On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence

Transcription

1 On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence David G. Dritschel a) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England Norman J. Zabusky Department of Mechanical and Aerospace Engineering, Laboratory for Visiometrics and Modeling, Rutgers University, Piscataway, New Jersey Received 8 February 1995; accepted 5 February 1996 A powerful feature-tracking tool is applied to several high-resolution, very long-duration, regularized contour dynamics contour surgery simulations of unforced nearly-inviscid two-dimensional turbulence 2DT on the surface of a sphere. Particularly low density gases of vortices i.e. on average, very widely separated vortices are examined to ascertain the nature of their interactions. The simplest minimal model system is studied, namely a set of vortex patches of just two vorticity values, 0, whose total circulation is zero. The areas of the patches are selected initially from a pre-assigned, stable nearly time invariant power-law distribution. When the vorticity occupation fraction f 0.01, often more than three vortices are found relatively close together at the onset of a strong interaction. But, when f 0.01, all such interactions involve only three nearby vortex patches, not all having the same sign of vorticity. This is related to the well-known collapse of three singular point vortices. Thus, under these conditions, isolated two-vortex interactions, which have figured in recent ad hoc theories and models for decaying 2DT, cannot occur. Taking into account these results, we propose an asymptotically-motivated and computationally-efficient reduced model American Institute of Physics. S I. INTRODUCTION Turbulence is a highly nonlinear, chaotic dynamical process which is difficult to quantify and model. Any theory for it would be a major accomplishment. The two-dimensional case is, in a curious way, less developed than the threedimensional case. This is due, it appears, to the notorious strong non-locality of vortex interactions in two dimensions. Spectral theories have been, as a result, much less successful at predicting statistical properties in two dimensions than in three. The insights gained from recent pseudospectral PS 1,2 and from regularized contour dynamics contour surgery, CS 3 simulations have motivated a physical-space view of unforced incompressible two-dimensional turbulence 2DT. Though the mechanism of vortex emergence from a random sea is not theoretically understood, vortex emergence does appear to be a robust feature of 2DT. Hence, vortices are robust features, and it is tempting to continue this line of reasoning by assuming that the evolution, often referred to as the decay, of 2DT is governed by vortex collisions. Then, given generic and simple vortex collision rules, one may develop a particle-like, vortex theory for the evolution of 2DT. Up until recently, it was widely believed that the merging of two like-signed vortices into one constitutes the generic vortex collision process in 2DT. Support had been found in the PS simulations 1,2 and led two research teams 4,5 a Corresponding author: Telephone: ; fax: ; electronic mail: dgd@amtp.cam.ac.uk to independently develop a similar reduced model and theory for 2DT based on a simple parameterization of vortex merging. In these models, vortices are circular for all time, and their centers move by the equations of point-vortex dynamics. However, when two like-signed vortices come closer than about 3.3 times the average of their radii, they are merged instantaneously. The radius of the post-collision vortex is chosen by a formula that purportedly conserves energy but causes a substantial loss of circulation four times more than observed. 6 Model simulations gave results for the temporal scaling behavior of various vortex properties e.g. number, enstrophy, etc.... which agreed well with a simple scaling theory based on conservation of vorticity and total energy. 5 Also, it was claimed that the results agreed with hyperviscous-ps, direct numerical simulations of 2DT. 5 More recently, simulations exhibiting a much weaker enstrophy decay dissipating at a much smaller scale were conducted using a fast CS algorithm 7 and gave greatly different results. 3 The effect of surgery was checked by a fourfold variation in spatial resolution, and the results were found to be insensitive to this variation. Further, the results were obtained from an ensemble of ten simulations, at each resolution. It was concluded 3 that the PS/CS difference was due to an unrecognized, significant amount of dissipation that occurred during vortex collisions. 1,2,5 Can a simple theory or model be found for very weaklydissipative conditions? In the inviscid limit, the simplest vortex interactions are certainly between localized vortex patches regions of uniform vorticity, described entirely by 1252 Phys. Fluids 8 (5), May /96/8(5)/1252/5/$ American Institute of Physics

2 their bounding contours. The merging of spatially-varying vortices is less well understood due to its dependence on the initial vortex profiles 8 as well as on the magnitude and form of the dissipative terms. 9 Much of our understanding has come from studies of vortex patch interactions, like the fact that two identical, initially circular vortex patches merge if they are separated by less than 3.3 vortex radii, 10 that even slightly different vortices evolve in a qualitatively different way see also Refs. 11 and 12, which emphasize other aspects of asymmetric vortex interactions resulting in not one but two final vortices, 6 and that two opposite-signed vortices may break up into three! 13 How do we take these facts into account when constructing a simplified vortex-based model that functions at very weak dissipation? Obviously, it is imperative that we carefully scrutinize the kinds of interactions that actually take place in 2DT. It is not an easy task, since examining vortex interactions in massive data sets requires refined techniques of feature extraction and tracking. II. RESULTS Our results are unique in two respects: 1 We examine extremely dilute 2DT, in which vortex patches are widely separated on average, and for which collisions are rare and assume their simplest forms; and 2 we employ a powerful, automatic vortex-tracking tool that allows us to rapidly scan the dataset for the space-time regions of greatest interest. 14 The rarity of collisions requires a very accurate numerical simulation method, and here, like in Ref. 3, we employ a fast CS algorithm to examine 2DT on the surface of an isotropic non-rotating sphere. We have conducted four simulations. In the first, 50 positive and 50 negative identical circular vortex patches were placed at random on the spherical surface so that 10% of its area was covered by vorticity, and so that no two vortex patches were closer than 4 radii measured along the line joining their centers. The purpose of this simulation was to demonstrate that, through repeated inelastic interactions, the vortices are rapidly redistributed in sizes and assume a number density distribution very well described by n(a) A 1.8, where A is the vortex area see Fig. 1. This distribution was used to start three other simulations with 5%, 2%, and 1% vorticity coverage see Table I for details. Significant collisions were identified by choosing a threshold magnitude of area circulation change and searching the data set for vortices experiencing a greater area change during their lifetimes. The vortex area can change as a result of the topological reconnection of contours at very small scales permitted by the numerical algorithm. Reducing the threshold causes more collisions to be identified; in fact the number of collisions has been found to be inversely proportional to the threshold, to a high degree of accuracy. Thresholds were chosen to pick out about the ten most significant collisions in each data set. A change in the character of vortex interactions was observed to occur between 2% and 1% vorticity coverage, from principally four-body and five-body interactions at 2% to a more elementary three-body interaction, analogous to threevortex collapse, 15 at 1% as conjectured earlier in Ref. 6. To FIG. 1. The evolution of the number density distribution n(a) in a contoursurgery simulation beginning with N positive and 50 negative equal-sized vortex patches covering 10% of the spherical surface ( f 0 0.1). Here, A refers to vortex area/4. Initially, n(a) is a delta function centered on A 10 3 dashed line. The thin solid curve shows n(a) in the time interval 5,15 time in units of vortex rotation periods, while the thick solid curve shows n(a) in the time interval 80,88. Note that a power law dependence n(a) A 1.8 is established for moderate to small-scale vortices early on and remains nearly invariant. appreciate the diluteness implied by these figures, note that conventional, PS simulations of 2DT normally exhibit 5% vortex coverage at late times. 16 We now give the evidence. Eight collisions were identified in the simulation for 1% vorticity coverage; each of these collisions caused an area change exceeding 4% of the area of the largest vortex at the initial time. To view these collisions, a small part of the spherical surface is expanded by making a polar-stereographic projection centered on a chosen vortex see Table II. Four representative collisions are shown in Figs. 2 a 2 d. The images displayed are separated in time by 1 vortex rotation period. Various diagnostics are displayed for the chosen vortex, 14 including the best fit ellipse, 7 the displacement vector of the vortex center in one vortex rotation period, and the compressional light line segment and extensional axes orthogonal dark line segment of external strain TABLE I. Calculation parameters and global diagnostics. a f 0 N 0 c A max /A min A max /4 t F f F / f 0 L a The initial vorticity coverage f 0, vortex number N 0, dimensionless minimum chord distance between vortex centers c, and maximum to minimum vortex area ratio A max /A min are specified. All other quantities and parameters are derived from these except the numerical parameter, defined below. t F is the duration of each simulation in units of a vortex rotation period, T 4 / max, f F is the vorticity coverage at t t F, L A max /, L is the maximum separation of computational nodes along a contour, and 2 L/4 is the scale of surgery. All simulations use a fourth-order Runge-Kutta time-integration scheme with time step T/40. Phys. Fluids, Vol. 8, No. 5, May 1996 D. G. Dritschel and N. J. Zabusky 1253

3 TABLE II. Collisions at 1% vorticity coverage. a t c Vortex 10 4 A i /4 % Area change 10 4 A f /4 Net % change v t v a t c is the time when a vortex experiences an area change of the value noted, A i /4 is the vortex area just before t c and A f /4 is the vortex area after all area changes have been accounted for. Four of these eight collisions are illustrated in Figs. 2 a 2 d all eight collisions, in postscript, can be obtained from ftp.amtp.cam.ac.uk/ pub/contour-surgery/2dturb/dz96/v*. In each, a polar stereographic projection is made about the center of the vortex being tracked; v is the co-latitude of the circle which just fits in the square domains shown. The area of the domain shown is given by 16sin 1 2 v tan 1 (sin 1 2 v ), so that 0.96% of the surface is visible for v 10, 2.16% for v 15, and 3.80% for v 20. t v is the time of the first frame shown in Figs. 2 a 2 d, and each successive frame is separated by unit time. deduced from the rate of change of the mean vortex aspect ratio and orientation using Kida s formula 17. In Fig. 2 a, for vortex 55, there are three close vortices of comparable size at the onset of the collision. A fourth, larger vortex at the edge of the domain exerts strain but does not directly participate. The positive vortices completely merge, aided by the third, negative vortex, which shows little deformation. In Fig. 2 b, for vortex 13, a vortex dipole impinges on a larger positive vortex, and the interaction tears the smaller positive vortex into two parts, one of which is then strained into a filament. The other part is later caught and strained between the two counter-rotating vortices, further reducing its size. In Fig. 2 c, for vortex 4, we again have three close vortices of comparable size at the onset of the collision, with a significant fourth further afield. As in Fig. 2 b, one of the like-signed vortices is torn apart between the other two, though more rapidly here. Only a tiny piece of this vortex survives and is seen circling around an opposite-signed vortex moving to the bottom right part of the domain of view. In Fig. 2 d, for vortex 31, once again three close vortices are found at the onset of the collision, and once again a fourth can be seen not very far away. In this case, the smallest, negative vortex mainly acts to bring the positive vortices into contact, and the result is a partial merger. We believe that this rapid cascade process seen in Figs. 2 b and 2 c is generic. It arises from a rapid stretching of the smaller of the two like-signed vortices when it passes through the high strain field in the region between the counter-rotating dipolar vortices. This process is not unrelated to what occurs in the inelastic interaction of two isolated, like-signed vortices, 6 where four regimes of interaction were identified: complete straining out, partial straining out, complete merger and partial merger. All of these can be seen here. However, they are considerably modified by the presence of the opposite-signed vortex, which acts to accelerate the process, cascading vorticity structures to extremely fine scales in as little as a single vortex rotation period. III. A REDUCED MODEL This paper has demonstrated the existence of a generic three-body interaction in sufficiently dilute 2DT. One might be tempted to construct a vortex-based model using a parameterization of this interaction, along the lines of previous models for two vortices. 4,5,18 However, the parameter space is much too great to map out the collision characteristics quantitatively. But there is an alternative. From the data analysis, it has emerged that vortices are almost always close to an elliptical shape this can be readily seen in Figs. 2 a 2 d, where the dashed fitting ellipse is often indistin Phys. Fluids, Vol. 8, No. 5, May 1996 D. G. Dritschel and N. J. Zabusky

4 FIG. 2. Evolution sequences for some of the collisions listed in Table II; the vortices shown are a 55, b 13, c 4, and d 31. In each frame shown, positive vortices are rendered in solid lines and negative vortices in dotted lines with one dot per computational point to give an idea of the resolution employed. A fitting ellipse is displayed as a dashed line for the vortex being tracked, a bold arrow shows where the vortex is going in the next unit time, and the cross displays the magnitude and the orientation of the external strain, with extension along the bold axis and compression along the orthogonal thin axis. The magnitude of the strain would be half of the vorticity within the vortex if the axes were the same length as the edge of the domain. guishable from the vortex it fits, despite the fact that the vortex being fit is one of the most disturbed in the flow in the short time period shown. Clearly, any reduced model for 2DT should exploit this observation. However, the complexity of vortex collisions prevents one from retaining an elliptical shape at all times, as this would require parameterized collision rules. Thus, we insist that there is no recourse but to resolve the vortex collisions explicitly. Phys. Fluids, Vol. 8, No. 5, May 1996 D. G. Dritschel and N. J. Zabusky 1255

5 The fast CS algorithm employed already exploits this observation in part. Interactions between distantly-separated vortices are done rapidly using a fast multipole method, 7 involving only a few operations per vortex interaction. The minimum separation distance required and the number of moments retained in the multipole expansion are both determined by a prescribed level of accuracy, which additionally controls the level of surgery, the point density on contours, and the time step, for consistency. For dilute 2DT, it is often sufficient to retain only the first two non-zero moments, and for this truncation, distant vortices act like ellipses having the same moments. For close-range interactions, the numerical algorithm uses direct contour integration. This can be approximately one hundred times more costly than the moment expansion. Of course, the moment expansion cannot be used, so one has to pay the price. However, included among these close-range interactions are self interactions, and it has materialized that these self interactions dominate the total simulation cost when the vorticity coverage is small; this is because collisions in dilute turbulence are rare. A significantly faster algorithm can thus be constructed by exploiting the observation that vortices are often close to an elliptical shape, for the self-induced velocity of an elliptical patch is simply a linear function of the x and y Cartesian coordinates relative to its center a result known since Kirchhoff in the last century. The only modification necessary is to use the departure of each vortex from an elliptical shape, 14 already computed in the present CS algorithm, in deciding when it is sufficiently accurate to use Kirchhoff s formula. Then, the algorithm would compute as if it were at a vortex level, except during vortex collisions. This, we believe, is the simplest, cogent reduced model for 2DT, and we propose to call it the Elliptical-Contour EC model. It maximally simplifies self and distant vortex interactions where possible, while explicitly and accurately computing vortex collisions. Its speed depends on the accuracy prescribed, as accuracy controls the volume of space time in which expensive computations are required. The model is asymptotically motivated in that it converges to contour dynamics in the limit of infinite accuracy. While a theory as such may never be achieved, we have the next best thing. We have reduced the bewildering complexity of nearly-inviscid 2DT to its minimal computational representation. ACKNOWLEDGMENTS DGD is supported by the UK Natural Environment Research Council. Additional support was provided by the Isaac Newton Trust, by NATO, by the European Community CHRX-CT , and by the UK Engineering and Physical Sciences Research Council for computing time. The work was accomplished during the visit of NJZ to the Department of Applied Mathematics and Theoretical Physics and St. Catharine s College at Cambridge University in the fall of NJZ acknowledges support from ONR under Contract No. N and from the Hypercomputing and Design project supported by ARPA under Contract No. DABT C The content of the information herein does not necessarily reflect the position of the U.S. Government, and official endorsement should not be inferred. 1 J. C. McWilliams, The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech. 146, J. C. McWilliams, The vortices of two-dimensional turbulence, J. Fluid Mech. 219, D. G. Dritschel, Vortex properties of two-dimensional turbulence, Phys. Fluids A 5, R. Benzi, M. Briscolini, M. Colella, and P. Santangelo, A simple point vortex model for two-dimensional decaying turbulence, Phys. Fluids A 4, G. F. Carnevale, J. C. McWilliams, Y. Pomeau, J. B. Weiss, and W. R. 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