Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows

Abstract

A central question in sea-state modeling is the role that various physical effects have on the evolution of the statistical properties of random sea states. This becomes a critical issue when one is concerned with the likelihood of rare events such as rogue, or freak, waves which can have significant destructive potential on deep sea ships and other offshore structures. In this paper then, using a recently derived higher-order model of deep water nonlinear waves, we examine the impact of constant vorticity currents on the statistical properties of nonlinearly evolving random sea states. As we show, these currents can both decrease and increase the kurtosis of the affiliated distributions of the sea states, thereby diminishing or enhancing the likelihood of rare events. We likewise numerically study the relationship between the kurtosis and a non-dimensional parameter, the Benjamin–Feir Index, which has proven to be a useful measure of when rare events are likely in oceanographic application.

Appendix

1.1 Shear Profiles over Infinitely Deep Fluids

As noted in the text, the assumption that the fluid velocity is to leading order given by

$$\begin{aligned} \mathbf{u} \approx \omega z \hat{\mathbf{i}} \end{aligned}$$

is clearly unrealistic insofar as it leads to currents of infinite speed as one descends through the fluid. A more realistic, though also more complicated, ansatz is to suppose that to leading order we have that $\mathbf{u}\approx u(z)\hat{\mathbf{i}}$ where

$$\begin{aligned} u(z) = \left\{ \begin{array}{rl} \omega z, &{} -h_{0} \le z< 0 \\ \frac{-\omega h_{0}}{h_{1}-h_{0}} (z+h_{1}), &{} -h_{1} \le z< -h_{0} \\ 0, &{} z < -h_{1}, \end{array} \right. \end{aligned}$$

where $h_{1}>h_{0}\gg 1$, so that we are looking at a deep, continuous shear profile which is zero at or near the surface $z=\epsilon \eta (x,t)$ and at the depth $z=-\,h_{1}$, after which the fluid is to leading order quiescent. Note, we can also see this profile as satisfying to leading order two ‘no-slip’ conditions, one near the free surface at $z=0$ and one near $z=-\,h_{1}$.

While a full, nonlinear description of the above shear profile would require two more freely evolving interfaces, and thus is beyond the scope of this paper, we can readily find the dispersion relationship affiliated with this profile. This then allows us to provide some analytic argument for why we study an otherwise unphysical velocity profile in the main body of the text. Likewise, information from the dispersion relationship provides us with a better understanding of how depth varying shear profiles induce both surface and internal waves.

Following relatively classical approaches, we introduce three fluid velocities:

$$\begin{aligned} \mathbf{u}_{1}&= \omega z \hat{\mathbf{i}} + \epsilon \nabla \phi _{1},\qquad \qquad \qquad \qquad \,\, ~-h_{0}+\epsilon \eta _{2}< z< \epsilon \eta _{1}\\ \mathbf{u}_{2}&= \frac{-\omega h_{0}}{h_{1}-h_{0}} (z+h_{1}) \hat{\mathbf{i}} + \epsilon \nabla \phi _{2},\quad ~-h_{1}+\epsilon \eta _{3}< z< -h_{0}+\epsilon \eta _{2}\\ \mathbf{u}_{3}&= \epsilon \nabla \phi _{3},\qquad \qquad \qquad \qquad \qquad \qquad \,\, ~z < -h_{1} + \epsilon \eta _{3} \end{aligned}$$

so that after linearizing around the small disturbances, we derive the dispersion relationship

$$\begin{aligned} \frac{gk}{\Omega } {+} \omega -\Omega \tanh (kh_{0}) {+} \left( \frac{1}{\Omega }\left( \frac{gk}{\Omega } {+} \omega \right) \tanh (kh_{0}) - 1\right) \left( \frac{\omega h_{1}}{\delta h} {+} \Omega \Omega _{1}(k\delta h)\right) = 0, \end{aligned}$$

where

$$\begin{aligned} \Omega _{1}(k\delta h) = \frac{s\left( \Omega -\frac{\omega h_{0}}{\delta h}s\right) +\Omega \tanh (k\delta h)}{\Omega +s\left( \Omega - \frac{\omega h_{0}}{\delta h}s\right) \tanh (k\delta h)}, \end{aligned}$$

and where $\delta h = h_{1}-h_{0}$ and $s=\text{ sgn }(k)$. While in general we would have to find the roots of a fifth-order polynomial to determine the values of $\Omega$, we see for $h_{1}\gg h_{0}$ that $\Omega _{1}\sim s$, and thus we get that the dispersion relationship simplifies to

$$\begin{aligned} \frac{gk}{\Omega } + \omega -\Omega \tanh (kh_{0}) + \left( \frac{1}{\Omega }\left( \frac{gk}{\Omega } + \omega \right) \tanh (kh_{0}) - 1\right) \left( \tilde{\omega } + s\Omega \right) \sim 0, \end{aligned}$$

where

$$\begin{aligned} \tilde{\omega } = \frac{\omega h_{1}}{\delta h}. \end{aligned}$$

Since $\tilde{\omega }$ only approaches $\omega$ at an algebraic rate, it seems appropriate to keep it included in the analysis. Letting

$$\begin{aligned} \left| \tanh (kh_{0}) \right| = 1 - \tilde{\epsilon }, \end{aligned}$$

and noting that $\tilde{\epsilon }$ vanishes to zero exponentially fast as we increase $h_{0}$, we see the reduced dispersion relationship factors into the form

$$\begin{aligned} \left( 2\Omega + s\tilde{\omega } \right) \left( g\left| k\right| + s\omega \Omega - \Omega ^{2} \right) + \tilde{\epsilon }\left( \Omega ^{3} - (s\tilde{\omega } + \Omega )(g|k|+\omega s\Omega ) \right) \sim 0. \end{aligned}$$

Thus, we have a completely regular perturbation problem for the roots, which we readily see are given by

$$\begin{aligned} \Omega \sim \frac{1}{2}\left( -\,s\omega \pm \sqrt{\omega ^{2}+4g|k|} \right) , ~ -\frac{s\tilde{\omega }}{2}. \end{aligned}$$

Thus, of the three roots we find, two give the dispersion relationship we find in the body of the text using $\mathbf{u}\sim \omega z \hat{\mathbf{i}}$. Looking at the affiliated disturbances of the relevant free surface and internal wave, we find that

$$\begin{aligned} \eta _{1} \sim \alpha _{12}\frac{ik}{\Omega }e^{i\theta } + \text{ cc }, ~ \eta _{2} \sim \alpha _{12}\frac{ik}{\Omega ^{3}}\cosh (kh_{0})\left( \Omega ^{2}-(1-\tilde{\epsilon })\left( g|k|+s\omega \Omega \right) \right) e^{i\theta }+ \text{ cc }. \end{aligned}$$

Thus, if we choose $\Omega$ so that $\Omega ^{2}-\left( g|k|+s\omega \Omega \right) = 0$, the magnitude of the internal wave essentially vanishes, thereby localizing dynamics along the free surface near $z=0$. Therefore, while not necessarily physically justifiable throughout the bulk of the fluid, our simplified shear profile assumption produces results which are asymptotically consistent with a more sophisticated treatment of the shear profile.

We note though that the more physically realistic shear profile shows that there is a choice of $\Omega$ which corresponds to a non-trivial internal mode near $z=-h_{0}$. We also note that this choice of $\Omega$ makes many of the terms in our modulation theory singular, thus showing that this case would require a markedly different treatment. While interesting, this issue is beyond the scope of the current paper and will be addressed in future research.

1.2 Higher Order-Dispersive Corrections to the VDE

To better understand why we must include the higher order corrections to the dispersion used in the VDE, we examine how such terms are found from first principles. To leading order from Eq. (1) over an infinitely deep fluid, we have that

$$\begin{aligned} \eta _{t} = -\mathcal {H}Q, \end{aligned}$$

where $q(x,t) = \phi (x,\eta (x,t),t)$, $Q = q_{x}$, and $\mathcal {H}$ is the Hilbert transform. Likewise, from Eq. (2) we have to leading order that

$$\begin{aligned} Q _{t}+ \omega \eta _{t} + \eta _{x}= 0, \end{aligned}$$

so that by combining the two expressions we have the wave equation in $\eta$ alone given by

$$\begin{aligned} \eta _{tt}- \omega \mathcal {H}\eta _{t} - \mathcal {H}\eta _{x}= 0. \end{aligned}$$

Using Fourier transforms, we can readily write a solution to the affiliated initial-value problem in the form

$$\begin{aligned} \eta (x,t) = \frac{1}{2\pi }\int _{\mathbb {R}}\hat{\eta }_{0}(k) e^{ikx+i\Omega _{+}(k)t} \mathrm{d}k + \frac{1}{2\pi }\int _{\mathbb {R}}\hat{\eta }_{0}(k) e^{ikx+i\Omega _{-}(k)t} \mathrm{d}k, \end{aligned}$$

where

$$\begin{aligned} \Omega _{\pm }(k) = \frac{1}{2}\left( \omega s \pm \sqrt{\omega ^{2}+4|k|} \right) . \end{aligned}$$

If we assume, as per usual when deriving NLS-type models, that

$$\begin{aligned} \hat{\eta }_{0}(k) = \frac{1}{\epsilon }\hat{A}\left( \frac{k-k_{0}}{\epsilon } \right) , \end{aligned}$$

representing the assumption that the initial conditions are narrowly banded around a carrier wave number $k_{0}$, then we see that, taking only the plus branch of $\Omega _{\pm }$,

$$\begin{aligned} \eta (x,t) =&\frac{e^{i\theta (x,t,k_{0})}}{2\pi }\int _{\mathbb {R}}\hat{A}(\tilde{k})e^{i\tilde{k}\xi } e^{i\tau \tilde{\Omega }(k_{0},\tilde{k})}\mathrm{d}\tilde{k}\\ \sim&\frac{e^{i\theta (x,t,k_{0})}}{2\pi }\int _{\mathbb {R}}\hat{A}(\tilde{k})e^{i\tilde{k}\xi }e^{i\tau \left( \frac{\Omega ''(k_{0})}{2}\tilde{k}^{2} + \frac{\epsilon }{6}\Omega '''(k_{0})\tilde{k}^{3} + \frac{\epsilon ^{2}}{24}\Omega ''''(k_{0})\tilde{k}^{4}\right) }\mathrm{d}\tilde{k}, \end{aligned}$$

where

$$\begin{aligned} \theta (x,t,k_{0}) = k_{0}x + \Omega (k_{0})t, ~c_{g}(k_{0}) = \Omega '(k_{0}), ~ \xi = \epsilon (x+c_{g}t), ~ \tau = \epsilon ^{2}t, \end{aligned}$$

and where

$$\begin{aligned} \tilde{\Omega }(k_{0},\tilde{k}) = \frac{1}{\epsilon ^{2}}\left( \Omega (k_{0}+\epsilon \tilde{k})-\Omega (k_{0})-\Omega '(k_{0})\epsilon \tilde{k}\right) \end{aligned}$$

and where we have expanded the dispersion relationship up to $\mathcal {O}(\epsilon ^{3})$ as is done in the main body of the text. This expansion corresponds to the affiliated linear-evolution equation for the slowly evolving envelope $A(\xi ,\tau )$

$$\begin{aligned} i\partial _{\tau }A = \left( \frac{\Omega ''(k_{0})}{2}\partial _{\xi }^{2}-\frac{i\epsilon }{6}\Omega '''(k_{0})\partial _{\xi }^{3} - \frac{\epsilon ^{2}}{24}\Omega ''''(k_{0})\partial _{\xi }^{4}\right) A. \end{aligned}$$

As to the issue of why to include the $\epsilon ^{2}$ term, we note that the affiliated group velocity of the slowly evolving envelope is given by

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tilde{k}}\tilde{\Omega } =&\frac{1}{\epsilon }\left( \Omega '(k_{0}+\epsilon \tilde{k}) - \Omega '(k_{0})\right) \\ \sim&\Omega ''(k_{0})\tilde{k}\left( 1 - \frac{3\epsilon c_{g}^{2}}{s}\tilde{k} + 10 \epsilon ^{2}c_{g}^{4}\tilde{k}^{2}\right) . \end{aligned}$$

We see that the full dispersion relationship has exactly one root at $\tilde{k}=0$, so that this root is the only point of stationary phase in the corresponding integral representation of the solution $\eta (x,t)$. However, using our Taylor series expansion, were we to ignore the $\epsilon ^{2}$ term, we would introduce a new stationary-phase point at $\tilde{k}_{*}$, where

$$\begin{aligned} \tilde{k}_{*} = \frac{s}{3\epsilon c_{g}^{2}}. \end{aligned}$$

The presence of this stationary-phase point allows for the accumulation of energy at high frequencies, which without a corresponding band-limiting requirement introduces non-physical effects into the model. By including the $\epsilon ^{2}$ term though, we remove this spurious high-frequency stationary-phase point, thereby avoiding the inclusion of an otherwise unnecessary bandwidth condition on the slowly evolving envelope.