# The Effect of a Variable Background Density Stratification and Current on Oceanic Internal Solitary Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Variable-Coefficient Korteweg–De Vries Equation

## 3. Estimation of the Non-Conservative Term from Oceanic Data

^{−1}, respectively.

#### 3.1. Cases Where the Horizontal Density Stratification Is the Major Effect

#### 3.1.1. A: Western Portugal

^{−1}, and has almost no effect on the linear long wave speed c, and hence also for all the derived coefficients $\alpha $, $\beta $, R. For example, the magnitude of $\beta $ is quite small, just $O\left({10}^{-12}\right)$, no matter with or without the current.

#### 3.1.2. B: South China Sea

^{−1}. The vKdV coefficients are shown in Figure 6. However, unlike the WP case, the background current now does have an discernible effect on the linear long wave speed c, the linear magnification factor Q and the coefficients $\mu $, $\delta $ and $\sigma $ as well.

#### 3.1.3. C: North West Shelf

^{−1}and has almost no effect on c, Q, $\mu $ and $\delta $, but does have some effects on $\beta $, of order ${10}^{-6}$ with the current and ${10}^{-8}$ without it.

#### 3.2. Cases Where the Horizontal Current Variation Is Significant

#### 3.2.1. D: Malvinas Current

^{−1}and can be almost up to 0.8 m s

^{−1}in the in situ observational data (Magalhaes and da Silva [17]), but here from the climatological data, the N–S velocity component of the current is only 0.1–0.25 m s

^{−1}. However, nevertheless, this signal is much stronger than the above three cases. Compared to the long wave speed c, whose whole range is around 0.2–0.6 m s

^{−1}in Figure 12, we can see that it has the same magnitude as the background current. Thus, here, for the two scenarios with and without the background current, the difference of the original coefficients between these two situations is much clearer compared to those in the former cases, especially for the linear long wave phase speed c, the magnification factor Q, and the derived coefficient $\beta $.

#### 3.2.2. E: Amazon River Mouth

^{−1}, according to a series of current meter observation (see Johns et al. [19]). Here again, based on the climatological data, the current we use is weaker than that, with a range of 0.4–0.6 m s

^{−1}(N–S component) and 0.2–0.5 m s

^{−1}(W–E component). Although it is smaller than the in situ value, the background current is much stronger than that in the Malvinas Current case. Again, the current now makes an obvious difference to all the original coefficients (c, Q, $\mu $, $\delta $), not only on c and Q as in the previous cases. For the derived coefficient $\beta $, the order of magnitude is $O\left({10}^{-5}\right)$ compared to that without the current $O\left({10}^{-8}\right)$.

^{−1}, much smaller than here. In terms of c, the magnitude of the background current contributes significantly. From Figure 16, we see that, although the initial wave amplitude is the same in these four panels, the wave evolution is clearly different in each, and both the wave structure and amplitude have been significantly modified due to the effects of the background current. The difference between the physical solution $\eta $ and the numerical solution $\tilde{A}$ is the joint influence of Q and R (see Formulas (2) and (7)), so, for the situation without the current, it mainly depends on Q, while, in the other scenario with a current, it mainly depends on R.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Bottom Friction

^{−1}, this yields $\gamma \left|\eta \right|\sim {10}^{-8}$ s

^{−1}even for a large amplitude wave where $\eta \sim 100$ m . Thus, especially in deep water, we expect this frictional term to be insignificant, but it could become comparable with the non-conservative term in shallow water. For instance, in shallow water where $h\sim 2{h}_{1}$, this same estimate becomes $\gamma \left|\eta \right|\sim {10}^{-5}$ s

^{−1}. To examine this in more detail in the present context, we plot the frictional index $\gamma \left|\eta \right|$ for each of our five case studies in Figure A1, where we set $\left|\eta \right|$ equal to the initial wave amplitude. In all cases, this index is at least an order of magnitude smaller than $\sigma $, indicating that, by comparison, this frictional term can usually be neglected. However, as indicated above, in very shallow water, the frictional index increases to a level where the frictional term should be taken into account. Note the curious exception of the SCS, where the frictional index decreases as the depth decreases.

**Figure A1.**The frictional index $\gamma \left|\eta \right|$ for the five cases, where $\left|\eta \right|$ is set to be the initial amplitude. From left to right, they are the cases, Western Portugal, the South China Sea, the North West Shelf, the Malvinas Current and the Amazon River mouth. For each panel, it represents the frictional index in summer with the background current (blue, solid), without the current (blue, dashed-dotted) and in winter with the current (red, dash), without the current (red, dot), respectively.

## Appendix B. Rotational Effects

Case | A | B | C | D | E | |||||
---|---|---|---|---|---|---|---|---|---|---|

Season | S | W | S | W | S | W | S | W | S | W |

O_{s} | 0.44 | 0.71 | 74 | 42 | 126 | 136 | 18 | 14 | 10,579 | 6106 |

${t}_{e}$(day) | 0.4 | 0.4 | 6.0 | 5.0 | 9.0 | 9.0 | 1.5 | 1.8 | 407.4 | 309.6 |

${t}_{f}$(day) | 2.8 | 2.8 | 2.3 | 2.6 | 5.8 | 4.9 | 3.8 | 3.0 | 0.9 | 1.0 |

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**Figure 2.**Background density and buoyancy frequency distribution along the transect of the Western Portugal case. (

**a**,

**b**) are situations in summer, while (

**c**,

**d**) are in winter. Five different lines in each panel represent situations on the chosen five points along the wave propagation transect.

**Figure 3.**Variation of the original coefficients (left panel and top of the right panel) and the derived coefficients (remainder of the right panel) of the vKdV equation for conditions of the Western Portugal case. The results show the coefficients in summer with background current ${u}_{0}$ (blue, solid), without ${u}_{0}$ (black, dash) and the coefficients in winter with ${u}_{0}$ (red, solid), without ${u}_{0}$ (blue, dashed-dotted).

**Figure 4.**A numerical simulation of the vKdV Equation (5) for the Western Portugal case for summer (

**a**,

**b**) and winter (

**c**,

**d**) conditions, with ${u}_{0}=0$ (

**a**,

**c**) or ${u}_{0}\ne 0$ (

**b**,

**d**), respectively. The initial condition is the solitary wave (black, dash), the numerical solution is $\tilde{A}$ (red, dot) and the physical solution is $\eta $ (blue, solid). From top to bottom, the distances from the initial point are (I) 0 km, (II) 110 km, (III) 220 km, (IV) 330 km, (V) 440 km, respectively.

**Figure 5.**Background density and buoyancy frequency distribution along the transect of the South China Sea case. (

**a**,

**b**) are situations in summer, while (

**c**,

**d**) are in winter. Five different lines in each panel represent situations on the chosen five points along the wave propagation transect.

**Figure 6.**Variation of the original coefficients (left panel and top of the right panel) and the derived coefficients (remainder of the right panel) of the vKdV equation for conditions of the South China Sea case. The results show the coefficients in summer with background current ${u}_{0}$ (blue, solid), without ${u}_{0}$ (black, dash) and the coefficients in winter with ${u}_{0}$ (red, solid), without ${u}_{0}$ (blue, dashed-dotted).

**Figure 7.**A numerical simulation of the vKdV Equation (5) for the South China Sea case for summer (

**a**,

**b**) and winter (

**c**,

**d**) conditions, with ${u}_{0}=0$ (

**a**,

**c**) or ${u}_{0}\ne 0$ (

**b**,

**d**), respectively. The initial condition is the solitary wave (black, dash), the numerical solution is $\tilde{A}$ (red, dot) and the physical solution is $\eta $ (blue, solid). From top to bottom, the distances from the initial point are (I) 0 km, (II) 110 km, (III) 220 km, (IV) 330 km, (V) 440 km, respectively.

**Figure 8.**Background density and buoyancy frequency distribution along the transect of the North West Shelf case. (

**a**,

**b**) are situations in summer, while (

**c**,

**d**) are in winter. Five different lines in each panel represent situations on the chosen five points along the wave propagation transect.

**Figure 9.**Variation of the original coefficients (left panel and top of the right panel) and the derived coefficients (remainder of the right panel) of the vKdV equation for conditions of the North West Shelf case. The results show the coefficients in summer with background current ${u}_{0}$ (blue, solid), without ${u}_{0}$ (black, dash) and the coefficients in winter with ${u}_{0}$ (red, solid), without ${u}_{0}$ (blue, dashed-dotted).

**Figure 10.**A numerical simulation of the vKdV Equation (5) for the North West Shelf case for summer (

**a**,

**b**) and winter (

**c**,

**d**) conditions, with ${u}_{0}=0$ (

**a**,

**c**) or ${u}_{0}\ne 0$ (

**b**,

**d**), respectively. The initial condition is the solitary wave (black, dash), the numerical solution is $\tilde{A}$ (red, dot) and the physical solution is $\eta $ (blue, solid). From top to bottom, the distances from the initial point are (I) 0 km, (II) 110 km, (III) 220 km, (IV) 330 km, (V) 440 km, respectively.

**Figure 11.**Background density and buoyancy frequency distribution along the transect of the Malvinas Current case. (

**a**,

**b**) are situations in summer, while (

**c**,

**d**) are in winter. Five different lines in each panel represent situations on the chosen five points along the wave propagation transect.

**Figure 12.**Variation of the original coefficients (left panel and top of the right panel) and the derived coefficients (remainder of the right panel) of the vKdV equation for conditions along the Malvinas Current case. The results show the coefficients in summer with background current ${u}_{0}$ (blue, solid), without ${u}_{0}$ (black, dash) and the coefficients in winter with ${u}_{0}$ (red, solid), without ${u}_{0}$ (blue, dashed-dotted).

**Figure 13.**A numerical simulation of the vKdV Equation (5) of a transect along the Malvinas Current case in summer (

**a**,

**b**) and winter (

**c**,

**d**) conditions, with ${u}_{0}=0$ (

**a**,

**c**) or ${u}_{0}\ne 0$ (

**b**,

**d**), respectively. The initial condition is the solitary wave (black, dash), the numerical solution is $\tilde{A}$ (red, dot) and the physical solution is $\eta $ (blue, solid). From top to bottom, the distances from the initial point are (I) 0 km, (II) 36.5 km, (III) 73 km, (IV) 109.5 km, (V) 146 km, respectively.

**Figure 14.**Background density and buoyancy frequency distribution along the transect of the Amazon River Mouth case. (

**a**,

**b**) are situations in summer, while (

**c**,

**d**) are in winter. Five different lines in each panel represent situations on the chosen five points along the wave propagation transect.

**Figure 15.**Variation of the original coefficients (left panel and top of the right panel) and derived coefficients (remainder of the right panel) of the vKdV equation for conditions of the Amazon River Mouth case. The results show the coefficients in summer with background current ${u}_{0}$ (blue, solid), without ${u}_{0}$ (black, dash) and the coefficients in winter with ${u}_{0}$ (red, solid), without ${u}_{0}$ (blue, dashed-dotted).

**Figure 16.**A numerical simulation of the vKdV Equation (5) of a transect close to the Amazon River Mouth case in summer (

**a**,

**b**) and winter (

**c**,

**d**) conditions, with ${u}_{0}=0$ (

**a**,

**c**) or ${u}_{0}\ne 0$ (

**b**,

**d**), respectively. The initial condition is the solitary wave (black, dash), the numerical solution is $\tilde{A}$ (red, dot) and the physical solution is $\eta $ (blue, solid). From top to bottom, the distances from the initial point are (I) 0 km, (II) 36.5 km, (III) 73 km, (IV) 109.5 km, (V) 146 km, respectively.

Case | A | B | C | D | E | |||||
---|---|---|---|---|---|---|---|---|---|---|

Season | S | W | S | W | S | W | S | W | S | W |

$(\overline{\eta}-\eta )/\eta (\%)$ | $2.74$ | $0.00$ | $2.43$ | $-1.80$ | $2.06$ | $2.52$ | $-1.41$ | $2.10$ | $44.42$ | $6.14$ |

$(\overline{\tilde{A}}-\tilde{A})/\tilde{A}(\%)$ | $1.40$ | $0.00$ | $2.50$ | $-0.60$ | $0.00$ | $0.16$ | $0.70$ | $1.35$ | $2.73$ | $-1.28$ |

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**MDPI and ACS Style**

Liu, Z.; Grimshaw, R.; Johnson, E. The Effect of a Variable Background Density Stratification and Current on Oceanic Internal Solitary Waves. *Fluids* **2018**, *3*, 96.
https://doi.org/10.3390/fluids3040096

**AMA Style**

Liu Z, Grimshaw R, Johnson E. The Effect of a Variable Background Density Stratification and Current on Oceanic Internal Solitary Waves. *Fluids*. 2018; 3(4):96.
https://doi.org/10.3390/fluids3040096

**Chicago/Turabian Style**

Liu, Zihua, Roger Grimshaw, and Edward Johnson. 2018. "The Effect of a Variable Background Density Stratification and Current on Oceanic Internal Solitary Waves" *Fluids* 3, no. 4: 96.
https://doi.org/10.3390/fluids3040096