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Don’t forget
• Keys• homework
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