Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing

Abstract

Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

Appendices

Appendix 1: Derivation of the forced nonlinear Schrödinger equation for time- and space-variant wind-induced pressure

It can be shown that weakly nonlinear solutions of the Euler equations can be reduced to solutions described by a complex envelope, which satisfies the NLS. Such reduction can be achieved by the method of multiple scales, cf. [22]. In the following the multiple scales analysis is presented for the case of forcing of water waves by a time- and space-variant wind field. For the forced Euler equations, an incompressible fluid with density $\rho _{\mathrm{w}}$, a finite depth h, and a free surface $\eta (x,t)$ is assumed, where x is the spatial variable and t is time, and the corresponding velocity field is irrotational. This leads to four equations, where the nonlinear gravity wave problem is governed by two linear equations, which are the Laplace equation and the kinematic condition at the sea bed, and by kinematic and dynamic free surface conditions, which are nonlinear [70]. If in addition a time- and space-variant wind-induced pressure $P_a(x,t)$ at the free surface is assumed, and if dissipative effects are included by means of the water kinematic viscosity $\nu$, as suggested by Dias et al. [71], then the forced Euler equations become

$$\begin{aligned}&\phi _{xx}+\phi _{zz}=0, \quad \text {for }-h\le z\le \eta (x,t), \end{aligned}$$

(21a)

$$\begin{aligned}&\eta _t+\phi _x\eta _x-\phi _z-2\nu \eta _{xx}=0,\quad \text {for } z=\eta (x,t), \end{aligned}$$

(21b)

$$\begin{aligned}&\phi _t+\frac{1}{2}\left( \phi _x^2+\phi _z^2\right) +g\,\eta =-\frac{1}{\rho _{\mathrm{w}}}P_a-2\nu \phi _{zz},\nonumber \\&\quad \text {for } z=\eta (x,t), \end{aligned}$$

(21c)

$$\begin{aligned}&\phi _{z}=0, \quad \text {for } z=-h, \end{aligned}$$

(21d)

where $\phi (x,z,t)$ is the velocity potential and g is acceleration due to gravity.

The nonlinear boundary conditions at the free surface $\eta (x,t)$ make the solution of Eq. (21) very difficult, since the nonlinear boundary $\eta (x,t)$ is unknown. Thus, one seeks to use simplifications.

Here, the method of multiple scales is used for the derivation of a nonlinear Schrödinger equation resulting from forced Euler Eq. (21). Following Dawey and Stewardson [72] and Hasimoto and Ono [73], the scalings

$$\begin{aligned} \xi =\varepsilon (x-c_{\mathrm{g}}\,t),\quad \tau =\varepsilon ^2\,t \end{aligned}$$

(22)

are used, whereby the small parameter $\varepsilon$ represents the wave steepness, $c_{\mathrm{g}}=\dfrac{g}{2\omega }(\tanh kh+kh(1-\tanh ^2kh))$ is the group velocity, $\omega$ is the frequency, and k is the wave number of the considered carrier wave. The velocity potential $\phi$ and the surface elevation $\eta$ are expanded in series of the form

$$\begin{aligned} \phi (x,z,t)&=\sum _{n=1}^\infty \varepsilon ^n\sum _{m=-n}^n \phi ^{n,m}(\xi ,z,\tau )\,E^m, \end{aligned}$$

(23)

$$\begin{aligned} \eta (x,t)&=\sum _{n=1}^\infty \varepsilon ^n\sum _{m=-n}^n \eta ^{n,m}(\xi ,\tau )\,E^m, \end{aligned}$$

(24)

where

$$\begin{aligned}&E=\exp \left( \mathrm {i}(kx-\omega t) \right) ,\quad \phi ^{(n,-m)}={\bar{\phi }}^{(n,m)},\\&\eta ^{(n,-m)} ={\bar{\eta }}^{(n,m)}, \end{aligned}$$

and a bar denotes the complex conjugate. As a next step, the velocity potential at the free surface $\eta$ is expanded in a Taylor series around $z=0$

$$\begin{aligned} \phi (x,\eta ,t)=\sum _{j=0}^\infty \frac{ \eta ^j}{j!} \frac{\partial ^j}{\partial z^j} \phi \bigg |_{z=0} \end{aligned}$$

(25)

Then expansions (23)–(25) are substituted into Eq. (21).

As in [41, 42] the wind-induced pressure $P_a$ is assumed to be of order ${\mathcal {O}}(\varepsilon ^3)$. Therefore, the wind-induced pressure evaluated at $z=0$ is expanded as

$$\begin{aligned} P_a(x,t)=\sum _{n=1}^\infty \varepsilon ^{n-1}\sum _{m=-n}^n p^{n,m}(\xi ,\tau )\,E^m. \end{aligned}$$

(26)

and substituted into boundary condition (21c). Then, terms of linear and quadratic order are not affected by wind forcing and the well-known results for $\phi ^{m,n}(\xi ,z,\tau )$ with $n\le 2$ can be used [72, 73]. These are given as

$$\begin{aligned} \begin{aligned}&\phi ^{1,1}=\psi \frac{\cosh k(z+h)}{\cosh kh},\\&\phi ^{1,0}_z=0,\quad \phi ^{2,0}_z=0,\\&\phi ^{2,1}=D\frac{\cosh k(z+h)}{\cosh kh}-\mathrm {i} \beta _1 \psi _{\xi }, \\&\phi ^{2,2}=\mathrm {i}\beta _2\frac{\cosh 2k(z+h)}{\cosh (2kh)},\quad \eta ^{1,0}=0, \\&g\eta ^{1,1}= \mathrm {i} \omega \psi , \\&g\eta ^{2,0}=c_{\mathrm{g}} \phi ^{1,0}_{\xi }-k^2(1-\theta ^2)|\psi |^2, \\&g\eta ^{2,1}=\mathrm {i}\omega D+ c_{\mathrm{g}}\psi _{\xi }, \quad g\eta ^{2,2}=\beta _3 \psi ^2, \end{aligned} \end{aligned}$$

(27)

where

$$\begin{aligned} \begin{aligned}&\beta _1= -\frac{(z+h)\sinh (k(z+h))-h\theta \cosh (k(z+h))}{\cosh (kh)},\\&\beta _2= \frac{3k^2(1-\theta ^4(kh))}{4\omega \theta ^2},\; \beta _3=\frac{k^2(\theta ^2-3)}{2\theta ^2},\\&\theta =\tanh (kh), \end{aligned} \end{aligned}$$

(28)

$\omega$ fulfills the dispersion relation $\omega =\sqrt{gk\theta }$, and $\psi$ and D are functions of the slow variables $\xi$, $\tau$.

Terms of the velocity potential $\phi$ of order $n=3$ and harmonic $m=0$ in boundary condition (21b) do not involve the wind-induced pressure $P_a$ or the viscosity $\nu$ and lead similarly as in [72] to

$$\begin{aligned} \phi _{\xi }^{1,0}= & {} \beta _4|\psi |^2,\; \mathrm {where}\nonumber \\ \beta _4= & {} -k^2\frac{ 2c_p+c_{\mathrm{g}}(1-\theta ^2) }{gh-c_{\mathrm{g}}^2},\quad \phi _{\xi z}^{1,0}=0, \end{aligned}$$

(29)

and $c_p=\omega /k$ is the phase velocity. From Laplace Eq. (21a) together with bottom boundary condition (21d), ${\phi }^{3,1}$ is obtained as

$$\begin{aligned} \begin{aligned} {\phi }^{3,1}&= \frac{\cosh k(z+h)}{\cosh kh}G+\frac{\beta _1}{2k}\left( 2kh\theta ^2\psi _{\xi \xi }-2\mathrm {i}k D_{\xi } \right) \\&\quad -\,\frac{\left( (z+h)^2-h^2\right) \cosh (k(z+h)}{2 \cosh (kh)}\psi _{\xi \xi }, \end{aligned} \end{aligned}$$

(30)

whereby G is a function of $\xi$, $\tau$. This is the same result as in [74] (without surface tension), since no wind-induced pressure is involved in the Laplace equation and the bottom boundary condition.

Collecting terms of the velocity potential $\phi$ of order $n=3$ and harmonic $m=1$ in boundary condition (21b), the equation

$$\begin{aligned} \begin{aligned}&\phi _{z}^{3,1}+\mathrm {i}\omega {\eta }^{3,1}\\&=-\,c_{g}\,\eta _{\xi }^{2,1}+\eta _{\tau }^{1,1}+\mathrm {i}k{ \eta }^{1,1}\phi _{\xi }^{1,0}\\&\quad +\,2\,\nu \,k^2\,{\eta }^{1,1}-{\eta }^{2,0}\phi _{ zz}^{1,1}-{\bar{\eta }}^{1,1}\phi _{zz}^{2,2}\\&\quad -\,\eta ^{2,2}{\bar{\phi }}_{zz}^{1,1}+\left( \eta ^{1,1}\right) ^2 \left( k^2 {\bar{\phi }}_{zz}^{1,1}-\frac{1}{2} {\bar{\phi }}_{zzz}^{1,1}\right) \\&\quad -\,\eta ^{1,1}{\bar{\eta }}^{1,1} \phi _{zzz}^{1,1} \\&\quad +\,2 k^2 {\bar{\eta }}^{1,1}\phi _{z}^{2,2}+2 k^2 {\eta }^{2,2}{\bar{\phi }}_{z}^{1,1} \end{aligned} \end{aligned}$$

(31)

is obtained. Thereby, terms that are zero in Eq. (27) have already been omitted. The corresponding equation for $n=3$ and $m=1$ in boundary condition (21c) is given by

$$\begin{aligned} \begin{aligned}&-\mathrm {i}{\phi }^{3,1}\omega +g{\eta }^{3,1}\\&=-{\frac{1}{\rho _{{\mathrm{w}}}}}{p}^{1,1}-2\,\nu \,{\phi }^{1,1}_{zz}\\&\quad -\,\frac{1}{2}\mathrm {i}\omega \,{\eta }^{2,2}{{\bar{\phi }}_{{{ zz}}}}^{1,1} -c_{g}\,{\phi }^{2,1}_{\xi }-{\phi }^{ 1,1}_{\tau }\\&\quad -\,\mathrm {i}k{\phi }^{1,1}\phi _{\xi }^{1,0}-\mathrm {i}\omega \,{\eta }^{2,2}{ {\bar{\phi }}_{{z}}}^{1,1}\\&\quad -\,2\,{k}^{2}{\bar{\phi }}^{1,1}{ \phi }^{2,2} -{k}^{2}{\eta }^{1,1}{\phi _{{z}}}^{1,1}{{\bar{\phi }}}^{1,1}\\&\quad -\,{k}^{2}{\eta }^{1,1}{{\bar{\phi }}_{{z}}}^{1,1}{\phi }^{1,1 }+{k}^{2}{{\bar{\eta }}}^{1,1}\phi _{{z}}^{1,1}{\phi }^{1,1}\\&\quad +\,2\,\mathrm {i} \omega \,{{\bar{\eta }}}^{1,1}\phi _{{z}}^{2,2}\\&\quad +\,\mathrm {i}\omega \,{\eta _{}}^{ 1,1}{{\bar{\eta }}}^{1,1}\phi _{{{ zz}}}^{1,1}-\phi _{{z}}^{ 2,2}{{\bar{\phi }}_{{z}}}^{1,1}\\&\quad +\,\mathrm {i}\omega \,{\eta }^{2,0}\phi _{z}^{1,1}\\&\quad -\,{\eta }^{1,1}\phi _{{z}}^{1,1}{{\bar{\phi }}_{{{ zz}}}}^{1,1}-{\eta }^{1,1}{{\bar{\phi }}_{{z}}}^{1,1}\phi _{{{ zz}}}^{1,1}\\&\quad -\,{{\bar{\eta }}}^{1,1}\phi _{{z}}^{1,1}\phi _{{{ zz}}}^{1,1}, \end{aligned} \end{aligned}$$

(32)

and includes the leading-order time- and space-varying wind-induced pressure term $p^{1,1}(\xi ,\tau )$. It is also possible to assume $P_a$ to be of order ${\mathcal {O}}(\varepsilon ^2)$, which involves additional terms due to the wind-induced pressure, as can be seen in [44], where a time- and space-invariant wind velocity is considered. Subtracting (31) from (32) such that the unknown ${\eta }^{3,1}$ vanishes leads finally to the evolution equation for the wave envelope $\psi (\xi ,\tau )$, which is subjected to a time- and space-variant wind-induced pressure. This equation is given by

$$\begin{aligned} \mathrm {i}\,\psi _{\tau }+\mu \,\psi _{\xi \xi }-\gamma \,|\psi |^2 \psi =-\mathrm {i}\,\frac{p^{1,1}}{2\rho _{{\mathrm{w}}}}-\mathrm {i}\,2\,\nu \,k^2\,\psi , \end{aligned}$$

(33)

where $\mu =\frac{1}{2}\frac{\partial ^2}{\partial k^2} \omega (k)$, and

$$\begin{aligned} \begin{aligned} \gamma&=\frac{k^4}{4\omega }\left( \frac{9}{\theta ^2}-12+13 \theta ^2-2\theta ^4\right. \\&\quad -\,\frac{2}{gh-c_{\mathrm{g}}^2}\left( 4c_p^2+4c_p c_{\mathrm{g}}(1-\theta ^2)\right. \\&\quad \left. \left. +\,c_{\mathrm{g}}^2(1-\theta ^2)^2 \right) \right) . \end{aligned} \end{aligned}$$

The dissipative term $-\mathrm {i}\,2\,\nu \,k^2\,\psi$ in Eq. (33) is the same as was for example previously obtained in [75]. It is noted that only the space- and time-variant pressure component $p^{1,1}(\xi ,\tau )$ from pressure expansion (26) is involved in this equation. For deep water ($h\rightarrow \infty$) we get $\gamma =\frac{2k^4}{\omega }$ and (33) reduces with $\theta \rightarrow 1$, $c_{\mathrm{g}} \rightarrow \frac{\omega }{2k}$ and a scaling of $\psi \rightarrow \frac{\omega }{2k} \psi$ to

$$\begin{aligned}&\mathrm {i}\frac{\omega }{2k}\,\psi _{\tau }-\frac{\omega }{8 k^2}\frac{\omega }{2k}\,\psi _{\xi \xi } -\frac{2k^4}{\omega }\frac{\omega ^3}{8k^3} \,|\psi |^2\psi \nonumber \\&\quad =-\mathrm {i}\,\frac{ p^{1,1}}{2\rho _{{\mathrm{w}}}}-\mathrm {i}\,2\,\nu \,k^2\frac{\omega }{2k}\,\psi \nonumber \\&\quad \Longleftrightarrow \mathrm {i}\,\psi _{\tau }-\frac{\omega }{8 k^2}\,\psi _{\xi \xi }-\frac{1}{2}\omega k^2\,|\psi |^2\psi \nonumber \\&\quad =-\mathrm {i}\,k\,\frac{ p^{1,1}}{\omega \rho _{{\mathrm{w}}}}-\mathrm {i}\,2\,\nu \,k^2\,\psi . \end{aligned}$$

(34)

Appendix 2: Relaxation method for nonlinear gravity waves

Using the relaxation scheme, deterministic NLS (6) is discretized by [68, 69]

$$\begin{aligned} \begin{aligned} \frac{\phi ^{n+1/2}+\phi ^{n-1/2}}{2}&=\frac{1}{2}\omega \,k^2|\psi ^n|^2,\\ \mathrm {i}\frac{\psi ^{n+1}-\psi ^n}{\delta t}&=\frac{\omega }{8k^2} \, \left( \frac{\psi ^{n+1}+\psi ^n}{2}\right) _{xx}\\&\quad +\,\phi ^{n+1/2} \frac{\psi ^{n+1}+\psi ^n}{2}\\&\quad +\,\frac{\Gamma ^{n+1}\,\psi ^{n+1}+\Gamma ^n\,\psi ^n}{2}. \end{aligned} \end{aligned}$$

(35)

Moreover, the space can be discretized by a pseudo-spectral approximation, which leads to a high accuracy, see for example [69]. For the white noise case, where the excitation $\zeta$ of stochastic NLS (8) is given by a scalar time-varying white noise $\zeta =\zeta (t)$ as specified in Definition 1, the stochastic integral is approximated in the sense of Stratonovich by

$$\begin{aligned}&\int _{t_n}^{t_{n+1}}\psi (s,x)\circ \mathrm {d}W(s) \nonumber \\&\quad \approx \frac{\psi (t_{n+1},x)+\psi (t_{n},x)}{2}\, ( W_{t_{n+1}}- W_{t_n}), \end{aligned}$$

(36)

where $\mathrm {d}W(t)=\zeta (t)\, \mathrm {d}t$ is the increment of the standard Wiener process $W_t$.

We set ${\chi ^{n+1/2}:=( W_{t_{n+1}}- W_{t_n})/\sqrt{\delta t}}$, which are distributed according to the normal distribution ${\mathcal {N}}(0,1)$. Then the relaxation scheme for the NLS, which is excited by white noise in time, reads

$$\begin{aligned} \begin{aligned} \frac{\phi ^{n+1/2}+\phi ^{n-1/2}}{2}&=\frac{1}{2}\omega \,k^2|\psi ^n|^2,\\ \mathrm {i}\frac{\psi ^{n+1}-\psi ^n}{\delta t}&=\frac{\omega }{8k^2} \, \left( \frac{\psi ^{n+1}+\psi ^n}{2}\right) _{xx}\\&\quad +\,\phi ^{n+1/2} \frac{\psi ^{n+1}+\psi ^n}{2}\\&\quad +\,\frac{1}{2\sqrt{\delta t}}\chi ^{n+1/2}(\psi ^{n+1}+\psi ^n). \end{aligned} \end{aligned}$$

(37)

For the above numerical calculations, the GPELab toolbox can be used [68]. In the case, where the NLS is excited by a random wind process in time, we first have to generate the nonwhite wind velocity process $U(z,t_n)$ by means of a CARMA process, as given in Definition 1. This can be done using for example the Euler–Maruyama scheme [76]. From the randomly time-varying wind velocity process $U(z,t_n)$ the friction velocity $u_*(t_n)$ is calculated by means of a fix point iteration at each instant of time using Eq. (9). The resulting random friction velocity $u_*(t)$ defines the stochastic process $\Gamma ^n$, which is then substituted into Eq. (35) where

$$\begin{aligned} \Gamma ^n:=\frac{k\omega }{2g}\frac{\rho _{\mathrm{a}}}{\rho _{{\mathrm{w}}}} \,\beta \,\left( \frac{u_*(t_n)}{\kappa }\right) ^2 -2\,\nu \,k^2. \end{aligned}$$

(38)

Appendix 3: CARMA process

A CARMA process is defined as follows, cf. [2, 3].

Definition 1

(CARMA(p,q) process) A CARMA(p,q) process y(t), $0\le q< p$, is defined as the stationary solution of

$$\begin{aligned} y={\mathbf {c}}\;{\mathbf {u}}(t), \end{aligned}$$

(39)

with the linear differential equation for the state vector ${\mathbf {u}}(t)\in \mathbb {R}^p$

$$\begin{aligned} {\dot{\mathbf{u}}}(t)=\mathrm {{\mathbf {A}}}\;{\mathbf {u}}(t)+{\mathbf {b}}\; \xi (t), \end{aligned}$$

(40)

where $\xi (t)$ is white noise with $E\{\xi (t)\}=0$ and $E\{\xi (t)\xi (t+\tau )\}=\sigma ^2\delta (\tau )$, $\;\sigma \in \mathbb {R}$, $\delta (\cdot )$ is the Dirac function,

$$\begin{aligned} \begin{aligned} \mathrm {{\mathbf {A}}}&=\left( \begin{matrix} -a_1&{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ -a_2&{}\quad 0 &{}\quad 1 &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad 0\\ -a_{p-1}&{}\quad 0 &{} \quad \cdots &{}\quad 0&{}\quad 1 \\ -a_p&{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \end{matrix}\right) ,\\ {\mathbf {b}}&=\left( \begin{array}{c} b_{p-1}\\ b_{p-2}\\ \vdots \\ b_1\\ b_0\\ \end{array}\right) ,\;\;\; {\mathbf {c}}= \left( \begin{array}{c} 1\\ 0\\ \vdots \\ 0\\ \end{array}\right) , \end{aligned} \end{aligned}$$

(41)

and $b_j=0$ if $j>q$.

Equation (40) can be expressed as the stochastic differential equation

$$\begin{aligned} \mathrm {d}{\mathbf {u}}(t)=\mathrm {{\mathbf {A}}}\,{\mathbf {u}}(t) \,\mathrm {d}t+{\mathbf {b}}\, \mathrm {d}W(t), \end{aligned}$$

(42)

using the relation $\mathrm {d}W(t)=\xi (t)\, \mathrm {d}t$ between the white noise $\xi (t)$ and the increment of the Wiener process $\mathrm {d}W(t)$. The above-defined CARMA(p,q) process has the transfer function

$$\begin{aligned} H_{\mathrm {carma}}(s)=\frac{b_0+b_1s+\cdots +b_qs^q}{s^p+a_1 s^{p-1}+\cdots +a_p}, \end{aligned}$$

(43)

where $s:=\mathrm {i}\omega$, and the spectral density

$$\begin{aligned} S_{\mathrm {carma}}(\omega )=\sigma ^2\frac{|b_0+b_1s+\cdots +b_qs^q|^2}{|s^p+a_1 s^{p-1}+\cdots +a_p|^2}. \end{aligned}$$

(44)

Thereby, $\omega$ is the angular frequency. It should be noted that the state space representation in Definition 1 is not unique.

Appendix 4: Forces on a cylindrical pile

In order to compute the wave-induced hydrodynamic force on a vertical cylindrical pile, the velocity and acceleration of water particles have to be computed. This can be done by determining the corresponding velocity potential. Given the complex-valued wave envelope $\psi =\psi (x,t)$, obtained from the solution of NLS (6), the velocity potential at a given depth z was computed by Carter et al. [47]

$$\begin{aligned} \begin{aligned} \phi (x,z,t)&=\biggl \lbrace \frac{\mathrm {i} \varepsilon \omega }{k} {\hat{B}} +\frac{\varepsilon ^2 \omega }{2k^2} {\hat{B}}_X + \varepsilon ^3 \\&\quad \left( -\frac{\mathrm {i} k \omega }{2} |{\hat{B}}|^2 {\hat{B}} - \frac{3 \mathrm {i} \omega }{8 k^3} {\hat{B}}_{XX} \right) \biggl \rbrace e ^{\mathrm {i} \omega t - \mathrm {i} kx + kz} \\&\quad +\, {\mathcal {O}}(\varepsilon ^4) + c.c., \end{aligned} \end{aligned}$$

(45)

where ${\hat{B}}=B(X+\mathrm {i}Z,T)$, $X=\varepsilon x$, $Z=\varepsilon z$, $T=\varepsilon ^2 t$ and c.c. represents the complex conjugate. The velocity u in x-direction, velocity v in z-direction, and acceleration a in x-direction can then be computed by

$$\begin{aligned} u(x,z,t)= & {} \frac{\partial \phi }{\partial x} (x,z,t),\quad v(x,z,t)=\frac{\partial \phi }{\partial z} (x,z,t),\nonumber \\ a(u,z,t)= & {} \frac{\partial u}{\partial t} (x,z,t). \end{aligned}$$

(46)

With that, the force $d F_x$ acting on a part $d z$ of a vertical cylindrical pile with diameter D can be computed by means of the Morison equation and reads [48]

$$\begin{aligned} d F_x=C_{\mathrm {m}}\rho _{\mathrm{w}} \frac{\pi D^2}{4} d z a + C_{\mathrm{d}} \frac{\rho _{\mathrm{w}}}{2} D d z |u|u, \end{aligned}$$

(47)

whereby $C_{\mathrm {m}}$ is the inertia coefficient, $C_{\mathrm{d}}$ is the drag coefficient, and $\rho _{\mathrm{w}}$ is the density of the incompressible fluid.