EDDY DYNAMICS IN THE EASTERN GULF

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```					                                Mathematics
in the Ocean
• Andrew Poje
Mathematics Department
College of Staten Island

•   M. Toner            U. Delaware
•   A. D. Kirwan, Jr.
•   G. Haller
•   C. K. R. T. Jones     Brown U.
•   L. Kuznetsov
•   … and many more!
April is Math Awareness Month
Why Study the Ocean?
• Fascinating!
• 70 % of the planet is
ocean
• Ocean currents control
climate
• Dumping ground -
Where does waste go?
Ocean Currents:
The Big Picture
• HUGE Flow Rates
(Football Fields/second!)
• Narrow and North in
West
•   Broad and South in East
•   Gulf Stream warms
Europe
•   Kuroshio warms Seattle

image from Unisys Inc.
(weather.unisys.com)
Drifters and Floats:
Measuring Ocean Currents
Particle (Sneaker) Motion in
the Ocean
Particle Motion in the Ocean:
Mathematically

• Particle locations:         dx
(x,y)                         u ( x, y )
•   Change in location is
dt
dy
given by velocity of          v ( x, y )
water: (u,v)              dt
•   Velocity depends on
x t  0   x0
position: (x,y)
•   Particles start at some
initial spot              y t  0   y0
Ocean Currents:
Time Dependence
• Global Ocean Models:
è   Math Modeling
è   Numerical Analysis
è   Scientific Programing
• Results:
è   Highly Variable Currents
è   Complex Flow Structures
• How do these effect
transport properties?
image from Southhampton Ocean Centre:.
http://www.soc.soton.ac.uk/JRD/OCCAM
Coherent Structures:
Eddies, Meddies, Rings & Jets
• Flow Structures
responsible for Transport
• Exchange:
è   Water
è   Heat
è   Pollution
è   Nutrients
è   Sea Life
• How Much?
• Which Parcels?
image from Southhampton Ocean Centre:.
http://www.soc.soton.ac.uk/JRD/OCCAM
Coherent Structures:
Eddies, Meddies, Rings & Jets
Mathematics in the Ocean:
Overview
• Mathematical Modeling:
è   Simple, Kinematic Models
(Functions or Math 130)
è   Simple, Dynamic Models
(Partial Differential Equations or Math 331)
è   ‘Full Blown’, Global Circulation Models
• Numerical Analysis: (a.k.a. Math 335)
• Dynamical Systems: (a.k.a. Math 330/340/435)
è   Ordinary Differential Equations
è   Where do particles (Nikes?) go in the ocean
Modeling Ocean Currents:
Simplest Models
• Abstract reality:
è   Look at real ocean currents
è   Extract important features
è   Dream up functions to mimic ocean

• Kinematic Model:
è   No dynamics, no forces
è   No ‘why’, just ‘what’
Modeling Ocean Currents:
Simplest Models
• Jets: Narrow, fast currents
• Meandering Jets: Oscillate in
time
•   Eddies: Strong circular currents

Jet :  jet ( x, y, t )  K tanh y  sin( kx  ct ) L 
 
Eddy : eddy ( x, y )  A exp   x  xeddy    y  yeddy 
2                 2

  jet  eddy                           jet  eddy 
u ( x, y , t )                          , v ( x, y , t ) 
y                                        x
Modeling Ocean Currents:
Simplest Models

Dutkiewicz & Paldor : JPO ‘94
Haller & Poje: NLPG ‘97
Particle Dynamics in a
Simple Model
Modeling Ocean Currents:
Dynamic Models
è   Wind blows on surface
è   F = ma
è   Earth is spinning
• Ocean is Thin Sheet (Shallow Water Equations)
• Partial Differential Equations for:
è   (u,v): Velocity in x and y directions
è   (h): Depth of the water layer
Modeling Ocean Currents:
Shallow Water Equations
D                 h        2u    2u 
u  fv   g '     e 
 xx  yy   Wx ( x, y, t )

Dt                x                   
ma = F:      D                 h        2v     2v 
v  fu   g '     e 
 xx  yy   W y ( x, y, t )

Dt                y                    

Mass Conserved:
D
h  hb   h  hb  u  v   0
 x y 
         
Dt                               

D        
Non-Linear:                           u     v
Dt       x    y
Modeling Ocean Currents:
Shallow Water Equations

• Channel with Bump
• Nonlinear PDE’s:
è   Solve Numerically
è   Discretize
è   Linear Algebra
è   (Math 335/338)
• Input Velocity: Jet
• More Realistic (?)
Modeling Ocean Currents:
Shallow Water Equations
Modeling Ocean Currents:
Complex/Global Models
è   Depth Dependence (many shallow layers)
è   Account for Salinity and Temperature
è   Ice formation/melting; Evaporation
è   Realistic Geometry
è   Outflow from Rivers
è   ‘Real’ Wind Forcing
• 100’s of coupled Partial Differential Equations
• 1,000’s of Hours of Super Computer Time
Complex Models:
North Atlantic in a Box

• Shallow Water
Model
•   b-plane (approx.
Sphere)
Winds and
Westerlies
Particle Motion in the Ocean:
Mathematically

• Particle locations:         dx
(x,y)                         u ( x, y )
•   Change in location is
dt
dy
given by velocity of          v ( x, y )
water: (u,v)              dt
•   Velocity depends on
x t  0   x0
position: (x,y)
•   Particles start at some
initial spot              y t  0   y0
Particle Motion in the Ocean:
Some Blobs S t r e t c h
Dynamical Systems Theory:
Geometry of Particle Paths
• Currents:              Simplest Example :
Characteristic Structures
dx
• Particles:                      y
Squeezed in one direction    dt
Stretched in another
dy
• Answer in Math 330 text!        x
dt
Dynamical Systems Theory:

Simplest Example:
 x (t ) 
 y (t ) 
X (t )          
        
dX  0 1 
  1 0X  
dt             

 1               1 
X (t )  c1   exp( t )  c2   exp( t )
 1                1
                 
Dynamical Systems Theory:
North Atlantic in a Box:

disappear

• … but they still affect
particle behavior
Dynamical Systems Theory:
The Theorem
è don’t move too fast
è don’t change shape too much
è are STRONG enough

• Then there are MANIFOLDS in the flow
• Manifolds dictate which particles go where
Dynamical Systems Theory:
Making Manifolds
UNSTABLE MANIFOLD:
A LINE SEGMENT
IS INITIALIZED ON DAY 15
ALONG THE EIGENVECTOR
ASSOCIATED WITH THE
POSITIVE EIGENVALUE
AND INTEGRATED
FORWARD IN TIME
STABLE MANIFOLD:
A LINE SEGMENT
IS INITIALIZED ON DAY 60
ALONG THE EIGENVECTOR
ASSOCIATED WITH THE
NEGATIVE EIGENVALUE
AND INTEGRATED
BACKWARD IN TIME
Dynamical Systems Theory:
Mixing via Manifolds
Dynamical Systems Theory:
Mixing via Manifolds
North Atlantic in a Box:
Manifold Geometry

• Each saddle has pair of
Manifolds
•   Particle flow:
IN on Stable
Out on Unstable
•   All one needs to know
BLOB HOP-SCOTCH

BLOB TRAVELS FROM HIGH MIXING REGION IN THE EAST TO HIGH
MIXING REGION IN THE WEST
BLOB HOP-SCOTCH:
Manifold Explanation
RING FORMATION

around day 159.5
• Eddy is formed mostly
from the meander water
• No direct interaction with
outside the jet structures
Summary:
Mathematics in the Ocean?
• ABSOLUTELY!
• Modeling + Numerical Analysis = ‘Ocean’ on
Anyone’s Desktop
•   Modeling + Analysis = Predictive Capability
(Just when is that Ice Age coming?)
•   Simple Analysis = Implications for
Understanding Transport of Ocean Stuff

• …. and that’s not the half of it ….
April is Math Awareness Month!

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