Choosing and implementation of the new advection scheme in the wave model WAM-4 Kostrykin S.V. Institute of Numerical Mathematics RAS 119333,ul. Gubkina, 8 Moscow, Russia Email:kostr@inm.ras.ru CITES-2007, Tomsk, Russia, 14-25 July 2007 1
The WAM model: Fourth generation wave model developed by the WAM group. The wave spectrum is computed by integration of the energy balance equation, without any prior restriction on the spectral shape. Parameterization of the exact non-linear interactions, Hasselmann (1985) Implicit integration of the source term First order upwind scheme for propagation The WAM model runs for any given regional or global grid with a prescribed topographic dataset. It runs for deep and shallow water, and depth and current refraction can be included. The grid resolution can be arbitrary in space and time (restricted by stability criteria) 2
Problem formulation Evolution equation for the wave action density on the sphere ∂ t ∂ ˙ ∂ ∂ ˙ ∂ ∂ ˙ ∂ S S S ∂ ∂ ˙ S S ∂ = F where l - longitude, f - latitude, b - wave direction, S ( t , l,f,b,w ) - wave action density. cos sin ˙ R cos , ˙ = c g = c g , R tan cos =−∂ ˙ =− c g , ˙ R ∂ t Here c g =g/(4 pw ) – group velocity, g – free-fall acceleration = k U wave dispersion relation, s – intrinsic frequency 3
Source terms describing wave physics F = F dis F nl F gen Where F dis - dissipation due to bottom friction and viscosity F nl - source term due to non-linear interactions F gen – source term due to wave input 4
Flux form of the advection scheme ∂ q ∂ t ∂ u q ∂ x = 0 Consider 1D advection problem u u=const>0 : f,u q f,u n 1 n 1 Flux form 2 ,c ≡ u t n 1 = q i n − c f 2 − f q i of the advection scheme: i 1 i − 1 x 2 2 5
Candidate schemes (I) - first order upwind n 1 n 2 = q i f i 1 2 (II) - modified van Leer ( Walcek, 1998 ) n 1 n 1 n − q i − 1 n 1 − c 2 = q i f 4 q i 1 i 1 2 = 1 - for every point, except ones joint to the extrema (III) third-order upwind (QUICK, Leonard 1979 ) n 1 n − 1 − c 2 2 = 1 n − c n − q i n − 2 q i n n q i 1 n q i − 1 f 2 q i 2 q i 1 q i 1 i 1 6 2 6
Candidate schemes (2) (IV) – second-order «cabaret», ( Goloviznin, Samarsky, 1998 ) 1 = 1 0 , 0 q i 1 2 f i 1 / 2 2 q i n 1 n − 1 n − f 2 = 2 q i 2 f i 1 i − 1 2 2 upwind interpolation (V) CIP-CSLR0 ( Xiao et al., 2002 ) by quadratic rational constrained interpolation - polynomial function conservative semi-lagrangian rational 2 n 1 a i c b i c 2 = f i 1 1 i c 2 n , i = 1 − ∣ f n ∣ n − 1 2 − q i i 1 n − 1 n − 1 2 ,b i = f 2 2 − 1 i q i a i = f ∣ f n ∣ i 1 i 1 n − 1 2 2 2 − q i i − 1 2 To the schemes (II-IV) the flux correction procedure was implemented to impose quasi-monotonicity (TVD-condition, Leonard, 1979 ). 7
Spectral properties of candidate-schemes q x ,t = e i kx − t For the sample function the exact solution of the advection equation with flux estimate via (I-IV) is n 1 = c , q i n q i where y = k D x , k - dimensionless wavenumber, r - complex transition factor, c = wD t/k D x spectral characteristics of advection scheme ∣ ∣ - damping factor = c num c arctan I m c =− 1 R e - phase factor 8
Spectral properties of candidate-schemes ∣ ∣ g (II) (III) (I) (IV) 9
Results of 1D-tests of the advection schemes initial condition - gaussian bell solid line – (IV), dashed – (I), dash-dotted – (V), circles – (II), pluses - (III) 10
Results of 1D-tests of the advection schemes initial condition – two-sided step function solid line – (IV), dashed – (I), dash-dotted – (V), circles – (II), pluses - (III) 11
Results of 4D-experiments with WAM-4 model Initial conditions are typical for wind swell S 0 ( l,f,b,w ) = E 0 ( w ) G 0 ( b ) R 0 ( l,f ) where E 0 ( w ) - JOHSWAP spectra, R 0 ( l,f ) – gaussian function, G 0 ( b ) = H ( cos ( b-b 0 )) cos 2 ( b-b 0 ) Evolution equation in the plain geometry ∂ t ∂ ˙ ∂ ∂ ˙ ∂ ∂ ˙ ∂ S S S S ∂ = 0 exact solution S , , , ,t = S 0 − ˙ t , − ˙ t , , analyzed quantity - significant wave height 1 h , ,t = 2 ∬ S , , , ,t d d 2 12
Results of 4D-experiments with WAM-4 model ° significant wave height, = 15 13
Results of 4D-experiments with WAM-4 model ° significant wave height spacial distribution, = 7.5 14
Results of 4D-experiments with WAM-4 model significant wave height RMS error ° ° = 15 = 7.5 15
Conclusions 1) from 1D experiments best ratio of accuracy to computational effectiveness show cabaret and CIP schemes 2) from 4D experiments with «cabaret» scheme implemented in WAM-4 model it is shown a large improvement of the SWH forecast on the short time range - 1-2 days and smaller improvement on the 3-d day 3) the benefit in accuracy of the new scheme is much better for the small steps in wave direction variable 4) this effect probably may be overcome by introducing additional diffusion term in the governing equation 16
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