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

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Lagrangian observations; single particle statistics

J. H. LaCasce

Norwegian Meteorological Institute Oslo, Norway

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History

Stommel and Arons (1960)

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History

John Swallow

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History

Swallow floats

Confirmed Deep Western Boundary Current
Did not find sluggish poleward flow
Discovered deep (mesoscale) eddies

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Instruments

1) Atmosphere

Balloons
Constant level (pressure)
Tracked by satellite

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Instruments

2) Ocean

Surface drifters

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Instruments

Subsurface floats

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Lagrangian data

SOFAR floats in meddies

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Data

North Atlantic (Richardson, 1981)

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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

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Statistics

1) Single particles 2) Multiple particles → Results in atmosphere and ocean

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Single particle statistics

Means, diffusivities
PDFs
Non-cartesian coordinates
Euler-Lagrange relation

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Single particle statistics

Taylor (1921), Batchelor and Townsend (1953) Probability of particle at position x is P(x, t) Displacement PDF: Q(x, t|x0, t0) then: P(x, t) =

P(x0, t0)Q(x, t|x0, t0)dx0

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1 particle statistics

If one particle (P(x0, t0) = 1 in volume, V) P(x, t) = V Q(x, t|x0, t0) If homogeneous: Q(x, t|x0, 0) = Q(x − x0, t) ≡ Q(X, t)

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1 particle statistics

First moment is the mean displacement: X(t) =

XQ(X, t)dX

Second is the dispersion: X2(t) =

X2Q(X, t)dX

The diffusivity: d dtX2 = 2Xu = 2 t u(t′)u(t)dt′

Lagrangian observations; single particle statistics – p.15/??

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1 particle statistics

If the flow is stationary: d dtX2 = 2u2 t R(t′)dt′ ≡ 2ν2 t R(t′)dt′ so that X2(t) = 2ν2 t (t − t′)R(t′)dt′

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1 particle statistics

At early times R(t′) ≈ 1 X2(t) ≈ ν2t2 At late times, if integrals converge X2(t) ≈ 2ν2t ∞ R(t′)dt′ − 2ν2 ∞ t′R(t′)dt′

r: X2(t) ∝ t

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1 particle statistics

The integral time is TL ≡ ∞

0 R(t′)dt′

Lagrangian frequency spectrum: T(ω) = 2 ∞ R(t)cos(2πωt) dt

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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)

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Means

Time average ← ensemble average instantaneous velocities in geographical bins:

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Means

North Atlantic (Richardson, 1981)

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Means

North Atlantic (Richardson, 1981)

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Means

Jakobsen et al., 2003

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Diffusivity

Davis, 1991 If velocities are quasi-Gaussian, can calculate κ as: κ(x, t) = 1 2 d dt < d′

j(x, t|x0, t0) d′ k(x, t|x0, t0) >

r

κ(x, t) = − < v′

j(x′, t|x, t0) d′ k(x′′, −t|x, t0) >

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Diffusivity

here v′(t0) = v(t0) − 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)

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Diffusivity

Swenson and Niiler, 1998 (California Current)

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Diffusivity

Zhurbas and Oh, 2004

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Diffusivity

Zhurbas and Oh, 2004

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Refinements

Improve estimates if use splines to determine means (Bauer et al., 1998)

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An easier way?

Use (eddy) diffusivities in models, but must sample entire ocean Alternately, relate diffusivity to the rms or mean square velocity

κ ∝ ν2T
κ ∝ νL

Assuming T or L is constant...

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Diffusivity

Zhang et al., 2001

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Diffusivity

Lumpkin et al., 2002 (LL ≡ νTL) Time, length scales not constant

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Stochastic models

Alternate to the A-D formalism is to write a random walk routine for particles Zeroth order model: random walk dx = nxdW dy = nydW

Einstein limit: observation time > particle step

time

Particle motion is diffusive

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Stochastic models

First order model: dx = (u + U) dt, dy = (v + V ) dt du = − 1 Tx udt + nxdW, dv = − 1 Ty vdt + nydW

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First order model

X2 ∝ t2 as t → 0
X2 ∝ t as t → ∞
include U(x, y), V (x, y), etc.

Determine parameters from data (e.g. Griffa et al., 1995)

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Summary A-D

Evaluate U, κ from data
Use for transport studies (e.g. plankton)
Large scale coverage
More and more refined

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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

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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 but ∞

0 u2P(u) du diverges logarithmically

→ Finite number vortices: PDF has extended tails

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PDFs

Bracco et al., 2000

280 300 320 340 360 −20 −10 10 20 30 40 50 60 70

z > −1000 m

280 300 320 340 360 −20 −10 10 20 30 40 50 60 70

z < −1000 m

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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

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Shallow west NA

280 290 300 310 320 330 340 15 20 25 30 35 40 45 50 55 60

Energetic events

Western North Atlantic (z > −1000 m)

−6 −4 −2 2 4 6 10

−5

10

−4

10

−3

10

−2

10

−1

10

Zonal velocity

−6 −4 −2 2 4 6 10

−5

10

−4

10

−3

10

−2

10

−1

10

Meridional velocity

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Deep east NA

−6 −4 −2 2 4 6 10

−5

10

−4

10

−3

10

−2

10

−1

10

Zonal velocity

−6 −4 −2 2 4 6 10

−5

10

−4

10

−3

10

−2

10

−1

10

Meridional velocity

315 320 325 330 335 340 345 350 355 360 15 20 25 30 35 40 45 50 55 60

Energetic events

Eastern North Atlantic (z < −1000 m)

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2-D turbulence

−6 −4 −2 2 4 6 10

−4

10

−3

10

−2

10

−1

10

ν = 5e−8

−6 −4 −2 2 4 6 10

−4

10

−3

10

−2

10

−1

10

ν = 5e−9

Velocity Lagrangian observations; single particle statistics – p.43/??

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Gaussianity

When is a PDF non-Gaussian? → Goodness-of-fit tests Kolmogorov-Smirnov Test (supremum statistic) Anderson-Darling Test (quadratic statistic)

C(u) u D 1

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NA stats

Data set # data points K α (merid) WA, deep 37,658 5/4.2 0.008/0.019 WA, up 54,372 4.3/4.5 0.000/0.000 EA, deep 38,759 6.6/5.8 0.001/0.002 EA, up 13,502 4.3/4.5 0.19/0.30 Eq 33,461 3.7/3.6 0.13/0.11 2-D Turb 1,024,000 4.2 0.000

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Gaussian?

Altimeter data: Gaussian PDFs (Gille and Llewellyn-Smith, 2000) Float experiment in NW Atlantic: Gaussian PDFs (Zhang et al., 2001)

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Gaussian?

No : North Atlantic floats Yes : Some floats, satellite altimeter Lagrangian and Eulerian velocity PDFs should be same (Tennekes and Lumley, 1972) But some magic required with both floats and satellites

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Current meters

LaCasce, 2004

−80 −75 −70 −65 −60 −55 −50 −45 −40 −35 10 15 20 25 30 35 40 45 50 55 60

Western Atlantic moorings, 500 m

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Shallow NA

−6 −4 −2 2 4 6 10

−4

10

−3

10

−2

10

−1

10

ku=3.6, kv=3.4

u/std(u) Western Atlantic moorings, 500 m

Zonal Meridional Gaussian

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Floats/meters

−80 −75 −70 −65 −60 −55 −50 −45 −40 −35 10 15 20 25 30 35 40 45 50 55 60

Western Atlantic floats, z < 1000 m

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Floats/meters

−6 −4 −2 2 4 6 10

−4

10

−3

10

−2

10

−1

10

α=0.915

Western Atlantic moorings + floats, 500 m

Meridional velocity

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Gaussianity

Why didn’t others see non-Gaussianity? Extreme events rare—need long time series

10

2

10

3

10

4

10

5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

50% 90% 99%

Anderson−Darling statistics for 500m Atlantic Current meters

Days

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Gaussianity

Bracco et al., 2003: MICOM NA simulation

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Summary, PDFs

Velocity PDFs weakly non-Gaussian
Lagrangian and Eulerian PDFs same
Can determine best bin size

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Alternate coordinates

Ocean is inhomogeneous, anisotropic. Transport might be impeded by:

β-effect
Topography
Other large scale PV gradients

How to detect sensitivity? What if it is sensitive?

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Alternate coordinates

Project displacements along and across specified field. If field constrains motion, dispersion should be anisotropic in the projected coordinates.

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Shallow water equations

∂ ∂tu − fv = −g ∂ ∂xη + τx − ru ∂ ∂tv + fu = −g ∂ ∂yη + τy − rv ∇ · (H u) = 0 Vorticity: ∂ ∂t∇ × u + J(ψ, f H ) = ∇ × τ − r∇ × u → f/H constrains flow

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f/H dispersion

LaCasce, 2000

−70 −60 −50 −40 −30 −20 −10 20 25 30 35 40 45 50 55 60 10 20 30 40 50 −10 10 20 30 40 50 60 70

Day Mean positions (km)

10 20 30 40 50 1 2 3 4 5 6 7 8 9 x 10

4

Day Dispersion (km2)

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f/H dispersion

10 20 30 40 50 −20 −15 −10 −5 5 10 15 20

Day Mean positions (km)

10 20 30 40 50 1000 2000 3000 4000 5000 6000 7000 8000

Day Dispersion (km2)

−70 −60 −50 −40 −30 −20 −10 20 25 30 35 40 45 50 55 60

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f/H dispersion

−70 −60 −50 −40 −30 −20 −10 20 25 30 35 40 45 50 55 60 10 20 30 40 50 −50 50 100 150 200

Day Mean positions (km)

10 20 30 40 50 2 4 6 8 10 x 10

4

Day Dispersion (km2)

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Test

Compare to stochastic “floats” dx = (u + U) dt, dy = (v + V ) dt du = − 1 Tx udt + nxdW, dv = − 1 Ty vdt + nydW Match the zonal/meridional dispersion to observed.

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Stoch. NW Atl.

10 20 30 40 50 −50 50 100 150 200

Day Mean positions (km)

10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 x 10

4

Day Dispersion (km2)

−70 −60 −50 −40 −30 −20 −10 20 25 30 35 40 45 50 55 60

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Alternate coordinates

NA dispersion sensitive to topography
Pacific dispersion zonal
Exploit with f/H stochastic models

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Euler-Lagrange

Helpful Lagrangian analysts want to give modellers diffusivities But Lagrangian and Eulerian diffusivities are generally different For example, typical Eulerian time scale in North Atlantic 10 days, Lagrangian time scale 5 days Can we relate κL to κE?

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Euler-Lagrange

General idea: floats experience Eulerian spatial and temporal decorrelations simultaneously

x t R

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Euler-Lagrange

Recall: d dt < X2 >= 2 t RL(t′)dt′ Corrsin’s (1959) conjecture: RL(t) = RE11(r, t) Q(r, t) dr RE11(r, t) = ∞ E(k)[1 − J0(kr)]D(k, t) dk

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Euler-Lagrange

Middleton (1985)

Assume PDF (Q = exp(−k2r2/2))
Assume form for D(k, t)
Assume E(k) ∝ k−n

Derive RL(t) (hence diffusivity)

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Euler-Lagrange

1 2 d2 dt2X2 =

E(k) D(k, t)×

[1 − J0(kX)] exp(−k2X2/2) dk Results depend on α ≡ TE/Ta where TE is the integral time Ta ≡ L/ν is the advective time

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Euler-Lagrange

If α << 1 (“fixed float”), TL ≈ TE If α >> 1 (“frozen field”), TL/TE ≈ q (q2 + α2)1/2, q = (π/8)1/2 Result is relatively insensitive to the choice of D(k, t)

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Euler-Lagrange

Lumpkin et al., 2002: 1/3 Degree NA model

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Summary

Means, diffusivities nearing a fine art
Lagrangian PDFs nearly (but not) Gaussian
Oceanic floats are sensitive to f/H
Corrsin’s conjecture promising

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