Journal of Marine Science and Engineering Article Wind–Wave Modeling: Where We Are, Where to Go Luigi Cavaleri * , Francesco Barbariol and Alvise Benetazzo Institute of Marine Sciences (ISMAR), National Research Council (CNR), 30122 Venice, Italy; [email protected] (F.B.); [email protected] (A.B.) * Correspondence: [email protected]; Tel.: +39-041-2407955 Received: 11 February 2020; Accepted: 20 March 2020; Published: 7 April 2020 Abstract: We perform a critical analysis of the present approach in wave modeling and of the related results. While acknowledging the good quality of the best present forecasts, we point out the limitations that appear when we focus on the corresponding spectra. Apart from the meteorological input, these are traced back to the spectral approach at the base of the present operational models, and the consequent approximations involved in properly modeling the various physical processes at work. Future alternatives are discussed. We then focus our attention on how, given the situation, to deal today with the estimate of the maximum wave heights, both in the long term and for a specific situation. For this, and within the above limits, a more precise evaluation of the wave spectrum is shown to be a mandatory condition. Keywords: wind–wave modeling; wave spectrum; wave maxima 1. Current State of the Field We all agree that, by and large, the present situation of the best operational models is quite satisfactory. Figure1 shows one of the latest comparisons between altimeter (in this case Sentinel 3A) significant wave heights Hs and the corresponding results of the European Centre for Medium-Range Weather Forecasts (ECMWF, Reading, UK). A 2.5 cm worldwide bias and 0.091 scatter index speak for themselves. It is significant that ESA (the European Space Agency) actually verifies (not calibrates) its altimeter Hs versus the ECMWF data. Similar results are reported by the National Center for Environmental Research (NOAA-NCEP, Maryland, USA), the UK Meteorological Office (UKMO, Exeter, UK), and Meteo France (Toulouse, France). It is instructive that a similar statement on the quality of the results [1] was offered ten years ago, pointing out however some substantial limitations concerning two main items: a frequent miss of the Hs peak values, and an often crude approximation of the spectra. If not quantitatively, on a qualitative basis this is still true today. The aim of this paper is to perform a (short) historical analysis of the development of the daily used wave models pointing out where the present problems come from. 1.1. A Brief History of Wave Modeling Development Granted the obvious relationship between wind and waves, the first quantification was achieved with the Sverdrup and Munk [2] pragmatic (but reasoned) relationships among wind speed, fetch, time, and wave height and period. This was later reconsidered by Bretschneider, leading to the worldwide and long used SMB abacus and method. The substantial breakthrough came in 1952 with the suggestion by Pierson and Marks to describe the sea conditions (in a given area and at a given time) as the superposition of sinusoidal waves, each specified in frequency, direction, and height (hence energy). This opened the door to mathematicians. Sinusoids are an easily handled function, suitable for a number of manipulations and physical assumptions. We still live with this hypothesis. J. Mar. Sci. Eng. 2020, 8, 260; doi:10.3390/jmse8040260 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 260 2 of 15 J. Mar.The Sci. contemporary,Eng. 2020, 8, x FOR and PEER complementary, REVIEW theories by Phillips [3] and Miles [4] on wave generation2 of 15 soon followed. On the basis of the Phillips framing of the problem, Hasselmann [5] quantified the interactionsquantified the and interactions conservative and exchange conservative of energy exch amongange of the energy spectral among components. the spectral White-capping, components. theWhite-capping, energy loss by the wave energy crests loss breaking by wave in crests deep water,breaking had in to deep wait water, 17 years had after to wait the formulation 17 years after of thethe inputformulation function of before the input Hasselmann function [6 before], in connection Hasselmann with [6], the JONSWAPin connection (JOint with North the SeaJONSWAP WAve Project) (JOint experimentsNorth Sea WAve and report Project) [7 ],experiments proposed an and empirical, report [7], but proposed at least quantified an empirical, and but spectrum-dependent, at least quantified usableand spectrum-dependent, expression suitable usable for practical expression applications. suitable for The practical adult ageapplications. of wave modelingThe adult age was of finally wave reachedmodeling in was 1988 finally [8] with reached the reduced in 1988 Discrete [8] with Interactionthe reduced Approximation Discrete Interaction (DIA) Approximation parameterization (DIA) [9] ofparameterization the non-linear interactions,[9] of the non-linear and the interactions, consequent and formulation the consequent of the firstformulation third-generation of the first model third- WAM.generation The overallmodel WAM. situation The was overall well situation pictured inwa thes well so-called pictured WAM in the book so-called by Komen WAM et book al. [10 by]. OtherKomen models et al. followed[10]. Other on models the same followed route: WAVEWATCH on the same route: [11], SWANWAVEWATCH [12], and the[11], one SWAN by the [12], Danish and Hydraulicthe one byInstitute. the Danish Hydraulic Institute. TodayToday wewe live in in a a slow slow evolutio evolutionn of of the the 1994 1994 situation. situation. Improvem Improvementsents have have beenbeen made made on some on someaspects aspects of the ofsource the source functions, functions, computers computers are faster, are faster,and we and have we gone have to gone progressively to progressively higher higherresolutions. resolutions. However, However, although although with witha rather a rather crude crude statement, statement, we wecan can say say that that most most of ourour improvementsimprovements inin thethe lastlast threethree decadesdecades havehave beenbeen associatedassociated withwith thethe parallelparallel improvementsimprovements inin thethe definitiondefinition andand accuracyaccuracy ofof thethe drivingdriving windwind fields.fields. Indeed,Indeed, duringduring thethe lastlast 3030 yearsyears thethe onlyonly basicbasic conceptualconceptual improvementimprovement hashas beenbeen thethe acknowledgementacknowledgement byby JanssenJanssen [[13]13] ofof thethe two-waytwo-way interactioninteraction withwith thethe atmosphere,atmosphere, and and thethe consequentconsequent need need to to useuse aa coupledcoupled model.model. IncludingIncluding thethe interactioninteraction withwith oceanocean currents,currents, thethe processprocess isis stillstill onon itsits way.way. Figure 1. Comparison between Sentinel 3A significant wave height and the European Centre for Figure 1. Comparison between Sentinel 3A significant wave height and the European Centre for Medium-Range Weather Forecasts (ECMWF) wave model results. Medium-Range Weather Forecasts (ECMWF) wave model results. 1.2. Analysis of the Present Situation 1.2. Analysis of the Present Situation As seen in Figure1, the Hs results are on average pretty good, at least in the best modeling centers.As However,seen in Figure we still 1, oftenthe H misss results the peaks are on of average the, especially pretty higher,good, at storms. least Figurein the 2best is indicative: modeling oncenters. a general However, good estimatewe still often of the miss whole, the peaks the peak of the, model especially value higher, is too low. storms. Of course, Figure 2 especially is indicative: for hurricaneson a general and good typhoons, estimate we of could the whole, blame thethe windpeak information,model value butis too it is low. correct Of tocourse, ponder especially how much for ourhurricanes basic physical and typhoons, hypotheses we cancould actually blame hold the wind in these info situations.rmation, but One it indeedis correct wonders to ponder when how Katrina, much oneour ofbasic the worstphysical hurricanes hypotheses in the can Gulf actually of Mexico hold (2005), in these has beensituations. simulated One hundreds indeed wonders of times tryingwhen toKatrina, get the one correct of the results. worst hurricanes in the Gulf of Mexico (2005), has been simulated hundreds of times trying to get the correct results. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 3 of 15 J. Mar. Sci. Eng. 2020, 8, 260 3 of 15 Figure 2. Wave model performances compared with buoy data. Figure3 (courtesy of Jesus Portilla Yandun) reports a frequent problem with spectra, especially in the so-called cross-sea conditionsFigure 2. Wave (one model of the performances worst conditions compared for with navigation). buoy data. Granted a very good fit in Hs (bias virtually null), the right panel shows a completely different situation for the spectrum, with aFigure strong underestimate3 (courtesy of Jesus of the Portilla wind sea Yandun) component reports (at 0.2a frequent Hz) and problem a compensating with spectra, overestimate especially of thein swellthe so-called (at 0.09 cross-sea Hz). Examples conditions in 2D (one spectra of the also worst abound, conditions typically for withnavigation). a shifted Granted peak frequency a very good or afit directional in Hs (bias distribution virtually null), that is the too right wide. panel The practicalshows a implicationscompletely