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Representer-based analyses in the coastalupwelling system
A.L. Kurapov ∗, G.D. Egbert, J.S. Allen, R.N. MillerCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, United States
The impact of surface velocity and SSH data assimilated in a modelof wind-driven upwelling over the shelf is studied using representerand observational array mode analyses and twin experiments, uti-lizing new tangent linear (TL) and adjoint (ADJ) codes. Bathymetry,forcing, and initial conditions are assumed to be alongshore uni-form reducing the problem to classical two-dimensional. The modelerror is attributed to uncertainty in the surface wind stress. The rep-resenters, analyzed in cross-shore sections, show how assimilatedobservations provide corrections to the wind stress and circulationfields, and give information on the structure of the multivariateprior model error covariance. Since these error covariance fields sat-isfy the dynamics of the TL model, they maintain dominant balances(Ekman transport, geostrophy, thermal wind). Solutions computedover a flat bottom are qualitatively similar to a known analyticalsolution. Representers obtained with long cross-shore decorrela-tion scale for the wind stress errors lx (compared to the distance tocoast) exhibit the classical upwelling structure. Solutions obtainedwith much smaller lx show structure associated with Ekman pump-ing, and are nearly singular if lx is smaller than the grid resolution.The zones of maximum influence of observations are sensitive tothe background ocean conditions and are not necessarily centeredaround the observation locations. Array mode analysis is utilizedto obtain model structures (combinations of representers) that aremost stably observed by a given array. This analysis reveals thatfor realistic measurement errors and observational configurations,surface velocities will be more effective than SSH in providing cor-rection to the wind stress on the scales of tens of km. In the DA test
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with synthetic observations, the prior nonlinear solution is obtainedwith spatially uniform alongshore wind stress and the true solutionwith the wind stress sharply reduced inshore of the upwelling front,simulating expected ocean–atmosphere interaction. Assimilation ofdaily-averaged alongshore surface currents provides improvementto both the wind stress and circulation fields.
Data assimilation (DA) has been implemented in oceanography to combine circulation models andobservations 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 stimu-lated 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). Avail-able 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), temperatureand salinity sections from autonomous underwater vehicles and gliders (Castelao et al., 2008), andtime-series of velocities, temperature and salinity from moorings. These data sets remain sparse com-pared to the small scale, rapidly evolving, and nonlinear circulation processes in the coastal ocean. Tofacilitate development of effective DA systems, it is important to understand the spatial and temporalscales of influence of the different data types in the coastal ocean. It is also important to learn howassimilation of observations of a given type affects estimation of unobserved oceanic fields. Given thenonlinear character of the coastal flows, answers to these questions can depend on the state of theocean.
A variational approach provides a convenient framework for the assessment of the impact of obser-vations in a DA system. Variational methods attempt to obtain ocean state estimates by minimizinga penalty functional that is a sum of quadratic terms on errors in observations and model inputsintegrated over space and a specified time interval. The inputs adjusted by DA may include initialand boundary conditions, forcing, model parameters, and space- and time-dependent errors in thedynamical equations. Minimization algorithms leading to the optimum solutions are complicated andcan require the repeated use of companion tangent linear (TL) and adjoint (ADJ) models. However, ifcomputational burdens of variational DA can be overcome, the resulting solutions bear many attrac-tive 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 con-sistent with the model dynamics and dependent on the ocean state. These covariance functions aremultivariate, such that observations of one type influence the correction to the prior model fieldsof different type (for instance, velocity measurements can be utilized to provide dynamically con-sistent corrections to velocity, temperature, and salinity fields). Although in practice the model stateerror covariance functions do not have to be provided or computed explicitly, it can still be instruc-tive to analyze them to understand the zones of influence and the multivariate impact of assimilatedobservations.
In this manuscript, we explore the utility of tools for observational array assessment in the frame-work of a particular variational algorithm, namely the representer method (Bennett, 1992, 2002; Chuaand Bennett, 2001), implemented for the case of wind-driven upwelling on the shelf. In the representermethodology the minimization of the penalty functional proceeds as a series of linearized optimiza-tion problems, with the correction to the prior model at each step obtained as a linear combination ofrepresenter functions, one for each observation. Based on the statistical interpretation of variationalDA, the representer is the prior model error covariance between the observed quantity and all theelements of the three-dimensional (3D), time-dependent, and multivariate ocean state vector. A rep-resenter shows zones of influence of a single assimilated observation in space and time. It satisfies thelinearized model dynamics and depends on the background ocean state. The representer method isparticularly explicit and flexible in the choice of error covariances in the model inputs, which define
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norms in the penalty functional. Thus, representer analysis can illuminate the dependency of the priormodel error covariance (i.e., error covariance in the model outputs) on the assumed input error statis-tics. For instance, one can compare the structure of the prior model error covariances correspondingto different error sources, such as boundary conditions or local forcing, or different error decorrelationscales in the forcing.
The variational representer formalism is not the only possible way to estimate prior model errorcovariances. Echevin and De Mey (2000) and Broquet et al. (2008) utilized ensembles of model runsto obtain covariances (representers) in regional applications. Le Hénaff and de Mey (2008, in prepa-ration) have used a similar approach to provide assessment for large arrays of observations such asproposed wide-swath altimetry sets (Fu, 2003). Their representer spectrum analysis is similar to thearray mode analysis (Bennett, 2002) implemented in Section 5 of this manuscript. Advantages of theensemble approach is that TL&ADJ models are not needed. However, with such an approach accurateestimation can be limited by the ensemble size. For a system with many degrees of freedom ensemblecalculations can result in spurious long-range correlations, which are often dealt with by localizingthe estimated representer. The variational representer approach avoids these complications, providinga more direct and reliable link between the assumed statistics of uncertainties in model inputs andcalculated 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 currentand upwelling during summer (Allen et al., 1995). In this system, it is plausible that uncertaintyin the wind stress is a dominant source of model error. In wind stress estimates obtained froman atmospheric model, errors in prediction of mesoscale features will have relatively long spatialdecorrelation scales [O (100 km)]. Errors with smaller scales [O (10 km)] can be associated withprocesses misrepresented in the atmospheric model, including orographic effects near coastal irreg-ularities (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 alongshorevelocities. Although the standard alongtrack satellite altimetry products are now not available closerthan 20–50 km from the coast, improving processing algorithms (P.T. Strub, pers. comm.) and develop-ing new observational technologies including the delay Doppler radar (Raney, 1988) and wide-swathaltimetry (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 aboutsubsurface 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 initialtest of new TL&ADJ codes that we have developed to implement the representer algorithm for shelfareas. Second, we illustrate fundamental features of the multivariate prior model error covariance forassimilation on the coastal shelf in the wind-driven upwelling regime. The prior model error is assumedto 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 performedin an idealized set-up assuming no variability in the alongshore direction, which reduces the problemfrom 3D to 2D (cross-shore and vertical coordinates). Despite this idealization, the analyses retainmany features essential for coastal upwelling, including baroclinicity, shelf slope effects, advection,and nontrivial background ocean conditions. Detailed nonlinear dynamical analyses of alongshoreand cross-shore transports and turbulence in the wind-driven regime have been done for a similarset-up, e.g., by Allen et al. (1995), Federiuk and Allen (1995), Allen and Newberger (1996), Austin andLentz (2002), Wijesekera et al. (2003), and Kuebel Cervantes et al. (2004). Here we look at this familiardynamical problem from the standpoint of adjoint-based DA.
In 3D applications, there has been a concern that alongshore instabilities, occurring in shelf circula-tion models on temporal scales of several days (e.g., Durski and Allen, 2005) and growing unconstrainedin 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 witha shallow-water model of circulation in the nearshore surf zone has provided insights on how vari-ational assimilation can be done effectively over time intervals exceeding the time scales associatedwith instability growth (Kurapov et al., 2007).
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In the following sections, after the description of the basics of representer methodology (Section2) 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 windstress, and background ocean conditions. Then, in Section 5, array mode analysis is utilized as a toolfor assessment of utility of a set of observations. An idealized DA test with synthetic observations ofsurface velocity (a twin experiment, Section 6) is performed to confirm that the DA performance isconsistent 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, (1)
q(0) = qprior0 + e0, (2)
where q(t) is the true state (multivariate, discrete in space, continuous in time), N is the nonlinearmodel operator, fprior(t) is the prior forcing vector (which in this formulation may include errors ininterior dynamics, surface forcing, and boundary conditions), qprior
0 is the prior initial condition, e(t)is the dynamical error, and e0 is the initial condition error.
The observations are written in the general form as
L(q) =∫ T
0
dt
⎛⎜⎝
g′1(t)
g′2(t)· · ·
g′K (t)
⎞⎟⎠q(t) = d + ed, (3)
where L is a linear operator matching the model state q and the observations, gk define the observa-tional functionals (sampling rules for each datum dk), d = {dk} is the vector of observations (of sizeK × 1), ed is the observation error, and the prime denotes matrix transpose.
The penalty functional to be minimized can be written as
J(q) =∫ T
0
dt1
∫ T
0
dt2 e′(t1)C−1(t1, t2)e(t2) + e′0C−1
0 e0 + e′dC−1
d ed, (4)
where e, e0, and ed are residuals satisfying (1)–(3) with q. C−1 and C−10 are inverse prior error covari-
ances in the corresponding inputs. C−1d is the inverse data error covariance. Note that C, C0, and Cd
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 therepresenter computation. In the following, the TL operator is defined as
A[q] = ∂N
∂q
∣∣∣∣q=q
. (5)
In our notation, it is a matrix with elements depending on the time-variable background ocean stateq. As noted in Section 1, the correction to the prior model can be written as a linear combination of therepresenter functions rk(t). To obtain each representer, the ADJ model is run backward in time forcedwith the kernel of the kth observational functional:
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The adjoint solution � is convolved with the input error covariances and then used to force the TLmodel:
∂rk
∂t= A[q]rk +
∫ T
0
dt1 C(t, t1)�(t1), (8)
rk(0) = C0�(0). (9)
For instance, the analysis of representers in Section 4 will be done for local and instantaneousobservations (SSH or surface alongshore velocity component), such that gk in (6) will be an impulse inthe corresponding field at the observational location and time. The prior model error will be assumedto be only due to that in the wind stress. Correspondingly, C0 in (9) will be set to 0 (no correction toinitial conditions), and the last term in (8) will represent a correction to wind forcing. Covariance Cwill provide spatial and temporal smoothing of the forcing correction.
As mentioned above, the representer is the prior error covariance between the observed value∫
g′kq
and all the elements of the model vector q(t). This covariance is computed given the assumptions oferror covariances in the inputs (C, C0). It satisfies the linearized model dynamics and is thus dependenton the background ocean state q(t). As a test of validity of the TL code, one can verify whether thecomponents of the multivariate representer rk satisfy the dominant linear dynamical balance relations.Note that the representer does not depend on the actual observational value dk, so representers canbe utilized for the assessment of the potential impact of observations.
The set of coefficients of the optimal combination of representers∑
kbkrk, providing correction tothe prior model state qprior, can be found as
b = {bk} = (R + Cd)−1[d − L(qprior)], (10)
where R = L[r1|r2| . . . |rK ] is the symmetric non-negative representer matrix, obtained by samplingthe representers at observational locations and times. Note that in practice, with a large number ofdata, it would be impractical and unnecessary to compute and store all the representers. Optimal linearcombinations can be obtained iteratively using the indirect representer method, involving a series ofADJ 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, http://www.myroms.org/; Shchepetkin andMcWilliams, 2005; Wilkin et al., 2005) is utilized to describe the nonlinear dynamics. ROMS is basedon the baroclinic, free surface, hydrostatic primitive equations discretized on a terrain following coor-dinate grid. To represent the physics on subgrid scales, we have used the Mellor-Yamada 2.5 turbulencescheme (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 developmenthas been influenced by the experiences reported by researchers using adjoint components of popularmodels based on the hydrostatic primitive equations, including ROMS (Moore et al., 2004; Di Lorenzoet al., 2007), the MIT General Circulation Model (e.g., Stammer et al., 2002; Gebbie et al., 2006) and NavyCoastal Ocean Model (NCOM; Ngodock, priv. comm.). Details making our code structure different fromTL&ADJ ROMS will be mentioned later in this section. In this paper we provide only a brief outline of ourmodel development and focus primarily on analyses that to our knowledge have not been previouslyreported for any model.
Our TL&ADJ codes have been developed manually following recipes for line-by-line automatic codegeneration (Giering and Kaminski, 1998). The algorithms for time-stepping and barotropic–baroclinicmode splitting have been adopted from ROMS. At every time step, the background ocean state isobtained by linear interpolation between fields provided at specified time instances (in our examples
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below, once every 4 h). Although a range of high-order advection schemes are available in ROMS, onlythe second-order centered scheme has been coded in AVRORA. This choice is the most straightforwardsince the second order scheme is differentiable (in particular, it does not include non-differentiableIF statements as in some higher order schemes). Also, boundary conditions for this scheme are morestraightforward compared to higher order schemes that have a larger footprint (stencil) on the grid andthus require extra, “numerical” boundary conditions. Similarly to the ROMS development, our TL&ADJcodes do not involve a linearized version of the turbulence equations. The vertical eddy dissipation anddiffusion coefficients, KM and KH , are known functions of space and time provided with the backgroundstate, e.g., obtained from the nonlinear model. In our applications, a nonlinear equation of state forseawater 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 ROMSruns and linear friction with the drag coefficient of 2.5 × 10−4 m s−1 in TL&ADJ AVRORA.
In DA, it is important to define clearly the states of inputs, since those are to be corrected, andoutputs, since those are to be matched to the observations. The need to attend to these details wasone of the motivational points for development of the AVRORA codes, which have been structured tosimplify 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 byMoore et al. (2004) and Di Lorenzo et al. (2007).
The vector of outputs of our TL model includes the sea surface height �, two orthogonal componentsof horizontal velocity (u, v), T, and S. The TL model outputs instantaneous fields every NHIS time stepsinto the “history” file. In applications considered here, the vector of inputs includes the initial valuesfor the elements of the state vector and the wind stress fields specified at selected time instances. As innonlinear ROMS, our TL code obtains the wind stress by linear interpolation between values providedat specified times. The ADJ model outputs sensitivities to the wind stress at the same specified timeinstances using the appropriate adjoint to the time-interpolation, similarly to (Kurapov et al., 2007). Inparticular, if the wind stress is specified in the TL model at times t = t1, t2, . . . , tN , then the ADJ model,executed backward in time, does not output sensitivity to the forcing at time tk until the model time tbecomes smaller than tk−1, since the TL solution at tk−1 < t < tk is sensitive to forcing values both attimes tk−1 and tk.
From (3), it follows that the kernels of the data functionals gk, 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 providethese as files of the same structure as the output (history) file from the TL model, with fields of thecoefficients of the linear combination of the state vector elements organized as a series of values everyNHIS time steps. As the ADJ model steps backward in time, those values are added to correspondinginterior adjoint variables at appropriate times. If the input of gk for forcing the ADJ model is organizedin this way, any linear combination of the elements of the state vector can be assimilated withoutmaking additional changes in the TL&ADJ codes (e.g, including time-averaged and low-pass filteredobservations, HF radar radial components of velocity, SSH alongtrack anomalies). This approach isconsistent 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 canbe viewed as the rule by which a rectangular matrix [TL] multiplies the vector of inputs. In the way theADJ model has been constructed, it provides the rule by which the transposed matrix [TL]′ multipliesa vector of the same dimension as the output vector in the TL model. A series of adjoint symmetrychecks (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 closedboundaries around the domain. The ADJ code provides sensitivities to the initial conditions and theatmospheric forcing. Inclusion of sensitivities to open boundary conditions is planned in the future.
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3.2. Implementation details
The 2D (cross-shore and vertical coordinates) solutions are obtained by running the model in a shortsouth-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 projectionson x and y directions, respectively. No-normal-flow and free-slip boundary conditions are applied atthe coast (x = 0) and offshore boundaries (x = −200 km). The resolution is 2 km in horizontal and 40layers in vertical, with relatively finer resolution near the surface and bottom. The Coriolis parameter isf = 10−4 s−1. In the TL&ADJ model, the parameters of the linear equation of state (11) are obtained basedon hydrographic data off Oregon: �◦ = 1025 kg m−3, T◦ = 10 ◦C, S◦ = 34 psu, ˛T = 1.7 × 10−4 ◦C−1, and˛S = 7.5 × 10−4 psu−1.
Representer solutions will be first considered for a flat bottom. In these cases the results of ourTL&ADJ model can be compared, at least qualitatively, to the analytical representer solutions obtainedin 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) is at rest: u = v = 0 and T and S are horizontally uniform and varying linearly with depth suchthat the background buoyancy frequency is N = 0.01 s−1. The resulting Rossby radius of deformationis NH/f = 20 km.
We also consider a case on the shelf slope, with the sea bottom profile obtained as the alongshoreaverage of bathymetric data off the Oregon coast between 44.9 and 45.1N and a maximum depth of1000 m. On the slope, we consider cases with two different sets of background ocean conditions. Thefirst 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 obtainedby running the nonlinear ROMS from a state of rest forced with the spatially uniform southwardalongshore 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 potentialdensity at t = 3 d are shown in Fig. 2. In these cases, the time-variable background fields are savedevery 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 areconstant, KM = KH = 10−3 m2 s−1. In cases corresponding to upwelling conditions, these coefficientsare 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.
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Fig. 2. The cross-shore section of alongshore velocity (line contours) and potential density (kg m−3, color) corresponding to theupwelling background ocean state at t = 3 d. Velocity contour interval is 0.05 m s−1. Bold contours correspond to v = −0.2 and−0.4 m s−1.
In all cases considered, we assume that the prior model error is due to uncertainty in the alongshorewind stress �(y)(x, t). There are two reasons for this choice. First, shelf flows respond strongly andquickly to variability in the wind stress such that the atmospheric forcing can be a dominant sourceof error in model forecasts. Inadequate resolution of atmospheric fields obtained from satellites ornumerical models, the effects of land topography (Samelson et al., 2002), diurnal wind cycle (Perlin etal., 2004), and ocean–atmosphere coupling (Chelton et al., 2007; Perlin et al., 2007) can all contributeto the uncertainty in the wind stress estimates over the shelf. Second, if we assumed error in the initialconditions, the dynamical structures in the representer would possibly be affected by the choice of theerror covariance C0 [see (9)]. This covariance would ideally be constructed to provide a dynamicallybalanced correction to the initial ocean state (Weaver et al., 2005). Since one of our goals here is totest dynamical consistency of the TL fields, we would like to avoid implying the dynamical constraintsvia C0. Thus, we choose C0 = 0. Stringent requirements of dynamical consistency are not required forerrors in the alongshore stress ı�(y) so that a bell-shaped error covariance can be assumed for thisunivariate field:
C = 〈ı�(y)(x1, t1) ı�(y)(x2, t2)〉 = �2� exp
(− (x1 − x2)2
2l2x− (t1 − t2)2
2l2t
), (12)
where 〈〉 denote the ensemble average, �� the forcing error standard deviation, and lx and lt the cross-shore and time error decorrelation scales, respectively.
4. Representers
Representer solutions discussed in this section correspond to observations of � or surface v sampledat x = −20 km, t = 3 d. Dynamical consistency among representer components, as well as dependenceof representers on data type, background ocean conditions, and wind stress error decorrelation lengthscale lx are analyzed (cases I–VII, Figs. 3–9 ; a summary of these cases is given in Table 1). In each
Table 1Parameters of representer cases I–VII
Case no. Observation lx (km) Bathymetry Background state Figure no.
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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 lx = 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/√
Rkk , where Rkk isthe expected prior error variance of the observed quantity. Shades of blue correspond to the negative values and yellow-red topositive 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: �; lx = 0.1 km; bathymetry:flat; background conditions: u, v = 0 and linear stratification. Other details similar to Fig. 3.
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Fig. 5. The components [(a)–(f)] of the representer in the cross-shore section, case III. Observation: surface v; lx = 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 theforcing term C� in (8)] and the �-, u-, v-, T- and S-components of the representer in the cross-shoresections at the time of observation. In (12), the standard deviation in the alongshore wind stress isassumed to be �� = 0.1 N m−2 and the decorrelation time scale lt = 2 d.
Fig. 6. The components [(a)–(f)] of the representer in the cross-shore section, case IV. Observation: �; lx = 50 km; bathymetry:slope; background conditions: u, v = 0 and horizontally uniform T and S (see Fig. 1). Other details similar to Fig. 3.
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Fig. 7. The components [(a)–(f)] of the representer in the cross-shore section, case V. Observation: �; lx = 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 compo-nent. To plot representer components in conventional units of the stress, SSH, velocity, T, and S andto obtain structures consistent with the traditional picture of upwelling, the representers are dividedby −
√Rkk, where Rkk =
∫gkrk 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: �; lx = 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.
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Fig. 9. The components [(a)–(f)] of the representer in the cross-shore section, case VII. Observation: �; lx = 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 typi-cal correction fields resulting from assimilation of a single observation that provides the reduction ofSSH 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 forcingdecorrelation length scale lx = 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 fieldsare 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 inthe plot for u (Fig. 3c), and negative, southward, sheared v near the coast (Fig. 3d). In this and othercases presented below, we have verified that correction fields are in quantitative dynamic balance(Ekman transport, geostrophy, thermal wind). This solution exhibits features qualitatively similar tothe analytical solution of Scott et al. (2000). For instance, the observation has the maximum influenceon v, T, and S in the coast-surface corner, where the analytical solution has a singularity. The firstmode structure is apparent in v within one radius of deformation of the coast (20 km). Note that inthis case, the maximum correction to any variable considered is obtained near the coast rather thanat the observation location. In other words, the maximum of the prior model error covariance is notat 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 scalelx, which is now nearly 0 (in computations, we have used lx = 0.1 km). The representer structure inthis 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 waterunder the observational location. The observation has little influence near the coast. The wind stresscorrection has a finite amplitude discontinuity and the corresponding correction to � is not smooth atthe 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 computedfor a surface velocity observation. In the long-wave and low-frequency limit (see Scott et al., 2000), vis proportional to the cross-shore derivative of the pressure. Consistent with this, the representer forthe surface v measurement computed using our model exhibits features that are qualitatively similar
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to the x-derivative of the representer for the � measurement (case II). For instance, the correction tothe wind stress (Fig. 5a) has a delta-function singularity while the correction to � (Fig. 5b) has a finiteamplitude 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 lx = 50 km.Similarly to case I on the flat bottom (Fig. 3), the multivariate representer structure is consistent withthe classical picture of upwelling. Note that v remains nearly singular near the coast. One of the dif-ferences from the flat bottom, linear stratification case is that a relatively larger correction to T andS is found near the bottom over a large stretch of the shelf slope. Note that the magnitude of T and Scorrection is not uniform across the slope. Areas of larger data influence are found where backgroundgradients dT/dz and dS/dz are relatively large (see Fig. 1), a manifestation of the importance of lin-earized 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 100–80 m, where the gradient of back-ground T is relatively lower (see Fig. 1c). In the inner-shelf zone near the coast, there is large correctionto T, but not to S (since dS/dz is small at shallow depths).
Case V (Fig. 7) differs from the previous one only by the background state, which now correspondsto 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 distancesof 7–12 km from the coast, over the upwelling jet in the background solution. Also, the maximumcorrection to v (Fig. 7d) now occurs in the area of the upwelling jet rather than at the coast. Themaximum 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 largersurface boundary layer (compare Figs. 6c and 7c).
Case VI (Fig. 8) corresponds to small lx = 0.1 km, shelf slope, observation of �, and backgroundocean at rest, differing from case IV (Fig. 6) by only the decorrelation length scale. Again, similarly tothe 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 notsingular anymore (case VII, Fig. 9). It appears that both cross-shore background current and the largerbackground vertical dissipation act to provide smoothing in the horizontal direction, with a scale near10 km.
5. Array mode analyses
As described in Section 2, the correction to the prior model solution can be obtained as a com-bination of representer functions. Some combinations of representers are constrained better thanothers by assimilation of a given set. Array mode analysis (Bennett, 2002, Section 2.5) computes anorthogonal basis in the data space and corresponding dynamical structures (linear combinations ofrepresenters) that are best constrained by the observing system. If the eigenvalue decomposition ofthe representer matrix is obtained, R = VSV′, elements vk,n of each eigenvector (the nth column of V)provide coefficients for a linear combination of observations dn =
∑kvk,ndk. The representer for this
“superobservation”, rn =∑
kvk,nrk, is called an array mode. The nth eigenvalue of R, sn, provides the
expected prior model error variance of dn. If Cd = �2d
I, where I is the identity matrix, the array modevariances can be compared to the assumed data variance �2
d. If the signal-to-noise ratio sn/�2
d� 1,
direction rn in the state space is well constrained by the linear combination of observations dn.For illustration, let us consider the array of 16 observations of surface v located between x = −99
and x = −9 km, spaced every 6 km, which is representative of an HF radar array. Assume the data aresampled once at t = 3 d. The background ocean state corresponds to upwelling conditions (as describedin Section 3.2; also, see Fig. 2). For the error covariance (12), we assume a relatively small, but non-zero,cross-shore decorrelation scale lx = 10 km, lt = 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 fourobservations from this array, scaled by −1/
√Rkk, are shown in Fig. 10 (�(y), �, and v components
are shown in top, middle and bottom plots, respectively). For each representer plotted, the zone ofmaximum influence on the velocity field is centered near the observation location. We note that the
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Fig. 10. Representer components for observations of surface v used in the array mode analysis of Fig. 11, shown in cross-shoresections at observation time (t = 3 d): (top) wind stress (N m −2), (middle) surface elevation (m), and (bottom) alongshorevelocity (m s −1). Each column corresponds to an observation at (left to right) x = −81, −57, −33, and −9 km, with the obser-vation location shown as the white circle. Wind stress error decorrelation scale lx = 10 km, upwelling background conditions.
Representers are scaled by −1/√
Rkk .
magnitude of the wind stress correction is reduced for observations taken closer to the coast. At thesame time, the magnitude of � and v components is relatively smaller for measurements taken overthe 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 ifnearly symmetric, with positive (negative) correction on the offshore (inshore) side of the observationlocation and decaying influence at a distance from that location. For shelf observations, the �-correctionis asymmetric, with relatively smaller magnitudes on the offshore side and larger magnitudes on theinshore 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 linedenotes 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 to50 km from the coast), with linearly decreased weighting coefficients. Contribution of the offshoremeasurements to this mode is relatively small. Higher eigenvectors are increasingly harmonic with aprogressively larger wave number. The first mode combination of representers (Fig. 12 a–c) providesmaximum 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 upwellingjet 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 similaranalysis for the array of � observations, sampled at the same locations and time as in the previous
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Fig. 11. (a) The eigenvalues of the representer matrix and (b–j) observational array modes for the array of observations of surfacev. 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 andupwelling 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/
√sk , where sk is the mode variance. Velocity contour offset
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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 ratherlow. For instance, the largest mode variance is below the reference value of �2
d= (0.05)2 m2shown as
the dashed line (see Fig. 13a). In this case, to constrain stably the model error associated with errors inthe wind stress on a scale of 10 km, SSH observations are required with 1 cm precision. Based on thisanalysis we conclude that assimilation of the HF radar observations would be expected to be moreeffective than assimilation of SSH to constrain model error associated with uncertainties in the windstress on small horizontal scales. This conclusion is consistent with the scaling analysis for geostrophicflows. For instance, if f v = g∂�/∂x, where g is gravitational acceleration, then 0.1 m s−1 changes in vwould correspond to 0.01 m changes in � on the horizontal scale of 10 km. In additional tests, we havefound that the SSH array under consideration would have more utility constraining larger scale errors.For instance, in the case with lx = 100 km, the largest eigenvalue of R is 0.0033 m2, larger than thereference level of �2
d= 0.0025 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 nearlysingular, with dominant weight given to the observation closest to the coast. Note that in this case thefirst 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 performedto provide an additional test of our TL&ADJ model and to demonstrate that assimilation of surfacecurrents from HF radars can effectively correct model error associated with uncertainty in the windstress, 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 alongshoresouthward wind stress that is ramped-up from 0 to −0.12 N m−2 during day 1 and then held constantfor 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 ofthe ocean to the atmosphere is such that the wind stress can be reduced inshore of the upwelling front
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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 boundarylayer 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 ofthe prior wind stress inshore of the upwelling front by a factor of 2 using the SST information fromthe prior solution. The width of the transition zone from the offshore zone (stress −0.12 N m−2) to theinshore zone (stress −0.06 N m−2) is approximately 10 km. This modified wind field, shown in Fig. 16a, is used to force the nonlinear model to obtain the true solution in the twin experiment. At the end ofday 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 thanthe single jet in the prior solution. Also, the depth of the surface boundary layer is reduced in the areabetween 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 is0.05 m s −1; the bold contour is −0.5 m s−1.
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Fig. 16. The (a) true and (b) inverse alongshore wind stress shown as a function of the cross-shore coordinate and time, in thetest 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 0.2 m s−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 observa-tional values with standard deviation �d = 0.05 m s−1. These observations are assimilated using thedirect representer approach [see (10)], with covariance parameters �� = 0.05 N m−2, lx = 10 km, and
Fig. 17. The time-averaged (days 1–10) RMS error of (top) the prior and (bottom) assimilation solutions, shown in cross-shoresections. Left to right: u, v, T, and S.
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lt = 2 d. The DA estimate of the wind stress is shown in Fig. 16 b. Although not perfect, the estimatereproduces 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 estimateof oceanic fields. The resulting solution at the end of day 10 is shown in Fig. 15 c. The double-jetstructure is reproduced, and the first jet, next to the coast, is of the correct magnitude and at theright location. The subsurface v in the area of the second jet, 30 km offshore, is somewhat weakerthan in the true solution. The reduction of the surface boundary layer depth at 20 km offshore is alsoreproduced in the DA estimate. The plots of the RMS error of the inverse solution averaged over 10days (Fig. 17, bottom) show that assimilation of the surface observations results in a sizable reductionof 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 furtherimprovement of the state or forcing.
7. Summary
Analyses of representer functions, array modes, and the DA test all confirm the validity of ourAVRORA 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, sat-isfy dominant balances such as Ekman transport, geostrophy, and thermal wind. Variability in the Tand S components of the representers computed over the shelf slope can be explained as the effect ofvertical advection of the background temperature and salinity. Representers computed on the flat bot-tom are qualitatively close to the analytical solutions for the linear baroclinic coastal ocean, utilizinglong-wave and low-frequency approximations (Scott et al., 2000; Kurapov et al., 1999, 2002). Arraymode analysis shows that for realistic data error levels, surface velocities will be considerably moreeffective in correcting coastal flows on the scale of tens of km than SSH. This result is consistent witha simple scale analysis for geostrophic flows. The spatial structure of the representers is influenced bytopographic details, assumptions about errors in the model inputs, and, most importantly, the state ofthe ocean. For instance, our representer computations suggest that during upwelling periods, the zoneof maximum influence of surface velocity observations will be in the area of the upwelling jet. Duringperiods of calm winds or relaxation from upwelling to downwelling, the same observational arrays willhave maximum impact near the coast. Although simple sequential schemes based on a time-invariantforecast error covariance estimate have shown some skill in coastal areas (Oke et al., 2002; Kurapovet al., 2005a, b; Li et al., 2008), more advanced methods that rely on the state-dependent model (fore-cast) error covariances would make a more effective use of observations, particularly in intermittentregimes (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 toguide the design of initial condition error covariances suitable for the coastal ocean. The representersolutions can also be compared to forecast error covariances generated in ensemble-based DA. Inparticular, such comparisons would help to assess whether ensembles of model solutions sample thestate space adequately, as well as the degree to which nonlinearity may impact the adjoint-basedapproaches.
Using variational methods in the coastal ocean, one can potentially not only correct the initialconditions at a sequence of intervals, but also provide correction to forcings, and learn somethingabout deficiencies of the model formulation. In particular, the results of the DA test with syntheticobservations suggest that surface currents from HF radar can provide information on the variabilityin the wind stress on small scales (10 km), as are expected to result from effects of atmosphere-oceancoupling 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 fea-tures associated with coastal trapped wave propagation, alongshore advection, instabilities, and eddyinteractions.
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Acknowledgments
The research was supported by the Office of Naval Research (ONR) Physical OceanographyProgram under Grant N00014-06-1-0257. Partial support was obtained from NOAA under grantNA03NES4400001 (US GLOBEC, North East Pacific Synthesis Phase).
References
Allen, J.S., Newberger, P.A., 1996. Downwelling circulation on the Oregon continental shelf. Part I. Response to idealized forcing.J. Phys. Oceanogr. 26, 2011–2035.
Allen, J.S., Newberger, P.A., Federiuk, J., 1995. Upwelling circulation on the Oregon continental shelf. Part I. Response to idealizedforcing. J. Phys. Oceanogr. 25, 1843–1866.
Austin, J.A., Lentz, S.J., 2002. The inner shelf response to wind-driven upwelling and downwelling. J. Phys. Oceanogr. 32,2171–2193.
Barth, A., Alvera-Azcàrate, A., Weisberg, R.H., 2008. Assimilation of high-frequency radar currents in a nested model of the WestFlorida Shelf. J. Geophys. Res. 113, C08033, doi:10.1029/2007JC004585.
Bennett, A.F., 1992. Inverse Methods in Physical Oceanography. Cambridge University Press, 346 pp.Bennett, A.F., 2002. Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, 234 pp.Bennett, A.F., Chua, B.S., Pflaum, B.L., Erwig, M., Fu, Z., Loft, R.D., Muccino, J.C., 2008. The Inverse Ocean Modeling System. I.
Implementation. J. Atmos. Ocean. Technol., doi:10.1175/2008JTECHO519.1, in press.Besiktepe, S.T., Lermusiaux, P.F.J., Robinson, A.R., 2003. Coupled physical and biochemical data driven simulations of Mas-
sachusetts Bay in late summer: real-time and post-cruise data assimilation. J. Mar. Syst. 40, 171–212.Broquet, G., Brasseur, P., Rozier, D., Brankart, J.-M., Verron, J., 2008. Estimation of model errors generated by atmospheric
forcings for ocean data assimilation: experiments in a regional model of the Bay of Biscay. Ocean Dynam. 58, 1–17,doi:10.1008/s10236-007-0128-z.
Castelao, R., Glenn, S., Schofield, O., Chant, R., Wilkin, J., Kohut, 2008. Seasonal evolution of hydrographic fields in the centralMiddle Atlantic Bight from glider observations. Geophys. Res. Lett., doi:10.1029/2007GL032335.
Chelton, D.B., Schlax, M.G., Samelson, R.M., 2007. Summertime coupling between sea surface temperature and wind stress inthe California current system. J. Phys. Oceanogr. 37, 495–517.
constraint data assimilation in the inverse regional ocean modeling system (ROMS): development and application for abaroclinic coastal upwelling system. Ocean Model. 16, 160–187, doi:10.1016/j.ocemod.2006.08.002.
Durski, S.M., Allen, J.S., 2005. Finite amplitude evolution of instabilities associated with the coastal upwelling front. J. Phys.Oceanogr. 35, 1606–1628.
Echevin, V., De Mey, P., 2000. Horizontal and vertical structure of the representer functions for sea surface measurements in acoastal circulation model. J. Phys. Oceanogr. 30, 2627–2635.
Egbert, G.D., Bennett, A.F., Foreman, M.G.G., 1994. TOPEX/POSEIDON tides estimated using a global inverse model. J. Geophys.Res. 99, 24821–24852.
Evensen, G., 2007. Data Assimilation: The Ensemble Kalman Filter. Springer, 208 pp.Federiuk, J., Allen, J.S., 1995. Upwelling circulation on the Oregon continental shelf. Part II. Simulations and comparisons with
Laboratory, Pasadena, CA, 67 pp. (available at online at ftp://ftp-oceans.jpl.nasa.gov/pub/llf/WSOAreportFinal2.pdf).Gebbie, G., Heimbach, P., Wunsch, C., 2006. Strategies for nested and eddy-permitting state estimation. J. Geophys. Res. 111,
C10073, doi:10.1029/2005JC003094.Giering, R., Kaminski, T., 1998. Recipes for adjoint code construction. ACM Trans. Math. Softw. 24 (4), 437–474.Kosro, P.M., 2005. On the spatial structure of coastal circulation off Newport, Oregon, during spring and summer 2001, in a region
of varying shelf width. J. Geophys. Res. 110, C10S06, doi:10.1029/2004JC002769.Kuebel Cervantes, B.T., Allen, J.S., Samelson, R.M., 2004. Lagrangian characteristics of continental shelf flows forced by periodic
wind stress. Nonlinear Process. Geophys. 11, 3–16.Kurapov, A.L., Allen, J.S., Miller, R.N., Egbert, G.D., 1999. Generalized inverse for baroclinic coastal flows. In: Proceedings of the
3rd Conference on Coastal Atmospheric and Oceanic Prediction and Processes, 3–5 November 1999, New Orleans, LA, pp.101–106.
Kurapov, A.L., Egbert, G.D., Miller, R.N., Allen, J.S., 2002. Data assimilation in a baroclinic coastal ocean model: ensemble statisticsand comparison of methods. Month. Weather Rev. 130, 1009–1025.
Kurapov, A.L., Allen, J.S., Egbert, G.D., Miller, R.N., Kosro, P.M., Levine, M., Boyd, T., 2005a. Distant effect of assimilation of mooredcurrents into a model of coastal wind-driven circulation off Oregon. J. Geophys. Res. 110, C02022, doi:10.1029/2003JC002195.
Kurapov, A.L., Allen, J.S., Egbert, G.D., Miller, R.N., Kosro, P.M., Levine, M., Boyd, T., Barth, J.A., 2005b. Assimilation of mooredvelocity data in a model of coastal wind-driven circulation off Oregon: multivariate capabilities. J. Geophys. Res. 110, C10S08,doi:10.1029/2004JC002493 (COAST Special Issue).
Kurapov, A.L., Egbert, G.D., Allen, J.S., Miller, R.N., 2007. Representer-based variational data assimilation in a nonlinear model ofnearshore circulation. J. Geophys. Res. 112, C11019, doi:10.1029/2007JC004117.
Lewis, J.K., Shulman, I., Blumberg, A.F., 1998. Assimilation of CODAR observations into ocean models. Cont. Shelf. Res., 541–559.Li, Z., Chao, Y., McWilliams, J.C., Ide, K., 2008. A three-dimensional variational data assimilation scheme for the Regional Ocean
Modeling System: implementation and basic experiments. J. Geophys. Res., 113, C05002, doi:10.1029/2006JC004042.
Please cite this article in press as: Kurapov, A.L., et al., Representer-basedanalyses in the coastal upwelling system. Dyn. Atmos. Oceans (2008),doi:10.1016/j.dynatmoce.2008.09.002
ARTICLE IN PRESSG ModelDYNAT-792; No. of Pages 21
A.L. Kurapov et al. / Dynamics of Atmospheres and Oceans xxx (2008) xxx–xxx 21
Mellor, G., Yamada, T., 1982. Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. SpacePhys. 20, 851–875.
Moore, A.M., Arango, H.G., Di Lorenzo, E., Cornuelle, B.D., Miller, A.J., Neilson, D.J., 2004. A comprehensive ocean prediction andanalysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Model. 7, 227–258.
Oke, P.R., Allen, J.S., Miller, R.N., Egbert, G.D., Kosro, P.M., 2002. Assimilation of surface velocity data into a primitive equationcoastal ocean model. J. Geophys. Res. 107 (C9), 3122, doi:10.1029/2000JC000511.
Perlin, N., Samelson, R.M., Chelton, D.B., 2004. Scatterometer and model wind and wind stress in the Oregon-Northern Californiacoastal zone. Month. Weather Rev. 132, 2110–2129.
Perlin, N., Skyllingstad, E.D., Samelson, R.M., Barbour, P.L., 2007. Numerical simulation of air–sea coupling during coastalupwelling. J. Phys. Oceanogr. 37, 2081–2093.
Raney, R.K., 1988. The delay/Doppler radar altimeter. IEEE Trans. Geosci. Remote Sens. 36, 1578–1588.Samelson, R., Barbour, P., Barth, J., Bielli, S., Boyd, T., Chelton, D., Kosro, P., Levine, M., Skyllingstad, E., Wilczak, J., 2002.
Wind stress forcing of the Oregon coastal ocean during the 1999 upwelling season. J. Geophys. Res. Oceans 107, 3034,doi:10.1029/2001JC000900.
Scott, R.K., Allen, J.S., Egbert, G.D., Miller, R.N., 2000. Assimilation of surface current measurements in a coastal ocean model. J.Phys. Oceanogr. 30, 2359–2378.
Shchepetkin, A.F., McWilliams, J.C., 2005. The regional oceanic modeling system (ROMS): a split-explicit, free-surface,topography-following-coordinate oceanic model. Ocean Model. 9, 347–404.
Smith, R.L., Huyer, A., Fleischbein, J., 2001. The coastal ocean off Oregon from 1961 to 2000: is there evidence of climate changeor only of Los Ninos? Progr. Oceanogr. 49, 63–93.
Stammer, D., Wunsch, C., Giering, R., Eckert, C., Heimbach, P., Marotzke, J., Adcroft, A., Hill, C., Marshall, J., 2002. The global oceancirculation during 1992–1997 estimated from ocean observations and a general circulation model. J. Geophys. Res. 107 (C9),3118, doi:10.1029/2001JC000888.
Venegas, R.M., Strub, P.T., Beier, E., Letelier, R., Thomas, A.C., Cowles, T., James, C., Soto-Mardones, L., Cabrera, C., 2008. Satellite-derived variability in chlorophyll, wind stress, sea surface height, and temperature in the northern California Current System.J. Geophys. Res. 113, C03015, doi:10.1029/2007JC004481.
Weaver, A.T., Deltel, C., Machu, E., Ricci, S., Daget, N., 2005. A multivariate balance operator for variational data assimilation. Q.J. R. Meteorol. Soc. 131, 3605–3625.
Wijesekera, H.W., Allen, J.S., Newberger, P.A., 2003. Modeling study of turbulent mixing over the continental shelf: comparisonof turbulent closure schemes. J. Geophys. Res. 108 (C3), 3103, doi:10.1029/2001JC001234.
Wilkin, J.L., Arango, H.G., Haidvogel, D.B., Lichtenwalner, C.S., Glenn, S.M., Hedstrom, K.S., 2005. A regional ocean modelingsystem for the long-term ecosystem laboratory. J. Geophys. Res. 110, C06S91, doi:10.1029/2003JC002218.
Wunsch, C., 1996. The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.
