ARTICLE IN PRESS. Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx. Contents lists available at ScienceDirect. Dynamics of Atmospheres and Oceans

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1 Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Contents lists available at ScienceDirect Dynamics of Atmospheres and Oceans journal homepage: Representer-based analyses in the coastal upwelling system A.L. Kurapov, G.D. Egbert, J.S. Allen, R.N. Miller College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, United States article info abstract Available online xxx Keywords: Data assimilation Coastal ocean dynamics Representers The impact of surface velocity and SSH data assimilated in a model of wind-driven upwelling over the shelf is studied using representer and observational array mode analyses and twin experiments, utilizing new tangent linear (TL) and adjoint (ADJ) codes. Bathymetry, forcing, and initial conditions are assumed to be alongshore uniform reducing the problem to classical two-dimensional. The model error is attributed to uncertainty in the surface wind stress. The representers, analyzed in cross-shore sections, show how assimilated observations provide corrections to the wind stress and circulation fields, and give information on the structure of the multivariate prior model error covariance. Since these error covariance fields satisfy the dynamics of the TL model, they maintain dominant balances (Ekman transport, geostrophy, thermal wind). Solutions computed over a flat bottom are qualitatively similar to a known analytical solution. Representers obtained with long cross-shore decorrelation scale for the wind stress errors l x (compared to the distance to coast) exhibit the classical upwelling structure. Solutions obtained with much smaller l x show structure associated with Ekman pumping, and are nearly singular if l x is smaller than the grid resolution. The zones of maximum influence of observations are sensitive to the background ocean conditions and are not necessarily centered around the observation locations. Array mode analysis is utilized to obtain model structures (combinations of representers) that are most stably observed by a given array. This analysis reveals that for realistic measurement errors and observational configurations, surface velocities will be more effective than SSH in providing correction to the wind stress on the scales of tens of km. In the DA test Corresponding author. address: kurapov@coas.oregonstate.edu (A.L. Kurapov) /$ see front matter 2008 Elsevier B.V. All rights reserved.

2 2 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx with synthetic observations, the prior nonlinear solution is obtained with spatially uniform alongshore wind stress and the true solution with the wind stress sharply reduced inshore of the upwelling front, simulating expected ocean atmosphere interaction. Assimilation of daily-averaged alongshore surface currents provides improvement to both the wind stress and circulation fields Elsevier B.V. All rights reserved. 1. Introduction Data assimilation (DA) has been implemented in oceanography to combine circulation models and observations with the primary goal of obtaining improved estimates of the ocean state (e.g., Bennett, 1992, 2002; Wunsch, 1996; Evensen, 2007). Recent advances in observational technology have stimulated the development of DA over coastal shelves (Lewis et al., 1998; Oke et al., 2002; Besiktepe et al., 2003; Kurapov et al., 2003, 2005a,b; Di Lorenzo et al., 2007; Li et al., 2008; Barth et al., 2008). Available observations include surface velocities from land-based high-frequency (HF) radars (Kosro, 2005), high-resolution alongtrack altimetry and SST maps from satellites (Venegas et al., 2008), temperature and salinity sections from autonomous underwater vehicles and gliders (Castelao et al., 2008), and time-series of velocities, temperature and salinity from moorings. These data sets remain sparse compared to the small scale, rapidly evolving, and nonlinear circulation processes in the coastal ocean. To facilitate development of effective DA systems, it is important to understand the spatial and temporal scales of influence of the different data types in the coastal ocean. It is also important to learn how assimilation of observations of a given type affects estimation of unobserved oceanic fields. Given the nonlinear character of the coastal flows, answers to these questions can depend on the state of the ocean. A variational approach provides a convenient framework for the assessment of the impact of observations in a DA system. Variational methods attempt to obtain ocean state estimates by minimizing a penalty functional that is a sum of quadratic terms on errors in observations and model inputs integrated over space and a specified time interval. The inputs adjusted by DA may include initial and boundary conditions, forcing, model parameters, and space- and time-dependent errors in the dynamical equations. Minimization algorithms leading to the optimum solutions are complicated and can require the repeated use of companion tangent linear (TL) and adjoint (ADJ) models. However, if computational burdens of variational DA can be overcome, the resulting solutions bear many attractive features. In particular, these estimates can be interpreted as a result of space time interpolation (objective mapping) of the data, in which interpolation (model error covariance) functions are consistent with the model dynamics and dependent on the ocean state. These covariance functions are multivariate, such that observations of one type influence the correction to the prior model fields of different type (for instance, velocity measurements can be utilized to provide dynamically consistent corrections to velocity, temperature, and salinity fields). Although in practice the model state error covariance functions do not have to be provided or computed explicitly, it can still be instructive to analyze them to understand the zones of influence and the multivariate impact of assimilated observations. In this manuscript, we explore the utility of tools for observational array assessment in the framework of a particular variational algorithm, namely the representer method (Bennett, 1992, 2002; Chua and Bennett, 2001), implemented for the case of wind-driven upwelling on the shelf. In the representer methodology the minimization of the penalty functional proceeds as a series of linearized optimization problems, with the correction to the prior model at each step obtained as a linear combination of representer functions, one for each observation. Based on the statistical interpretation of variational DA, the representer is the prior model error covariance between the observed quantity and all the elements of the three-dimensional (3D), time-dependent, and multivariate ocean state vector. A representer shows zones of influence of a single assimilated observation in space and time. It satisfies the linearized model dynamics and depends on the background ocean state. The representer method is particularly explicit and flexible in the choice of error covariances in the model inputs, which define

3 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 3 norms in the penalty functional. Thus, representer analysis can illuminate the dependency of the prior model error covariance (i.e., error covariance in the model outputs) on the assumed input error statistics. For instance, one can compare the structure of the prior model error covariances corresponding to different error sources, such as boundary conditions or local forcing, or different error decorrelation scales in the forcing. The variational representer formalism is not the only possible way to estimate prior model error covariances. Echevin and De Mey (2000) and Broquet et al. (2008) utilized ensembles of model runs to obtain covariances (representers) in regional applications. Le Hénaff and de Mey (2008, in preparation) have used a similar approach to provide assessment for large arrays of observations such as proposed wide-swath altimetry sets (Fu, 2003). Their representer spectrum analysis is similar to the array mode analysis (Bennett, 2002) implemented in Section 5 of this manuscript. Advantages of the ensemble approach is that TL&ADJ models are not needed. However, with such an approach accurate estimation can be limited by the ensemble size. For a system with many degrees of freedom ensemble calculations can result in spurious long-range correlations, which are often dealt with by localizing the estimated representer. The variational representer approach avoids these complications, providing a more direct and reliable link between the assumed statistics of uncertainties in model inputs and calculated statistics of output errors. Study cases in this manuscript are designed to be relevant to the dynamics on the Oregon shelf (U.S. West Coast) where predominantly southward winds drive a baroclinic alongshore current and upwelling during summer (Allen et al., 1995). In this system, it is plausible that uncertainty in the wind stress is a dominant source of model error. In wind stress estimates obtained from an atmospheric model, errors in prediction of mesoscale features will have relatively long spatial decorrelation scales [O (100 km)]. Errors with smaller scales [O (10 km)] can be associated with processes misrepresented in the atmospheric model, including orographic effects near coastal irregularities (Samelson et al., 2002; Perlin et al., 2004) or coupled ocean atmosphere effects (Perlin et al., 2007). Our paper provides analyses for two types of observations, namely, SSH and surface alongshore velocities. Although the standard alongtrack satellite altimetry products are now not available closer than km from the coast, improving processing algorithms (P.T. Strub, pers. comm.) and developing new observational technologies including the delay Doppler radar (Raney, 1988) and wide-swath altimetry (Fu, 2003) hold promise for making these data available over ocean shelves in the future. Surface currents from HF radars have already proven to be a valuable source of information about subsurface flows (e.g., Oke et al., 2002; Barth et al., 2008). The study presented here has two goals. First, we pursue representer-based analyses as an initial test of new TL&ADJ codes that we have developed to implement the representer algorithm for shelf areas. Second, we illustrate fundamental features of the multivariate prior model error covariance for assimilation on the coastal shelf in the wind-driven upwelling regime. The prior model error is assumed to arise from uncertainties in the wind stress, with decorrelation scales ranging from large to small (compared to the baroclinic Rossby radius of deformation). All the tests presented below are performed in an idealized set-up assuming no variability in the alongshore direction, which reduces the problem from 3D to 2D (cross-shore and vertical coordinates). Despite this idealization, the analyses retain many features essential for coastal upwelling, including baroclinicity, shelf slope effects, advection, and nontrivial background ocean conditions. Detailed nonlinear dynamical analyses of alongshore and cross-shore transports and turbulence in the wind-driven regime have been done for a similar set-up, e.g., by Allen et al. (1995), Federiuk and Allen (1995), Allen and Newberger (1996), Austin and Lentz (2002), Wijesekera et al. (2003), and Kuebel Cervantes et al. (2004). Here we look at this familiar dynamical problem from the standpoint of adjoint-based DA. In 3D applications, there has been a concern that alongshore instabilities, occurring in shelf circulation models on temporal scales of several days (e.g., Durski and Allen, 2005) and growing unconstrained in the TL model, may potentially limit applicability of variational DA in the coastal ocean. Admittedly, in our idealized set-up, the problem of these alongshore instabilities is avoided. Our recent study with a shallow-water model of circulation in the nearshore surf zone has provided insights on how variational assimilation can be done effectively over time intervals exceeding the time scales associated with instability growth (Kurapov et al., 2007).

4 4 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx In the following sections, after the description of the basics of representer methodology (Section 2) and the model (Section 3), we provide analysis of single representers (Section 4), in particular, their dependence on the details of model topography, statistical assumptions about errors in the wind stress, and background ocean conditions. Then, in Section 5, array mode analysis is utilized as a tool for assessment of utility of a set of observations. An idealized DA test with synthetic observations of surface velocity (a twin experiment, Section 6) is performed to confirm that the DA performance is consistent with inferences from the representer and array modes analyses. 2. Basics of the representer methodology Let us write the nonlinear model symbolically as q t = N(q) + fprior + e, q(0) = q prior 0 + e 0, (1) (2) where q(t) is the true state (multivariate, discrete in space, continuous in time), N is the nonlinear model operator, f prior (t) is the prior forcing vector (which in this formulation may include errors in interior dynamics, surface forcing, and boundary conditions), q prior is the prior initial condition, e(t) 0 is the dynamical error, and e 0 is the initial condition error. The observations are written in the general form as g T 1 (t) g L(q) = dt 2 (t) q(t) = d + e d, (3) 0 g K (t) where L is a linear operator matching the model state q and the observations, g k define the observational functionals (sampling rules for each datum d k ), d ={d k } is the vector of observations (of size K 1), e d is the observation error, and the prime denotes matrix transpose. The penalty functional to be minimized can be written as T T J(ˆq) = dt 1 dt 2 ê (t 1 )C 1 (t 1,t 2 )ê(t 2 ) + ê 0 C 1 0 ê0 + ê d C 1 d êd, (4) 0 0 where ê, ê 0, and ê d are residuals satisfying (1) (3) with ˆq. C 1 and C 1 0 are inverse prior error covariances in the corresponding inputs. C 1 is the inverse data error covariance. Note that C, C d 0, and C d must be specified prior to assimilation. For the derivation of equations for the extremum of (4) and for details of linearization, see (Chua and Bennett, 2001; Bennett, 2002; also Kurapov et al., 2007). Here, we only outline details of the representer computation. In the following, the TL operator is defined as A[ q] = N q. (5) q= q In our notation, it is a matrix with elements depending on the time-variable background ocean state q. As noted in Section 1, the correction to the prior model can be written as a linear combination of the representer functions r k (t). To obtain each representer, the ADJ model is run backward in time forced with the kernel of the kth observational functional: t A [ q] = g k, (T) = 0. (6) (7)

5 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 5 The adjoint solution is convolved with the input error covariances and then used to force the TL model: T r k = A[ q]r k + dt 1 C(t, t 1 )(t 1 ), (8) t 0 r k (0) = C 0 (0). (9) For instance, the analysis of representers in Section 4 will be done for local and instantaneous observations (SSH or surface alongshore velocity component), such that g k in (6) will be an impulse in the corresponding field at the observational location and time. The prior model error will be assumed to be only due to that in the wind stress. Correspondingly, C 0 in (9) will be set to 0 (no correction to initial conditions), and the last term in (8) will represent a correction to wind forcing. Covariance C will provide spatial and temporal smoothing of the forcing correction. As mentioned above, the representer is the prior error covariance between the observed value g k q and all the elements of the model vector q(t). This covariance is computed given the assumptions of error covariances in the inputs (C, C 0 ). It satisfies the linearized model dynamics and is thus dependent on the background ocean state q(t). As a test of validity of the TL code, one can verify whether the components of the multivariate representer r k satisfy the dominant linear dynamical balance relations. Note that the representer does not depend on the actual observational value d k, so representers can be utilized for the assessment of the potential impact of observations. The set of coefficients of the optimal combination of representers k b kr k, providing correction to the prior model state q prior, can be found as b ={b k }=(R + C d ) 1 [d L(q prior )], (10) where R = L[r 1 r 2... r K ] is the symmetric non-negative representer matrix, obtained by sampling the representers at observational locations and times. Note that in practice, with a large number of data, it would be impractical and unnecessary to compute and store all the representers. Optimal linear combinations can be obtained iteratively using the indirect representer method, involving a series of ADJ and TL model computations (Egbert et al., 1994; Chua and Bennett, 2001). 3. Model 3.1. General model details The Regional Ocean Modeling System (ROMS, Shchepetkin and McWilliams, 2005; Wilkin et al., 2005) is utilized to describe the nonlinear dynamics. ROMS is based on the baroclinic, free surface, hydrostatic primitive equations discretized on a terrain following coordinate grid. To represent the physics on subgrid scales, we have used the Mellor-Yamada 2.5 turbulence scheme (Mellor and Yamada, 1982; see also Wijesekera et al., 2003). In this study, we have utilized our own, newly developed TL&ADJ codes, AVRORA. 1 This development has been influenced by the experiences reported by researchers using adjoint components of popular models based on the hydrostatic primitive equations, including ROMS (Moore et al., 2004; Di Lorenzo et al., 2007), the MIT General Circulation Model (e.g., Stammer et al., 2002; Gebbie et al., 2006) and Navy Coastal Ocean Model (NCOM; Ngodock, priv. comm.). Details making our code structure different from TL&ADJ ROMS will be mentioned later in this section. In this paper we provide only a brief outline of our model development and focus primarily on analyses that to our knowledge have not been previously reported for any model. Our TL&ADJ codes have been developed manually following recipes for line-by-line automatic code generation (Giering and Kaminski, 1998). The algorithms for time-stepping and barotropic baroclinic mode splitting have been adopted from ROMS. At every time step, the background ocean state is obtained by linear interpolation between fields provided at specified time instances (in our examples 1 Advanced Variational Regional Ocean Representer Analyzer.

6 6 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx below, once every 4 h). Although a range of high-order advection schemes are available in ROMS, only the second-order centered scheme has been coded in AVRORA. This choice is the most straightforward since the second order scheme is differentiable (in particular, it does not include non-differentiable IF statements as in some higher order schemes). Also, boundary conditions for this scheme are more straightforward compared to higher order schemes that have a larger footprint (stencil) on the grid and thus require extra, numerical boundary conditions. Similarly to the ROMS development, our TL&ADJ codes do not involve a linearized version of the turbulence equations. The vertical eddy dissipation and diffusion coefficients, K M and K H, are known functions of space and time provided with the background state, e.g., obtained from the nonlinear model. In our applications, a nonlinear equation of state for seawater is utilized in nonlinear runs with ROMS, and a linear equation of state is used in AVRORA: = (1 T (T T ) + S(S S )), (11) where is the in situ density, T is the potential temperature, S is the salinity, and variables with subscript are their respective reference values. Quadratic bottom friction has been used in the nonlinear ROMS runs and linear friction with the drag coefficient of ms 1 in TL&ADJ AVRORA. In DA, it is important to define clearly the states of inputs, since those are to be corrected, and outputs, since those are to be matched to the observations. The need to attend to these details was one of the motivational points for development of the AVRORA codes, which have been structured to simplify implementation for a variety of data functionals and model error assumptions. In these details, discussed in the next two paragraphs, our model differs from TL&ADJ ROMS applications described by Moore et al. (2004) and Di Lorenzo et al. (2007). The vector of outputs of our TL model includes the sea surface height, two orthogonal components of horizontal velocity (u, v), T, and S. The TL model outputs instantaneous fields every N HIS time steps into the history file. In applications considered here, the vector of inputs includes the initial values for the elements of the state vector and the wind stress fields specified at selected time instances. As in nonlinear ROMS, our TL code obtains the wind stress by linear interpolation between values provided at specified times. The ADJ model outputs sensitivities to the wind stress at the same specified time instances using the appropriate adjoint to the time-interpolation, similarly to (Kurapov et al., 2007). In particular, if the wind stress is specified in the TL model at times t = t 1,t 2,...,t N, then the ADJ model, executed backward in time, does not output sensitivity to the forcing at time t k until the model time t becomes smaller than t k 1, since the TL solution at t k 1 <t<t k is sensitive to forcing values both at times t k 1 and t k. From (3), it follows that the kernels of the data functionals g k, providing the forcing of the ADJ model, have to be defined as vectors in the same space as the TL outputs q. Then, it is convenient to provide these as files of the same structure as the output (history) file from the TL model, with fields of the coefficients of the linear combination of the state vector elements organized as a series of values every N HIS time steps. As the ADJ model steps backward in time, those values are added to corresponding interior adjoint variables at appropriate times. If the input of g k for forcing the ADJ model is organized in this way, any linear combination of the elements of the state vector can be assimilated without making additional changes in the TL&ADJ codes (e.g, including time-averaged and low-pass filtered observations, HF radar radial components of velocity, SSH alongtrack anomalies). This approach is consistent with the methodology of the modular Inverse Ocean Modeling system (Bennett et al., 2008), but has not always been used in previous developments of TL&ADJ codes. Note that the dimensions of discrete spaces of inputs and outputs are different. The TL model can be viewed as the rule by which a rectangular matrix [TL] multiplies the vector of inputs. In the way the ADJ model has been constructed, it provides the rule by which the transposed matrix [TL] multiplies a vector of the same dimension as the output vector in the TL model. A series of adjoint symmetry checks (Kurapov et al., 2007) has been performed throughout the model development to ensure that [TL] [TL] is symmetric and positive definite within computer accuracy. In its present form, AVRORA has been tested with periodic boundary conditions and closed boundaries around the domain. The ADJ code provides sensitivities to the initial conditions and the atmospheric forcing. Inclusion of sensitivities to open boundary conditions is planned in the future.

7 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Implementation details The 2D (cross-shore and vertical coordinates) solutions are obtained by running the model in a short south-to-north periodic channel with alongshore uniform bathymetry, forcing, and other conditions. Cartesian coordinates are introduced with the x-axis directed toward the coast, y-axis north, and z- axis upward. The cross-shore u and alongshore v components of horizontal velocity are projections on x and y directions, respectively. No-normal-flow and free-slip boundary conditions are applied at the coast (x = 0) and offshore boundaries (x = 200 km). The resolution is 2 km in horizontal and 40 layers in vertical, with relatively finer resolution near the surface and bottom. The Coriolis parameter is f = 10 4 s 1. In the TL&ADJ model, the parameters of the linear equation of state (11) are obtained based on hydrographic data off Oregon: = 1025 kg m 3, T = 10 C, S = 34 psu, T = C 1, and S = psu 1. Representer solutions will be first considered for a flat bottom. In these cases the results of our TL&ADJ model can be compared, at least qualitatively, to the analytical representer solutions obtained in the near-coast boundary layer using long-wave and low-frequency approximations (Scott et al., 2000; Kurapov et al., 1999, 2002). In our examples, the depth is H = 200 m. The background ocean ( q) isatrest:ū = v = 0 and T and S are horizontally uniform and varying linearly with depth such that the background buoyancy frequency is N = 0.01 s 1. The resulting Rossby radius of deformation is NH/f = 20 km. We also consider a case on the shelf slope, with the sea bottom profile obtained as the alongshore average of bathymetric data off the Oregon coast between 44.9 and 45.1N and a maximum depth of 1000 m. On the slope, we consider cases with two different sets of background ocean conditions. The first is the ocean at rest with T and S profiles (Fig. 1) corresponding to average conditions off the mid- Oregon shelf in April (Smith et al., 2001). The second corresponds to upwelling conditions obtained by running the nonlinear ROMS from a state of rest forced with the spatially uniform southward alongshore wind stress that is ramped up from 0 to 0.12 N m 2 during day 1, and is then held constant. No surface heat flux is applied. The resulting cross-shore sections of alongshore velocity and potential density at t = 3 d are shown in Fig. 2. In these cases, the time-variable background fields are saved every 4 h, nearly a quarter of the inertial period (2/f 17 h). In cases with the background ocean at rest the vertical dissipation and diffusion coefficients are constant, K M = K H = 10 3 m 2 s 1. In cases corresponding to upwelling conditions, these coefficients are space- and time-dependent, obtained from the nonlinear ROMS. Fig. 1. Profiles of background (a) T, (b) S, and (c) dt/dz used in computations on the slope bathymetry.

8 8 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 2. The cross-shore section of alongshore velocity (line contours) and potential density (kg m 3, color) corresponding to the upwelling background ocean state at t = 3 d. Velocity contour interval is 0.05 m s 1. Bold contours correspond to v = 0.2 and 0.4ms 1. In all cases considered, we assume that the prior model error is due to uncertainty in the alongshore wind stress (y) (x, t). There are two reasons for this choice. First, shelf flows respond strongly and quickly to variability in the wind stress such that the atmospheric forcing can be a dominant source of error in model forecasts. Inadequate resolution of atmospheric fields obtained from satellites or numerical models, the effects of land topography (Samelson et al., 2002), diurnal wind cycle (Perlin et al., 2004), and ocean atmosphere coupling (Chelton et al., 2007; Perlin et al., 2007) can all contribute to the uncertainty in the wind stress estimates over the shelf. Second, if we assumed error in the initial conditions, the dynamical structures in the representer would possibly be affected by the choice of the error covariance C 0 [see (9)]. This covariance would ideally be constructed to provide a dynamically balanced correction to the initial ocean state (Weaver et al., 2005). Since one of our goals here is to test dynamical consistency of the TL fields, we would like to avoid implying the dynamical constraints via C 0. Thus, we choose C 0 = 0. Stringent requirements of dynamical consistency are not required for errors in the alongshore stress ı (y) so that a bell-shaped error covariance can be assumed for this univariate field: C = ı (y) (x 1,t 1 ) ı (y) (x 2,t 2 ) = 2 exp ( (x 1 x 2 ) 2 2l 2 x ) (t 1 t 2 ) 2 2lt 2, (12) where denote the ensemble average, the forcing error standard deviation, and l x and l t the crossshore and time error decorrelation scales, respectively. 4. Representers Representer solutions discussed in this section correspond to observations of or surface v sampled at x = 20 km, t = 3 d. Dynamical consistency among representer components, as well as dependence of representers on data type, background ocean conditions, and wind stress error decorrelation length scale l x are analyzed (cases I VII, Figs. 3 9 ; a summary of these cases is given in Table 1). In each Table 1 Parameters of representer cases I VII Case no. Observation l x (km) Bathymetry Background state Figure no. I 50 Flat Rest 3 II 0.1 Flat Rest 4 III v 0.1 Flat Rest 5 IV 50 Slope Rest 6 V 50 Slope Upwelling 7 VI 0.1 Slope Rest 8 VII 0.1 Slope Upwelling 9

9 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 9 Fig. 3. The components [(a) (f)] of the representer in the cross-shore section, case I. Observation: at (x, t) = ( 20 km,3 d); alongshore wind stress cross-shore decorrelation length scale l x = 50 km; bathymetry: flat; background conditions: u, v = 0 and linear stratification. The representer is shown at the time of observation. Every component is scaled by 1/ R kk, where R kk is the expected prior error variance of the observed quantity. Shades of blue correspond to the negative values and yellow-red to positive values. Contour intervals are provided in the titles for each plot. (a) wind stress (N m 2 ); (b) SSH (m); (c) u; (d) v; (e) T; (f) S. Fig. 4. The components [(a) (f)] of the representer in the cross-shore section, case II. Observation: ; l x = 0.1 km; bathymetry: flat; background conditions: u, v = 0 and linear stratification. Other details similar to Fig. 3.

10 10 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 5. The components [(a) (f)] of the representer in the cross-shore section, case III. Observation: surface v; l x = 0.1 km; bathymetry: flat; background conditions: u, v = 0 and linear stratification. Other details similar to Fig. 3. case, we plot the correction to the wind stress forcing of the TL model [which would correspond to the forcing term C in (8)] and the -, u-, v-, T- and S-components of the representer in the cross-shore sections at the time of observation. In (12), the standard deviation in the alongshore wind stress is assumed to be = 0.1Nm 2 and the decorrelation time scale l t = 2d. Fig. 6. The components [(a) (f)] of the representer in the cross-shore section, case IV. Observation: ; l x = 50 km; bathymetry: slope; background conditions: u, v = 0 and horizontally uniform T and S (see Fig. 1). Other details similar to Fig. 3.

11 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 11 Fig. 7. The components [(a) (f)] of the representer in the cross-shore section, case V. Observation: ; l x = 50 km; bathymetry: slope; background conditions: upwelling (nonlinear model solution). Other details similar to Fig. 3. Representer fields have units of a covariance between the observed quantity and the field component. To plot representer components in conventional units of the stress, SSH, velocity, T, and S and to obtain structures consistent with the traditional picture of upwelling, the representers are divided by R kk, where R kk = g k r k is the representer value, or the expected prior error variance, of the Fig. 8. The components [(a) (f)] of the representer in the cross-shore section, case VI. Observation: ; l x = 0.1 km; bathymetry: slope; background conditions: u, v = 0 and horizontally uniform T and S (see Fig. 1). Other details similar to Fig. 3.

12 12 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 9. The components [(a) (f)] of the representer in the cross-shore section, case VII. Observation: ; l x = 0.1 km; bathymetry: slope; background conditions: upwelling (nonlinear solution). Other details similar to Fig. 3. observed quantity. As follows from (10), representers scaled in this way can be interpreted as typical correction fields resulting from assimilation of a single observation that provides the reduction of SSH or enhancement of the southward flow at the observation location in case of or v observation, respectively. In case I (Fig. 3), the representer is computed for a -observation, flat bottom, and the forcing decorrelation length scale l x = 50 km, which is longer than the Rossby radius of deformation. Recall, in this case, the background corresponds to the linearly stratified ocean at rest. The representer fields are consistent with the generic picture of upwelling: the negative, large-scale correction to (Fig. 3b) is consistent with intensification of southward stress (Fig. 3a), cross-shore Ekman transport as seen in the plot for u (Fig. 3c), and negative, southward, sheared v near the coast (Fig. 3d). In this and other cases presented below, we have verified that correction fields are in quantitative dynamic balance (Ekman transport, geostrophy, thermal wind). This solution exhibits features qualitatively similar to the analytical solution of Scott et al. (2000). For instance, the observation has the maximum influence on v, T, and S in the coast-surface corner, where the analytical solution has a singularity. The first mode structure is apparent in v within one radius of deformation of the coast (20 km). Note that in this case, the maximum correction to any variable considered is obtained near the coast rather than at the observation location. In other words, the maximum of the prior model error covariance is not at the observation location, as often assumed in sequential OI schemes (e.g., Li et al., 2008). Case II (Fig. 4) differs from the previous one only by the assumed forcing error decorrelation scale l x, which is now nearly 0 (in computations, we have used l x = 0.1 km). The representer structure in this limiting case is consistent with the picture of Ekman pumping, including cross-shore divergence (convergence) in the surface (bottom) boundary layer, and upwelling of colder and more saline water under the observational location. The observation has little influence near the coast. The wind stress correction has a finite amplitude discontinuity and the corresponding correction to is not smooth at the observation location (note that dots in Fig. 4a and b show the grid resolution). In case III (Fig. 5), we retain all the features from case II, except that the representer is now computed for a surface velocity observation. In the long-wave and low-frequency limit (see Scott et al., 2000), v is proportional to the cross-shore derivative of the pressure. Consistent with this, the representer for the surface v measurement computed using our model exhibits features that are qualitatively similar

13 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 13 to the x-derivative of the representer for the measurement (case II). For instance, the correction to the wind stress (Fig. 5a) has a delta-function singularity while the correction to (Fig. 5b) has a finite amplitude step at the observation location. Next, we consider cases over the sloping bottom, for which an analytical solution is not available. Representer case IV (Fig. 6) is computed for a -observation, background ocean at rest, and l x = 50 km. Similarly to case I on the flat bottom (Fig. 3), the multivariate representer structure is consistent with the classical picture of upwelling. Note that v remains nearly singular near the coast. One of the differences from the flat bottom, linear stratification case is that a relatively larger correction to T and S is found near the bottom over a large stretch of the shelf slope. Note that the magnitude of T and S correction is not uniform across the slope. Areas of larger data influence are found where background gradients d T/dz and d S/dz are relatively large (see Fig. 1), a manifestation of the importance of linearized vertical advection w d{ T, S}/dz (where w is the tangent linear vertical velocity). For instance, the representer T field (Fig. 6e) is relatively smaller at depths of m, where the gradient of background T is relatively lower (see Fig. 1c). In the inner-shelf zone near the coast, there is large correction to T, but not to S (since d S/dz is small at shallow depths). Case V (Fig. 7) differs from the previous one only by the background state, which now corresponds to upwelling conditions. The correction to the wind stress is nearly the same (compare Figs. 6a and 7a). However, the component of the representer (Fig. 7b) has a relatively stronger gradient at distances of 7 12 km from the coast, over the upwelling jet in the background solution. Also, the maximum correction to v (Fig. 7d) now occurs in the area of the upwelling jet rather than at the coast. The maximum influence of the observation on T and S fields is on the inshore side of the upwelling jet (Fig. 6e and f). Also note that in this case the cross-shore Ekman transport is distributed over a larger surface boundary layer (compare Figs. 6c and 7c). Case VI (Fig. 8) corresponds to small l x = 0.1 km, shelf slope, observation of, and background ocean at rest, differing from case IV (Fig. 6) by only the decorrelation length scale. Again, similarly to the case on the flat bottom (case II, see Fig. 4), the representer is singular near the observation location. However, if we do a similar computation with upwelling background conditions, the representer is not singular anymore (case VII, Fig. 9). It appears that both cross-shore background current and the larger background vertical dissipation act to provide smoothing in the horizontal direction, with a scale near 10 km. 5. Array mode analyses As described in Section 2, the correction to the prior model solution can be obtained as a combination of representer functions. Some combinations of representers are constrained better than others by assimilation of a given set. Array mode analysis (Bennett, 2002, Section 2.5) computes an orthogonal basis in the data space and corresponding dynamical structures (linear combinations of representers) that are best constrained by the observing system. If the eigenvalue decomposition of the representer matrix is obtained, R = VSV, elements v k,n of each eigenvector (the nth column of V) provide coefficients for a linear combination of observations d n = v k k,nd k. The representer for this superobservation, r n = v k k,nr k, is called an array mode. The nth eigenvalue of R, s n, provides the expected prior model error variance of d n.ifc d = 2 I, where I is the identity matrix, the array mode d variances can be compared to the assumed data variance 2 d. If the signal-to-noise ratio s n/ 2 d 1, direction r n in the state space is well constrained by the linear combination of observations d n. For illustration, let us consider the array of 16 observations of surface v located between x = 99 and x = 9km, spaced every 6 km, which is representative of an HF radar array. Assume the data are sampled once at t = 3 d. The background ocean state corresponds to upwelling conditions (as described in Section 3.2; also, see Fig. 2). For the error covariance (12), we assume a relatively small, but non-zero, cross-shore decorrelation scale l x = 10 km, l t = 1 d, and wind stress standard error of = 0.05 N m 2 (these values are relevant to the DA experiment described in the next section). Representers for four observations from this array, scaled by 1/ R kk, are shown in Fig. 10 ( (y),, and v components are shown in top, middle and bottom plots, respectively). For each representer plotted, the zone of maximum influence on the velocity field is centered near the observation location. We note that the

14 14 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 10. Representer components for observations of surface v used in the array mode analysis of Fig. 11, shown in cross-shore sections at observation time (t = 3 d): (top) wind stress (N m 2 ), (middle) surface elevation (m), and (bottom) alongshore velocity (m s 1 ). Each column corresponds to an observation at (left to right) x = 81, 57, 33, and 9 km, with the observation location shown as the white circle. Wind stress error decorrelation scale l x = 10 km, upwelling background conditions. Representers are scaled by 1/ R kk. magnitude of the wind stress correction is reduced for observations taken closer to the coast. At the same time, the magnitude of and v components is relatively smaller for measurements taken over the shelf break (Fig. 10e and f) and mid-shelf (Fig. 10h and i), but larger for those over deep ocean (Fig. 10b and c) and inner-shelf (Fig. 10k and l). For a measurement over deep ocean, the correction if nearly symmetric, with positive (negative) correction on the offshore (inshore) side of the observation location and decaying influence at a distance from that location. For shelf observations, the -correction is asymmetric, with relatively smaller magnitudes on the offshore side and larger magnitudes on the inshore side of the observation where the influence on extends all the way to the coast. The mode error variances for this array are shown in Fig. 11 a. In the same plot, the dashed line denotes the assumed data variance of 2 d = (0.05 m s 1 ) 2. Ten array modes have error variance above that threshold. The modal coefficients (columns of V) for the first 9 modes are shown in Fig. 11(b) (j). In the first eigenvector, having the largest variance, observations are coupled over the shelf (up to 50 km from the coast), with linearly decreased weighting coefficients. Contribution of the offshore measurements to this mode is relatively small. Higher eigenvectors are increasingly harmonic with a progressively larger wave number. The first mode combination of representers (Fig. 12 a c) provides maximum correction to v in the area of the background alongshore jet. The second mode (Fig. 12 d f), also giving relatively larger weights to coastal observations, influences the intensity of the upwelling jet and also increases horizontal shear on the offshore side of the upwelling jet. Encouraged by the prospect of obtaining satellite altimetry close to the coast, we provide a similar analysis for the array of observations, sampled at the same locations and time as in the previous

15 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 15 Fig. 11. (a) The eigenvalues of the representer matrix and (b j) observational array modes for the array of observations of surface v. Background ocean state: upwelling. The dashed line corresponds to the assumed data error variance of (0.05 m s 1 ) 2. Fig. 12. Components of the (left) 1st and (right) 2nd array modes, computed for the array of surface v-observations and upwelling background conditions, shown in cross-shore sections at t = 3 d, (top) wind stress, (middle) surface elevation, and (bottom) alongshore velocity. These combinations are scaled by 1/ s k, where s k is the mode variance. Velocity contour offset is 0.02 m s 1.

16 16 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 13. (a) The eigenvalues of the representer matrix and (b j) observational array modes for the array of observations of SSH (). Background ocean state: upwelling. The dashed line corresponds to the assumed data error variance of (0.05 m) 2. example, and utilizing the same assumptions about the background ocean and forcing error (Fig. 13). Array modes are similar to the previous case. However, the prior error variance of the modes is rather low. For instance, the largest mode variance is below the reference value of 2 d = (0.05)2 m 2 shown as the dashed line (see Fig. 13a). In this case, to constrain stably the model error associated with errors in the wind stress on a scale of 10 km, SSH observations are required with 1 cm precision. Based on this analysis we conclude that assimilation of the HF radar observations would be expected to be more effective than assimilation of SSH to constrain model error associated with uncertainties in the wind stress on small horizontal scales. This conclusion is consistent with the scaling analysis for geostrophic flows. For instance, if f v = g / x, where g is gravitational acceleration, then 0.1ms 1 changes in v would correspond to 0.01 m changes in on the horizontal scale of 10 km. In additional tests, we have found that the SSH array under consideration would have more utility constraining larger scale errors. For instance, in the case with l x = 100 km, the largest eigenvalue of R is m 2, larger than the reference level of 2 d = m2 used here. The final example in this section is a variant of the first case, for a cross-shore array of v- measurements, but now with the background ocean at rest (Fig. 14). The first eigenvector is nearly singular, with dominant weight given to the observation closest to the coast. Note that in this case the first mode variance is much larger than in the case corresponding to upwelling conditions (see Fig. 11). The second mode variance in this case is close to the first mode variance in the first example. 6. Wind stress error correction by assimilation of surface velocities A DA test with synthetic observations (a twin experiment ) described in this section is performed to provide an additional test of our TL&ADJ model and to demonstrate that assimilation of surface currents from HF radars can effectively correct model error associated with uncertainty in the wind stress, which would be consistent with the result of array mode analysis (Section 5). The prior solution is obtained by forcing the nonlinear model with spatially uniform alongshore southward wind stress that is ramped-up from 0 to 0.12 N m 2 during day 1 and then held constant for 9 more days. The cross-section plot, Fig. 15 a, shows the resulting alongshore current (line contours) and the potential density (color) at the end of day 10. Based on the analysis of a coupled atmosphere-ocean model (Perlin et al., 2007), the feedback of the ocean to the atmosphere is such that the wind stress can be reduced inshore of the upwelling front

17 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 17 Fig. 14. (a) The eigenvalues of the representer matrix and (b j) observational array modes for the array of observations of (v). Background ocean state: at rest. The dashed line corresponds to the assumed data error variance of (0.05 m) 2. by a factor of 2. This effect is associated in part with the stabilization of the atmospheric boundary layer over the region of cold upwelled water. To approximate this situation, we have constructed the true, spatially and temporally variable wind stress in an ad hoc way by reducing the magnitude of the prior wind stress inshore of the upwelling front by a factor of 2 using the SST information from the prior solution. The width of the transition zone from the offshore zone (stress 0.12 N m 2 )tothe inshore zone (stress 0.06 N m 2 ) is approximately 10 km. This modified wind field, shown in Fig. 16 a, is used to force the nonlinear model to obtain the true solution in the twin experiment. At the end of day 10, the true solution (Fig. 15b) is qualitatively different from the prior (see Fig. 15a). For instance, the alongshore current has a double-jet structure with the first, inshore, jet closer to the coast than the single jet in the prior solution. Also, the depth of the surface boundary layer is reduced in the area between the jets. The root mean square (RMS) differences of the true and prior solutions obtained by Fig. 15. The alongshore velocity (line contours) and potential density (color) in the cross-shore section at the end of day 10, in the DA test with the synthetic observations: (a) prior solution (forced with spatially uniform wind stress), (b) true solution, (c) assimilation solution (nonlinear model run with the estimated wind stress, see Fig. 16b). The velocity contour interval is 0.05 m s 1 ; the bold contour is 0.5ms 1.

18 18 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx Fig. 16. The (a) true and (b) inverse alongshore wind stress shown as a function of the cross-shore coordinate and time, in the test with synthetic observations. The bold contour is 0.12 N m 2. The contour interval is 0.01 N m 2. averaging over the 10 days are shown in Fig. 17 (upper plots), separately for u, v, T, and S. In particular, the RMS difference for v reaches 0.2ms 1 near the surface. The daily-averaged observations of v are sampled from the true solution, between x = 51 and 9 km, every 6 km, on days 2 10 (a total of K = 72 observations). Random noise is added to observational values with standard deviation d = 0.05 m s 1. These observations are assimilated using the direct representer approach [see (10)], with covariance parameters = 0.05 N m 2, l x = 10 km, and Fig. 17. The time-averaged (days 1 10) RMS error of (top) the prior and (bottom) assimilation solutions, shown in cross-shore sections. Left to right: u, v, T, and S.

19 A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx xxx 19 l t = 2 d. The DA estimate of the wind stress is shown in Fig. 16 b. Although not perfect, the estimate reproduces the progressive reduction of the stress magnitude as the front moves offshore. The DA wind forcing estimate was then used to force nonlinear ROMS to obtain the DA estimate of oceanic fields. The resulting solution at the end of day 10 is shown in Fig. 15 c. The double-jet structure is reproduced, and the first jet, next to the coast, is of the correct magnitude and at the right location. The subsurface v in the area of the second jet, 30 km offshore, is somewhat weaker than in the true solution. The reduction of the surface boundary layer depth at 20 km offshore is also reproduced in the DA estimate. The plots of the RMS error of the inverse solution averaged over 10 days (Fig. 17, bottom) show that assimilation of the surface observations results in a sizable reduction of the averaged subsurface error for all of u, v, T, and S. We also find that the nonlinear DA estimate fits the observations with accuracy comparable to d. So using this estimate as the background for model linearization, and assimilating the data again [i.e., doing a second iteration in the scheme of Chua and Bennett (2001)], does not result in further improvement of the state or forcing. 7. Summary Analyses of representer functions, array modes, and the DA test all confirm the validity of our AVRORA TL&ADJ codes developed recently with focus on coastal ocean assimilation and analysis. The components of representer functions, which are linearized prior model error covariances, satisfy dominant balances such as Ekman transport, geostrophy, and thermal wind. Variability in the T and S components of the representers computed over the shelf slope can be explained as the effect of vertical advection of the background temperature and salinity. Representers computed on the flat bottom are qualitatively close to the analytical solutions for the linear baroclinic coastal ocean, utilizing long-wave and low-frequency approximations (Scott et al., 2000; Kurapov et al., 1999, 2002). Array mode analysis shows that for realistic data error levels, surface velocities will be considerably more effective in correcting coastal flows on the scale of tens of km than SSH. This result is consistent with a simple scale analysis for geostrophic flows. The spatial structure of the representers is influenced by topographic details, assumptions about errors in the model inputs, and, most importantly, the state of the ocean. For instance, our representer computations suggest that during upwelling periods, the zone of maximum influence of surface velocity observations will be in the area of the upwelling jet. During periods of calm winds or relaxation from upwelling to downwelling, the same observational arrays will have maximum impact near the coast. Although simple sequential schemes based on a time-invariant forecast error covariance estimate have shown some skill in coastal areas (Oke et al., 2002; Kurapov et al., 2005a, b; Li et al., 2008), more advanced methods that rely on the state-dependent model (forecast) error covariances would make a more effective use of observations, particularly in intermittent regimes (wind direction changes, frontal instabilities and eddy interactions). DA correction to initial conditions requires specification of a dynamically balanced error covariance. Our representer solutions, in which dynamical balances are a result of TL computation, can be used to guide the design of initial condition error covariances suitable for the coastal ocean. The representer solutions can also be compared to forecast error covariances generated in ensemble-based DA. In particular, such comparisons would help to assess whether ensembles of model solutions sample the state space adequately, as well as the degree to which nonlinearity may impact the adjoint-based approaches. Using variational methods in the coastal ocean, one can potentially not only correct the initial conditions at a sequence of intervals, but also provide correction to forcings, and learn something about deficiencies of the model formulation. In particular, the results of the DA test with synthetic observations suggest that surface currents from HF radar can provide information on the variability in the wind stress on small scales (10 km), as are expected to result from effects of atmosphere-ocean coupling over areas of coastal upwelling. In this manuscript, all results were obtained in an alongshore-uniform 2D (vertical vs. cross-shore) set-up. Analyses using our TL&ADJ model in 3D configurations are underway, with focus on the features associated with coastal trapped wave propagation, alongshore advection, instabilities, and eddy interactions.