A Numerical and Experimental Study on Infragravity Waves

PRESENTATION(OUTLINE( ( I.(MoLvaLon( ( II.(Nonlinear(Wave(PropagaLon(in(the(Nearshore( ( III.(A(Numerical(and(Experimental(Study(on(Infragravity(Waves( ( IV.(Chile(as(an(InternaLonal(Natural(Coastal(Laboratory( III.(INFRAGRAVITY(WAVES( Different methods to cope & understand water waves Source: P. Bonneton, 2010 III.(INFRAGRAVITY(WAVES( NonQdimensional(depthQaveraged(mass(and(momentum(equaLons( ht + ε [ hu ] x = 0 [( ε [ hu ] t + h ε u 2 + p )] x + ξx p ξ = 0 Nonlinearity( ε = a / h0 Dispersion( σ = h0 / L Where(the(pressure(field(is(wri`en(as(:( εη p(x,z,t) = (εη − z) + εσ 2 ∫ Γ( x,z,t ) dz z and(the(funcLon(Γ(represents(the(verLcal(acceleraLon(of(fluid(parLcles(:( Dw 1 & # Γ ( x, z , t ) = = wt + ε $ uwx + 2 wwz ! Dt σ % " III.(INFRAGRAVITY(WAVES( Using(approximate(expressions(for(the(velocity(field((potenLal(flow(theory)(we(get( the(fully(nonlinear(weakly(dispersive(Serre/GreenQNaghdi(EquaLons((Cienfuegos(et( al.,(2006;(Lannes(and(Bonneton,(2009)( ht + ε [hu ] x = 0 & 2, 1 1 )# (hu ) t + ε hu x + h(h + ξ )x + σ %h * P(x, t ) + Q(x, t )'" + 2 (! x $ +3 ( ) 2 2 (1 % + σ ξ x h& P(x, t ) + Q(x, t )# = O σ 4 $ '2 2 Where(:( ( P(x, t ) = −h u xt + εuu xx − εu x2 ) Q(x, t ) = ξ x (ut + εuu x ) + εξ xx u 2 ( ) Dropping(overbars(for(simplicity( III.(INFRAGRAVITY(WAVES( A(4th(Order(FV(S/GQN(Model((Cienfuegos(et(al.,(2006,(2007)( •(A(strongly(nonlinear(and(weakly(dispersive(Boussinesq(system(O(σ4)! #ht + Fx = 0 " !qt + Gx = 0 With(the(following(definiLons(for(flux(funcLons(and(auxiliary(variables( F = hu 1 2 1 2 2 G = uq + g (h + ξ ) − u 1 + ξ x + hu xξ x u − (hu x ) 2 2 1 3 q = (1 + r )u − h ux x 3h 1 r = (hx + ξ x )ξ x + hξ xx 2 ( ( ) ) III.(INFRAGRAVITY(WAVES( IncorporaLng(Breaking(Energy(DissipaLon((Mignot(and(Cienfuegos,( 2009;(Cienfuegos(et(al.,(2010)( •(DiffusiveQlike(terms(in(both,(mass(and(momentum(conservaLon(equaLons( Dh = (ν h hx ) x [ Dhu = ν hu ( hu) x ] x Diffusivity function $ht + Fx + Dh = 0 ! # 1 !"qt + Gx + h Dhu = 0 Local redistribution of mass and momentum below breakers Similar to Kennedy et al., 2000 III.(INFRAGRAVITY(WAVES( nonlinear processes L.H.(Holthuijsen,(2007( •(Typical(windQgenerated(waves(:(0.05Hz((20s)(–(1Hz((1s)( •(Nonlinear(processes((shoaling,(breaking,(swash)(generate( infragravity((IG)(waves(:(0.001Hz((1000s)(–(0.05Hz((20s)( III.(INFRAGRAVITY(WAVES( Experimental(InvesLgaLon(at(the(InsLtuto(Nacional(de(Hidráulica( • 70m long x 1.5m wide x 1.5m deep wave tank • Piston wave paddle with active absorption (DHI) • Resistive probes for free surface measurements • Sontek MicroADV for velocity measurements • Very mild slope beach 1/80 III.(INFRAGRAVITY(WAVES( Random(Incident(Wave(Field( • Jonswap spectrum (Hasselmann, 1973) for incident wave field Tp=4 s (0.25Hz) ; Hmo=18 cm ; h0=52cm • Without phase structure (uniform random distribution [0,2π]) • Highly nonlinear shallow water regime ⇒(Although(the(forcing(wave(field(does( not(contain(energy(in(the(infragravity( band,(experiments(show(important( energy(content(at(these(frequencies(( III.(INFRAGRAVITY(WAVES( SeparaLon(of(Incident(and(Reflected(Waves( • Radon transform (Deans, 1983) in time domain to separate incident and reflected signals (Almar et al., 2013) Total( Free(surface( Lme(stacks( Free(surface( at(x=20m( Amplitude( evoluLon( Incident( Reflected( III.(INFRAGRAVITY(WAVES( Model(Performance(on(High(Frequency(Waves( Total/ 1st/harmonic/ Wave(height( and(energy( distribuLon( CrestQtrough(wave( asymmetry((Sk)( ( LetQright(wave( symmetry((As)( 2nd/harmonic/ III.(INFRAGRAVITY(WAVES( Spectral(Energy(EvoluLon(Along(the(Experimental(Flume( x =-9[m] x=-9m E(f) [m2/Hz] E(f) [m2/Hz] x =-22[m] x=-22m -5 10 -2 -5 10 -1 10 -2 10 E(f) [m2/Hz] x =0[m] x=0m 10 x =9[m] x=9m -5 10 -5 10 -2 -1 10 0 10 -2 10 -1 10 0 10 x =20[m] x=20m E(f) [m2/Hz] -1 10 10 x =32[m] x=32m -5 -5 10 10 -2 -1 10 10 f(Hz) -2 -1 10 10 f(Hz) Partial reflection at the wave paddle produces low frequency selection in the flume Natural modes f1=0.013Hz f2=0.023Hz f3=0.031Hz III.(INFRAGRAVITY(WAVES( Model(Performance(in(Spectral(Energy(EvoluLon( x =-9[m] E(f) [m2/Hz] E(f) [m2/Hz] x =-22[m] -5 10 -2 -5 10 -1 10 -2 10 E(f) [m2/Hz] x =0[m] 10 Blue: Experiments Red : Model x =9[m] -5 10 -5 10 -2 -1 10 0 10 -2 10 -1 10 0 10 x =20[m] E(f) [m2/Hz] -1 10 10 x =32[m] -5 -5 10 10 -2 -1 10 10 f(Hz) -2 -1 10 10 f(Hz) ⇒(The(model(succeeds(in(transferring(energy(to(higher(harmonics(and(IG( waves(but(no(clear(modal(selecLon((open(boundary(condiLon)( III.(INFRAGRAVITY(WAVES( CrossQcorrelaLon(maps(between(group(amplitude(and(IG(wave( amplitude(at(each(locaLon( (Experimental(( (Numerical(( III.(INFRAGRAVITY(WAVES( CrossQcorrelaLon(maps(between(group(amplitude(at(the(first(gauge( and(IG(wave(amplitude(at(each(locaLon( (Experimental(( (Numerical(( III.(INFRAGRAVITY(WAVES( • A bound IG appears rapidly in the shoaling zone forced by short wave groups (negative correlation slightly lagging the group) • A positive long wave surge preceding the group develops as the short waves propagate towards the shore (positive correlation running ahead of the group) • There is a phase shift for the positively correlated IG wave in the inner surf zone • Both IG waves appear to be partially reflected on run-up (some dissipation by breaking might occur) − þ η ðx; 0Þ + η ðx; 0Þ + η forced ðx; 0Þ = 0 ð3Þ and the long wave, zero initial velocity condition + ηþ ðx; 0Þ pffiffiffiffiffi η− ðx; 0Þ pffiffiffiffiffi η ðx; 0Þ gh $ gh + ∞ cg = 0 h h h ð4Þ III.(INFRAGRAVITY(WAVES( This defines the two free wave amplitudes þ $ ðA free ; A free Þ = $ " # " #! cg cg A forced A 1 + pffiffiffiffiffi ; $ forced 1− pffiffiffiffiffi 2 2 gh gh ð5Þ The non-resonant growth of the surface shape (η = η forced + η+ + away from the forcing at η−) then results from pffiffimoving ffiffiffi pffiffiffiffiffi the free waves relative speeds gh $ cg and $ gh $ cg respectively. The water surface shapes occurring during this growth process enable energy transfer from the forcing to the growing long wave in the form $ ∂S∂xxx u e $ ∂S∂xxx η. + The nature of the combination η forced(x,t) + η free(x,t) of the forced wave and the forward moving free wave, is illustrated in Fig. 2 for a Gaussian-shaped forcing. The initial appearance is that of an И-wave which stretches as it grows. Eventually it becomes two separate Gaussian surges of “opposite sign”. Fig. 2. ηforced(x,t) + η free(x,t), thin lines, at different stages after the switch-on of the + forcing, – × –. Eventually η free(x,t) disappears out of the picture to the right and the steady state represented by Eq. (2) alone is reached. very large but almost each other) merge into a solution of pffiffiffifficancelling ffi ∂ f ðx $ ghtÞ, eg, a Gaussian-shaped forcing generates an the form t ∂x И-shaped long wave, which grows with time (until the linearity assumptions are violated). See Fig. 3. i pffiffiffiffiffih Using the linear wave theory expression: cg = gh 1 $ 12 ko h + Oðko hÞ2 , and initially keeping So fixed for simplicity, the details are as follows " # ! pffiffiffiffiffi cg 1 1 + pffiffiffiffiffi f ðx $ ghtÞ 2 gh $ $ #% pffiffiffiffiffi" pffiffiffiffiffi % $So = ρgh ko h → f x $ gh 1 $ t $ f ðx $ ghtÞ ko h→o ko h 2 ð6Þ $pffiffiffiffiffi p ffiffiffiffiffi % $So = ρgh k h ′ → gh o tf ðx $ ghtÞ ko h→o ko h 2 p ffi ffi ffi ffi ffi $So ′ pffiffiffiffiffi t f ðx $ ghtÞ = 2ρ gh $So = ρgh 1 $ c2g = gh • The shallow water regime (Cg~C) produces a nearly resonant condition for IG wave forcing (Nielsen and Baldock, 2010) ηðx; tÞ = f ðx $ cg tÞ $ ⇒(The(soQcalled(NQshaped(long(wave(is(forced(as(a(bound(wave(in(the(shoaling( zone( 2.2. The resonant case The forced wave Expression (2) becomes singular in the shallow pffiffiffiffiffi water limit where cg = gh. The linear problem then has no steady asymptotic solution. Instead, η+ free(x,t) and η forced(x,t), (both becoming Non-resonant condition Fig. 1. Steady, bowl-shaped long wave (solid line) driven by a steady, non-resonant wave group (thin line) with Sxx varying as indicated by – × –. This result illustrates the similarity with the familiar resonant scenario of the forced oscillation of a mass on a spring, see eg, Kreyszig (2006), p 87. The fact that the strength So of the forcing may initially grow due to shoaling and subsequently decrease due topwave ffiffiffiffiffi breaking, does not alter the qualitative result ηðx; tÞ e f ′ ðx $ ghtÞ, Resonant condition pffiffiffiffiffi Fig. 3. At resonance, cg = gh, which happens in the shallow water limit, the Gaussian shaped Sxx forcing – × – from the wave pulse generates an И-shaped long wave which grows linearly with time. III.(INFRAGRAVITY(WAVES( • The phase shift in the positively correlated IG wave in the inner surf zone is due to a strong interaction between short waves and IG motions in very shallow waters (merging of short frequency waves) C02011 Free surface time series at different locations along the numercal wave flume VAN DONGEREN ET AL.: SHOALING AND SHORELINE DISSIPATION OF WAVES C02011 Figure 4. Time series of the filtered (low-frequency component and its higher harmonics) long wave Merging waves bi(thick solid line), andof total high measured timefrequency series (thin solid line) for consecutive gauge on positionsa around the still water line for case A-1. Tracks of converging wave crests are shown as dash-dotted lines. chromatic experiment (Van Dongeren et energy transfer to higher frequencies in the inneral., surf zone2007) be difficult to interpret because of the sparseness of the very near to the shoreline. We will discuss this mechanism measurement array. The above shows that while reflection in the next section. Their finding of full reflection from the shoreline seems inconsistent with the present findings. However, we define our reflection coefficient in about 3 centimeters of water depth, which would represent about 30 centimeters of depth or 15 meters from the shoreline, coefficients less than unity have not in fact been measured, they may occur in the field. 7. Shoreline Dissipation Mechanism [31] Figure 3 shows that the reflection coefficient at the III.(INFRAGRAVITY(WAVES( Model(Performance(for(Incident(and(Reflected(IG(Waves( Long(Wave( Time(Stacks( Long(Wave( Amplitude( Total( ⇒(Under(esLmaLon(of( incident(IG(wave( Incident( ⇒(Good(representaLon( of(incident(IG(wave( generaLon(and(shoaling( Reflected( ⇒(Under(esLmaLon(of( reflected(IG(wave( Black : Experiments Red : Model Dashed line : theoretical free long wave amplitude shoaling III.(INFRAGRAVITY(WAVES( Summary( • Maximum run-ups under highly nonlinear waves are strongly dependent on infragravity wave motions (strong under estimation of maximum run-ups using linear wave theories) • Fully nonlinear Boussinesq-type equations including breaking and run-up are able to cope with the complex energy transfer processes within the nearshore producing accurate estimations of short wave propagation and IG generation and dynamics • More research is needed to (fully) understand nearshore wave processes and its (un)expected consequences on morphodynamics and coastal hazards