122 L.J.A. van Bokhoven et al. 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 8 t S ∂ i u i (a) Aref A Cref C3 Dref D 10 -1 10 -1 10 0 τ S ω 3 (b) ∝t 0.75 A C3 D Fig. 3. (a) Time evolution of the velocity derivative skewness S ∂ i u i during and after the isotropic precalculation for different viscosities ν. Background rotation in casesA,C3andDisappliedatt ini =5.0, 4.0and2.0, respectively. For reference, the isotropic precalculations have been prolonged. (b) Log-log plot showing the vorticity skewness S ω 3 as a function of the scaled, shifted time τ for cases A, C3 and D. S ω 3 appears to depend inversely on time t ini , i.e. shorter precalculations yield higher final values of S ω 3 . The behavior observed in Fig. 3(b) may partly be ascribed to slight differences in S ∂ i u i at time t ini . Refined vorticity statistics of decaying rotating 3D turbulence 123 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 8 9 10 t S ∂ i u i (a) Bref B1 B2 B3 B4 10 -1 10 -1 10 0 τ S ω 3 (b) ∝t 0.75 B1 B2 B3 B4 Fig. 4. As Fig. 3, but for different durations of the isotropic precalculation. Back- ground rotation in cases B1-B4 is applied at t ini =2.0, 4.0, 6.0and8.0, respectively. Finally, Fig. 5 shows the time evolution of S ∂ i u i and S ω 3 for various back- ground rotation rates, viz. f =0.5, 2.5, 5.0and10.0 (cases C1-C4). Clearly, a lower background rotation rate results in a larger final value of S ω 3 .This result expresses the fact that the asymmetry between cyclonic and anticyc- lonic structures is more pronounced at low rotation rates than at high rotation rates. It is remarked that similar results were extracted from lower resolution (N = 144) calculations. 124 L.J.A. van Bokhoven et al. 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 8 9 10 t S ∂ i u i (a) Cref C1 C2 C3 C4 10 -1 10 -1 10 0 τ S ω 3 (b) ∝t 0.75 C1 C2 C3 C4 Fig. 5. As Fig. 3, but for different background rotation rates. Background rotation in cases C1-C4 is applied at t ini =4.0. Third Order Vorticity Correlations Figure 6 shows the time evolution of all nontrivial VTCs for various back- ground rotation rates. The following three observations are made: 1) ω 3 1 , ω 1 ω 2 3 and ω 1 ω 2 ω 3 are much smaller than unity and fluctuate around zero; 2) ω 2 1 ω 3 , ω 2 2 ω 3 and ω 3 3 are clearly nonzero; and 3) the ratio ω 2 1 ω 3 /ω 2 2 ω 3 (not shown) is found to fluctuate around unity. These results are consistent with relationship (3). Refined vorticity statistics of decaying rotating 3D turbulence 125 -0.1 0 0.1 0.2 0.3 0.4 〈ω 1 3 〉/〈ω 3 2 〉 3/2 C1 C2 C3 C4 -0.1 0 0.1 0.2 0.3 0.4 〈ω 1 ω 3 2 〉/〈ω 3 2 〉 3/2 C1 C2 C3 C4 -0.1 0 0.1 0.2 0.3 0.4 〈ω 1 ω 2 ω 3 〉/〈ω 3 2 〉 3/2 C1 C2 C3 C4 -0.1 0 0.1 0.2 0.3 0.4 〈ω 1 2 ω 3 〉/〈ω 3 2 〉 3/2 C1 C2 C3 C4 -0.1 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 τ 〈ω 2 2 ω 3 〉/〈ω 3 2 〉 3/2 C1 C2 C3 C4 -0.1 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 τ S ω 3 C1 C2 C3 C4 Fig. 6. Time evolution of the minimal set of VTCs in axisymmetric turbulence for various background rotation rates. All VTCs are normalized by ω 2 3 3/2 . 4 Discussion Our numerical results show that in most of the considered cases S ω 3 initially grows at a rate proportional to t 0.75±0.1 . The latter power-law exponent is in good agreement with the 0.7 obtained from recent laboratory experiments [14, 15]. However, the amplitude of maximum S ω 3 and the (scaled) time at 126 L.J.A. van Bokhoven et al. which this maximum occurs are significantly smaller in our DNS calculations than in the mentioned experiment. Morize et al. [15] already showed that the maximum of S ω 3 depends on the experimental configuration. Based on our results, a better agreement between DNS and experiment is expected for very low rotation rates (e.g. f =0.10π), viz. a weak background rotation will not succeed to immediately destroy the triple correlation of the velocity derivative ∂ 3 u 3 . Furthermore, we have investigated how the vertical vorticity skewness is affected by the viscosity, the value of the velocity derivative skewness at time t ini , and the background rotation rate. The first two parameters affect the initial Taylor-based Reynolds number Re λ (t ini ) while all parameters affect the initial Taylor-based Rossby number Ro λ (t ini ). The obtained results lead to the following general conclusion: lower Re λ (t ini ) and/or lower Ro λ (t ini ) – implying a higher degree of linearity – yield a lower final vorticity skewness. This result confirms that the asymmetry in terms of cyclonic and anticyclonic vorticity is most prominently present in an intermediate range of Rossby numbers, as also discussed by Jacquin et al. [11] for the anisotropic development of integral length-scales, and more recently by Bartello [2] in the context of VTCs. If the Rossby number is too small, nonlinearity is not important enough – even if pure linear dynamics can induce a transient growth of S ω 3 ,thatsame dynamics results in damping S ω 3 at later times. The opposite case of very large Rossby number is not addressed here, but recall that isotropy is conserved, and therefore asymmetry excluded, at macroscopic Rossby numbers larger than one [11]. Another surprising result is the different behavior of different triple cor- relations. Even if the velocity derivative skewness and the vorticity skewness look similar as statistical descriptors, their evolution in presence of solid-body rotation is far from similar, the former always being damped while the latter is showing transient (linear) growth. Initial Gaussianity and isotropy are also very important, especially if linear terms are dominant. For instance, the velo- city derivative skewness is zero only if Gaussianity holds, whereas the vorticity skewness is zero either because of isotropy or because of Gaussianity. The multi-fold behavior of various triple correlations in the non-isotropic case suggests to revisit elaborated EDQNM theories in order to derive any rel- evant three-point triple velocity and vorticity correlation, which are difficult to extract experimentally and numerically. In previous studies, the EDQNM2-3 formalisms were used to derive a nonlinear energy transfer, but much more in- formation, including VTCs, can be obtained. At least isotropic basic EDQNM can be used for initializing vorticity correlations in the general linear solution applied to VTCs, but more can be done. In this sense, anisotropic multi-point statistical theory remains a relevant alternative to DNS, allowing much higher Reynolds numbers and elapsed times (with in counterpart, less flexibility and need for statistical assumptions). In addition, the subtle interplay between linear and nonlinear processes is altered in the presence of boundaries, Ekman pumping, or initially coherent Refined vorticity statistics of decaying rotating 3D turbulence 127 structures: interesting insights to these effects can be found in recent studies by Zavala Sans´on and Van Heijst [18], Morize et al. [14], and Davidson et al. [7]. References [1] Bartello P, M´etais O, Lesieur M (1994), J. Fluid Mech. 273:1–29. [2] Bartello P (2006), private communication. [3] Batchelor GK, Townsend AA (1947), Proc. Roy. Soc. A 191:534–550. [4] Cambon C, Jacquin L (1989), J. Fluid Mech. 202:295–317. [5] Cambon C, Mansour NN, Godeferd FS (1997), J. Fluid Mech. 337:303– 332. [6] Cambon C, Scott JF (1999), Annu. Rev. Fluid Mech. 31:1–53. [7] Davidson PA, Staplehurst PJ, Dalziel SB (2006), J. Fluid Mech. 557:135– 144. [8] Greenspan HP (1968) The theory of rotating fluids. Cambridge Univer- sity Press. [9] Gence J-N, Frick C (2001), C. R. Acad. Sci. Paris S´erie IIB 329:351–356. [10] Godeferd FS, Lollini L (1999), J. Fluid Mech. 393:257–308. [11] Jacquin L, Leuchter O, Cambon C, Mathieu J (1990), J. Fluid Mech. 220:1–52. [12] Liechtenstein L, Godeferd FS, Cambon C (2005), JoT 6:1–21. [13] McComb WD (1990) The Physics of Fluid Turbulence, Oxford Engin- eering Science Series 25. Clarendon Press, Oxford. [14] Morize C, Moisy F, Rabaud M (2005), Phys. Fluids 17:095105. [15] Morize C, Moisy F, Rabaud M, Sommeria J (2006), Conf. Proc. TI2006, Porquerolles, France. [16] Rogallo RS (1981), NASA Tech. Mem. 81315. [17] Vincent A, Meneguzzi M (1991), J. Fluid Mech. 225:1–20. [18] Zavala Sans´on L, Van Heijst GJF (2000), J. Fluid Mech. 412:75–91. Lagrangian passive scalar intermittency in marine waters: theory and data analysis Fran¸cois G. Schmitt 1 and Laurent Seuront 1,2 1 CNRS, FRE 2816 ELICO, Wimereux Marine Station, Universite des Sciences et Technologies de Lille - Lille 1, 28 av Foch, 62930 Wimereux, France francois.schmitt@univ-lille1.fr 2 School of Biological Sciences, Flinders University, GPO Box 2100, Adelaide 5001, South Australia Laurent.Seuront@flinders.edu.au Intermittency is a basic feature of fully developed turbulence, for both ve- locity and passive scalars. We consider here intermittency in a Lagrangian framework, which is also a natural representation for marine organisms. We characterize intermittency using multi-fractal power-law scaling exponents. In this paper we recall four theoretical relations previously obtained to link Lagrangian and Eulerian passive scalar multi-fractal functions. We then exper- imentally estimate these exponents and compare the result to the theoretical relations. Section 1 describes the non intermittent Lagrangian passive scalar scaling laws; section 2 introduces the multi-fractal generalization, and gives the four theoretical relations ; section 3 presents experimental results. 1 Non-intermittent Lagrangian passive scalar scaling laws Marine particle dynamics is an important area in turbulence studies. Particles sampling is most easily achieved in the Eulerian sense, that is, in a refer- ence frame fixed with respect to the moving fluid, such as moored buoy or a pier. However, plankton organisms such as viruses, bacteria, phytoplankton and copepods, perceive their surrounding environment in a Lagrangian way. Those are mostly advected by the flows. The related Lagrangian turbulent fluctuations in the flow velocity and passive scalars perceived by individual plankton organisms have critical implications for foraging, growth and pop- ulations dynamics, and ultimately for a better understanding of the struc- ture and functioning of the pelagic realm. An absolute pre-requisite to the analysis of e.g. behavioral response to the fluctuations of purely passive scal- ars (e.g. temperature and salinity) and potentially biologically active scalars in their Lagrangian environment is the characterization of Lagrangian pass- ive scalar intermittency. The main objective of the present work is thus to Bernard J. Geurts et al. (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 129–138. © 2007 Springer. Printed in the Netherlands. 130 Fran¸cois G. Schmitt and Laurent Seuront provide baseline information on passive scalar Lagrangian intermittency that could be compared to biological scalars (e.g. prey/mate abundance) in future studies. In the following we will consider scales belonging to the inertial range; i.e. larger than the Kolmogorov scale. For phytoplankters of smaller size, the influence of turbulence is still important, but limited to inertial range scales. In this section, we will recall the basic scaling properties for Lagrangian passive scalar turbulence. We consider the inertial convective subrange, associ- ated to large Peclet and Reynolds numbers, and hypothesize an homogeneous and isotropic turbulence, which is generally the case at small scales for 3D oceanic turbulence. In the Eulerian framework, velocity and passive scalar fluctuations in ho- mogeneous turbulence are classically characterized using Kolmogorov-Obu- khov-Corrsin (KOC) [1, 2, 3] scaling laws (see [4] for details). For passive scalar scaling exponents, let us mention the important result indicating that even in case of uncorrelated velocity field, the passive scalar field is multi- scaling (see [5] and [6] for a review). However such scaling exponents are quite far from experimental estimates, indicating that intermittency in velocity fluc- tuations has influence on temperature scaling exponents. This framework has been extended to the Lagrangian framework for velo- city fluctuations by Landau [7] and for passive scalar fluctuations by Inoue [8]. Let us note V (x 0 ,t)andΘ(x 0 ,t) the velocity and passive scalar concentration of an element of fluid at time t, initially at a position x(0) = x 0 . Hereafter these will be simply referred to as V (t)andΘ(t) since we assume statistical homogeneity. We note also for the Lagrangian velocity and passive scalar time increments ∆V τ = |V (t + τ) − V (t)| and ∆Θ τ = |Θ(t + τ) −Θ(t)|.Thisgives Landau’s relation for the velocity [7]: ∆V τ ∼ 1/2 τ 1/2 (1) and Inoue’s law for passive scalars [8] ∆Θ τ ∼ χ 1/2 τ 1/2 (2) where is the dissipation, χ = Γ θ |∇θ| 2 is the scalar variance dissipation rate and Γ θ is the scalar diffusivity of the fluid. The Eulerian power spectra are of the form E(k) ∼ k −5/3 for velocity and passive scalars (k is the wave number). In contrast, for Lagrangian fields, the power spectra are also scaling, with a different exponent: E(f) ∼ f −2 for both velocity and passive scalars (f is the frequency). These laws provide velocity and passive scalar fluctuations in time, assuming constant and homogeneous values for the fields and χ. In reality, one of the characteristic features of fully developed turbulence is the intermittent nature of the fluctuations of as- sociated fields, providing intermittent corrections for Eulerian and Lagrangian fields (see reviews in [4]). This is discussed in the next section. Lagrangian passive scalar intermittency in marine waters 131 2 Intermittent Lagrangian passive scalar multi-fractal relations: four predictions 2.1 Intermittent multi-fractal generalization Previous scaling relations describe only the mean behavior of passive scalar fluctuations. Fully developed turbulence is now known to be associated to in- termittency: fluctuations are random variables, whose scale dependence is usu- ally characterized using statistical moments of various order q>0. Following developments obtained in an Eulerian framework, Lagrangian intermittency has been characterized using scaling moments functions as: ∆Θ q τ ∼τ ξ θ (q) (3) where Θ is the passive scalar concentration, ∆Θ τ = Θ(t + τ) − Θ(t)isthe passive scalar increment, and ξ θ (q) is the Lagrangian passive scalar scaling moment function [9]. Without intermittency the latter is linear: ξ θ (q)=q/2. In case of intermittency ξ θ (q) is nonlinear and concave, and the non-intermittent value is valid only for q =2:ξ θ (2) = 1, indicating also that there is no intermittency correction for the power spectrum exponent. Fig. 1. The passive scalar Eulerian scaling exponent function ζ θ (q)estimatedby various authors, and with an average fit (see Table 1). It is interesting here to compare this Lagrangian scaling exponent ξ θ (q)to the more classical Eulerian ζ θ (q) defined by: (∆θ ) q ∼ ζ θ (q) (4) In the following we will also need another Eulerian quantity, which depends only on the passive scalar flux χ , and which is called “mixed moment func- tion” and is denoted here ζ m (q). This may be written in the following way 132 Fran¸cois G. Schmitt and Laurent Seuront (see [10, 11]): ∆θ 2 ∆U q/3 ∼ ζ m (q) (5) We will also need experimental or numerical estimates of the functions ζ θ (q) and ζ m (q). Several values have been provided in the literature, and an average estimate has been provided in [12]. These values are provided in Table 1, and the corresponding curves are displayed in Figure 1 and Figure 2. Table 1. Average values of ζ θ (q), estimated from several published estimates [10, 13, 11, 14, 15, 16, 17, 18] (Column 1). Average values of ζ m (q), estimated from several published estimates [11, 14, 19, 20, 18, 15, 21] (Column 2). qζ θ (q) ζ m (q) 0.5 0.21 1 .365 0.39 1.5 0.56 2 .65 0.72 2.5 0.87 3.85 1 3.5 1.12 4 .99 1.24 4.5 1.35 5 1.10 1.45 6 1.20 1.65 7 1.30 1.83 2.2 Four relations linking Eulerian and Lagrangian passive scalar scaling exponents We recently obtained four theoretical relations relating ζ θ (q)andξ θ (q)based on different sets of hypotheses [12]. All these relations verify ξ θ (2) = 1, and differ for other moments. We only provide here the four theoretical relations and refer the reader to Ref. [12] for the detailed description of how they have been derived. The first and simplest relation was obtained assuming a “characteristic time” relation for the de correlation of eddies, and a non-intermittent space- time relation: ξ Θ (q)= 3 2 ζ θ (q) Case I (6) The second choice was to assume an “ergodic” hypothesis corresponding to an equality of the statistics of the passive scalar flux in Eulerian and Lagrangian frame, and a non-intermittent space-time relation: ξ Θ (q)= 3 2 ζ m 3q 2 − q 4 Case II (7) [...]... dissipation in the light Gravity Currents: A Computational Investigation 145 0.5 z 0.25 0 -0 .25 -0 .5 0 4 8 12 16 x 20 24 28 32 20 24 28 32 20 24 28 32 (a) γ = 0.92 0.5 z 0.25 0 -0 .25 -0 .5 0 4 8 12 16 x (b) γ = 0.7 0.5 z 0.25 0 -0 .25 -0 .5 0 4 8 12 16 x (c) γ = 0.2 Fig 3 Concentration contours from a non-Boussinesq simulation for Re = 4, 000 at time t = 10 for different density ratios γ, clearly showing the... differentiation with respect to z 0.5 0.25 0 -0 .25 -0 .5 0 4 8 12 16 20 24 28 32 20 24 28 32 20 24 28 32 (a) t = 2.0 0.5 0.25 0 -0 .25 -0 .5 0 4 8 12 16 (b) t = 10.0 0.5 0.25 0 -0 .25 -0 .5 0 4 8 12 16 (c) t = 16.0 Fig 2 Concentration contours from a Boussinesq simulation for Re = 4, 000, at different times, showing the evolution of a typical Boussinesq gravity current 4 Boussinesq gravity currents The density... 0.955 9.55 0.57 0. 14 0.06 0.03 9.5 12 11.9 9 9 1.1 · 10−2 1.0 · 10−2 1.06 · 10−2 1.17 · 10−2 1 .4 · 10−2 w 2 (ms−1 ) 1. 04 · 10−2 8.06 · 10−3 4. 36 · 10−3 4. 24 · 10−3 6 .48 · 10−3 by the Froude number, which is defined as F r = urms /N Lz , and it gives the ratio of inertial forces to buoyancy forces The root-mean-square velocity is given by u2 = 2 Ekin and Lz is the vertical integral length-scale given by... 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M van Aartrijk and H.J.H Clercx 6 10 t 4 10 4 10 t 2 10 2 10 0 10 0 10 t2 t2 −2 −2 10 10 −1 10 0 10 1 10 2 10 3 10 −1 10 0 10 1 10 2 10 3 10 Fig 3 Horizontal (left) and vertical (right) single-particle dispersion as function of time t for five different cases N0: ——, N1: · · · · ·, N10: − · − · −, N100: - - - -, N1000: —— Horizontal dispersion is averaged over x- and ydirection, 1 (X(t) − X(0))2 + (Y... equation 5 Time integration is performed using the same third-order Adams-Bashforth technique as for the Eulerian velocity field Velocity and position time series of 163 84 particles are collected These particles are grouped in triangular pyramid structures, to be able to study both single-particle and particle-pair statistics The initial position of one-quarter of the particles is uniformly spread over the... 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