Wave forecasting at ECMWF

Wave forecasting at ECMWF Peter Janssen, ECMWF <peter.janssen@ecmwf.int> 1.

. Freak Waves. INTRODUCTION I will briefly discuss progress in ocean wave forecasting at ECMWF during the past years or so, by studying verification results of forecast wave height against verifying analysis, buoy data and Altimeter data. The quality of the analysis is judged by verification against independent buoy data for wave height and wave spectrum. considerable improvement in forecasting parameters such as and. large improvements in the wave model, but main reason is better quality of analyzed and forecast wind. I will close the talk with a discussion of a new topic, namely the prediction of extreme events, such as for example freak waves..

. Freak Waves. The programme of the talk is as follows. Laws of ocean wave forecasting Ocean wave forecasting is about forecasting of the mean sea state in a grid box. The fundamental evolution equation for the wave spectrum is the energy balance equation. Recently, it is realized that also fluctuations around the mean can be predicted as they depend on the mean sea state. Verification Discuss validation of forecast against own analysis and buoy data. Quantify accuracy of analysis. Wave Model Improvements Wave model improvements are important, of course, but the major improvement of ocean wave forecasting skill comes from improved surface winds. 3.

. Freak Waves. Forecasting Extreme Sea States: On the open ocean extreme waves are generated by nonlinear focussing, a four-wave interaction process that also causes the Benjamin-Feir Instability. Theoretical developments in the past 10 years enable us to obtain, for given wave spectrum, the probability distribution function (pdf) of the wave height for surface gravity waves. In other words we can make statements about the probability of occurence of freak waves. An operational freak wave prediction scheme was introduced in October 003, but extensive validation in the field is still required. 4.

. Freak Waves. $ # Forecasting of the mean sea state Ocean waves obey a set of deterministic evolution equations. For operational forecasting, solving these equations is not practical because, apart from the initial amplitudes, knowledge of the phase of the waves is required. This information is not available. Furthermore, just as in the atmospheric problem, there is chaotic behaviour. Therefore, consider the evolution of the mean sea state in a box with width location. The ensemble average is essentially an average over the phases of the waves. The mean sea state is then given by the wavenumber spectrum action density spectrum is defined as at, while the with the energy given by %. The action density is the number density of waves, hence of the waves is given by, while the wave momentum is. "! $ 5. %

J 5. Freak Waves. :, : 9 5, 9 1, 6 9 1 : Averaging the deterministic evolution equations then gives for waves on a slowly varying current the energy balance equation &(' 5* -.0/ -.431 687 5* -.43 -.0/1 )* )+ Here, represents the dispersion relation,=<?> &' &; The source function on the right hand side represents the physics of wind-wave generation ( ), dissipation by wave breaking and other causes ( four-wave interactions ( :A@CB I@CB B H : B ). In other words, :ED @CF F @CG ) and @CG1 : D @ F F I@ B B H : B : @CB In the 1980 s there was a dedicated effort to develop efficient parametrisations of all the source functions, which still is the basis of present day wave forecasting and the two-way interaction of ocean waves and atmosphere. Time evolution of the wave spectrum and source function balance. 6.

LK 150 100 T = h T = 4h T = 8h T = 1h T = 4h F(f) (m /Hz) 50 0 0 0. 0.4 f (Hz) Evolution in time of the one-dimensional frequency spectrum for a wind speed of m/s. 7.

NM 0.010 0.005 Sin Snl Sds S(f) 0 0.005 0.010 0 0.1 0. 0.3 0.4 0.5 frequency (Hz) The plot shows the energy balance of wind-generated ocean waves for a duration of 4 hrs, and a wind speed of m/s. 8.

O P ]. Freak Waves. VU ^ ^ XW [f ] ^ e Z Y Operational Configuration Ocean wave forecasting at ECMWF is based on WAM cy4. Discuss global implementation only. GLOBAL MODEL (81 deg S to 81 deg N) coupled to atmospheric model [two-way interaction with feedback of ocean waves on ocean surface roughness (since June 9, 1998) thus giving a sea-state dependent momentum and heat flux] Deterministic forecasts : has frequencies and directions runs on an irregular lat-lon grid, km. assimilation of ENVISAT Altimeter and ERS- SAR spectra. 10 day forecasts from 00Z and 1 Z. coupled to 10 m winds from ATM model every timestep min. QRTS [\ _a` e e^ bdc 9.

HS 0040151 (FC4) 1 - - 3 3-4 4-5 5-6 6-7 7-8 8-9 9-9.9708 MAGICS 6.7 icarus - dax Mon Jan 6 1:3:34 004 * 1 ERROR(S) FOUND * 10.

Sunday 5 January 004 1UTC ECMWF Forecast t+4 VT: Monday 6 January 004 1UTC Surface: -d wave spectrum EXP: 0001 16.5 W 15 W 4 13.5 W 1 W 4.5 10.5 W 9 W 7.5 W 6 W 3 4.5 W 3 W 1.5 W 0.5 1.5 E 57 N 57 N 4 55.5 N 5 1.5 55.5 N 5 4 54 N 54 N 4.5 5.5 N 5.5 N 4 3.5 3 3 51 N 51 N 49.5 N.5.5 49.5 N 48 N 48 N SWH [m] 3.5 3 3.5 4 4.5 5 16.5 W 15 W 13.5 W 1 W 10.5 W MAGICS 6.7 icarus - waa Mon Jan 6 03:4:49 004 Bjorn Hansen 9 W 7.5 W 6 W 4.5 W 3 W 1.5 W 0 1.5 E 11.

g. Freak Waves. h Probabilistic Forecasts : (50+1) from the 10-day wave ensemble forecasts coupled to the 10 m winds ATM model. l ikj h m l ikj q qp ndo Potential use: Probability of high sea state and Ship Routing. For example, error in forecast ship route may be obtained from the 50 shiproutes generated by the winds and waves of the 50 member ensemble. 1.

Brest -> New York 003115 1 0031 00 80 W 60 W 40 W 0 W Det. fc. 70 N 70 N 0 60 N 60 N 50 N 50 N 40 N 40 N 30 N 30 N 80 W 60 W 40 W 0 W 0 MAGICS 6.8 leda - was Tue Dec 16 01:7:09 003 13.

r r r. Freak Waves. Verification Over the years we have done extensive verifications of the skill of the wave forecasting system. Here we concentrate on one application only, namely the global deterministic model: Verification against analysis. Shown are timeseries of standard deviation of error of forecast wind and ocean wave height for Northern Hemisphere, Tropics and Southern Hemisphere. A substantial improvement is seen. Verification of first-guess wave height against altimeter data from ERS-, which confirms the improvement Verification of wave model data against independent buoy observations. Both analysed and forecast wave height, and also analysed spectra. 14.

WAVE_G FORECAST VERIFICATION 1UTC WIND SPEED AT 10 M SURFACE LEVEL STANDARD DEVIATION OF ERROR FORECAST N.HEM LAT 0.000 TO 90.000 LON -180.000 TO 180.000 T+ 4 T+ 4 MA T+ 7 T+ 7 MA T+10 T+10 MA T+168 T+168 MA 4 M/S MA = 1 Month Moving Average 3.5 3.5 1.5 1 0.5 1995 1996 1997 1998 1999 000 001 00 003 004 1.4 WAVE_G FORECAST VERIFICATION 1UTC HEIGHT OF WAVES SURFACE LEVEL STANDARD DEVIATION OF ERROR FORECAST N.HEM LAT 0.000 TO 90.000 LON -180.000 TO 180.000 M T+ 4 T+ 4 MA T+ 7 T+ 7 MA T+10 T+10 MA T+168 T+168 MA MA = 1 Month Moving Average 1. 1 0.8 0.6 0.4 0. 0 1995 1996 1997 1998 1999 000 001 00 003 004 15

RMS error of first-guess wave height against ERS- Altimeter data for the whole globe over the period of August 1997 until August 003. 16.

1.75 Wave height comparison with buoy data in winter (October to March) R.M.S.E. (m) 1.5 1.5 1 1996-1997 1997-1998 1998-1999 1999-000 000-001 001-00 00-003 003-004 004-005 0.75 0.5 0.5 0 1 3 4 5 6 7 8 9 10 Forecast range (days) (forecasts from 1 UTC) RMS error of analysed and forecast wave height against Buoy wave height data for all winters from 1996 onwards. 17.

Equivalent wave height bias as function of wave period at all US and Canadian buoy locations for the period December 000 to May 005. 18

s s. Freak Waves. Wave model Improvements In the past few years we have introduced a number of improvements to our wave forecasting system. The main ones are: Effects of unresolved bathymetry Analysis increments in the summertime show large scale systematic errors in the whole North Pacific. Caused by small islands and atols which are not resolved by coarse resolution models. Introduce a blocking factor which is proportional to the fraction of land inside a grid box in the wave propagation direction. Major impact on tropical scores. Changes to wave dissipation New definition of steepness parameter in wave dissipation. Large impact on spectral shape, hence on parameters such as the mean frequency, mean wave direction, directional spread. 19.

Mean wave height increments for July 001 0 E 40 E 60 E 80 E 100 E 10 E 140 E 160 E 180 160 W 140 W 10 W 100 W 80 W 60 W 40 W 0 W 70 N 70 N 60 N 60 N 50 N 50 N 40 N 40 N 30 N 30 N 0 N 0 N 10 N 10 N 0 0 10 S 10 S 0 S 0 S 30 S 30 S 40 S 40 S 50 S 50 S 60 S 60 S 70 S 70 S 0 E 40 E 60 E 80 E 100 E 10 E 140 E 160 E 180 160 W 140 W 10 W 100 W 80 W 60 W 40 W 0 W 0.0770 0.07 0.06 0.05 0.04 0.03 0.0 0.01-0.01-0.0-0.03-0.04-0.05-0.06-0.07-0.0715 0.

100 % 90 80 70 60 FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL ANOMALY CORRELATION FORECAST AREA=TROPICS TIME=1 MEAN OVER 7 CASES DATE = 0030401 TO 003047 UNRESOLVED WAVE_G FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL STANDARD DEVIATION OF ERROR FORECAST AREA=TROPICS TIME=1 MEAN OVER 7 CASES DATE = 0030401 TO 003047 0.45 0.4 0.35 0.3 UNRESOLVED WAVE_G 50 0.5 40 30 0 10 0. 0.15 0.1 0 0 1 3 4 5 6 7 8 9 10 Forecast Day 0.05 0 1 3 4 5 6 7 8 9 10 Forecast Day MAGICS 6.9.1 leda - dax Wed Jul 13 13:35:36 005 Verify SCOCOM MAGICS 6.9.1 leda - dax Wed Jul 13 13:35:36 005 Verify SCOCOM * 1 ERROR(S) FOUND * 0.18 FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL MEAN ERROR FORECAST AREA=TROPICS TIME=1 MEAN OVER 7 CASES DATE = 0030401 TO 003047 UNRESOLVED WAVE_G 1.88 FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL MEAN VERIFYING ANALYSIS AREA=TROPICS TIME=1 MEAN OVER 7 CASES DATE = 0030401 TO 003047 UNRESOLVED WAVE_G 0.16 1.87 0.14 0.1 1.86 0.1 1.85 0.08 1.84 0.06 1.83 0.04 0.0 1.8 0 0 1 3 4 5 6 7 8 9 10 Forecast Day 1.81 0 1 3 4 5 6 7 8 9 10 Forecast Day MAGICS 6.9.1 leda - dax Wed Jul 13 13:35:36 005 Verify SCOCOM * 1 ERROR(S) FOUND * MAGICS 6.9.1 leda - dax Wed Jul 13 13:35:36 005 Verify SCOCOM * 1 ERROR(S) FOUND * 1.

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Importance of forecast skill in wind These wave model improvements have, however, only a limited impact on forecast skill of wave height in the Extra-Tropics during winter time. Quality of the surface wind is important. We have shown this by running the latest wave model on the period Jan-Feb 1998 using operational winds, and compared with operational wave height results. Using these old winds we only see improvements with the new wave model up to day 4. From day 5 onwards the skill is completely determined by the quality of the wind. 3.

100 Comparison with GTS buoys January-February 1998 forecasts from 1UTC Normalised standard deviation of error 90 80 70 60 50 40 30 ops (0001) CY9R1 (en9o) 0 0 4 48 7 96 10 144 168 19 16 40 Forecast range in hours 4.

t t. Freak Waves. w u u v New development: Freak Waves Linear theory No wave-wave interaction. Focussing of wave energy only occurs when the phases of the waves are favourable ( constructive interference ). Gives at best a doubling of wave height Gaussian pdf for elevation. Nonlinear Waves Now there are four-wave interactions. Thus, a wave may borrow energy from its neighbours. Because of this extra focussing wave height may become at most times as large in 1D, while it D it becomes 4.5-5 times as large as the average wave height Large deviations from Gaussian. The fun is that these deviations can be obtained by means of the usual statistical mechanics approach for wave-wave interactions. 5.

Evolution of surface elevation in space and time from the big wave tank in Trondheim (from Onorato et al, 004). The formation of Freak Waves is clearly seen. 6

0.0 Comparison of wave height distribution Obs Kurtosis K4= 4 (C4=K4/3 1=0.33) 1.0 log10(p(h/h_s)).0 3.0 4.0 Nonlinear Theory OBS (Onorato et al) Rayleigh 5.0 0.0 1.0.0 H/H_S Comparison of theoretical and observed (Onorato et al, 004) wave height distribution. For reference, the linear Rayleigh result is shown as well. 7.

x x x. Freak Waves. OPERATIONAL IMPLEMENTATION Thus, there is nowadays an improved understanding why extreme sea states do occur and deviations from the Normal Distribution can be expressed in terms of a spectral shape parameter, called the Benjamin-Feir Index : this is basically the ratio of the steepness of the waves and the width of the spectrum. Hence, we have implemented the following scheme: From the predicted wave spectrum we infer the B.F. Index. From the B.F Index we obtain the deviations from the Normal distribution, e.g. as measured by the Kurtosis. Given the kurtosis and the significant wave height, we are able to answer question such as what is the enhanced probability on extreme events. 8.

ENH FC on 00508600 STEP 90 0.5-1 1-1.5 1.5 - -.5.5-3 3-3.5 3.5-4 4-4.5 4.5-5 1 1 1 1 1 1 1 1 Enhancement of extreme events caused by nonlinear focussing for the 90 hr forecast of the landfall of hurricane Katrina. 9.

HS FC on 00508600 STEP 90 1 - - 3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 4 4 Significant wave height for the 90 hr forecast of the landfall of hurricane Katrina. 30.

y y. Freak Waves. CONCLUSIONS In the past few years we have seen considerable improvements in the modelling of the shape of the wave spectrum. Delicate parameters such as the Benjamin-Feir Index can now be determined accurately. Now is the time ripe to concentrate on the coastal zone problem and to study issues such as the interaction of ocean waves and tidal motion, effects of bottom-induced wave breaking and nonlinear interactions. Recent observations of freak waves have confirmed the theoretical picture of freak waves generation that already existed in 1965! Using well-established methods one can, for given average sea state, obtain estimates of the enhanced probability of extreme events. The theoretical results are confirmed by laboratory results. Hence, in this sense, freak wave prediction is feasible. However, validation of all this in the field is of course desirable. 31.