1. Introduction
This study examines the effect of subsurface velocity measurements across the Yucatan Channel (YC) on the state estimates and forecasts of the Loop Current (LC) circulation in the Gulf of Mexico (GoM). It uses a regional implementation of the Massachusetts Institute of Technology general circulation model (MITgcm) (Marshall et al. 1997)–Estimating the Circulation and the Climate of the Ocean (ECCO) (Stammer et al. 2002) four-dimensional variational assimilation (MITgcm–ECCO 4DVAR) system for the GoM. The circulation in the GoM is part of the North Atlantic subtropical gyre and is dominated by the LC. The LC enters the GoM through the YC and exits to the Atlantic Ocean through the Straits of Florida. The LC extends northward into the GoM Basin, sometimes farther, approaching the shelf break south of the Louisiana–Mississippi continental shelf. An extended LC often forms an anticyclonic Loop Current eddy (LCE) that separates from the LC and travels westward until it dissipates in the western GoM, usually near the shelf off the coast of Mexico (Maul 1977; Sturges and Evans 1983; Molinari and Morrison 1988; Candela et al. 2002). Sometimes there are several LCEs in the GoM at a given time because the LCE separation occurs at irregular intervals ranging from 6 to 11 months (Sturges and Leben 2000) with a mean separation period of about 8 months (Hall and Leben 2016).
Understanding and predicting the GoM circulation is essential for scientific and socioeconomic reasons. For example, the GoM is an important region for the exploration and production of oil and gas. The LC and LCEs are major circulation features in the GoM that can strongly impact oil and gas operations and influence severe weather, including hurricanes, in the United States (Molina et al. 2016) and Mexico. The positions of LC and LCE fronts are also crucial for fisheries (Roffer et al. 2006).
The selection of the numerical ocean model and assimilation method plays an important role in producing realistic and dynamically consistent state estimates and forecasts of the regional ocean circulation. The MITgcm–ECCO 4DVAR is a widely used assimilation system for producing global (Wunsch and Heimback 2007; Forget et al. 2015), as well as basin-scale, and regional, eddy-permitting (Hoteit et al. 2005; Gebbie et al. 2006; Mazloff et al. 2010; Hoteit et al. 2009, 2010; Verdy et al. 2017) ocean state estimates. A regional implementation of the MITgcm–ECCO 4DVAR system for the GoM (Gopalakrishnan et al. 2013a) using satellite-derived sea surface height (SSH) and sea surface temperature (SST) has produced short-term (2 month) analysis and forecasts of the LC circulation (Rudnick et al. 2015; Gopalakrishnan et al. 2019). The state estimation framework used in this study is a modified version of the GoM MITgcm–ECCO 4DVAR system by Gopalakrishnan et al. (2013a).
The 4DVAR state estimation minimizes a cost function subject to the nonlinear model equations by adjusting the model control variables (Le Dimet and Talagrand 1986; Wunsch 1996) through iterative optimization. The cost function is defined as a weighted sum of quadratic norms of the model–data misfit and corrections to the control variables between the initial and final time of the assimilation window. The estimates of the cost function gradients with respect to model controls are obtained by integrating the adjoint of the tangent linear model backward in time (Le Dimet and Talagrand 1986). The adjoint model is relinearized around a new nonlinear forward model simulation at each iteration of the optimization process. The iterative optimization process starts from the first-guess solution (called the “reference” solution). The length of the assimilation window is a compromise between maximizing the amount of data within the window and limitations due to model error and nonlinearity of the assimilation system. Because the nonlinearity of the 4DVAR assimilation system and model error grows with time (Yaremchuk and Martin 2014), a shorter assimilation window will be closer to a linear system; however, it may not have enough observations to constrain the model. A longer assimilation window synthesizes more observations but will be less linear, making it difficult to find an optimized solution matching the observations. In this study, we continue to use 2-month assimilation windows as previously used for the GoM MITgcm–ECCO 4DVAR system (Gopalakrishnan et al. 2013a), which is long enough to constrain the model and produce skillful estimates and forecasts (Gopalakrishnan et al. 2013a; Rudnick et al. 2015; Gopalakrishnan et al. 2019).
The type and location of the observations used in the assimilation, and their assumed uncertainties, may need to be chosen carefully depending on the model capabilities and the oceanic features one wants to resolve and represent (Edwards et al. 2015; Lermusiaux 2007). Multiple hypotheses describe how the upstream ocean state near the YC critically affects the LC system and LCE separation. The direct ocean current measurements across the YC made by the Canek program played an important role in advancing our understanding of the flow exchange between the Caribbean Sea and the GoM (Sheinbaum et al. 2002). Using Canek observations, Candela et al. (2002) reported the effect of potential vorticity (PV) flux through the YC on the LC dynamics. They reported that periods of negative (anticyclonic) cumulative PV influx through the YC results in LC extension, whereas periods of positive (cyclonic) cumulative PV influx results in LC retraction and sometimes leading to LCE separation. On the contrary, a numerical study by Oey (2004) reported that influx of cyclonic vorticity flux anomaly (VFA) through the YC results in LC extension into the GoM, while influx of anticyclonic VFA may trigger retraction or eddy shedding. The relationship between deep YC flow and LC extension were explored by Bunge et al. (2002) based on Canek data and showed a high correlation between LC extension and deep YC flows using a conceptual box model that included the YC and the Florida strait transport components. A numerical study by Ezer et al. (2003) reported that the deep return transport below 800 m at the YC correlates with changes in the LC extension. Overall, the YC transport is considered as a preconditioner to the LC extension, retraction, and eddy separation, and the YC is considered to be a strategically important location to study and constrain the LC dynamics. This study attempts to understand the impact of assimilation of an 8-month-long moored velocity data taken across the YC during different phases of the LC, including LC intrusion, growth/extension, and LCE separation on both LC hindcasts and forecasts. We constrained the GoM ocean model using combinations of the subsurface moored velocity data and satellite-derived SSH and SST data to produce state estimates with closed budgets of mass and momentum. We used those estimates as a tool for model diagnostics and process studies to improve our understanding of the four-dimensional oceanic circulation.
This study is organized as follows: section 2 briefly describes the data, ocean model setup, and assimilation experiments. Section 3 describes the results and discussion, and finally, section 4 provides a summary and the main conclusions of this work. Appendixes A and B provide technical details.
2. Methods
a. Observations
Satellite-derived data along with/without moored velocity data were assimilated into the model. Satellite-derived data included the daily optimally interpolated SST product derived from the Tropical Rainfall Measuring Mission’s (TRMM) Microwave Imager (TMI) and the Advanced Microwave Scanning Radiometer for the Earth Observing System (AMSR-E) instruments produced by Remote Sensing Systems Inc. (http://www.remss.com/) on a 0.25° longitude × 0.25° latitude grid (TMI-AMSRE OI-SST) and along-track SSH obtained from the Radar Altimetry Database System (RADS; http://rads.tudelft.nl/rads/index.shtml; Scharroo et al. 2013) from three satellites (Jason-1, Jason-2, and Envisat). The SST data coverage and the superposed satellite tracks for SSH from the three satellites are shown in Figs. 1a and 1b, respectively. For more details about the satellite-derived data and their preprocessing, please refer to Gopalakrishnan et al. (2013a).
Observations used in the assimilation experiments. (a) The shading shows the mean satellite-derived SST over the mooring observation period (May–December 2010). (b) The superposed satellite SSH tracks from three satellites [Jason-1 (red tracks), Jason-2 (blue tracks), and Envisat-1 (green tracks)]. The region enclosed by the blue outline shown in (a) and (b) is defined as the GoM Basin. (c) The locations of the moorings in the YC with the black dots marking the horizontal positions together with the bathymetry (color) of the YC region, and (d) the depths of the current meter measurements, marked as red dots.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Moored velocity data were taken in the YC at different depths along a transect from Yucatan Peninsula to Cuba. The horizontal and vertical distribution of the mooring array is shown in Figs. 1c and 1d, respectively. The mooring array consisted of eight moorings (M1–M8) deployed from May to December 2010. Moorings on the Mexican side of the YC are part of the Bureau of Energy Management (BOEM) funded Loop Current Dynamics Experiment (Lugo-Fernandez et al. 2016; Sheinbaum et al. 2016) and the whole array is part of the Yucatan Channel Canek observing program funded by the Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE) in Ensenada, Mexico. Since moorings 1 and 2 were deployed very close and their geographic location falls within the same GoM MITgcm model grid cell, only moorings 2–8 (M2–M8) were used in the assimilation. The mooring observations were filtered using a Lanczos filter with a 48-h cutoff period. A standard quality control procedure was also applied to the data to remove outliers. The data were further subsampled daily before assimilating them into the model. This preprocessing retains the low-frequency features of the GoM circulation (mainly LC evolution), which are the focus of this study.
We used gridded SSH from the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) to evaluate model hindcasts and forecasts. The AVISO SSH maps are an estimation of the true field. They are produced by mapping along-track satellite SSH observations using space–time covariances (Le Traon et al. 1998; Ducet et al. 2000). The AVISO product is widely used by the oceanographic community doing research in the GoM (Weisberg and Liu 2017; Rudnick et al. 2015; Gopalakrishnan et al. 2013a) and has proven to give good SSH estimates in this highly energetic environment. Although SSH variability occurring on time scales of less than approximately 20 days will not be well represented by AVISO (Donohue et al. 2016), much of the ocean mesoscale variability we seek to reproduce here is captured by this product. The daily gridded SSH fields from AVISO analysis were obtained from the SSALTO/Duacs altimeter product on a 0.25°× 0.25° grid produced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS) (http://marine.copernicus.eu/).
b. Ocean model and assimilation
The MITgcm (Marshall et al. 1997) integrates the primitive equations on a sphere under the Boussinesq approximation. The equations are written in z coordinates and are discretized using the third-order direct-space–time approximation in a staggered Arakawa C grid. The MITgcm code is designed to enable automatic generation of its adjoint through the transformation of algorithms in Fortran (TAF) (Giering and Kaminski 1998; Heimbach et al. 2002). The MITgcm and its adjoint model have been used in numerous ocean state estimation and adjoint sensitivity studies at global, basin, and regional scales (Stammer et al. 2002; Fukumori et al. 2004; Menemenlis et al. 2005; Hoteit et al. 2009, 2010, 2013; Köhl et al. 2007; Mazloff et al. 2010; Gopalakrishnan et al. 2013a,b).
The model configuration and the 4DVAR assimilation setup used in this study are the same as the one described in Gopalakrishnan et al. (2013a, 2019). It covers a region that extends from 8.5° to 31°N and from 98° to 72.5°W. This region encompasses the GoM Basin, part of the Caribbean Sea, and the Gulf Stream. The horizontal resolution of the model is 1/10° in both zonal and meridional directions. The model vertical grid has 40 z levels, with level spacing gradually increasing with depth from 5 m near the surface to 500 m near the bottom. The model bathymetry is extracted from the 2-min gridded global topography (ETOPOV2; https://www.ngdc.noaa.gov/mgg/global/etopo2.html). Vertical mixing is based on the K-profile parameterization (KPP; Large et al. 1994) with background values of 1 × 10−6 and 1 × 10−4 m2 s−1 for the second-order vertical diffusivity and viscosity coefficients, respectively. The model uses both second-order and fourth-order horizontal diffusive and viscous operators with coefficients 1 × 102 m2 s−1 and 1 × 1010 m4 s−1, respectively in all forward model simulations. For the adjoint model simulations, the viscosity and diffusivity parameters were increased by factors of 10 and 5 for the second-order and fourth-order coefficients, respectively, to remove the small-scale features in the sensitivities that trigger nonlinear instabilities. In addition to increased viscosity and diffusivity, the KPP mixing parameterization was also disabled to minimize its contribution to the adjoint nonlinearity, permitting longer assimilation windows (Hoteit et al. 2005; Köhl et al. 2007; Gopalakrishnan et al. 2013a).
The open-ocean boundary conditions for the model were obtained from the Hybrid Coordinate Ocean Model (HYCOM) 1/12° global daily analysis (Chassignet et al. 2007) (http://hycom.org/dataserver/glb-analysis). This system uses the Navy Coupled Ocean Data Assimilation (NCODA), which is an oceanographic implementation of the Multi-Variate Optimal Interpolation (MVOI; Cummings 2005). HYCOM/NCODA estimates of temperature, salinity, and horizontal velocities (u: zonal component; υ: meridional component) sampled at 7-day intervals were spatially interpolated onto the MITgcm grid for initial conditions and specified along the open-ocean boundaries for boundary conditions. The surface winds and atmospheric fluxes are from the National Centers for Environmental Prediction–National Center for Atmospheric Research Reanalysis (NCEP–NCAR; Kalnay et al. 1996) sampled every 6 hours, with the latter estimated using the bulk formulation (Large and Pond 1981). Annual climatological runoff (freshwater) fluxes (Fekete et al. 2002) were used as well. The MITgcm is run in its hydrostatic mode configuration and is not forced with tidal or atmospheric pressure fields.
Details of the state estimation procedure used in this study, including satellite-derived SSH and SST observations, model control variables, cost functions for observations and controls, observation and model background uncertainty covariances, and the iterative optimization procedure can be found in Gopalakrishnan et al. (2013a). For completeness, a brief description is provided in appendix A.
We study the effect of moored YC velocity data on the GoM estimates through three assimilation experiments. The experiments use a combination of SSH and SST and subsurface moored velocity data as follows: 1) SSH and SST data assimilation [experiment 1 (Exp1)]; 2) SSH, SST, and moored velocity data assimilation [experiment 2 (Exp2)]; and 3) only moored velocity data assimilation [experiment 3 (Exp3)]. The reference simulation (REF; i.e., the first-guess solution with no data assimilation) for Exp1–Exp3 is the same. A series of four independent 2-month-long state estimates were produced for Exp1–Exp3, covering the eight months of the mooring deployment. The iterative optimization minimizes the model–data misfit of the REF solutions over each 2-month assimilation window by adjusting the model controls, which are temperature and salinity initial conditions, atmospheric forcings, and open-ocean boundary conditions.
A 2-month (60 days) forecast was produced using the optimized state from the end of each 2-month assimilation window for Exp1–Exp3. To evaluate the forecasts as in a real-time ocean forecasting system, model forecasts were forced using monthly climatologies of open-ocean boundary conditions from HYCOM/NCODA 1/12° global daily analysis, atmospheric forcings from NCEP–NCAR reanalysis, and runoff fluxes (Fekete et al. 2002). This forecast does not take advantage of the predictability of the realistic boundary conditions and atmospheric forcings.
The forecasts examine practical LC predictability and cross validate the state estimates by comparing them to future independent observations. Reference forecasts (called “F–REF”) for each of the four 2-month experiments are initialized using the REF solutions from the end of the assimilation period. The F–REF forecasts are compared with corresponding forecasts initialized using the state estimates (called “SE”) from Exp1 to Exp3 (F–SE1, F–SE2, F–SE3). Additional forecasts (called “F–HY”), using the GoM MITgcm model initialized from the HYCOM/NCODA 1/12° global daily analysis for the start date of each forecast period are also compared. The F–HY forecasts are designed to compare MITgcm forecasts with different initial conditions (MITgcm–ECCO 4DVAR state estimate and HYCOM/NCODA 1/12° global daily analysis) and do not represent a direct comparison between MITgcm and HYCOM/NCODA model forecasts. Note that F–HY forecast and REF hindcast for the overlapping hindcast/forecast periods use the same HYCOM/NCODA 1/12° global daily analysis initial conditions, but different forcings and boundary conditions. The F–HY forecast uses climatological forcings and boundary conditions, while the REF hindcast uses realistic forcings and boundary conditions. A summary of the experiments is presented in Table 1.
Model hindcast and forecast abbreviations. IC: Initial conditions. OBCS: Open-ocean boundary conditions. AF: Atmospheric forcings. SE1: State estimate from Exp1 (SSH + SST data assimilation). SE2: State estimate from Exp2 (SSH + SST + moored velocity data assimilation). SE3: State estimate from Exp3 (moored velocity data assimilation). REF: Reference solution (first guess) for Exp1, Exp2, and Exp3. All hindcasts and forecasts used annual climatology of continental runoff.
3. Results and discussion
The assimilation experiments over May–December 2010 cover different stages of LC evolution, including before, during, and after the LCE (Eddy Franklin: Eddy-F) separation. Eddy-F initially started detaching in late May 2010, after the Deepwater Horizon (DwH) oil spill disaster in the GoM. It was reattached and detached several times during the summer before its final separation from the LC in early September 2010 (Hamilton et al. 2011; Sheinbaum et al. 2016). In the series of four 2-month state estimates, each one has an LC behaving differently, allowing us to analyze the assimilation effect on different stages of the LC. We define the phases of the LC heuristically as follows: 1) May–June 2010 (phase 1, LCE detachment, 05–06/2010), 2) July–August 2010 (phase 2, LC growth, 07–08/2010), 3) September–October 2010 (phase 3, LC growth and retraction, 09–10/2010), and 4) November–December 2010 (phase 4, LC retraction, 11–12/2010).
For guidance, daily averaged SSH estimates at the beginning of each two months from Exp2 assimilation (SE2) are compared with corresponding AVISO and HYCOM/NCODA analysis in Fig. 2. The SE2 SSH follows AVISO and HYCOM/NCODA analysis closely at the beginning of all four experiment periods. The spatial distribution of the SSH (Fig. 3) at the time Eddy-F first started detaching from the LC (toward the end of May 2010) shows that estimates from Exp1 (SE1), Exp2 (SE2), and Exp3 (SE3) initiate an LCE separation, which is not present in the REF SSH. Out of the SSH estimates from Exp1 to Exp3, SE2 SSH shows a closer agreement with AVISO for the LC structure and the LCE size, shape, and location.
The SSH from (left) state estimate using satellite and mooring data, (center) AVISO, and (right) HYCOM/NCODA daily global analysis at the start date of the four 2-month assimilation periods: (first row) 05–06/2010, (second row) 07–08/2010, (third row) 09–10/2010, and (fourth row) 11–12/2010. The colors show SSH (m) with values ranging from −0.5 m (blue) to 0.5 m (red), and the arrows show surface currents. The black curve marks the 17-cm SSH contour, and the black dot marks the DwH oil spill location.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The SSH from REF and estimates from three experiments (SE1, SE2, and SE3) of phase 1 are shown for the approximate time (25 May 2010) of the first detachment of LCE (Eddy-F) from the LC. The SSH fields from AVISO and HYCOM/NCODA analysis for the corresponding time are also plotted for comparison. The colors show SSH in m with values ranging from −0.5 m (blue) to 0.5 m (red), and the arrows show surface currents. The black curve marks the 17-cm SSH contour, and the black dot marks the DwH oil spill location.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
a. Model quality metrics
The GoM state estimates and forecasts are evaluated using two model quality metrics widely used in previous GoM studies. One is based on point-to-point differences and the other on the LC state (phenomenon oriented). The first metric is the SSH root-mean-square-difference (rmsd) averaged over the GoM Basin between the model (hindcasts and forecasts) and the AVISO SSH analysis maps. The region over which model–data SSH rmsd is averaged is marked by the blue outline in Fig. 1a. Although AVISO SSH fields are produced by mapping multiple satellite along-track data using space–time covariances (Le Traon et al. 1998; Ducet et al. 2000), and the state estimate fit along-track SSH, the comparison of the optimized SSH with AVISO absolute SSH can be considered as an independent validation of the estimate. Likewise, the assimilated HYCOM/NCODA 1/12° global daily analysis SSH was also compared with AVISO to provide another benchmark for the model–data comparison. The second metric is the maximum northward LC extension, where the LC is delimited by the 17-cm SSH contour, representing the maximum gradient in the LC velocity (Leben 2005). This 17-cm SSH contour also represents the maximum gradient in the LC velocity for our model configuration after calibrating the SSH by removing the spatial mean within the GoM Basin (region marked by the blue outline in Fig. 1a) (Gopalakrishnan et al. 2013b; Rudnick et al. 2015; Gopalakrishnan et al. 2019).
1) SSH rmsd
The Fig. 4 shows the rmsd with mapped AVISO SSH for both hindcasts and forecasts. The SSH estimates always improve over the REF solution for all phases of the LC. The REF rmsd grows most rapidly in the first 2-month window when the LC is extended, with multiple detachments and reattachments of Eddy-F. In contrast, the REF rmsd is only about 2 cm higher than that for the state estimates at the end of the two months for less energetic LC phases of 09–10/2010 and 11–12/2010 periods. It should be noted that the SSH hindcast rmsd from all experiments are smoother than those from HYCOM/NCODA analysis. The roughness of HYCOM/NCODA 1/12° global analysis rmsd is likely due to its daily update cycle, introducing minor discontinuities between the solutions at update times. In contrast, the state estimates are forward model simulations forced by adjusted model controls resulting in a dynamically consistent (constrained by the equations of motion) solution over the 2-month assimilation period.
The SSH point-to-point metric (rmsd) comparison. (top) Model–data SSH rmsd (m) for the hindcast period. The rmsd is computed with respect to AVISO gridded SSH data and averaged over the GoM Basin (region marked by the blue outline in Fig. 1a) for the daily averaged SSH fields from the optimized state estimates from 3 assimilation experiments (SE1: red, SE2: green, SE3: blue), reference model solution (REF: black), and HYCOM/NCODA daily global analysis (HYCOM-GLOBAL: golden). The vertical gray dashed lines separate the state estimates for four different periods. (bottom) Model–data SSH rmsd (m) for the forecast period from 3 assimilation experiments (F–SE1: red, F–SE2: green, F–SE3: blue), reference forecast (F–REF: black), and MITgcm forecast initialized using HYCOM/NCODA daily global analysis (F–HY: magenta). The rmsd for HYCOM/NCODA daily global analysis (HYCOM-GLOBAL: golden) is also shown for the forecast period.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The estimates from assimilating only satellite data (SE1) reduces the SSH rmsd compared to the REF for all the 2-month periods. An overall improvement in the model hindcasts was observed when moored subsurface velocity data were included in the assimilation with satellite data (SE2). The SE2 SSH hindcast rmsd is the lowest of all the MITgcm estimates for all the periods except toward the end of the 09–10/2010. There, the rmsd of SE2 is slightly higher than SE1. The SSH hindcast rmsd, when assimilating only mooring data (SE3), closely follows the REF, except toward the end of 05–06/2010 and 09–10/2010 periods that correspond to the times when the LC was in the extended and active phase (see Fig. 5 for the LC northward extension metric).
The SSH phenomena oriented metric (LC northward extent). (top) The hindcast comparison and (bottom) the forecast comparison. The LC northward extent metric is computed by tracking the 17-cm contour of SSH calibrated for the GoM Basin (region marked by the blue outline in Fig. 1a). The LC extent metric is shown for AVISO (AVISO: gray), optimized state estimates from three assimilation experiments (SE1: red, SE2: green, SE3: blue), reference model solution (REF: black), and HYCOM/NCODA daily global analysis (HYCOM-GLOBAL: golden). The vertical gray dashed lines separate the state estimates for four different periods. (bottom) The LC extension metric for the optimized forecast from 3 assimilation experiments (F–SE1: red, F–SE2: green, F–SE3: blue), reference forecast (F–REF: black), and MITgcm forecast initialized using HYCOM/NCODA daily global analysis (F–HY: magenta). The LC extent metric is also shown for AVISO (AVISO: gray) and HYCOM/NCODA (HYCOM-GLOBAL: golden) daily analysis for the forecast period.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Even though SSH rmsd, a point-to-point metric, is not significantly changed from REF by the assimilation of mooring data alone (SE3). The maximum northward LC extension metric that focuses specifically on the LC structure for SE3 shows considerable improvement over REF in phase 1 (see Fig. 5). These results agree with previous findings (Bunge et al. 2002; Candela et al. 2002; Oey 2004) suggesting the importance of subsurface velocity observations across the YC on the LC evolution in the GoM, especially for the extended LC phase. The SSH hindcast rmsd for SE1 and SE2 are slightly higher by about 2 cm than that for the HYCOM/NCODA analysis for most of the periods (Fig. 4, top), except for August and October 2010, and are at times lower or equal to that of HYCOM/NCODA. This result suggests that the optimized solutions from GoM MITgcm–ECCO 4DVAR system using 2-month assimilation windows can often remain at least as close to the mapped AVISO SSH as the HYCOM/NCODA daily analysis.
The SSH hindcast results suggest an overall improvement of the estimates with the inclusion of more data. This improvement is not always the case for forecasts. The forecast initialized from the state estimate using satellite data (F–SE1) shows a lower SSH rmsd for the 07–08/2010 and 11–12/2010 periods while showing slightly higher values than for the reference forecasts (F–REF) toward the end of periods 09–10/2010 and 01–02/2011 (January–February 2011). The SSH forecasts initialized from the state estimates that use both satellite and mooring data (F–SE2) show a consistent improvement over all of the 2-month periods except toward the end of 07–08/2010. This result is interesting because moored subsurface velocity data samples smaller scales than SSH and SST and tend to be more nonlinear (Hoteit et al. 2009). This nonlinearity makes it more challenging to assimilate them over extended periods. For those cases, a shorter assimilation window might be preferred (i.e., days to weeks), which are not explored in the present study. Improvements in the state estimates and forecasts with moored velocity data highlight the effect of subsurface observations across the YC on the surface circulation in the GoM.
The SSH forecasts initialized from the state estimates that used only mooring data (F–SE3) generally show higher SSH rmsd than F–SE1 and F–SE2 (Fig. 4, bottom). This result is not surprising. Subsurface velocities across the YC alone are not sufficient to constrain the surface circulation in the GoM, based on the SSH rmsd metric. Nevertheless, F–SE3 SSH forecasts perform better than F–REF, especially during the 07–08/2010 and 11–12/2010 forecast periods.
Additional MITgcm forecasts (called F–HY), initialized using HYCOM/NCODA 1/12° global daily analysis, often show lower SSH rmsd than F–SE1, F–SE2, F–SE3, and F–REF forecasts. Especially during the 09–10/2010 and 01–02/2011 forecast periods, which correspond to the assimilation windows of 07–08/2010 and 11–12/2010, respectively. These results suggest that HYCOM/NCODA analysis often provides a better forecast initialization for the MITgcm than the other estimates (REF, SE1, SE2, and SE3). We hypothesize that this behavior is due to the frequent update cycle of the HYCOM/NCODA 1/12° global analysis as well as the extra data being assimilated, like expendable bathythermographs, Argo floats, moored buoys, and downward projection of surface information using the Modular Ocean Data Assimilation System (Fox et al. 2002). The daily updated HYCOM/NCODA analysis stays closer to the observations than the GoM MITgcm–ECCO 4DVAR 2-month state estimates, which constrains the model dynamics over extended periods and may initialize an SSH forecast with larger rmsd.
We remind the reader that REF and F–HY use the same initial conditions for overlapping hindcast/forecast periods. They differ only in the forcing and boundary conditions. The F–HY forecast uses climatological forcing and boundary conditions, while the REF hindcast uses realistic forcing and boundary conditions. SSH rmsd (Fig. 4) shows that in general, F–HY exhibits a lower rmsd than REF for all the periods except toward the end of 11–12/2010. This result is counterintuitive, suggesting that forecasts using climatological forcing and boundary conditions perform better than those using realistic forcing and boundary conditions. Based on additional forecast experiments (not shown) using a combination of realistic and climatological forcing and boundary conditions, we found that this discrepancy is due to perturbations generated by the realistic forcing and boundary conditions that grow with time in the semienclosed GoM Basin. These perturbations are less smooth than the perturbations introduced by climatological forcing and boundary conditions.
2) Maximum northward LC extension
The maximum northward LC extension metric (Leben 2005) shows irregularities in the LC behavior with abrupt changes, as shown in the AVISO SSH (Fig. 5). This metric is not smooth, and sudden changes reflect LC events such as detachments/reattachments and separations of an LCE. An LCE detachment corresponds to a steep decrease in the northward LC extent, whereas an LCE reattachment corresponds to a steep increase. Although this metric is meant to be a stringent test of the state estimates because of the substantial variability of the LC frontal extent, all the experiments reproduced the upper and lower bound of variation and the growths and retractions, not associated with LCE detachment/reattachment processes, of the AVISO SSH maximum northward LC extension.
During phase 1, SE1 through SE3 hindcasts show an LCE detachment, while the REF shows no LCE detachment. SE1 and SE2 also show a reattachment, and another detachment, whereas SE3 shows no reattachment. This result implies that SE1 and SE2 improve REF since it contains detachment/reattachment as shown in the observations but with incorrect timing. However, the LCE detachment shown by the SE3 hindcast coincides with that of the first detachment of LCE in AVISO. This result highlights the possible impact of the moored velocity data on the GoM state estimate, particularly for the LCE detachment process. Although the assimilation of only moored data (SE3) matches the LCE detachment timing as that of AVISO on 25 May 2010, both satellite and moored data assimilated together (SE2) results in a slightly earlier LCE detachment by about five days (20 May 2010). We speculate that this can be due to SE2 trying to fit more data simultaneously than SE3.
The LC extent metric from all experiments for phase 2–4 shows similar behavior to AVISO except for the reattachment shown in AVISO during phase 2 (07–08/2010), which corresponds to a LC growing phase in the estimates. The LC growth in phase 3 shows a stepping increase in the LC northward extent metric for all experiments, but the ones assimilating mooring observations (SE2 and SE3) show a steeper increase and retraction. In phase 4, AVISO shows a smooth LC evolution, while all assimilation experiments show a faster LC extension and retraction. This more rapid retraction and extension of the LC might be due to the higher resolution of the model (1/10°) compared to AVISO (1/4°), resolving small-scale features that can contribute to these differences. The LC extension metric for HYCOM/NCODA does not capture the first reattachment in the middle of June 2010, but it does capture the second reattachment in early August 2010 (Fig. 5). For reference, the daily averaged SSH from SE2 is compared with AVISO and HYCOM/NCODA analysis in Fig. 6 for the first and second reattachment of LCE (Eddy-F) in the middle of June 2010 and early August 2010, respectively. The LC structure of the SE2 SSH estimate is comparable to AVISO and HYCOM/NCODA for both reattachment periods. The LC/LCE frontal features of SE2 for the first reattachment period are similar to HYCOM/NCODA. However, they both differ from that of AVISO. For the second reattachment period, LC/LCE frontal features of HYCOM/NCODA and AVISO are similar, whereas the SE2 estimate shows slightly different features. We tested the sensitivity of this LC extension metric based on the 17-cm SSH contour with other contour values ranging from 15 to 20 cm. However, we found that the overall behavior of this metric of the estimates is not dependent on the choice of the contour value (not shown).
The SSH from the Exp2 (SE2) is shown for the first and second reattachment of LCE (Eddy-F) (top) in the middle of June 2010 and (bottom) in early August 2010. The SSH from AVISO and HYCOM/NCODA analysis for the corresponding times are also plotted for comparison. The colors show SSH (m) with values ranging from −0.5 m (blue) to 0.5 m (red), and the arrows show surface currents. The black curve marks the 17-cm SSH contour, and the black dot marks the DwH oil spill location.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The lower panel in Fig. 5 is similar to the upper panel, but for model forecasts. Although the forecasts show less variability than their corresponding hindcasts, they show similar patterns in general. However, the sudden steeper LC growth and retraction during phase 3 hindcast and the faster LC retraction during phase 4 hindcast are not reproduced in their respective forecasts.
b. Impact of moored velocity data
The observation cost function when assimilating only moored velocity data (SE3) shows that the influence of moorings gets smaller as the LC phase progresses (see different y axis on Fig. 7) from phase 1 (highest cost function) to phase 4 (lowest cost function). This behavior is due to the first guess solutions being close to the observations in later phases. In phase 1, the most significant reduction in the moored velocity cost contributions is the meridional velocities of moorings close to the Yucatan Peninsula (M2 and M3). In SE3, the synthesis of these moored observations modified the LC state from the REF solution with no LCE detachment to an LCE detachment, matching the timing as observed in AVISO data. As a zero-order result, we can say that flow field information from the moorings closer to the Yucatan Peninsula could have the largest impact on the state estimation for LCE separation. A different cost distribution is observed in phase 2 of the LC (LC growth, 07–08/2010: Fig. 7b), where all the mooring cost contributions are important, and the cost contributions of zonal velocities become noticeable. During phase 3 and phase 4, intermittent intervals of LC growth and retraction are shown (Fig. 5, 09–10/2010 and 11–12/2010), and the meridional velocity cost components are again dominant (Figs. 7c,d). It should be noted that in Figs. 7c and 7d, the M8 (close to Cuba) cost contribution is the second most important, next to the M2 cost, even in Fig. 7b the mooring close to Cuba is important. This result suggests that the flow dynamics across the YC during these LC phases are different from east to west. This result is consistent with the recent work by Androulidakis et al. (2021), which highlights that during phases of a retracted LC, anticyclones from the Caribbean Sea intrude into the GoM Basin through the YC close to Cuba. The passage of Caribbean anticyclones close to Cuba was first discussed by Oey et al. (2003) and Abascal et al. (2003). Overall, the LC growth/retraction appears to be associated with the inflow/outflow processes at M2 and M8 locations.
The observational cost function of Exp3 for each one of the phases of the LC, (a) phase 1, (b) phase 2, (c) phase 3, and (d) phase 4. The bottom x axis marks the moorings (M2–M8) for each one of the two velocity components (u and υ) with the number of observations (N) labeled at the bottom. The left y axis is the observational cost for each one of the moorings, and the number on the top of the bars is the normalized observational cost: observational cost divided by the number of observations (N)
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
c. FTLEs
The dynamically consistent MITgcm–ECCO 4DVAR GoM solutions provide an excellent resource for process studies to understand the LC dynamics and the LCE shedding mechanisms. Since the processes involved in the system are nonlinear, linear measures like the difference between the two model states may not highlight the dynamical features and behavior in each experiment. Likewise, calculating the divergence or curl of the velocity field (not shown) of the estimates from different assimilation experiments of this study does not seem to reveal patterns between the experiments. Alternatively, an analysis tool designed for advection-dominated fields, finite-time Lyapunov exponents (FTLEs), can be effectively used to visualize the flow field that helps highlight the differences between the solutions for each experiment (REF and SE1–SE3) for the four phases of the LC.
The FTLEs are a way to unveil the Lagrangian skeleton (Mathur et al. 2007) of the flow by measuring how fast a pair of nearby water parcels separate either forward or backward in time. Regions in the flow field where the water parcels separate at high rates are represented as ridges (i.e., local maxima) in the FTLE maps. Those ridges delineate structures (Haller 2001) that help visualize similarities and differences among the different state estimates. FTLEs have been previously used in GoM studies for understanding submesoscale motions (Beron-Vera et al. 2018), the LCE detachment (Andrade-Canto et al. 2013), the behavior of Lagrangian drifters (Olascoaga et al. 2013), and as a base to identify and delimit the genesis and apocalypse dates of LCEs (Andrade-Canto et al. 2020). A brief description of the FTLE method is provided in appendix B.
Our analysis focus on a particular time of interest, from which we integrate 30 days forward to obtain the FTLEs. To understand the key dynamics in the LC region, the time of interest chosen for the FTLE analysis is based on the northward extent of the LC (i.e., day 21 in phase 1 that corresponds to the first detachment in SE1 and SE2, or day 31 in phase 2 that corresponds to the point where the differences between experiments start to grow). For the FTLE analysis, we selected areas around the YC that are hypothesized to be influencing the LC dynamics. These areas are the Campeche Bank, the Cuban offshore basin, the upstream flow features in the Yucatan Basin, and the Cayman Sea region.
In phase 1, we selected day 21, corresponding to the first LCE detachment in SE1 and SE2. The LCE detachment occurs farther north for SE1 and SE2 (both assimilate satellite data) than SE3 (assimilating only the mooring data). SE1 and SE2 have a coherent structure between the LC and the Cuban offshore basin (marked by the red box in Fig. 8), which is similar to the anticyclonic eddies along the northern Cuban coast (CubANs) of type A described in Kourafalou et al. (2017). That CubAN is nonexistent in SE3. This structure seems to preclude the LC from retracting farther south after the LCE detachment and might promote a reattachment. An experiment releasing particles in the location of the coherent structure for SE1 confirms that the structure stays there for ∼12 days and particles are advected away from day 37 onward. Contrary to the SE1 particle release experiment, particles released for SE3 in the same location as for SE1 (yellow dots in Fig. 9) are less stationary and move away from the region shortly after being released. An LCE detachment is absent in the REF simulation, while SE3 shows an LCE detachment matching the timing in AVISO data. The difference between REF and SE3 FTLEs is the presence of a coherent structure on the east side of the LC in SE3 (marked by the green box in Fig. 8). This coherent structure in SE3 is similar to the Loop Current frontal eddies (LCFEs), which are thought to have a controlling influence on the LCE separation (Schmitz 2005; Zavala-Hidalgo et al. 2003; Le Henaff et al. 2012; Huang et al. 2013; Jouanno et al. 2016). An experiment releasing particles in this coherent structure for SE3 shows that the particles stay as a coherent structure prior to the LCE detachment but then, some of the particles are ejected to the west through the separation between the LCE and the LC at day 37 (see green dots in Figs. 9i–p).
The FTLEs for day 21 of phase 1 correspond to the time of the LCE detachment in Exp3. It shows the difference in the LC state among all the experiments. The region marked in the red box is where an anticyclonic eddy (CubAN) is spotted when satellite data are assimilated (SE1 and SE2), and the green box is where an LC frontal eddy is observed in SE3 and not in REF.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Particle release experiments to validate the structures identified from the FTLEs of phase 1. (a)–(h) The 29 days of the estimate SE1, which shows an eddy inside the LC (CubAN eddy, red particles). (i)–(p) As in (a)–(h), but for SE3 with the LCFE (green particles). The yellow particles in (i)–(p) correspond to particles released in the location identified in SE1 [CubAN eddy, red particles in (a)–(h)]. The background corresponds to the FTLEs used for structure identification and is kept unchanged as time evolves. The particles can cross the FTLE isolines and are allowed to leave the region.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
We selected day 31 as a target in phase 2 because the differences in the northward extent of the LC between experiments grow after that day. The LC begins to grow in SE1 and SE2, whereas the growth in REF occurs later, and SE3 seems to have minimal LC growth and stays mostly retracted. The FTLEs for day 31 show that SE1 and SE2 have a semiorganized structure (an anticyclonic eddy) that orients itself penetrating the LC and merging with an eddylike structure in the LC (Fig. 10, red box). Particle release experiments for SE1 show that these eddylike structures merge and although some particles are lost through the Straits of Florida in the process (see Figs. 11i–p) the structure remains there until the end of the integration period. The structures in SE3 are of a smaller scale relative to SE1 and SE2. Conversely, REF seems to have a larger coherent structure slowly rotating clockwise. The particle release experiment in this large structure for REF shows that only a few of those particles made their way into the GoM after 29 days (see Figs. 11a–h), suggesting this as the reason why there is no growth of the LC in REF.
The FTLEs for day 31 of phase 2. The red box region encloses structures inside and makes its way into the GoM when the satellite data are assimilated, leading to an LC growth in SE1 and SE2, whereas the green box region in SE3 shows structures considerably smaller than those in the REF experiment, making them easier to be washed out of the system by the LC. The green box region in REF shows a large structure that cannot make its way through the YC until later.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Particle release experiments to validate the structures identified from the FTLEs of phase 2. (a)–(h) The 29 days of the REF experiment that shows the large structure (green particles) from the Cayman Sea that slowly makes its way into the GoM. (i)–(p) As in (a)–(h), but for SE1 and marking the eddy inside the LC (red particles). The yellow particles in the bottom panels correspond to particles released in the structure making their way into the GoM. The background corresponds to the FTLEs used for structure identification and is kept unchanged as time evolves. The particles can cross the FTLE isolines and are allowed to leave the region.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The phase 3 of the LC is quite active, characterized by steady growth with times of sudden extension and retraction without detachment or reattachment of an LCE. Although all experiments captured these features, they are larger in those that use mooring data (SE2 and SE3). The biggest differences in the LC northward extension metric occur around day 31 of the 09–10/2010 period. There, a steep growth of the LC is initiated, and SE2 and SE3 have an unorganized structure south of the YC (see Fig. 12 green boxes). This structure is composed of small-scale structures marked with several ridges, suggesting that it will eventually rip apart into small structures while interacting with the LC system. In REF and SE1, there is a small structure with a well-defined ridge (red boxes in Fig. 12), implying that the structure will stay as a whole when interacting with the LC with relatively fewer elongations. There is a difference between the reference run and the experiments with assimilation. The structure north of the Yucatan Peninsula (marked with arrows in Fig. 12) gets more stable with the assimilation of data (fewer ridges inside the structure). The particle release experiments (Fig. 13) contrast the structures from the Cayman Sea and the inside of the LC as well as the structure north of the Yucatan Peninsula. We observe that in REF, the Cayman Sea anticyclonic structure makes its way into the LC system merging with the LC structure and releasing some particles through the Florida Straits. So, the intrusion of the structure into the GoM results in a sudden LC growth but in the merging process, some particles are released from the LC resulting in a retraction. On the other hand, although the Cayman Sea structure in SE3 is larger than in REF, it is composed of small structures separated by less sharp edges (see Fig. 12 SE3). In this particular scenario, the Cayman Sea structure intrudes into the LC in stages, and a big chunk of particles is released into the Gulf Stream (promoting a retraction), while the Cayman Sea structure is still not fully through the YC. In all the state estimates there is a cyclonic eddy north of Campeche Bank that is affected similarly by the assimilation of the different data (red and green arrows in Fig. 12). The assimilation makes the cyclonic eddy to stay coherent as a single structure longer (until the end of the simulation), whereas in the REF run the cyclonic eddy is separated after 15 days. The effect of this eddy on the LC is described in the adjoint sensitivity analysis section (section 3d).
The FTLEs for day 31 of phase 3. The LC is extended, and a CubAN can be spotted inside the LC, which is similar in all the experiments; however, the structure south of the YC is different among experiments with mooring data assimilated (green box) from experiments without (red box). Regions in the red box mark a more compact structure, whereas in the green box, the structure is bigger and composed of several small structures. The assimilation experiments affect the cyclonic eddy located north of the Yucatan Peninsula (marked with a red arrow for REF and a green arrow for SE1–SE3). In general, the assimilation makes the cyclonic eddy more stable.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Particle release experiments to validate the structures identified from the FTLEs of phase 3. (a)–(h) The 29 days of the REF experiment that shows an eddy in the Cayman Sea (red particles), the cyclonic eddy north of the Yucatan Peninsula (magenta particles), and the structure inside the LC (yellow particles). (i)–(p) As in (a)–(h), but for SE3, marking the big structure in the Cayman Sea (green particles), the cyclonic eddy north of the Yucatan Peninsula (magenta particles), and the structure inside the LC (yellow particles). The background corresponds to the FTLEs used for structure identification and is kept unchanged as time evolves. The particles can cross the FTLE isolines and are allowed to leave the region.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The mooring cost function for phase 4 was the smallest of the four 2-month experiments, which means that the REF solution for phase 4 was closer to the observations compared to other phases of the LC. In general, all experiments (SE1–SE3) resulted in a minimum observation cost function for phase 4 (not shown). However, the LC northward extension metric shows a different behavior at the beginning of the 2-month period for phase 4, with REF and SE3 showing a decrease and SE1 and SE2 showing an increase in their values. From the FTLE analysis, we observe that the structures south of Cuba (marked with arrows in Fig. 14) are different between experiments assimilating satellite data (SE1 and SE2, red boxes) and REF/assimilating mooring data (REF/SE3, green boxes). The particle release experiment explored these two structures for REF and SE2 (Fig. 15). The particles released at the location of structures in REF tend to go inside the LC during the 28-day integration, leaving some particles in the Cayman Sea. For SE2, most of the particles released in the two structures remained in the Cayman Sea. Moreover, the structure closer to Cuba is an anticyclonic eddy generated by the assimilation process. The structure farther south in SE2 is elongated and the particles remain in the Cayman Sea. Note that in REF the particles made their way into the GoM close to Cuba; hence the data assimilated from the mooring close to Cuba appears to play an important role during this phase, which is consistent with the previous section.
The FTLEs for day 5 of phase 4. The general structures for experiments assimilating data are similar. They are also similar to the REF experiment except for the structures in the Cayman Sea south of Cuba (marked with arrows). These structures seem to be affected differently by assimilating satellite data (red box region) than in REF/assimilating only mooring data (green box region).
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
Particle release experiments to validate the structures identified from the FTLEs in phase 4. (a)–(h) The 29 days of the REF experiment that shows a couple of structures in the Cayman Sea (yellow and red particles). (i)–(p) As in (a)–(h), but for SE2, marking a couple of structures in the Cayman Sea (magenta and green particles). The background corresponds to the FTLEs used for structure identification and is kept unchanged as time evolves. The particles can cross the FTLE isolines and are allowed to leave the region.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The FTLE analysis and particle release experiments showed that the structures identified to be different among state estimates are eddylike structures in the Cayman Sea that interact with the LC, topography, and with other eddies in the LC. There is a high concentration of eddy kinetic energy (EKE) southwest of Cuba (Jouanno et al. 2012) and the Cayman Sea has been considered a breeding ground for many eddies that have been locally generated or resulted from finite perturbations induced by eddies coming from the Colombia and Venezuela Basins that were possibly generated by local instabilities or perturbations from north Brazilian rings when interacting with the Lesser Antilles (Jouanno et al. 2009). Many of those Cayman Sea eddies could reach the YC region and enter the GoM Basin (Duran-Matute and Velasco-Fuentes 2008; Athié et al. 2012). A numerical modeling study by Oey et al. (2003) reported the passage of anticyclonic eddies through the YC close to Cuba. Using mooring observations across the YC, a mode of variability resembling the passage of eddies into the GoM was reported (Abascal et al. 2003). Recently, Androulidakis et al. (2021) highlighted the effects of Caribbean anticyclonic eddies (CARAs) on the LC variability. These CARAs are common during retracted stages of the LC and absent during fully extended LC phases.
It is important to note that in the particle release experiments, particles are advected to the Caribbean Sea leaving the GoM passing through the YC close to Cuba. This is in agreement with the results showed by Sheinbaum et al. (2002) and Candela et al. (2002, 2019) where an average flow leaving the GoM close to Cuba is obtained from mooring data. This result is also in agreement with the apparent disconnection between the flow dynamics at the east and west side of the YC as observed in the observational cost function (Fig. 7).
d. Adjoint sensitivity experiment
The FTLE analysis discussed above does not determine causality. However, the adjoint model can be used to trace the dynamical relationships of circulation features by computing the linearized gradient or “sensitivity” of a scalar measure of interest to model state and controls at earlier times (Fukumori et al. 2004). The scalar measure, called the “cost function” here, but also called the “objective function” among other names, can be defined based on a specific feature as long as it is differentiable with respect to the control variables (Errico 1997).
The adjoint sensitivities are useful when the linearity assumption is valid. We could validate the linearity assumption by doing adjoint-model tests (Heimbach et al. 2010) or through data assimilation (Gopalakrishnan et al. 2021). Here, we tested the linearity assumption through data assimilation, by minimizing the model–data misfit (cost function) through iterative optimization using the GoM MITgcm–ECCO 4DVAR system (discussed above). The maximum time over which the linearity assumption of the GoM MITgcm–ECCO 4DVAR system is valid is about two months based on model perturbation experiments (Gopalakrishnan et al. 2013b). Here, we use an adjoint sensitivity experiment to identify regions and dynamic controls that affect the LC extension/retraction mechanism. Based on the FTLE analysis of phase 3, we propose that the abrupt LC growth is related to the passage of an eddylike structure from the Cayman Sea through the YC. One approach to understanding this process is by analyzing the adjoint sensitivities for a cost function that represents the LC growth and tracking it backward in time for positive sensitivity within the eddy in the Cayman Sea. Since joint assimilation of moored data and satellite-derived data improved the state estimates and forecasts, SE2 estimates were used for initializing the forward model in the adjoint sensitivity study. A detailed description of GoM MITgcm adjoint sensitivity studies can be found in Gopalakrishnan et al. (2013b). In this work, we use the same methodology but for a different cost function.
The cost function is based on the mean LC SSH (JSSH) within a region of interest defined by a 1.5° × 1.1° box (83.8°–85.3°W, 24.9°–26°N, called the “target region”) located at the northern tip of the LC (black box on Fig. 16) and over a day (called the “target day”). An increase in the cost function implies an LC extension, whereas a decrease in the cost function implies a retraction from its original position. The target region is chosen so that it should be small enough to exclude any spurious sensitivities but, at the same time, big enough to capture the features of interest. This target region captures the sudden growth in the LC that we are interested in, similar to the LC northward extension metric (not shown).
The snapshots of the sensitivity of JSSH with respect to SSH for phase 3. Forward and corresponding backward days of integration are indicated within the panels. The 17-cm SSH contour is marked as a reference for the LC (magenta curve), the 59-cm SSH contour encompasses the anticyclonic eddy down south (green curve), and the target region is marked as a black box. The colors correspond to the sensitivities: blue and red show negative and positive sensitivity, respectively. The sensitivities shown here are nondimensionalized.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-21-0213.1
The target day for this adjoint experiment is day 44 of the phase 3 SE2, corresponding to the maximum northward LC extension for that phase (see Fig. 5). Figure 16 shows a series of snapshots of the sensitivity of JSSH with respect to SSH (∂JSSH/∂SSH) as well as the 17-cm SSH contour (magenta curve) marking the LC and the 59-cm SSH contour (green curve) marking the anticyclonic eddy south of the YC. The 59-cm SSH contour encompassed the eddy south of YC and was defined heuristically after analyzing the FTLEs and the SSH fields. A positive adjoint sensitivity quantifies an increase in the averaged SSH in the target region whereas negative adjoint sensitivity quantifies a decrease in the averaged SSH in the target region for a unit SSH perturbation at earlier times. The series of snapshots shows a large region of positive sensitivity located on the edge of the eddy south of YC, which can be tracked backward in time from day 44 to day 24 (Figs. 16a–e). The sign of the sensitivity changes from positive to negative within the eddy during days 24–14. Later on, the eddy disappears, and the LC exhibits a sudden northward extension of about 0.88 at day 1 backward in time. The tracking of adjoint sensitivities supports the hypothesis that the propagation of an eddy from upstream Cayman Sea and its penetration into the GoM modifies the LC growth. However, the distribution of the sensitivity is not only inside the eddy south of YC. We speculate that this is due to several processes that affect the LC simultaneously and some of them seem to be generated by the interaction of the anticyclonic eddy south of YC with the Yucatan Current, topography, and other features.
In the adjoint sensitivity analysis, when the eddy passes through the YC (i.e., Fig. 16g), a set of different sensitivities are at play, and all of those can contribute to the LC evolution. Here we summarize different structures and how their interactions with one another impact the LC northward extension. The structures are the LC itself, with sensitivity propagating downstream; the anticyclonic eddy inside the LC system (the CubAN); coastally trapped waves (CTWs) traveling counterclockwise along the Gulf coast to reach the base of the LC (Jouanno et al. 2016); the cyclonic eddy on the Campeche Bank, north of the CTWs, showing negative sensitivity; and the eddy mentioned above in the Cayman Sea, south of the YC. All these structures have a strong signal in the velocity field and can be identified on the FTLE maps (except for the CTW signal) corresponding to SE2 of phase 3 (Fig. 12), where FTLE analysis time matches the time of adjoint sensitivity (Fig. 16g). The interaction of these sensitivities is hypothesized to affect the LC evolution as follows: 1) An intensification of the anticyclonic eddy inside the LC (CubAN) will lead to an LC retraction (decrease in the cost function). This eddy will be strong enough preventing the incoming flow from expanding inside the GoM Basin and diverting it toward the Straits of Florida. 2) An intensification of the cyclonic eddy on the Campeche Bank will increase the LC northward extension (increase in the cost function). The adjoint sensitivity fields shown for phase 3 in Fig. 16g correspond to the FTLE analysis for day 31 of phase 3 shown in Fig. 12. The cyclonic eddy located north of Campeche Bank (identified in Fig. 12 with arrows and Fig. 13 with the pink particles) is shown as an eddy feature in Fig. 16g with negative (blue) sensitivity. This negative sensitivity to SSH implies that if we apply a negative perturbation in SSH it will result in a positive effect, increasing the cost function in the target region, and hence, the LC extension. Because the identified eddy is cyclonic, a negative SSH perturbation will decrease the SSH further inside the eddy, increasing the intensity of such eddy, and this corresponds to a positive effect on the cost function, increasing the LC extension. The presence of cyclonic eddy, north of Campeche Bank, can thus propel the incoming Yucatan Current/LC farther north and promote LC extension. This cyclonic eddy can also grow stronger and penetrate the LC and favor an LCE separation on the western side of the neck of the LC as reported in previous observational and modeling studies (Huang et al. 2013; Hiron et al. 2020). 3) The negative sensitivity directly south of the YC corresponds to an anticyclonic eddy. This eddy will merge with the CubAN eddy, increasing the flow to the east and pushing the inflow to exit the GoM through the Straits of Florida. 4) Last, the expected LC contribution is straightforward, an increase in the LC SSH will favor an LC northward extension.
4. Summary and conclusions
This study examines the impact of subsurface velocity observations from a tall mooring array across the YC on the state estimates of the circulation in the GoM. The moored velocity observations across the YC were used, for the first time, to constrain a regional implementation of the MITgcm–ECCO 4DVAR system for the GoM that assimilates satellite-derived SSH and SST data (Gopalakrishnan et al. 2013a). State estimates were produced for eight months (May–December 2010), covering the length of the mooring observations. This hindcast period also covered several phases of the LC, including the separation of LCE (Eddy-F) immediately after the DwH oil spill disaster in the GoM. We evaluated the model performance using the SSH rmsd and the maximum LC northward extent metrics. By taking advantage of the dynamically consistent optimized solutions, we analyzed the effect of the data assimilation on the flow structure by computing the FTLEs. Further, we used an adjoint sensitivity experiment to study the dynamic linkage (causal) of the processes that contribute to the sudden LC northward extension.
The results suggest that the assimilation of subsurface moored velocities and satellite-derived SSH and SST improves the SSH hindcasts and forecasts as quantified by the model metrics for all the four phases of the LC. Assimilation experiments using only mooring data showed marginal improvement for SSH hindcasts, and their respective SSH forecasts were less skillful than forecasts from other experiments. This is due to the definition of the rmsd metric as the point-to-point difference of the SSH circulation features averaged over the whole GoM Basin. Since the moorings are local, deployed across the YC, they only provide fewer, sparse, and deep velocity information that is insufficient to constrain the surface circulation and LC frontal features within the whole GoM Basin. However, the moorings are deployed in a strategically important location across the YC, therefore affecting the LC enough to be noticeable even with this nonfavorable SSH rmsd metric during the active and extended phases of the LC (i.e., phase 1 and phase 3). On the other hand, when using the maximum LC northward extent metric, the assimilation of only mooring data (SE3) precisely matches the timing of the first LCE detachment (end of May 2010) as that of AVISO, while the reference simulation (REF) fails to simulate the timing of the LCE detachment. A caveat of this result is that it is from one state estimation realization. More subsurface YC velocity data assimilation experiments focusing on LCE separation events would be needed to validate this result statistically. However, it highlights the potential importance of the assimilation of subsurface moored velocity data across the YC. The results of the LC northward extent metric showed that the model estimates from all experiments reasonably captured the range of the LC growing and retraction. This is the upper and lower bound of the LC northward extent metric as that of AVISO and HYCOM SSH, especially during the LC growing and retraction in phases 3 and 4. Also, the model estimates were able to reproduce the first LCE detachment of Eddy-F during phase 1 either within a few days difference (Exp1 and Exp2) or at the same time (Exp3) as that of AVISO. However, the model estimates showed pronounced differences during the active LC phases 1 and 2 when compared to AVISO and failed to reproduce the series of reattachment/detachment and final separation events of Eddy-F.
The moored velocity cost contribution for SE3 has the largest misfit during 05–06/2010 and was reduced the most for the meridional velocity of the two moorings close to the Yucatan Peninsula. The concurrency of LCE (Eddy-F) detachment shown by SE3 and AVISO SSH suggests that fitting the model to the flow observations close to the Yucatan Peninsula can help the estimation of an LCE separation. More experiments are required to corroborate this result. The cost contribution of the meridional velocity component of the moorings close to Cuba becomes important for the LC phases with intermittent growth (phases 2 through 4), which seems to be related mostly to flow features incoming from the Cayman/Caribbean Sea and passing through the YC into the GoM Basin as well as filaments leaving the GoM and advecting back into the Cayman Sea as shown by the FTLE analysis and particle release experiments (Figs. 10–15). These findings are in line with the YC flow structure obtained from observations by Sheinbaum et al. (2002) and Candela et al. (2002, 2019) showing a southward near-surface flow close to Cuba.
A stepping LC growth and retraction were observed during the last six months of the experiment period (July–December 2010). We hypothesize this vacillating LC behavior is due to the presence of eddies that are advected from the Cayman Sea through the YC into the GoM Basin and exits through the Straits of Florida. The application of FTLEs to the GoM estimates has proven useful to visualize and highlight these features. Moreover, the FTLEs together with the particle release experiments helped to examine the possible dynamical processes in the region. We identified anticyclonic eddies entering the GoM from the Cayman Sea, the presence of CubAN eddies in the LC that precludes the LC from retracting farther south, and the presence of features like the LCFEs on the east side of the neck of the LC when an LCE detachment was about to happen. Those features were affected differently in different assimilation experiments.
The adjoint sensitivity experiment for phase 3 of the LC helped track the dynamic link of sudden LC growth to the passage of an anticyclonic eddy through the YC into the GoM. The adjoint sensitivity analysis also confirms several interactions that potentially affect the LC northward extent. These complex dynamical interactions make the LC a highly variable phenomenon that is difficult to predict.
The results from this study support the previous recommendations for a sustained monitoring system of the flow structure across the YC as an essential constituent of the GoM observation and modeling system. These results form the basis for optimizing the number of moorings required to effectively constrain the GoM model solution. It is also important to consider that velocity data is more nonlinear than SSH and SST. Therefore, it requires additional studies addressing the optimal length of the assimilation window.
Acknowledgments.
Research reported in this publication was supported by the Gulf Research Program of the National Academies of Sciences, Engineering, and Medicine under Award 2000013149. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Gulf Research Program or the National Academies of Sciences, Engineering, and Medicine. We thank all the members of the Canek program for providing the 2010 mooring data supported by CICESE’s nongovernment funds and BOEM contract M09PC00017 as part of the Loop Current Dynamics experiment (2009–11). We gratefully acknowledge the ECCO consortium, including MIT, JPL, and the University of Hamburg. We also thank the two anonymous reviewers for their insightful comments that helped to improve the final version of the manuscript.
Data availability statement.
The MITgcm code used in this study is checkpoint 64Y and can be obtained from http://mitgcm.org/. The SSALTO/Duacs altimeter product AVISO is produced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS) (http://marine.copernicus.eu/). The HYCOM/NCODA 1/12° global daily analysis can be obtained from the HYCOM Consortium (http://hycom.org/dataserver/). The along-track altimetry data can be obtained from RADS (http://rads.tudelft.nl/rads/index.shtml). The SST data can be obtained from Remote Sensing Systems Inc. (http://www.remss.com/). The NCEP–NCAR-Reanalysis-1 atmospheric forcings can be obtained from http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.html. The mooring data and ocean state estimates presented in this paper are available from the authors upon request (hvazquezperalta@ucsd.edu).
APPENDIX A
State Estimation Procedure
The background uncertainty covariances of initial conditions for temperature and salinity, open-ocean boundary conditions, and atmospheric forcings, as well as the observation uncertainty covariances for satellite SST and SSH data, are the same as in Gopalakrishnan et al. (2013a). Subsurface mooring velocities (u: zonal component; υ: meridional component) are the new observations introduced in this study.
As described above, the MITgcm–ECCO 4DVAR system minimizes the total cost function, composed of observation and control cost, using an iterative optimization procedure by adjusting the model controls. We analyzed the relative importance and the contribution of each type of data and model control to the total cost function at each iterative step of the optimization. The control adjustments for the first iteration are zero by definition and are cumulatively integrated over the successive iterations of the state estimation. The cost function from the forward model solution of the first iteration (iteration 1) or the background state assesses the model performance, that is, identifying the model–data mismatch in the four-dimensional space–time. The estimates of the cost function [Eq. (A1)] gradients are obtained by integrating the adjoint of the tangent linear model backward in time (Le Dimet and Talagrand 1986). To determine the adjustments to the controls that minimize Eq. (A1) through iterative optimization, we use this gradient information and a variable-storage quasi-Newton M1QN3 software (https://who.rocq.inria.fr/Jean-Charles.Gilbert/modulopt/optimization-routines/m1qn3/m1qn3.html), which is a publicly available Fortran code based on the algorithm described in Gilbert and Lemaréchal (1989). Ideally, in this iterative optimization process, the relative contribution of the observation cost decreases over the iterations. At the same time, the model fits the data by adjusting the control variables and increasing the control cost contribution. The iterative minimization procedure yields an optimized solution when two conditions are met. First, the cost function descent reaches a plateau with a relatively small slope (less than 1% per iteration). Second, the sum of the model–data squared misfits normalized by the number of observations and the assumed observation variance was less than 1 for all data types. Because of the relatively arbitrary nature of the error assumptions, exact values of 1 for the normalized cost function are not expected, and the iteration will be stopped when the gradients become small, not because the normalized cost function approaches 1.
APPENDIX B
Finite-Time Lyapunov Exponents
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