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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 )

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SLIDE 1

Lagrangian observations; multi-particle statistics

  • J. H. LaCasce

Norwegian Meteorological Institute Oslo, Norway

Lagrangian observations; multi-particle statistics – p.1/58

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Single particle stats

Previously noted that, as t → 0: X2(t) ≈ ν2t2 and as t → ∞: X2(t) ∝ 2ν2TLt So single particle dispersion is relatively insensitive to details of the flow

Lagrangian observations; multi-particle statistics – p.2/58

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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

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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

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Relative separations

Two particles, of volume V , at x1 and x2 Define x ≡ x1 and y ≡ x2 − x1. Then P(x, y, t) dxdy = V 2 The probability of separation, y, is then: p(y, t) = 1 V

  • P(x, y, t) dx

Lagrangian observations; multi-particle statistics – p.5/58

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Relative separations

As before, can relate to a (joint) displacement PDF: P(x, y, t) =

  • P(x′, y′, t)Q(x, y, t|x′, y′, t0)dx′dy′

Integrating over space: q(y, t|y′, t0) =

  • Q(x, y, t|x′, y′, t0) dx

Lagrangian observations; multi-particle statistics – p.6/58

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Relative separations

So p(y, t) =

  • p(y′, t0)q(y, t|y′, t0) dy′

Richardson (1926) called q(y, t|y′, t0) the “distance-neighbour function” Can define dispersion: y2 =

  • y2q(y, t|y0, t0)dy

And similarly relative diffusivity, κ2 ≡ d

dty2, etc.

Lagrangian observations; multi-particle statistics – p.7/58

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Structure function

How does the dispersion evolve? Assume Eulerian flow and examine consequences. Lagrangian second order structure function: ( d dty)2 = (v1 − v2)2 = v2

1 + v2 2 − 2v1v2

Lagrangian observations; multi-particle statistics – p.8/58

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Large scales

At scales larger than the “energy-containing eddies”: ( d dty)2 → 2ν2, κ2 = d dty2 = 2κ1

Lagrangian observations; multi-particle statistics – p.9/58

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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 − J0(ky)] dk Assume E(k) ∝ k−α

Lagrangian observations; multi-particle statistics – p.10/58

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Intermediate scales

For larger scales: 1 − J0(ky) ≈ 1 4k2y2, ky ≪ 1 For smaller scales: 1 − J0(ky) ≈ 1 + O(ky)−1/2, ky ≫ 1

Lagrangian observations; multi-particle statistics – p.11/58

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Intermediate scales

So: ( d dty)2 ≈ 2 1/y k−α(1 4k2y2) dk + 2 ∞

1/y

k−α dk

  • r

= 1 2y2 1 3 − αk3−α|1/y + 2 1 − αk1−α|∞

1/y

First diverges if α ≥ 3, second if α ≤ 1

Lagrangian observations; multi-particle statistics – p.12/58

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Local dispersion

if 1 < α < 3 ( d dty)2 ∝ yα−1 The corresponding diffusivity is: κ2 = d dty2 ∝ y(α+1)/2 “Richardson’s Law” obtains if α = 5

3:

d dty2 ∝ y4/3

Lagrangian observations; multi-particle statistics – p.13/58

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Non-local dispersion

if α > 3 ( d dty)2 ≈ 1 2y2 ∞ k2E(k) dk = c1Ωy2 with diffusivity: κ2 = d dty2 ≈ c2T −1y2 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

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PDFs

Richardson proposed: ∂ ∂tp(y, t) = y−1 ∂ ∂y(yκ2 ∂ ∂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

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Kurtoses

Then can show that the kurtosis: ku(y) ≡

  • y4p(y) dy

(

  • y2p(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

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2-D turbulence

Energy injected Enstrophy Energy k^(−3) k^(−5/3) E(k) k

Lagrangian observations; multi-particle statistics – p.17/58

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2-D turbulence

Energy cascade: y2 ∝ t3, κ2 ∝ y4/3, ku(y) = C Enstrophy cascade: y2 ∝ exp(c3η1/3t), κ2 ∝ y2, ku(y) ∝ exp(c4T −1t)

Lagrangian observations; multi-particle statistics – p.18/58

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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

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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

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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

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Diffusivity

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

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Dispersion

Morel and Larcheveque, 1974

Lagrangian observations; multi-particle statistics – p.23/58

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Dispersion

Er-El and Peskin, 1981

Lagrangian observations; multi-particle statistics – p.24/58

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Dispersion

Er-El and Peskin, 1981

Lagrangian observations; multi-particle statistics – p.25/58

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Isotropy

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

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Relative velocity

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

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Relative kurtosis

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

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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

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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): κ2 ∝ Dn with 4/3 ≤ n ≤ 2 with floats in the western North Atlantic

Lagrangian observations; multi-particle statistics – p.30/58

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Subsurface ocean

LaCasce and Bower (2000): North Atlantic

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

NAC AMUSE ACCE SiteL/LDE

Lagrangian observations; multi-particle statistics – p.31/58

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Floats

Experiment D0 ≤ 7.5 D0 ≤ 15 D0 ≤ 30 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

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Eastern Atlantic

10 20 30 40 50 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 x 10

4

Day

  • Rel. Disp. (km

2)

10 20 30 40 50 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 x 10

4

Day

  • Abs. Disp. (km

2)

10 10

1

10

2

10

3

10

2

10

3

10

4

10

5

D1 2 K(1)

Distance (km) Diffusivity (m2/sec) AMUSE Lagrangian observations; multi-particle statistics – p.33/58

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SLIDE 34

Western Atlantic

10 10

1

10

2

10

3

10

2

10

3

10

4

10

5

2 K(1) D4/3

Distance (km) Diffusivity (m2/sec) NAC

10 20 30 40 50 −4 −2 2 4 6 8 10 12 14 16 x 10

4

Day

  • Abs. Disp. (km

2)

10 20 30 40 50 −4 −2 2 4 6 8 10 12 14 16 x 10

4

Day

  • Rel. Disp. (km2)

Lagrangian observations; multi-particle statistics – p.34/58

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Relative velocities

10 20 30 40 50 0.5 1 1.5

AMUSE <u2>,<v2>

10 20 30 40 50 0.5 1 1.5

ACCE

10 20 30 40 50 0.5 1 1.5

NAC <u2>,<v2>

10 20 30 40 50 0.5 1 1.5

SiteL Day

10 20 30 40 50 0.5 1 1.5

LDE 1300 <u2>,<v2> Day Lagrangian observations; multi-particle statistics – p.35/58

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Relative kurtoses

5 10 15 20 25 30 35 40 45 50 1 2 3 4 5 6 7 8 9 10

Kurtosis Day Kurtoses of relative displacements

AMUSE NAC ACCE LDE1300 SiteL

Lagrangian observations; multi-particle statistics – p.36/58

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Subsurface ocean

  • Richardson growth in west
  • Diffusive growth in east
  • Small scales (r < LD ≈ 20km) not well-resolved

Lagrangian observations; multi-particle statistics – p.37/58

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Surface ocean

LaCasce and Ohlmann (2003): SCULP (Niiler) 140 pairs with r0 ≤ 1 km

260 262 264 266 268 270 272 274 276 278 280 24 25 26 27 28 29 30 31 32

SCULP2 pairs

260 262 264 266 268 270 272 274 276 278 280 24 25 26 27 28 29 30 31 32

SCULP1 pairs

Lagrangian observations; multi-particle statistics – p.38/58

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Relative dispersion

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

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Isotropy

Lagrangian observations; multi-particle statistics – p.40/58

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Relative dispersion

Lagrangian observations; multi-particle statistics – p.41/58

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Relative velocities

Lagrangian observations; multi-particle statistics – p.42/58

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Relative kurtoses

Lagrangian observations; multi-particle statistics – p.43/58

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Surface of Gulf

  • Exponential growth r < 50km
  • Ballistic growth 50 < r <?km
  • No diffusion
  • But uncorrelated velocities later on

Lagrangian observations; multi-particle statistics – p.44/58

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Turbulence?

Tempting to attribute results to 2-D turbulence

Energy injected Enstrophy Energy k^(−3) k^(−5/3) E(k) k

Lagrangian observations; multi-particle statistics – p.45/58

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Not so fast

But similar statistics from other causes

  • Renovating Wave Model—exponential growth
  • Shear dispersion—cubic growth

So more studies required...

Lagrangian observations; multi-particle statistics – p.46/58

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Three particles

Number of drifters large enough (nearly) to consider triplets: what should they do? Small scale, constant strain (enstrophy cascade): exponential stretching (Batchelor, 1952)

Lagrangian observations; multi-particle statistics – p.47/58

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Three particles

Energy cascade: areal growth (mean square leg length increases as t3) and aspect ratio towards unity (Celani and Vergassola, 2001)

Lagrangian observations; multi-particle statistics – p.48/58

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Drifters

Lagrangian observations; multi-particle statistics – p.49/58

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Triangles: early

Lagrangian observations; multi-particle statistics – p.50/58

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Triangles: early

Lagrangian observations; multi-particle statistics – p.51/58

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Triangles: late

Lagrangian observations; multi-particle statistics – p.52/58

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Triangles: late

Lagrangian observations; multi-particle statistics – p.53/58

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Triangle sum

Phase 1

  • Exponential growth of long leg
  • Non-conservation of area
  • Aspect ratio increases

Phase 2

  • Power law growth of long leg
  • Aspect ratio decreases

Lagrangian observations; multi-particle statistics – p.54/58

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FSLEs

Aurell et al., 1997; Boffetta and Sokolov, 2002 Relative dispersion involves averaging separations at fixed times. An alternate approach is the Finite Scale Lyapunov Exponent, in which exit times are averaged for predefined distances. Advantage is that uses all available pairs—can greatly increase degrees of freedom

Lagrangian observations; multi-particle statistics – p.55/58

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Gulf drifters

Lagrangian observations; multi-particle statistics – p.56/58

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EOLE balloons

Lacorata et al. (in press)

Lagrangian observations; multi-particle statistics – p.57/58

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Summary

  • Exponential stretching in atmosphere,

L < 1000km

  • Exponential stretching in ocean, L < LD
  • Non-Gaussian separations (increasing kurtosis)
  • Large scale behavior uncertain: inhomogeneous,

anisotropic

  • Triple point correlations interesting future work

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