Mathematics in the Ocean Andrew Poje Mathematics Department College of Staten Island M. Toner A. D....
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Transcript of Mathematics in the Ocean Andrew Poje Mathematics Department College of Staten Island M. Toner A. D....
Mathematics Mathematics in the Oceanin the Ocean• Andrew Poje
Mathematics Department College of Staten Island
• M. Toner
• A. D. Kirwan, Jr.
• G. Haller
• C. K. R. T. Jones
• L. Kuznetsov
• … and many more!
April is Math Awareness Month
U. Delaware
Brown U.
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: (x,y)
• Change in location is given by velocity of water: (u,v)
• Velocity depends on position: (x,y)
• Particles start at some initial spot
( )( ) 0
0
0
0
),(
),(
yty
xtx
yxvdt
dy
yxudt
dx
==
==
=
=
Ocean Currents: Time Dependence
• Global Ocean Models: Math Modeling Numerical Analysis Scientific Programing
• Results: Highly Variable Currents 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: Water Heat Pollution Nutrients 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: Simple, Kinematic Models
(Functions or Math 130) Simple, Dynamic Models
(Partial Differential Equations or Math 331) ‘Full Blown’, Global Circulation Models
• Numerical Analysis: (a.k.a. Math 335)
• Dynamical Systems: (a.k.a. Math 330/340/435) Ordinary Differential Equations Where do particles (Nikes?) go in the ocean
Modeling Ocean Currents:Simplest Models
• Abstract reality: Look at real ocean currents
Extract important features
Dream up functions to mimic ocean
• Kinematic Model:
No dynamics, no forces
No ‘why’, just ‘what’
Modeling Ocean Currents:Simplest Models
• Jets: Narrow, fast currents
• Meandering Jets: Oscillate in time
• Eddies: Strong circular currents
{ }( )( ) ( ){ }( )
( ) ( )x
tyxvy
tyxu
yyxxAyx
LctkxyKtyx
eddyjeteddyjet
eddyeddyeddy
jet
∂
Ψ+Ψ∂=
∂
Ψ+Ψ∂−=
−+−−=Ψ
−−=Ψ
),,(,),,(
exp),( :Eddy
)sin(tanh),,( :Jet22α
Modeling Ocean Currents:Simplest Models
Dutkiewicz & Paldor : JPO ‘94
Haller & Poje: NLPG ‘97
Particle Dynamics in a Simple Model
Modeling Ocean Currents:Dynamic Models
• Add Physics: Wind blows on surface F = ma Earth is spinning
• Ocean is Thin Sheet (Shallow Water Equations)
• Partial Differential Equations for: (u,v): Velocity in x and y directions (h): Depth of the water layer
Modeling Ocean Currents:Shallow Water Equations
( ) ( )
( ) ( ) ( )y
vx
uDt
D
y
v
x
uhhhh
Dt
D
tyxWyy
v
xx
v
y
hgfuv
Dt
D
tyxWyy
u
xx
u
x
hgfvu
Dt
D
bb
ye
xe
∂
∂+
∂
∂=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂
∂−+−
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂+
∂∂
∂+
∂
∂−=+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂+
∂∂
∂+
∂
∂−=−
0
),,('
),,('
22
22
υ
υ
ma = F:
Mass Conserved:
Non-Linear:
Modeling Ocean Currents:Shallow Water Equations
• Channel with Bump
• Nonlinear PDE’s: Solve Numerically Discretize Linear Algebra (Math 335/338)
• Input Velocity: Jet
• More Realistic (?)
Modeling Ocean Currents:Shallow Water Equations
Modeling Ocean Currents:Complex/Global Models
• Add More Physics: Depth Dependence (many shallow layers) Account for Salinity and Temperature Ice formation/melting; Evaporation
• Add More Realism: Realistic Geometry Outflow from Rivers ‘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
• -plane (approx. Sphere)
• Forced by Trade Winds and Westerlies
Particle Motion in the Ocean:Mathematically
• Particle locations: (x,y)
• Change in location is given by velocity of water: (u,v)
• Velocity depends on position: (x,y)
• Particles start at some initial spot
( )( ) 0
0
0
0
),(
),(
yty
xtx
yxvdt
dy
yxudt
dx
==
==
=
=
Particle Motion in the Ocean:Some Blobs S t r e t c h
Dynamical Systems Theory:Geometry of Particle Paths
• Currents: Characteristic Structures
• Particles: Squeezed in one direction
Stretched in another
• Answer in Math 330 text! xdt
dy
ydt
dx
=
=
:Example Simplest
Dynamical Systems Theory:Hyperbolic Saddle Points
)exp(1
1)exp(
1
1)(
01
10
)(
)()(
21 tctctX
Xdt
dX
ty
txtX
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
Simplest Example:
Dynamical Systems Theory:Hyperbolic Saddle Points
North Atlantic in a Box:Saddles Move!
• Saddle points appear
• Saddle points disappear
• Saddle points move
• … but they still affect particle behavior
Dynamical Systems Theory:The Theorem
• As long as saddles:don’t move too fastdon’t change shape too much are STRONG enough
• Then there are MANIFOLDS in the flow
• Manifolds dictate which particles go where
UNSTABLE MANIFOLD:A LINE SEGMENT IS INITIALIZED ON DAY 15ALONG THE EIGENVECTOR ASSOCIATED WITH THEPOSITIVE EIGENVALUE AND INTEGRATED FORWARD IN TIME
STABLE MANIFOLD:A LINE SEGMENT IS INITIALIZED ON DAY 60ALONG THE EIGENVECTOR ASSOCIATED WITH THENEGATIVE EIGENVALUE AND INTEGRATED BACKWARD IN TIME
Dynamical Systems Theory:Making Manifolds
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 about particle paths (?)
BLOB HOP-SCOTCH
BLOB TRAVELS FROM HIGH MIXING REGION IN THE EAST TO HIGH MIXING REGION IN THE WEST
BLOB TRAVELS FROM HIGH MIXING REGION IN THE EAST TO HIGH MIXING REGION IN THE WEST
BLOB HOP-SCOTCH:Manifold Explanation
RING FORMATION
• A saddle region appears around day 159.5• Eddy is formed mostly from the meander water• No direct interaction withoutside 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!