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

Lagrangian observations; single particle statistics J. H. LaCasce Norwegian Meteorological Institute Oslo, Norway Lagrangian observations; single particle statistics – p.1/ ??

History Stommel and Arons (1960) Lagrangian observations; single particle statistics – p.2/ ??

History John Swallow Lagrangian observations; single particle statistics – p.3/ ??

History Swallow floats • Confirmed Deep Western Boundary Current • Did not find sluggish poleward flow • Discovered deep (mesoscale) eddies Lagrangian observations; single particle statistics – p.4/ ??

Instruments 1) Atmosphere • Balloons • Constant level (pressure) • Tracked by satellite Lagrangian observations; single particle statistics – p.5/ ??

Instruments 2) Ocean • Surface drifters Lagrangian observations; single particle statistics – p.6/ ??

Instruments • Subsurface floats Lagrangian observations; single particle statistics – p.7/ ??

Lagrangian data SOFAR floats in meddies Lagrangian observations; single particle statistics – p.8/ ??

Data North Atlantic (Richardson, 1981) Lagrangian observations; single particle statistics – p.9/ ??

Statistics → Ocean is undersampled, time dependent In chaotic systems, trajectories depend sensitively on initial conditions → Statistical description is preferable Averages (positions, velocities, etc.) over many particles Lagrangian observations; single particle statistics – p.10/ ??

Statistics 1) Single particles 2) Multiple particles → Results in atmosphere and ocean Lagrangian observations; single particle statistics – p.11/ ??

Single particle statistics • Means, diffusivities • PDFs • Non-cartesian coordinates • Euler-Lagrange relation Lagrangian observations; single particle statistics – p.12/ ??

Single particle statistics Taylor (1921), Batchelor and Townsend (1953) Probability of particle at position x is P ( x, t ) Displacement PDF: Q ( x, t | x 0 , t 0 ) then: � P ( x, t ) = P ( x 0 , t 0 ) Q ( x, t | x 0 , t 0 ) dx 0 Lagrangian observations; single particle statistics – p.13/ ??

1 particle statistics If one particle ( P ( x 0 , t 0 ) = 1 in volume, V) P ( x, t ) = V Q ( x, t | x 0 , t 0 ) If homogeneous: Q ( x, t | x 0 , 0) = Q ( x − x 0 , t ) ≡ Q ( X, t ) Lagrangian observations; single particle statistics – p.14/ ??

1 particle statistics First moment is the mean displacement: � X ( t ) = XQ ( X, t ) dX Second is the dispersion: � X 2 Q ( X, t ) dX X 2 ( t ) = The diffusivity: � t d dtX 2 = 2 Xu = 2 u ( t ′ ) u ( t ) dt ′ 0 Lagrangian observations; single particle statistics – p.15/ ??

1 particle statistics If the flow is stationary: � t � t d dtX 2 = 2 u 2 R ( t ′ ) dt ′ ≡ 2 ν 2 R ( t ′ ) dt ′ 0 0 so that � t X 2 ( t ) = 2 ν 2 ( t − t ′ ) R ( t ′ ) dt ′ 0 Lagrangian observations; single particle statistics – p.16/ ??

1 particle statistics At early times R ( t ′ ) ≈ 1 X 2 ( t ) ≈ ν 2 t 2 At late times, if integrals converge � ∞ � ∞ R ( t ′ ) dt ′ − 2 ν 2 X 2 ( t ) ≈ 2 ν 2 t t ′ R ( t ′ ) dt ′ 0 0 or: X 2 ( t ) ∝ t Lagrangian observations; single particle statistics – p.17/ ??

1 particle statistics � ∞ The integral time is T L ≡ 0 R ( t ′ ) dt ′ Lagrangian frequency spectrum: � ∞ T ( ω ) = 2 R ( t ) cos (2 πωt ) dt 0 Lagrangian observations; single particle statistics – p.18/ ??

Advection-diffusion Davis (1987, 1991) ∂ ∂tC + U · ∇ C = −∇· < u ′ C ′ > = ∇ · ( κ ∇ C ) If have particle statistics, can derive U , κ Because ocean is inhomogeneous: U = U ( x, y, z ) , κ ( x, y, z ) Lagrangian observations; single particle statistics – p.19/ ??

Means Time average ← ensemble average instantaneous velocities in geographical bins: Lagrangian observations; single particle statistics – p.20/ ??

Means North Atlantic (Richardson, 1981) Lagrangian observations; single particle statistics – p.21/ ??

Means North Atlantic (Richardson, 1981) Lagrangian observations; single particle statistics – p.22/ ??

Means Jakobsen et al., 2003 Lagrangian observations; single particle statistics – p.23/ ??

Diffusivity Davis, 1991 If velocities are quasi-Gaussian, can calculate κ as: κ ( x , t ) = 1 d dt < d ′ j ( x , t | x 0 , t 0 ) d ′ k ( x , t | x 0 , t 0 ) > 2 or κ ( x , t ) = − < v ′ j ( x ′ , t | x , t 0 ) d ′ k ( x ′′ , − t | x , t 0 ) > Lagrangian observations; single particle statistics – p.24/ ??

Diffusivity here v ′ ( t 0 ) = v ( t 0 ) − U, d ′ ( t ) = d ( t ) − D ( t ) are the residual velocity and displacement relative to the (local) means. Note: assume quantities constant (at least for experiment, e.g. 2 years) Lagrangian observations; single particle statistics – p.25/ ??

Diffusivity Swenson and Niiler, 1998 (California Current) Lagrangian observations; single particle statistics – p.26/ ??

Diffusivity Zhurbas and Oh, 2004 Lagrangian observations; single particle statistics – p.27/ ??

Diffusivity Zhurbas and Oh, 2004 Lagrangian observations; single particle statistics – p.28/ ??

Refinements Improve estimates if use splines to determine means (Bauer et al., 1998) Lagrangian observations; single particle statistics – p.29/ ??

An easier way? Use (eddy) diffusivities in models, but must sample entire ocean Alternately, relate diffusivity to the rms or mean square velocity • κ ∝ ν 2 T • κ ∝ νL Assuming T or L is constant... Lagrangian observations; single particle statistics – p.30/ ??

Diffusivity Zhang et al., 2001 Lagrangian observations; single particle statistics – p.31/ ??

Diffusivity Lumpkin et al., 2002 ( L L ≡ νT L ) Time, length scales not constant Lagrangian observations; single particle statistics – p.32/ ??

Stochastic models Alternate to the A-D formalism is to write a random walk routine for particles Zeroth order model: random walk dx = n x dW dy = n y dW • Einstein limit: observation time > particle step time • Particle motion is diffusive Lagrangian observations; single particle statistics – p.33/ ??

Stochastic models First order model: dx = ( u + U ) dt, dy = ( v + V ) dt du = − 1 dv = − 1 udt + n x dW, vdt + n y dW T x T y Lagrangian observations; single particle statistics – p.34/ ??

First order model • X 2 ∝ t 2 as t → 0 • X 2 ∝ t as t → ∞ • include U ( x, y ) , V ( x, y ) , etc. Determine parameters from data (e.g. Griffa et al., 1995) Lagrangian observations; single particle statistics – p.35/ ??

Summary A-D • Evaluate U , κ from data • Use for transport studies (e.g. plankton) • Large scale coverage • More and more refined Lagrangian observations; single particle statistics – p.36/ ??

PDFs Central quantity to single particle statistics is displacement PDF, Q ( X, t ) Closely related is velocity PDF, P ( u ′ , t ) The advective-diffusive formalism of Davis (1991) assumes that Q ( X, t ) and P ( u ′ , t ) are approximately Gaussian Lagrangian observations; single particle statistics – p.37/ ??

Example: point vortices Jimenez, 1996; Weiss et al., 1998 One point vortex: P ( u ) ∝ u − 3 With many vortices, expect Gaussian PDF from central limit theorem � ∞ 0 u 2 P ( u ) du diverges logarithmically but → Finite number vortices: PDF has extended tails Lagrangian observations; single particle statistics – p.38/ ??

PDFs Bracco et al., 2000 z > −1000 m 70 60 50 40 30 20 10 0 −10 −20 280 300 320 340 360 z < −1000 m 70 60 50 40 30 20 10 0 −10 −20 280 300 320 340 360 Lagrangian observations; single particle statistics – p.39/ ??

PDFs How to calculate PDFs? Floats drift over large areas with different means and variances. • Divide region into geographic bins • De-mean and normalize locally • Recombine the normalized velocities regionally to generate single PDFs Lagrangian observations; single particle statistics – p.40/ ??

Shallow west NA Western North Atlantic (z > −1000 m) 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 Zonal velocity Meridional velocity Energetic events 60 55 50 45 40 35 30 25 20 15 280 290 300 310 320 330 340 Lagrangian observations; single particle statistics – p.41/ ??

Deep east NA Eastern North Atlantic (z < −1000 m) 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 Zonal velocity Meridional velocity Energetic events 60 55 50 45 40 35 30 25 20 15 315 320 325 330 335 340 345 350 355 360 Lagrangian observations; single particle statistics – p.42/ ??

2-D turbulence ν = 5e−8 0 10 −1 10 −2 10 −3 10 −4 10 −6 −4 −2 0 2 4 6 ν = 5e−9 0 10 −1 10 −2 10 −3 10 −4 10 −6 −4 −2 0 2 4 6 Velocity Lagrangian observations; single particle statistics – p.43/ ??