Nonlinear internal wave spirals in the northern East China Sea

Oceanic internal waves are known to be important to the understanding of underwater acoustics, marine biogeochemistry, submarine navigation and engineering, and the Earth’s climate. In spite of the importance and increased knowledge of their ubiquity, the wave generation is still poorly understood in most parts of the world’s oceans. Here, we use satellite synthetic aperture radar images, in-situ observations, and numerical models to (1) show that wave energy (having relatively high amplitude) radiates from a shallow sill in the East China Sea in all directions, but with a significant time lag dependent on background conditions, (2) reveal that wave fronts are locally formed with often favorable conditions for re-initiation, and (3) demonstrate the resulting variety of wave patterns. These findings would be the case for any broad shelf having shallow sills with time-varying conditions, and therefore have significant implications on the redistribution of energy and materials in the global as well as regional ocean.

not been unraveled until the present study. Thus, here we present the results of an analysis on the NIW features in SAR images recently taken around the IORS (Fig. 1, Table 1) in accordance with the analysis of modeled conditions of regional circulation.

Results and Discussion
Regional ocean circulation model. Tidal oscillations (tidal elevation and tidal currents) in addition to regional circulation and their seasonal changes were reasonably simulated using a Regional Ocean Modeling System (ROMS) 25 , which provides a realistic background condition for NIWs -vertical density stratification and horizontal currents (tidal plus subtidal) at multiple layers of the IORS (validated with in-situ observations; see Methods section for details; e.g., Fig. 2). Two seasons -September 2014 and May to June 2015 -were selected based on when the SAR images were taken around the IORS. Specifications of SAR images including sea surface wind and wave conditions are provided in Table 2 (see Methods section for details). Modeled sea surface temperature and volume transport across the straits in the region had consistent features with observations of sea surface temperature 25 and volume transport 25 . Flow regime and directional wave evolution. The Froude number (Fr), used as the key parameter analyzed here to determine the timing of NIW generation, is given as the ratio between advective flow speed and free wave speed. To consider strong time-varying background horizontal flow (primarily tidal current here) as well as density stratification, the composite Froude number for water column can be defined as below 26 : where N defines the number of layers, and i are the advective flow speed and apparent free wave speed incorporating an effective Doppler shift due to the advective flow at each layer, respectively. ρ ρ ′ = ∆ g g / i i is reduced gravity determined by the density ratios between layers, ( ) /2 and h i is the thickness of each layer. The effective Doppler shift due to the background current field is considered by the Fr analysis, in other words, the flow is steady (c i = 0) and the Fr becomes unity (F i = 1) when the intrinsic phase speed and the opposite current speed match (zero apparent speed) 2,3 . Both the surface Fr (F i ) and the composite Fr for the water column (G) obtained from the model are of primary interest here to test consistency with the estimated timing of NIW generation prior to the time of each SAR observation.
Then, the distance D(θ, t) over which the NIWs propagate following generated (t = 0) in the direction θ was calculated by time-integrating the apparent free wave speed, which is the sum of the intrinsic speed and the directional component of the advective speed as a function of direction and time, as shown below: Here, θ 0 is set to the flow direction at the time when the free propagation of NIWs was favorably initiated (t = 0) to best fit the SAR observations. Surface-layer current u 1 , v 1 and density stratification such as reduced gravity (g′) to estimate the apparent free wave speed at the surface layer (c 1 ) were used to calculate the distance.  Fig. 1) are not interpreted as the ones locally formed at the sill but rather propagated from a remote source, i.e., somewhere east of the sill when considering the curvature. In contrast, the other NIW fronts observed in the SAR images #1, #3, #4, and #5 (features from ' A' to 'H' and from 'J' to 'M' in Fig. 1) are reasonably assumed to be locally generated at the sill when the latest flow regimes become near-critical (F i ∼ 1 and/or G ∼ 1) due to changes in advective flows and density stratification (Fig. 3a,c,d and e). The near-critical flow regime, however, does not mean there is no NIWs of remote source. There are other NIW features than the ones labeled in the SAR images ( Fig. 1) which may or may not be generated at remote locations. Our focus here is on the leading wave fronts labeled from ' A' to 'H' where all but 'I' are interpreted as the ones locally generated at the sill. After the Fr becomes small enough (subcritical) to allow the NIWs propagate (e.g., vertical arrows in Fig. 3a,c,d and e), the NIW fronts formed locally can be seen in the SAR images, experiencing severe refraction as apparent free wave speed (c i ) significantly varies in space and time (Fig. 4). Patterns of the NIW fronts are figured using the estimated directional distance D and compared to the SAR observations. Short arc-like patterns of NIW fronts in the SAR images #3 and #4 (features 'J' and 'K' in Fig. 1) are reproduced with NIW generations 1.72 and 1.78 hours prior to the SAR acquisition times respectively ( Table 1). Note that the initial (t = 0) directions of NIW propagation from the sill (origin) are θ 0 = 273.9 for image #3 and θ 0 = 327.2 for image #4, following the tidal flow (red vectors in Fig. 5b and c). The propagation of NIW fronts from the initial times to the SAR acquisition times (Table 1) rotating clockwise (from red to orange vectors in Fig. 5b and c) around and away from the sill yields the short arc-like patterns similar to the SAR observations (blue vs orange in Fig. 5b and c). This is consistent with results of the Fr number analysis, as Fr is close to unity (near-critical flow, e.g., F i ∼ 1 and/or G ∼ 1) with advection speed (U i ) comparable to the intrinsic free wave speed (c i − U i ) and decreases to yield subcritical condition (F i < 1 and G < 1) during the time of decelerating tidal flow (shaded in Fig. 3c and d).
Long and spiral patterns originating from the sill are found in SAR image #5. The features 'L' and 'M' appear to be linked to NIW generation 15.7 and 4.9 hours prior to the SAR image acquisition times, respectively (Table 1). Similarly, the flow was subcritical with decelerating advective flow and Fr is sufficiently less than unity (arrows in Fig. 3e). Interestingly, the Froude number at the surface layer (F i ) approached local minimum first at around 06:00 ('L') and once more at around 17:00 ('M') on June 12, 2015, re-initiating the NIW generation with an approximate semidiurnal period (between the first and second arrows in Fig. 3e). The resultant distance D of the NIW fronts, with θ 0 = 245.3 and θ 0 = 212.6 before and after the re-initiation, respectively (the latter is shown with a red arrow in Fig. 5d), yields two long spiral patterns where the one formed before the re-initiation was arbitrarily shifted in the southwest direction for better comparison to the SAR observations (cyan vs orange in Fig. 5d). Thus, the two long spiral patterns in SAR image #5 are successfully reproduced using the estimated D with the re-initiation due to changes in advective flow speed and stratification or Fr conditions (Fig. 5d).
Here, our focus is on the leading wave front yet a number of parallel waves in a packet are seen from multiple wave fronts. It is typical that such a nonlinear internal soliton disintegrates into rank-ordered multiple solitary waves (fission process) while propagating and evolving 18,27 . Changing stratification and bathymetry following the propagating and evolving NIWs account for the disintegration and fission processes [28][29][30] .
More similar to what is found in SAR images #3 and #4 (features 'J' and 'K' in Fig. 1) than those in the SAR image #5 ('L' and 'M'), multiple short arc-like or wedge patterns were observed in the SAR image #1 (features ' A' to 'H' in Fig. 1). Four pairs of wedge patterns (' A' and 'B' , 'C' and 'D' , 'E' and 'F' , and 'G' and 'H') are generated at intervals of quasi-semidiurnal periods with θ 0 ranging from 251.8 to 347.2 (Fig. 5a), where the latest one ('H') is shown in Fig. 5a (red arrow). Following the semidiurnal tidal cycle and diurnal inequality, the Fr reaches the minimum twice a day with a subcritical flow condition when the NIWs re-initiate and start to propagate (arrows in Fig. 5a). Thus, diurnal inequality in tidal current speed and consequent changes in Fr account for the wedge features found in the SAR image #1 (Fig. 1a,f, or pink in Fig. 5a), reproducing shorter features 'B' , 'D' , 'F' , and 'H' (cyan in Fig. 5a) and longer features ' A' , 'C' , 'E' , and 'G' (green in Fig. 5a). The longer features of the wedge patterns 'C' , 'E' , and 'G' are more comparable to the SAR observation when the intrinsic free wave speed or reduced gravity arbitrarily doubled (thick green shaded lines in Fig. 5a), implying that better agreements would be possible with advancements to future modeling skills. The time intervals between ' A'-'B' and 'C'-'D' , between 'C'-'D' and 'E'-'F' , and between 'E'-'F' and 'G'-'H' do not exactly match but close to semidiurnal lunar tidal cycle following the dominant tidal forcing. It is because the flow regime is not only function of the tidal current but also free wave speed (c i ) imposed by stratification, non-tidal component of U i , apparent free wave speed (c i − U i ), and consequently Fr condition. Note that the stratification may also vary significantly particularly when internal tides become significant (occasionally true but not always).

Conclusions
In summary, arc-like, wedge, and spiral patterns were found in the surface expression of oceanic NIWs in the vicinity of small topographic sills, IEODO, in the northern ECS in September 2014 and May to June 2015. Series of analyses of satellite SAR images taken in the region along with background conditions on horizontal flow and density stratification reproduced by the ROMS reveal that the NIWs were locally generated when the flow approached near-critical Froude number conditions (Fr is close to unity) and started to propagate when the flow became subcritical (Fr < 1), and that the NIW energies radiated from the sill in all directions but with a significant time lag primarily dependent on the conditions of tidal flow and stratification. The position of the NIW fronts as a function of direction and time is simply modeled and compared with the SAR observations, as determined by the time-varying advective flow speed (mostly semidiurnal tidal currents) and density stratification (intrinsic free wave speed). The modeled NIW fronts are surprisingly similar to the observations, providing all the short arc-like, wedge, and spiral patterns. How the NIW fronts generated at and radiated from the sill will ultimately dissipate in the ECS remains an open question. The abundance of topographic features like sills, islands, and shelf breaks in the ECS support more complex NIW processes including vigorous wave-wave interaction. These mechanisms, in addition to the one presented here, will be realized for any broad shelf having a shallow sill or island with strong time-varying and/or rotary (such as tidal) flows. The significance of this finding lies in the unprecedented observations of a new type (wedge and spiral patterns) of NIW fronts radiating from a shallow sill   Table 2. Specification of SAR images with sea surface wind and wave conditions. Vertical arrow with labels corresponding to the NIW fronts shown in Fig. 1f indicates when the free wave propagation was favorably initiated or NIW front is presumably formed to fit the SAR observations, which generally correspond to periods of small Froude number (G; thick solid black). Significant decreases in the Froude number to yield subcritical conditions are remarked with pink and orange shades. topography on the broad shelf, which has important implications for processes transporting and redistributing heat, momentum, energy, and materials in the global coastal ocean as well as the ECS.

Methods
The IEODO, also known as Socotra rock, is a submerged rock 4.6 m below mean sea level located in the northern ECS where an ocean research station, the IORS was built in 2003. Long and continuous time-series of oceanographic and meteorological parameters (wind speed and wave height) have been collected at the station since then. Here, we mainly used the time-series data collected in 2014 and 2015. Other data, such as vertical profiles of water temperature and salinity intermittently collected (with use of SBE Conductivity-Temperature-Depth; CTD) flow velocities at multiple levels (with use of the bottom-mounted TRDI acoustic Doppler current profiler (ADCP) near the station; no data are available for the periods of 2014 and 2015 but 2008), sea surface temperature remotely sensed from the station (with use of thermal infrared radiometer), were used to verify general characteristics of the vertical density stratification, free wave speed, and characteristics of tidal currents as well as the internal wave fronts. Five SAR images taken in the vicinity of IORS in September 2014 and May to June 2015 were used to find frontal features associated with NIWs. All SAR images were firstly geolocated using the ancillary information provided with the SAR data and a small translation was applied to coincide the IORS signature (bright point) between SAR images. The NIW fronts were delineated along the maximum gradient of leading NIW seen in the SAR images. Specification of the SAR images such as frequency band, orbit direction, polarization, incident angle, and sea surface wind and wave conditions are listed in Table 2. Time series data collected at the IORS were used to extract the sea surface wind and wave conditions at the SAR image times.
A research cruise, named the shallow-water acoustic variability experiment (SAVEX-15) was conducted using the Korean R/V Onnuri during 14-28 May 2015 in the northern ECS, ~50 km east of the IORS. Under the goal to obtain simultaneous oceanographic and underwater acoustic data appropriate for investigating the coupling of oceanography and underwater communications in the region 31 , the vessel mounted 150 kHz ADCP and both stationary and underway CTDs (with 26 and 1,062 casts respectively) were used to measure temporal and spatial variations of flow velocities at multiple depths (water depth is about 100 m) and density stratification, respectively. An abundance of isotherm or isopycnal undulations at semidiurnal and higher frequencies was confirmed from the CTD data. Horizontal currents at the upper (25 m) and lower (65 m) depths, which are dominated by semidiurnal tidal currents, observed from the ADCP during the SAVEX-15, were used to verify the reproduced currents (Fig. 2).
Regional scale circulations with background conditions (density stratification and horizontal currents) were simulated with use of the ROMS. The modeled area covers the Yellow Sea and the ECS (117.5-130.3°E, 24.8-41°N). Ocean circulation in the Yellow Sea and ECS was spun-up using open boundary conditions imposed with the data from the Northwest Pacific model 32 . The horizontal grid size is ~3 km and number of vertical levels are