scholarly journals Towards mapping turbulence in the intra-cluster medium

2019 ◽  
Vol 629 ◽  
pp. A144 ◽  
Author(s):  
E. Cucchetti ◽  
N. Clerc ◽  
E. Pointecouteau ◽  
P. Peille ◽  
F. Pajot
Keyword(s):  
Structure Function ◽  
Spatial Scales ◽  
Companion Paper ◽  
Structure Functions ◽  
Order Structure ◽  
Sampling Variance ◽  
Function Estimation ◽  
X Ray ◽  
Spatially Resolved Spectroscopy ◽  
Field Unit

X-ray observations of the hot gas filling the intra-cluster medium (ICM) provide a wealth of information on the dynamics of clusters of galaxies. The global equilibrium of the ICM is believed to be ensured by non-thermal and thermal pressure support sources, among which gas movements and the dissipation of energy through turbulent motions. Accurate mapping of turbulence using X-ray emission lines is challenging due to the lack of spatially resolved spectroscopy. Only future instruments such as the X-ray Integral Field Unit (X-IFU) on Athena will have the spatial and spectral resolution to quantitatively investigate the ICM turbulence over a broad range of spatial scales. Powerful diagnostics for these studies are line shift and the line broadening maps, and the second-order structure function. When estimating these quantities, instruments will be limited by uncertainties of their measurements, and by the sampling variance (also known as cosmic variance) of the observation. Here, we extend the formalism started in our companion Paper I to include the effect of statistical uncertainties of measurements in the estimation of these line diagnostics, in particular for structure functions. We demonstrate that statistics contribute to the total variance through different terms, which depend on the geometry of the detector, the spatial binning and the nature of the turbulent field. These terms are particularly important when probing the small scales of the turbulence. An application of these equations is performed for the X-IFU, using synthetic turbulent velocity maps of a Coma-like cluster. Results are in excellent agreement with the formulas both for the structure function estimation (≤3%) and its variance (≤10%). The expressions derived here and in Paper I are generic, and ensure an estimation of the total errors in any X-ray measurement of turbulent structure functions. They also open the way for optimisations in the upcoming instrumentation and in observational strategies.

Deep Chandra observations of the core of the Perseus cluster

2018 ◽  
Vol 14 (S342) ◽  
pp. 127-132
Author(s):  
Jeremy S. Sanders
Keyword(s):  
Spatial Scales ◽  
High Quality ◽  
X Ray ◽  
The Core ◽  
Spatially Resolved ◽  
Spatially Resolved Spectroscopy ◽  
Perseus Cluster ◽  
Metallicity Distribution

AbstractThe Perseus cluster is the X-ray brightest cluster in the sky and with deep Chandra observations we are able to map its central structure on very short spatial scales. In addition, the high quality of X-ray data allows detailed spatially-resolved spectroscopy. In this paper I review what these deep observations have told us about AGN feedback in clusters, sloshing and instabilities, and the metallicity distribution.

scholarly journals Shallow water wave turbulence

2019 ◽  
Vol 874 ◽  
pp. 1169-1196 ◽  
Author(s):  
Pierre Augier ◽  
Ashwin Vishnu Mohanan ◽  
Erik Lindborg
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Shallow Water ◽  
Water Wave ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Shallow Water Wave ◽  
The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

Applicability of Kolmogorov's and Monin's equations of turbulence

1997 ◽  
Vol 353 ◽  
pp. 67-81 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Inertial Range ◽  
Homogeneous Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Local Isotropy ◽  
The Third ◽  
Moderate Reynolds Number ◽  
Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

Third-order structure function relations for quasi-geostrophic turbulence

2007 ◽  
Vol 572 ◽  
pp. 255-260 ◽  
Author(s):  
ERIK LINDBORG
Keyword(s):  
Structure Function ◽  
Three Dimensional ◽  
Structure Functions ◽  
Order Structure ◽  
Third Order ◽  
Longitudinal Structure ◽  
The Third ◽  
Inverse Cascade ◽  
Geostrophic Turbulence ◽  
Potential Enstrophy

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

Triplet and Higher Correlations in the Long Wavelength Limit

1975 ◽  
Vol 53 (4) ◽  
pp. 372-376 ◽  
Author(s):  
L. E. Ballentine ◽  
A. Lakshmi
Keyword(s):  
Structure Function ◽  
Electronic Properties ◽  
Lower Order ◽  
Liquid Metals ◽  
Structure Functions ◽  
Order Structure ◽  
Long Wavelength ◽  
Wavelength Limit ◽  
Ionic Potential ◽  
Superposition Approximation

The nth order structure function of a liquid, defined as the ensemble average of a product of n Fourier components of the atomic density, is studied in the limit of long wavelength for one or more Fourier components. It is shown by means of fluctuation theory that these limits are simply related to lower order structure functions and their derivatives with respect to pressure, and that they will be small in magnitude for normal liquids. This conclusion is important for the study of electronic properties of liquid metals because the screened ionic potential is always large in the long wavelength limit. Thus large spurious contributions may be obtained from the use of approximate structure functions that do not satisfy the correct long wavelength limits. In this respect the Kirkwood superposition approximation is very unsatisfactory.

scholarly journals Exact second-order structure-function relationships

2002 ◽  
Vol 468 ◽  
pp. 317-326 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Scaling Laws ◽  
Velocity Structure ◽  
Structure Functions ◽  
Energy Dissipation Rate ◽  
Second Order ◽  
Order Structure ◽  
Third Order ◽  
Navier Stokes Equation ◽  
Local Isotropy

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

scholarly journals The Vertical Structure of Open-Ocean Submesoscale Variability during a Full Seasonal Cycle

2020 ◽  
Vol 50 (1) ◽  
pp. 145-160 ◽  
Author(s):  
Zachary K. Erickson ◽  
Andrew F. Thompson ◽  
Jörn Callies ◽  
Xiaolong Yu ◽  
Alberto Naveira Garabato ◽  
...  
Keyword(s):  
Structure Function ◽  
Mixed Layer ◽  
Seasonal Cycle ◽  
Spatial Scales ◽  
Horizontal Resolution ◽  
Ocean Model ◽  
Open Ocean ◽  
Order Structure ◽  
Numerical Ocean Model ◽  
Ocean Region

AbstractSubmesoscale dynamics are typically intensified at boundaries and assumed to weaken below the mixed layer in the open ocean. Here, we assess both the seasonality and the vertical distribution of submesoscale motions in an open-ocean region of the northeast Atlantic. Second-order structure functions, or variance in properties separated by distance, are calculated from submesoscale-resolving ocean glider and mooring observations, as well as a 1/48° numerical ocean model. This dataset combines a temporal coverage that extends through a full seasonal cycle, a horizontal resolution that captures spatial scales as small as 1 km, and vertical sampling that provides near-continuous coverage over the upper 1000 m. While kinetic and potential energies undergo a seasonal cycle, being largest during the winter, structure function slopes, influenced by dynamical characteristics, do not exhibit a strong seasonality. Furthermore, structure function slopes show weak vertical variations; there is not a strong change in properties across the base of the mixed layer. Additionally, we compare the observations to output from a high-resolution numerical model. The model does not represent variability associated with superinertial motions and does not capture an observed reduction in submesoscale kinetic energy that occurs throughout the water column in spring. Overall, these results suggest that the transfer of mixed layer submesoscale variability down to depths below the traditionally defined mixed layer is important throughout the weakly stratified subpolar mode waters.

scholarly journals Impacts of Convergence on Structure Functions from Surface Drifters in the Gulf of Mexico

2019 ◽  
Vol 49 (3) ◽  
pp. 675-690 ◽  
Author(s):  
Jenna Pearson ◽  
Baylor Fox-Kemper ◽  
Roy Barkan ◽  
Jun Choi ◽  
Annalisa Bracco ◽  
...  
Keyword(s):  
Gulf Of Mexico ◽  
Structure Function ◽  
Structure Functions ◽  
Second Order ◽  
Order Structure ◽  
Third Order ◽  
Second Order Statistics ◽  
Surface Drifter ◽  
Sign Change ◽  
Surface Drifters

AbstractThere are limitations in approximating Eulerian statistics from surface drifters, due to biases from surface convergences. By contrasting second- and third-order Eulerian and surface drifter structure functions obtained from a model of the Gulf of Mexico, the consequences of the semi-Lagrangian nature of observations during the summer Grand Lagrangian Deployment (GLAD) and winter Lagrangian Submesoscale Experiment (LASER) are estimated. By varying launch pattern and location, the robustness and sensitivity of these statistics are evaluated. Over scales less than 10 km, second-order structure functions of surface drifters consistently have shallower slopes (~r2/3) than Eulerian statistics (~r), suggesting that surface drifter structure functions differ systematically and do not reproduce the scalings of the Eulerian fields. Medians of Eulerian and cluster release second-order statistics are also significantly different across all scales. Synthetic cluster release statistics depend on launch location and weakly on launch pattern. The observations suggest little seasonal difference in the second-order statistics, but the LASER third-order structure function shows a sign change around 1 km, while GLAD and the synthetic cluster releases show a third-order structure function sign change around 10 km. Further, synthetic surface drifter cluster releases (and therefore likely the GLAD observations) show robust biases in the negative third-order structure functions, which may lead to significant overestimation of the spectral energy flux and underestimation of the transition scale to a forward energy cascade. The Helmholtz decomposition, and curl and divergence statistics, of Eulerian and cluster releases differ, particularly on scales less than 10 km, in agreement with observations of drifters preferentially sampling convergences in coherent structures.

scholarly journals First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

2012 ◽  
Vol 25 (16) ◽  
pp. 5538-5555 ◽  
Author(s):  
Kyle G. Pressel ◽  
William D. Collins
Keyword(s):  
Water Vapor ◽  
Structure Function ◽  
Power Law ◽  
Scale Invariance ◽  
Structure Functions ◽  
Scale Dependence ◽  
Global Maximum ◽  
Order Structure ◽  
Scaling Exponents ◽  
First Order

Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence?

1999 ◽  
Vol 388 ◽  
pp. 259-288 ◽  
Author(s):  
ERIK LINDBORG
Keyword(s):  
Energy Spectrum ◽  
Structure Function ◽  
Structure Functions ◽  
Separation Distance ◽  
Inertial Range ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order ◽  
Kinetic Energy Spectrum ◽  
The Third

The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r2/3, giving a k−5/3-range and another, including a logarithmic dependence, giving a k−3-range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as Πω≈1.8×10−13 s−3 and the constant in the k−3-law is measured as [Kscr ]≈0.19. It is argued that the k−3-range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k−5/3-range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.

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