*2.2. Methods*

2.2.1. Two-Layered KdV (Korteweg-de Vries) Theories

In classical KdV theory [49], a leading-order weak non-linearity and dispersion are competing but comparable to each other. For the two-layered KdV theory, the thicknesses (*h*1, *h*2) and densities (*ρ*1, *ρ*2) of the upper and lower layers can be used to estimate the parameters of mode-1 NLIWs, including linear phase speed *cKdV*.*<sup>l</sup>* , theoretical propagation speed *cKdV*.*iw*, characteristic width 2∆*KdV*.*iw*, nonlinear parameter *α*, and dispersion parameter *β*, yielding the wave equation as follows [14,50]:

$$\frac{\partial \eta}{\partial t} + c\_{\text{KdV}1} \frac{\partial \eta}{\partial \chi} + \alpha \eta \frac{\partial \eta}{\partial \chi} + \beta \frac{\partial^3 \eta}{\partial x^3} = 0,\tag{1}$$

where *η*, *t*, and *x* are the vertical displacement of the isopycnals (or isotherms), time, and horizontal coordinates, respectively. The *cKdV*.*<sup>l</sup>* , *α*, and *β* can be estimated using the density stratification parameters (*ρ*1, *ρ*2, *h*1, and *h*2) in a two-layered system as follows:

$$c\_{KdV.l} = \sqrt{g \frac{\rho\_2 - \rho\_1}{(\rho\_1 + |\rho\_2)/2} \frac{h\_1 h\_2}{(h\_1 + h\_2)}},\tag{2}$$

$$\alpha = \frac{3}{2} \frac{c\_{\rm KdV,l}}{h\_1 h\_2} \frac{\rho\_2 h\_1^2 - \rho\_1 h\_2^2}{\rho\_2 h\_1 + \rho\_1 h\_2},\tag{3}$$

$$\beta = \frac{c\_{KdV,l}h\_1h\_2}{6} \frac{\rho\_1h\_1 + [\rho\_2h\_2}{\rho\_2h\_1 + \rho\_1h\_2}.\tag{4}$$

Here, *g* is the gravity acceleration set to 9.80 m s−<sup>2</sup> . The thicknesses of the upper and lower layers (*h*<sup>1</sup> and *h*2) were determined based on the depth of the maximum density gradient from the density profiles obtained from the UCTD. The densities at the upper and lower layers (*ρ*<sup>1</sup> and *ρ*2, respectively) were determined as the minimum density within the upper layer and the maximum density within the lower layer, respectively. The solution of Equation (1) for the displacement *η*(*x*, *t*) yields the nonlinear soliton as follows:

$$\eta(\mathbf{x},t) = \eta\_0 \text{sech}^2\left(\frac{\mathbf{x} - \mathbf{c}\_{KdV.iw}t}{\Delta\_{KdV.iw}}\right). \tag{5}$$

Here, the theoretical propagation speed *cKdV*.*iw* and characteristic width 2∆*KdV*.*iw* were calculated from the linear phase speed *cKdV*. *<sup>l</sup>* and the amplitude (*η*0) of the vertical displacement of *η* are as follows:

$$\mathbf{c}\_{\rm KdV.inv} = \mathbf{c}\_{\rm KdV.I} + \frac{|\mathbf{a}|\eta\_0}{\mathbf{3}},\tag{6}$$

$$2\Delta\_{KdV.iw} = 2\left(\frac{12\beta}{|\alpha|\eta\_0}\right)^{1/2}.\tag{7}$$

By considering cubic nonlinearity, Equation (1) becomes as follows, yielding the extended KdV (eKdV) theory [51]:

$$\frac{\partial \eta}{\partial t} + c\_{\text{KdV},l} \frac{\partial \eta}{\partial \mathbf{x}} + a \eta \frac{\partial \eta}{\partial \mathbf{x}} + a\_1 \eta^2 \frac{\partial \eta}{\partial \mathbf{x}} + \beta \frac{\partial^3 \eta}{\partial \mathbf{x}^3} = 0. \tag{8}$$

Here, *α*<sup>1</sup> = 3*cKdV*.*<sup>l</sup>* (*h*1*h*2) 2 7 8 *ρ*2*h*<sup>1</sup> <sup>2</sup>−*ρ*1*h*<sup>2</sup> 2 *ρ*2*h*1+*ρ*1*h*<sup>2</sup> 2 − *ρ*2*h*<sup>1</sup> <sup>3</sup>+*ρ*1*h*<sup>2</sup> 3 *ρ*2*h*1+*ρ*1*h*<sup>2</sup> is a cubic nonlinear parameter in the two-layer system. The theoretical propagation speed *ceKdV*.*iw* and characteristic width 2∆*eKdV*.*iw* based on the eKdV theory in the two-layered system are as follows:

$$c\_{\epsilon KdV.iw} = c\_{KdV.l} + \frac{|\alpha|\eta\_0}{\mathfrak{3}} + \frac{\alpha\_1 \eta\_0}{\mathfrak{6}},\tag{9}$$

$$2\Delta\_{\epsilon KdV.iw} = 2\left(\frac{12\beta}{|\alpha|\eta\_0 + 0.5\alpha\_1\eta\_0^2}\right)^{1/2}.\tag{10}$$

#### 2.2.2. *Doppler Shift Method* 2.2.2. *Doppler Shift Method*

To estimate the propagation direction of NLIWs using the Doppler shift caused by propagating NLIWs observed from a moving ship, the theoretical propagation speed *cKdV*.*iw* and ship speed *vsh* were assumed to be constant during the measurement period, and the propagation direction was assumed to be orthogonal to the constant phase lines parallel to the wavefront lines (Figure 2a,b). As the estimated propagation direction *φds* is in reference to the ship course *φsh*, the apparent propagation speed *cap* can be represented as the difference between *cKdV*.*iw* and the ship speed in direction *θds* as follows: To estimate the propagation direction of NLIWs using the Doppler shift caused by propagating NLIWs observed from a moving ship, the theoretical propagation speed ௗ.௪ and ship speed ௦ were assumed to be constant during the measurement period, and the propagation direction was assumed to be orthogonal to the constant phase lines parallel to the wavefront lines (Figure 2a,b). As the estimated propagation direction ௗ௦ is in reference to the ship course ௦, the apparent propagation speed can be repre‐ sented as the difference between ௗ.௪ and the ship speed in direction ௗ௦ as follows:

*J. Mar. Sci. Eng.* **2021**, *9*, 1089 5 of 16

$$\mathcal{L}\_{ap} = \mathcal{c}\_{\text{KdV.inv}} - v\_{\text{sh}} \cos(\theta\_{\text{ds}}). \tag{11}$$

Here, *θds* = |*φsh* − *φds*| is the angular difference between the ship course and the propagation direction of the NLIWs. Because the Doppler-shifted apparent frequency *fap* or the inverse of the apparent period *Tap* can be represented by *cap* and *ciw* = *λKdV*.*iw fiw*, where *λKdV*.*iw* is the wavelength of the NLIWs and the Doppler equation *fap* = *fiw cap ciw* [52], the following equation can be used to estimate the *φds*: gation direction of the NLIWs. Because the Doppler‐shifted apparent frequency or the inverse of the apparent period can be represented by and ௪ ൌ ௗ.௪௪, where ௗ.௪ is the wavelength of the NLIWs and the Doppler equation ൌ ௪ ೌ ೢ [52], the following equation can be used to estimate the ௗ௦: ଵ ൌ ൌ ௪ ೌ ൌ ೌ

Here, ௗ௦ ൌ |௦ െ ௗ௦| is the angular difference between the ship course and the propa‐

$$\frac{1}{T\_{ap}} = f\_{ap} = f\_{iw} \frac{c\_{ap}}{c\_{iw}} = \frac{c\_{ap}}{\lambda\_{KdV.iw}}.\tag{12}$$

Here, *Tap* is determined from measurements (Table 1), while *λKdV*.*iw* is determined by the Cnoidal model [50] as ௗ.௪ ൌ 2∆ௗ.௪ሺሻ ൎ 3.7∆ௗ.௪, (13) where ሺሻ is a complete elliptic integral of the first kind and parameter is set to 0.5.

$$
\lambda\_{\text{KdV.iw}} = 2\Lambda\_{\text{KdV.iw}}\text{K}(s) \approx 3.7\Delta\_{\text{KdV.iw}}\tag{13}
$$

where *K*(*s*) is a complete elliptic integral of the first kind and parameter *s* is set to 0.5. Equation (11) can then be rewritten using Equation (12) as follows: ௗ.௪ െ ௦ cos|௦ െ ௗ௦| ൌ ௗ.௪. (14) Further, the ௗ௦ is obtained as follows:

has an angular ambiguity caused by the sign of the arccosine part. Thus, a physically rea‐

$$|\mathbf{c}\_{\rm KdV.iv} - \upsilon\_{\rm sh} \cos|\phi\_{\rm sh} - \phi\_{\rm ds}| = f\_{ap} \lambda\_{\rm KdV.iv}. \tag{14}$$

Further, the *φds* is obtained as follows: The propagation direction of the NLIWs estimated using the method described above

the Cnoidal model [50] as

$$\phi\_{\rm ds} = \phi\_{\rm sh} \pm \cos^{-1}\left(\frac{c\_{\rm KdV.iv} - f\_{\rm ap}\lambda\_{\rm KdV.iv}}{v\_{\rm sh}}\right). \tag{15}$$

**Figure 2.** Schematic description of (**a**) the definition of parameters and estimation of the NLIW propagating direction based on the (**b**) Doppler shift and (**c**) time lag. **Figure 2.** Schematic description of (**a**) the definition of parameters and estimation of the NLIW propagating direction based on the (**b**) Doppler shift and (**c**) time lag.

The propagation direction of the NLIWs estimated using the method described above has an angular ambiguity caused by the sign of the arccosine part. Thus, a physically reasonable direction between the two was selected. The *φds* and *φsh* are angles in degrees

measured counter-clockwise from the east (for example, 180◦ and 270◦ correspond to the westward and southward directions, respectively).
