A global wave parameter database for geophysical applications. Part 2: Model validation with improved source term parameterization

TL;DR: Ardhuin et al. as discussed by the authors presented a multi-scale global hindcast of ocean waves that covers the years 1994-2012, based on recently published parameterizations for wind sea and swell dissipation.

About: This article is published in Ocean Modelling.The article was published on 2013-10-01 and is currently open access. It has received 332 citations till now. The article focuses on the topics: Wind wave & Significant wave height.

Summary

Introduction

• Even the dynamics of all-to-all coupled networks of identical oscillators can be extremely rich, including synchronization,31,38 clustering and slow switching,25 and chaotic dynamics.
• For N¼ 4 phase oscillators—the smallest number where chaos can oc- cur—no sign of chaos was found for coupling functions with two harmonics5 and the only known example of a cou- pling function giving rise to chaotic attractors has four nontrivial harmonics.

II. NONPAIRWISE INTERACTION IN PHASE REDUCTION NEAR HOPF BIFURCATIONS

• So, gkðz1; z2;…; zNÞ is symmetric under all permutations of the last N 1 arguments that fix the first.
• (8) for some constant coefficients nji and v j i.
• Consider system (II) with SN-symmetry (for fixed N) such that the N uncoupled systems ( ¼ 0) undergo a generic supercritical Hopf bifurcation on k passing through 0.

A. Symmetric phase oscillator dynamics

• TN 1 that maps the T-orbits onto points, the generalized system (7) (the authors assume ¼ 1 from hereon) reduces to phase differences on TN 1.
• For any partition of N ¼ m1 þ þ m‘; ‘ 2, there are ‘-cluster states with isotropy Sm1 Sm‘ with ‘ clusters of mk oscillators at the same phase.
• This structure is instructive in that it places limits on where any chaotic behavior can be found.

B. Fully synchronized and splay phase: Existence and stability

• The authors recall from Refs. 5 and 10 two important periodic sol- utions that are guaranteed to exist for (7).
• The authors do not compute these eigenvalues in their full generality but give them for the special cases N¼ 2, 3, 4 in Section IV.

IV. DYNAMICS OF SMALL NETWORKS WITH NONPAIRWISE COUPLING

• Since the nonpairwise coupling involves combinations of three and four phases, the dynamics for N¼ 2 and N¼ 3 oscillators reduce to the coupling of simpler form.
• In particular, for N¼ 2 oscillators, (7) reduces to pairwise coupling (1) with coupling function gðhÞ ¼ n1 cosðhþ v1Þ þ n2 cosð2hþ v2Þ.
• Similarly, for N¼ 3, the authors can express the contributions of Fð4Þj to the dynamics in terms of just pairwise and three-phase interactions.
• The cases N¼ 2 and 3 are special case as the authors now discuss.

V. CHAOS IN SMALL NETWORKS WITH NONPAIRWISE COUPLING

• Since the reduced system of N oscillators evolves on TN 1, only networks of N 4 oscillators can exhibit chaotic dynamics.
• The authors calculate the expansion of a perturbation along a trajectory by integrating the variational equations _vj ¼ XN k¼1 JjkðhðtÞÞvk numerically along a solution h(t) of (7) with Jacobian (11): see for example11,13 for more details.

A. Chaos in networks of N 5 4 oscillators

• For appropriately chosen parameters, networks of gener- ically coupled phase oscillators (7) with coupling functions (14) give rise to positive maximal Lyapunov exponents.
• More specifically, numerical con- tinuation of the branch of periodic solutions which arises in the Hopf bifurcation of Hsplay in AUTO for fixed v1¼ 0.1 and decreasing v2 from 0.3 towards the parameter values of Figure 2 shows subsequent period doubling bifurcations (not shown).
• The bifurcation of a equilibrium on the boundary of C induces bistability with the attractors in the interior of C. Trajectories with positive maximal Lyapunov exponents kmax also appear close to the boundary of C. Figure 3(a) shows a solution h in C for v¼ (0.154, 0.318, 0, 1.74, 0).

B. Chaos in networks of N 5 5 oscillators

• Positive maximal Lyapunov exponents also for networks of N¼ 5 oscillators.
• Note that for the same parameter range as in Figure 4, positive maximal Lyapunov exponents also arise for the dynamics of N¼ 4 oscillators (not shown).

VI. DISCUSSION

• The authors show that symmetrical phase oscillator networks with coupling that involves nonpairwise interaction can ex- hibit chaotic dynamics with coupling functions that only contain two nontrivial harmonics.
• By contrast, for networks of four oscillators with pairwise interaction, the only known example of a coupling function that gives rise to chaotic dynamics has four nontrivial harmonics.
• Moreover, positive Lyapunov exponents arise in the same region of parameter space.
• Nonpairwise interaction between phase oscillators also facilitates the emergence of chaotic weak chimeras, i.e., dy- namically invariant sets on which oscillators are locally fre- quency synchronized.
• Thus, the authors anticipate that a detailed understanding of the dynamical effects in- duced by general nonpairwise coupling will give additional insights into the analysis of real-world networks of oscillato- ry units with generic coupling.

ACKNOWLEDGMENTS

• The authors would like to thank P. Glendinning for helpful conversations and M. Timme, Network Dynamics, and the Department of Nonlinear Dynamics at the Max Planck Institute for Dynamics and Self-Organization in G€ottingen, Germany, for generous access to their high performance computing facilities to perform large scale computations.
• Ashwin, C. Bick, and O. Burylko, “Identical phase oscillator networks: Bifurcations, symmetry and reversibility for generalized coupling,” Front.
• 33A. Pikovsky and M. Rosenblum, “Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators,” Physica D 238, 27–37 (2009).

Figures

Figure 3: Normalized bias forHs against altimeter data, as a function ofHs, for the years 1993 to 2009. The new parameterization TEST451 is used with the CFSR wind forcing.

Figure 7: Annual mean of the ratioUss/U10 for the year 2005, whereUss is cut at 0.72 Hz. Model results are shown for the new parameterization TEST451 using CFSR winds (top panel) and using ECMWF winds (middle panel). The results with the BJA parameterization (and using ECMWF winds) are shown in the bottom panel.

Table A.4: Gridded parameters archived in the database. WBBL means Wave Bottom Boundary Layer.

Figure 4: (a) Bias and (b) normalized RMS error for the modelledsignificant wave heights for the year 2005, against data from Jason-1, Jason-2 and Envisat altimeters. The new parameterization TEST451 is used with the CFSR wind forcing. Both model and altimeter data are averaged along the satellite tracks over 1 degree in latitude. There are 2.4 million averaged values.

Figure 6: Average modelled ratiosUss/U10 from the global 0.5-degree resolution model domain.Uss is estimated using eq. (6) and cut here at 0.72 Hz. (a) uses the new TEST451 parameterization whereas (b) uses the BJA parameterization as in part I, both forced by ECMWF winds. Averages are taken after the model output has been binned with bins of 1 m significant wave height and 1 m/s wind speed.

Figure 9: (a) Time series of vertical ground motion variance reco ded and modelled at the Hawaiian Kipapa station during May 2008. (b) Power spectral density of recorded vertical ground displacement at Kipapa. The colors give the spectral level in dB relative to 1 m2/Hz. (c) and (d) Modelled spectra of ground displacement at Kipapa based on the numerical wave model using either the BJA or TEST451 parameterizations, both forced by ECMWF winds.

Figure 8: Time series of observed and modelled wave height, mean fr quency, mean direction and directional spread at the buoy number 51001, located to the north-west of Hawaii. The model uses the new TEST451 parameterization with the CFSR winds.

Figure 1: Dispersion of CFSR versus ECMWF analyses for the month f January 2005. The color scale is the base-10 logarithm of the number of data points in each 0.1 m/s bin.

Figure 12: (a) Drag coefficientCD for the new parameterization TEST451 with ECMWF winds. (b) For the BJA parameterization used in Part I. (c) Expected with the empirical formulation of Drennan et al. (2005, their eq. 4 ). For the latter, theCD is a function ofHs and the wave age, thus we suppose pure wind-sea conditions and we vary the wind and the fetch, with the wind-wave growth taken from Elfouhaily (1997). .

Figure 11: Global fluxes of energyΦoc from waves to ocean in 2005, for the three datasets BJA, TEST451 ECMWF and TEST451 CFSR. The BJA parameterization is 20% larger. For TEST451 ECMWF, the distribution between Northern and Southern Hemisphere is shown. All time series have been low-pass filtered at 30 days. .

Table 1: The three different model runs compared in this paper.

Table 2: Model accuracy forUss estimated by the model and from wave buoy spectra for the year 2008. The models are forced with ECMWF winds and use either the new TEST451 parameterization or the BJA parameterization. Buoys are referenced by their WMO identification code. Details on the instrument packages and locations can be found at www.ndbc.noaa.gov . The normalized bias (NB) is defined as the bias divided by the r.m.s. observed value, the scatter index (SI) is defined by equation (2), andr is Pearson’s correlation coefficient. All buoy spectra were averaged over 3 hours (except for 1 hour at 62069) before integrating in frequency from 0.037 to 0.405 Hz and computing these statistics. Wind speeds were averaged quadratically over the same time frame after a 10% increase, which is an approximate bias correction for the 5 m anemometer height of the buoys. For directional buoys 46013, 46214, 41048, the Stokes drift includes the reduction due to the directional spreading.

Table 3: Average over the year 2005 of the global fluxes of energyfrom atmosphere to wavesΦaw and from waves to oceanΦoc, for the BJA parameterization and for the new TEST451 parameterizations with ECMWF and CFSR winds.

Figure 5: (a) Bias and (b) scatter index for the modelled mean square slopes for the year 2005, against data from Jason-1, Jason-2 and Envisat altimeters. The new parameterization TEST451 is used with the CFSR wind forcing. Both model and altimeter data are averaged along the satellite tracks over 1 degree in latitude. There are 2.4 million averaged values.

Figure 2: (a) Normalized bias and (b) scatter index forHs in 2008, against altimeter data, as a function ofHs, for different wind forcings (ECMWF or CFSR) and parameterizations (TEST451 or BJA). In this case we use a single grid of 0.5 degree resolution.

Figure 10: Craig and Banner (1994)’s parameterαCB of waves to ocean energy flux as a function ofu∗ andHs. (a) For the new parameterization TEST451 with ECMWF. (b) For the parameterization BJA. (c) As a function ofu∗ only, and for the different models and wind forcings. .

Frequently asked questions

Q1. What are the contributions in "A global wave parameter database for geophysical applications. part 2: model validation with improved source term parameterization" ?

A multi-scale global hindcast of ocean waves is presented that covers the years 1994–2012, based on recently published parameterizations for wind sea and swell dissipation [ Ardhuin, F., Rogers, E., Babanin, A., Filipot, J. -F., Magne, R., Roland, A., van der Westhuysen, A., Queffeulou, P., Lefevre, J. -M., Aouf, L., Collard, F., 2010. Results from this hindcast include traditional wave parameters, like the significant wave height and mean periods, and the authors particularly consider the accuracy of the results for phenomenal sea states, with significant heights above 14 m. 

Q2. What is the effect of the CFSR on the model?

A reduced wind input at high frequencies compared to Janssen (1991), and an intermediate input level at the peak, compared to the higher values with Janssen (1991) and much lower values with Tolman and Chalikov (1996). 

Q3. What is the purpose of the paper?

For the purpose of coherent comparisons, three model runs have thus been performed, two using the model with the new parameterization TEST451 and forced with two different wind fields, CFSR and ECMWF, and one run with the model with the BJA parameterization forced with ECMWF winds. 

Q4. What is the effect of the new parameterization on the wind-wave growth?

Given the large bias between the two data sets, the authors adjusted the wind-wave growth parameter from βmax = 1.52 with ECMWF winds, to βmax = 1.33 for CFSR winds (see eq. 19 in Ardhuin et al., 2010). 

Q5. Why are the hindcasts in the Southern Ocean different?

As a result of their forcing choices, the hindcasts contain some discontinuities in the Southern Ocean, due to the effects of icebergs. 

Q6. What was the main effect of the CFSR on the model?

For years 2005 to the present the authors also used ECMWF (European Centre for Medium-Range Weather Forecasts) operational analyses from their Integrated Forecast System. 

Q7. What is the effect of the CFSR?

This effect is parameterized as a sheltering term, reducing the effective winds for the shorter waves (Chen and Belcher, 2000; Banner and Morison, 2010).•