A time series analysis of the sea surface height anomaly (SSHa) was conducted in the Kuril Basin of the Sea of Okhotsk. The mapping of the satellite-derived SSHa data was optimized to mitigate the effects of sea ice on the SSHa field during winter and early spring. Complex empirical orthogonal functions (CEOFs) were then used to analyze the SSHa field, revealing that the first three modes account for 55% of the signal variance. Mode 1 mainly represents the coherent variability trapped over the shelves all along the coastal regions and the Kuril Islands. Both seasonal and interannual variations are strongly correlated with the alongshore wind stress and can be well explained by an arrested topographic wave. Mode 3 is a propagating mode that exhibits trains of southwestward-propagating, high-amplitude anomalies. One possible mechanism for this is first-mode baroclinic Rossby waves, whose energy propagates from the Kuril Straits toward the Kuril Basin. However, mode 3 can be better interpreted as barotropic Rossby normal modes generated in the deep Kuril Basin. Mode 2 is a standing mode that may encompass the baroclinic variability in the basin. The monthly mean of the SSHa in the Kuril Basin is primarily governed by variability in mode 1, with mode 2 contributing to a lesser extent, and mode 3 being insignificant.
The Kuril Basin is the deepest basin in the Sea of Okhotsk, a large semiclosed marginal sea with a seasonal sea ice cover, surrounded by the Sea of Japan on its western side and the North Pacific on its southeastern side (Fig. 1a). The Kuril Basin is separated from the North Pacific by the Kuril Islands, and its depth exceeds 3000 m. A notable water mass in the basin is the Okhotsk Sea Intermediate Water (OSIW) (Freeland et al. 1998; Wong et al. 1998; You et al. 2000; Itoh et al. 2003), which is the main ventilation source of the North Pacific Intermediate Water (Talley 1991; Warner et al. 1996; Shcherbina et al. 2003). OSIW, also called Okhotsk Sea Mode Water (Yasuda 1997) is a cold, fresh, and oxygen-rich water mass, with a high concentration of iron (Nishioka et al. 2013). The Kuril Basin is the region where water masses from the northwestern shelf of the Sea of Okhotsk [dense shelf water (DSW)], the Sea of Japan [Soya Current Water (SCW)], and the North Pacific [Western Subarctic Water (WSAW)] mix to form OSIW (Itoh et al. 2003; Mensah et al. 2019, hereafter M2019). As such, it is a key region for the overturning and ventilation of the North Pacific, which has been shown to be weakening over the past four decades (Ohshima et al. 2014). In particular, Nakanowatari et al. (2007) demonstrated that a multidecadal trend of isopycnal warming and decreasing oxygen in the North Pacific Intermediate Water resulted from changing properties in the Sea of Okhotsk. These changes were later attributed to a reduced DSW production caused by the decreased production of sea ice in this sea (Kashiwase et al. 2014).
The Sea of Okhotsk is characterized by two distinct circulations. In the central basin a clear cyclonic gyre exists, well understood as a wind-driven circulation (Ohshima et al. 2004; Simizu and Ohshima 2006), while the Kuril Basin has a less well defined and understood anticyclonic circulation (Wakatsuchi and Martin 1991; Ohshima et al. 2004). The western boundary current of the cyclonic gyre is the East Sakhalin Current, which flows southward off the Sakhalin shelf and bifurcates eastward just north of the Kuril Basin. Its maximum transport occurs during winter and its minimum is in summer (Mizuta et al. 2003). It transports cold, fresh waters (including DSW) from the northwestern shelf toward the Kuril Basin (Fukamachi et al. 2004). The Soya Warm Current is another major current in the Sea of Okhotsk. It originates from the Sea of Japan and transports warm, saline SCW via the Soya Strait to the Kuril Basin. Its intensity depends on the sea level difference between the Sea of Japan and the Sea of Okhotsk (Ohshima 1994). Consequently, its maximum volume transport occurs in summer and its minimum transport occurs in winter (Ebuchi 2006; Ohshima et al. 2017). The circulation in or around the Kuril Basin may also be affected by the maximum outflow of OSIW via the Bussol Strait and the inflow of WSAW via the northern Kuril Straits in winter (Ohshima et al. 2010).
Mesoscale eddy activity is a dominant factor characterizing the flow field in the Kuril Basin (Wakatsuchi and Martin 1991; Ohshima et al. 2002; Rabinovich et al. 2002; Rogachev 2000). Possible mechanisms for the generation of these eddies include the enhanced inflow of the Soya Warm Current in summer (Uchimoto et al. 2007), or the generation of baroclinic instabilities around the Kuril Straits, which would develop and propagate westward via the β effect (Ohshima et al. 2005). M2019 suggested the importance of these mesoscale eddies for the formation of OSIW, as they noticed that the three main water masses in the Okhotsk Sea homogenize into the OSIW within 6 months in the western part of the Kuril Basin, in spite of the absence of strong tidal mixing in this region. This work was a climatology study that did not seek to further clarify the role of the eddies in the basin. Many of the other studies of the Kuril Basin were based on synoptic surveys or semi-idealized models, so no comprehensive assessment of these mesoscale eddies’ characteristics has been conducted to this day.
The coastal dynamics of the shallow regions of the Sakhalin and Hokkaido shelves and the Kuril Islands may also affect the Kuril Basin water properties and circulation. Nakanowatari and Ohshima (2014) described coherent sea level changes along the coastal regions of the Sea of Okhotsk. These variations were related to the seasonal cycle of an arrested topographic wave (ATW; Csanady 1978), which results from the variation of the alongshore wind stress over the shelf in the western part of the Sea of Okhotsk (Simizu and Ohshima 2006). An ATW is the arrested version of the coastally trapped wave (CTW), where the time derivative term is replaced by the frictional term. In the seasonal time scale, the coastal current or disturbances trapped over the shelf can be interpreted as an ATW, i.e., the response to a steady alongshore wind stress. Physically, onshore Ekman transport caused by the enhanced northwesterly during winter enhances the sea level and coastal current, trapped along the shelf. As a result, the ATW is characterized by a maximum along-shelf sea level and volume transport in winter, when the wind stress is the strongest, and a minimum during summer. The ATW can be considered as the nearshore branch of the East Sakhalin Current in winter and reaches as far as the coast of Hokkaido. Its high sea level anomaly contributes to the low volume transport of the Soya Warm Current during this season. The ATW in the Sea of Okhotsk also includes a baroclinic component which is explained by the dynamic response of isopycnals to seasonal wind variations (M2019). The deepening of isopycnals in winter followed by their shoaling in spring and summer create a positive and negative dynamic height anomaly, respectively.
In addition to these basin-scale, mesoscale, and coastal processes, the last major factor characterizing the Kuril Basin is tidal mixing near the Kuril Straits. The strong tidal currents in this region are known to generate intense mixing (Nakamura and Awaji 2004; Yagi and Yasuda 2012; Ono et al. 2013), which contribute to the ventilation of the North Pacific (Yamamoto-Kawai et al. 2004). Correspondingly, variations in the 18.6-yr nodal tidal cycle can impact the hydrographic properties of the waters exiting through the Kuril Straits via enhanced or reduced tidal mixing, and the intensity of the outflow of OSIW toward the North Pacific (Osafune and Yasuda 2006, 2013; Sasano et al. 2018).
The overlapping between the various phenomena described above makes the circulation in and around the Kuril Basin a complex system that is not fully understood to this day. A study of sea surface height in a limited area of the Kuril Basin revealed that the anticyclonic circulation in the basin reaches its maximum intensity in summer (Uchimoto et al. 2008). Further, the dynamic height climatology study of M2019 in the basin demonstrated that this anticyclonic circulation is highly seasonal. This conclusion was based on numerous data collected by profiling floats during winter. The anticyclonic circulation typically only exists between May and October, while the dynamic height maximum is concentrated along the coastal regions from Sakhalin to the southern Kuril Islands between November and February. This local maximum is the direct result of the ATW-related deepening of the isopycnals described above. As this positive dynamic height anomaly is likely released toward the Kuril Basin in spring, the anticyclonic gyre forms in May and is enhanced throughout the summer before decaying quickly around November. M2019 also demonstrated that the seasonal variations of dynamic height in the Kuril Basin are mainly related to the vertical displacement of isopycnals. However, this work did not address the issue of interannual variability in both the ATW and the dynamic height and sea surface height anomaly (SSHa) in the basin. This, along with a more exhaustive assessment of the roles of mesoscale eddy activity in shaping the Kuril Basin circulation are major scientific issues that merit further study.
This study aimed to document the time variability in satellite-derived SSHa in this region, determine the importance of interannual and long-term variability in the data record, and better understand the origin and evolution of mesoscale eddy activity in the basin. Emphasis was given to the role of the ATW, mesoscale eddies, and other phenomena in shaping the SSHa in the region. Additionally, time variability in the anticyclonic circulation in the basin was further assessed. Prior to conducting a complex empirical orthogonal function (CEOF) analysis on the data, a gridded SSHa dataset from along-track altimeter data was optimized to mitigate the effect of sea ice on the data. Section 2 delineates the methodology used to process and map the SSHa data, while the first CEOF mode is discussed in section 3, and the second and third CEOF modes are discussed in section 4. Emphasis is given to the nature of the phenomena causing the first and third CEOF mode, and how the three CEOF modes contribute to the formation, enhancement, and decay of the anticyclonic circulation. Concluding remarks are provided in section 5.
2. Data and methodology
For this SSHa-based study, we first considered using the level 4 gridded SSHa product with a 0.25° resolution provided by Copernicus Marine Environment Monitoring Service (CMEMS). To generate this product, the along-track data from the various satellite altimeter missions were gridded via an objective mapping methodology (Le Traon et al. 1998; Ducet et al. 2000) further described in section 2a. As the gridding process does not discriminate between data acquired in different basins, contamination between the Sea of Okhotsk and the North Pacific or the Sea of Japan occurs. This means, for example, that the SSHa values in the Okhotsk Sea regions close to the Pacific Ocean are calculated based on data from both the Okhotsk Sea and the North Pacific sides of the Kuril Islands. This is undesirable as different sides of the islands may be governed by different dynamics. The seasonal sea ice cover occurring in the Sea of Okhotsk during winter is also problematic as large regions of the Kuril Basin are covered by sea ice for approximately 20–80 days of each year (Fig. 1b). In coastal regions such as the Sakhalin shelf, the sea ice cover is prevalent for more than 120 days yr−1. The data acquired by satellite altimeters are not useable when the sea is covered with ice, leading to missing data for large swaths of ocean for long periods of time. This reduces the quality of the gridded data. In an effort to mitigate the effects of basin contamination and sea ice cover on the SSHa dataset, the along-track altimeter data were gridded via the standard objective mapping procedure, but with parameters optimized for the study area.
a. Optimizing objective mapping parameters to mitigate the effects of sea ice
The SSHa data used in this study were the level 3 processed along-track data provided by CMEMS. These consisted of data acquired along the track of each of the 13 satellite altimeter missions between 1993 and 2018, which were corrected for land and ocean tide, residual orbit errors, long wavelength errors and the inverse barometric effect. During the first 7 years of this period, only data from two satellite altimeter missions were available (TOPEX/Poseidon and ERS-1/-2), while three satellites missions provided data from 2000 to 2014, and four to five satellite missions provided data thereafter. Only data located within the Sea of Okhotsk were used to avoid basin contamination and the along-track data were subsequently gridded following the objective mapping technique described by Bretherton et al. (1976). The weights were determined via a space–time correlation function devised by Le Traon et al. (1998) and improved by Le Traon et al. (2003):
Here, a is a coefficient with the value of 3.337; Lx and Ly represent the correlation length scale in the x and y direction, respectively, T represents the correlation time; and dx, dy, and t stand for the distances (in the direction of each axis) and the time difference between a given altimeter data point and the grid point, respectively. Last, Cpx and Cpy are the propagation velocities in the x and y direction. To obtain the level 4 product in the Sea of Okhotsk, CMEMS sets the correlation time and space scales, and propagation velocities to standard values of 10 days, 100 km, and 0 m s−1, respectively (CMEMS Service Desk 2018, personal communication). In this study, however, the correlation scales were determined for each individual grid point to assess whether time and length scales adapted to the region could mitigate the effects of sea ice on the gridded dataset. The domain was rotated by 38° so that the x axis was aligned in the direction of the Kuril Islands and the y axis in the direction of the Hokkaido coast, and gridded onto a 20-km-resolution grid at a 10-day time interval. The correlation time and length scales were calculated on this rotated grid, for each year between 2009 and 2017, and these yearly correlation scales are then averaged together. This period was chosen because 2009 marked the beginning of coverage by interleaved TOPEX/Poseidon tracks, which ensures a better spatial distribution of the data. Last, to prevent the gridded correlation length scales from yielding sharp and unphysical SSHa gradients near locations where the length scales vary greatly (e.g., between the shelf and the deep basin), the correlation scales were smoothed with a 300 km window.
The smoothed correlation scales Lx, Ly, and T are displayed in Fig. 2 and show large differences associated with the bathymetry of the region. The Kuril Basin generally presented shorter length and time scales, with average values of 60–100 km and 20–25 days, whereas the shelf regions presented considerably larger scales, at 120–220 km and 30–35 days. Both the length and time scales in the Kuril Basin tended to increase from the Kuril Islands toward the basin interior, which implies that the regions surrounding the Kuril Islands are subjected to mesoscale eddy activity. Conversely, the long time and length scales on the shelf indicate that these regions are mostly affected by well-correlated movements at the monthly scale, which is consistent with the coherent sea level variations observed by Nakanowatari and Ohshima (2014) and attributed to an ATW cycle. The larger time and length scales, especially along the shelf regions, should considerably limit the effects of sea ice, as more data points become available in both time and space to calculate each gridded SSHa value. The first step for generating the SSHa dataset was to map the data from June 1993 to June 2018 onto the 20 km grid rotated by 38° according to (1) at 10-day intervals, with the correlation scales shown in Fig. 2. This dataset will hereafter be referred to as the optimized dataset.
To estimate the effect of the adapted correlation scales on the data, the optimized SSHa dataset was compared with the standard SSHa dataset produced using uniformed correlation length and time scales (100 km in both directions and 10 days). The mapping error, which is an estimate of the difference between the gridded values and the satellite observations, was calculated following Le Traon et al. (1998) for both sets and expressed as a percentage of the SSHa variance. This mapping error was averaged monthly and over the whole domain so that monthly time series of the error could be drawn for the two datasets, as shown in Fig. 3. Both time series show a relatively small error between May and December and a peak in the error between January and April, owing to the presence of sea ice, which reduces the number of points available for mapping. The time series of error for the standard and optimized datasets exhibited large differences. In the standard case, the error decreased from 22% between 1993 and 2000 to 7% between 2014 and 2018, which highlights the improved data quality resulting from increased satellites. The peak error in winter also decreased over time, but frequently exceeded 40% even toward the latter part of the time series.
In contrast, the adapted dataset error from May to December was essentially stable for the whole period, and never exceeded 5%, which suggests that the quality of data is somewhat independent of the number of satellites. Over the 25 years of the dataset, the mapping error between May and December was 77% smaller for the optimized than for the standard dataset (3.6% of the signal variance versus 15.9%). The winter error was also considerably reduced, but showed large variations, due to both the changes in satellite coverage (i.e., dependency on the number of satellite in winter) and the interannual variability in the sea ice cover. The peak error in winter tended to decrease over time, decreasing to less than 30% almost every year after 2003. The 25-yr averaged January–April error for the updated dataset was 57% smaller than that of the standard dataset (16.6% of the signal variance versus 38.7%), and these results demonstrate the relevance of using adapted correlation scales for optimizing the quality of SSHa mapping.
Figures 4a and 4b display the average distribution of the mapping error between 1993 and 2018 for the standard and adapted datasets, respectively. Figure 4c shows the error difference between the adapted and standard dataset. Figures 4d–f exhibit the results averaged only for the winter months (defined as January–April in this section only). The results of Fig. 4 clearly show how the error decreased over the whole study region when using the adapted correlation scales. The decrease was especially large during the winter months for the shelf off Sakhalin, which is the area most affected by sea ice (Fig. 1b).
c. Absolute error estimate
To further limit the negative impact of winter sea ice on the data, at each time steps, all data with a mapping error greater than 50% were excluded. The dataset then included several periods with missing data, and Fig. 5a displays the average number of days per year lacking data. This showed that the number of missing days was significantly smaller than the number of days when the ice concentration was greater than 10% (Fig. 1b). The shelf region off Sakhalin still had the highest number of missing days, whereas considerable improvement was seen in the Soya Strait, which now averages less than 10 days of missing data per year. The western part of the Kuril Basin usually included 15–25 days without data, compared to the 40–50 days of Fig. 1b. To fill in these blank periods, a temporal linear interpolation was conducted for each grid point. This interpolation process itself can introduce error. To estimate this error, the missing data mask of each winter (January–April) day from 1994 to 2018 was shifted by 6 months so that the points from July–September with known values were artificially masked. Linear interpolation was used to fill in these masked summer values. The error was then estimated as the mean square difference between the original dataset and the artificially masked dataset filled with interpolated data:
with x and y as the grid coordinates; ηint and η as the interpolated and actual gridded SSHa values, respectively; and (j = 1, …, N) indicating the number of interpolated points between July and October (equivalent to January–April) during the 25-yr period. Note that this estimate of the interpolation error would only be accurate if propagating phenomena such as Rossby waves do not have a strong seasonality in the basin. The seasonal amplitude of the monthly mean signal for the propagating mode (mode 3) in the Kuril Basin is ~1.5 cm (section 4a, Fig. 12a), which is smaller than the standard deviation of the mode (Fig. 12b). We therefore assume that the time interpolation and estimation of its error are valid in our study. The mapping error (ϵmap) and the interpolation error (ϵint) are independent. Thus, the total error (ϵ) can be obtained by adding the two terms. The interpolation error between January and April is presented as a value in cm (Fig. 5b), while Fig. 5c and Fig. 5d respectively present the total error (ϵ) for the period of January–April as a percentage of the variance and as an absolute SSHa value. Last, Figs. 5e and 5f show the total error (ϵ), for the period from May to December expressed in the same way. The latter two panels include very little interpolation error as the sea ice has mostly vanished by May in the Kuril Basin. These, furthermore, demonstrate the low rate of error in the absence of sea ice. The interpolation error (Fig. 5b) was largest on the Sakhalin shelf, and was also significant in the western part of the Kuril Basin. This error was mostly below 2 cm, which represents less than 20% of the local variance. As a result, the total error averaged for the winter months was kept well below 50% for most of the Kuril Basin and Soya Strait, with average values of 17% for the Soya Strait, 30% for western Kuril Basin, and <10%, for the eastern Kuril Basin. The error largely exceeded 50% off the Sakhalin coast, and for this reason, this region was not included in the following analysis.
3. Mode 1: Coherent shelf-trapped mode and seasonal cycle of the Kuril Basin
CEOF analysis was conducted on the 10-day interval SSHa data from June 1993 until June 2018, after minimal processing (detrending and 1-month low-pass filtering). The first three modes contributed 55.5% of the variance, with the first mode accounting for 40.2%, the second mode 8.3%, and the third mode 7.0%. As the contribution of the fourth mode was nearly twice as small (3.8%) as that of the third mode, only the first three modes were analyzed, which are presented in sections 3 and 4.
a. Amplitude, phase, and climatology
The spatial and temporal phases and amplitudes of mode 1 are displayed in Fig. 6. A large part of its energy was concentrated along the shelf regions from Sakhalin Island to the Kuril Islands, while strong energy was also seen around the center of the western Kuril Basin (Fig. 6a). The spatial phase along the coast was uniform, indicating that variations in the mode 1 SSHa occurred in concert along the shelf (Fig. 6b). Similarly, mode 1 SSHa should vary nearly uniformly in the Kuril Basin as the spatial phase there shows little variation. The −60° to −90° phase difference between the shelf and basin indicated a lag of 2–3 months between variations occurring in the shelf and in the basin. The temporal phase (Fig. 6d) indicated a very clear annual cycle whose maximum amplitude (Fig. 6c) was usually reached in January and February.
The annual repeatability of the first mode allows for a bimonthly climatology to be drawn (Fig. 7). The bimonthly interval was chosen so that this climatology could be easily compared with the dynamic height climatology of M2019. The mode 1 climatology revealed a clear seasonal pattern, with the alongshore SSHa reaching a maximum in November–December and January–February, and a minimum in May–June and July–August. These alongshore variations seemed consistent with the seasonal cycle of the ATW (maximum in winter, minimum in summer), which is discussed further in the next section. The mode 1 SSHa in the Kuril Basin also followed a seasonal cycle, but reversed and possibly lagging that of the nearshore. The SSHa was negative in the basin for the first half of the year, and positive in the second half of the year. The maximum negative value was obtained in March–April and the maximum positive in September–October, in an area centered around 46°N, 146°E. M2019 analysis suggested that the barotropic component of the SSHa around this region showed similar temporal variations. This barotropic component was strongly negative from January to June whereas the dynamic height (i.e., baroclinic) anomaly became positive in May–June. This results in a positive SSHa in July–August, i.e., 2 months after the dynamic height anomaly. In light of the appearance of a positive mode 1 SSHa in July rather than May in the basin, we suggest that mode 1 comprises a significant barotropic component.
As both the mode 1 (Fig. 7) and SSHa climatologies represent anomalies from the yearly mean rather than absolute signals, these anomalies should be interpreted cautiously. In the Sea of Okhotsk, the absolute levels of sea surface height are not well defined due to the relatively poor definition of the geoid in this region. For this reason, no accurate estimate of the barotropic component of sea surface height exists, either. However, the biyearly climatology of surface velocity drawn from the combination of dynamic height and float trajectories by M2019 in the Kuril Basin provides reliable estimates of the absolute velocity in the basin to some extent. This climatology exhibited a clear anticyclonic circulation during the second half of the year. However, no cyclonic circulation existed during the first 6 months of the year, which differs from Figs. 7a–c. Therefore, the negative values for SSHa and mode 1 seen in the present study should not be interpreted as signs of a cyclonic circulation in the basin, but rather as the weakening or disappearance of the anticyclonic gyre. By the same logic, the anticyclonic circulation during the second half of the year should be stronger than what was seen in Figs. 7d–f.
b. Relationship with the ATW
The climatology of mode 1 suggests that this mode could be related to the cycle of the ATW. To assess this possibility, the monthly ATW transport was calculated from the alongshore wind stress integrated along a path running from the north of Sakhalin Island to the Hokkaido coast (black line in Fig. 1), according to Csanady (1978) and Ohshima et al. (2017):
In a right-handed coordinate system, x is positive toward the offshore and y is oriented along the coastline. Parameter τy is the alongshore component of the wind stress, calculated as described in Large and Pond (1981). We used 6-hourly wind data obtained from the European Centre for Medium-Range Weather Forecasts interim reanalysis (ERA-Interim) at a resolution of 0.5° × 0.5° to calculate the daily mean wind stress; ρw is the density of seawater, and f is the Coriolis parameter. The alongshore volume transport at the final integration point, y1 (southernmost point on the black line in Fig. 1) corresponds to the Ekman transport to or from the coast, integrated southward from the origin (gray dot in Fig. 1). The time series of ATW transport and of the mode 1 SSHa off the shore of Hokkaido (area within the yellow rectangle in Figs. 8c,d) from 1993 to 2018 are displayed in Fig. 8a. Both curves exhibited similar seasonal patterns, without phase difference and with matching interannual anomalies. For example, the period from 2007 to 2010 was marked by low amplitudes for both time series, while 2006 exhibited a large positive peak during winter in both cases. As the seasonal cycle represented an overwhelming part of both the mode 1 and ATW signals, these time series were further compared after removing the monthly mean. The monthly mean was calculated as the 25-yr average of the data for each calendar month. Removal of this monthly signal yielded time series of mode 1 and ATW monthly residuals (Fig. 8b). These anomaly time series also showed some similarities, and the cross-correlation between both series was maximal at zero lag, 0.44 versus 0.88 for the raw time series (99% significance level was 0.15 for the anomaly time series and 0.49 for the raw time series).
Last, the cross correlations between the ATW and the time series of mode 1 for each grid cell were calculated, both for the raw (Fig. 8c) and monthly residual (Fig. 8d) cases. The raw time series correlation results were striking in that the shore regions were uniformly and strongly positive, whereas most of the basin regions showed a strong and negative correlation. The lag (not shown) was 0 for the alongshore regions and +2 or 3 months for the basin, and the positive correlation ended abruptly at the Bussol Strait (Fig. 8c), implying that this deep strait may prevent the shelf anomalies from propagating further. The separation between shelf (positive) and basin (negative) regions was still evident for the anomaly correlations, albeit with reduced values of 0.4 (lag = 0) on the shelf and −0.3 (lag = +1 month) in the Kuril Basin. Both of these values were significant for the large degree of freedom (N > 300) of these time series.
The results of the analysis on the first mode demonstrated that mode 1 is a standing mode, which probably comprises a significant barotropic signature. On the shelf, mode 1 is governed by variations in the alongshore wind stress in the Sea of Okhotsk, at both the seasonal and interannual scales, as indicated by the high correlation with the ATW. The variations within the basin were negatively correlated and lagged by 2–3 months behind the alongshore variations. Contrary to the shelf regions however, the ATW does not affect deep basins, which leaves the possibility of another phenomenon generating variations in mode 1 in the Kuril Basin. The detailed mechanism responsible for these variations merits further study.
4. Mode 2 and 3: Standing and Rossby-wave-like propagating modes
a. General properties
Contrary to the periodic pattern of mode 1, the temporal phase of modes 2 and 3 exhibit irregularities, with both rapid (3–4 months) and slow (1–2 years) phase variations (Figs. 9d and 10d ). The energy of the second mode was located mainly in the Kuril Basin and, to a lesser extent on the Soya Strait (Fig. 9a). The areas of high energy were distinct and could represent a standing wave. In contrast, the energy of mode 3 was strongly confined to the central part of the Kuril Basin, roughly corresponding to the area enclosed by the 3200 m isobath (Fig. 10a). Within this area, the maximum energy formed a clear dipole-like pattern.
To better understand the spatial and temporal variability of these modes, Hovmöller diagrams, which present the time evolution of deseasoned SSHa, mode 2, and mode 3 SSHa along a section oriented from the center to the southwestern part of the Kuril Basin (gray line in Fig. 1a), are drawn in Fig. 11. Here, deseasoned SSHa is defined as the raw SSHa minus the mode 1 SSHa, which can be considered a seasonal signal. The deseasoned SSHa exhibited several instances of anomalies propagating from the northeast toward the southwest, e.g., from September 2010 to June 2011 and from March 2015 to April 2016 (Fig. 11a). The Hovmöller diagram of mode 3 showed similar high amplitude anomalies (Fig. 11c). In contrast, in 2006 and 2008, the deseasoned SSHa showed no propagating anomalies, which coincided with a low mode 3 amplitude (Figs. 10c and 11c). Therefore, mode 3 appears to be a clear propagating mode, with its high amplitude denoting the presence of coherently southwestward-propagating positive and negative anomalies. Around the middle part of the section (~200 km), the level of the SSHa for mode 3 was often close to zero and showed little variation. This gap in energy appeared to separate large anomalies and could also be seen between the two maximum amplitude areas in Fig. 10a. This could indicate excitation of a Rossby normal mode that includes a node in the middle (mode21). The further investigation of this is presented in section 4c.
The mode 2 amplitude was particularly high during the period from 2015 to 2017 (Fig. 9c) and its Hovmöller diagram (Fig. 11b) presented standing features, which were not obvious from the deseasoned SSHa (Fig. 11a). In addition to the dipole area of concentrated energy in the Kuril Basin, two more anomalies existed outside the section area, one toward the northeast of the basin, and the other one on the Hokkaido shelf (Fig. 9a). None of these showed any propagative characteristics. mode 2 then appears to be a standing wave mode in the basin.
To understand how the three modes contribute to the annual cycle of SSHa in the basin, especially with regard to the formation and decay of the anticyclonic gyre, the monthly mean value and standard deviation of each mode and of the SSHa were calculated and shown in Fig. 12a and Fig. 12b, respectively. The monthly raw SSHa mainly followed the cycle of mode 1 and exhibited very close amplitudes. The mode 2 monthly mean exhibited a biyearly pattern with two maxima in February–April and September–October, and two minima in June–July and December–January. The first maximum reduces the absolute value of the negative SSHa while the second maximum enhanced the anticyclonic circulation in summer. The biyearly cycle of mode 2 was consistent with that of the dynamic height anomaly integrated between depths of 1500 and 50 m by M2019. It is therefore reasonable to assume that mode 2 represents some of the baroclinic changes in the Kuril Basin described by M2019. As these baroclinic variations include highly nonlinear processes, and due to the failure to reproduce the same mode 2 pattern when the CEOF was conducted over shorter periods of time (1993–2005, 1999–2011, and 2005–17), the dynamics of mode 2 will not be discussed further in this study.
The mode 3 monthly mean was lower than the two other modes, never exceeding ±1 cm, but its standard deviation was the highest among the three modes, consistent with the propagative nature of alternating negative and positive anomalies. Therefore, while mode 2 significantly enhances the anticyclonic circulation in late summer to early fall, mode 3 may have a near-zero contribution on average. Its high standard deviation means that it should episodically affect the SSHa field, though.
b. Mode 3: Rossby wave propagation determined from event analysis
The validity of the interpretation of mode 3 crucially depends on the robustness of this mode as a propagating mode. To verify the veracity of mode 3, we conducted a CEOF analysis for three 12-yr time periods (1993–2005, 1999–2011, and 2005–17). For each period, the CEOF output consistently reproduced a propagating mode (the third-highest energy mode for the first period, second-highest energy mode for the latter two periods) with properties in term of spatial and temporal amplitude and phase very close to those shown in Fig. 10. Hovmöller diagram of the propagating mode for each of the three periods (not shown) also yielded results similar to those displayed in Fig. 11. Mode 3 was therefore deemed valid, which allows for its interpretation as a physical phenomenon. To evaluate the mode 3 properties, we first selected all possible events manually from the mode’s 10-day interval SSHa field. Strict criteria were then used to extract a qualified event, i.e., an anomaly, lasting for at least 9 consecutive time steps (80 days), with an amplitude consistently exceeding the maximum standard deviation of the mode within the basin (7 cm). Once an event was detected, its properties were recorded until the event vanished, i.e., no significant amplitude coherent with the previous time step was detected in the basin. The properties retained were the starting and ending time, duration, distance traveled, average phase speed, wavelength, and maximum amplitude. In all, between June 1993 and June 2018, 12 events were thus detected, and their characteristics are summarized and averaged in Table 1. The duration-weighted averages are also indicated in Table 1, so the influence of longer events can be better accounted for. The weighted average mode 3 event has a wavelength of 255 ±6 km oriented toward 249° ± 2° (not shown in Table 1), a maximum amplitude (in absolute value) of 15.5 ± 2.9 cm, a duration of 203 ± 82 days, and travels 121 ± 29 km at a phase speed of 0.70 ± 0.21 km day−1. The averaged maximum amplitude of the mode 3 events was significantly larger than the averaged maximum amplitude of mode 1 in the Kuril Basin, which was 8 ± 3 cm.
The propagation characteristics of the mode 3 properties suggest that these eddies could be part of a train of Rossby waves, and an analysis of these properties with regard to the Rossby wave dispersion relation was conducted based on all the events. The zonal and meridional wavenumbers are determined to be kobs = −2.3 × 10−5 and lobs = −0.9 × 10−5 from the events averaged properties. The westward propagation speed was determined from the weighted averaged zonal phase speed: Cx_obs = −0.79 km day−1 (0.0091 m s−1). The properties of Rossby waves were calculated for both barotropic and baroclinic cases and compared with the characteristics of the mode 3 events. The dispersion relation for a barotropic Rossby wave is
where ω is the wave frequency, β0 = 1.6 × 10−11 m−1 s−1 at 47°N, and k and l are the respective zonal and meridional wavenumbers. Parameter R = 1661 km is the external radius of deformation (obtained for a depth H of 3200 m).
The dispersion relation for a baroclinic Rossby wave in a stratified structure is
where λp is the pth eigenvalue solving the vertical structure eigenproblem set as
Here, f0 is the reference Coriolis parameter, N(z) represents the buoyancy frequency profile, and Φ is the vertical mode structure (eigenvector). The N profile was obtained from a yearly climatology of hydrographic properties in the Kuril Basin (following M2019) and is shown in Fig. 13 (right panel) together with the first (p = 1) and second (p = 2) vertical mode structure (left panel). The zonal phase speed was then calculated via Cx = ω/k. Substituting k and l with kobs and lobs, and solving (5b) using the profile of N (Fig. 13) the theoretical properties of the first and second baroclinic mode Rossby waves were calculated and listed in Table 2, together with those of the barotropic mode.
The properties of the first baroclinic mode Rossby wave were of the same order as the mode 3 events properties (Table 2), but present a zonal phase speed half that of mode 3 (−0.40 vs −0.79 km day−1) and a period about twice as large as mode 3 (712 vs 382 days). If the observed CEOF mode 3 is due to first baroclinic Rossby waves, then the energy propagation direction (279°) would indicate that the wave energy originates from the Kuril Straits and propagates toward the basin interior. This suggests that the Rossby waves would be excited from the Kuril Straits. One possible energy source is the baroclinic instability caused by the intense tidal mixing near the straits (Nakamura and Awaji 2004; Ohshima et al. 2005). Although a first baroclinic Rossby wave may be a possible interpretation of the CEOF mode 3, the relatively large differences in phase speed and period cannot be overlooked and call for an alternative interpretation to be investigated. This is presented in the following section.
c. Interpretation as Rossby normal modes
The amplitude of mode 3 is strictly confined to the Kuril Basin, and fits well with the geometry formed by the 3200 m isobath (Fig. 10a). Within these limits, mode 3 accounts for 17% of the total variance and has the second-highest variance after mode 1 (19%). Mode 3 represents two maximum cores evenly separated as a dipole-like feature. These characteristics along with its westward propagation suggest the excitation of Rossby normal modes in an enclosed basin (Fig. 11c). Our subsequent analysis examined this possibility. The properties of such modes (frequency, wavenumber, phase speed, etc.) depend on the characteristics of the basin rather than the forcing mechanism. The theoretical frame for barotropic Rossby normal modes in a tilted basin was first established by Longuet-Higgins (1964), and the solution for wind-forced baroclinic waves in a tilted basin was presented by LaCasce and Pedlosky (2002). The solution for a barotropic Rossby normal mode was first evaluated. The approximate solution for the free-oscillating barotropic Rossby normal mode in a rectangular tilted basin is (see the appendix for a full derivation):
where L and l are the length and width of the basin, respectively; Amn is the wave amplitude; α is the angle between the zonal direction and the orientation of the basin; σmn is the frequency of the wave; θ is an arbitrary phase lag; x and y are the plane coordinates; and m and n represent the modes in the x and y direction, respectively. The wave frequency for a barotropic wave is represented by
and the zonal phase speed by
Equation (6) demonstrates that Rossby normal modes are characterized by a stationary envelope (the two sines) whose shape depends only on the value of the modes (n and m), modulated by a westward-propagating carrier wave (the cosine portion) whose properties depend solely on the wave frequency and the orientation of the basin.
The characteristics of the Rossby normal modes for m = 1 and 2 and n = 1 can be obtained by idealizing the Kuril Basin as a rectangle of dimensions L = 290 km and l = 157 km, titled by an angle α = 27° to the zonal direction, with R = 1661 km and λ1 = 3.1 × 10−9 (estimated in section 4a). These results can then be compared to the basic properties of the CEOF mode 3. The barotropic Rossby normal mode wavelength, and period listed in Table 2 suggest that the properties of the barotropic Rossby normal modes, rather than the baroclinic Rossby waves are better matched with the mode 3 events’ characteristics. The amplitude of CEOF mode 3 (Fig. 10a) shows a structure similar to both the dipole envelope of M21 (m = 2, n = 1; Fig. 14d) and the M11 envelope (Fig. 14a). The possible influence of M21 is also suggested by the Hovmöller diagram where lower SSHa amplitudes were consistently seen at a distance of 200 km (Figs. 11a,c). We therefore surmise that the propagating events observed in Fig. 11 may be a combination of Rossby normal modes M11 and M21. Here, we fit a M11 + M21 model to the deseasoned SSHa (dSSHa) data during each of the event listed in Table 1. This is because the dSSHa contains the full spectrum of energy of the propagating features, while the CEOF truncates some energy out of the main propagation mode. For each separate event, the least squares fitting problem is set as
with φmn = sin(mπx/L) sin(nπy/l).
Here, C1, C2, C3, and C4 is the set of four coefficients that minimizes the sum of the errors ε(x, y, t). The coefficients C1 and C3 respectively stand for the amplitude A11 and A21, while C2 and C4 represent the phase lags θ11 and θ21. Besides these four coefficients, all terms are fixed and determined by the dimensions of our idealized basin. For the calculation of the fitting we use t = 0, 10, 20, …, n days, with n corresponding to the duration of each event (Table 1) for events shorter than 140 days, and n = 140 for longer events. Setting n to a larger number leads to poorer fitting as dSSHa is likely to be affected by other modes of variability at longer time scales.
The results of the least squares fitting experiment for each event are listed in Table 3, and yield an event-averaged adjusted coefficient of determination R2 = 0.42. This value is significant, especially considering that dSSHa is also affected by other modes of variability. To test the significance of this result, we also conducted the fitting over 20 randomly selected periods during which no propagating event is observed. On average, we obtained a much lower R2 (=0.14). Further, we repeated the least squares fitting experiment for the plane Rossby wave model (with amplitude, zonal and meridional wavenumbers, and phase lag as the four coefficients), yielding an event-averaged coefficient of determination R2RWbt = 0.21 for the barotropic case, and R2RWbc = 0.31 for the baroclinic case. These lower values demonstrate that the barotropic Rossby normal mode rather than the plane Rossby wave is the most likely process explaining the existence of westward-propagating events in the Kuril Basin. The larger amplitude of M11 over M21 (Table 3) suggests that the former dominates the wave pattern. This assumption is confirmed by the results of an experiment where a model including only M11 is fitted to the data, which yields on average . The dSSHa during five time steps of event 6 is shown in Figs. 15a–e, together with its fitted barotropic Rossby normal mode counterpart in Figs. 15f–j. At day 10 (D10), two large eddies, one anticyclonic and one cyclonic, can be clearly seen from the dSSHa (Fig. 15a). As the eddies progress westward toward the edge of the basin (3200 m isobath), the amplitude of the westernmost eddy gradually reduced, and another cyclonic eddy appeared to the northeastern side of the basin (Figs. 15b–d). The fitted model reproduces well all these steps in term of amplitude, size of the eddies, and propagation speed (Figs. 15f–j). The propagation direction is somewhat different, with the dSSHa showing isolines slanted across the basin direction, whereas the theory requires isolines to be oriented along the meridional direction.
Besides the barotropic case, we also examined the possibility of a baroclinic Rossby normal mode in the basin. We first considered the case with the boundary conditions ϕ = 0 (ϕ is the spatial component of the streamfunction φ, see appendix for details). This solution does not conserve mass in the baroclinic case, and is therefore incorrect for a fully enclosed basin. However, it remains an acceptable approximation when considering an inviscid fluid (Flierl 1977), or that the Kuril Basin is not fully enclosed in reality. If ϕ = 0 were applied, the baroclinic Rossby normal mode would have properties with wavelengths ~100 km, and zonal phase speeds ~−0.20 m s−1 for both the M11 and M21 waves. These properties differ largely from the mode 3 and dSSHa characteristics.
We further considered the case ϕ = Γ(t), where Γ is a time varying nonzero value, constant all along the boundary. This boundary condition is part of a correct solution which conserves mass (LaCasce 2000). When the boundary is allowed to oscillate, low frequency, large scale modes (instead of Rossby normal modes) are selected through dissipation (Cessi and Primeau 2001). These modes propagate at a zonal phase speed close to the long Rossby wave speed, CR = −β0R2. At 47°N, CR = −0.43 km day−1, which is similar to that of the first baroclinic Rossby wave. The faster phase speed of the events observed from the dSSHa may then be explained by the superposition of such low-frequency free mode with wind-forced modes (LaCasce 2000).
To further examine whether barotropic or baroclinic waves better represent the westward propagating events, the actual setting of the Kuril Basin needs to be considered. The Rossby normal mode model assumes that the Kuril Basin is enclosed by lateral boundaries (walls) on all sides. In reality, while the western and southern boundaries of the basin can be considered as steep walls fitting our idealized model, the northern boundary has a gentler slope, as can be seen from the distance separating the 3200 m isobath from the 1500 m isobath (Fig. 15), and no shallow topography exists in the vicinity of this boundary. This implies that only the barotropic Rossby normal mode may have its energy well confined within the limits of the deepest bathymetry as shown in Fig. 10a. In contrast, the baroclinic waves present a large amplitude only within the upper 1500 m (Fig. 13) and should have some energy leaking northward well beyond the limits of the 3200 m isobath. The confinement of the mode’s energy within the 3200 m isobath limits, and the good match in properties displayed by the barotropic Rossby normal modes imply that the westward-propagating events are better represented by the barotropic case.
The solution of Rossby normal modes requires an exactly westward propagation, as seen in Figs. 15f–j, and thus α in (6) and (9) is equivalent to the orientation (tilting angle) of the basin to the horizontal. However, the dSSHa tends to show a propagation direction along the direction of the basin (Figs. 15a–e). To confirm this discrepancy, another least squares fitting experiment was conducted by slightly modifying (9). For this experiment, we relaxed the tilting angle condition by making α an additional variable in the least squares problem (i.e., α becomes coefficient C5). This experiment yielded a higher averaged coefficient of determination , where the averaged tilting angle is 6°. On the rotated domain, a tilting angle closer to zero means that the wave propagation is more along the basin direction. The discrepancy between actual and theoretical propagation direction could be due to the possible influence of a topographic β on the realistic situation, which is not represented in our idealized case. We acknowledge here the limitation of our model which does not enable us to evaluate the influence of a more complex topography on the wave propagation. More strictly, whether Rossby normal modes can be excited or not in such an imperfect condition should be examined with theoretical and modeling approaches, which remains a topic for future investigations.
d. Possible effect of the 18.6-yr cycle on the SSHa field
No significant mode 3 events occurred during a period from 1996 to 2003 (Table 1) whereas numerous long events took place before and after this period. These interannual variations of mode 3 may be related to the 18.6-yr tidal cycle proposed by Osafune and Yasuda (2006). In Fig. 10c, the lowest tidal amplitude between 1996 and 2003 at Abashiri, Hokkaido, coincides with the absence of events occurring between 1996 and 2003. In contrast, the long, high-amplitude, long-distance events occurring during 1993–94, and 2003–10 match the high-amplitude periods of the 18.6-yr tidal cycle (Fig. 10c). However, the large number of long events in 2014–17, does not match this tidal cycle. More intense tidal current and mixing near the Kuril Straits might then affect the generation of baroclinic instabilities, and hence, westward-propagating eddies. Based on this assumption, we went back to the raw SSHa data and calculated the eddy kinetic energy (EKE) from the SSHa-derived surface velocity during the two periods. As the spectrum of eddy energy likely spreads onto several CEOF modes, the analysis was conducted based on the raw SSHa rather than the mode 3 data. The high period was defined as the data prior to spring 1994 and from summer 2003 to fall 2012, while the low period encompasses spring 1994 to summer 2003 and data after fall 2012. The EKE during the two periods differs notably around the region of the Bussol Strait, which exhibited a much larger EKE level during the high portion of the 18.6-yr period (Fig. 16). This suggests that tidal mixing significantly affects the characteristics of the SSHa field in the Kuril Straits region. As the available SSHa data only encompasses one full nodal-tide cycle, no definite conclusion on the effects of the 18.6-yr tide on the SSHa or generation of Rossby waves can be drawn. However, the mode 3 statistics and the EKE data suggest that this tidal cycle may play a significant role.
The temporal variability in the circulation in and around the Kuril Basin was studied via an analysis of satellite-derived SSHa. A dedicated processing of along-track SSHa data was conducted to avoid cross-basin contamination and mitigate the impact of sea ice on data quality. The use of correlation length and time scales adapted to each grid point during the mapping process enabled the mapping error to be reduced by 77% for periods without sea ice and 57% for periods with sea ice cover. The total error, which included the mapping error and the error induced by the time interpolation of missing data points, was kept below 20% of the signal variance in most parts of the Kuril Basin and never exceeded 50% even in winter in the basin, whereas it was well below 10% outside of winter. The improvements in the quality of the SSHa dataset allowed for a detailed analysis of these data to be conducted.
The CEOF analysis of 25 years of SSHa data from 1993 to 2018 revealed that the first three CEOF modes accounted for 55% of the total SSHa variance. Mode 1 was a seasonal cycle caused by variations in the alongshore wind stress in the Sea of Okhotsk. It manifested as a maximum SSHa in the shelf regions in late fall and winter and a minimum in late spring and summer, consistent with the cycle of an ATW. The mode 1 SSHa in the Kuril Basin was characterized by negative values from January to June, and positive values from July to December. Interannual variability in the wind stress or ATW is crucial for changing the SSHa in the Kuril Basin and on the shelf as it directly affects the mode 1 amplitude.
Mode 2 consisted of a series of standing waves (of similar signs) in the Kuril Basin. Its mean monthly cycle was well defined. This cycle enhanced the anticyclonic circulation in September–October, and was consistent with the subsurface (50–1500 m) cycle of baroclinic anomaly determined by M2019. It may therefore represent the formation of the baroclinic anticyclonic circulation in the Kuril Basin at the subsurface, which is enhanced in late summer and decays in fall.
Mode 3 was characterized by southwestward-propagating anomalies in the Kuril Basin which were clearly visible from the Hovmöller diagram of the deseasoned SSHa data. A total of 12 separate, long (>80 days) propagating events were found to occur in the basin from 1993 to 2018. An event analysis revealed that, on average, these anomalies lasted for approximately half a year (187 days), had a wavelength of 255 km, traveled about 120 km at a phase speed of 0.70 km day−1 (zonal phase speed = −0.79 km day−1), and had a period of 365 days. The first baroclinic Rossby waves were first considered as the mechanism explaining these events. The mode 3 properties were of the same order but somewhat different from first baroclinic Rossby waves (plane waves), whose phase speed was 0.33 km day−1 (zonal phase speed = −0.37 km day−1), and period was 712 days.
We propose that mode 3 can be best interpreted as a barotropic Rossby normal modes, whose characteristics depend on the basin properties. Properties of Rossby normal modes in a tilted rectangular basin fitted to the deep Kuril Basin were estimated for the M11 and M21 waves. Rossby normal modes M11 and M21, respectively had zonal phase speeds of −1.29 and −0.77 km day−1, wavelengths of 275 and 212 km, and periods of 214 and 277 days. As the energy of the lower CEOF modes may be truncated by the CEOF process, we used the deseasoned SSHa (dSSHa) to further analyze the data. Both the plane Rossby wave and the M11 + M21 Rossby normal mode model were least-squares fitted onto the dSSHa data, and the Rossby normal mode model was found to be more consistent with the dSSHa observations. One possible cause for the resonance of a basin leading to the appearance of Rossby normal mode is wind stress variability, such as in the case observed by Warren et al. (2002) in the Mascarene Basin, where bimonthly current undulations were triggered by intraseasonal variations in the wind stress curl forcing. Another possible cause for the resonance of Rossby normal modes could be variability in water exchange between the North Pacific Ocean and the Sea of Okhotsk, which is forced by the North Pacific wind stress variability (Ohshima et al. 2010). The generation of Rossby normal modes warrants further study.
The Kuril Basin is located west of the Kuril Straits, where water exchange with the North Pacific, strong tidal mixing, and the associated baroclinic instability could generate disturbances or eddies. Such disturbances or eddies can propagate westward to the Kuril Basin via β effect or Rossby waves. In addition, the geometry of the deep Kuril Basin can potentially provide a resonant occurrence of barotropic Rossby normal modes, M11 and M21. Based on these conditions, we characterize the Kuril Basin as an area with high eddy activity, and these eddies play key roles in the mixing of different water masses in the basin to form the OSIW (M2019). The CEOF analysis, Rossby waves, and Rossby normal modes used in this study are all based on a linear framework. The relatively high amplitude of CEOF mode 2 and mode 3 implies that nonlinearity in the eddies or Rossby waves cannot be ignored. Further, seasonal changes in the anticyclonic circulation in the western Kuril Basin, which is related to the migration of high sea surface height from the shelf region near the Kuril Islands (M2019), might interact nonlinearly with the Rossby waves. This is, therefore, a limitation to our linear framework approach. Mode 1 is influenced by interannual variations in the wind stress over the Sea of Okhotsk, and mode 3 is likely subjected to various sources of interannual and multidecadal variability. How the variabilities described in this study affect the properties and overturning of the North Pacific via changes in the Kuril Basin hydrography merits further study.
This work is supported by Grants-in-Aid for Scientific Research 17H01157 from the Ministry of Education, Science, Sports, and Culture of Japan. This study was conducted using E.U. Copernicus Marine Service Information. Data analyses were conducted using the Pan-Okhotsk Information System of Hokkaido University. Discussions with Drs. Genta Mizuta, Humio Mitsudera, and Naoto Ebuchi greatly improved this study. We would like to extend our thanks to two anonymous reviewers and the editor Dr. Joseph LaCasce for their valuable comments which contributed to significant improvements of this work.
Derivation of Rossby normal modes Fitted to the Kuril Basin
The solution described below starts from the quasigeostrophic potential vorticity equation (Pedlosky 1987) in a tilted basin (LaCasce and Pedlosky 2002), and the derivation then follows Xie et al. (2018). In the following, the basin with a flat bottom of 3200 m has a length L = 300 km and a width l = 160 km, and is tilted at an angle α = 27° to the zonal direction (α is positive counterclockwise). The coordinates x and y are transformed, following LaCasce and Pedlosky (2002), so that the axes are not oriented toward the horizontal and vertical directions, but toward α and α + π/2, respectively.
The barotropic quasigeostrophic potential vorticity equation in a tilted basin is
where ψ(x, y, t) is a streamfunction representing the oscillation of a free surface; β = ∂f/∂y is the latitudinal gradient of the Coriolis parameter f and is a constant in a limited geographical domain; and R is the external radius of deformation. Note that in the transformed coordinate, β is now tilted by the angle α. Contrary to the wind-forced and damped wave case, the right-hand side of (A1) equals zero, as we investigate the free-wave case here. The streamfunction can be further decomposed as
It follows that
where σ is the frequency of the system.
The spatial function ϕ is further decomposed into
To solve (A1), φ(x, y) must satisfy
with γ2 = (β2/4σ2) − (1/R2).
For the barotropic case, the boundary conditions are ϕ = 0 at x = 0, L and y = 0 and l. The solution to (A5) is then of the form
Equation (A9) shows that the variations of the streamfunction ψ depend on a westward-propagating wave [first term on the RHS of (A9)] whose properties depend on the wave frequency and the inclination of the basin, which modulates a stationary envelope (A7) depending solely on the zonal and meridional mode number.