Dataset Preview
The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 15
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 82801)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 15
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text
string | meta
dict |
|---|---|
\section{Introduction}
The dipole anisotropy seen in the cosmic microwave background (CMB) temperature map
\citep{1996ApJ...473..576F} is compelling evidence that the solar system has a large peculiar motion with respect to the overall cosmic expansion. There are known local components to this motion, including the orbital velocity of the Sun in the Milky Way Galaxy and the attraction of our Galaxy toward M31. Once these components are taken into account, it is found that the Local Group of galaxies has a peculiar motion of over 600 km~s$^{-1}$\ in a well established direction.
Soon after the discovery of the CMB dipole the coincidence in direction of our motion with prominent large scale structure was noted \citep{1984ApJ...280..470S} and then evidence was found for flows of nearby galaxies toward this direction \citep{1988ApJ...326...19L}. There has been great interest in trying to identify the dominant source, or at least the characteristic distance, of the `great attractor' causing our large scale motion. This interest is well summarized in several conference proceedings
\citep{1988lsmu.book.....R, 1993cvf..conf.....B, 2000ASPC..201.....C}.
The issue has been complicated by the observation that two important structures lie in the general direction of our motion: the Norma-Hydra-Centaurus complex in the foreground and the
enormous Shapley Concentration in the background \citep{1989Natur.338..562S, 1989Natur.342..251R}. The debate continues regarding the relative importance of these structures on our motion
\citep{2006ApJ...645.1043K, 2006MNRAS.368.1515E}.
There has been a long-standing appreciation that there are significant dynamical influences on intermediate scales within what has traditionally been called the Local Supercluster. Our galaxy is known to experience a pull toward the Virgo Cluster at the heart of the Local Supercluster
\citep{1981ApJ...246..680T, 1982ApJ...258...64A, 1982ApJ...263..485H, 1984ApJ...281...31T, 2000ApJ...530..625T}. However, the story on intermediate scales is more complicated than just an attraction centered on or near the Virgo Cluster. The Numerical Action Method (NAM) models of \citet{1995ApJ...454...15S} assign mass according to the complex distribution of light and provide a reasonable description of galaxy motions. Still, NAM reconstructions have not yet provided a fully satisfactory explanation of the `local velocity anomaly' \citep{1988lsmu.book..169T, 1988lsmu.book..115F,
1992ApJS...80..479T}. We use the term to describe the pattern of negative motions with respect to Hubble expansion of galaxies in a neighboring filament called the Leo Spur in the Nearby Galaxies Atlas \citep{1987nga..book.....T}.
In the present paper we return to the problem of the local velocity anomaly. Imaging with Hubble Space Telescope (HST) has provided a wealth of accurate distances to nearby galaxies based on measures of the luminosity of stars at the tip of the red giant branch, the TRGB method \citep{2004AJ....127.2031K, 2006AJ....131.1361K}. Additionally, over the years many other good distances have become available. Those that are particularly important for this work include those provided by the HST Cepheid Key Project \citep{2001ApJ...553...47F}, the Surface Brightness Fluctuation (SBF) study of
\citet{2001ApJ...546..681T}, and two catalogs of luminosity--linewidth distances, one a sample of extreme edge-on galaxies with 2MASS magnitudes that has been discussed by \citet{2002A&A...396..431K} and the other an extension of the sample discussed by \citet{2000ApJ...533..744T}. These new observations of distances have clarified that the phenomenon referred to as the `local velocity anomaly' definitely exists but it is so much more extensive than previously suspected that the adjective `local' may not be appropriate. It will be shown that the observed anomalous motion has nothing to do with the known pull toward the Virgo Cluster nor to the large--scale great attractor(s).
\section{A Catalog of Galaxy Distances}
Our database is an outgrowth of the Nearby Galaxies Catalog \citep{1988ngc..book.....T} and for the
current discussion has the same limit of 3,000 km~s$^{-1}$. We presently have distance estimates for 1791 galaxies in 743 groups in this volume derived from four different methods.
The reference scale for our distances is set by the HST Cepheid Key Project observations \citep{2001ApJ...553...47F}. Including all sources, we have 51 distances by the Cepheid method. Next we add galaxies with TRGB distance estimates. Individual TRGB distances are of comparable quality to the Cepheid values and are demonstrated to be on a consistent scale \citep{2003AJ....125.1261D,
2004ApJ...608...42S, 2007ApJ...661..815R}. Procedures for measuring the TRGB are discussed by \cite{1996ApJ...461..713S} and \citet{2006AJ....132.2729M}. Distance moduli are directly compared in the top panel of Figure~\ref{compare} for 14 galaxies with both Cepheid and TRGB measurements. There are 221 TRGB estimates in the present sample. Third, we accept the SBF measures of \citet{2001ApJ...546..681T} and \citet{2007ApJ...655..144M}. The Tonry measures are available for 299 galaxies around the sky while the Mei sample of 84 galaxies is restricted to the Virgo Cluster and a projected group. The claimed accuracies with SBF are comparable with the Cepheid and TRGB accuracies. The zero point for SBF distances is confirmed to agree with the Cepheid scale. The top panel of Figure~\ref{compare} compares distance moduli for 7 galaxies with both Cepheid and SBF determinations. Of course, the TRGB and SBF methods are intimately related; both use the standard candle nature of the brightest red giant stars. The TRGB method requires that the stars be individually resolved and hence can only be applied to nearby galaxies. The SBF method uses the statistical properties of these stars as they blend together in an image and can be applied to larger distances. Some galaxies have been observed by more than one of the above three methods. In total, 601 galaxies in our sample have been observed by at least one of the Cepheid, TRGB, or SBF methods. We assign an uncertainty of 10\% to a distance obtained by one of these methods. Generally, the galaxies with TRGB estimates lie within 7 Mpc and the galaxies with SBF estimates lie beyond this distance.
\begin{figure}
\figurenum{1}
\centering
\includegraphics[scale=0.3]{ceph_trgb_sbf_2.jpg}
\includegraphics[scale=0.3]{reference_tf_2.jpg}
\includegraphics[scale=0.3]{ceph_trgb_sbf_tf_2.jpg}
\caption{Comparison of distance moduli determined by different methods. {\bf Top:} Comparison of TRGB moduli (red squares) and SBF moduli (green circles) with moduli determined with Cepheid variables. {\bf Middle:} Comparison of luminosity--linewidth moduli with moduli determined by either Cepheids or the TRGB. Source: Tully (blue triangles) and Karachentsev (black open circles) {\bf Bottom:} Combination of the top two panels. }
\label{compare}
\end{figure}
On top of these, we add two luminosity--linewidth samples. The larger of these involves 1030 distance measures derived from the correlation between galaxy luminosity and rotation rate as measured from the width of an HI line profile \citep{1977A&A....54..661T}. The calibration is that of \citet{2000ApJ...533..744T} shifted slightly to be consistent in zero point with the HST Cepheid Key Project results. This zero point is set by 40 galaxies with Cepheid or TRGB distance measures. Here is our current calibration:
\begin{equation}
M_B^{b,i,k} = -19.99 -7.27 (W_R^i - 2.5)
\end{equation}
\begin{equation}
M_R^{b,i,k} = -21.00 -7.65 (W_R^i - 2.5)
\end{equation}
\begin{equation}
M_I^{b,i,k} = -21.43 -8.11 (W_R^i - 2.5)
\end{equation}
\begin{equation}
M_H^{b,i,k} = -22.17 -9.55 (W_R^i - 2.5)
\end{equation}
where the superscripts on the $B,R,I,H$ absolute magnitudes indicate corrections have been made for obscuration within our Galaxy ($b$) and due to the inclination of the target galaxy ($i$) and for redshift effects ($k$) \citep{2000ApJ...533..744T}. The parameter $W_R^i$ is a measure of the inclination corrected neutral Hydrogen linewidth \citep{1985ApJS...58...67T}. The optical band magnitudes $B,R,I$ are `total' values. The near infrared $H$ magnitudes are aperture values in the system of \citet{1986ApJ...302..536A}.
The other luminosity--linewidth sample is composed of edge-on galaxies with 2MASS $K$-band photometry \citep{2002A&A...396..431K} restricted to less than 3,000~km~s$^{-1}$. This sample contributes 402 distances, 178 already included. The substantial overlap between the two luminosity--linewidth samples provides
confirmation that the zero points are the same and gives rms agreement per measure of 0.39 mag. The excellent agreement in distance moduli between the luminosity--linewidth and other measures is shown in the middle panel of Figure~\ref{compare}. The bottom panel of Figure~\ref{compare} compares all cases in the current sample with distance measurements by more than one method.
The luminosity--linewidth distance estimates are considered to have an accuracy of 20\% rms for a single observation. They are less accurate than those obtained with the procedures previously discussed but are much more numerous. SBF observations are restricted to early--type galaxies that tend to reside together in high density environments. Luminosity--linewidth observations are restricted to spiral galaxies that are more widely distributed. The combination of the two provides a rich sampling of the distribution of galaxies and their motions throughout the Local Supercluster.
Our current database of distances for galaxies within 3,000~km~s$^{-1}$\ is provided in two tables that can be accessed in their entirety in electronic form. Table~1 identifies the 1791 individual galaxies with measured distances. The column information is as follows. (1) J2000 equatorial coordinates.
(2) Principal Galaxies Catalogue (PGC) name from the Lyon Extragalactic Database (LEDA:
http://leda.univ-lyon1.fr/). (3) Common name. (4) Group ID for cross-reference with Table 2. (5) NBG ID, the group ID in the Nearby Galaxies Catalog \citep{1988ngc..book.....T}. (6,7) Galactic longitude and latitude. (8,9) Supergalactic longitude and latitude. (10) Numeric morphological type code. (11) Differential Galactic reddening $E(B-V)$ \citep{1998ApJ...500..525S}. (12) Total blue magnitudes, mostly from the Third Reference Catalogue \citep{1991trcb.book.....D}. (13,14,15,16) Velocities in the reference frames of the Sun, the Galactic center, the Local Sheet (defined later), and the CMB, in km~s$^{-1}$. The columns 17 to 29 are filled if the galaxy has a luminosity--linewidth distance estimate based on the revised \citet{2000ApJ...533..744T} calibration. (17) Photometrically derived ratio of minor to major axes $b/a$, related to the galaxy inclination $i$ by ${\rm cos}~i = [((b/a)^2 - q_0^2))/(1-q_0^2)]^{1/2}$ where $q_0=0.2$ is taken as the axial ratio of a spiral galaxy seen edge on. (18) Number of sources for the measurement of axial ratio. (19) Total $B$ magnitude from CCD area photometry. (20,21) Total $R$ magnitude and number of sources for $R$ magnitude. (22,23) Total $I$ magnitude and number of sources for $I$ magnitude. (24,25) $H_{-0.5}$ aperture magnitude and number of sources of $H$ photometry. (26,27) Heliocentric velocity and linewidth based on HI observations; the linewidth is the parameter $W_R^i$ defined by \citet{1985ApJS...58...67T}, including rectification from the viewing inclination to edge on, defined to agree statistically with twice the maximum rotation velocity. (28,29) Distance modulus and uncertainty determined from the luminosity--linewidth method, where the uncertainty reflects a weighting of the separate bandpasses. (30) Distance modulus determined in the case of a galaxy from the flat galaxy--2MASS sample; an uncertainty of
0.40 mag is accepted in these cases. (31,32) Distance modulus given by either the Surface Brightness Fluctuation method ($s$), the brightest Red Giant Branch stars ($r$), or the Cepheid period--luminosity relation ($c$), and indication of the source, $s$, $r$, or $c$; an uncertainty of 0.2 mag is accepted in these cases. Table~1 is available at
\noindent
ifa.hawaii.edu/$\sim$tully/voidtable1.
In Table~2 information is reassembled and averaged within 743 groups (including groups of one). The columns are described below. (1) A unique group identification number; appears in column 4 of Table~1 for individual galaxies. (2) NBG ID, as in column 5 of Table~1. (3,4) Galactic longitude and latitude of group. (5,6) Supergalactic longitude and latitude of group. (7) Logarithm of B absolute luminosity summed over group and based on observed distance. (8,9,10,11) Group averaged velocities in
reference frames of the Sun, the Galactic center, the Local Sheet, and the CMB, in km~s$^{-1}$. (12,13) Distance modulus and uncertainty, averaged over all estimates for group members. (14,15,16,17) Distance, and components of distance in the Supergalactic SGX, SGY, and SGZ directions, in Mpc. (18) Peculiar velocity if H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$, $V_{pec} = V_{LS} - {\rm H}_0 d$, in km~s$^{-1}$, where $V_{LS}$ is the velocity in column 10 and $d$ is the distance in column 14. (19,20,21) Number of galaxies in group with luminosity--linewidth distance measures from the extended Tully--Pierce sample, the averaged distance modulus from luminosity--linewidth measures, and the assigned uncertainty. (22,23,24) Number of galaxies in group with distances measures from the flat galaxies--2MASS sample, the averaged modulus, and uncertainty. (25,26,27) The sum of the number of galaxies in the group with Surface Brightness Fluctuation, Tip of the Red Giant Branch, or Cepheid distance measures, the averaged modulus, and uncertainty.
Table~2 is available at
\noindent
ifa.hawaii.edu/$\sim$tully/voidtable2-743groups.
\clearpage
\begin{deluxetable}{lrlrlrrrrrrrrrrrrrrrrrrrrrrrrrrc}
\tabletypesize{\scriptsize}
\rotate
\tablewidth{0pt}
\tablenum{1}
\tablecolumns{32}
\tablecaption{Distance Estimates for 1791 Galaxies}
\label{tbl:dist_indiv}
\tablehead{\colhead{1}&\colhead{2}&\colhead{3}&\colhead{4}&\colhead{5}&\colhead{6}&\colhead{7}&\colhead{8}&\colhead{9}&\colhead{10}&\colhead{11}&\colhead{12}&\colhead{13}&\colhead{14}&\colhead{15}&\colhead{16}&\colhead{17}&\colhead{18}&\colhead{19}&\colhead{20}&\colhead{21}&\colhead{22}&\colhead{23}&\colhead{24}&\colhead{25}&\colhead{26}&\colhead{27}&\colhead{28}&\colhead{29}&\colhead{30}&\colhead{31}&\colhead{32}}
\tablehead{\colhead{RA (J2000) Dec}&\colhead{PGC}&\colhead{Name}&\colhead{Gp ID}&\colhead{NBG ID}&\colhead{$\ell$}&\colhead{$b$}&\colhead{SGL}&\colhead{SGB}&\colhead{T}&\colhead{$E(B-V)$}&\colhead{$B_T$}&\colhead{$V_{\odot}$}&\colhead{$V_{GSR}$}&\colhead{$V_{LS}$}&\colhead{$V_{CMB}$}&\colhead{$b/a$}&\colhead{N$_i$}&\colhead{$B$}&\colhead{$R$}&\colhead{N$_R$}&\colhead{$I$}&\colhead{N$_I$}&\colhead{$H$}&\colhead{N$_H$}&\colhead{$V_{21}$}&\colhead{$W_R^i$}&\colhead{$\mu_{LL}$}&\colhead{$\epsilon_{LL}$}&\colhead{$\mu_{FG}$}&\colhead{$\mu_{src}$}&\colhead{$src$}}
\startdata
000158.5-152741 & 143& WLM & 222& 14-12 12& 75.8655& -73.6256& 277.8076& 8.0847& 10& 0.036& 11.04& -127& -74& -33& -462& & & & & & & & & & & & & & & 24.89& ~rc\\
000315.0+160843 & 218& NGC7814 & 1211& 65 ~-6 ~6& 106.4094& -45.1749& 309.0612& 16.4021& 2& 0.045& 11.57& 1054& 1194& 1279& 696& 0.19& 1& 11.72& 10.01& 1& 9.40& 1& 0.00& 0& 1054& 521& 31.29& 0.36& & 30.60& s~~\\
000358.7+204506 & 279& NGC7817 & 1178& 64 ~-8 ~8& 108.2271& -40.7610& 313.8132& 17.1426& 4& 0.058& 12.74& 2308& 2457& 2547& 1956& 0.27& 3& 0.00& 0.00& 0& 10.60& 3& 9.25& 1& 2308& 432& 32.10& 0.38& 32.01 & & \\
000620.1-412945 & 474& ESO293-034& 1088& 61 ~~0 16& 332.8271& -72.9123& 253.5419& -1.5693& 6& 0.017& 13.64& 1516& 1482& 1474& 1278& & & & & & & & & & & & & & 31.60 & & \\
000813.9-343445 & 621& ESO349-031& 233& 14-13 13& 351.4707& -78.1179& 260.1831& 0.4018& 10& 0.012& 15.81& 229& 217& 222& -40& & & & & & & & & & & & & & & 27.48& ~r~\\
000820.7-295458 & 627& NGC0007 & 1096& 61-18 18& 13.9903& -80.1369& 264.5891& 1.9321& 5& 0.014& 14.35& 1496& 1499& 1513& 1209& 0.21& 1& 0.00& 0.00& 0& 12.83& 1& 0.00& 0& 1496& 212& 31.81& 0.40& & & \\
000956.4-245748 & 701& NGC0024 & 355& 19 ~-8 ~7& 43.6887& -80.4344& 269.3877& 3.2260& 5& 0.019& 12.10& 553& 572& 594& 249& 0.26& 2& 12.03& 11.05& 1& 10.56& 2& 9.75& 1& 553& 223& 29.79& 0.35& & & \\
001124.7-412353 & 800& ESO293-045& 1088& 61 ~~0 16& 330.3100& -73.5343& 253.9490& -2.4348& 8& 0.011& 15.25& 1466& 1430& 1423& 1229& 0.23& 1& 0.00& 0.00& 0& 14.20& 1& 0.00& 0& 1466& 156& 32.17& 0.40& & & \\
001508.4-391313 & 1014& NGC0055 & 234& 14 13 13& 332.6677& -75.7388& 256.2418& -2.4123& 9& 0.013& 8.47& 125& 95& 91& -121& 0.21& 1& 8.42& 7.57& 1& 7.21& 1& 0.00& 0& 125& 203& 25.98& 0.36& & 26.70& ~r~\\
001531.5-321051 & 1038& ESO410-005& 234& 14 13 13& 357.8407& -80.7103& 262.9460& -0.2577& -5& 0.014& 15.17& 0& 0& 0& 0& & & & & & & & & & & & & & & 26.39& ~r~\\
001745.5+112701 & 1160& NGC0063 & 1213& 65 ~~6 ~6& 109.8744& -50.5655& 305.1594& 11.9146& 5& 0.111& 12.73& 1160& 1282& 1361& 805& & & & & & & & & & & & & & & 31.36& s~~\\
002023.1+591735 & 1305& IC0010 & 222& 14-12 12& 118.9699& -3.3395& 354.4176& 17.8657& 10& 1.560& 11.78& -346& -161& -55& -565& & & & & & & & & & & & & & & 24.10& ~~c\\
\enddata
\end{deluxetable}
\begin{deluxetable}{rlrrrrrrrrrrrrrrrrrrrrrrrrr}
\tabletypesize{\scriptsize}
\rotate
\tablewidth{0pt}
\tablenum{2}
\tablecolumns{27}
\tablecaption{Averaged Distance Estimates for 743 Groups}
\label{tbl:dist_groups}
\tablehead{\colhead{Gp ID}&\colhead{NBG ID}&\colhead{$\ell$}&\colhead{$b$}&\colhead{SGL}&\colhead{SGB}&\colhead{log $M_B$}&\colhead{$V_{\odot}$}&\colhead{$V_{GSR}$}&\colhead{$V_{LS}$}&\colhead{$V_{CMB}$}&\colhead{$\mu$}&\colhead{$\epsilon_{\mu}$}&\colhead{$d$}&\colhead{SGX}&\colhead{SGY}&\colhead{SGZ}&\colhead{$V_{pec}$}&\colhead{N$_{LL}$}&\colhead{$\mu_{LL}$}&\colhead{$\epsilon_{LL}$}&\colhead{N$_{FG}$}&\colhead{$\mu_{FG}$}&\colhead{$\epsilon_{FG}$}&\colhead{N$_{src}$}&\colhead{$\mu_{src}$}&\colhead{$epsilon_{src}$}}
\startdata
1& 11 -1 1& 282.93& 74.45& 102.70& -2.35& 12.26& 1091& 1042& 999& 1421& 31.13& 0.10& 16.8& -3.7& 16.4& -0.7& -246.& 51& 31.33& 0.11& 2& 30.92& 0.30& 91& 31.09& 0.10 \\
2& 11 ~2 1& 299.61& 66.04& 112.68& -1.26& 11.11& 900& 832& 776& 1235& 30.96& 0.13& 15.6& -6.0& 14.4& -0.3& -376.& 9& 31.42& 0.16& 2& 31.45& 0.30& 2& 30.08& 0.17 \\
3& 11 -3 1& 291.06& 68.94& 108.83& -3.18& 9.77& 2011& 1946& 1894& 2349& 30.70& 0.36& 13.8& -4.4& 13.0& -0.8& 872.& 1& 30.70& 0.36& & & & & & \\
4& 11 -4 1& 289.24& 65.45& 111.57& -5.36& 11.12& 1615& 1538& 1480& 1960& 31.23& 0.11& 17.6& -6.4& 16.3& -1.6& 176.& 7& 31.47& 0.17& 1& 31.97& 0.41& 3& 30.94& 0.15 \\
5& 11 -5 1& 283.70& 69.13& 107.33& -5.36& 11.25& 1079& 1011& 960& 1421& 31.85& 0.11& 23.4& -7.0& 22.3& -2.2& -775.& 7& 32.31& 0.16& 1& 31.69& 0.41& 5& 31.80& 0.11 \\
8& 11 ~7 1& 245.25& 76.24& 94.22& -6.75& 9.53& 1147& 1106& 1073& 1472& 29.64& 0.36& 8.5& -0.6& 8.4& -1.0& 446.& 1& 29.64& 0.36& & & & & & \\
9& 11 -8 1& 299.14& 62.46& 116.08& -2.64& 10.38& 1565& 1486& 1424& 1904& 30.98& 0.37& 15.7& -6.9& 14.1& -0.7& 262.& 1& 30.98& 0.37& & & & & & \\
10& 11 ~9 1& 304.29& 62.04& 117.17& -0.48& 10.23& 1233& 1158& 1096& 1566& 31.81& 0.15& 23.0& -10.5& 20.5& -0.2& -607.& 1& 31.27& 0.41& & & & 1& 31.86& 0.16 \\
15& 11 ~0 1& 283.19& 68.68& 107.56& -5.78& 10.09& 748& 679& 626& 1091& 32.15& 0.28& 26.9& -8.1& 25.5& -2.7& -1366.& 2& 32.15& 0.28& & & & & & \\
20& 11 ~0 1& 304.60& 78.71& 101.03& 3.72& 9.36& 1244& 1217& 1182& 1553& 30.87& 0.36& 14.9& -2.8& 14.6& 1.0& 78.& 1& 30.87& 0.36& & & & & & \\
\enddata
\end{deluxetable}
\clearpage
\section{The Peculiar Velocity Field Within 3000 km~s$^{-1}$}
Knowledge of distances, $d$, permits a subtraction of cosmic expansion velocities, H$_0 d$, from observed velocities, $V_{obs}$, to give $V_{pec}$, the radial component of what are referred to as peculiar velocities:
\begin{equation}
V_{pec} = V_{obs} - {\rm H}_0 d
\end{equation}
where H$_0$ is the Hubble Constant.
The decomposition of observed velocities into cosmic expansion and peculiar velocity terms is seen to require knowledge of the Hubble Constant which is defined as
\begin{equation}
{\rm H}_0 = < V_{obs} / d>
\end{equation}
that is, a measure of the expansion rate over a sufficiently large domain of the Universe that peculiar motions cancel and have a negligible impact.
Imagine that observers make a zero point error in the determination of distances; i.e., on average, distances are off by a factor $f_e$ from true values, $d_{true} = f_e d_{measured}$. Then the product H$_0 d$ has terms $f_e$ in the numerator and denominator that cancel. The consequence is the well--known result that peculiar velocity measures are insensitive to a zero-point error in the distance scale as long as the assumed value of H$_0$ is consistent with the scale of measured distances.
Yet there is a problem. We are not guaranteed that peculiar motions are negligible in the volume we sample to establish H$_0$. For example, we live in the Local Supercluster which is an overdense part of the Universe. It would not be surprising if there was a net infall within this region. As a general statement, most observers in the Universe must live in overdense places, with a local retardation of the cosmic expansion, and will tend to measure a value of H$_0$ locally that is smaller than the cosmic value. Or as another example, an observer might live on the outskirts of a large concentration and the preponderance of nearby galaxies in the direction of the concentration might be rushing away, toward the concentration. The large number of these receding objects might cause H$_0$ to be overestimated.
In the present case, it is rather clear that the volume of our sample, limited to 3000 km~s$^{-1}$, is too small to define H$_0$ without bias. It might be tempting to assert that H$_0$ is known, for example from CMB measurements \citep{2003ApJS..148..175S}. However such a value might not be consistent with the zero point scale of the distance measures. Here we avoid the issue of which scale might be `correct', if there should be an inconsistency. We simply note that H$_0$ on the scale of our present sample is not well defined because it does not extend in a self--consistent manner to large enough distances.
These caveats regarding H$_0$ are mentioned because, as will be seen, there are large deviations from cosmic expansion seen within the 3000 km~s$^{-1}$\ region {\it whatever} reasonable value is assumed for H$_0$. Selecting a larger value of H$_0$ enhances a pattern in co-moving coordinates of overall infall while selecting a smaller value of H$_0$ creates a trend toward outflow.
The patterns of peculiar motion can best be seen in video animations posted at
\noindent
ifa.hawaii.edu/$\sim$tully.
\subsection{The pattern of peculiar velocities and a choice of H$_0$}
With specification of H$_0$, peculiar velocities can be found through Eq. (1) for all galaxies with measured distances. Although there is uncertainty in H$_0$, observations constrain it to lie roughly within $70 < {\rm H}_0 < 80$ km~s$^{-1}$~Mpc$^{-1}$. The HST Key Project best estimate is toward the low end of this range \citep{2001ApJ...553...47F} while our own best estimate is toward the high side \citep{2000ApJ...533..744T}.
The series of panels in Figure~\ref{varyH} illustrates the effect of varying the choice of H$_0$ from 70, through 75, to 80 km~s$^{-1}$~Mpc$^{-1}$. Negative peculiar velocities are coded blue while positive peculiar velocities are red. Large symbols are given in cases with Cepheid, TRGB, or SBF distances and small symbols are given in cases with the numerous but individually less accurate luminosity--linewidth distances. The tiny black dots locate galaxies positioned according to their observed velocities but lacking distance measures. There are 8795 galaxies in the $V<3500$~km~s$^{-1}$\ cube, part of a compilation drawn mainly from the Center for Astrophysics Redshift Catalog (http://www.cfa.harvard.edu/~huchra/zcat/zcom.htm) circa 2002.
\onecolumn
\begin{figure}[htbp]
\figurenum{2}
\centering
\includegraphics[scale=1.0]{vpec_xyz_3H_red-blue.jpg}
\caption{Peculiar velocities from 2 views and with 3 choices for H$_0$. {\bf Left:} SGX vs. SGZ in velocity units. {\bf Right:} SGY vs. SGZ. {\bf Top:} H$_0=70$~km~s$^{-1}$~Mpc$^{-1}$. {\bf Middle:} H$_0=75$~km~s$^{-1}$~Mpc$^{-1}$. {\bf Bottom:} H$_0=80$~km~s$^{-1}$~Mpc$^{-1}$. Large symbols: distances determined from Cepheids, TRGB, or SBF. Small symbols: distances determined by the correlation between luminosity and linewidth. Black dots: no distance available. {\it Red}: peculiar velocities away from us. {\it Blue}: peculiar velocities toward us.
}
\label{varyH}
\end{figure}
\twocolumn
Patterns of average positive and negative peculiar velocities can be seen in large swaths across these figures. There is a prominent overall pattern of infall in the maps with H$_0=80$ km~s$^{-1}$~Mpc$^{-1}$\ which
successively diminishes in the H$_0=75$ and H$_0=70$ maps. These negative velocities dominate the map in a large sector toward the Virgo Cluster, the most populated region, and almost everywhere at negative SGZ; i.e., below the equatorial plane in the supergalactic coordinate system. By contrast, peculiar velocities tend to be positive in the quadrant south of the Galactic plane (SGY negative) and above the supergalactic equator (SGZ positive). Peculiar velocities also tend to swing positive at greater distances in the general direction of the motion indicated by the CMB dipole (near the supergalactic equator toward SGX negative). The positive velocities are most pronounced in maps with H$_0=70$ but the trends persist with H$_0=75$ and $80$.
The major point we would make with this part of the discussion is that the overall patterns in the peculiar velocity field are similar whatever value for H$_0$ is considered in the range of reasonable values between 70 and 80 km~s$^{-1}$~Mpc$^{-1}$. The direction and amplitude of inferred peculiar velocity of our Galaxy is insensitive to the choice of H$_0$ over this range. In Section 3.4, a weak preference will be found for H$_0=74$ km~s$^{-1}$~Mpc$^{-1}$. The amplitudes of peculiar velocities of individual galaxies other than our own depend on the choice of H$_0$. The fundamental results of this paper are based on the well determined motion of our Galaxy and the patterns, but not critically the amplitudes, of other galaxies in our sample.
\subsection{Galactic and Local Group standards of rest}
As a preliminary step, we review the status of the Solar motion with respect to the galaxies of the Local Group. Here, as in the subsequent discussion, the amplitude and direction of our motion is determined by minimizing a condition of the following form:
\begin{equation}
{\rm min} [\sum_{i=1}^N ( V^i -{\rm H} d^i +\hat x_i V_x +\hat y_i V_y +\hat z_i V_z )^2]
\end{equation}
The $N$ galaxies to be considered with measured distances $d^i$ and observed velocities $V^i$ have Galactic coordinates $\ell_i$,$b_i$ which decompose along cardinal axes as
\begin{equation}
\hat x_i = {\rm cos} ~\ell_i ~{\rm cos}~ b_i
\end{equation}
\begin{equation}
\hat y_i = {\rm sin} ~\ell_i ~{\rm cos} ~b_i
\end{equation}
\begin{equation}
\hat z_i = {\rm sin} ~b_i
\end{equation}
(or the equivalent $\hat X$, $\hat Y$, $\hat Z$ in Supergalactic coordinates $L, B$).
One solves for the expansion component H and the cardinal components of our motion with respect to the chosen rest frame, $V_x, V_y, V_z$. The term H can alternatively be fixed at a reasonable value or left as a free parameter. In general, the solutions are more stable if H is fixed.
In the first step of the analysis of motions within the Local Group, heliocentric velocities are considered and the reference sample is $N=40$ galaxies within 1.1~Mpc, hence within roughly the zero--velocity surface or radius of first turnaround to infall in the Local Group \citep{2002A&A...389..812K}. For this gravitationally bound sample the Hubble parameter is set to H$=0$~km~s$^{-1}$~Mpc$^{-1}$. The Sun is found to have a motion of $(V_x^2+V_y^2+V_z^2)^{1/2}=318 \pm 20$~km~s$^{-1}$\ toward $\ell=106 \pm 4, b=-6 \pm 4$ ($L=349, B=+30$). This solution is in good agreement with previous results \citep{1977ApJ...217..903Y, 1996AJ....111..794K, 1999AJ....118..337C}.\footnote{Courteau (private communication) points out a misprint in his paper with van den Bergh. They intended to report $V_{LG}^{\odot}=306$~km~s$^{-1}$\ toward $\ell=99, b=-3$.}
The close agreement with earlier work is expected since the Local Group reference information has only been augmented incrementally. It is instructive to note that the amplitude of 318~km~s$^{-1}$\ is 12~km/s greater than found by \citet{1999AJ....118..337C} because we include 14 additional galaxies, 5 of them dwarfs around M31 which turn out to have negative velocities larger than any previously known. Though we now have 40 galaxies for the analysis they are strongly clustered on the sky and in distance. If the sample is split between the 16 galaxies nearer than 500 kpc (the Milky Way companions) and the 24 more distant than 500 kpc (mostly the M31 sub-group) then the amplitude of the Solar motion with respect to these separate samples varies by $\pm 20$~km~s$^{-1}$\ (342 and 299 km~s$^{-1}$\ respectively). The assigned error attempts to account for the effects of poor sampling. Bootstrap resampling gives errors less than half what we quote. The direction of the Sun's motion with respect to the Local Group has small errors because it is stabilized by the dominant component: the orbital motion of the Sun in the Galaxy.
Our solution provides the transform from the heliocentric rest frame, $V_{\odot}$, to the Local Group rest frame, $V_{LG}$
\begin{equation}
V_{LG} = V_{\odot} -86 \hat x +305 \hat y -33 \hat z
\end{equation}
\begin{equation}
V_{LG} = V_{\odot} +270 \hat X -52 \hat Y +159 \hat Z
\end{equation}
The largest component of this motion is due to the rotation of the Sun within the disk of the Milky Way.
\citet{1997MNRAS.291..683F}
claim the angular velocity at the Solar position is $27.19 \pm 0.87$~km~s$^{-1}$~kpc$^{-1}$ and review the details of the local motion of the Sun, while
\citet{2005ApJ...628..246E}
report a distance to the Galactic center of $R_0=7.62\pm0.32$~kpc. The resultant transform of velocities from the Solar to the Galactic Standard of Rest is
\begin{equation}
V_{GSR} = V_{\odot} +9.3 \hat x +218 \hat y +7.6 \hat z
\end{equation}
corresponding to a motion of the Sun of $V_{GSR}^{\odot}=219\pm12$~km~s$^{-1}$\ toward $\ell=87.6, b=+2.0$ ($L=356, B=+50$).
Then the motion of the Galaxy within the Local Group is
\begin{equation}
V_{LG}^{GSR}=V_{LG}-V_{GSR} = -95 \hat x +87 \hat y -41 \hat z
\end{equation}
or $135 \pm 25$~km~s$^{-1}$\ toward {$\ell=137 \pm 10, b=-18 \pm 10$ ($L=342, B=-3$). This direction is $17^{\circ}$ removed from the position of M31, offset toward the Maffei--IC~342 Group. The projected positions of these features are shown in Figure~\ref{localmotion}. The offset of the vector of our motion from M31 is sufficiently uncertain that a direct hit on M31 is not precluded.
\footnote{Throughout this discussion, subscripts on velocities will identify the reference frame and superscripts
will identify the object with the velocity. If there is no superscript it will be understood that the velocity pertains to object $i$ in an ensemble $N$.}
\begin{figure}[htbp]
\figurenum{3}
\centering
\plotone{lb_localmotion.jpg}
\caption{The direction of solar motion. The Sun has a motion $V_{GSR}^{\odot} = 219$~km~s$^{-1}$\ about the center of our Galaxy and a motion $V_{LG}^{\odot} = 318$~km~s$^{-1}$\ with respect to the centroid of the Local Group. The vector difference $V_{LG}^{GSR}=V_{LG}-V_{GSR}=135$~km~s$^{-1}$\ reflects the motion of the center of the Milky Way Galaxy in the Local Group rest frame. The other vector direction that is marked locates the motion $V_{LS}^{\odot} = 318$~km~s$^{-1}$\ of the Sun with respect to galaxies beyond the Local Group but within 7~Mpc, within the Local Sheet. The dominant local attractor beyond the Galaxy is M31. Ranked by apparent luminosity, and accepting luminosity as a stand--in for mass (both luminosity and gravitational attraction fall as $d^2$), M33 is an order of magnitude dynamically less important and IC~342 is a further factor of two less important. The other galaxies identified in the figure are less important by further factors of 2 to 3. The dashed line locates the Supergalactic equatorial plane.}
\label{localmotion}
\end{figure}
This Local Group rest frame may not deserve much attention. Within this rest frame the Milky Way has a motion of 135~km~s$^{-1}$\ essentially toward M31 while M31 has a motion of {\it zero} toward us. With respect to the center of mass of the Local Group, in the absence of other forces, the Milky Way and M31 will have motions of approach that partition the observed 135~km~s$^{-1}$\ in proportion to their masses. The evidence from the motions of satellites suggest the two systems have comparable masses \citep{2000MNRAS.316..929E}. Based on their relative luminosities, M31 could be expected to be 50\% more massive. Certainly, the Milky Way is not a negligible test particle compared with M31. The so-called Local Group rest frame is essentially the M31 rest frame. In the next section we will investigate a more useful frame of reference.
\subsection{Peculiar velocities within the Local Sheet}
It is at this second step that things get interesting. Figure~\ref{local} zooms in from Figure~\ref{varyH} (now for the case
H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$) to highlight the local neighborhood. The color coding of velocities is more detailed. We see a remarkable discontinuity in peculiar velocities between the galaxies that lie in our filament and the regions just beyond. In the Nearby Galaxies Atlas \citep{1987nga..book.....T}, the structure we live in is called the Coma--Sculptor Cloud because it creates a band from Galactic north pole to Galactic south pole. Our neighbors are tightly confined to the equatorial plane of the supergalactic coordinate system, so fall within the slice only $\pm 1.5$~Mpc thick about SGZ=0 shown in Figure~\ref{localsheet}. The structure we live in and that has now been reasonably sampled with accurate distances has comparable dimensions in SGX and SGY. This region is not quite synonymous with the Coma--Sculptor Cloud so we will refer to it as the `Local Sheet'.
The nearest adjacent structure lies in a layer at negative SGZ with respect to the Local Sheet and in the Nearby Galaxies Atlas is called the Leo Spur. The abrupt step in peculiar velocities at the edge of the Local Sheet was called the `local anomaly' by \citet{1988lsmu.book..115F} and we called that step in conjunction with the apparent motion toward us of the Leo Spur the `local velocity anomaly'
\citep{1988lsmu.book..169T, 1992ApJS...80..479T}. The large number of good TRGB distances available today place the local velocity anomaly in glaring relief.
The motion of the Local Group within the Local Sheet can be determined by realizing the minimization of Eq.~(7) for $N=158$ galaxies with measured distances in the range $1.1 < d_i<7$~Mpc; that is, beyond those in the Local Group and nearer than those in the adjacent structures. The best fit is found with H$=67$~km~s$^{-1}$~Mpc$^{-1}$\ and a motion of the Local Group of $66 \pm 24$~km~s$^{-1}$\ toward $L=150 \pm 37, B=+53 \pm 20$ ($\ell=349, b=+22$). This motion differs from zero with only marginal significance and is consistent with the observation by \citet{2003A&A...398..479K} that nearby groups and individual isolated galaxies adhere to the local expansion with a dispersion of only 40~km~s$^{-1}$. Note that the local expansion need not be, and is probably not, the same as the cosmic expansion.
The very low relative peculiar velocities within the Local Sheet and our small, marginally significant peculiar motion within this structure suggest we consider a frame of reference that is at rest with respect to this structure. The motion of the Sun with respect to 158 galaxies with accurate distances at $1.1 < d_i < 7$~Mpc in the Local Sheet defines the following relations:
\begin{equation}
V_{LS} = V_{\odot} -26 \hat x +317 \hat y -8 \hat z
\end{equation}
\begin{equation}
V_{LS} = V_{\odot} +234 \hat X -31 \hat Y +214 \hat Z
\end{equation}
or a motion of the Sun of $V_{LS}^{\odot}=318 \pm 20$~km~s$^{-1}$\ toward $\ell=95 \pm 4$, $b=-1 \pm 4$ ($L=353$, $B=+42$).\footnote{The best fit is found with the local expansion term H=67~km~s$^{-1}$~Mpc$^{-1}$. Rigorously, there should be a velocity correction to the centroid of the reference frame by the term $({\rm H}_0 - {\rm H})*d^{centroid} \sim 10$~km~s$^{-1}$.} The distance constraints are from Cepheid and TRGB methods and equally weighted.
This direction is shown in Fig.~\ref{localmotion}. It is seen to be close in amplitude and direction with $V_{LG}^{\odot}$. In fact, it is as close to our value of $V_{LG}^{\odot}$ as our value is to other estimates of the Local Group motion given in the literature. Given both the uncertainties in $V_{LG}^{\odot}$ and the ambiguity in the meaning of this reference frame, we will base the rest of our discussion on the rest frame established by galaxies beyond the Local Group but within 7~Mpc, velocities we designate $V_{LS}$. This reference frame is established by a completely independent sample than those that define $V_{LG}$ and is more robust, based on five times more galaxies and with good sky coverage.
\onecolumn
\figurenum{4}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.85]{vpec_xyz_loc.jpg}
\caption{Peculiar velocities in the local neighborhood. Zoom into the central region of the panels of Fig.~\ref{varyH}, now with H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$. In this figure and all the subsequent figures showing peculiar velocities, the color code is more refined than in Fig.~2. Here: {\it purple} $V_{pec} \leq -400$~km~s$^{-1}$; {\it dark blue} $-400 < V_{pec} \leq -200$~km~s$^{-1}$; {\it light blue} $-200 < V_{pec} \leq -100$~km~s$^{-1}$; {\it green} $-100 < V_{pec} < 0$~km~s$^{-1}$; {\it yellow} $0 \leq V_{pec} < 100$~km~s$^{-1}$; {\it orange} $100 \leq V_{pec} < 200$~km~s$^{-1}$; {\it red} $200 \leq V_{pec} < 400$~km~s$^{-1}$; {\it brown} $400 \leq V_{pec}$~km~s$^{-1}$. The left panel has a depth $-500<SGY<+1500$~km~s$^{-1}$; the right panel: $-500<SGX<+1000$~km~s$^{-1}$.}
\label{local}
\end{figure}
\twocolumn
\begin{figure}[htbp]
\figurenum{5}
\centering
\includegraphics[scale=0.56]{vpec_xy_localsheet.jpg}
\caption{
Peculiar velocities in and around the Local Sheet. The region displayed is a slice 3 Mpc thick
centered on the supergalactic equator. Symbols and colors have the same meaning as in previous plot (peculiar velocities are calculated assuming H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$). The 7~Mpc radius circle centered on the Milky Way defines the region referred to as the Local Sheet. The Virgo Cluster is at the upper left.
}
\label{localsheet}
\end{figure}
\subsection{The velocity discontinuity beyond the Local Sheet}
The availability of TRGB distances to objects beyond the Local Sheet have strongly confirmed the existence of a velocity discontinuity at $\sim 7$~Mpc. The effect is unambiguously seen in the nearest part of the Leo Spur at large negative supergalactic latitudes. The 14+19 association of dwarf galaxies \citep{2006AJ....132..729T} involves 4 galaxies with well established distances ($7.8 \pm 0.3$~Mpc) and velocities ($195 \pm 26$~km~s$^{-1}$\ in the Local Sheet frame). The derivation of peculiar velocities requires an assumption of the value of the Hubble Constant and, as will be justified later, we take
H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$. In this case, members of the 14+19 association have peculiar velocities of $-382 \pm 47$~km~s$^{-1}$. In addition, two companions of NGC~2903, D564-08 and D565-06, have reliable TRGB distances of 8.4 and 8.5~Mpc and velocities of 385 and 394~km~s$^{-1}$\ respectively which imply peculiar velocities of $-237$ and $-235$~km~s$^{-1}$, and there is the extreme case of the relatively isolated Leo Spur galaxy D634-03 at 9.3~Mpc with $V_{LS}=186$~km~s$^{-1}$\ and $V_{pec}=-502$~km~s$^{-1}$\ \citep{2006AJ....131.1361K}. These new high precision distances to galaxies with significant SGZ components in the line of sight confirm what had earlier been suspected: that the Leo Spur and our Local Sheet have peculiar motions of several hundred km~s$^{-1}$\ toward each other. The peculiar motions of these galaxies in the line-of-sight are indicated in Figure~\ref{leo}.
\begin{figure}[htbp]
\figurenum{6}
\centering
\plotone{vpec_xyz_leo.jpg}
\caption{Peculiar velocities in the local neighborhood. Same projection as right panel of Fig.~\ref{local}. Special attention is given to seven galaxies in three groups in the Leo Spur.
These galaxies have well determined TRGB distances that place them at $8-9$~Mpc and imply $V_{pec} \sim -325$~km~s$^{-1}$. The straight line vectors show the line-of-sight peculiar velocities averaged over each group assuming H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$.}
\label{leo}
\end{figure}
It is evident from the peculiar velocity patterns in Figure~\ref{varyH} that the anomaly is not restricted to the Leo Spur. Negative peculiar velocities are seen all over in the region around the Virgo Cluster near the +SGY axis and generally at all $-$SGZ. The negative peculiar velocities in the direction of the Virgo Cluster have long been seen as a reflex of the pull of the cluster on us \citep{1982ApJ...258...64A}. This is part of the story but not all of it. Another general feature is the trend of positive peculiar velocities in the quadrant with +SGZ and $-$SGY.
The occurrence of the velocity anomaly is manifested in an abrupt break in the amplitude and direction of galaxy motions relative to our motion found through the condition imposed by Eq.~(7). The vector of our relative motion can be determined in shells of either distance or velocity to look for systematic drifts that would be indicative of the depth of perturbations or for erratic bounces that would indicate instability in the solution. It is found that the vector of our motion is quite stable, achieving a direction and amplitude in the immediate shells beyond 7~Mpc that changes very little out to $V_{LS} \sim 3000$~km~s$^{-1}$. The global solution over this range with $N=683$ distance measures after averaging in groups provides the solution for transformation from Local Sheet referenced motions $V_{LS}$ to a reference frame established from objects within the general region of the Local Supercluster, $V_{LSC}$
\begin{equation}
V_{LSC} = V_{LS} -211 \hat x -178 \hat y +169 \hat z
\end{equation}
or
\begin{equation}
V_{LSC} = V_{LS} +35 \hat X +196 \hat Y -255 \hat Z .
\end{equation}
With respect to the general Local Supercluster reference frame, the Local Sheet has a motion of $V_{LSC}^{LS}=323 \pm 25$~km~s$^{-1}$\ toward $\ell=220 \pm 7, b=+32 \pm 6$ ($L=80, B=-52$). This best solution is achieved with H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$. The solution is remarkably insensitive to the choice of H$_0$. Variations from 60 to 90 km~s$^{-1}$~Mpc$^{-1}$\ result in variations in the velocity amplitude of only $\pm 2$~km~s$^{-1}$\ and variations in direction of only $\pm 5$ degrees. With a choice of H$_0$ less than 74 there is an overall expansion and with H$_0$ greater than 74 there is an overall compression. The value of H$_0=74$ is accepted for the rest of the discussion although it rests on the weak hypothesis that there is neither expansion nor compression centered on our location.
Figure~\ref{aitoff_pecvel} provides a display of the currently available sample of peculiar velocities in an equal area projection on the sky. Galaxies with $\vert V_{pec} \vert < 100$~km~s$^{-1}$\ are not shown in order to make clear the separation on the sky between galaxies with large positive peculiar velocities and those with large negative values. The crosses in the figure labeled LSC and CMB indicate vectors of motion that will be discussed in later sections.
In the following discussion, the volume beyond 7 Mpc and with $V_{LS}<3000$~km~s$^{-1}$\ will frequently be referred to as simply the Local Supercluster (LSC).
A significant component of the peculiar motion of the Local Sheet comes from the pull of matter in and near the Virgo Cluster. The cluster itself has a mass approaching $1 \times 10^{15}~M_{\odot}$ \citep{2005ApJ...635L.113M}. Numerical Action Method models demonstrate that this much mass in the cluster and a comparable amount of mass in the north Galactic hemisphere within the Local Supercluster generates a peculiar motion of $\sim 200$~km~s$^{-1}$\ in the Virgo direction at our location -- as has long been implicated; eg \citet{1982ApJ...258...64A}. The vector representing the motion of the Local Sheet with respect to the Local Supercluster, $V_{LSC}^{LS}$, has a component directed toward the Virgo Cluster of $V_{LSC;V}^{LS}=185 \pm20$~km~s$^{-1}$. If this vector toward Virgo is subtracted off the vector toward the overall Local Supercluster, the result is the vector $V_{LSC;LV}^{LS}$, where $LV$ stands for Local Void for reasons that will soon be described. Coordinate frame transforms obey
\begin{equation}
V_{LV} = V_{LS} -222 \hat x -130 \hat y -10 \hat z
\end{equation}
\begin{equation}
V_{LV} = V_{LS} +77 \hat X +16 \hat Y -248 \hat Z
\end{equation}
corresponding to a Local Sheet motion of $259 \pm 25$~km~s$^{-1}$\ toward $\ell=210 \pm 7$, $b=-2 \pm 6$ ($L=11, B=-72$). Since the Virgo and LV vectors are almost orthogonal, the decomposition has only a weak dependence on the amplitude of the Virgo component. A variation of $\pm 50$~km~s$^{-1}$\ in velocity toward Virgo affects $V_{LV}$ at the level of 10~km~s$^{-1}$\ in amplitude and $15^{\circ}$ in direction. The direction of the motion $V_{LSC}$ is shown in Figures~\ref{lsc_vectors_big} and \ref{lsc_vectors} along with the decomposition vectors $V_{LSC;V}^{LS}$ and $V_{LSC;LV}^{LS}$.
\onecolumn
\begin{figure}[htbp]
\figurenum{7}
\begin{center}
\includegraphics[scale=0.85]{pecvel_aitoff_sglb.jpg}
\caption{Aitoff projection of observed peculiar velocities. Blue symbols: $V_{pec}<-100$~km~s$^{-1}$; red symbols: $V_{pec}>+100$~km~s$^{-1}$. The Local sheet has a motion with respect to this sample toward the orange cross labeled LSC and a motion toward the apex of the Cosmic Microwave Background dipole at the position of the cyan cross labeled CMB. The heavy blue line defines the plane of our Galaxy. The knot of blue symbols at $L=103$, $B=-2$ is the Virgo Cluster.}
\label{aitoff_pecvel}
\end{center}
\end{figure}
\twocolumn
\begin{figure}[htbp]
\figurenum{8}
\begin{center}
\plotone{vpec_xyz_yz.jpg}
\caption{Motion within the Local Supercluster in the rest frame of the Local Group. Galaxies with $-2000<SGX<2000$~km~s$^{-1}$\ are plotted. Peculiar velocities are color coded as in previous figures. The vectors emanating from our position at the origin indicate our motion relative to these galaxies. They are described in the following figure which is an enlargement of the central region of this figure.}
\label{lsc_vectors_big}
\end{center}
\end{figure}
\begin{figure}[htbp]
\figurenum{9}
\centering
\plotone{vpec_xyz_locyz.jpg}
\caption{Motion within and around our home structure, the Local Sheet, with $-500<SGX<1000$~km~s$^{-1}$. The orange vector represents $V_{LSC}^{LS}$ with an amplitude of 323~km~s$^{-1}$\ in the rest frame of the Local Sheet. The blue vector has an amplitude of 185~km~s$^{-1}$\ and is directed toward the Virgo Cluster at the right edge of the figure. The red vector is the residual of these two, called $V_{LSC;LV}^{LS}$, and has an amplitude of 259~km~s$^{-1}$.}
\label{lsc_vectors}
\end{figure}
\subsection{The Local Void}
The vector defined by Eqs.~(19,20) is not pointing at anything prominent but it is directed {\it away} from the Local Void. This negative feature was identified in the Nearby Galaxies Atlas. The possible influence of the Local Void has been anticipated \citep{1988lsmu.book..115F, 1988MNRAS.234..677L}. There is the claim that the far side of the void is in expansion away from us \citep{2005ASPC..329...59I}. The significance of the Local Void has been difficult to evaluate because it is intersected by the zone of obscuration but the neutral Hydrogen survey HIPASS substantiates its importance \citep{2004MNRAS.350.1195M}.
Figure~\ref{void} attempts a visualization of the Local Void. This absence of galaxies begins at the edge of the Local Group at positive SGZ. It appears to consist of a void within larger voids; i.e., a smaller void shares an interior wall of a larger almost empty region. We lie on a filament that serves as a wall for both the smaller and larger voids. Even the smaller void is not so small, with a long dimension of $\sim 35$~Mpc. The geometry of the larger enclosing void is quite uncertain. It appears to be bisected by a filament into north and south parts. The long dimension may be as large as 5,000~km~s$^{-1}$\ $\sim 70$~Mpc. In the entire region, but especially with the larger component, aspects of the voids are poorly defined because of interruption by the zone of obscuration (roughly coincident with SGY=0). The near and split far underdense regions will be referred to as the Inner, North, and South Local Voids, or in the ensemble as just the Local Void.
Motions on the far walls of the Local Void are poorly documented because of their distance and problems caused by obscuration.
Current distance estimates for galaxies at 25--30~Mpc bounding the Local Void have peculiar velocities $\sim +300$~km~s$^{-1}$.
For the moment, these distances do not have sufficient quality to distinguish peculiar motions at the far wall of the Local Void from the reflex of the motion of the Local Sheet.
\onecolumn
\begin{figure}[htbp]
\figurenum{10}
\centering
\includegraphics[scale=0.85]{vpec_xyz_void_CMYK.jpg}
\caption{The region of the Local Void. The ellipses outline the 3 apparent sectors of the Local Void. The solid dark blue ellipses show two projections of the Inner Local Void bounded on one edge by the Local Sheet. The North and South extensions of the Local Void are identified by the light blue short-dashed ellipses and the green long-dashed ellipses, respectively. These separate sectors are separated by bridges of wispy filaments.
The red vector indicates the direction and amplitude of our motion away from the void.}
\label{void}
\end{figure}
\twocolumn
The Local Void being a void, there is not much opportunity to measure motions {\it within} the void, but we are offered at least one chance. An HST observation provides a TRGB measurement for the lonely galaxy ESO~461-36 = KK~246 \citep{2006AJ....131.1361K}. The distance given in that reference is probably too great, primarily because the reddening estimate that was used \citep{1998ApJ...500..525S} is too low. Using the procedures described by \citet{2007ApJ...661..815R} we find a distance of 6.4~Mpc. Though closer than previously suspected, the galaxy still lies well into the Local Void. This galaxy has an observed $V_{LS}=443$~km~s$^{-1}$\ resulting in $V_{pec}=-30$~km~s$^{-1}$\ with H$_0=74$. However, ESO~461-36 is at almost the opposite pole from the Local Sheet motion described by Eqs.~(17,18). Its motion with respect to the Local Supercluster is roughly the sum of our motion and its additional motion in the same direction (discounting proper motion components). Hence this galaxy is trying to escape from the void with a deviant velocity of at least 350~km~s$^{-1}$. The situation is seen in Figure~\ref{lvdw}. ESO~461-36 has a peculiar velocity toward us in the Local Sheet rest frame as do galaxies on almost the opposite side of the sky in the Leo Spur. However in the rest frame established by galaxies with known distances in the Local Supercluster we are moving toward the Leo Spur and away from ESO~461-36. With respect to the LSC, ESO~461-36 has a very high peculiar velocity.
\begin{figure}[htbp]
\figurenum{11}
\begin{center}
\plotone{pecv_leospur_cmyk.jpg}
\caption{Motions of galaxies with accurate distances in the Leo Spur and Local Void. Horizontal axis: vector sum of SGX and SGY components of distance. Vertical axis: SGZ component of distance. Filled circles: TRGB distances. Open circles: SBF distances. Black vectors: peculiar velocity assuming H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$. Orange vector: motion of Local Sheet with respect to galaxies with measured distances within the Local Supercluster ($V_{LS}<3000$~km~s$^{-1}$\ sample), $V_{LSC}^{LS}=323$~km~s$^{-1}$\ toward $L=80, B=-52$. The blue and red vectors are the residuals of the black vectors after vector addition of the component of the orange vector in their lines of sight (blue: residual toward our position; red: residual away from our position). In the case of the isolated galaxy ESO 461-36 in the Local Void the components add to a velocity of 349~km~s$^{-1}$\ toward us in the Local Supercluster reference frame.}
\label{lvdw}
\end{center}
\end{figure}
Figure~\ref{lvdw} provides more details than Fig.~\ref{leo} concerning the motions of galaxies below the supergalactic equatorial plane. All the galaxies indicated in the plot lie in the Leo Spur and have accurately known distances and systemic velocities. Assuming H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$,
all galaxies in this sector have substantial peculiar velocities toward us. The average motion for the 10 good cases in the figure is $-335$~km~s$^{-1}$\ in the Local Sheet rest frame. Upon cancelation of the motion of the Local Sheet with respect to the Local Supercluster , the average residual for these 10 galaxies is $-34$~km~s$^{-1}$\ with a standard deviation of 29~km~s$^{-1}$. To within the errors, velocities in the Leo Spur can be viewed as simply the reflex of our motion in that direction. We should be reminded, though, of the continuing uncertainty in the parameter H$_0$. The average residual for these 10 galaxies would be nil if the assumed value of the Hubble Constant is reduced by $\Delta {\rm H} = -3$. By the same token, the residual would be significant if $\Delta {\rm H} >+3$.
\begin{figure}[htbp]
\figurenum{12}
\begin{center}
\plotone{vpec_sgz_h74.jpg}
\caption{Peculiar velocity as a function of distance SGZ from the supergalactic plane. Red symbols:
distances of 1 to 6 Mpc; crosses: galaxies in the equatorial plane, and small dots: galaxies in a
flare to negative SGZ in the vicinity of $\ell \sim 270$. Blue circles: distances between 6 and 10 Mpc. The galaxies with the most deviant negative peculiar velocities at the most negative values of SGZ lie within the Leo Spur. The galaxy in the Local Void, ESO 461--36, is identified by the large filled circle.
}
\label{sgz}
\end{center}
\end{figure}
Returning nearer to home, we can ask if there is a gradient of peculiar velocity with SGZ {\it within} the Local Sheet. We look for this possibility with Figure~\ref{sgz}. The distance from the supergalactic equatorial plane is plotted against peculiar velocity. Galaxies inside and outside the filament are distinguished by color and symbol shape. The general trend of negative velocities can be interpreted as an overall local retardation from the mean cosmic expansion. The largest negative $V_{pec}$ are seen in the Leo Spur. Restricting attention to the galaxies within $\pm 1.5$~Mpc of the plane of our filament, one finds a marginal offset in peculiar velocities between positive and negative SGZ: $<V_{pec}>^{+SGZ} = -33 \pm 10$~km~s$^{-1}$\ for 80 cases and $<V_{pec}>^{-SGZ} = 0 \pm 13$~km~s$^{-1}$\ for 54 cases, a difference of $33 \pm 16$~km~s$^{-1}$.
The flare of galaxies at $-5<SGZ<-1.5$ Mpc off the Local Sheet seen in Fig.~\ref{sgz} is a minor feature that includes NGC~1313 and intrinsically smaller galaxies. For 14 cases, $<V_{pec}>^{flare} = -63 \pm 12$~km~s$^{-1}$. These galaxies are moving toward positive SGZ with respect to the Local Sheet. However in the LSC frame they are moving toward negative SGZ, like us but not as rapidly.
\subsection{The large--scale component of our peculiar velocity}
The Local Supercluster motion expressed by Eqs.~(17,18) fails in both amplitude and direction to explain the motion indicated by the cosmic microwave background. The principal sources of that motion are suspected to lie at distances in velocity of 3,000 $-$ 6,000~km~s$^{-1}$\ \citep{1988ApJ...326...19L,2006MNRAS.368.1515E} if not out at 10,000 $-$ 15,000~km~s$^{-1}$\ \citep{1989Natur.338..562S, 2006ApJ...645.1043K}. The sample of distances used in this paper reaches only to the inner edge of the nearer of these domains. Perturbations consistent with the large--scale flows discussed by others \citep{2000ApJ...530..625T} are seen at the edge of our field of study at large $-$SGX. The Centaurus Cluster with $V_{LS}^{Cen} = 3152$~km~s$^{-1}$\ at $d^{Cen} = 37$~Mpc has $V_{pec}^{Cen} = +429$~km~s$^{-1}$\ if H$_0 = 74$~km~s$^{-1}$~Mpc$^{-1}$.
To a first approximation, the local and large-scale components of our motion can be treated as decoupled. Let us determine the properties of the large--scale component upon subtraction of the local component from the CMB vector. The transform between our Local Sheet frame and the reference frame of the CMB \citep{1996ApJ...473..576F} is given by
\begin{equation}
V_{CMB} = V_{LS} +1 \hat x -563 \hat y +285 \hat z
\end{equation}
\begin{equation}
V_{CMB} = V_{LS} -381 \hat X +331 \hat Y -380 \hat Z
\end{equation}
which describes a motion of the Local Sheet of $631 \pm 20$~km~s$^{-1}$\ toward $\ell = 270 \pm 3, b = +27 \pm 3$ ($L=139, B=-37$). Subtraction of the Local Supercluster motion of Eqs.~(17,18) from the CMB motion:
\begin{equation}
V_{CMB} - V_{LSC} = +212 \hat x -385 \hat y +116 \hat z
\end{equation}
\begin{equation}
V_{CMB} - V_{LSC} = -416 \hat X +135 \hat Y -125 \hat Z
\end{equation}
describes a motion of $455 \pm 15$~km~s$^{-1}$\ toward $\ell = 299 \pm 3, b = +15 \pm 3$ ($L=162, B=-16$). The vector of motion of the Local Sheet indicated by the CMB dipole, $V_{CMB}^{LS}$, and the residual to this vector after the locally generated component $V_{LSC}^{LS}$ is subtracted off are shown in Figure~\ref{allv}. The direction of this large--scale component is closely aligned with the Norma--Hydra--Centaurus supercluster complex and background Shapley Concentration, lying within $7^{\circ}$ of the direction of the Centaurus Cluster.
There is a recapitulation of the various reference frames and vectors in Table~3, and Figure~\ref{aitoffv} provides a visual summary. The projected locations of the various vectors are indicated on this plot. The CMB vector can be decomposed into the vector determined by motions within 3000~km~s$^{-1}$\ (the Local Supercluster component) and a residual attributed to structure on large scales. The Local Supercluster component can be decomposed in turn into the components toward the Virgo Cluster and away from the Local Void.
It is known that the distribution of various populations of galaxies peak in roughly the direction of the CMB dipole maximum. Two recent studies are considered here. \citet{2006MNRAS.368.1515E} have calculated the dipole in the distribution of sources brighter than $K_s=11.25$ from the Two-Micron All-Sky Redshift Survey (2MRS). \citet{2006ApJ...645.1043K} have made the equivalent determination based on the distribution of X-ray selected clusters of galaxies. These alternatively derived dipole directions are plotted in Fig,~\ref{aitoffv}. In both cases, these dipole directions are within 20 degrees of the CMB dipole direction. However, they are offset in revealingly different directions. The 2MRS dipole is offset toward the Local Supercluster component of our motion and the X-ray dipole is offset toward the large scale component of our motion.
These distinctive offsets can be understood by giving consideration to Figure~\ref{dipoles} adapted from \citet{2006ApJ...645.1043K}. The bottom panel gives histograms of the run of sources with velocity in the 2MRS and X-ray samples. The 2MRS sample peaks at 5,000~km~s$^{-1}$\ while the X-ray sample peaks at 18,000~km~s$^{-1}$. In the top panel we see that the 2MRS dipole amplitude crests at 4,000~km~s$^{-1}$\ and then is flat \citep{2006MNRAS.368.1515E}. By contrast, the X-ray dipole peaks twice, once at 4,000~km~s$^{-1}$\ and then again around 20,000~km~s$^{-1}$\ \citep{2006ApJ...645.1043K}. The shot noise is greater with the X-ray sample. However, it is rather convincing that the two dipole investigations are sensitive to two separate features in the distribution of matter. The 2MRS dipole is strongly influenced by nearby structure and insensitive to structure beyond 13,000~km~s$^{-1}$. We can appreciate why the dipole in the objects mapped with the 2MRS is pulled from the CMB direction toward the vector of Local Supercluster motion. By contrast, the X-ray dipole only starts to build at 3,000~km~s$^{-1}$\ so is strictly a reflection of the distribution of matter on large scales. It is not a surprise that the direction of the X-ray dipole is pulled from the CMB direction toward the direction of the large scale component of our motion.
The 2MRS sample is attractive because redshifts are available for almost all the galaxies. This information can be used to construct dynamical models \citep{2006MNRAS.373...45E}. However if mass is distributed like light then, since both luminosity and gravity diminish as the square of distance, the net attraction on the Galaxy should be described by the full, deep 2MASS sample, without recourse to redshifts. The analysis by \citet{2003ApJ...598L...1M} produced a dipole that moves from the 2MRS position $22^{\circ}$ W of the CMB position to $8^{\circ}$ W in Galactic longitude. The Maller et al. dipole position is $15^{\circ}$ N of the CMB position in Galactic latitude but this displacement may be due to the way the mask of the obscured region of the Galactic plane is filled. This is the region of the Local Void. If the region of the Local Void were given far fewer sources in the Maller et al. mask, the full 2MASS dipole would be pushed close to the CMB target. It can be noted that the 2MRS analysis uses more information at low Galactic latitudes and gets a closer fit to the CMB in latitude (though it was mentioned that this shallower survey gets a worse fit in longitude).
\onecolumn
\begin{figure}[htbp]
\figurenum{13}
\begin{center}
\includegraphics[scale=0.8]{vpec_xyz_all.jpg}
\caption{Decomposition of the vectors of the motion of the Local Sheet. Two orthogonal views are shown and peculiar velocities of galaxies with observed distances are shown with the same color code introduced in Fig.~\ref{local}. The orange vector indicates the motion of the Local Sheet with respect to this sample. The blue vector indicates the motion of the Local Sheet with respect to the rest frame established by the CMB. The brown vector is the vector difference between these two and is attributed to influences on scales greater than 3,000~km~s$^{-1}$. For clarity, the lengths of these three vectors are doubled compared with the scales on the axes.}
\label{allv}
\end{center}
\end{figure}
\begin{figure}[htbp]
\figurenum{14}
\begin{center}
\includegraphics[scale=0.85]{pecvel_vectors.jpg}
\caption{Components of the motion of the Local Sheet projected onto the sky. The motion of 631~km~s$^{-1}$\ given by the CMB dipole in the direction indicated by the cyan cross can be decomposed into the
323~km~s$^{-1}$\ component defined by the distance measures discussed in this paper, confined to the traditional Local Supercluster, and labeled LSC in orange and the 455~km~s$^{-1}$\ residual located by the brown cross that can be ascribed to large scale structures in Norma--Hydra--Centaurus and to the background. The Local Supercluster $<3000$~km~s$^{-1}$\ component can in turn be separated into a component of 185~km~s$^{-1}$\ toward the Virgo Cluster, at the blue cross, and a component of 259~km~s$^{-1}$\ away from the Local Void, at the (severely distorted) red cross. The Local Void reflex, Virgo, and large scale Cen/Shapley attractions are almost orthogonal to one another, toward the --SGZ, +SGY, and --SGX axes respectively. The X-ray dipole direction is found to lie close to the direction of the large scale Cen/Shapley vector which can be understood since the characteristic distances of the X-ray cluster sample are large. By contrast the 2MASS dipole from the 2MRS lies midway between the CMB and Local Supercluster vector directions. It is inferred that the 2MRS dipole is determined relatively locally.}
\label{aitoffv}
\end{center}
\end{figure}
\twocolumn
\begin{figure}[htbp]
\figurenum{15}
\begin{center}
\includegraphics[scale=0.5]{dipole_zhist2.jpg}
\caption{X-ray and near infrared dipole amplitudes. Top panel: The solid blue line shows the
development of the number-weighted X-ray cluster dipole amplitude with redshift. The dashed red line shows the equivalent information for 2 micron selected sources. Bottom panel: Histograms of the
redshift distributions of the X-ray and 2 micron selected samples.
}
\label{dipoles}
\end{center}
\end{figure}
\section{Discussion}
In the future, the database of galaxy distances and peculiar velocities will be used for detailed studies of the distribution of matter using non-parametric Numerical Action Methods \citep{1995ApJ...454...15S}, techniques that can be used on small scales and at high densities. For the moment, the discussion is restricted to first order effects. It has been emphasized that the motion of the Local Sheet reflected in the CMB dipole can be decomposed into three main components. Of course this is a simplification and taken to the other extreme of complexity this motion can be broken into an arbitrarily large number of influences.
The particular interest in this study is the influence of the Local Void on our motion. First, though, a few words are in order regarding the other two principal components. Concerning scales larger than 3000~km~s$^{-1}$, we would only point out here that a larger local contribution to the CMB motion implies a smaller value of $\beta = \Omega_m^{0.6} / b$ as derived from the amplitude of the dipole of large scale features. Here, $\Omega_m$ is the mean density of matter compared with the density of matter that would give a closed universe and $b$ is the bias between the distribution of matter and the distribution of observable tracers. For example, the values of $\beta$ calculated by \citet{2006ApJ...645.1043K} from the X-ray cluster sample, which has the dependence $\beta = V_{pec} / D_{cl}$, where $D_{cl}$ is the dipole amplitude found from the X-ray clusters, is reduced by the lower large scale component of $V_{pec}$ found here by 11\% from the values given by Kocevski \& Ebeling.
The component of our motion attributed to infall toward the Virgo Cluster was discussed by \citet{2005ApJ...635L.113M}. This component is particularly amenable to modeling by Numerical Action Methods with the large number of distance constraints that are becoming available. We reserve further discussion for another paper but emphasize the relative decoupling from the motion away from the Local Void because (a) the two components are almost orthogonal, and (b) the scale of the Virgo Cluster influence is governed by the cluster distance of 17~Mpc while there are sharp gradients attributed to Local Void effects on scales of only a few Mpc.
\subsection{Expansion of Voids}
We turn now to consider the reflex motion from the Local Void. First, what can be expected on theoretical grounds?
The Friedmann Equation can be written as:
\begin{equation}
{\rm H}^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G \rho}{3} + \frac{\Lambda}{
3} - K\frac{c^2}{a^2} .
\end{equation}
where $a$ is the radial scale factor normalized to $a=1$ today and the three terms on the right describe contributions from the mean density of matter, $\rho$, the vacuum energy, $\Lambda$, and spatial curvature, $K$.
Within a completely empty void:
\begin{equation}
\dot{a}^2 = (\Lambda/3) a^2 - K c^2 .
\end{equation}
This expression can be related to global parameters with $\alpha = {\rm H_v / H}_0$ where ${\rm H_v} = \dot{a}/a$ inside the void and $\Omega_{\Lambda} = \Lambda / 3 {\rm H}_0^2$.
\begin{equation}
\dot{a}={\rm H_v} \left(\frac{\Lambda a^2}{3 {\rm H_v}^2} - \frac{K c^2}{{\rm H_v}^2} \right)^{1/2}
\end{equation}
then since the curvature term in the void is
\begin{equation}
\frac{-K c^2}{{\rm H_v}^2} = 1 - \frac{\Lambda}{3 {\rm H_v}^2}
\end{equation}
we arrive at
\begin{equation}
\dot{a}=\alpha {\rm H}_0 \left(\frac{\Omega_{\Lambda} a^2}{\alpha^2} +(1- \frac{\Omega_{\Lambda}}{\alpha^2}) \right)^{1/2} .
\end{equation}
We solve for the value of $\alpha$ that takes $a$ from 0 to 1 in time $t_0$ for the case $\Omega_m=0.24$, $\Omega_{\Lambda}=0.76$, H$_0=74$~km~s$^{-1}$~Mpc$^{-1}$, and $t_0=13.7$~Gyr. We find $\alpha=1.22$, which gives an expansion in the void relative to the universal flow of
\begin{equation}
{\rm H_v - H}_0 = 16 ~{\rm km~s^{-1}~Mpc^{-1}}.
\end{equation}
For comparison, with $\Omega_m=0.3$, $\Omega_{\Lambda}=0.7$ one finds $\alpha=1.25$ and ${\rm H_v - H}_0 = 18$~km~s$^{-1}$.
\begin{figure}[htbp]
\figurenum{16}
\begin{center}
\includegraphics[scale=0.4]{void_veloc.jpg}
\caption{
Velocities of expansion generated in each Mpc in a void as a function of the density in the void.
The solid and dashed lines corresponds to the cosmologies with H$_0=73$, $\Omega_m=0.24$ (WMAP) and H$_0=70$, $\Omega_m=0.3$ (LCDM), respectively.
}
\label{void_veloc}
\end{center}
\end{figure}
These values are in good agreement with results from simulations reported by \citet{2007arXiv0708.1441V}. Those simulations show that in models with $\Lambda = 0$ the voids are not fully evacuated at the present epoch and motions out of voids are consequently lower than if the voids were empty. However in the simulations with $\Omega_{\Lambda} \sim 0.7$ the voids are quite empty at $z=0$, suggesting that we can give serious consideration to this possibility in the case of the Local Void. Figure~\ref{void_veloc} illustrates the dependence of outflow velocities on the residual density within a void for two cosmological models.
In Section~3.4 it was determined that the Local Sheet has a bulk motion of 259~km~s$^{-1}$\ away from the Local Void. Simplistically, it could be inferred that the radius of a completely empty Local Void is at least 16 Mpc.
It is not out of the question that the entire Local Void including Inner, North, and South components could have this dimension. The geometry of the Local Void is poorly delineated because of the unfortunate intersection of the plane of the Milky Way. Certainly, this region is not entirely empty. Several wispy filaments lace through the volume. If the void is not empty then a larger size is required to generate the observed expansion velocities.
We can raise a couple of layers of complexity. Recovery of velocity fields from simulations provide a guide. Figure~\ref{dtfe} is extracted from \citet{2007arXiv0708.1441V}. The example we show involves the reasonably symmetric convergence of material onto a filament. Streaming motions grow approximately linearly away from the centers of the low density regions on each side of the filament, reaching maxima at the interface with the filament. This behavior is consistent with expectations based on the formulae given above. However, although particles impinging on the filament have large velocities in the example seen here, the filament does not have a large lateral bulk motion. In this particular case, there is considerable symmetry with the influx from the opposite sides of the filament. In other circumstances, there might be an asymmetry. An example is provided by Figure~\ref{horizon}. The sequence of 4 time steps is drawn from an N-body GADGET simulation \citep{2005MNRAS.364.1105S} produced by the HORIZON collaboration (www.projet-horizon.fr). Attention is drawn to a filament that has the combined properties seen in the Local Sheet of lateral
motion due to the dominance of a void on one side and a flow toward a nearby cluster in an orthogonal direction. Two adjacent filaments displace laterally toward convergence, reproducing the behavior seen between the Local Sheet and the Leo Spur.
\begin{figure}[htbp]
\figurenum{17}
\begin{center}
\plotone{schaap_dtfe.jpg}
\caption{Example of void dynamics extracted from thesis by Schaap. Top left: particle distribution in a thin slice through simulation box. Top right: 2D slice through the 3D Delaunay Tessellation Field Estimator (DTFE) density field reconstruction of the simulation. Bottom left: 2D slice through the 3D DTFE velocity field reconstruction. Bottom right: reconstructions along the thick line shown in the other frames; solid line is the density reconstruction and dot--dashed line is the velocity reconstruction. Velocities reach large amplitudes toward the edges of the voids.
}
\label{dtfe}
\end{center}
\end{figure}
\begin{figure}[htbp]
\figurenum{18}
\begin{center}
\plotone{skel_4z.jpg}
\caption{Four snapshots in time of an N-body simulation with conditions resembling the observations. The dark matter GADGET simulation is of a 20 Mpc box with $256^3$ particles, $\Lambda$CDM, with $\Omega_m=0.3$ and $\Omega_{\Lambda}=0.7$. The region shown is 10 Mpc across in co-moving coordinates at the redshift steps $z=1$, 0.5, 0.25, and 0. The skeleton method \citep{2006astro.ph..2628S} delineates the backbone of the filaments at each step. The large circle in each panel identifies the progression of a location that ends up with properties resembling those of the Milky Way: on a filament, with motions that are both lateral to the filament away from a void, and along the filament toward a cluster. The large triangle in each panel tracks a location that comes to resemble the Leo Spur, with upward motions headed toward a future closure with the filament bearing the circle. For animation, see
ifa.hawaii.edu/$\sim$tully/skeleton-060421.mpg.
}
\label{horizon}
\end{center}
\end{figure}
The observed motion of ESO~461--36 gives a useful constraint. This dwarf galaxy is still well within the Local Void, still within the unfettered flow toward our filament. With reference to the flow pattern seen in the lower right panel of Fig.~\ref{dtfe}, a galaxy such as ESO~461-36 might be anticipated to be near the maximum of the swing of peculiar velocities. We measure a peculiar velocity for this galaxy of $-30$~km~s$^{-1}$. As was noted in Section 3.5, this motion is additive with our velocity in the reference frame of the Local Supercluster, which implies a peculiar velocity of at least 350~km~s$^{-1}$\ with respect to that reference frame. The peculiar velocity is higher if the galaxy has a significant tangential component. This galaxy should have a motion of $\sim 120$~km~s$^{-1}$\ due to the influence of the Virgo Cluster, leaving an additional $\sim 230$~km~s$^{-1}$\ attributable to evacuation from the void. Equation (30) requires
that ESO~461--36 be at least 17 Mpc from the void center. Since we are 6 Mpc farther back, that would put us 23 Mpc from the void center.
Maybe so. However, this requires that the void be very big and very empty. The situation invites consideration of more radical alternatives. \citet{2007PhRvD..75f3507D} argue that dark energy with a varying equation of state might have enhanced density in places with lower mass density; i.e., in voids. A consequence could be an enhanced expansion rate of space in voids resulting in increased velocities away from void centers. Potentially, observations of the motions of galaxies within voids could give important information about the fundamental properties of the universe.
\section{Summary}
Our motion inferred by the CMB dipole anisotropy of 631~km~s$^{-1}$\ in the Local Sheet reference frame decomposes into three main contributions that are almost orthogonal. The three main components conveniently lie near the cardinal axes of the Supergalactic coordinate system (only partially by chance). Two of these components decouple from the third because two are local, the closest seen abruptly in a peculiar velocity discontinuity at 7~Mpc, while the third is large scale, acquiring an importance at $V_{LS} > 3000$~km~s$^{-1}$.
One of the local components is caused by the Virgo Cluster and its dense surroundings. The motion of the Local Sheet with respect to galaxies within 3000~km~s$^{-1}$\ has a component of 185~km~s$^{-1}$\ toward this cluster at $L=103, B=-2$. The residual from the local component of CMB motion is a Local Sheet velocity of 259~km~s$^{-1}$\ toward $L=11, B=-72$, toward nothing of importance but {\it away} from the Local Void. Subtraction of these two local components from the CMB vector leaves the third component attributed to large scale attractors of 455~km~s$^{-1}$\ toward $L=162, B=-16$, close to the direction of the Centaurus Cluster. These three components cause motions roughly aligned with the +SGY, $-$SGZ, and $-$SGX axes respectively, providing a decomposition that is gruntling.
The availability of a large number of accurate TRGB distances has clarified details about the `local velocity anomaly'. Our Local Sheet is participating in the cosmic expansion (though probably somewhat retarded) but simultaneously moving in bulk toward the Virgo Cluster and away from the Local Void. Our Local Group has only a small peculiar velocity (66~km~s$^{-1}$) with respect to other galaxies of the Local Sheet which, internally, has only small random motions (40~km~s$^{-1}$\ in the radial direction averaged over groups). We advocate the use of the Local Sheet as a frame of reference in preference to the Local Group because the reference sample is five times larger and more widely distributed. The local velocity anomaly is given emphasis because there is a sharp discontinuity in velocities as we look beyond the Local Sheet toward $-$SGZ. Galaxies in the Leo Spur with well measured distances all have large negative peculiar velocities. Much, if not all, of these motions are a reflex of the motion of the Local Sheet toward $-$SGZ and away from the Local Void.
Our distances are not yet numerous or accurate enough to demonstrate if other filaments have similar bulk motions. However we do see clearly that a galaxy within but near our edge of the Local Void, ESO~461$-$36, has a peculiar velocity of at least 230~km~s$^{-1}$\ away from the center of the void. Our Local Sheet and that galaxy are participating in the evacuation of the Local Void.
The large expulsion velocities imply a dimension of the radius to the center of the Local Void at our position of at least 23~Mpc. We lie on the boundary of a major void. The evidence for expansion is unambiguous at our privileged position on the void wall. Voids have few galaxies but are they really empty? Yes, to create such a large outflow, our Local Void must be really empty.
\vskip 1cm
\noindent
Figure~\ref{dtfe} is originally from the Ph.D. thesis by W.E. Schaap and was provided by Rien van de Weygaert.
This research has been sponsored by Space Telescope Science Institute awards GO-9771, GO-9950, GO-10210, GO-10905, and the National Science Foundation award AST 03-07706. IK is partially supported by the programs DFG-RFBR 06-02-04017 and RFBR 07-02-00005 in Russia.
|
{
"timestamp": "2007-12-14T03:40:53",
"yymm": "0705",
"arxiv_id": "0705.4139",
"language": "en",
"url": "https://arxiv.org/abs/0705.4139"
}
|
\section*{I. Introduction}
The damage spreading method has been widely used to study the critical
properties of Ising-like systems
\cite{derrS,stan,grass1,montani,wang,vojta,lima1,lima2} as well
as to spin glass \cite{derrW,argo}. The method is based on the
synchronous Monte Carlo update of two distinct spin configurations
that are evolving from an almost identical initial state \cite{herr}.
Since the Ising model lacks of an intrinsic dynamic, it has to be
chosen a particular one. Among others, the more frequently used
are Glauber and heat-bath dynamics. In contrast to usual statistical
Monte Carlo studies, damage spreading results depends on the used
dynamic \cite{herr}. This behavior has been used to evaluate the
dynamical exponent $z$ of the 2D and 3D Ising models with
different dynamics \cite{grass1,wang1,sta}. Also, very recently,
it has been shown that damage spreading is a powerful and useful
technique for the numerical study of the role of the interfaces
between magnetic domains on the propagation of
perturbations in magnetic materials \cite{we1,we2}.
Within this context, we have reported that the presence of interfaces
act as a ''catalyst'' of the damage in at least two different ways:
speeding up the propagation and causing an enhancement of the
total damaged area \cite{we1,we2}.
The basic idea for the implementation of the damage
spreading method \cite{herr}
is to start from an equilibrium configuration
of the system at temperature $T$, which is generically called \( S^{A}(T) \).
Subsequently, a very small perturbation is applied to that configuration
in order to obtain a new one, i.e. the so called perturbed
configuration \( S^{B}(T) \). Usually the perturbed configuration
is obtained by flipping a small number of spins of the unperturbed
configuration. Then one has to study the time evolution of the
perturbation in order to investigate under which conditions such
a small perturbation will grow up indefinitely or
eventually (hopefully) it will vanish and become healed.
In order to follow the time evolution of the perturbation an useful method is
to measure the ``Hamming distance'' or damage between the unperturbed
and the perturbed configurations. The total damage \( D(t) \) is defined as
the fraction of spins with different orientations, that is
\begin{equation}
D(t)=\frac{1}{2N}\sum ^{N}_{l}\left| S^{A}_{l}(t,T)-S^{B}_{l}(t,T)]\right|,
\label{eq:dam}
\end{equation}
\noindent where the summation runs over the total number of spins N and the
index \( l\, (1\leq l\leq N) \) is the label that identifies the
spins of the configurations. Starting from a vanishing small
perturbation \( D(t=0)\rightarrow 0 \) one can expects at least
two scenarios, namely: a) \( D(t\rightarrow \infty )\rightarrow 0 \)
and the perturbation is irrelevant because it become healed
and b) \( D(t\rightarrow \infty ) \) goes to some well defined finite
value. In the latter, frequently undesired case, the perturbation is
relevant because it can not be healed out. For further details see the
review \cite{herr}.
The aim of this work is propose and study a model for damage spreading
with damage healing, which is based in the two dimensional ($2D$) Ising model.
It is well known that the $2D$ Ising magnet undergoes a second-order
order-disorder transition when the temperature is raised from a relatively low
initial value. The location of the critical point is know exactly and it is
given by the so called Onsager critical
temperature \( \frac{kT_{C}}{J}=2.269... \).
According to previous studies of damage spreading using Glauber dynamics
it is known that there is a critical damage temperature given by
\( T_{D}\cong 0.992\, T_{C} \) \cite{grass2,grass3}, such as
for \( T>T_{D} \) the damage spreads out over the whole sample while
for \( T<T_{D} \) the damage becomes healed after some finite time.
Consequently, the proposed model, that incorporates a healing probability,
is suitable for the study of damage spreading above \( T_{D} \), as well
as to gain insight on the robustness of the damage behavior in the
Ising model.
The manuscript is organized as follows: in Section II we propose the
model and describe the numerical procedure for the simulation of
damage spreading. Section III is devoted to the presentation and
discussion of the results, while our conclusions are stated in Section IV.
\section*{II. A Model for Damage Spreading with Healing and the
Monte Carlo Simulation Method.}
The model is based on the $2D$ ferromagnetic Ising model, which for a
square lattice of side $L$ can
be described by the following Hamiltonian \( H \):
\begin{equation}
H = -J \sum ^{L,L}_{<ij,mn>} \sigma _{ij}\sigma _{mn}
\label{eq:ham}
\end{equation}
\noindent where \( \sigma _{ij} \) is the spin variable,
corresponding to the site
of coordinates (i, j), that may assume two different values,
namely \( \sigma _{ij}=\pm 1 \), \( J>0 \) is the coupling constant of
the ferromagnet and the summation of (\ref{eq:ham}) runs over all
the nearest-neighbor pairs of spins.
For the purpose of the simulations, periodic boundary conditions are
always used. Furthermore, we have used the Glauber dynamics. In
order to implement this dynamic a randomly selected spin is flipped
with probability \( p(flip) \) given by:
\begin{equation}
p(flip)=\frac{\exp (-\beta \cdot \bigtriangleup H)}{1+\exp (-\beta \cdot \bigtriangleup H)}
\label{eq:rates}
\end{equation}
\noindent where \( \bigtriangleup H \) is the difference between the
energy of the would-be new configuration and the old configuration,
and \( \beta =1/kT \) is the usual Boltzmann factor. The temperature
is measured in units of the Onsager critical temperature of the $2D$
Ising model.
In order to study the time evolution of the damage spreading a meaningful
definition of the Monte Carlo time step (mcs) is necessary. For this
purpose the standard definition is adopted according to that during one
mcs all \( L\times L \) spins of the sample are flipped once, in the average.
For the practical implementation of equation (\ref{eq:dam}), first
an equilibrium configuration (say configuration \( S^{A}_{e} \))
is generated. For this purpose one starts from an initial configuration
with all \( N = L\times L\) spins oriented at random. The application
of the Glauber dynamics leads to the desired equilibrium
configuration after \( 10^{4} \)mcs. Subsequently, a fully damaged
replica \( S^{B}_{e} \) of such configuration is created, such as
\( S^{B}_{e} \) is the mirror image of \( S^{A}_{e} \) and consequently
$D(0) = 1$. Of course, due to the spin flip symmetry, the replica
configuration is also equilibrated.
In standard damage spreading studies, the standard Monte Carlo procedure
is then applied to both configurations, where the same sites are randomly
selected and the same random numbers are used in both systems in order
to perform the updates. For the purpose of the proposed model we first
define the healing probability $q$ and we used the
same method but according the following rules:
i) A sample site of coordinates $(i,j)$ is selected at random. Then,
iia) If $S^A (i,j) = S^B (i,j)$, which means that the selected site is not
damaged, the Glauber dynamics is followed according to the standard
procedure \cite{herr}. However,
iib) If $S^A (i,j) \neq S^B (i,j)$, i.e. for a damaged site, a new
random number $h$ is generated and one proceeds as follows:
iib1) If $h < q$, the damage is healed, so that one sets
$S^A (i,j)=S^B (i,j)$. However,
iib2. If $h \geq q$, the standard dynamics is applied.
According to the defined rules and taking $p = 1 -q$, one has
that for $p \rightarrow 0$, the damage would become healed for
all temperatures, while in the limit $p \rightarrow 1$, the usual damage spreading dynamics in the Ising model is recovered.
Using the above described procedure, we have followed the time evolution
of the damage $D(t)$, which is evaluated according to equation (\ref{eq:dam}),
for different values of $p$, $1.1 \leq T \leq 50$ and the lattice
size $50 \leq L \leq 1000$. Simulations are stopped when the damage
is healed, otherwise they are performed up to $t = 65000$ mcs.
\section*{III. Results and discussion}
Figure 1 shows log-log plots of $D(t)$ versus $t$ obtained at $T = 2$
for different values of $p$. It is found that: i) for \( p > p_{c} \)
the damage becomes healed for finite times and the log-log plots of $D(t)$
versus $t$ exhibit a downward curvature. ii) For \( p <p_{c} \)
the damage quickly propagates and the log-log plots of $D(t)$ versus $t$
show that the damage reaches a stationary value.
Finally, just at \( p = p_{c} = 0.1895(5) \)
the damage decays according to a power law. So, the change of the
curvature of the plots shown in figure 1 allows us to identify the critical
probability \( p_{c} \) for damage spreading.
The straight line observed in figure 1 for \( p_{c} \) suggests
a power-law behavior given by
\begin{equation}
D(t)\propto t^{-\delta}
\label{pwlaw}
\end{equation}
\noindent where \( \delta \) is an exponent. The best fit of the data
gives \( \delta = 0.450 \pm 0.005 \). Similar plots obtained for $T > T_{C}$
(not shown here for the sake of space) exhibit the same behavior as that
observed in figure 1 and the typical slope, averaged within the range
$2 \leq T \leq 50$, is given by \( \delta = 0.45 \pm 0.01 \).
On the other hand, for $p > p_{c}$ figure 1 shows a fast decay
of the damage. As suggested by the study of spreading
behavior \cite{gradela}, we proposed an exponential decay given by
\begin{equation}
D(t) \propto exp(-t/ \tau)
\label{expodeca}
\end{equation}
\noindent where $\tau$ is the characteristic time of decay.
In order to check the scaling Ansatz given by equation (\ref{expodeca}),
figure 2 shows log-lineal plots of $D(t)$ versus $t$ obtained for
$p > p_{c}$. The obtained straight lines for the long time
behavior allow us to evaluate the dependence of $\tau$ on $p$.
It is found that $\tau$ increases when approaching the critical probability.
Furthermore, plots performed at different temperatures (no shown here for the
sake of space) also exhibit the same behavior, independent of $T$.
Again, using the spreading formalism \cite{gradela}, the following
power-law can be proposed
\begin{equation}
\tau \propto (p-p_{c})^{-\nu_{\parallel}}
\label{nu}
\end{equation}
\noindent where $\nu_{\parallel}$ is the correlation length exponent
for damage propagation along the time direction. Figure 3 shows log-log
plots of $\tau$ versus $p - p_{c}$ obtained at different
temperatures. The typical averaged value resulting from the fits
is given by $\nu_{\parallel} = 1.25(5)$.
It is worth mentioning that our estimations for the exponents, given by
$\delta = 0.45(1)$ and $\nu_{\parallel} = 1.25(5)$ are in agreement with
the well known critical exponents of the universality class of
Directed Percolation (DP), namely $\delta^{DP} = 0.4505(10)$ \cite{h34}
and $\nu_{\parallel}^{DP}=1.295(6)$ \cite{f33}. So, these results strongly
suggest that the the proposed model of damage spreading with damage healing
belongs to the DP universality class. This statement is further supported
by the fact that the limit $p \rightarrow 1$ corresponds to the well known
Ising model where the damage spreading transition is known to lie
within the DP universality class \cite{grass2,grass3}.
On the other hand, as follows from figure 1,
for $p < p_{c}^{D}$ and after a short decay period
($t \approx 10^{2}$ mcs), the damage becomes stabilized (or saturated)
with typical average values $D_{sat}$ that depend on $p$. This trend
has been checked by means of extensive simulations run
up to $t=10^6$ mcs. Furthermore, we have also checked that
the stationary values of $D_{sat}$ exhibit negligible finite-size effects for
$L \ge 200$. Figure \ref{Fig4} shows the dependence of $D_{sat}$
on $(p_{c}^{D}-p)$ in a lineal-lineal plot. It follows that
$D_{sat}$ increases lineally as a function of $(p_{c}^{D}-p)$,
and that the slope $F(T)$ depends on temperature.
This assumption is valid only in the limit of $p\rightarrow p_{c}^{D}$.
Accordingly, we propose
\begin{equation}
D_{sat} = D_0 + F(T) (p_{c}^{D} - p) ,
\label{dsat}
\end{equation}
\noindent where $D_0 \approx 0.05$ is almost a constant independent of
$T$ (see figure \ref{Fig4}). The slope of the lines can be
obtained by fitting the data of figure \ref{Fig4} and it is found that
$F(T)$ also depends lineally with $T$ (see figure \ref{Fig5}) according to
\begin{equation}
F(T)=F_0 + A \left( \frac{T-T_C}{T_C}\right)
\label{fdeT}
\end{equation}
\noindent where $F_0 \approx 0$ and $A \approx 1.5$ are also constants.
So, by inserting equation (\ref{fdeT}) into equation (\ref{dsat}),
one obtains
\begin{eqnarray}
D_{sat} = D_0 + \left( F_0 + A \frac{T-T_C}{T_C)}\right) \cdot (p-p_{c}^{D}) ,
\label{Dsatscal}
\end{eqnarray}
\noindent that gives the full dependence of $D_{sat}$ on both $p$ and $T$.
The scaling plot suggested by equation (\ref{Dsatscal}) holds acceptably as
shown in figure \ref{Fig6}.
Performing plots of the time evolution of the damage using different values of
$p$ (as shown in figure \ref{Fig1}) and changing $T$, it is finally possible
to evaluate the phase diagram of the model, i.e. a plot of $p_{c}^{D}$ as a
function of $(T-T_C)/T_C$, as shown in figure \ref{Fig7}. It is found that
$p_{c}^{D} \rightarrow 0$ for very large values of $T$, as expected from the
definition of the model. For the asymptotic decay of $p_{c}^{D}$ we propose
\begin{equation}
p_{c}^{D} \propto \left( \frac{T-T_C}{T_C}\right) ^{-\alpha}
\label{fasedia}
\end{equation}
\noindent where the best fit of the data, within the range
$T \gg 1.5 T_C \sim 3.4J$, gives $\alpha=1$ for the exponent of equation
(\ref{fasedia}). It is worth mentioning that, for the lattices used in the
simulations, the phase diagram exhibits negligible deviations due to
finite size effects, as shown in figure \ref{Fig7}. On the other hand,
close to $T_C$, it is observed that $p_{c}^{D}$ reaches a maximum value of
the order of $p_{c}^{D} \sim 0.22$ for $ T = 1.5 T_C \sim 3.4J$. We have
carefully checked that this behavior, which remains to be understood,
is not due to the operation of finite size effects.
Finally, let us remark that all phase transitions at
the critical curve at the border between
damage healing and damage spreading, shown in figure \ref{Fig7},
belong to the universality class of directed percolation.
\section*{IV. Conclusions}
Based on the fact that in the 2D Ising model the damage
is not healed above the critical temperature, we proposed
a model that introduces a healing probability. It is found
that for $T > T_{C}$ the damage becomes healed for critical
values of $p$. The phase diagram of the model is obtained and
it is shown that the transitions between states of damage
spreading and healing belong to the universality class of
directed percolation in $(2 + 1)-$dimensions.
Our results thus support the conjecture that damage spreading
transitions are generically in the universality class of directed
percolation \cite{grass2}, provided the fact that such transitions
do not coincide with the critical point of the undamaged system.
|
{
"timestamp": "2007-05-30T22:02:49",
"yymm": "0705",
"arxiv_id": "0705.4466",
"language": "en",
"url": "https://arxiv.org/abs/0705.4466"
}
|
\section{Introduction}
Recently considerable attention has been devoted to the study of local isotropy
of the
high Reynolds number turbulence suggested by \cite{k41}. The turbulence is stirred at
the large scales, and this energy is transported into the small scales, where, after
intense nonlinear interaction, it becomes isotropic. Deviation from isotropy would
mean that there is direct interaction between large scales containing non-universal anisotropy
and small scales, leading to non-universal behavior of small scale spectral properties.
It was indeed shown experimentally that in a sheared turbulence the
isotropy is not
sufficiently restored for both scalar and velocity fields, see \cite{001}, \cite{02}, \cite{1r},
\cite{2r}, \cite{new2}, \cite{03}, \cite{new3}.
High Reynolds number (atmospheric) turbulence also shows deviations from isotropy. It was
shown that the large scale shear does contribute into the scaling of the structure
functions, see \cite{new1}. Anisotropic scaling of high-order structure functions was studied
by \cite{Susan}. It was shown that the anisotropy in small scales
remains stronger than expected before. The SO(3) decomposition was used to describe the
anisotropy, see \cite{Susan}, \cite{Biferale1}, \cite{Biferale3}, \cite{Biferale4}. The persistent
anisotropy in small scale turbulence was found to be related to the intermittency corrections,
\cite{Biferale2}.
In numerical simulations, the failure to
return to isotropy was linked to both asymmetry of the probability distribution function
(PDF) and to the vortex sheets, \cite{3r}. It became clear that the shear in the integral
scale induces asymmetry
down to the small scales, where it is manifested by intermittent structures like cliffs, etc., \cite{4r}.
On the other hand, the asymmetry PDF was found to be related to the intermittency, \cite{94}.
Thus, these three items, i.e., anisotropy, asymmetry and intermittency seem to be related.
Note, however, that, in principle, these items are independent of each other.
For example, the asymmetry of the PDF appears naturally in turbulence, even without any
anisotropy.
Denote $u$ - longitudinal and $v$ - transverse (vertical) components of the velocity,
and $u_r=u(x+r)-u(x)$, $v_r=v(x+r)-v(x)$, the velocity increments.
Then, we have,
\begin{equation}
\langle u_r\rangle=0, ~~~ {\rm and} ~~~ \langle v_r\rangle=0,
\label{first_m}
\end{equation}
and (in inertial range)
\begin{equation}
\langle u_r^3\rangle= -\frac{4}{5}\varepsilon r,
\label{kolm}
\end{equation}
the so-called 4/5-Kolmogorov law \cite{law}. Besides,
\begin{equation}
\langle u_rv_r^2\rangle=\frac{1}{6}\frac{d\langle u_r^3\rangle}{dr},
\label{mixed0}
\end{equation}
\cite{book}, \cite{Monin}. The fact that the first moments vanish, see (\ref{first_m}),
whereas the two third moments
(\ref{kolm}) and (\ref{mixed0}) do not clearly indicate that both the PDF for $u_r$ and
the joint PDF for $u_r$ and $v_r$ are asymmetric.
As the Kolmogorov law is derived assuming isotropic turbulence, this asymmetry in principle exists
without anisotropy. Besides, the scaling defined by the Kolmogorov law (\ref{kolm}) does not have any intermittency
corrections. Nevertheless, as mentioned above,
the asymmetry (even without anisotropy) may be related to the intermittency of turbulence,
as suggested by \cite{94}, \cite{96a} and \cite{asym}. Still, this connection does not seem to present
the whole
picture. Presumably, as in sheared flows, the anisotropy should also be taken into
considerations. As mentioned above, even very high-Reynolds-number turbulence manifests
anisotropy in small scales, which is higher than predicted from dimensional arguments by
\cite{Lumley}. This suggests that there is additional to Kolmogorov cascade transfer of
energy from large scales directly to small, leading to anomalous events like intermittency and anisotropy.
In this paper we focus on these events. In particular, we construct experimental joint
PDF for $u_r$ and $v_r$ to see directly if anisotropy and asymmetry is present in rare
violent events (responsible for the intermittency). Note that constructing
the joint 2D PDF's proved to be useful in turbulence, see for example \cite{joint1},
\cite{joint2}, \cite{joint3}. Even more information about the connection between anisotropy,
asymmetry and intermittency we obtain from several conditional and cumulative averages described
in the next section.
\section{Problem description}
We will work with dimensionless variables,
$$
u'_r=\frac{u_r}{\langle u_r^2\rangle^{1/2}}, ~~~~
v'_r=\frac{v_r}{\langle v_r^2\rangle^{1/2}},
$$
and construct experimental joint PDF, $p(u'_r,v'_r)$ to study both asymmetry and
anisotropy. This 2D distribution is useful to compare with 2D Gaussian anisotropic
distribution, $p_G$, see Appendix, (\ref{GaussA}), (\ref{Gauss1A}).
The joint 2D PDF gives general information about the distribution. More detailed information
which is easier to analyze are provided by different 1D distributions and cumulative moments.
In particular, we are interested in studding the third mixed moment, (\ref{mixed0}),
\begin{equation}
\langle u_rv_r^2\rangle=\int u_rv_r^2p(u_r,v_r)du_rdv_r,
\label{mixed}
\end{equation}
that is, it is important to find out what part of the distribution contributes most into this moment.
It was shown before, see \cite{jfm}, that the tail parts of 1D distributions satisfactory recover the
moment.
Additional information is given by conditional average,
\begin{equation}
\langle v_r^2|u_r\rangle=\int v_r^2 p(v_r|u_r) dv_r=\frac{\int v_r^2 p(v_r,u_r)dv_r}{p(u_r)}=
\frac{\Phi(u_r)}{p(u_r)},
\label{mixed_c}
\end{equation}
or, in dimensionless variables,
\begin{equation}
\langle {v'}_r^2|{u'}_r\rangle=\int p({v'}_r|{u'}_r){v'}_r^2 d{v'}_r=\frac{\Phi({u'}_r)}{p({u'}_r)},
\label{mixed_c_d}.
\end{equation}
In these expressions we introduced
\begin{equation}
\Phi(u_r)=\int v_r^2p(u_r,v_r)dv_r, ~~~\Phi({u'}_r)=\int {v'}_r^2p({u'}_r,{v'}_r)d{v'}_r
\label{Phi}
\end{equation}
Therefore,
\begin{equation}
\langle u_rv_r^2\rangle=\int u_r\langle v_r^2|u_r\rangle p(u_r)du_r=\int u_r\Phi(u_r)du_r,
\label{mixed1}
\end{equation}
\begin{equation}
\langle {u'}_r{v'}_r^2\rangle=k_a=
\int{u'}_r\langle{v'}_r^2|{u'}_r\rangle p({u'}_r)d{u'}_r=\int {u'}_r\Phi({u'}_r)d{u'}_r
\label{mixed2}
\end{equation}
The function $\Phi(u_r)$ deserves special attention. It follows from (\ref{Phi}) that
\begin{equation}
\int \Phi(u_r)du_r=\langle v_r^2\rangle,
\label{Phi0}
\end{equation}
that is, in a way, the function is a $v_r^2$-distribution versus $u_r$. The first moment
of this distribution should not vanish, as it follows from (\ref{mixed1}). This means
that the $v_r^2$-distribution should be be asymmetric.
For Gaussian distribution, the function $\Phi_G$ can be
easily calculated, see Appendix, (\ref{GaussPhiA}).
We used x-wire data acquired at Brookhaven National Lab.
Distance of probe above the ground: 35m; Number of samples: 40960000 per component, that
is, for longitudinal ($u$) and transverse ($v$) components. Sampling frequency: 5 kHz.
Mean velocity: 5.15076224 m/s;
rms $u$-velocity: 1.81617371 m/s;
rms $v$-velocity: 1.3646025 m/s;
Taylor Reynolds number: 10680 (courtesy of Sreenivasan). As usual, the data are interpreted
using the Taylor's hypothesis.
All throughout the paper we process data for closest two samples, that is,
$r$ corresponds to the smallest distance between two samples.
\begin{figure}
\includegraphics[height=8cm,width=16cm]{fig2.ps}
\caption{ Experimental joint PDF, core part. The indicated levels correspond to
$e^{-2},~e^{-4},~e^{-6},~e^{-8}$ of the maximum of the PDF, inside out correspondingly.
The contours a) depict experimental PDF, and b) -- Gaussian anisotropic (\ref{Gauss1A}).
}
\vspace{1cm}
\label{fig1}
\end{figure}
\section{Joint PDF}
Figures \ref{fig1} and \ref{fig2}
present this PDF in different ranges. We (loosely) define these regions as core part,
Fig. \ref{fig1}, and tail part, Fig. \ref{fig2}. Clearly, Fig. \ref{fig1} corresponds to
the main events, while Fig. \ref{fig2} - to the rare and violent events, -- as indicated by
the levels given in the captures.
It is clear that the main events, Fig. \ref{fig1}, are not much different from Gaussian.
Both PDF's are noticeably anisotropic, that is the levels are roughly ellipses with big
axises inclined at some angles to the x-axis. The positions of the levels are roughly the same.
The only difference is asymmetry, the latter being absent in the Gaussian PDF by construction.
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig1.ps}}
\caption{ Experimental joint PDF, tail part. The indicated levels correspond to
$e^{-8},~e^{-10},~e^{-12},~e^{-14}$ of the maximum of the PDF, inside out correspondingly.
The contours a) depict experimental PDF, and b) -- Gaussian anisotropic (\ref{Gauss1A}).
}
\vspace{1cm}
\label{fig2}
\end{figure}
The asymmetry of the experimental PDF is evident from the following observations.
The left edge of the outer level is at $u'_r=-4.1$, while the right
edge
of it is at $u'_r=3.8$; the left edge of the next level is at $u'_r=-2.5$, while the right
edge is at $u'_r=2.4$.
Note that both anisotropy and asymmetry are expected to be manifested
by the main events.
Consider now Fig. \ref{fig2}. We note first that now there is dramatic difference between
the experimental contours and Gaussian. Namely, the experimental contours are much further
away from the core values than the corresponding Gaussian. This feature is however anticipated,
simply corresponding to the presence of tails, that is, to intermittency
-- in both longitudinal and transverse velocity component
increments.
On the other hand, other features of the rare events PDF are more surprising.
The experimental contours in the Fig. \ref{fig2}(a) look roughly similar to those in Fig.
\ref{fig1}(a), only rescaled to different values, and -- naturally -- more ragged. Indeed, we
see anisotropy -- ellipses with axes inclined roughly the same way in both figures. And, what
is more important, we notice asymmetry in the rare events PDF. The left edge of the outer
level corresponds to $u'_r=-15.4$, while the right
edge
of it is at $u'_r=13.8$; the left edge of the next level corresponds to $u'_r=-10.7$, while the right
edge is at $u'_r=10.2$.
If we characterize the asymmetry by the ratio of the distance from
the left edge of the level to zero to the distance from the right edge of the level to zero, we
will get for both figures \ref{fig1} and \ref{fig2} -- for the external levels the number $1.1$,
while for the next levels (again for the both figures) the value of $1.05$. Thus, approximately,
the asymmetry is the same in both typical and rare events.
\section{The ${v'}_r^2$ versus ${u'}_r$ distribution: Function $\Phi$, (\ref{Phi})}
\subsection{Global features}
\label{global}
Figure \ref{fig3} presents experimental ${v'}_r^2$ versus ${u'}_r$ distribution. It is compared
with its Gaussian analog on one hand, and with regular experimental distribution $p({u'}_r)$
on the other.
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig3.ps}}
\caption{ Experimental ${v'}_r^2$ distribution, compared with its Gaussian analog, and
with $p({u'}_r)$.
}
\vspace{1cm}
\label{fig3}
\end{figure}
All three distributions in the figure are normalized on unity, i.e.,
$$
\int \Phi({u'}_r)d{u'}_r=\int\tilde{\Phi}_G({u'}_r)d{u'}_r=\int p({u'}_r)d{u'}_r=1,
$$
and therefore their direct comparison makes sense.
It is clear that ${v'}_r^2$-distribution has quite extensive tails.
Naturally, the Gaussian distribution is much lower outside
its core part. Moreover, the tail parts
of $\Phi({u'}_r)$ are much above corresponding parts of $p({u'}_r)$. This is because
the ${v'}_r^2$-distribution is a second moment, see definition (\ref{Phi}), while $p({u'}_r)$
is a zeroth moment of the same distribution,
$$
p({u'}_r)=\int p({u'}_r,{v'}_r)d{v'}_r.
$$
Note that the
${v'}_r^2$-distribution is much more dispersed at the tail parts as compared with $p({u'}_r)$
in the same areas. Analogous (huge) dispersion of data is observed in conditional average, see below
Fig. \ref{fig7}, and is discussed in Subsection \ref{global_cond}.
We note here only that this dispersion is attributed to the presence of intermittency.
\subsection{Contribution of the rare violent events}
\label{rare}
Additional information about the tails of the ${v'}_r^2$ versus ${u'}_r$
distribution can be obtained from
cumulative moments,
\begin{equation}
\langle v_r^2\rangle\rule[-2mm]{.5mm}{6mm}_{|u_r| \ge t}=\int_{-\infty}^{-t}
\Phi(u_r)du_r+\int_t^\infty \Phi(u_r)du_r,
\label{Phi0t}
\end{equation}
that is, the contribution of events with $|u_r| \ge t$ into the moment $\langle v_r^2\rangle$,
see definition of this moment in (\ref{Phi0}). These moments will be plotted against
\begin{equation}
P(t)=\int_{-\infty}^{-t}
p(u_r)du_r+\int_t^\infty p(u_r)du_r,
\end{equation}
the probabilities of these events.
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig3a.ps}}
\caption{ Experimental ${v'}_r^2$-distribution for $|{u'}_r| \ge t$ events versus the
probability of these events $P(t)$ (solid line), compared with its Gaussian counterpart (dashed
line).
}
\vspace{1cm}
\label{fig3a}
\end{figure}
Figure \ref{fig3a} presents such a plot. The experimental distribution is compared with
its Gaussian counterpart. For $t=0$, all events are presented, and therefore $P(t=0)=1$ and
$\langle {v'}_r^2\rangle\rule[-2mm]{.5mm}{6mm}_{|{u'}_r| \ge 0}=1$. If, on the other hand,
$t\to\infty$, then both distributions go to zero.
The difference between experimental and
Gaussian cases is quite substantial. As an example (depicted by dashed-dotted straight lines),
we see that
$\langle {v'}_r^2\rangle$ reaches $10^{-3}$ fraction of its final value with only $2.8\times
10^{-6}$ part of all events. According to our estimate, these events correspond to quite
violent outbursts with $t\ge 27.4$ As seen from Fig. \ref{fig3}, these events are almost at
the very end of the measured tails.
The Gaussian counterpart reaches the same value of $10^{-3}$
with $4.7\times 10^{-3}$ part of events; both these numbers are comparable for
the Gaussian distribution.
On the other hand, the $\langle {v'}_r^2\rangle$ of the Gaussian counterpart with
the same probability $2.8\times 10^{-6}$ as the experimental reaches only
$5.8\times 10^{-8}$ fraction of
its final value, these numbers being again more or less comparable.
\subsection{Asymmetry}
Note that all three distributions depicted in Fig. \ref{fig3} are constructed with the same
resolution, that is, the bin-size for all three distribution was the same and equal to $0.1$.
This makes it possible to compare the dispersion of the data for different
distributions. On the other hand, huge
dispersion of the ${v'}_r^2$-distribution does not make it possible to compare its tails. In
order to do this, we have to have a more smooth PDF, and that can be done by constructing the
distribution with a larger bin-size. Figure \ref{fig4} presents the ${v'}_r^2$-distribution
constructed with a bin-size $=1.$
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig4.ps}}
\caption{ Experimental ${v'}_r^2$-distribution. Direct comparison of the left and right
wings.
}
\vspace{1cm}
\label{fig4}
\end{figure}
Figure \ref{fig4}(a) compares the positive and negative parts of the distribution directly. It
is obvious that the negative tail prevails over the positive for rather
big ${u'}_r$. That is, the rare
stormy events are definitely asymmetric.
The ${v'}_r^2$-distribution can be considered as a PDF
centered at ${u'}_r=k_a$, see (\ref{mixed2}). That is, the distribution is definitely
asymmetric, and theoretically, this asymmetry {\it should be} present at the tails as well.
This actually
can be seen from the shifted Gaussian distribution $\tilde{\Phi}$,
see definition (\ref{GaussPhiA}) and (\ref{GaussPhi1}), and
from the Fig. \ref{fig4}(a), that the negative part of the distribution
is above the positive for any ${u'}_r$.
In order to see if the ${v'}_r^2$-distribution is just a shifted distribution, or not, we make
direct comparison of
its part where ${u'}_r-k_a$ is positive with the part where ${u'}_r-k_a$ is negative, see
\ref{fig4}(b). The two parts of the corresponding Gaussian distribution $\tilde{\Phi}$, of course, coincide
(and cannot be distinguished in the plot),
while the experimental distribution still shows asymmetry: The centered negative part still prevails.
This confirms the above conclusion that the rare violent events are asymmetric.
Another test for asymmetry is
measuring the contribution of rare events, as in Subsection \ref{rare}.
Namely, we will consider cumulative moments,
\begin{equation}
\langle u_r v_r^2\rangle\rule[-2mm]{.5mm}{6mm}_{|u_r| \ge t}=\int_{-\infty}^{-t}
u_r\Phi(u_r)du_r+\int_t^\infty u_r\Phi(u_r)du_r,
\label{kat}
\end{equation}
cf. (\ref{Phi0t}). Or, in dimensionless form,
\begin{equation}
k_a(t)=\langle {u'}_r {v'}_r^2\rangle\rule[-2mm]{.5mm}{6mm}_{|{u'}_r| \ge t}=\int_{-\infty}^{-t}
{u'}_r\Phi({u'}_r)d{u'}_r+\int_t^\infty {u'}_r\Phi({u'}_r)d{u'}_r,
\label{kat1}
\end{equation}
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig6.ps}}
\caption{ Experimental $k_a(t)$ (defined in (\ref{kat1})) for $|{u'}_r| \ge t$ events versus
the probability of these events $P(t)$.
}
\vspace{1cm}
\label{fig6}
\end{figure}
The case $t=0$ corresponds to all events, so that $k_a(t=0)$ assumes its final value, and at $t>0$ it
approaches
it. The case $t\to\infty$ corresponds to $k_a=0$. Figure \ref{fig6} illustrates $k_a(t)/k_a(0)$, or
actually it shows how the the third moment (\ref{mixed2}) is formed.
Obviously, there is a big difference between the experimental $k_a(t)/k_a(0)$ and its Gaussian
counterpart, as seen from Fig. \ref{fig6}. As an example (see the dashed-dotted
straight lines in the figure), we took the same probability $2.8\times
10^{-6}$ as in Fig. \ref{fig3a}, corresponding to extremely violent events. This time, it
reaches $0.03$th fraction of its final value. In contrast, its Gaussian counterpart reaches the
same fraction with $0.01$ part of events, -- these two numbers ($0.03$ and
$0.01$) being comparable (cf. Subsection \ref{rare}).
On the other hand, the Gaussian counterpart with probability $2.8\times 10^{-6}$ reaches the value of
$1.8\times 10^{-6}$. Again, these two numbers are comparable.
Thus, the contribution of the rare violent events into the experimental moment (\ref{mixed2}) is
substantial, as opposed to the "regular" situation presumably illustrated by Gaussian distribution.
Recall that this (odd) moment does not vanish due to asymmetry of the distributions, and therefore
this substantial contribution of the rare violent events into $k_a$-moment means that these
events also possess asymmetry.
\section{Conditional averages}
\subsection{Global properties}
\label{global_cond}
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig7.ps}}
\caption{ Experimental conditional average $\langle {v'}_r^2|{u'}_r\rangle$, compared
with Gaussian defined in (\ref{cond1}), and with "ideal" correlation, depicted by
dashed-dotted line.
}
\vspace{1cm}
\label{fig7}
\end{figure}
It is interesting to note that the the experimental conditional average
$\langle {v'}_r^2|{u'}_r\rangle$ is much above the
Gaussian defined in (\ref{cond1}), see Fig. \ref{fig7}.
If these two variables, ${u'}_r$ and ${v'}_r$ would
be statistically independent, then the conditional average is unity, well below
experimental average. The anisotropic Gaussian distribution (for which ${u'}_r$ and ${v'}_r$
variables are related, and therefore so are ${u'}_r$ and ${v'}_r^2$), is still much lower
than experimental, as seen from the figure.
In another extreme case (as opposed to statistically independent variables) we have an "ideal"
correlation, ${v'}_r=\alpha{u'}_r$, where $\alpha$ is a constant, which is, due to our
normalization, equal to unity. Then, simply,
$$\langle {v'}_r^2|{u'}_r\rangle={u'}_r^2$$
(cf. with Gaussian conditional average with maximal correlation coefficient $C=\pm 1$, from
(\ref{cond1})). As seen from the Fig. \ref{fig7}, this conditional average is even higher than
the experimental value: Of course, any connection between two variables is less than "ideal".
Another remarkable feature of this average is its gigantic dispersion,
cf. Subsection \ref{global}. Perhaps, the simplest way to explain
it is to consider the two variables ${u'}_r$ and ${v'}_r$ statistically independent.
As mentioned above, in this case, theoretically $\langle {v'}_r^2|{u'}_r\rangle=1$.
However, experimental measurements would give dispersed values, the statistics being defined
by the number of events. To be more specific, for small and moderate values of $|{u'}_r|$
with huge number of events, the data of $\langle {v}_r^2\rangle$ would be quite close
to unity. For big values of
$|{u'}_r|$, with only few events the data would be strongly dispersed around unity.
In particular,
if there is only one event in some bins, then the values of $\langle {v}_r^2\rangle$ would
coincide with $ {v}_r^2$ themselves, and they would be quite dispersed if the process is intermittent.
Qualitatively, the experimental conditional average in the Fig. \ref{fig7} looks as
described above, that is, it is smooth for small and moderate values of $|{u'}_r|$,
and strongly dispersed for big values. Returning to ${v'}_r^2$ vs ${u'}_r$ distribution, i.e.,
function $\Phi({u'}_r)$, we recall that $\Phi({u'}_r)=\langle {v'}_r^2|{u'}_r\rangle p({u'}_r)$,
see (\ref{mixed_c_d}),
where, $p({u'}_r)$ is not very dispersed function, as seen from Fig. \ref{fig3}. Therefore,
the dispersion of the ${v'}_r^2$ vs ${u'}_r$ distribution is explained analogously to
the dispersion of conditional average $\langle {v'}_r^2|{u'}_r\rangle$.
\subsection{Asymmetry}
\begin{figure}
\centerline{\includegraphics[height=8cm,width=16cm]{fig8.ps}}
\caption{ Direct comparison of negative and positive parts of conditional average
$\langle {v'}_r^2|{u'}_r\rangle$. Dashed-dotted line corresponds to "ideal" correlation.
}
\vspace{1cm}
\label{fig8}
\end{figure}
The data in Fig. \ref{fig7} are presented with relatively high resolution, the bin-size being
equal to $0.1$. This makes it possible to see the dispersion and to interpret it as in
Subsection \ref{global_cond}. On the other hand, it is difficult to study the asymmetry with
these dispersed data. Another processing of data with bin-size equal to $1$ are presented
in Fig. \ref{fig8}, where we compare the negative and positive parts of this conditional average
directly.
We note here that Gaussian expression for conditional average (\ref{cond1}) calculated from
the joint 2D distribution (\ref{Gauss1A}) and used in Fig.
\ref{fig8} is symmetric (recall that it is not known how to include asymmetry into Gaussian
or some other simple test 2D distribution). For this reason, the difference between the
right and left Gaussian wings in both figures (a) and (b) is spurious. Still, putting the Gaussian
conditional average into these two plots seems to be useful in order to compare
the experimental data with something "regular".
Both quantities, $\langle {v'}_r^2\rangle$ vs ${u'}_r$ distribution, Fig. \ref{fig4}, and
$\langle {v'}_r^2|{u'}_r\rangle$,
Fig. \ref{fig8}, are related. For that reason in the latter figure we also check the symmetry
in respect to the ${u'}_r=k_a$ point (in \ref{fig8}(b), analogously to \ref{fig4}(b)), in addition
to the symmetry in respect to ${u'}_r=0$ point, given in \ref{fig4}(a) and \ref{fig8}(a). In other
words, we check if the conditional average $\langle {v'}_r^2|{u'}_r\rangle$ is simply a shifted
(but still symmetric) distribution, or not.
Close examination of Fig. \ref{fig8} suggests that there is no substantial asymmetry
{\it as a systematic trend} in this conditional average. Although this is an unexpected
conclusion, there might be an explanation as follows.
As can be seen from Fig. \ref{fig4}, the $\langle{v'}_r^2\rangle$ distribution, i.e., the
function $\Phi({u'}_r)$, is asymmetric, and so is the ${u'}_r$ distribution, $p({u'}_r)$, see
\cite{asym}. This
means that, in particular, the negative wings of both quantities are
elevated above the right-hand wings. According
to (\ref{mixed_c_d}), the conditional average $\langle{v'}_r^2|{u'}_r\rangle$ is defined as
a ratio of these two quantities. This suggests that increased value of $\Phi({u'}_r)$ in
numerator is balanced by increased value of $p({u'}_r)$ in denominator, decreasing the asymmetry
of $\langle{v'}_r^2|{u'}_r\rangle$. In other words, the latter quantity appears to be less
sensitive to the asymmetry than the former two.
\section{Discussion}
The asymmetry of turbulence has been studied for a long time. Its relation to the vorticity
production was pointed out
by \cite{Betchov}.
The asymmetry of $u_r$-distribution resulting in the 4/5-Kolmogorov law was interpreted with
the help of the ramp-model, see \cite{94}, \cite{96a}: As $\langle u_r\rangle=0$, any compression,
with $u_r<0$, is as efficient as expansion, with $u_r>0$. However, the compression appears
stronger but rarer
than decompression, the latter being weaker and longer. Therefore, $\langle u_r^3\rangle\not= 0$.
This model immediately suggests that this asymmetry is related to the intermittency: The
compressed rare but strong events are supposed to be intermittent.
The ramp-model is only heuristic, however. It was shown by \cite{1}, \cite{2}, \cite{111}
that Burgers vortex,
embedded into a converging motion, acquires negative skewness, this picture containing both
asymmetry and intermittency. As the ramp-model does not exactly correspond to this picture,
it had to be modified. This modification was called the bump-model, see \cite{jfm}. Actually,
``the bump" corresponds to the Burgers vortex. This model is now two-dimensional, -- to
include transverse velocity increments $v_r$ into the picture. In particular, it
automatically explains why the
mixed third moment $\langle u_rv_r^2\rangle$, (\ref{mixed0}), does not vanish.
Fully two-dimensional PDF describing all three items, namely asymmetry, anisotropy and
intermittency,
was discussed in this paper. The PDF, and also some additional conditional and cumulative averages
seem to support the idea about the connection between these three features. In the whole picture,
there is
additional (to the Kolmogorov cascade) energy transport from the large non-universal scales
directly to the small scales. We note that the Burgers vortex, with a small radius, is generated
by a relatively large-scale motion, thus making it possible to transfer the energy directly from
large scales to small. In the framework of the bump-model, it is ``the bump" which is generated
analogously to the Burgers vortex. The scale of the bump is definitely much smaller than the
scale of the generating it motion (which can be seen from Fig. 5(c) by \cite{jfm}), and that could
explain the direct interaction between the non-universal large scales and small scales.
I thank K. R. Sreenivasan and S. Kurien for the data and for useful discussions.
\newpage
|
{
"timestamp": "2007-05-20T22:22:10",
"yymm": "0705",
"arxiv_id": "0705.2890",
"language": "en",
"url": "https://arxiv.org/abs/0705.2890"
}
|
"\n\\section{Executive Summary}\n\nThis report provides the results of an extensive and important st(...TRUNCATED)
| {"timestamp":"2007-05-30T16:13:11","yymm":"0705","arxiv_id":"0705.4396","language":"en","url":"https(...TRUNCATED)
|
"\\section*{Acknowledgments}\n\nWe thank Y. Mimura, M.~ur Rehman and V.~N.~Senoguz for very\nhelpf(...TRUNCATED)
| {"timestamp":"2007-08-18T22:44:09","yymm":"0705","arxiv_id":"0705.3035","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction }\n1.We consider the Knizhnik-Zamolodchikov differential system (see\n[3])\n(...TRUNCATED)
| {"timestamp":"2007-05-30T11:35:31","yymm":"0705","arxiv_id":"0705.4362","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:introduction}\n\nSecurity protocols, in particular those for f(...TRUNCATED)
| {"timestamp":"2007-05-24T06:28:47","yymm":"0705","arxiv_id":"0705.3503","language":"en","url":"https(...TRUNCATED)
|
"\\subsection{Introduction}\n\nFrustrating antiferromagnetic (AF) interaction in quantum\nmagnets~\\(...TRUNCATED)
| {"timestamp":"2007-05-29T13:43:49","yymm":"0705","arxiv_id":"0705.4198","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nIn the global approach to quantum field theory \\cite{DeWi03}, the physi(...TRUNCATED)
| {"timestamp":"2007-05-18T17:34:02","yymm":"0705","arxiv_id":"0705.2723","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\nThe representations of 2d $\\mathcal{N}=(2,2)$ superfields play an importa(...TRUNCATED)
| {"timestamp":"2007-05-22T18:15:43","yymm":"0705","arxiv_id":"0705.3207","language":"en","url":"https(...TRUNCATED)
|
End of preview.
No dataset card yet
- Downloads last month
- 3