Fig. 19.9 Modelled mean age (colour scale in days) for a simulation commencing on 13-March increase produces a spectral shift in the remote sensing reflectance. Zhang et al. (2010a) found a robust empirical relationship between simulated age and observed reflectance ratio that has promise for estimating river water age in the NYB—a property of relevance to rates of biogeochemical transformation of river source organic matter and pollutants (Moline et al. 2008).

19.3.3 Data Assimilation and Observing System Design

The NYB is among the most densely observed coastal oceans in the world, having been the target of pioneering deployments of new observing instruments including a cabled observatory (Glenn and Schofield 2003), surface current measuring high-frequency radar (CODAR) (Kohut et al. 2006) and autonomous underwater vehicles (gliders) (Schofield et al. 2007). To these systems and regular satellite imagery, LaTTE added moorings, surface drifters, and towed undulating CTD instruments deployed from the research vessels Cape Hatteras and Oceanus. These data and the sustained operation of much of the instrumentation make the NYB an attractive location to explore the integration of observation and modelling capabilities through advanced data assimilation.

The locations of LaTTE 2006 in situ observations are shown in Fig. 19.1. CO-DAR coverage was near complete from Long Island to Delaware Bay and out to the 40 m isobath, with some gaps in the apex of NYB. There were satellite SST data from approximately four passes each day, cloudiness permitting.

Here we use data assimilation (DA) for state estimation; namely, to obtain an analysis for initializing subsequent forecasts so as to enhance short-term forecast skill. This approach is common practise in Numerical Weather Prediction (NWP). We use a 4-dimensional (time-dependent) variational (4DVAR) method for DA, which is one among many possible approaches but again one that draws on experience in advanced NWP. We use the so-called Incremental Strong Constraint (IS4DVAR) formulation (Courtier et al. 1994) whose implementation in ROMS is described in detail elsewhere (Broquet et al. 2009; Powell et al. 2008; Zhang et al. 2010b).

IS4DVAR minimizes a cost function expressing the mismatch between observations and the model state at each observation location and time, summed over an analysis interval. Our implementation uses a 3-day interval—short enough for the linearization assumption of the incremental formulation to hold, but long enough for the model physics (embodied in the adjoint and tangent linear models) to exert the strong constraint interconnection (covariance) of model state variables. The control variables of the DA are the initial conditions to each 3-day analysis, with the intervals overlapped so as to generate initial conditions each day to launch a new 72-h forecast.

IS4DVAR does not explicitly allow for model error as would, for example, rep-resenter-based or weak constraint 4DVAR (Bennett 2002; Courtier 1997). Errors in model physics, numerics, meteorological forcing and boundary conditions are incorporated into the model background error covariance. The observations are assigned error variances appropriate to the observation source.

Temperature Salinity

Fig. 19.10 Added skill introduced by data assimilation for analysis and forecast periods for individual forecast variables. Results are ensemble average of 60 forecast cycles. Vertical bars on symbols indicate 95% confidence intervals. Vertical dashed lines denote the boundary between analysis window and forecast window

Our reanalysis was conducted after the data were gathered, but we describe a DA and forecast system that could have operated in real-time because glider and vessel data are telemetered to shore. Lessons learned from this study on practical issues of data timeliness, quality control, and configuration of the IS4DVAR algorithm on a broad, shallow shelf, with significant tides have been incorporated in the Experimental System for Predicting Shelf and Slope Optics (ESPreSSO1) that currently runs operationally for the Mid-Atlantic Bight and encompasses the LaTTE domain.

The value that DA adds to the forecast system can be evaluated by considering how well observations are forecast prior to their assimilation on later analysis cycles. We quantify this with a DA skill metric

where RMS is the root-mean-square of model-observation mismatch weighted by observational error. For 60 days of simulation spanning LaTTE 2006 we have multiple 1-day, 2-day, etc. forecasts that may be combined into ensemble estimates for increasing forecast window. Figure 19.10 shows the skill for different variables

Fig. 19.10 Added skill introduced by data assimilation for analysis and forecast periods for individual forecast variables. Results are ensemble average of 60 forecast cycles. Vertical bars on symbols indicate 95% confidence intervals. Vertical dashed lines denote the boundary between analysis window and forecast window

forecast day number

forecast day number

1 ESPreSSO results may be viewed at www.myroms.org/applications/espresso.

when all available data are assimilated (black lines), and when selected data categories are withdrawn from the analysis step (coloured lines). Forecast times less than zero are in the analysis interval, and show the ability of the system to match observations and model prior to launching the forecast. As forecast time proceeds the skill declines, but note that S = 0 does not say the model has no utility at all, merely that assimilation no longer adds any advantage to the model predictive skill. For temperature, DA adds skill to the forecast out to some 10-15 days, for salinity 5-10 days, and for velocity about 2-3 days. The more rapid decline in skill for velocity compared to tracers reflects the shorter autocorrelation timescales for velocity and that it is inherently less predictable.

Not surprisingly, withdrawing data diminishes skill for that variable, i.e. without HF-radar data the velocity skill falls, and without satellite SST the temperature skill falls. However, there can be a modest increase in skill for other variables, e.g. salinity forecast skill is slightly higher when SST are not assimilated. We interpret this as the DA system not needing to reconcile glider and satellite temperatures and having rather more freedom to adjust initial salinity to improve the salinity analysis; recall that all the variables are dynamically linked through the strong constraint of the adjoint and tangent linear models. Overall, skill is best when all data are included, and therefore diversity in the data sources is to be preferred.

Details of the ROMS IS4DVAR configuration for LaTTE with respect to background error covariance and the pre-processing of observations are discussed by Zhang et al. (2010b), who also examine surface versus subsurface skill, and the influence of errors in surface forcing on system performance.

A further application of variational methods in ocean modelling is adjoint sensitivity analysis, which allows some inference of observation locations that are likely to have greater impact on the DA analysis. Studies using adjoint sensitivity in coastal oceanography are still relatively few compared to meteorology and mesoscale and gyre-scale oceanography, but Moore et al. (2009) examine how upwelling, eddy kinetic energy and baroclinic instability in the California Current are affected by surface forcing on seasonal timescales. Here we present some results due to Zhang et al. (2009b) who use the adjoint of the LaTTE model to reveal the spatial and temporal distribution of ocean model state variables that are "dynamically upstream" to features of coastal circulation.

A characteristic of New Jersey coastal ocean dynamics is that significant SST variability is driven by along-shore winds (Chant 2001; Munchow and Chant 2000). Zhang et al. (2009b) considered this process by introducing a scalar function that expresses SST anomaly variance averaged over a localized area adjacent to the coast where Ts is SST and Ts is its temporal mean; this definition considers temperature anomaly within an area A during a set time interval. Here, the time period is chosen to be the last three hours of the simulation time window. Defining J in quadratic form prevents the cancellation of positive and negative anomalies.

Temperature, salinity and velocity outside region A affect J through transport (advection and diffusion) and dynamics (baroclinic pressure gradients, stratification, turbulent mixing). Denoting the 4-dimensional ocean state (T, S, u, v, Z) by a vector O, it can be shown that dJ/d $—representing the dependence of J on the ocean state—is the solution of the ROMS adjoint model integrated backward in time and forced by dJ/dT computed from the forward model. See Zhang et al. (2009b) for details. Although J is a scalar, dJ/d$ has the same dimension as O, i.e. the entire ocean state through time, which emphasizes that all the surrounding ocean can potentially project on to SST variance in A. This adjoint sensitivity concept can be grasped, qualitatively, from an example: Fig. 19.11 maps the sensitivity of J to surface temperature, i.e. dJ/dT at z=0, over the 3 days that precede the interval tj to t2 over which J is defined, for the cases of downwelling and upwelling winds. The sequence proceeds backwards in time from day 3 to day 0. We have already demonstrated that southward (downwelling) winds favour coastal current formation, and for this case (Fig. 19.11, top row) the adjoint sensitivity advances from region A (delineated by the black box) back along the trajectory of the coastal current to New York Harbor. In the upwelling wind case (Fig. 19.11, bottom row), surface temperatures in preceding times have very little impact of SST variance in A. This is because the coastal current is not dynamically upstream in this situation; rather, surface temperatures depend more on source waters drawn from below the surface. The final panel on the right shows dJ/dT at t = 0 along a vertical cross-section slightly south of region A, and confirms that J is sensitive to remote subsurface temperatures during upwelling. While these results have a ready qualitative interpretation, adjoint sensitivity quantifies the dependence and immediately indicates where "upstream" is. Zhang et al. (2009b) further quantify the relative importance of other state variables by contrasting the magnitude of dJ/dT with dJ/dS, dJ/du, etc.

One can immediately see the potential for this information to assist observing system operation. By identifying the timing and location of ocean conditions having significant influence on the subsequent evolution of specific circulation features (characterized by some chosen J), adjoint sensitivity indicates where, when and what observations are likely to have greater impact in a 4DVAR assimilation system. In a companion paper, Zhang et al. (2010c) extend this approach using so-called representers, also based on variational methods, to examine the information content of a set of observations such as might be gathered routinely on a repeat transect occupied by an autonomous vehicle, or by a sustained cabled observatory.

19.4 Processes and Dynamics for Further Study 19.4.1 Air-Sea and Wave-Current Interaction

The results described above all utilize essentially the same model configuration options emphasized in Sect. 19.2, but the LaTTE program identified roles for some o> £

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