Universal Multifractals: Theory and Observations for Rain and Clouds

Tessier, Yves; Lovejoy, S.; Schertzer, D.

doi:10.1175/1520-0450(1993)032<0223:umtaof>2.0.co;2

Cited by 358 publications

(303 citation statements)

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“…It is interesting to note that the fractional integration approach in Tessier et al [21] characterizes cloud-field variability by a Dq for the gradient-magnitude field and a power spectrum H for underlying field; the implication of (14) is that the fractionally integrated fields obtained from their procedure should have the correct structure function, despite the fact that only the behavior of the variance scaling H is built into the construction.…”

Section: (11)mentioning

confidence: 99%

Tracer microstructure in the large-eddy dominated regime

Pierrehumbert

1994

Chaos, Solitons & Fractals

257

301

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Abstract--We have re-examined the inertial range behavior of a passive scalar which is advected by a large-scale velocity field causing a cascade of tracer variance to small scales, where it is dissipated by diffusion. This has been done within the context of an idealized model based on mixing by a 2D area-preserving map alternating with a weak diffusion step; the model is a special case of the general advection-diffusion problem. Both freely decaying and forced equilibrium systems were considered. Our main interest in this concerns the validity of Batchelor's theory predicting a k -I tracer variance spectrum, but the tracer microstructure has been diagnosed in terms of concentration probability distribution functions, generalized dimensions of the dissipation field, structure functions, and cancellation exponents. 2D simulations carried out at 10242 resolution show that i;, the decaying case the evolution settles into a 'fractal eigenmode' in which the variance decays exponentially with time at a rate dependent on the Lyapunov exponent but independent of the diffusion coefficient. Although the concentration pattern is self-similar with time, the power spectrum is not algebraic. Concentration PDFs have exponential tails. The dissipation field is not multifractal, and formally has Dq = 2 for all q. The convergence of the squared-gradient PDFs under coarse-graining indicates some underlying fractal behavior, however, and we have introduced the notion of 'fractal degree of freedom' systems to describe such entities. Cancellation exponents and structure functions were also considered, and have a self-similarity which is compatible with a non-intermittent behavior of the dissipation field. These matters have also been addressed for the equilibrium case. The main difference is that the power spectrum of concentration variance in equilibrium exhibits a power-law inertial range, though it is steeper than k -1, but not as steep as k -2. Other features are similar to the decaying case. Very high resolution simulations of the undiffused problem indicate that a k -1 spectrum is approached asymptotically, but only at resolutions corresponding to 106 x 106. The passive scalar behavior is compared and contrasted with the behavior of scalar pseudo-vorticity (an 'active scalar') in the family of generalized 2D turbulence models introduced elsewhere in this issue.

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Section: (11)mentioning

confidence: 99%

Tracer microstructure in the large-eddy dominated regime

Pierrehumbert

1994

Chaos, Solitons & Fractals

257

301

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“…Yet a fourth approach to the statistics of precipitation is that taken by Tessier et al (1993), and Hubert et al (1993), who apply the methods of fractal analysis to rainfall data. In this context, Hubert et al (1993) examine precipitation records as a function of duration.…”

Section: Introductionmentioning

confidence: 99%

The scaling law relating world point‐precipitation records to duration

Galmarini

Steyn

Ainslie

2004

Intl Journal of Climatology

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In order to understand the remarkable six-orders of magnitude scaling law underlying worldwide point-precipitation records, we analyse precipitation data from a wide range of stations worldwide. The analysis shows that single-exponent scaling laws exist only for single stations experiencing extremely high precipitation. This analysis, and a consideration of the sequence of earlier published precipitation records, leads us to conclude that record precipitation exists because of an optimization of all factors leading to precipitation. This idea is incorporated into a scheme for simulating the record-duration curve for precipitation, which utilizes only the probability distribution function for precipitation amounts, the temporal autocorrelation of precipitation and a starting record-duration point. The simulation suggests record precipitation is asymptotically independent of most underlying physical processes.

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“…Lovejoy and Schertzer, 1990) on rainfall time series (e.g. Hubert et al, 1993;Olsson et al, 1993;Tessier et al, 1993Tessier et al, , 1996de Lima and Grasman, 1999), but other random cascade models have also been employed (e.g. Menabde et al, 1997Menabde et al, , 1999Deidda et al, 1999).…”

Section: Introductionmentioning

confidence: 99%

Cascade-based disaggregation of continuous rainfall time series: the influence of climate

Güntner¹,

Olsson

Calver

et al. 2001

Hydrol. Earth Syst. Sci.

119

123

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Rainfall data of high temporal resolution are required in a multitude of hydrological applications. In the present paper, a temporal rainfall disaggregation model is applied to convert daily time series into an hourly resolution. The model is based on the principles of random multiplicative cascade processes. Its parameters are dependent on (1) the volume and (2) the position in the rainfall sequence of the time interval with rainfall to be disaggregated. The aim is to compare parameters and performance of the model between two contrasting climates with different rainfall generating mechanisms, a semi-arid tropical (Brazil) and a temperate (United Kingdom) climate. In the range of time scales studied, the scale-invariant assumptions of the model are approximately equally well fulfilled for both climates. The model parameters differ distinctly between climates, reflecting the dominance of convective processes in the Brazilian rainfall and of advective processes associated with frontal passages in the British rainfall. In the British case, the parameters exhibit a slight seasonal variation consistent with the higher frequency of convection during summer. When applied for disaggregation, the model reproduces a range of hourly rainfall characteristics with a high accuracy in both climates. However, the overall model performance is somewhat better for the semi-arid tropical rainfall. In particular, extreme rainfall in the UK is overestimated whereas extreme rainfall in Brazil is well reproduced. Transferability of parameters in time is associated with larger uncertainty in the semi-arid climate due to its higher interannual variability and lower percentage of rainy intervals. For parameter transferability in space, no restrictions are found between the Brazilian stations whereas in the UK regional differences are more pronounced. The overall high accuracy of disaggregated data supports the potential usefulness of the model in hydrological applications.

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Universal Multifractals: Theory and Observations for Rain and Clouds

Cited by 358 publications

References 0 publications

Tracer microstructure in the large-eddy dominated regime

Tracer microstructure in the large-eddy dominated regime

The scaling law relating world point‐precipitation records to duration

Cascade-based disaggregation of continuous rainfall time series: the influence of climate

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