Tropical Pacific Ocean model error covariances from Monte Carlo simulations

Transcription

1 Q. J. R. Meteorol. Soc. (2005), 131, pp doi: /qj Tropical Pacific Ocean model error covariances from Monte Carlo simulations By O. ALVES 1 and C. ROBERT 2 1 BMRC, Melbourne, Australia 2 Météo-France, Toulouse, France (Received 2 June 2005; revised 16 March 2006) SUMMARY As a first step towards the development of an Ensemble Kalman Filter (EnKF) for ocean data assimilation in the tropical oceans, this article investigates a novel technique for explicitly perturbing the model error in Monte Carlo simulations. The perturbation technique involves perturbing the surface zonal stress. Estimates of the characteristics of the wind stress errors were obtained from the difference between zonal wind fields from the NCEP and ECMWF re-analyses. In order to create random zonal wind stress perturbations, an EOF analysis was performed on the intraseasonally time-filtered difference between the two re-analysis products. The first 50 EOFs were retained and random wind stress fields for each ensemble member were created by combining random amounts of each EOF. Ensemble runs were performed using a shallow-water model, with both short forecasts and long simulations. Results show covariance patterns characteristic of Kelvin wave and Rossby wave dynamics. There are interesting differences between covariances using short forecasts and those using long simulations. The use of the long simulations produced non-local covariances (e.g. negative covariances between east and west Pacific), whereas short forecasts produced covariances that were localized by the time it takes Kevin and Rossby waves to travel over the forecast period and the scales of spatial covariance in the wind stress errors. The ensembles of short forecasts produced covariances and cross-covariances that can be explained by the dynamics of equatorial Rossby and Kevin waves forced by wind stress errors. The results suggest that the ensemble generation technique to explicitly represent the model error term can be used in an EnKF. KEYWORDS: Assimilation Ensemble Kalman Filter 1. INTRODUCTION The Bureau of Meteorology Research Centre (BMRC) is planning to develop a data assimilation scheme for seasonal prediction based on the Ensemble Kalman Filter (EnKF; Evensen 1997; Burgers et al. 1998). Model error covariances will be calculated from a Monte-Carlo-generated ensemble of forecasts. Such a scheme has been developed for the NASA seasonal forecast system (Keppenne and Rienecker 2002). In the formulation of the EnKF, the background-error covariance matrix is calculated from a forecast ensemble. The forecast ensemble starts from an ensemble of analyses (from the previous assimilation), thus perturbing the forecast evolution due to errors in the analysis. Errors in the forecast ensemble due to model (or surface forcing) errors need to be explicitly represented. In this paper we develop a technique for explicitly generating model error perturbations and evaluate the technique using idealized experiments. Ocean model errors are due to errors in the model physics and/or in the surface forcing fields. The equatorial Pacific is strongly driven by the wind stress forcing. Therefore, in this study we assumed that most of the model error is coming from the wind stress error (although the technique can be generalized to include other forcing fields) and we used estimates of zonal wind stress error to produce an ensemble of Monte Carlo simulations. These were used to estimate the model error covariance and cross-covariance for different variables. The wind stress error perturbations are generated using a novel technique that creates random wind stress forcing fields from the difference between stresses from re-analyses by the National Centers for Environmental Prediction/National Center Corresponding author: Bureau of Meteorology Research Centre, 700 Collins Street, Melbourne, Victoria 3000, Australia. o.alves@bom.gov.au c Royal Meteorological Society,

2 3644 O. ALVES and C. ROBERT for Atmospheric Research (NCEP/NCAR) and by the European Centre for Mediumrange Weather Forecasts (ECMWF). Borovikov et al. (2005) investigated the usefulness of a model ensemble for calculating error covariances. They generated an ensemble by forcing an ocean model with different sets of surface forcing fields obtained from different runs of the atmospheric model forced with observed SST. Their ensemble perturbations are therefore generated by using surface forcing fields from different intraseasonal events produced by their atmospheric model. In our approach we develop a technique that generates ensembles using random intraseasonal surface forcing errors, rather than intraseasonal anomalies. Borovikov et al. (2005) investigate the covariance patterns produced from a long continuous run, but in our approach we additionally evaluate the covariance patterns from short forecast runs. The aim of this study is to describe an ensemble perturbation technique for representing model error and to use this technique to demonstrate the potential use of Monte Carlo ensemble runs to estimate ocean model error covariances. The article is structured as follows. Section 2 describes the model, forcing data, ensemble generation technique and experiments. Section 3 describes the results. A summary is presented in section EXPERIMENTS (a) Model and forcing data A one-and-a-half-layer shallow-water ocean model (Burgers et al. 2002) was used to investigate the impact of wind stress perturbations. The model was written in perturbation form and has prognostic variables for layer thickness and zonal and meridional currents. The model spans the equatorial Pacific from 30 Sto30 N and has a horizontal resolution of 1 in latitude and longitude. Surface wind stress fields from two of the most recent re-analysis projects are used in this study. The first wind stress product was from the ECMWF ERA-40 project (Kållberg et al. 2004). The second was from the NCEP/NCAR 50-year re-analysis (Kistler et al. 2001). These two re-analysis products represent the latest state-of-theart estimates of the atmospheric state. Differences between them give an estimate of the errors in the re-analyses. A basic assumption in this paper is that the spatial and temporal differences in the wind stresses between these two re-analyses are characteristic of the errors in the re-analysis wind stresses. Two years (1996 and 1997) of zonal wind stress analyses were used in this study. This period was chosen because it is known to have strong intraseasonal activity associated with the onset of the 1997 El Niño. Differences between NCEP and ERA-40 were calculated as a function of time and then filtered in time to focus on intraseasonal differences (a 10-day running mean was applied to remove synoptic variability and then a 90-day running mean was removed). This was because wind stress variability on intraseasonal scales projects strongly onto ocean thermocline anomalies (e.g. Kessler et al. 1995). The intraseasonally filtered zonal wind stress difference will hereafter be referred to as the wind stress error. Only wind stress errors within 20 of the equator were considered in this study. (b) Ensemble generation technique An empirical orthogonal function (EOF) analysis was performed on the two years of intraseasonal stress error. The first few leading EOFs explain most of the variance

3 MONTE CARLO ERROR COVARIANCES 3645 (a) (b) Figure 1. Empirical orthogonal functions (a) EOF1 and (b) EOF2 of the wind stress difference between NCEP and ERA-40 (filtered for intraseasonal time scales). EOFs are normalized, the contour interval is 0.25, and negative areas are shaded. and can usually be related to physical phenomena. For example, the first two patterns (Figs. 1(a) and (b)) are characteristic of wind anomalies associated with largescale atmospheric convection, such as the Madden Julian Oscillation. The first EOF (Fig. 1(a)) shows a pattern with strong loadings in the west Pacific, centred on the equator and with a latitudinal scale of around This is characteristic of wind bursts in the west Pacific associated with intraseasonal variability, such as the Madden Julian Oscillation (see, for example, Harrison and Vecchi 1997). The second EOF pattern (Fig. 1(b)) shows positive loadings in the central equatorial Pacific and negative loadings in the western and eastern equatorial Pacific and is also characteristic of the longitudinal extent of wind bursts in the central and western Pacific. Wind stress perturbations were constructed by randomly combining the first 50 EOF patterns. (Together these explained more than 99% of the variance.) A random set

4 3646 O. ALVES and C. ROBERT Figure 2. Example from one of the ensemble members of wind stress error along the equator shown as a function of time (days since the beginning of 1996). The contour interval is 0.01 N m 2 and negative values are shaded. of wind stress perturbations were created and for ensemble member n (n = 1, 100), the perturbation wind stress was given by τn (x, y, t) 50 Ri,n Pi (x, y)ti (t), (1) i=1 where Pi (x, y) is the eigenvector representing the spatial pattern of the errors, Ti (t) is the associated time series and Ri,n is a random number taken from a Gaussian distribution with a standard deviation of 1. A different random number was used for each EOF and each ensemble member. Thus τn (x, y, t) represents the wind stress perturbation over the two-year period for ensemble member n. An example of the wind stress error forcing field used for one of the ensemble members is shown in Fig. 2. The largest error variance is in the central and west Pacific. The figure shows a series of westerly and easterly wind errors, varying on intraseasonal scales, with amplitudes reaching up to 0.03 N m 2. Note that our approach is very different from that of Borovikov et al. (2005). They use different realizations of modelled intraseasonal forcing anomalies, whereas we use forcing errors (as opposed to physical anomalies) based on differences between re-analysis products. (c) Ensemble integrations Two types of ensemble integrations were performed. The first type, hereafter referred to as continuous ensemble, involved running an ensemble of 100 members. Each member was run continuously for two years (1996 and 1997) and started from a state of rest (i.e. zero anomalies since the model is only an anomaly model). Each member was forced by a random wind stress given by Eq. (1). This allowed errors at one point in time to influence errors in the future. The second type, hereafter referred to as forecast ensemble, was designed to imitate short (10-day) forecasts. The runs were performed similarly to those of the

5 MONTE CARLO ERROR COVARIANCES 3647 continuous ensemble, except every 10 days the model variables were set to zero. Fields at the end of each 10-day period (before being reset) formed an ensemble of 100 ten-day forecasts, every 10 days throughout the two-year period. This set of forecasts represented the error growth in 10 days, assuming that the error was set to zero during the previous assimilation. In a practical data assimilation application, it is unlikely that the analysis error will be zero for each assimilation, and so some of the errors will propagate in time between assimilation cycles. Therefore, the background-error covariance will be somewhere between the error covariance from the continuous ensemble and the error covariance from the forecast ensemble. In relatively data-rich areas (e.g. the Tropical Ocean Global Atmosphere (TOGA) project s TAO array region in the equatorial Pacific), one might expect the error covariance to be closer to the errors from the forecast ensemble, whereas in data-sparse regions (e.g. Indian Ocean) it may be closer to the errors from the continuous ensemble. 3. DESCRIPTION OF COVARIANCES (a) Covariance calculation technique In this paper we investigate how the variability of errors at a reference point, say x 0,ofvariableT(x) depends on the variability of errors of variable U(x) at every point x, wherex spans the spatial domain of the model. We therefore use a normalized covariance or regression coefficient defined as c = T(x 0)U(x) var{t(x 0 )}, (2) where var{t(x 0 )} is the variance of variable T at the reference point x 0, and the overbar represents an average over the ensemble. For the rest of this paper, the term covariance will refer to this normalized covariance. It provides a covariance between two points in units of U per unit of T. Covariance fields were calculated every 10 days by averaging over the ensemble members. A mean covariance over the two-year period was calculated by averaging the covariances calculated every 10 days. A reference point on the equator at the Date Line is used in this article to illustrate the covariance patterns obtained using Eq. (2). The approach here is also significantly different from Borovikov et al. (2005). Firstly, they only performed ensemble simulations similar to our continuous ensemble and did not perform short forecast runs. Secondly, because they only had a 32-member ensemble, they increased the ensemble size by sampling each run five times one year apart, therefore generating a 160-member ensemble. This means that their covariances depend both on the ensemble covariance (due to different intraseasonal anomalies) and the covariance of the time evolution of the anomalies. (b) Wind stress covariances The covariance pattern (for the reference point at 0 N, 180 W) calculated from the ensemble of wind stresses used to force the ensemble integrations (τ n givenbyeq.(1))is shown in Fig. 3. Positive covariance is mainly concentrated in the west Pacific. Positive covariance extends some 10 to the east of the reference point, but around 40 to the west of the reference point. There is significant positive covariance in the west Pacific reaching up to 17 off the equator. In the central and eastern Pacific, the covariance with the point on the Date Line is in general small and less than 0.2 N m 2 (N m 2 ) 1.

6 3648 O. ALVES and C. ROBERT Figure 3. Covariance for wind stress error using a reference point at 0 N, 180 W (marked with a cross). The contour interval is 0.2 N m 2 (N m 2 ) 1 and negative areas are shaded. Wind stress perturbations were used only between 20 S and 20 N. (c) Thermocline covariances (i) Forecast ensemble. Ensemble covariances are calculated from the forecast ensemble using values every ten days over the two-year period, representing a mean covariance of 10-day forecasts over the whole period. The forecast covariances for layer thickness are shown in Fig. 4(a) for the reference point at 0 N, 180 W. Centred on the reference point is an area of positive covariance. The east west length-scale is approximately symmetric about the equator with a value of around 20 at the equator. To the east it is longer than the wind stress covariance length-scale, but to the west it is slightly shorter (compare to Fig. 3). The north south length-scale is also symmetric about the equator but depends on longitudinal distance from the reference point. At the longitude of the reference point, it is about 3 and decreases towards the west and increases towards the east. Off the equator and slightly to the west are two areas of negative covariance associated with Rossby waves. The centres of the negative covariances are around 10 to 20 to the west of the reference point. Figure 4(b) shows the covariance of zonal wind stress with layer thickness at the 0 N, 180 W reference point. This shows that thermocline variability at the Date Line depends on wind stress variability to the west, with the covariance peaking some 15 to the west of the reference point. The covariance pattern is characteristic of forced equatorial Kelvin and Rossby wave dynamics. To illustrate this, we have carried out a simple experiment with the shallow-water model. We run the shallow-water model for 60 days, with a westerly wind burst forcing the model in the first 10 days and no forcing after that. This was to simulate the impact of a wind stress error. The wind stress forcing applied to the model was Gaussian shaped, centred on the equator and Date Line with an east west length-scale of 15 and a north south length-scale of 6. The impact of the wind burst is shown in Fig. 5. Figure 5(a) shows the layer thickness after 10 days. This shows a pattern similar to the covariance pattern in Fig. 4 with a Kelvin wave to the right and a pair of Rossby waves to the left of the centre of the westerly wind burst. Figure 5(b)

7 MONTE CARLO ERROR COVARIANCES 3649 (a) (b) Figure 4. (a) Covariance for layer thickness using a reference point at 0 N, 180 W (marked by a cross) for the forecast ensemble, with contour interval 0.1 m m 1. (b) Cross-covariance between zonal stress and layer thickness at the reference point 0 N, 180 W for the forecast ensemble, with contour interval 0.2 (N m 2 )m 1. In both panels negative areas are shaded. shows a longitude time plot of the layer thickness along the equator. This shows the build-up of the Kevin and Rossby waves (although the amplitude of the Rossby wave is small along the equator) in the first 10 days during the westerly wind forcing. After the first 10 days, the waves propagate as free Rossby and Kelvin waves as there is no longer any wind forcing. The free Kelvin wave propagates with a speed of approximately 1.8 per day and the Rossby wave at approximate one third of this. During the first 10 days, the propagation speed is much smaller since the waves are being forced by the wind anomalies. After 10 days, the separation of the Kelvin and Rossby wave peaks is approximately 15 which is similar to that seen in the covariance pattern of Fig. 4(a). Note that, in the covariance pattern, the Rossby waves are not symmetric about the equator because the wind patterns are also not symmetric (see Fig. 3).

8 3650 O. ALVES and C. ROBERT (a) (b) Time (days) (East) Figure 5. Plots from shallow-water model forced with a westerly wind burst for the first 10 days: (a) layer thickness after day 10 with contour interval 3 m, and (b) longitude time plot of layer thickness at the equator with contour interval 4 m. In both panels negative areas are shaded. (ii) Continuous ensemble. Fields of model layer thickness were extracted every 10 days from each of the continuous ensemble members and the covariances calculated. These were then averaged over the 2-year period as done for the forecast ensemble in the previous section. The covariance pattern for layer thickness using a reference point at 0 N, 180 W is shown in Fig. 6. Centred on the reference point is a Gaussianshaped region of positive covariance with a north south length-scale of about 3 4 and a symmetric east west scale of about 20. In the far west Pacific is a band of negative covariance between about 8 Sand8 N, with peaks near the coastline at around 3 Nand 4 S. This pattern has a boomerang pattern symmetric about the equator and straddling the positive covariance. Another region of negative covariance with the point at 0 N, 180 W is centred on the equator to the east at 110 W with relatively large spatial scales, similar to that of the positive covariance centred on the reference point itself.

9 MONTE CARLO ERROR COVARIANCES 3651 Figure 6. Covariance for layer thickness using a reference point at 0 N, 180 W (marked by a cross) for the continuous ensemble. The contour interval is 0.1 m m 1 and negative areas are shaded. The covariance pattern from the continuous ensemble is similar to that from the forecast ensemble near the reference point, but significantly different far from the reference point. The forecast ensemble (Fig. 4(a)) only shows significant covariances local to the reference point, limited to the distance that the Kevin and Rossby waves can travel in 10 days and the length-scales of the wind stress covariance. However, the continuous ensemble shows significant and negative covariance on the opposite side of the Pacific basin. The covariance pattern seen in Fig. 6 can also be explained by Kelvin and Rossby wave dynamics generated by wind anomalies in the western Pacific. Unlike the forecast ensemble where these waves only propagate for 10 days, the waves in the continuous ensemble are allowed to propagate freely once generated, similar to the propagation after the first 10 days (see Fig. 5(b)) in the simple westerly wind burst experiment described in the previous section. The negative covariance between the east Pacific and the reference point on the Date Line can be explained by Kelvin wave propagation and time correlation of the wind errors. Figure 7(a) shows the time decorrelation of wind stress errors in the west Pacific (averaged between 150 E and 180 E along the equator). Wind stress errors have a time decorrelation scale of around days. Furthermore, wind stress errors are negatively correlated on a time-scale of days. Figure 7(b) shows the correlation between layer thickness at a reference point on the equator at 130 W and layer thickness at all other points on the equator at different lag times. The correlation is calculated across the ensemble members and averaged for all reference-point times (except the first 100 and last 100 days of the integration to allow for the lagged correlation). This figure shows that the variability at the reference point in the east Pacific is associated with the propagation of a Kelvin wave from the west Pacific, originating in the west Pacific some days earlier. The propagation speed is around 1 per day, about half the speed of a free Kelvin wave (see previous section), which means that the Kelvin wave is being forced by the wind stress. At zero lag, the layer thickness at the reference point in the east Pacific is negatively correlated with the layer thickness in the west Pacific, as seen in the covariance plot of Fig. 6, but this negative correlation is due to a Kelvin wave

10 3652 O. ALVES and C. ROBERT (a) (b) Time lag (days) (degrees east) Figure 7. (a) Lag correlation of zonal stress in the west Pacific (150 E 180 E along the equator). (b) time correlation of layer thickness along the equator with layer thickness at reference point 130 W at zero lag (indicated by a cross). The contour interval is 0.2 and negative values are shaded. Correlations were calculated across ensemble members and averaged over different reference-point times. of the opposite sign originating in the west Pacific some days earlier. The timescale of days that the wind forcing changes sign is similar to the time-scale for the Kelvin wave to travel halfway across the Pacific. Therefore, the covariance pattern seen in Fig. 6, in particular the strong negative covariance between the east and west Pacific, is due to variability in the west Pacific associated with a Kelvin wave being negatively correlated with a Kelvin wave of the opposite sign generated in the opposite phase of the intraseasonal variability.

11 MONTE CARLO ERROR COVARIANCES 3653 Note that the covariance patterns from the continuous ensemble are similar to those of Borovikov et al. (2005), except that they have applied a localization function that reduces the covariances far from the reference point. Borovikov et al. generated their ensemble by using different realizations of intraseasonal forcing from an atmospheric model. In a practical application of data assimilation, one would expect, at least in datarich regions, that the background error would be significantly reduced each assimilation time and that errors would not propagate over several assimilation cycles. In this case, the covariance patterns from the forecast ensemble may be more representative of the background errors than (say) the covariance pattern using a continuous ensemble. The use of a localization function, for example, as used by Borovikov et al. (2005), would help reduce the impact of non-local covariances. (d) Current covariances (i) Zonal current. For simplicity we will limit our discussion of the multivariate covariances (between layer thickness and currents) to the forecast ensemble experiments. The cross-covariance between layer thickness at a reference point and the zonal current at all other points in the model grid is shown in Fig. 8(a) for the forecast ensemble. The pattern shows a Gaussian-type distribution centred about 8 to the west of the reference point with peak value of 0.02 (m s 1 )m 1 and with a zonal length-scale of approximately 20 and a meridional length-scale of 4. Positive layer thickness anomalies correspond to eastward current anomalies. This cross-covariance pattern can be explained using the simple experiment discussed in section 3(c)(i) where the shallow-water model was forced for 10 days with a westerly wind burst. The zonal current at the end of the 10 days is shown in Fig. 9(a). The pattern shows an equatorially symmetric zonal current anomaly shifted slightly eastwards of the wind forcing and elongated along the equator. The meridional scales are shorter to the west than to the east. As discussed earlier, the westerly wind burst forces Kevin and Rossby waves, both of which have eastward currents along the equator. The elongated pattern is because the Rossby and Kelvin waves are moving away from each other. The maximum current lies to the east of the maximum wind forcing (Date Line) and to the west of the layer thickness anomaly associated with the forced Kelvin wave. The covariance pattern from the forecast ensemble is similar to that generated by a westerly wind burst. The zonal current cross-covariance is shifted westwards relative to the layer thickness at the reference point (Fig. 8(a)) but shifted eastwards compared to the wind stress cross-covariance associated with layer thickness at the same reference point (Fig. 4(b)). Burgers et al. (2002) showed that for wind errors in the west Pacific, the zonal current errors near the equator were close to being in geostrophic balance with the temperature errors. They recommended making geostrophic current corrections based on the temperature corrections. We used the forecast ensemble results to check how closely the cross-covariance between layer thickness errors and zonal current errors was to geostrophic balance. We calculated a geostrophic zonal current error covariance from the gradient of the layer thickness covariance function, i.e. c u = g f c h y, (3)

12 3654 O. ALVES and C. ROBERT (a) (b) Figure 8. (a) Cross-covariance of zonal current with layer thickness at 0 N, 180 W (indicated by a cross). (b) Geostrophic relation applied to layer thickness covariance using Eq. (3). In both panels, the contour interval is (m s 1 )m 1 and negative values are shaded. where c h is the covariance of layer thickness (as in Fig. 4(a)), f is the Coriolis parameter and g is the reduced gravity. For points on the equator where f = 0, c u was calculated each side of the equator and averaged onto the equator in order to produce a continuous geostrophic current across the equator, as suggested by Burgers et al. (2002). The resulting current covariance field is shown Fig. 8(b) and can be compared with Fig. 8(a). The two figures are very similar, indicating that for wind stress errors the zonal current errors are in close geostrophic balance with the layer thickness errors and that this relationship is captured by the forecast ensemble. (ii) Meridional current. The meridional current error cross-covariance with layer thickness error at the reference point of 0 N, 180 W is shown in Fig. 10(a). It shows equatorial convergence associated with thermocline depression. The area of maximum

13 MONTE CARLO ERROR COVARIANCES 3655 (a) (b) Figure 9. Plots after day 10 from the simple model forced with a westerly wind burst for the first 10 days: (a) zonal current with contour interval 0.05 (m s 1 )m 1, and (b) meridional current with contour interval 0.01 (m s 1 )m 1. In both panels negative areas are shaded. convergence is shifted some 20 to the west of the reference point and extends up to 20 from the equator. This meridional circulation, together with the westward portion of the eastward current along the equator (Fig. 8(a)), is due to cyclonic gyres associated with the Rossby waves. A similar pattern can be seen in the simple experiment where the model was forced for 10 days with a westerly wind burst (Fig. 9(b)). A geostrophic current error covariance was calculated in the same way as for the zonal component, using c v = g f c h x. (4) This is shown in Fig. 10(b). The geostrophic covariance shows equatorially concentrated divergence to the west and convergence to the east of the reference point. Clearly the meridional current error covariances (Fig. 10(a)) from the forecast ensemble are not in

14 3656 O. ALVES and C. ROBERT (a) (b) Figure 10. (a) Cross-covariance of meridional current with layer thickness at 0 N, 180 W (indicated by a cross). (b) Geostrophic relation applied to layer thickness covariance using Eq. (4). In both panels the contour interval is (m s 1 )m 1 and negative values are shaded. geostrophic balance with the layer thickness errors. Therefore, the ensemble covariances have the advantage of providing dynamical consistency in situations where geostrophy does not dominate. 4. CONCLUSIONS Monte Carlo simulations using a shallow-water model are used to estimate model error covariances. The simulations are perturbed with a novel technique that uses random zonal wind stress error estimates. These stress perturbations are formed by randomly combining EOFs of the intraseasonally filtered difference between NCEP and ERA-40 wind stresses. Two ensemble experiments were carried out, one using an ensemble of two-year-long continuous simulations and one imitating 10-day forecasts throughout the same two-year period.

15 MONTE CARLO ERROR COVARIANCES 3657 The layer thickness error covariances showed patterns in the west Pacific characteristic of the generation of Kelvin and Rossby waves associated with wind stress bursts. For example, a westerly wind stress error would generate a downwelling Kelvin wave on the equator that subsequently propagated eastwards, and upwelling Rossby waves each side of the equator that propagated westwards. The forecast ensemble showed local covariance structures, limited by the distance that the Kevin wave travelled in the forecast period. However, the covariance patterns from the continuous ensemble showed significant non-local covariances, e.g. negative covariance between layer thickness variability in the east and west Pacific. This nonlocal covariance was explained as being a result of the time decorrelation of wind stress errors being similar to the time it takes a Kevin wave to cross halfway across the Pacific, so that a downwelling Kelvin wave succeeds an upwelling Kelvin wave, and so on. Therefore, at any point in time, a wave of one sign would be crossing the east Pacific, with a wave of the opposite sign being generated in the west Pacific. The cross-covariance between ocean currents and layer thickness was examined using the forecast ensemble. The zonal current covariance was found to be close to geostrophic balance. The same was not true for the meridional current. The current layer thickness cross-covariances from the forecast ensemble were dynamically consistent with Kelvin/Rossby wave dynamics and provide a method of generating current increments from temperature observations (even when geostrophy does not dominate). In a practical data assimilation application, the background-error covariance will be somewhere between the error covariance from the continuous ensemble and the error covariance from the forecast ensemble. In relatively data-rich areas (e.g. the TOGA TAO array region in the equatorial Pacific), one might expect the error covariance to be closer to the errors from the forecast ensemble as the data significantly reduce the background error each analysis time, whereas in data-sparse regions (e.g. Indian Ocean), it may be closer to the errors from the continuous ensemble. Our results demonstrate a novel technique for generating forecast ensembles using different surface forcing errors. When used to generate forecast ensembles, in particular short forecast ensembles, the error covariances and cross-covariances produced are consistent with equatorial Kelvin and Rossby wave dynamics. This approach shows potential for its use in explicitly generating model error perturbations in an EnKF framework. In the future we will repeat these experiments with a full ocean generalcirculation model and implement the ensemble generation approach in an EnKF. ACKNOWLEDGEMENTS We would like to thank Gerrit Burgers of KNMI for providing the shallow-water model used in this study. We would also like to thank ECWMF and NCEP for providing their respective re-analysis data. We are grateful to Diana Greenslade for providing useful comments that improved the quality of this paper. We are very grateful to ENM of Météo-France for allowing Christelle Robert to spend a five-month training period at BMRC, during which most of this work was done. REFERENCES Borovikov, A., Rienecker, M. M., Keppenne, C. L. and Johnson, G. C. Burgers, G., van Leeuwen, P. J. and Evensen, G Multivariate error covariance estimates by Monte Carlo simulation for assimilation studies in the Pacific Ocean. Mon. Weather Rev., 133, Analysis scheme in the Ensemble Kalman Filter. Mon. Weather Rev., 126,

16 3658 O. ALVES and C. ROBERT Burgers, G., Balmaseda, M. A., Vossepoel, F. C., van Oldenborgh, G. and van Leeuwen, P. J Balanced ocean-data assimilation near the equator. J. Phys. Oceanogr., 32, Evensen, G Advanced data assimilation for strongly linear dynamics. Mon. Weather Rev., 125, Harrison, D. E. and Vecchi, G. A Westerly wind events in the tropical Pacific, J. Climate, 10, Kållberg, P., Simmons, A., Uppala, S. and Fuentes, M. Keppenne, C. L. and Rienecker, M. M. Kessler, W., McPhaden, M. and Weickmann, K. Kistler, R., Kalnay, E., Collins, W., Saha, S., White, G., Woollen, J., Chelliah, M., Ebisuzaki, W., Kanamitsu, M., Kousky, V., van den Dool, H., Jenne, R. and Fiorino, M The ERA-40 archive. ERA-40 Project report series no. 17, ECMWF, Reading, UK 2002 Initial testing of a massively parallel Ensemble Kalman Filter with the Poseidon isopycnal ocean general circulation model. Mon. Weather Rev., 130, Forcing of intraseasonal Kelvin waves in the equatorial Pacific. J. Geophys. Res., 100, The NCEP NCAR 50-year re-analysis: Monthly means CD- ROM and documentation. Bull. Am. Meteorol. Soc., 82,