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The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 1
1.3. Energy Balance Eqn (Physics):
Discuss wind input and nonlinear transfer in some detail. Dissipation is just given.
Common feature: Resonant Interaction
Wind: Critical layer: c(k) = Uo(zc).
Resonant interaction betweenair at zc and wave
Nonlinear: i = 0 3 and 4 wave interaction
zc
Uo=c
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 2
Transfer from Wind:
Instability of plane parallel shear flow (2D)
Perturb equilibrium: Displacement of streamlinesW = Uo - c (c = /k)
)()(,)(
ˆ,ˆ)(:um”“Equilibri
zdzgzPz
eggezUU
oooo
zxoo
0d
d
1
d
d0
:fluid Adiabatic
t
gPut
u
Ww /
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 3
give Im(c) possible growth of the wave
Simplify by taking no current and constant density in water and air.
Result:
Here, =a /w ~ 10-3 << 1, hence for 0, !
Perturbation expansion
growth rate = Im(k c1)
)()d
d(
d
d 222ooo gWk
zW
z
)(d
d;,0 z
zz oo
0,
)]0(/[)1(,1)0(
0,d
d
d
d
2
222
a
aa
aa
z
kgc
zWkz
Wz
c2 = g/k
)1(,/ 121
11 akooo cckgcccc
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 4
Further ‘simplification’ gives for = w/w(0):
Growth rate:
Wronskian:
0,
)(lim,1)0(
)0at(singular Eq.Rayleigh d
d 22
2
z
cUW
WWkz
W
occ
o
ooo
o
0
*),(4
1
z
o
Wk
)( ** iW
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 5
1. Wronskian W is related to wave-induced stress
Indeed, with and the normal mode formulation for u1, w1: (e.g. )
2. Wronskian is a simple function, namely constant except at critical height zc
To see this, calculate dW/dz using Rayleigh equation with proper treatment of the singularity at z=zc
where subscript “c” refers to evaluation at critical height zc (Wo = 0)
11 wuw 0 u
..1 cceuu i
)( ** wwwwk
iw
zcz
w
)(2d
d 2
ococ WWWz
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 6
This finally gives for the growth rate (by integrating dW/dz to get W(z=0) )
Miles (1957): waves grow for which the curvature of wind profile at zc is negative (e.g. log profile).
Consequence: waves grow slowing down wind: Force = dw / dz ~ (z) (step
function)For a single wave, this is singular! Nonlinear theory.
2
2 coc
oc
W
W
k
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 7
Linear stability calculation
Choose a logarithmic wind profile (neutral stability)
= 0.41 (von Karman), u* = friction velocity, =u*
Roughness lengrth, zo : Charnock (1955) zo = u*
2 / g , 0.015
Note: Growth rate, , of the waves ~ = a / w
and depends on
so, short waves have the largest growth.
Action balance equation
oo z
zuzU 1ln)( *
g
u
c
u **
2
*~,2
c
uNSN
t windwind
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 8
Wave growth versus phase speed(comparison of Miles' theory and observations)
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 9
Nonlinear effect: slowing down of wind
Continuum: w is nice function, because of continuum
of critical layers
w is wave induced stress
(u , w): wave induced velocity in air (from Rayleigh Eqn).
. . .
with (sea-state dep. through N!).
Dw > 0 slowing down the wind.
wwaves
o zU
t
wuw
owwaves
o Uz
DUt 2
2
)(233
kNvc
kcD
g
w
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 10
Example:
Young wind sea steep waves. Old wind sea gentle waves.
Charnock parameter depends on sea-state!
(variation of a factor of 5 or so).
Wind speed as function of height for young & old wind sea
old windsea
young windsea
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 11
Non-linear Transfer (finite steepness effects)
Briefly describe procedure how to obtain:
1.
2. Express in terms of canonical variables and = (z =) by solving
iteratively using Fourier transformation.
3. Introduce complex action-variable
nonlint
N
22
212
21
zdzg
0,
,
0
z
zz
z
z
)(ka
)()(2
1ˆ *
2/1
kakak
)()(2
ˆ *2/1
kakak
i
kg
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 12
gives energy of wave system:
with , … etc.
Hamilton equations: become
Result:
Here, V and W are known functions of
....].[)(ddd
d
d
32*13,2,1321321
*1111
ccaaaVkkkkkk
aak
xE
)( 11 kaa
E
t
E
t*
)(a
Eika
t
43*243214,3,2,1432
321323,2,132111
)(ddd
...)(dd
aaakkkkWkkki
kkkaaVkkiaiat
k
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 13
Three-wave interactions:
Four-wave interactions:
Gravity waves: No three-wave interactions possible.
Sum of two waves does not end up on dispersion curve.
0
0
321
321
kkk
4321
4321
kkkk
dispersion curve
2, k2
1, k1
k
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 14
Phillips (1960) has shown that 4-wave interactions do exist!
Phillips’ figure of 8
Next step is to derive the statistical evolution equation for with N1 is the action density.
Nonlinear Evolution Equation
)( 211*21 kkNaa
hierarchyInfinite
&
&
543214321321
*4
*32132121
aaaaaaaaaaaat
aaaaaaaaat
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 15
Closure is achieved by consistently utilising the assumption of Gaussian probability
Near-Gaussian
Here, R is zero for a Gaussian.
Eventual result:
obtained by Hasselmann (1962).
Raaaaaaaaaaaa *32
*41
*42
*31
*4
*321
)]()([
)(
)(ddd4
32414132
1432
14322
4,3,2,14321
NNNNNNNN
kkkkTkkkNt nonlin
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 16
Properties:
1. N never becomes negative.
2. Conservation laws:
action:
momentum:
energy:
Wave field cannot gain or loose energy through four-wave interactions.
)(d kNk
)(d kNkk
)(d kNk
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 17
Properties: (Cont’d)
3. Energy transfer
Conservation of two scalar quantities has implications
for energy transfer
Two lobe structure is impossible because if action is conserved, energy ~ N cannot be conserved!
)(d
d Nt
The Wave Model: 1.3. Energy Balance Eqn (Physics) Slide 18
Dissipation due to Wave Breaking
Define:
with
Quasi-linear source term: dissipation increases with
increasing integral wave steepness
)(d
)(d
kNk
kNk
)(d
)(d
kNk
kNkkk
NXXmkS odissip2
22)1(
2,/ omkkX
omk2
