Choosing and implementation of the new advection scheme in the wave - - PowerPoint PPT Presentation

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Choosing and implementation

• f 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

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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)

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∂ S ∂t ∂ ˙ S ∂ ∂ ˙  S ∂  ∂ ˙ S ∂ ∂ ˙  S ∂ =F Evolution equation for the wave action density on the sphere where l - longitude, f - latitude, b - wave direction, S(t,l,f,b,w) - wave action density. ˙ =cg cos R cos , ˙ =cg sin  R , ˙ =−c g tancos R , ˙ =−∂ ∂t Here cg=g/(4pw) – group velocity, g – free-fall acceleration

Problem formulation

=k U wave dispersion relation, s – intrinsic frequency

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F=F disF nlF gen

Source terms describing wave physics

Where Fdis - dissipation due to bottom friction and viscosity Fnl - source term due to non-linear interactions Fgen – source term due to wave input

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Flux form of the advection scheme

∂q ∂t  ∂u q ∂ x =0 Consider 1D advection problem Flux form

• f the advection scheme:

f,u f,u q

qi

n1=qi n−c f i1 2 n1 2− f i−1 2 n 1 2 ,c≡ut

 x u=const>0:

u

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Candidate schemes

(III) third-order upwind (QUICK, Leonard 1979) f

i1 2 n1 2=1

2 qi

nqi1 n − c

2 qi1

n −qi n−1−c2

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qi1

n −2qi nqi−1 n 

(II) - modified van Leer (Walcek, 1998)

• for every point, except ones joint to the extrema

f

i1 2 n1 2=qi n 1

4 qi1

n −qi−1 n 1−c

f

i1 2 n1 2=qi n

(I) - first order upwind =1

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(V) CIP-CSLR0 (Xiao et al., 2002) constrained interpolation -

conservative semi-lagrangian rational upwind interpolation by quadratic rational polynomial function

To the schemes (II-IV) the flux correction procedure was implemented to impose quasi-monotonicity (TVD-condition, Leonard, 1979). (IV) – second-order «cabaret», (Goloviznin, Samarsky, 1998) f i1/2

1 2

=1

2 qi

0qi1 0  ,

f

i1 2 n1 2=2qi n− f i−1 2 n−1 2

f

i1 2 n1 2=

ai cbic

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1ic ai= f

i1 2 n−1 2 ,bi= f i1 2 n−1 2−1iqi n ,i=1−∣f i1 2 n−1 2−qi n∣

∣f

i−1 2 n−1 2−qi n∣

Candidate schemes (2)

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Spectral properties of candidate-schemes

q x ,t=eikx−t qi

n1=c ,qi n

For the sample function the exact solution of the advection equation with flux estimate via (I-IV) is where y = kDx, k - dimensionless wavenumber, r - complex transition factor, c = wDt/kDx ∣∣ - damping factor spectral characteristics of advection scheme = cnum c =− 1 c arctan I m  R e   - phase factor

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Spectral properties of candidate-schemes

∣∣

g (II) (III) (I) (IV)

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Results of 1D-tests of the advection schemes

initial condition - gaussian bell

solid line – (IV), dashed – (I), dash-dotted – (V), circles – (II), pluses - (III)

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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)

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Results of 4D-experiments with WAM-4 model

Initial conditions are typical for wind swell S0(l,f,b,w)=E0(w)G0(b)R0(l,f) where E0(w) - JOHSWAP spectra, R0(l,f) – gaussian function, G0(b) = H(cos(b-b0))cos2(b-b0) S  , ,  , ,t=S 0− ˙ t ,− ˙ t ,  , ∂ S ∂t ∂ ˙ S ∂ ∂ ˙  S ∂  ∂ ˙ S ∂ =0 Evolution equation in the plain geometry

h , ,t =2∬ S  , ,  , ,t d d 

1 2

analyzed quantity - significant wave height exact solution

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Results of 4D-experiments with WAM-4 model

significant wave height,  =15

°

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Results of 4D-experiments with WAM-4 model

significant wave height spacial distribution,  =7.5

°

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Results of 4D-experiments with WAM-4 model

significant wave height RMS error  =7.5

°

 =15

°

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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