Skip to main content

Advertisement

SpringerLink
Log in

Deducing an upper bound to the horizontal eddy diffusivity using a stochastic Lagrangian model

  • Original Article
  • Published:
Environmental Fluid Mechanics Aims and scope Submit manuscript

Abstract

We present a method for estimating the upper bound of the horizontal eddy diffusivity using a non-stationary Lagrangian stochastic model. First, we identify a mixing barrier using a priori evidence (e.g., aerial photographs or satellite imagery) and using a Lagrangian diagnostic calculated from observed or modeled spatially non-trivial, time-dependent velocities [for instance, the relative dispersion (RD) or finite time Lyapunov exponent (FDLE)]. Second, we add a stochastic component to the observed (or modeled) velocity field. The stochastic component represents sub-grid stochastic diffusion and its mean magnitude is related to the eddy diffusivity. The RD of Lagrangian trajectories is computed for increasing values of the eddy diffusivity until the mixing barrier is no longer present. The value at which the mixing barrier disappears provides a dynamical estimate of the upper bound of the eddy diffusivity. The erosion of the mixing barrier is visually observed in numerical simulations, and is quantified by computing the kurtosis of the RD at each value of the eddy diffusivity. We demonstrate our method using the double gyre circulation model and apply it to high frequency (HF) radar observations of surface currents in the Gulf of Eilat.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Stommel H (1949) Horizontal diffusion due to oceanic turbulence. J Mar Res 13: 199–225

    Google Scholar 

  2. Burchard H, Bolding K (2001) Comparative analysis of four second-moment turbulence closure models for the oceanic mixed layer. J Phys Oceanogr 31: 1943–1967

    Article  Google Scholar 

  3. Majda AJ, Kramer PR (1999) Simplified models for turbulent diffusion: theory, numerical modeling, and physical phenomena. Phys Rep 314: 237–574

    Article  Google Scholar 

  4. Ross ON, Sharples J (2004) Recipe for 1-D Lagrangian particle tracking models in space-varying diffusivity. Limnol Oceanogr Methods 2: 289–302

    Google Scholar 

  5. Rueda FJ, Schladow SG, Clark JF (2008) Mechanisms of contaminant transport in a multi-basin lake. Ecol Appl 18: A72–A87

    Article  Google Scholar 

  6. Chen W, Liu W, Kimura N, Hsu M (2009) Particle release transport in Danshuei River estuarine system and adjacent coastal ocean: a modeling assessment. Environ Monit Assess. doi:10.1007/s10661-009-1123-2

  7. Marinone SG, Ulloa MJ, Parés-Sierra A, Lavín MF, Cudney-Bueon R (2008) Connectivity in the northern Gulf of California from particle tracking in a three-dimensional numerical model. J Mar Syst 71: 149–158

    Article  Google Scholar 

  8. Tilburg CE, Reager JT, Whitney MM (2005) The physics of blue crab larvae recruitment in Delaware Bay: a model study. J Mar Res 63: 471–495

    Article  Google Scholar 

  9. Xue H, Incze L, Xu D, Wolff N, Pettigrew N (2008) Connectivity of lobster (Homarus americanus) populations in the coastal Gulf of Maine: part I: circulation and larval transport potential. Ecol Model 210: 193–211

    Article  Google Scholar 

  10. Incze L, Xue H, Wolff N, Xu D, Wilson C, Steneck R, Wahle R, Lawton P, Pettigrew N, Chen Y (2010) Connectivity of lobster (Homarus americanus) populations in the coastal Gulf of Maine: part II. Coupled biophysical dynamics. Fish Ocean 19: 1–20

    Article  Google Scholar 

  11. Blumberg AF, Dunning DJ, Li H, Heimbuch D, Geyer WR (2004) Use of a particle-tracking model for predicting entrainment at power plants on the Hudson River. Estuaries 27: 515–526

    Article  Google Scholar 

  12. Davis RE (1991) Observing the general circulation with floats. Deep Sea Res 38: 531–571

    Article  Google Scholar 

  13. Bogucki DJ, Jones BH, Carr M (2005) Remote measurements of horizontal eddy diffusivity. J Atmos Ocean Technol 22: 1373–1380

    Article  Google Scholar 

  14. Ledwell JR, Watson AJ, Law CS (1998) Mixing of a tracer in the pycnocline. J Geophys Res 103(C10): 21,499–421,529

    Article  Google Scholar 

  15. Sundermeyer MA, Ledwell JR (2001) Lateral dispersion over the continental shelf: analysis of dye release experiments. J Geophys Res 106(C5): 9603–9621

    Article  Google Scholar 

  16. List J, Gartrell G, Winant CD (1990) Diffusion and dispersion in coastal waters. J Hydraul Eng 116: 1158–1179

    Article  Google Scholar 

  17. Sanderson BG, Booth DA (1991) The fractal dimension of drifter trajectories and estimates of horizontal eddy-diffusivity. Tellus 43A: 334–349

    Google Scholar 

  18. Zhurbas V, Sang Oh I (2003) Lateral diffusivity and Lagrangian scales in the Pacific Ocean as derived from drifter data. J Geophys Res. doi:10.1029/2002JC001596

  19. Kawabe M (2008) Vertical and horizontal eddy diffusivities and oxygen dissipation rate in the subtropical northwest Pacific. Deep Sea Res I 55: 247–260

    Article  Google Scholar 

  20. Haller G, Yuan G (2000) Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys D 147: 352–370

    Article  Google Scholar 

  21. Haller G (2001) Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys Fluids 13: 3365–3385

    Article  CAS  Google Scholar 

  22. Lipphardt BL, Small G, Kirwan AD, Wiggins S, Ide K, Grosch CE, Paduan JD (2006) Synoptic Lagrangian maps: application to surface transport in Monterey Bay. J Mar Res 64: 221–247

    Article  Google Scholar 

  23. Olascoaga MJ, Rypina II, Brown MG, Beron-Vera FJ, Kocak H, Brand LE, Halliwell GR, Shay LK (2006) Persistent transport barrier on the west Florida shelf. Geophys Res Lett. doi:10.1029/2006GL027800

  24. Boffetta G, Lacorata G, Radaelli G, Vulpiani A (2001) Detecting barriers to transport: a review of different techniques. Phys D 159(1–2): 58–70

    Article  Google Scholar 

  25. Berloff PS, McWilliams JC, Bracco A (2002) Material transport in oceanic gyres. Part I: Phenomenology. J Phys Oceanogr 32: 764–796

    Article  Google Scholar 

  26. Shadden SC, Lekien F, Marsden JE (2005) Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys D 212: 271–304

    Article  Google Scholar 

  27. Orre S, Gjevik B, Lacasce JH (2006) Characterizing chaotic dispersion in a coastal tidal model. Cont Shelf Res 26: 1360–1374

    Article  Google Scholar 

  28. d’Ovidio F, Fernandez V, Hernandez-Garcia E, Lopez C (2004) Mixing structure in the Mediterranean Sea from finite-size Lyapunov exponents. Geophys Res Lett. doi:10.1029/2004GL020328

  29. Lehahn Y, d’Ovidio F, Levy M, Heifetz E (2007) Stirring of the northeast Atlantic spring bloom: a Lagrangian analysis based on multisatellite data. J Geophys Res. doi:10.1029/2006JC03927

  30. Waugh DW, Abraham ER, Bowen MM (2006) Spatial variations of stirring in the surface ocean: a case study of the Tasman Sea. J Phys Oceanogr 36: 526–542

    Article  Google Scholar 

  31. Garcia-Olivares A, Isern-Fontanet J, Garcia-Ladona E (2007) Dispersion of passive tracers and finite-scale Lyapunov exponents in the Western Mediterranean Sea. Deep Sea Res 54: 253–268

    Article  Google Scholar 

  32. Samelson R (1992) Fluid exchange across a meandering jet. J Phys Oceanogr 22: 431–440

    Article  Google Scholar 

  33. Ridderinkhof H, Zimmerman J (1992) Chaotic stirring in a tidal system. Science 258: 1107–1111

    Article  CAS  Google Scholar 

  34. Joseph B, Legras B (2002) Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex. J Atmos Sci 59: 1198–1212

    Article  Google Scholar 

  35. Gildor H, Fredj E, Steinbuck J, Monismith S (2009) Evidence for submesoscale barriers to mixing in the ocean from current measurements and aerial photographs. J Phys Oceanogr 39: 1975–1983

    Article  Google Scholar 

  36. Lekien F, Coulliette C, Mariano AJ, Ryan EH, Shay LK, Haller G, Marsden J (2005) Pollution release tied to invariant manifolds: a case study for the coast of Florida. Phys D 210: 1–20

    Article  Google Scholar 

  37. Nakamura N (2008) Quantifying inhomogeneous, instantaneous, irreversible transport using passive tracer field as a coordinate. Lect Notes Phys 744: 137–164

    Article  CAS  Google Scholar 

  38. Haller G (2002) Lagrangian coherent structures from approximate velocity data. Phys Fluids 14: 1851–1861

    Article  CAS  Google Scholar 

  39. Lichtenberg AJ, Lieberman MA (1992) Regular and chaotic dynamics, 2nd edn, Applied Mathematical Sciences, vol 38. Springer, New York

    Google Scholar 

  40. Wiggins S (2005) The dynamical systems approach to Lagrangian transport in oceanic flows. Annu Rev Fluid Mech. doi:10.1146/annurev.fluid.37.061903.175815

  41. Koshel KV, Prants SV (2006) Chaotic advection in the ocean. Phys-Uspekhi. doi:10.1070/PU2006v049nl1ABEH006066

  42. Garrett C (2006) Turbulent dispersion in the ocean. Prog Oceanogr 70: 113–125

    Article  Google Scholar 

  43. Bauer S, Swenson MS, Griffa A, Mariano AJ, Owens K (1998) Eddy-mean flow decomposition and eddy-diffusivity estimates in the tropical Pacific Ocean, 1. Methodology. J Geophys Res 103: 30855–30871

    Article  Google Scholar 

  44. Pedlosky J (1987) Geophysical fluid dynamics. Springer, New York

    Google Scholar 

  45. LaCasce JH (2008) Statistics from Lagrangian observations. Prog Oceanogr 77: 1–29

    Article  Google Scholar 

  46. Ullman DS, O’Donnell J, Kohut J, Fake T, Allen A (2006) Trajectory prediction using HF radar surface currents: Monte Carlo simulations of prediction uncertainties. J Geophys Res. doi:10.1029/2006JC003715

  47. Gard TC (1988) Introduction to stochastic differential equations, pure and applied mathematics, vol 114. Marcel Dekker, New York

    Google Scholar 

  48. Yang H, Liu Z (1994) Chaotic transport in a double gyre ocean. Geophys Res Lett 21: 545–548

    Article  Google Scholar 

  49. Monismith SG, Genin A (2004) Tides and sea level in the Gulf of Aqaba (Eilat). J Geophys Res. doi:10.1029/2003JC002069

  50. Biton E, Silverman J, Gildor H (2008) Observations and modeling of a pulsating density current. Geophys Res Lett. doi:10.1029/2008GL034123

  51. Monismith SG, Genin A, Reidenbach MA, Yahel G, Koseff JR (2006) Thermally driven exchanges between a coral reef and the adjoining ocean. J Phys Oceanogr 36: 1332–1347

    Article  Google Scholar 

  52. Berman T, Paldor N, Brenner S (2000) Simulation of wind-driven circulation in the Gulf of Elat (Aqaba). J Mar Syst 26: 349–365

    Article  Google Scholar 

  53. Gildor H, Fredj E, Kostinski A (2010) The Gulf of Eilat/Aqaba: a natural driven cavity? Geophys Astron Fluid Dyn. doi:10.1080/03091921003712842

  54. Lekien, F, Coulliette C, Bank R, Marsden J (2004) Open-boundary modal analysis: interpolation, extrapolation, and filtering. J Geophys Res. doi:10.1029/2004JC002323

  55. Lekien F, Gildor H (2009) Computation and approximation of the length scales of harmonic modes with application to the mapping of surface currents in the Gulf of Eilat. J Geophys Res. doi:10.1029/2008JC004742

  56. Hathorn WE (1996) A second look at the method of random walks. Stoch Hydrol Hydraul 10: 319–329

    Article  Google Scholar 

  57. Tel T, de Moura A, Grebogi C, Karolyi G (2005) Chemical and biological activity in open flows: a dynamical system approach. Phys Rep 413: 91–196

    Article  CAS  Google Scholar 

  58. Smagorinsky J (1963) General circulation experiments with the primitive equations I: the basic experiment. Mon Weather Rev 91: 99–164

    Article  Google Scholar 

  59. Leith CE (1996) Large eddy simulation of complex engineering and geophysical flows. Phys D 98: 481–491

    Article  Google Scholar 

  60. Figueroa HA, Olson DB (1994) Eddy resolution versus eddy diffusion in a double gyre GCM. Part I: the Lagrangian and Eulerian description. J Phys Oceanogr. doi:10.1016/j.dsr.2006.10.009

  61. Hunter JR, Craig PD, Phillips HE (1993) On the use of random walk models with spatially variable diffusivity. J Comput Phys 106: 366–376

    Google Scholar 

  62. Nakamura N (1996) Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J Atmos Sci 53: 1524–1537

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Environmental Science and Energy Research, Weizmann Institute of Science, 76100, Rehovot, Israel

    Daniel F. Carlson

  2. Department of Computer Science, Jerusalem College of Technology, 21 Haavad Haleumi St., P.O. Box 16031, 91160, Jerusalem, Israel

    Erick Fredj

  3. Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel

    Hezi Gildor

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Vered Rom-Kedar

Authors
  1. Daniel F. Carlson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erick Fredj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hezi Gildor
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vered Rom-Kedar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hezi Gildor.

Additional information

Hezi Gildor—Formerly at The Environmental Sciences, Weizmann Institute of Science, Rehovot, Israel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carlson, D.F., Fredj, E., Gildor, H. et al. Deducing an upper bound to the horizontal eddy diffusivity using a stochastic Lagrangian model. Environ Fluid Mech 10, 499–520 (2010). https://doi.org/10.1007/s10652-010-9181-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10652-010-9181-0

Keywords

Search

Navigation