## Interp k e [1 K

The cell average values are then recovered as,

The exact integrals of the discrete integral variable are advantageous when Ax is chosen to be an integer multiple of AX (i.e., Ax=nAX) such that a subset of ¥k are equivalent to This formulation has been expressed for a uniform grid in Cartesian coordinates but this method is readily extended to non-uniform grids and other orthogonal curvilinear coordinate systems. It is also noteworthy that a centre point value is a second-order accurate estimate of a averaged over the cell volume, (Sanderson and Brassington 1998) so that the above method can be applied to most ocean general circulation models.

An example of longwave radiation flux for the region surrounding Tasmania (Fig. 18.12a) for GASP. This field shows significant variability in the small scales

as some of the physics is computed by a 1D radiation scheme but also shows front systems that have a step-like structure in the coarse resolution model. Regridding of coarse resolution information to finer resolution needs an algorithm that de-aliases. The integral variable can be used to de-aliase by performing interpolation for a subset of ^ such that j e [0:m:J] where J is an integer multiple of m. De-aliasing in multiple dimensions can be achieved by iteratively applying the integral variable interpolation in each dimension. The regridded longwave radiation onto the OFAM grid (Fig. 18.12b) is performed by applying the integral variable method for grid refinements of n=~2 and a dealiasing parameter of m=2 successively alternating in each dimension. The target grid resolutions are Ax=0.4°, 0.2° and 0.1°.

Accurate direct observations of fluxes and flux budgets are sparse in time and space. The scatterometer provide an instantaneous estimate of stress which in practice has a limited weighting and impact to atmospheric analyses and forecasts. The monitoring of SST from multiple satellites and sensors together with the Argo and mooring arrays provide a basis for diagnosing errors in flux parameterisations. Atmospheric forecast errors grow rapidly and are constrained through data assimilation commonly on a 6 h cycle.

Numerical weather prediction systems provide three alternative strategies for computing fluxes for ocean forecasting: (a) prescribed fluxes, (b) re-estimate the fluxes and (c) coupling. Numerical weather prediction systems currently persist SST analyses, may or may not have a dynamic surface roughness from a wave model and assume the ocean currents are negligible which will lead to a deterioration in skill in the surface fluxes in the forecasts. The next level of sophistication is to use the prescribed atmospheric state variables and replace the ocean boundary conditions by the forecast conditions using a bulk formula method (e.g., Large et al. 1997). Two specific flaws to this approach include (a) the near-surface atmospheric state variables in the forecast have been forecasted using boundary layer turbulence models based on the original boundary conditions and (b) the ocean boundary conditions for SST may be less accurate or have greater bias than persisted SST. This is presently the case for the BLUElink OceanMAPS system compared with the RAMSSA (Beggs et al. 2006). In part this is because the background errors from a forecast model are more difficult to define and the analysis for OceanMAPS is multi-variate and by definition will not fit the same SST observations as a univari-ate analysis. As the ocean forecasting systems continue to mature the performance gap is expected to close. This is also expected to be critical to achieve before more complex solutions of earth system coupling will yield the performance gains in operations (Brassington 2009).

### 18.7 Modelling

The governing equations for the ocean are an extension of the Navier-Stokes for a thin layer on a rotating planet. The ocean state equation is an empirical formulae dependent on temperature, salinity and pressure. There are a number of assumptions that can be introduced to simplify the governing equations that exploit the properties of the ocean such as incompressible, hydrostatic which are either convenient for analytical, numerical or data analysis. Software designed to solve these governing equations are referred to as ocean general circulation models (OGCMs). A summary of the design choices in OGCM's is summarised in Table 18.6. The prevalence of

Table 18.6 Properties of ocean modelling that result in unique design choices in ocean forecasting systems Ocean modelling

Selection of model code

Non-eddy, eddy permitting and eddy resolving

Coastal and bathymetric control

Boundary conditions

Numerical methods and computational performance

Turbulent parameterizations

Compressible/incompressible Hydrostatic/Nonhydrostatic, Non-Boussinesq/Non-Boussinesq Vertical coordinate system

Community models NEMO, HYCOM, ROMS, MOM, ...

eddies are ubiquitous in global ocean

Focii, horizontal mesh 0.1 a minimum

Geostrophic turbulent closure and submesoscale

High order, conservative advection schemes

Vertical/Horizontal

Bathymetry products

### Practical bathymetry tuning

Explicit tides or parameterised (more an assimilation challenge) Open boundaries, radiation conditions Nonhydrostatic/Hydrostatic (Lattice-Boltzmann methods) Nesting 3:1, alignment of grids, common interfacial bathymetry Explicit/Implicit

A-grid, B-grid, C-grid (Arakawa) Order accuracy of methods Numerical stability Parallelism and scalability Surface and bottom boundary layers Tidal mixing Diapycnal mixing community ocean models mean that the first design choice is to select a community model. Community models have already made several design choices on the governing equations as well as to make some choices optional. Many ocean models could be categorised by their primary applications, climate modelling, coastal modelling however many community models aspire to be applicable to multi-scale modelling. It is important to be aware of these design choices and their potential impact to performance and range of applications.

Starting from the position that a community model (e.g., Modular Ocean Model version 4, MOM4; Griffies et al. 2003) has been selected the first step to implementation is to compile and configure the environment for the software on the system architecture. It is then important to optimise the performance and diagnose the scaling. This is a specialist area that can be architecture and compiler specific and is not discussed further. The next step in development is to define the model grid making use of the latest bathymetry products (e.g., Smith and Sandwell 1997). The target resolution in ocean forecasting is for eddy-resolving which is approximately <1/8°. On a global scale this is expensive and would require the latest in high performance computing systems. Alternative approaches are a nested strategy (coarse global, fine regional) or an adapted grid in a single model. The horizontal grid for the Ocean Forecast Australia Model (OFAM; Schiller et al. 2008) version 2, uses a single global model with higher resolution (0.1° * 0.1°) in the Australian region, 90E-180E and 75S-16N (see Fig. 18.13). Provided the grid transitions are performed smoothly to minimise gravity wave reflections and energy accumulation this strategy avoids the problems of nesting and open boundary conditions. The convergence of meridions towards the poles means a Mercator projection should be used to retain local aspect ratio of spatial resolution at unity. At the Artic, the north pole introduces a grid singularity which can be resolved through displaced pole projections (Murray 1996).

Fig. 18.14 Anticyclonic (red) and Cyclonic (blue) vortices in the Tasman Sea identified by a pattern matching method and their coherent circulation in the vertical (Brassington et al. 2010b). The analysis was applied to a day average velocity field from behind real-time analysis of OceanMAPS (Brassington et al. 2007) for the 4th April 2009

### Longitude

Fig. 18.14 Anticyclonic (red) and Cyclonic (blue) vortices in the Tasman Sea identified by a pattern matching method and their coherent circulation in the vertical (Brassington et al. 2010b). The analysis was applied to a day average velocity field from behind real-time analysis of OceanMAPS (Brassington et al. 2007) for the 4th April 2009

The vertical coordinates of z- (or geopotential), a- (sigma or terrain following) and p- (isopycnal or density following) are believed to offer favourable properties for different parts of the ocean. The turbulent surface mixed layer for z-, the continental shelf for a- and a-z and the thermocline and deeper ocean p-. Generalised (or hybrid) coordinate systems provide the flexibility to apply these different grid types in the favourable areas.

A pattern matching method applied to the daily mean velocity from Ocean-MAPSv1 to locate approximately shear free rotating motion reveals coherent deep vortices in the Tasman Sea (see Fig. 18.14). Some of these vortices extend from the surface to full depth, some are shallow and other mid-depth and bottom vortices. An animation of this eddy tracking visualisation reveals that there are stratified (Reinaud and Dritschel 2002) and unstratified vortex interactions taking place in the model. The deep vortices correspond to a weak density anomaly that could be represented by the three coordinate systems. However, in all three coordinate systems the emphasis is on concentrating the vertical grid points into the surface layer (i.e., where the variability is greatest) and reduce the resolution with depth. There is minimal observational evidence to support the existence of the deeper features (e.g., Johnson and McTaggart 2010) but it is likely the model representation of space and timescales are biased by the choice of vertical grid. The importance of these deep features is primarily through their influence on the upper ocean vortices. An animation of Fig. 18.14 reveal stratified interactions.

The resolution in the surface mixed layer is critical to the representation of the physical processes and improving the accuracy where the majority of applications occur. The resolution of the top cells determine the resolvable scales of the bathymetry. In OFAM, water columns require a minimum of two cells for numerical stability resulting in a minimum column depth of 20 m. This impacts the representation of bays, straits and gulfs. The representation of Torres Strait by OFAM is shown in Fig. 18.15 in blue and compared against the best bathymetry in red. The cross-sections at 142.1E-142.4E show that OFAM is too deep and results in a bias in mass transport through the strait. There are several strategies to controlling the transport such as narrowing the opening to calibrate the total volume or to add boundary drag to reduce the flow rate.

It is noteworthy that steric anomalies are not a volume conserving process. Therefore apriori it is not clear whether an ocean general circulation model that makes the Boussinesq approximation (i.e., assumes volume conservation, V-u=0) should represent the sea surface anomalies of mesoscale eddies. Experience has shown that Boussinesq models such as MOM4 (Griffies et al. 2003) do indeed have sea level anomalies corresponding to eddies within an eddy-resolving simulation and these have been developed into successful ocean forecast systems (Brassington et al. 2007; Oke et al. 2008). It is reasonable however to pose the question why? The inroduction of the Boussinesq approximation to models of geophysical fluids can be traced to Spiegel and Veronis (1960). The climate modelling community have also been studying this problem to interpret other large scale processes (see Greatbatch 1994; Ducowicz 1997; McDougall et al. 2002). It has been determined that there is a duality between the Boussinesq and Non-Boussinesq model equations for a hydrostatic fluid (De Szoeke and Samelson 2002). However it is important to note that in the dual formulation the prognostic variable reverts from sea surface height (which can be remotely observed) to bottom pressure which is poorly observed and has a complex surface. The engineering community have also noted this problem in the context of Benard convection for rigid lid models (Zeytounian 2003). Zeytonian (2003) performs an asymptotic analysis and demonstrates the Boussinesq approximation remains valid with the addition of surface pressure perturbation. An initial value problem for a temperature anomaly is used to demonstrate the behaviour of a strictly volume conserving model with a free-surface formulation Eqs. 3a-e.

Po n

where u=ui+vj + wk, U=ui + vj and p = p(T, S, p). The shallow water equation derived for a free-surface model is given by ^ + V ■ U = 0. For the initial value problem, T(0) = 25°C for x * 0 and y * 0 and T(0)=26°C for x=0, y = 0, z = 1, S(0) = 35 psu, u = 0, q = 0, Az = 100 m, H = 1000 m, At=20 s. After 20 min of elapsed time the ocean responds to the temperature anomaly by adjusting the local sea level for the small expansion corresponding to the temperature anomaly. This volume however is obtained through a barotropic adjustment where gravity waves radiate from the source (see Fig. 18.16). This response can be detected in all variables for example sea level (Fig. 18.16a), pressure gradient (Fig. 18.16b), temperature (Fig. 18.16c) and vertical velocity (Fig. 18.16d).

In the corrected model we assume that the compressible terms are small for any

small volume of seawater I i.e, f — dpdz & 0 1 . This ensures that conservative z-Az/2

Distance (km) Distance ikm)

Fig. 18.16 Response after 20 min to the initial value problem of a temperature perturbation using a strictly volume conserving model formulation Eq. 18.1 a-e. a Sea level anomaly. b Pressure gradient. c Surface temperature and d Vertical velocity

Distance (km) Distance ikm)

Fig. 18.16 Response after 20 min to the initial value problem of a temperature perturbation using a strictly volume conserving model formulation Eq. 18.1 a-e. a Sea level anomaly. b Pressure gradient. c Surface temperature and d Vertical velocity numerical schemes remain valid for cell to cell interfacial fluxes (i.e., temperature and salinity remain conserved). However, the vertical integral of the compressible anomalies can be non negligible and measurable (e.g., ocean eddies). Therefore the shallow water equation is formulated to include a compressible correction term, n

dt J po dt

This is then reflected in a perturbation to the free-surface that feeds back to the momentum equations through the pressure gradient term. The impact of this correction to the same initial value problem is to reduce the barotropic response by an order of magnitude which is reflected in all variables (see Fig. 18.17). The correction term provides approximately the required volume for the water column that remains local to the temperature anomaly and does not require the volume to be sourced globally. A small residual barotropic response remains due to the discrete computation of the vertical integral is not exact.

Distance (km) Distance [Krn)

Fig. 18.17 Response after 20 min to the initial value problem of a temperature perturbation using a modified volume conserving model formulation Eq. 18.1a-e to include a compression term in the shallow water equation Eq. 18.2 a Sea level anomaly. b Pressure gradient. c Surface temperature and d Vertical velocity

Distance (km) Distance [Krn)

Fig. 18.17 Response after 20 min to the initial value problem of a temperature perturbation using a modified volume conserving model formulation Eq. 18.1a-e to include a compression term in the shallow water equation Eq. 18.2 a Sea level anomaly. b Pressure gradient. c Surface temperature and d Vertical velocity

### 18.8 Data Assimilation

The statistical machinery for combining background fields with observations based on a least squares approach has been established for some time and successfully applied to objective analyses, weather prediction and seasonal prediction. The GO-DAE initiated in 1999 coinciding with OceanObs'99 targeted the application of the same methods to the problem of ocean forecasting. The fundamentals of ocean data assimilation and its application to the ocean are presented in this volume by Zaron (2011) and Moore (2011). The principle challenges in ocean data assimilation are prescribing the background error covariance, an observing system that is skewed toward surface observations and the scale of basin and global scale ocean model state space. Unlike ocean modelling, there are few community software that satisfy the essential requirement of model independence although previous attempts have been made to develop modules that could be shared, Chua and Bennett (2001). GODAE itself was developed to support collaborative development amongst the participants and GODAE OceanView aims to maintain that legacy. A summary of

Table 18.7 Properties of ocean data assimilation sys-

Analysis formulation

Data assimilation tems that result in unique design choices in ocean forecasting systems

Computational efficiency

Background error covariance

Localisation

Observation error covariance

3D—OI, 3DVar, EnOI, 4D—4DVar, EnKF Stationary, non-stationary Multi-variate Error model

Statistical significance and sample space Explicit control of far-field covariances Uniform scale Parallelism

Rank and condition of inversion

Uncorrelated/correlated error

Instrument error

Representation error

Age-error/FGAT

Super-obs

### Localisation

Inversion the design choices are given in Table 18.7 which include the use of 3D or 4D approaches or the use of variational, ensemble or some form of hybrid approach. Four dimensional data assimilation is the formal generalisation of optimal interpolation for a 4D dynamical model (Bennett 2002). However the computational expense of 4D methods, 4DVar or ensemble Kalman Filter, are prohibitive. All operational forecast systems on a basin/global scale use 3D approaches as a practical design choice. At present, only on a regional context have 4D approaches been successfully implemented.

The background error covariance in FOAM (Martin et al. 2007) uses a second-order autoregressive functional form which includes a synoptic component and me-soscale component. This functional form has similar practical advantage for computations. The NCODA system also uses a second-order autocovariance approach and extends this to include a flow dependent covariance function (Cummings 2005). The implementation of the SEEK filter to the operational Mercator system uses a reduced -order EOF method that is stationary but specified in four seasons (Brasseur et al. 2005). The BLUElink ocean data assimilation system (BODAS; Oke et al. 2008) uses an ensemble optimal interpolation approach. The background error covariances (BEC's) are specified as a stationary ensemble of model anomalies from an ocean model simulation forced by reanalysis fluxes. The use of model anomalies from the seasonal cycle is based on the assumption that the background errors scale with the mesoscale variability. An advantage of this physical based approach is the ability to capture anisotropic BEC's that mimic the actual covariances. For example, sea level at a point along the coast (e.g., Thevanard, Australia) exhibits anisotropic covariances extending along the coastline and negligible covariance beyond the shelf break, Fig. 18.18a. The local and far field model anomaly correlations are validated by the anomaly correlations of Australian tide gauges with the Thevanard tide gauge (see Fig. 18.18a). The anisotropic correlations are further

Fig. 18.18 a Correlation coefficient of the ensemble of sea level anomalies at Thevanard (blue circle) and the ensemble of sea level anomalies at all other points in the Australian domain. The correlation coefficients of sea level anomalies of the Thevanard tide gauge (TG) and all other Australian coastal (TG's) is shown a circles coloured according to the correlation coefficient. The correlation coefficient of SLA at Thevanard TG with SLA from satellite altimetry is shown for b Jason and c GFO

Fig. 18.18 a Correlation coefficient of the ensemble of sea level anomalies at Thevanard (blue circle) and the ensemble of sea level anomalies at all other points in the Australian domain. The correlation coefficients of sea level anomalies of the Thevanard tide gauge (TG) and all other Australian coastal (TG's) is shown a circles coloured according to the correlation coefficient. The correlation coefficient of SLA at Thevanard TG with SLA from satellite altimetry is shown for b Jason and c GFO

validated by those of the SLA obtained from satellite altimetry of Jason and GFO in Fig. 18.18b, c respectively.

The specification of BECs based on a reduced-rank approach such as ensemble anomalies can exhibit spurious far field covariances due to undersampling. For example the positive correlations in the Coral Sea at ~150E, 17S shown in Fig. 18.18a are assumed to be spurious such that their magnitude become negligible as the ensemble size is increased. Localisation is frequently introduced for a Gaussian (or similar function) distribution as a function of distance from the target. A single e-folding scale is commonly implemented to preserve the symmetry of the inversion, however the spatial scale of the BECs based on the mesoscale variability will scale with latitude or internal Rossby radius as well as impacted by boundaries. A single length scale is therefore a compromise and will typically be sub-optimal for low- and high-latitudes. A formal approach for the detection of optimal localisation scales can be performed with two independent ensembles by determining the length scale where the RMSE of the two increment fields converges indicating the random far field noise is negligible. Standard localisation introduces imbalances to the analysis that lead to initialization shock. This can be improved through the use of a transformation to streamfunction-velocity potential (Kepert 2009) or the use of an adaptive initialization scheme that uses the model to filter imbalances in the target field (Sandery et al. 2010). A convenient parametric formulae that approximates a Gaussian distribution but has the property that it smoothly converges to zero at a

Fig. 18.19 BODAS-MPI (blue curve) and BODAS-serial (redcurve) performance. Data assimilation task was divided into 48 independent computational sub-domains. In the PETSc parallel case each case was run on 8 cores giving a total core usage of 384 cores. The BODAS-MPI software is 8 times faster (for domain 9) and averages ~6.5 times better performance than the serial version

Fig. 18.19 BODAS-MPI (blue curve) and BODAS-serial (redcurve) performance. Data assimilation task was divided into 48 independent computational sub-domains. In the PETSc parallel case each case was run on 8 cores giving a total core usage of 384 cores. The BODAS-MPI software is 8 times faster (for domain 9) and averages ~6.5 times better performance than the serial version finite length scale is given by Gaspari and Cohn (1999). In this form the analyses beyond a specified localisation length scale is independent which is convenient for parallelisation as is used in BODAS.

The computational performance of the assimilation is the critical determinant to optimising the ensemble size, localisation length scale and super-obs and other strategies to reduce the observation space and inversion. The impact of the choice of inversion solver for the OceanMAPS system is shown in Fig. 18.19. An SVD solver is robust for near singular matrices but has a computational cost that scales as N3. The maximum wall clock time for OceanMAPS analysis exceeds 2000s. The PETSc parallel conjugate gradient solver improves the parallel performance by ~8 times (see Fig. 18.19). This reduction in wall clock will permit several performance upgrades in the next version.

### 18.9 Initialization

Integrating an ocean model from a specified ocean state requires an initialisation procedure as a priori the target state may not be a balanced model state. There are many sources and applications for initialisation such as climatological target states,

Table 18.8 Properties of ocean initialization that result in unique design choices in ocean forecasting systems

Analysis initialization

### Balancing Restoring

Linear restoration/nudging Incremental Analysis Updating Adaptive restoration Dynamical balancing Climatological data Spectral nudging nesting/downscaling and data assimilation increments. In many instances the introduction of the whole target state can result in model shock that degrades the model state, particularly if the model is starting from rest and or with an unperturbed free-surface. A summary of some of the types of initialization/nudging that can be performed and the potential choices are given in Table 18.8.

A common approach is to use a relaxation scheme or "nudging" where a forcing term is added that is proportional to the difference between the model state and the target state. In the absence of other forcing terms the model state will follow a natural decay with an e-folding timescale. In practice, the other forcing terms in the model are not negligible everywhere in space and time reducing the effectiveness of the relaxation. The restoring timescale can be modified to increase the dominance of the forcing term however, this must remain bounded to minimise model shock and maintain numerical stability for an operational system. A relaxation initialization procedure was implemented into the BLUElink OceaMAPS version 1 system over a period of 24 h and for the state variable eta, temperature and salinity (Brassington et al. 2007). An example of the initialized ocean model state is shown in Fig. 18.20b based on the analysed target state Fig. 18.20a for the 1st August 2009

Fig. 18.20 Daily mean sea level anomaly for the 1st August 2009 in the Tasman Sea. a Ocean-MAPS BODAS behind real-time analysis. b OceanMAPS near real-time initialised ocean model state after initialization with nudging for 24 h of eta, temperature and salinity and c Ocean model state after adaptive initialisation for 24 h of eta, temperature and salinity. Sea level anomaly is represented for the range ±0.5 m and the largest velocity magnitude is 2 m/s

Fig. 18.20 Daily mean sea level anomaly for the 1st August 2009 in the Tasman Sea. a Ocean-MAPS BODAS behind real-time analysis. b OceanMAPS near real-time initialised ocean model state after initialization with nudging for 24 h of eta, temperature and salinity and c Ocean model state after adaptive initialisation for 24 h of eta, temperature and salinity. Sea level anomaly is represented for the range ±0.5 m and the largest velocity magnitude is 2 m/s in the Tasman Sea. The Tasman Sea is a region with active geostrophic turbulence with a high eddy kinetic energy to total kinetic energy ratios (Schiller et al. 2008), and identified as one of the largest in the world ocean (Stammer 1997). By inspection the initialised ocean state poorly represents the analysed state leading to a large initial state RMSE in all fields. This is a particularly extreme example but illustrates that the other model forcing terms can prevent the ocean model from reaching the target state within the initialization period. An alternative approach to more efficiently introduce analysis fields is the Incremental Analysis Updating (IAU; Bloom et al. 1996) which has been applied in the ocean prediction context (Ourmieres et al. 2006; Martin et al. 2007) with positive results compared with relaxation.

Another important feature of any initialisation procedure is that the forcing term becomes negligible over the finite period the scheme is applied to minimise residual shock after the forcing term is set to zero. In a relaxation scheme the forcing term becomes negligible only if the model state approaches the target state. In an IAU scheme a specific fraction (1/N) of the analysis increment is introduced over N sequential updates which by design reduces the amplitude of the update but remains non-zero at the end of the initialisation procedure. A modified or adaptive relaxation procedure has been developed to inflate the relaxation when the model-target differences are large (at initial time) by making the relaxation timescale also a function of model target differences (Sandery et al. 2010). An important feature of the scheme is to introduce a threshold on the relaxation to satisfy numerical stability. An example of the adaptive scheme applied to the OceanMAPS BODAS analysis fields is shown in Fig. 18.20c and demonstrates an improvement of 50% RMSE for sea level anomaly and 90% for sea surface temperature (Sandery et al. 2010).

Atmospheric science have put considerable effort into assimilation target states that are dynamically balanced (i.e., do not generate a spurious vertical transport during the initialization). So-called dynamical initialization procedures (Daley 1991) impose dynamical based constraints on the analysis target fields prior to initialization. These procedures are crucial to atmospheric models as errors in vertical transports can produce spurious precipitation and convection leading to a loss of mass to the system. Some ocean systems have implemented similar procedures which minimises spurious gravity waves although it is worth noting that the physical state is less sensitive to these errors compared with the atmosphere. However, this is not the case for coupled bio-geo-chem models where vertical transport errors can result in a sensitive ecosystem response. Both balanced analysis fields and convergent initialization schemes will be important for this application.

A common feature of forced model integrations is model drift leading to model bias where the long term average significantly departs from the observed long-term average or climatological state. The drift of the model can result from the accumulation of flux errors as well as errors in the physical model and numerical representation of the physical model. These fundamental problems are the subject of continuous improvement however at any instance there remain outstanding sources of error. Data assimilation based on least squares assume the system is unbiased. In order to address this bias correction schemes have been implemented (e.g., Dee 2005). Alternative approaches include introducing relaxation procedures often referred to as restoring schemes into the ocean model to reduce this effect. The schemes use a climatological or seasonally evolving reference state and use a large relaxation timescale to produce a forcing term that opposes long period departures from the reference state. Although a large timescale is prescribed the size of the relaxation is also a function of the model-reference differences and therefore will be maximal for extreme model states that can occur transiently. An alternative approach is to control the model-reference state difference by estimating the long-term average of the model and remove the influence of the higher frequencies, so called spectral nudging (Thompson et al. 2006).

### 18.10 Forecasting Cycle

Robust delivery of ocean forecast services in real-time requires a defined schedule for each sequential step with dependencies and wallclock completion times. A summary of the dependencies for each component of the forecast cycle is summarised in Table 18.9.

The in situ and satellite SST observations are provided in near real-time via the GTS and space agencies. However, satellite altimetry IGDR products are provided 3 days behind real-time (Jason-series) and 4 days behind real-time for Envisat. The quality of the ocean analysis is critically dependent on the projection of altimetry and require near a full cycle to improve the least squares analysis. The best estimate ocean state from BODAS is achieved when a symmetric observation window to the analysis date is used for satellite altimetry. In this case, the IGDR products arrive 3 days behind real-time and extend a further 5 days behind real-time for half the period of the Jason series altimeters to complete a cycle. Therefore Ocean-MAPS performs the best analysis 8 days behind real-time as shown in Fig. 18.21. To fit within a 7 day schedule, the best analysis cycles every 3 and 4 days adjusting the analysis to 8 and 9 days behind real-time. For each forecast cycle a near real-time analysis is performed as close to real-time (5 days behind real-time) with near full coverage of altimetry and an asymmetric observation window. A hindcast using analysis fluxes is brought up to real-time which is further integrated with forecast fluxes.

Table 18.9 Properties of the design of the forecast cycle for ocean forecasting systems

Forecast cycle

Hindcast

Delayed mode observations Duplicate checking Quality control

Analysis NWP flux regridding

Analysis

Initialization

Ocean model hindcast

Forecast NWP flux regridding

Ocean model forecast

Forecast

Base Date

April L May

Base Date

April L May

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Fig. 18.21 Schematic representation of the operational schedule for 0ceanMAPSv1.0b. Each cycle is composed of an analysis cycle (orange), a near real-time cycle (green) and forecast cycle (blue). The behind real-time analysis is performed 9 days behind real-time

Fig. 18.21 Schematic representation of the operational schedule for 0ceanMAPSv1.0b. Each cycle is composed of an analysis cycle (orange), a near real-time cycle (green) and forecast cycle (blue). The behind real-time analysis is performed 9 days behind real-time

The mean and 90 percentile range in RMSE for SLA forecast cycles between January 2009 and October 2009 is shown Fig. 18.22. There is a consistent deterioration in mean and range in RMSE performance between the behind real-time and near real-time analyses. The RMSE continues to grow from the near real-time analysis with increasing forecast period. The statistics on the 5th and 6th day indicates the RMSE growth has saturated and there is no further skill in the system.

### 18.11 System Performance

Ocean forecasting has yet to develop an internationally agreed standard or consensus on the metrics to monitor system performance or estimate forecast errors. Much can and should be borrowed from the numerical weather prediction community which have developed a wide range of general methods (e.g., http://www.cawcr. gov.au/staff/eee/verif/verif_web_page.html and http://cawcr.gov.au/bmrc/wefor/ staff/eee/verif/Stanski_et_al/Stanski_et_al.html, Stanski et al. 1989). The first international intercomparison experiment (Hernandez et al. 2009) has developed a framework defining a series of metric classes. Each operational system were to provide daily average ocean state variables for pre-defined regions of the global ocean including the Indian Ocean for the period 1st February 2008-30th April 2008. The intention was to provide a common forecast period, however the data provided

OceanMAPSvl.ob, Jan09-0ct 09, (55S - 10S,100E - 170E)

OceanMAPSvl.ob, Jan09-0ct 09, (55S - 10S,100E - 170E)

Time (days)

Fig. 18.22 Distribution of OceanMAPSvl.Ob RMSE for sea level anomaly for the 9 day behind real-time analysis, 5 day behind real-time analysis and 2, 5 and 6 day forecasts. The mean RMSE is shown by the horizontal black line and the 95th percentile RMSE are shown by the coloured bars. The lines represent a linear estimate of the RMSE performance

Time (days)

Fig. 18.22 Distribution of OceanMAPSvl.Ob RMSE for sea level anomaly for the 9 day behind real-time analysis, 5 day behind real-time analysis and 2, 5 and 6 day forecasts. The mean RMSE is shown by the horizontal black line and the 95th percentile RMSE are shown by the coloured bars. The lines represent a linear estimate of the RMSE performance corresponded to HYCOM-NCODA (5 day forecast), Mercator (14 day hindcast), UKMetOffice FOAM (Real-time analysis), BLUElink OceanMAPS (9 day hind-cast and 3 day forecast). The inhomogeneity of the time periods and the limited period preclude a definitive comparison of performance. Nonetheless, the summary of RMSE, anomaly correlation and model standard deviation against observational data is summarised in the Taylor diagrams (Taylor 2001) in Fig. 18.23a for sea level anomaly and Fig. 18.23b for sea surface temperature. The greyscale background is based on the skill score,

where R is the anomaly correlation, R0=1, of = Of/ar, and o> are the forecast and observation standard deviations (Taylor 2001). The skill score provides further guidance for interpreting performance for systems with different model variance. In the Timor Sea, the prediction systems are achieving anomaly correlations of 0.7-0.8

Timor Sea 01-Feb-2008 - 30-Apr-2006 Sea Level Anomaly

Cbs9rvall0ns(i0l>5) JASONiEMVISAT (22?2B)

System HYC MER UKW OM(an) OMffc)

Timor Sea 01-Feb-2008 - 30-Apr-2006 Sea Level Anomaly

Cbs9rvall0ns(i0l>5) JASONiEMVISAT (22?2B)

System HYC MER UKW OM(an) OMffc)

Timor Sea 01-Feb-2008 - 30-Apr-2008

Sea SurTacö Temperature

066ervatitms(#cti6): AMSRE (72772)

System HYC MER UK M OM(anJ OM(K}

Timor Sea 01-Feb-2008 - 30-Apr-2008

Sea SurTacö Temperature

066ervatitms(#cti6): AMSRE (72772)

System HYC MER UK M OM(anJ OM(K}

Fig. 18.23 Taylor diagram representation of the performance of the HYCOM-NCODA 5 day forecast (red), Mercator 14 day hindcast (yellow), UKMetOffice FOAM analysis (green) and BLUELink OceanMAPS analysis (blue), BLUELink OceanMAPS 3 day forecast (aqua) operational systems during the GODAE intercomparison period for the Timor Sea (100E-120E, 22S-8S) a SLA and b SST. The bias and observations used are summarised above for analyses and forecasts of SLA and SST indicating the products have useful signal during the Austral-autumn.

Further development of these metrics will continue within GODAE OceanView with robust and proven metrics included in "the Guide" to operational ocean forecasting being developed by JCOMM Expert Team on Operational Ocean Forecasting Systems (ET-OOFS). The intercomparison of metrics or concensus amongst the forecast systems is of greatest value when the systems are based on independent components to maximise the variance between models. A review of present operational systems (Dombrowsky et al. 2009) shows unique ocean models, unique data assimilation approaches and unique numerical weather prediction fluxes.

The expected performance of any system configuration can be diagnosed over a hindcast period to achieve a sample space sufficient to estimate statistics. Therefore the forecast system can be monitored using simple metrics based on the background innovations or analysis increments. We will use the Montara wellhead Oil Spill, which took place between the 21st August 2009 and 3rd November 2009, as an example where the performance of the prediction system was important. All of the increments for previous analyses (2nd January 2008-7th November 2009) from the OceanMAPS system were used to form a statistical distribution at each grid point in the Timor Sea. The increments for 22 August 2009 were then compared with this distribution to determine if any were in the 95th or 99th percentile as an indicator of performance that was a statistical outlier (see Fig. 18.24). The region surrounding the Montara wellhead show increments for the analysis exceeded the 99th percen-tile, indicating the model adjustment was a maximum and potentially unreliable.

SLA increment outliers, 20090822

SLA increment outliers, 20090822

### 110E 120E 130E 140E

Fig. 18.24 BLUElink OceanMAPS SLA analysis increments on the 22nd August 2009 that exceed the 95th (green) and 99th (red) percentile of all increments (2nd January 2008 to 7th November 2009) at each ocean model grid point

### 110E 120E 130E 140E

Fig. 18.24 BLUElink OceanMAPS SLA analysis increments on the 22nd August 2009 that exceed the 95th (green) and 99th (red) percentile of all increments (2nd January 2008 to 7th November 2009) at each ocean model grid point

We can extend this analysis by taking the distribution for all grid points within the region 123.5E-125.5E, 11.6S-13.6S, surrounding the Montara well and determine the normalized frequency for each grid point into the increment bins [-0.15:0.01:0.15]. The median of the 400 grid point normalized frequency is shown as the black line in Fig. 18.25 with the 90th percentiles shown shaded in grey. The distribution of increments for the analyses 20nd, 29th August and 5th, 12th, 19th September 2009 are shown in Fig. 18.25 in colour normalised for visualisation. The increments shown in Fig. 18.24 are shown in dark blue and are a statistical outlier. The subsequent analyses, with 7 day separation, have increments that are within the higher frequency range and indicate the system recovered and behaved normally.

An important distinction should be made between methods that monitor performance of the system (e.g., skill scores, Murphy 1988) and methods that estimate the statistics of forecast errors to estimate the expected error. The expected error estimates are however by definition applicable to the most frequently occurring states. These estimates do not apply to events that are rare, for example extreme events. This is an important class of events because designs and operational decisions that are based on the expected conditions can fail if unlikely events do occur and in some instance might result in loss of life or property. Specific methods are required to address this class of problem (e.g., Garrett and Muller 2008). The physical processes of the extreme event may need to be considered to improve the estimated

SLA increment (m)

Fig. 18.25 Normalized frequency of all BLUElink OceanMAPS SLA increments (2nd January 2008 to 7th November 2009) and limited to the region (123.5E-125.5E, 11.6S-13.6S) surrounding the Montara Oil Well binned for increments [-0.15:0.01:0.15]. The median normalized frequency of all the grid points (20 x 20) (solid line) and 90th percentile (shaded grey). The distribution of increments for the 22nd, 29th August and 5th, 12th, 19th September 2009 shown in color normalized to a scale of 0.01 (i.e., freq x (0.01/max(freq)) for visualisation

SLA increment (m)

Fig. 18.25 Normalized frequency of all BLUElink OceanMAPS SLA increments (2nd January 2008 to 7th November 2009) and limited to the region (123.5E-125.5E, 11.6S-13.6S) surrounding the Montara Oil Well binned for increments [-0.15:0.01:0.15]. The median normalized frequency of all the grid points (20 x 20) (solid line) and 90th percentile (shaded grey). The distribution of increments for the 22nd, 29th August and 5th, 12th, 19th September 2009 shown in color normalized to a scale of 0.01 (i.e., freq x (0.01/max(freq)) for visualisation likelihood. For example, consider coastal fog that results from cool sea surface temperatures from a coastal upwelling. The presence of fog prevents the AVHRR from observing the cool SST's and the microwave resolution does not observe close to the coast. An SST analysis will persist the background state and depart from the true state. As cloud frequently occurs, a simple statistic will not separate fog events from other clouds and the expected SST error would be low as there is skill in persisted SST's. However, if other factors such as the upwelling favourable winds and cloud type of fog are included a higher error might be estimated.

### 18.12 Conclusion

State estimation and forecasting for the ocean's mesoscale is a grand challenge. In particular the ocean state at these scales continues to be under-observed and our understanding of the dynamics is at the frontier of ocean science. Despite this there is now ample and growing evidence that the first generation systems have demonstrated that the existing global ocean observing system is sufficient to constraint the mesoscale variability. These systems have achieved a performance that is positively impacting real applications. At the same time there is evidence that the performance is patchy in time and in space and sensitive to the quality and coverage of the observing system and forcing in real-time. None the less this provides a solid foundation for continued advances and improved performance.

A complex system such as an ocean prediction system is composed of several components each of which have critical design choices that impact the performance and cost of the total system. In the first generation forecast system there are numerous choices taken that are scientifically robust given the constraints imposed by the observing system and computational costs available at the time of development. There are also many decisions that are taken that compromise the performance for practical constraints of completion of integrations within a finite schedule. These decisions and methods will continue to be revised as the constraints are reduced and new methods and models are developed.

There are numerous directions for which ocean forecasting will be extended to optimise forecast skill including: 4D data assimilation, ensemble forecasting, coastal ocean forecasting, coupled ocean-wave-atmosphere modelling (e.g., Fan et al. 2009), coupled ocean-wave-atmospheric data assimilation and more just within the physical modelling space. The challenge moving forward will be the development of methods that continue to abstract the complexity to make it more manageable. It is very likely unavoidable that "black boxes" become more prevalent. However, this must be done with sufficient rigor that it is readily verified that the component, sub-system and system are solving the right problem to a known precision. This development will need to be undertaken in parallel to improvements in the ocean observing systems and computational hardware and software technologies.

Acknowledgements OceanMAPS was developed by the BLUElink a joint project of the Bureau of Meteorology, CSIRO and Royal Australian Navy and the BLUElink science team. The author gratefully acknowledges Claire Spillman, Nicholas Summons, Paul Sandery, Justin Freeman, Leon Majewski for contributions to the figures in this manuscript.

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