Download - Theory of wind-driven sea
Theory of wind-driven sea
by V.E. Zakharov
S. BadulinA.DyachenkoV.GeogdjaevN.IvenskykhA.KorotkevichA.Pushkarev
In collaboration with:
Plan of the lecture:
1.Weak-turbulent theory
2.Kolmogorov-type spectra
3.Self-similar solutions
4.Experimental verification of weak-turbulent theory
5.Numerical verification of weak-turbulent theory
6.Freak-waves solitons and modulational instability
),( zrZ ),( yxr
V 0divV 0
hz |
H
t H
t
UTH
sdsdssssGdzdrTr
)()(),(2
12
),(),( ssGssG - Green function of the Dirichlet-Neuman problem
hz | 0z
z
...210 HHHH432
k -- average steepness
Normal variables:
*
*
||2
2
kkk
k
kkk
k
aak
i
aag
*a
Hi
t
ak
][ˆ]ˆ[])ˆ[ˆ(ˆ]ˆ[ˆ))((ˆ 2212
21 kkkkkkkkt
]ˆ[]ˆ[ˆ]ˆ[])ˆ()[( 2221 kkkkkgt
Truncated equations:
),,,(),,,(
2
1
3213
321
***
321321321
kkkkTkkkkT
bbbbTdkbbH kkkkkkkkkkkkkkk
)( 41233210
*3
*2
*1
)4(012312332103
*2
*1
)3(0123
123321032*1
)2(01231233210321
)1(0123
12210*2
*1
)3(012122102
*1
)2(0121221021
)1(012
00
bOdkbbbBdkbbbB
dkbbbBdkbbbB
dkbbAdkbbAdkbbA
ba
Canonical transformation - eliminating three-wave interactions:
24132
32324141241
23131
42423131231
22121
43432121221
32324141
42423131
43432121
3232414141322
41
4242313131422
31
4343212121432
21
43214
1
4321
21234
)(
))(()(4
)(
))(()(4
)(
))(()(4
))((
))((
))((
)()()(2
)()()(2
)()()(2
12)(
1
32
1
q
qqkkqqkk
q
qqkkqqkk
q
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqqqqqqq
T
|| kq where
Statistical description: )(* kknbb kkk
Hasselmann equation: nlkrkk
k SNt
N
3213213210
310210320321
2
012325
)()(
)(16
dkdkdkkkkk
NNNNNNNNNNNNTgSnl
)4/()()( 2knkN
Kinetic equation for deep water waves (the Hasselmann equation, 1962)
32132103210
3102103213202
0123
)()(
)(||2
kkkkkkk ddd
nnnnnnnnnnnnTSnl
dissS,inputS - empirical dependences
dt
dnknlS inputS dissS
),,,(),,,( 3216
321 kkkkkkkk TT
Conservative KE has formal constants of motion
wave action
energy
momentum
kdnN k
kkkk ndE ;k
kkM dnk
,divQ
nlk St
n
,divP
nlkk St
Q – flux of action
P – flux of energy
For isotropic spectra n=n(|k|) Q and P are scalars
let n ~ k-x, then Snl ~ k19/2-3xF(x), 3 < x < 9/2
Energy spectrum
ddndd ||||)()(),( kkkk
))((2),( 2
4 kn
g
311
31
34
)2(623
)2(2
4
31
34
)1(4)1(1
21
)(;~;623
;)(;~;4
when,0)(
QgCknx
PgCknx
,xxxxF
q
p
k
k
F(x)=0, when x=23/6, x=4 – Kolmogorov-Zakharov solutions
Kolmogorov’s constants are expressed in terms of F(y), where
31
3/11
231
4
2
38
;3
8
yqyp y
FC
yF
C
42x-y exponent for
yn ~)()(y
F(y)
Kolmogorov’s cascades Snl=0 (Zakharov, PhD thesis 1966)
4 / 3 1/ 3(1)
4( ) p
g PC
4 / 3 1/ 3(2)
11/ 3( ) q
g QC
Direct cascade (Zakharov PhD thesis,1966; Zakharov & Filonenko 1966)
Inverse cascade (Zakharov PhD thesis,1966)
Numerical experiment with “artificial” pumping (grey). Solution is close to Kolmogorov-Zakharov solutions in the corresponding “inertial” intervals
Phillips, O.M., JFM. V.156,505-531, 1985.
Snl >> Sinput , Sdiss
Nonlinear transfer dominates!
Just a hypothesis to check
kdissdiss
kinin
nS
nS
Existence of inertial intervals for wind-driven waves is a key point of critics of the weak turbulence approach for water waves
Wave input term Sin for U10p/g=1
Non-dimensional wave input rates
Dispersion of different estimates of wave input Sin and dissipation Sdiss is of the same magnitude as the terms themselves !!!
Term-to-term comparison of Snl and Sin. Algorithm by N. Ivenskikh (modified Webb-Resio-Tracy). Young waves, standard JONSWAP spectrum
Mean-over-angle
Down-wind
The approximation procedure splits wave balance into two parts when Snl dominates
• We do not ignore input and dissipation, we put them into appropriate place !
• Self-similar solutions (duration-limited) can be found for (*) for power-law dependence of net wave input on time
(*)k
nl
kin diss
dn Sdtd n
S Sdt
2;~when
),(
rtn
tbUatNr
k
k
We have two-parametric family of self-similar solutions where relationships between parameters are determined
by property of homogeneity of collision integral Snl
4219
;4/19 ba
and function of self-similar variable Uobeys integro-differential equation
(**))]([ USUU nlStationary Kolmogorov-Zakharov solutions appear to be particular
cases of the family of non-stationary (or spatially non-homogeneous) self-similar solutions when left-hand and right-hand sides of (**)
vanish simultaneously !!!
),( 11/120
11/2 tUtn
Self-similar solutions for wave swell (no input and dissipation)
Quasi-universality of wind-wave spectra
Spatial down-wind spectra spectra
Dependence of spectral shapes on indexes of self-similarity is weak
Numerical solutions for duration-limited case vs non-dimensional frequency U/g
*
1. Duration-limited growth
2. Fetch-limited growth
qpEE 00~~;~~
qpEE 00~~;~~
g
U
U
EgE h
h
~;~
4
2
Time-(fetch-) independent spectra grow as power-law functions of time (fetch) but experimental wind speed scaling
is not consistent with our “spectral flux approach”
Experimental dependencies use 4 parameters. Our two-parameteric self-similar solutions dictate two relationships between these 4 parameters
For case 2
2
110;
2 31
100
20
q
pp
Ess
ss – self-similarity parameter
Thanks to Paul HwangExperimental power-law fits of wind-wave growth.
Something more than an idealization?
Exponents are not arbitrary, not “universal”, they are linked to each other. Numerical results (blue – “realistic” wave inputs)
Total energy and total frequency
Energy and frequency of spectral “core”
219
q
p
qpEE 00~~;~~
Exponents p (energy growth) vs q (frequency downshift) for 24 fetch-
limited experimental dependencies. Hard line – theoretical dependence
p=(10q-1)/2
1. “Cleanest” fetch-limited
2. Fetch-limited composite data sets
3. One-point measurements converted to fetch-limited one
4. Laboratory data included
Self-similarity parameter ss vs exponent p for 24 experimental
fetc-limited dependencies
1. “Cleanest” fetch-limited
2. Fetch-limited composite data sets
3. One-point measurements converted to fetch-limited one
4. Laboratory data included
Numerical verification of the
Hasselmann equation
ˆ][ˆ]ˆ[])ˆ[ˆ(ˆ]ˆ[ˆ))((ˆ 2212
21 kkkkkkkkt
ˆ]ˆ[]ˆ[ˆ]ˆ[])ˆ()[( 2221 kkkkkgt
Dynamical equations :
Hasselmann (kinetic) equation :
yxrki
kdkdkekk
2
1ˆ
kkkkkk ndkdkdkkkkknnnnnnnnTt
n
321321321321132
2
123
Two reasons why the weak turbulent theory could fail:
1.Presence of the coherent events -- solitons, quasi - solitons, wave collapses or wave-breakings
2.Finite size of the system – discrete Fourier space:
Quazi-resonances
4321
4321
kkkk
Dynamic equations:
domain of 4096x512 point in real space
Hasselmann equation:
domain of 71x36 points in frequency-angle space
22
Four damping terms:
1. Hyper-viscous damping
2. WAM cycle 3 white-capping damping
3. WAM cycle 4 white-capping damping
4. New damping term
2)1024( kCk
),(~~
~
~
~1),(
4
Ek
k
S
S
k
kCS
PMdsds
totEkS~~
2/13)1002.3(
~ PMS
4,,1036.2 5 P0. Cds
4,,1010.4 5 P0.5 Cds
WAM Dissipation Function:
WAM cycle 3:
WAM cycle 4:
Komen 1984
Janssen 1992 Gunter 1992Komen 1994
),(~~
~
~
~1),( Ek
k
S
S
k
kCS
P
PM
dsds
totEkS~~
2/13)1002.3(~ PMS
12,,1000.1 6 P0 Cds
New Dissipation Function:
Freak-waves solitons and modulational instability