Relative dispersion in the Gulf Stream and its recirculation

Relative dispersion in the Gulf Stream and its recirculation

Rick LumpkinRick Lumpkin(Rick.Lumpkin@noaa.gov)(Rick.Lumpkin@noaa.gov)

National Oceanic and Atmospheric Administration (NOAA)Atlantic Oceanographic and Meteorological Laboratory (AOML)

Miami, Florida USA

CLIMODE PI workshop, 6-7 August 2008

.)('2 tx

Ensemble average

x

U t

x’

Dispersion:

Richardson’s 4/3 law

Richardson (1926): observed smoke spreading from a stack. Realized that diffusion must be scale-dependent (bigger at larger separation distances). Proposed

Obukov (1941): Richardson’s law is a result of energy cascade from large to small scales (inertial subrange) in 3D turbulence.

.3/4rmsx

En

erg

y in

pu

t

Wavenumber k

En

erg

y E

(k)

Energy cascade

Enstrophy cascade

3k

3/5k2D turbulence: energy cascade to large scale, enstrophy cascade to small scale (Kraichnan, 1967). Richardson’s law followed in energy cascade range; exponential growth of particle separation in enstrophy cascade range (Lin, 1972).

Finite Scale Lyapunov Exponents (FSLEs)

x

U t

Separation distance

Pick such that growth of is given by ).exp(0 t

(Exponential growth if is constant, but more generally can vary with .)

FSLEs, continued.

Over interval (n, n+1) in which is approximately constant,

)./ln()( 11 nnnn tt

For n+1 = n, this becomes:

nn t

ln

)(

where tn is the mean time for the separation distance to grow from n to n.

Unlike dispersion, which averages the data in time, this approach averages the data in separation distance.

Dispersion regimes

D2(t) Regime

exp(0t) 0 exponential

t2 ballistic

t3 Richardson

t diffusive

Relative Dispersion FSLE Dispersion

From Haza et al., 2007

Relative dispersion observations in the oceanLaCasce and Bower, 2000: float pairs in the western North Atlantic. Dispersion follows Richardson’s law from the smallest resolvable distance (>deformation radius of 20km) to 60—100km.

LaCasce and Ohlmann, 2003: drifter pairs in the Gulf of Mexico.

Separation is exponential at scales smaller than deformation

radius(~45km). Richardson law behavior at larger scales.

)(

)ln()(

t

Limitations of earlier data LaCasce and Bower (2000), LaCasce and Ohlmann (2003) were forced to rely on chance pairs. Floats: not enough chance pairs at distances smaller than 1st Baroclinic Rossby radius.Drifters: Dense array allowed resolution at smaller scales, but Argos positioning system provided only a few fixes per day on average, with gaps as long as a day common.

Do chance pairs present an unbiased sample of the statistics of the turbulent field? This cannot be tested without intentional pairs: pairs launched close to each other at various points in the turbulent field.

What is the effect of undersampling in time? Higher frequency data is needed. Argos multisatellite processing: introduced January 2005. Mean time between fixes decreased from 6 hours to 1 hour.

Drifter observations during February—March 2007 cruise, R/V Knorr

Goal: measure dispersion, eddy fluxes

CLIMODE observations

60 drifters deployed: 16 trios, 6 pairs.

Median spacing of satellite fixes: 1.2 hours

60 drifters deployed: 16 trios, 6 pairs. One drifter failed.

Median spacing of satellite fixes: 1.2 hours

dispersion

rms displacement: 1.5km

300-500km

55 pairs with earliest fixes <700m

Solid black: zonal. Dashed black: meridional.

Grey dashed: D2=(3.5109 m2 s3 )t3

(Richardson’s Law)

Noise level of Argos positioning

Ro2

(5.8104 m2 /s)

(2.9104 m2 /s)

Evidence of exponential behavior at short times?

Dashed black:.

95% confidence

Ro2

FSLEs

Stars: methodology of LaCasce (first crossing approach).

Circles: methodology of Haza, Özgökmen (fastest crossing approach).

Methodologies converge at large scales. Slopes very different at intermediate scales.

nn t

ln

)(

Neither approach indicates exponential behavior (constant ) from the smallest scales to the first baroclinic Rossby radius, ~30 km (Chelton et al., 1998).

Early behavior (<1.5km)

.2urmseff Tu

~rmsu 1—2 m/s

~uT 5—20 s

eff 25 m2/s

Long time behavior (>300km)

Diffusive behavior, governed by a two-particle diffusivity of K=3—6

104 m2/s at separation scales greater than 300—500 km. This is consistent with a single-particle effective diffusivity of eff=1.5—3 104 m2/s.

Single-particle diffusivities

Davis (1991):

Zhurbas and Oh (2003): Use minor principle component for robust scalar lateral diffusivity in presence of mean shear.

.),|(),|(),( 00'

00' tttdttvt kjjk xxx

Left: single-particle diffusivity from 1500 unique drifters in the Gulf Stream and recirculation region, 1989—present.

Pair dispersion: eff=1.5—3 104 m2/s. Comparison suggests that mean shear amplifies zonal pair spreading, but not meridional spreading, to lowest order.

Mean interpolated onto CLIMODE drifter positions:

1.6104 m2/s (std.dev. 7103)

Mean semimajor axis:

5.8 104 m2/s.

1.5 km—300 km:

Then diffusion .2

3'

d

d

2

1 3/43/12rmsxax

t

,' 2/32 axxrms .1043

29

s

ma

Lagrangian structure function vs. separation distance for 55 CLIMODE drifter pairs. Inertial range behavior is seen for separations from 1.5-300km.

2

21 )(d

d

xx

t

Intermediate behavior

Why no enstrophy cascade in Gulf Stream recirculation? (Why so different from Gulf of Mexico drifters of LaCasce and Ohlmann, 2003?)

Hypothesis 1: there isn’t an observable enstrophy cascade in CLIMODE region at these scales (with respect to dispersion).

• Significant energy input at a scale of 1-2 km (2—4x mixed layer depth) to the first baroclinic Rossby radius. Mixed layer submesoscale turbulence. This is overwhelming an enstrophy cascade from larger scales.

• Richardson’s Law scaling may not be due to energy cascade. E.g., Bowden, 1965: 4/3 law behavior can be caused by small-scale mixing superimposed on large-scale shear.

Test of hypothesis 1a: in a more quiet part of the ocean, away from the energetic Gulf Stream region, drifters will behave more like LaCasce and Ohlmann’s Gulf of Mexico drifters and demonstrate enstrophy cascade-like behavior at scales smaller than 1st BC.

Eastern subtropical Atlantic drifters

Drifters deployed as part of a 2005—2006 comparison study of drifters from different manufacturers.

All drifters deployed within a few meters of each other.

18 drifter pairs had initial separation distances less than 700m (accuracy of Argos positioning).

Eastern subtropical Atlantic drifters

Why no enstrophy cascade in Gulf Stream recirculation? (Why so different from Gulf of Mexico drifters of LaCasce and Ohlmann, 2003?)

Hypothesis 2: chance pairs (like in LaCasce and Ohlmann) present a biased sampling of the statistics of the turbulent field.

• Where energetic submesoscale features exist, they may prevent chance encounters. Convergent regions may be characterized by a steeper wavenumber spectral slope.

Test of hypothesis 2: repeat study for chance pairs in the Gulf Stream region.

Gulf Stream chance pairs

29 chance pairs in the region 25—45°N, 40—80°W, 2005—2007, that came within 10 km of each other (bullets). Trajectories before (light grey) and after (dark grey) closest approach are also shown.

Only 9 pairs came within 700m of each other.

Gulf Stream chance pairs

Why so different from Gulf of Mexico drifters of LaCasce and Ohlmann, 2003?

Hypothesis 3: Increased temporal resolution of these data, due to multisatellite processing introduced since LaCasce and Ohlmann (2003).

• Some transitions from to are extremely fast, even for relatively large . These would be missed at daily resolution, and lead to smaller FSLEs.

Test of hypothesis 3: repeat study for CLIMODE drifters subsampled to daily resolution.

LaCasce (2008, in press): original Gulf of Mexico data, daily resolution (open white stars). Interpolated to higher resolution (stars, triangles): plateau of constant shifts to very small scales.

CLIMDE drifter FSLEs, daily resolution

Conclusions

• As part of CLIMODE, an array of 60 drifters were deployed in February and March 2007 to resolve relative dispersion, mixing and stirring in the Gulf Stream and its recirculation.

• Drifters collected velocity and SST measurements at ~hourly resolution.

• Relative dispersion consistent with Richardson’s Law behavior at separation of 1.5—300 km. At larger separation, pairs exhibit diffusive spreading with effective eddy diffusivities of 1—3 x 104 m2/s.

• No evidence of enstrophy cascade at sub-deformation scales.

• Most likely reason: significant energy input at submesoscale, via frontal and mixed layer instabilities.

• This appears to be a ubiquitous characteristic of the ocean, even away from the Gulf Stream front, as suggested by eastern subtropical Atlantic drifters.

• Earlier results consistent with QG turbulence expectations at sub-mesoscale were affected by temporal resolution of those data.