Dynamical Systems Tools for Ocean Studies:

Chaotic Advection versus Turbulence

Reza Malek-Madani

Monterey Bay Surface Currents - August 1999

Observed Eulerian Fields

• Vector field is known at discrete points at discrete times – interpolation becomes a major mathematical issue

• VF is known for a finite time only – How are we to prove theorems when invariance is defined w.r.t. continuous time and for all time?

• VF is often known in parts of the domain – this knowledge may be inhomogeneous in time. How should we be filling the gaps?

• Normal Mode Analysis (NMA) is one way to fill in the gaps. (Kirwan, Lipphardt, Toner)

@u@x +

@v@y +

@w@z = 0

@u@t +

@u2@x + @uv

@y + @uw@z = f v ¡ 1

½0@p@x+

@@x(AH

@u@x) + @

@y(AH@u@y) +

@@z(AV

@u@z)

@v@t +

@uv@x + @v2

@y + @vw@z = ¡ f u ¡ 1

½0@p@y+

@@x(AH

@v@x) + @

@y(AH@v@y) +

@@z(AV

@v@z)

@p@z = ¡ ½g

@T@t +

@uT@x + @vT

@y + @wT@z = @

@x(K H@T@x )

+ @@y(K H

@T@y ) +

@@z(K V

@T@z )

@S@t +

@uS@x + @vS

@y + @wS@z = @

@x(K H@S@x)

+ @@y(K H

@S@y) +

@@z(K V

@S@z )

½= ½(T;S) = P®+0:698P

where

P = 5890+ 38T ¡ 0:375T2 + 3S;

®= 1779:5+11:25T¡ 0:0745T 2¡ (3:8+0:01T)S

Chesapeake Bay (Tom Gross- NOAA)

-77.4 -77.2 -77 -76.8 -76.6 -76.4 -76.2 -76 -75.8 -75.6 -75.436.5

37

37.5

38

38.5

39

39.5

40

long

lat

velocity profile of Bay at one level(?) at one instant in time

Close-up of VF

-76.02 -76 -75.98 -75.96 -75.94 -75.92

36.92

36.94

36.96

36.98

37

37.02

37.04

37.06

37.08

37.1

37.12

longitude

latit

ude

Modeled Eulerian Data

• Very expensive (CPU, personnel)

• Data may not be on rectangular grid (from FEM code)

But …

No gaps in data, either in time or space

Nonlinear PDEs

• Do not have adequate knowledge of the exact solution.

• Need to know if a solution exists – if so, in which function space? (Clay Institute’s $ 1M prize for the NS equations)

• Need to know if solution is unique. Otherwise why do numerics?

• How does one choose the approximating basis functions? Convergence?

Typical Setting

• A velocity field is available, either analytically (only for “toy” problems), or from a model (Navier-Stokes) or from real data, or a combination – VF is typically Eulerian

• To understand transport, mixing, exchange of fluids, we need to solve the set of differential equations ……

Lagrangian Perspective

dxdt

= v(x; t)

Steady Flow

• Stagnation (Fixed) Points … Hyperbolic (saddle) points

• Directions of stretching and compressions ( stable and unstable manifolds)

• Linearization about fixed points; spatial concept; always done about a trajectory; time enters nonlinearly in unsteady problems.

• Instantaneous stream functions are particle trajectories. Trajectories provide obstacle to transport and mixing.

A “toy” problem

Du± ng's E quation

dxdt

= y;dydt

= x ¡ x3 + ² sin t

T he °ow is incompressible (div v = 0) sothere is a streamfunction Ã(x;y; t) such that

v1 = ¡@Ã@y

; v2 =@Ã@x

à = ¡12y2 +

12x2 ¡

14x4 + ²x sin t

Duffing with eps = 0

Streamlines

versus Particle Paths,eps= 0 -1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Streamlines

versus Particle Paths,eps= 0.01

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Streamlines

versus Particle Paths,eps= 0.1

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Ã(x;y; t) =Aksin¼ysinkx+ k² cos ! t coskx

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1

A =0.1 , omg =0.2 , k =1, eps =10

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1A = 0.1, omg = 0.6, k = 10, eps = 0

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1A = 0.1 , omg = 0.6 , k = 10, eps = 0

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1A = 0.1 , omg = 0.6 , k = 10, eps = 5

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1A = 0.1 , omg = 0.6 , k = 10, eps = 10

Unsteady Flows

• The basic concepts of stagnation points and poincare section as tools to quantify transport and mixing fail when the flow is aperiodic.

• How does one define stable and unstable manifolds of a solution in an unsteady flow? How does one compute these manifolds numerically?

– Mancho, Small, Wiggins, Ide, Physica D, 182, 2003, pp. 188 -- 222

Can’t we just integrate the VF?

• Is it worth to simply

integrate the velocity

field to gain insight

about the flow?

Where are the coherent structures?

(Kirwan, Toner,

Lipphardt, 2003)

New Methodologies for Unsteady Flows

• Chaotic oceanic systems seem to have stable coherent structures. However, Poincare map idea does not work for unsteady flows.

• Distinguished Hyperbolic Trajectories – “moving saddle points” Their stable/unstable manifolds play the role of separatrices of saddle point stagnation points in steady flows

• These manifolds are material curves, made of fluid particles, so other fluid particles cannot cross them. It is often difficult to observe these curves by simply studying a sequence of Eulerian velocity snapshots.

• Wiggins’ group has devised an iterative algorithm that converges to a DHT.– Exponential dichotomy– The algorithm starts with identifying the Instantaneous Stagantion points (ISP), i.e., solutions to

v(x,t) = 0 for a fixed t. Unlike steady flows, ISP are not generally solutions to the dynamical system.

– The algorithm then uses a set of integral equations to iterate to the next approximation of the DHT– In real data sets (and, in general in unsteady flows) ISP may appear and disappear as time goes on – Stable and unstable manifolds are then determined by (very careful) time integration of the vector

field (using an algorithm by Dritschel and Ambaum)– Have applied this method to the wind-driven quasigeostrophic double-gyre model.

Exponential Dichotomy

A system of linear ODEs

»0= A(t)»; » 2 R2

is said to have an expentential dichotomy ifthere is a projection P such that

jjX (t)P X ¡ 1(¿)jj · K 1e¡ ¸1(t¡ ¿); t¸ ¿;

jjX (t)(I ¡ P )X ¡ 1(¿)jj · K 2e¡ ¸2(t¡ ¿); t· ¿;

for some constants ¸1; ¸2;K 1;K 2 > 0. HereX (t) is a fundamantal solution matrix ofthe system.

A solution °(t) of dxdt = v(x; t) is called hyper-bolic if the linearization about °, i.e.,

d»dt

= Dxv(°(t); t)»

has an exponential dichotomy.

Double Gyre Flow

Governing equations:

ut+(u¢r )u+f0(1+¯y)k£u = ¡ g0r h+F+º¢ u;

ht + r (hu) = 0;

W ind

F = h¿0

½0H0sin

2¼yL y

;0i

Wiggins, Small and Mancho

Double Gyre with

Large (turbulent)

Wind Stress

Summary

• Dynamical Systems tools have been extended to discrete data.

• Concepts of stable and unstable manifolds have been tested on numerically generated aperiodic vector fields.

• What about stochasticity? Data Assimilation?• Our goal is to determine the relevant manifolds for

the Chesapeake Bay Model– Major obstacle: VF is given on a triangular grid.

Dynamical Systems and Data AssimilationChris Jones

• Computing stable and unstable manifolds requires knowing the Eulerian vector field backward and forward in time. But we lack that information in a typical operational setting. We do, however, have access to Lagrangian data (drifter, etc.)

• Integrate Dynamical Systems Theory into Lagrangian data assimilation (LaDA) strategy – develop computationally efficient DA methods

• Key Idea: The position data by a Lagrangian instrument is assimilated directly into the model, not through a velocity approximation.

• Behavior near chaotic saddle points in vortex models showed the need for “patches” for this technique. Ensemble Kalman Filtering.

Point vortex flows

ff * f *

1,

tt * t *

1

"Truth"

State vector:

Mode

, number of

l (deterministic)

vortices,

2 2

Nj

mi j m

Nm m

mm

j

i m

m

j j

iz

z

N z x

i

z

iy

zzz

z C

,

N

mj m

f

f * f *1,

ff *

tt * t *

1,

tt* t *

1

( ) ( )

f *1

I

2

nclude

2

tracers: , , , =

2

2

Nj

L x yl

Nj

mj j m m j

Nj

lj l j

m mj j m m j

Nj

l lj l j

l l

iz

z z

iz

z

i

z

i

z z

i

x z ζ ζ C

,

F D x z x ζ

Stream function in the co-rotating frame

Two vortices, N=2, one tracer, L=1

1 2

1,2

1, 1

2

1

0

0.04

.6 0.3

0.02

z z

T

i