Fractal contours of scalar in a 2D smooth random flow Marija Vucelja Gregory Falkovich Konstantin S. Turitsyn Courant Institute of Mathematical Sciences, Weizmann Institute of Sciences, MIT Fluorescent dye in a turbulent jet “Mathematics of particles and Flows” May 2012, WPI of Reynolds number Re = 4000 ( K. Sreenivasan, 1991) Monday, June 18, 12
Mixing and transport • The basic understanding of turbulent mixing and transport: Taylor, Richardson, Kolmogorov, Obukhov and Corrsin... from dimensional arguments: the average rate of spreading or mixing of e.g. a smoke plume. • The statistics of the large fluctuations is a more difficult problem.The practical relevance: • the probability of a pollutant concentration exceeding some tolerable level as it spreads from a source. • the role of large concentration fluctuations in controlling the rate of slow (high order) chemical reactions (e.g. in the process of atmospheric ozone destruction). Monday, June 18, 12
Zero modes multipoint correlation functions of scalar and gradient: Schraiman, Siggia, Chertkov, Falkovich, Kolokolov, Lebedev, Gawedzki, Bernard, Kupiainen (1994-2000) Geometry of points that started close by... 1 3 2 d θ d t = ∂θ ∂ t + ( v · r ) θ = ϕ Z t ˙ R ( t ) = v ( R ( t ) , t ) ϕ ( R ( t 0 ) , t 0 ) d t 0 θ ( R ( t ) , t ) = �1 Z t h ϕ ( R ( t 0 1 ) , t 0 1 ) . . . ϕ ( R ( t 0 1 ) , t 0 N ) i d t 0 1 . . . d t 0 h θ ( r 1 , t ) . . . θ ( r N , t ) i = N �1 Monday, June 18, 12
Passive scalar evolution We studied a passive scalar under the action of pumping, diffusion, advection in smooth 2D flow with Lagrangian chaos. ∂ t θ ( t, r ) + ( v ( t, r ) · r ) θ ( t, r ) = κ d r 2 θ ( t, r ) + ϕ ( t, r ) pumping ϕ ( t, r ) Poisson process of independently adding size- L blobs of passive scalar at random positions and with random amplitudes pumping scale L diffusion molecular diffusion coefficient κ d Monday, June 18, 12
Smooth 2d flow field Batchelor regime Batchelor 1959 Kraichnan model Kraichnan 1968 The velocity field: random, smooth and incompressible white in time and of spatially linear velocity profile tensor of the local σ ij = ∂ v i (Lagrangian) velocity ∂ r j gradients h σ ij ( t ) σ kl ( t 0 ) i = λ [3 δ ik δ jl � δ ij δ kl � δ il δ jk ] δ ( t � t 0 ) Lyapunov exponent λ Monday, June 18, 12
Batchelor regime v ( t, R 1 ( t )) − v ( R 2 ( t )) = σ ( t, R 2 ( t ))( R 1 ( t ) − R 2 ( t )) valid when | R 1 ( t ) − R 2 ( t ) | < η ⌘ ≡ ( ⌫ 3 / ✏ ) 1 / 4 Kolmogorov (viscous) scale kinematic viscosity ✏ energy dissipation ν Pr ⌘ ν / κ d � 1 Prandtl number Spatial scales pumping scale L diffusion scale p r d ≡ κ d / λ Monday, June 18, 12
Scalar blob evolution L 2 r d L r d Velocity Forcing Monday, June 18, 12
Smooth flows - a simplest example of mixing • large-scale mixing in the Earth atmosphere where the turbulence spectrum (the velocity gradients are then k − 3 well-defined and the flow can be locally considered smooth). • flows in the phase space of dynamical systems. Mixing in smooth flows is provided by exponential separation of trajectories and Lagrangian chaos. We consider an incompressible fluid flows which correspond to Hamiltonian flows in phase space. Monday, June 18, 12
NEW computational method • exploiting the linearity of the reaction diffusion equation • method of characteristic to solve for a single spherical blob of scalar initial form of blob θ ( t 0 , r 0 ) = Θ 0 exp[ − ( r 0 − r c ) 2 / (2 L 2 )] at a later time Θ 0 L 2 2 ( r ( t ) − r c ) I − 1 ( t,t 0 )( r ( t ) − r c ) e − 1 θ ( t, r 0 ) = p det I ( t, t 0 ) r ( t ) ≡ W ( t ) r ( t 0 ) defines evolution operator W ( t ) moment of inertia Z t WW T � d t 0 W ( t ) W ( t 0 ) � 1 [ W ( t ) W � 1 ( t 0 )] T � I ( t, t 0 ) ≡ ( t ) + κ d 0 Monday, June 18, 12
Semi-analytic method allows HUGE resolutions Θ 0 L 2 2 ( r ( t ) − r c ) I − 1 ( t,t 0 )( r ( t ) − r c ) e − 1 θ ( t, r 0 ) = p det I ( t, t 0 ) seven values specify a blob at time t : I 11 , I 12 , I 21 , Θ 0 , r c , t 0 Summing over the blobs can get passive scalar field. Large enough number of blobs assures the Gaussianity of pumping statistics. Conventional methods out used are pseudospectral and thus always one is limited by the slow decay (logarithmic) of the passive scalar pair correlation function. Monday, June 18, 12
A long iso-contour top: 2400 L x 360 L (L pumping scale) bottom: zoomed inset Monday, June 18, 12
Appearance of fjords of isolines of scalar fjords L l λ 1 creation of fjords snapshot scalar field 75 L x 75 L ϕ ϕ v v v IMPORTANT: diffusion randomly reconnects the curves Monday, June 18, 12
Statistics of nodal lines Expectations - “trivial statistics” • pumping alone would produce a Gaussian field whose zero isolines θ are smooth at the scales below L , while at larger scales they are equivalent to critical percolation ( ). SLE 6 , D 0 = 7 / 4 • does not change the statistics of as it just rearranges it; the flow θ v stretches isolines uniformly at the direction of and contracts them λ 1 transversal to it . Actually: Non-trivial statistics Non-trivial statistics and its isolines arises from an interplay of velocity, pumping and finite diffusivity or finite resolution, which leads to the dissipation of scalar and reconnection of isolines that came closer than the resolution scale. Monday, June 18, 12
Non-gaussian field substituting PDF in FP equation first term Indeed we know that correlation functions include cumulants. Our scalar is a non-gaussian field... Monday, June 18, 12
No conformal invariance four point correlation function F 4 = h θ 1 θ 2 θ 3 θ 4 i Conformal symmetry restricts a four point correlation function: ✓ r 12 r 34 ◆ , r 12 r 34 F 4 = f ( r 12 r 34 r 14 r 23 r 13 r 24 ) a r 13 r 24 r 14 r 23 For passive scalar we know that F 4 = F ( r 12 , r 34 ) + F ( r 13 , r 24 ) + F ( r 14 , r 23 ) Balkovsky et al 1995 NOTE: this is not Wick theorem! BOTH equations satisfied by F ( x, y ) = ( xy ) a/ 2 + ( xy − 2 ) a/ 2 + ( x − 2 y ) a/ 2 F 4 = ( r 12 r 34 ) 2 a + ( r 13 r 24 ) 2 a + ( r 14 r 23 ) 2 a i.e. to a Gaussian statistics, which is not the case, as we have just shown. Passive scalar is not in any way close to a free field and its statistics is not conformally invariant. Monday, June 18, 12
P - perimeter PDF of contour perimeters � / � = 0.05 P [log( P/L )] � / � = 0.02 − 2 � / � = 0.01 (1.04 ± 0.02) 10 � / � = 0.005 r d /L = 0.06 3/2 r d /L = 0.03 − 4 10 r d /L = 0.015 (1.04 ± 0.02) 3/2 0 2 4 10 10 10 P/L pumping frequency ν / λ resolution r d /L All collapse on top of one another (lower three curves shifted down by dividing by 10) Monday, June 18, 12
PDF of sizes (mean radius R) � / � = 0.05 0.1 � / � = 0.02 P [log( R/L )] � / � = 0.01 0.01 � / � = 0.005 r d /L = 0.06 r d /L = 0.03 3/2 0.001 r d /L = 0.015 3/2 0.0001 − 2 0 2 10 10 10 R/L p R ⌘ h ( ρ � h ρ i ) 2 i All collapse on top of one another (lower three curves shifted down by dividing by 10) Monday, June 18, 12
Salient features of PDFs of P and R � / � = 0.05 � / � = 0.05 0.1 P [log( P/L )] � / � = 0.02 � / � = 0.02 P [log( R/L )] − 2 � / � = 0.01 (1.04 ± 0.02) � / � = 0.01 10 � / � = 0.005 0.01 � / � = 0.005 r d /L = 0.06 r d /L = 0.06 3/2 r d /L = 0.03 3/2 r d /L = 0.03 0.001 − 4 r d /L = 0.015 10 r d /L = 0.015 3/2 (1.04 ± 0.02) 0.0001 3/2 − 2 0 2 10 10 10 R/L 0 2 4 10 10 10 P/L Left tails: 3/2. Contours shorter than L must appear when pumping cuts a piece off a thin long contour, the probability of such √ √ a cut is . Extra factor in the PDF may appear P ∝ R P ∝ R because to be observed small contours need to survive without being swallowed by further pumping events. Since creation and survival are independent events, their probabilities are multiplied. Right tail PDF of P: In log coordinates the tail is close to P − 1 P ( P ) ∝ P − 2 (no theoretical explanation so far) Monday, June 18, 12
generalized box counting fractal dimension q = 0 1e+07 q = 2 1 . 0 q = 4 1 1e+06 i ( � )]1/(1-q) 1.6 1e+05 1.8 1.7 [ � i p q D q 1e+04 1.6 1 . 55 ÷ 1 . 7 1.5 1e+03 0 1 2 3 4 q 0.1 1 10 100 � /L fractal dimension is scale dependent 0 1 N ( ε ) X p q A [( q − 1) log( ε )] − 1 D q ≡ lim ε → 0 log i ( ε ) @ i Monday, June 18, 12
box counting fractal dimension 1.6 1 . 55 ÷ 1 . 7 1.4 D 0 1.2 1 . 0 1 1 10 100 � /L ln( N ( ✏ )) box counting fractal dimension D 0 = lim ln( ✏ − 1 ) ✏ → 0 estimated from d log N ( ε ) / d log( L/ ε ) It is scale dependent: bellow forcing scale it is smooth, above it is fractal. Monday, June 18, 12
Mono fractals Monday, June 18, 12
Characterize curves with SLE (Schramm-Loewner Evolution) d g t ( z ) 2 = d t g t ( z ) − a t Loewner , 1923 Schramm , 2000 • conformal map with which the tip of the curve is g t mapped into the real axis • driving function a t • for Markovian successive maps the driving function at is a t = √ κ B t a standard Brownian motion ( Schramm , 2000): Monday, June 18, 12
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