Particle motions beneath irrotational water waves

Abstract

Neutral and buoyant particle motions in an irrotational flow are investigated under the passage of linear, nonlinear gravity, and weakly nonlinear solitary waves at a constant water depth. The developed numerical models for the particle trajectories in a non-turbulent flow incorporate particle momentum, size, and mass (i.e., inertial particles) under the influence of various surface waves such as Korteweg-de Vries waves which admit a three parameter family of periodic cnoidal wave solutions. We then formulate expressions of mass-transport velocities for the neutral and buoyant particles. A series of test cases suggests that the inertial particles possess a combined horizontal and vertical drifts from the locations of their release, with a fall velocity as a function of particle material properties, ambient flow, and wave parameters. The estimated solutions exhibit good agreement with previously explained particle behavior beneath progressive surface gravity waves. We further investigate the response of a neutrally buoyant water parcel trajectories in a rotating fluid when subjected to a series of wind and wave events. The results confirm the importance of the wave-induced Coriolis-Stokes force effect in both amplifying (destroying) the pre-existing inertial oscillations and in modulating the direction of the flow particles. Although this work has mainly focused on wave-current-particle interaction in the absence of turbulence stochastic forcing effects, the exercise of the suggested numerical models provides additional insights into the mechanisms of wave effects on the passive trajectories for both living and nonliving particles such as swimming trajectories of plankton in non-turbulent flows.

Appendices

Appendix A: Derivation of the KdV Eq. 10

The KdV equation describes the propagation of weakly nonlinear water waves in a dispersive media. The equation is derived from the Euler equations under the assumptions that the amplitude is small compared to the depth and the depth is small compared to the wavelength. Although rich mathematical studies on different aspects of this equation have been carried out, for the matter of completeness, we present the derivation of KdV equation using the approach proposed by Whitham (1974).

To provide governing equations more tractable, and to split them into different regions of interest, we take the standard step of nondimensionalization as follows

$$ x = L\hat{x}; z = H\hat{z}; t =\frac{L}{c_{0}}\hat{t}; \varphi=aL\sqrt{\frac{g}{H}}\hat{\varphi}; \eta=a\hat{\eta}\-, $$

(29)

where $c_{0}=\sqrt {gH}$, a is wave amplitude, and L denotes the characteristic wavelength. Using the definition of a potential function, and substituting Eq. 29 into the Euler Eqs. 1–3, and then dropping the hats, we obtain the following nondimensionalized equations:

$$\begin{array}{@{}rcl@{}} \delta \frac{\partial^{2} \varphi}{\partial x^{2}} + \frac{\partial^{2} \varphi}{\partial z^{2}}&=& 0, z\in (-1,1+\varepsilon \eta)\-, \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} \eta_{t}+\varepsilon\varphi-\frac{1}{\delta}\frac{\partial \varphi}{\partial z} &=& 0, z=1+\varepsilon\eta\-, \end{array} $$

(31)

$$\begin{array}{@{}rcl@{}} \eta+\frac{\partial \varphi}{\partial z}+\frac{1}{2}\varepsilon\frac{\varepsilon}{\delta}\frac{\partial^{2} \varphi}{\partial z^{2}} &=& 0, z=1+\varepsilon\eta\-, \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} \frac{\partial \varphi}{\partial z} &=& 0, z=-1\-. \end{array} $$

(33)

Without loss of generality, we neglected the effect of pressure at the free surface in Eqs. 31–32. We also applied the following dimensionless parameters:

$$ \varepsilon=\frac{a}{H}, \text{and}~\delta=\frac{H^{2}}{L^{2}}\-, $$

(34)

where ε gives the measure of nonlinearity, and δ describes the limit of shallowness. For weak nonlinearity assumption, we should have ε≪1 (but not zero), and for shallowness assumption, we should have δ≪1. In the case that the depth is small compared to the wave characteristic length scale, we can expand the nondimensional potential function φ in terms of δ without imposing any assumption on ε:

$$\varphi=\sum\limits_{n} \varphi_{n}(x,t)(z\delta)^{n}\-. $$

By substituting this series into Eq. 33, and since velocity vanishes at z=−1, we get φ ₁=0. By recurrence, it results in φ _2n+1=0. Consequently,

$$ \varphi=\sum\limits_{n} (-1)^{n} \frac{\delta^{n}z^{2n}}{(2n)!}\frac{\partial^{2n}\varphi_{0}(x,t)}{\partial x^{2n}}\-. $$

(35)

By substituting Eq. 35 into the Eqs. 31–32, we obtain the following estimates up to order $\mathcal {O} (\varepsilon ^{2},\delta \varepsilon ,\delta ^{2})$:

$$ \eta_{t}+\frac{\partial^{2} \phi_{0}}{\partial x^{2}}+\varepsilon \left( \eta\frac{\partial\phi_{0}}{\partial x}\right)_{x}-\frac{1}{6}\delta\left( \frac{\partial \phi_{0}}{\partial x}\right)_{xxx}= 0\-, $$

(36)

$$ \eta+\frac{\partial \phi_{0}}{\partial t}+\frac{1}{2}\delta\left( \frac{\partial \phi_{0}}{\partial x}\right)^{2}-\frac{1}{2}\delta\frac{\partial^{4} \phi_{0}}{\partial x^{3}\partial t}= 0\-. $$

(37)

Furthermore, using Eq. 33, the lowest order term in the power series satisfies (φ ₀) _z =0, for all z, meaning that the horizontal velocity components at the bottom (z=−1) can be determined independent of the vertical coordinate z. Therefore, using prior relations and defining f(x,t)=∂ _x φ ₀, the horizontal and vertical components of velocity at the bottom are determined as follows

$$\begin{array}{@{}rcl@{}} u(x,t)&=&f(x,t)-\frac{1}{2}z^{2}\delta f_{xx}(x,t)\-, \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} w(x,t)&=&-zf_{x}(x,t)+\frac{1}{2}z^{3} f_{xx}(x,t)\-. \end{array} $$

(39)

Following Whitham (1974), f can be estimated as a function of η:

$$f(x,t)=\eta-\frac{1}{4}\varepsilon\eta^{4}+\frac{1}{4}\varepsilon\eta_{xx}+\mathcal{O} (\varepsilon^{2},\delta\varepsilon,\delta^{2})\-. $$

Hence, the nondimentionlized KdV equation, Eq. 36, can be written using above relations as

$$\eta_{t}+\eta_{x}+\frac{3}{2}\varepsilon\eta\eta_{x}+\frac{1}{6}\delta\eta_{xxx}+\mathcal{O} (\varepsilon^{2},\delta\varepsilon,\delta^{2})\-. $$

This expression is then converted to the physical dimensional space:

$$\eta_{t}+c_{0}\left( 1+\frac{3}{2}\frac{\eta}{H}\right)\eta_{x}+\frac{1}{6}c_{0}H^{2}\eta_{xxx}=0\-. $$

To evaluate the horizontal and vertical velocity components at level 𝜃 H, where 0≤𝜃≤1 (13–14), we use Taylor expansion about 𝜃 H up to third order as $f(x,t)=f^{\theta }+\frac {1}{2}(\theta H)^{2}f_{xx}$ by using Eqs. 38 and 39.

Appendix B: Estimation of the Jacobian elliptic parameter

Determination of the Jacobian elliptic parameter is a crucial step in cnoidal wave theory. Here, we briefly give a recipe on how to determine m accurately.

In cnoidal waves, the relationships between associated parameters are given by

$$\begin{array}{@{}rcl@{}} \lambda &=& H\sqrt{\frac{16mH}{3\mathcal{A}}} K\-, \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} T^{2} &=&\frac{16H^{3}m^{2}K^{2}}{3g\mathcal{A}\left[mH+\mathcal{A}\left( 2-m-\frac{3E}{K}\right)\right]}\-, \end{array} $$

(41)

where K is the complete elliptic integral of second kind, and $\mathcal {A}$ is the cnoidal wave amplitude. Using Eqs. 12 and 40–41, the elliptic parameter m is related to the wavelength, λ, and cnoidal wave period, T, through the following relation:

$$c-\frac{\lambda}{T}=0\-. $$

Here, c is the cnoidal wave phase speed. The root of this equation, which varies from 0 to 1, is called cnoidal elliptic parameter. Here, we use the bisection technique to find modulus m.