Error Calculation of Large-Amplitude Internal Solitary Waves Within the Pycnocline Introduced by the Strong Stratification Approximation
https://doi.org/10.1007/s11804-023-00312-2
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Abstract
At present, studies on large-amplitude internal solitary waves mostly adopt strong stratification models, such as the twoand three-layer Miyata–Choi–Camassa (MCC) internal wave models, which omit the pycnocline or treat it as another fluid layer with a constant density. Because the pycnocline exists in real oceans and cannot be omitted sometimes, the computational error of a large-amplitude internal solitary wave within the pycnocline introduced by the strong stratification approximation is unclear. In this study, the two- and three-layer MCC internal wave models are used to calculate the wave profile and wave speed of large-amplitude internal solitary waves. By comparing these results with the results provided by the Dubreil–Jacotin–Long (DJL) equation, which accurately describes large-amplitude internal solitary waves in a continuous density stratification, the computational errors of large-amplitude internal solitary waves at different pycnocline depths introduced by the strong stratification approximation are assessed. Although the pycnocline thicknesses are relatively large (accounting for 8%–10% of the total water depth), the error is much smaller under the three-layer approximation than under the two-layer approximation.Article Highlights• The two- and three-layer MCC internal wave models and the DJL equation are used to study large-amplitude internal solitary waves.• The error on profiles and speed of the internal solitary waves introduced by the strong stratification approximation are obtained.• Three pycnocline thicknesses are considered to study the error of strong stratification approximation on describing internal solitary waves. -
1 Introduction
Internal waves always exist in density-stratified oceans. The South China Sea is an area of frequent large-amplitude internal waves. Huang et al. (2016) observed a large-amplitude internal solitary wave with an amplitude of 240 m in the South China Sea. Large-amplitude internal solitary waves pose a great threat to the safety of underwater structures. For example, the KRI Nanggala-402 submarine crashed during an exercise in 2021, causing the deaths of 53 sailors (Gong et al. 2022). The main reason for the crash is believed to be that the submarine encountered large internal waves. Therefore, great importance is associated with accurately describing the characteristics of large-amplitude internal waves.
For the internal wave problem, most studies neglect the thickness of the pycnocline and approximate it to a two-layer fluid problem. The two-layer Miyata–Choi–Camassa (MCC) (Miyata 1985, 1988; Choi and Camassa 1999) internal wave model is widely used to study large-amplitude internal solitary waves. In the two-layer MCC model, only the depth-averaged horizontal velocity of the particles in each layer is considered. Many studies have shown that the two-layer MCC model can accurately describe the large-amplitude internal waves in a two-layer fluid system for shallow configurations (h1/λ ≪ 1 and h2/λ ≪ 1, where h1 and h2 are the undisturbed thicknesses of the upper- and lower-fluid layers, respectively, and λ is the characteristic wavelength of the internal wave) (Choi and Camassa 1999; Camassa et al. 2006; Xie et al. 2010; Gao et al. 2012; Huang et al. 2013; la Forgia and Sciortino 2019; Du et al. 2019; Zou et al. 2020; Cui et al. 2021).
Compared with the two-layer strong stratification approximation, a more suitable approach is to approximate the practical fluid system as a three-layer fluid system where the pycnocline is treated as another fluid layer with a constant density (Jo and Choi 2014). Barros et al. (2020) derived a three-layer internal wave model in which a depth-averaged horizontal velocity is introduced for each layer. This model can be considered a generalized two-layer MCC model, i. e., a three-layer MCC model. Zhang et al. (2020) demonstrated that the three-layer MCC internal wave model can accurately describe the large-amplitude internal solitary waves in a three-layer fluid system for shallow configurations (h1/λ ≪ 1, h2/λ ≪ 1, and h3/λ ≪ 1, where h1, h2, and h3 are the undisturbed thicknesses of the upper-, middle-, and lower-fluid layers, respectively, and λ is the characteristic wavelength of the internal wave) by comparing with laboratory measurements.
When the pycnocline thickness is small or negligible, the two- and three-layer internal wave models can accurately describe large-amplitude internal waves (Grue et al. 1999; Huang et al. 2013; la Forgia and Sciortino 2021). However, pycnocline thickness in real oceans cannot be ignored sometimes. Under such conditions, using the strong stratification approximation to calculate large-amplitude internal solitary waves will introduce certain errors. Cheng and Hsu (2014) and Lu et al. (2021) noted that the internal wave speed decreases with increasing pycnocline thickness. Nevertheless, the computational error of a large-amplitude internal wave within the pycnocline introduced by the strong stratification approximation should be further studied.
In this paper, we apply the two- and three-layer MCC internal wave models to calculate the wave profile and wave speed of large-amplitude internal solitary waves. Meanwhile, we also use the open-source solver (Dunphy et al. 2011) to obtain the results of the Dubreil–Jacotin–Long (DJL) equation (Dubreil-Jacotin 1934; Long 1953), given that this equation is formally equivalent to the full set of Euler equations that can accurately describe the large-amplitude mode-1 internal solitary waves in continuous density stratification when the environment is unchanging (Stastna and Lamb 2002; Dunphy et al. 2011; Stastna and Lamb 2020). Through comparison, the errors introduced by the strong stratification approximation under different pycnocline thicknesses are obtained.
2 Theoretical models
In this section, we will briefly introduce the theoretical models that are applied in this paper for describing large-amplitude internal waves.
2.1 Two-layer MCC internal wave model
Figure 1 shows a sketch of an internal solitary wave in a two-layer fluid system.
Figure 1 Internal solitary wave in a two-layer fluid systemThe fluids are assumed to be inviscid, incompressible, and immiscible. The density and undisturbed thickness of each layer are ρi and hi, respectively, where the subscripts i = 1 and i = 2 represent the variables for the upper- and lower-fluid layers, respectively. A wave-coordinate system OXZ, which travels with the internal wave at the same speed c, is established. The OX-axis is set at the undisturbed interface and positive to the right, and OZ is positive up. The internal solitary wave crest is set at X = 0. The upper surface of the upper-fluid layer, the interface between the two-fluid layers, and the lower surface of the lower-fluid layer are expressed as Z = h1, Z = ζ(X), and Z = −h2, respectively. The local thicknesses of the upper- and lower-fluid layers are expressed as η1(X) and η2(X), respectively.
In the two-layer MCC internal wave model, for a given internal wave amplitude a, the internal wave speed c can be obtained as
$$ c=\sqrt{\frac{\left(h_1-a\right)\left(h_2+a\right)}{h_1 h_2-\left(c_0^2-g\right) a}} c_0 $$ (1) where g is the acceleration of gravity, and c0 is the linear long wave speed, given by
$$ c_0=\sqrt{\frac{g h_1 h_2\left(\rho_2-\rho_1\right)}{\rho_1 h_2+\rho_2 h_1}} $$ (2) Then, the wave profile can be obtained by solving the following equation:
$$ \left(\zeta_X\right)^2-\frac{3 \zeta^2\left[\rho_1 c^2 \eta_2+\rho_2 c^2 \eta_1-g\left(\rho_2-\rho_1\right) \eta_1 \eta_2\right]}{\rho_1 c^2 h_1^2 \eta_2+\rho_2 c^2 h_2^2 \eta_1}=0 $$ (3) where
$$ \eta_1=h_1-\zeta $$ (4) $$ \eta_2=h_2+\zeta $$ (5) A detailed derivation of the two-layer MCC internal wave model and the related numerical algorithm is provided by Choi and Camassa (1999).
2.2 Three-layer MCC internal wave model
Figure 2 shows a sketch of an internal solitary wave in a three-layer fluid system.
Figure 2 Internal solitary wave in a three-layer fluid systemThe assumptions of the fluid layers and the establishment of the coordinate axes are similar to those in Section 2.1. The upper surface of the upper-fluid layer, the interface between the upper- and middle-fluid layers, the interface between the middle- and lower-fluid layers, and the lower surface of the lower-fluid layer are expressed as Z = h1, Z = ζ1(X), Z = ζ2(X) − h2, and Z = −(h2 + h3), respectively. The local thicknesses of the upper-, middle-, and lower-fluid layers are expressed as η1(X), η2(X), and η3(X), respectively.
In the three-layer MCC internal wave model, the wave profile and wave speed can be obtained by solving the following equations (Barros et al. 2020):
$$ \begin{aligned} & c^2\left\{\frac{1}{3}\left(\frac{h_1^2}{\eta_1}+\frac{h_2^2}{\eta_2}\right) \zeta_{1 X X}+\frac{1}{6} \frac{h_2^2}{\eta_2} \zeta_{2 X X}+\frac{1}{6}\left(\frac{h_1^2}{\eta_1^2}-\frac{h_2^2}{\eta_2^2}\right) \zeta_{1 X}^2\right. \\ & \left.+\frac{1}{3} \frac{h_2^2}{\eta_2^2} \zeta_{1 X} \zeta_{2 X}+\frac{1}{3} \frac{h_2^2}{\eta_2^2} \zeta_{2 X}^2\right\}=\frac{1}{2} c^2\left(\frac{h_1^2}{\eta_1^2}-\frac{h_2^2}{\eta_2^2}\right)-g\left(1-\frac{\rho_1}{\rho_2}\right) \zeta_1 \end{aligned} $$ (6) $$ \begin{aligned} & c^2\left\{\frac{1}{6} \frac{h_2^2}{\eta_2} \zeta_{1 X X}+\frac{1}{3}\left(\frac{h_2^2}{\eta_2}+\frac{h_3^2}{\eta_3}\right) \zeta_{2 X X}-\frac{1}{3} \frac{h_2^2}{\eta_2^2} \zeta_{1 X}^2-\frac{1}{3} \frac{h_2^2}{\eta_2^2} \zeta_{1 X} \zeta_{2 X}\right. \\ & \left.+\frac{1}{6}\left(\frac{h_2^2}{\eta_2^2}-\frac{h_3^2}{\eta_3^2}\right) \zeta_{2 X}^2\right\}=\frac{1}{2} c^2\left(\frac{h_2^2}{\eta_2^2}-\frac{h_3^2}{\eta_3^2}\right)-g\left(\frac{\rho_3}{\rho_2}-1\right) \zeta_2 \end{aligned} $$ (7) where
$$ \eta_1=h_1-\zeta_1 $$ (8) $$ \eta_2=\zeta_1-\zeta_2+h_2 $$ (9) $$\eta_3=h_3+\zeta_2 $$ (10) For a detailed derivation of the three-layer MCC internal wave model, we refer the reader to Barros et al. (2020). The numerical algorithm used for the steady-state solution of the three-layer internal wave model is given in Wang (2021).
2.3 DJL equation
The DJL equation describes steady large-amplitude internal solitary waves in a continuously stratified fluid. The density profile of the fluid is described by ρ(Z) = ρ0 + ρr(Z), where ρ0 is the reference density. Under the Boussinesq and rigid-lid assumptions, the DJL equation is written as (Dubreil-Jacotin 1934; Long 1953):
$$ \nabla^2 \xi+\frac{N^2(Z-\xi)}{c^2} \xi=0 $$ (11) where ξ(X, Z) represents the isopycnic displacement (internal wave elevation),
$$ N(Z)=\sqrt{-\frac{g}{\rho_0} \frac{\mathrm{d} \rho_r}{\mathrm{~d} Z}} $$ (12) is the buoyancy frequency, and c is the speed of the internal solitary wave, which is obtained together with ξ(X, Z).
For a detailed derivation of the DJL equation, we refer the reader to Dubreil-Jacotin (1934) and Long (1953). A detailed solution algorithm and the solution to the DJL equation are given by Dunphy et al. (2011).
In this paper, we focus on the steady solutions of internal solitary waves in a flat bottom (i. e., the internal solitary waves propagate through an unchanging environment), which can be accurately described by the DJL equation (Stastna and Lamb 2020). Hence, the results provided by the DJL equation are treated as the benchmark solutions.
3 Numerical test cases
In this section, we consider the wave profile and wave speed of large-amplitude internal solitary waves under three types of pycnoclines. The results of the two- and three-layer MCC internal wave models are compared with the results of the DJL equation, and the errors introduced by the strong stratification approximation are obtained. Relevant case parameters are shown in Table 1. We note that Case A, Case B, and Case C belong to shallow configurations and apply to the MCC internal wave model. H and d represent the total water depth and the pycnocline thickness, respectively. The parameters of Case A are selected from the physical experiments conducted by Grue et al. (1999). The parameters of Case B and Case C are selected from the measurements in real oceans by Kuang (1986).
Table 1 Parameters of the test cases considered in this studyCase Two-layer Three-layer Continuous Amplitude (m) d/H (%) Density (kg/m3) Thickness (m) Density (kg/m3) Thickness (m) Density (kg/m3) Thickness (m) ρ1: ρ2 h1: h2 ρ1: ρ2: ρ3 h1: h2: h3 ρmin–ρmax H Case A 999:1 022 0.15:0.62 999:1 010.5:1 022 0.14:0.02:0.61 999–1022 0.77 0.136 5 3 0.184 5 Case B 1 025:1 028 125:525 1 025:1 026.5:1 028 100:50:500 1 025–1 028 650 120 8 160 Case C 1 025:1 028 16.6:83.4 1 025:1 026.5:1 028 11.4:10.4:78.2 1 025–1 028 100 15 10 20 3.1 Description of the pycnocline
In this paper, the pycnocline is described as (Camassa and Tiron 2011):
$$ \rho(Z)=\rho_{\min }+\frac{\rho_{\max }-\rho_{\min }}{2}\left\{1+\tanh \left[\alpha\left(Z_p-Z\right)\right]\right\} $$ (13) where ρmin and ρmax are the minimum and maximum values of the density, respectively, and Zp is the vertical position of the center of the pycnocline. α is a parameter related to the pycnocline thickness. By defining the upper and lower boundaries of the pycnocline corresponding to the vertical positions where the density is ρmin + 0.1(ρmax − ρmin) and ρmax − 0.1(ρmax − ρmin), respectively, the pynocline thickness can be obtained, and the value of α can be further determined.
Figure 3 shows the pycnocline profiles for Cases A, B, and C, where we have translated the vertical position of the pycnocline center Zp – Z = 0.
Figure 3 Density profiles of the three cases considered in this study3.2 Results and discussions
We use the two- and three-layer MCC internal wave models and the DJL equation to calculate the profile and speed of large-amplitude internal solitary waves for Cases A, B, and C. For the two-layer MCC internal wave model, the internal wave profile to be compared with is ζ(X), presented in Section 2.1; for the three-layer MCC internal wave model, (ζ1(X) + ζ2(X))/2, presented in Section 2.2; for the DJL equation, the isopycnic displacement for the density (ρmax − ρmin)/2.
3.2.1 Case A (d/H = 3%)
The wave profiles of the two- and three-layer MCC internal wave models and the DJL equation are compared in Figure 4. The experimental results of Grue et al. (1999) are also included in this figure. We find that the three numerical results agree well with the experimental results.
Figure 4 Comparison of the internal solitary wave profiles obtained by different models and laboratory measurements for Case A (d/H = 3%)The results of the internal solitary wave speed of the two amplitudes are shown in Table 2. Considering the results of the DJL equation as the benchmark solution, the results of the two- and three-layer MCC internal wave models are found to be very close to the results of the DJL equation, and the relative errors are within 2%.
Table 2 Internal solitary wave speeds for Case A (d/H = 3%)Model a/H = 0.177 a/H = 0.240 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 7.20 × 10−2 - 7.42 × 10−2 - Two-layer MCC 7.31 × 10−2 +1.53 7.50 × 10−2 +1.08 Three-layer MCC 7.20 × 10−2 0 7.39 × 10−2 −0.40 It is observed that when the pycnocline thickness is small (d/H = 3%), the relative error introduced by the strong stratification approximation for calculating the wave profile and wave speed is also very small.
3.2.2 Case B (d/H = 8%)
The wave profiles of the two- and three-layer MCC internal wave models and the DJL equation are compared in Figure 5. Since experimental results are unavailable for comparison, we select the DJL results as the benchmark solution. From Figure 5, we find that the wave profile obtained by the two-layer MCC internal wave model is slightly wider, while the wave profiles obtained by the three-layer MCC internal wave model and the DJL equation agree well with each other.
Figure 5 Comparison of the internal solitary wave profiles for Case B (d/H = 8%)To further investigate the difference between results, we consider the half-profile width λ1/2, which is defined by Koop and Butler (1981) as the distance between the horizontal position of the peak and the horizontal position at half-wave amplitude. We set the results of the DJL equation as the benchmark values. Table 3 shows the results of this comparison.
Table 3 Internal solitary wave half-profile widths for Case B (d/H = 8%)Model a/H = 0.185 a/H = 0.246 Half profile λ1/2/H Relative error (%) Half profile λ1/2/H Relative error (%) DJL 0.991 - 1.191 - Two-layer MCC 1.079 +8.88 1.326 +11.34 Three-layer MCC 0.989 −0.20 1.168 −1.93 We find that the relative errors introduced by the two-and three-layer MCC internal wave models are approximately 10% and within 2%, respectively.
The internal solitary wave speeds of the two amplitudes are shown in Table 4. The relative error is less than 4% and 2% between the two- and three-layer MCC internal wave models, respectively, and the DJL equation.
Table 4 Internal solitary wave speeds for Case B (d/H = 8%)Model a/H = 0.185 a/H = 0.246 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 2.53 × 10−2 - 2.6 × 10−2 - Two-layer MCC 2.62 × 10−2 +3.56 2.68 × 10−2 +3.08 Three-layer MCC 2.50 × 10−2 −1.19 2.58 × 10−2 −0.77 From these results, it is concluded that when the ratio of the pycnocline thickness to the total water depth increases (d/H = 8%), using the two-layer strong stratification approximation will introduce certain errors. Meanwhile, using the three-layer strong stratification approximation to calculate the wave profile and wave speed of large internal solitary waves does not introduce significant errors.
3.2.3 Case C (d/H = 10%)
The results of the two- and three-layer MCC internal wave models and the DJL equation are shown in Figure 6. We observe that the wave profiles are obviously wider when obtained by the two-layer MCC internal wave model and generally consistent with the DJL equation when obtained by the three-layer MCC internal wave model.
Figure 6 Comparison of the internal solitary wave profiles for Case C (d/H = 10%)The half-profile widths are compared in Table 5. The introduced relative errors of the two- and three-layer MCC internal wave models are approximately 12% and approximately 2%, respectively.
Table 5 Internal solitary wave profiles for Case C (d/H = 10%)Model a/H = 0.150 a/H = 0.200 Half profile λ1/2/H Relative error (%) Half profile λ1/2/H Relative error (%) DJL 0.816 - 0.875 - Two-layer MCC 0.909 +11.40 0.986 +12.69 Three-layer MCC 0.806 −1.23 0.853 −2.51 The internal solitary wave speeds of the two amplitudes are shown in Table 6. We find that the relative error between the two- and three-layer internal wave models and the DJL equation reaches 6% and less than only 1%, respectively.
Table 6 Internal solitary wave speeds for Case C (d/H = 10%)Model a/H = 0.150 a/H = 0.200 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 2.36 × 10−2 - 2.46 × 10−2 - two-layer MCC 2.51 × 10−2 +6.36 2.61 × 10−2 +6.10 three-layer MCC 2.34 × 10−2 −0.85 2.44 × 10−2 −0.81 Hence, when the ratio of pycnocline thickness to the total water depth further increases (d/H = 10%), calculating the wave profile and wave speed of large internal solitary waves will lead to significant errors when the two-layer strong stratification approximation is used but can be effectively accomplished using the three-layer strong stratification approximation.
4 Conclusions
To study the influence of the strong stratification approximation on describing large-amplitude internal solitary waves with pycnocline, this paper uses the two- and three-layer MCC internal wave models to calculate the wave profile and wave speed of large-amplitude internal solitary waves and compares them with the DJL equation, which considers continuous density stratification. The conclusions are as follows.
1) When pycnocline thickness is small (d/H = 3%), the results of the two- and three-layer MCC internal wave models and the DJL equation are in good agreement with the experimental values, indicating that introducing the strong stratification approximation does not lead to significant errors.
2) When the pycnocline thickness is relatively large (d/H = 8% or 10%), significant differences between the results of the two-layer MCC internal wave model and the DJL equation are observed. Wider wave profiles and larger wave speeds are obtained by the two-layer MCC internal wave model compared with those obtained by the DJL equation. When the pycnocline thickness increases from 8% to 10%, the error introduced by the two-layer approximation becomes larger. In contrast, the three-layer MCC internal wave model remains in good agreement with the DJL equation regarding wave profile and wave speed. As demonstrated, although the pycnocline thickness is large, the error introduced by the three-layer approximation, rather than the two-layer approximation, remains small in the section considered here but should be noted in general.
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Figure 1 Internal solitary wave in a two-layer fluid system
Figure 2 Internal solitary wave in a three-layer fluid system
Figure 3 Density profiles of the three cases considered in this study
Figure 4 Comparison of the internal solitary wave profiles obtained by different models and laboratory measurements for Case A (d/H = 3%)
Figure 5 Comparison of the internal solitary wave profiles for Case B (d/H = 8%)
Figure 6 Comparison of the internal solitary wave profiles for Case C (d/H = 10%)
Table 1 Parameters of the test cases considered in this study
Case Two-layer Three-layer Continuous Amplitude (m) d/H (%) Density (kg/m3) Thickness (m) Density (kg/m3) Thickness (m) Density (kg/m3) Thickness (m) ρ1: ρ2 h1: h2 ρ1: ρ2: ρ3 h1: h2: h3 ρmin–ρmax H Case A 999:1 022 0.15:0.62 999:1 010.5:1 022 0.14:0.02:0.61 999–1022 0.77 0.136 5 3 0.184 5 Case B 1 025:1 028 125:525 1 025:1 026.5:1 028 100:50:500 1 025–1 028 650 120 8 160 Case C 1 025:1 028 16.6:83.4 1 025:1 026.5:1 028 11.4:10.4:78.2 1 025–1 028 100 15 10 20 Table 2 Internal solitary wave speeds for Case A (d/H = 3%)
Model a/H = 0.177 a/H = 0.240 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 7.20 × 10−2 - 7.42 × 10−2 - Two-layer MCC 7.31 × 10−2 +1.53 7.50 × 10−2 +1.08 Three-layer MCC 7.20 × 10−2 0 7.39 × 10−2 −0.40 Table 3 Internal solitary wave half-profile widths for Case B (d/H = 8%)
Model a/H = 0.185 a/H = 0.246 Half profile λ1/2/H Relative error (%) Half profile λ1/2/H Relative error (%) DJL 0.991 - 1.191 - Two-layer MCC 1.079 +8.88 1.326 +11.34 Three-layer MCC 0.989 −0.20 1.168 −1.93 Table 4 Internal solitary wave speeds for Case B (d/H = 8%)
Model a/H = 0.185 a/H = 0.246 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 2.53 × 10−2 - 2.6 × 10−2 - Two-layer MCC 2.62 × 10−2 +3.56 2.68 × 10−2 +3.08 Three-layer MCC 2.50 × 10−2 −1.19 2.58 × 10−2 −0.77 Table 5 Internal solitary wave profiles for Case C (d/H = 10%)
Model a/H = 0.150 a/H = 0.200 Half profile λ1/2/H Relative error (%) Half profile λ1/2/H Relative error (%) DJL 0.816 - 0.875 - Two-layer MCC 0.909 +11.40 0.986 +12.69 Three-layer MCC 0.806 −1.23 0.853 −2.51 Table 6 Internal solitary wave speeds for Case C (d/H = 10%)
Model a/H = 0.150 a/H = 0.200 Speed c/(gH)1/2 Relative error (%) Speed c/(gH)1/2 Relative error (%) DJL 2.36 × 10−2 - 2.46 × 10−2 - two-layer MCC 2.51 × 10−2 +6.36 2.61 × 10−2 +6.10 three-layer MCC 2.34 × 10−2 −0.85 2.44 × 10−2 −0.81 -
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