Don’t forget

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Sunday Sept 20th: Summary

•Main threads•Equations of geophysical fluid dynamics•Eigenfunctions of the Laplacian

Threads•TechniquesoEigenvalues/vectors/functionsoDifferentiating scalar and vector functionsoDifferential equations

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Threads•TechniquesoEigenvalues/vectors/functionsoDifferentiating scalar and vector functionsoDifferential equations

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Recall:Finding eigvals and eigvecs

( ) ( ) ( ) ( ) ( )

( )

th

( ) 0

homogeneous | | 0

order polynomial equation for , solutions.

n n n n n

n

Av v A v

A

NN

( )f x

x

'( )f x

( )y f x

0( ) ( )'( ) lim

hf x h f xf x

h

The derivative as a limit

Taylor series

0

0 0 0 0 0(3) ( )2 31 1 1( ) ( ) '( ) ''( ) ( ) ( )

2! 3! !n nf x f x f x h f x h f x h f x h

n

h x x

Factorial function: ! ( 1)( 2) 1 1!=1 2!=2 1=2 3!=3 2 1=6

n n n n

4!=4 3 2 1=24

(3) ( )0 0 0 0

2 30

1 1 12

( ) '( ) ''( ) (! 3! !

( () ) ) nnh h h hf x f x f x f x xn

f fx

constant derivativeat x0

powerof h=x-x0

Rules of differentiation 1

Sum rule: ( )' ' '

Product rule: ( )' ' '

Multiplicati

Pow

on by a constant: ( )' '

Lineari

er ru

ty

l

:

:

e d x xdx

f g f g

fg f g fg

af af

( )' ' ' af bg af bg

Properties of the exponential function

1

2 31 12! 3!

1 , 2.71828

,

( ) , with special case  1/ ,

, 

.

x

x y yx

x x x x

x x

x x

e x x x e

e e e

e e e e

d e edx

e dx e c

Sum rule:

Power rule:

Taylor series:

Derivative

Indefinite integral

All implicit in this: '( ) ( ); (0) 1E x E x E

The Gradient( , , , ) or ( , )

, ,

f f x y z t f f x t

f f ffx y z

The Laplacian

2 2 22

2 2 2f f ff

x y z

Divergence of a vector field

( , ) ( , ) ( , ) ( , )

· ,

, ,

, ,,

V x t xu v w

u v

t x t x t

u v wVx y z x y z

w

Curl

ˆˆ ˆˆˆ ˆ( ) ( ) ( )zy x x x yx y z

i j k

V i w v j w u k v u

u v w

Partial differential equationsAlgebraic equation: involves functions; solutions are numbers.

Ordinary differential equation (ODE): involves total derivatives; solutions are univariate functions.

Partial differential equation (PDE): involves partial derivatives; solutions are multivariate functions.

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Separation of VariablesManipulate PDE into the form ( ) ( ).

Result is 2 ODES: ( ) ; ( ) .

f x g y

f x c g y c

Separation/superposition in linear systems

0( ) t

dv Avdt

v t v e

1 2(1) (2) ( )1 0 2 0 0( ) Nt t tN

Nv t a v e a v e a v e superposition

separation

Wave superposition, Beats

1 2 1 21 2sin(2 ) sin(2 ) 2sin 2 cos 2

2 2f f f f

f t f t t t

wave beats

Fri 18Wed16Mon 14Sat 12 Sun 20Sun 13 Tue 15 Thu 17 Sat 19

Tides at Newport, Sept 12-20 2009

Spring tides

Superposition(linear, homogeneous equations)

( ), ( ) solutions

( ) ( ) solution

f x g x

af x bg x

Can build a complex solution from the sum of two or more simpler solutions.

Superposition in PDEs

0 0

0

0

1

ANY FUNCTION ( ) that obeys the boundary conditions (0) 0 and ( ) 0

can be represented as a Fourier serie

FOUR

(

IER'S THEOREM:

) si

s:

The c

nnn

xT x A

T x T

L

T

n

L

2

1

orresponding solution for the diffusion problem is:

( , ) sinn tL

nn

xT x t A n eL

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Perturbations about equilibrium

(3) ( )22

32 1 1 1''(0) (0) (0)

2! 3! ! = (0) '(0) n nF x F x F x

nd xd

Fm F xt

Equilibrium:F=0 ~0

2

2n'(0) 0 oscillatio

= '(0)

What if '(

0) 0 ??

d x Fx

F

m Fdt

x

0x

mF

(LINEARIZATION)

Lunar tides

Interpreting σ

b. repellora. attractor c. saddle

d. limit cycle e. unstable spiral f. stable spiral

Interpreting two σ’s

3D system: chaos

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

The equations of geophysical fluid dynamics

2

2

( , , , )

Provided , ,

zx yt

x z y t

d dydx dzdt t x dt y dt z dt

dydx dzu v wdt dt dt

u v w

flowvelocity

diffusion

density

0

21zx yt xu uu vu wu ufp v

2zx yt u v w

Pressuregradient

Coriolis Viscosity(velocity diffusion)

0

21zx yt yv uv vv wv vfp u

00

21zx yt zw uw vw ww p wg

Buoyancy

Motionless equilibrium

0

21zx y xtu uu vu wu p fv u

0

21zx y ytv uv vv wv p fu v

00

21z zx ytw uw vw ww p g w

0u v w

Motionless (hydrostatic) equilibrium

00

21zx yt zw uw vw ww p wg

0u v w

zp g

Geostrophic equilibrium

0

21zx yt xu uu vu wu ufp v

Pressuregradient

Coriolis

0

21zx yt yv uv vv wv vfp u

•Neglect viscosity•Assume velocity is small, so that products are negligible•Set time derivatives to zero.

Geostrophic equilibrium

0

1yf

u p0

1xf

v p

dp p pdydxdt x dt y dt

Flow is along isopycnals.

Geostrophic equilibrium

0

1yf

u p0

1xf

v p

0 0

0 0

=

1 1 =

1 1 = 0

dp p pdydxdt x dt y dt

p pu vx y

p p p px f y y f x

p p p pf x y f y x

Flow is along isopycnals.

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Eigenfunctions of the Laplacian:diffusion

2f ft

2 2

2

2

( , ) ( ) ( )

( )

( ) ( )

1 ~

eigenvalue of the Laplacian

eigenfunction ~ sin,cos

t

f x t F x T t

f dTF xt dt

f T t F x

dT F T eT Fdt

F F

F

Eigenfunctions of the Laplacian: diffusion

2f ft

2 2

2

( , ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )

Divide by ( ) ( )

f x t F x T t

f dTF xt dt

f T t F x

dTF x T t F xdt

F x T t

Eigenfunctions of the Laplacian: diffusion

2f ft

2 2

2

2

2

( , ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )

Divide by ( ) ( )

1 ~

eigenvalue of the Laplacian

eigenfunction ~ sin,cos

t

f x t F x T t

f dTF xt dt

f T t F x

dTF x T t F xdt

F x T t

dT F T eT Fdt

F F

F

Eigenfunctions of the Laplacian:waves

22 22f c f

t

2

2 2

2 2

2 2

2

2

22

22

( , ) ( ) ( )

( )

( ) ( )

1

eigenvalue of the Laplacian

eigenfunction~ sin,cos

f x t F x T t

f d TF xt dt

f T t F x

d T c FT Fdt

F Fc

c

F

Cylindrical coordinates

r

z

cylindrical

22

1 1( )r r zzf frf fr r Chain rule:

Waves on a circular basin

22

22 2 2

2 2

22 2

2 2 22

1 1( , , ) ( )

1 1( )

( , , ) ( ) ( ) ( )

( )1  

1Time

~: sin( ),cos  ( )

r r

r r

tt r r

tttt

f f r t f rf fr r

f c f c rf frt r

f r t R r P T t

PT rR kT r

T kc

Rc r P

Tk T k c T

Tt k t

cc

Waves on a circular basin

22

22 2

2 2 2

2

( )

( )

( )

Azimuthal:

:

~sin ,cos

r r

r r

r r

PrR krR r P

Pr rR k rR P

PrRr k r nR P

Pn

P

r

P n n

Waves on a circular basin

2 2 2

2

2 2 2

2 2 2

2 2 2 2

( )

Azimuthal:

( )Radial:

(

~sin ,cos

~ ( ),

)

( ) ( )0 n n

r r

r r

r r

rr r

PrRr k r nR P

Pn

P

rRr k r nR

r rR k r n

r R rR k r n R

P n n

R R

R J kr Y kr

Analyze just like exp, sin, cos.

Bessel’s equation Bessel functions

Bessel functions

Y singular at r=0.

To fit initial conditions: n nR a J (superposition)

Spherical coordinates

r

2 22 2 22

)s

1 1 1( ) (sinin sinr r ff r f f

r r r

Threads•TechniquesoDifferentiating scalar and vector functionsoDifferential equationsoEigenvalues/vectors/functions

•PrinciplesoSeparationoSuperposition

•ObjectivesoEquilibrium, oPerturbations – oscillations, instabilities

Don’t forget

• Keys• Homework• Mart!n’s coat, stapler