Evolution of a narrowband spectrum of random surface gravity waves


 Baldwin Hopkins
 1 years ago
 Views:
Transcription
1 J. Fluid Mech. (3), vol. 478, pp.. c 3 Cambridge University Press DOI:.7/S66 Printed in the United Kingdom Evolution of a narrowband spectrum of random surface gravity waves By KRISTIAN B. DYSTHE, KARSTEN TRULSEN, HARALD E. KROGSTAD 3 AND HERVÉ SOCQUETJUGLARD Department of Mathematics, University of Bergen, Johs. Brunsgt., N58 Bergen, Norway SINTEF Applied Mathematics, P.O. Box 4 Blindern, N34 Oslo, Norway 3 Department of Mathematical Sciences, NTNU, N749 Trondheim, Norway (Received April and in revised form 7 August ) Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (978) in the twodimensional but not in the threedimensional case. Using a modified NLS equation (Trulsen et al. ) the spectra relax towards a quasistationary state on a timescale (ɛ ω ).Inthisstate the lowfrequency face is steepened and the spectral peak is downshifted. The threedimensional simulations show a powerlaw behaviour ω 4 on the highfrequency side of the (angularly integrated) spectrum.. Introduction It is wellknown that a uniform train of surface gravity waves is unstable to the socalled modulational instability or Benjamin Feir (BF) instability (Lighthill 965, 967; Whitham 967; Benjamin & Feir 967; Zakharov 968). The stability of nearly Gaussian random wave fields of narrow bandwidth was considered by Alber & Saffman (978), Alber (978) and Crawford, Saffman & Yuen (98). For waves on deep water they found that the modulational instability occurs provided that the relative spectral width, σ,isless than twice the average steepness, ɛ k a,wherek is the central wavenumber, and ā =(η ) / is the r.m.s. value of the amplitude. For deep water waves the results of Alber & Saffman (978) and Alber (978) are based on the nonlinear Schrödinger (NLS) equation, which is the lowestorder model equation in the relative spectral width and wave steepness that takes into account both dispersion and nonlinearity. It can be written ( ) B B i + v g + ( dvg B t x dk x + v ) g B ω k k y B B =, () where the surface elevation η is given by η = ( Be i(k x ω t) + B e i( ω t) + +c.c. ) () where B and B are slowly varying functions of x =(x,y) andt. Current address: Department of Mathematics, University of Oslo, P.O. Box 53 Blindern, N36 Oslo, Norway.
2 K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. SocquetJuglard Alber (978) considers the case where B is a slowly varying, nearly Gaussian random function of x and t. Hedevelops a nonlinear transport equation for the ensembleaveraged envelope spectrum F (K, x,t)(where F dk =η = a ) centred around k =(k, ), where the wave vector is k = k + k K and K =(K x,k y ). The stability of a random homogeneous and stationary wave field with spectrum F (K) is then investigated, where F (K) istaken to have the Gaussian shape [ ( )] F (K) = η exp K x K y, (3) πσ x σ y σx σy and where σ x and σ y are the relative bandwidths in the x and ydirections, respectively. Alber s result for three dimensions is unfortunately somewhat difficult to interpret except for the case when the spectrum is symmetric around k (i.e. σ x = σ y σ ). In that case, a necessary and sufficient condition for stability is found to be σ>ɛ, (4) which is formally identical to the twodimensional condition of Alber & Saffman (978) and Alber (978). Crawford et al. (98) extended the work of Alber. Their starting point is the socalled Zakharov integral equation, which does not have any spectral limitations. They derive the corresponding transport equation governing the evolution of a nonhomogeneous random wave field, assuming the zerofourthorder cumulant hypotheses. By further taking the narrowband limit, i.e. evaluating the coupling coefficients in their integrodifferential equation at the central wave vector, they arrive at the same transport equation as Alber. To investigate the stability of narrowband random wavetrains, they choose a different initial spectrum, namely η F (K) = σ x σ y π ( )( ), (5) Kx + σx K y + σy where σ x and σ y are the halfwidths in the x andydirections, respectively. They find that the spectrum in (5) is stable provided that σ x > ɛ. (6) For the symmetric case σ x = σ y,the condition (6) is formally the same as Alber s condition (4) although the definitions of the σ,aswellastheshapeof the spectra, are quite different. Note also that the relative spectral width transversal to the main wave direction, σ y, does not enter the condition (6). These results suggest that narrowband spectra do not change on the timescale (ɛ ω ),which is the typical timescale for the BF instability and henceforth referred to as the BF timescale, provided that the relative spectral width exceeds a certain threshold. If this is the case, the spectral change should only occur on the much longer timescale (ɛ 4 ω ),i.e. the Hasselmann timescale (Hasselmann 96; Crawford et al. 98). To our knowledge these important results, though more than years old, have not been compared to experiments or numerical simulations. Previous simulations using the NLS equation have to some extent concentrated on the stability of Stokes waves, or the evolution of a few Fourier modes. Here the phenomena of recurrence or near recurrence appear, as thoroughly discussed by Yuen &Ferguson (978) for the twodimensional case. As the number of Fourier modes
3 Narrowband spectrum of surface gravity waves 3 increases (although they may be very small initially see Trulsen & Dysthe 997), the recurrence phenomenon disappears from the computational time horizon. In the threedimensional simulations to be presented here with more than 4 modes, the probability of seeing a near recurrence becomes vanishingly small. In the present paper we report numerical simulations on the evolution of narrowband spectra in two and three dimensions with four different numerical models: (i) the NLS equation (); (ii) the modified nonlinear Schrödinger (MNLS) equation of Dysthe (979) which includes terms to the next order in nonlinearity and bandwidth, assuming the relative spectral width to be of order ɛ; (iii) the broader bandwidth equation of Trulsen & Dysthe (996) which relaxes the bandwidth in the MNLS equation to be of order ɛ / ; (iv) the equation of Trulsen et al. () which extends the MNLS equation with exact linear dispersion. For the present applications we have found that the two extensions (iii) and (iv) of the MNLS equation give practically the same results as the MNLS equation, (ii), for the behaviour near the spectral peak, hence these results will collectively be referred to as MNLS in the following. The MNLS has been compared with wave tank experiments (Trulsen & Stansberg ) and shown to give excellent agreement up to times one order of magnitude larger than the BF timescale. The twodimensional results using the NLS equation only approximately confirm the stability conditions of Alber & Saffman (978) and Alber (978). They also show that an unstable spectrum relaxes, on the timescale of the BF instability, to a wider symmetrical spectrum that appears to be stable within the time horizon for the equation. The threedimensional NLS simulations, however, do not agree with the theoretical predictions. The initial spectral distributions spread out in both horizontal directions, regardless of whether they satisfy the Alber stability criterion or not. Also, for the threedimensional case, the spectral evolution takes place on the BF timescale. The MNLS simulations, on the other hand, show a somewhat different evolution both in two and three dimensions. Regardless of the initial spectral width, the spectra evolve on the BF timescale from an initial symmetrical form into a skewed shape with a downshift of the peak frequency. In three dimensions, the angularly integrated spectrum shows an evolution towards a powerlaw behaviour ω 4 on the highfrequency side.. The numerical simulations The NLS equation is given in (), while the higherorder equations can be found in Dysthe (979), Trulsen & Dysthe (996) and Trulsen et al. (), and are not reproduced here. Length and time are normalized with the characteristic wavenumber and frequency, k and ω = gk,respectively. We use the numerical method described by Lo & Mei (985, 987) with periodic boundary conditions in both horizontal directions. A uniform grid in the physical and wave vector planes with N x = N y = 8 points is employed. With a computational domain of 5 characteristic wavelengths in each horizontal direction, we set the discretization of the wave vector plane to be K x = K y =/5. However, we only employ the modes with relative bandwidth less than unity, K <. The physical plane has dimensions L x =π/ K x and L y =π/ K y,andhasthegrid points x j = L x j/n x, j =,,...,N x andy l = L y l/n y, l =,,...,N y.
4 4 K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. SocquetJuglard In all simulations we start with the initial Gaussian bellshaped spectrum (3) with σ x = σ y = σ,orthecorresponding onedimensional form. Note that in the threedimensional simulations the spectrum has a small angular spread around the main wave direction, with a halfangle spread θ σ (approximately 6 and for σ =. and., respectively). For each sample simulation the Fourier components are initialized with random phases and a modulus proportional to the square root of the spectrum Kx K y ˆB mn () = ɛ exp ( (m K ) x) +(n K y ) exp iθ πσ 4σ mn, (7) where the random variables θ mn are uniformly distributed on the interval [, π). The physical amplitude B is obtained through the discrete Fourier transform B(x j,y l,t)= m,n ˆB mn (t) exp(i(m K x x j + n K y y l )). (8) In order to have a truly Gaussian initial condition, each Fourier coefficient should be chosen as an independent complex Gaussian variable with a variance proportional to the corresponding value of the spectrum. However, numerical experiments using complex Gaussian Fourier coefficients, unlike when using a uniformly distributed phase only, yielded virtually identical time evolutions of the spectrum. The spectra shown below are obtained as the squared modulus of the Fourier coefficients and, in addition, are ensemble averaged and smoothed. In the twodimensional simulations an ensemble of 5 independent simulations has been used, whereas for the threedimensional simulations only are used due to the long CPUtime involved. The averaged wavenumber spectra from the threedimensional simulations have therefore been smoothed with a moving average Gaussian bell with standard deviation.7 K x. In the simulations presented here we have taken the average steepness ɛ to be.. We have performed simulations with other values of the steepness and the results were quite similar. In particular, the relevant timescale for the spectral evolution was still found to be the BF scale, (ɛ ω ). Our computations conserve the total energy and momentum within the bandwidth constraint with high accuracy. We do not account for coupling with freewave Fourier modes outside the bandwidth constraint. In Trulsen & Stansberg () quantitative comparisons were made between simulations and wave tank experiments of longcrested bichromatic waves. It was found that while the NLS equation gave good quantitative comparisons with the experiments up to x (ɛ k ),thecorresponding range for the MNLS equation was x 5(ɛ k ).Translated to temporal, rather than spatial evolution, these numbers are t (ɛ ω ) and t (ɛ ω ),respectively. This suggests that the NLS equation can be reliably used up to the BF scale, while the MNLS equation can be reliably used an order of magnitude longer. We have taken the present MNLS simulations up to a maximum time t = (ɛ ω ),whichshould be in the domain in which the equations are reliable... Twodimensional simulations In the NLS simulations the Alber criterion σ>ɛforsuppression of the modulational instability is only approximately verified. Starting with the case σ<ɛ, wefindthat the spectrum broadens symmetrically until it reaches a quasi steady width. This
5 Narrowband spectrum of surface gravity waves 5 ( ) (a) 4 T p 4 T p 8 T p. (b) F (k) ε/σ (final) k.5..5 ε/σ (initial). Figure. Spectral evolution of the twodimensional NLS equation starting with an initial Gaussian spectrum. (a) The spectrum at four different times with initial width σ =. and steepness ɛ =.. (b) The relation between initial and final spectral widths:, prediction of Alber; +, numerical result. Here ɛ =.5. ( ) (a) 4 T p 35 T p 5 T p 8 ( ) 6 (b) T p 5 T p F (k) F (k) k k Figure. Spectral evolution of solutions of the MNLS equation. Gaussian initial spectrum with (a) σ =., ɛ =., and (b) σ =.3, ɛ =.. relaxation of the spectrum clearly takes place on the BFtimescale (ɛ ω ).This is illustrated in figure (a), where the initial parameters are ɛ =. andσ =.. Even for the case σ > ɛ there is still a small widening of the spectrum. This is shown in figure (b) illustrating the relation between the final spectral width (after the relaxation to a quasisteady state) and the initial width. The relative spectral width is here defined as { K F dk/ F dk} /. In the MNLS simulations, the initially symmetric Gaussian shape of the spectrum does not persist regardless of the initial spectral width. The spectrum evolves, on the timescale (ɛ ω ),towards a quasisteady state which has an asymmetrical shape with a steepening of the lowfrequency side and a widening of the highfrequency side and, consequently, a downshift of the spectral peak. This is shown in figure for the initial widths σ =. andσ =.3. This was overlooked in the initial version of the paper, and brought to our attention by Dr P Janssen.
6 6 K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. SocquetJuglard Time = 95T p 7T p k y k y k y Figure 3. Evolution of an initially Gaussian spectrum for the threedimensional NLS equation. The contour interval is logarithmic (.5 db) covering 5 db. The cross indicates the directions of the most unstable BF modes, K y = ± / K x. Mori & Yasuda (, ) have found results qualitatively similar to ours. They use a numerical twodimensional model based on the full Euler equations. Starting from a Wallopstype spectrum they show that a widening takes place if the initial width is small... Threedimensional simulations In three dimensions the NLS simulations do not support the theoretical result of Alber (978). A considerable widening of the spectrum is seen in all our simulations regardless of the initial spectral width σ.whenσ < ɛ the spectrum flattens out almost to a plateau shape. When σ ɛ, the spectrum extends predominantly in the directions K y /K x = ± /,coinciding with the directions for the most unstable BF modulations (see also LonguetHiggins 976). This is shown in figure 3 where σ =. andɛ =.. For large times, we see that energy accumulates near the unit bandwidth constraint K =,and eventually renders the simulation invalid. Without the bandwidth constraint, Martin & Yuen (98) found that the NLS equation could suffer energy leakage to arbitrarily large Fourier modes along the diagonal directions. Figure 4 shows results of similar computations carried out using the MNLS equation. The initial spectral widths are σ =. intheupper row, and σ =. inthe lower row. In this case we see an asymmetric development with a downshift of the spectral peak, k p (i.e. k p <k )andanangular widening, mainly for k>k p. For the corresponding angularly integrated spectra shown in figure 5 some steepening is seen on the lowfrequency side, k<k p,while a powerlaw behaviour k.5 (corresponding to a power law ω 4 for the frequency spectrum) is seen on the highfrequency side, k>k p.itisalso interesting to note that the level of the k.5 line is apparently the same for the two cases. 3. Discussion The original intention of this work was to verify the theoretical results of Alber (978), that wave spectra broader than a certain threshold value (4) would not evolve on the BF timescale (ɛ ω ).Weanticipated that these results based on the NLS equation would also be valid for the MNLS equation. The NLS simulations only approximately verified Alber s result in two dimensions, while in three dimensions a widening of the spectra was seen regardless of the initial
7 Narrowband spectrum of surface gravity waves 7. Time =. 95T p. 7T p k y k y Figure 4. Spectral evolution with the MNLS equation. The contour interval.5 db down to 5 db. The steepness ε is equal to., and the spectral width is σ =. intheupper row, and σ =. inthelowerrow. Time = 95T p 7T p log (F) log (F).5 log (k).5.5 log (k).5.5 log (k).5 Figure 5. Evolution of the angularly integrated spectra corresponding to those in figure 4. The lines have slopes.5.
8 8 K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. SocquetJuglard width, as explained above. The MNLS simulations were rather surprising. Regardless of the initial spectral width, the simulations show that narrow Gaussianshaped spectra quickly evolve into a quasistationary state with a downshifted spectral peak. In the threedimensional case a powerlaw behaviour k.5 (or ω 4 )abovethepeak is observed for the angularly integrated spectrum. The relaxation takes place on the BF timescale (ɛ ω ).Thistimescale is much faster than the (ɛ 4 ω ) timescale implied by the Hasselmann spectral evolution equation (Hasselmann 96). For arealistic ocean wave spectrum, the effects of wind input and dissipation by wave breaking are of course important for its evolution. For young waves these effects change the spectrum on a comparable or even faster timescale than the BF scale. Our simulations are restricted to fairly narrow spectra corresponding to a region around the spectral peak (carrying perhaps 8% or more of the total wave energy). For these waves the ratio of the efolding time, T e,duetowindinput, to the peak period T p depends strongly on the wave age. From a semiempirical formula for the wind growth due to Hsiao & Shemdin (983) we find that T e T p 3 (.85(U /C p )cosθ ) (9) where θ is the angle between the phase velocity and the wind. Using (9) we find that T e /T p > when the wave age C p /U >.4. For long storm waves on the open ocean T e may thus be considerably longer than the relaxation time of the order of T p that we find. Thus, the fact that the wind input is absent in our simulations seems consistent provided that the wave age is not too small. The spectral evolution also shows realistic features like the steepening of the lowfrequency part (ω <ω p )andthepowerlaw behaviour for the highfrequency part (ω >ω p ), with acorresponding frequency downshift. For the longterm consistent downshift of a realistic wave spectrum, however, we believe that the wind input and dissipation are essential. After the submission of our paper, our results were presented at the WISE (Waves In Shallowwater Environments) meeting in Bergen, Norway, May 3,. Interestingly, similar twodimensional results were also presented by Peter Janssen (see Janssen ) using the NLS and the Zakharov integral equations, and by Miguel Onorato et al. (see Onorato et al. ) in three dimensions using the full Euler equations (they demonstrated the powerlaw behaviour k 5/ to occur for a much wider spectral range than shown in the present paper). As some of the same conclusions have been reached through three independent efforts using three different model equations, we believe that these results are important and not the consequence of a particular choice of wave model. It is not obvious how these results can be reconciled with the Hasselmann picture, however. Janssen (983) showed that the approximation to the Hasselmann equation given by Dungey & Hui (979) for narrowband spectra can be derived from the MNLS equations using the same statistical assumptions as Hasselmann. It is perhaps worth noting that the zerocumulant hypotheses implied by the approximate multivariate complex Gaussian property of the Fourier coefficients, and necessary for the derivation of Hasselmann s equation, may not be realistic in the present case. Even if the initial Fourier coefficients are chosen to be independent complex Gaussian variables, the nonlinear deterministic simulation will couple the Fourier coefficients and destroy their Gaussian property. Also, the homogeneity in time and space assumed
9 Narrowband spectrum of surface gravity waves 9 in the Hasselmann treatment may not be realistic. The twodimensional theoretical and simulation results by Janssen referred to above seem to indicate that the deviation from the Gaussian property is more important than nonhomogeneity. We find, by making longer simulations than those shown above, that even after the spectrum has relaxed to the new shape there is still a very slow change. It is not possible for us to decide whether this indicates an evolution as predicted by the Hasselmann equation, as the time horizon for our numerical model is too limited. This research has been supported by the BeMatA program of the Research Council of Norway (3977/43), by Norsk Hydro, by the European Union through the EnviWave project (EVG CT ) and by Statoil. REFERENCES Alber, I. E. 978 The effects of randomness on the stability of twodimensional surface wavetrains. Proc. R. Soc. Lond. A 363, Alber, I. E. & Saffman, P. G. 978 Stability of Random nonlinear deep water waves with finite bandwidth spectrum.twr Defense and Spacesystems Group Rep RU. Benjamin, T. B. &Feir,J.E.967 The disintegration of wave trains on deep water. Part. Theory. J. Fluid Mech. 7, 473. Crawford, D. R., Saffman, P. G. & Yuen, H. C. 98 Evolution of a random inhomogeneous field of nonlinear deepwater gravity waves. Wave Motion,. Dungey, J. C. & Hui, W. H. 979 Nonlinear energy transfer in a narrow gravitywave spectrum. Proc. R. Soc. Lond. A 368, 395. Dysthe, K. B. 979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 5. Hasselmann, K. 96 On the nonlinear energy transfer in a gravitywave spectrum. Part. General theory. J. Fluid Mech., 48. Hsiao, G. C. & Shemdin, O. H. 983 Measurements of wind velocity and pressure with a wave follower during MARSEN. J. Geophys. Res. 88, Janssen, P. A. E. M. 983 On a fourthorder envelope equation for deepwater waves. J. Fluid Mech. 6,. Janssen, P. A. E. M. Nonlinear four wave interaction and freak waves. ECMWF Tech. Mem. Lighthill, M. J. 965 Contribution to the theory of waves in nonlinear dispersive systems. J. Inst. Maths Applics., 696. Lighthill, M. J. 967 Some special cases treated by the Whitham theory. Proc. R. Soc. Lond. A 99, 83. Lo, E. & Mei, C. C. 985 A numerical study of waterwave modulation based on a higherorder nonlinear Schrödinger equation. J. Fluid Mech. 5, Lo, E. Y.& Mei, C. C. 987 Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study. Wave Motion 9, 459. LonguetHiggins, M. S. 976 On the nonlinear transfer of energy in the peak of a gravity wave spectrum: a simplified model. Proc. R. Soc. Lond. A 347, 38. Martin, D. U.& Yuen, H. C. 98 Quasirecurring energy leakage in the twospacedimensional nonlinear Schrödinger equation. Phys. Fluids 3, Mori, D. & Yasuda, T. Effects of highorder nonlinear interactions on gravity waves. Proc. Rogue Waves (ed. M. Olagnon & G. Athanassoulis). Mori, D. & Yasuda, T. Effects of highorder nonlinear interactions on unidirectional wave trains. Ocean Engng 9, 335. Onorato, M., Osborne, A. R., Serio, M., Resio, D., Pushkarev, A., Zakharov, V. & Brandini, C. Freely decaying weak turbulence for sea surface waves. Phys. Rev. Lett. 89(4), art. no Trulsen, K. & Dysthe, K. B. 996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 4, 8 89.
10 K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. SocquetJuglard Trulsen, K., Kliakhandler, I., Dysthe, K. B. & Velarde, M. G. On weakly nonlinear modulation of waves on deep water. Phys. Fluids, Trulsen, K. & Stansberg, C. T. Spatial evolution of water surface waves: Numerical simulation and experiment of bichromatic waves. In Proc. ISOPE 3, Whitham, G. B. 967 Nonlinear dispersion of water waves. Proc. R. Soc. Lond. A 99, 6. Yuen, H. C. & Ferguson, W. E. 978 Relationship between Benjamin Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids, Zakharov, V. E. 968 Stability of periodic waves of finite amplitude on the surface of deep water. Zh. Prikl. Mekh. Tekh. Fiz. 9, (English transl. J. Appl. Mech. Tech. Phys. 9, 9 94.)
On deviations from Gaussian statistics for surface gravity waves
On deviations from Gaussian statistics for surface gravity waves M. Onorato, A. R. Osborne, and M. Serio Dip. Fisica Generale, Università di Torino, Torino  10125  Italy Abstract. Here we discuss some
More informationExtreme waves, modulational instability and second order theory: wave flume experiments on irregular waves
European Journal of Mechanics B/Fluids 25 (2006) 586 601 Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves M. Onorato a,, A.R. Osborne a,m.serio
More informationModeling and predicting rogue waves in deep water
Modeling and predicting rogue waves in deep water C M Schober University of Central Florida, Orlando, Florida  USA Abstract We investigate rogue waves in the framework of the nonlinear Schrödinger (NLS)
More informationExperimental and numerical study of spatial and temporal evolution of nonlinear wave groups
Nonlin. Processes Geophys., 15, 91 942, 28 www.nonlinprocessesgeophys.net/15/91/28/ Author(s) 28. This work is distributed under the Creative Commons Attribution. License. Nonlinear Processes in Geophysics
More informationWave statistics in unimodal and bimodal seas from a secondorder model
European Journal of Mechanics B/Fluids 25 (2006) 649 661 Wave statistics in unimodal and bimodal seas from a secondorder model Alessandro Toffoli a,, Miguel Onorato b, Jaak Monbaliu a a Hydraulics Laboratory,
More informationarxiv: v1 [nlin.cd] 21 Mar 2012
Approximate rogue wave solutions of the forced and damped Nonlinear Schrödinger equation for water waves arxiv:1203.4735v1 [nlin.cd] 21 Mar 2012 Miguel Onorato and Davide Proment Dipartimento di Fisica,
More informationSimulations and experiments of short intense envelope
Simulations and experiments of short intense envelope solitons of surface water waves A. Slunyaev 1,), G.F. Clauss 3), M. Klein 3), M. Onorato 4) 1) Institute of Applied Physics, Nizhny Novgorod, Russia,
More informationPhysical mechanisms of the Rogue Wave phenomenon
Physical mechanisms of the Rogue Wave phenomenon Manuel A. Andrade. Mentor: Dr. Ildar Gabitov. Math 585. 1 "We were in a storm and the tanker was running before the sea. This amazing wave came from the
More informationFreakish sea state and swellwindsea coupling: Numerical study of the SuwaMaru incident
Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L01607, doi:10.1029/2008gl036280, 2009 Freakish sea state and swellwindsea coupling: Numerical study of the SuwaMaru incident H. Tamura,
More informationInfluence of varying bathymetry in rogue wave occurrence within unidirectional and directional seastates
J. Ocean Eng. Mar. Energy (217) 3:39 324 DOI 1.17/s472217866 RESEARCH ARTICLE Influence of varying bathymetry in rogue wave occurrence within unidirectional and directional seastates Guillaume Ducrozet
More informationWaves on deep water, I Lecture 13
Waves on deep water, I Lecture 13 Main question: Are there stable wave patterns that propagate with permanent form (or nearly so) on deep water? Main approximate model: i" # A + $" % 2 A + &" ' 2 A + (
More informationModulational Instabilities and Breaking Strength for DeepWater Wave Groups
OCTOBER 2010 G A L C H E N K O E T A L. 2313 Modulational Instabilities and Breaking Strength for DeepWater Wave Groups ALINA GALCHENKO, ALEXANDER V. BABANIN, DMITRY CHALIKOV, AND I. R. YOUNG Swinburne
More informationLinear and Nonlinear Rogue Wave Statistics in the Presence of Random Currents
Linear and Nonlinear Rogue Wave Statistics in the Presence of Random Currents Lev Kaplan and Linghang Ying (Tulane University) In collaboration with Alex Dahlen and Eric Heller (Harvard) Zhouheng Zhuang
More informationStrongly nonlinear long gravity waves in uniform shear flows
Strongly nonlinear long gravity waves in uniform shear flows Wooyoung Choi Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Received 14 January
More informationRogue Waves. Alex Andrade Mentor: Dr. Ildar Gabitov. Physical Mechanisms of the Rogue Wave Phenomenon Christian Kharif, Efim Pelinovsky
Rogue Waves Alex Andrade Mentor: Dr. Ildar Gabitov Physical Mechanisms of the Rogue Wave Phenomenon Christian Kharif, Efim Pelinovsky Rogue Waves in History Rogue Waves have been part of marine folklore
More informationUniversity of Bristol  Explore Bristol Research. Link to publication record in Explore Bristol Research PDFdocument.
Dobra, T., Lawrie, A., & Dalziel, S. B. (2016). Nonlinear Interactions of Two Incident Internal Waves. 18. Paper presented at VIIIth International Symposium on Stratified Flows, San Diego, United States.
More informationProgress in ocean wave forecasting
529 Progress in ocean wave forecasting Peter A.E.M. Janssen Research Department Accepted for publication in J. Comp. Phys. 19 June 2007 Series: ECMWF Technical Memoranda A full list of ECMWF Publications
More informationWhat Is a Soliton? by Peter S. Lomdahl. Solitons in Biology
What Is a Soliton? by Peter S. Lomdahl A bout thirty years ago a remarkable discovery was made here in Los Alamos. Enrico Fermi, John Pasta, and Stan Ulam were calculating the flow of energy in a onedimensional
More informationB 2 P 2, which implies that g B should be
Enhanced Summary of G.P. Agrawal Nonlinear Fiber Optics (3rd ed) Chapter 9 on SBS Stimulated Brillouin scattering is a nonlinear threewave interaction between a forwardgoing laser pump beam P, a forwardgoing
More informationGeneration of undular bores and solitary wave trains in fully nonlinear shallow water theory
Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Gennady El 1, Roger Grimshaw 1 and Noel Smyth 2 1 Loughborough University, UK, 2 University of Edinburgh, UK
More informationLongtime solutions of the Ostrovsky equation
Longtime solutions of the Ostrovsky equation Roger Grimshaw Centre for Nonlinear Mathematics and Applications, Department of Mathematical Sciences, Loughborough University, U.K. Karl Helfrich Woods Hole
More informationLOCAL ANALYSIS OF WAVE FIELDS PRODUCED FROM HINDCASTED ROGUE WAVE SEA STATES
Proceedings of the ASME 215 34th International Conference on Ocean, Offshore and Arctic Engineering OMAE215 May 31June 5, 215, St. John's, Newfoundland, Canada OMAE21541458 LOCAL ANALYSIS OF WAVE FIELDS
More informationNote the diverse scales of eddy motion and selfsimilar appearance at different lengthscales of the turbulence in this water jet. Only eddies of size
L Note the diverse scales of eddy motion and selfsimilar appearance at different lengthscales of the turbulence in this water jet. Only eddies of size 0.01L or smaller are subject to substantial viscous
More informationOTG13. Prediction of air gap for column stabilised units. Won Ho Lee 01 February Ungraded. 01 February 2017 SAFER, SMARTER, GREENER
OTG13 Prediction of air gap for column stabilised units Won Ho Lee 1 SAFER, SMARTER, GREENER Contents Air gap design requirements Purpose of OTG13 OTG13 vs. OTG14 Contributions to air gap Linear analysis
More informationITTC Recommended Procedures and Guidelines. Testing and Extrapolation Methods Loads and Responses, Ocean Engineering,
Multidirectional Irregular Wave Spectra Page 1 of 8 25 Table of Contents Multidirectional Irregular Wave Spectra... 2 1. PURPOSE OF GUIDELINE... 2 2. SCOPE... 2 2.1 Use of directional spectra... 2 2.2
More informationLinearly forced isotropic turbulence
Center for Turbulence Research Annual Research Briefs 23 461 Linearly forced isotropic turbulence By T. S. Lundgren Stationary isotropic turbulence is often studied numerically by adding a forcing term
More informationOscillatory spatially periodic weakly nonlinear gravity waves on deep water
J. Fluid Mech. (1988), vol. 191, pp. 341372 Printed in Great Britain 34 1 Oscillatory spatially periodic weakly nonlinear gravity waves on deep water By JUAN A. ZUFIRIA Applied Mathematics, California
More informationTHE problem of phase noise and its influence on oscillators
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 54, NO. 5, MAY 2007 435 Phase Diffusion Coefficient for Oscillators Perturbed by Colored Noise Fergal O Doherty and James P. Gleeson Abstract
More informationTIME DOMAIN COMPARISONS OF MEASURED AND SPECTRALLY SIMULATED BREAKING WAVES
TIME DOMAIN COMPARISONS OF MEASRED AND SPECTRAY SIMATED BREAKING WAVES Mustafa Kemal Özalp 1 and Serdar Beji 1 For realistic wave simulations in the nearshore zone besides nonlinear interactions the dissipative
More informationMax Planck Institut für Plasmaphysik
ASDEX Upgrade Max Planck Institut für Plasmaphysik 2D Fluid Turbulence Florian Merz Seminar on Turbulence, 08.09.05 2D turbulence? strictly speaking, there are no twodimensional flows in nature approximately
More informationDeterministic aspects of Nonlinear Modulation Instability
Deterministic aspects of Nonlinear Modulation Instability E van Groesen, Andonowati & N. Karjanto Applied Analysis & Mathematical Physics, University of Twente, POBox 217, 75 AE, Netherlands & Jurusan
More informationThe infrared properties of the energy spectrum in freely decaying isotropic turbulence
The infrared properties of the energy spectrum in freely decaying isotropic turbulence W.D. McComb and M.F. Linkmann SUPA, School of Physics and Astronomy, University of Edinburgh, UK arxiv:148.1287v1
More informationWave Forecasting using computer models Results from a onedimensional model
Proceedings of the World Congress on Engineering and Computer Science 007 WCECS 007, October 6, 007, San Francisco, USA Wave Forecasting using computer models Results from a onedimensional model S.K.
More informationOn Vertical Variations of WaveInduced Radiation Stress Tensor
Archives of HydroEngineering and Environmental Mechanics Vol. 55 (2008), No. 3 4, pp. 83 93 IBW PAN, ISSN 1231 3726 On Vertical Variations of WaveInduced Radiation Stress Tensor Włodzimierz Chybicki
More informationFrom time series to superstatistics
From time series to superstatistics Christian Beck School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS, United Kingdom Ezechiel G. D. Cohen The Rockefeller University,
More informationOn the Whitham Equation
On the Whitham Equation Henrik Kalisch Department of Mathematics University of Bergen, Norway Joint work with: Handan Borluk, Denys Dutykh, Mats Ehrnström, Daulet Moldabayev, David Nicholls Research partially
More informationPredicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices
ILASS Americas 2th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 27 Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices M. Rohani,
More informationThe Details of Detailed Balance. Don Resio University of North Florida Chuck Long Retired Will Perrie Bedford Institute of Oceanography
The Details of Detailed Balance Don Resio University of North Florida Chuck Long Retired Will Perrie Bedford Institute of Oceanography MOTIVATION Spectral shape provides critical information for understanding
More informationTHE EXCEEDENCE PROBABILITY OF WAVE CRESTS CALCULATED BY THE SPECTRAL RESPONSE SURFACE METHOD. R.Gibson, P.Tromans and C.Swan
THE EXCEEDENCE PROBABILITY OF WAVE CRESTS CALCULATED BY THE SPECTRAL RESPONSE SURFACE METHOD R.Gibson, P.Tromans and C.Swan Department of Civil and Environmental Engineering, Imperial College London. SW7
More informationModelling Interactions Between Weather and Climate
Modelling Interactions Between Weather and Climate p. 1/33 Modelling Interactions Between Weather and Climate Adam Monahan & Joel Culina monahana@uvic.ca & culinaj@uvic.ca School of Earth and Ocean Sciences,
More informationYanlin Shao 1 Odd M. Faltinsen 2
Yanlin Shao 1 Odd M. Faltinsen 1 Ship Hydrodynamics & Stability, Det Norsk Veritas, Norway Centre for Ships and Ocean Structures (CeSOS), NTNU, Norway 1 The stateoftheart potential flow analysis: Boundary
More informationThe Canadian approach to ensemble prediction
The Canadian approach to ensemble prediction ECMWF 2017 Annual seminar: Ensemble prediction : past, present and future. Pieter Houtekamer Montreal, Canada Overview. The Canadian approach. What are the
More informationLEVEL REPULSION IN INTEGRABLE SYSTEMS
LEVEL REPULSION IN INTEGRABLE SYSTEMS Tao Ma and R. A. Serota Department of Physics University of Cincinnati Cincinnati, OH 452440011 serota@ucmail.uc.edu Abstract Contrary to conventional wisdom, level
More informationAnalysis of the 500 mb height fields and waves: testing Rossby wave theory
Analysis of the 500 mb height fields and waves: testing Rossby wave theory Jeffrey D. Duda, Suzanne Morris, Michelle Werness, and Benjamin H. McNeill Department of Geologic and Atmospheric Sciences, Iowa
More informationA new adaptive windowing algorithm for Gabor depth imaging
A new adaptive windowing algorithm A new adaptive windowing algorithm for Gabor depth imaging Yongwang Ma and Gary F. Margrave ABSTRACT Adaptive windowing algorithms are critical for improving imaging
More informationThe behaviour of high Reynolds flows in a driven cavity
The behaviour of high Reynolds flows in a driven cavity CharlesHenri BRUNEAU and Mazen SAAD Mathématiques Appliquées de Bordeaux, Université Bordeaux 1 CNRS UMR 5466, INRIA team MC 351 cours de la Libération,
More informationEnergy dissipation and transfer processes during the breaking of modulated wave trains
Journal of Physics: Conference Series PAPER OPEN ACCESS Energy dissipation and transfer processes during the breaking of modulated wave trains To cite this article: F De Vita et al 2015 J. Phys.: Conf.
More informationA Truncated Model for Finite Amplitude Baroclinic Waves in a Channel
A Truncated Model for Finite Amplitude Baroclinic Waves in a Channel Zhiming Kuang 1 Introduction To date, studies of finite amplitude baroclinic waves have been mostly numerical. The numerical models,
More informationA twolayer approach to wave modelling
1.198/rspa.4.135 A twolayer approach to wave modelling By Patrick Lynett and Philip L.F. Liu School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA (plynett@civil.tamu.edu)
More informationOcean Wave Speed in the Antarctic MIZ
1 Ocean Wave Speed in the Antarctic MIZ Colin Fox Mathematics Department, University of Auckland, Private Bag 9219, Auckland, New Zealand Tim Haskell Industrial Research Limited, Lower Hutt, NewZealand
More informationEliassenPalm Cross Sections Edmon et al. (1980)
EliassenPalm Cross Sections Edmon et al. (1980) Cecily Keppel November 14 2014 EliassenPalm Flux For βplane Coordinates (y, p) in northward, vertical directions Zonal means F = v u f (y) v θ θ p F will
More informationUltranarrowband tunable laserline notch filter
Appl Phys B (2009) 95: 597 601 DOI 10.1007/s0034000934476 Ultranarrowband tunable laserline notch filter C. Moser F. Havermeyer Received: 5 December 2008 / Revised version: 2 February 2009 / Published
More informationSpectral evolution and probability distributions of surface ocean gravity waves and extreme waves. Hervé SocquetJuglard
Spectral evolution and probability distributions of surface ocean gravity waves and extreme waves. Doctor Scientarum Thesis Hervé SocquetJuglard Department of Mathematics University of Bergen November
More informationEffects of Interactive Function Forms in a SelfOrganized Critical Model Based on Neural Networks
Commun. Theor. Phys. (Beijing, China) 40 (2003) pp. 607 613 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Effects of Interactive Function Forms in a SelfOrganized Critical Model
More informationBoussinesq dynamics of an idealized tropopause
Abstract Boussinesq dynamics of an idealized tropopause Olivier Asselin, Peter Bartello and David Straub McGill University olivier.asselin@mail.mcgill.ca The neartropopause flow has been interpreted using
More informationPHYS 432 Physics of Fluids: Instabilities
PHYS 432 Physics of Fluids: Instabilities 1. Internal gravity waves Background state being perturbed: A stratified fluid in hydrostatic balance. It can be constant density like the ocean or compressible
More informationModeling For Both Deep Ocean And Coastal WaveWave. Implications For WAM. Hydromechanics Directorate (Code 5500) Carderock Division
Modeling For Both Deep Ocean And Coastal WaveWave Interactions: Implications For WAM Ray Q.Lin Hydromechanics Directorate (Code 5500) Carderock Division Naval Surface Warfare Center (David Taylor Model
More informationTHE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT.
ISTP6, 5, PRAGUE 6 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. MeiJiau Huang Mechanical Engineering Department, National Taiwan University
More informationInternational Workshop on the Frontiers of Modern Plasma Physics July 2008
195341 International Workshop on the Frontiers of Modern Plasma Physics 145 July 008 Patterns of sound radiation behind pointlike charged obstacles in plasma flows. H. Pecseli University of Oslo Institute
More informationOptical solitons and its applications
Physics 568 (Nonlinear optics) 04/30/007 Final report Optical solitons and its applications 04/30/007 1 1 Introduction to optical soliton. (temporal soliton) The optical pulses which propagate in the lossless
More informationddhead Where we stand
Contents ddhead Where we stand objectives Examples in movies of cg water Navier Stokes, potential flow, and approximations FFT solution Oceanography Random surface generation High resoluton example Video
More informationAnalyticallyoriented approaches to nonlinear sloshing in moving smooth tanks
Analyticallyoriented approaches to nonlinear sloshing in moving smooth tanks by Alexander Timokha (JenaKievLeipzigTrondheim) & Ivan Gavrilyuk (EisenachLeipzig) Overview Motivation: coupling with rigid
More informationNOTE. Application of Contour Dynamics to Systems with Cylindrical Boundaries
JOURNAL OF COMPUTATIONAL PHYSICS 145, 462 468 (1998) ARTICLE NO. CP986024 NOTE Application of Contour Dynamics to Systems with Cylindrical Boundaries 1. INTRODUCTION Contour dynamics (CD) is a widely used
More informationHarnessing and control of optical rogue waves in. supercontinuum generation
Harnessing and control of optical rogue waves in supercontinuum generation John. M. Dudley, 1* Goëry Genty 2 and Benjamin J. Eggleton 3 1 Département d Optique P. M. Duffieux, Institut FEMTOST, UMR 6174
More informationPoint Vortex Dynamics in Two Dimensions
Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei
More informationLet us consider a typical Michelson interferometer, where a broadband source is used for illumination (Fig. 1a).
7.1. LowCoherence Interferometry (LCI) Let us consider a typical Michelson interferometer, where a broadband source is used for illumination (Fig. 1a). The light is split by the beam splitter (BS) and
More informationRadiation Effect on MHD Casson Fluid Flow over a PowerLaw Stretching Sheet with Chemical Reaction
Radiation Effect on MHD Casson Fluid Flow over a PowerLaw Stretching Sheet with Chemical Reaction Motahar Reza, Rajni Chahal, Neha Sharma Abstract This article addresses the boundary layer flow and heat
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationEXCITATION OF GÖRTLERINSTABILITY MODES IN CONCAVEWALL BOUNDARY LAYER BY LONGITUDINAL FREESTREAM VORTICES
ICMAR 2014 EXCITATION OF GÖRTLERINSTABILITY MODES IN CONCAVEWALL BOUNDARY LAYER BY LONGITUDINAL FREESTREAM VORTICES Introduction A.V. Ivanov, Y.S. Kachanov, D.A. Mischenko Khristianovich Institute of
More informationProgress in ocean wave forecasting
Available online at www.sciencedirect.com Journal of Computational Physics 227 (2008) 3572 3594 www.elsevier.com/locate/jcp Progress in ocean wave forecasting Peter A.E.M. Janssen * European Centre for
More informationOn the interaction of wind and steep gravity wave groups using Miles and Jeffreys mechanisms
Nonlin. Processes Geophys.,,, 8 www.nonlinprocessesgeophys.net///8/ Author(s) 8. This work is distributed under the Creative Commons Attribution. License. Nonlinear Processes in Geophysics On the interaction
More informationIN RECENT years, the observation and analysis of microwave
2334 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 4, JULY 1998 Calculation of the Formation Time for Microwave Magnetic Envelope Solitons Reinhold A. Staudinger, Pavel Kabos, Senior Member, IEEE, Hua Xia,
More informationSpatioTemporal Chaos in PatternForming Systems: Defects and Bursts
SpatioTemporal Chaos in PatternForming Systems: Defects and Bursts with Santiago Madruga, MPIPKS Dresden Werner Pesch, U. Bayreuth YuanNan Young, New Jersey Inst. Techn. DPG Frühjahrstagung 31.3.2006
More informationGRATINGS and SPECTRAL RESOLUTION
Lecture Notes A La Rosa APPLIED OPTICS GRATINGS and SPECTRAL RESOLUTION 1 Calculation of the maxima of interference by the method of phasors 11 Case: Phasor addition of two waves 12 Case: Phasor addition
More informationPhysics 115/242 Comparison of methods for integrating the simple harmonic oscillator.
Physics 115/4 Comparison of methods for integrating the simple harmonic oscillator. Peter Young I. THE SIMPLE HARMONIC OSCILLATOR The energy (sometimes called the Hamiltonian ) of the simple harmonic oscillator
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG ChuanJian ( ), 1 DAI ZhengDe (à ), 2, and MU Gui (½ ) 3 1 Department
More informationThe Restricted 3Body Problem
The Restricted 3Body Problem John Bremseth and John Grasel 12/10/2010 Abstract Though the 3body problem is difficult to solve, it can be modeled if one mass is so small that its effect on the other two
More informationRaman Spectra of Amorphous Silicon
Chapter 6 Raman Spectra of Amorphous Silicon In 1985, Beeman, Tsu and Thorpe established an almost linear relation between the Raman transverseoptic (TO) peak width Ɣ and the spread in mean bond angle
More informationSHORT WAVE INSTABILITIES OF COUNTERROTATING BATCHELOR VORTEX PAIRS
Fifth International Conference on CFD in the Process Industries CSIRO, Melbourne, Australia 1315 December 6 SHORT WAVE INSTABILITIES OF COUNTERROTATING BATCHELOR VORTEX PAIRS Kris RYAN, Gregory J. SHEARD
More informationSURFACE WAVE DISPERSION PRACTICAL (Keith Priestley)
SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley) This practical deals with surface waves, which are usually the largest amplitude arrivals on the seismogram. The velocity at which surface waves propagate
More informationVector dark domain wall solitons in a fiber ring laser
Vector dark domain wall solitons in a fiber ring laser H. Zhang, D. Y. Tang*, L. M. Zhao and R. J. Knize 1 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
More informationIntermittent distribution of heavy particles in a turbulent flow. Abstract
Intermittent distribution of heavy particles in a turbulent flow Gregory Falkovich 1 and Alain Pumir 2 1 Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100 Israel 2 I.N.L.N., 1361
More informationSimulated and Observed Scaling in Earthquakes Kasey Schultz Physics 219B Final Project December 6, 2013
Simulated and Observed Scaling in Earthquakes Kasey Schultz Physics 219B Final Project December 6, 2013 Abstract Earthquakes do not fit into the class of models we discussed in Physics 219B. Earthquakes
More informationLecture 4 Fiber Optical Communication Lecture 4, Slide 1
ecture 4 Dispersion in singlemode fibers Material dispersion Waveguide dispersion imitations from dispersion Propagation equations Gaussian pulse broadening Bitrate limitations Fiber losses Fiber Optical
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More informationDavydov Soliton Collisions
Davydov Soliton Collisions Benkui Tan Department of Geophysics Peking University Beijing 100871 People s Republic of China Telephone: 861062755041 email:dqgchw@ibmstone.pku.edu.cn John P. Boyd Dept.
More informationPhase ramping and modulation of reflectometer signals
4th Intl. Reflectometry Workshop  IRW4, Cadarache, March 22nd  24th 1999 1 Phase ramping and modulation of reflectometer signals G.D.Conway, D.V.Bartlett, P.E.Stott JET Joint Undertaking, Abingdon, Oxon,
More informationConductance and resistance jumps in finitesize random resistor networks
J. Phys. A: Math. Gen. 21 (1988) L23L29. Printed in the UK LETTER TO THE EDITOR Conductance and resistance jumps in finitesize random resistor networks G G Batrounit, B Kahngt and S Rednert t Department
More informationarxiv:physics/ v1 [physics.fludyn] 28 Feb 2003
Experimental Lagrangian Acceleration Probability Density Function Measurement arxiv:physics/0303003v1 [physics.fludyn] 28 Feb 2003 N. Mordant, A. M. Crawford and E. Bodenschatz Laboratory of Atomic and
More informationDoublediffusive lockexchange gravity currents
Abstract Doublediffusive lockexchange gravity currents Nathan Konopliv, Presenting Author and Eckart Meiburg Department of Mechanical Engineering, University of California Santa Barbara meiburg@engineering.ucsb.edu
More informationNote on Group Velocity and Energy Propagation
Note on Group Velocity and Energy Propagation Abraham Bers Department of Electrical Engineering & Computer Science and Plasma Science & Fusion Center Massachusetts Institute of Technology, Cambridge, MA
More informationSelfExcited Vibration in Hydraulic Ball Check Valve
SelfExcited Vibration in Hydraulic Ball Check Valve L. Grinis, V. Haslavsky, U. Tzadka Abstract This paper describes an experimental, theoretical model and numerical study of concentrated vortex flow
More informationFIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 07
FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 07 Analysis of WaveModel of Light Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of
More informationInterfacial waves in steady and oscillatory, twolayer Couette flows
Interfacial waves in steady and oscillatory, twolayer Couette flows M. J. McCready Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 Page 1 Acknowledgments Students: M.
More informationsine wave fit algorithm
TECHNICAL REPORT IRS3SB9 1 Properties of the IEEESTD57 four parameter sine wave fit algorithm Peter Händel, Senior Member, IEEE Abstract The IEEE Standard 57 (IEEESTD57) provides algorithms for
More informationPlumes and jets with timedependent sources in stratified and unstratified environments
Plumes and jets with timedependent sources in stratified and unstratified environments Abstract Matthew Scase 1, Colm Caulfield 2,1, Stuart Dalziel 1 & Julian Hunt 3 1 DAMTP, Centre for Mathematical Sciences,
More informationOn fully developed mixed convection with viscous dissipation in a vertical channel and its stability
ZAMM Z. Angew. Math. Mech. 96, No. 12, 1457 1466 (2016) / DOI 10.1002/zamm.201500266 On fully developed mixed convection with viscous dissipation in a vertical channel and its stability A. Barletta 1,
More informationImproving the stability of longitudinal and transverse laser modes
Optics Communications 239 (24) 147 151 www.elsevier.com/locate/optcom Improving the stability of longitudinal and transverse laser modes G. Machavariani *, N. Davidson, A.A. Ishaaya, A.A. Friesem Department
More informationFourwave mixing in PCF s and tapered fibers
Chapter 4 Fourwave mixing in PCF s and tapered fibers The dramatically increased nonlinear response in PCF s in combination with singlemode behavior over almost the entire transmission range and the
More informationThe Fifth Order Peregrine Breather and Its EightParameter Deformations Solutions of the NLS Equation
Commun. Theor. Phys. 61 (2014) 365 369 Vol. 61, No. 3, March 1, 2014 The Fifth Order Peregrine Breather and Its EightParameter Deformations Solutions of the NLS Equation Pierre Gaillard Unversité de Bourgogne,
More information