class: center, middle .title[Opportunities and challenges in Lagrangian observations of turbulence ] .author[Jonathan Lilly
1
] .coauthor[Sofia Olhede
2
, Adam Sykulski
1,2
, Jeffrey Early
1
] .institution[
1
NorthWest Research Associates,
2
University College London] .date[January 16, 2017] .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 ## The Global Surface Drifter Dataset
--- class: center ## Mean Surface Current Speed
Formed by binning in latitude and longitude, then averaging. Easy to compute maps of low-order statistics: mean, variance, etc. Does not capture full richness of dataset. What else can be done? We will examine two new methods, the first pertaining to
diffusive
transport and the second to
ballistic
transport by coherent eddies. -- --- class: left ## Perspective Lagrangian trajectories offer a greater potential for studying transport in ocean turbulence than is generally appreciated. -- This is significant because there is a lot of such data—both surface drifters and RAFOS-type deep floats. -- This talk will give an overview of two promising methods for rigorous and objective analysis of massive Lagrangian datasets: -- 1. Stochastic modeling of dispersive Lagrangian trajectories 1. Time/frequency extraction of coherent eddy properties -- The development of these methodologies involved adapting and advancing modern ideas from both signal processing and statistics. -- These methods complement more traditional tools such as pseudo-Eulerian maps, dispersion estimates, FTLE's, etc. -- There are outstanding physical questions that would make promising directions for further research. --- class: center, middle #Part I: A Stochastic Model for Turbulent Dispersion --- class: left ## Modeling Dispersion in 2D Turbulence
A very simple stochastic model for Lagrangian trajectories in geostrophic turbulence is
damped
Fractional Brownian motion. .cite[Lilly, J. M., A. M. Sykulski, J. J. Early, and S. C. Olhede (2016).
Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion.
Submitted. Available at
{arXiv}.
] --- class: left ## A Hierarchy of Stochastic Models With advance apologies for notation: the following SDEs should be written as the corresponding stochastic integral equations. Note that `\(z(t)=u(t)+iv(t)\)` represents a complex-valued
velocity
. Brownian motion: `\[\frac{dz}{d t} = A\,\frac{d w}{d t} \quad\longrightarrow\quad S_{zz}(\omega) = \frac{A^2}{\omega^2}\]` Fractional Brownian motion (fBm): `\[\frac{d^\alpha z}{d t^\alpha} = A\,\frac{d w}{d t} \quad\longrightarrow\quad S_{zz}(\omega) = \frac{A^2}{\omega^{2\alpha}}\]` Damped Fractional Brownian motion a.k.a. the Matérn process: `\[\left[\frac{d}{d t} + \lambda \right]^\alpha z = A\,\frac{d w}{d t} \quad\longrightarrow\quad S_{zz}(\omega) = \frac{A^2}{(\omega^2+\lambda^2)^\alpha}\]` In Lilly et al. (2016), we showed that the relatively little-known Matérn process is in fact equivalent to adding a damping to fBm. --- class: left ##Diffusivity is the Spectrum's Value at Zero The velocity autocovariance and spectrum are defined as `\[R_{zz} (\tau)\equiv\left\langle z(t+\tau)\,z^*(t)\right\rangle = \frac{1}{2\pi}\int_{-\infty}^\infty e^{i \omega \tau} S_{zz}(\omega) \, d\omega\]` The corresponding diffusivity is given by `\[\kappa \equiv \lim_{t\longrightarrow\infty} \frac{1}{4} \frac{d}{d t} \left\langle x^2(t) + y^2(t)\right\rangle =\frac{1}{4} \int_{-\infty}^\infty R_{zz}(\tau)\, d \tau =\frac{1}{4}S_{zz}(0)\]` with the last equality following from the Fourier transform `\(\int_{-\infty}^\infty e^{-i \omega \tau} R_{zz}(\tau) \, d\tau= S_{zz}(\omega)\)` evaluated at zero frequency. Matérn: `\( \,\,\,\,S_{zz}(\omega) = \frac{A^2}{(\omega^2+\lambda^2)^\alpha} \quad\longrightarrow\quad \kappa = \frac{1}{4}\frac{A^2}{\lambda^{2\alpha}}\)` fBm: `\( \quad\quad S_{zz}(\omega)= \frac{A^2}{\omega^{2\alpha}} \quad\quad\,\,\longrightarrow\quad \kappa = \infty\)` Thus the Matérn, unlike fBm, can model diffusive processes. --- class: center ##Comparison with Other Models
--- class: center ##Application to the Global Drifter Dataset
The nondimensionalized damping parameter from a parametric fit to the Matérn, only on the anti-inertial side. Shorter ‘memory’ in more energetic regions. We are not aware of any theory for this. --- class: center ##Application to the Global Drifter Dataset
The first global map of the slope parameter `\(\alpha\)`, with `\(|\omega|^{-2\alpha}\)`. Slopes vary from `\(|\omega|^{-1}\)` to `\(|\omega|^{-3}\)`, with `\(|\omega|^{-2}\)` over major currents. Does not fit the conventional wisdom of only `\(|\omega|^{-2n}\)` slopes, as in e.g. .cite[Berloff and McWilliams (2002b)] --- class: left ##Perspective on Stochastic Modeling The Matérn process is the simplest random process that can simultaneously reproduce (i) the velocity variance (ii) the diffusivity and (iii) the spectral slope or degree of small-scale roughness. -- This is a promising stochastic model for Lagrangian trajectories, and diffusive processes in general. -- It is useful for parameteric spectral modeling, as model for synthetic trajectories, as a null hypothesis in eddy studies. -- Work to date has simply focused on finding a stocastic model for the
observed form
of Lagrangian frequency spectra. -- A very promising direction for future research is to connect this stochastic model to the turbulence theory by linking Eulerian
wavenumber
spectra to Lagrangian
frequency
spectra! --- class: left ### Details of the Stochastic Modeling Approach We have developed a method for stochastic modeling of trajectories in ocean turbulence, and inferring parameters from large datasets. * Create an appropriate stochastic model for particle trajectories. .cite[Sykulski, Olhede, Lilly, and Danioux (2015)] * A key ingredient is a *damped* version of fractional Brownian motion. .cite[Lilly, Sykulski, Early, and Olhede (2016), submitted] * Parameter estimation is best done in the frequency domain. .cite[Whittle (1953)] * Parameter estimation must be adjusted to handle *complex-valued* time series. .cite[Sykulski, Olhede, Lilly, and Early (2016a), accepted] * Parameter estimation must be corrected for bias due to small sample size effects. .cite[Sykulski, Olhede, Lilly, and Early (2016b), submitted] --- class: center, middle # Part II: Extracting Coherent Eddy Motions --- class: center ### A Numerical Simulation of an Unstable Current
A highly idealized version of the Gulf Stream. Note formation of vortices or “coherent eddies”. --- class: center ### Identifying Vortices from Particle Trajectories
Vortices (loops) are identified using only particle trajectories (dots). --- class: center ###Deep Connection to Vortex Dynamics
Blue = time-varying integrated vorticity, red = one-point estimate. From a simulation by Jeffrey Early. --- class: center ##Example of Multivariate Ridge Analysis
The
ridge
is the curve made by tracing out the maximum modulus of the wavelet transform. --- class: center ##Example of Multivariate Ridge Analysis
Ridge analysis separates modulated elliptical signal from low-frequency meandering and higher-frequency variability. --- class: center ## Application to the Global Drifter Dataset
Apply eddy detection algorithm to 20K time series. Use a range of frequencies compared to the Coriolis frequency `\(f\)`. --- class: center ### Eddy Detection in the Global Drifter Dataset
Red = cyclonic rotation, blue = anticyclonic. --- class: center ### Statistically Significant Eddies
After comparison with a null hypothesis of red noise. --- class: left ###Perspective on Eddy Extraction Eddy currents can be objectively and rigorously extracted from the “background” flow by modeling them as oscillations with properties that vary slowly in time. -- A wavelet-based method is then used to locate and extract
time-varying
oscillation properties through a “best fit” with the principle of maximizing energy density. -- There is a remarkable connection to integral vortex properties: -- Given eddy geometry varying strictly more slowly than the orbital period, and assuming a constant background flow, the integral properties along a material ellipse are
exactly
recovered by the instantenous properities of a single particle along that ellipse. -- A promising direction for future research is to connect this work, which derives from signal processing theory, to Lagrangian Coherent Structures. -- Intuitively this approach seems closely related to the vortex transport barriers of .cite[Haller and Beron-Vera (2012)]. --- class: left ### Details of the Eddy Extraction Method We have developed a general method for extracting and analyzing *time-varying* and quasi-periodic signals, such as eddy currents. * The notion of the *analytic signal* lets an *instantaneous* amplitude and frequency be associated with a given time series. .cite[Gabor (1946), Vakman and Vainshtein (1977), Picinbono (1997)] * Similarly, a bivariate (x,y) signal defines the geometrical properties of a time-varying ellipse. .cite[Lilly and Olhede (2010a)] * This may be extended to 3D (e.g. seismographs) or N-D signals. .cite[Lilly (2011), Lilly and Olhede (2012a)] * Modulated oscillations can be extracted using *wavelet ridge analysis*, a local best fit onto an oscillatory test function. .cite[Delprat et al. (1992), Mallat (1999), Lilly and Olhede (2010b)] * Modulated multivariate oscillations can be treated similarly .cite[Lilly and Olhede (2012a)] * Errors are proportional to modulation strength, and are minimized by a suitable choice of wavelet. .cite[Lilly and Olhede (2010b), Lilly and Olhede (2012a)] * Best choice of wavelet is found by considering a superfamily encompassing all other continuous analytic wavelets. .cite[Lilly and Olhede (2009, 2012b)] --- class: left ## Concluding Perspective Two promising directions for future research are: 1. Connecting the observed form of Lagrangian velocity spectra to the theory of Eulerian wavenumber spectra, and 2. Connecting the vortex extraction method to Lagrangian Coherent Structure theory -- Together these directions would contribute to understanding and quantifying two very different contributions to turbulent transport: the diffusive transport, and ballistic transport due to eddy motions. --- class: center, middle # Thank you! .center[Visit [{www.jmlilly.net}](http://www.jmlilly.net) for papers, this talk, and a Matlab toolbox of all numerical code.] .center[.footnote[P.S. Like the way this presentation looks? Check out [{Liminal}](https://github.com/jonathanlilly/liminal).]]