Spectra of Gravity Wave Turbulence in a Laboratory Flume

S Lukaschuk1, P Denissenko1, S Nazarenko2

1 Fluid Dynamics Laboratory, University of Hull2 Mathematics Institute, University of Warwick

WTS workshop, Warwick- Hull, 17-21 September

1. Phillips (JFM 1958, 1985)

sharp wave crests

strong nonlinearity

dimensional analysis

1K. Kuznetsov (JETP Letters, 2004)

slope breaks occurs in 1D lines

wave crests are propagating with a preserved shape

Theoretical prediction forenergy spectra of surface gravity waves

rdttxtxeE

tdttxtxeE

rki

k

ti

,,

,,

gk

353 kEgE k

) of instead( gkk 44 kEE k

2. Weak turbulence theory(Theory and numerical experiment - Hasselman, Zakharov, Lvov,

Falkovich, Newell, Hasselman, Nazarenko … 1962 - 2006)

• Kinetic equation approach for WT in an ensemble of weakly interacted low amplitude waves (Hasselmann)

• Assumptions: weak nonlinearity

random phase (or short correlation length) spatial homogeneity stationary energy flow from large to small scales

• Zakharov – Filonenko spectrum for gravity waves in infinite spaceis an exact solution of Hasselmann equation which describes a steady state with energy cascading through an inertial range from large to small scale (Kolmogorov - like spectrum): for gravity waves in infinite space

2

743

12 kEgE k

3. Finite size effects (mesoscopic wave turbulence)Theory: Kartashova (1998), Zakharov (2005), Nazarenko (2006) et al

• For the WTT mechanisms to work in a finite box, the wave intensity should be strong enough so that non-linear resonance broadening is much greater than the spacing of the k-grid (2/L ). This implies a condition on the minimal angle of the surface elevation

• Discrete scenario (Nazarenko, 2005)For weaker waves the number of four-wave resonances is depleted. This arrests the energy cascade and leads to accumulation of energy near the forcing interval. Such accumulation will proceed until the wave intensity is strong enough to the nonlinear broadening to become comparable to the k-grid spacing. At this point the four-wave resonances will get engage and the energy will propagate towards lower k. Mean spectrum settles at a critical slope determined by δk ~2/L:

411 kL

62127 LgE

Numerical experiments:

Phillips spectrum:

could not be expected in direct numerical simulations because • nonlinearity truncation at cubic terms, • artificial numerical dissipation at high k to prevent numerical blowups.

Confirmation of ZF spectra:

• Zakahrov et al (2002-5), • Onorato (2002),• Yokyama (2004), • Nazarenko (2005).

Results are not 100% satisfying because no greater than 1 decade inertial range

Field experiments: P.A. Hwang, D.W.Wang, Airborne Measurements of surface elevation k-spectra, (2000)

6 metres12

met

res

90 c

m8 Panel Wave Generator

Laser

Capacity Probes

Rain Generator

Small amplitude

400 405 410 415 420 425 430

-5

0

5

Ele

vatio

n, c

m Elevation as function of time: Ch 1(red), Ch 2(blue), (file 81)

400 405 410 415 420 425 430

-5

0

5

ch 2

time, [s]

Ele

vatio

n, c

m

Large amplitudes

400 405 410 415 420 425 430

-10

0

10

20

Ele

vati

on

, cm

Elevation as finction of time: Ch. 1(red), Ch. 1(blue) (file 88)

400 405 410 415 420 425 430

-10

0

10

20

time, sec

Ele

vati

on

, cm

Typical spectra E for small and large wave amplitudes

A=1.85 cm (=0.074)

mk

Ak

tA

m

m

6.1,m4

2.0,052.0,1-

2

A=3.95 cm (=0.16)

Spectrum slopes vs the wave spectral density Ef

(f is from the inertial interval)

Inset:spectral density Efvs the energy dissipation rate

fE f

=0 “avalanches”and also Phillips

=1/3 WTT

Estimation of the Dissipation Rate

0

0

Edt

dE

eEE t

PDF of the wave crests

Tayfun M.A. J Geophys. Res. (1980)

PDF of the spectral intensity band-pass filtered at f = 6 Hzwith f = 1 Hz

cm85.1,074.0 A cm85.1,074.0 A

cm73.3,15.0 Acm85.1,074.0 A

PDF of the spectral intensity Ef (f=6 Hz, f=1Hz)

ConclusionRandom gravity waves were generated in the laboratory flumewith the inertial interval up to 1m - 1cm.

The spectra slopes are not universal: they increase monotonically from about -6 to -4 with the amplitude of forcing.

At low forcing level the character of wave spectra is defined by the nonlinearity and discreteness effects, at high and intermediate forcing - by the wave breaking.

PDFs of surface elevation are non-gaussian at high wave nonlinearity.

PDF of the squared wave elevation filtered in a narrow frequencyinterval (spectral energy density) always has an intermittent tail.

Acknowledgements: Hull Environmental Research Institute

References: P. Denissenko, S. Lukaschuk and S. Nazarenko, PRL, July 2007

Cross-section images water boundary detection

Boundary detection

k-spectrum