Lagrangian observations; multi-particle statistics J. H. LaCasce - PowerPoint PPT Presentation

Lagrangian observations; multi-particle statistics J. H. LaCasce Norwegian Meteorological Institute Oslo, Norway Lagrangian observations; multi-particle statistics – p.1/58

Single particle stats Previously noted that, as t → 0 : X 2 ( t ) ≈ ν 2 t 2 and as t → ∞ : X 2 ( t ) ∝ 2 ν 2 T L t So single particle dispersion is relatively insensitive to details of the flow Lagrangian observations; multi-particle statistics – p.2/58

Multiple particles In contrast, the behavior of groups (pairs, triplets) of particles depends more on the specific flow → Dispersion depends on Eulerian energy spectrum So multi-particle statistics possibly more revealing about ocean dynamics Lagrangian observations; multi-particle statistics – p.3/58

Multi-particle Statistics concern the distortion of marked clouds Richardson, Obukhov, Batchelor, Corrsin, Kraichnan, Monin and Yaglom, etc. Lagrangian observations; multi-particle statistics – p.4/58

Relative separations Two particles, of volume V , at x 1 and x 2 Define x ≡ x 1 and y ≡ x 2 − x 1 . Then � � P ( x, y, t ) dxdy = V 2 The probability of separation, y , is then: p ( y, t ) = 1 � P ( x, y, t ) dx V Lagrangian observations; multi-particle statistics – p.5/58

Relative separations As before, can relate to a (joint) displacement PDF: � P ( x ′ , y ′ , t ) Q ( x, y, t | x ′ , y ′ , t 0 ) dx ′ dy ′ P ( x, y, t ) = Integrating over space: � q ( y, t | y ′ , t 0 ) = Q ( x, y, t | x ′ , y ′ , t 0 ) dx Lagrangian observations; multi-particle statistics – p.6/58

Relative separations So � p ( y ′ , t 0 ) q ( y, t | y ′ , t 0 ) dy ′ p ( y, t ) = Richardson (1926) called q ( y, t | y ′ , t 0 ) the “distance-neighbour function” Can define dispersion: � y 2 = y 2 q ( y, t | y 0 , t 0 ) dy And similarly relative diffusivity, κ 2 ≡ d dt y 2 , etc. Lagrangian observations; multi-particle statistics – p.7/58

Structure function How does the dispersion evolve? Assume Eulerian flow and examine consequences. Lagrangian second order structure function: ( d dty ) 2 = ( v 1 − v 2 ) 2 = v 2 1 + v 2 2 − 2 v 1 v 2 Lagrangian observations; multi-particle statistics – p.8/58

Large scales At scales larger than the “energy-containing eddies”: ( d κ 2 = d dty ) 2 → 2 ν 2 , dty 2 = 2 κ 1 Lagrangian observations; multi-particle statistics – p.9/58

Intermediate scales Relate Focus on isotropic, stationary (2-D) turbulence (Bennett, 1984): ( d dty ) 2 = ( u ( x + y, t ) − u ( x, t )) 2 � ∞ = 2 E ( k ) [1 − J 0 ( ky )] dk 0 Assume E ( k ) ∝ k − α Lagrangian observations; multi-particle statistics – p.10/58

Intermediate scales For larger scales: 1 − J 0 ( ky ) ≈ 1 4 k 2 y 2 , ky ≪ 1 For smaller scales: 1 − J 0 ( ky ) ≈ 1 + O ( ky ) − 1 / 2 , ky ≫ 1 Lagrangian observations; multi-particle statistics – p.11/58

Intermediate scales So: � 1 /y � ∞ ( d k − α (1 k − α dk dty ) 2 ≈ 2 4 k 2 y 2 ) dk + 2 0 1 /y or = 1 1 2 3 − αk 3 − α | 1 /y 2 y 2 1 − αk 1 − α | ∞ + 0 1 /y First diverges if α ≥ 3 , second if α ≤ 1 Lagrangian observations; multi-particle statistics – p.12/58

Local dispersion if 1 < α < 3 ( d dty ) 2 ∝ y α − 1 The corresponding diffusivity is: κ 2 = d dty 2 ∝ y ( α +1) / 2 “Richardson’s Law” obtains if α = 5 3 : d dty 2 ∝ y 4 / 3 Lagrangian observations; multi-particle statistics – p.13/58

Non-local dispersion if α > 3 � ∞ ( d dty ) 2 ≈ 1 2 y 2 k 2 E ( k ) dk = c 1 Ω y 2 0 with diffusivity: κ 2 = d dty 2 ≈ c 2 T − 1 y 2 which implies exponential growth of pair separations T is a constant time scale (Batchelor’s strain rate) Lagrangian observations; multi-particle statistics – p.14/58

PDFs Richardson proposed: ∂tp ( y, t ) = y − 1 ∂ ∂ ∂ ∂y ( yκ 2 ∂yp ) which can be exploited to predict the evolution of the separation PDF (Lundgren, 1981; Bennett, 1984) Lagrangian observations; multi-particle statistics – p.15/58

Kurtoses Then can show that the kurtosis: y 4 p ( y ) dy � ku ( y ) ≡ y 2 p ( y ) dy ) 2 � ( is constant for local dynamics (and dependent on α ) and grows exponentially with non-local dynamics. Lagrangian observations; multi-particle statistics – p.16/58

2-D turbulence k^(−5/3) Energy E(k) k^(−3) Enstrophy k Energy injected Lagrangian observations; multi-particle statistics – p.17/58

2-D turbulence Energy cascade: y 2 ∝ t 3 , κ 2 ∝ y 4 / 3 , ku ( y ) = C Enstrophy cascade: y 2 ∝ exp ( c 3 η 1 / 3 t ) , κ 2 ∝ y 2 , ku ( y ) ∝ exp ( c 4 T − 1 t ) Lagrangian observations; multi-particle statistics – p.18/58

Recap • With stationarity and isotropy, can equate Lagrangian and Eulerian structure functions • Kinematic derivation • Associate relative dispersion with the Eulerian kinetic energy • Distinct behavior for local and non-local mixing • Distinct behavior for 2-D inertial ranges Lagrangian observations; multi-particle statistics – p.19/58

Atmosphere EOLE experiment (Morel and Larcheveque, 1974) 483 constant level balloons at 200 mb in the Southern Hemisphere; TWERLE experiment (Er-El and Peskin, 1981) 393 constant level balloons at 150 mb in the Southern Hemisphere Lagrangian observations; multi-particle statistics – p.20/58

Chance pairs Statistics depend greatly on number of pairs. Increase if use “chance pairs” in addition to deployed pairs Lagrangian observations; multi-particle statistics – p.21/58

Diffusivity Lagrangian observations; multi-particle statistics – p.22/58

Dispersion Morel and Larcheveque, 1974 Lagrangian observations; multi-particle statistics – p.23/58

Dispersion Er-El and Peskin, 1981 Lagrangian observations; multi-particle statistics – p.24/58

Dispersion Er-El and Peskin, 1981 Lagrangian observations; multi-particle statistics – p.25/58

Isotropy Lagrangian observations; multi-particle statistics – p.26/58

Relative velocity Lagrangian observations; multi-particle statistics – p.27/58

Relative kurtosis Lagrangian observations; multi-particle statistics – p.28/58

Atmosphere • Exponential growth at L < 1000 km • Large scale behavior unclear • Relative velocities inconsistent • Non-Gaussian separations during exponential growth Lagrangian observations; multi-particle statistics – p.29/58

Ocean Richardson and Stommel (1948): Richardson’s law on a pond (with parsnips) Okubo (1971): 4/3 law at the surface in the North Sea (up to 100 km) Davis (1985): No clear spatial dependence from drifters in California Current Price (1983): κ 2 ∝ D n with 4 / 3 ≤ n ≤ 2 with floats in the western North Atlantic Lagrangian observations; multi-particle statistics – p.30/58

Subsurface ocean LaCasce and Bower (2000): North Atlantic 65 60 55 50 45 ACCE 40 NAC 35 AMUSE 30 SiteL/LDE 25 20 15 −80 −70 −60 −50 −40 −30 −20 −10 0 Lagrangian observations; multi-particle statistics – p.31/58

Floats D 0 ≤ 7 . 5 D 0 ≤ 15 D 0 ≤ 30 Experiment AMUSE 28 54 89 ACCE 14 22 50 NAC 19 38 81 Site L (+ LDE) 14 33 75 LDE1300 4 14 37 Lagrangian observations; multi-particle statistics – p.32/58

Eastern Atlantic 4 4 x 10 x 10 4 4 3.5 3.5 3 3 2.5 2 ) 2.5 2 ) Abs. Disp. (km Rel. Disp. (km 2 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 0 10 20 30 40 50 0 10 20 30 40 50 Day Day AMUSE 5 10 Diffusivity (m 2 /sec) 4 10 2 K (1) D 1 3 10 2 10 0 1 2 3 10 10 10 10 Distance (km) Lagrangian observations; multi-particle statistics – p.33/58

Western Atlantic 4 4 x 10 x 10 16 16 14 14 12 12 Rel. Disp. (km 2 ) 10 2 ) 10 Abs. Disp. (km 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 0 10 20 30 40 50 0 10 20 30 40 50 Day Day NAC 5 10 2 K (1) Diffusivity (m 2 /sec) 4 10 D 4/3 3 10 2 10 0 1 2 3 10 10 10 10 Distance (km) Lagrangian observations; multi-particle statistics – p.34/58

Relative velocities AMUSE ACCE 1.5 1.5 <u 2 >,<v 2 > 1 1 0.5 0.5 0 0 0 10 20 30 40 50 0 10 20 30 40 50 NAC SiteL 1.5 1.5 <u 2 >,<v 2 > 1 1 0.5 0.5 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Day LDE 1300 1.5 <u 2 >,<v 2 > 1 0.5 0 0 10 20 30 40 50 Day Lagrangian observations; multi-particle statistics – p.35/58

Relative kurtoses Kurtoses of relative displacements 10 9 AMUSE NAC ACCE 8 LDE1300 SiteL 7 Kurtosis 6 5 4 3 2 1 0 5 10 15 20 25 30 35 40 45 50 Day Lagrangian observations; multi-particle statistics – p.36/58

Subsurface ocean • Richardson growth in west • Diffusive growth in east • Small scales ( r < L D ≈ 20 km ) not well-resolved Lagrangian observations; multi-particle statistics – p.37/58

Surface ocean LaCasce and Ohlmann (2003): SCULP (Niiler) 140 pairs with r 0 ≤ 1 km SCULP1 pairs 32 31 30 29 28 27 26 25 24 260 262 264 266 268 270 272 274 276 278 280 SCULP2 pairs 32 31 30 29 28 27 26 25 24 260 262 264 266 268 270 272 274 276 278 280 Lagrangian observations; multi-particle statistics – p.38/58

Relative dispersion Lagrangian observations; multi-particle statistics – p.39/58