AMSC 664 Final Presentation: Lagrangian Analysis of 2D and ...rvbalan/TEACHING/AMSC663Fall2015...AMSC 664 Final Presentation: Lagrangian Analysis of 2D and 3D Ocean Flows from Eulerian

AMSC 664Final

Presentation:Lagrangian

Analysis of 2Dand 3D Ocean

Flows fromEulerian

Velocity Data

David Russell

Introduction

Methods

Implementation

Validation

Application

Conclusions

Timeline

References

AMSC 664 Final Presentation:Lagrangian Analysis of 2D and 3D Ocean Flows

from Eulerian Velocity Data

David RussellSecond-year Ph.D. student, Applied Math and Scientific Computing

Project Advisor: Kayo IdeDepartment of Atmospheric and Oceanic Science

Center for Scientific Computation and Mathematical ModelingEarth System Science Interdisciplinary CenterInstitute for Physical Science and Technology

May 11th, 2016

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Flows fromEulerian

Velocity Data

David Russell

Introduction

Methods

Implementation

Validation

Application

Conclusions

Timeline

References

Project Goals

Project goals:

Develop tools for Lagrangian (particle-based) analysis of a2D or 3D ocean flow from Eulerian (grid-based) velocitydata

Validate on a series of simple ODE systems withwell-understood dynamics

Validate against existing tools from ROMS (RegionalOcean Modeling System)

Test on a dataset from the Chesapeake Bay

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Introduction

Dynamics background:

Given an ODE system x = u(t, x) (u = (u, v ,w) in R3)with corresponding particle trajectories X(t,X0)

A hyperbolic equilibrium point is a point x∗ for whichu(t, x∗) ≡ 0 and all eigenvalues of the Jacobian ∂u

∂x

∣∣x=x∗

have nonzero real part

Stable manifold of x∗: {x ∈ Rn | limt→∞X(t, x) = x∗}Unstable manifold of x∗: {x ∈ Rn | limt→−∞X(t, x) = x∗}

Corresponding concept when x∗ is moving is that of adistinguished hyperbolic trajectory

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Introduction

Stable and unstable manifolds:

Act as boundaries between coherent structures in the flow(e.g. as rough barriers to pollutant transport in oceans)

Can be found by one of two numerical methods:finite-time Lyapunov exponents (traditional) and arclengths, also called M-function (newer)

Each of these requires a thick soup of particle trajectoriesfor calculation

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Introduction

Example: M-function applied to dataset from the Kuroshiocurrent, northwest Pacific Ocean [? ]:

Red fast, blue slow (on average)

Thin yellow lines: stable and unstable manifolds

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Methods: Overview

Three main numerical tasks:

Interpolate velocities to off-grid particle locationsIntegrate particle velocities to obtain trajectoriesAnalyze trajectories using numerical tools

Implement and compare various versions of each of these

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Methods: Interpolation

2D horizontal interpolation methods:

Bilinear splines: Fit function of form

f (x , y) =1∑

i ,j=0

cijxiy j = c00 + c10x + c01y + c11xy

to given values of u at box corners

Bicubic splines: Fit function of form

f (x , y) =3∑

i ,j=0

cijxiy j

to given values of u, approximate values of ux , uy , uxy atbox corners (using second-order finite differences:two-sided for interior, one-sided for boundary)

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Methods: Interpolation

Vertical and temporal interpolation methods:

Linear: Fit linear function of z through two nearestvertical neighbors

Cubic: Fit cubic polynomial in z through four nearestvertical neighbors

Same algorithms for temporal interpolation

Intergration method:

4th-order Runge-Kutta

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Methods: Analysis

Approaches:

M-function (arc length over fixed time interval)

Forward: M(τ,X0) =∫ τ

0|u(t,X(t,X0))| dt

Backward: M(τ,X0) =∫ 0

−τ |u(t,X(t,X0))| dtBoth: M(τ,X0) =

∫ τ−τ |u(t,X(t,X0))| dt

Maximal finite-time Lyapunov exponent (FTLE) (maximalgrowth rate of distance between close trajectories):

FTLE(τ,X0) =1

τln (σmax (L(τ,X0)))

where L(τ,X0) = ∂X(τ,X0)∂X0

is the transition matrix andσmax is the largest singular value

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Methods: Analysis

Calculating M and FTLE:

M-function: Consider M as a state variable like u and v ,integrate alongside u and v

FTLE:

Track particles directly next to X0 up to time τApproximate L = ∂X

∂X0using centered differences for

∆X0,∆Y0,∆Z0 and final locations for ∆X ,∆Y ,∆ZCalculate σmax(L) as

√λmax(LTL), where eigenvalues of

LTL ∈ (R2×2 or R3×3) are roots a quadratic or cubicpolynomial

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Implementation

Implementation details:

Software: MATLAB

Hardware:

Shorter runs (up to 6-hour dataset): MacBook Pro laptop,2.6 GHz Intel Core i5, 8 GB 1600 MHz DDR3Longer runs: Deepthought2

Vectorization but no parallelization across particles

Trajectories that leave domain set to NaN

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Validation: 2D Interpolation

2D accuracy: ‖finterp − fsemi-interp‖2 vs. ∆x (interpolation meshsize) for f (x , y) = ex cos(2πy) on [−1, 1]2 for particle grid at6x resolution

Convergence rate(‖e‖2 ∼ hp)

Bilinear p ≈ 2.0064Bicubic p ≈ 3.0175

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3D horizontal accuracy: ‖fapprox − f ‖2 vs. ∆x = ∆y , ∆z fixed,for f (x , y , z) = ex cos 2πy cos 2πz on [−1, 1]2

Horizontal(-vertical)

method

Convergence rate(‖e‖2 ∼ hp)

Bilinear(-linear)

p ≈ 2.0253

Bicubic(-linear)

p ≈ 3.2675

Bicubic(-cubic)

p ≈ 2.9192

MATLABCubic

p ≈ 3.2675

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3D vertical accuracy: ‖finterp − fsemi-interp‖2 vs. ∆z , ∆x = ∆yfixed

(Horizontal-)vertical Convergence ratemethod (‖e‖2 ∼ hp)

(Bilinear-)linear p ≈ 2.0101(Bicubic-)cubic p ≈ 4.0484MATLAB cubic p ≈ 3.2836

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Validation: 2D Trajectory Calculation

2D test system: Undamped Duffing oscillator (1918) [? ]:

Simple nonlinear oscillator

General case (forcing parameter ε):

x = y

y = x − x3 + ε sin t

Autonomous case (ε = 0):

Hyperbolic equilibrium point at x = (0, 0)

Stable and unstable manifolds both given by y2 = x2 − x4

2(figure-eight shape centered at origin)

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Autonomous Duffing oscillator:

Hamiltonian system with H(x , y) = 12y

2 − 12x

2 + 14x

4

conserved along trajectories

x(t) and y(t) can be computed explicitly via Jacobielliptic functions [? ], e.g. for y0 = 0 we have

x(t) = x0 · cn

t√

x20 − 1,

√√√√ x20

2(x20 − 1)

y(t) = −x0

√x2

0 − 1 · sn

t√

x20 − 1,

√√√√ x20

2(x20 − 1)

· dn

t√

x20 − 1,

√√√√ x20

2(x20 − 1)

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Exact trajectories of autonomous Duffing (level sets ofH(x , y):)

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Visual validation: A few computed vs. exact trajectories forDuffing oscillator (RK4, bilinear, ∆x = ∆y = 0.1, ∆t = 2−3,tf = 6):

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RMSE vs. interpolation ∆x for individual trajectory withX0 = (−1.5, 0), ∆t = 0.1, tf = 6:

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Hamiltonian vs. time for X0 = (−1.5, 0), tf = 20:

Hamiltonian becomes constant as ∆x → 0

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Visual Validation: 2D Lagrangian Tools

Autonomous Duffing oscillator: M-function, forwards andbackwards, τ = 10

Figure: My results

tened’’ versions of short segments of manifolds of the unstable manifold of the hyperbolic trajectory. For the longer time thecontours of the finite time average of the horizontal component of velocity develop a more complex spatial structure thatobscures the underlying unstable manifold structure of the hyperbolic trajectory.

3. Applications to time dependent 3D flows

In this section we show that Lagrangian descriptors can also provide accurate information on the stable and unstable man-ifolds of hyperbolic trajectories in three dimensional (3D) time dependent flows. Computation of the stable and unstable man-ifolds of hyperbolic trajectories in aperiodically time dependent flows was discussed in [5], where an algorithm for theircalculation was developed and several ‘‘benchmark’’ examples were considered. The particular example that we will consideris the perturbed Hill’s spherical vortex, which we will take as a benchmark for the performance of our methods in 3D. We givea brief description of the velocity field. More details on the background of Hill’s spherical vortex can be found in [5].

The velocity field, v, has the general form:

v ¼ Hðx; y; zÞ þ Sðx; y; z; tÞ

where Hðx; y; zÞ is given by:

Hx ¼ ður sin Hþ uH cos HÞ cos U; ð31ÞHy ¼ ður sin Hþ uH cos HÞ sin U; ð32ÞHz ¼ ður cos H% uH sin HÞ: ð33Þ

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

x

Fig. 13. For the integrable Duffing equation: (a) the stable and unstable manifolds of the hyperbolic fixed point; (b) forward FTLE for s ¼ 10; (c) contours ofM1 for s = 10. For the periodically forced Duffing equation: (d) segments of the stable and unstable manifolds of the hyperbolic trajectory near the origincomputed for s ¼ 10 (and displayed at t ¼ 0); (e) forward FTLE for s ¼ 10; (f) contours of M1 for s = 10. For the aperiodically forced Duffing equation: (g)segments of the stable and unstable manifolds of the hyperbolic trajectory near the origin computed for s ¼ 10 (and displayed at t ¼ 0); (h) forward FTLE fors ¼ 10; (i) contours of M1 for s = 10.

A.M. Mancho et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3530–3557 3545

Figure: Mancho et al results [? ]

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Flows fromEulerian

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Introduction

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Autonomous Duffing oscillator: FTLE, forwards, τ = 10

Figure: My results








(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

x




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Visual Validation: 2D + Time Lagrangian Tools

Forced Duffing oscillator (ε = 0.1): M-function, forwards andbackwards, τ = 10

Figure: My results








(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

x




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Forced Duffing oscillator: FTLE, forwards, τ = 10

Figure: My results








(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

x




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Finding manifolds: M vs. FTLE, forwards, τ = 10

Figure: M-function Figure: FTLE

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3D test case: Hill’s spherical vortex (1894) [? ]

Simple model of axisymmetric flow around and inside asphere of radius a

Cartesian ODE system:

x = −3Uxz

2a2

y = −3Uyz

2a2

z =3U(2x2 + 2y2 + z2 − a2)

2a2

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Hyperbolic fixed points: x∗1 = (0, 0,−a), x∗2 = (0, 0, a)

Stable/unstable manifolds consist of sphere and verticalline through center of sphere

Streamfunction (conserved along trajectories):

ψ(r , θ) = −3

4Ur2

(1− r2

a2

)sin2 θ

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Hill’s spherical vortex: Streamfunction ψ vs. time for variousinterpolation ∆x , X0 = (0.4, 0.3, 0), ∆t = 0.1, tf = 10

ψ becomes more constant as ∆x → 0

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Hill’s vortex: M-function, tf = 10:

Spherical + z-axis manifold clearly visible

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Hill’s vortex: FTLE, tf = 10:

Spherical manifold clearly visible

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Application: Chesapeake Bay ROMS Dataset

Chesapeake Bay ROMS (Ches-ROMS) domain:

Dimensions: ∼ 84 mi x300 mi x 2.5-50 ft

Discretized to 150 x 480 x20 grid (terrain-followingvertical coordinate)

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ChesROMS velocity data:

Grid indexed by (ξ, η) instead of (i , j)

Arakawa C-grid, so (ξu, ηu) staggered from (ξv , ηv )

Physical velocity components uξ, uη, vξ and vη proscribedat grid points (in m/s), also longitude (λ) and latitude (φ)

Velocity given every two minutes on spatial grid

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ROMS implementation details:

Interpolation done in (ξ, η)-space (linear and bilinear only)

Velocities normalized to index space (s−1), integrationdone there

M-function calculated in physical space (m)

FTLE calculated from physical x and y displacements,estimated from λ and φ

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Validating Chesapeake trajectories: Mine vs. ROMS-generated,backwards (left) and forwards (right) 60 hrs, static flow field

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Trajectory error vs. time (2-norm in index space), backwardsand forwards 60 hrs

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Average speed in m/s (= Mτ ) and FTLE, τ ≈ 6 hr, forward in

time

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Average speed in m/s and FTLE, τ ≈ 6 hr, forward in time

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Flows fromEulerian

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David Russell

Introduction

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Average speed in m/s and FTLE, τ ≈ 6 hr, backward in time

AMSC 664Final



Flows fromEulerian

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David Russell

Introduction

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Average speed in m/s and FTLE, τ ≈ 6 hr, backward in time

AMSC 664Final



Flows fromEulerian

Velocity Data

David Russell

Introduction

Methods

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Conclusions

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Average speed in m/s and FTLE, τ ≈ 6 hr, forward andbackward in time

AMSC 664Final



Flows fromEulerian

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David Russell

Introduction

Methods

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Average speed in m/s and FTLE, τ ≈ 6 hr, forward andbackward in time

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Average speed in m/s, τ ≈ 1 (left), 2 (center), and 4 (right)days, backward in time

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Average speed in m/s, τ ≈ 1 (left), 2 (center), and 4 (right)days, backward in time (changing scale)

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Flows fromEulerian

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Conclusions

Conclusions:

Both M-function and FTLE reveal coherent structures inan ocean flow

Boundaries of coherent structures given by large gradientsin M, ridges and troughs of FTLE

Boundary conditions important: some version of no-slip orfree-slip preferable to nothing

Interpretation somewhat sensitive to color scale, especiallyfor FTLE

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Flows fromEulerian

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Introduction

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Timeline

October - Mid-November

Project proposal presentation and paper X2D and 3D interpolation X

Mid-November - December

2D trajectory implementation and validation XM function implementation and validation XMid-year report and presentation X

January - February

3D trajectory implementation and validation (mostly)FTLE implementation XTailor all existing code to work with ROMS data (mostly)

March - April

Apply tools to Chesapeake ROMS dataset (mostly)Final presentation and paper (in progress)

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Flows fromEulerian

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References

[1] Georg Duffing. Erzwungene Schwingungen bei veranderlicherEigenfrequenz und ihre technische Bedeutung. Number 41-42. R,Vieweg & Sohn, 1918.

[2] Micaiah John Muller Hill. On a spherical vortex. Proceedings of theRoyal Society of London, 55(331-335):219–224, 1894.

[3] Ana M Mancho, Stephen Wiggins, Jezabel Curbelo, and CarolinaMendoza. Lagrangian descriptors: A method for revealing phase spacestructures of general time dependent dynamical systems.Communications in Nonlinear Science and Numerical Simulation,18(12):3530–3557, 2013.

[4] Carolina Mendoza and Ana M Mancho. Hidden geometry of oceanflows. Physical review letters, 105(3):038501, 2010.

[5] Alvaro H Salas. Exact solution to duffing equation and the pendulumequation. Applied Mathematical Sciences, 8(176):8781–8789, 2014.

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AMSC 664 Final Presentation: Lagrangian Analysis of 2D and ...rvbalan/TEACHING/AMSC663Fall2015...AMSC 664 Final Presentation: Lagrangian Analysis of 2D and 3D Ocean Flows from Eulerian