Intense solitonsoliton--likelike groups in laboratory experiments and numerical simulations. PiblifttiithPossible manifestations in the oceani c wave fi fildelds

,QVWLWXWHRI$SSOLHG3K\VLFV1L]KQ\1RYJRURG5XVVLD Alexey Slunyaev 1L]KQ\1RYJRURG6WDWH7HFKQLFDO8QLYHUVLW\1L]KQ\1RYJRURG5XVVLDHPDLO6OXQ\DHY#DSSOVFLQQRYUX in collaboration with: G. Clauss, H.-C. Hsu, M. Klein, A. Kokorina, M. Onorato, E. Pelinovsky Free propagation of intense envelope solitons in the facility of the Technical Univ. of Berlin

Several methods for the producing the signal for the wave maker were tested. The best efficiency was 1 2 3 8 9 achieved when the signal was taken from the auxiliary numerical simulation of the Euler equations. However, the analytic solution of the nonlinear Schrodinger equation is sufficient for the generation of the solitary wave groups with the peak steepness kA = 0.30 as well. cr cos  0t kA = 0.30 AtA 0 2 kAcr = 0.14 cr 2cosh  0.2 0.2 00 grtCxkA η η m m

Absorbing beach or rigid wall k 0 k 0 Snapshots of two solitary groups −0.2 −0.2 with different peak wave steepness $VWURQJJ\SJJSO\GLVSHUVLQJJURXS −5 0 5 −5 0 5 x/λ x/λ (numerical simulations of the Euler eqseqs)) m m 10 9 &RPSDULVRQEHWZHHQWKHODERUDWRU\PHDVXUHPHQWVDQGQXPHULFDOVLPXODWLRQV 8 7 :DYHJURXSVKDSHV 6 ges Individual stationary groups of gauges 2−10 gauge 3 numerics 5 Surface elevation records from laboratory tests gau unidirectional waves over deep 4 and fitted envelopes from numerical simulations water are observed in numerical 0.2 3 and laboratory simulations η c 2 k 0 &UHVWWURXJKDV\PPHWU\ QXPVLPV The groups correspond to the envelope 1 solitons of the nonlinear Schödinger −0.2 solitons equation, when the waves are of 0 20 40 60 80 100 120 140 160 −60 −40 −20 0 20 40 60 1.4 ω Stokes wave small amplitude and there are many

t tr t, s c A

/ individual waves in a group

gauges 2−10 gauge 3 numerics cr

A 1.2 Very short and intense (but non- $VWDEOHJURXSದ D VROLWRQOLNHJURXS breaking) wave groups are observed 0.2 1 with peak wave height up to η

c 0.15 0.2 0.25 0.3 0.35 0.4 10 k 0 kH ~ 0.57 in HOSM simulations, A ω 2/g and 9 cr p −0.2 kH ~ 0.52 in laboratory runs 8 −60 −40 −20 0 20 40 60 (k = 2/g). ω t :DYHJURXSYHORFLWLHV 7 c Good agreement between the wave 6 group shapes is observed, when the gauges 2−10 gauge 3 numerics best lab groups 5 1.2 results of laboratory and numerical gauges

g numerics

/ simulations are compared 4 Stokes wave 0.2 V m

3 peak steepness ω 1.1 The solitary wave groups possess the η 2 c

k same vertical asymmetry as the Stokes 2 kAcr = 0.30 0 waves, but move noticeably faster 1 −0.2 1 0.15 0.2 0.25 0.3 0.35 0.4 than corresponding Stokes waves. −60 −40 −20 0 20 40 60 0 20 40 60 80 100 120 140 160 2 ω t A ω / g t, s c cr m

Collisions of the solitary groups: lab vs numerical sims Reflection from the vertical wall (numerical simulations)

95.03 m 72.51 m 9 kA = 0.30 The effect of the wall on 94.04 m 61.68 m kAcr = 0.14 115 cr 8 8 waves is equivalent to a 93.54 m 59.66 m 7 7 mirror reflection, which 93.04 m 57.65 m 6 6 s leads to the appearance 80 m 53.43 m 110 ge ges 5 5 of a standing wave

gau 50.19 m gau 79.8 m 4 surface envelope m surface envelope structure of the wave 62.5 m 38.16 m 3 3 is shown t/T 105 envelope with maximum 27.5 m 27.5 m is shown 2 2 on the wall and the 10 m measurement 10 m measurement 1 1 nearest waist at about simulation simulation one fourth of the 0 50 100 150 200 0 20 40 60 80 100 120 140 100 t, s t, s dominant wave length from the wall. Reasonable agreement between the instrumental measurements and the numerical −5 −4 −3 −2 −1 0 simulations of the potential Euler equations (HOSM6) is obtained (x−L)/λ m Collision of solitons with different frequencies (num simssims)) The dynamic pressure under intense groups (analytic solution)

0 0.6 0.6 −2 2QHVROLWDU\JURXS &ROOLVLRQRIWZRJURXSV 0.3 head−on 0.3 −4 −3 η head−on η −6 x 10 m 0 m 0 k overtaking k 0 0 −8 −0.3 overtaking −0.3 0 −0.02 z 0 −10 −5

k −5 VWHHSQHVV  −0.04 YV −0.6 −0.6 −12 z p   −2 z 0 −5 0 5 −10 −5 0 5 10 −10 0 −10 −0.06 k t k x/λ t/T −14 m m −0.08 −16 −15 −4 −15 two groups −0.1 0.6 0.6 one group (×23) kA = 0.25 −18 cr kAcr = 0.25 progressive wave −20 −20 −0.12 0.3 head−on 0.3 −20 −20 0 20 −20 0 20 0 0.05 0.1 k x k x η head−on η k p/g 0 0

m 0 m VW 0 0 k DOODQG QGVORZ overtaking k D D I P −0.3 overtaking −0.3 Due to thliiddhe nonlinear induced current t he pressure fildbhilifield beneath intense solitary groups can penetrate deep into the sea. It may me comparable in magnituge with the pressure induced by a linear propagating wave. −0.6 −0.6 The induced pressure structure is complicated. It is drastically different for the cases of one travelling group YVV −5 0 5 −10 −5 0 5 10

ODUJH and of two head-on colliding groups. x/λ t/T m m Steep solitary groups in stochastic numerical simulations 0.6 0.6 VW G 0 0 10 10 D Q 0.3 0.3 −1 −1 head−on unidirectional Realization No 295 10 10 η η head−on JONSWAP m −2 m 0 0 −2

k 10 10 k overtaking Tp = 10s

VORZ −3 −0.3 overtaking −0.3 10 −3 Hs = 3.5m 10

−4 −4 −0.6 −0.6 = 3 10 10 YVVPDOOD −10 −5 0 5 10 Rayleigh ODUJHDQGI −5 0 5 100 realiz−s −5 −5 ce probability distribution λ t/T 10 ce probability distribution 10 P±σ an x/ an m m P±Δ −6 −6 300 realiz−s 10 10 Exceed 0.6 0.6 Rayleigh Exceed P±σ 20 realiz−s P±Δ −7 100 realiz−s −7 10 10 realiz. No 295 0.3 0.3 300 realiz−s P±σ head−on realiz. No 295 P±Δ −8 −8 10 10 η head−on η 0 2 4 6 8 10 0.2 0 2 4 6 8 10 m 0 m 0 (H/H )2 2 k (H/H ) overtaking k 1/3 1/3 overtaking 0.15 YV −0.3 HSQHVV −0.3 The occu rr en ce of an abn orm all y in ten se l on g-living  k H 0.1 −0.6 −0.6 wave group from the waves of moderate steepness VW −5 0 5 −10 −5 0 5 10 0.05  exhibits the features of the ‘black swan’ effect. It x/λ t/T m m 0 could not be expected from the previous consideration −20 −10 0 10 20 30 (x−x − C (t−t ))/λ of smaller statistical ensembles, but when happens, * gr * p Overtaking wave groups may result in pronouncing set-ups of the snapshots of extremely large waves. produces an enormous effect on the wave statistics. In head-on collisions the setup is more evident for the snapshots rather than for the time series. AgustAgustAgust,, 20172017,