Exact relation between Eulerian and Lagrangian velocity increment statistics. O. Kamps,1 R. Friedrich,1 and R. Grauer2. 1Theoretische Physik, Universität ...
PHYSICAL REVIEW E 79, 066301 共2009兲
Exact relation between Eulerian and Lagrangian velocity increment statistics 1
O. Kamps,1 R. Friedrich,1 and R. Grauer2
Theoretische Physik, Universität Münster, 48149 Münster, Germany Theoretische Physik I, Ruhr-Universität Bochum, 44780 Bochum, Germany 共Received 10 June 2008; revised manuscript received 5 January 2009; published 3 June 2009兲 2
We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments. DOI: 10.1103/PhysRevE.79.066301
PACS number共s兲: 47.27.E⫺, 02.50.Fz, 47.10.ad
I. INTRODUCTION
ue = v共y + x,t兲 − v共y,t兲,
The relation between Eulerian and Lagrangian statistical quantities is a fundamental question in turbulence research. It is of crucial interest for the understanding and modeling of transport and mixing processes in a broad range of research fields spanning from cloud formation in atmospheric physics over the dispersion of microorganisms in oceans to research on combustion processes and the understanding of heat transport in fusion plasmas. The recent possibility to assess the statistics of Lagrangian velocity increments by experimental means 关1,2兴 has stimulated investigations of relations between the Eulerian and the Lagrangian two-point velocity statistics. Especially, the emergence of intermittency 共i.e., the anomalous scaling of the moments of the velocity increment distributions 关3兴兲 in both descriptions and its interrelationship is of great importance for our understanding of the spatiotemporal patterns underlying turbulence. A first attempt to relate Eulerian and Lagrangian statistics has been undertaken by Corrsin 关4兴, who investigated Eulerian and Lagrangian velocity correlation functions. Recently, the Corrsin approximation has been reconsidered in 关5兴, where it has become evident that it is a too crude approximation and cannot deal with the question of the connection between Eulerian and Lagrangian intermittencies. Further approaches to characterize Lagrangian velocity increment statistics are based on multifractal models 关6,7兴 which also have been extended to the dissipation range 关8兴. In 关7兴 a direct translation of the Eulerian multifractal statistics to the Lagrangian picture is presented. In this approach a nonintermittent Eulerian velocity field cannot lead to Lagrangian intermittency. This statement is in contrast with the experimental results of Rivera and Ecke 关9兴, as well as numerical calculations performed for two-dimensional turbulence 关10兴. Motivated by this fact we have derived an exact relation between the Eulerian and the Lagrangian velocity increment distributions, which allows us to study the emergence of Lagrangian intermittency from a statistical point of view.
where the velocity difference is measured at the time t between two points that are separated by the distance x, and the Lagrangian velocity increment
II. CONNECTING THE INCREMENT PROBABILITY DENSITY FUNCTIONS
The quantities of interest are the Eulerian velocity increments 1539-3755/2009/79共6兲/066301共5兲
ul = v„y + ˜x共y, ,t兲,t… − v共y,t − 兲.
共1兲
共2兲
In the latter case the velocity difference is measured between two points connected by the distance ˜x共y , , t兲 traveled by a tracer particle during the time interval . In both cases v is defined as the projection v · eˆ i of the velocity vector on one of the axes 共i = x , y , z in three dimensions and i = x , y in two dimensions兲 of the coordinate system 共see, e.g., 关11,12兴兲. In the case of an isotropic flow the results do not depend on the chosen axis; however, we do not have to make this assumption yet. Additionally, we define the velocity increment uel = v„y + ˜x共y, ,t兲,t… − v共y,t兲,
共3兲
which is a mixed Eulerian-Lagrangian quantity because the points are separated by ˜x but the velocities are measured at the same time. The properties of this quantity have been investigated in 关13兴. Finally, we introduce u p = v共y,t兲 − v共y,t − 兲,
共4兲
measuring the velocity difference over the time at the starting point of the tracer. Following 关14,15兴 we define the socalled fine-grained probability density function 共PDF兲 for ue as ˆf 共v ;x,y,t兲 = ␦共u − v 兲, e e e e
共5兲
where ue is the random variable and ve is the independent sample-space variable. The fine-grained PDF describes the elementary event of finding the value ve given the measured ue = v共y + x , t兲 − v共y , t兲. The relation to the PDF is determined by f e共ve ;x,y,t兲 = 具fˆ e共ve ;x,y,t兲典,
共6兲
where the angular brackets denote ensemble averaging. The quantity f e is a function with respect to the variables x , y , t and a PDF with respect to the variable ve. In analogy to Eq. 共5兲 we can define fine-grained PDFs for all other quantities. Now we want to derive an exact relation between the fine-
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KAMPS, FRIEDRICH, AND GRAUER
grained PDFs ˆf e and ˆf l. This task can be split up into two steps. First we have to replace the distance x with the trajectory ˜x共y , , t兲 of a tracer in order to translate from ue to uel. This is done by ˆf 共v ;y, ,t兲 = el e
冕
˜ 共y, ,t兲 − x…fˆ e共ve ;x,y,t兲. dx ␦„x
共7兲
We see that during this operation the sample-space variable is not affected. The subscript in ˆf el denotes the fact that we now have ve = uel instead of ve = ue. In the second step we have to connect ˆf l and ˆf el. From the definitions of increments 共2兲–共4兲 we see that ul = uel + u p and therefore the fine-grained PDF for ul is given by the fine-grained distribution of the sum of uel and u p. To that end we have to multiply ˆf 共v ; y , , t兲 with ˆf 共v ; y , , t兲 = ␦共u − v 兲 to get the fineel e el e p p grained joint probability of finding uel and u p at the same time. Subsequent application of 兰dve兰dv p ␦关vl − 共ve + v p兲兴 leads to ˆf 共v ;y, ,t兲 = l l
冕
dve ˆf p共vl − ve ;y, ,t兲fˆ el共ve ;y, ,t兲.
共8兲
To derive the corresponding PDFs we have to perform the ensemble average. In the case of Eq. 共8兲 we obtain f l共vl ;y, ,t兲 = =
冓冕
冕
dve ˆf p共vl − ve ;y, ,t兲fˆ el共ve ;y, ,t兲
冔
dve f p共vl − ve兩ve ;y, ,t兲f el共ve ;y, ,t兲. 共9兲
f l共 v l ; 兲 =
冕
d v e p b共 v l − v e兩 v e ; 兲
冕
⬁
dr pa共r兩ve ; 兲f e共ve ;r兲 .
0 f el共ve;兲
共11兲 For convenience, we included the integrated functional determinant 共2r in two and 4r2 in three dimensions兲 in pa. In this case pa共r 兩 ve ; 兲 is a measure for the turbulent transport and gives the probability of finding a tracer traveling the absolute distance r within the time interval . Multiplication by f e共ve ; r兲 and subsequent integration over the whole r range mix the Eulerian statistics from different length scales r weighted by pa to form the PDF f el共ve ; 兲 for a fixed time delay . The occurrence of the condition in pa shows that this weighing depends on ue. The transition probability pb incorporates the fact that during the motion of the tracer particle the velocity at the starting point changes by u p. Multiplying f el共ve ; 兲 with pb and integrating over ve sort all events where uel + u p = ul into the corresponding bin of the histogram for ul. We want to stress that Eqs. 共9兲 and 共10兲 and except from symmetry considerations also Eq. 共11兲 are of a purely statistical nature and, as a consequence, hold for quite different turbulent fields. They are valid for two-dimensional as well as three-dimensional incompressible turbulence but can also be applied to magnetohydrodynamic turbulence. The differences in the details of these turbulent systems, which are connected to the presence of different types of coherent structures like localized vortices in the case of incompressible fluid turbulence or sheetlike structures in magnetohydrodynamic turbulence, are therefore closely related to the functional form of pa and pb. III. TWO-DIMENSIONAL TURBULENCE
In the last line we used the general relation p共a , b兲 = p共a 兩 b兲p共b兲 valid for two random variables in order to extract f el from the average. We can treat Eq. 共7兲 in a similar manner. This leads us to f el共ve ;y, ,t兲 = =
冓冕
冕
˜ 共y, ,t兲 − x…fˆ e共ve ;x,y,t兲 dx ␦„x
冔
˜ 共y, ,t兲 − x…兩ve典fˆ e共ve ;x,y,t兲. dx具␦„x 共10兲
Inserting Eq. 共9兲 into Eq. 共10兲 shows that the Eulerian and the Lagrangian velocity increment PDFs are connected via ˜ 共y , , t兲 − x) 兩 ve典 and pb the transition probabilities pa = 具␦(x = f p共vl − ve 兩 ve ; y , , t兲 = f p共v p 兩 ve ; y , , t兲. Before we connect both equations we introduce some simplifications. In most experiments and numerical simulations dealing with Lagrangian statistics, the flow is assumed to be stationary and homogeneous. In this case we may average with respect to y and t and hence the dependence on this parameter in Eqs. 共9兲 and 共10兲 drops. Under the assumption of isotropy f el depends only on r = 兩x兩. Therefore, we can introduce spherical coordinates in Eq. 共10兲 and integrate with respect to the angles. As mentioned before, in the case of isotropy the statistical quantities do not depend on the chosen axis. Finally, we arrive at
In this section we want to estimate numerically the two transition PDFs in Eq. 共11兲 for the case of the inverse energy cascade of two-dimensional turbulence. The data are taken from a pseudospectral simulation of the inverse energy cascade in a periodic box with box length of 2 关10兴. Recapitulating the derivation of Eq. 共11兲 we see that we need the velocity at the start and the end points of a tracer trajectory at the same time to estimate the transition PDFs. Therefore we have to record the velocity at the starting points of the tracers additionally to their current position and their current velocity. In Fig. 1 the transition probabilities pa and pb are depicted for = 0.09TI, where TI denotes the Lagrangian integral time scale 关16兴. We have chosen this rather small time lag as an example because in this case the deviation of the Lagrangian increment PDF from a Gaussian is significant. The transition probability pa can be approximated by pa共r兩ve ; 兲 = N共ve, 兲r exp兵− 关r − m共ve, 兲兴2/2共ve, 兲其. 共12兲 For small ve we have m共ve , 兲 ⬃ ␣共兲兩ve兩 and 2共ve , 兲 ⬃ 共兲. From the functional form of pa one can see that the transport of the tracer particles is of probabilistic nature. For any fixed ve the tracers travel different distances during the same time . We also see a strong dependence on the condition ve which can be interpreted as the deterministic part of
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sponds to an idealized situation but it gives a hint at the cause of the negative correlations in pb. For larger ve the correlation decreases. This is captured by the fact that tanh关共兲ve兴 ⬃ const for large ve. For both transition PDFs we observe that for increasing the dependence on their conditions vanishes. Now we want to turn to the question of how the transition PDFs transform the Eulerian PDF f e共ve ; r兲 into the Lagrangian PDF f l共ul ; 兲. This process is depicted in Fig. 2. The left part of the figure shows several examples of f e共ve ; r兲. Applying 兰dr pa共r 兩 ve ; 兲 关see Eq. 共11兲兴 superposes different Eulerian PDFs f e共ve ; r兲 with different variances leading to the triangular shape in the semilogarithmic plot of the new PDF f el共ve ; 兲 共middle of Fig. 2兲. During the transition from f el共ve ; 兲 to f l共vl ; 兲 the variables v p and ve are added to form vl. The previously described observation that v p and ve tend to cancel each other for small ve leads to a stronger weighting of very small vl, so that the new PDF f l共vl ; 兲 is strongly peaked around zero 共right part of Fig. 2兲. In contrast to the center of the distribution the tails seem not to be influenced significantly by pb共v p 兩 ve ; 兲. This is in agreement with the fact that for large ve the correlation between ve and vl decreases.
FIG. 1. 共Color online兲 The plot shows pa共r 兩 ve ; 兲 共top兲 and pa共v p 兩 ve ; 兲 共bottom兲 for = 0.09TI.
IV. THREE-DIMENSIONAL TURBULENCE
To get an impression of the transition probabilities in three-dimensional turbulence we used the data provided by 关7,17兴 to calculate the PDF pa共r ; 兲 for different . The result is depicted in Fig. 3. We see that, as in the two-dimensional case, the Lagrangian time scale is related to the Eulerian length scales by a PDF. This well-known result shows that in principle it is not possible to connect them by a Kolmogorov-type relation such as ⬃ r / ␦ur 关7兴. In this relation is a typical eddy turnover time connected to eddies of length scale r. In our example we have chosen three different values of the time delay taken from the inertial range. Even when the maximum of pa is at a distance r, which lies in the Eulerian inertial range, there are significant contributions from very small and very large r. From this observation we can conclude that due to the turbulent transport the Lagrangian statistics is influenced by contributions from the Eulerian integral and dissipative length scales.
the turbulent transport. This distinguishes it from pure diffusion and directly shows that the widely used Corrsin approximation is violated. In the case of deterministic transport pa would be proportional to ␦(r − m共ve , 兲). A good approximation for pb is given by pb共v p兩ve ; 兲 = N共ve, 兲exp兵− 关u p − m共ve, 兲兴2/2共ve, 兲其 共13兲 and 2共 v e , 兲 = ␥ 共 兲 with m共ve , 兲 = ␣共兲tanh关共兲ve兴 ⫻关1 + ␦共兲兩ve兩兴. For small ve we see a strong negative correlation between ve and v p 关here tanh共ve兲 ⬃ ve兴. Both quantities tend to cancel in this case. This negative correlation between the sample-space variables in pb is connected with the sweeping effect. For a tracer starting with v1 traveling at the time without changing its velocity, we have uel = v1 − v2 when the velocity at the starting point changes during from v1 to v2. In this case we have u p = v2 − v1 = −uel. This corre1
1
pa
-1
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-2
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10
-4
10
II
0
pb
10
fel (ve , τ )
fe (ve ; r )
10
-1
10
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10
-3
10
-4
10
-5
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-3
-2
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0 ve
1
2
3
4
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III
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-2
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10-3 10-4
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fl (vl , τ )
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-1
0 ve
1
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0 vl
1
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FIG. 2. The figure shows the impact of the transition probabilities on the Eulerian PDF. The left part of the figure shows f e共ve ; r兲 for r = 0.06, 0.12, 0.3. These PDFs are transformed into f共ve ; 兲 共middle兲 by the transition PDF presented in the upper half of Fig. 1. Subsequently f共ve ; 兲 is converted into f l共vl ; 兲 共right picture兲 by the second transition PDF from Fig. 1. In both cases = 0.09TI. In plots II and III the points denote the reconstructed PDFs based on Eqs. 共9兲 and 共10兲, and the lines denote the PDFs directly computed from the Lagrangian data. 066301-3
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p (r; τ)
KAMPS, FRIEDRICH, AND GRAUER 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
10 8 6 4 2 0
Sl共n兲 =
0.5
⬁
dvl vnl f l共vl ; 兲
0
=
⬁
1 dvl vnl vl
0
0
0
冕 冕
0.4
1
1.5 r
0.8
2
1.2
2.5
3
Sl共n兲 =
FIG. 3. The figure shows pa共r ; 兲 for = 3.5 , 14 , 28 for three-dimensional turbulence. In the inset pa is depicted for the two-dimensional case with = 0.22TI , 0.44TI , 0.9TI.
The characterization of the relation between Eulerian and Lagrangian PDFs with the help of the conditional probabilities pa共r 兩 ve ; 兲 共Eulerian to semi-Lagrangian transition兲 and pb共v p 兩 ve ; 兲 共semi-Lagrangian to Lagrangian transition兲 allows one to recover a well-known multifractal approach for relating Eulerian and Lagrangian structure function exponents 关7兴. As a side product, a simpler formula for the Lagrangian structure function exponents of this multifractal approach will be obtained. Regarding the translation rule in 关7兴 ˜ 兩 ⬃ ve 共␦ur corresponds we would have a fixed relationship 兩x to ue in our notation兲 between the time lag and the distance traveled by a tracer particle during this time. This would correspond to choosing pa ⬃ ␦共r − ve兲 in Eq. 共11兲. The additional assumption 关7兴 ␦v ⬃ ␦ur 共ul ⬃ ue in our notation兲 that the velocity fluctuations on the time scale are proportional to the fluctuations on the length scale r could be incorporated in our framework by choosing pb ⬃ ␦共vl − ve兲 leading to d v e ␦ 共 v l − v e兲
冕
⬁
0
共14兲
The exponents for the Lagrangian structure function exponents can now be obtained by making use of the Mellin transform 1 f e共ve,r兲 = ve
冕
⬁
dvl
冕
dn
共15兲
−i⬁
with Se共n兲 = Ae共n兲re共n兲. Here, we use the same notation as in 关18兴. Please note that it will not be necessary to know the amplitudes Ae共n兲 but only the Eulerian scaling exponents e共n兲. Using Eq. 共14兲 and the Mellin transform we obtain f l共 v l ; 兲 =
1 vl
冕
i⬁
dn Ae共n兲e共n兲vle共n兲−n .
1 n v ␦vl l
冕
i⬁
dj⬘ Sl共j⬘兲共␦vl兲−j⬘
共18兲
−i⬁
A 共j兲
with Sl共j⬘兲 = 1−eje共j兲 e共j兲 and j⬘ = j − e共j兲, and we obtain for the exponents 共19兲
It remains to show that this relation 共19兲 is identical to the formulas derived by Biferale et al. 关7兴. To see this, we shortly repeat the multrifractal approach which starts with the Eulerian structure function exponents
e共p兲 = inf关ph + 3 − De共h兲兴 = phⴱe + 3 − De共hⴱe 兲 h
共20兲
and p = De⬘共hⴱe 兲. Here De共h兲 is the Eulerian singularity spectrum and hⴱe is the value where the infimum is achieved. The assumption r = ve appears now in the denominator of the expression of the Lagrangian structure function exponents
冉
l共p兲 = inf h
冊
ph + 3 − De共h兲 . 1−h
From this it follows
冉
l关p − e共p兲兴 = inf h
共21兲
冊
关p − phⴱe − 3 + De共hⴱe 兲兴h + 3 − De共h兲 . 1−h
In order to find the infimum we differentiate with respect to h, ! De⬘共hⴱe 兲 − De⬘共h兲 + De共hⴱe 兲 − De共h兲 − De⬘共hⴱe 兲hⴱe + De⬘共h兲h= 0. 共23兲 From this it follows that hLⴱ = hEⴱ and
L„p − e共p兲… = phⴱe + 3 − De共hⴱe 兲 = e共p兲,
i⬁
Se共n兲v−n e
共17兲
−i⬁
共22兲
dr ␦共r − ve兲f e共ve ;r兲
= f e共vl ; vl兲.
dj Ae共j兲e共j兲vle共j兲−j .
l关n − E共n兲兴 = e共n兲
V. RELATION TO THE MULTIFRACTAL APPROACH
冕
i⬁
Now we substitute j⬘共j兲 = j − e共j兲, dj⬘ = 关1 − je共j兲兴dj, and denote the inverse function by j = j共j⬘兲. Thus we have
0
f l共 v l ; 兲 =
冕
which recovers Eq. 共19兲. A consequence of Eqs. 共14兲 and 共19兲 is that for a self-similar Eulerian velocity field 共as we can find it in the two-dimensional inverse energy cascade兲 we should find self-similar Lagrangian PDFs. As mentioned above this is in contradiction to recent experiments 关9兴 and our own numerical simulations 关10兴. VI. CONCLUSION AND OUTLOOK
共16兲
−i⬁
This Lagrangian PDF is now inserted into the inverse Mellin transform to obtain the Lagrangian structure functions
共24兲
We presented a straightforward derivation of an exact relationship between Eulerian and Lagrangian velocity increment PDFs. For the example of two-dimensional forced turbulence we were able to explain how it is possible to observe
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strongly non-Gaussian intermittent distributions for the Lagrangian velocity increments. The two mechanisms in this context are the turbulent transport of the tracers leading to the mixing of statistics from different length scales and the velocity change at the starting point of the tracers leading to a further deformation of the increment PDF. In comparison we analyzed data from simulations of three-dimensional turbulence. Similar to the two-dimensional case we demonstrated that Lagrangian time and Eulerian length scales are connected via a transition PDF that varies with the time scale. We were also able to show that the well-known multifractal model for the Lagrangian structure functions is a limiting case of the presented translation rule. The next step is to estimate the transition probabilities for threedimensional turbulence as well as magnetohydrodynamic turbulence in order to get a deeper understanding of the in-
fluence of the underlying physical mechanisms, especially the presence of coherent structures on the translation process. In this context the question of why intermittency in the Lagrangian picture is stronger in magnetohydrodynamics than in fluid turbulence 关19兴, although the situation is reversed in the Eulerian picture, will be addressed.
We are grateful to H. Homann, M. Wilczek, and D. Kleinhans for fruitful discussions and we acknowledge support from the Deutsche Forschungsgesellschaft 共Grants No. FR 1003/8-1 and No. FG 1048兲. We also thank the supercomputing center Cineca 共Bologna, Italy兲 for providing and hosting the data for 3D turbulence.
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ACKNOWLEDGMENTS
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