1 Advantages of data assimilation in coastal ocean circulation models: Oregon perspective Alexander L. Kurapov, J. S. Allen, G. D. Egbert, R. N. Miller COAS/Oregon State University In cooperation with P. M. Kosro, M. D. Levine, T. Boyd, J. A. Barth, J. N. Moum, P. T. Strub, S. Erofeeva 29 January 2004, AGU/Ocean Sciences

2 wind stress (upwelling favorable) is dominant forcing strong effects of flow- topography interactions energetic internal tide Summer circulation on the Oregon shelf: HF radars Moorings (ADP, T, S) currents: 3D+time density: 3D+time Summer 2001: DA system is implemented with data from COAST observational program Data assimilation:  improves prediction of the ocean state,  provides solution error estimates,  is used as a tool for data synthesis,  helps to design an observational system (e.g., suggests optimal observational locations)

3 Dual approach: Variational (generalized inverse)  DA method  Simpler, sequential (optimal interpolation) Linearized  Dynamics  Fully non-linear Internal tides  Application  Wind-driven circulation Objectives: to develop practical, but still nearly optimal methods for the assimilation of data into coastal circulation models to apply these methods to measurements from the Oregon shelf to utilize DA to increase scientific understanding of shelf circulation

4 Model of of M 2 internal tide [Kurapov et al., JPO 33, 2003] - linearized, primitive eqns, 3D, periodic in time [~exp(i  t)] - terrain following coordinates e.g., momentum equations: HF (P. M. Kosro) HF ADP Model domain: 40  60 km,  x=1 km, 21  - layers - Zone of coverage of 2 HF radars (May- July 1998) - Efficient model solver (direct factorization of the model operator) - Address open boundary issues Most internal tide comes from outside the computational domain DA: corrects open boundary baroclinic flux

5 Generalized Inverse Method (GIM): Solution minimizes a cost function: Cost Function = || Model error || 2 + || BCond error || 2 + || Obs error || 2  min Explicit statistical assumptions about errors in the inputs - Explicit statistical assumptions about errors in the inputs - Statistics in the output (prior model and inverse solutions) are computed [] - Statistics in the output (prior model and inverse solutions) are computed [ Bennett, 1992, 2002 ] State vector: v = {velocity, sea surface elevation, density} Model+BCond: S v = f + e m Data: L v = d + e d errors in model forcing and data specified prior to assimilation

6 Use of Representers: Model+BCond: S v = f + e m Data: L v = d + e d Adjoint solverFwd solver Reduce burden of representer computation with: - reduced basis representer approach - indirect representer approach [Egbert et al., JGR, 1994] HF radars: K=900 locations where radial velocity components are available Standard feature in Inverse Ocean Modeling system [IOM, Chua and Bennett, Ocean Modeling, 2001] vovo Strongly constrained dynamics:

7 Solution sensitivity to the choice of model error covariance C OB (in an experiment with synthetic data) -”true” solution: forced at open boundary (OB) with a significantly baroclinic flux -synthetic data (velocity harmonic constants) are sampled from true solution -prior model: forced with depth-averaged OB current -DA: corrects OB baroclinic fluxes Depth-ave RMS error with respect to true solution Prior DA, C OB (Type I) DA, C OB (Type II) these two solutions allow for OB b/clinic correction of the same magnitude (but different correlation structure)

8 DA C OB (Type I) is obtained by nesting approach: In a large domain, compute representers for small domain boundary data then sample these representers along the OB of small domain  C OB (covariance for the errors on the OB of the small domain, with a dynamically consistent spatial structure) C OB controls radiation at an open boundary representer  column of prior solution error covariance matrix C OB (Type II): our best guess w/out nesting

9 A series of M 2 tidal solutions, May-July 1998 Internal tide intermittence: analysis in 2-week overlapping time windows DA: in each time window Validation ADP DA solution No DA deviations from depth- ave. (CW) depth-ave (rotating CCW) Assimilation of HF surface currents improves prediction at depth Tidal ellipses of horizontal currents at ADP location, vs depth: (a) observed, (b) prior model, (c) DA. ADP

10 M 2 tidal ellipses on the surface: internal tide velocities can be twice as large as barotropic tidal velocities CCW rotation CW rotation Depth-ave Deviations from depth-ave (time window centered on day 139)

11 Energy balance is closed : Data assimilation corrects only boundary inputs 40 W m -1 Most baroclinic signal comes into the computational domain from outside Some persistent features are found: e.g., baroclinic phase and energy propagation is from NW. Terms in the baroclinic energy equation (time and space averaged) Baroclinic energy flux (depth- integrated and time-ave.) day, 1998

12 Baroclinic KE averaged over a series of days : a) surface, b) bottom, c) cross-section north of Stonewall Bank, d) cross-section through Stonewall Bank. Zones of higher KE variability are aligned along the coast, consistent with energetic of a internal Poincare wave interaction with bathymetry Dominance of 1st baroclinic mode beams over Stonewall B A series of tidal solutions (constrained by HF radar data) provides a uniquely detailed description of spatial and temporal variability of M 2 internal tide

13 Model of wind-driven circulation: AVHRR SST, o C [courtesy P.T. Strub] -Princeton Ocean Model: 220  350 km, periodic OB conditions (south-north),  x~2 km, 31  -layers -Forcing: alongshore wind stress, heat flux -Data assimilation: Optimal Interpolation -Initial implementation (summer 1998): assimilation of HF radar data improves modeled circulation at depth [ Oke et al., JGR- Oceans, 2002 ] -Data from COAST program (summer 2001): assimilate moored ADP currents

14 Optimal Interpolation (3DVAR): matrix matching observations to state vector ||Error|| Time model w/out DA DA forecast analysis Forecast error covariance (stationary in OI): P f = P m  F (lagged P m, C d ) where P m is the covariance of errors in the model solution not constrained by the data (in contrast, P f is conditioned upon previously assimilated data) [ Kurapov et al., Mon. Wea Rev., 2002 ] P f has a shorter horizontal scale in the alongshore direction than P m (effect of propagation) P m : could be obtained as representer calculation, if an adjoint model were available Presently, P m is computed from an ensemble of model solutions Incremental approach: correction is applied gradually over the analysis time window (1/4 of inertial period)

15 Spatial structure of P f : NMS, 12m SSB, 16m [cm 2 s -2 ]

16 Time- and depth-ave terms in the momentum eqn. (along-jet direction) no DA DA (ADPs in south) Dominant dynamical balance is preserved Smooth, large scale correction (in this case, DA tends to reduce upwelling intensity)

17 Assimilation of moored ADP velocities (May-Aug 2001): 90 km Central part of model domain with mooring locations, Bathymetry each 100 m (black) and 10 m (half-tone, from 0 to 200 m) Moorings: Lines N and S – COAST (Kosro, Levine, Boyd), NH10 – GLOBEC (Kosro) Study is focused on: -Distant effect of data assimilation - Multivariate capabilities (effect on SSH, isopycnals, temperature, salinity transport, turbulent dissipation rate 

18 Case 1: assimilate currents at Northern Line  improve currents at NH10, SSB Correction can be advected by a predominantly southward current 90 km ADP sites, May-Aug 2001 Assimilated ADP sites Sites where DA is better than model only solution (smaller model-data rms error, larger correlation) NH10 SSB rmse: 7.8  5.8 cm s  1, corr: 0.18  0.71 rmse: 9.6  7.1 cm s  1, corr: 0.36  0.70 Alongshore depth-ave current: obs, no DA, DA

19 Case 2: assimilate ADP currents at Southern Line  improve currents up North Correction can be propagated northward with coastal trapped waves NMS NH10 rmse: 11.3  7.9 cm s  1, corr: 0.46  0.79 rmse: 7.8  6.9 cm s  1, corr: 0.18  0.63 Alongshore depth-ave current: obs, no DA, DA ADP sites, May-Aug 2001 Assimilated ADP sites Sites where DA is better than model only solution (smaller model-data rms error, larger correlation)

20 Posterior error statistics analysis E.g., compare expected and actual analysis rms error as a consistency test for P f Expected performance diag (P m ) and (P a ) are compared, where P a = P f – G H P f is the analysis error covariance Actual performance Assimilated site DA is better than model only solution DA is worse than model only solution Discrepancy between expected and actual outcome when assimilating inner- shelf data : artificially large decorrelation scale in P f  inclusion of a more realistic spatially varying wind stress is a necessity

21 Multivariate capabilities no DA DA (South) SeaSoar measurements (Barth et al.) e.g., effect on SSH (validation - tide gauge data): effect on isopycnal slope: Model-data Corr.: 0.51  0.78, rmse: 5.4  3.8 cm SSH: obs, model only, DA (Lines N+S) ( white contours are measured    24, 25, and 26 kg m -3 ) + improvement in temperature correlations, surface salinity transport

22 Turbulent Dissipation rate (  )  Microstructure data [J. Moum, A. Perlin] No DADA (North) 12 transects on Line N yearday, 2001 Time series of  averaged near bottom (in box area) DA correction in near-bottom velocity field yields improvement in  Analysis of BBL dynamics is extended for the whole study period – presentation OS52I-08

23 SUMMARY: Progress has been made on both aspects of the dual approach to coastal ocean DA Linearized dynamics, variational DA (internal tides) -has provided unique information on spatial and temporal variability of internal tide from HF radar measurements of surface currents -has given us experience in open boundary DA Nonlinear dynamics, sequential OI DA (wind-driven circulation) - has shown the value of assimilation of currents from HF radar and from moored ADPs (distant effect, multivariate capabilities, BBL analysis) -has provided information on optimal ADP mooring locations and on effective alongshore scales of ADP current measurements In both cases, formulation of error hypotheses is the science and art of DA DA is utilized to increase scientific understanding of shelf circulation

24 PLANNED RESEARCH:  Merger of approaches: use tangent linear and adjoint codes for a fully non-linear ocean circulation model (ROMS)  Use data assimilation to help provide open boundary conditions for high-resolution limited-area coastal models  Tidal research: study effect of wind-forced subinertial flows on internal tide propagation  Study of wind-forced upwelling circulation: analyze cross-shelf transport, bottom boundary layer processes, dynamical balances