eurm10 msam10 1–30

# Growth of Stokes Waves Induced by Wind on a Viscous Liquid of Infinite Depth

SHAHRDADG.SAJJADI  Department of Mathematics, ERAU, Florida, USA,
and Trinity College, Cambridge, UK.

(2003; April 2016)

###### Abstract

The original investigation of Lamb (1932, §349) for the effect of viscosity on
monochromatic surface waves is extended to account for second-order Stokes
surface waves on deep water in the presence of surface tension. This extension
is used to evaluate interfacial impedance for Stokes waves under the
assumption that the waves are growing and hence the surface waves are
unsteady. Thus, the previous investigation of Sajjadi et al. (2014) is further
explored in that (i) the surface wave is unsteady and nonlinear, and (ii) the
effect of the water viscosity, which affects surface stresses, is taken into
account. The determination of energy-transfer parameter, from wind to waves,
are calculated through a turbulence closure model but it is shown the
contribution due to turbulent shear flow is some 20% lower than that obtained
previously. A derivation leading to an expression for the closed streamlines
(Kelvin cat-eyes), which arise in the vicinity of the critical height, is
found for unsteady surface waves. From this expression it is deduced that as
waves grow or decay, the cats-eye are no longer symmetrical. Also investigated
is the energy transfer from wind to short Stokes waves through the viscous
Reynolds stresses in the immediate neighborhood of the water surface. It is
shown that the resonance between the Tollmien-Schlichting waves for a given
turbulent wind velocity profile and the free-surface Stokes waves give rise to
an additional contribution to the growth of nonlinear surface waves.

††volume: 268

##  1 Introduction

The energy exchange from wind to waves crucially depends on accurate
determination of stresses on the water surface. The energy-transfer parameter
(as is commonly known in the literature) is determined from the complex part
of the interfacial impedance (as is termed by Miles), see (11) below. John
Miles made several analytical attempts to improve upon the energy-transfer
parameter, beginning with his pioneering work in 1957 and his final
contribution in 1996. In all his contributions he assumed the initial surface
is composed of a monochromatic surface wave of small steepness. Miles (2004)
remarked ”… it will be interesting to see the extension of my 1996
contribution to Stokes waves, and its comparison with some numerical studies.”
Although it was not explicitly mentioned by Miles, we assume he was referring
to numerical contributions, for example, by Al-Zanaidi & Hui (1984) and
Mastenbroek et al. (1996). However, he had recognized a major obstacle for
this task, and further commented ”… the proper determination of the
interfacial impedance for an a priori assumed nonlinear waves is by no means
straightforward…”. In this note we offer a way to resolve this anomaly.

In a recent study by Sajjadi, Hunt, and Druillion (2014), it was shown that
the growth rate, $kc_{i}$, where $k$ is the wave number and $c_{i}$ is the
wave complex part of the wave phase speed, for growing waves critically depend
on the energy-transfer parameter $\beta$. Moreover, Sajjadi and Hunt (2003)
(SHD therein) have suggested that wave steepness (for nonlinear waves, such as
those observed in the sea) are also a contributing factor for the momentum
transfer from wind to surface waves, see also Sajjadi (2015). Thus, one goal
of the present study is the accurate determination of $\beta$, and this
requires the calculation of the complex amplitude of the wave-induced pressure
at the surface.

To achieve our objective we must extend the effect of viscosity on
monochromatic waves (Lamb 1932, §349) to a nonlinear surface wave, here we
shall consider the simplest case, i.e. that of the Stokes wave. Thus, we shall
adopt the bicrohomic assumption for the mean motion which admits the
representation of the form (see section 4)

$(\sigma,\chi,\tau)=\mbox{Re}\left\\{({\mathscr{P}}_{1},\Xi_{1},{\mathscr{T}}_{1})e^{ik(x-ct)}+ka({\mathscr{P}}_{2},\Xi_{2},{\mathscr{T}}_{2})e^{2ik(x-ct)}\right\\}$

With ${\mathscr{P}}_{1},\cdots$, representing complex amplitude of stresses,
and where $c=c_{r}+ic_{i}$ ($c_{r}$ is the wave speed) and $ka$ is the wave
steepness.

For an unsteady monochromatic surface wave SHD showed that the total energy
transfer comprises the sum of two components $\beta_{c}+\beta_{T}$,

$\displaystyle\beta_{c}=-\pi(U_{c}^{\prime\prime}/kU_{c}^{\prime})(\overline{{\mathscr{W}}_{c}^{2}}/U_{1}^{2}h^{2}_{0x}),\qquad(\,\,\,)^{\prime}\equiv\displaystyle\frac{d}{dz}$
(1)

is the contribution associated with the singularity at the critical layer
Miles (1957), but due to the unsteadiness of the surface wave $\beta_{c}$ is
additionally a function of $c_{i}$ (see SHD). In (1.1), the overbar signifies
an average over $x,U_{1}$ is the kinematic friction velocity, ${\cal W}$ is
the wave-induced vertical velocity, and the subscript $c$ denotes evaluation
at the critical point $z=z_{c}$, where $U(z_{c})=c_{r}$. The second component,
$\beta_{T}$ is the rate of energy transfer to the surface due to the turbulent
shear flow blowing over it.

Thus, we determine the energy-transfer parameter through calculating the
pressure ${\mathscr{P}}_{\ell}$, and the shear stress ${\mathscr{T}}_{\ell}$,
$(\ell=1,2)$ at the surface. As mentioned above, this requires generalization
of the interfaced impedance by extending the monochromatic viscous theory of
water waves (Lamb 1932 §349) to account for the effect of viscosity for Stokes
waves. Then, the extended Lamb’s solution to Stokes wave in a viscous liquid,
with prescribed stresses at the surface can be adopted to evaluate expressions
for $\beta_{c}$ and $\beta_{T}$ in the limit as
$s\equiv\rho_{a}/\rho_{w}\downarrow 0$, as outlined in section 3 below.

Hence, following the procedure adopted by SHD, for unsteady monochromatic
waves, which led through evaluation of ${\mathscr{P}}_{0}$ and
${\mathscr{T}}_{0}$, to the following expressions

$\displaystyle\beta_{c}=\pi\xi_{c}^{3}L_{0}^{4}\left[1+(4-\textstyle\frac{1}{3}\pi^{2}+10\hat{c}_{i}^{2})\Lambda^{2}+O(\Lambda^{2})\right]$
(2) $\displaystyle\beta_{T}=5\kappa^{2}L_{0}+O(\Lambda)$ (3)

is obtained for an unsteady Stokes wave in sections 4 and 5. In equations
(1.2) and (1.3) $\xi_{c}=kz_{c}$, $L_{0}=\gamma-\ln(2\xi_{c})$ and
$\gamma=0.5772$ is Euler’s constant.

In section 6, we derive an expression for closed loop streamlines, namely
Kelvin’s cat-eye, and the significance of which is explored and explained.
Finally, the results and discussion is given in section 7.

##  2 Interfacial impedance

Miles-Sajjadi (Miles 1996, Sajjadi 1998, hereafter M96 and S98 respectively)
theory of surface wave generation considers the role of wave-induced Reynolds
stresses in the transfer of energy from a turbulent shear flow to gravity
waves on deep water. In their theories the Reynolds-averaged equations for
turbulent flow over a deep-water sinusoidal gravity wave, $z=a\cos kx\equiv
h_{0}(x)$ (M96) and fully nonlinear surface gravity wave
$z=\sum^{\infty}_{n=1}a_{n}\cos k_{n}x$ (S98), are formulated. Their
formulations uses the wave-following coordinates $\xi,\eta$, where
$x=\xi,z=\eta+h(\xi,\eta),h(\xi,0)=h_{0}(\xi)$ and $h$ is exponentially small
for $k\eta\gg 1$. The turbulent Reynolds stress equations are closed by a
viscoelastic constituent equation-a mixing-length model with relaxation (M96)
and by the rapid-distortion theory (S98). Both derive their evolution equation
on the assumption that: (i) the basic velocity profile is logarithmic in
$\eta+z_{0}$, where $z_{0}$ is a roughness length; (ii) the lateral transport
of turbulent energy in the perturbed flow is negligible; and (iii) the
dissipation length is proportional to $\eta+z_{0}$. In both theories an
inhomogeneous counterpart of the Orr-Sommerfeld equation is derived for the
complex amplitude of the perturbation streamfunction and then used to
construct a quadratic functional for the energy transfer to the wave. A
corresponding Galerkin approximation that is based on independent variational
approximation for outer (quasi-laminar) and inner (shear-stress) domains
yields the interfacial impedance $\alpha+\mbox{i}\beta$ (defined by Miles
1957) in the limit $s\equiv\rho_{\rm a}/\rho_{\rm w}\downarrow 0$. The
calculation of the interfacial impedance requires the solution of the
linearized equations of motion of water bounded above by a monochromatic
surface wave (Lamb 1932, §349).

However, for Stokes surface wave $z=a\cos kx+\textstyle\frac{1}{2}ka^{2}\cos
2kx$ the extension of Lamb’s solution is not immediately obvious (and Lamb, as
well as other researchers to date, did not address this problem). However, in
the case of a shear flow over a sinusoidal wave (for the application to air-
sea interactions), we may consider the solution of the linearized Navier-
Stokes equations in the semi-infinite body of water bounded above by the
surface wave

$\displaystyle z=a{\rm e}^{{\rm i}k(x-ct)}\equiv h_{0}(x,t)\qquad(ka\ll 1)$
(4)

in a fixed frame of reference gives, after renaming variables, that is after
letting $A=\mbox{i}ac(1+\mathscr{C}),C=ac\mathscr{C},n=-\mbox{i}kc$, (where
the letters on the left-hand side denote those used by Lamb), and neglecting
surface tension therein; following Lamb and adopting complex dependent
variables, we obtain

$\displaystyle u_{\rm w}=[k+\mathscr{C}(k-m)]ch_{0}$ (5)
$\displaystyle\tau_{\rm w}=(\tau_{13})_{\rm w}=(2\nu_{\rm w}kc+{\rm
i}\mathscr{C}c^{2})kh_{0}$ (6)

and

$\displaystyle\left(-\displaystyle\frac{p}{\rho}+\tau_{33}\right)_{\rm
w}=\left\\{\displaystyle\frac{g}{k}-c^{2}-2{\rm i}k\nu_{\rm
w}c-\mathscr{C}[c^{2}-2{\rm i}(k-m)\nu_{\rm w}c]\right\\}kh_{0}$ (7)

for the tangential velocity, tangential stress, and normal stress,
respectively, at the surface. The subscript $w$ refers to water,
$m\equiv[k^{2}-\mbox{i}(kc/\nu_{\rm
w})]^{\mbox{\tiny$\textstyle\frac{1}{2}$}}$, and $s=\rho_{a}/\rho_{w}$ being
the ratio of air to water density.

Invoking continuity of the perturbation velocity $u=c\partial h/\partial\eta$
and $\tau$ and $-\sigma$, the tangential and normal stresses, eliminating
$\mathscr{C}$, and letting $k\nu_{\rm w}/c\downarrow 0$, we obtain the
interfacial conditions111We emphasize that, $u$ is perturbation to tangential
wind velocity and $c$ is the complex phase speed induced by the presence of
the wind.

$\displaystyle u-{\rm i}(kc\nu_{\rm
w})^{-\mbox{\tiny$\textstyle\frac{1}{2}$}}s\tau=kch_{0}$ (8) $\displaystyle
s(\sigma+{\rm i}\tau)=(c^{2}-c^{2}_{\rm w})kh_{0}$ (9)

where

$\displaystyle c_{\rm
w}\equiv(g/k)^{\mbox{\tiny$\textstyle\frac{1}{2}$}}-2{\rm i}k\nu_{\rm
w}\qquad(|k\nu_{\rm w}/c|\ll 1)$ (10)

is the complex phase speed in the absence of the air comprehends (through its
imaginary part, which may be replaced by an empirical equivalent) the
dissipation in water. The ratio of the second term to the first term on the
left-hand side of (8) is typically smaller than $10^{-2}$; accordingly (8) may
be approximated by $u=kch_{0}$. However, we note that (8) does not reduce to
$c=kch_{0}$ in the limit of air inviscid liquid $(k\nu_{\rm w}\rightarrow 0)$.

Finally by replacing $\sigma$ and $\tau$ with their complex amplitudes
$\mathscr{P}$ and $\mathscr{T}$ (defined as in Miles 1957) the interfacial
impedance is obtained which may be expressed in the form

$\displaystyle\alpha+{\rm i}\beta\equiv\displaystyle\frac{c^{2}-c_{\rm
w}^{2}}{sU^{2}_{1}}=\displaystyle\frac{({\mathscr{P}}+{\rm
i}{\mathscr{T}})_{0}}{kaU^{2}_{1}}$ (11)

where $U_{1}=U_{*}/\kappa,U_{*}$ is the kinematic shear stress, $\kappa\approx
0.4$ is von Kármán’s constant, and the suffix zero indicates evaluation on
$\eta=0$, which to $O(ka)\ll 1$ is the same as evaluation at $z=0$.

Sajjadi (1998) followed Miles (1996) and calculated the interfacial impedance
for every harmonic of the fully nonlinear surface wave. We remark that
although M96 and S98 formulations are basically different for the turbulent
flow over a surface wave, nevertheless the form of the interfacial impedance
adopted are the same, provided Sajjadi’s series, for the representation of a
fully nonlinear surface wave, is truncated after the first harmonic. Moreover,
in S98, the inclusion of surface tension leads to an ambiguous results, even
for the second-order Stokes wave, when his series is truncated after the
second harmonic.

This ambiguity can readily be seen from equation (30) below in which

$\displaystyle(\phi,\zeta,w)=\sum_{s}(\Phi_{s},\Xi_{s},W_{s})$

where

$\displaystyle\Phi_{s}=A_{s}{\rm e}^{k_{s}z},\qquad\Xi_{s}=\left(A_{s}{\rm
e}^{k_{s}z}-{\rm i}C_{s}{\rm e}^{m_{s}z}\right),\qquad W_{s}=-k_{s}\Xi_{s},$

then substituting into (30) then for the $s$-harmonic we have

$\displaystyle\mbox{[LHS
(\ref{12})]}=-\displaystyle\frac{1}{n_{s}}\left\\{[n_{s}^{2}+k_{s}(g+{\cal
T}k^{2})+2\nu_{w}k_{s}^{2}n_{s}]A_{s}-{\rm i}k_{s}[2\nu_{w}m_{s}n_{s}+g+{\cal
T}k^{2}]C_{s}\right\\}$

Here distinction has to be made between $k$, the wavenumber of the entire
wave, and $k_{s}$, the wavenumber associated with the $s$-harmonic. One may
make the seemingly obvious assumption that $k_{s}=sk$, however, this will lead
to a result which appears to be wrong. Note incidently, this ambiguity can be
circumvented if surface tension is neglected as in S98.

The purpose of this note is to extend Lamb’s original investigation, for
monochromatic waves, to Stokes waves in the presence of surface tension but
under the assumption that the wave steepness $ka\ll 1$.

##  3 Stokes waves on a viscous liquid

We consider the effect of viscosity on Stokes waves on deep water whose
profile is given by

$\displaystyle z=a{\rm e}^{{\rm i}k(x-ct)}+\textstyle\frac{1}{2}ka^{2}{\rm
e}^{2{\rm i}k(x-ct)}\equiv h_{0}+kah_{1}=h(x,t),\qquad(ka\ll 1)$ (12)

see figure 1.

Figure 1: Schematic diagram showing a Stokes wave.

If we take the $z$-axis to be vertically upwards, and if we assume a two-
dimensional motion with velocities ${\sf u}$ and ${\sf w}$ being confined to
the $x,z$-coordinates and pressure ${\sf p}$, then ignoring the inertia terms,
the equations of motion may be cast as

$\displaystyle\displaystyle\frac{\partial\sf u}{\partial
t}=-\displaystyle\frac{1}{\rho_{\rm w}}\displaystyle\frac{\partial\sf
p}{\partial x}+\nu_{\rm w}\nabla^{2}{\sf
u},\qquad\displaystyle\frac{\partial\sf w}{\partial
t}=-\displaystyle\frac{1}{\rho_{\rm w}}\displaystyle\frac{\partial\sf
p}{\partial z}-g+\nu_{\rm w}\nabla^{2}{\sf w},$ (13)

together with the continuity equation

$\displaystyle\displaystyle\frac{\partial\sf u}{\partial
x}+\displaystyle\frac{\partial\sf w}{\partial z}=0$ (14)

where $\rho_{\rm w},\nu_{\rm w}$ and $g$ are the water density, the kinematic
viscosity of water and acceleration due to gravity, respectively.

Equations (13) and (14) are satisfied by

$\displaystyle{\sf u}=-\displaystyle\frac{\partial\phi}{\partial
x}-\displaystyle\frac{\partial\psi}{\partial z},\qquad{\sf
w}=-\displaystyle\frac{\partial\phi}{\partial
z}+\displaystyle\frac{\partial\psi}{\partial x},$ (15)

and the linearized dynamic condition

$\displaystyle\displaystyle\frac{\sf p}{\rho_{\rm
w}}=\displaystyle\frac{\partial\phi}{\partial t}-gz$ (16)

provided

$\displaystyle\nabla_{1}^{2}\phi=0,\qquad\displaystyle\frac{\partial\psi}{\partial
t}=\nu_{\rm w}\nabla^{2}_{1}\psi,$ (17)

where

$\nabla^{2}_{1}\equiv\displaystyle\frac{\partial^{2}}{\partial
x^{2}}+\displaystyle\frac{\partial^{2}}{\partial z^{2}}.$

We consider the solutions in normal mode by assuming that they are periodic in
$x$ with a prescribed wavelength $\lambda=2\pi/k$. Thus, assuming transient
factors ${\rm e}^{nt},{\rm e}^{2nt}$ and spacial factors ${\rm e}^{{\rm
i}kx},{\rm e}^{2{\rm i}kx}$, for first and second harmonics, respectively. The
solution of (17) may therefore be expressed in the following form:

$\displaystyle\left.\begin{array}[]{l}\phi=(A_{1}{\rm e}^{kz}+B_{1}{\rm
e}^{-kz}){\rm e}^{{\rm i}kx+nt}+\textstyle\frac{1}{4}ka(A_{2}{\rm
e}^{2kz}+B_{2}{\rm e}^{-2kz}){\rm e}^{2({\rm i}kx+nt)}\\\ \\\ \psi=(C_{1}{\rm
e}^{mz}+D_{1}{\rm e}^{-mz}){\rm e}^{{\rm
i}kx+nt}+\textstyle\frac{1}{4}ka(C_{2}{\rm e}^{2mz}+D_{2}{\rm e}^{-2mz}){\rm
e}^{2({\rm i}kx+nt)}\\\ \end{array}\right\\}$ (21)

with

$\displaystyle m^{2}=k^{2}+n/\nu_{\rm w}$ (22)

The boundary conditions will provide equations which are sufficient to
determine the nature of the various modes, and the corresponding values of
$n$.

In the case of infinite depth one of these conditions takes the form that the
motion must be finite at $z=-\infty$. Excluding for the present case where $m$
is purely imaginary, this requires that $B_{i}=D_{i}=0$ for $i=1,2$ provided
$m$ denote the root of equation (22) with $\mbox{Re}\\{m\\}>0$. Hence

$\displaystyle\left.\begin{array}[]{l}{\sf u}=-({\rm i}kA_{1}{\rm
e}^{kz}+mC_{1}{\rm e}^{mz}){\rm e}^{{\rm i}kx+nt}-\textstyle\frac{1}{2}ka({\rm
i}kA_{2}{\rm e}^{2kz}+mC_{2}{\rm e}^{2mz}){\rm e}^{2({\rm i}kx+nt)}\\\ \\\
{\sf w}=-k(A_{1}{\rm e}^{kz}-{\rm i}C_{1}{\rm e}^{mz}){\rm e}^{{\rm
i}kx+nt}-\textstyle\frac{1}{2}k^{2}a(A_{2}{\rm e}^{2kz}-{\rm i}C_{2}{\rm
e}^{2mz}){\rm e}^{2({\rm i}kx+nt)}\\\ \end{array}\right\\}$ (26)

Since $h$ denotes the elevation at the free surface, then the linearized
kinematic condition is $\partial h/\partial t={\sf w}$. Taking the origin of
$z$ at the undisturbed level, this condition gives

$\displaystyle h=-\displaystyle\frac{k}{n}(A_{1}-{\rm i}C_{1}){\rm e}^{{\rm
i}kx+nt}-\displaystyle\frac{k^{2}a}{4n}(A_{2}-{\rm i}C_{2}){\rm e}^{2({\rm
i}kx+nt)}.$ (27)

Let $T$ be the surface tension, then the stress conditions at the free surface
are given by

$\displaystyle p^{(zz)}=T\displaystyle\frac{\partial^{2}h}{\partial
x^{2}},\qquad p^{(xz)}=0$ (28)

to the first order, since we have assumed the inclination of the surface to
the horizontal is sufficiently small, so that $ka\ll 1$.

Now, if $\mu_{\rm w}$ denotes the dynamic viscosity of the water,

$\displaystyle p^{(zz)}=-{\sf p}+2\mu_{\rm
w}\displaystyle\frac{\partial^{2}\sf w}{\partial z^{2}},\qquad
p^{(xz)}=\mu_{\rm w}\left(\displaystyle\frac{\partial\sf w}{\partial
x}+\displaystyle\frac{\partial\sf u}{\partial z}\right)$ (29)

whence, by (16) and (27) we find, at the surface

$\displaystyle\displaystyle\frac{p^{(zz)}}{\rho_{\rm w}}-{\cal
T}\displaystyle\frac{\partial^{2}\zeta}{\partial
x^{2}}=-\displaystyle\frac{\partial\phi}{\partial t}+(g+{\cal
T}k^{2})\zeta+2\nu_{\rm w}\displaystyle\frac{\partial\sf w}{\partial z}$ (30)

where ${\cal T}=T/\rho_{\rm w}$. Next, writing
$p^{(zz)}=p_{1}^{(zz)}+kap_{2}^{(zz)}$, $h=h_{0}+kah_{1}$,
$\phi=\phi_{1}+ka\phi_{2}$ and ${\sf w}={\sf w}_{1}+ka{\sf w}_{2}$, then from
(30), (27), (26) and (21) we have, after equating coefficients for ${\rm
e}^{{\rm i}kx+nt}$ and ${\rm e}^{2({\rm i}kx+nt)}$, respectively,

$\displaystyle\displaystyle\frac{p_{1}^{(zz)}}{\rho_{\rm w}}-{\cal
T}\displaystyle\frac{\partial^{2}h_{0}}{\partial
x^{2}}=-\displaystyle\frac{1}{n}\left\\{[n^{2}+k(g+{\cal T}k^{2})+2\nu_{\rm
w}nk^{2}]A_{1}-{\rm i}k[2\nu_{\rm w}mn+g+{\cal T}k^{2}]C_{1}\right\\}$
$\displaystyle\displaystyle\frac{p_{2}^{(zz)}}{\rho_{\rm w}}-{\cal
T}\displaystyle\frac{\partial^{2}h_{1}}{\partial
x^{2}}=-\displaystyle\frac{1}{n}\left\\{\left[\displaystyle\frac{n^{2}}{2}+\displaystyle\frac{k}{4}(g+{\cal
T}k^{2})+2\nu_{\rm w}nk^{2}\right]A_{2}-{\rm i}k\left[2\nu_{\rm
w}mn+\textstyle\frac{1}{4}(g+{\cal T}k^{2})\right]C_{2}\right\\}$

Similarly, writing $p^{(xz)}=p_{1}^{(xz)}+akp_{2}^{(xz)}$ and
$u=u_{1}+aku_{2}$, we obtain from (29) and (26)

$\displaystyle\displaystyle\frac{p_{1}^{(xz)}}{\rho_{\rm w}}=-\\{2{\rm
i}\nu_{\rm w}k^{2}A_{1}+(n+2\nu_{\rm w}k^{2})C_{1}\\}$ (31)
$\displaystyle\displaystyle\frac{p_{2}^{(xz)}}{\rho_{\rm w}}=-\\{2{\rm
i}\nu_{\rm w}k^{2}A_{2}+(n+2\nu_{\rm w}k^{2})C_{2}\\}$ (32)

Substituting (31) into the first equation of (28), and eliminating the ratio
$A_{1}:C_{1}$, we see to $O(1)$ that

$\displaystyle(n+2\nu_{\rm w}k^{2})^{2}+gk+{\cal T}k^{3}=4\nu_{\rm
w}^{2}k^{2}m$ (33)

Similarly, substitution of (32) into the second equation of (28), and
eliminating the ratio $A_{2}:C_{2}$, yields, to $O(ka)$

$\displaystyle(n+4\nu_{\rm w}k^{2})(n+2\nu_{\rm w}k^{2})+gk+{\cal
T}k^{3}=16\nu_{\rm w}^{2}k^{2}m$ (34)

Eliminating $m$ between (33) and (34) then gives

$n=-2\nu_{\rm w}k^{2}\pm\sqrt{4\nu_{\rm w}^{2}k^{4}-6\sigma^{2}}$

and by virtue of the fact that $\nu_{\rm w}k\ll 1$, we obtain

$\displaystyle n=-2\nu_{\rm w}k^{2}\pm{\rm i}\sigma^{*}$ (35)

where $\sigma^{*}=\sqrt{6}\sigma$ and $\sigma^{2}=gk+{\cal T}k^{3}$.

The condition $p^{(xz)}=0$ shows that

$\displaystyle\displaystyle\frac{C_{1}}{A_{1}}=\displaystyle\frac{C_{2}}{A_{2}}=-\displaystyle\frac{2{\rm
i}\nu_{\rm w}k^{2}}{n+2\nu_{\rm w}k^{2}}=\mp\displaystyle\frac{2\nu_{\rm
w}k^{2}}{\sigma^{*}}$ (36)

which is, under the same circumstances, very small. Hence the motion is
approximately irrotational, with a velocity potential

$\displaystyle\phi=A_{1}{\rm e}^{-2\nu_{\rm w}k^{2}t+kz+{\rm
i}(kx\pm\sigma^{*}t)}+\textstyle\frac{1}{4}kaA_{2}{\rm e}^{-4\nu_{\rm
w}k^{2}t+2kz+2{\rm i}(kx\pm\sigma^{*}t)}$ (37)

If we put

$A_{1}=A_{2}=\mp\displaystyle\frac{{\rm i}\sigma^{*}a}{k}$

the equation (27) of the free surface becomes, on taking the real parts

$\displaystyle h=a{\rm e}^{-2\nu_{\rm
w}k^{2}t}\cos(kx\pm\sigma^{*}t)+\textstyle\frac{1}{4}a^{2}k{\rm e}^{-4\nu_{\rm
w}k^{2}t}\cos 2(kx\pm\sigma^{*}t)$ (38)

Since the motion is nearly (but not exactly) irrotational, there is vorticity
present whose magnitude is given by

$\omega=\displaystyle\frac{\partial\sf w}{\partial
x}-\displaystyle\frac{\partial\sf u}{\partial z}\equiv\nabla_{1}^{2}\psi$

Thus, from (22) and (35), we have approximately

$m=(1\pm{\rm i})b\quad{\rm where}\quad b=(\sigma^{*}/2\nu_{\rm w})^{1/2}$

Hence, with the same notation as before, we find

$\displaystyle\omega=$ $\displaystyle\mp$ $\displaystyle 2\sigma^{*}ka{\rm
e}^{-2\nu_{\rm w}k^{2}t+bz}\cos\\{kx\pm(\sigma^{*}t+bz)\\}$ (39)
$\displaystyle\mp$ $\displaystyle 2\sigma^{*}k^{2}a^{2}{\rm e}^{-4\nu_{\rm
w}k^{2}t+2bz}\cos 2\\{kx\pm(\sigma^{*}t+bz)\\}$

From equation (39) it can be seen that the vorticity diminishes rapidly from
the surface downwards. Moreover, since the motion has an oscillatory
character, the sign of the vorticity which is being diffused inwards from the
surface is continually reversing, such that (paraphrasing Lamb) ‘beyond a
stratum’ of thickness of $O(2\pi/b)$ the effect diminishes.

The above analysis gives results for the first two components of the normal
modes of the prescribed wavelength. For a fully nonlinear Stokes wave, there
are an infinitely more of these modes exist and they correspond to pure-
imaginary values of $m$, which are less persistent in character.

It is interesting to note that, if we now, in place of (21), assume

$\displaystyle\left.\begin{array}[]{l}\phi=A_{1}{\rm e}^{kz}{\rm e}^{{\rm
i}kx+nt}+\textstyle\frac{1}{4}kaA_{2}{\rm e}^{2kz}{\rm e}^{2({\rm i}kx+nt)}\\\
\\\ \psi=(C_{1}\cos\ell z+D_{1}\sin\ell z){\rm e}^{{\rm
i}kx+nt}+\textstyle\frac{1}{4}ka(C_{2}\cos 2\ell z+D_{2}\sin\ell z){\rm
e}^{2({\rm i}kx+nt)}\\\ \end{array}\right\\}$ (43)

and carrying out the previous analysis, we find to $O(1)$

$\displaystyle\left.\begin{array}[]{r}(n^{2}+2\nu_{\rm w}k^{2}n+gk+{\cal
T}k^{3})A_{1}-{\rm i}(gk+{\cal T}k^{3})C_{1}+2{\rm i}\nu_{\rm w}k\ell
nD_{1}=0\\\ \\\ 2{\rm i}k^{2}A_{1}+(k^{2}-\ell^{2})C_{1}=0\\\
\end{array}\right\\}$ (47)

and to $O(ka)$

$\displaystyle\left.\begin{array}[]{r}(2n^{2}+8\nu_{\rm w}k^{2}n+gk+{\cal
T}k^{3})A_{2}-{\rm i}(gk+{\cal T}k^{3})C_{2}+8{\rm i}\nu_{\rm w}k\ell
nD_{2}=0\\\ \\\ 2{\rm i}k^{2}A_{2}+(k^{2}-\ell^{2})C_{2}=0\\\
\end{array}\right\\}$ (51)

We note that now any value of $\ell$ is admissible in these equations for
determining the ratios $A_{i}:C_{i}:D_{i},(i=1,2)$; and the corresponding
value of $n$ is

$n=-\nu_{\rm w}(k^{2}+\ell^{2})$

We remark the extension of the above analysis to third or higher order Stokes
waves, (if at all analytically tractable) is by no means an easy task.

##  4 Energy transfer to unsteady Stokes waves

We consider a turbulent shear flow of air whose density is $\rho_{a}$ blowing
over an unsteady second-order Stokes wave of the form (12) with a complex
phase speed $c=c_{r}+ic_{i}$, where $c_{r}=\sqrt{g/k(1+k^{2}a^{2})}$ is the
wave speed and $kc_{i}$ is the growth ($>0$) or decay ($<0$) rate.

We shall neglect the molecular viscosity of the air by virtue of the fact that
$\nu_{a}\ll\nu_{w}$ and thus the viscous forces in the airflow becomes
negligible. Then, the governing Reynolds-averaged equations are given by S98

$\partial_{i}\langle u_{i}\rangle=0,\qquad\mathscr{D}\langle
u_{i}\rangle=-\partial_{i}\langle p/\rho_{a}\rangle-\partial_{j}\langle
u_{i}^{\prime}u_{j}^{\prime}\rangle,$

where

$\mathscr{D}=\partial_{t}+\langle u_{j}\rangle\partial_{j}.$

Hence, the horizontal and vertical momentum equations may be expressed,
respectively, as

$\displaystyle\mathscr{D}\langle u\rangle=-\sigma_{x}+\chi_{x}+\tau_{z}$ (52)

and

$\displaystyle\mathscr{D}\langle w\rangle=-\sigma_{z}+\tau_{x}$ (53)

where

$\sigma\equiv\langle p/\rho_{a}+w^{\prime 2}\rangle,\qquad\chi\equiv\langle
u^{\prime 2}-w^{\prime 2}\rangle,\qquad\tau\equiv-\langle
u^{\prime}w^{\prime}\rangle.$

Here $\sigma$ and $\tau$ are the mean normal and shear stresses.

Following Townsend’s scaling argument (Townsend 1972), we may further neglect
the components $\chi_{x}$ and $\tau_{x}$ without affecting the solution
significantly. Accordingly, equations (52) and (53) reduce to

$\mathscr{D}\langle u\rangle=-\sigma_{x}+\tau_{z}\qquad\mathscr{D}\langle
w\rangle=-\sigma_{z}.$

The continuity of the air-water at the surface requires

$u=c\displaystyle\frac{\partial h}{\partial\eta},\qquad
s\tau=\tau_{w}\quad{\rm and}\quad s\sigma=\sigma_{w}$

where $s=\rho_{a}/\rho_{w}\ll 1$. Note we have used the same transformations
given in section 2. However, we note that if the wave steepness $ka\ll 1$ the
by virtue of which $z\approx\eta+O(ka)$.

Using the expression for the horizontal velocity, given by the first of
equations (26), in the curvilinear coordinates, namely

$\displaystyle{\sf u}=-({\rm i}kA_{1}{\rm e}^{k\eta}+mC_{1}{\rm
e}^{m\eta}){\rm e}^{{\rm i}k\xi+nt}-\textstyle\frac{1}{2}ka({\rm i}kA_{2}{\rm
e}^{2k\eta}+mC_{2}{\rm e}^{2m\eta}){\rm e}^{2({\rm i}k\xi+nt)},$

and using the following transformation:

$(A_{1},A_{2},C_{1},C_{2},n)=[iac(1+{\mathscr{C}}_{1}),2iac(1+{\mathscr{C}}_{2}),ac{\mathscr{C}}_{1},2ac{\mathscr{C}}_{2},-ikc]$

we obtain

$u_{w}=c[k+{\mathscr{C}}_{1}(k-m)]h_{0}+2kac[k+{\mathscr{C}}_{2}(k-m)]h_{1},$

where the subscript $w$ refers to water.

Neglecting the surface tension, for Stokes waves the total mean normal stress
at the surface is (see section 3)

$\sigma_{w}\equiv
p^{(\eta\eta)}=p_{1}^{(\eta\eta)}+kap^{(\eta\eta)}\equiv\left(-\displaystyle\frac{{\sf
p}}{\rho}+p_{33}\right)_{w}.$

Thus, referring to the previous section, we have

$\displaystyle
p_{1}^{(\eta\eta)}=ka\left\\{\displaystyle\frac{g}{k}-c^{2}-2ik\nu_{w}c-{\mathscr{C}}_{1}[c^{2}-2i(k-m)\nu_{w}c]\right\\}$
$\displaystyle
p_{2}^{(\eta\eta)}=\textstyle\frac{1}{2}ka\left\\{\displaystyle\frac{g}{k}-2c^{2}-8ik\nu_{w}c-2{\mathscr{C}}_{2}[c^{2}-4i(k-m)\nu_{w}c]\right\\}$

and similarly,

$\tau_{w}=p^{(\xi\eta)}=(2\nu_{w}k+i{\mathscr{C}}_{1}c)ckh_{0}+2ka(2\nu_{w}k+i{\mathscr{C}}_{2}c)ckh_{1}.$

We next express the mean surface stresses in the bichromatic perturbation form

$(\sigma_{w},\tau_{w})=({\mathscr{P}}_{1}h_{0}+ka{\mathscr{P}}_{2}h_{1},{\mathscr{T}}_{1}h_{0}+ka{\mathscr{T}}_{2}h_{1})$

where ${\mathscr{P}}_{i}$ and ${\mathscr{T}}_{i},(i=1,2)$ represent the
complex amplitudes of the normal and shear stresses, respectively. Invoking
continuity of the perturbation velocity ($u=c\partial h/\partial\eta$),
eliminating ${\mathscr{C}}_{1}$ and letting $k\nu_{w}/c\downarrow 0$ (as
$m\rightarrow k$), we obtain the interfacial conditions:

$u_{1}-is(kc\nu_{w})^{-\mbox{\footnotesize$\textstyle\frac{1}{2}$}}{\mathscr{T}}_{1}h_{0}+\textstyle\frac{1}{2}ka\left[2u_{2}-is(kc\nu_{w})^{-\mbox{\footnotesize$\textstyle\frac{1}{2}$}}{\mathscr{T}}_{2}h_{1}\right]$
$\displaystyle
s[({\mathscr{P}}_{1}+i{\mathscr{T}}_{1})h_{0}+ka({\mathscr{P}}_{2}+i{\mathscr{T}}_{2})h_{1}]=(c^{2}-c^{2}_{w})kh_{0}+ka(2c^{2}-c_{w}^{2})kh_{1}$
(54)

where $c_{w}$ is given by (10). Hence from (54) we see that

$\displaystyle\displaystyle\frac{c^{2}(1+2ka)-c_{w}^{2}(1+ka)}{sU_{1}^{2}}$
$\displaystyle=$
$\displaystyle\displaystyle\frac{({\mathscr{P}}_{1}+i{\mathscr{T}}_{1})+ka({\mathscr{P}}_{2}+i{\mathscr{T}}_{2})}{kaU_{1}^{2}}$
(55) $\displaystyle=$
$\displaystyle(\alpha_{1}+i\beta_{1})+ka(\alpha_{2}+i\beta_{2}).$

##  5 Determination of energy-transfer parameter

The energy-transfer parameter, $\beta$ defined as in (55), requires the
calculation of the complex amplitudes of the wave-induced pressure and shear
stress at the surface of Stokes waves. We may obtain these from the solution
of the Orr-Sommerfeld equations (for details see S98 or SHD),

$[\nu_{e}(\Phi_{1}^{\prime\prime}+U^{\prime\prime}{\mathscr{H}}^{\prime\prime}_{1})]^{\prime\prime}={\rm
i}k[{\mathscr{U}}(\Phi_{1}^{\prime\prime}-k^{2}\Phi_{1})-U^{\prime\prime}\Phi_{1}]$
$[\nu_{e}(\Phi_{2}^{\prime\prime}+U^{\prime\prime}{\mathscr{H}}^{\prime\prime}_{2})]^{\prime\prime}=2{\rm
i}k[{\mathscr{U}}(\Phi_{2}^{\prime\prime}-4k^{2}\Phi_{2})-U^{\prime\prime}\Phi_{2}]$

where $(^{\prime}\equiv d/d\eta)$, subject to boundary conditions at $\eta=0$
and $\eta=\infty$. In the above equations $\nu_{e}$ is the eddy viscosity,
$\mathscr{U}\equiv U-c_{r}$, and $\mathscr{H}(\eta)$ and $\Phi(\eta)$ are,
respectively, the complex amplitudes of $h$ and the perturbation stream
function $\phi$, given by

$\displaystyle\Psi=\int_{0}^{\eta}\mathscr{U}(\eta)\,d\eta+\mathscr{U}h(\xi,\eta)+\phi(\xi,\eta)$
(56)

with understanding that $h=h_{1}+kah_{2}$, etc.

Alternatively we can follow generalization of M96 for Stokes waves and
evaluate $\beta$ by taking the real part of the quadratic functional

$\displaystyle\beta=(kaU_{1})^{-2}\int^{\infty}_{0}\\{\nu_{e}[{\mathscr{U}}{\mathscr{Z}}_{1}^{\prime\prime
2}+2U^{\prime}{\mathscr{Z}}_{1}^{\prime}{\mathscr{Z}}_{1}^{\prime\prime}+U^{\prime\prime}({\mathscr{Z}}_{1}-{\mathscr{H}}_{1}){\mathscr{Z}}_{1}^{\prime\prime}]+$
$\displaystyle+ik{\mathscr{U}}^{2}({\mathscr{Z}}_{1}^{\prime
2}+k^{2}{\mathscr{Z}}_{1}^{2})\\}d\eta$
$\displaystyle+ka(2kaU_{1})^{-2}\int^{\infty}_{0}\\{\nu_{e}[{\mathscr{U}}{\mathscr{Z}}_{2}^{\prime\prime
2}+2U^{\prime}{\mathscr{Z}}_{2}^{\prime}{\mathscr{Z}}_{2}^{\prime\prime}+U^{\prime\prime}({\mathscr{Z}}_{2}-{\mathscr{H}}_{2}){\mathscr{Z}}_{2}^{\prime\prime}]$
$\displaystyle+2ik{\mathscr{U}}^{2}({\mathscr{Z}}_{2}^{\prime
2}+4k^{2}{\mathscr{Z}}_{2}^{2})\\}d\eta,$ (57)

which provides a Galerkin approximation for suitable approximations to
${\mathscr{H}}_{n}$ and ${\mathscr{Z}}_{n}$ for $(n=1,2)$. We note that

$h(\xi,\eta)=h_{0}(\xi)e^{-k\eta}+kah_{1}(\xi)e^{-2k\eta}$

with

$[h,\zeta]=[{\mathscr{H}}_{1},{\mathscr{Z}}_{1}]e^{ik(\xi-
ct)}+ka[{\mathscr{H}}_{2},{\mathscr{Z}}_{2}]e^{2ik(\xi-ct)}.$

Note further, $\zeta$ is obtained from the linear approximation
${\mathscr{U}}\zeta_{\xi}=-\phi_{\xi}$ to the kinematic surface condition
${\mathscr{D}}\zeta=\langle w\rangle$, which in turn yields
$\zeta=-\phi/{\mathscr{U}}$, or

$\zeta_{1}+ka\zeta_{2}=-{\mathscr{U}}^{-1}(\phi_{1}+ka\phi_{2}).$

Following M96, S98 and SHD, and generalizing the former and the latter to the
second-order Stokes waves, we arrive at

$\displaystyle
a{\mathscr{P}}_{0}=-\int_{0}^{\infty}{\mathscr{U}}^{2}({\mathscr{Z}}_{1}^{\prime
2}+k^{2}{\mathscr{Z}}_{1}^{2})\,d\eta-
ka\int_{0}^{\infty}{\mathscr{U}}^{2}({\mathscr{Z}}_{2}^{\prime
2}+4k^{2}{\mathscr{Z}}_{2}^{2})\,d\eta$ (58)

Choosing the simplest trial function for the variational integral (58), namely

${\mathscr{Z}}_{n}=ae^{-k_{n}\eta/b}\quad\mbox{with $k_{1}\equiv k$ and
$k_{2}\equiv 2k$},$

where $b$ is a free parameter, we obtain

$\mathscr{P}_{n0}/k_{n}aU_{1}^{2}=\displaystyle\frac{b}{2}(b^{-2}+1)\left\\{\displaystyle\frac{\pi^{2}}{6}+\ln^{2}\left(\displaystyle\frac{2k_{n}\gamma\eta_{c}}{b}\right)-2i\hat{c}_{i}\ln\left(\displaystyle\frac{2k_{n}\gamma\eta_{c}}{b}\right)+\hat{c}_{i}^{2}\right\\}$

where $\hat{c}_{i}=c_{i}/U_{1}$, and $\gamma=0.5772$ being the Euler’s number.
Proceeding as in SHD, we may cast the above expression as

$\beta_{nc}=\pi
k_{n}^{3}\eta_{c}^{3}L_{n0}^{4}\left\\{1+\left(4-\displaystyle\frac{\pi^{2}}{3}+10\hat{c}_{i}^{2}\right)\Lambda_{n}^{2}+O(\Lambda_{n}^{3})\right\\}$

with

$\Lambda_{n}^{-1}=L_{n0}=\gamma-\ln(2k_{n}\eta_{c}).$

Hence the energy-transfer contribution due to the initial critial layer,
$\beta_{c}$, becomes

$\displaystyle\beta_{c}$ $\displaystyle=$
$\displaystyle\beta_{1c}+ka\beta_{2c}$ $\displaystyle=$ $\displaystyle\pi
k^{3}\eta_{c}^{3}\left\\{\left(L_{10}^{4}+8kaL_{20}^{4}\right)+\left(4-\displaystyle\frac{\pi^{3}}{3}+10\hat{c}_{i}^{2}\right)\left(L_{10}^{2}+8kaL_{20}^{2}\right)\right\\}$

Similarly the contribution of the energy-transfer parameter due to turbulence
may be evaluated from the integral (58) with the extra contribution

$\beta_{T}=-4k{\rm
i}\int_{0}^{\infty}\lambda^{\prime}\mathscr{W}\mathscr{Z}_{1}\mathscr{Z}^{\prime}_{1}\,d\eta-8k^{2}a{\rm
i}\int_{0}^{\infty}\lambda^{\prime}\mathscr{W}\mathscr{Z}_{2}\mathscr{Z}^{\prime}_{2}\,d\eta$

where $\lambda^{\prime}=U_{1}\kappa^{2}$, and
$\mathscr{W}=U_{1}\ln(\eta/\eta_{c})$. Evaluating the integral it can be shown
that the result may be put in the form222SHD obtained a very similar results
for monochromatic waves, namely $\beta_{T}=5\kappa^{2}L_{01}$, using a
different approach.

$\beta_{T}=4\kappa^{2}(L_{01}+2kaL_{02}).$

We note taht the above expression (for a monochromatic wave) is some 20% lower
than that given by SHD, (cf. equation (1.3) above).

##  6 Kelvin cats-eye

Closed streamlines, commonly known as Kelvin cats-eye, or simply cats-eye,
occur in the neighbourhood of the critical point $\eta=\eta_{c}$, where
${\mathscr{U}}(\eta)=0$. The stream function $\psi_{b}$, for the basic flow,
there

$\psi_{b}\equiv\int^{\eta}_{0}{\mathscr{U}}(\eta)\,d\eta\rightarrow\psi_{c}+\textstyle\frac{1}{2}U^{\prime}_{c}(\eta-\eta_{c})^{2}\qquad{\rm
as}\quad\eta\rightarrow\eta_{c}$

has minimum when $U^{\prime}_{c}>0$ where

$\psi_{c}\equiv\displaystyle\int_{0}^{\eta_{c}}{\mathscr{U}}(\eta)\,d\eta\simeq-U^{\prime}_{c}\eta^{2}_{c}$
$None$

when ${\mathscr{U}}=U_{1}\log(\eta/\eta_{c}))$. The stream function for the
perturbed flow, (56), in the neighbourhood of $\eta_{c}$ has the following
expansion (cf. Lighthill 1962 and Phillips 1977 §4.3)

$\psi=\psi_{c}+\textstyle\frac{1}{2}U^{\prime}_{c}(\eta-\eta_{c})^{2}+U^{\prime}_{c}(\eta-\eta_{c})h_{c}(\xi)+\phi_{c}(\xi),$
$None$

wherein the subscript $c$ implies evaluation at $\eta=\eta_{c}$, and an error
factor of $1+O(ka)$ is implicit in (6.2).

Assuming $c=c_{r}+{\rm i}c_{i}$,

$\displaystyle\phi_{c}=\mbox{Re}\\{\Phi_{c}e^{ik(\xi-
ct)}\\}\equiv-\textstyle\frac{1}{4}{\mathscr{A}}^{2}e^{kc_{i}t}U^{\prime}_{c}\cos
k(\xi-c_{r}t-x_{0})$
$=\textstyle\frac{1}{4}{\mathscr{A}}^{2}e^{kc_{i}t}U_{c}^{\prime}\left\\{1-2\sin^{2}\left[\textstyle\frac{1}{2}k(\xi-
x_{0})\right]\right\\}$ $None$

and

$\psi_{0}\equiv\psi_{c}\\-\textstyle\frac{1}{4}{\mathscr{A}}^{2}e^{kc_{i}t}U^{\prime}_{c},\qquad\psi=\psi_{c}+\textstyle\frac{1}{4}{\mathscr{A}}^{2}e^{kc_{i}t}U^{\prime}_{c}$
$None$

where

$\Phi_{c}\equiv\textstyle\frac{1}{4}{\mathscr{A}}^{2}e^{kc_{i}t}U^{\prime}_{c}e^{i(\pi-
kx_{0}-\omega_{r}t)}$ $None$

with $\omega_{r}=kc_{r}$, then $\phi_{c}$ can determined from the outer
solution (6.2). Subtracting (6.4a) from (6.4b) and substituting the result
together with (6.1) and (6.6) into (6.2), we obtain

$(\eta-\eta_{c}+h_{c})^{2}+{\mathscr{A}}^{2}e^{kc_{i}t}\sin^{2}[\textstyle\frac{1}{2}k(\xi-
x_{0})]=h^{2}_{c}+(2/U^{\prime}_{c})(\psi-\psi_{0})$ $None$

where $\psi_{0}\leq\psi\leq\psi_{1}$.

Figure 2: Formation of Kelvin’s cats-eye for a logarithmic mean velocity
profile over a Stokes wave.

Equation (6.6) describes a periodic sequence of nested sets of closed
streamlines, or unsteady cats-eye (on which $z=\eta+h\simeq\eta+h_{c}$, so
that the displacement of a particle form its ambient position $z=\eta_{c}$ is
$\zeta=\eta-\eta_{c}+h_{c}$) with $\psi$ as a family parameter. If
$h^{2}_{c}\ll{\mathscr{A}}^{2}e^{kc_{t}}$, which is usually the case, and we
assume for the sake of algebraic simplicity the centers are located at
$k(\zeta-x_{0})\,\mbox{mod}\,2\pi$ and $\eta=\eta_{c}$, where $\psi=\psi_{0}$.
Note that, the separatrix, outside of which the streamlines are sinuous, is at
$\psi=\psi_{1}$, and the maximum thickness of the separatrix is
$2{\mathscr{A}}e^{kc_{t}}$. We infer from the development of the previous
section that ${\mathscr{A}}e^{kc_{t}}=O[(ka)^{1/2}\eta_{c}/\delta_{c}]$, where
$\delta_{c}$ is the critical-layer thickness. These closed streamlines
bifurcate from the minimum of $\psi_{b}$, and the entire set may be regarded
as descending from the basic streamlines $z=\eta=\eta_{c}$, while the sinuous
streamlines above/below the separatrix may be regarded as descending from the
basic streamlines $z=\eta\gtrless\eta_{c}$ (but note that $\psi>\psi_{1}$ on
all sinuous streamlines).

For the purpose of demonstration, we consider an idealized case when $c_{i}\ll
1$ (but not exactly equal to zero). In this case the factor
$e^{kc_{i}t}\approx 1$ for a short interval over two cyclic wave. The result
of the closed streamlines, and those above and below it are plotted in figure
2.

##  7 Results and discussion

The results of the foregoing theory warrants the following discussion. We
consider the shear flow instability where energy is transferred to surface
waves by viscous mechanisms.

We point out the appliciability of the present model requires that

$c_{w}<2.3(U_{*}/\nu_{a}k)^{1/3}U_{*}$ $None$

where $c_{w}\equiv c_{r}$ is the speed of the free surface waves given by

$c_{r}=\left[\displaystyle\frac{g}{k}(1+k^{2}a^{2})+k{\cal T}\right]^{1/2}$
$None$

and ${\cal T}$ is the surface tension between air and water (divided by the
density of the water). For all calculations reported here the crtical height
is calculated from the relations

$\eta_{c}=\Omega(U_{1}/c)^{2}e^{c/U_{1}}\qquad{\rm with}\quad\Omega=2.3\times
10^{-3}$

The growth or decay of the Stokes surface wave initial disturbances depends on
whether $c_{i}>0$ or $c_{i}<0$, respectively. Moreover, the net growth rate
$\zeta=kc_{i}$ consists of a sum of two terms: a growth rate $\zeta_{a}$ due
to the wind blowing over the surface of the water, and a damping rate
$\zeta_{w}$ resulting from the viscous dissipation of the wave energy in the
water.

The damping rate (for a small amplitude wave) is given by

$\zeta_{w}=-2k^{2}\nu_{w}-(2k^{3}\nu_{w}c_{r})^{1/2}e^{-2kd}$ $None$

but for deep waters (considered here) $d\gg 1$ and thus the second term in
(7.3) rapidly tends to zero.

However, the growth rate due to the air, at the air-sea interface, is obtained
from the solution of the Orr-Sommerfeld equation,333The mathematical detail
see the appendix of Sajjadi and Drullion (2015). see for example Benjamin
(1959) by writing the solution as a sum of inviscid and viscous solutions,
namely ${\it\Phi}_{n}(\eta)=\phi_{n}(\eta)+f_{n}(\eta)$, for $n=1,2$. Since we
have assumed $ka\ll 1$, we shall only consider the effect of the dominant
harmonic (the second harmonic will have very minor effect to the overal
solution), namely $n=1$, and for brevity we drop the suffix 1.

The inviscid solution is obtained using the method originally proposed by
Miles (1962). Thus, we write

${\it\Omega}(\eta)=(U-c)^{-1}[U^{\prime}\phi-(U-c)\phi^{\prime}]^{-1}\phi$

where ${\it\Omega}$ satisfies the Riccati equation

${\it\Omega}^{\prime}=k^{2}(U-c)^{2}{\it\Omega}^{2}-(U-c)^{-2}$

Miles (1962) showed that

$w=U_{0}^{\prime}c\left[{\it\Omega}_{1}-\displaystyle\frac{1}{U_{1}^{\prime}(U_{1}-c)}-\int_{0}^{\eta_{1}}\displaystyle\frac{U^{\prime\prime}\,d\eta}{U^{\prime
2}(U-c)}\right]\left[1+O(k\eta_{1})^{2}\right]$ $None$

where the supscript 1 now implies evaluation at the point $\eta=\eta_{1}$
defined such that

$k|\eta_{c}|<k\eta_{1}\ll 1,\qquad|U_{1}-c|\gg c_{i}$

Upon integrating (7.4), and taking the path of integration under the
singularity at $\eta=\eta_{c}$, we obtain

$w_{i}=-\pi cU_{0}^{\prime}\displaystyle\frac{U^{\prime\prime}}{U_{c}^{\prime
3}}\left[1+O(k|\eta_{c}|)^{2}\right]$

Following Benjamin (1959) the viscous solution satisfies

$if^{\rm iv}+\eta f^{\prime\prime}=0$

for which the only solution that vanishes as $\eta\uparrow\infty$ is

$f(\eta)=\int_{\infty}^{(\eta/\delta_{c})-z}\int_{\infty}^{\varpi}\vartheta^{1/2}H^{(1)}_{1/3}\left[\textstyle\frac{2}{3}(i\vartheta)^{3/2}\right]\,d\vartheta\,d\varpi$

where $H^{(1)}_{1/3}$ denotes the Hankel function of the first kind. Then we
may construct the following complex functions

$\displaystyle\begin{array}[]{lcccl}w(k,c)=[1+(U_{0}^{\prime}/c)(\phi_{0}/\phi_{0}^{\prime})]^{-1},&&&&{\mathscr{F}}(z)=[1+(U_{0}^{\prime}/c)(f_{0}/f_{0}^{\prime})]^{-1},\\\
G(z)=f^{\prime\prime}_{0}/i\delta_{c}f^{\prime\prime\prime}_{0},&&&{\rm
and}&H(z)=z({\mathscr{F}}-1)-iG\\\ \end{array}$ (61)

The asymptotic forms of these functions, for small and large values of
$z=\eta_{c}/\delta_{c}\doteqdot c/U_{0}^{\prime}\delta_{c}$, is given by Miles
(1960).

Thus, the growth rate, $\zeta_{a}$, may be determined from the viscous
solution and can be expressed as

$\zeta_{a}=\displaystyle\frac{1}{2}sU_{0}^{\prime}\left[\displaystyle\frac{w_{i}-{\mathscr{F}}_{i}-k\delta_{c}(w_{r}-{\mathscr{F}}_{r})H_{i}}{|{\mathscr{F}}-w|^{2}}\right]_{c=c_{r}}$
$None$

where $\delta_{c}=(\nu_{a}/U_{0}^{\prime}k)^{1/3}$ and $s=1.2\times 10^{-2}$.

Using the common practice, we approximate the slope of the mean wind velocity
profile at the surface to be

$U_{0}^{\prime}\thickapprox U_{*}^{2}/\nu_{a}$

Furthermore, since $w_{i}\ll 1$ the denominator on the left-hand side of
equation (7.5) may be approximated as

$|{\mathscr{F}}-w|^{2}=({\mathscr{F}}_{r}-w_{r})^{2}+{\mathscr{F}}_{i}^{2}$

In this case, the function $w$ as given by Miles (1962), is estimated from the
relation

$w=\displaystyle\frac{\kappa c_{r}}{U_{*}}W(R_{a},A)=2.3$

where

$R_{a}=\displaystyle\frac{\kappa U_{*}}{k\nu_{a}}\qquad{\rm and}\qquad
A=\displaystyle\frac{\kappa(u_{s}-c_{r})}{U_{*}}$

Note that, the velocity $u_{s}$ at the edge of the viscous sublayer,
$\eta=\eta_{1}$, can be found from the logarithmic law of the mean velocity
profile, as described by Miles (1957). Thus, the asymptotic approximation to
(7.5) becomes

$\zeta_{a}=\displaystyle\frac{1}{2}sU_{0}^{\prime}\left\\{\displaystyle\frac{w_{i}-(\nu_{a}k/2c_{r})^{1/2}[(U_{c}^{\prime}/kc_{r})+2(w_{r}-1)]}{(w_{r}-1)^{2}+w_{i}^{2}}\right\\}$
$None$

Having obtained the necessary formulation for the growth rates, we shall now
consider the resonance between the free-surface waves in the water and the
Tollmien-Schlichting waves associated with the air flow above the surface. In
figure 3 we have plotted the dispersion curve $c_{r}/U_{\infty}$ as a function
of $kR$ for $g=980,U_{\infty}=100,T=73,\nu_{a}=0.154$ and $\nu_{w}=10^{-2}$,
all in c.g.s units. The Tollmien-Schlichting wave was obtained from the
solution of the eigenvalue problem

$c=c_{2}+\displaystyle\frac{sU_{0}^{\prime}}{k}\left[\displaystyle\frac{1+ikH_{i}(z)}{{\mathscr{F}}^{\prime}/U_{0}\delta_{c}-\partial
w/\partial c}\right]\left\\{(c_{2}+2ik\nu_{w})^{2}-c_{r}^{2}\right\\}^{-1}$
$None$

Figure 3: The neutral dispersion curve in boundary layer above the surface
wave (red curve), and the neutral curve for Stokes wave on deep-water (green
curve).

by equating $c_{2}$ to $c_{r}(k)$, as given by equation (7.2). The result of
this graph gives a first approximation to the resonance condition between
neutral ocillations associated with the mean wind velocity profile and that of
deep-water waves on the free surface.

Figure 4 shows the variation of $\zeta_{a}$ with the wavelength $\lambda$.
Also plotted is the damping rate $-\zeta_{w}$ for comparison. We note the peak
at $\lambda\approx 3$ cm and $U_{*}=5$ cm/s is a close approximation to the
resonant point $\lambda=2.5$ cm for $U_{*}=4.5$ cm/s.

Figure 4: The growth rate for Stokes waves on deep water due to a turbulent
with with logarithmic mean velocity with $U_{1}=5U_{*}$.

In figure 5 we depict the variation of $c_{i}$ with $\lambda$ for five values
of $U_{*}$ in the range $20\leq U_{*}\leq 40$. The results show the peak value
of $c_{i}=\zeta_{a}/k$ occur at lower wavelength for smaller $U_{*}$. From
these results we deduce (i) in the absence of wind $c_{i}=c_{r}$, and (ii) the
resonance appears faster for slower wind friction velocity, $U_{*}$, and
slower for faster wind friction velocity. This condition can also be seen by
plotting $c_{i}/U_{\infty}$ as a function $c_{r}/U_{\infty}$, as shown in
figure 6. At very small and very large values of $c_{r}/U_{\infty}$ we observe
that $\mbox{Re}\\{c\\}\approx c_{w}$ which is characteristic of the Kelvin-
Helmholtz instability.

Figure 5: The variation of the complex part of the wave phase speed with the
wavelength for five values of $U_{*}$. Figure 6: The variation of the
normalized complex part of the phase speed with the normalized speed of a
Stokes wave. The wave steepness is 0.01 and the reference wind speed is 100
cm/s.

Finally, in figure 7 we have plotted the variation of
$\zeta=\zeta_{a}+\zeta_{w}$ as a function of $c_{r}/U_{\infty}$. From this
figure we see that for a very slow moving wave, the growth rate is negative.
This is indicative of the fact that for very small values of $c_{r}$ the water
damping rate is dominant. As the wind, which is blowing over the wave, become
stronger, which in turn increases $c_{r}$, the growth rate due to the air flow
overcomes the viscous damping of the water. Hence, the initial surface ripples
begin to grow. The maximum growth rate occurs at the resonant point, for a
given value of $U_{*}$, and then a further increase of the wind shear
(resulting to larger values of $c_{r}$) begings to reduce the wave growth. At
this point the Tollmien-Schlichting instability will be overcome by the
Kelvin-Helmholtz instability.

Figure 7: The variation of the net growth rate with normalized speed of Stokes
wave. The wave steepness is 0.01 and the reference wind speed is 100 cm/s.

Since the growth rate $\zeta=kc_{i}$ is related to the energy transfer
parameter $\beta$ through

$\zeta=\displaystyle\frac{1}{2}s\beta\omega_{r}\left(\displaystyle\frac{U_{*}}{c_{r}}\right)^{2},\qquad\omega_{r}\equiv
kc_{r}$

and that the variation of the wave amplitude with time is given by

$a(t)=a_{0}e^{\zeta t}$

Where $a_{0}$ is the amplitude of the wave initially (i.e. at $t=0$) we see
that for the fixed value of $c_{r}/U_{*}$ as $\beta$ increases or decreases
then accordingly $\zeta$ increases or decreases. Hence, the wave amplitude
accordingly grows or decays as to whether $\beta$ increases or decreases. It
should be noted that this scenario does not hold when $c_{i}=0$. Thus, as a
wave grows, the inertial critical layer moves up to the inner region of the
flow above the wave. Conversely, when the wave decays, the inertial critical
layer moves down and for small enough values of $c_{i}$ it becomes confined in
the inner shear layer which is very close to the wave surface.

As the growth rate $\zeta=kc_{i}$ enters in the expression (6.6) we see that
the position of the critical point $\eta_{c}$ rises or drops depending on
whether $\zeta\gtrless 0$. Hence when $c_{i}\neq 0$ the cats-eye are not
symmetrical above the critical point above the progressive wave. Moreover, the
thickness of the critical layer $\delta_{c}$ will also vary, becoming thinner
when $\zeta\ll 1$ and thicker when $\zeta$ becomes larger.

##  8 Conclusions

In our investigation here, we have considered the extension of evaluation of
interfaced impedance, given originally by Miles (1957), to Stokes waves. This
task required the development of a theory of Stokes waves on a viscous fluid,
which is in fact based on the original investigation of Lamb (1932). The
former which has never been previously reported in the literature shows how
the interfacial impedance is affected when the surface water waves consist of
multiple harmonics. Here, the effect of the surface tension is also taken into
account.

In the theory presented here, we have assumed that the waves are growing and
hence it has been assumed that the surface waves are unsteady. In such cases,
the wave phase speed is complex, when the real part represents the wave speed
and the complex part is related to the growth or decay of originally formed
Stokes waves by a shear flow.

Thus, the present work extends to the previous investigation of SHD in that
(i) the surface wave is unsteady and nonlinear, and (ii) the effect of the
water viscosity, which affects surface stresses, is taken into account.

For the determination of energy-transfer parameter $\beta$, we have invoked a
similar turbulence closure model (the details of which will be reported in a
subsequent paper), and we have shown the component of $\beta$ arising from the
critical layer, namely $\beta_{c}$, is identical to that discovered by SHD.
However, the contribution due to turbulent shear flow $\beta_{T}$, is some 20%
lower than that found by SHD. Note that $\beta_{c}$ essentially arises under
the assumptions that (i) the critical layer is well within the inner surface
layer, and (ii) the flow above the surface of water waves is inviscid (Miles
1957). But, the interaction of turbulent shear flow with viscous water waves
(which was neglected by SHD) reduces the contribution of $\beta_{T}$ due to
viscous damping at the surface, see below.

In section 6, we have derived an expression for the closed streamlines
(namely, Kelvin cats-eye, which arises in the vicinity of the critical height)
when the surface wave is unsteady. From this expression, it is clear as waves
grow or decay, the cats-eye is no longer symmetrical. We remark the symmetry
only arises when the waves are steady, or more precisely, when the wave
amplitude remains constant.

Finally, we explored the energy transfer from wind to short Stokes waves
through the viscous Reynolds stresses in the immediate neighborhood of the
water surface. We have conjected that the resonance between the Tollmien-
Schlichting waves for a given turbulent wave profile and the free-surface
Stokes waves are an additional factor that contributes to the growth of
surface waves.

## Acknowledgment

This paper is dedicated to the memory of a friend and a colleague John W.
Miles 1920–2007.

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