page_title: Lagrangian coherent structure
section_title: Rotational coherence from the Lagrangian-averaged vorticity deviation (LAVD)
hierarchical_section_title: Lagrangian coherent structure / Elliptic LCSs / Rotational coherence from the Lagrangian-averaged vorticity deviation (LAVD)
caption_attribution_description: English: Representative level surfaces of LAVD field at t0 = May 15, 2006 with T = 120 days.
context_page_description: Lagrangian coherent structures are distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest. The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one readily identifies coherent features, but it is often the underlying structure creating these features that is of interest.
  As illustrated on the right, individual tracer trajectories forming coherent patterns are generally sensitive with respect to changes in their initial conditions and the system parameters. In contrast, the LCSs creating these trajectory patterns turn out to be robust and provide a simplified skeleton of the overall dynamics of the system. The robustness of this skeleton makes LCSs ideal tools for model validation, model comparison and benchmarking. LCSs can also be used for now-casting and even short-term forecasting of pattern evolution in complex dynamical systems.
context_section_description: The level sets of the PRA are objective in two dimensions but not in three dimensions. An additional shortcoming of the polar rotation tensor is its dynamical inconsistency:  polar rotations computed over adjacent sub-intervals of a total deformation do not sum up to the rotation computed for the full-time interval of the same deformation.  Therefore, while  is the closest rotation tensor to   in the  norm over a fixed time interval , these piecewise best fits do not form a family of rigid-body rotations as  and  are varied. For this reason, rotations predicted by the polar rotation tensor over varying time intervals divert from the experimentally observed mean material rotation of fluid elements.
  An alternative to the classic polar decomposition provides a resolution to both the non-objectivity and the dynamic inconsistency issue. Specifically, the Dynamic Polar Decomposition (DPD) of the deformation gradient is also of the form
  where the proper orthogonal tensor  is the dynamic rotation tensor and the non-singular tensors  are the left dynamic stretch tensor and right dynamic stretch tensor, respectively. Just as the classic polar decomposition, the DPD is valid in any finite dimension. Unlike the classic polar decomposition, however, the dynamic rotation and stretch tensors are obtained from solving linear differential equations, rather than from matrix manipulations. In particular,   is the deformation gradient of the purely rotational flow
  and   is the deformation gradient of the purely straining flow
  .
  The dynamic rotation tensor  can further be factorized into two deformation gradients: one for a spatially uniform (rigid-body) rotation, and one that deviates from this uniform rotation:
  As a spatially independent rigid-body rotation, the proper orthogonal relative rotation tensor  is dynamically consistent, serving as the deformation gradient of the relative rotation flow
  In contrast, the proper orthogonal mean rotation tensor  is the deformation gradient of the mean-rotation flow
  The dynamic consistency of  implies that the total angle swept by   around its own axis of rotation is dynamically consistent. This intrinsic rotation angle  is also objective, and turns out to equal to one half of the Lagrangian-averaged vorticity deviation (LAVD). The LAVD is defined as the trajectory-averaged magnitude of the deviation of the vorticity from its spatial mean. With the vorticity  and its spatial mean
  the LAVD over a time interval  therefore takes the form
  with  denoting the (possibly time-varying) domain of definition of the velocity field . This result applies both in two-  and three dimensions, and enables the computation of a well-defined, objective and dynamically consistent material rotation angle along any trajectory.
  Outermost complex tubular level curves of the LAVD define initial positions of rotationally coherent material vortex boundaries in two-dimensional unsteady flows (see Fig. 11a). By construction, these boundaries may exhibit transverse filamentation, but any developing filament keeps rotating with the boundary, without global transverse departure form the material vortex. (Exceptions are inviscid flows where such a global departure of LAVD level surfaces from a vortex is possible as fluid elements preserve their material rotation rate for all times). Remarkably, centers of rotationally coherent vortices (defined by local maxima of the LAVD field) can be proven to be the observed centers of attraction or repulsion for finite-size (inertial) particle motion in geophysical flows (see Fig. 11b). In three-dimensional flows, tubular level surfaces of the LAVD define initial positions of two-dimensional eddy boundary surfaces  (see Fig. 11c) that remain rotationally coherent over a time intcenter|erval  (see Fig. 11d).
caption: Figure 11c: A rotationally coherent mesoscale eddy (yellow) in the Southern Ocean State Estimate (SOSE) ocean model at t0 = May 15, 2006, computed as a tubular LAVD level surface over t1-t0=120 days. Also shown are nearby LAVD level surfaces to illustrate the rotational incoherence outside the eddy. (Image: Alireza Hadjighasem)[33]
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