**1. Introduction**

Nonlinear internal waves (NLIWs) are ubiquitous in stratified seas and are accompanied by isopycnal fluctuations with a sharp vertical density gradient. They play an important role in underwater acoustics, regional circulation, local biogeochemistry, and energetics, mostly via vertical mixing in the stages of generation, propagation, evolution, and dissipation. NLIWs affect the transportation of momentum, heat, and energy via turbulent dissipation and mixing [1–5]. Marine ecosystems are significantly influenced by vertical nutrient supply, chlorophyll bloom, and biological redistribution, which can be modulated by NLIWs [6–9]. The NLIWs drive sediment resuspension and transportation; thus, they affect marine geophysics and underwater acoustics [10–15]. Vertical isopycnal displacements, which allow the wave amplitude to be defined, and propagation speed and direction, are fundamental parameters of NLIWs that are useful, but cannot be directly measured from in situ sampling, for a clear understanding of their generation, propagation, evolution, and dissipation. Estimating the propagation speed and direction can be important for assessing regional ocean circulation, biogeochemical cycles, energetics, underwater acoustics, and the dynamics of NLIWs.

Methods to estimate the propagation speed and direction have been suggested but are mostly limited by sampling strategies that have not yet been validated. The most common method using multiple moorings aligned in the propagation direction of NLIWs

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**Citation:** Lee, S.-W.; Nam, S. Estimation of Propagation Speed and Direction of Nonlinear Internal Waves from Underway and Moored Measurements. *J. Mar. Sci. Eng.* **2021**, *9*, 1089. https://doi.org/10.3390/ jmse9101089

Academic Editor: Shuqun Cai

Received: 1 September 2021 Accepted: 2 October 2021 Published: 6 October 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

aims to divide the distance between the mooring locations by the arrival time differences [16–22]. However, it is not practical to deploy many moorings along the ray of NLIWs, particularly where the continental shelf is wide and multiple NLIWs are generated from multiple sources with different unknown propagation directions. Another method is to use the principal direction of the wave-induced horizontal velocity and the temporal difference of enhanced echo intensity from acoustic Doppler current profiler (ADCP) measurements [23–27]. This method is useful, but not very practical, as extracting the propagation speed and direction is not straightforward. Using remote sensors, such as synthetic aperture radar (SAR) and spectroradiometer, the propagation speed and direction can be estimated from the horizontal curvature of satellite images [28–32]; however, the limited spatiotemporal satellite sampling from polar orbits does not allow NLIWs to be easily detected. Therefore, it is necessary to develop a method to estimate the propagation speed and direction of NLIWs from widely used ship-based in situ measurements. mon method using multiple moorings aligned in the propagation direction of NLIWs aims to divide the distance between the mooring locations by the arrival time differences [16–22]. However, it is not practical to deploy many moorings along the ray of NLIWs, particularly where the continental shelf is wide and multiple NLIWs are generated from multiple sources with different unknown propagation directions. Another method is to use the principal direction of the wave‐induced horizontal velocity and the temporal dif‐ ference of enhanced echo intensity from acoustic Doppler current profiler (ADCP) meas‐ urements [23–27]. This method is useful, but not very practical, as extracting the propaga‐ tion speed and direction is not straightforward. Using remote sensors, such as synthetic aperture radar (SAR) and spectroradiometer, the propagation speed and direction can be estimated from the horizontal curvature of satellite images [28–32]; however, the limited spatiotemporal satellite sampling from polar orbits does not allow NLIWs to be easily detected. Therefore, it is necessary to develop a method to estimate the propagation speed and direction of NLIWs from widely used ship‐based in situ measurements.

Methods to estimate the propagation speed and direction have been suggested but are mostly limited by sampling strategies that have not yet been validated. The most com‐

In the northern East China Sea (ECS), NLIWs are mainly formed by strong tidal forces that interact with bathymetric features. NLIWs in this region have been observed in association with strong semidiurnal internal tides over slope areas in the southern and southeastern parts of the ECS [33] and local lee-wave generation by small islands and seamounts near Jeju Island and the Ieodo Ocean Research Station (IORS) in the northern ECS [34–36] (Figure 1). Unlike the typical setting where dominant first-mode NLIWs in a two-layered condition propagate from the shelf break towards the coast, high NLIW modes propagating in multiple directions from multiple sources have been identified in the northern ECS [34,37]. In the northern East China Sea (ECS), NLIWs are mainly formed by strong tidal forces that interact with bathymetric features. NLIWs in this region have been observed in asso‐ ciation with strong semidiurnal internal tides over slope areas in the southern and south‐ eastern parts of the ECS [33] and local lee‐wave generation by small islands and sea‐ mounts near Jeju Island and the Ieodo Ocean Research Station (IORS) in the northern ECS [34–36] (Figure 1). Unlike the typical setting where dominant first‐mode NLIWs in a two‐ layered condition propagate from the shelf break towards the coast, high NLIW modes propagating in multiple directions from multiple sources have been identified in the northern ECS [34,37].

*J. Mar. Sci. Eng.* **2021**, *9*, 1089 2 of 16

**Figure 1.** (**a**) Map showing the geographic region of this study for two areas of experiments (black dashed boxes), bathymetry (grey lines), and distribution of surface manifestation of NLIWs de‐ scribed by Alpers et al. [35] (blue lines) and Nam et al. [36] (green lines). Two stations conducting historical hydrographic data sampling for the National Institute of Fisheries Science (NIFS) used in this study are marked by red open circles. Zoomed‐in maps of the two areas of (**b**) Shallow‐water Acoustic Variability EXperiment 2015 (SAVEX15) and (**c**) Ieodo Ocean Research Station 2018 (IORS18). Locations of underway conductivity–temperature–depth (UCTD) data collection, moored **Figure 1.** (**a**) Map showing the geographic region of this study for two areas of experiments (black dashed boxes), bathymetry (grey lines), and distribution of surface manifestation of NLIWs described by Alpers et al. [35] (blue lines) and Nam et al. [36] (green lines). Two stations conducting historical hydrographic data sampling for the National Institute of Fisheries Science (NIFS) used in this study are marked by red open circles. Zoomed-in maps of the two areas of (**b**) Shallow-water Acoustic Variability EXperiment 2015 (SAVEX15) and (**c**) Ieodo Ocean Research Station 2018 (IORS18). Locations of underway conductivity–temperature–depth (UCTD) data collection, moored observations (VLA1 and VLA2), and Ieodo Ocean Research Station (IORS) are marked by blue open squares in (**b**,**c**), purple triangles in (**b**), and yellow stars in (**c**).

Herein, we present a new method for estimating the propagation speed and direction of NLIWs using both moored and underway measurements, and the results of applying the method to two cases of NLIWs observed in the northern ECS in May 2015 and August 2018 (Figure 1).
