Coastal Engineering 58 (2011) 806–811

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Coastal Engineering

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Short communication

Morphodynamic upscaling with the MORFAC approach: Dependenciesand sensitivities

Roshanka Ranasinghe a,b,⁎, Cilia Swinkels c, Arjen Luijendijk b,c, Dano Roelvink a,c, Judith Bosboom b,Marcel Stive b, DirkJan Walstra b,c

a Department of Water Engineering, UNESCO-IHE, PO Box 3015, 2601 DA Delft, The Netherlandsb Civil Engineering and Geosciences, Technical University of Delft, PO Box 5048, 2600 GA Delft, The Netherlandsc Harbour, Coastal and Offshore Engineering, Deltares, PO Box 177, 2600 MH Delft, The Netherlands

⁎ Corresponding author. Department of WaterPO Box 3015, 2601 DA Delft, The Netherlands. Tel.: +646 465 915 (mobile); fax: +31 15 2122 921.

E-mail address: r.ranasinghe@unesco-ihe.org (R. Ran

0378-3839/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.coastaleng.2011.03.010

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 July 2010Received in revised form 14 December 2010Accepted 23 March 2011Available online 22 April 2011

Keywords:Morphodynamic modelingMORFACDelft3DLong term coastal evolution

The recently developed Morphological Acceleration Factor (MORFAC) approach for morphodynamicupscaling enables numerical model simulations of coastal evolution at decadal to millennial time scales.Primarily due to the massive increase in modeling time scales it affords, the MORFAC approach is nowstandard in state-of-the-art commercially available coastal morphodynamic modeling suites. However, thegeneral validity of the MORFAC concept for coastal applications has not yet been comprehensivelyinvestigated. Furthermore, a robust and objective method (as opposed to the subjective and inelegant trialand error method) for the a priori determination of the highest MORFAC that is suitable for a given simulation(i.e. critical MORFAC) does not currently exist. This communication presents some initial results of anongoing, long-term study that attempts to rigorously and methodically investigate the limitations andstrengths of the MORFAC approach. Based on the results of a strategically designed numerical modelingexercise using the morphodynamic model Delft3D, two main outcomes are presented. First, the maindependencies and sensitivities of the MORFAC approach to fundamental forcing conditions and modelparameters are elucidated. Second, a criterion based on the Courant–Friedrichs–Levy (CFL) condition for bedform propagation that maybe used as a guide to determine the critical MORFAC a priori is proposed.

Engineering, UNESCO-IHE,31 15 2151 805 (office), +31

asinghe).

ll rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The ability to numerically simulate coastal morphological evolu-tion at decadal and millennial time scales is a need long felt by coastalengineers and managers worldwide. This is nowmore the case due tothe potential impacts of climate change. However, until recently itwas only possible to numerically simulate coastal evolution at timescales of up to a couple of years using traditional morphodynamicupscaling techniques such as the ‘continuity correction’ method. Theintroduction of the morphological acceleration factor (MORFAC)concept to coastal morphodynamic modeling by Lesser et al. (2004)and Roelvink (2006) has changed this. TheMORFAC (referred to asMFfor convenience hereon) approach enables numerical simulations ofcoastal morphological evolution due to waves and currents at timescales of decades (Jones et al., 2007; Lesser, 2009; Lesser et al., 2004;Tonnon et al., 2006), and under very uniform forcing conditions (e.g.tides only), for centuries (Dissanayake et al., 2009a,b; Van der Wegenet al., 2008; van der Wegen and Roelvink, 2008).

In general, coastal morphodynamic models consist of a computa-tional control shell that sequentially calls hydrodynamic, sedimenttransport and bed level update modules that are linked via amorphodynamic feedback loop (Fig. 1). Hydrodynamics are resolvedusing, for example, the spectral wave action balance equations todescribe waves (Booij et al., 1993) and the unsteady shallow waterequations to describe currents (Lesser et al., 2004). Sedimenttransport rates are calculated using one of the many availablesediment transport equations (e.g. Bijker, 1971; Deigaard et al.,1986; Engelund and Hansen, 1967; Van Rijn, 1993). Bed level updateis facilitated via the sediment continuity equation. However as thetime scales associated with bed level changes are generally muchgreater than those associated with hydrodynamic forcing, to enablereasonably fast computations, coastal morphodynamic modelshave until recently adopted the approach of updating bed levelsand feeding them back into the hydrodynamic calculations only everyfew hydrodynamic time steps. The MF approach departs fromthis traditional way of thinking and essentially multiplies the bedlevels computed after each hydrodynamic time step by a factor (MF)to enable much faster computations. The significantly upscalednew bathymetry is then used in the next hydrodynamic step(Fig. 1). A fundamental assumption in this approach is that themorphological response to the hydrodynamic forcing is linear during

Hydrodynamic processes

Sediment transport processes

Bed level change calculations

Morphodynamicfeedback

Bottom change multiplied by MORFAC

Fig. 1. General structure of coastal morphodynamic models and the implementation ofthe MF concept.

807R. Ranasinghe et al. / Coastal Engineering 58 (2011) 806–811

one morphological time step; somewhat akin the elongated tideapproach described by Latteux (1995).

A number of recent studies have compared morphologies predictedwhen using low (but not equal to unity) and highMFs,mostlywith tidalforcing only, and concluded that high MFs produce sufficiently realisticmorphological predictions (Lesser et al., 2004; Roelvink, 2006). Severalother studies have simply concluded that MFs as high as 1000 arecapable of producing realistic results based on qualitative comparisonswithmeasured bathymetry (Dissanayake et al., 2009a,b;VanderWegenet al., 2008; Van der Wegen and Roelvink, 2008). However, toconclusively prove the veracity of the MF as a concept that does notviolate geophysical reality, long term (50–100 years) hindcasts ofmorphological evolution produced when using a range of different MFsshould be quantitatively compared with corresponding observedbathymetric changes, preferably at a site (or sites) with dynamicmorphology. This is unfortunately very difficult to achieve due to thedearth of high quality bathymetric surveys at suitable locations.

The next best approach to assess the MF concept is to accept thatthe most accurate numerical model simulation currently possible isone undertaken with MF=1 and use that as the benchmark to assessthe relative accuracy of higher MF model predictions. In the least, asimulation undertaken with MF=1 is not intuitively difficult toaccept and is likely to be a better representation of the physics ofnature than results obtained with previously adopted morphologicalmodeling philosophies. A few studies have adopted this philosophyto determine the highest MF that can be used for the specificissue investigated in those studies (Jones et al., 2007; Lesser, 2009;Roelvink, 2006; Van der Wegen and Roelvink, 2008). Although thesestudies may have verified the validity of the MF concept for someconditions, the general validity of the MF concept for coastalapplications at engineering time scales has not yet been comprehen-sively investigated. Furthermore, no attempts have so far been madeto develop a method for the a priori determination of the highest MFthat is suitable for a given simulation (or, in other words, the lowestMF at which the model results start to depart from an acceptable levelof accuracy: critical MF). To date, the determination of the critical MFhas been via the cumbersome and inelegant trial and error approach.

Although it is indeed very tempting to simply accept the MFconcept due to the massive increase in modeling time scales it affords,such an innovative concept should be rigorously and methodicallyassessed prior to its general acceptance in coastal morphodynamicmodeling. In recognition of this need, a comprehensive, multi-yearresearch program was recently initiated via a strategic alliance

among Deltares, UNESCO-IHE and the Technical University of Delft(Netherlands). The overarching aim of this initiative is to compre-hensively investigate the validity and limitations of the MORFACapproach in coastal morphodynamic simulations at typical engineer-ing time/length scales, with the ultimate goal of developing aneffective criterion to determine the critical MF a priori for any givenreal-life situation. The overall initiative comprises several individualstudies that focus on issues such as the relationship between thecritical MF and (i) the type of simulation (i.e. 2DV, 2DH or 3D),(ii) forcing (uniform flow, tides, waves, combined tides and waves),(iii) temporal and spatial resolution (i.e. hydrodynamic time step, gridsize) and extent (length of simulation, size of domain), (iv) criticalmodel parameters (e.g. eddy viscosity, diffusivity, friction, turbulenceclosure scheme, wave parameters, sediment transport parameters,numerical parameters), (v) physical processes included (e.g. type ofsediment transport equation, wave–current interaction), (vi) numer-ical schemes adopted for hydrodynamics and bed level update,(vii) numerical error, and (viii) forcing-response characteristics ofsimulation (e.g. short term response to extreme events vs. near-equilibrium response under averaged/dominant forcing conditions).Model simulations are being undertaken with several commonly usedmodels such as Delft3D and Xbeach as well as other simpler modelsthat have been/are being custom-built for this study. This communi-cation presents some exciting initial insights which demonstratesome of the main dependencies and sensitivities of the MF approach,and suggests a preliminary method for the a priori determination ofthe critical MF.

2. Methods

The fundamental assumption in the investigation described hereinis that the most accurate numerical model simulation currentlypossible is one undertaken with MF=1 (i.e. benchmark simulation).In order to commence this analysis at a very fundamental level, twosimple and opposing idealized cases were considered: the morpho-logical evolution of a symmetrical protrusion (hump) and adepression (trench), one being the direct opposite of the other, bothmorphologically and hydrodynamically. The hump is expected torepresent features such as sand bars, while the trench representschannel-like features; both being commonly found features in thecoastal zone. In both cases, the bed perturbation was initially locatedon a flat bed and subjected to uniform unidirectional flow. Theambient water depth was set to 4 m above the plane bed, while theamplitude of the bed perturbation was set to 2 m, resulting in aminimumwater depth of 2 m above the hump, and amaximumwaterdepth of 6 m in the trench (Fig. 2).

Two series of Delft3D (for a detailed description of Delft3D, seeLesser et al., 2004) simulations were undertaken in profile mode forthe two bed perturbations considered. Uniform unidirectional flowwas introduced from the left hand side (LHS) boundary and a zero-gradient flow boundary condition was imposed at the downstream(RHS) boundary. The Engelund andHansen total load formulationwasused for sediment transport calculations. In each series of simulations,the flow velocity, grid size, hydrodynamic time step, and the MF werevaried systematically to investigate the dependencies and sensitivitiesof model predictions to these variables.

The critical MF (MFcrit) is defined as the highest MF that results inbed level predictions that are similar to those predicted by abenchmark case at the same morphological time (MT). Or defined inanother way, MFcrit is the lowest MF at which the model results startto depart from the benchmark case at the same MT. Therefore, first,the benchmark conditions have to be determined. As described above,for the purposes of this study, the benchmark case is taken as theMF=1 case. The benchmark conditions for each forcing conditionwere thus obtained by continuing the respective MF=1 casesuntil morphodynamic equilibrium was reached (say at MT=Te).

Fig. 2. Initial test bathymetries: (A) hump on a flat bed, and (B) trench on a flat bed. Uni-directional flow from left to right.

Fig. 3. Definition of morphological equilibrium for (A) the hump case, and (B) thetrench case. In each case, the top, middle and bottom panels show the morphologicalevolution, change in bed form amplitude, and change in bed form propagation speedrespectively. The dashed vertical lines indicate the equilibriummorphological time (Te)thus obtained for the two cases.

808 R. Ranasinghe et al. / Coastal Engineering 58 (2011) 806–811

Morphodynamic equilibrium was defined when both the change inamplitude and the change in propagation speed of the bedperturbation approached zero (Fig. 3). The bed levels predicted atMT=Te by subsequent MFN1 simulations were then compared withthe equilibrium morphology predicted by the benchmark simulation.

In a model with an explicit numerical scheme, the hydrodynamic

Courant number (Cr = Udtdx

, where U=flow velocity, dt=hydrody-

namic time step, and dx=grid size) could be used to define modelfailure. That is, if CrN1 at any location in the model domain, the modelwill ‘blow up’. However, this criterion is not valid for models withimplicit numerical schemes. This is because, in the latter case, modelstability is not only a function of Cr but also of the accuracy/errorassociated with the numerical scheme adopted in the model. Ashigher order numerical schemes are more accurate than lowerorder schemes, models with higher order implicit numericalschemes can be relied upon to produce accurate results for Crvalues greater than one. Delft3D is such a higher order implicit model,where the alternating direction implicit (ADI) method is adopted tosolve the horizontal momentum and continuity equations (Leendertse,1987). Stelling and Leendertse (1991) extended the ADI method suchthat the third-order upwind finite-difference scheme for the firstderivative was split into two second-order consistent discretizations, acentral discretization, and an upwind discretization, that are succes-sively used in both stages of the ADI scheme. Thismethod is expected tobe at least second-order accurate, and stable at Cr values of up to 10.Therefore, a simple Crb1 criterion cannot be used to define modelfailure (due to instability and/or inaccuracy) in the simulations

undertaken herein, and thus the Brier Skill Score (BSS) is used as analternate measure of model performance.

Following Van Rijn et al. (2003), the Brier Skill Score is defined as:

BSS = 1−zb−zb;mf1

� �2� �

z1−zb;mf1

� �2� � ð1Þ

Where zb=final bed level predicted by simulation withMFN1, zb,mf1=final bed level predicted by benchmark simulation(MF=1), and z1=initial bed level.

According to Eq. (1), a BSS value of 1 indicates a perfect matchbetween the bed levels predicted by the benchmark case and a casewith a higher MF. BSS values lower than unity indicate a differencebetween bed levels predicted by the two cases; the lower the BSS thegreater the divergence between the predicted bed levels. Thenormalization of the 2nd term on the RHS of Eq. (1) by the totalbed level change during the benchmark simulation ensures that very

Fig. 5. MFcrit vs. ambient Froude number for the hump and trench cases for a constantgrid size (15 m) and hydrodynamic time step (3 s).

809R. Ranasinghe et al. / Coastal Engineering 58 (2011) 806–811

small local differences in bed level between the two cases beingcompared do not result in an artificially low BSS value.

In complex engineeringmodel applications, BSS values as low as 0.5may be considered sufficiently accurate (van Rijn et al., 2003). However,the aim of the preliminary investigations undertaken herein was todetermine the lowestMF value atwhichmodel results started to departfrom the benchmark case (i.e. worst case estimate). Therefore, for thepurposes of this analysis, a stringent cut-off BSS value of 0.99 wasadopted. Thus, for a given set of model conditions (flow velocity, gridsize, and time step), as the MF was gradually increased from unity, thefirst MF at which the BSS dropped below 0.99, before reaching atMT=Te, was considered as MFcrit. The determination of MFcrit for thetwo bed forms using this method (with flow velocity=0.9 m/s, gridsize=15m, and time step=3 s) is exemplified in Fig. 4.

3. Dependencies and sensitivities

3.1. Hump vs. trench

Two separate sets of simulations were undertaken for the humpand trench cases with flow velocities (U) of 0.5 m/s, 0.7 m/s, 0.9 m/s,1.1 m/s, 1.3 m/s and 1.5 m/s. In these first simulation sets, the grid sizeand hydrodynamic time step were kept constant at 15 m and 3 srespectively (resulting in a constant Courant number of 2.5) while theMF was systematically increased from 1 to MFcrit. The dependency ofMFcrit on the ambient Froude number (Fr) (i.e. Fr at the upstreamboundary) is shown in Fig. 5 (note: similar dependencies are seen atthe bed perturbation but are not shown here to avoid repetition ofFig. 5). Two phenomena are clearly visible in Fig. 5. First, for both cases,MFcrit decreases exponentially with ambient Fr. This is intuitivelycorrect as higher velocitieswill result in higher sediment transport andthus larger bed level variations which will eventually lead tohydrodynamic instabilities in the model. Second, for a given Froudenumber, the trench case can accommodate a significantly higherMFcrit.This is also intuitively correct as velocities (and thus sedimenttransport) will increase over the hump while they will decrease overthe trench. This indicates that, for a real-world application,MFcrit for agiven application will be governed by morphodynamics of bedprotrusions in the domain rather than those of bed depressions.Therefore, further analysis is restricted to the hump case.

3.2. MFcrit vs. grid size (dx), hydrodynamic time step (dt), and Courantnumber (Cr)

A second series of simulations was undertaken to investigate thedependency and sensitivity of MFcrit on grid size dx, hydrodynamic

Fig. 4. Brier Skill Score vs. MF for the hump case (dashed line) and trench case (solidline). When the BSS falls below 0.99 the simulation is considered to have failed.

time step dt and Courant number Cr for the hump case. Thesesimulations were undertaken for flow velocities (U) of 0.9 m/s and1.3 m/s (i.e. Fr values of 0.35 and 0.42 respectively at the hump crest).For eachU, first dtwas kept constant at 3 swhile dxwas systematicallydoubled from 3.75 m to 60 m (~1/5th of hump width). Then dx waskept constant at 15 mwhile dtwas systematically doubled from 0.75 sto 12 s. This set of simulations resulted in Cr values that variedbetween 0.04 and 0.72.MFcrit vs. Cr for this set of simulations is plottedin Fig. 6 which shows that MFcrit can vary by up to 1500 for the sameCr. Thus MFcrit appears not to be directly governed by Cr.

To investigate the individual dependencies betweenMFcrit, dx, anddt, in addition to the above set of simulations where Crwas allowed tofreely vary while dx and dt were varied, another set of simulationswhere Cr was forced to remain constant at 0.18 while dx and dt werevaried was undertaken. Fig. 7 shows the resulting variations of MFcritwith dx and dt for both constant and varying Cr. An almost linearcorrelation between MFcrit and dx can be seen, regardless of whetherCr is constant or varying. However, the dependency of MFcrit on dt isless clear. For the cases where Cr was allowed to vary with dt (i.e. bykeeping dx constant — solid circles), MFcrit increases with decreasingdt, which is intuitively correct as numerical stability increases withdecreasing dt. However, when Cr is forced to be constant whileincreasing/decreasing dt (i.e. by simultaneously decreasing/increasingdx), MFcrit decreases with decreasing dt (unfilled circles), which iscounter-intuitive. This indicates that the dependency of MFcrit on dxover-rides that on dt.

Fig. 6.MFcrit vs. Cr for U=0.9 m/s. dxwas varied between 3.75 m and 60 mwhile dtwasvaried between 0.75 s and 12 s. Cr was allowed to vary freely between 0.04 and 0.72.

Fig. 7. Dependency ofMFcrit on grid size (dx), hydrodynamic time step (dt) and Courantnumber (Cr).

Fig. 8. MF inclusive CFL for bed form propagation (CFLMF) for the last successfulsimulation (i.e. the simulation associated withMFcrit) (shown by asterisks) and the firstunsuccessful simulation (shown by solid circles) for each simulated combination of U,dx and dt. The proposed preliminary criterion of CFLMFb0.05 for a safe first estimate ofMFcrit is also shown (dashed vertical line).

810 R. Ranasinghe et al. / Coastal Engineering 58 (2011) 806–811

4. Preliminary criterion for the a priori determination of MFcrit

4.1. Courant–Friedrichs–Levy (CFL) criterion for bed form propagation

An absolute numerical criterion in the case of an explicit bedlevel update scheme (which is commonly used) requires that thepropagation of bed forms in one morphological time step should notexceed the grid cell size (CFL criterion). The propagation speed (orcelerity) of the bed forms Cbed is commonly estimated as

Cbed =b→S

��� ���1−εð Þh ð2Þ

where b is the power of the sediment transport formulation used, ε isthe porosity, h is the water depth and

→S

��� ��� is the sediment transportmagnitude.

For accurate morphodynamic representation Cbed should be suchthat bed forms do not propagate more than one grid cell (dx) in onemorphological time step (MF×dt). Thus,

Cbed bdx

MF × dtð3Þ

which gives the MF inclusive CFL criterion as:

CFLMF =CbedMFdt

dxb 1 ð4Þ

To examine whether the simulations undertaken here satisfy theabove criterion, the CFLMF values for the last successful simulation (i.e.the simulation associated with MFcrit) and the first unsuccessfulsimulation for each simulated combination of U, dx and dt are plottedas a binary plot (Fig. 8). Successful and unsuccessful simulations areindicated by values of 1 and −1 respectively on the Y-axis of Fig. 8.The most obvious feature of Fig. 8 is that the CFLMF values associatedwith both successful and unsuccessful simulations fall below thecritical value of 1. This essentially indicates that while the MFcritsimulations do satisfy the CFLMF criterion given by Eq. (4), simulationsbecome unsuccessful at lowerMF values than they should (else all thecircles (i.e. unsuccessful cases) in Fig. 8 would have been locatedfarther to the right on the X-axis such that the associated CFLMF valueswould be N1). There could be numerous reasons for this phenomenonincluding; failure of the assumed linear relationship betweenmorphological response and hydrodynamic response, MF inducederrors in bed form celerity and/or amplitude, the potential depen-dency between CFLMF and model accuracy. Furthermore, it is not

unlikely that CFLMF values determined in this study are to some degreedependent on the specific numerical scheme adopted in Delft3D.However, the CFLMF values discussed above are expected to begenerally applicable to some degree. This is because CFLMF is largelydependent on bed perturbation propagation speed (CbedMF) that isultimately an output of the bed level update scheme, which is explicitin Delft3D as is the case with most other commonly used coastalnumerical models. Nevertheless, similar investigations need to beundertaken using different numerical schemes to obtain a robustcriterion for the a priori determination ofMFcrit. The above mentionedissues (among many others) are currently being investigated inseveral sub-projects within the overall initiative.

It should also be noted that the above CFLMF analysis is entirelybased on conditions at the beginning of a simulation (i.e. Cbed wascalculated at the start of each simulation). However, Cbed, and thusCFLMF, will vary over the duration of the simulation. Therefore MFcritcan also vary during a given simulation. For constant forcing con-ditions, such as those used in the simulations described herein, MFcritis likely to increase as a stable simulation progresses in time.

Based on the above analysis and discussion, it is clear that, at thisstage, the development of a simple and definitive criterion to predictMFcrit for even this simplest case of uni-directional flow over asymmetric bed feature is non-trivial. Nevertheless, Fig. 8 indicatesthat, at least for the range of conditions tested here, CFLMFb0.05 maybe used as a preliminary guide to obtain a safe first estimate ofMFcrit .This returns a MFcrit value of about 100 when typical values usedin nearshore coastal applications are substituted in Eq. (4) for dt (3–10 s), dx (5–20 m) and Cbed (~0.001 m/s). This critical value is,however, very likely to be significantly lower where the forcingconditions and/or initial bathymetry is/are more complex than thoseconsidered herein.

5. Concluding remarks

A preliminary investigation into the dependencies and sensitivitiesof theMF approach for morphodynamic upscaling was undertaken viaa strategically designed numerical modeling exercise using themorphodynamic model Delft3D. Based on the results, the followinggeneral observations are made:

• The criticalMF (MFcrit), which is considered as the highest MF that islikely to result in reliable morphodynamic predictions, has a strong

811R. Ranasinghe et al. / Coastal Engineering 58 (2011) 806–811

dependency on the Froude number Fr and the grid size dx. MFcritdecreases exponentially as Fr increases, while it increases almostlinearly with dx.

• MFcrit does not appear to be directly governed by the (hydrody-namic) Courant number (Cr).

• MFcrit has a dependency on dt that is secondary to MFcritdependencies on Fr and dx.

• The criterion CFLMFb0.05 may provide a safe first estimate of MFcritunder conditions similar to those considered herein.

It should be noted that the above analysis and conclusions arebased on a few, strategic model simulations of the simple case of uni-directional flow over symmetric bed perturbations. Therefore, theresults and conclusions presented herein may not be directlyapplicable to complex real-life situations which are likely toincorporate highly non-uniform morphology and time varying non-linear forcingwhichmay include tides, waves, wind etc. It is likely thatMFcrit for such complex applications would be significantly smallerthan those discussed herein. IndeedMF values used in reportedmodelsimulations to date, even for schematized bathymetries with constantsingle constituent tidal forcing, have not exceeded values of about500. Similarly, the highest reported MF values used in simulationsincluding wave forcing (on highly schematized bathymetries) arelimited to about 50. In fact, in medium to long term simulations thatinclude extreme wave events (i.e. storms) in the forcing, MFcritmust also be by necessity time varying with a relatively small MFcritduring the storm. This is because themorphological changes (erosion/accretion) that occur during storms will be large and theirmultiplication by even a moderate MF may result in an unrealisticpositive feedback mechanism which will lead to erroneous results.However, a relatively large MF may be defined during the milder andrelatively constant forcing in between storms as the associatedmorphological changes will be minimal in comparison. Obviously, ifthe ratio of total duration of extreme events to total duration offorcing is high, the computational gain afforded by the MF approachwill be rather limited.

In complex, real-life applications, although the CFLMF criterionpresented here may provide some guidance in selecting a firstestimate of a reliable MF, it is imperative to assess the validity of theselected MF via a sensitivity test to ensure that model predictions areindependent of the MF value (i.e. similar to the manner in which thecritical hydrodynamic time step is determined in hydrodynamicmodels). While strategic research is currently being undertaken tofurther investigate the many dependencies and sensitivities of the MFapproachwith the ultimate, and highly challenging goal of developingan effective criterion for the a priori determination of MFcrit for anygiven real-life situation, it is our hope that this short communication

will act as a catalyst for robust discussion and more independentinvestigations on this intriguing topic which is of great relevance forcoastal numerical modeling at engineering time scales.

Acknowledgments

The authors wish to thank both reviewers for their excellent andvery constructive comments on the original manuscript.

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