The Regional Ocean Modeling System (ROMS v3.9 ROMS/TOMS Framework: 3 March 2020) is a 3-D finite-difference model solving the primitive equations on a horizontal grid with a terrain-following vertical coordinate . The model application used here focusses on the EAC and has been used in several previous studies (e.g. ). Along the coastline, the model domain extends from 27–38∘ S and over ∼700 km offshore (Fig. a). Bathymetry is sourced from the Geoscience Australia 50 m multibeam survey . The grid discretisation has a spatial resolution of 2.5 to 6 km, linearly increasing in the off-shelf direction, and is rotated 20∘ clockwise from the north to approximately align the model grid with the along-continental shelf and off-continental shelf directions. There are 30 vertical s-coordinate layers, with the sigmoidal distribution tuned for finer spacing and resolution in the surface layer.

The EAC model application is used with two different configurations: a free-running simulation and the DA configuration. The free-running simulation uses lateral forcing conditions of currents, temperature, and salinity from BRAN2020 (2020 version of the Bluelink Reanalysis, ) and daily surface forcing conditions from BARRA-R (Bureau of Meteorology Atmospheric high-resolution Regional Reanalysis for Australia; ). We refer to this free-running configuration as the “Ref state”.

The DA configuration used in these OSSEs is an Incremental Strong Constraint 4-Dimensional Variational scheme (IS4D-VAR; e.g. ), which has been applied to the EAC region previously . This 4DVar scheme considers the difference between a free-running forecast and observations (each with associated error fields) over an assimilation window (in our case, 5 d). Adjusted initial and boundary conditions are then generated, such that a new analysis simulation, using these adjusted forcing conditions, has minimised differences (in a least-squares sense) between the analysis simulation and the observations. The assimilation cycle then increments forward using the previous analysis to initialise a new forecast, and the process repeats.

The DA configuration uses lateral forcing conditions from BRAN2020 and surface forcing conditions from a bulk flux formulation with daily atmospheric conditions from the Australian Bureau of Meteorology's ACCESS reanalysis . Vertical and horizontal mixing parameters have also been modified between the free-run and data-assimilating configurations. The different surface forcing conditions and mixing parameters in the assimilating and free-running configurations (see Table ) are appropriate, as they lead to a source of error that the DA system must reduce, as required in an OSSE. Both the free-running and DA simulations are performed over the period November 2011–January 2013.

The performance and configuration options of the DA system were extensively tested and were shown to produce relatively low error in estimates and forecasts of the EAC . The system uses 14 inner loops with one outer loop, set following testing of how many loops were required to achieve acceptable reduction in the cost function (see ). The background error covariances are static and computed by factorisation based on , as described in detail in .

OSSEs compare a free-running reference simulation (the Ref state) against data-constrained simulations, where the data to be assimilated are sourced from the Ref state with the addition of errors. The OSSE that is simulating the same period as the Ref state is perturbed to introduce error and initiate divergent evolution through the use of different initial conditions. These initial conditions are similar to those used to initialise the Ref state but are extracted from a point 8 d later (the OSSE begins at 2 December 2011 with conditions from 10 December 2011). This offset is chosen so as to fairly test the DA system (see , for further information about this choice of perturbation).

The assimilation of the synthetic observations, which are selected to represent a chosen observation type, location, and time, should then converge the resulting analyses towards the Ref state. Comparing the OSSE to the Ref state will show the improvement in the data-constrained reanalysis for the synthetic observation platform tested in each OSSE. For a more detailed description of the procedure, readers are directed to and the schematic outlining the process in their Fig. 2.

The Ref state to which the OSSE is compared should be quasi-realistic of the true ocean, and, as a result, the impact of assimilating synthetic observations into the OSSEs should translate to the real ocean. Our Ref state simulation has been rigorously shown to produce an accurate representation of the EAC, including eddy field structure , EAC separation latitude , and long-term conditions . The OSSEs should also be simulated with a sufficient amount of difference to the Ref state such that the ocean state will tend to evolve differently to the Ref state. These differences could arise from different initialisation, model parameterisations, grid resolution, and different forcing . We employed a “fraternal twin” approach for our OSSEs, where we use different initialisation conditions, different model parameterisations (e.g. vertical mixing parameters), and different boundary forcing conditions between the free-running and DA simulations. However, it is also important to ensure that the different configurations of the Ref state and OSSEs (e.g. in this case, surface forcing and some mixing parameters) do not cause such an impact as to introduce a large long-term bias. To assess this, a “baseline” experiment was conducted using the OSSE configuration but without assimilating any observations. Comparison of this against the Ref state showed a warm bias in the surface waters, which is likely to be corrected by assimilating sea surface temperature (SST). More importantly, there is no strong bias in the subsurface ocean, which would otherwise be difficult to correct with assimilation (see Fig. ).

In this study, we assess the performance of four OSSEs by comparing them against the free-running Ref state. These experiments are designed to test the impact of surface-only observations of sea surface height (SSH) and SST as measured by satellites (the “Surf” OSSE), the additional impact of surface observations with XBT-like subsurface temperature measurements (e.g. Scripps PX30 and PX34 XBT lines) in a long transect in the north of the domain (the “XBT-N” OSSE; see Fig. a for transect location), the same surface observations together with an XBT-like transect of subsurface temperature measurements in the south of the domain (the “XBT-S” OSSE; see Fig. a for transect location), and lastly, the surface observations together with the transect of subsurface temperature observations in the north and south of the domain (the “XBT-N+S” OSSE). Example coverage from SSH, XBT, and SST is shown in Fig. . Some configuration details and differences between the Ref state and the OSSEs are given in Table . Otherwise, we direct readers to , where details of the synthetic observations are given, including how their timing and locations are sourced from satellite observations and the applied observation errors. Background error covariances are set following , and the reader is directed there for details.

We first explore the representation of the immediate subsurface, through consideration of the mixed layer depth and thermocline depth. The MLD for all OSSEs (Fig. b–e) is represented as too shallow when compared to the Ref state (Fig. a), despite the presence of approximately six subsurface observations within the ∼ 40 m thick mixed layer. Note that the mean MLD (shown in each panel in Fig. ) in each OSSE still falls within 1 standard deviation of the mean MLD of the Ref state. The metric we use here to diagnose a proxy for the MLD indicates that all OSSEs have a near-surface vertical temperature structure that displays a more rapid decrease in temperature with depth compared to the Ref state. The shallower MLD of all OSSEs indicates that, in these experiments, there is minimal improvement in the near-surface temperature structure offered by the assimilation of observations in this region.

The thermocline depth, which is deepest in the EAC eddy region (33–34∘ S, 154.5∘ E) in the Ref state (Fig. f), is relatively poorly represented in all OSSEs (Fig. g–j). The Surf OSSE has a thermocline which is not deep enough in the eddy-rich region (120–140 m deep, compared to over 160 m deep in the Ref state) but too deep for most of the rest of the domain (over 120 m deep, in contrast to the Ref state which outside of the eddy-rich region is 80–100 m deep) as shown in Fig. g. The presence of subsurface observations improves the spatial pattern of thermocline depth (Fig. h–j). The thermocline is deepest in the upstream region of the EAC core and in the eddy-dominated region between 32.5–36∘ S. In these deep thermocline regions, all of the XBT OSSEs represent the bottom of the thermocline as too shallow, while the regions of shallower thermocline (outside of the EAC and its eddies) are fairly well represented (and represented considerably better than in the Surf OSSE). This likely points to a poor representation of the EAC core and eddy vertical structure in all OSSEs.

While surface EKE can be reasonably estimated in the presence of surface (particularly SSH) observations, subsurface EKE, and hence the three-dimensional structure of eddy variability, has generally poor spatial and temporal representation. We integrate subsurface EKE over two depth ranges, from 0 to 250 m and from 250 to 2000 m, which were chosen to capture the upper (higher-energy) and deeper (lower-energy) regions of eddy depth structure.

As expected, the mean EKE in the top 250 m is strongest in the EAC eddy region (Fig. a). The representation of upper-ocean EKE in all OSSEs is relatively similar (Fig. b–e), with the difference from the Ref state being small and having a similar spatial pattern for all OSSEs. At depth, however, the representation of the depth-averaged EKE (Fig. f) is represented differently in each OSSE. The Surf OSSE overestimates EKE through the highest-EKE region (Fig. g). The OSSEs with a single transect of observations perform better, with lower error in the representation of the depth-averaged EKE (Fig. h–i). The XBT-N+S OSSE has a slightly higher EKE difference than XBT-N or XBT-S but performs better than Surf (cf. Fig. j and g). As discussed in , the XBT-N+S OSSE sometimes displays higher error than the single XBT transect OSSEs, which is likely because the DA scheme is forced to minimise errors at both the northern and southern subsurface observation locations. This leads to a degraded fit to either observation transect individually. This has also been demonstrated by others, for example, , who found that additional data streams (mooring data and high-frequency (HF) radar currents) degraded the representation of SSH and SST, and , who showed that assimilating HF radar currents increased the error in the subsurface temperature forecast.

The performance of each OSSE can be compared in the vertical profiles of EKE averaged over the high-EKE variability region (Fig. k). EKE in the Ref state and all OSSEs is surface-intensified, while EKE in the upper 250 m is relatively well represented in all OSSEs. The EKE in the Ref state continues to decrease with depth, until approximately 1000 m. The XBT-S OSSE matches the Ref state in the monotonic decrease in EKE with depth and provides the best fit; however, the decrease with depth is less compared to the Ref state. All other OSSEs display subsurface EKE maxima between ∼ 500–1100 m.

We now consider another measure of the three-dimensional structure of eddies in these simulations – the along-shelf and across-shelf slope of the temperature fields. The mean temperatures and isotherm slopes at 250 m are shown in Fig. . This depth is chosen as it represents the transition where the OSSEs begin to display enhanced EKE, compared to the Ref state, as shown in Fig. k.

The presence of subsurface observations improves the representation of mean temperature (cf. Fig. b and c–e), with the XBT-S OSSE having the best representation of the higher temperature region at 33∘ S, 154.5∘ E. The across-shelf isotherm slope is characterised by a strong, negative slope along the coast, a weaker, broader region of negative slope in the off-shelf region of the western Tasman Sea (e.g. at 36∘ S, 151.5∘ E), and, further eastwards (36∘ S, 153∘ E), a weak but broad region of positive slope (Fig. f). These features represent, respectively, the sloping isotherms associated with the southward flowing EAC jet, the western edge of the EAC eddy field, and the northward return flow. While all OSSEs broadly contain these features (Fig. g–j), the representation is most accurate in the presence of subsurface observations in the southern region (XBT-S and XBT-N+S; Fig. i, j). The relatively high vertical and horizontal spatial resolution of the subsurface observations improves the representation of the isotherm slope near the observation transect but also the magnitude and distribution of the positive slope in the return flow and the broad, weakly negative slope in the eddy region. However, all OSSEs still represent weaker sloping across-shelf isotherms.

The most notable feature in the Ref state along-shelf slope is the zonal band of negative slope at 35∘ S, associated with the eastwards extension of the EAC (Fig. k). This band of negative slope is poorly represented in the Surf OSSE (Fig. l), improved with subsurface observations (Fig. m–o) and best represented with the XBT-S observations (Fig. n). This indicates that subsurface observations through the eddy region are key for improving the representation of the EAC eastern extension.

While the 250 m isotherm represented the transition from the upper region of eddies and EKE, the deeper eddy region (which we defined as the region of increased EKE between 250–2000 m) can be captured by the 5 ∘C isotherm. The depth and slope of this isotherm indicates the degree of vertical motion in the eddy field and the potential to which water in this region could display baroclinicity.

The 5 ∘C isotherm in the Ref state is relatively flat with a mean depth of 1150 m (Fig. ), meaning most of the eddy variability (e.g. upwelling) is above this depth. All OSSEs overestimate the depth of this isotherm in the high-EKE region (35∘ S, 153∘ E) and underestimate the depth outside of this region (Fig. b–e). The across-shelf (Fig. f) and along-shelf (Fig. k) isotherm slopes display features related to the density structure that contributes to driving or maintaining the EAC jet, southward extension, return flow, and eastern extension, as outlined above.

Like the isotherm tilting at 250 m (e.g. Fig. f–j), the slope, both positive and negative, of the 5 ∘C isotherm is over estimated in the across-shelf direction (Fig. g–j) and the along-shelf direction (Fig. l–o).

The above sections have focussed on time-mean metrics; however, a case study analysis is useful for providing insight into the model performance of these dynamic features. We have chosen to focus on two example eddies, where these events represent “best case” or “worst case” scenarios for the accurate simulation of subsurface conditions, namely, eddies in the vicinity of or distant from subsurface observations.

We have established that the time-mean subsurface conditions are poorly represented especially in the high-EKE region, indicating that these DA simulations are struggling to capture the structure of eddies at depth. Indeed, our case studies show that the vertical structure of individual eddies is also poorly represented, whether they are far from or close to observations. We now consider mechanisms that could be inhibiting a more accurate vertical eddy structure.

The barotropic conversion rate (KmKe) captures the energy pathway by which depth-mean horizontal velocity shear instability forms eddies. In contrast, the baroclinic conversion rate (PeKe) captures eddy formation that results from unstable vertical density structure and baroclinic instability. Here we average both quantities over the top 450 m to capture the region of highest EKE (Fig. k).

Comparing KmKe to PeKe we see that barotropic conversion is approximately an order of magnitude larger than baroclinic conversion (cf. Fig. a and f), which agrees with results from , who suggest KmKe is the dominant mode of eddy production in the EAC. All OSSEs produce a good representation of barotropic conversion rate, with most of the high KmKe hotspots (e.g. 32.5∘ S, 152∘ E) being captured (Fig. b–e). This explains why all OSSEs have a good representation of the time-mean surface EKE field (see Fig. b–e).

However, baroclinic production is poorly represented in all OSSEs, with PeKe being too strong and extending too far to the east (Fig. g–j). This suggests that the vertical density structure in the eddy field of all OSSEs is such that baroclinic instability is too active and generates too much conversion to eddy kinetic energy.

In an effort to explore an alternative manifestation of incorrect eddy representation, we employ a normal-mode analysis, whereby the barotropic and baroclinic modes are computed (e.g. ). This gives insight into how the OSSEs are simulating the vertical partitioning of kinetic energy in each baroclinic mode throughout the water column.

The normal modes are derived from the stratification profile (see Sect.  and Eq. ) using the numerical implementation described in . For case study A (Ref state; Fig. a), there is a relatively smooth increase in density with depth. In contrast, the Ref state in case study B has a sharper thermocline at ∼ 200 m, weak change in density between ∼ 200 and 350 m, and below that, a smooth increase in density with depth (Ref state; Fig. g).

The amplitudes of the first three baroclinic modes, calculated for the centre of the case study A eddy, are shown in Fig. b–d. The barotropic mode, ϕ0, is normalised to have unity value at all depths, and so we focus attention on the first three baroclinic modes ϕ1, ϕ2, and ϕ3. For case study A, XBT-S and XBT-N+S have the most accurate baroclinic mode structure (e.g. rms values of 0.09–0.2 compared to 0.20–0.46 for Surf and XBT-N; Fig. b, c); the other OSSEs have errors in the amplitude with depth of all baroclinic modes (Fig. b–d). The improved mode structure in XBT-S and XBT-N+S, particularly of ϕ2 (Fig. c) likely corresponds to a better representation of the weaker, smoothly sloping thermocline – the other OSSEs display a sharper thermocline at ∼ 200 m.

For case study B, with a more complex density structure (Ref state; Fig. e), all OSSEs fail to represent accurate baroclinic mode structure, with higher rms values. The shape of ϕ1 is poorly represented in all OSSEs (rms errors ranging from 0.3–0.4; Fig. f) which likely indicates a failure to capture the increase in density with depth associated with the primary thermocline below ∼ 350 m. The section of near-constant ϕ1 amplitude with depth that is present in Surf, XBT-S, and XBT-N+S from 250–1250 m likely represents the isothermal lens (Fig. g, j). This feature is present in the Ref state but at a much shallower depth (from 100–250 m; Fig. f), with the signature showing in ϕ1, ϕ2, and ϕ3 (Fig. f–h).

For case study B, ϕ2 is represented differently in all OSSEs compared to the Ref state, having either a portion (∼ 500–1500 m) which stays relatively constant in amplitude with depth (Surf, XBT-S, and XBT-N+S; Fig. g) or by displaying an overly deep maximum (XBT-N; green line in Fig. g). The first maximum in ϕ3 is too shallow and too weak in all OSSEs (Fig. h). The second, deeper maximum in ϕ3 is too deep in all OSSEs but particularly in Surf (Fig. h), which shows that all OSSEs represent this eddy as too deep.

Together, these results show that the baroclinic mode structure is poorly represented in all OSSEs but especially in the absence of nearby observations (case study B). The second baroclinic mode is particularly susceptible to erroneous shape, which corresponds to a poor representation of any deviations in the primary thermocline. The third baroclinic mode also has poor representation in all the DA experiments, which captures how eddies here are simulated with an overly deep vertical extent.

By projecting these baroclinic modes onto velocity anomalies in the meridional and zonal directions, we can decompose the vertical partitioning of kinetic energy into the components resulting from the different baroclinic modes. This kinetic energy decomposition shows how energy is vertically distributed between the different modes.

In case study A (Fig. a), the XBT-N and XBT-N+S OSSEs poorly estimate the energy associated with either the barotropic mode and one or more baroclinic modes. The XBT-S OSSE displays a fair estimate of energy in ϕ1 and ϕ4 but underestimates ϕ0 and overestimates ϕ2–ϕ3. The Surf OSSE represents ϕ3 well but overestimates ϕ0, ϕ1, and ϕ2.

In case study B (Fig. b) all OSSEs overestimate the energy distribution in the barotropic and baroclinic modes. In particular, ϕ2 is best represented by XBT-N, XBT-S, and XBT-N+S; all other modes are represented as too energetic in all OSSEs.