Extended self-similarity works for the Burgers equation and why

Chakraborty, Sagar; Frisch, U.; Ray, Samriddhi Sankar

doi:10.1017/s0022112010000595

Cited by 43 publications

(56 citation statements)

References 32 publications

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“…4(b) for completeness. Similar to what is observed for the scaling behavior evidenced above through the MM, we consider also these types of results to be consistent with the interpretation proposed by [27], which can explain the ESS of variables that do not necessarily satisfy Burger's equation [34].…”

Section: A Bentheimer Sandstonesupporting

confidence: 90%

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Statistical scaling of pore-scale Lagrangian velocities in natural porous media

et al. 2014

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We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way qth-order sample structure functions (statistical moments of order q of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders q, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.

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Section: A Bentheimer Sandstonesupporting

confidence: 90%

“…With reference to ESS, a theoretical basis for Eq. (3) and its validity for all investigated lags has been provided with reference to (a) the one-dimensional Burger equation [34], (b) TFBM or TFGN [18], and (c) sub-Gaussian random processes subordinated to TFBM or TFGN [27].…”

Section: Scaling Of Statisticsmentioning

confidence: 99%

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

et al. 2014

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“…"In spite of several attempts to explain the success of ESS" cited by Chakraborty et al (2010) the authors note that "the latter is still not fully understood and we do not know how much we can trust scaling exponents derived by ESS. It would be nice to have at least one instance for which ESS not only works, but does so for reasons we can rationally understand."…”

Section: Introductionmentioning

confidence: 99%

“…Chakraborty et al (2010) cite the success of ESS in extending observed scaling ranges, and thus allowing more accurate empirical determinations of the functional exponent ξ(q) for turbulent velocities. ESS has been reported to achieve similar results for diffusion-limited aggregates, natural images, kinetic surface roughening, fluvial turbulence, sand wave dynamics, Martian topography, river morphometry, gravel-bed mobility and atmospheric barometric pressure, low-energy cosmic rays, cosmic microwave background radiation, metalinsulator transition, irregularities in human heartbeat time series, turbulence in edge magnetized plasma of fusion devices and turbulent boundary layers of the Earth's magnetosphere .…”

Section: Introductionmentioning

confidence: 99%

Extended power-law scaling of air permeabilities measured on a block of tuff

Siena

Guadagnini

Riva

et al. 2012

Hydrol. Earth Syst. Sci.

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Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

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“…2) breaks down, it nevertheless includes numerous examples demonstrating this to be the case. In addition to the classic case of turbulent velocities (Chakraborty et al, 2010), these examples include geographical (e.g. Earth and Mars topographic profiles), hydraulic (e.g.…”

Section: Introductionmentioning

confidence: 99%

Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks

Guadagnini

Riva

Neuman

2012

Hydrol. Earth Syst. Sci.

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Abstract. We analyze the scaling behaviors of two fieldscale log permeability data sets showing heavy-tailed frequency distributions in three and two spatial dimensions, respectively. One set consists of 1-m scale pneumatic packer test data from six vertical and inclined boreholes spanning a decameters scale block of unsaturated fractured tuffs near Superior, Arizona, the other of pneumatic minipermeameter data measured at a spacing of 15 cm along three horizontal transects on a 21 m long and 6 m high outcrop of the Upper Cretaceous Straight Cliffs Formation, including lowershoreface bioturbated and cross-bedded sandstone near Escalante, Utah. Order q sample structure functions of each data set scale as a power ξ(q) of separation scale or lag, s, over limited ranges of s. A procedure known as extended self-similarity (ESS) extends this range to all lags and yields a nonlinear (concave) functional relationship between ξ(q) and q. Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) ESS of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm), the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm or truncated fractional Gaussian noise (tfGn), stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts. Here we (i) demonstrate that the above two data sets are consistent with sub-Gaussian random fields subordinated to tfBm or tfGn and (ii) provide maximum likelihood estimates of parameters characterizing the corresponding Lévy stable subordinators and tfBm or tfGn functions.

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Extended self-similarity works for the Burgers equation and why

Cited by 43 publications

References 32 publications

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

Extended power-law scaling of air permeabilities measured on a block of tuff

Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks

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