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Diagnosing Ocean Stirring: Comparison of Relative Dispersion and Finite-Time Lyapunov Exponents

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  • 1 Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland
  • | 2 Courant Institute of Mathematical Sciences, New York University, New York, New York
  • | 3 NOAA/National Oceanographic Data Center, Silver Spring, Maryland
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Abstract

The relationship between two commonly used diagnostics of stirring in ocean and atmospheric flows, the finite-time Lyapunov exponents λ and relative dispersion R2, is examined for a simple uniform strain flow and ocean flow inferred from altimetry. Although both diagnostics are based on the separation of initially close particles, the two diagnostics measure different aspects of the flow and, in general, there is not a one-to-one relationship between the diagnostics. For a two-dimensional flow with time-independent uniform strain, there is a single time-independent λ, but there is a wide range of values of R2 for individual particle pairs. However, it is shown that the upper and lower limits of R2 for individual pairs, the mean value over a large ensemble of pairs, and the probability distribution function (PDF) of R2 have simple relationships with λ. Furthermore, these analytical expressions provide a reasonable approximation for the R2λ relationship in the surface ocean flow based on geostrophic velocities derived from satellite altimeter measurements. In particular, the bimodal distribution, upper and lower bounds, and mean values from the ocean flow are similar to the analytical expressions for a uniform strain flow. How well, as well as over what integration time scale, this holds depends on the spatial and temporal variations within the ocean region being considered.

Corresponding author address: Darryn W. Waugh, Earth and Planetary Sciences, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218. E-mail: waugh@jhu.edu

Abstract

The relationship between two commonly used diagnostics of stirring in ocean and atmospheric flows, the finite-time Lyapunov exponents λ and relative dispersion R2, is examined for a simple uniform strain flow and ocean flow inferred from altimetry. Although both diagnostics are based on the separation of initially close particles, the two diagnostics measure different aspects of the flow and, in general, there is not a one-to-one relationship between the diagnostics. For a two-dimensional flow with time-independent uniform strain, there is a single time-independent λ, but there is a wide range of values of R2 for individual particle pairs. However, it is shown that the upper and lower limits of R2 for individual pairs, the mean value over a large ensemble of pairs, and the probability distribution function (PDF) of R2 have simple relationships with λ. Furthermore, these analytical expressions provide a reasonable approximation for the R2λ relationship in the surface ocean flow based on geostrophic velocities derived from satellite altimeter measurements. In particular, the bimodal distribution, upper and lower bounds, and mean values from the ocean flow are similar to the analytical expressions for a uniform strain flow. How well, as well as over what integration time scale, this holds depends on the spatial and temporal variations within the ocean region being considered.

Corresponding author address: Darryn W. Waugh, Earth and Planetary Sciences, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218. E-mail: waugh@jhu.edu
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