Fluid geoscience, which includes the study of the atmosphere, the oceans, hydrology, and climate, is a special topic in science: it is the application, in some sense, of the Navier-Stokes and transport equations, as well as other fluid mechanics theoretical concepts, into the “dirty real-world”. In the theoretical realm and in the laboratory, where all conditions are controlled, a desired phenomenon can be isolated and studied. In the field, however, conditions are not controlled, leading to a multitude of mixed phenomena, and measurements are the result of a multiscale complex system. In such a system, stochastic fluctuations are often superposed onto deterministic forcing associated with astronomic cycles. Scaling concepts are very important when approaching and studying geophysical fields, but the stochastic scaling properties are perturbed by forcing of, say, the day-night cycle, the annual cycle, or the tidal cycle in oceanography. In such cases, scaling methodologies may fail. In this book, we shall discuss such issues, and provide new tools that are more adapted to retrieving stochastic scaling information from different turbulence and real-world situations.

Fluid geoscience is also a special topic, in the sense that it is extremely difficult to find adequate and solid theories. The underlying fundamental equations (e.g., Navier-Stokes) are nonlinear, and turbulence belongs to the last field of classical physics, which is still unsolved. The resolution of Navier-Stokes equations belongs to the Clay Institute's one-million-dollar problems. This explains why the field of fluid geosciences has only progressed, in some sense, so slowly. Nowadays, many models used are based on crude large-scale approximations of dynamics, based on averages of Navier-Stokes equations. These models are inaccurate and no theory really tackles the nonlinearities and intermittencies of the natural (fluid) turbulent-like world. New theories and new models are needed. The solution may reside in generalizations and continuations of Kolmogorov's seminal 1941 work, when a scaling law was proposed to describe velocity fluctuations at different scales. Multifractal models are continuations of these works from the 1940s, and their aim has been to find statistical laws (such as thermodynamics in statistical physics for an equilibrium system) able to describe and model, at different scales, the fluctuations of fields of interest.

Such a general objective explains why we have devoted the first chapters of this book to historical developments on scaling and multiscaling issues in the field of turbulence and passive scalar turbulent transport.

In this chapter, the previous chapter's methods will be applied to several data sets obtained from the field of turbulence. The experimental velocity data was obtained from a wind tunnel experiment at the Johns Hopkins University, Baltimore, Maryland, USA. It includes also Lagrangian data obtained from a high-resolution direct numerical simulation (DNS), temperature field obtained from the Rayleigh-Bénard convection systems, and a vorticity field from a very high-resolution DNS.

Homogeneous turbulence

The experimental velocity data obtained from a wind tunnel experiment was performed by Prof. Meneveau's group with an active grid technique to achieve a high Reynolds number. The measurement locations were x/M = 20, x/M = 30, x/M = 40, and x/M = 48, where M was the mesh size and x was the downstream location. An X-type hotwire was used to measure the velocity, each time for a duration of 30 seconds (there were 30 realizations of time series measurements), with a sampling frequency of 40 kHz. In this book, we shall only consider the data obtained at the location x/M = 20. We recall here the main parameters of this data set. The Taylor's microscale based Reynolds number is Reλ ≃ 720, the mean velocity is 12.0 ms−1, and the turbulent intensity is about 15.4%. We perform the analysis of this database in the temporal domain with the implicit use of the Taylor's frozen hypothesis to convert the temporal data to the spatial one.More detail of this database can be found in Kang et al. (2003).

Fourier power spectrum and second-order structure function

Figure 5.1 shows the experimental Fourier power spectrum E(f) for both the longitudinal (solid line) and transverse (dashed line) velocity components. The Fourier power spectrum E(f) is estimated with a realization of 215 = 32768 data points each without overlap. Power-law behavior is visible for both curves on nearly two decades of frequency range 10 < f < 1000 Hz with the scaling exponents β = 1.63 ± 0.01 for the longitudinal velocity, and β = 1.61 ± 0.01 for the transverse velocity. Note that these two values are slightly smaller than the Kolmogorov value of 5/3, which could be considered as a finite Reynolds number effect. Power-law behavior is thus expected for at least the second-order structure functions on the time scales 0.001 < τ < 0.1 second; see the mathematical relation in Equation (4.5). A measurement noise is visible for the longitudinal velocity around fN ≃ 20,000Hz.

As mentioned previously, turbulent flows or turbulent-like phenomena possess in many aspects self-similarity or multiscaling properties and multiscale statistics. The classical way to characterize this scaling behavior is to estimate the qth-order structure function Sq(l) for different scales. Power-law behavior is then expected, e.g., Sq(l) ~ lζ(q), in which Sq(l) represents the qth-order structure function (see Chapter 2). The structure function method was first proposed by Kolmogorov in 1941 in his famous homogeneous and isotropic turbulence paper. There are also to be found other methodologies to characterize the multiscale statistics of such fields. Such examples include: wavelet-based approaches; the Hilbert-Huang transform; detrended fluctuation analysis; and detrended structure function analysis, to mention a few. Each method has its own advantages and disadvantages as it might show different performances for different types of data. Before applying one of these methods to real data, it is necessary to know whether the method is suitable for that data. For example, in the fluid geosciences, the annual cycle is present in many observation data, and therefore the structure function analysis, as we show below, might not be suitable. Here, other methodologies which might constrain the large-scale influence should be considered.

In this chapter, we will present more details about the classical structure-function analysis and new approaches that have emerged in recent years. This chapter is organized as follows. The increment-based methods are introduced in Sections 4.1 to 4.3. The wavelet-based methods are presented in Section 4.7. The Hilbert-based method is described in Section 4.8.

Structure function analysis

Second-order structure function

We first consider here the second-order structure functions (SFs). The second-order SF is defined as,

in which l is a separation scale (Figure 4.1). Originally, such an approach was done for spatial scales; however, as explained in Chapter 2, we are dealing here with time series and transform spatial fluctuation into time fluctuation using Taylor's frozen hypothesis, hence, here l is a time scale.

In this chapter, we shall consider several case studies from the ocean and atmospheric sciences. Geophysical fluid dynamics is characterized by important variations over a large range of scales. This is true also of ocean sciences, which are often taken as typical example of complex systems, with a huge number of constitutive elements (fluid particles and their scalar properties) interacting through physical, chemical, and biological relations. In the ocean there has been historically one of the first field validation of Kolmogorov's theory, using data from a channel with strong currents between islands (Grant et al., 1962). Since then, other studies have considered oceanic passive scalars, as well as the fluorescence fluctuations as proxy of phytoplankton concentrations. These have compared fluorescence and temperature using Fourier spectra and correlations (Platt and Denman, 1975; Denman, 1976; Denman et al., 1977) and using multifractal approaches (Seuront et al., 1996a,b, 1999). Multifractal studies in the field of oceanology have also been done also using satellite images (Tessier et al., 1993; Lovejoy et al., 2001a,b; Nieves et al., 2007; Pottier et al., 2008; Turiel et al., 2009; de Karman et al., 2011; Renosh et al., 2015).

Superposed to the stochastic variability related to turbulence, coastal marine fields are also subject to deterministic periodic forcing, at fixed scales, such as tidal, diurnal, and annual forcing scales. Hence, time series in the marine sciences can be quite complex and their analysis nontrivial. In the present chapter, we focus on several coastal time series. We first present a coastal marine turbulence time series recorded at 1 Hz in a megatidal sea. We then present a long-term series of water level dynamics recorded every hour, and a database of water quality automatic monitoring recorded at fixed point every 20 minutes, with emphasis on three quantities: temperature, dissolved oxygen, and fluorescence. We finally consider atmospheric examples: atmospheric wind velocity and wind power time series. In each case we perform power spectra, identify some scaling ranges and forcing, and apply several methods among the methods described in the methodological chapters.

Coastal marine turbulence

We first consider coastal marine turbulence, the interface between land and sea. Close to the shore, waves break and provide a surplus of energy to the marine system. The zone where the bottom has influence on waves is the surf zone (Svendsen, 1987; Battjes, 1988; Schmitt et al., 2009).

Data from the real world

Data analysis is an essential part of scientific research and engineering applications. A proper data analysis method will provide a better understanding of the process under consideration, for example, extracting useful parameters to validate an existing theory or to inspire a new theory. However, data from the real world, such as field-observation data, well-controlled laboratory experiments, or numerical simulation, generally possess several problems. For instance, the data length may be too short to satisfy stationary or ergodic conditions, or the mechanism behind the data may be nonlinear, etc. In Figures 1.1 through 1.3 are several examples from the real world, demonstrating common problems of real datasets.

Figure 1.1a displays a collected oxygen saturation index obtained from a MAREL network (Automatic Monitoring Network for Littoral Environment, Ifremer, France) from the period January 1 to December 31, 2010. The large variation of the measured oxygen saturation index shows the nonstationarity of the data. For example, a high intensity of oxygen saturation index was observed at September 3, 2010. As shown in Section 1.3, to mimic this nonstationary event, high-order Fourier harmonic components are required. Moreover, sometimes the sensor fails to collect data, due to maintenance problems or failure of the system. This missing data problem is typical of field observation data. To emphasize this, we replotted a 10-day portion of the data in Figure 1.1b, which shows a discontinuous curve due to the problem of the data missing. This imposes a difficulty for the Fourier-based data analysis method, for which a uniform time step is often required. To see the irregular time step more clearly, the time interval δt has been shown between two successful measurements in Figure 1.1c. Visually, the time step δt demonstrates a strong intermittent distribution with large values of δt. Another aspect of this data set is that the underlying physical mechanism is nonlinear. It means that if one can write a governing equation for the oxygen saturation index, this equation must be nonlinear. Therefore, the difficulties of this type of data set are missing data/irregular time steps, both nonstationarity and nonlinearity.

Turning to another aspect of the real world data, namely varying sample size, considered here is an example from the global drifter program. Figure 1.2a illustrates several Lagrangian trajectories obtained from the global drifter program.

Introduction

In this book, we will consider stochastic time series with scaling properties and how to analyze them. The general inspiration, and the theoretical foundation of our approach, is the field of fully developed turbulence. Many properties of fields and time series, and many concepts, have been found and developed in the field of turbulence, and later exported to other complex fields such as engineering and natural sciences. Hence in this introduction chapter, we have revised some basic concepts and results in the field of locally homogeneous and isotropic three-dimensional turbulence. Older, more general references in this respect are the following: Batchelor (1953); Monin and Yaglom (1971); Tennekes and Lumley (1972); Frisch (1995); Pope (2000). With our consideration of time series, turbulence is first presented as a 3D problem, from which time series are extracted, and which inherit mainly nonlinear properties from the 3D world.

Turbulence is a phenomenon that is still only partly understood. However it concerns directly or indirectly a large number of scientific fields: meteorology, oceanography, astronomy, hydraulics, aeronautics, environment, chemical industry, automotive engineering, renewable energy, etc. A better understanding and theoretical description of turbulence is badly needed for practical applications in all these fields.

The first use of “turbulence” for sinuous motions in fluid mechanics was apparently due to Lord Kelvin in a paper published in 1887 (Thomson, 1887; Schmitt, 2007a). There are several definitions of turbulence. A generalist definition would indicate “instability or irregular movements in a fluid.” In physics the definition uses the dimensionless Reynolds number Re = UL/ν, where U is a typical velocity scale, L a typical length scale, and ν the kinematic viscosity of the fluid. When Re is small, the flow is laminar, and the fluid motion is regular. There is a transition value where instabilities occur, and when Re is very large. Here the flow is said to be in a state of fully developed turbulence. Such flow is also chaotic due to the nonlinear nature of Navier-Stokes equations. Another characteristic of turbulence is its very efficient mixing property. Tennekes and Lumley (1972) insisted on the three dimensional nature of turbulence, and also on the fact that it is a dissipative system, needing, to be in a stationary state, a continuous input of energy.

An important issue is what happens after a singularity has occurred. A very simple example was considered in Section 3.5; now we will turn to some physical examples in more detail. First, we consider the breakup of a liquid drop, which marks the transition from one piece to two separate pieces of liquid. To know how the two pieces evolve after breakup one needs to continue across the singularity. In particular, is the continuation unique or are there several possible ways for the solution to evolve after the singularity? As in Section 3.5 we will consider two very different approaches to continuation. In the first approach we construct a similarity solution that applies directly after the singularity, by solving two separate problems corresponding to the two pieces resulting from breakup. We show that they are determined uniquely by the pre-breakup dynamics. In the second approach we regularize the equations on a small scale, so that a true singularity never occurs.

Post-breakup solution: viscous thread

In Chapter 7 we saw that the asymptotics of drop breakup is described by the similarity solution (7.41). We described in some detail how a unique, stable, similarity solution is selected; this is shown in Fig. 7.3. Now we show how the knowledge of the pre-breakup solution permits us to construct a unique post-breakup solution. The strategy has already been laid out and hinges on the matching conditions (3.84), which transfer information about the behave-ior of the pre-breakup solution far from the pinch point to the post-breakup solution.

However, there remains a technical challenge, illustrated in Fig. 10.1. In the immediate neighborhood of the tip, where the slope h′ of the profile diverges, the solution can no longer be considered slender. As a result the derivation of the slender jet equations (6.57), (6.58) is not valid; the equations break down at the tip. Instead, we must consider a separate tip region, whose width tip is comparable with its radial extension. In the language of matched asymptotics the tip is the inner region, while the receding thread, over which the slender jet approximation applies, is the outer region.