TY - JOUR

T1 - Leaving flatland

T2 - Diagnostics for Lagrangian coherent structures in three-dimensional flows

AU - Sulman, Mohamed H.M.

AU - Huntley, Helga S.

AU - Lipphardt, B. L.

AU - Kirwan, A. D.

N1 - Funding Information:
Support for this research came from the Office of Naval Research through grants N00014-10-1-0522 and N00014-11-10081 to the University of Delaware, the Office of Naval Research MURI OCEAN 3D + 1 grant N00014-11-1-0087 , a grant from BP/The Gulf of Mexico Research Initiative , and the Mary A.S. Lighthipe endowment to the University of Delaware.

PY - 2013/9/1

Y1 - 2013/9/1

N2 - Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian Coherent Structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical (u,v) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical (u,v) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.

AB - Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian Coherent Structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical (u,v) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical (u,v) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.

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U2 - 10.1016/j.physd.2013.05.005

DO - 10.1016/j.physd.2013.05.005

M3 - Article

AN - SCOPUS:84879965704

SN - 0167-2789

VL - 258

SP - 77

EP - 92

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -