class: center, middle .title[Recovery of vortex properties from Lagrangian trajectories] .author[Jonathan Lilly
1
] .coauthor[Jeffrey Early
1
, Sofia Olhede
2
] .institution[
1
NorthWest Research Associates,
2
University College London] .date[February 15, 2018] .center[[{www.jmlilly.net}](http://www.jmlilly.net)] .footnote[Created with [{Remark.js}](http://remarkjs.com/) using [{Markdown}](https://daringfireball.net/projects/markdown/) + [{MathJax}](https://www.mathjax.org/)] --- class: center ##Lagrangian Analysis of a Nonlinear Eddy
An eddy at 24°N tracked by 512×256=131,072 particles, by J. Early. Color is estimated enclosed vorticity in inferred ellipses. --- class: center ##Estimated vs. Actual Enclosed Vorticity
Log10 histogram of estimated vs. true enclosed vorticity. --- class: center ##Estimated vs. Actual Normal Strain
Log10 histogram of estimated vs. true enclosed normal strain. --- class: center ##An Extended Version of Stokes' Theorem `\[\oint_C \mathbf{u}^T\mathrm{d}\mathbf{x} = \iint_A \mathbf{k} \cdot \nabla \times \mathbf{u}\,\mathrm{d} A \]` -- `\[\oint_C \mathbf{u}\,\mathrm{d}\mathbf{x}^T = \iint_A \nabla \mathbf{u}\,\mathrm{d} A \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix} \]` Contains four relations between contour integrals of velocity (left) and spatial integrals of velocity gradient (right): (i) Kelvin-Stokes theorem, (ii) divergence theorem, and similar results for (iii) normal strain and (iv) shear strain. An alternate presentation. `\[\oint_C \mathbf{u}\,\mathrm{d}\mathbf{n}^T = \iint_A \nabla \mathbf{u}\,\mathrm{d} A \]` Lilly (2018),
{Kinematics of a fluid ellipse in a linear flow}
,
Fluids
. --- class: left ##Ellipse / Flow Equivalence The principle of *ellipse / flow equivalence* states that, for any ellipse of fluid particles marked out within a linear flow: * (i) The linear flow determines ellipse rates of change * (ii) The ellipse rates of change imply a unique linear flow The ellipse rates of change include three geometry parameters (size, shape, orientation), plus the flow of particles around the periphery. This builds on the well known result that ‘a linear flow maps an ellipse into another ellipse’, e.g. .cite[(Kida, 1981)]. Ellipse/flow equivalence can be shown to be a consequence of the extended Stokes' theorem. Thus, if we know the rates of change of a fluid ellipse, we know the spatially averaged vorticity, divergence, and strain terms within it. This implies a deep connection between Lagrangian and Eulerian perspectives, and suggests untapped potential from Lagrangian data. Lilly (2018),
{Kinematics of a fluid ellipse in a linear flow}
,
Fluids
. --- class: left ##The Link to Lagrangian Observations A model for Lagrangian trajectories `$z(t)=x(t) + \mathrm{i} y(t)$` with eddies: `\[z(t)=x(t) + \mathrm{i} y(t) =z_o(t) + \boxed{z_*(t)}\]` Turbulent background `$z_o(t)$` (stochastic) plus `$z_*(t)$` (quasi-oscillatory). `\[z_*(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos \phi(t)+\mathrm{i}b(t)\sin \phi(t)\right]\]` Oscillatory signal `$z_*(t)$` is modeled as a *modulated ellipse*. --- class: left ##Recovery of Modulated Signals Q. *How* can `$a(t)$` and `$\phi(t)$` be recovered from `$x_*(t)=a(t)\cos \phi(t)$`? A. Using the analytic signal method of .cite[Gabor (1946)]. There is not actually any choice about this, see .cite[Vakman (1996)]. Q. *When* can `$a(t)$` and `$\phi(t)$` be recovered from `$x_*(t)=a(t)\cos \phi(t)$`? A. Provided `$a(t)$` are more slowly varying than `$\cos\phi(t)$`. *Bedrosian's Theorem*, .cite[Bedrosian, (1963)] Q. When can `$a(t)$`, `$b(t)$`, `$\theta(t)$`, and `$\phi(t)$` be recovered from the *modulated ellipse* `$z_*(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos \phi(t)+\mathrm{i}b(t)\sin \phi(t)\right]$`? A. Provided `$a(t)$`, `$b(t)$`, and `$\mathrm{e}^{\mathrm{i}\theta(t)}$` are more slowly varying than `$\mathrm{e}^{\mathrm{i}\phi(t)}$`. .cite[Lilly and Olhede, in prep.] An exact result. But... For `$f(t)$` to be ‘more slowly varying’ than `$g(t)$` is rather strict. It means the spectrum of `$f(t)$` has energy only at lower frequencies than does the spectrum of `$g(t)$`. Will this be true for eddies? --- class: left ##Separating Oscillation & Background With noise present, applying the analytic signal method directly fails. *Wavelet ridge analysis* combines filtering and the analytic signal method into one step. .cite[Delprat (1992), Lilly and Olhede (2010b)] *Optimal* in that separation is done as well as is possible subject to the constraints of the uncertainty principle (time vs. frequency). Important extensions: * Bias estimate for univariate case .cite[Lilly and Olhede (2010b)] * Interpretation for multivariate signals .cite[Lilly and Olhede (2010a)] * Bias estimate for multivariate signals .cite[Lilly and Olhede (2012)] Bias here refers to intrinsic error associated with the signal itself, as opposed to variance arising from additive noise. The first-order departure from a pure sinusoid is called the *bandwidth*, which has terms like `$a'(t)/a(t)$` (amplitude modulation). The *second-order* departure has terms like `$a''(t)/a(t)$`, which we call the *curvature*. This turns out to control the bias of the estimate. --- class: center ##Estimated vs. Actual Enclosed Vorticity
Computed for all inferred ellipses, prior to thresholding. --- class: center ##Estimated vs. Actual Normal Strain
Computed for all inferred ellipses, prior to thresholding. --- class: center ##Distribution on Radius / Velocity Plane
Distribution of radius and velocity of all inferred ellipses. --- class: center ##Estimated Bias on Radius / Velocity Plane
Bias is estimated within each recovered oscillation using a curvature condition, equation (62) of .cite[Lilly and Olhede (2012)]. --- class: center ##Vorticity Error on Radius / Velocity Plane
Actual error between estimated enclosed vorticity, and actual enlosed vorticity, normalizd by the actual enclosed vorticity. --- class: center ##Distribution After Threshholding
Including only those portions of inferred signals that fall below the value 1/15`$\approx 6.6\%$` for the estimated bias. --- class: center ##Effect of Thresholding on Vorticity Error
Including only those portions of inferred signals that fall below the value 1/15`$\approx 6.6\%$` for the estimated bias. --- class: center ##Effect of Thresholding on Strain Error
Including only those portions of inferred signals that fall below the value 1/15`$\approx 6.6\%$` for the estimated bias. --- class: center ##Evolution on the Radius/Velocity Plane
Color is vorticity error. Instantaneous radial profile is shown. --- class: center ##Interpretation of Low-Frequency Band
The right-hand side shows duration in units of the estimated period. Short-duration signals are difficult to estimate due to edge effects. The low-frequency band is associated with signals that are so brief that edge effects dominate their entire duration. This suggests that large bias is due to edge effects associated with entrainment and detrainment. --- class: center ##Physical Meaning of Estimated Bias
The estimated bias is a property of an inferred or estimated oscillatory signal. What does this correspond to physically? Very good agreement is found with the Lagrangian average speed deviation, that is, `$\sqrt{\langle V-\langle V\rangle\rangle}$`, normalized by `$\langle V\rangle$`. Here `$\langle \cdot\rangle$` is an average over the actual particle trajectory during `$\pm$` one half-period of the inferred signal. --- class: center ##Particle Trapping in a Nonlinear Eddy
Color is vorticity gradient strength. Black = stable, pink = unstable. --- class: left ## Three Types of Particles Combining vortex signal extraction with classical analysis, one finds particles may be classified into one of three types at each moment. 1. *Trapped and Stable:* These particles are moving with the vortex, and can be used to accurately infer vortex properties. 1. *Trapped and Unstable:* These particles are also moving with the vortex, but are too variable to accurately infer vortex properties. 1. *Untrapped and Unstable:* These particles do not move with the vortex, and vortex properties cannot be inferred from them—but oscillatory signals linked to the vortex are still detected. All trapped particles are either stable or unstable ridge points, i.e. there are no non-oscillatory trapped particles. Untrapped and *stable* particles occur very rarely and are short-lived. The distinction between trapping and recoverability is important. It means not all particles trapped in vortices contain useful information about the vortex (when each particle is considered in isolation). An implication of this is that not all vortices are equally observable. Highly unstable or variable vortices (e.g. rapidly precessing ellipses) are more difficult to recover information about. --- class: center ##Composite Eddy in First and Second Half
The eddy evolves to a smaller size from the first half to the second. --- class: center ##Vorticity Error in First and Second Half
Error is controlled by the zero vorticity line in the second half only. Particles in the trapping region, but outside the zero vorticity line, are able to accurately estimate vorticity during the first half only. This represents a qualitative change in the boundary between stable and unstable trapping, arising from the onset of vortex instability. Distinction between an entrainment region and a stirring region? --- class: left ## Conclusions These results represent progress in bridging the gap between Lagrangian data analysis and vortex dynamics. 3. An extended version of Stokes' theorem allows the estimated rate of change of an ellipse to be converted to *spatial averages*, for all four components of the velocity gradient tensor. 1. The application of a bias, or error, metric allows enclosed vorticity to be estimated with very high accuracy, and enclosed strain to be estimated fairly well also. 2. In this approach, we asses the analysis method with respect to elliptical curves *that the particles believe they are observing.* 1. Application to a model of a nonlinear eddy shows very good agreement with classical particle trapping analysis. Further work: application to other simple systems; idealized vortex solutions; and global datasets. A very interesting direction would be to connect the single-particle Lagrangian signal theory to the LCS work of Haller & Beron-Vera. Nonlinear mesoscale eddies are not well understood at large times! --- class: center, middle # Thank you! .center[This talk is available at].center[[http://www.jmlilly.net/talks/lilly18-os.html](http://www.jmlilly.net/talks/lilly18-os.html)] .center[.footnote[P.S. Like the way this presentation looks? Check out [{Liminal}](https://github.com/jonathanlilly/liminal).]] --- class: center ##Comparison with the Naive Estimator
Right-hand side is estimated from *all* Lagrangian data points by locating vortex center, then computing azimuthal velocity. --- class: center ##Comparison with the Naive Estimator
As on the previous slide and assuming solid-body rotation. Right-hand side gives estimated vorticity within a circle. --- class: center ##Trajectories on the Radius/Velocity Plane
Each line represents a detected signal in a different particle. --- class: center ##Trajectories on the Radius/Velocity Plane
Zoomed in to see details of eddy core evolution. --- class: center ##Extraction of Oscillatory Signals
All trajectory segments that contain a quasi-oscillatory signal. `\[z(t)=x(t) + \mathrm{i} y(t) =z_o(t) + \boxed{z_*(t)}\]` `$z_o(t) =$` background, `$z_*(t) =$` oscillatory (e.g. eddy) At each moment, the oscillatory signal is represented by an ellipse. This ellipse is *assigned to* the particle by the analysis method. It is a material ring that the particle is inferred to belong to. --- ##Net Result of the Analysis Method From a Lagrangian dataset, we obtain the following: * *Detection*: time intervals when oscillations are detected (“ridges”) * *Inference*: estimates of the signal properties (ellipse parameters) * *Quality*: a bias metric measuring the *quality* of the inference Very few free parameters (filter bandwidth, ridge duration cutoff). -- Loosely speaking, a particle in a ridge *believes* it is connected to a set of other particles in an elliptical ring. -- Properties of the inferred ring are estimated at each moment. This is *nonlocal*. Spatial averages are then obtained via Stokes theorem. -- The method is subject to “false positives” and other artifacts. -- Thus there is a need to apply some kind of quality estimate, that can be evaluated from *within* the trajectories themselves. -- Essentially, we want know whether the particles ‘believe’ that they are likely observing a signal that can be accurately recovered.