Impact of Correction Target Selection on Long-Term Spectral Nudging in Luzon Strait and Its Adjacent Regions

Abstract

Previous studies have pointed out that spectral nudging is still insufficient in improving the long-term simulation ability of numerical models. In response to this problem, this study started with the Luzon Strait and its adjacent areas and discussed the influence of the selection of correction targets on its long-term spectral nudging. We established two sets of numerical experiments with the same parameter configuration except for the correction target: one was the monthly climatological target, and the other was the monthly real-time. The results showed that, compared with the climatology, the real-time target improved the consistency with the observations in large-scale variability on the premise of ensuring the correction of the climatological bias of the model. Further verification of the real-time scheme better simulated the meso- and small-scale characteristics, especially more accurately reproducing the position, intensity, and movement trend of eddies when the Kuroshio intrusion event occurred. Multi-scale energy analysis revealed the significance of adjusting large-scale potential energy to improve the overall simulation ability. The premise is that the correction target needs to fully contain these effective large-scale signals and non-stationary features, and then introduce them into the numerical integration of the regional model through appropriate band-pass filter parameter settings, driving a more reasonable large-scale background state thereby.

1. Introduction

The interaction of various scales in the ocean forms a complex dynamic process and motion system, which poses greater challenges and higher requirements for the accurate prediction of small-scale ocean processes [1,2]. At present, the tempo and spatial resolution of ocean numerical models are constantly improving, which provides favorable conditions for the refined forecast and research of ocean processes [3,4].

For a long time, people have carried out research on improving the simulation ability of regional models from the perspectives of refining horizontal resolution, optimizing parameterization schemes, and developing new assimilation methods [5,6,7,8]. Due to the limitation of computing resources, Dickinson et al. [9] developed a regional climate model (RCM) nested in the global circulation model (GCM), which continuously provides lateral boundary conditions to obtain high-resolution regional simulation information. RCM can improve the simulation ability of small-scale ocean processes to a certain extent, but the realization effect of this method strongly depends on the scope of regional models, boundary conditions, and parameterization scheme configuration [10]. In particular, the information from boundary conditions is difficult to directly affect the interior of the regional model and even decoupled from the coarse-resolution global model, which makes it impossible to achieve effective dynamic downscaling through one-way nesting in short-term forecasting models [11].

In this regard, Von Storch et al. [12] pioneered the “spectral nudging” method, and Thompson et al. [13] subsequently proposed an improved scheme applicable to ocean numerical models. The principle of this bias correction method is to use the filter to constrain the model in a number of specific frequency bands, while still allowing the free evolution outside. Its revolutionary innovation is that it provides a simple recursive method to filter any input time series. They first applied this method to the North Atlantic regional model and approximated the temperature-salinity to the seasonal climatology of observation, which indirectly optimized the simulation results of meso and small scales while suppressing large-scale bias and drifts. Since then, the idea of scale-selective correction has also been extended to the study of continental shelf circulation, mixed layer thickness, mesoscale eddy simulation, and tidal prediction, and its effectiveness has also been proved [14,15,16,17,18].

Although spectral nudging has a decent performance in many application scenarios, especially in the short-term simulation (the model runs for less than 1 year), it is satisfactory [19,20,21,22]. However, previous studies have also pointed out that the scheme has defects and shortcomings in the long-term numerical simulation process. Specifically, Lu et al. [23] found that the model cannot reproduce the low-frequency interannual temperature variation (this variability exists objectively in in-situ observations) when it approaches the seasonal climatology for a long period. After that, Zhu et al. [24] argued that the application of spectral nudging throughout the model did drive the simulated temperature and salinity to be significantly closer to the climatological observations, but it also weakens the convection in the subsurface and middle layers, making it underestimate the convection depth in the simulation area. In addition, the experiments of He et al. [25] and Shore et al. [26] have also illustrated the limitation of the nudging scheme based on the climatological correction target on the variation outside the frequency band.

Returning to this study, we took the Luzon Strait and its adjacent area as an example to study the long-term spectral nudging-related issues. For example, is spectral nudging sensitive to the selection of correction targets? In particular, can the non-climatological correction target effectively improve the long-term simulation ability of the regional model? Spectral nudging is how to affect the numerical simulation effect of meso- and small-scale phenomena. The rest of this article is structured as follows. Section 2 describes the spectral nudging method and analyzes its application scenarios in the Luzon Strait. The numerical model, experimental setup (including the features of correction targets), and the description of the observation data are described in Section 3. The comparison and analysis of large-scale simulation results are in Section 4. In Section 5, the influence of the correction target on the meso- and small-scale simulation is discussed. Section 6 is the main conclusion of the study.

2. Method and Application Scenario Analysis

2.1. Spectral Nudging

As described in Section 1, the core idea of spectral nudging is to constrain the numerical model in the selected frequency band so that the model tends to a certain distribution state at a specific scale, and then adjust the evolution of the small-scale by the nonlinear effect of the model itself. Its basic form is as follows:

$$x_{t+1}=x_{t+1}^f+\gamma \langle c_{t+1}-x_{t+1}^f\rangle ,$$

(1)

where x_f is the model background field, x is the analysis field, c denotes the correction target, and the nudging constant is γ. The angle brackets in the formula represent a filtering operator. The principle and details of this method are described in Thompson et al. [13]. The setting method of the filter operator has a direct impact on the dynamic downscaling effect. We intend to use the following filters:

$$\langle c_{t+1}-x_{t+1}\rangle =H{s}_{t+1},$$

(2)

$$s_{t+1}=\left(I-KH\right)M{s}_t+K\left(c_{t+1}-x_{t+1}\right),$$

(3)

The common forms of vectors K, H, and matrix M are:

$$K=\kappa \left[\begin{array}{c}1\\ 2\\ 0\\ 2\\ 0\end{array}\right],H=\left[\begin{array}{ccccc}1& 1& 0& 1& 0\end{array}\right],M=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& \cos \left({\omega }_1\right)& -\sin \left({\omega }_1\right)& 0& 0\\ 0& \sin \left({\omega }_1\right)& \cos \left({\omega }_1\right)& 0& 0\\ 0& 0& 0& \cos \left({\omega }_2\right)& -\sin \left({\omega }_2\right)\\ 0& 0& 0& \sin \left({\omega }_2\right)& \cos \left({\omega }_2\right)\end{array}\right]$$

(4)

where the parameter κ here determines the bandwidth and the spin-up time of the filter, and ω₁ and ω₂ represent the frequency of the selected constraint. Suppose x_t = e^iωt. The transfer function of the filter is:

$$\Gamma =H{\left[I-\left(I-KH\right)M{e}^{i\omega }\right]}^{-1}K,$$

(5)

We took the annual cycle and semi-annual cycle commonly used in previous studies as examples to draw the amplitude-frequency characteristic curve, as shown in Figure 1.

The frequency bands with the strongest filtering effect were concentrated near 0, 1, and 2 cycles per year. By setting different filter parameters κ, the bandwidth of the filter became narrower with the decrease of κ (i.e., the increase of κ⁻¹). It is proved that the advantage of this filter lies in the band-pass filtering characteristics of specific scales and frequencies, and the numerical integration is not constrained outside these frequency bands.

2.2. Application Scenario Analysis

As a large strait along the western boundary of the Pacific Ocean, the Luzon Strait is also the main channel connecting the Northwest Pacific Ocean and the South China Sea [27,28]. When the Kuroshio passes through this area, the westward invasion branch contributes to the heat, salinity, circulation, eddy activity, and even energy budget of the South China Sea [29,30,31,32,33,34].

Previous studies have shown that the Kuroshio intrusion is influenced by the East Asian monsoon and therefore has clear seasonal variations [35,36]. With the maturity of marine technology and the abundance of observational data, Chen et al. [37] noticed that the Kuroshio intrusion intensity in southwestern Taiwan still has interannual modulation and variation. Subsequently, more studies on the controlling factors affecting the Kuroshio intrusion were carried out to explore the possible mechanisms behind it [38,39,40]. These also include some research work from the perspective of multi-scale energy and interaction [41,42,43,44].

Due to the limited spatio-temporal coverage, it is not realistic to carry out detailed research on specific areas by employing observation data. Therefore, high-resolution numerical simulation of the Luzon Strait and its adjacent areas has become an indispensable tool [4,45,46,47,48]. The simulated Kuroshio morphology, however, in the Luzon Strait is still quite different from the actual situation [46,49,50,51]. Inspired by the analysis of multi-scale interactions, He et al. [42] proposed that if the temperature and salinity can be constrained in the process of model integration, it is beneficial to the improvement of numerical simulation results, especially the bias correction at the climatological scale and seasonal scale. This coincides with the idea of spectral nudging. In this regard, we directly constrained the temperature and salinity (T and S) of the regional model through spectral nudging (placed in the integration process of the model rather than separated from it), and the corresponding parameters γ, κ⁻¹, ω₁ and ω₂ were set to 0.05, 1000 days, annual cycle and semi-annual cycle (referring to the configuration scheme of Thompson et al. [13]; Urrego-Blanco and Sheng [17]).

3. Model and Data

3.1. Model Configuration

In this study, the Princeton Ocean Model with the generalized coordinate system (POMgcs; Ezer and Mellor [52]) developed a regional ocean model. The model domain was located at 13° N–30° N and 105.5° E–130.5° E, covering the northern South China Sea, the Luzon Strait, and parts of the Northwest Pacific Ocean. The horizontal resolution was 1/12°, and the z-σ mixed coordinate system was applied vertically. The layer was divided into 35 layers (the maximum depth is 5035 m). The internal and external modal time steps were 450 s and 7.5 s, respectively.

The initial conditions and open boundary conditions of the model were derived from the monthly mean version of the global ocean data assimilation and reanalysis system GLORYS2V4 of the European Union’s Global Observing and Monitoring Program (CEMES). The horizontal resolution was 1/4°, the vertical stratification was 75 layers, and the maximum depth layer was 5902 m. GLORYS2V4 also provided daily mean data sets. The system used a reduced-order Kalman filter and 3DVAR assimilation scheme to assimilate data including in-situ observations, satellite altimeter data, and sea surface temperature [53].

The forcing field of the surface layer of the model was obtained from the ERA-5 reanalysis data of the ECMWF [54], including surface wind speed, total cloud cover, and 10 m air temperature. The horizontal resolution was the same as GLORYS2V4, and the time resolution was 1 h. In addition, we matched the topography and depth of the experimental area from the ETOPO5 dataset.

3.2. Numerical Experiments and Correction Target

The model integration time was from 1 January 1993 to 31 December 2013, a total of 21 years, of which from 1993 to 1999 was for spin-up adjustment, and the next 14 years (1 January 2000–31 December 2013) were to carry out numerical experiments. All experimental results were output and stored in the form of daily mean. In order to explore the influence of the correction target on the long-term simulation of the regional model, we completed several numerical experiments under different spectral nudging conditions. One was the control experiment (CTR), which had no nudging during the operation. Next, in the first spectral nudging experiment, the monthly climatological was used for large-scale constraints, which was denoted as SPN1. Finally, in the second experiment, the monthly real-time data in the same period as the model integration time was applied, which was called SPN2.

The correction targets of the above two spectral nudging experiments were calculated and derived from the monthly mean GLORYS2V4 from 1993 to 2017 (Figure 2). Two sets of targets in the northwestern Pacific near the Luzon Strait were taken as an example (Figure 3). Figure 3a shows the comparison of the temperature correction targets at this point. It is not difficult to notice that they had relatively stable seasonal signals near the mixing layer, and the variability was also consistent. However, the real-time target located in the thermocline contained more abundant information than the climatology, including intraseasonal variations with periods of half a year and below, and the dominance of seasonal changes decreased. As for the real-time monthly salinity, it had significant non-stationary characteristics in the mixed layer. The case of the thermocline was even more so, and there were signal components with a period of more than 10 years (Figure 3b, purple solid line). The climatological only retained the annual periodic signal that is stable and decreased with the increase of depth (Figure 3b, green solid line).

3.3. Observation Data

To verify the simulation results of each experiment, we used the global ocean satellite observation grid data of the Copernicus Climate Change Service (C3S), with a spatial resolution of 1/4° and a temporal resolution of the daily average. It integrated the high-resolution absolute dynamic topography (ADT) and sea surface height anomaly (SLA) of multiple satellites. C3S also released the Group for High-Resolution Sea Surface Temperature Multi-product Ensemble (GMPE). The spatial and temporal resolution of the product was the same as that of the satellite altimeter. The SST analysis data from OSTIA, ESA, and HadISST were integrated with AVHRR observations.

We also applied the profile data EN4.2.2 from the British Met Office’s Hadley Centre. The observed data sources of EN4 were the World Ocean Database (WOD09), Coriolis Ocean Dataset for Reanalysis (CORA), Global Temperature and Salinity Profile Program (GTSPP), and the Argo data set of the Argo global data center since 2000 [55]. The current version provided two types of temperature and salinity data sets, which were global profile sets and reconstructed gridded data.

4. Impact of Correction Target Selection on Large-Scale

4.1. Direct Impact

In this section, we first evaluated the simulation results of temperature and salinity. Figure 4 shows the mean and standard deviation of SST. Each numerical experiment generally conformed to the observed spatial distribution pattern, but the warm water in the Luzon Strait simulated by CTR continued to extend to the northeast of the South China Sea (Figure 4a) and there was a weak long-term variation of SST (Figure 4b). After adding spectral nudging, both SPN1 and SPN2 corrected the warm bias of the surface temperature in the Luzon Strait and improved its variability. The results here depicted that the two spectral nudging schemes had little difference in the ability to control SST deviation, and they could drive SST to a stable large-scale condition.

Next, we screened the in-situ observations of EN4 in the model domain from January 2002 to December 2013, and counted the mean error and root mean square error of each model, as detailed in

Table 1. The mean error (Mean) and root mean square error (RMSE) between the simulated temperature below the surface layer and the in-situ observation of EN4, unit: °C.

	Error				RMSE
Depth (m)	GLO	CTR	SPN1	SPN2	GLO	CTR	SPN1	SPN2
20	−0.0556	−0.5599	−0.1268	−0.0722	1.1735	1.5667	1.3884	1.3457
50	0.0188	−1.3048	−0.1884	−0.1195	1.3555	2.4529	1.6341	1.6382
100	0.1237	−1.3044	0.0191	0.0350	1.3494	2.5859	1.7990	1.7428
200	−0.1473	−0.5586	−0.1368	−0.0532	1.0451	1.3836	1.2354	1.1405
300	−0.1135	−0.1143	−0.0247	−0.0219	0.8583	1.0909	0.9893	0.9109
400	−0.3672	0.1973	0.1627	0.0019	0.8477	1.0101	1.0176	0.8948
500	−0.2354	0.5834	0.1181	0.0139	0.7015	1.0293	0.9083	0.7952

and

Table 2. Same as Table 1 but salinity (unit: psu).

	Error				RMSE
Depth (m)	GLO	CTR	SPN1	SPN2	GLO	CTR	SPN1	SPN2
20	−0.0443	0.1784	0.0387	0.0332	0.2407	0.3580	0.2956	0.2823
50	0.0132	0.1734	0.0517	0.0132	0.1847	0.3067	0.2946	0.2400
100	0.0203	0.0371	0.1499	0.0442	0.1331	0.1840	0.2292	0.1659
200	0.0021	−0.0603	0.0560	0.0054	0.0771	0.1210	0.1189	0.0870
300	−0.0087	−0.0337	0.0219	0.0002	0.0709	0.0860	0.0823	0.0733
400	−0.0160	0.0215	0.0419	0.0113	0.0694	0.0812	0.0946	0.0795
500	−0.0037	0.0334	0.0214	0.0084	0.0556	0.0858	0.0818	0.0704

. The deviations of all numerical models and experiments mainly occurred in the 50–200 m thermocline. Considering that GLO assimilated multi-source observations, its error had been controlled within a small range. However, our CTR experiment showed a large degree of cold (salty) characteristics in temperature (salinity). Through large-scale correction, both schemes effectively constrained the drift of the mode in the thermocline, and the effect was in line with expectations. In particular, SPN2 was superior to SPN1. However, it can be observed that SPN1 continued to develop in a higher direction in the 100 m halocline. After excluding the possibility of filtering algorithm errors, we reviewed the salinity time series in Figure 3b and believed that the cause of this anomaly was related to the deviation of climatological salinity. During the model operation, the mean value of the climatological salinity in this layer significantly exceeded the observations of the same period (including the real-time value of the global model), so the model results were pushed in the opposite direction of expectations.

According to the description of Table 1 and Table 2, we were sure that the simulation results of the global model at this layer were reliable. However, due to the limitation of the number of field observations, it cannot evaluate the role of the spectral nudging method in the filter band. The two sub-regions of LS-SCS and TW-WPO here were taken as samples (Figure 4a), compared with GLO, and the Taylor diagram of each frequency band of the filter was drawn. The figure also shows the standard deviation of the experimental results, the correlation between the experiment and the reference value, and the standard deviation of the difference between them [56], as shown in Figure 5. In the climatology and low-frequency components, except for the temperature of LS-SCS, the correlation between the other elements of CTR and the reference value was higher than 0.8, and the deviation was within 0.75 (Figure 5a, purple point). After the climatological information was nudged to the model, the numerical points were pulled back to the vicinity of the coordinate origin, which means that the low-frequency variations of temperature and salinity were forced to weaken (Figure 5a green point). It is noted that the correction target used by SPN1 not only failed to filter out information other than the average value inside the frequency band but also constrained the low-frequency fluctuation signal originally driven by the regional model.

For the annual cycle (Figure 5b), CTR successfully reproduced the annual variation of the Luzon Strait temperature, but the other elements were far from GLO. When the model was constrained by the annual cycle information of the climatology, the correlation between the SPN1 and GLO was enhanced and had a certain positive effect. However, it also suppressed the salinity variability. A similar situation also occurred in the frequency band centered on the semi-annual cycle, as shown in Figure 4c. When we replaced the correction target with a real-time value, the result of SPN2 was much closer to the reference value (blue dots in Figure 5a–c). So far, the results and phenomena described in this section showed that the long-term spectral nudging of thermocline temperature and salinity was more sensitive to the correction target than the surface layer. More importantly, compared with the traditional nudging scheme, SPN2 allowed its long-term variability to be consistent with the global model while controlling the temperature-salinity bias.

Through Section 3.2, the characteristics of the two correction targets in the time dimension were clarified. On this basis, we further discussed the role of the time filter. Here, we set x^f in Equation (1) to 0, so that it only filtered the correction target. In order to ensure the filtering effect, it was necessary to normalize the mean of the target data and only retain the variation trend:

$$\{\begin{array}{l}{X^{\prime }}_{Real}=\frac{X_{Real}-{\overline{X}}_{Real}}{\max \left(X_{Real}\right)-\min \left(X_{Real}\right)}\\ {X^{\prime }}_{Clim}=\frac{X_{Clim}-{\overline{X}}_{Clim}}{\max \left(X_{Real}\right)-\min \left(X_{Real}\right)}\end{array},$$

(6)

where X represents T or S, and X_Real and X_Clim refer to real-time monthly mean and climatology monthly mean, respectively. Here, the climatology was regarded as a component of the real-time monthly mean. The filtering results of all frequency bands are shown in Figure 6.

In the case of Figure 6a,b, the difference was mainly reflected in the frequency band centered on the number of annual cycles 0. The real-time monthly data filtered out a low-frequency component with a period greater than 10 years, which was not available in the monthly average of climatology. It fully showed that even if the bandwidth parameter κ of the filter was set very small, there would still be a small number of harmonics that can be introduced into the regional model. They were concentrated near the constrained frequency, which contained low-frequency components with a period of more than 10 years. Furthermore, the filtering results of the annual cycle and the semi-annual cycle also revealed that the amplitude and phase between them were not completely consistent, and the stationarity difference was significant (Figure 6c,d). That is to say, this type of filter also had the ability to extract non-stationary signals. These large-scale information and non-stationary seasonal variations, however, were forcibly erased in the process of obtaining the climatology monthly mean (Figure 3), resulting in unreasonable constraints on the model in the process of long-term spectral nudging. This fully explained the cause of the difference in the simulation effect of the thermocline.

4.2. Indirect Impact

In this section, satellite altimetry data were used to verify the indirect effect of the correction target on long-term simulation. Figure 7 depicts the correlation distribution and root mean square error distribution between the simulated sea surface height and the observation. Here, we subtracted their respective spatio-temporal averages from SSH/ADT to obtain the anomaly values. The CTR had a good correlation (>0.75) with observations offshore and near the open boundary of the model but decayed rapidly in areas such as the east of Taiwan and the Luzon Strait (Figure 7a). Correspondingly, their RMSE reached or even exceeded 0.2 m (Figure 7c). It fully confirmed that, with the current model parameterization conditions, it was difficult for the information from the open boundary to effectively act on the core region and decouple it from the global model. More specifically, the distortion of the T-S large-scale variation within the model led to the deviation of the density layer position [57].

Peng et al. [58] pointed out that by adjusting the temperature and salinity of the entire water column, the variability of density layer and spatial height can be constrained, which indirectly affects the simulation of sea surface height. In the SPN1 experiment, the SSH simulation of the Luzon Strait and the northeastern South China Sea had indeed been improved de facto (Figure 7b,e). However, the extremely stable climatological correction target also made the effect of the Northwest Pacific region less ideal, or even worse. The results of SPN2 were highly consistent with the altimeter observations, which effectively reduced the error and alleviated the decoupling between the regional and the global (Figure 7f). This effect was inspiring. Real-time nudging targets effectively promoted the transfer of open boundary information to the inside of the model, standardized the unreasonable variations of SSH in the time dimension, and benefitted its long-term evolution.

To reveal the effect of different nudging schemes on the low-frequency components of SSH, the time series were decomposed into low-frequency signals with a period greater than or equal to half a year (the remaining part is all regarded as high-frequency components, see Section 5.2), as shown in Figure 8. The altimetry data showed that the low-frequency variation of the Northwest Pacific was stronger than that of the Kuroshio axis and the continental shelf in the northern South China Sea (Figure 8a). The distribution patterns of CTR and SPN1 were consistent with observations but were 30% to 60% lower in the western North Pacific (Figure 8b,c). SPN2 recovered in the above area compared to other experiments (Figure 8d). Furthermore, although both CTR and SPN1 had successfully captured the significant annual variation characteristics in this region, especially the SSH annual variation amplitude of SPN1 was even closer to the observation, the low-frequency interannual variation (e.g., >10 years) and seasonal fluctuation (the period between 8 months and 1 year) of SPN1 were seriously weakened synchronously (Figure 8e). A similar situation also occurred in the Luzon Strait (Figure 8f). The results of the spectrum corresponded well to the case in Figure 8c, exposing the suppression of the climatological correction target on the low-frequency variability in the long-term spectral nudging process.

The satellite observation data released by C3S not only provided global ADT and SLA but also gave the calculation results of surface geostrophic current. Figure 9 shows the comparison of surface geostrophic current obtained by model calculation and altimetry data. In winter, the Kuroshio intruded into the South China Sea and generated an anticyclonic circulation over southwestern Taiwan [31,40]. At the same time, a large-scale cyclonic circulation formed along the northern and western continental shelves of the South China Sea [59]. However, the anticyclonic circulation intensity of CTR was overestimated, and its surface current field in the northern South China Sea and eastern Taiwan was not well reproduced (Figure 9a). In contrast, under the constraint of spectral nudging, the model successfully suppressed the over-intrusion and made it closer to the observation (Figure 9b,c).

The summer observations of the surface flow field displayed a northward boundary current along the western South China Sea, which turned eastward to the interior of the South China Sea at about 18° N. At this time, the Kuroshio did not continue to intrude westward but directly returned to the Northwest Pacific [60]. These seasonal properties were not accurately simulated when the model ran freely, and even false anticyclones persisted in southwestern Taiwan (Figure 9d). The above results were not consistent with the observed average geostrophic current distribution, but we noted that some previous numerical simulations had similar phenomena [46,49,51]. Through the indirect effect of spectral nudging, the simulated summer large-scale flow field has also been greatly optimized, which was more in line with regional characteristics and enhanced the summer coastal current along the southeast coast of China (Figure 9e,f).

In addition to qualitative analysis, we applied the ε² value defined by Urrego-Blanco and Sheng [17] to quantify the fitting effect between simulated geostrophic current and altimetry data:

$${\epsilon }^2=\frac{{\displaystyle \sum _{i=1}^N\left[{\left({\overline{U}}_i^O-{\overline{U}}_i^M\right)}^2+{\left({\overline{V}}_i^O-{\overline{V}}_i^M\right)}^2\right]}}{{\displaystyle \sum _{i=1}^N\left[{\left({\overline{U}}_i^O+{\overline{U}}_i^M\right)}^2+{\left({\overline{V}}_i^O+{\overline{V}}_i^M\right)}^2\right]}},$$

(7)

where $\overline{U}$ and $\overline{V}$ are the mean values of zonal and meridional geostrophic current components, respectively, and the superscript O and M represent observations and models, respectively. The subscript i represents the ith grid point of the altimetry data, and N is the total number of observation positions. It was not difficult to derive the value range of ε² as 0~1. When ε² = 0, the model was completely consistent with the observed value, and the simulation did not have any skill if ε² = 1. According to the calculation results of Figure 9, the two nudging schemes were superior to CTR in both winter and summer, indicating that the regional model with spectral nudging had good skill in simulating the time-mean surface geostrophic circulation. In the long-term simulation, spectral nudging played a positive role in large-scale flow fields, but the differences between different schemes were not significant, which needs to be further compared with the special phenomena of the region. We will give a detailed description later.

5. Impact on Meso- and Small-Scale Variations

5.1. Subsurface Temperature and Salinity

In Figure 5a–c, we present the comparison of the temperature and salinity of the 100 m layer in the specified frequency band of the filter with the reference value and also constructed the Taylor diagram of the high-frequency signal components (the number of annual cycles is greater than 2, Figure 5d). The nudging-free model had limited small-scale simulation capabilities, and even after correcting the annual and semi-annual cycles, it only improved the high-frequency variability of temperature in the LS-SCS, while other elements had no substantial improvement, even worse than CTR.

Although the small-scale component variations of the SPN2 experiment were still far from the global model, from the blue scatters in Figure 5d, we still noted that SPN2 had more severe volatility than other numerical experiments and stronger consistency with the global model. Combined with the scenarios in Figure 5a–c, we deduced that the reason for the difference in the simulation results of the high-frequency components of the two spectral nudging experiments was not limited to the constraints on the seasonal components, but whether the low-frequency interannual signals near the climatological mean could be nudged during the integration process of the model. As expected, reasonably constraining the large-scale components of temperature and salinity was conducive to enhancing the ability to reproduce its high-frequency information.

5.2. Sea Surface Height

The standard deviation and spectrum of SSH high-frequency signals are presented in Figure 10. On the whole, the simulation results were consistent with the observed spatial pattern (Figure 10a–d). However, in the northern South China Sea, the Luzon Strait, and the Northwest Pacific, the short-term fluctuations of the CTR and SPN1 experiments were severely underestimated, with only 50% and 20% of the observed amplitudes, respectively (Figure 10b,c). SPN2 reflected the excellent simulation ability of small and medium scale compared with other experiments (Figure 10d). For the spectrum of high-frequency components, there were several peaks in the altimetry data between 2 months and half a year (Figure 10e,f red solid line). Closer to the observation than SPN1 at the same position, SPN2 even exceeded the observation at the position with a period of about 4 months, and the fluctuation is more intense (Figure 10e,f green and blue solid lines). By introducing more realistic large-scale features, the long-term simulation of the model accurately reproduced the abundant intraseasonal variations in the northeastern South China Sea and the northwestern Pacific (including the Luzon Strait and east of Taiwan [61,62]).

When looking back at the results of Figure 8, it should be emphasized that the SPN1 experiment, which had the weakest fluctuation in the low-frequency component, was also less active in high-frequency variations than CTR. While SPN2 could avoid the suppression of large-scale and long-period variability, it could also promote the recovery of high-frequency signals in specific regions. Katavouta and Thompson [11,63] proposed that the large-scale dynamic process of the adjustment mode affects the simulation ability of the meso- and small-scale, but we tend to believe that the direction and degree of this influence were related to the richness of information contained in the correction target closely (Figure 3 and Figure 6).

5.3. Kuroshio Intrusion in Luzon Strait

It is difficult to simulate the Kuroshio intrusion in the Luzon Strait area, which can be employed as an important basis for evaluating the meso- and small-scale simulation ability of the regional model. Due to the variability of the intrusion path, it is impossible to accurately distinguish only by sea surface temperature, temperature, and salinity profiles. Therefore, Nan et al. [64] proposed for the first time to determine the type of Kuroshio intrusion path by calculating the vorticity integral of geostrophic current in southwest Taiwan: looping path, leaping path, and leaking path. On this basis, to avoid the influence of positive and negative geostrophic vorticity offset each other, Huang et al. [65] separated the positive and negative vorticity values and derived the ‘Double Index’ (DI):

$$\{\begin{array}{l}{I}_{KC}={\displaystyle ∯sign\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)dA}\\ I_{KW}={\displaystyle ∯sign\left(-\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)dA}\end{array},$$

(8)

where

$$sign\left(x\right)=\{\begin{array}{l}1,x\ge 0\\ 0,x<0\end{array},$$

(9)

The I_KW (I_KC) of the above equation is the warm (cold) eddy coefficient, and u and v are the zonal and meridional components of the geostrophic current, respectively, which can be calculated from SSH or ADT. According to Huang et al. [65], the integral region A was located at 20° N–22° N, 119° E–121° E. DI takes the sum (difference) of the mean μ and the standard deviation σ as the threshold, and is defined as the looping path (anticyclonic structure) when I_KW was less than μ−σ. When I_KC was greater than μ + σ, the leaping path is detected (the cyclone eddy hinders the Kuroshio from deepening into the Luzon Strait) and the rest is the Leaking path. The time series of double decision parameters is shown in Figure 11.

In the winters of 2004–2005, 2006–2007, 2009–2010, 2011–2012, and 2012–2013, the I_KW of satellite altimetry was below the threshold (Figure 11a), that is, winter was the high incidence period of the looping path [31]. The I_KW of CTR also had the same seasonal features as the observations, but its magnitude was 2–3 times that of the observation (Figure 11b). It was confirmed that the simulated anticyclonic structure existed for a long time and its intensity was overestimated once again (Figure 9). In addition, the events that CTR was judged to be ‘Looping’ concentrated before 2008, and then due to the rising trend of I_KW, the number of eligible events decreased significantly, which was also greatly different from the OBS. On the premise of ensuring the seasonal variation of I_KW, the interannual distribution of ‘Looping’ events had been effectively improved in SPN1 and SPN2, but no events corresponding to the observation had been identified in some years (Figure 11c,d), such as 2004–2005 winter and 2007–2008 winter.

For I_KC, the calculated results reflect that the Kuroshio had more frequent Leaping paths from 2008 to 2013 than in previous years (Figure 11a, solid blue line). Our numerical model experiment was fundamentally consistent with the trend of altimeter data during this period. The difference was that CTR almost did not extract the Leaping path before 2005, and the number of events determined after 2009 was too large (Figure 11b, solid blue line). These anomalies were resolved in the spectral nudging experiment, but there was still a considerable gap between the observations in terms of the duration of the event and the intensity of cyclone eddies in the integral region during the ‘Leaping’ event (Figure 11c,d, blue solid line).

The DI statistical results of the model and observation are demonstrated in

Table 3. Statistical comparison of DI between satellite observation and numerical experiments.

	rkw	rkw	Looping (%)	Leaping (%)	Leaking (%)
OBS	1.0000	1.0000	13.2	13.7	73.1
CTR	0.1128	0.0913	17.3	17.8	64.9
SPN1	0.0457	0.2151	13.8	13.1	73.1
SPN2	0.2128	0.3146	13.3	10.4	76.3

. From the perspective of the correlation coefficient, the simulation effect of SPN2 on the leaping path (cyclone eddy) was the best in all simulation experiments, and the ability to reproduce the Kuroshio anticyclone structure was also superior to CTR and SPN1. However, their correlation with observations was not more than 0.4, and there was still much room for improvement. From the perspective of the proportion of the three intrusion paths, leaking had the highest proportion, which was consistent with the previous research conclusions [64,65,66]. The number of looping and leaping events of CTR was significantly higher than that of observation, while the proportion of leaking was low. The proportion of each type of SPN1 was close to the observation, and the Kuroshio ‘Leaping’ event determined by SPN2 was the lowest among the four groups of statistical results. In order to clarify the change of Kuroshio intrusion morphology caused by spectral nudging, we selected two typical events based on the intrusion index determination results of Figure 11 and drew a snapshot of instantaneous SSH and surface geostrophic current in Figure 12.

In late August 2005, a cyclonic structure appeared in the integration area of OBS, blocking the Kuroshio from moving westward into the South China Sea (Figure 12a). On the contrary, CTR simulated an anticyclonic center (Figure 12b), and the integral value at this time did not reach the I_KW threshold, so it was classified as a leaking path (as shown in Figure 12c). This anticyclonic structure continued to exist in SPN1, but SPN2 had successfully reproduced the cyclone eddy (Figure 12c,d). At the same time, SPN2 also simulated an anticyclone on the west side of the cyclone structure, which was close to the altimetry data. In December 2012, an anticyclonic structure was generated in the southwest of Taiwan under the strong intrusion of the Kuroshio and there was a tendency to shedding from the Kuroshio into the South China Sea (Figure 12e). Our simulation experiments captured this phenomenon. The distinction was that the anticyclonic intensity and range of CTR significantly exceeded the observations, while SPN1 inhibited its extension and limited its shedding trend (Figure 12f,g). Unlike previous experiments, SPN2 controlled the strength and location of the anticyclone, and the motion trend was reproduced (Figure 12h).

5.4. Multiscale Energy Analysis

In Section 5.1, Section 5.2 and Section 5.3, the influence of the selection of correction targets on the simulation of meso- and small-scale phenomena was clarified. On this basis, we applied the multi-scale energy and vorticity analysis (MS-EVA) method proposed by Liang and Robinson [67,68] to diagnose energy transfer and explore the mechanism of spectral nudging on the simulation ability of meso- and small-scale information.

The first step towards MS-EVA is the multi-scale window transform (MWT) developed by Liang and Anderson [69]. It can divide a function space into the direct sum of several mutually orthogonal subspaces, and each subspace has a relatively independent time scale range. Such subspaces are called scale windows. Given a time series a(t), the transformation coefficient ${\widehat{a}}_n^{~\varpi }$ of the corresponding scale window ϖ at time step n was generated in MWT. In the framework of MWT, kinetic energy (K^ϖ) and potential energy (A^ϖ) can be expressed as:

$$\{\begin{array}{l}{K}_n^{\varpi }=\frac{1}{2}{\widehat{V}}_{h,n}^{\sim \varpi }\cdot {\widehat{V}}_{h,n}^{\sim \varpi }\\ A_n^{\varpi }=\frac{1}{2}c{\left({\widehat{\rho }}_n^{\sim \varpi }\right)}^2\end{array},$$

(10)

where $V_h$ and ρ are horizontal velocity and density anomaly, respectively, the coefficient ρ₀ is the reference density value, which can be taken as 1025 kg m⁻³, and N² is the square of buoyancy frequency. This equation and the governing equations (Navier-Stokes equations) can derive the multi-scale evolution equations of kinetic energy and potential energy:

$$\frac{\partial K^{\varpi }}{\partial t}+\underset{\nabla \cdot Q_K^{\varpi }}{\underbrace{\nabla \cdot \left[\frac{1}{2}{\left(\widehat{V{V}_h}\right)}^{~\varpi }\cdot {\widehat{V}}_h^{~\varpi }\right]}}+\underset{\nabla \cdot Q_P^{\varpi }}{\underbrace{\nabla \cdot \left(\frac{1}{{\rho }_0}{\widehat{V}}_h^{~\varpi }{\widehat{P}}^{~\varpi }\right)}}=\underset{{\Gamma }_K^{\varpi }}{\underbrace{\frac{1}{2}\left[{\left({\widehat{VV}}_h\right)}^{~\varpi }:\nabla {\widehat{V}}_h^{~\varpi }-\nabla \cdot {\left({\widehat{VV}}_h\right)}^{~\varpi }\cdot {\widehat{V}}_h^{~\varpi }\right]}}-\underset{b^{\varpi }}{\underbrace{\frac{g}{{\rho }_0}{\widehat{\rho }}^{~\varpi }{\widehat{w}}^{~\varpi }}}+F_K^{\varpi }$$

(11)

$$\frac{\partial A^{\varpi }}{\partial t}+\underset{\nabla \cdot Q_A^{\varpi }}{\underbrace{\nabla \cdot \left[\frac{1}{2}c{\widehat{\rho }}^{~\varpi }{\left(\widehat{V\rho }\right)}^{~\varpi }\right]}}=\underset{{\Gamma }_A^{\varpi }}{\underbrace{\frac{1}{2}c\left[{\left(\widehat{V\rho }\right)}^{~\varpi }\cdot \nabla {\widehat{\rho }}^{~\varpi }-{\widehat{\rho }}^{~\varpi }\nabla \cdot {\left(\widehat{V\rho }\right)}^{~\varpi }\right]}}+\underset{b^{\varpi }}{\underbrace{\frac{g}{{\rho }_0}{\widehat{\rho }}^{~\varpi }{\widehat{w}}^{~\varpi }}}+\underset{S_A^{\varpi }}{\underbrace{\frac{1}{2}{\widehat{\rho }}^{~\varpi }{\left(\widehat{w\rho }\right)}^{~\varpi }\frac{\partial c}{\partial z}}}+F_A^{\varpi }$$

(12)

where ${\Gamma }_K^{\varpi }$ (${\Gamma }_A^{\varpi }$) denotes the sum of kinetic energy (potential energy) transfer within the window ω, respectively (according to Liang and Robinson 2005, they cover several cross-scale transfer components, such as ${\Gamma }_K^{0\to 2}$, ${\Gamma }_A^{0\to 2}$, ${\Gamma }_K^{1\to 2}$ and ${\Gamma }_A^{1\to 2}$, etc.), and $b^{\varpi }$ represents the conversion term of kinetic energy and potential energy in the same scale window. These terms are all local transfer terms of energy, and the following analysis work was carried out around them. The other parts of equations, their physical meaning, and the derivation process are detailed in Liang [70].

Referring to the amplitude-frequency characteristics of the transfer function and filter settings in Section 2, the model results were decomposed into a climatological scale window (period greater than 512 days; the corresponding kinetic energy and potential energy are K⁰ and A⁰, respectively), the seasonal scale window (between 128 days and 512 days; K¹, A¹), and eddy window (less than 128 days; K², A²).

The volume integral of each energy component in the sub-region of Figure 4a was obtained by using Equation (10) (

Table 4. Volume integral of multi-scale energy components in two sub-regions (Figure 4a), unit: 10¹¹ m⁵ s⁻².

		A0	A1	A2	K0	K1	K2
TW-WPO	GLO	67.6545	9.4935	9.4059	16.8163	2.3802	12.3906
	CTR	60.2448	4.6805	3.9007	17.3433	1.9502	4.2440
	SPN1	58.6134	6.4047	2.5815	13.4960	0.6998	2.6758
	SPN2	63.6513	6.9630	5.0244	14.8814	1.2572	5.2186
LS-SCS	GLO	16.0355	9.5464	5.7908	1.2325	1.5966	6.6361
	CTR	14.4205	5.1643	4.7856	15.0402	2.1236	5.6545
	SPN1	14.0683	6.7944	3.2963	2.3211	1.0899	3.0826
	SPN2	15.6068	7.3791	4.8273	2.2272	1.9128	5.4373

). Through the horizontal comparison of different models and their numerical experiments, we found that the high (low) potential energy of eddy scale A² in the two sub-regions had a favorable correspondence with the strong (weak) potential energy of large scale but had little relationship with the variation of seasonal scale A¹. Recalling Section 4, we were convinced that such a contribution relationship was inseparable from the low-frequency component introduced by the time filter. At the same time, it was also observed that the eddy kinetic energy K² intensity and A² were also synchronously adjusted. The above phenomena imply that there was a clear and intense cross-scale energy transfer and same-scale conversion in the study area. In this regard, Equations (11) and (12) were used to calculate the local transfer of energy in the climatological eddy and seasonal eddy, respectively, as shown in Figure 13 and Figure 14.

The ${\Gamma }_A^{0\to 2}$ in the whole model domain was dominated by the positive direction, which corresponded to the situation in Table 4, that is, the transfer of potential energy from A⁰ to A² (Figure 13a–d). It is undeniable that the potential energy cascade strength of CTR was seriously overestimated in the Luzon Strait (Figure 13b). The spectral nudging had a constraining effect on the large-scale potential energy cascade in the Luzon Strait, which can be found in Figure 13c,d. However, the introduction of inappropriate large-scale information also suppressed the transfer intensity of the Northwest Pacific. The large-scale kinetic energy cascade was also positive in the Luzon Strait (i.e., K⁰ → K²), and the ${\Gamma }_K^{0\to 2}$ of the CTR tended to extend to the South China Sea (Figure 13e–h), which was not unrelated to the abnormal morphology of the Kuroshio principal axis (i.e., the long-term stranded anticyclonic structure, Figure 9). ${\Gamma }_K^{0\to 2}$ was also regulated by spectral nudging, and its change was consistent with the intensity of the potential energy cascade ${\Gamma }_A^{0\to 2}$, which was most typical in the Luzon Strait and east of Taiwan.

Similarly, Figure 14 also depicts the seasonal-scale nudging that regulates the seasonal potential energy transfer in the Luzon Strait, making its intensity and distribution close to GLO (A¹ → A², Figure 14a). Due to the large difference in the stability of the seasonal scale information introduced from the correction target (Figure 6), the potential energy transfer intensity between the seasonal scale and the eddy scale on both sides of the Luzon Strait was affected (Figure 14b–d). Specifically, the positive cascade of the seasonal-scale potential energy of SPN1 was weaker than that of CTR and SPN2. By comparing the distribution of Figure 14f–h, it was not difficult to observe that increasing the intensity of the potential energy cascade still played a positive role in enhancing the local kinetic energy cascade in the South China Sea and the Northwest Pacific.

At the end of this section, Figure 15 reveals the magnitude and path of local energy transfers (conversions) of the model in each sub-region. In the global model, the intensities of ${\Gamma }_A^{0\to 2}$ and ${\Gamma }_A^{1\to 2}$ in TW-WPO exceeded the corresponding kinetic energy transfer and converged to the eddy kinetic energy K² with the help of buoyancy conversion b² (Figure 15a). The source of K² in the spectral nudging experiment was consistent with that of GLO, but the local energy transfer and eddy-scale buoyancy conversion of SPN1 was much lower than in other experiments, and the energy imported into K² was less than the loss through the inverse cascade of kinetic energy (Figure 15c). This can explain the reason why the simulation ability of the meso- and small-scale phenomenon of SPN1 in Section 5.1, Section 5.2 and Section 5.3 became worse, and accordingly confirmed the relationship between the eddy scale energy A² and K² in Table 4.

As for LS-SCS, its large-scale K⁰ was first converted into large-scale potential energy (K⁰ → A⁰) by positive b⁰ and then converted into K² by potential energy cascade and negative buoyancy conversion b² (Figure 15e). The b⁰ direction of CTR was opposite to that of GLO, and the transfer intensity of ${\Gamma }_A^{0\to 2}$ and ${\Gamma }_K^{0\to 2}$ was overestimated (Figure 15f), which was consistent with the situation in Figure 13 and Figure 13. Although SPN1 recovered the negative b⁰, it underestimated the ${\Gamma }_A^{0\to 2}$ and b², thus slowing down the accumulation of eddy kinetic energy in this region, and the transfer direction of its multi-scale kinetic energy does not completely agree with the GLO (Figure 15g). Under the action of real-time correction target, not only the magnitude of each transfer term between K⁰ and K² was restored to a certain extent, but also the direction of other energy paths was guaranteed (Figure 15h).

So far, based on the above research work, we were more convinced that the overall improvement of the simulation effect of the regional model was closely related to the constraint of large-scale temperature and salinity field. This implied a strong local multi-scale interaction within the study area and its continuous energy transfer direction (forward cascade of potential energy and reverse buoyancy conversion). The premise of all this was that the data used to constrain the model needed to fully contain these important large-scale information components, and under reasonable time filter parameter settings, they were appropriately introduced into the integral operation of the regional model.

6. Conclusions

Based on the numerical model of the Luzon Strait and its adjacent areas, this study explored the influence of the selection of spectral nudging correction targets on its long-term numerical simulation. In the first spectral nudging experiment SPN1, the model used the climatological monthly mean temperature and salinity conditions for large-scale constraints. For the second experiment, SPN2 was used to correct the real-time monthly mean of the target replaced by GLORY2V4.

Firstly, we investigated the effect of large-scale simulation. Each experiment directly constrained the climatological drift of long-term simulated temperature and salinity, indirectly reduced the sea surface height deviation in the Luzon Strait and the South China Sea, and adjusted the unreasonable distribution of the surface flow field. In particular, it effectively alleviated the long-term overestimation of the Kuroshio intrusion intensity. However, in the SPN1 experiment, the above effects were achieved at the expense of suppressing the long-period variation of the simulation results and reducing the correlation with the observations. On the premise of ensuring that the simulation results do not deviate seriously from the climatological state, SPN2 improved the consistency with the observed data in a specific frequency band and allowed free integration of uncorrected frequency bands (e.g., interannual variation and interseasonal variation). These results fully demonstrated that the large-scale simulation of the regional model was sensitive to the selection of correction targets.

Next, we focused on the simulation ability of meso- and small-scale. Similar to the large scale, the simulation of meso- and small-scale phenomena in the Luzon Strait and its adjacent areas was extremely sensitive to the selection of correction targets. Specifically, compared with SPN1, the high-frequency fluctuations and changes of T, S, and SSH of SPN2 below the semi-annual cycle were more significant, and the consistency of the trend and the location of the spectral peak were closer to the observation. Through the evaluation of a special phenomenon in the study area–Kuroshio intrusion, it was further confirmed that SPN2 and its corresponding spectral nudging scheme could better simulate the long-term change of Kuroshio intrusion path, and even more accurately reproduce the transient distribution of SSH and its geostrophic current during specific Kuroshio intrusion events, including the position, intensity and movement trend of eddies.

The reason is that the multi-scale energy diagnosis for each spectral nudging experiment revealed that the local forward cascade of large-scale potential energy and the stable local reverse buoyancy conversion were the main factors driving the small-scale phenomena in the region. This fully confirmed the view of He et al. [42] that adjusting large-scale potential energy plays a crucial role in improving the simulation ability of the model. The premise of all this is that the temperature and salinity data used to constrain the model need to fully contain effective large-scale information and non-stationary variations, and the integration of the regional model was constrained under the appropriate time filter parameter settings, so as to obtain a more reasonable background state.

In this study, the spectral nudging scheme based on real-time correction targets further improved the ability of regional models. However, it is undeniable that the climatology of the traditional scheme can also characterize the seasonal variation and climate average state of the region to a certain extent. The key is that its easy access is incomparable with any other dataset (e.g., WOA dataset, Levitus et al. [71]). Furthermore, this study still followed the configuration of previous studies on the parameter of spectral nudging and did not take into account the influence of more time-scale information. Therefore, there is still much room for improvement in the actual effect (such as the simulation of the Kuroshio intrusion flow path). Nevertheless, this new scheme provided a new idea for our future research work, which helps us to use high-resolution numerical simulation to explore the mechanism of Kuroshio intrusion more effectively.