We use the Regional Ocean Modeling System (ROMS) to simulate the eddying ocean circulation off the southeastern coast of Australia between January 2012 and December 2013. This modelling suite is named the South East Australian Coastal Ocean Forecast System (SEA-COFS, ). ROMS is a widely used free-surface, hydrostatic, terrain-following, primitive equation ocean model and is described by , , and . The model configuration used in this study has been used in various past studies of the EAC and is described in detail in and .

The study domain covers SE Australia from 25.25 to 41.55° S and approximately 1000 km offshore (Fig. a). The domain covers the latitudinal extent of the EAC system from where the current jet is most coherent, the EAC separation region, the region of high eddy activity associated with the EAC eastern extension, and the EAC southern extension. The grid is rotated 20° clockwise such that the domain y axis is oriented roughly parallel with the coastline. The cross-shore horizontal resolution varies from 2.5 km over the continental shelf and gradually increases to 6 km offshore. The horizontal resolution is 5 km in the along-shore direction. Higher resolution over the shelf allows the steep topography to be maintained while minimising pressure gradient errors that emerge in terrain-following coordinate schemes, which otherwise may result in artificial along-slope flow for steep topography . As such, less topographic smoothing is required to ensure low horizontal pressure gradient errors while still representing the shelf and seamount structures in the model. The model utilises 30 vertical s layers with higher resolution in the upper 500 m to resolve mesoscale dynamics and higher resolution near the seabed for improved representation of the bottom boundary layer. To better resolve surface currents, a near-constant-depth surface layer is provided by applying the vertical stretching scheme of .

Initial conditions and boundary forcing are derived from the Bluelink ReANalysis version 3 (BRAN3; ). The boundary forcing is applied daily, and misfits in baroclinic energy to the BRAN3 condition are absorbed at the boundary via a flow-relaxation scheme. The model is forced at the surface with realistic atmospheric forcing derived from the 12 km resolution Bureau of Meteorology (BOM) Australian Community Climate and Earth-System Simulation (ACCESS) analysis . The atmospheric forcing fields are applied every 6 h and used to compute the surface wind stress and surface net heat and freshwater fluxes using the bulk flux parameterisation of .

The free-running configuration, while unable to reproduce the temporal evolution of the mesoscale eddies, has been shown to accurately represent the mean dynamical features of the EAC and both the surface and subsurface (0–2000 m) variability . Specifically, they show that the model accurately represents the mesoscale eddy-related variability in sea surface height (SSH), the frequency in occurrence of EAC separation latitude, the seasonal cycle in sea surface temperature (SST), the ocean's subsurface structure based on data from Argo profiling floats, EAC transport, and the temperature depth structure across the EAC. Thus, using data assimilation, we aim to constrain the model to reproduce the temporal evolution of the mesoscale eddies and examine the forecast skill achieved.

The same set of observations are assimilated into the ROMS model configuration using the two DA systems (EnOI and 4D-Var) for comparison in this study. These include satellite-derived SSH, SST, sea surface salinity (SSS), vertical profiles of temperature and salinity from profiling Argo floats, and vertical profiles of temperature from expendable bathythermographs (XBTs) (refer to Fig. b). The number of processed observations assimilated for each 5 d assimilation window is shown in Fig. d and e. These observations are referred to as the “traditionally” available observations (TRAD) . We describe the observations used and the observation uncertainties specified below. For a detailed description of the observations, the processing performed prior to assimilation, and the prescribed observation uncertainties, the reader is referred to .

In this paper, we refer to three different configurations of the SEA-COFS model which differ in DA type and/or the observations assimilated. Each case is performed over the 2-year period from January 2012 and December 2013 and is described below.

4D-Var TRAD refers to the 4D-Var system that assimilates “traditionally” available observations (SSH, SST, SSS, Argo, and XBT). This system is similar to the system described in expect that it only assimilates the TRAD observations.

The classic state estimation problem can be given by 1Xa=Xf+K(y-H(Xf)), where X is the state estimate; subscripts f and a refer to forecast and analysis, respectively; K is the Kalman gain; y is the observation vector; H is the observation operator that samples the background circulation to observation points in space and time. The y-H(Xf) term is referred to as the innovation vector and describes the difference between the observations and the forecast model mapped to observation space. The difference in DA techniques lies in the formulation of K, which determines how the forecast innovations are mapped into model space to produce the new state estimate (Xa). For the standard analysis equation that is solved by the Kalman filter and the dual form of 4D-Var, K can be expressed as 2K=BGT(GBGT+R)-1, where B is the background covariance, R is the observation error covariance, and G performs the mapping from model space to observation space.

For time-dependent methods (4D-Var and EnKF), observations are assimilated over a time window respecting the dynamics of the model. The observation operator H samples the nonlinear forecast model Xf at the observation locations in space and time over an assimilation cycle time interval. In 4D-Var, the background error covariance matrix B is typically assumed to be unchanging in time, so there is no explicit flow dependence of the B. Flow dependence is implicit via the terms BGT and GBGT in Eq. (), since G is the operator that maps the tangent linear model solution to the observation points and GT is the adjoint ocean model forced at observation points . In EnKF, the background error covariance matrix and its evolution in time is estimated from an ensemble of nonlinear model solutions . For 3D-Var and EnOI, observations are all centred at a single time and, rather than using the model physics to constrain the model versus observation error, time-invariant covariances are prescribed.

We begin by assessing the performance of the EnOI and 4D-Var systems relative to the observations that the systems assimilate. The 5 d model forecast is compared to the observations that become available over those 5 d (that is, they have not yet been assimilated) to quantitatively assess the performance of the model forecasts over time. Comparing forecasts against observations provides objective assessment of the system performance.

Table  presents the mean innovation (mean absolute difference, MAD), innovation bias (mean difference, MD), and number of observations for the 2-year period. Both systems have an identical number of observations. Compared to the EnOI, the 4D-Var improves the SST forecast error from 0.42 to 0.36 °C, the SSH forecast error from 10.3 to 8.3 cm, in situ temperature from 0.90 to 0.71 °C, in situ salinity from 0.079 to 0.056 PSU, and SSS from 0.214 to 0.183 PSU. Overall, the improvement of the MAD for the 4D-Var over the EnOI is 9 %–21 %. The percentage differences in forecast error between the two systems are less for the surface observations (SLA, SST, and SSS) compared to the in situ observations, indicating that the advantages of 4D-Var extend through the water column. In WBC regions, the parent model displayed MADs between reanalysed and observed SST values on day 1 of each assimilation of 0.2–0.6 °C and MADs of 6–12 cm for SSH .

The performance of the two systems relative to SSH, SST, and Argo observations is presented in more detail using the root-mean-square difference (RMSD) between the model forecasts at the observation locations and the observation values. Figure a and b show the RMSD between the forecasts (4D-Var and EnOI, respectively) and observations for SSH across the model domain, averaged over the 2-year period. The EnOI forecasts display higher SSH errors across the model domain, with both systems showing higher errors in the eddy-dominated region compared to the rest of the domain. Figure c shows that the spatially averaged RMSD between the forecast and the observations is consistently higher for the EnOI forecasts over the 2-year period.

As each forecast is initialised from the previous analysis, forecast errors typically increase over the forecast horizon. SSH forecast errors are averaged across the model domain (Fig. d) and for the eddy-dominated region (Fig. e) for each day of the 5 d forecast horizon. With SSH, the forecast errors are consistently lower for the 4D-Var system due to lower errors in the initial conditions, while the rate of error increase is similar between the 4D-Var and EnOI systems. At day 5, the domain-averaged (eddy-dominated region averaged) root-mean-squared (rms) SSH forecast errors are 61 % (64 %) higher for the EnOI system compared to the 4D-Var system.

In a similar manner to the SSH forecast errors in Fig. , the forecast errors relative to SST observations are presented in Fig. . Both systems display higher errors in the core of the EAC upstream of the typical separation region and in the eddy-dominated region. The EnOI forecasts display higher SST errors across the model domain, with the most pronounced difference in the eddy-dominated region (Fig. a, b). The time series of RMSD for EnOI and 4D-Var (Fig. c) are highly correlated as the statistics are sensitive to the number of observations and the coverage in the high variability area. While the EnOI analyses provide a slightly improved fit to SST (Fig. d, e at day 0), SST forecast errors grow more quickly than in the 4D-Var system and the 4D-Var system outperforms the EnOI system for SST forecasts after 1 d. At day 5, the domain-averaged (eddy-dominated region averaged) rms SST forecast errors are 21 % (29 %) higher for the EnOI system compared to the 4D-Var system.

To assess the subsurface predictive skill, we extract the 5 d model forecast values at the observation times and locations for all Argo floats that were observed in the region over the forecast window. Binning these observations with depth, we present profiles for temperature and salinity of the mean (Fig. a, e), bias (Fig. b, f), and the RMSD between the forecasts and the observations for all observations that fall on the first day of the forecasts (Fig. c, g) and all observations that fall on day 5 of the forecasts (Fig. d, h). The magnitude of the RMSDs can be compared to the root-mean-squared (rms) observation anomaly, which describes the variability of the observations within each depth bin. For in situ temperature, both the 4D-Var and EnOI forecasts display similar skill on the first day of the forecasts (Fig. c); however by day 5 the 4D-Var forecasts display lower errors compared to the EnOI forecasts over the upper 600 m, with a maximum difference in RMSD (bias-corrected RMSD) of 0.56 °C (0.34 °C) at 200 m (Fig. d). For salinity, forecast errors at day 5 are of similar magnitude throughout the water column for the two systems (Fig. h). Both systems have rms errors considerably less that the rms observation anomaly. Salinity bias dominates the RMSD deeper than 600 m, so bias-corrected RMSD values are less that the total RMSD (Fig. g, h).

As described in Sect. , a number of observations were withheld from the 4D-Var and EnOI DA systems presented in this paper, allowing the system performances to be assessed against independent observations. In this section, forecasts from the 4D-Var and EnOI systems (which assimilate the traditional suite of observations, TRAD) are compared to the analyses and forecasts produced by assimilating the full suite of observations (FULL). Comparisons are made between the observations and the model solutions extracted at the observation times and locations, and predictive skill is assessed for days 1 to 5 of the forecast horizons (and analysis windows in the case of the FULL analysis).

Under the HF radar footprint at 30° S, surface radial velocity observations from two sources are combined to compute surface velocities to about 100 km offshore, covering the shelf and shelf slope circulation. This coverage typically includes the EAC as a coherent jet and the intermittent formation of cyclonic frontal eddies inshore of the EAC . The complex correlations between the observed and model velocities are presented in Fig. . At forecast day 5, the 4D-Var TRAD displays similar predictive skill to the FULL forecasts. The EnOI forecasts are worse than the 4D-Var TRAD across the 5 d, showing that the 4D-Var system provides better representation of the circulation under the HF radar footprint in the analyses and forecasts.

Glider data over the study period (2012–2013) were predominantly available over the NSW continental shelf in water depths <200 m; however, from May–July 2012, several glider missions extended offshore into eddies and sampled down to below 1000 m. These glider observations were shown to be particularly impactful in constraining transport and EKE estimates in the FULL simulation . These observations represent independent data for the 4D-Var and EnOI TRAD systems, and Fig.  shows how the simulations represent temperature and salinity as measured by the gliders.

Errors are lowest near the surface compared to over the thermocline region due to the assimilation of SST and SSS data in all three systems (4D-Var TRAD, EnOI TRAD, and 4D-Var FULL). The 4D-Var TRAD has rms forecast errors for temperature of a similar magnitude and depth structure as the rms observation anomalies, and the errors do not considerably change from day 1 to day 5 of the forecast window. The EnOI errors are of similar magnitude to the 4D-Var near the surface (∼ 1 °C), but they are 20 % greater between 100–200 m for day 1 and 40 % greater for that depth range at day 5 (Fig. c, d). Temperature bias plays a considerable part in the EnOI RMSD values below 100 m, but the bias-corrected RMSD for EnOI still exceeds the bias-corrected RMSD for 4D-Var TRAD at both day 1 and day 5 (Fig. c, d).

For salinity, the 4D-Var and EnOI display similar forecast errors in the upper 200 m. This depth range corresponds to where the many shelf glider observations exist. Below 200 m (the off-shelf missions into the Tasman Sea), forecast errors peak at 300 m reaching 0.30 for EnOI at day 5, compared to 0.23 for 4D-Var. Similar to the Argo-observed salinity (Fig. f, g, h), salinity bias dominates the errors associated with glider observed salinity below 500 m for 4D-Var TRAD and below 200 m for EnOI.

Subsurface velocities are measured by acoustic Doppler current profilers mounted on moorings in the EAC array, the SEQ shelf and slope, and on the NSW shelf (Fig. c). In Fig.  we present the complex correlation between the modelled and observed velocities for selected moorings extending from 28° S to 34° S. The mooring locations are shown in Fig. c, with EAC2 and SEQ400 being in 1500 and 400 m water depth at 28° S, CH100 being in 100 m water depth at 30° S and SYD100 being in 100 m water depth at 34° S. At EAC2 and SEQ400, the 4D-Var TRAD displays similar predictive skill to the FULL after 5 d and considerably outperforms the EnOI system throughout the water column. This indicates the benefit of 4D-Var including the northern boundary conditions in the cost function. On the shelf at 30° S (CH100) and 34° S (SYD100), the EnOI and 4D-Var systems show similar predictive skill.

As shown in both Figs.  and , the 4D-Var FULL complex correlations display a rapid reduction in correlation by day 3–5 of the forecast. As discussed in , while the analysis fits the velocity observations along the continental shelf, the forecast model is unable to resolve the complexities of the shelf circulation such as the cyclonic vorticity inshore of the EAC. As such, the forecast skill of the TRAD system is similar to that of the FULL system for 5 d forecast horizons.

We have shown that the 4D-Var TRAD system outperforms the EnOI TRAD system at the surface and subsurface when compared against both assimilated and independent observations. Improvements to temperature forecasts with 4D-Var are more pronounced in the subsurface (the upper ∼ 400 m) compared to at the surface (Figs.  and ). We now examine the model forecasts to elucidate the differences between the representation of the ocean state (in model space, rather than observation space) across the two DA systems.

The model forecast, Xf, is adjusted by the assimilation of observations (as per Eq. ) to produce an analysis, Xa. This model state estimate should provide a better representation of the observations and provides updated (improved) initial conditions for the subsequent model forecast. In the 4D-Var system used in this study we perform a 5 d forecast and a 5 d analysis every 4 d, such that the initial conditions for the subsequent forecast are taken from day 4 of the previous analysis. For the EnOI system, an analysis is generated every day. For consistent comparison across the two systems, we take the analysis every 4 d as initial conditions and perform a 5 d forecast. In both cases there are discontinuities in the ocean state between day 4 of the previous forecast and the beginning of the subsequent forecast (which correspond to concurrent times). This is illustrated in Fig. i, which shows a time series of temperature at the surface at 34° S. Assimilated (SST) and independent (SYD140 mooring near-surface temperature data) are shown for reference. The discontinuities between the forecasts are less pronounced for the 4D-Var system compared to the EnOI system. Over the entire 2-year test period, the RMSD between the initial conditions (from the analysis) and the previous forecast field at that time illustrates greater discontinuities for the EnOI system compared to the 4D-Var system for SSH, SST, and subsurface temperature (Fig. a–h).

The discontinuities presented here do not exactly correspond to the analysis increments. We have presented the differences in the ocean state between day 4 of the previous (5 d) forecast and the beginning of the subsequent forecast (which correspond to concurrent times). For 4D-Var, the ocean state at the beginning of the forecast is taken from the previous cycle analysis, and so the difference presented here represents the difference between the forecast (or the background) at day 4 and analysis at day 4 (once data assimilation has been performed on that assimilation cycle). This is essentially the “analysis increment at day 4”. However for a 4D-Var system the analysis increments typically refer to the adjustments to the initial conditions, boundary forcing, and surface forcing that are made to generate the analysis. For EnOI, the analysis increments refer to the difference between the background model and the analysis (both centred on a single time and computed daily in this case). However, here we take the analyses every 4 d and perform 5 d forecasts, and the differences presented here refer to the difference between day 4 of the forecast and the analysis that provides initial conditions for the subsequent forecast.

With 4D-Var we are able to represent the entirety of the observations collected over a time window (in this case 5 d), placing them in dynamical context using the (linearised) model equations. In contrast, EnOI performs discrete minimisations with observations centred on a single time (in this case every day). The estimate of the ocean over the observation window that is created with the 4D-Var assimilation system results is smaller discontinuities between forecast cycles, on average, compared to the EnOI system, as a continuous field evolves by the nonlinear primitive equations as opposed to starting a forecast from a discrete estimate, which can “shock” the system. Our results of the improved predictability achieved by the 4D-Var system support the understanding that a continual and dynamically balanced analysis field is advantageous to the quality of future predictions.

The modelled velocities are used to compute eddy kinetic energy (EKE) and mean kinetic energy (MKE) over the 2012–2013 simulation period. MKE is given by MKE=12(U‾2+V‾2), where U‾ and V‾ are the time mean velocity components, and the EKE is given by EKE=12(U′2+V′2), where U′ and V′ are the velocity anomalies. The MKE describes the energy associated with the mean currents, and the EKE describes the energy associated with the perturbations from the mean. Figure  shows the MKE and EKE averaged over the upper 400 m and from 400–1200 m.

Comparisons of MKE above 400 m show that the EAC core is narrower and more confined to the slope in the 4D-Var system, while MKE for the EnOI system is more spread out and with higher MKE directly over the continental shelf (Fig. a, e, i). This difference is despite the identical SSH observations being assimilated, noting that SSH observations in water depth <1000 m are not assimilated, and the identical forward numerical model. In the 4D-Var simulation, the MKE is greater below 400 m than the EnOI simulation downstream of 27.5° S to the typical EAC separation zone (Fig. b, f, j). This is consistent with , who use a long-term integration of the free-running simulation to describe a downstream deepening of the EAC before separation.

The spatial structure of the EKE is similar across the two systems. Above 400 m, the EnOI system has elevated EKE over the EAC jet (Fig. k, blue regions), while the 4D-Var system has elevated EKE in the eddy-dominated regions (Fig. k, red regions). The elevated EKE for the EnOI system (in the more coherent region) relates to the greater discontinuities between the subsequent forecasts, which manifests itself as greater low-frequency >1 d variability over the 5 d forecasts as the 5 d model run adjusts to the “shocks” to the system. In contrast, the elevated EKE in the 4D-Var system outside of the coherent jet relates to the greater near-inertial variability. This is explored in Sect.  and Fig. . At depth (400–1200 m), EKE is elevated for EnOI compared to 4D-Var in the EAC southern extension.

Eddies can form through barotropic instability in the mean flow or baroclinic instability in the vertical density structure. It is important for a model to correctly represent these instabilities, as they represent the pathways by which eddies are generated. Following , we calculate the barotropic conversion rate (KmKe) as 9KmKe=ρ0U′U′‾∂U‾∂x+U′V′‾∂U‾∂y+V′U′‾∂V‾∂x+V′V′‾∂V‾∂y, where ρ0=1025 kg m-3. The baroclinic conversion rate (PeKe), from eddy potential energy to EKE, is calculated as 10PeKe=-gρ′W′‾, where the acceleration due to gravity is g=9.81 m s-2, and ρ′ and W′ are the density and vertical velocity anomalies. KmKe and PeKe have been previously used to explore eddy generation rates in the EAC (e.g. ).

Barotropic and baroclinic energy conversions are computed from the model forecast fields and averaged over the 2-year period (Fig. ). Both the 4D-Var and EnOI systems show similar magnitude and overall spatial structure of the barotropic and baroclinic energy conversions, as well as similar partitioning between barotropic and baroclinic instabilities. The similarities are likely due to the common model and atmospheric forcing. The barotropic conversion (compare Fig. a, c) represents instabilities in the depth-mean flow, which 4D-Var and EnOI represent similarly. The baroclinic conversion (compare Fig. b–d) is also similar between the DA configurations in overall spatial structure and the zonally integrated magnitudes (Fig. e), although the EnOI baroclinic conversion rate contains more high-wavenumber spatial patterns, which likely relate to unbalanced adjustments upon assimilation. This is further explored in Fig.  and the associated discussion.

When observations are assimilated the goal is to provide an improved fit to the observations while retaining a dynamically consistent ocean state that can be used as initial conditions for the subsequent forecast. The background numerical model produces an estimate of the ocean state whose frequency and wavenumber spectra are limited by the resolution of the model and the processes resolved. If the observations sample time and space scales that cannot be resolved by the model, it is standard DA practice to either remove these scales of variability from the observations or account for them in the observation error terms (e.g. ). If the model background is deficient at some space scale and/or timescale (which it is able to resolve), then these may be corrected by DA so that the analyses and forecasts are better. However, if the assimilation process introduces energy at different, non-physical scales, this may negatively impact the forecast skill. By presenting the temporal and spatial scales of variability of the forecast ocean state, we can understand how the assimilation has changed the ocean's energy distribution and understand the differences in error growth across the two DA systems.

The subsurface structure of the model fields and their variability is shown in Fig. . The EnOI system has more temperature variability near the surface (upper 200–500 m) compared to 4D-Var. The greater near-surface temperature variability in EnOI compared to 4D-Var is greater in the eddy-dominated region (34° S), where adjustments are greater (Fig. d, f) compared to more coherent, upstream region (28° S). For velocity variability, 4D-Var shows elevated variability almost everywhere except in the upper 250 m near the shelf at 34° S. Both data-assimilating configurations show elevated variability in temperature and velocity over the upper ∼ 1000 m compared to the non-data assimilating simulation (hereafter referred to as the Free-run). The differences are further illustrated in Figs.  and , where the frequencies of the variability are revealed.

Frequency spectral analysis is first performed for all 5 d forecast windows and then averaged (Fig. ). A 5 d window with the model output 4-hourly gives a frequency range from 1/5 d to 1/8 h, with 15 points in frequency space due to the short time series (31 points). Surface velocity (but not temperature) and subsurface temperature and velocity display elevated energy in the 16–24 h band for the 4D-Var system compared to EnOI and the Free-run (Fig. ), corresponding to the near-inertial band. This inertial energy is introduced through the assimilation adjustments which, due to the nature of 4D-Var, must satisfy the model equations. Increased near-inertial variability upon 4D-Var data assimilation was also shown in and . found that the increased inertial energy had minimal impact on the mesoscale circulation. Using observing system simulation experiments, showed that, while the 4D-Var system displayed elevated near-inertial variability (compared to their free running truth simulation), near-inertial frequencies did not influence energy at other frequencies, and predictability at both higher frequencies (in their case internal tides) and lower frequencies (associated with the mesoscale circulation) was good.

The differences between EnOI and the Free-run and EnOI and 4D-Var (as revealed in Fig. ) are difficult to decipher from Fig.  as they exist at low frequencies (periods greater than 1 d). In order to resolve the low frequencies, we concatenate the forecast cycles in time to produce a full 2-year time series. As a longer time series allows a higher resolution in frequency space, Fig.  show a higher frequency resolution compared to Fig. , for which the spectra are computed for all 5 d periods and averaged. Concatenation of the time series requires removal of the 1 d overlap (the last day of each cycle is excluded) such that time is monotonously increasing. Because of the assimilation updates, discontinuities exist between the cycles every 4 d (this is not the case for the Free-run). These discontinuities (displayed in Fig.  and discussed in Sect. ) manifest as harmonics of the 1/4 d frequency and are most pronounced for EnOI in temperature at 34° S. For example, the surface temperature spectra for the EnOI system at 34° S show spikes at harmonics of 0.25 (0.5, 0.75 ,1.0 ,1.25, 1.5, etc. cycles per day). Nevertheless, the spectra are useful in showing the differences in variability across the DA systems and the Free-run particularly for low frequencies. The Free-run displays less energy than both DA systems in both temperature and velocity in the eddy-dominated region, consistent with the reduced variability shown in Fig. . This relates to less variability at low frequencies (periods greater than 1 d) in the Free-run compared to the DA systems, and for the 4D-Var system, less variability at inertial frequencies also.

The elevated energy in the EnOI system compared to 4D-Var and the Free-run relates to periods greater than 1 d for both temperature and velocity (Fig. ). Greater variability at low frequencies in the EnOI system compared to 4D-Var exists in both temperature and velocity and is most pronounced in the upper 500 m and in the eddy-dominated region (34° S compared to the more coherent region at 28° S, Fig. ). This increased low-frequency variability in EnOI compared to 4D-Var dominates the total variability (displayed in Fig. ) for near-surface temperature, but it is masked by greater inertial-period variability in the 4D-Var system for velocity. That is, despite the low-frequency velocity variability being greater in EnOI (Fig. ), the total velocity variability is greater for 4D-Var (Fig. ). We find that the greater low-frequency variability for EnOI compared to 4D-Var is associated with greater discontinuities between the subsequent forecasts. The discontinuities also exist in 4D-Var, but they are less pronounced (Fig. ).

The spatial scales of the forecast ocean state can be represented by wavenumber spectra. Here we present cross-shore wavenumber kinetic energy spectra through sections at 28 and 34° S (Fig. a) for days 1 and 5 of the forecasts, for the Free-run and for AVISO gridded geostrophic velocities (Fig. ). The observational data product used is the AVISO gridded velocities from altimetry and drifters using multiscale interpolation, version 0100 , with a 1/10° spatial resolution and temporal coverage from 1 July 2016–30 June 2020. Note that the spatial resolution is the same as that of the assimilated SSH observations (Sect. ). For the model forecasts, wavenumber kinetic energy spectra are computed for days 1 and 5 of all (186) cycles, and the averages are plotted. For the AVISO observations, wavenumber kinetic energy spectra are computed for every day of the available time period and the average plotted. Model spectra are shown at the surface and at depths of 400 and 1000 m; AVISO data provide geostrophic velocities, and the corresponding spectra are plotted on the surface velocity panels.

At the surface all systems, except the EnOI at day 1, display consistent kinetic energy spectra at 28° S. The AVISO velocities show less energy at spatial scales between 15–80 km compared to the Free-run, the 4D-Var system across all forecast days, and the EnOI system at day 5. At 34° S, where eddy variability is high, the Free-run underrepresents the kinetic energy across all spatial scales at all depths. At the surface, the 4D-Var system across all forecast days and the EnOI system at day 5 represent the AVISO spectrum well, with the AVISO velocities again showing slightly lower energy at spatial scales between 15–80 km.

For the first day of the EnOI forecasts (representative of the analyses), there is elevated kinetic energy at finer length scales and this energy dissipates by day 5 of the forecast. This elevated energy is most pronounced at the surface and near-surface (upper 200 m, not shown). Specifically, elevated kinetic energy exists in the EnOI initial states at length scales less than 100 km at 28° S and between 20–80 km at 34° S. For the 4D-Var system the wavenumber kinetic energy spectra remain relatively unchanged over the forecast window, with the day 1 and day 5 wavenumber spectra tracking closely. Compared to the Free-run, both the 4D-Var and EnOI assimilation systems introduce more kinetic energy across all spatial scales throughout the water column in the eddy-dominated region (illustrated by the sections through 34° S in Fig. ).

We include the idealised spectral slopes of k-5/3 and k-3 in Fig.  for reference. The wavenumber kinetic energy spectra approximately match the k-5/3 slope for the mesoscale range, the k-3 slope for the submesoscale range for the 4D-Var ocean state on day 1 and day 5, and the EnOI forecasts on day 5. However, we note that the submesoscale range is only partially resolved by the 2.5–6 km resolution model and even less so by the AVISO observations. The k-5/3 and k-3 slopes have been shown to represent surface quasi-geostrophic and quasi-geostrophic dynamics, respectively; however realistic simulations show that other slopes are possible .

We have shown that energy is elevated for shorter (less than 100 km) length scales in the EnOI analyses, and upon integration of the forecast model this energy dissipates to match the energy associated with the 4D-Var system. Wavenumber kinetic energy analysis of the atmosphere by showed the contrary: the increase of energy at small scales upon integration of the forecast model. They showed that the initial states of high-resolution NWP model forecasts lacked the fine-scale (mesoscale in the case of the atmosphere) energy because “observations to initialise the fine scales are not generally available and data assimilation methods that can use high-resolution observations are not yet mature”. The fine-scale portion of the kinetic energy spectrum was spun up in the forecasts in 6–12 h, providing increased value to the NWP forecasts. In our study we observe the introduction of energy at small spatial scales upon DA with EnOI, and this elevated small-scale energy is lost by day 5. This implies that the small-scale energy dissipates over the 5 d forecast. The elevated kinetic energy at scales less than 100 km is not a physical space scale that is resolved by the observations, as shown by the AVISO kinetic energy spectra, and does not exist in the 4D-Var system (Fig. ). Rather it comes about due to EnOI's adjustments upon assimilation. It is likely that the increased error growth (hence poorer forecast skill) for the EnOI system (compared to 4D-Var) in this study relates to these adjustments. The consistency of the wavenumber spectra over the 4D-Var 5 d forecast windows likely relates to the constraint that the analysis is a complete solution of the model nonlinear equations, requiring dynamically balanced adjustments.