The nonlinear nature of bottom friction in shallow flow complicates its analysis, particularly in idealised models. For tidal flows, Lorentz’ linearisation has been widely applied, using an energy criterion to specify the friction coefficient. Here we propose an extension of this approach to storm surges, leading to a friction coefficient that may gradually vary over a storm event. The derivation is provided along with first results for a single channel

Hulscher, Suzanne J.M.H.

Lipari, Giordano

Pitzalis, Chris

Reef, Koen R.G.

Roos, Pieter C.

English

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Time-dependent linearisation of bottom friction for storm surge modelling in the Wadden Sea

University of Twente Research Information

4th International Symposium of Shallow Flows, Eindhoven University of Technology, NL, 26-28 June 2017Time-dependent linearisation of bottom frictionfor storm surge modelling in the Wadden SeaPieter C. Roos1, Chris Pitzalis1, Giordano Lipari2, Koen R.G. Reef1, and Suzanne J.M.H. Hulscher11Water Engineering and Management, University of Twente, Enschede, NetherlandsE-mail (corresponding author): p.c.roos@utwente.nl2Watermotion| Waterbeweging: Consultancy Marketing Outreach Research,Zwolle, NetherlandsKeywords: storm surges, idealised modelling, bottom friction, Lorentz’ linearisation.AbstractThe nonlinear nature of bottom friction in shallow flow complicates its analysis, particularly in idealised models. Fortidalflows, Lorentz’ linearisation has been widely applied, using a energy criterion to specify the friction coefficient. Herewe propose an extension of this approach to storm surges, leading to a friction coefficient that may gradually vary over astorm event. The derivation is provided along with first results for a single channel.1. IntroductionBottom friction is important in shallow water flows, such as tides and storm surges in the marine environment. However,the nonlinear (quadratic) dependency of the bottom stress on flow speed complicates its implementation, analysis andinterpretation. In his pioneering channel network model for the impact of the closure of the Zuiderzee (Fig.1a), Lorentzintroduced a linearised stress parameterisation, with a coeffi ient r specified on the basis of equivalence of dissipatedenergy over a tidal period (Lorentz, 1922; StaatscommissieZu derzee, 1926). This coefficient is (proportional to) avelocity scale, which in the tidal context has the clear interpr tation of being the tidal flow velocity amplitude of asingle constituent. An iterative procedure can be applied to ensure that the velocity scale used to specifyr matches thevelocity scale in the model results (Reef et al., 2016). Lorentz’ linearisation has been verified experimentally (Terrat l.,2005), and the effect of a dominant tidal constituent on the friction experienced by weaker components has been treatedanalytically (Inoue & Garrett, 2007).For storm surges, however, such a linearisation approach isless straightforward as the velocity scale is harder tointerpret. Restricting to a time-invariant approach and retaining the quadratic formulation, Lorentz algebraicallynalysedthe equilibrium response to a representative value of the wind stress (Staatscommissie Zuiderzee, 1926). Clearly, thisapproach does not capture the time-varying nature of forcing (wind stress, atmospheric pressure) and response (surge).Here we propose a linearisation of the bottom stressτb involving a linear friction coefficient that adjusts to the temporaldevelopment of a storm event. Assuming one-dimensional depth-averaged flowu, this can be summarised asτbρ= cd|u|u︸ ︷︷ ︸quadratic≈ r(t)u︸︷︷︸linearised, (1)with cd the dimensionless drag coefficient of the original quadratic parameterisation andr(t) the time-dependent linearfriction coefficient (in m s−1). Further,ρ is the water density. The proposed linearisation enables usto simplify calcula-tions while still capturing the (nonlinear) variation in bottom stress over the various stages of a storm event (Fig.1b). Inparticular,r(t) will be large whenever flow is strong and small when it is weak.To specify the linear friction coefficient, we adopt an energy criterion analogous to that of Lorentz, but now in aninstantaneous rather than tide-averaged sense. Specifically, r(t) must be such that the instantaneous energy dissipation bybottom friction, i.e. thepower τbu, averaged over the channel is identical for both parameterisations in Eq.(1):1ℓ∫ ℓ0cd|u|u2dx =1ℓ∫ ℓ0r(t)u2dx ⇒ r(t) =cd∫ ℓ0 |u|u2dx∫ ℓ0 u2dx. (2)4th International Symposium of Shallow Flows, Eindhoven University of Technology, NL, 26-28 June 2017(a) network model0 1 2 3 4water level (m)0 2 4 6 800.511.5w (N/m2 )(b) wind stress0 2 4 6 800.050.10.15U (m/s)(c) velocity scale0 2 4 6 8time t (days)012r (m/s)10-3 (d) friction coefficientFigure 1. (a) Example simulation with network model applied by Lorentz andStaatscommissie Zuiderzee (1926), as reproduced byReef et al. (2016). Then, for an artificial storm event, as a function oftime (solid line): (b) wind stressτw(t), (c) channel-averagedvelocity scale, defined asU(t) = 1ℓ∫ ℓ0 |u|dx, (d) time-varying friction coefficientr(t). The dashed lines show an example with constantfriction r0, for the same wind event.Here we have considered depth-averaged flowu(x, t) in a channel ranging fromx = 0 to x = ℓ. Two remarks are in order:• Just like the tidal case, specifyingr(t) requires knowledge of the flow solutionu(x, t), which, in turn, depends onr(t) again. To deal with this cyclic dependency, the iterative procedure referred to above should be extended, nowconverging to a time-varying friction coefficient (rather than a scalar).• The friction coefficientr(t) depends ont but not onx. It is thus meant to represent the channel as a whole, whichremains an important simplification compared to the original qu dratic parameterisation in Eq.(1).In the remainder of this extended abstract, we present an idelised one-dimensional storm surge model (§2.1) along withits solution procedure (§2.2), analyse and discuss some first results (§3), before summarising the conclusions (§4).2. Methods2.1 Model formulationConsider a one-dimensional basin of lengthℓ and uniform depthh with closed boundaries, subject to time-dependentwind stress. The free surface elevation is denoted byζ (x, t), the depth-averaged flow velocity byu(x, t). Assuming that|ζ | ≪ h, conservation of mass and momentum is expressed in linearised form according to∂ζ∂ t+h∂u∂x= 0,∂u∂ t+r(t)uh=−g∂ζ∂x+τw(t)ρh, u(0, t) = 0, u(ℓ, t) = 0, (3)where also the boundary conditions are shown. In Eq.(3), we hav incorporated the linear stress parameterisation withcoefficientr(t) already presented in Eq.(1). The wind stressτw(t) serves as the forcing of the system, assumed spatiallyuniform yet time-dependent (Fig.1b). Bothr(t) andτw(t) are expressed as a discrete Fourier series according tor(t) =M∑m=−MRm exp(iωmt),τw(t)ρ=M∑m=−MTm exp(iωmt), ωm =2mπTrecur, (4)with truncation numberM and complex coefficientsRm andTm (satisfyingR−m = Rm andT−m = Tm becauser andτw arereal-valued, with an overbar denoting complex conjugation). Furthermore, we have introduced the angular frequencyωm4th International Symposium of Shallow Flows, Eindhoven University of Technology, NL, 26-28 June 2017and fictitious recurrence periodTrecur, over which the storm event recurs in our model (due to the discrete Fourier series).Initial conditions can be safely ignored by choosingTrecur sufficiently large (e.g., in the order of 10 days) compared tothefrictional decay time scaleTfric (typically less than a day). The dynamic equilibrium solution obtained will then displaystill water conditions at the beginning of the storm event (Chen, 2015; Reef et al., 2016).The above model equations and boundary conditions are supplemented with the energy criterion in Eq.(2), which —through the flow field — specifiesr(t) as a function of time. This dependency ofr(t) on the solution is in fact the onlynonlinear element in an otherwise linear model.2.2 Solution procedureOur solution method consists of a calculation ofζ (x, t) andu(x, t), nested in an iterative procedure to findr(t).First, the model is expressed in terms of the free surface elevation amplitudeZm(x) only. Then, analogous to Eq.(4), thefree surface elevation is written as a discrete Fourier serie , introducing complex Fourier coefficientsZm(x) as a functionof space. This gives the following coupled Helmholtz-type of b undary value problem forZm(x):d2Zmdx2−1gh2[∑nRm−niωnZn]+ω2mghZm = 0,dZmdx(0) =dZmdx(ℓ) =Tmgh, (for m = 0,±1, · · · ,±M). (5)The convolution sum between square brackets reflects the coupling among Fourier modes due to the time-dependency ofr(t). Assuming givenRm-values, Eq.(5) is analytically solved for the eigenvectors f the corresponding(2M+1)×(2M+1)-matrix. This providesZm(x) and, hence,ζ (x, t). We finally use the continuity equation in Eq.(3) to findu(x, t), as well.The iterative procedure starts with a constant friction coeffici nt, sayr(t) = r0, as first guess. For thisr(t), wedetermine the solutionζ (x, t) and u(x, t) as outlined above. With the aid of a Fast Fourier Transform, Eq.(2) is thenapplied to update theRm-values in Eq.(4), used as input for the next iteration. Thisprocedure is repeated until theRm-values and consequently alsor(t) have converged, by which the energy criterion in Eq.(2) is indeed satisfied.3. Results and discussionFirst model results for a synthetic storm event (Fig.1b) demonstrate that our new model approach works, displayingconvergence to a time-dependent friction coefficient (Fig.1d), which indeed follows the evolution of the channel-averg dflow velocity (Fig.1c). These results have been obtained with a channel of lengthℓ= 100 km and undisturbed water depthh = 4 m, subject to a storm event with a maximum wind stress of 1 N m−2, with Tevent= 1 day and a recurrence period ofTrecur= 8 days, usingM = 96. The drag coefficient for bottom friction has been set at a valuecd = 1×10−2. The sloshingobserved in this new simulation with time-varyingr(t) (solid line in Fig.1cd), disappears in the example with a consta tfriction coefficientr0 (dashed line), which overestimates bottom friction when flow is weak.Importantly, our new approach takes away the possible arbitr iness associated with choosing such a constant valuer0. Finally, we remark that our new method can also be applied toinclude tides, e.g. for tide-surge interactions or for atidal signal with several constituents.4. ConclusionsWe have explored a novel linearisation of the bottom friction f rmulation, appropriate for idealised storm surge models.The associated friction coefficientr(t), which adjusts to the temporal development of storm events,follows from an energycriterion that extends Lorentz’ (1922) approach. First results, obtained for an idealised single channel model in which r(t)is determined iteratively, show the expected qualitative behaviour. Ongoing research focuses on quantitative sensitivities,the role of tides as well as the extension to a channel networkrepresenting the Wadden Sea.ReferencesChen, W.L., P.C. Roos, H.M. Schuttelaars, & S.J.M.H. Hulscher (2015). Resonance properties of a closed rotating rectangular basinsubject to space- and time-dependent wind forcing.Ocean Dyn. 65 (3), 325-339. doi: 10.1007/s10236-015-0809-yInoue, R. & C. Garrett (2007) Fourier representation of quadratic friction. J. Phys. Oceanogr. 37, 593-610. doi: 10.1175/JPO2999.1Lorentz, H.A. (1922). Het in rekening brengen van den weerstand bij schommelende vloeistofbewegingen.De Ingenieur 37, p.695.InDutch.Reef, K.R.G., P.C. Roos, G. Lipari & S.J.M.H. Hulscher (2016).In the footsteps of Lorentz: extending the network model of theWadden Sea, PECS2016 conference, 9-14 October, 2016, Scheveningen, Netherlands.Staatscommissie Zuiderzee (1926).Verslag van de Staatscommissie Zuiderzee 1918-1926. Report. Algemene Landsdrukkerij. ’s-Gravenhage.In Dutch.Terra, G.M., W.J. van den Berg & L.R.M. Maas (2005). Experimental verification of Lorentz’ linearization procedure for quadraticfriction. Fluid Dyn. Res., 36 (3), 175–188. doi: 10.1016/j.fluiddyn.2005.01.005

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Time-dependent linearisation of bottom friction for storm surge modelling in the Wadden Sea

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