1. Slide 1 The Wave Model ECMWF, Reading, UK

2. Slide 2The Wave Model (ECWAM) Resources: Lecture notes available at: http://www.ecmwf.int/newsevents/training/lecture_notes/ pdf_files/NUMERIC/Wave_mod.pdf Recommended literature can be found at: http://www.ecmwf.int/newsevents/training/2012/nwp/timetable_resources_nwp.html Model documentation available at: http://www.ecmwf.int/research/ifsdocs/CY36r1/index.html

3. Slide 3The Wave Model (ECWAM) Directly Related Books: Dynamics and Modelling of Ocean Waves. by: G.J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, P.A.E.M. Janssen. Cambridge University Press, 1996. The Interaction of Ocean Waves and Wind. By: Peter Janssen Cambridge University Press, 2004.

4. Slide 4 INTRODUCTION

5. Slide 5The Wave Model (ECWAM) Introduction: State of the art in wave modelling. Energy balance equation from first principles. Wave forecasting. Validation with satellite and buoy data. Benefits for atmospheric modelling.

6. Slide 6The Wave Model (ECWAM) What we are dealing with 1.010.00.033x10 -3 2x10 -5 1x10 -5 Frequency (Hz) Forcing: wind earthquake moon/sun Restoring: gravity surface tensionCoriolis force

7. Slide 7The Wave Model (ECWAM) What we are dealing with Wave Period, T Wave Length, Wave Height, H Water surface elevation, Wave number: k = 2 π / λ Wave (angular) frequency: ω = 2 π / T

8. Slide 8The Wave Model (ECWAM) What we are dealing with

9. Slide 9The Wave Model (ECWAM) What we are dealing with

10. Slide 10The Wave Model (ECWAM) What we are dealing with

11. Slide 11The Wave Model (ECWAM) What we are dealing with

12. Slide 12The Wave Model (ECWAM) What we are dealing with?

13. Slide 13The Wave Model (ECWAM) What we are dealing with

14. Slide 14The Wave Model (ECWAM) What we are dealing with

15. Slide 15The Wave Model (ECWAM) What we are dealing with

16. Slide 16The Wave Model (ECWAM) What we are dealing with

17. Slide 17The Wave Model (ECWAM) What we are dealing with

18. Slide 18The Wave Model (ECWAM) What we are dealing with

19. Slide 19The Wave Model (ECWAM) What we are dealing with

20. Slide 20The Wave Model (ECWAM) What we are (not) dealing with (this week)

21. Slide 21The Wave Model (ECWAM) What we are dealing with Fetch, X, or Duration, t Simplified sketch! Energy, E, of a single component

22. Slide 22 PROGRAM OF THE LECTURES

23. Slide 23The Wave Model (ECWAM) Program of the lectures: 1. Derivation of energy balance equation 1.1. Preliminaries - Basic Equations - Dispersion relation in deep & shallow water. - Group velocity. - Energy density. - Hamiltonian & Lagrangian for potential flow. - Average Lagrangian. - Wave groups and their evolution.

24. Slide 24The Wave Model (ECWAM) Program of the lectures: 1. Derivation of energy balance equation 1.2. Energy balance Eq. - Adiabatic Part - Need of a statistical description of waves: the wave spectrum. - Energy balance equation is obtained from averaged Lagrangian. - Advection and refraction.

25. Slide 25The Wave Model (ECWAM) Program of the lectures: 1. Derivation of energy balance equation 1.3. Energy balance Eq. - Physics Diabetic rate of change of the spectrum determined by: - energy transfer from wind (S in ) - non-linear wave-wave interactions (S nonlin ) - dissipation by white capping (S dis ).

26. Slide 26The Wave Model (ECWAM) Program of the lectures: 2. The WAM Model WAM model solves energy balance Eq. 2.1. Energy balance for wind sea - Wind sea and swell. - Empirical growth curves. - Energy balance for wind sea. - Evolution of wave spectrum. - Comparison with observations (JONSWAP).

27. Slide 27The Wave Model (ECWAM) Program of the lectures: 2. The WAM Model 2.2. Wave forecasting - Quality of wind field (SWADE). - Validation of wind and wave analysis using ERS-1/2, ENVISAT and Jason-1/2 altimeter data and buoy data. - Quality of wave forecast: forecast skill depends on sea state (wind sea or swell).

28. Slide 28The Wave Model (ECWAM) Program of the lectures: 3. Benefits for Atmospheric Modelling 3.1. Use as a diagnostic tool Over-activity of atmospheric forecast is studied by comparing monthly mean wave forecast with verifying analysis. 3.2. Coupled wind-wave modelling Energy transfer from atmosphere to ocean is sea state dependent. Coupled wind-wave modelling. Impact on depression and on atmospheric climate.

29. Slide 29The Wave Model (ECWAM) Program of the lecture: 4. Error Estimation 4.1. Introduction Definition of measurement errors. Types of errors. Accuracy and precision. Data sources. Error estimation when the truth is known. 4.2. Triple Collocation Technique Error model. Derivation. Assumptions. Practical considerations. 4.3. Examples Estimation of wind and wave errors of various measuring systems: buoys, altimeters and model.

30. Slide 30 1 DERIVATION OF THE ENERGY BALANCE EQUATION

31. Slide 31The Wave Model (ECWAM) 1. Derivation of the Energy Balance Eqn: Solve problem with perturbation methods: (i) a / w << 1 (ii) s << 1 Lowest order: free gravity waves. Higher order effects: wind input, nonlinear transfer & dissipation

32. Slide 32The Wave Model (ECWAM) Result: Application to wave forecasting is a problem: 1. Do not know the phase of waves Spectrum F(k) ~ a*(k) a(k) Statistical description 2. Direct Fourier Analysis gives too many scales: Wavelength: 1 - 250 m Ocean basin: 10 7 m 2D 10 14 equations Multiple scale approach short-scale, O( ), solved analytically long-scale related to physics. Result: Energy balance equation that describes large-scale evolution of the wave spectrum. Deterministic Evolution Equations

33. Slide 33 1.1 PRELIMINARIES

34. Slide 34The Wave Model (ECWAM) 1.1. Preliminaries: Interface between air and ocean: Incompressible two-layer fluid Navier-Stokes: Here and surface elevation follows from: Oscillations should vanish for: z ± and z = -D (bottom) No stresses; a 0 ; irrotational potential flow: (velocity potential ) Then, obeys potential equation.

35. Slide 35The Wave Model (ECWAM) Conditions at surface: Conditions at the bottom: Conservation of total energy: with Laplaces Equation

36. Slide 36The Wave Model (ECWAM) Hamilton Equations Choose as variables: and Boundary conditions then follow from Hamiltons equations: Homework: Show this! Advantage of this approach: Solve Laplace equation with boundary conditions: = (, ). Then evolution in time follows from Hamilton equations.

37. Slide 37The Wave Model (ECWAM) Lagrange Formulation Variational principle: with: gives Laplaces equation & boundary conditions.

38. Slide 38The Wave Model (ECWAM) Intermezzo Classical mechanics: Particle (p,q) in potential well V(q) Total energy: Regard p and q as canonical variables. Hamiltons equations are: Eliminate p Newtons law = Force V(q)V(q)

39. Slide 39The Wave Model (ECWAM) Principle of least action. Lagrangian: Newtons law Action is extreme, where Action is extreme if (action) = 0, where This is applicable for arbitrary q hence (Euler-Lagrange equation)

40. Slide 40The Wave Model (ECWAM) Define momentum, p, as: and regard now p and q are independent, the Hamiltonian, H, is given as: Differentiate H with respect to q gives The other Hamilton equation: All this is less straightforward to do for a continuum. Nevertheless, Miles obtained the Hamilton equations from the variational principle. Homework: Derive the governing equations for surface gravity waves from the variational principle.

41. Slide 41The Wave Model (ECWAM) Linear Theory Linearized equations become: Elementary sines where a is the wave amplitude, is the wave phase. Laplace:

42. Slide 42The Wave Model (ECWAM) Constant depth: z = -D Z'(-D) = 0 Z ~ cosh [ k (z+D)] with Satisfying Dispersion Relation Deep water Shallow water D D 0 dispersion relation: phase speed: group speed: Note: low freq. waves faster! No dispersion. energy:

43. Slide 43The Wave Model (ECWAM) Slight generalisation: Slowly varying depth and current,, = intrinsic frequency Wave Groups So far a single wave. However, waves come in groups. Long-wave groups may be described with geometrical optics approach: Amplitude and phase vary slowly: (t) t

44. Slide 44The Wave Model (ECWAM) Local wave number and phase (recall wave phase !) Consistency: conservation of number of wave crests Slow time evolution of amplitude is obtained by averaging the Lagrangian over rapid phase,. Average L: For water waves we get with

45. Slide 45The Wave Model (ECWAM) In other words, we have (omitting the brackets ) Evolution equations then follow from the average variational principle We obtain: a : (dispersion relation) : (evolution of amplitude) plus consistency: (conservation of crests) Finally, introduce a transport velocity then L is called the action density.

46. Slide 46The Wave Model (ECWAM) Apply our findings to gravity waves. Linear theory, write L as where Dispersion relation follows from hence, with, Equation for action density, N, becomes with Closed by

47. Slide 47The Wave Model (ECWAM) Consequencies Zero flux through boundaries hence, in case of slowly varying bottom and currents, the wave energy is not constant. Of course, the total energy of the system, including currents,... etc., is constant. However, when waves are considered in isolation (regarded as the system), energy is not conserved because of interaction with current (and bottom). The total action density is N tot is conserved. The action density is called an adiabatic invariant.

48. Slide 48The Wave Model (ECWAM) Homework: Adiabatic Invariants (study this) Consider once more the particle in potential well. Externally imposed change (t). We have Variational equation is Calculate average Lagrangian with fixed. If period is = 2 /, then For periodic motion ( = const.) we have conservation of energy thus also momentum is The average Lagrangian becomes

49. Slide 49The Wave Model (ECWAM) Allow now slow variations of which give consequent changes in E and. Average variational principle Define again Variation with respect to E & gives The first corresponds to the dispersion relation while the second corresponds to the action density equation. Thus which is just the classical result of an adiabatic invariant. As the system is modulated, and E vary individually but remains constant: Analogy L L waves. Example: Pendulum with varying length!

50. Slide 50 1.2 ENERGY BALANCE EQUATION: THE ADIABATIC PART

51. Slide 51The Wave Model (ECWAM) 1.2. Energy Balance Eqn (Adiabatic Part): Together with other perturbations where wave number and frequency of wave packet follow from Statistical description of waves: The wave spectrum Random phase Gaussian surface Correlation homogeneous with = ensemble average depends only on

52. Slide 52The Wave Model (ECWAM) By definition, wave number spectrum is Fourier transform of correlation R Connection with Fourier transform of surface elevation. Two modes is real Use this in homogeneous correlation and correlation becomes

53. Slide 53The Wave Model (ECWAM) The wave spectrum becomes Normalisation: for = 0 For linear, propagating waves: potential and kinetic energy are equal wave energy E is Note: Wave height is the distance between crest and trough : given wave height. Let us now derive the evolution equation for the action density, defined as

54. Slide 54The Wave Model (ECWAM) Starting point is the discrete case. One pitfall: continuum:, t and independent. discrete:, local wave number. Connection between discrete and continuous case: Define then Note: ! O k

55. Slide 55The Wave Model (ECWAM) Then, evaluate using the discrete action balance and the connection. The result: This is the action balance equation. Note that refraction term stems from time and space dependence of local wave number! Furthermore, and

56. Slide 56The Wave Model (ECWAM) Further discussion of adiabatic part of action balance: Slight generalisation: N(x 1, x 2, k 1, k 2, t) writing then the most fundamental form of the transport equation for N in the absence of source terms: where is the propagation velocity of wave groups in z-space. This equation holds for any rectangular coordinate system. For the special case that & are canonical the propagation equations (Hamiltons equations) read For this special case

57. Slide 57The Wave Model (ECWAM) hence; thus N is conserved along a path in z-space. Analogy between particles (H) and wave groups ( ): If is independent of t : (Hamilton equations used for the last equality above) Hence is conserved following a wave group!

58. Slide 58The Wave Model (ECWAM) Lets go back to the general case and apply to spherical geometry (non-canonical) Choosing ( : angular frequency, : direction, : latitude, :longitude) then For simplification. Normally, we consider action density in local Cartesian frame (x, y): R is the radius of the earth.

59. Slide 59The Wave Model (ECWAM) Result: With v g the group speed, we have

60. Slide 60The Wave Model (ECWAM) Properties: 1.Waves propagate along great circle. 2.Shoaling: Piling up of energy when waves, which slow down in shallower water, approach the coast. 3.Refraction: * The Hamilton equations define a ray light waves * Waves bend towards shallower water! * Sea mountains (shoals) act as lenses. 4.Current effects: Blocking v g vanishes for k = g / (4 U o 2 ) !

61. Slide 61 1.3 ENERGY BALANCE EQUATION: PHYSICS

62. Slide 62The Wave Model (ECWAM) 1.3. Energy Balance Eqn (Physics): Discuss wind input and nonlinear transfer in some detail. Dissipation is just given. Common feature: Resonant Interaction Wind: Critical layer: c(k) = U o (z c ). Resonant interaction between air at z c and wave Nonlinear: i = 0 3 and 4 wave interaction zczc U o =c

63. Slide 63The Wave Model (ECWAM) Transfer from Wind: Instability of plane parallel shear flow (2D) Perturb equilibrium: Displacement of streamlines W = U o - c (c = /k)

64. Slide 64The Wave Model (ECWAM) give Im(c) possible growth of the wave Simplify by taking no current and constant density in water and air. Result: Here, = a / w ~ 10 -3 << 1, hence for 0, ! Perturbation expansion growth rate = Im(k c 1 ) c 2 = g/k

65. Slide 65The Wave Model (ECWAM) Further simplification gives for = w/w(0): Growth rate: Wronskian:

66. Slide 66The Wave Model (ECWAM) 1.Wronskian W is related to wave-induced stress Indeed, with and the normal mode formulation for u 1, w 1 : (e.g. ) 2.Wronskian is a simple function, namely constant except at critical height z c To see this, calculate dW/dz using Rayleigh equation with proper treatment of the singularity at z=z c where subscript c refers to evaluation at critical height z c (W o = 0) zczc z w

67. Slide 67The Wave Model (ECWAM) This finally gives for the growth rate (by integrating dW/dz to get W (z=0) ) Miles (1957): waves grow for which the curvature of wind profile at z c is negative (e.g. log profile). Consequence:waves grow slowing down wind: Force = d w / dz ~ (z) (step function) For a single wave, this is singular! Nonlinear theory.

68. Slide 68The Wave Model (ECWAM) Linear stability calculation Choose a logarithmic wind profile (neutral stability) = 0.41 (von Karman), u * = friction velocity, = u * 2 Roughness length, z o : Charnock (1955) z o = u * 2 / g, 0.015 (for now) Note: Growth rate,, of the waves ~ = a / w and depends on so, short waves have the largest growth. Action balance equation

69. Slide 69The Wave Model (ECWAM) Wave growth versus phase speed (comparison of Miles' theory and observations)

70. Slide 70The Wave Model (ECWAM) Nonlinear effect: slowing down of wind Continuum: w is nice function, because of continuum of critical layers w is wave induced stress ( u, w): wave induced velocity in air (from Rayleigh Eqn).... with (sea-state dep. through N!). D w > 0 slowing down the wind.

71. Slide 71The Wave Model (ECWAM) Example: Young wind sea steep waves. Old wind sea gentle waves. Charnock parameter depends on sea-state! (variation of a factor of 5 or so). Wind speed as function of height for young & old wind sea old windsea young windsea

72. Slide 72The Wave Model (ECWAM) Non-linear Transfer (finite steepness effects) Briefly describe procedure how to obtain: 1. 2.Express in terms of canonical variables and = (z = ) by solving iteratively using Fourier transformation. 3.Introduce complex action-variable

73. Slide 73The Wave Model (ECWAM) gives energy of wave system: with, … etc. Hamilton equations: become Result: Here, V and W are known functions of

74. Slide 74The Wave Model (ECWAM) Three-wave interactions: Four-wave interactions: Gravity waves: No three-wave interactions possible. Sum of two waves does not end up on dispersion curve. dispersion curve 2, k 2 1, k 1 k

75. Slide 75The Wave Model (ECWAM) Phillips (1960) has shown that 4-wave interactions do exist! Phillips figure of 8 Next step is to derive the statistical evolution equation for with N 1 is the action density. Nonlinear Evolution Equation

76. Slide 76The Wave Model (ECWAM) Closure is achieved by consistently utilising the assumption of Gaussian probability Near-Gaussian Here, R is zero for a Gaussian. Eventual result: obtained by Hasselmann (1962).

77. Slide 77The Wave Model (ECWAM) Properties: 1.N never becomes negative. 2.Conservation laws: action: momentum: energy: Wave field cannot gain or loose energy through four- wave interactions.

78. Slide 78The Wave Model (ECWAM) Properties: (Contd) 3.Energy transfer Conservation of two scalar quantities has implications for energy transfer Two lobe structure is impossible because if action is conserved, energy ~ N cannot be conserved!

79. Slide 79The Wave Model (ECWAM) R = (S nl ) fd /(S nl ) deep

80. Slide 80The Wave Model (ECWAM) Wave Breaking

81. Slide 81The Wave Model (ECWAM) Dissipation due to Wave Breaking Define: with Quasi-linear source term: dissipation increases with increasing integral wave steepness

82. Slide 82 2 THE WAM MODEL

83. Slide 83The Wave Model (ECWAM) 2. The WAM Model: Solves energy balance equation, including S nonlin WAM Group (early 1980s) State of the art models could not handle rapidly varying conditions. Super-computers. Satellite observations: Radar Altimeter, Scatterometer, Synthetic Aperture Radar (SAR). Two implementations at ECMWF: Global (0.25 0.25 reduced latitude-longitude). Coupled with the atmospheric model (2-way). Historically: 3, 1.5, 0.5 then 0.36. Limited Area = N. Atlantic + European Seas (0.1 0.1 ) Historically: 0.5 (Mediterranean only), then the current coverage with 0.25. Both implementations use a discretisation of 36 frequencies 36 directions (used to be 25 freq. 12 dir., then 30 freq. 24 dir.)

84. Slide 84The Wave Model (ECWAM) The Global Model:

85. Slide 85The Wave Model (ECWAM) Global model analysis significant wave height (in metres) for 12 UTC on 1 June 2005. Wave Model Domain

86. Slide 86The Wave Model (ECWAM) Limited Area Model:

87. Slide 87The Wave Model (ECWAM) Limited area model mean wave period (in seconds) from the second moment for 12 UTC on 1 June 2005.

88. Slide 88The Wave Model (ECWAM) ECMWF global wave model configurations Deterministic model 28-km grid, 36 frequencies and 36 directions, coupled to the TL1279 atmospheric model, analysis every 6 hrs and 10-day forecasts from 0 and 12Z. Probabilistic forecasts 55-km grid, 30 frequencies and 24 directions, coupled to the T639/T319 (up to 7/15 days) atmospheric model, (50+1 members) 15 (7+8)-day forecasts from 0 and 12Z. Monthly forecasts 1.5°x1.5° grid, 25 frequencies and 12 directions, coupled to the TL159 model, deep water physics only. Seasonal forecasts 3.0°x3.0°, 25 frequencies and 12 directions, coupled to the T95 model, deep water physics only.

89. Slide 89 2.1 Energy Balance for Wind Sea

90. Slide 90The Wave Model (ECWAM) 2.1. Energy Balance for Wind Sea Summarise knowledge in terms of empirical growth curves. Idealised situation of duration-limited wavesRelevant parameters:, u 10 (u * ), g,, surface tension, a, w, f o, t Physics of waves: reduction to Duration-limited growth not feasible: In practice fully- developed and fetch-limited situations are more relevant., u 10 (u * ), g, t

91. Slide 91The Wave Model (ECWAM) Connection between theory and experiments: wave-number spectrum: Old days H 1/3 H 1/3 H s (exact for narrow-band spectrum) 2-D wave number spectrum is hard to observe frequency spectrum One-dimensional frequency spectrum Use same symbol, F, for:

92. Slide 92The Wave Model (ECWAM) Let us now return to analysis of wave evolution: Fully-developed: Fetch-limited: However, scaling with friction velocity u is to be preferred over u 10 since u 10 introduces an additional length scale, z = 10 m, which is not relevant. In practice, we use u 10 (as u is not available).

93. Slide 93The Wave Model (ECWAM) JONSWAP fetch relations (1973): Empirical growth relations against measurements Evolution of spectra with fetch (JONSWAP) Hz km

94. Slide 94The Wave Model (ECWAM) Distinction between wind sea and swell wind sea: A term used for waves that are under the effect of their generating wind. Occurs in storm tracks of NH and SH. Nonlinear. swell: A term used for wave energy that propagates out of storm area. Dominant in the Tropics. Nearly linear. Results with WAM model: fetch-limited. duration-limited [wind-wave interaction].

95. Slide 95The Wave Model (ECWAM) Fetch-limited growth Remarks: 1.WAM model scales with u Drag coefficient, Using wind profile C D depends on wind: Hence, scaling with u gives different results compared to scaling with u 10. In terms of u 10, WAM model gives a family of growth curves! [u was not observed during JONSWAP.]

96. Slide 96The Wave Model (ECWAM) Dimensionless energy: Note: JONSWAP mean wind ~ 8 - 9 m/s Dimensionless peak frequency:

97. Slide 97The Wave Model (ECWAM) 2.Phillips constant is a measure of the steepness of high-frequency waves. Young waves have large steepness. 1000 p

98. Slide 98The Wave Model (ECWAM) Duration-limited growth Infinite ocean, no advection single grid point. Wind speed, u 10 = 18 m/s u = 0.85 m/s Two experiments: 1.uncoupled: no slowing-down of air flow Charnock parameter is constant ( = 0.0185) 2.coupled: slowing-down of air flow is taken into account by a parameterisation of Charnock parameter that depends on w : = { w / } (Komen et al., 1994)

99. Slide 99The Wave Model (ECWAM) Time dependence of wave height for a reference run and a coupled run.

100. Slide 100The Wave Model (ECWAM) Time dependence of Phillips parameter, p, for a reference run and a coupled run.

101. Slide 101The Wave Model (ECWAM) Time dependence of wave-induced stress for a reference run and a coupled run.

102. Slide 102The Wave Model (ECWAM) Time dependence of drag coefficient, C D, for a reference run and a coupled run.

103. Slide 103The Wave Model (ECWAM) Evolution in time of the one-dimensional frequency spectrum for the coupled run. spectral density (m 2 /Hz)

104. Slide 104The Wave Model (ECWAM) The energy balance for young wind sea.

105. Slide 105The Wave Model (ECWAM) The energy balance for old wind sea. 0 -2

106. Slide 106 2.2 Wave Forecasting

107. Slide 107The Wave Model (ECWAM) 2.2. Wave Forecasting Sensitivity to wind-field errors. For fully developed wind sea: H s = 0.22 u 10 2 / g 10% error in u 10 20% error in H s from observed H s Atmospheric state needs reliable wave model. SWADE case WAM model is a reliable tool.

108. Slide 108The Wave Model (ECWAM) Verification of model wind speeds with observations + + + observations OW/AES winds …… ECMWF winds (OW/AES: Ocean Weather/Atmospheric Environment Service)

109. Slide 109The Wave Model (ECWAM) Verification of WAM wave heights with observations + + + observations OW/AES winds …… ECMWF winds (OW/AES: Ocean Weather/Atmospheric Environment Service)

110. Slide 110The Wave Model (ECWAM) Validation of wind & wave analysis using satellite & buoy. Altimeters onboard ERS-1/2, ENVISAT and Jason-1/2 Quality is monitored daily. Monthly collocation plots SD 0.5 0.3 m for waves (recent SI 12-15%) SD 2.0 1.2 m/s for wind (recent SI 16-18%) Wave Buoys (and other in-situ instruments) Monthly collocation plots SD 0.85 0.45 m for waves (recent SI 16-20%) SD 2.6 1.2 m/s for wind (recent SI 16-21%)

111. Slide 111The Wave Model (ECWAM) WAM first-guess wave height against ENVISAT Altimeter measurements (June 2003 – May 2004)

112. Slide 112The Wave Model (ECWAM) Global comparison of WAM first-guess sig. wave height against ENVISAT Alt. measurements (plot & statistics) as well as Jason-1/2 (statistics only) (02 February 2010 to 01 February 2011)

113. Slide 113The Wave Model (ECWAM) Global wave height RMSE between ERS-2 Altimeter and WAM FG (thin navy line is 5-day running mean ….. thick red line is 30-day running mean)

114. Slide 114 Model improvements detected by RA-2 monitoring Change in RA-2 processing The Wave Model (ECWAM) Global wave height SDD between Envisat Alt. (RA-2) and WAM FG (Less along-track averaging of Envisat compared to ERS-2)

115. Slide 115The Wave Model (ECWAM) Analysed wave height and periods against buoy measurements for February to April 2002 2 4 6 8 10 12 14 H S (m) model 02468101214 H S (m) buoy 0 HS ENTRIES: 1 - 3 3 - 8 8 - 22 22 - 62 62 - 173 173 - 482 482 - 1350 02468101214161820 Peak Period Tp (sec.) buoy 0 2 4 6 8 10 12 14 16 18 20 Peak Period Tp (sec.) model TP ENTRIES: 1 - 3 3 - 6 6 - 14 14 - 33 33 - 79 79 - 188 188 - 450 SYMMETRIC SLOPE = 0.932 CORR COEF = 0.956 SI = 0.175 RMSE = 0.438 BIAS = -0.150 LSQ FIT: SLOPE = 0.879 INTR = 0.134 BUOY MEAN = 2.354 STDEV = 1.389 MODEL MEAN = 2.204 STDEV = 1.278 ENTRIES = 29470 SYMMETRIC SLOPE = 1.008 CORR COEF = 0.801 SI = 0.190 RMSE = 1.746 BIAS = 0.072 LSQ FIT: SLOPE = 0.813 INTR = 1.785 BUOY MEAN = 9.160 STDEV = 2.743 MODEL MEAN = 9.232 STDEV = 2.784 ENTRIES = 17212 02468101214 Mean Period Tz (sec.) buoy 0 2 4 6 8 10 12 14 Mean Period Tz (sec.) model TZ ENTRIES: 1 - 3 3 - 6 6 - 12 12 - 26 26 - 59 59 - 133 133 - 300 SYMMETRIC SLOPE = 0.985 CORR COEF = 0.931 SI = 0.105 RMSE = 0.779 BIAS = -0.149 LSQ FIT: SLOPE = 0.989 INTR = -0.069 BUOY MEAN = 7.306 STDEV = 1.979 MODEL MEAN = 7.157 STDEV = 2.102 ENTRIES = 6105 Wave height Peak periodMean period

116. Slide 116The Wave Model (ECWAM) Analysed wave height and periods against buoy measurements for March to May 2011

117. Slide 117The Wave Model (ECWAM) Global wave height SI (=SDD/Mean) between buoys and WAM analysis

118. Slide 118The Wave Model (ECWAM) Problems with buoys: 1.Atmospheric model assimilates winds from buoys (sometimes with unknown height but regards them as 10 m winds [10% maximum error]). 2.Buoy observations are not representative for aerial average. 3.Limited coverage (parts of NH) seasonal cycles. Problems with satellite Altimeters: 1.They have some difficulties at very low SWH values. 2.Satellite observations may not be available when or where needed. 3.Some Altimeters tend to overestimate SWH values.

119. Slide 119The Wave Model (ECWAM) Quality of wave forecast Compare forecast with verifying analysis. Forecast error, standard deviation of error ( ), persistence. Period: three months (January-March 1995). Tropics is better predictable because of swell: Daily errors for July-September 1994 Note the start of Autumn. New: 1. Anomaly correlation 2. Verification of forecast against buoy data.

120. Slide 120The Wave Model (ECWAM) Significant wave height anomaly correlation and st. deviation of error over 365 days for years 1997-2004 Northern Hemisphere (NH)

121. Slide 121The Wave Model (ECWAM) Significant wave height anomaly correlation and st. deviation of error over 365 days for years 1997-2004 Tropics

122. Slide 122The Wave Model (ECWAM) Significant wave height anomaly correlation and st. deviation of error over 365 days for years 1997-2004 Southern Hemisphere (SH)

123. Slide 123The Wave Model (ECWAM) RMSE of * significant wave height, * 10m wind speed and * peak wave period of different models as compared to buoy measurements for February to April 2005

124. Slide 124 3 BENEFITS FOR ATMOSPHERIC MODELLING

125. Slide 125 3.1 Use as Diagnostic Tool

126. Slide 126The Wave Model (ECWAM) 3.1. Use as Diagnostic Tool Discovered inconsistency between wind speed and stress and resolved it. Overactivity of atmospheric model during the forecast: mean forecast error versus time.

127. Slide 127 3.2 Coupled Wind-Wave Modelling

128. Slide 128The Wave Model (ECWAM) 3.2. Coupled Wind-Wave Modelling Coupling scheme: Impact on depression (Doyle). Impact on climate [extra tropic]. Impact on tropical wind field ocean circulation. Impact on weather forecasting. ATM WAM t u 10 t u 10 time

129. Slide 129The Wave Model (ECWAM) WAM – IFS Interface A t m o s p h e r i c M o d e l qTP air z i / L (U 10,V ) W a v e M o d e l

130. Slide 130The Wave Model (ECWAM) Simulated sea-level pressure for uncoupled and coupled simulations for the 60 h time uncoupledcoupled 956.4 mb963.0 mb U lml > 25 m/s 1000 km

131. Slide 131The Wave Model (ECWAM) Scores of FC 1000 and 500 mb geopotential for SH (28 cases in ~ December 1997) coupled uncoupled

132. Slide 132The Wave Model (ECWAM) Standard deviation of error and systematic error of forecast wave height for Tropics (74 cases: 16 April until 28 June 1998). coupled

133. Slide 133The Wave Model (ECWAM) Global RMS difference between ECMWF and ERS-2 scatterometer winds (8 June – 14 July 1998) coupling ~20 cm/s (~10%) reduction

134. Slide 134The Wave Model (ECWAM) Change from 12 to 24 directional bins: Scores of 500 mb geopotential for NH and SH (last 24 days in August 2000)