Hyperbolic Covariant Coherent Structures in two dimensional flows
HHyperbolic Covariant Coherent Structures in two dimensional flows
Giovanni Conti a) and Gualtiero Badin b) Institute of Oceanography, Center for Earth System Researchand Sustainability (CEN), University of Hamburg, Hamburg,Germany (Dated: September 11, 2018)
A new method to describe hyperbolic patterns in two dimensional flows is proposed.The method is based on the Covariant Lyapunov Vectors (CLVs), which have theproperties to be covariant with the dynamics, and thus being mapped by the tan-gent linear operator into another CLVs basis, they are norm independent, invariantunder time reversal and can be not orthonormal. CLVs can thus give a more de-tailed information on the expansion and contraction directions of the flow than theLyapunov Vector bases, that are instead always orthogonal. We suggest a definitionof Hyperbolic Covariant Coherent Structures (HCCSs), that can be defined on thescalar field representing the angle between the CLVs. HCCSs can be defined for everytime instant and could be useful to understand the long term behaviour of particletracers. We consider three examples: a simple autonomous Hamiltonian system, aswell as the non-autonomous “double gyre” and Bickley jet, to see how well the angleis able to describe particular patterns and barriers. We compare the results fromthe HCCSs with other coherent patterns defined on finite time by the Finite TimeLyapunov Exponents (FTLEs), to see how the behaviour of these structures changeasymptotically.PACS numbers: 05.45.-a, 05.90.+m, 92.10.Ty, 92.10.Lq, 92.10.Lf, 47.20.De, 47.27.edKeywords: Covariant Lyapunov Vectors, Dynamical Systems, Mixing, Hyperbolicity,Ergodic Theory a) Electronic mail: [email protected] b) Electronic mail: [email protected] a r X i v : . [ n li n . C D ] S e p . INTRODUCTION In the paradigm of chaotic advection, the trajectories of passive tracers can be complexeven when the velocity field of the flow is simple. This is the case, for example, for timedependent two-dimensional flows or even steady three-dimensional flows, like the celebratedABC flow . However, even flows with complicated time dependent structure allow for theformation of coherent patterns that influence the evolution of tracers. These structures arecommon in nature, appearing both at short and long time scales as well as small and largespatial scales. Remarkable examples of these structures are eddies and jets in the ocean andatmosphere, the Gulf Stream current, and ring clouds . These patterns can influence, forexample, the evolution of nutrients as well as oil spills and other pollutants. Furthermore,these coherent structures could act as local inhibitor for the energy transfer between scales .They also appear in other planets and other astrophysical systems, such as the jets on thesurface of Jupiter, Saturn and other gaseous planets, and in the solar photospheric flows .These structures, often referred as Lagrangian Coherent Structures (LCSs), can shed lighton the mixing and transport properties of a particular system on a finite time interval. Theterm “Lagrangian” in their name is motivated by the fact that they evolve as material lineswith the flow . A particular kind of LCS is called Hyperbolic LCS (HLCS), and can be seenas the locally most attracting or repelling material lines that characterize the dynamicalsystem over a finite time interval. The term hyperbolic is just an analogy with the stabilityof fixed points in dynamical systems, since usually one wants to study non-autonomoussystems for which entities such as fixed points or stable and unstable manifolds are notdefined. Moreover these systems are studied for finite time.Early attempts to detect HLCSs were based on Finite Time Lyapunov Exponents(FTLEs), that measure the rate at which initial conditions (or, equivalently, tracer par-ticles) separate locally after a given interval of time . One of the first rigorous definitionsof the LCSs was based on ridges of the FTLEs . FTLEs and the so defined LCSs wereused to study for example Lagrangian dynamics of atmospheric jets and oceanic stirringby mesoscale eddies , describing for example the chaotic advection emerging by mixedlayer instabilities and its sensitivity to the vertical shear . A similar method to detectcoherent structures uses the Finite Size Lyapunov Exponents (FSLEs), that represents the2eparation rate of particles given a specific final distance . However, several counterexamples are available in which both FTLEs and FSLEs ridges fail in characterizing theLCSs . Although the FTLEs field remain a popular diagnostics of chaotic stirring, othermethods now are available to detect LCSs, which include for example the so called La-grangian descriptors , which are based on integration along trajectories, for a finite time,of an intrinsic bounded positive geometrical and/or physical property of the trajectoriesthemselves. Notice however that the method of Lagrangian descriptors is not objective ;the connection between the Perron-Frobenius operator and almost invariant coherent setsof non-autonomous dynamical systems defined over infinite times , the use of braids , andthe extrema of trajectories . Other Fast Indicators (FIs) besides FTLEs and FSLEs, i.e.computational diagnostics that characterize chaos quickly and can be used to determinecoherent structures, are the Smaller (SALI) and Generalized (GALI) Alignment Indices ,the Mean Exponential Growth rate of Nearby Orbits (MEGNO) , and the Finite TimeRotation Number (FTRN) .Particularly promising for the detection of coherent structures is a variational theorythat considers the extremum properties of a specific repulsion rate function . Usingthis theory it has been further shown that HLCSs can be described using geodesics . In thisway, HLCSs can be represented as minimizers of a material length, with specific boundaryconditions for the variation function. Another geodesic theory describes HLCSs in terms ofshearless transport barrier that minimize the average shear functional . These theoriesare based on the computation of shrink/stretch-lines (tensorlines), thus trajectories alongthe eigenvectors of the deformation tensor also called Cauchy Green Tensor (CGT). Geodesictheories are also able to detect other two kinds of LCSs called Parabolic and Elliptic, whichare however of no interest for the present work.All these methods such as FTLEs, FSLEs, or the variational and geodesic theories, aimto find particular structures on the flow, among these the most repelling, or attracting,structures in the flow on a finite time interval, the HLCS. Particle tracers around thesemost influential material structures in finite time are maximally repelled or attracted. How-ever, changing the finite time interval under study also changes the dynamical system andthe correspondent structures emerging from the flow. Recent effort has been done in theunderstanding the instantaneous most influential coherent structures, Objective EulerianCoherent Structures (OECSs), using a method that is not based on a finite time interval3f evolution . The tracer particles could be maximally repelled for a short time, buton long time they could have a different behaviour. Is it natural to wonder what happenasymptotically to the tracer particles? Is it possible to find some coherent structures thatsuggest the asymptotic behaviour of the tracer particles?In this work we thus propose an alternative method to detect coherent patterns emergingin chaotic advection, which is based on the Covariant Lyapunov Vectors (CLVs). CLVswere first introduced by Oseledec and Ruelle , but for a long time they have receivedvery little attention due to the lack of an efficient algorithm to compute them. Only inthe last decade the computation of such vectors has become possible , and CLVs havebeen used to investigate e.g. the motion of rigid disk systems , convection , and otheratmospheric phenomena . For other theoretical discussions or for reviews on CLVs, seee.g. Refs. . Unlike the Lyapunov Exponents (LEs), which are time independent, thecorrespondent vectors, also known as forward and backward Lyapunov Vectors (LVs), dodepend on time. The LVs are orthonormal and their direction can thus give only limitedinformation about the local structure of the attractor. LVs also depend on the chosen norm,they are not invariant under time inversion and they are not covariant, where covariance ishere defined as the property of the forward (backward) LVs to be mapped by the tangentdynamics in forward (backward) LVs at the image point . Differently from the LVs, CLVsare norm independent, invariant for temporal inversion and covariant with the dynamics,making them thus mapped by the tangent linear operator into another CLVs bases duringthe evolution of the system. They are not orthonormal and their directions can thus probebetter the tangent structure of the system.All this intrinsic information can be summarized using the angle between the CLVs, ascalar field that allows to investigate the spatial structures of the system. The need topay more attention to the directions between LVs, backward and forward, has also beensuggested for the study of turbulence and for the definition of a diagnostic quantity forthe study of mixing, the Lyapunov’s diffusion . Using three simple examples, we showthat the attracting and repelling barriers tend to align along the paths on which the CLVsare orthogonal. The directions of the CLVs along these maxima provide thus informationon the attracting or repelling nature of the barriers and can be related to the geometryof the system. Using the CLVs we can define structures, at a given time instant, that areasymptotically the most attractive or repelling. Furthermore, since these structures can be4efined for every time instant, it is possible to follow the formation of coherent structuresduring the evolution of the flow.In section II we discuss the theory behind the CLVs, and suggest a definition for coherentstructures that give asymptotic information. The strategy will be to make use of the scalarquantity defined by the angle between the CLVs to locally identify the structures thatasymptotically are maximally attractive or repulsive. In section III we use the CLVs toidentify particular patterns in three different systems and we compare the results withFTLEs fields. Finally, in section IV we summarize the conclusions. II. COVARIANT LYAPUNOV VECTORS
In this section we summarize the theory behind CLVs for two dimensional flows. For amore general and detailed review see for example Ref. . Let the open set D ⊂ R be thedomain of the flow, t ∈ R the time and v ( x , t ) a vector velocity field in D . The dynamicalsystem that describes the motion of a tracer advected by the flow is thus d x ( x , t ; t ) dt = v ( x ( x , t ; t ) , t ) , (1a) x ( x , t ; t ) = x , (1b)where x ( x , t , t ) ∈ D is the trajectory of the tracer starting at the point x at time t . To(1) is associated the flow map φ tt ( x ) φ tt : D → D, x → x ( x , t , t ) , (2)that maps the initial position x at time t to the position x ( x , t , t ) at time t . It should benoted that the dependence on the initial condition is very important here, since the vectorswill be considered as function of time and of the initial positions. In the following, thecontracted form x = x ( x , t , t ) will be used.At each point x ∈ D we can identify the tangent space T x D ⊂ R . Infinitesimal per-turbations, u ( t ) ∈ T x D , to a trajectory of this system can be described by the linearizedsystem d u ( t ) dt = J ( t ) u ( t ) , (3a) u ( t ) = u , (3b)5here J ( t ) ∈ R × is the Jacobian matrix composed by the derivatives of the vector field v ( x , t ) with respect to the component of the vector x . Using the fundamental matrix M ( t ) ∈ R × , of (3), that satisfies d M ( t ) dt = J ( t ) M ( t ) , (4a) M (0) = I (4b)we define the so called tangent linear propagator F ( t , t ) = M ( t ) M ( t ) − . (5) F ( t , t ) maps a vector in x at time t into a vector in x at time t along the same trajectoryof the starting system (1), that is u ( t ) = F ( t , t ) u ( t ) . (6)According to (5), the propagator is always nonsingular. In terms of the flow map, thetangent linear propagator is F ( t , t ) = ∇ φ tt . (7)Exploiting Oseledec’s Theorem , it is possible to characterize the system using quan-tities that are independent on t or t . By virtue of this theorem the far-future operator O + ( t ) = lim t → + ∞ (cid:0) F ( t , t ) (cid:62) F ( t , t ) (cid:1) / t − t ) (8)and the far-past operator O − ( t ) = lim t →−∞ (cid:0) F ( t , t ) −(cid:62) F ( t , t ) − (cid:1) / t − t ) (9)are well defined quantities. Note that the product F (cid:62) ( t , t ) F ( t , t ) determines the Euclideannorm of the tangent vectors in the forward-time dynamics (a similar role it is played by F −(cid:62) ( t , t ) F − ( t , t ) for the backward-time dynamics), in fact, || u ( t ) || = (cid:2) u ( t ) (cid:62) (cid:0) F (cid:62) ( t , t ) F ( t , t ) (cid:1) u ( t ) (cid:3) / . (10)Operators (8) and (9) probe respectively the future and past dynamics of a certain point,and share the same eigenvalues λ ≥ λ , (11)6hat, assuming ergodicity, are independent on time and space. Each eigenvalue has multi-plicity m i ( m + m = 2). Their logarithms correspond to the LEs of the dynamical system(1). If the limits in (8) and (9) are not considered, the resulting eigenvalues are time andspace dependent and are called FTLEs.The two operators (8) and (9) can be evaluated at the same point in space at a given time t . The correspondent eigenvectors, { l +1 ( t ) , l +2 ( t ) } , { l − ( t ) , l − ( t ) } will define thus the forwardand backward Lyapunov basis computed at the same time. Conversely to the respectiveeigenvalues those bases are time dependent, depend on the chosen scalar product and arenot invariant under time reversal. Furthermore, these vectors are always orthogonal andgive thus limited information on the spatial structure of the configuration space.To overcome these issues, one can build particular spaces, the backward and forwardOseledec subspaces, defined as L − i ( t ) = span { l − j ( t ) | j = 1 , i } , i = 1 , , L − ( t ) = ∅ , (12)and L + i ( t ) = span { l + j ( t ) | j = i, } , i = 1 , , L +0 ( t ) = ∅ (13)In the forward dynamics, the generic vector l + i ( t ) grows or decays exponentially with anaverage rate λ i . If the system is evolved forward in time, by means of the tangent linearpropagator the evolution of the vector l +1 ( t ) will have a non-zero projection inside the spacegenerated by l +1 ( t (cid:48) ), but it will also have a non-zero projection onto the space generated by l +2 ( t (cid:48) ). On the other hand, l +2 ( t ) will be transported onto the space generated by l +2 ( t (cid:48) ) andwill have zero projection onto the space generated by l +1 ( t (cid:48) ). Repeating similar argumentsfor the backward Lyapunov Vectors leads to the observation that L − i and L + i are covariantsubspaces, L − i ( t (cid:48) ) = F ( t, t (cid:48) ) L − i ( t ) , (14a) L + i ( t (cid:48) ) = F ( t, t (cid:48) ) L + i ( t ) . (14b)Vectors that are covariant with the dynamics and invariant with respect to time reversalwill be found at the intersection of the Oseledec subspaces, i.e. at w i ( t ) = L − i ( t ) ∩ L + i ( t ) . (15)7hese spaces, often referred as Oseledec splitting , are not empty , and their vectors,also called covariant Lyapunov vectors (CLVs), are covariant with the dynamics, i.e. F ( t, t (cid:48) ) w i ( t ) = w i ( t (cid:48) ) . (16)It should be noted that CLVs posses thus the properties of a semi-group. These vectors havean asymptotic grow or decay with an average rate λ i , so that their asymptotic behaviourcan be summarized as (cid:107) F ( t, t ± τ ) w i ( t ) (cid:107) ≈ e ± λ i τ , (17)where τ = | t (cid:48) − t | , which shows their invariance under time reversal. Note also that thesevectors do not depend on any particular norm.Equations (12), (13) and (15) imply a simple relation between CLVs and LVs in a twodimensional system, w ≡ l − and w ≡ l +2 , (18)the first CLV corresponds to the first backward Lyapunov vector, and the second CLV withthe second forward Lyapunov vector.Furthermore, if the Jacobian matrix appearing in (3) is constant, the CLVs are not justcovariant, but invariant with the dynamics. In this particular case in fact they correspondto the eigenvectors of the Jacobian and the eigenvalues of this matrix coincide with the LEs.Since the CLVs highlight particular expansion and contraction directions at each point ofthe coordinate space, and these directions are not necessarily orthogonal, they can be usedto understand the geometrical structure of the tangent space. This geometric informationcan be summarized by the scalar field of the angle θ ( t ) between the CLVs. Because the CLVsidentify asymptotically the expansion and contraction directions sets of the tangent spaceassociated at every point of the domain, the θ field represents a measure of the hyperbolicityof the system, that is a measure of the orthogonality between these two directions. Notethat the orientation of the CLVs is defined as arbitrary. The angle between the CLVs, θ ( t ) ∈ (cid:104) , π (cid:105) , (19)is thus θ ( t ) = cos − ( | w ( t ) · w ( t ) | ) , (20)where { w ( t ) , w ( t ) } are the first and the second CLVs . It is interesting to point outthat when θ ( t ) = π/
2, the CLVs reduce to LVs in two dimensions, and the backward and8orward Lyapunov basis coincide. If the computation of the angle is done at every point ofthe domain, and if we consider the system (1) in which the initial conditions are varied insuch a way that all the domain is spanned, we can build a field of the orthogonality betweenthe expansion and contraction directions of the system.In the following section we will show how CLVs, with their capability of probing thegeometric structure of the tangent space, can be used to determine coherent structures thatgive asymptotic information on the tracer and then, how the CLVs highlights the mixingtemplate of the flow.
A. Hyperbolic Covariant Coherent Structures
Several frameworks are available to study passive scalar mixing. Although all these meth-ods aim to describe the mechanism underlying the chaotic advection, they have significantlydifferent approaches. However, most of them share a fundamental feature called objectivity.Objectivity is a fundamental requirement to define structures emerging from a flow. In prac-tical applications, objectivity can be used e.g. to move the reference frame into the referenceframe of a coherent structure. In particular a structure can be considered objective if it isinvariant under a coordinate changes of the form ˆx ( t ) = Q ( t ) x ( t ) + P ( t ) , (21)where Q denotes a time-dependent orthogonal matrix and P a time-dependent translation .The FTLEs, FSLEs and geodesic theory define objective quantities. Sala et all. haveshown that, generally, also for a linear transformation of coordinates the new angle non-linearly depends on the angle of the old reference system. However, for the particular classof transformation we are interested in, (21), the angle θ is an invariant. This means thatstructures highlighted by θ are objective.To study the asimptotically most attractive or repelling behaviour of tracer particles neara particular structure identified at a given instant of time, one can consider the lines r ( s, t ),with s representing the length parameter, defined by r (cid:48) ( s, t ) = w ( r ( s, t ) , t ) , (22a) r (cid:48) ( s, t ) = w ( r ( s, t ) , t ) , (22b)9here the primes indicates the derivative with respect to s , and characterized by θ = π/ θ ( t ) = π ⇒ w (cid:107) l and w (cid:107) l , (23)where, l and l are Lyapunov vectors. In these circumstances the backward and the forwardbasis are coincident. A sphere of initial conditions around a point of the path will bedeformed in an ellipsoid whose axis are aligned with these Lyapunov vectors. If (22) arealigned along a ridge of the hyperbolicity field θ , characterized by θ = π/
2, then they arealso pointwise the most attractive or repelling lines in terms of the asymptotic behaviour oftracer particles. Consider in fact line (22a) and its tangent CLV w as depicted in Figure 1. Figure 1. Orientation of CLVs on a most attracting material line (upper line) and on a neighbouringmaterial line (lower line). The distance (cid:107) P − P (cid:107) decreases faster than (cid:107) P − P (cid:107) since in the secondcurve the normal vector connecting the points P and P is expressed as a linear combination ofboth the CLVs. Along this curve, for any point P we can chose a point P which is on the normal vectorto the curve in P , n . Because of (11), the distance between these two points decreases as (cid:107) P − P (cid:107) ∝ (cid:107) w (cid:107) ≈ e λ ( t (cid:48) − t ) . (24)10onsider now a point P on a nearby curve, and a point P laying on the normal vector tothe curve in P . The normal vector n at P can be written as a linear combination of w and w and, considering again (11), one has thus (cid:107) P − P (cid:107) = (cid:107) α w + β w (cid:107) ≈ e λ ( t (cid:48) − t ) > e λ ( t (cid:48) − t ) . (25)The previous computation can be repeated analogously for w . The same reasoning holdsfor w .It must be remarked that (22b) gives asymptotic information, but is defined for a partic-ular instant of time. For every time instant the lines (22), together with the orthogonalitycondition θ ( t ) = π/
2, describe the so called tensorlines, and can be seen as the asymptoticversion of shearless barriers .Taking into account these properties of the CLVs, we propose a definition to describethese asymptotic coherent patterns: Definition II.1 (Hyperbolic Covariant Coherent Structures (HCCSs)) . At each time t , a Hyperbolic Covariant Coherent Structure , is an isoline of the hyperbolicity scalar field θ at the level θ = π/
2. Its attractive or repelling nature is determined by the CLVs alignedwith it.It is important to emphasize the difference between the information provided by FTLEsand the one provided by CLVs. FTLEs are the finite time version of Lyapunov’s exponentsand, as shown in the appendix, are calculated as a mean time of the logarithms of theseparation of two trajectories that start from near points. The FTLEs therefore give anaverage information over the time interval considered. Moreover, if the time interval islong enough, and the analyzed region can be considered ergodic, the dependence on initialconditions is lost and therefore the possibility of displaying possible structures. CLVs, on thecontrary, as well as LVs, depend on a time instant and do not converge to the same value.CLVs therefore allow the definition of instant structures, which, however, give asymptoticinformation.This approach presents some numerical challenges, CLVs are not fast to be computed asthe FTLEs, and numerically, in order to identify the isolines, it is necessary to consider aninterval of values for the angle. The HCCSs are thus computed here considering isoregionof θ ∈ [ π − δ, π ], where δ is 0 .
087 rads. 11
II. NUMERICAL EXAMPLES
The algorithm used in this work is the same presented in Ref. , so only a schematic ofits structure will be presented here. This algorithm can compute a large number of CLVsconverging exponentially fast when invertible dynamics is considered. The basic idea isthat, if the backward Lyapunov vectors basis { l − ( t ) , l − ( t ) } is evolved by means the theoperator F , it is always possible to keep it orthonormalized with a QR decomposition andstore the corresponding upper triangular projection matrix. It is then possible to exploitthe informations contained in these upper triangular matrices to evolve backward in timearbitrary vectors that will converge to the CLVs . This is done in five different phases,which are described in Appendix A.In the next we investigate the hyperbolicity field θ for three different examples, whichinclude one autonomous Hamiltonian flow and two non-autonomous two dimensional flows.The HCCSs emerging from the angle between the CLVs are compared with the FTLEsfield, while their attractive or repelling nature will be discussed in terms of the CLVs. Thealgorithm for the search of the CLVs and the computation of the FTLEs explore the samedynamics, because it is used the same time interval. However, the final CLVs are found justfor a temporal subinterval of the whole time used, see Appendix A and Appendix B.Technical details about the three different examples are included in the Appendix B fora clearer description of the physical results in the following. A. A simple autonomous Hamiltonian system
In this preliminary example we investigate the HCCSs in an autonomous Hamiltonianflow map. The time-independent Hamiltonian, corresponding to the streamfunction of theflow, is H = x y , (26)so that the dynamical equations of motion are dxdt = − x , (27a) dydt = xy, (27b)12ith x ( t ) = x , and y ( t ) = y . Integration of Eq. (27) yields the trajectories of thesystem, shown in Figure 2a, with flow map φ tt ( x , y ) = x ( t ) = − x x t y ( t ) = y (cid:0) x t (cid:1) . (28)13 igure 2. Panel (a): trajectories of the Hamiltonian system (28) (solid line). The trajectory y = 0represents a repulsive barrier for the system. Notice that the arrows indicate the flow velocity field,which is expressed on the right hand side of (27), and not the CLVs that are shown separately.Panels (b) and (c): streamlines for the double gyre and the Bickley jet, respectively. x > y = 0 axis clearly divides the dynamics of the systeminto two different regions. There is no flux across this line, and the positive and negative y regions are completely separated at each time. Furthermore, y = 0 is the only materialrepelling line. Notice that in the Figure, the arrows do not indicate the CLVs but are justindicators of the magnitude of the flow. The trajectory y = 0 behaves thus exactly as amaterial barrier dividing the system in two different regions, remains coherent at all time,and repels every close trajectories. We apply the CLVs theory to see if the angle betweenthat vectors is able to detect such kind of structure.15 igure 3. FTLEs and CLVs fields for the Hamiltonian system (27). (a) Maximum FTLEs fieldcomputed from (A1); (b) θ field; (c) zoom of the domain to show the CLVs fields. Arrows are notused due to the arbitrariness of the orientation of the CLVs. The red lines are associated to w ,the expansion direction, while the blue lines are associated to with w , the contraction direction. t = [5 , y = 0 barrier. Notice that if a bigger domain in the y direction would have been selected,the FTLEs field would have showed a large scale modulation with larger values aligned along y = 0. Figure 3b, shows the distribution of the CLVs angles at t = 5. It is important tostress the fact that the FTLEs depend on a finite time interval while CLVs, and then θ , ona particular instant of the time. The hyperbolicity field clearly shows values close to π/ y = 0 axis. The direction of the first (red) and second (blue) CLV is shown for anenlargement of a region close to this structure in Figure 3c. It is visible that the second CLV,that characterize the contraction direction, aside for numerical fluctuations, are aligned withthe θ = π/ x = 0, the velocity field is close to zero, sothat the barrier there is not well defined. This suggests that the dynamics near the originevolves with a different time scale.Along the x axis, θ = π/ F ( t, t (cid:48) ) = /B Bt (cid:48) y B , (29)where B = 1 + t (cid:48) x / . (30)Along y = 0, both the tangent linear propagator F ( t, t (cid:48) ) = /B B . (31)and the CGT (A2) C ( t, t (cid:48) ) = /B B , y = 0 , (32)are diagonal. If at time t one has θ ( t ) = π/ y = 0, then w ( t ) = ξ = and w ( t ) = ξ = , (33)17here ξ and ξ are the eigenvectors of the CGT, and at time t (cid:48) > t cos( θ ( t (cid:48) )) ∝ F ( t, t (cid:48) ) w ( t ) · F ( t, t (cid:48) ) w ( t )= w ( t ) · C ( t, t (cid:48) ) w ( t )= 0 . (34)This shows that y = 0 is a repelling HCCSs. B. Double gyre
The previous analysis is now applied to a time dependent system called double gyreflow , which is a simplification of observed geophysical flows consisting of two counterrotating vortices that expand and contract periodically. The tracers moving in this flowsatisfy the following dynamical equations dxdt = − πA sin( πf ( x, t )) cos( πy ) , (35a) dydt = πA cos( πf ( x, t )) sin( πy ) df ( x, t ) dx , (35b) f ( x, t ) = (cid:15) sin( ωt ) x + (1 − (cid:15) sin( ωt )) x, (35c)with initial condition x (0) = x , y (0) = y . The streamline of the flow at t = 5 are shownin Figure 2b.We consider two numerical experiments, DV1 and
DV2 , in which the CLVs are computedin two different time intervals with a non-null intersection. During
DV1 we compute theCLVs in the interval t = [5 ,
10] and during
DV2 for t = [10 , θ is a quantity that depends on a particular timeinstant and not on the time interval considered for its computation, as long as the intervalconsidered is sufficiently long. Figure 4 shows a comparison between the angle θ and theFTLEs for this system. The first column is relative to DV1 , while the second column isrelative to
DV2 . 18 igure 4. FTLEs, CLVs and joint probability plot for the experiments DV1 (left panels) and DV2(right panels) of the double gyre system. The first column is referred to the experiment DV1 andthe second column to DV2. Panels (a) and (b) show the maximum FTLE fields computed from(A1). Panels (c) and (d) show the angle between the CLVs for the two experiments. Panels (e)and (f) show the joint probability between FTLEs and angles shown in the previous panels.
The λ fields, the maximum FTLEs, are shown in Figures 4a and 4b for DV1 and
DV2 respectively. The regions characterized by white color correspond to maximum exponentialgrowth rate. Ridges of these regions correspond to the LCSs of the system according to19he definition by Shadden et al . It is interesting to notice how, although in Figure 4bthe ridges of the FTLEs are more developed, the λ field converges to the same structures,and the maximum values do not change. The FTLEs give an overall information of thestretching and folding over the whole time interval considered, they can not say anythingabout the state of the coherent structures at a particular instant of time. Figures 4c and 4dshow the hyperbolicity field θ computed for a particular time instant. The white colour hererepresents the maximum values of the hyperbolicity of the system, which means θ = π/ θ fields exhibitsimilar structures it is interesting to point out that the ridges of the FTLEs do not necessarilycorrespond to the regions of maximum hyperbolicity. Near the central regions of the twovortices, the hyperbolicity field has a noisy appearance, due the fact already discussed thatin there the expansion and contraction directions are not well defined, and the CLVs becometangent to each other. Notice that this effect does not influence the detection of hyperbolicregions. Figures 4e and 4f highlight the differences between FTLEs and the θ fields shownrespectively for the two experiments DV1 and
DV2 . As already mentioned, high values ofthe FTLEs do not necessarily correspond to high values of the θ field. The joint probabilityplots underline the fact that in general there is not a one to one relation between the θ and FTLE fields. In particular, small values of FTLEs are related to the broadest range ofpossible values for the angle between CLVs. This can be understood considering that theFTLEs measure the exponential growth rate of divergence of nearby trajectories. Wherethe FTLEs are smaller, the expansion and contraction directions are not well defined anda broader range for the possible values of the angle is obtained. Although there is not aclear relation between the two quantities it is interesting to point out that, for this example,the maximum values of FTLEs correspond to a reduced range of possible values of θ , andfor the DV2 experiment there is a univocal correspondence. The joint probabilities areasymmetric and their peaks towards the highest values of θ highlight the fact that thesystem is hyperbolic. In this case, in fact, it is possible to find, almost for every point ofthe domain, well defined expansion and contraction directions. It is interesting to note thatthe joint probability of Figure 4f is less asymmetric with respect to the correspondent jointprobability in Figures 4e.Figure 5 compares the HCCSs at time t = 10 for the experiment DV2 , found with thedefinition II.1, and the FTLEs field shown in Figure 4b. In the right panel there is a blow up20f a region containing HCCSs. These pictures show that the instant structures highlightedby the HCCSs do not always correspond to the ridges of the FTLEs. The contours of theregions characterized by θ = π/ w , that define the contractiondirection. Every particle of the tracer near these HCCSs will tend asymptotically to getaway from them. Figure 5. Comparison between the HCCSs at time t = 5, green lines, computed for the experiment DV2 and the FTLEs field shown in Figure 4b. In the right panel there is a zoom of a portion of thedomain in which are also shown the CLVs characterizing that region. Blue direction is contractingwhile red is expanding. Since the contour in this region is aligned with the second CLVs, the blueone, the character of the HCCSs here is repulsive. igure 6. Comparison of the angle between CLVs, θ , computed for the two experiments DV1 andDV2 at the same time t = 10. Panel (a) show the evolution of the angle obtained in DV1 at t = 10.Panel (b) shows the difference between the angle computed at the beginning of the experiment DV2 and the one computed at the end of the interval of the CLVs computation for the experiment
DV1 . Figure 6 compares the evolution of θ field in DV1 , computed at t = 10, with the anglescomputed at the same time for the experiment DV2 . The region of maximum hyperbolicitythat appears in Figure 6a, corresponds to the same region that is found in Figure 4d.The difference between the two fields (Figure 6b) shows large values in regions with lowhyperbolicity, and values close to zero for θ = π/ θ field. Itshould also be noted that the structure highlighted by the maximum hyperbolicity in the leftpart of the domain is pretty similar to the shown in Figure 4a. Once again, this underlinethat the FTLEs field gives an overall information and not an instantaneous one. The HLCSs22ound by using the FTLEs are, by definition, strongly related to the time interval chosenfor the study, while the HCCSs, which correspond to instantaneous structures of maximumhyperbolicity, are independent of the time interval considered for their computation, asshown in these two experiments. The short-term behavior of passive tracer analyzed byHLCSs is strongly influenced by the time window used with respect to long-term behaviorhighlighted by HCCSs. It should also be noted that the HLCSs, once they have beenfound, are advected with the flow so as to ensure that they act as barriers for the passivetracer. However, this precludes the study of any new barriers that may arise in the flow in atime subinterval. These constraints are not present in the long-term study performed withHCCSs. Figure 7. PDF of θ for the DV2 experiment computed at t = 10 (dashed line) and at t = 20 (fullline). igure 8. Evolution of the first four moments of the PDF for θ for the DV2 experiments in theinterval T It is also interesting to consider the evolution of the PDF of the hyperbolicity field, P ( θ, t ).If the distribution is peaked around π/
2, the expansion and contraction directions are almosteverywhere perpendicular between each other. If the distribution is peaked around zero, theexpansion and contraction directions are almost tangent everywhere. If the distributionis flat there is not a clear correlation between the expansion and contraction directions.Figures 7 shows the PDF of θ computed for the initial and final time of CLVs computationfor the experiment DV2 . The final PDF (full line) highlight an increasing of points withhight hyperbolicity, and an increment of points in the tail of the distribution near zero, withrespect to the PDF computed at the beginning of the interval (dashed line). During the24volution in the interval t = [10 ,
20] the distribution becomes more asymmetric. The changein shape is visible also looking at the first four moments of the PDF computed at every timestep in this time interval (Figure 8). All the moments of the distribution display onlysmall changes in time during the evolution. This time interval is characterized by a meanvalue of θ oscillating in the range [1 . , . t ≈
16. It should be noted however that the range of thischange is very small. The s.t.d. of the angles distributions is characterized by values thatchange within the interval [0 . , . t ≈
16 the s.t.d. shows a monotonic increase,as a signature that, if the change in time is given by mixing, this must be of adiabaticnature and thus not resulting in a homogeneization of the field. The third moment of thePDF decreases in time from − .
57 to − .
91, showing thus a negatively skewed distribution.Finally, the kurtosis initially increases from 2 .
81 to 4 .
03, and then shows a slight decrease invalues, generally indicating a less flat distribution than the normal distribution. Althoughthe motion of the velocity field is periodic, the motion of the tracers in this flow is chaotic.This is the reason why we do not see in this interval a periodicity in the moments of thePDF for the variable θ . C. Bickley jet
The Bickley jet is an idealized model of a jet perturbed by a Rossby wave . Thevelocity field is given in terms of the stream function, ψ ( x, y, t ), that can be decomposed asthe sum of a mean flow ψ ( x, y, t ) and a perturbation ψ ( x, y, t ) ψ ( x, y, t ) = ψ ( x, y ) + ψ ( x, y, t ) , (36)where ψ ( x, y ) = c y − U L y tanh (cid:18) yL y (cid:19) + (cid:15) U L y sech (cid:18) yL y (cid:19) cos( k x ) , (37)and ψ ( x, y, t ) = U L y sech (cid:18) yL y (cid:19) R e (cid:34) (cid:88) n =1 (cid:15) n f n ( t ) e ik n x (cid:35) . (38)25ollowing the work by Onu et al. , the forcing is chosen as a solution that runs on thechaotic attractor of the Duffing Oscillator dϕ dt = ϕ (39a) dϕ dt = − . ϕ − ϕ + 11 cos( t ) , (39b) f ( t ) = f ( t ) = 2 .
625 10 − ϕ ( t/ . × ) . (39c)From now on the time will be scaled with the quantity L x /U . The streamline of the flowat t = 1 .
89 are shown in Figure 2c.As for the double gyre, we show two experiments, BJ BJ
2, for which the end ofthe CLVs computation in the first experiment correspond with the first CLVs computationfor the second experiment. 26 igure 9. As in Figure 4, but for the two experiments
BJ1 and
BJ2 . The first column of Figure 9 shows the analysis for
BJ1 and the second column for
BJ2 . Figures 9a and 9b, show the field of the maximum FTLEs, λ . Results show theconvergence for the FTLE fields. Maximum values of λ are reached in the central jet andaround the boundaries of the vortices. Figures 9c and 9d show the angle between CLVs,where the white color indicates θ = π/
2. The θ fields clearly provide more details of theflow patterns than the FTLE fields. In agreement with the FTLE fields, values of θ close to π/ θ fields show also spiraling patterns within the vortices, which are insteadnot visible in the FTLE fields. The spiral patterns are particularly evident in the uppervortices, where the perturbation ψ acts as a positive feedback to the flow. In the bottomvortices, the perturbation acts instead to weaken the flow. Figures 9e and 9f show the jointprobability plots of the angles and the FTLEs just presented. These plots show that thereis no clear relation between FTLEs and angle. For a given FTLE correspond many valuesof the angle between CLVs. As for the double gyre example, smaller values of the FTLEscorrespond to the maximum range of possible values for θ . However, differently from thedouble gyre, also high values of FTLEs correspond to a broad range of θ values. This canbe explained considering that almost all the points corresponding to the highest values ofFTLEs are contained in the central jet. In this region, the FTLEs converge to the samevalue, independently from the space position along the jet or time. Due to this convergencewe do not have much information about smaller structures that can characterize the systemin the jet region as instead shown by the θ field. For this reason, the joint probability plotsdo not exhibit a reduced range of θ values in correspondence of the highest FTLEs. Thecomparison between the BJ1 and
BJ2 integrations shows that for the Bickley jet both thePDFs of the FTLEs and θ field are more stationary in time with respect the double gyre. Figure 10. As in Figure 5, but for the two experiments
BJ1 and
BJ2 . The left panel of Figure 10 shows the superposition of HCCSs computed for t = 3 .
79 for28he experiment
BJ2 , green lines, and the FTLEs computed in the same experiments (Figure9b). This figure remarks the fact that the central jet is not completely a HCCSs. HCCSsare also present at the border of the vortices and in the center part of the upper vortices.The right panel of Figure 10 show an enlargment into a portion of the central jet containingHCCSs and the CLVs in that part of the domain. In this picture it is possible to appreciatethe repelling nature of the HCCSs looking the alignment of the contour with the secondCLVs. It is interesting to point out that, looking at the FTLEs field, it is not possible tosee ridges in the central jet. In this case it is not possible to find LCSs, if we use definitionof LCSs based on the FTLEs field , and this once again remark the difference betweenthe HCCSs and LCSs.Figure 11 shows the comparison between the angles at the end of the CLVs computationinterval of the experiment BJ BJ θ = π/
2, i.e. the HCCSs, havethe same structure. The difference between the two fields (Figure 11b), shows that theseregions are characterized by smaller errors. In contrast, the regions with low hyperbolicityof Figure 11a appear to be more noisy with respect to the same regions computed for the BJ igure 11. As in Figure 7, but for the BJ2 experiment.Figure 12. As in Figure 7, but for the
BJ2 experiment. θ for the experiment BJ2 at the beginning and at the endof the CLVs computational interval. Results show that the PDF of the angle for this thisinterval remains stable. This is confirmed by the analysis of the moments of the distribution(not shown), which appear to be constant in time. With respect to the double gyre, thePDF for the Bickley jet is more flat indicating the presence of a larger number of points inwhich the CLVs are not orthogonal.
IV. CONCLUSIONS
We have here proposed a new definition and a new computational framework to determinehyperbolic structures in a two dimensional flow based on Covariant Lyapunov Vectors. CLVsare covariant with the dynamics, invariant for temporal inversion and norm independent.These vectors are the natural mathematical entity to probe the asymptotic behaviour ofthe tangent space of a dynamical system. All these properties allow an exploration of thespatial structures of the flow, which can not be done using the Lyapunov Vectors bases dueto their orthogonality.CLVs are related to the contraction and expansion directions passing through a point ofthe tangent space, and the angle between them can be thus considered as a measure of thehyperbolicity of the system. This information can be summarized in a scalar field, the angle θ , between the CLVs referred to the initial grid conditions, and used to define hyperbolicstructures. The structures identified with the isolines of this field, characterized by θ = π/ in terms of the asymptotic behaviour of tracer particles , with respectto nearby structures at a given time. In terms of practical applications, this has importantconsequences, as it will provide an indicator for the long time transport of passive tracerssuch as for example oil spills in the ocean.CLVs, and the correspondent θ field, have been computed for three numerical examples tocompare how the behaviour of the particles tracer near HLCSs, highlighted by the FTLEs,can change asymptotically in time. The three examples include an Hamiltonian autonomoussystem, and two non-autonomous systems that are bounded or periodic. For all theseexamples it is possible to compute CLVs, HCCSs, and compare them with the HLCSs.Since the FTLEs tend to converge to LEs and lose their dependence on the initial conditions31he angle between the CLVs could give more detailed information about possible structuresthat can emerge from the flow. This feature has been highlighted in particular for theHamiltonian autonomous system, in which θ is able to detect the central barrier in contrastwith the FTLEs field, and in the Bickley jet in which the FTLEs converge to the same valuein the jet region and it is not possible to see any kind of particular finest structure. The useof θ provides information on the structures appearing at each time of the evolution of theflow, and the three examples underline that not always the HCCSs correspond to HLCSsand vice versa. So, particles tracer, such as chlorophyll or oil in water, can be maximallyattracted or repelled by some HLCSs, but if we consider a different time interval and inparticular the asymptotic behaviour of these particles, we can obtain a distribution that iscompletely different. Note that, in practice, the asymptotic time length can be consideredas the time taken by two random initial basis to converge to the same BLVs basis.It should be noticed that, while no fluxes can be present across the HLCSs the samedoes not necessary hold for all the structures appearing characterized by θ = π/
2. HCCSscan be found for every instant of time but they give the asymptotic information about thebehaviour of the particles tracer near the HCCS at that instant of time and, for this reason,the zero flux requirement of the HLCSs is not necessary.
HCCSs are not necessarily barrier,their meaning is different from the one of HLCSs .For the three examples considered we have also computed HLCSs with the geodesictheory using the LCS tool (not shown). The results are in agreement with the discussionabove. Looking at asymptotic time it is still possible to find particular structures emergingfrom flow, but clearly these structures not necessarily correspond to HLCSs computed for aparticular time interval.For the two non-autonomous systems we have considered the evolution of the PDFs ofthe θ field and the evolution of its first four moments. For the Bickley jet the probabilitydistribution of the angle is stationary in time, and so its moments, but for the double gyreit is possible to appreciate a small variation in time for the PDF. The information derivingby the evolution of the θ field, related to the variation of the strength of the hyperbolicityfield, could be used to characterize the dynamical mixing of the system.Finally, future studies will have to address the detection of hyperbolic structures beyondanalytical systems, i.e. for two-dimensional turbulent flows. This will be particularly in-teresting for flows at the transition between balance and lack of balance , where the32etection of HCCSs can shed light both on the structure of mixing and on the forwardcascade of energy to dissipation, or the lack of thereof. In particular, the evolution of themoments of the PDF of θ could be used to define an index of dynamical mixing for thesystem under study. Particularly interesting will be the behaviour of the HCCSs in presenceof intermittency. We can conjecture that in particular cases, such as e.g. the merging of twovortices, the instantaneous structures underlined by the HCCS can give reliable informationof the asymptotic tracer dynamics, that is the dynamics after the merging event. This ishowever left for future studies. ACKNOWLEDGMENTS
We would like to thank two anonymous referees for comments that helped to improvethe manuscript. This study was partially funded by the research grant DFG1740. GB waspartially funded also by the research grants DFG TRR181 and DFG BA 5068/8-1. Theauthors would also like to thanks Sergiy Vasylkevych and Sebastian Schubert for interestingdiscussions on the subject.
Appendix A: Description of the algorithm
The algorithm for the identification of CLVs is based on five different phases:1.
Initialization ( T1 ): this preliminary step is used to find the initial backward Lya-punov vector bases { l − ( t ) , l − ( t ) } for a whole set of initial conditions . A set of initialcondition { x } ∈ D and two sets of initial orthonormal random bases are defined inthe tangent spaces at every point at time t . The second set is necessary to checkthe convergence to the backward Lyapunov vectors bases. The initial conditions andthe random bases are evolved respectively with (1) and (6) until the convergence ofthe two bases is reached with the desired accuracy. The convergence toward the Lya-punov vectors is typically exponential in time . At every time step, for every initialcondition, the evolved vectors are stored as column of a matrix that is decomposedwith a QR decomposition. The last passage is implemented in order to find the neworthogonal basis at every time step, and the upper triangular matrix containing thecoefficients that allow to express the old basis in terms of the new one.33. Forward Transition ( T2 ): the backward Lyapunov bases are evolved from time t to t (cid:48) . The evolution is done with the help of (1), (6) and the QR decomposition forevery evolution step. We indicate with X ( t k ) the matrix which columns contain thenew bases at time t k , and with R ( t k − , t k ) the correspondent upper triangular matrix.During this step both the local Lyapunov bases and upper triangular matrices arestored. The diagonal elements ( R ( t k − , t k )) ii of the upper triangular matrices giveinformation about the local growth rates of the bases vectors at a given time t k , andthey are used to compute the FTLEs as a time average λ i = 1 t (cid:48) − t N − (cid:88) k =0 log( R ( t k − , t k )) ii , (A1)where N time step are considered between t and t (cid:48) . If the LEs exist, (A1) will convergeto them for a sufficiently long evolution. It is worth to note that sometimes the FTLEsare computed using a different method, that is using the so called Cauchy Green Tensor(CGT) defined as G ( t, t (cid:48) ) = F ( t, t (cid:48) ) (cid:62) F ( t, t (cid:48) ) . (A2)This operator is also known as deformation tensor, whose eigenvalues µ i ( t , t ), andeigenvectors ξ i ( t , t ), satisfy G ( t, t (cid:48) ) ξ i ( t, t (cid:48) ) = µ i ( t, t (cid:48) ) ξ i ( t, t (cid:48) ) , (A3a) µ ( t, t (cid:48) ) > µ ( t, t (cid:48) ) > , (A3b) ξ ( t, t (cid:48) ) ⊥ ξ ( t, t (cid:48) ) . (A3c)From a geometric point of view, a set of initial conditions corresponding to the unitsphere is mapped by the dynamics into an ellipsoid, with principal axis aligned in thedirection of the eigenvectors of the CGT and with length determined by the corre-spondent eigenvalues. The eigenvalues of the CGT determine the FTLEs as λ i ( t, t (cid:48) ) = 12( t (cid:48) − t ) log( µ i ( t, t (cid:48) )) , (A4)where the dependence on the starting position has here been suppressed. This secondmethod for the computation of the FTLEs exhibits some problems, for example, ifjust one finite local growth rate is taken into account considering a large time interval,(A4) teds to zero and not to the LEs. The first method should be preferred.34. Forward Dynamics ( T3 ): in this step the trajectories and the bases are further evolvedfrom time t (cid:48) to time t (cid:48)(cid:48) using (1) and (6). This time interval should grant the conver-gence, during the backward dynamic, to the CLVs. During this step only the uppertriangular matrices are stored and are used to continue the computation of the FTLEsusing (A4).4. Backward Transition ( T4 ): in this step random upper triangular matrices are gener-ated for every point of the grid, C . These matrices contain the expansion coefficientsof a set of two generic vectors (expressed as column of a matrix) in terms of the L − bases. Using the stored matrices R of the step 3, these matrices are evolved backwardin time, until time t (cid:48) , using the following relation: C ( t n ) = R − ( t n , t n +1 ) C ( t n +1 ) D ( t n , t n +1 ) , (A5)where t n and t n +1 are time step between t (cid:48) and t (cid:48)(cid:48) . This method uses all the informationcontained in R and not just the diagonal part of the matrix. The D diagonal matricescontain the column norm of C . Using Eq. (A5) it is possible to show that thegeneric vectors chosen will be aligned with the CLVs. Note that when a trajectorypasses close to tangency of an invariant manifold, the matrices C can be ill-defined,and so a little amount of noise on the diagonal element of these matrices, or an averageon the diagonal elements of the neighbor matrices is used to correct the problem.5. Backward Dynamics ( T5 ): in this final part of the algorithm, (A5) is used with the R matrices of the step 2 to evolve backward the upper triangular matrices C , from time t (cid:48) to time t . In this phase the backward Lyapunov bases stored can be used to writethe CLVs. The matrix containing in each columns the different CLVs at a given pointin space and time, W , can thus be written as W ( t n ) = C ( t n ) X ( t n ) . (A6)For a two dimensional system, the algorithm could be optimized making use of (18).One can follow the first step ( T1 ) of the previous algorithm to find the convergencetoward the backward Lyapunov bases in the time interval [ t , t ]. After this firststep it is possible to carry on the evolution of the vectors, as in the second step( T2 ) during the time interval [ t, t (cid:48) ], without saving the triangular matrices. In the35ame way it is possible to repeat the step ( T1 ) but for a backward evolution duringthe time interval [ t (cid:48)(cid:48) , t (cid:48) ] to find the forward Lyapunov bases and then continue thebackward evolution as in the step ( T2 ) during the time interval [ t (cid:48) , t ]. At this stageit is possible to consider directly the first backward Lyapunov vectors and the secondforward Lyapunov vectors in the time interval [ t, t (cid:48) ] as the CLVs. In this algorithmthere are just four steps and not five as in the one presented above, but for two timesone has to consider the convergence step ( T1 ) that is more time consuming in respectto the steps ( T4 ) or ( T5 ). It should be noted that in this algorithm, the forwardand the backward evolutions could be done in parallel. The comparison between thisalgorithm and the one used for this study is left for future studies. Appendix B: Technical details of the numerical examples1. Hamiltonian system
For the simple Hamiltonian flow we consider the domain x = [0 . , y = [ − . , . ×
300 grid points and a time step dt = 0 .
01. The CLVs are computed justfor t = 5, so the phase T2 and T5 includes just a few time steps. The forward and backwardevolution is done in the time interval T = [5 , t = 0 to t = 5 the algorithm passesthrough the initialization phase (see Appendix A) to find the backward Lyapunov vectorsbases. In Figure 13 it is shown the convergence for the spatial average of the scalar productbetween the starting random bases chosen. 36 → →
10 10 →
15 10 ←
15 5 ← DV2 →
10 10 →
20 20 →
30 20 ←
30 10 ← DV1 and
DV2 .Figure 13. Convergence of the averaged scalar product for the two random bases chosen for theinitialization of the numerical algorithm for the Hamiltonian system (27).
2. Double gyre
The parameters used for the double gyre example are A = 0 . (cid:15) = 0 . ω = π/
5. Thespatial domain is x = [0 , y = [0 , ×
200 points and timestep dt = 0 .
02. Two experiments,
DV1 and
DV2 , involving different CLVs computationaltime interval are considered and summarized in Tab. I. During the initial phase of thenumerical algorithm for the calculation of the CLVs, the average of the scalar product ofthe two initial random bases does not converge exactly to one (not shown here). This isbecause near the central regions of the two vortices of the double gyre, the expansion andcontraction directions are not well defined, the CLVs become tangent to each other, andtheir separation is difficult to attain. 37 → .
89 1 . → .
79 3 . → .
68 3 . ← .
68 1 . ← . BJ2 → .
79 3 . → .
57 7 . → .
36 7 . ← .
36 3 . ← . BJ1 and
BJ2 .Times have been rescaled with L x /U .
3. Bickley jet
The parameters used for this example are U = 62 . c = 0 . U , c = 0 . U , L y =1 . × , (cid:15) = 0 . (cid:15) = 0 . L x = 6 . π × , k n = 2 πn/L x . The spatial domainconsidered is x = [0 , L x ], y = [ − . , . L y , with a resolution of 500 ×
250 grid pointsand a time step of dt = 1800. Two different time windows evolutions are considered,experiments BJ BJ
2, and summarized in Tab. II.
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