1. Introduction
The Kuroshio Extension (KE) is an inertial jet formed after the Kuroshio separates from the east coast of Japan. It brings an amount of heat to the midlatitudes and thus leaves profound effects on the local and even the basin-scale atmosphere–ocean system (e.g., Nonaka and Xie 2003; Putrasahan et al. 2013; Ma et al. 2015; Nakamura et al. 2004; Kwon et al. 2010; Wills and Thompson 2018). The bimodality is one of the most significant features of the KE: its path state alternates between a stable and an unstable mode with a roughly 10-yr cycle (Qiu and Chen 2005; Taguchi et al. 2007; Qiu et al. 2014). Under the stable state, the KE is characterized by regular quasi-stationary meanders, powerful jet, and weak eddy kinetic energy. On the contrary, convoluted meanders, weaker jet, and vigorous eddy kinetic energy can be observed when the KE is in an unstable state.
The KE bimodality leaves significant imprints on the dynamical environment of the ocean and atmosphere. Fundamentally, the strength of eddy kinetic energy in the KE domain is closely related to the state of the KE, as evidenced by significant interannual variations (Yang et al. 2017; Yang et al. 2018; Wang and Pierini 2020; Zhang et al. 2022). Besides, the KE state influences the temporal and spatial variations of ocean heat content, temperature, and salinity (Vivier et al. 2002; Seager et al. 2001; Geng et al. 2018). The activities of North Pacific cyclones, storm tracks, cloud amount, and large-scale atmospheric circulations are also remarkably coupled with the KE bimodality (Tokinaga et al. 2009; Frankignoul et al. 2011; Kwon and Joyce 2013; Masunaga et al. 2016; Révelard et al. 2016; Wills and Thompson 2018; Yu et al. 2019). Moreover, the KE state has been found to yield great impacts on marine ecosystems and fisheries production (Miller et al. 2004; Nishikawa et al. 2011; Oka et al. 2012). As such, the dynamic mechanisms and predictability of the KE bimodality have gained extensive attention over the past decades.
Different physical processes have been explored to uncover the dynamics of transitions between KE bimodality. Some studies linked the transitions to the westward-propagating Rossby wave excited by the basin-scale wind stress curl anomalies associated with Pacific decadal oscillation (PDO) or North Pacific Gyre Oscillation (NPGO; e.g., Qiu and Chen 2005; Taguchi et al. 2007; Ceballos et al. 2009). Nevertheless, some others attributed the KE state transitions to local nonlinear internal oscillation (Pierini 2006; Pierini et al. 2009; Gentile et al. 2018). However, it is unlikely that KE variability is completely unrelated to remote signals because the Rossby linear model does reproduce large-scale KE variabilities but not small-scale ones (Taguchi et al. 2007). Recent studies reconciled the above views and suggested that the KE bimodality is caused by the local intrinsic variability which is triggered by the large-scale external wind stress forcing (Taguchi et al. 2007; Pierini 2014; Pierini et al. 2014; Nonaka et al. 2016; Zhang et al. 2022). In this way, both the good correspondence of the KE regime change with the wind anomalies and the KE local nonlinear variability can be well explained.
Over the past decade, KE predictability has been investigated based on the dynamical mechanism of the KE bimodality. Taking into account of the linkage between PDO (NPGO) and the KE state, Ceballos et al. (2009) and Qiu et al. (2014) demonstrated that the KE state transitional events can be predicted 3–6 years in advance. However, even in the same KE decadal dynamic state, the distribution of the ocean dynamic and environment fields in the KE region can still be very different, which is associated with the short-term variability of the KE. Nonaka et al. (2012, 2016) found that large-scale and long-term variability in the KE domain is more predictable than the front-scale and short-term variability, and the factors limiting the local potential predictability are the oceanic internal processes. Moreover, the nonlinear intrinsic mechanism reflects that the uncertainties in the initial condition may have remarkable influences on the numerical short-term prediction of the KE, as reported by Kramer et al. (2012), Wang et al. (2020), and Geng et al. (2020).
As mentioned previously, large-scale wind stress anomalies are crucial to the KE decadal variability. Nevertheless, the local wind stress anomalies also have a nonnegligible effect on the state of the KE and short-term transitions between the KE bimodality. For example, local anticyclonic wind stress anomalies in the KE region can intensify the KE southern recirculation gyre and the KE jet and make its path more stable (Sakamoto et al. 2005; Li et al. 2017). Besides, positive wind stress curl overlying the KE jet in early summer exacerbates the meridional temperature gradient and maintains the stable KE front (Sato et al. 2016). From an energetic perspective, local wind stress can damp the generation of eddy energy and facilitates its dissipation (Renault et al. 2016, 2017; Yang and Liang 2018; Yang et al. 2019), favoring the path transition processes from the unstable to stable state while inhibiting the reverse processes (Zhang et al. 2022). In this situation, the uncertainties of local wind stress are likely to yield an effect on the prediction of the KE transitions.
Lorenz (1975) classified predictability into two categories. The first kind predictability refers to the effects of uncertainties in the initial state on the prediction result, while the second kind refers to the impacts of uncertainties in the boundary condition, forcing field, and numerical model on the prediction (Schneider and Griffies 1999). Previous studies mainly focused on the first kind predictability of the KE state transitions (Wang et al. 2017, 2020; Geng et al. 2020). To what extent does local wind stress uncertainty affect prediction, and what kind of dynamic mechanisms are involved? These issues belong to the second category of predictability problem, which has not been investigated yet and is the focus of this study.
To explore the effect of wind stress uncertainty on the short-term prediction of the KE state transition process, we employ the conditional nonlinear optimal perturbation (CNOP; Mu et al. 2003) method to find the optimal wind stress error leading to the largest impact on the prediction. The CNOP method is advantageous for addressing nonlinear physical processes in the ocean and atmosphere. It has therefore been applied to predictability studies of many high-impact ocean and atmosphere events, such as Kuroshio path variation (Liang et al. 2019; Liu et al. 2018, 2022), Antarctic Circumpolar Current transport sudden shifts (Zhou et al. 2021), El Niño–Southern oscillation (Yu et al. 2012; Yang et al. 2020), tropical cyclone (Zhou and Mu 2012; Huo et al. 2018), and Ural blocking (Ma et al. 2022). These successful applications of the CNOP method motivate the present study.
The rest of the paper is organized as follows. Section 2 introduces the numerical model used in this study and validates the simulation. The CNOP method and related settings are described in section 3. Section 4 displays the temporal and spatial structures of the optimal boundary errors (OBE), their effects on the KE prediction, and the internal dynamics modulating the growth of oceanic errors. Finally, the conclusions and discussion are presented in section 5.
2. Numerical model and simulation
Regional Ocean Modeling System (ROMS) is a terrain-following, free-surface ocean general circulation model which subjects the primitive equations with Boussinesq, hydrostatic, and incompressible approximations (e.g., Song and Haidvogel 1994). In this study, the ROMS nonlinear module (NLM) and adjoint module (ADM) are employed to establish an optimization system. The NLM performs numerical simulations and prediction experiments, and the ADM provides the gradient information of the cost function, which is required in the calculation of the OBE (see section 3; Moore et al. 2004).
a. Model settings
The model domain is bounded within 14°–57°N and 117°E–125°W, covering a large part of the North Pacific Ocean (Fig. 1a). The black box in Fig. 1 denotes the KE region (33°–37°N, 140°–153°E). The horizontal resolution of the model is 1/10° and the vertical direction is divided into 30 sigma levels with upper-layer mesh refinement. The Mellor–Yamada 2.5-level turbulent closure scheme is adopted as the vertical parameterization scheme (Mellor and Yamada 1982).
The climatological mean sea surface height (SSH; m) in the simulation domain (14°–57°N, 117°E–125°W) for (a) ROMS from 1990 to 2012 and (b) AVISO from 1993 to 2012. The black box donates the KE region (31°–37°N, 140°–153°E).
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
The ocean model adopts warm starts and is integrated from the year 1980 to 2012. The initial condition and lateral boundary condition, including flow velocity, temperature, salinity, and sea surface height, are extracted from the Simple Ocean Data Assimilation dataset (SODA 3.4.2; Carton et al. 2018). The model topography is interpolated from 2-arc-min global relief model ocean bathymetry (ETOPO2). To reproduce the realistic KE low-frequency state oscillations, the model is driven and forced by monthly time-varying reanalysis data. Sea surface wind stress is obtained from the National Centers for Environmental Prediction–National Center for Atmospheric Research reanalysis 1 (NCEP–NCAR R1; Kalnay et al. 1996) dataset. The other surface forced fields, comprising shortwave radiation, net heat flux, freshwater flux, sea surface salinity (SSS), sea surface temperature (SST), etc., adopt the climatological mean of the Comprehensive Ocean–Atmosphere Data Set (COADS; Woodruff et al. 1987). Here, SSS and SST data are used to restore the ROMS output with a 30-day interval and correct the surface net heat and freshwater flux during the restoration process. The monthly atmospheric forcing is applied to the model at 0000 on the fifteenth day of each corresponding month, with other steps interpolating the forcing of two adjacent fifteenth days.
During the total integrated 33 years, the model in the first 7 years (1980–86) is nudged with SODA’s velocity, temperature, and salinity to prevent drift and speed up the process of spinup. The model integration after the year 1986 is free of nudging but it does not drift afterward. The next 3 years (1987–89) are taken for the model to self-adapt and achieve stability. The model outputs of the last 23 years (1990–2012) are employed for the following study.
b. Model validation
Before investigating the impacts of wind stress errors on the numerical prediction of KE transitions, the capability of ROMS to simulate and hindcast the KE low-frequency variability is validated with Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO). As shown in Fig. 1, the climatological distribution of simulated sea surface height (SSH) in the North Pacific is in good agreement with the satellite observations. Simultaneously, the simulation captures the essential characteristics of the KE observed: the Kuroshio separates from the Japanese coast at 35°N and two quasi-stationary meanders are located upstream of the KE. We also note that, compared with the observations, the simulated KE jet meanders are less pronounced and the axis east of 150°E migrates slightly northward. These discrepancies in the KE jet pattern may stem from several factors: the restraint of the simulation domain size, the limited resolution of forcing field (non-air–sea coupling model), the absence of nudging or restoring, and errors introduced by inaccurate topography and terrain-following coordinate system (e.g., Taguchi et al. 2007; Nonaka et al. 2016; Putrasahan et al. 2013; Ma et al. 2015; Shan et al. 2020; Tsujino et al. 2006; Nakano and Ishikawa 2010; Delman et al. 2015).
Additionally, the decadal oscillation of the path state is reproduced by ROMS, though the simulated KE state variations do not exactly correspond to the observations (cf. Fig. 2 with Fig. S1 in the online supplemental material). The differences may be attributed to nonlinear internal variability (e.g., Taguchi et al. 2010; Nonaka et al. 2020). As many previous studies exhibited, the KE has gone through four bimodal transitions since 1993 when altimeter data started recording: two transitions from the stable-to-unstable state (SU; roughly in 1995 and 2006; Fig. S1b) and two transitions from the unstable-to-stable state (US; roughly in 2002 and 2009). Fortunately, our model captures all four transition processes though there is a time discrepancy of phase transitions with the observations [the general timing of modeled transitions can be recognized in the years 1994, 2000, 2004, and 2011 according to the KE pathlength (KPL; Fig. 2b)]. Here, the KPL is defined by the length of 30-cm SSH contours between 141° and 153°E for simulation (90 cm for AVISO). Therefore, it validates the capability of ROMS to hindcast KE bimodal transitions and motivates the subsequent predictability study.
(a) Time series of SSH anomaly (cm) averaged in the KE region (31°–36°N, 140°–165°E) derived from simulation (dark line: 15-day interval data; red line: 720-day low-pass filtered) and time series of integrated kinetic energy (×1013 m2 s−2) over 1000 m in the KE domain (31°–37°N, 140°–153°E; light blue line: 15-day interval data; dark blue dashed line: 720-day low-pass filtered). (b) The time series of KPL derived from simulation (dark line: 15-day interval data; red line: 720-day low-pass filtered). Blue lines indicate KPLs of selected SU and US transitions derived from the nonlinear simulation without any correction. (c) The tendency of KPL with a 720-day low-pass filter. Blue dashed lines denote the one-time positive and negative standard deviation. Orange and blue shadows in (b) and (c) denote the selected SU and US transitional events, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
The remarkable KE bimodality can be observed from the time series of SSH anomaly, kinetic energy integrated in the KE domain, and the KPL (Figs. 2a,b). When the KE is in a stable (unstable) state, the SSH anomaly is positive (negative), the kinetic energy in the KE region is relatively stronger (weaker), and the KPL is shorter (longer). The short-term transitions between the adjacent stable and unstable stages are defined as the KE transitional events. Considering that the KPL is a common index to judge the dynamic state of the KE, we distinguish the transitional events based on the KPL. According to the low-pass filtered KPL (Fig. 2b), it can confirm whether the large-scale SSH anomaly signal has west propagated and reached the KE region. If the signal has arrived, it would trigger a rapid shift of the KPL. Since the KE path state exhibits a faster change during transition periods compared to other phases, the concrete temporal phase of the modal transitions can be further distinguished by the KPL tendency (Fig. 2c). With a threshold of one standard deviation of 720-day low-pass filtered KPL tendency, selected transitional events last approximately one year, consistent with previous findings (Qiu et al. 2014; Wang et al. 2020). For simplicity, the duration times of the four events are all set to 360 days, which start on 15 April 1994, May 2000, October 2003, and June 2011 (denoted by SU1, US1, SU2, and US2, respectively). Note that the transitional events can also be selected out when the time window is set in the range of 500–1000 days. In such a range, the obtained results are insensitive to the choices of the time window. Besides, although large KPL tendencies can also be observed in other years, such as the year 1997, they are attributed to high-frequency signals and do not fulfill the criteria for large-scale variability as indicated by the low-pass filtered KPL. This predictability study only focuses on the four events induced by low-frequency variability.
3. CNOP method and its settings
The CNOP method is a useful tool for examining the predictability of numerical prediction, aiming at identifying the uncertainties which maximize the prediction error (Mu et al. 2003). The method was first applied to explore the effects of initial condition uncertainty and model parameter uncertainty on prediction (e.g., Mu et al. 2010; Wang et al. 2012; Zhang et al. 2016; Sun and Mu 2022). Subsequently, Wang and Mu (2015) extended the method to investigate the impacts of boundary condition uncertainty, in which the extended CNOP is called CNOP-B. Given that wind stress serves as an upper boundary condition for the ocean model, the CNOP-B method can be employed to study the influence of wind stress uncertainty on the prediction of the KE state transition process.
To obtain the OBE, an optimization framework is built based on the spectral projected gradient 2 (SPG2) optimization algorithm (Birgin et al. 2000) and ROMS NLM and ADM modules. The detailed operational steps are shown in the appendix. The necessary settings are listed as follows:
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Selecting the reference states. In this study, we focus on the predictability of the KE state transitional events. Four transition processes are extracted as the reference states (REF), including two SU and two US transitions. The process of selecting the transitional events has been described in detail in section 2b. Notably, the REF is equivalent to the nonlinear integration of the model from the initial time to the end time of prediction tp without any error superimposed. In fact, it should be the reference trajectory, which denotes a series of continuous states with time. Moreover, the corrections of SST and SSS have been switched off in REF experiments, because the corrections may introduce additional prediction errors into the model integration and cause the final results not to only reflect the prediction error growth induced by wind stress error. In this way, the REF is able to replicate the transitional processes (Fig. 2b).
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Determining the constraint of boundary errors. The boundary error b concerned by this study refers to wind stress errors, comprising zonal and meridional wind stress errors (
). Previous second kind predictability studies focused on the impact of time-independent boundary condition errors on atmospheric blocking prediction (Ma et al. 2022; Dai et al. 2023). In this study, we further consider time-varying (monthly) boundary condition errors, which ensures the reality and rationality of errors. The constraint of wind stress errors is defined as where Nt and Ng represent the number of months (Nt = 12) and model grids at the sea surface. The boundary condition errors are constrained in the whole simulation domain with Ng = 657 260 surface grids. Equation (4) indicates the square of spatial and temporal averaged wind stress error. This error should be less than the threshold estimated by the wind field observation error at each grid at each time. Currently, the observational wind speed error can reach 2 m s−1 (e.g., Tomita et al. 2015; Bourassa et al. 2019). Converted into wind stress error, it is about 5 × 10−3 −1 × 10−2 N m−2 according to the bulk formula (Trenberth et al. 1990). Too large errors are meaningless for predictability studies since they do not exist in reality. As such, different values of δ have been tested to calculate the OBE using the optimization algorithm. It is found that when δ is taken as 1 × 10−6 N2 m−4, the obtained wind stress error (OBE) is close to the error threshold 5 × 10−3 N m−2 at some grids at certain times. Therefore, δ is eventually set to 1 × 10−6 N2 m−4. Of course, smaller δ would yield smaller errors that did not significantly contribute to large prediction uncertainties. -
Setting the cost function. The cost function is used to measure the prediction uncertainty caused by the boundary condition error. In this study, the kinetic energy of the prediction error is defined as the cost function [Eq. (5)], where VKE is the volume of the KE domain above 1000 m. The terms
and are the zonal and meridional velocity errors at the end time of prediction, caused by the boundary condition error. Here, indicates the volume integral of error kinetic energy (ERKE) in the KE domain (31°–37°N, 140°–153°E) above 1000 m. As mentioned in section 2b, the SSH anomaly, kinetic energy, and KPL are all good indicators to distinguish the dynamic state of the KE. However, the form of 2 norm is beneficial to get the gradient of the cost function and reach convergence for the optimization algorithm. Therefore, kinetic energy with a form of 2 norm is selected as the cost function. -
Obtaining the gradients of boundary errors for the SPG2. SPG2 is an optimization algorithm that utilizes the gradient descent method to find the minimum of a nonlinear function with constraint conditions (Birgin et al. 2000). In the framework of solving CNOP-B (Fig. A1), we iterate the SPG2 algorithm along the decrease direction of
(gradient of the cost function with respect to the wind stress error) to optimize the wind stress error step by step. The detailed procedures of the SPG2 algorithm for solving the CNOP problem have been clarified by Wang et al. (2012) and Liu et al. (2018). In this study, the ROMS ADM is used to calculate the gradient of the cost function with respect to the wind stress error. The effectiveness of the ADM has been verified in various studies such as sensitivity analysis, data assimilation, stability analysis, and ensemble forecasting (Moore et al. 2004, 2009; Veneziani et al. 2009; Zhang et al. 2009).
4. Results
As aforementioned, previous predictability studies on the KE state transitions have focused on the effects of the initial condition uncertainties (Wang et al. 2017, 2020; Geng et al. 2020). Before exploring the boundary condition errors based on the CNOP-B method, a set of experiments is conducted to compare the relative importance of the uncertainties of boundary conditions and initial conditions. Taking SU1 as an example, the wind stress field differences between the same months of adjacent years during 1995–99 are used to create the five groups of wind stress errors, which are then superimposed on the boundary conditions of REF. Similarly, the ocean state differences (including sea surface height, zonal velocity, meridional velocity, temperature, and salinity) on the fifteenth day of the same months of adjacent years during 1995–99 are used to create the five groups of initial errors, which are superimposed on the initial conditions of REF. The prediction errors caused by these wind stress errors and initial errors are displayed in Fig. 3. It can be seen that the SSH error (SSHe) induced by boundary and initial conditions are similar, ranging from 0.15 to 0.35 m. The other three cases exhibit resemble results (figure not shown). The experiments demonstrate that wind stress errors can also introduce significant errors in simulations or numerical predictions, highlighting the comparable significance of uncertainty in boundary conditions as initial conditions. Therefore, the uncertainty in boundary conditions is worth investigating.
The prediction error (absolute value of SSHe; m) at day 360 caused by boundary and initial errors averaged in the KE region for SU1.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
The OBEs for the four transition events are obtained based on the aforementioned model settings and optimization system. In this section, we will explain how these wind stress errors affect the prediction of the KE state from the following three aspects: the structure of OBE, the influence of OBE on the KE prediction, and the oceanic error growth dynamics.
a. Spatial and temporal structures of OBE
Figure 4 shows the average patterns of OBEs and their temporal variations for each transitional process (the unprocessed time-varying OBE structures are shown in Figs. S2–5). The wind stress is perturbed in the whole simulated North Pacific domain at the beginning step of the optimization iteration of the CNOP method. However, when the optimization algorithm converges, the large amplitudes of the obtained OBE are mainly located in the KE region, while they are much smaller in the other regions. Such a localization of the OBE is automatically obtained by the CNOP method, reflecting that the wind stress error in the local KE region has a more significant impact on the short-term prediction.
(a) The time-averaged OBE vector (black arrows) and its magnitude (color shading; ×10−3 N m−2) during the SU1 in the whole simulated North Pacific domain (top-right-corner panel shows the OBE in the KE region). (c),(e),(g) As in (a), but for the US1, the SU2, and the US2, respectively. (b) The time series of the OBE’s magnitude averaged over the whole simulated region during the SU1. (d),(f),(h) As in (b), but for the US1, the SU2, and the US2, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
Besides, the spatial distribution of the OBE for each transition process shows a multieddies structure with a stronger averaged magnitude in the US transition and a relatively weaker one in the SU transition. Temporally, the OBE for the SU decays exponentially with time: the initial decline is faster. In contrast, the OBE for the US does not exhibit such an exponential decay feature: its decay at the initial time is much slower than that for the SU. Therefore, it indicates that the US requires larger wind stress errors and takes a longer time to maximize the prediction errors. In other words, prediction errors are more likely to arise and develop during the SU transitions. The reason for the temporal distribution of OBE will be further explained in section 4d.
b. Effects of OBE on the KE prediction
To investigate the effects of the OBE on the short-term prediction of the KE transitions, we superimpose the OBE on the wind stress field and then integrate the model for 360 days, which are called EXP experiments. Figure 5 shows the predicted state of the KE path in the EXP experiments. At day 360, the KE path of the SU (Figs. 5d,n) is found to be more stable compared to the REF (Figs. 5b,l). Throughout the 1-yr prediction period, the EXP KE path cannot transform from a stable state into a highly unstable state as observed in the REF, especially after day 180 when a significant discrepancy emerges between the REF and EXP (Figs. 5e,o). By contrast, the KE path of US (Figs. 5i,s) exhibits more unstable than that of the REF (Figs. 5g,q) at day 360. From day 180 to day 360, the EXP KE path is notably less stable compared to the REF (Figs. 5j,t). In brief, the OBE always makes the prediction of the KE state transitions worse and leaves the KE transitional events much harder to be predicted, no matter whether it is SU or US transitions.
The REF SSH (color shading; m) and KE axis (black contour) at (a) day 0 and (b) day 360 for the SU1. (c),(d) As in (a) and (b), but for EXP. (e) The 180-day low-pass filtered KPL (×103 km) for REF (blue line) and EXP (red line) during the SU1. (f)–(t) As in (a)–(e), but for the US1, the SU2, and the US2, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
(a) The OBE curl (contour: solid for positive and dashed for negative; ×10−8 N m−3) and SSHe (color shading; m) for the first month of the SU1. (b)–(d) As in (a), but for the US1, the SU2, and the US2, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
Given the importance of wind stress error curl, Fig. 7 further displays the OBE curl throughout the four KE transition processes, in which the curl for the SU and US exhibit almost opposite structures. For the SU transitions, a tripole structure of curl with a positive–negative–positive distribution is evident along the KE axis from upstream to downstream with the largest negative curl in the middle (Figs. 7a,b). On the zonal mean, the negative curl signal presents on the southern side of the KE axis (Figs. 7e,f). Note that the curl signals on the north side of the KE axis are different for the two SU transitions, which is due to the different offset results of the positive curl near the Japanese coast and the negative curl downstream. For the reverse transitions, there is a negative–positive–negative curl tripole along the axis with the middle positive curl as the most evident error (Figs. 7c,d). Meanwhile, the south side of the KE axis is dominated by the positive curl error (Figs. 7g,h).
(a) The time-averaged OBE curl (color shading; ×10−9 N m−3) during the SU1. The black contour indicates the KE axis at day 360 for REF. The time–latitude map of (e) OBE curl (color shading; ×10−9 N m−3) and (i) SSHe (color shading; m) during the SU1 averaged over the KE domain (140°–153°E). The red (blue) contours indicate positive (negative) SSHe in the first two months. The black lines denote the latitude position of the KE axis for REF. (m) The difference of the SSH meridional gradients along the KE axis between EXP and REF. The second to the fourth columns are the same as the first column, but for the SU2, the US1, and the US2, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
According to Eq. (6), the negative (positive) OBE curl can generate the positive (negative) SSHe. As such, Figs. 7i–l display the zonal mean SSHe in the KE region, which is the difference between EXP and REF. The positive (negative) SSHe along or on the south side of the KE axis basically aligns with the negative (positive) OBE curl in Figs. 7e and 7f (Figs. 7g,h). Previous studies have recognized that the KE decadal variability is triggered by external SSH anomaly signals associated with westward-propagated Rossby wave (e.g., Qiu and Chen 2005; Taguchi et al. 2007; Ceballos et al. 2009). Specifically, the arrival of a negative (positive) SSH anomaly signal in the KE region will excite the SU (US) transitions. Here the local SSHe induced by superimposing OBE is opposite in sign to these SSH anomaly signals. Therefore, the OBE seems to be able to disrupt or delay the short-term KE state transitions controlled by large-scale SSH anomaly signals.
More specifically, during the SU transitions, the positive SSHe south of the KE can lead to the overestimation of the meridional SSH gradient across the KE axis (see gradient difference between EXP and REF in Figs. 7m,n), making the KE path tend to be regular, thereby hindering the KE path to transit from a stable to an unstable state. For the reverse transitions, the meridional SSH gradient across the KE axis is underestimated (Figs. 7o,p), inhibiting the transition of the KE from an unstable to a stable state. Note that the SSHe mentioned above, which affects the stability of the KE path, is not directly caused by Ekman pumping, as the magnitude of the SSHe induced by Ekman pumping is much smaller than that of the total SSHe (cf. Fig. 6 with Figs. 7i–l). Furthermore, there is no strict linear correspondence between OBE curl and SSHe (cf. Figs. 7e–h with Figs. 7i–l). The increase in magnitude and the nonlinear changes of SSHe arise due to ocean internal variability. Further insight into internal dynamic processes will be revealed in the following section.
c. Analysis of the oceanic error growth mechanism
Previous studies have shown that internal dynamical processes play a significant role in the growth and development of oceanic errors (e.g., Wang et al. 2012; Liu et al. 2018; Geng et al. 2020). This section focuses on the ocean internal variability and further explores the generation and growth process of the prediction errors caused by OBE and the associated physical mechanisms.
Figure 8 displays the evolution of SSHe for the SU1 and US1 transitions (SSHe for the SU2 and US2 is shown in Fig. S6). During the SU transitions, the visible error first appears along the KE axis east of 153°E (day 120 in Fig. 8 and Fig. S6) and then intensifies in the upstream region. In the first 180 days, the error along the KE axis is weak and chaotic. However, at day 360, a negative–positive–negative SSHe tripole structure forms along the KE axis, which causes the KE path to be remarkably different from that in REF. The tripole structure almost corresponds to the time-averaged tripole wind stress curl error along the axis (Figs. 7a,b), implying that the curl of OBE does have a significant imprint on the formation of oceanic errors.
(top three rows) The SSHe (color shading; m) patterns at day 60, 120, 180, 240, 270, 300, 330, and 360 and the temporal evolution of squared SSHe (m2) during the SU1 transition induced by superimposing OBE. The green and black contours indicate the KE axis for REF and EXP, respectively. The dashed box denotes the KE region. (bottom three rows) As in the top three rows, but for the US1 transition.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
For the US transitions, the nonnegligible error initially appears in the KE upstream (west of 153°E; day 120 in Fig. 8 and Fig. S6), which is different from that of the SU transitions. The reason for the difference is that errors are more likely to arise and develop in the less stable jet domain. Subject to the westward SSH anomaly signal associated with the Rossby wave, the dynamic state of the KE jet shifts gradually from downstream to upstream (Zhang et al. 2022). At the beginning of the SU transitions, the KE upstream is still stable, but the unstable signal has already arrived in the KE downstream. Hence, SSHe is more likely to become stronger in the unstable KE downstream. Conversely, at the start of the US transitions, the KE upstream is unstable and the downstream has already become stable, thereby allowing SSHe to first grow up in the KE upstream. Furthermore, during the US transitions, the SSHe grows before day 180, and then positive and negative errors alternate along the axis. From day 180 to 360, the SSHe structures make the KE path more convoluted at day 360 than in REF.
From the temporal development of SSHe throughout the entire period (time series in Fig. 8 and Fig. S6), we can see that SSHe of SU transitions exhibits a more rapid growth in the early and late stages, while SSHe of US experiences faster growth in the middle stage. The discrepancy could be attributed to the different OBE temporal structures (US OBE is stronger in the middle stage compared with SU). In addition, the predicted SSHe at day 360 of the SU is much greater than that of the US, reflecting that prediction errors are more likely to arise and develop in the SU process than in the US process, as mentioned in section 4a. This is consistent with the previous findings of Wang et al. (2017): the predictability of the SU transitions is lower than that of the US transitions.
(a) The time series of ERKE (m2 s−2) during the SU1 averaged over the KE region above 500 m depth. (c) The time series of the terms that possibly induce the variation of ERKE and the time tendency of ERKE during the SU1 (×10−8 m2 s−3). (e) The 1-yr-averaged pattern of budget diagnosis of ERKE for the SU1, and only the term dominating the energy growth is drawn in each grid. Contours indicate the SSH for EXP at day 360, and the white dashed line frames the KE area. (b),(d),(f) As in (a), (c), and (e), but for the US1 transition.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
During the SU transitions (Fig. 9a and Fig. S7a), ERKE experiences rapid growth in the early stage (phase I; SU1: month 1–5; SU2: month 1–6), slows down in the middle stage (phase II; SU1: month 6–8; SU2: month 7–10), and then grows fast again in the last stage (phase III; SU1: month 9–12; SU2: month 11–12). The time series for the ERKE budget items in the whole year is exhibited in Fig. 9c and Fig. S7c. During the initial two months, EWW predominantly facilitates the error accumulation. In this stage, the errors are relatively small and the ocean internal dynamic processes have not taken effect. In the later stage, BT and BC tend to promote error growth, while ADV suppresses it. Besides, the effects of EWW are much more random than in the first two months. Notably, BT dominates the fast growth of oceanic errors in phase III. Moreover, the correlation coefficients between BT and the time tendency of ERKE are 0.79 and 0.60 for SU1 and SU2, respectively, while the other three terms share lower correlations. This good correspondence between BT and the tendency of ERKE indicates that BT is intensified when ERKE experiences rapid growth, suggesting that barotropic instability modulates the evolution of oceanic errors. In terms of the spatial distribution (Fig. 9e and Fig. S7e), BC mainly affects the area along the KE axis, especially upstream west of 143°E, while BT and ADV take more significant effects downstream with a broader latitude range. On the whole, the entire KE region is dominated by BT, which again emphasizes the significance of barotropic instability in the growth of prediction errors during the SU processes.
The ERKE growth process during the US transitions is distinct from that of the SU (Fig. 9b and Fig. S7b) due to the difference in the temporal structure of the OBE and the distinct underlying physical mechanisms. ERKE grows moderately at the beginning (phase I; US1: month 1–6; US2: month 1–5) and the end (phase III; US1: month 11–12; US2: month 9–12), but experiences sharp growth in the middle stage (phase II; US1: month 7–10; US2: month 6–8). Barotropic and baroclinic instabilities promote ERKE growth during the entire transitions (Fig. 9d and Fig. S7d), while ADV damps the growth, similar to the SU transitions. Furthermore, the time series of BT also has the highest correlation with the time tendency of ERKE (0.70 for US1 and 0.52 for US2), indicating the importance of BT. More significantly, wind stress yields a direct and powerful effect on the entire process of ERKE growth, which dominates the whole KE region, especially on the KE axis and the southern recirculation gyre region (Fig. 9f and Fig. S7f). This is the largest difference from the SU process and highlights the critical role of EWW in the US process.
As aforementioned, OBE decays while oceanic error increases with time. According to Eq. (11), powerful EWW easily occurs when both wind stress error and oceanic error are remarkable. For the SU processes, the OBE decays exponentially (Figs. 4b,f), resulting in a weak (strong) oceanic error matched with a strong (weak) wind stress error in the early (later) stage (Fig. 9a and Fig. S7a). By contrast, the OBE decays much slower during the US processes (Figs. 4d,h), resulting in the simultaneous presence of relatively significant wind stress error and oceanic error in the middle period (Fig. 9b and Fig. S7b), thereby amplifying the effect of EWW. In short, the synergistic effect of strong wind stress error and strong oceanic error leads to a more pronounced impact of EWW during the US compared to the SU.
The above analysis indicates that barotropic instability plays a dominant role in the growth of oceanic error during the KE transitional processes. Meanwhile, baroclinic instability also facilitates error growth. Considering that the barotropic and baroclinic instabilities are closely related to the KE jet velocity and its shear (Pedlosky 1987), we examine these fields for both REF and EXP. Specifically, the focus is on the last six months of the prediction period, during which oceanic errors can make a distinct change in the KE path state (Fig. 5).
For the SU (US) transitions, the KE jet of EXP is faster (slower) than that of REF, especially in the KE downstream (Fig. 10 and Fig. S8). Furthermore, there is a slightly northward (southward) shift of the KE axis, a narrowing (widening) of the flow amplitude, and a mild strengthening (weakening) of the southern recirculation gyre after superimposing OBE. Previous studies have proposed that negative wind stress curl overlying the KE region can accelerate the KE jet and its southern recirculation gyre, making the KE path more stable (Sakamoto et al. 2005; Li et al. 2017). As aforementioned, the OBE curl for the SU (US) is dominated by negative (positive) error (Figs. 7a–d). Hence, this study confirms the previous findings and reveals that positive local wind stress curl can correspondingly lead to the deceleration of the KE jet and its southern recirculation gyre, leaving the KE path less stable. Furthermore, wind stress usually works for deflecting energy from the geostrophic current in the KE region (e.g., Renault et al. 2016, 2017). Therefore, during the US processes (Fig. 9f and Fig. S7f), EWW plays the key facilitating role in the KE jet and southern recirculation gyre regions where flow field is weakened (Figs. 10g–l and Figs. S8g–l).
(a) The patterns of velocity
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
In the following, the horizontal and vertical shears of flow velocity are investigated. Figure 11 exhibits the horizontal and vertical shear of the KE zonal jet in REF and EXP. Here, only zonal velocity shear is illustrated, because the KE jet west of 153°E can be approximated as a zonal jet. It is evident that the shear strength is positively correlated with the jet intensity: a stronger jet corresponds to a stronger shear. However, the distribution of regions with significant horizontal shear differs from that of vertical shear. Horizontal shear is powerful on both sides of the KE axis, while strong vertical shear is confined to the KE axis, particularly upstream. Overall, strong horizontal shear is more widespread in latitude compared to vertical shear, which is consistent with the spatial distribution where BT and BC play the roles in facilitating error growth (cf. Fig. 11 with Figs. 9e,f). This means that velocity shear contributes to the occurrences of barotropic and baroclinic instabilities.
(a) The absolute pattern of the horizontal shear of zonal velocity
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
d. Sensitivity experiments
1) Experiments on temporal distribution
As shown in section 4a, the OBE gets smaller as it approaches the end time of the prediction, which is consistent with Wang and Mu (2015). The analysis of the ERKE growth mechanism (section 4c) shows that the oceanic error directly caused by Ekman pumping is very small, and these small oceanic errors will grow through oceanic internal processes, eventually becoming large enough to affect the KE path state. Hence, the oceanic errors induced by the wind stress uncertainties in the previous stage have more time to develop into larger prediction errors, while the uncertainties closer to the end time leave less influence on the prediction results. The optimization algorithm is intelligent, allocating the larger wind stress error in the early stage of the prediction to obtain larger prediction errors.
To investigate the influence of wind stress uncertainty at different time stages before the end time, a series of sensitivity experiments are conducted with the first N months’ OBE (SU: N = 4, 5, …, 8; US: N = 4, 5, …, 10). In detail, we only superimpose the first N months of OBE onto the wind stress field to examine the evolution of oceanic error. Figure 12 depicts the evolution of SSHe in the KE region for sensitivity experiments during four transitions. When the first 7, 8, 10, and 9 months of OBE are superimposed onto the wind stress fields of SU1, SU2, US1, and US2, respectively, a similar SSHe to that of the complete OBE can be reproduced at day 360 (pattern not shown). When the first four months of OBE are superimposed, the prediction error is nearly half of the largest prediction error. As the number of months of OBE increases, the prediction error is enhanced accordingly. Hence, small wind stress errors during the last 2–5 months do not have a significant impact on the prediction results.
(a) The temporal evolution of the square of SSHe (m2) integrated in the KE region when superimposing the entire OBE (black) and first 4–8 months’ OBE on the boundary condition of REF for the SU1 transitions. (c) As in (a), but for the SU2 transition. (b),(d) The US1 and US2 transitions with the additional results of superimposing the first 9 and 10 months’ OBE.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
To further understand why the prediction error cannot reach a higher level when superimposing the first few month’s OBE, an ERKE budget analysis is carried out on the 4-month experiments. Figure 13 demonstrates that the suppression of BT is the main reason hindering the growth of oceanic error during the SU transitions, while the absence of EWW and BT are the top two factors impeding the error growth during the US transitions. It again highlights the importance of BT to error growth in the SU transitions and the significance of both BT and EWW to the US transitions. After the initial four months, EWW becomes ineffective as no wind stress error is superimposed during this period. In addition, compared to EXP (under the influence of the whole OBE), the flow field and its horizontal shear undergo fewer changes, leading to the suppression of the process by which barotropic instability facilitates the growth of error energy.
(a) The time series of the difference between the ERKE (m2 s−2) obtained by superimposing the total OBE and the ERKE obtained by superimposing the OBE of the first four months during the SU1. (b) The time series of the budget terms’ difference between superimposing the total OBE and superimposing the OBE of the first four months during the SU1 (×10−8 m2 s−3). (c),(d), (e),(f), (g),(h) As in (a) and (b), but for the US1, the SU2, and the US2, respectively.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
2) Experiments on spatial distribution
To see if reducing vital wind stress errors can improve prediction skills, rectangular sensitive areas (SAs) are identified for four cases based on the large-value region of OBE (Fig. 14). The ranked top 1% grid points of Nt × Ng (12 × 657 260) are identified as sensitive grids (SG). The SA, determined by the SG with the largest area occupation in the first month, is set as a rectangular region bounded by adjacent integer latitude and longitude lines. The SA for SU1 (SU2, US1, US2) covers approximately 8% (7%, 5%, 6%) of the whole simulation domain. For comparison, we also set three additional rectangular areas (A1–A3) with an equal number of grid points.
The locations of the sensitive areas (SAs) and three other additional areas (A1–A3) for four cases. The yellow shadow indicates the sensitive grids (SG).
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
The improvements to prediction skills have been examined after removing wind stress errors in SA or three additional areas A1–A3. The formula for prediction skill improvement is
The prediction skill improvement (%) caused by removing OBE wind stress errors in the SA and A1–A3 regions.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
Moreover, another set of experiments has been conducted to further confirm the position of the SA regions. The key OBE structure in SA region is superimposed onto other regions A1, A2, and A3. Then the wind stress errors are removed from the regions SA, A1, A2, and A3 successively and their prediction skill improvements are respectively examined. The greatest improvement in prediction skills occurs when errors within the SA region are removed (SU1: 42%; SU2: 52%; US1: 33%; US2: 45%), reaffirming the effectiveness of the sensitive area.
5. Conclusions and discussion
This study utilizes the CNOP-B method to explore the impact of wind stress uncertainty on the short-term prediction of the KE state transitions in an eddy-resolving regional ocean model. The time-varying OBEs for two SU and two US transitions are obtained through an optimization framework based on the nonlinear and adjoint modules of ROMS.
The OBE exhibits localized characteristics near the KE domain, implying that the local wind stress perturbation has important effects on the KE short-term prediction. Furthermore, the OBE displays multieddies structures with decreasing magnitude as approaches the end time of prediction. The large amplitude of OBE in the SU is concentrated in the early stage, whereas it distributes more homogeneously over time in the reverse transition, suggesting that the prediction results of the SU are sensitive to early stage wind stress errors, while the US predict biases prefer the persistent wind stress errors.
After superimposing the time-varying OBE onto the 12-month REF wind stress field, the small oceanic errors are generated through Ekman pumping. Subsequently, the small oceanic errors gradually amplify due to oceanic internal processes. As the magnitudes of oceanic errors reach a high level (after day 180), they will affect the KE state and result in the failure of the forecast.
Specifically, the patterns of OBE curl for the SU and US are opposite: in the zonal direction, it shows a positive–negative–positive (negative–positive–negative) tripole along the KE axis during the SU (US) transitions. In the meridional direction, the SU (US) OBE curl is negative (positive) on the south side of the KE axis. Anticyclonic (cyclonic) wind stress error can instantly trigger positive (negative) small-amplitude SSHe through Ekman pumping. Once the small oceanic error generates, nonlinear internal dynamics modulate subsequent error growth. During the SU transitions, EWW first plays a leading role (first two months), followed by BT dominating the entire growth process. It is horizontal velocity shear that contributes to energy cascade through barotropic instability. In contrast, the synergistic effect of strong wind stress error and strong oceanic error results in a much essential influence of EWW throughout the whole US, with the role of BT also being significant. As the oceanic error is large enough, it induces an overestimation (underestimation) of the local meridional SSH gradient, resulting in a more (less) stable predicted path state during the SU (US) transitions.
Regarding the difference between the SU and US predictability: on the one hand, the US processes require larger and more prolonged wind stress errors to generate maximum prediction errors, in contrast to the SU processes. On the other hand, the prediction errors induced by OBE at day 360 are smaller for the US compared to the SU. These two aspects suggest that prediction errors are more likely to arise and develop in the SU transitions, indicating potentially lower predictability of the SU transitions compared to the US transitions.
It should be mentioned that the present study mainly focuses on the short-term prediction of KE state transitions, during which the roles of internal dynamics and local wind stress are important. As introduced in section 1, the long-term (decadal) variability of KE is mainly controlled by the propagation of Rossby wave triggered by wind stress anomaly in the northeast Pacific (e.g., Qiu and Chen 2005; Taguchi et al. 2007; Ceballos et al. 2009). Therefore, if considering a long-term prediction, such as extending the prediction time from 1 year to 3–6 years, the large values of the wind stress error may not appear in the local KE region, but in the northeast Pacific. Conducting such a long-term prediction in the future is highly meaningful, although it demands enormous computational resources.
In addition, prediction errors of the KE transitions can be caused by uncertainty in both initial conditions and boundary conditions. Previous studies have investigated the short-term predictability associated with initial conditions (Wang et al. 2017, 2020; Geng et al. 2020), highlighting the significant influence of initial errors on accurate KE state prediction. The present study shows that boundary condition errors, such as wind stress uncertainty, can also result in remarkable prediction errors. In the future, it will be interesting to explore the synergistic effects of these two factors.
Furthermore, the SA has been identified based on the spatial patterns of the OBE in this study. It is found that reducing wind stress errors in the SA can lead to large improvements in prediction skills, which is a significant guidance for devising observational strategies. Nevertheless, conducting observing system experiments (OSE) will verify whether the SA is effective in the real prediction of KE transitions and enhance the forecast accuracy from the perspective of reducing wind stress uncertainty. This issue needs to be further studied and implemented.
Acknowledgments.
This study was supported by the National Natural Science Foundation of China (42076017). It was also supported by the National Key Scientific and Technological Infrastructure project “Earth System Numerical Simulation Facility” (EarthLab), Oceanographic Data Center, the Institute of Oceanology, Chinese Academy of Sciences, and the Key Scientific Research Projects Plan in Henan Higher Education Institutions (24A170031).
Data availability statement.
The ETOPO2 topographic data can be obtained from NOAA National Centers for Environmental Information (https://www.ngdc.noaa.gov/mgg/global/etopo2.html). The SODA3.4.2 oceanic reanalysis data are available at the UMD Ocean Climate Lab (https://www2.atmos.umd.edu/∼ocean/index_files/soda3.4.2_mn_download_b.html). The NCEP–NCAR R1 atmosphere-forcing data are provided by the National Center for Atmospheric Research (https://psl.noaa.gov/data/gridded/data.ncep.reanalysis.html). The COADS climatological atmospheric fields are downloaded from the University of Colorado/NOAA (https://iridl.ldeo.columbia.edu/SOURCES/.COADS). The AVISO altimeter products are acquired from the Copernicus Marine and Environment Monitoring Service (https://resources.marine.copernicus.eu/).
APPENDIX
Detailed Steps for CNOP-B Calculation
As shown in Fig. A1, the OBE is obtained using the CNOP-B method through an iterative optimization framework. The detailed steps are listed as follows:
The flowchart of the detailed steps for CNOP-B calculation.
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0047.1
Step 1: Run the NLM from the initial time (t0) to the end time (tn) of prediction under the background boundary conditions (τ1, τ2, τ3, …, τn). Then we will obtain the reference state (yb) at the end time (day 360). Note that n is equal to 12 because the transitional events are defined within 12 months and the wind forcing is monthly time varied.
Step 2: Superimpose the first-guess boundary errors (e01, e02, e03, …, e012) on the background boundary conditions. Driving by the erroneous boundary field, NLM is integrated to get the state at the end time (y) of prediction. The prediction errors (y′) are acquired by subtracting yb from y.
Step 3: Calculate the objective function (f) and its gradient to prediction errors (
Step 4: Run ADM backward with
Step 5: Iterate the spectral projected gradient 2 (SPG2) optimization algorithm initialized by f and
Hereafter, steps 3, 4, and 5 are repeated in a loop until the objective functions converge. In other words, the SPG2 algorithm attempting to decrease the value of
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