# On the Generalized Kinetic Equation for Surface Gravity Waves, Blow-Up and Its Restraint

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Formulation for a Degenerate Quartet

## 4. Analytic Solution for the GKE

#### 4.1. Analytical Solution for Z

#### 4.1.1. Case $D>0$

#### 4.1.2. Case $D\le 0$

## 5. Analytic Solution of JKE

## 6. List of Examples

**a**and

**b**we initially assume that the waves are collinear ($q=0$) and fix the steepness ${\epsilon}_{a}=0.15$ of the wave ${\mathbf{k}}_{a}$. Then we take $\u03f5={\epsilon}_{b}={\epsilon}_{c}$ to be the steepness of the waves ${\mathbf{k}}_{b}$ and ${\mathbf{k}}_{c}$. In cases

**c**and

**d**the wave slopes are fixed ${\epsilon}_{a}=0.15$, ${\epsilon}_{b}=0.05$ and ${\epsilon}_{c}=0$ and we explore the nature of the solutions in the $(p,q)$-plane. Notice that $q\ne 0$ implies a two-dimensional disturbance. We recall that while solutions to the JKE are periodic, the behavior of the GKE is determined by the initial conditions via the discriminant D given in (48). To visualize how wave slope and wavenumber impact the solutions of the GKE, the sign of D is plotted in Figure 2. It establishes when the solutions remain bounded $(D>0)$ or tend to infinity in finite time $(D\le 0)$.

## 7. Comparison between GKE and JKE—Blow Up

**a**and

**c**(see Table 1), shown as triangles in Figure 2. Figure 3 shows the time evolution of case

**a**. In the upper panel we see the solution of the GKE that tends to infinity as time approaches ${T}_{\infty}=100.02$ s, see (63). The solution becomes physically meaningless as ${C}_{b}$ and ${C}_{c}$ become negative, prior to the critical time. On the other hand the solution of the JKE, on the lower panel of Figure 3 remains bounded, and is periodic with period $T=3207.19$ s. Adopting the notation ${t}_{n}=(2\pi /{\omega}_{a})/{\epsilon}_{a}^{n}$ for the time scales we see that the blow-up occurs at $\mathcal{O}\left({t}_{2}\right)$. The corresponding evolution for the JKE looks initially similar, but settles down to near equipartition of wave action, before eventually exhibiting periodicity on a time scale $\mathcal{O}\left({t}_{4}\right).$

**c**is shown in Figure 4. The upper panel shows blow-up for the solution of the GKE. The critical time is ${T}_{\infty}=89.80$ s. This non-physical behavior is similar to case

**a**(above) and occurs on the same time scale $\mathcal{O}\left({t}_{2}\right)$.

**c**and

**d**it is possible to have triads that are in exact resonance. The dashed line on the right panel of Figure 2 shows such triads. According to the GKE the time evolution of exact resonant triads blows up.

## 8. Comparison between GKE and JKE—Bounded Solutions

**b**and

**d**where both are periodic and bounded. These cases are depicted by squares $(p=0.2,\u03f5=0.01)$ and $(p=0.32,q=0.05)$ in Figure 2.

**b**. The upper panel shows the solution of the GKE with period $T=192.51$ s, see (56), while the lower panel shows the solution of the JKE with a period of $T=198.07$ s. In both cases the period of the solution is $\mathcal{O}\left({t}_{2}\right)$ and the qualitative behavior of both solutions is rather similar.

**d**, depicted in Figure 6, shows considerable similarity between the solutions of the GKE and solutions of the JKE. The periods are $T=94.22$ s for the GKE and $T=80.42$ s for JKE, both of $\mathcal{O}\left({t}_{2}\right)$.

## 9. Discussion and Concluding Remarks

#### 9.1. Time Scales

**a**, $\mathcal{O}\left({t}_{3}\right)$ for case

**c**and $\mathcal{O}\left({t}_{2}\right)$ for cases

**b**and

**d**.

#### 9.2. Assumption (i), Weak Non-Gaussianity

#### 9.3. Assumption (ii), Slow Temporal Evolution

**a**and

**c**the period of ${C}_{a}{C}_{b}{C}_{c}-{C}_{a}^{2}({C}_{b}+{C}_{c})/2$ is of the order of $\mathcal{O}\left({t}_{4}\right)$ and $\mathcal{O}\left({t}_{3}\right)$ respectively, although there are some rapid changes on the scale $\mathcal{O}\left({t}_{2}\right)$.

**b**and

**d**${C}_{a}{C}_{b}{C}_{c}-{C}_{a}^{2}({C}_{b}+{C}_{c})/2$ has variations on the $\mathcal{O}\left({t}_{2}\right)$ scale, which is also the time scale of their period. Note that the solution of the JKE has the same period as the oscillatory term ${e}^{i{\mathsf{\Delta}}_{a,a}^{b,c}t}$, see formula (71). This contradicts assumption (ii) for the JKE. Nevertheless, the surprising agreement between the GKE and the JKE in cases

**b**and

**d**suggests that the approximation may be valid, but it is not clear at this stage how it could be justified.

#### 9.4. Numerical Computations

**c**, (71) was used as a consistency check.

**b**and

**d**we obtained a relative error between the exact and the numerical solutions below $2.8\times {10}^{-8}$ and $6.1\times {10}^{-10}$ at each time step from 0 to 1000 s, respectively. For the unbounded cases

**a**and

**c**we measured the difference at each time step between 0 and ${t}_{f}={T}_{\infty}-1$ s, which exhibited a relative error of $1.9\times {10}^{-8}$ for case

**a**and $2.8\times {10}^{-9}$ for case

**c**.

#### 9.5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**a**and

**c**only one reinitialization is needed to obtain bounded solutions, as shown in Figure A1 and Figure A2. Subsequent to the phase-mixing at ${t}_{1}$ the solutions are bounded and periodic with periods $T=502.3$ s and $T=536.3$ s respectively, both within the time scale $\mathcal{O}\left({t}_{3}\right)$.

**Figure A1.**Upper panel: Solution of the GKE showing ${C}_{a}$ in blue, ${C}_{b}$ in yellow (circles) and ${C}_{c}$ in red (asterisks) for case

**a**$(p=0.05,\u03f5=0.01),$ extended by phase-mixing at time ${t}_{1}=72.38$ s. Lower panel: Time evolution of ${\kappa}_{a,a,b,c}$. The real part of ${\kappa}_{a,a,b,c}$ is shown in blue and the imaginary part of ${\kappa}_{a,a,b,c}$ in red (asterisks). In all cases, solid lines denote the solution before phase-mixing, dashed lines subsequent to phase mixing. The time t is in seconds.

**Figure A2.**Upper panel: Solution of the GKE showing ${C}_{a}$ in blue, ${C}_{b}$ in yellow (circles) and ${C}_{c}$ in red (asterisks) for case

**c**$(p=0.16,\phantom{\rule{0.166667em}{0ex}}q=0.05),$ extended by phase-mixing at time ${t}_{1}=56.66$ s. Lower panel: Time evolution of ${\kappa}_{a,a,b,c}$. The real part of ${\kappa}_{a,a,b,c}$ is shown in blue and the imaginary part of ${\kappa}_{a,a,b,c}$ in red (asterisks). In all cases, solid lines denote the solution before phase-mixing, dashed lines subsequent to phase mixing. The time t is in seconds.

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**Figure 1.**Schematic representation of the polynomial ${P}_{4}$. The red region (dashed line) represents the range of the function $Z\left(t\right)$. The yellow dot is the initial condition $Z\left(0\right)=0$. Left panel: ${P}_{4}$ has four real roots, $D>0$. Right panel: ${P}_{4}$ has two real roots and two complex roots, $D\le 0$.

**Figure 2.**Plot of the discriminant (48) for varying initial conditions. The solid lines correspond to $D=0,$ and the region they enclose about the origin is where $D<0,$ and solutions of the GKE will diverge. The outer region is where $D>0$ and solutions of the GKE are periodic and bounded. Left panel: collinear waves, with varying p and $\u03f5.$ Case

**a**has $(p=0.05,\u03f5=0.01)$ and case

**b**has $(p=0.2,\u03f5=0.01)$. Right panel: fixed wave slopes, with varying p and $q.$ Case

**c**has $(p=0.16,q=0.05)$ and case

**d**has $(p=0.32,q=0.05)$. The dashed line shows the waves that are in exact resonance $2{\omega}_{a}={\omega}_{b}+{\omega}_{c}$.

**Figure 3.**Time evolution of ${C}_{a}$ in blue (solid), ${C}_{b}$ in yellow (circles) and ${C}_{c}$ in red (asterisks) for $(p=0.05,\u03f5=0.01)$. Upper panel: Solution of the GKE. The blow-up time is ${T}_{\infty}=100.02$ s. Lower panel: Solution of the JKE. The period is $T=3207.19$ s. The time t is in seconds.

**Figure 4.**Time evolution of ${C}_{a}$ in blue, ${C}_{b}$ in yellow and ${C}_{c}$ in red for $(p=0.16,q=0.05)$. Upper panel: Solution of the GKE. The blow-up time is ${T}_{\infty}=89.80$ s. Lower panel: Solution of the JKE. The period is $T=390.19$ s. The time t is in seconds.

**Figure 5.**Time evolution of ${C}_{a}$ in blue (solid), ${C}_{b}$ in yellow (circles) and ${C}_{c}$ in red (asterisks) for $(p=0.2,\u03f5=0.01)$. Upper panel: Solution of the GKE. The period is $T=192.51$ s. Lower panel: Solution of the JKE. The period is $T=198.07$ s. The time t is in seconds.

**Figure 6.**Time evolution of ${C}_{a}$ in blue (solid), ${C}_{b}$ in yellow (circles) and ${C}_{c}$ in red (asterisks) for $(p=0.32,q=0.05)$. Upper panel: Solution of the GKE. The period is $T=94.22$ s. Lower panel: Solution of the JKE. The period is $T=80.42$ s. The time t is in seconds.

Wave Numbers | Steepness | Nature of Solution | Detuning | |||||
---|---|---|---|---|---|---|---|---|

case | $\mathit{p}$ | $\mathit{q}$ | ${\mathit{\epsilon}}_{\mathit{a}}$ | ${\mathit{\epsilon}}_{\mathit{b}}$ | ${\mathit{\epsilon}}_{\mathit{c}}$ | GKE | JKE | ${\mathsf{\Delta}}_{\mathit{a},\mathit{a}}^{\mathit{b},\mathit{c}}$ (s${}^{-\mathbf{1}}$) |

a | $0.05$ | 0 | $0.15$ | $0.01$ | $0.01$ | blow-up | periodic | $0.002$ |

b | $0.2$ | 0 | $0.15$ | $0.01$ | $0.01$ | periodic | periodic | $0.0317$ |

c | $0.16$ | $0.05$ | $0.15$ | $0.05$ | 0 | blow-up | periodic | $0.0161$ |

d | $0.32$ | $0.05$ | $0.15$ | $0.05$ | 0 | periodic | periodic | $0.0781$ |

**Table 2.**Values of $\delta =\underset{0\le t\le 1000}{max}\left|{\kappa}_{a,a,b,c}\left(t\right)\right|/{C}_{a}^{2}\left(0\right)$ for computed solutions to the GKE and JKE in the four cases

**a**–

**d**.

GKE | JKE | |
---|---|---|

case a | blow up | $0.2516$ |

case b | $0.2268$ | $0.2478$ |

case c | blow up | $0.2097$ |

case d | $0.1142$ | $0.1460$ |

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**MDPI and ACS Style**

Andrade, D.; Stuhlmeier, R.; Stiassnie, M.
On the Generalized Kinetic Equation for Surface Gravity Waves, Blow-Up and Its Restraint. *Fluids* **2019**, *4*, 2.
https://doi.org/10.3390/fluids4010002

**AMA Style**

Andrade D, Stuhlmeier R, Stiassnie M.
On the Generalized Kinetic Equation for Surface Gravity Waves, Blow-Up and Its Restraint. *Fluids*. 2019; 4(1):2.
https://doi.org/10.3390/fluids4010002

**Chicago/Turabian Style**

Andrade, David, Raphael Stuhlmeier, and Michael Stiassnie.
2019. "On the Generalized Kinetic Equation for Surface Gravity Waves, Blow-Up and Its Restraint" *Fluids* 4, no. 1: 2.
https://doi.org/10.3390/fluids4010002