The future of the DIA

The future of the DIA

Gerbrant van Vledder Delft University of Technology

Contents

• Purpose

• Non-linear four-wave interactions

• History of the DIA

• Shortcomings

• Improvements

• Inter-comparison of methods

Purpose

• Discuss the future of the Discrete Interaction Approximation

(DIA) for computing non-linear four-wave interactions S_nl4 in

discrete spectral models

• Provide an overview of developments and requirements in

the development of efficient algorithms for S_nl4 in

operational wave models

• Discuss the concept ‘efficient algorithm’ in relation to

computational costs, wave model performance and types of application

Spectral wave modelling

Action balance equation,

N

=

N(



,



,x,y,t)



 





 



 

, ,

4 3

g x g y

wind wcap nl brk fric nl

N

c N

c N

c N

c N

S

t

x

y

S

S

S

S

S

S

S









































Importance of S

_nl4

• Phillips (1960) showed basic principle of non-linear four-wave

interaction

• Theory extended to random surface gravity waves by

Hasselmann (1962) and Zakharov (1968)

• JONSWAP experiment (1973) concluded that S_nl4 is mainly

responsible for forward shift of peak frequency

• Shape stabilization and influence on spectral shape, both in

frequency and direction space (Young and Van Vledder, 1993)

Basic equation of S

_nl4

• Rate of change of action density in wavenumber k₁ function of four

wave numbers involved in a resonant interaction

• Six-fold integral over the wave numbers k_2, k₃ and k₄

• Delta functions reflect resonance conditions and ensure

conservation of wave energy, action and momentum

• G complicated coupling coefficient



 

 











,

,

,

n

G

t

n n n

n

n n n

n

d d d



    









 















_









_



k k k k

k

k

k

k

k k k

Discrete Interaction

Approximation (DIA)

• Discrete Interaction Approximation (DIA) evaluates one subset of

one subset of possible interactions configurations

•

• DIA quite successful in development of 3G wave models WAM,

Wavewatch, SWAN, …

















 



 

Basic equation of DIA



 







2

1

1

2

1

1

₁

S

S

C

g

f

S

E

E

EE E

E











_





 





 

_



_



 





 





 





_

_





_

_

_

_









_

_

_

_

_







Evaluation in a discrete spectrum

E

E

E

E

E

Central density E loops

over all spectral bins

E

_{and E}

_{obtained by}

bi-linear interpolation

S

_{and S}

similarly

distributed over

surrounding bins

DIA is just bi-linear

interpolation !!

Variations in the DIA

0.25





0.05





0.35





Medium range

Short range

Long range

Comparison of DIA with exact

solution (Xnl)

Deficiencies of DIA

• too much transfer towards higher frequencies

• too broad spectra, both in direction and in frequencies

Figure by Erick Rogers using 1-point model

after 12 hours simulation

with U₁₀=12 m/s

• deficiencies of DIA usually compensated by tuning of

DIA and spectral resolution

• Increasing spectral resolution does NOT improve accuracy of

DIA, as DIA is tuned to a certain spectral resolution

• Optimal resolution f/f=10%, =10

• Experiment of fetch-limited wave growth with f=4%

actually led to double peaked spectra

• Only solution is to add, additional configuration covering

Degenerate results of DIA

Van Vledder, et al. ICCE 2000

A simple case going wrong

• Luigi playing with SWAN; reproducing wave flume experiment

• Dimensions of grid: xlen=70 m, ylen 50 m, depth 3 m

• Uni-model wave propagation in a wave flume, D()=cos840_()

Wave direction upwards, in y-direction

• Wave boundary conditions: H_s=0.068 m, T_p=1 s,

JONSWAP spectrum with =6

Diagnosis

• DIA spreads energy because its single wave number configuration is calibrated against required transfer

• The equivalent X_nl wave number configuration is much weaker

as it is one of thousands others

• Growth of energy due to non-conservative behavior of DIA for extremely narrow directional spectra

• Also occurs for mDIA, will it also occur in GMD?

• Will it affect narrow directional spectra, implications for swell propagation?

How to improve on the DIA

• Fundamental difference between Xnl and DIA

is its number of wave number configurations

• The way to go is to increase the number of

configurations to include more scales of

interaction

• Either by developing a multiple DIA or by

reducing workload Xnl

Accurate Incorrect

Time consuming Fast

Exact methods

Discrete Interactions

Full Reduced Extended Classic

X

GMD, mDIA DIA

Considerations how to proceed

• Challenge is find balance between accuracy

and computational requirements

• What is an optimal method for computing

S

?

• Good representation of S

for given spectra,

or, good model performance in which S

cooperates with other source terms?

• Exact methods are used as benchmark, but

which ?

Exact methods for Snl4

•

• Rewrite transfer integral to eliminate delta-functions and to make

transfer integral computationally feasible

• At least three basic analytical transformations exist in literature:

Webb (1978) - Masuda (1980) - Lavrenov (2001)

• Methods differ in various ways:

• choice of integration variables, i.e. Webb uses k₁ and k₃,

Masuda uses k₁ and k₂

• treatment of singularities

• internal transformations and approximations



 

 













• WRT – RIAM – GQM: Like their analytical masters these

computational methods differ in various (hidden, unknown) ways

• Do they provide the same answers? Which one to trust?

• They are used as reference in development of approximate

methods

• To resolve these issues an inter-comparison study for S_nl4 is

now being carried out (Van Vledder, Benoit, Hashimoto, Resio, Tolman, …) in the spirit of the SWAMP study (1983)

Inter-comparison study for Snl4

• Comparison against individual spectra

• Sensitivity to spectral resolution, spectral shape, directional

properties, symmetries, depth, ….

• Reveal internal hidden features like quadrature methods,

integration ranges, assumptions (e.g. smooth spectra)

• Perform dynamic wave model runs in combination with other

source terms to find out about stability and overall model performance (Tolman developed such a set for his GMD)

• Methodology can also be applied to approximate methods like

Extension of the DIA

• Adding additional -configurations

• Van Vledder et al. (2000), 2 configurations

• Hashimoto & Kawaguchi (2001), up to 5 configurations

• The original Discrete Interaction Approximation (DIA) of Hasselmann

had two configurations: ₁=0.25 and ₂=0.15 with weights of 3000

and 375

• The second configuration was dropped because it’s added value in

terms of wave model performance was insufficient with respect to model efficiency !

Generalized Multiple DIA

• Generalized DIA with arbitrary configuration proposed by Van Vledder (2001); cast in symmetric form by

Tolman (2003)

• MDIA in principle able to represent

full transfer using multiple configurations

• Final GMD (Tolman, 2011): _{ }   1 2 2 1 1 2 3 4 1 1 1                            k k         1 2 2 1 1 2 3 4 1 1 1 1                            k k      _{   }  _{   }  _{   }      _{  }             _{ }        _{     }              _{ } nl,1 nl,2

deep deep shal shal nl,3 d s nl,4 g1 1 g2 2 g3 3 g4 4 1 1 2 2 3 3 4 4 g1 1 g2 2 δS -1 δS -1 _{1 1} = C B + C B × δS 1 n n δS 1 c E c E c E c E + -σ k σ k σ k σ k × c E c E + σ k σ k                     _{ } c E_{σ k}g3 3 _c E_{σ k}g4 4  

Results of optimized GMD

• Results obtained by Tolman (2012) are good. It is a major

improvement over the classic DIA

• Tolman (2012) used X_nl based on WRT method as ground truth

• Optimal GMD configuration(s) depend on choice of other source terms,

characteristics of host model, spectral resolution, spatial discretization and set of model runs

Conclusions

• More and more shortcomings of DIA are being discovered

• Efficient and accurate algorithms are being developed, notably the GMD

• The concept ‘efficient algorithm’ for S_nl4 must be viewed in relation to

model performance and computational requirements, … not only against

its ability to approximate X_nl

• Efficiency should also be considered in relation to types of model

application, host model including settings, and choice of other source terms

• Inter-comparison framework needed for objective judgment of exact as

The future of the DIA (epilog)

• The DIA has served us well

• Accuracy of Snl4 (DIA) is lagging behind other source terms

• Increased computational power makes other methods more attractive

• More attention to shape of spectrum