of large--scale accretion shocks, produced by gravitationally
driven structure formation, at the virial radius and beyond, does not exist as of today.
A feasibility study for observations of such shocks around clusters of galaxies,
using their SZ signature, with the
next generation high-spatial-resolution, high-sensitivity interferometer, the Atacama Large
Millimeter Array ({\it ALMA})\footnote{See http://www.alma.nrao.edu.},
was carried out by Kocsis, Haiman \& Frei (2005).
The advantage of using SZ, instead of X-ray, observations is that
the SZ signal is proportional to the gas density, while the X--ray flux is proportional to
the square of the density, and therefore
the SZ surface brightness drops considerably less rapidly with radius.
Assuming spherical models for the cluster gas and temperature profiles,
\citet{Kocset05} concluded that strong shocks around the virial radii of clusters of
galaxies should be detectable with high significance using {\it ALMA}\ mosaic observations.

In this paper, motivated by the proposal of \citet{Kocset05}, we report a study of
non-spherical accretion to quantify the 3D properties of accretion shocks
around massive clusters of galaxies.
We have used two sets of clusters, one drawn from adaptive mesh refinement (AMR)
and another drawn from SPH simulations of {$\Lambda$CDM}\ models.
First, we describe the cluster samples, the methods used to locate and analyze
accretion shocks, and the properties of the shocks found in the AMR (\S\ref{S:AMR}) and the
SPH (\S\ref{S:SPH}) simulations.
In the discussion section (\S\ref{S:Discussion}), first we study the feasibility of observing
shocks in clusters of galaxies with the future high resolution radio interferometer, {\it ALMA}\
using their SZ signature.
We then discuss the consequences of our results for the interpretation of large scale
X-ray and SZ signals associated with clusters of galaxies.
Finally in this section, we provide a comparison of our results with those from other studies,
as well as between our own results from AMR versus SPH simulations.
We briefly summarize our main results and offer our conclusions in \S\ref{S:Conclusion}.

\section{Accretion Shocks in AMR Simulations}
\label{S:AMR}

\subsection{Locating and Analyzing Accretion Shocks in AMR Clusters}
\label{SS:AMRShocks}

We use a sample of 10 clusters of galaxies drawn from AMR simulations,
each located at redshift $z=0$. We refer to them as AMRCL1 through AMRCL10, sorted by
their virial mass
(i.e. the total gravitational mass within the virial radius, \ensuremath{R\rmsub{vir}}, defined as in \citealt{BryaNorm98}),
AMRCL1 being the most massive.
The AMR simulations were performed with the cosmological code {\sc ENZO}\
 assuming a spatially flat {$\Lambda$CDM}\ cosmology with
($\Omega_m$, $\Omega_\Lambda$, $\Omega_b$, h, $\sigma_8$) = (0.3, 0.7, 0.047, 0.7, 0.92),
where the Hubble constant is $H_0$ = 100 h km s$^{-1}$ Mpc$^{-1}$,
and $\sigma_8$ is the power spectrum normalization on 8 h$^{-1}$ Mpc scales.
This cosmological model is close to the {\it WMAP}\ 5 year results, except the simulations
used a somewhat larger value of $\sigma_8$ \citep{Dunk08}.
The box size of the original simulation is 300 h$^{-1}$ Mpc.
The AMR simulations were adiabatic (in the sense that no radiative heating, cooling,
star formation, or feedback were taken into account).
The clusters of galaxies in our sample were re-simulated with high resolution
using the same  technique as described in \cite{YounBria07}.
The virial masses of the clusters in our AMR sample are in the range of
9.1$\times 10^{14}$ {\ensuremath{\mbox{\rm M}_{\odot}}}\ - 2.3$\times 10^{15}$ {\ensuremath{\mbox{\rm M}_{\odot}}}; in Table~\ref{T:Table1}, we list  the
masses, radii, and dynamical state (see below) of each cluster.

We are interested in accretion shocks around each cluster of galaxies;
therefore, we use spherical coordinates with the origin always placed at the
center of mass of each cluster.
We sample physical variables from the simulation output at spatial positions
$\vec{x} = (R, P)$, described by the radial distance $R$ from the cluster center, and by
the 2D angular position specified by a pixel number $P$, based on the {\sc HEALPix}\
pixelization scheme with 3072 pixels \citep{Gorset05}.
This coordinate system provides a convenient way of describing physical
variables around collapsed objects.
A 3D volume element in our coordinate system has the shape of a two-dimensional (2D)
{\sc HEALPix}\ pixel with a thickness of the corresponding radial bin, $\Delta R$.
The physical size of the 3D volume elements are larger at larger radii, but, in our case,
that is desired because the resolution of the simulations also decreases with
distance from the cluster center (except for substructures, which are not
important for our analysis).
As an example, we show the resolution (spherically averaged AMR cell size)
as a function of radius in a typical relaxed cluster, AMRCL9
({\ensuremath{M\rmsub{vir}}} =1.1$\times 10^{15}$ {\ensuremath{\mbox{\rm M}_{\odot}}}), in Figure~\ref{F:Fig1}
(points with error bars connected with solid line).
The error bars represent the dispersion due to spherical averaging.
We estimate the dispersion by calculating the standard deviations separately
for data points that are larger and smaller than the mean (upper and lower error bars).
With the choice of 3072 pixels, the {\sc HEALPix}\
pixel sizes as a function of radius track the resolution of the simulations well
(within a factor of two; compare the solid line with error bars and the straight
solid line in Figure~\ref{F:Fig1}).
The median resolution (AMR cell size) of the high resolution simulations
at $R/R_{vir}$ = 0.1, 1 and 4 (where $R$ is the distance from the center of the cluster)
was about 25, 80 and 300 kpc, respectively.

We carry out our analysis of shocks in all clusters, but we quote results
for relaxed clusters separately.
We define relaxed clusters as clusters without a sign of a recent merger.
Recent mergers are identified by visual inspection of the 3D
distribution of dark matter, the spherically averaged density,
and temperature distributions within the virial radius, \ensuremath{R\rmsub{vir}}.
Head-on mergers in a stage close to pericentric passage
would not be identified by this method; however, we ignore this possibility, since
such events are exceedingly rare, and are not expected to occur in our small sample.
In total, we find three relaxed clusters in our AMR sample.

In AMR simulations, the initial cells are adaptively divided into sub-cells, and the
physical parameters are represented by values assigned to the center of the cells.
We use a piecewise constant interpolation scheme to determine the physical variables
at the center of our 3D volume elements.
Shocks in our AMR sample of clusters of galaxies are identified based on regions of large
compression, {\it i.e.}, where the velocity convergence, $\kappa = -\nabla \cdot \vec{v}$, is large.
The large peak in the convergence accurately determines the shock position
as demonstrated in Figure~\ref{F:Fig2}.
We use this figure to illustrate the features of different shocks as well.
In Figure~\ref{F:Fig2}
we show the distribution of physical variables as a function of radius in three directions
(fixed pixels: $P = A$, $B$ and $C$; right, middle and left panels) in a relaxed AMR cluster, AMRCL9.
Similar features can be seen on all other relaxed clusters.
We plot the gas density (in units of the critical density, $\rho_c$),
temperature (in keV), pressure (in units of the central pressure, $P_0$), radial velocity (in km/s),
and convergence profiles (in units of $\kappa_0 = \mbox{\rm Max} \{ \kappa \}$) with error bars.
The error bars represent the dispersion (standard deviation) of a given parameter
evaluated in all AMR cells located within the 3D sampling volume element.
Note, however, that these values are only rough estimates because some
volume elements contain only a few points
(we omitted error bars for those 3D volume elements that contain only one point).
The dot-dashed lines on the density plots represent the universal background baryon
mass density ($\Omega_b = 0.047$).
Pixels $A$ and $B$ are chosen toward directions that avoid any overdense filaments,
Pixel $C$ points toward a filament.

The left panels (Pixel A) represent an individual sight--line from the cluster center
to a low--density,  no--filament region with only one accretion shock.
For reference, the dashed curves on the density and radial velocity plots show the
self-similar collapse solution of \cite{Bert85} for the pressureless  infall region.
We fit the functional form of the radial velocity, {\ensuremath{V\rmsub{rad}}},
obtained by Bertschinger (Equation 2.13 of Bertschinger 1985) to the
spherically averaged distribution of {\ensuremath{V\rmsub{rad}}}\ in this direction
allowing the amplitude and the turn--around radius, {\ensuremath{R\rmsub{ta}}}, to change (dashed curve).
From this, remarkably good fit, extrapolating the curve out to the point where ${\ensuremath{V\rmsub{rad}}} = 0$,
we derive ${\ensuremath{R\rmsub{ta}}} = 22 \ensuremath{R\rmsub{vir}}$
(note that this large value is the turn--around radius at $z=0$ and not the turn--around radius of
the shells presently located near the virial radius).
We use this {\ensuremath{R\rmsub{ta}}}, and fit for the amplitude of the gas density distribution in this direction
using Equation 2.15 of \cite{Bert85} (dashed curve).
We obtain good fits for both the radial velocity and density profiles, thus we conclude that,
in the undisturbed pre-shock (infall) regions, both distributions
are consistent with the functional forms derived by \cite{Bert85}.
Note, that the radial velocity is not zero within the virial radius due to internal flows.
In this direction, we can identify a virial shock at 1.8\ensuremath{R\rmsub{vir}}; at this radius,
we see a large peak in the convergence associated with a significant decrease in pressure with radius,
and the radial velocity of the gas falling toward the cluster center drops to zero.
The peak in the convergence at 0.35\ensuremath{R\rmsub{vir}}\ is due to an internal flow shock.
We see no other accretion shock, therefore, in this direction, the virial shock is also an external shock.
The density, temperature, and pressure drop significantly
between 3--4~\ensuremath{R\rmsub{vir}}, well into the free collapse region, breaking the self-similarity.
This drop is not due to a shock, since the radial velocity is continuous, still consistent with
the self-similar solution, and there is no peak in the convergence.
Rather, this feature is due to voids between massive clusters of galaxies;
the infall toward our cluster simply runs out of gas.

Pixel B in the middle panels of Figure~\ref{F:Fig2} shows a representative
line of sight with two accretion shocks. Most lines of sight toward low--density no--filament
regions (80--90\%) have two or more accretion shocks.
The dashed curves on the density and radial velocity plots toward this region
show the self-similar collapse solutions of \cite{Bert85} with parameters fixed at their
values derived from the profiles toward Pixel A.
In this direction, the viral shock is located at 1.1\ensuremath{R\rmsub{vir}},
where there is a peak in the convergence, a drop in the pressure, and the radial infall is halted.
Outside of the virial radius, the density profile does not deviate significantly from that predicted
by the self--similar solution of Bertschinger.
The radial velocity profile is close to the self--similar solution in the free infall region.
At 2.5\ensuremath{R\rmsub{vir}}\ the velocity of the infall becomes smaller than that of the free fall,
there is a peak in the convergence, and a large jump in the pressure:
this is the radius where the external shock is located in this direction.
An internal flow shock due to an outward bulk flow is located at 0.5\ensuremath{R\rmsub{vir}},
where we see a small peak in the convergence, and a drop in the pressure.

Pixel $C$ in the right panel of Figure~\ref{F:Fig2}
shows a representative sight-line toward a direction where accretion occurs along a
filament, and where the infalling gas
forms a shock inside the virial radius (in this case at 0.8\ensuremath{R\rmsub{vir}}).
We see multiple peaks in the convergence in this direction.
Density and temperature bumps can be seen at the location of
collapsed objects falling toward the cluster along the filaments.
The dashed curves on the density and radial velocity plots toward this region
show the self-similar collapse solutions with parameters fixed as before.
The density is much larger than that predicted by the self similar collapse
model (the property we use to identify filaments).
As expected, the self--similar solutions of Bertschinger are not good
descriptions of the collapse in this direction that includes a filament.

Since we expect more coherent accretion shocks to occur in the low-density diffuse gas
(in regions where the infall is not disturbed by overdense filaments or sheets along the infall
path), we identify and remove pixels in the directions of filaments.
Note that because filaments bend and become more radial near the
virial radius, and because they miss the cluster center, the projected surface area
(number of pixels) of filaments as viewed from the cluster center will be larger than the area
actually covered by filaments at a fixed distance
(which is usually around one-third of total surface area; see below).
It is well known that the morphology of the density distribution in cosmological simulations
depends on the density threshold chosen: high density thresholds results in a set of
collapsed halos; at lower density thresholds filamentary structures emerge, and at even lower
density thresholds we see a network of sheets.
Colberg, Krughoff \& Connolly (2005) identified and classified filaments in {$\Lambda$CDM}\ cosmological
numerical simulations using the dark matter density distribution between clusters of galaxies.
In order to identify filaments, we visually inspect the gas density distribution around each cluster
and choose the gas density threshold that results in a filamentary structure
(typically 1-1.5 $\rho_c$).
While this threshold changes with distance from the cluster center, we verified that
varying it within reasonable bounds (a factor of 5) does not affect our results.

We illustrate the results of this selection for the relaxed cluster AMRCL9 in Figure~\ref{F:Fig3}
(other relaxed clusters show very similar characteristics). The figure shows spherically averaged profiles,
along radial sight--lines that either explicitly avoid or explicitly include filaments.
From top to bottom, we plot profiles of the density, temperature,
pressure, radial velocity, and convergence as a function of radius
(we use the same units as in Figure~\ref{F:Fig2})
We show mean values over all directions (pixels) with error bars which
represent the dispersion due to spherical averaging (left panel),
and mean values of physical quantities toward filaments (thick solid curves),
no--filaments regions (long dashed curves),
and in all directions (thin solid curves) for reference (right panel).
The upper and lower error bars are calculated as standard deviations for data points that are
larger and smaller than the mean.
The average universal baryon background density, $\Omega_b$, is plotted as a dot-dashed line.
The curve representing the density distributions for filaments and no--filament regions splits
at about 1.2\ensuremath{R\rmsub{vir}}, where the radial velocity, {\ensuremath{V\rmsub{rad}}}, reaches zero; this is, on average,
the location of the innermost accretion shock, the ``virial'' shock, in this cluster.
The large dispersion in the density, temperature, and thus in the pressure data greater
than the mean (upper error bars) outside of the virial shock is
caused by clumps and collapsed halos falling toward the cluster center along filaments.
The density is much lower in the no--filament regions, and the spherically averaged
density over all directions is very bumpy in $R$ due to clumps in the filaments.
However, the spherically averaged density distribution in the low-density diffuse
gas is smooth out to large radii, and slowly drops to the background level at about 4\ensuremath{R\rmsub{vir}}.
We conclude that, outside of the virial radius, spherical averaging is not a good
approximation for the distribution of baryons.
The radial velocity average toward filaments reaches zero at about 0.6\ensuremath{R\rmsub{vir}},
much closer to the cluster center than the virial radius.

We expect coherent accretion shocks primarily in gas that falls nearly radially, unaffected
by filaments and sheets. Therefore, as a second (and last) step in locating such coherent
accretion shocks, among the regions we identified as non--filamentary based on their density
(in our first step), we further select pixels where the infalling gas velocity is nearly radial
($-{\ensuremath{V\rmsub{rad}}} / V \ge 0.85$).
This second step assures us that we isolate truly infalling
low-density regions, as opposed to low-density regions that might be disturbed by incoherent,
non-radial motions.  In practice, we found that non--radial accretion shocks in the non--filamentary
regions are rare, and this second step provides only a small correction.
This is reassuring, since it may be difficult to disentangle radial and non--radial shocks in observations.

The determination of Mach numbers for shocks in numerical simulations in general is not a trivial
task (see, for example, \citealt{McCaret07,Pfroet06,Skilet08}).
However, since we found the gas velocities through the accretion shocks in the non--filamentary directions
to be close to radial, we assume that the shock surface is tangential to the radial direction.
At any fixed direction (pixel number), we use the Rankine-Hugoniot jump conditions
to determine the upstream Mach number from the pressure ratio taken from the simulations:
\begin{equation}
 M_1^2 = {(\gamma + 1) P_2/P_1 + (\gamma - 1) \over  2 \gamma}
,
\end{equation}
where $P_1$ and $P_2$ are the upstream (pre-shock) and downstream (post-shock)
pressures at the position of the shock,
and $\gamma = 5/3$.
Although we assume radial shocks, our method still gives a good approximation
for the Mach numbers of shocks as long as the shock surface is not in the
radial direction (the number of pixels with the shock surface close to
radial is indeed negligible).

\subsection{Results: AMR Clusters}
\label{SS:AMREsults}

We next describe the properties of the shocks we identified around the
AMR clusters. As will become clear below, there are two distinct sets
of accretion shocks, which can be distinguished by their Mach numbers, and also
by their radial location.  We first discuss the ``virial'' shocks,
which are located closer in, and have low Mach numbers.
We then discuss the ``external'' shocks, which
are located farther out, with large Mach numbers, and where infalling
gas is shocked for the first time.

\subsubsection{Virial Shocks}
\label{SSS:VShocks}

We fit Gaussian functions to the peaks of the convergence profile for the sight--lines in
no--filament directions (excluding the few pixels with non--radial infall).
We determine the position of the virial shock as
the radial position of the shock near the virial radius toward no--filament regions, where the radial infall
is halted (the radial velocity drops to zero the first time moving toward the cluster center).
We illustrate our results for AMR clusters using the relaxed cluster AMRCL5, but other
relaxed clusters show similar features.
In Figures~\ref{F:Fig4}a and \ref{F:Fig4}b we show histograms of the radial position
and Mach number of virial shocks in AMRCL5.
These figures show that the shock position distribution in this cluster peaks at about
${\cal R}\rmsub{vsh} = \mbox{\rm Mode} \{R\rmsub{vsh} \}= 1.1 \ensuremath{R\rmsub{vir}}$, and has a width of about
$\Delta {\cal R}\rmsub{vsh}\approx 0.5 \ensuremath{R\rmsub{vir}}$; the Mach number distribution peaks at
${\cal M}\rmsub{vsh} = \mbox{\rm Mode} \{M\rmsub{vsh} \}\approx 3.5$,
and has a broad distribution with a width of about $\Delta {\cal M}\rmsub{vsh}\approx 3$.
A map of the 2D angular distribution of the virial shocks in this cluster is shown in Figure~\ref{F:Fig5}a.

The parameters of the virial shocks in all of our AMR clusters are summarized in Table~\ref{T:Table1}.
Columns $e$, $f$ and $g$ show the characteristic radial position of the virial shocks in different clusters,
${\cal R}\rmsub{vsh} = \mbox{\rm Mode} \{ R\rmsub{vsh} \}$, the characteristic Mach numbers,
${\cal M}\rmsub{vsh} = \mbox{\rm Mode} \{ M\rmsub{vsh} \}$, and the surface areas covered by virial shocks, ${\rm A}\rmsub{vsh}$.
The positions of the virial shocks in these clusters fall between 1\ensuremath{R\rmsub{vir}}\ and 1.5\ensuremath{R\rmsub{vir}}.
The characteristic Mach number of virial shocks for all clusters
in our AMR sample is about 3.
The distribution of the surface area covered by virial shocks is broad,
falling between 18--46\% of the total surface area.
In the relaxed clusters (AMRCL2, 5, 9), virial shocks cover 37--46\% of the total area.

\subsubsection{External Shocks}
\label{SSS:XShocks}

External shocks have high Mach numbers due to the fact that the un-shocked gas is very cold.
Since our high--resolution cluster simulation data are in a box cut out from a larger simulation,
we can not determine directly, based on geometry, whether a given shock is internal or external.
Therefore, we use a two--step procedure to locate external shocks.
First, we remove shocks in filaments using the method we described in Section~\ref{SS:AMRShocks}.
Then, based on the Mach number distribution of shocks, we determine a cut--off Mach number
for external shocks and identify external shocks as shocks with Mach number greater than this
cut--off value.

As an example of how this process works, in Figures~\ref{F:Fig4}c and \ref{F:Fig4}d
we show the histogram of the distributions of shock radii, $R_{x}$, and Mach numbers, $M_x$,
for external and internal shocks in one relaxed cluster, AMRCL5
(other clusters show similar characteristics).
Based on the distribution of Mach numbers, shown in Figure~\ref{F:Fig4}d,
we can identify two populations of shocks: external (solid line) and internal (dashed line) shocks.
In this cluster, the characteristic Mach number for external shocks (solid line)
${\cal M}_x \approx 80$, but the Mach number distribution is broad,
extending up to very high Mach numbers (10$^4$),
while the characteristic Mach number for internal shocks (dashed line) is about 3.
External shocks can be clearly identified by using a lower cut--off Mach number of 10 in this cluster.
The distribution of radial positions of external shocks (including pixels with Mach number
above 10) in this cluster peaks at about ${\cal R}_x = 3$\ensuremath{R\rmsub{vir}}, with a width of 1.5\ensuremath{R\rmsub{vir}}.
We determine the lower cut--off of external shocks for each cluster by locating the transition
from internal shocks to external shocks using their Mach number
distribution (typically 10 - 15).
This is clearly an approximation, but it is adequate for our statistical study.

We show the 2D angular distribution of external shocks
for the relaxed AMR cluster AMRCL5 in Figure~\ref{F:Fig5}b.
The large green and light yellow areas in this figure correspond to
pixels with $R_x$ around 3 (${\cal R}_x = 3$; see Figure~\ref{F:Fig4}c).
Pixels covered by internal and external shocks are only in rough alignment with one another
due to two reasons:
i) filaments bend, and become more radial near \ensuremath{R\rmsub{vir}};
ii) the filaments miss the cluster center.

The properties of the external shocks in all AMR clusters are summarized in Table~\ref{T:Table1}, including
their characteristic radial positions, ${\cal R}_x$, and Mach numbers, ${\cal M}_x$, and the
surface area covered, ${\rm A}_x$ (columns $h$, $i$, and $j$, respectively).
This distribution of the radial position of external shocks peaks at about 3\ensuremath{R\rmsub{vir}},
ranging from 2\ensuremath{R\rmsub{vir}}\ to 3.5\ensuremath{R\rmsub{vir}}.
The characteristic Mach numbers of external shocks are very high, around 100 or higher,
due to the low sound speed in the cold, un-shocked gas.
We note, however, that UV background photons, radiation from high--temperature
post-shock gas, and/or other forms of energy injection, which are not included in the adiabatic simulations,
would warm the un-shocked gas to about 10$\,^4$ K, and reduce these Mach numbers.
About half of the surface area around clusters (40\%-60\%) is covered by
these external shocks with very large Mach numbers.

\section{Accretion Shocks in SPH Simulations}
\label{S:SPH}

\subsection{Locating and Analyzing Accretion Shocks in SPH Clusters}
\label{SS:Shocks}

We use a sample of 10 clusters of galaxies drawn from Lagrangian (SPH)
simulations (SPHCL1-10, sorted by total virial mass, SPHCL1 being the most massive),
each located at redshift $z=0$, to analyze accretion shocks.
The properties of these clusters are summarized in Table~\ref{T:Table2}.
The SPH simulations used preheating (2 keV per particle) to match the observed scalings of
luminosity and intra-cluster medium mass with temperature
(Bialek, Evrard \& Mohr, 2001). The underlying cosmology for this
simulation was a spatially flat, {$\Lambda$CDM}\ model with
($\Omega_{\Lambda}$, $\Omega_m$, $\Omega_b$, $h$, $\sigma_8$) = (0.7, 0.3, 0.03, 0.7, 1.0).
These cosmological parameters are in the range of the parameter values allowed by the {\it WMAP}\ 5 year results
except that the simulations have a somewhat low $\Omega_b$ and a high $\sigma_8$ \citep{Dunk08}.
The clusters of galaxies were identified in a large, low--resolution simulation
with a box size of 366 Mpc, using dark matter only, and then the regions identified as seeds of
cluster formation were re-simulated at high resolution, including a full treatment of gas
dynamics coupled to dark matter by gravity (see Bialek et al. 2001 for details).
The total virial mass of the clusters in our SPH sample is in the range of 1.1 - 2.1$\times 10^{15}$ {\ensuremath{\mbox{\rm M}_{\odot}}}.
The average resolution (SPH smoothing length) of the high resolution cluster simulations
at $R/\ensuremath{R\rmsub{vir}}$ = 0.1, 1, and 4 were about 40 kpc, 200 kpc, and 1 Mpc, respectively.
As an example, in Figure~\ref{F:Fig1}, we show the spherically averaged resolution
(SPH smoothing length) in a typical relaxed cluster, SPHCL4
(${\ensuremath{M\rmsub{vir}}} = 1.4\times 10^{15}~{\ensuremath{\mbox{\rm M}_{\odot}}}$; points with error bars connected with the dashed curve).
We use 768 {\sc HEALPix}\ pixels to represent our SPH cluster data.
With this choice  the {\sc HEALPix}\ pixel sizes as a function of radius track the resolution of the
simulations well (within a factor of two; compare the dashed curve with error bars and the straight
dashed line in Figure~\ref{F:Fig1}).
We apply the same visual method to identify relaxed clusters as in the AMR case above (\S\ref{SS:AMRShocks}).
We find four relaxed clusters in our SPH sample.

In SPH simulations, Lagrangian particles are used as interpolation points to produce
a continuum representation of the gas.
Given a compact distribution function $W(\vec{x}, h)$, also called a ``smoothing kernel''
(usually similar to a Gaussian distribution), a physical quantity, $A$,
can be expressed at an arbitrary position $\vec{x}$ as
\begin{equation}
 A(\vec{x}) = \sum_{i} A_{i} \frac{m_{i}}{\rho_{i}} W(\vec{x} - \vec{x}_{i} ; h_{i})
,
\end{equation}
where $\rho_{i}$, $\vec{x}_{i}$, $m_i$ and $h_{i}$ are the local mass density, position,
mass and smoothing length, respectively, associated with particle $i$,
and the summation is over all particles.

Similarly to our method used to locate shocks in the AMR cluster sample,
we locate shocks in our SPH clusters using the convergence, $\kappa$.
However, the discreteness of the SPH particles and the unstructured nature of their
distribution can make regions of strong shear
(large $\left| \nabla \times \vec{v} \right|$) appear as regions of compression.
Therefore we used the shear correction factor of Balsara (1995):
\begin{equation}
f = \frac{\left| \nabla \cdot \vec{v} \right|}{\left| \nabla \cdot
                     \vec{v} \right| + \left| \nabla \times \vec{v} \right| + \epsilon^{2} c_s / h_{sc}}
\end{equation}
where $\epsilon$ is a small arbitrary constant used to prevent division by zero,
$c_s$ is the local sound speed,
and $h_{sc}$ is the scale width of the smoothing kernel.  The factor $f$ is near unity where there is
no shear, and near zero where the shear is large relative to the compression.
Thus, shocks should only appear in regions where the corrected convergence,
$\kappa = - f \, \nabla \cdot \vec{v}$ has large values.
We also looked at the derivative of the internal energy generated by the artificial viscosity (AV),
${D u_{AV}/D t}$.
In order to prevent the unphysical ``post-shock oscillations'' and particle
inter-penetration, SPH codes typically employ AV to mimic energy dissipation
and smooth the thermal energy distribution behind a shock front.
These algorithms are designed to convert kinetic energy into thermal energy only
around regions of compression, and AV is large only where shocks appear.
We use the {\sc Tillamook}\ SPH code (Hearn 2002)
to reconstruct the AV energy derivatives for our SPH cluster data.
The {\sc Tillamook}\ code is an extensible, parallel program written in
C++, which provides the user with significant flexibility in the simulation
components used, including artificial viscosity.
Here, we employ the AV algorithm of Balsara (1995),
and express the contribution of AV to the thermal energy derivative as
\begin{equation}
 { D u_{AV} \over D t} \propto \frac{p}{\rho^2} \left(-\alpha \mu + \beta \mu^2 \right)
,
\end{equation}
where $p$ is the pressure, $\rho$ is the mass density, and $\alpha$ and $\beta$
are the viscosity parameters.
The factor $\mu$ is similar to a velocity divergence;
the contribution of particle $j$ to the viscosity of particle $i$ is
\begin{equation}
\mu_{i,j} = \left\{ \begin{array}{ll}
                   f_{i,j} \,h_{sc} {\vec{x}_{i,j} \, \cdot \vec{v}_{i,j} \over \left( \left|\vec{x}_{i,j}\right|^{2}
                                                                    + \epsilon^{2} \right) c_s} & \vec{x}_{i,j} \cdot \vec{v}_{i,j} < 0 \\
                                                                                0    & \vec{x}_{i,j} \cdot \vec{v}_{i,j} \ge 0
\end{array} \right.
,
\end{equation}
where $\vec{x}_{i,j} = \vec{x}_{i} - \vec{x}_{j}$, $\vec{v}_{i,j} = \vec{v}_{i} - \vec{v}_{j}$,
and $f_{i,j}$ is the mean value of the shear correction for particles $i$ and $j$.

We illustrate our results using the relaxed cluster SPHCL4. Other relaxed clusters
show similar characteristics.
This cluster has a total mass of $M = 1.4 \times 10^{15}~{\ensuremath{\mbox{\rm M}_{\odot}}}$,
and a virial radius of $\ensuremath{R\rmsub{vir}} = 2.38$ Mpc.
In Figure~\ref{F:Fig6} we show the distribution of the physical parameters
toward two fixed pixel directions representing no--filament and filament regions.
In the no--filament regions the largest compression (the maximum of the convergence or
$D u_{AV}/D t$), located at about 3.0\ensuremath{R\rmsub{vir}}, is due to the the virial shock.
However, the radial velocity, {\ensuremath{V\rmsub{rad}}}, becomes zero at around 2.5\ensuremath{R\rmsub{vir}}, about 0.5\ensuremath{R\rmsub{vir}}\
closer to the cluster center. This offset is due to numerical effects in SPH simulations.
The infalling gas has a very low density, therefore the shock regions in it are smeared out due
to the large smoothing length and artificial viscosity \citep{AbadBN00}.
We find that in the direction of filaments (represented by Pixel B), as we would expect,
the gas plunges deep into the cluster and the radial velocity reaches zero at about 0.6\ensuremath{R\rmsub{vir}}.

In Figure~\ref{F:Fig7}, we show spherical averages of physical parameters
toward all pixel directions in SPHCL4.
The left panel shows mean values over all pixel directions with error bars
(the error bars represent the dispersion due to spherical averaging
calculated as for Figure~\ref{F:Fig3} left panel),
while the right panel shows averages toward filaments (thick solid curves), no--filaments
(dashed curves) and, for reference, toward all directions (thin solid curves).
From top to bottom, we plot profiles of the gas density, temperature, pressure,
radial velocity, convergence (units are the same as in Figure~\ref{F:Fig2})
and ``compression'' $D u_{AV}/D t$ (in arbitrary units, normalized to its maximum value),
as a function of distance from the cluster center in units of \ensuremath{R\rmsub{vir}}.
The large dispersion on the gas density at large radii of this relaxed cluster
is due to clumps falling into the cluster via filaments, and show that the
spherically averaged density is not a good approximation far from the cluster center.
The large dispersion between 1\ensuremath{R\rmsub{vir}}\ and 3\ensuremath{R\rmsub{vir}}\ is due to the fact that
the shock location is a function of direction from cluster center
(most visible in the temperature and the $D u_{AV}/D t$ plots).
The temperature floor at about 2 keV is due to the 2 keV preheating
assigned to every particle according to the preheating scheme.

From the distribution of the gas radial velocity, {\ensuremath{V\rmsub{rad}}}, (right panel of Figure~\ref{F:Fig7}),
we can see that along the no--filament directions, the gas infall slows down
at about 3\ensuremath{R\rmsub{vir}}, reaching zero velocity at about 2.1\ensuremath{R\rmsub{vir}}, whereas in directions toward filaments,
the gas plunges into the cluster to about $0.5~\ensuremath{R\rmsub{vir}}$.
The density distribution in the no--filament regions (dashed line) drops below the
average density (solid line) at around 2.1\ensuremath{R\rmsub{vir}}.
Based on our simulations, this should be the position of the virial shock in this cluster.
The position of the maximum of the convergence and $D u_{AV}/D t$
(last two figures on the right panel) is at about 2.7\ensuremath{R\rmsub{vir}}.
This shows that the maximum compression happens at about 0.6\ensuremath{R\rmsub{vir}}\ farther
from the cluster center than where the virial shock position should be
based on the density and velocity distributions.
This is due to the large smoothing length (about 1 Mpc at this distance) and the artificial viscosity.

Since the density is very low in the pre-shock regions, the large smoothing length and the artificial
viscosity make it difficult to estimate the pre-shock values of density, temperature, radial velocity,
at the shock reliably, therefore we omit the analysis of Mach numbers for our SPH cluster sample.

\subsection{Results: SPH Clusters}
\label{SS:SPHResults}

\subsubsection{Virial Shocks}
\label{SSS:VShocksSPH}

The method we apply to locate virial shocks in our SPH sample is similar to that
applied in the AMR sample.
We use the convergence maximum and the radial velocity to identify the position of
the virial shocks to be consistent with our analysis of the AMR simulations.
We illustrate our SPH results on one relaxed cluster, SPHCL4, but other relaxed
clusters show similar characteristics.
In Figure~\ref{F:Fig8}, we show the radial distribution of virial shocks toward different pixels
expressed as a percentage of surface area covered by shocks
(number of pixels with shocks over total number of pixels).
The distribution peaks at about 2.7\ensuremath{R\rmsub{vir}}\ with a width of 1\ensuremath{R\rmsub{vir}}.
The 2D angular distribution of the virial shocks in SPHCL4 is shown in
Figure~\ref{F:Fig9}.

The characteristic positions, ${\cal R}\rmsub{vsh}$,
and the fractional areas covered by virial shocks, ${\rm A}\rmsub{vsh}$, for clusters in our SPH sample are listed
in Table 2 (columns $e$ and $f$).
The positions of the virial shocks in most preheated SPH clusters fall between
2\ensuremath{R\rmsub{vir}}\ and  3\ensuremath{R\rmsub{vir}}.
In relaxed clusters, the virial shocks cover a fraction 36--56\% of the total $4\pi$ solid angle.

\subsubsection{External Shocks}
\label{SSS:XShocksSPH}

Unlike in the AMR case, in the no--filament directions, the SPH clusters always show
only a single accretion shock in each line of sight.  Since there is no sign of a shock farther out,
and the kinetic energy from radial infall is being converted to thermal energy by these shocks,
we refer to these as virial shocks.

However, at about 3\ensuremath{R\rmsub{vir}}, which is where we would expect to find the
external shocks in our AMR simulations, the pre-shock gas in the SPH simulations
experiences strong numerical smoothing (as already emphasized in \citealt{AbadBN00}).
The smoothing length at these large distances is greater than 1 Mpc, thus even if
external shocks were formed at these locations, the SPH simulations could not resolve it.
Therefore we do not analyze external shocks in our SPH cluster sample.

\section{Discussion}
\label{S:Discussion}

\subsection{Observing Virial Shocks Using Their SZ Signature}
\label{SS:ObserveSZ}

The primary motivation in this paper for studying accretion shocks -- besides characterizing their basic properties --
is to assess whether they could be detectable by {\it ALMA}\ through the SZ effect.
As stated in the Introduction,
the advantage of SZ observations is that the SZ surface brightness is proportional to the gas
density (rather than its square, such as the X--ray flux), and therefore
drops considerably less rapidly with radius.
Using spherically symmetric models for the cluster gas and temperature profiles,
\cite{Kocset05} calculated the signal-to-noise ratio ($S/N$) for detecting
shocks near the virial radius of clusters, using the high spatial resolution ($2\,\arcsec$), high sensitivity
($\approx 10\mu$K, with a day of integration) next generation interferometer, {\it ALMA}.
They defined the $S/N$ ratio as the difference in the predicted surface brightness near the
location of the shock, in the presence and absence of a strong discontinuity in gas pressure, divided by
the instrumental noise. The Kocsis et al. study was ``semi--analytical'',
based on a spherically symmetric, self-similar, polytropic intra-cluster gas model, in hydrostatic
equilibrium in a Navarro, Frenk \& White (1997; NFW) dark matter halo.
Different virial shock positions were assumed between 1 and 2\ensuremath{R\rmsub{vir}}, where \ensuremath{R\rmsub{vir}}\ is the
fiducial virial radius.
Non-spherical accretion and projection effects were taken into account
by assuming that the shock fronts are spread over a finite radial distance,
with a width of $0 < \Delta R < \ensuremath{R\rmsub{vir}}$.

For the sake of completeness, we briefly recapitulate the pertinent aspects of
the \citet{Kocset05} model, including their calculation of the SZ surface brightness.
The interested reader should consult their work for more details.
The SZ surface brightness at the impact parameter $y$ relative to the center
of a cluster with a given mass {\ensuremath{M\rmsub{vir}}}\ and redshift $z$ is expressed as
\begin{equation}
   \Delta T(y) =  \Delta T_s \, Y(y),
\end{equation}
where $y$ is the dimensionless distance from the cluster's center at
the closest approach for a given sightline (in units of the virial radius),
and $\Delta T_s$ is a constant normalization,
\begin{equation}
   \Delta T_s = p(x_{\nu}) \, { \sigma_T \, T\rmsub{CMB} \over m_e \, c^2} \, { G {\ensuremath{M\rmsub{vir}}} \over 3 \, c\rmsub{NFW} } \,
                        \eta \, \rho_g(0).
\end{equation}
Here, the frequency dependence is given by $p(x_{\nu}) = x_{\nu} \coth x_{\nu}/2 - 4$
(e.g. Birkinshaw 1999), where the dimensionless frequency is $x_{\nu} = h_P \nu / k_B T\rmsub{CMB}$
($h_P$ and $k_B$ are the Planck and Boltzmann constants; relativistic corrections are
ignored since they are negligible for {\it ALMA}\ observations at $\nu = 100$ GHz of clusters
with $T_g \simless 10$ keV); $T\rmsub{CMB}$ is the temperature of the CMB today;
$c\rmsub{NFW}$ is the concentration parameter for the NFW halo profile;
$\rho_g(0)$ is a normalization constant for the gas density, determined by the universal baryon
fraction; and $\eta$ is a normalization constant for the mass--temperature relation.
Assuming a polytropic equation of state $P\rmsub{gas}\propto\rho^\gamma$,
and also that the gas density profile tracks that of the dark matter
at large radii, determines the polytropic index, $\gamma$, and the
normalization, $\eta$ (Komatsu \& Seljak 2001).

The shape of the surface brightness profile is contained in
\begin{equation}
    Y(y) =  2 \, \int_0^{\ell\rmsub{cut}(y)} W(x) \, y_g(x)^\gamma d\ell,
\end{equation}
where the integral is performed along the line of sight $\ell$; $x =
\sqrt{y^2 + \ell^2}$ is the dimensionless radial coordinate (we use
\ensuremath{R\rmsub{vir}}\ as a unit for distance, thus $x = r / \ensuremath{R\rmsub{vir}}$); $\ell\rmsub{cut}(y) =
\sqrt{ ({\ensuremath{x\rmsub{max}}} + D)^2 - \ell^2}$, where {\ensuremath{x\rmsub{max}}}\ is the assumed radial
position of the virial shock with a width of $2 D$, $y_g(x)$ is the
solution of the equation of hydrostatic equilibrium with a polytropic
gas model (Komatsu \& Seljak 2001), and $W(x)$ is a linear
cut--off function that allows for the edge of the cluster to be ``blurred'',
mimicking shocks that are spread over a finite distance:

\begin{equation}
     W(x) = \left\{ \begin{array}{ll}
                  1                                          &   x < {\ensuremath{x\rmsub{max}}} - D                            \\
                  { {\ensuremath{x\rmsub{max}}} + D - x \over 2 D} &   {\ensuremath{x\rmsub{max}}} - D \le x \le {\ensuremath{x\rmsub{max}}} + D \\
                  0                                          &   x > {\ensuremath{x\rmsub{max}}} + D
\end{array} \right.
.
\end{equation}
\citet{Kocset05} used the above model to compute the difference $\delta
\Delta T$ in surface brightness profiles for pairs of clusters, one
with no edge (${\ensuremath{x\rmsub{max}}}\rightarrow\infty$), and another cluster that
was identical, except that it has an edge near the virial radius
(${\ensuremath{x\rmsub{max}}}\approx 1$).  By comparing $\delta \Delta T$ to the noise
expected from {\it ALMA}, they showed that, in theory, virial shocks
should be detectable with high significance.

We find that the positions of virial shocks in our simulated AMR
clusters fall in the distance interval of 0.9-1.4\ensuremath{R\rmsub{vir}}\ (see Table~\ref{T:Table1}).
Here we focus on these virial shocks, rather than any external shocks,
because the SZ surface brightness at $\approx 3 \ensuremath{R\rmsub{vir}}$ is likely to be
too low to yield a detection of any external shocks.
From Figure~\ref{F:Fig4} we can see that over different
directions, the radial position of the virial shocks varies by about
$\pm 0.5\ensuremath{R\rmsub{vir}}$.  The effective width $D$ of these shocks (which do not
include the directions along filaments) will be narrower, since the 2D
projected SZ surface brightness in a given direction on the sky
depends only on a small fraction of the entire 4$\pi$ steradian solid
angle around the cluster.  Virial shocks cover about 40\% of the total
surface area of clusters (see Table~\ref{T:Table1}), rather than the 100\% coverage
assumed by \cite{Kocset05}.

Most importantly, the 2D projection of the surface area covered by
virial shocks can change substantially, depending on the viewing angle
relative to the active filament plane of the cluster.  Here ``active
filament plane'' refers to the plane crossing the cluster center that
contains most of the filaments around the cluster. The orientation of
this plane can be determined unambiguously by visual inspection in almost
all clusters in our sample, and we expect that this plane can indeed be identified
in most relaxed massive clusters (but our sample of relaxed clusters is not large
enough to verify this).
For example, 50\% of the 2$\pi$ azimuth angle measured from the
cluster center is covered by virial shocks in the XZ-projection of
AMRCL5 (Figure~\ref{F:Fig10}a; area within dot-dashed lines) whereas this
fraction is only 10\% in the YZ-projection of AMRCL7 (Figure~\ref{F:Fig11}a; area
within dot-dashed lines).  The regions contaminated by filaments in
these 2D images were identified via visual inspection.  Since the
majority of filaments around rich clusters of galaxies are located in
the active filament plane of the cluster, the contamination by
filaments can be minimized quite effectively by choosing clusters that
we happen to observe ``edge on'', i.e. we are located in the active
filament plane. Filaments might be identifiable using galaxy redshift
information (see Pimbblet, Drinkwater \& Hawkrigg 2004), which would help in picking the
most promising clusters for observations with {\it ALMA}.

We next use two massive clusters, AMRCL5 and AMRCL7, to make
explicit comparisons to the spherical model predictions in \citet{Kocset05},
and to illustrate some issues related to observing virial shocks based
on their SZ signature (Figures~\ref{F:Fig10} and \ref{F:Fig11}).
Relaxed cluster AMRCL5 has a total mass of $1.2 \times 10^{15}~{\ensuremath{\mbox{\rm M}_{\odot}}}$,
and AMRCL7, a merging cluster, has a mass of $1.1
\times 10^{15}~{\ensuremath{\mbox{\rm M}_{\odot}}}$ (see Table~\ref{T:Table1}).

We first calculate the 2D SZ surface brightness distribution from our
simulation data, integrating along the line of sight ($\ell$) over the
extent of the cluster along the line of sight (from $\ell_1$ to
$\ell_2$) using

\begin{equation}
           \Delta T(x,y) =  p(x_{\nu}) \, T\rmsub{CMB} \,{k_B \sigma_T \over m_e c^2}
                                                     \int_{\ell_1}^{\ell_2} n_e(x,y,\ell) \, T_e(x,y,\ell) \, d\ell
.
\end{equation}
Here $(x,y)$ are spatial coordinates in the plane of the sky,
i.e. perpendicular to the line--of--sight coordinate $\ell$; $n_e =
f_g \rho_g / \mu_e m_p$ is the electron density, where $\mu_e$ is the
mean molecular weight per electron and $m_p$ is the proton mass;
$\rho_g$ is the gas density; $f_g$ is the gas mass fraction (we adopt
$f_g=0.9$), and we use the standard assumption that the electron
temperature equals the gas temperature, $T_e = T_g$ (similar
expressions were used for the XZ and YZ projections).  In practice, we
pixelize $x,y$ and $\ell$, and approximate the integral with a sum
over the line of sight from $\ell_1 = -10$ Mpc to $\ell_2 = 10$ Mpc.
\cite{Kocset05} found that the $S/N$ ratio is the lowest for clusters
at a redshift of $z=0.3$, thus we generated images of clusters
assuming they are located at this redshift. This choice makes our
$S/N$ ratio estimates conservative.

We first examine the XZ projection of AMRCL5, which, as mentioned
above, is close to an edge on projection, and therefore has the least
amount of contamination from filaments (Figure~\ref{F:Fig10}).  In this
projection, the line of filaments is stretching from the upper left
corner to the lower right corner of the image (as revealed by the
shape of the contours in Figure~\ref{F:Fig10}a).  We find that other views of
even relaxed clusters are often much more contaminated by filaments.
The azimuthally averaged SZ surface brightness profile, excluding the
angles contaminated by filaments (i.e. the area within the dot-dashed
lines, containing $\approx$50\% of the 2$\pi$ azimuth angles from the
cluster center) is shown in Figure~\ref{F:Fig10}a. In this figure,
the error bars represent the dispersion (standard deviation) due to azimuthal averaging
at each radius (instrument noise and contamination from other astrophysical effects are
not included).

In Figure~\ref{F:Fig10}b, we see that the virial shock appears to
leave a relatively sharp feature in the average SZ profile, i.e. a
sharp drop near \ensuremath{R\rmsub{vir}}\, with a width of $\approx$ 0.1\ensuremath{R\rmsub{vir}}.  In
addition, there is a distinct break in the slope of the surface
brightness profile at $\approx$0.6\ensuremath{R\rmsub{vir}}. This break is not caused by
the change in the slope of the NFW density profile, since it is not
present in the spherical models.  We suspect it is caused by the
breaking of the self-- similarity of the pressure profiles due to the
finite entropy of the cluster gas.  As a consequence, the spherical
models discussed above, shown by the dotted and dot--dashed curves in
Figure~\ref{F:Fig10}b (with and without an edge, respectively), do
not fit the simulated SZ profiles over the whole region inside the
virial shocks.

To quantify the significance at which the presence of the virial shock
can be detected, we fit a power--law model to the SZ surface
brightness profile near the virial shock, as might be done to an
actual observation (dashed curve).  The difference between the
simulated SZ profile and the power--law model ($\Delta T\rmsub{pow} \propto
r^{-\alpha}$, $\alpha$ = 2.45) with no cut--off, $\delta \Delta T =
\Delta T_{cl} - \Delta T\rmsub{pow}$, is shown in
Figure~\ref{F:Fig10}c.  We also show the difference (dot--dashed
curve) between the SZ profiles in the Kocsis et al. spherical models for a
cluster with the same mass, located at a redshift of $z=0.3$, with and
without an edge (dotted and dot-dashed curves in
Fig.~\ref{F:Fig10}b; cf. Figure 2 of Kocsis et al. 2005).  For the
cluster with an edge, we chose the position and width of the virial
shock to be similar to those of our simulated cluster (${\ensuremath{x\rmsub{max}}} =
1\ensuremath{R\rmsub{vir}}$ and $D = 0.1\ensuremath{R\rmsub{vir}}$).

The central SZ decrement of our simulated cluster is $\Delta T_0 =
-1360 \mu$K.  The difference in the SZ decrement of our simulated
cluster, compared to the power--law extrapolation, is about $\delta
\Delta T\approx 4\mu$K within a 0.2\ensuremath{R\rmsub{vir}}\ wide annulus around the
1\ensuremath{R\rmsub{vir}}\ (evaluated in the no--filament regions).  Assuming an angular
resolution of $2\,\arcsec$, and taking into account the 50\% coverage,
this corresponds to $3.4 \times 10^4$ independent pixels with a virial
radius of $490\,\arcsec$, at z=0.3.  Following \cite{Kocset05}, we
assume 20 hour on-source integration time, and estimate the $S/N$
ratio as $S/N \approx N\rmsub{pix}^{1/2} (S/N)_1$, where $N\rmsub{pix}$ is the
number of independent pixels and $(S/N)_1$ is the signal to noise for
one pixel, and we obtain a high $S/N$ ratio of about 70.  The Kocsis
et al. model has a somewhat lower central decrement of -921$\mu$K, but
it has a significantly flatter SZ profile, and it predicts a factor of
$\approx 5$ higher $S/N$ than our analysis based on the XZ projection
of AMRCL5.

For comparison, we next estimate the $S/N$ ratio for detecting virial
shocks in a cluster which is {\em not} optimally selected to be edge
on.  This case is illustrated in Figure~11a, showing the YZ projection
of the 2D SZ surface brightness map of AMRCL7. Again, the dashed
circle represents the projected \ensuremath{R\rmsub{vir}}, and the area enclosed by the
dot-dashed wedge show the region where virial shocks can be seen (no
projected filament contamination).  As we expect, only a small
fraction, in this case 10\% of the 2$\pi$ azimuth around the cluster
center, is not contaminated by filaments.  The azimuthally averaged SZ
profile of this cluster in this no--filament region is shown in
Figure~\ref{F:Fig11}b.  The central SZ decrement for this cluster
in this projection is $\Delta T_0 = -736 \mu$K.  This cluster also
shows a sharp drop in the surface brightness profile near \ensuremath{R\rmsub{vir}}\, and
also a discernible break in the slope at around 0.4\ensuremath{R\rmsub{vir}}, somewhat
closer to the cluster center than the break in AMRCL5.  The difference
between the simulated surface brightness and the extrapolation of a
power--law fit (with slope $\alpha$ = 2.15), $\delta \Delta T$, is
shown in Figure~\ref{F:Fig11}c.  This shows $\approx$2$\,\mu$K
difference in a 0.2 \ensuremath{R\rmsub{vir}}\ wide annulus.  Again, assuming an angular
resolution of $2\,\arcsec$, and taking into account the 10\% coverage,
this corresponds to $6.2 \times 10^3$ independent pixels with a virial
radius of $470\,\arcsec$, at a redshift of $z=0.3$.  Using a simple
estimate as above, we obtain $S/N\approx 16$ for the detection of the
virial shock in this cluster.  The Kocsis et al. model with the same
mass and ${\ensuremath{x\rmsub{max}}} = 1.1\ensuremath{R\rmsub{vir}}$, $D = 0.05\ensuremath{R\rmsub{vir}}$ located at a redshift
of $z=0.3$ has a central SZ decrement of -809$\mu$K, and yields a
$S/N$ ratio about 10 times higher than our model.

Based on our analysis of the SZ profiles of simulated clusters, we
have found that the $S/N$ ratios for detecting virial shocks in
different clusters are quite significant, $S/N \approx 16-70$, but
reduced by a factor of 5-10 relative to the predictions of the
spherically symmetric, self-similar models used by \cite{Kocset05}.
There are two reasons for this reduction: (i) the simulations show a
generally steeper SZ profile, especially at radii beyond 0.5 \ensuremath{R\rmsub{vir}};
and (ii) only a fraction (10-50\% in our two cases) of the
2$\pi$azimuthal angle around the cluster's perimeter is covered by
virial shocks near \ensuremath{R\rmsub{vir}}\ in our simulated clusters.

The second of these effects is simply caused by the smearing of the
shocks in the 2D projection of the three--dimensional, non-spherical
features, over a significant fraction of the cluster's perimeter.  The
first effect is attributable to two factors. First, we find that the
concentration parameter for our massive relaxed clusters is about
$c\rmsub{NFW}=5.5$, much larger than the value $c\rmsub{NFW}=2.4$
adopted by Kocsis et al. Second, the drop in the surface density
profile might be related to the excess entropy in the core of the
cluster. (The slope of the SZ profile could thus plausibly provide
information on the entropy structure of clusters; a break in its slope
may be verified by {\it ALMA}\ relatively easily, since the SZ decrement is
much larger at around 0.5 \ensuremath{R\rmsub{vir}}\ than at the virial radius.)  Finally,
we note that the toy model used in Kocsis et al. adopted a somewhat
different outer boundary condition than was used in the original work
of Komatsu \& Seljak (2001), on which the Kocsis et al. toy models are
based. In particular, while Komatsu \& Seljak require the {\it slope}
of the gas profile to match that of the DM halo at the virial radius,
Kocsis et al. require the baryon/DM mass ratio interior to a large
radius (=200 $R_{\rm vir}$) to equal the universal mean
$\Omega_b/\Omega_{\rm DM}$.  As a result, the SZ profile in the Kocsis
et al. toy model is significantly flatter near the virial radius than
that of the Komatsu \& Seljak toy model (e.g. the drop in the surface
brightness profile from the cluster core to the virial radius,
predicted in Figure 1 in Kocsis et al., is a factor of $\sim 50$,
which is about a factor of $\sim 2$ smaller than the factor of $\sim
100$ drop predicted for the same cluster parameters in the upper left
panel in Figure 15 in Komatsu \& Seljak).

Our simplified treatment suggests that even if we do not select
clusters optimally (to minimize contamination by filaments), {\it ALMA}\
may still be able to detect virial shocks with
mosaic observations covering
a significant portion of the annulus near \ensuremath{R\rmsub{vir}}\ for a suitably bright cluster.
If in real clusters the surface brightness drops below 10$\mu$K around
the virial radius and $\delta \Delta T$ is only about 2-3$\,\mu$K,
then the detection of virial shocks will become much more difficult,
due to contamination from secondary effects in the CMB at the $\mu$K
level from unrelated background or foreground galaxy clusters, dust
emission, and point source contamination.  Additional secondary
effects which can be important at around \ensuremath{R\rmsub{vir}}\ may be generated by
clusters of galaxies via gravitational lensing, and by the thermal SZ
effect from unresolved clusters.  A bipolar pattern with an amplitude
of a few $\mu$K is produced by clusters when they lens CMB
fluctuations with a large--scale gradient \citep{SeljZald00}. A
similar pattern is generated by the so--called moving cluster effect
\citep{BirkGull83}, but the amplitude of this effect at the virial
radius is most likely below $1\mu$K \citep{MolnBirk00}.  The
theoretical sensitivity of interferometers is also reduced for
large--scale smooth surface density distributions, therefore we may
have overestimated the $S/N$ ratio.  On the other hand, using
nonlinear de-convolution, much larger scale structures can be
recovered from mosaic observations than an analytical treatment would
suggest \citep{Helfet02}.
 A redshift of 0.3 was used to obtain a conservative estimate of the $S/N$ ratio;
 we expect a significant increase in $S/N$ if the redshift of the clusters were selected optimally.
An improved estimate for the significance
of detecting virial shocks in clusters of galaxies can be done using
simulated mock {\it ALMA}\ observations and analyzing the simulated
visibilities with software packages used to analyze real data
(e.g. {\it MIRIAD}, {\it AIPS}). A detailed analysis would also make it
possible to find the optimal redshift for clusters to observe their
virial shocks.  We defer this detailed analysis to future work.

\subsection{Large Scale X-ray and SZ Signal Associated with Clusters of Galaxies}
\label{SS:LargeScaleSZX}

Soltan, Freyberg \& Hasinger (2002) have found evidence for large-scale extended X-ray
emission out to about 2 degrees around rich Abell clusters using the {\it ROSAT}\ All-Sky Survey.
Assuming that this emission is originated from thermal bremsstrahlung within diffuse gas
around clusters, Soltan et al.\ estimated the gas temperature to be below 1 keV, and
concluded that this signal may originate from super-cluster gas -- gas left over from
galaxy cluster formation and trapped by the gravitational potential well of super-clusters.

\cite{Myeret04} searched for the SZ decrement associated
with clusters and groups of galaxies in the {\it WMAP}\ data by cross-correlating it
with the APM Galaxy Survey and the Abell-Corwin-Olowin catalog.
They found evidence for an SZ signal around clusters of galaxies
extending out to about 3\ensuremath{R\rmsub{vir}}\ (assuming that most of the signal is originated from
clusters with about $\ensuremath{R\rmsub{vir}} \approx 1.75$ Mpc),
and suggested that this signal is due to the hot super-cluster gas.
\cite{Afshet07} used WMAP 3 year data to stack images of 193 massive clusters of
galaxies. Similar to the results of Myers et al., Afshordi et al. found that
the distribution of the spherically averaged SZ signal of the stacked clusters
extends out to at least about 3\ensuremath{R\rmsub{vir}}\ (about 4 - 5 Mpc; see their Figure 2).

Most direct searches for super-cluster gas in individual objects -- as opposed to
the stacked searches discussed in the preceding paragraph -- have not been successful
\citep{Geno05,Boug99,MoBi98,Perset90}.
However, recently, \cite{Geno08} found evidence
for SZ signal associated with the Corona Borealis super-cluster.
\cite{Zappet05} have found tentative evidence for diffuse,
large scale emission from the Sculptor super-cluster. The estimated temperature of this
super-cluster was less then 0.5 keV.

Studies of the gas in superclusters can offer vital clues to the baryon evolution in strongly
clustered environments.
Based on our cluster sample from the AMR simulations, we find that, in relaxed clusters,
the external shocks are located at about 3\ensuremath{R\rmsub{vir}}, and the region between the virial shocks and
the external shocks are filled with gas falling into the cluster.
Thus, our results suggest that a significant fraction of the large scale signal found around
clusters of galaxies might be associated with infalling gas around individual massive clusters,
and not with super-cluster gas
(i.e. the signal is coming from gas bound to the individual massive clusters).

\subsection{Comparison with Other Results Based on Simulations}
\label{SS:Compare}

A comparison of our results to those based on other cosmological simulations is
 difficult, because we focus here on the high--resolution details of accretion shocks around
 individual, massive clusters, whereas other studies focused on deriving global properties,
using a much larger simulation volume at lower resolution.
Nevertheless, we find that external shocks around massive clusters of galaxies form at about
3\ensuremath{R\rmsub{vir}}\ ($\pm 1\ensuremath{R\rmsub{vir}}$), with large Mach numbers, extending to values as high as
$M = 10^4$.
Studies based on other simulations also found very high Mach numbers for external shocks
\citep{Miniet00,Ryu_et03,Keshet03,Pfroet06}.
Taking into account the photo-heating of the pre-shocked gas, \cite{Miniet00} found that the
Mach numbers of external shocks are less than a few hundred.
Based on the self-similar solution of \cite{Bert85}, scaled to our AMR simulations,
we estimate that external shocks at about 3\ensuremath{R\rmsub{vir}}\ from the centers of massive
clusters of galaxies should have Mach numbers of $\approx 90$ depending on
the actual distance of the shock from cluster center.

Quantitative comparisons between our results for the AMR and SPH cluster samples
are hindered by the fact that the
AMR simulations were adiabatic, whereas the SPH simulations assumed that the gas was pre--heated.
The two simulations also used different cosmological parameters: $\sigma_8$=0.92 (AMR)
versus 1.0 (SPH), and $\Omega_b$=0.047 (AMR) versus 0.03 (SPH).
Overall, we find that the position of the virial shocks are much farther out in our
SPH simulations (located at about 2-3\ensuremath{R\rmsub{vir}})
than in the AMR simulations ($\approx 1$\ensuremath{R\rmsub{vir}}); this is
in spite of the fact that the SPH clusters have a lower baryon fraction.
This difference is probably not due to preheating: high--resolution
AMR simulations of clusters of galaxies with and without preheating
show no noticeable difference in the radial positions of the virial shocks
(see Figure 1 of Younger \& Bryan 2007).
Therefore, we attribute the large difference in the radial position of the virial shocks between our
AMR and SPH simulations to the artificial viscosity and large smoothing
length far from the cluster center.
(We note that the problem may be indirectly exacerbated by the presence
of pre--heating, which keeps gas densities lower, and the smoothing length larger).
However, we expect the other features of virial shocks (such as the covering area and mass ratios)
in our SPH simulations to be reliable. Indeed, these quantities agree with
 those we found in the AMR cluster sample.

A comparison of the amount of mass locked up in filaments versus mass
in no--filament regions is also of interest, because it tells us
roughly how much mass will be accreted via filaments in the form of
dense clumps, collapsed halos and gas collapsed into filaments versus
via accretion of low--density diffuse gas. We determined this mass ratio in
regions outside of the smoothed virial radius and thus should be free from
the numerical problems mentioned in the previous paragraph.
We show the total mass in filaments as a percentage of the total mass for
all clusters in our AMR and SPH cluster sample in Tables 1 and 2.
About 80\% to 90\% of the mass resides in the filaments falling toward the
cluster centers both in our AMR and SPH sample.  The slightly larger
percentage of mass in filaments in our SPH simulations is most likely
due to the higher value of $\sigma_8$ used in those simulations.
Based on our simulations, we conclude that massive clusters of
galaxies accrete most of their mass via filaments, rather than via
low--density diffuse gas, even at the later stages of their evolution.

\section{Conclusion}
\label{S:Conclusion}

We have studied large-scale accretion shocks in a sample of massive
($\approx 10^{15}~{\ensuremath{\mbox{\rm M}_{\odot}}}$) galaxy clusters at redshift $z=0$ drawn from
AMR and SPH cosmological {$\Lambda$CDM}\ simulations.  The properties of shocks
in clusters with a larger range of masses and redshifts will be
studied in a forthcoming paper.  Most of our quantitative results are
based on the cluster sample drawn from our AMR simulations, which did
not take feedback (heating, cooling, star--formation, and supernovae)
into account.  However, the properties of the shocks we derived from
these simulations agree with those in an independent set of SPH
simulations, with the exception of the radial location of the virial
shocks (which are found to be farther out in the SPH runs, due to
numerical effects).

The main results of this study can be summarized as follows.
(1) Clusters are surrounded by two distinct sets of accretion shocks,
located near $\sim 1\ensuremath{R\rmsub{vir}}$, and $\sim 3\ensuremath{R\rmsub{vir}}$, respectively.
(2) These shocks cover $\sim 50\%$ of the surface area around each
cluster.
(3) The shocks near $\sim 1\ensuremath{R\rmsub{vir}}$ cause a sharp drop in
the projected SZ surface brightness, tracing out a circular
arc in projection over a non--negligible fraction (10--50\%; depending
strongly on both the dynamical state of the cluster, and on the viewing angle) of the 2$\pi$
azimuthal angle around the cluster center.
(4) The significance of the SZ feature due to these shocks is reduced by a factor
of 5-10 relative to earlier predictions based on spherical models \citep{Kocset05},
because of the partial extent of the sharp arc, and because the simulations
predict steeper SZ surface brightness profiles.  Nevertheless, the features may
be detectable with dedicated mosaic observations, by
 the next generation high angular resolution and sensitivity radio interferometer, {\it ALMA}.

Our results should motivate a more detailed follow--up study of the
effects of non-spherical accretion, astrophysical and instrumental
noise, and image processing techniques on the detectability of virial
shocks in massive clusters of galaxies using realistic mock {\it ALMA}\
observations of clusters drawn from high--resolution cosmological
simulations.  Such studies will be necessary for a more robust
estimate of the feasibility of the detection of these shocks, as well
as to find the optimal redshift for the observations, and to determine
the best target selection method and observation strategy.

\acknowledgments

We thank M. Birkinshaw, F. Miniati, and P. Ricker for useful
discussions, and the anonymous referee for comments that significantly
improved this manuscript.  We acknowledge the use of {\sc HEALPix}\ (Gorski
et al., 2005; http://healpix.jpl.nasa.gov).  AEE acknowledges support
from NSF grant AST-0708150 and NASA ATP Grant NAG5-13378.  GB
acknowledges support from NSF grants AST-05-07161, AST-05-47823 and
supercomputing resources from the National Center for Supercomputing
Applications.  ZH was supported by the NSF grant AST-05-07161 and by
the Pol\'anyi Program of the Hungarian National Office for Research
and Technology (NKTH).

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