# An asymptotic analysis for generation of unsteady surface waves on deep
water by turbulence

S.G. Sajjadi  Department of Mathematics, ERAU, FL, USA.
and Trinity College, Cambridge, UK.

###### Abstract

The detailed mathematical study of the recent paper by Sajjadi, Hunt and
Drullion (2014) is presented. The mathematical developement considered by
them, for unsteady growing monochromatic waves is also extended to Stokes
waves. The present contribution also demonstrates agreement with the
pioneering work of Belcher and Hunt (1993) which is valid in the limit of the
complex part of the wave phase speed $c_{i}\downarrow 0$. It is further shown
that the energy-transfer parameter and the surface shear stress for a Stokes
wave reverts to a monochromatic wave when the second harmonic is excluded.
Furthermore, the present theory can be used to estimate the amount of energy
transferred to each component of nonlinear surface waves on deep water from a
turbulent shear flow blowing over it. Finally, it is demonstrated that in the
presence of turbulent eddy viscosity the Miles (1957) critical layer does not
play an important role. Thus, it is concluded that in the limit of zero growth
rate the effect of the wave growth arises from the elevated critical layer by
finite turbulent diffusivity, so that the perturbed flow and the drag force is
determined by the asymmetric and sheltering flow in the surface shear layer
and its matched interaction with the upper region.

##  1 Introduction

It is well known that a surface wave travelling along a water surface can
force a couple motion in the air and water, both propagating at the same
speed, namely the eigenvalue $c_{r}$, being the real part of the wave speed
$c$. Hence the surface wave could force an unstable shear mode in the air,
which then grows and induces growth of the water wave. In a pioneering work,
Miles (1957) constructed a model for generation of waves by shear flows by
assuming that the critical height is sufficiently high that the turbulent
stresses could be neglected. Given this assumption, he argued that the airflow
perturbations are described by Rayleigh equation

$\displaystyle(U-c)(\phi^{\prime\prime}-k^{2}\phi)-U^{\prime\prime}\phi=0$ (1)

for the non-dimensional perturbation stream function $\phi$. In (1), $U(z)$ is
the undisturbed velocity profile for the wind, blowing over the waves, and $k$
is the wavenumber. Clearly, unless the wave amplitude varies with time, i.e.
$c_{i}\neq 0$, the equation (1) is singular at critical point $z_{c}$. By
solving (1) above and below the air-water interface and by matching the
vertical velocity and pressure at $z_{c}$, Miles (1957) calculated $c_{i}$ in
the limit as $c_{i}/U_{*}\downarrow 0$ from the resulting eigenvalue
relationship.

Miles (1957) showed

$\displaystyle c^{2}=c_{w}^{2}+s(\alpha+i\beta)U_{1}^{2}$ (2)

where $c_{w}=\sqrt{g/k}$ is the free surface wave speed,
$s=\rho_{a}/\rho_{w}\ll 1$ and $U_{1}=U_{*}/\kappa$, with $U_{*}$ representing
the wind friction velocity, $\rho_{a}$ and $\rho_{w}$ are air and water
densities, respectively, and $\kappa$ is the von Kármán’s constant. He then
deduced the growth of a monochromatic wave is given by the following
expression

$\displaystyle\zeta_{a}=2{\mathscr{I}}\\{c\\}/{\mathscr{R}}\\{c\\}=s\beta(U_{1}/c)$
(3)

The very important aspect of (2) and (3) is that, for a steady wave (in which
the wave amplitude $a$ remains constant), $c$ must have a non-zero imaginary
part, i.e. such a wave will only grow if $c_{i}\neq 0$, This is quite evident
from equations (2) and (3).

In his paper Miles derived an integral expression for, what is commonly known
as, the ‘energy-transfer parameter’ $\beta$,

$\displaystyle\beta=-{\mathscr{I}}\left\\{\int_{\eta_{c}}^{\infty}|\phi|^{2}(w^{\prime\prime}/w)\,d\eta\right\\}$
(4)

where $w$ is the dimensionless wind velocity profile and the suffix $c$
indicating evaluation at the critical point $\eta=\eta_{c}$ where the wind
velocity equals the wave speed. However, in evaluation the integral in (4) and
hence arriving at his ‘well known’ inviscid expression for $\beta$ at
$\eta=\eta_{c}$, for $w_{c}=0$ and $c_{i}\neq 0$,

$\displaystyle\beta=-\pi|\phi_{c}|^{2}(w_{c}^{\prime\prime}/w_{c}^{\prime}),$
(5)

he states “The path of integration in (5) must be indented either over or
under the singularity at $\eta=\eta_{c}$, where $w(\eta_{c})=0$, and on this
choice depends the sign of $\beta...$”. At the bottom of the same paragraph he
concludes that “… the path must be indented under the singularity∗.” Note the
asterisks attached to the word ‘singularity’ refers to a crucial footnote
which holds the key to Miles’ (1957) critical-layer theory. In this footnote
he states “∗ This assumes $c=c_{w}$ is real. In the next approximation
${\mathscr{I}}\\{c\\}>0$, so that the singularity lies slightly above the real
axis (assuming $w_{c}^{\prime\prime}/w_{c}<0$), and the path of integration in
(4) passes under the singularity without the necessity of indentation.”

This important footnote is generally overlooked by many who refer to Miles’
(1957) critical-layer mechanism. This footnote is very significant in that (a)
has a physical consequence which contradicts the results in (2) and (3) which
shows clearly if $c_{i}=0$ then $\beta=0$ and hence waves cannot grow; and (b)
has a mathematical consequence which indicates that equation (5) is valid if
$c_{i}\neq 0$.

###  1.1 Physical mechanisms

We now present two alternative physical arguments that not only their results
agree with each other but proves rigoriously that Miles (1957) critical-layer
mechanism is valid only for slowly-growing waves which does not apply to
growth of surface waves by strong or turbulent shear flows in open ocean.

Belcher & Hunt (1993) (referred to as BH therein) considered a fully developed
boundary layer over a two-dimensional monochromatic wave of small steepness
$ak$ propagating with small wave speed $c$ and calculated the perturbations in
the asymptotic limit ${\cal U}\equiv(U_{*}+c)/U\downarrow 0$. In this limit,
the critical height $z_{c}$ lies within the inner surface layer, where the
perturbation Reynolds shear stress varies slowly. Then, by considering the
equation for the shear stress, they constructed solutions across the critical
layer and demonstrated that the shear stress perturbation plays an important
role at the critical height which in turn implies Miles’ (1957) inviscid
theory is not the dominating mechanism for the wave growth in this parameter
range. Note that the perturbations above the inner surface layer are not
directly influenced by the critical height and the region below $z_{c}$ where
the flow reversal occurs (see figure 1). In fact this is very similar to the
perturbations due to a static undulation, but with the difference that the
effective roughness length, that determines the shape of the unperturbed
velocity profile, is modified according to

$z_{c}=z_{0}\exp(\kappa c/U_{*}).$

BH then used the solutions for the perturbations to the boundary layer and
calculated the wave growth, which is determined, in the leading order of
perturbation, by the asymmetric pressure perturbation induced by the
thickening of the perturbed boundary layer on the leeside of the wave crest.
To the first order in ${\cal U}$, BH discovered that there are new effects
that contribute significantly to the rate of growth: (a) the asymmetries in
both the normal and shear Reynolds shear stresses associated with the leeside
thickening of the boundary layer, this they termed the non-separated
sheltering (cf. Jeffreys 1925); (b) asymmetrical perturbations which are
induced by the varying surface velocity associated with the fluid motion in
the wave; and, (c) asymmetries induced by the variation in the surface
roughness along the wave. The theoretical value, predicted by their theory,
for the shear stress perturbation at the crest of the wave on the wave surface
as well as on the top of the inner region is in good agreement with laboratory
measurements. Hence, despite the restriction that ${\cal U}\ll 1$, their
theory describes a large portion of the experimental observations of the wave
growth rate made at sea and in the laboratory.

Figure 1: Schematic diagram for flow geometry and asymptotic multi-layer
structure for analysing turbulent shear flow over steady and unsteady
monochromatic waves. Taken from Sajjadi et al. (2014).

In a subsequent study, Belcher et al.(1999) (hereafter will be referred to as
BHC) considered turbulent flow over growing waves, using triple-deck boundary-
layer theory originally developed by Lighthill (1957) and Stewartson (1974),
to analyse the sheltering mechanism described above. They suggested that
Miles’ (1957) critical-layer theory generates growth of waves but this was not
demonstrated in data. In this model they assumed the atmospheric mean flow is
neutrally stable and is logarithmic and the surface wave, moving in the
positive $x$-direction, is monochromatic whose profile is given by

$z^{\prime}=\mbox{Re}\\{ae^{ik(x^{\prime}-ct)}\\}$

where $c=c_{r}+ic_{i}$ is a complex wave speed, such that $c_{r}$ is the phase
speed and the wave amplitude, $a$, grows exponentially at the rate $kc_{i}$.
They considered a frame of reference moving with the wave at a speed $c_{r}$
so that the mean velocity profile can be expressed as

$U(z)=U_{1}\log(z/z_{0})-c_{r}$

where $U_{1}=U_{*}/\kappa$, and that this wave speed vanishes at the critical
height $z_{c}$. In this frame of reference the surface wave is described by

$z=\mbox{Re}\\{ae^{ik(x-ic_{i}t)}\\}.$

The boundary conditions imposed at the wave surface is that the wind velocity
is equal to the surface velocity of the water flow, being approximated by the
surface value of the orbital velocity of an irrotational wave on deep water.
The other boundary condition is that perturbations in the basic flow vanish as
$kz\uparrow\infty.$

BHC modelled the turbulent shear stress in the inner region, adjacent to the
wave surface, using a mixing length model, and in the region above this, the
outer region, they invoked rapid distortion theory to describe the turbulence.
They showed that the depth of the inner region, $\ell_{i}$, may be obtained
from the following implicit relation

$k\ell_{i}=\displaystyle\frac{2\kappa^{2}}{|\ln(\ell_{i}/z_{0})-\kappa
c_{r}/U_{*}|}$

where the variation of solution to this equation for $\ell_{i}$ as a function
of $c_{r}/U_{*}$ for $kz_{0}=10^{-4}$ is shown in figure 2.

They further assumed the perturbations to the air flow are governed by
equation (1.7) where the turbulent stresses on the right-hand side is modelled
by an eddy viscosity. The vertical component of the velocity perturbation
$\boldsymbol{\mathscr{U}}\equiv\mathscr{U}_{i}=(\mathscr{U},0,\mathscr{W})$ is
expanded in the normal form

${\mathscr{W}}(x,z,t)=\hat{\mathscr{W}}(k,z)e^{i(kx-ic_{i}t)},$

where the amplitude of the perturbation, $\hat{\mathscr{W}}$, satisfies the
inhomogeneous Rayleigh equation

$\displaystyle\displaystyle\frac{\partial^{2}\hat{\mathscr{W}}}{\partial
z^{2}}-\left(k^{2}+\displaystyle\frac{U^{\prime\prime}}{U-ic_{i}}\right)\hat{\mathscr{W}}=\displaystyle\frac{i}{U-ic_{i}}\displaystyle\frac{\partial^{2}}{\partial
z^{2}}\left(\nu_{e}\displaystyle\frac{\partial^{2}\hat{\mathscr{W}}}{\partial
z^{2}}\right)$ (6)

where $\nu_{e}$ is an eddy viscosity.

Figure 2: Variation with $c_{r}/U_{1}$ of solutions for the normalized inner-
region height, $k\ell_{i}$, and critical height, $kz_{c}$, when
$kz_{0}=10^{-4}$ . For given $c_{r}/U1$ , an inner, local equilibrium region
lies between $kz=0$ and the smallest value of $k\ell_{i}$, and an outer,
rapid-distortion, region lies above. Solid lines: $k\ell_{i}$; dotted lines:
$kz_{c}$.

BHC showed that in the middle layer, $\ell_{i}\ll z\ll\ell_{m}$, the advection
term is negligible compared with the curvature term and the thus (6) reduces
to

$\displaystyle\displaystyle\frac{\partial^{2}\hat{\mathscr{W}}}{\partial
z^{2}}-\displaystyle\frac{U^{\prime\prime}}{U-ic_{i}}\hat{\mathscr{W}}\sim 0$
(7)

whose leading-order solution may be expressed as

$\displaystyle\hat{\mathscr{W}}\sim\left\\{U(z)-ic_{i}\right\\}\left\\{A+B\int\displaystyle\frac{d\zeta}{[U(\zeta)-ic_{i}]^{2}}\right\\}$
(8)

In (8) $A$ and $B$ are constants which may be determined by matching the inner
and upper layer solutions. Now, for slow waves, the critical layer lies close
to the surface in the inner region. Hence, the solution (8) is regular, since
$U(z)>0$ and does not vanish there. Therefore, the integrand (8) is regular
throughout the middle layer. Note, the same argument applies to the moderate
waves, but there are differences in its application.

Suppose now for a range of intermediate waves the critical layer lies in the
outer region. Thus, we can expect, for a particular range of $c_{r}/U_{*}$,
the critical layer lies within $\ell_{i}\ll z\ll\ell_{m}$. Neglecting the
Reynolds stresses in the vicinity of the critical layer, $\hat{\mathscr{W}}$
satisfies (7) with the solution given by (8).

However, in this senario, the critical layer lies within the range of the
integral in (8) and if $c_{i}=0$ then there is a singularity in this integrand
at the critical layer, where $U(z_{c})=0$. This singularity may be resolved by
inertial effects, i.e. inviscid processes that control its dynamics.

If we now suppose the wave grows so that $c_{i}>0$, then the integral (1.8) is
regular at $z=z_{c}$. If we further assume the waves grows slowly, such that
$c_{i}\ll U_{c}^{\prime 2}/U^{\prime\prime}_{c}$, where the suffix $c$
indicates evaluation at $z=z_{c}$, then the integral in (8) can be evaluated
approximately. To do this, $U(z)$ is Taylor expanded in the vicinity of the
critical layer, i.e. $U(z)\sim\zeta
U_{c}^{\prime}+\textstyle\frac{1}{2}\zeta^{2}U_{c}^{\prime\prime}$, where
$\zeta=z-z_{c}$ and thus the integral becomes

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int^{z-z_{c}}\displaystyle\frac{d\zeta}{[U(\zeta)-ic_{i}]^{2}}\sim\int\displaystyle\frac{d\zeta}{\left\\{\zeta
U_{c}^{\prime}+\textstyle\frac{1}{2}\zeta^{2}U_{c}^{\prime\prime}-ic_{i}\right\\}^{2}}$
(9) $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{c_{i}U_{c}^{\prime}}\int\displaystyle\frac{d\xi}{(\xi
U_{c}^{\prime}-i)^{2}}\left\\{1+\displaystyle\frac{\varepsilon}{2}\displaystyle\frac{\xi^{2}}{\xi-i}\right\\}^{-2}$

where $\xi=\zeta U_{c}^{\prime}/c_{i}$ and
$\varepsilon=c_{i}U_{c}^{\prime\prime}/U_{c}^{\prime 2}\ll 1$. Note that, if
$U(z)$ has a logarithmic profile then $\varepsilon=c_{i}/U_{*}\ll 1$ which
confirms that these are slowly-growing waves. In the limit of slow-growing
waves the factor in braces can be expanded for $\varepsilon\ll 1$ to give

$\displaystyle I$ $\displaystyle\sim$
$\displaystyle\displaystyle\frac{1}{c_{i}U_{c}^{\prime}}\int\displaystyle\frac{d\xi}{(\xi
U_{c}^{\prime}-i)^{2}}\left\\{1-\varepsilon\displaystyle\frac{\xi^{2}}{\xi-i}+O(\varepsilon^{2})\right\\}$
(10) $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{c_{i}U_{c}^{\prime}}\left[\displaystyle\frac{1}{\xi-i}+\varepsilon\left\\{\displaystyle\frac{1}{2(\xi-i)^{2}}-\displaystyle\frac{2i}{\xi-i}+\ln(\xi-i)\right\\}+O(\varepsilon^{2})\right]$

From (10) we can deduce that far from the critical height $I$ is dominated by
the logarithmic term and hence (10) reduces to

$\displaystyle I$ $\displaystyle\sim$
$\displaystyle\displaystyle\frac{\varepsilon}{c_{i}U_{c}^{\prime}}\ln(\xi-i)\hskip
28.45274pt\mbox{as $\xi\rightarrow\pm\infty$}$ (11) $\displaystyle=$
$\displaystyle\displaystyle\frac{\varepsilon}{2c_{i}U_{c}^{\prime}}\ln(\xi^{2}+1)+i\theta$

where $\theta$ is given by

$\displaystyle\tan\theta=-\xi^{-1}=-c_{i}/U_{c}^{\prime}(z-z_{c})$ (12)

For a logarithmic velocity profile $\tan\theta=\varepsilon z_{c}/(z-z_{c})$
and hence $\theta$ varies between

$\displaystyle\theta\rightarrow 0\hskip 14.22636pt\mbox{as
$(z-z_{c})/\ell_{c}\rightarrow\infty$}\hskip 14.22636pt{\rm and}\hskip
14.22636pt\theta\rightarrow\pi\hskip 14.22636pt\mbox{as
$(z-z_{c})/\ell_{c}\rightarrow\infty$}$ (13)

The imaginary part of the integral for $z\gg z_{c}$ is then

$\displaystyle{\mathscr{I}}\\{I\\}\sim\varepsilon/c_{i}U_{c}^{\prime}\theta=U_{c}^{\prime\prime}/U_{c}^{\prime
3}{\sf H}(z-z_{c})\hskip 14.22636pt\mbox{as $\xi\rightarrow\pm\infty$}$ (14)

where ${\sf H}(z-z_{c})$ is the Heaviside step function. The result given by
(14) is remarkable since it is independent of $c_{i}$ which means even for a
slowly-growing wave it leads to an out of phase component of the motion that
is independent of the growth rate.

The significance of the term $iU^{\prime\prime}_{c}/U_{c}^{\prime 3}$ in the
solution for $I$ is that it yields an out of phase contribution to the
vertical velocity that ultimately leads to the same contribution to the wave
growth from the critical layer as found by Miles (1957). This result shows the
solution found by Miles (1957) is valid only when the waves grow sufficiently
slowly such that

$\displaystyle c_{i}\ll U_{c}^{\prime}z_{c}\sim U_{*}$ (15)

and hence the effects of the critical layer calculated by Miles (1957) are
valid only in the limit $c_{i}/U_{*}\downarrow 0$.

BHC further argued that in the vicinity of the critical height the turbulent
shear stress perturbation, $\Delta\tau$, can be modelled by an eddy viscosity
model via

$\displaystyle\Delta\tau=\nu_{e}\displaystyle\frac{\partial\Delta u}{\partial
z}$ (16)

where $\nu_{e}=\alpha U_{*}z$ is an eddy viscosity and $\alpha=2\kappa$ is a
parameter which reduces (16) to the mixing length model used in the inner
region. Thus, there is a layer around the critical height whose thickness is
$\ell_{r}$ in which the stresses cannot be neglected. By balancing the shear
stress term in (6) with the gradient term,
$\partial^{2}\hat{\mathscr{W}}/\partial z^{2}$, they then estimated

$\ell_{r}\sim z_{c}(\alpha/kz_{c})^{{\scriptstyle\frac{1}{2}}}.$

Hence they found that the thickness of the shear stress layer surrounding the
critical height is much smaller than the critical-layer thickness, i.e.
$\ell_{r}\ll\ell_{c}$, provided

$(\alpha/kz_{c})^{{\scriptstyle\frac{1}{2}}}\ll c_{i}/U_{*}.$

The main conclusions arrived by BHC, which has motivated the present
investigation, is summarized below.

(a) From various studies and experimental data it is well known that $c_{i}$
varies with governing parameters according to

$c_{i}/U_{*}=s(U_{*}/c_{r})\beta$

where $s\ll 1$ is the ratio of the air density, $\rho_{a}$, to that of water
density, $\rho_{w}$, and $\beta$ is the energy-transfer parameter for the
wind-wave interaction. Using this expression, the ratio of the thickness of an
inertial critical layer, $\ell_{c}$ to the thickness of a stress-dominated
critical layer, $\ell_{r}$, varies according to

$\ell_{c}/\ell_{r}\sim(kz_{c}/\alpha)^{{\scriptstyle\frac{1}{3}}}s(U_{*}/c_{r})\beta.$

For growing waves in the ocean $s\sim 1/800$ and $\beta\approx O(25-30)$ then
for intermediate waves when $kz_{c}\sim 1$ and $c_{r}/U_{*}\approx O(15-25)$,
we see that $\ell_{c}/\ell_{r}\approx
1/(500\alpha^{{\scriptstyle\frac{1}{3}}})$. We remark that for the mixing-
length model $\alpha=0.8$ and thus $\ell_{c}/\ell_{r}$ is likely to be small
for a fully developed turbulent flow. Hence, one is led to conclusion that in
the intermediate regime the critical layer is dominated by effects of Reynolds
stress and, contrary to Miles’ (1957) conclusion, there will not be any
contribution from the critical layer to the wave growth. We further remark
that for those ocean waves that are rapidly growing in time, such that
$c_{i}/U_{*}\sim 1$, the inertial effects are dominant in the critical layer
and thus the critical-layer mechanism (c.f. Miles 1957). This suggests that
such ‘rapidly-growing’ waves might occur as a wave crest moves through a wave
group. This is currently being investigated by the present authors.

(b) For slow waves, $c_{r}/U_{*}\leq 15$, the critical height $z_{c}$ lies
within the lower part of the inner region and thus the reverse-flow region,
situated below $z_{c}$ and the inner layer itself plays no significant
dynamical role. In such circumstances, asymmetry in the flow is generated by
the frictional effect of the shear stress through the inner region, resulting
to lower wind speeds on the downwind side of the wave and consequently leads
to a sheltering in the lee of the wave crest. This asymmetry results to an
out-of-phase pressure perturbation which subsequently yields wave growth. We
emphasize that in this case the air flow perturbations are similar to those
over a stationary undulation, but in the range $1\leq c_{r}/U_{*}\leq 15$ the
flow is similar to that over a rough surface except now the surface roughness
is now effectively $z_{c}$, which increases the value of $\beta$. It is to be
noted that small corrections to the velocity of $O(akc_{r}k\ell_{i})$ due to
the orbital motions at the wave surface reduce $\beta$.

For fast waves, on the other hand, the critical layer is far above the
surface, $kz_{c}>1$, and again there is no significant dynamical role for the
wave growth. In this scenario, the air above the wave flows largely against
the wave which induces a ‘negative’ asymmetry from sheltering. Furthermore,
orbital motions at the water surface generate additional air flow
perturbations that contribute comparable ‘negative’ asymmetries. This
’negative’ asymmetry causes an out-of-phase pressure perturbation which makes
waves to decay.

Finally, between the two regions discussed above, there is also an
intermediate region in which $15\leq c_{r}/U_{*}\leq 30$ and $z_{c}\sim\ell$.
In this region, numerical simulations show that as $c_{r}/U_{*}$ increases
from slow to the fast region, the reverse flow below the critical height
becomes stronger and produces a ‘negative’ asymmetric displacement of
streamlines upwind of the crest. However, above the critical height the
asymmetric displacement is ‘positive’ downwind of the crest, similar to that
for slow waves. Moreover, the critical-layer mechanism also displaces
streamlines downwind of the crest. Therefore, as $c_{r}/U_{*}$ increases
across the intermediate region, the asymmetric component of the flow peaks to
its maximum and then decreases to zero, with the wave growth following the
same trends as that of the asymmetric component of the flow.

##  2 Shear stress model for unsteady wave growth

In this section we consider the perturbation Reynolds stresses in the flow of
a turbulent wind over the surface wave

$\displaystyle z=a\cos[k(x-ct)]\equiv{h}(x,{t}),\quad(ak\ll 1)\qquad
c=c_{r}+ic_{i}$ (17)

through an interpolation between inner, mixing-length approximation and an
outer, rapid-distortion approximation. We will show that the wind-to-wave
energy transfer predicted by this model is substantially larger than that
predicted by either Miles’ (1957) quasi-laminar model (in which the
perturbation Reynolds stresses are neglected) or Townsend’s viscoelastic model
(Townsend 1972) and is in very good agreement with the model proposed by
Belcher & Hunt (1993). Here we will point out, Townsend’s inner approximation
differs from the conventional mixing-length approximation and yields a ratio
of perturbation shear stress to perturbation shear that is negative for his
choice of parameters.

The equations of motion for a viscous incompressible fluid may be cast in
Cartesian tensor form as

$\displaystyle\displaystyle\frac{\partial u_{i}}{\partial
t}+u_{j}\displaystyle\frac{\partial u_{i}}{\partial
x_{j}}=\displaystyle\frac{1}{\rho}\displaystyle\frac{\partial p_{ij}}{\partial
x_{j}}$ (18) $\displaystyle\displaystyle\frac{\partial u_{i}}{\partial
x_{i}}=0$ (19)

where $x_{i}$ denotes the Cartesian coordinate, $u_{i}$ a velocity component,
$p_{ij}$ component of a stress tensor and $\rho$ the fluid density.
Decomposing the variables according to

$\displaystyle u_{i}=U_{i}+u_{i}^{\prime}+u_{i}^{\prime\prime},\hskip
28.45274ptp_{ij}=P_{ij}+p_{ij}^{\prime}+p_{ij}^{\prime\prime}$ (20)

where $U_{i}+u_{i}^{\prime}$ and $P_{ij}+p_{ij}^{\prime}$ represent a solution
to (18), being functions of coordinates $x_{1}$ and $x_{3}$ and having mean
values with respect to $x_{2}$. Note that, $U_{i}$ and $P_{ij}$ represent the
mean components, and $u_{i}^{\prime}$ and $p_{ij}^{\prime}$ the turbulent
fluctuations. In (20) $u_{i}^{\prime\prime}$ and $p_{ij}^{\prime\prime}$
represent a small perturbation with respect to the solution of (18)–(19).

Substituting (20) in (18), neglecting second order terms in the perturbation
flow, and noting the fact that the unperturbed flow satisfies (18), we obtain

$\displaystyle\displaystyle\frac{\partial u_{i}^{\prime\prime}}{\partial
t}+(U_{j}+u_{j}^{\prime})\displaystyle\frac{\partial
u_{i}^{\prime\prime}}{\partial
x_{j}}+u_{j}^{\prime\prime}\displaystyle\frac{\partial(U_{i}+u_{i}^{\prime})}{\partial
x_{j}}=\displaystyle\frac{1}{\rho}\displaystyle\frac{\partial
p_{ij}^{\prime\prime}}{\partial x_{j}}$ (21)
$\displaystyle\displaystyle\frac{\partial u_{i}^{\prime\prime}}{\partial
x_{i}}=0$ (22)

Taking the mean values with respect to $x_{3}$, the results can be expressed
in the following form

$\displaystyle\displaystyle\frac{\partial\overline{u_{i}^{\prime\prime}}}{\partial
t}+U_{j}\displaystyle\frac{\partial\overline{u_{i}^{\prime\prime}}}{\partial
x_{j}}+\overline{u_{j}^{\prime\prime}}\displaystyle\frac{\partial
U_{i}}{\partial
x_{j}}=\displaystyle\frac{1}{\rho}\displaystyle\frac{\partial}{\partial
x_{j}}(\overline{p_{ij}^{\prime\prime}}-\overline{r_{ij}^{\prime\prime}}),$
(23) $\displaystyle\displaystyle\frac{\partial u_{i}^{\prime\prime}}{\partial
x_{i}}=0$ (24)

Invoking the equations of continuity for both $u_{i}^{\prime}$ and
$u_{i}^{\prime\prime}$ the perturbation Reynolds stress may be written as

$\displaystyle\overline{r_{ij}^{\prime\prime}}$ $\displaystyle=$
$\displaystyle\rho(\overline{u_{i}^{\prime}u_{j}^{\prime\prime}}+\overline{u_{j}^{\prime}u_{i}^{\prime\prime}})$
(25) $\displaystyle=$
$\displaystyle\rho[\overline{u_{i}^{\prime}(u_{j}^{\prime\prime}-\overline{u_{j}^{\prime\prime}})}+\overline{u_{j}^{\prime}(u_{i}^{\prime\prime}-\overline{u_{i}^{\prime\prime}})}]$
(26)

with (26) follows from (25) by virtue of $\overline{u_{i}^{\prime}}=0$.

If we now set $x_{1}=x$, $x_{3}=z$, $U_{1}=U(z)$, $U_{2}=U_{3}=0$,
$\overline{u_{1}^{\prime\prime}}=u$, $\overline{u_{3}^{\prime\prime}}=w$,
$\overline{p_{ij}^{\prime\prime}}=-\delta_{ij}\wp$, and
$\overline{r_{ij}^{\prime\prime}}=\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}$,
after taking the time average, we obtain the linearized, Reynolds-averaged
equations governing $u$ and $w$, the $x$ (horizontal) and ${z}$ (vertical)
components of the mean perturbation velocity, and the kinematic perturbation
pressure ${\sf p}$ as

$\displaystyle u_{x}+w_{z}=0,$ (27)
$\displaystyle(U-c)u_{x}+U^{\prime}w=-\wp_{x}+\sigma_{x}+\tau_{z},$ (28)
$\displaystyle(U-c)w_{x}=\wp_{z}+\tau_{x},$ (29)

where the subscript $x$ and ${z}$ signify partial differentiation,
$U^{\prime}\equiv dU/d{z}$,

$\displaystyle\wp\equiv{\sf p}+\overline{w^{\prime 2}}-(\overline{w^{\prime
2}})_{0},\hskip 9.95863pt\sigma\equiv-(\overline{u^{\prime
2}}-\overline{w^{\prime 2}})-\sigma_{0},\hskip
9.95863pt\tau\equiv-\overline{u^{\prime}w^{\prime}}-\tau_{0}$ (30)

$(\overline{w^{\prime 2}})_{0},\sigma_{0}$ and $\tau_{0}$ are the unperturbed
values of $\overline{w^{\prime 2}},-(\overline{u^{\prime
2}}-\overline{w^{\prime 2}})$ and $-\overline{u^{\prime}w^{\prime}}$, and
$\sigma$ is Townsend’s $\tau_{n}$.

In this paper we consider a turbulent shear flow blowing over the surface wave
(17) whose mean velocity profile is given by

$\displaystyle U({z})=(\tau_{0}^{1/2}/\kappa)\log({z}/z_{0})$ (31)

where $\tau_{0}$ is the kinematic shear stress in the basic flow, $\kappa$ is
Kármán’s constant, ${z}$ is the elevation, $c$ is the complex phase speed of
the surface wave (17).

###  2.1 Energy-transport equation

As a first step toward a Reynolds-stress closure, Townsend points out the
transport equation for the turbulent kinetic energy
$\textstyle\frac{1}{2}\overline{\sf{q}^{2}}$ may be expressed in the form

$\displaystyle(U-c)\partial_{x}\left(\textstyle\frac{1}{2}\overline{\sf{q}^{2}}\right)=D+G-\varepsilon^{\prime},$
(32)

where

$\displaystyle
D=\varrho\kappa\tau^{1/2}_{0}\partial_{z}\left[{z}\partial_{z}\left(\textstyle\frac{1}{2}\overline{{\sf
q}^{2}}\right)\right]$ (33)

represents diffusion111Townsend chooses $\varrho=0.3$ but states that ‘the
value … is not critical’.

$\displaystyle G$ $\displaystyle=$ $\displaystyle-\overline{u^{\prime
2}}u_{x}-\overline{w^{\prime
2}}w_{z}-\overline{u^{\prime}w^{\prime}}(U^{\prime}+u_{z}+w_{x})-\tau
U^{\prime}$ (34) $\displaystyle=$
$\displaystyle\sigma_{0}u_{x}+\tau_{0}(u_{z}+w_{x})+U^{\prime}\tau$ (35)

represents generation (Launder et al.1975), (35) follows (34) through (27),
(30) and linearization, and $\varepsilon^{\prime}$ repesents dissipation (see
below). Townsend neglects $w_{x}$ in $G$, although this appears to be
inconsistent with his subsequent rapid-distortion approximation (see below).

Townsend’s approximation to the dissipation rate $\varepsilon^{\prime}$
involves ${h}$, the surface displacement (17), but this may be eliminated
through a transformation to wave-following coordinates, in which (Miles 1996,
equation (3.4))

$\displaystyle\varepsilon^{\prime}=\textstyle\frac{3}{2}\tau_{0}U^{\prime}(e/e_{0}),\hskip
14.22636pte\equiv\overline{{\sf q}^{2}}-e_{0},$ (36)

and $e_{0}\equiv\overline{{\sf q^{2}}}$ in the basic flow. Substituting (35)
and (36) into (32), neglecting diffusion (see Miles 1996, §3), and multiplying
the result by $2a_{1}$, we obtain

$\displaystyle({\mathscr{D}}+\lambda)a_{1}e-2\lambda(\tau-
a_{1}e)=2\tau_{0}[a_{1}(u_{z}+w_{x})+a_{n}u_{x}]\equiv{\mathscr{A}}_{i}$ (37)

where

$\displaystyle{\mathscr{D}}\equiv(U-c)\partial_{x},\hskip
7.11317pt\lambda\equiv a_{1}U^{\prime},\hskip
7.11317pta_{1}\equiv\tau_{0}/e_{0},\hskip 7.11317pta_{n}=\sigma_{0}/e_{0}$
(38)

The relaxation rate $\lambda$ is a reciprocal measure of eddy life and
dominates (is dominated by) ${\mathscr{D}}$ in the inner (outer) domain.

###  2.2 Rapid-distortion approximation

To determine the outer departure of $\tau/e$ and $\sigma/e$ from their
equilibrium values $a_{1}$ and $a_{n}$, Townsend posits the rapid-distortion
approximations

$\displaystyle{\mathscr{D}}\left\\{\displaystyle\frac{\tau_{0}+\tau}{e_{0}+e}\right\\}\simeq{\mathscr{D}}\left\\{\displaystyle\frac{\tau-
a_{1}e}{e_{0}}\right\\}\sim A_{1}u_{z}+A_{2}w_{x}+A_{3}u_{x}$ (39)

and

$\displaystyle{\mathscr{D}}\left\\{\displaystyle\frac{\sigma_{0}+\sigma}{e_{0}+e}\right\\}\simeq{\mathscr{D}}\left\\{\displaystyle\frac{\sigma-
a_{n}e}{e_{0}}\right\\}\sim B_{1}u_{z}+B_{2}w_{x}+B_{3}u_{x}$ (40)

where $A_{1,2,3}$ and $B_{1,2,3}$ are ‘the incremental rates of change for
suddenly imposed additional distortions’. He then ‘interpolates’ between
(39)–(40) and the inner domain, in which (by hypothesis) $\tau\rightarrow
a_{1}e$ and $\sigma\rightarrow a_{n}e$, by replacing ${\mathscr{D}}$ by
${\mathscr{D}}+\lambda$:

$\displaystyle({\mathscr{D}}+\lambda)(\tau-
a_{1}e)=e_{0}(A_{1}u_{z}+A_{2}w_{x}+A_{3}u_{x})\equiv{\mathscr{A}}_{0}$ (41)

and

$\displaystyle({\mathscr{D}}+\lambda)(\sigma-
a_{n}e)=e_{0}(B_{1}u_{z}+B_{2}w_{x}+B_{3}u_{x}\equiv){\mathscr{B}}_{0}$ (42)

However, the elimination of $a_{1}e$ between (37) and (41) in the inner domain
(in which $|{\mathscr{D}}|\ll|\lambda|$) yields

$\displaystyle\tau\rightarrow\lambda^{-1}({\mathscr{A}}_{i}+3{\mathscr{A}}_{0}),$
(43)

which differs from the mixing-length approximation obtained by invoking
$|{\mathscr{D}}|\ll\lambda$ in (37):

$\displaystyle\tau\rightarrow
a_{1}e\rightarrow\lambda^{-1}{\mathscr{A}}_{i}=2\nu_{0}(u_{z}+w_{x}+{\mathscr{H}}u_{x}),$
(44)

where

$\displaystyle{\mathscr{H}}\equiv a_{n}/a_{1},\hskip
14.22636pt\nu_{0}\equiv\tau_{0}/U^{\prime}=\kappa\tau_{0}^{1/2}{z}.$ (45)

Indeed, if (as typically assumed in the mixing-length approximation)
$|u_{x},w_{x}|\ll|u_{z}|$, (43) implies

$\displaystyle\displaystyle\frac{\tau}{2\nu_{0}u_{z}}\rightarrow
1+\displaystyle\frac{3A_{1}}{2a_{1}^{2}},$ (46)

which reduces to the conventional mixing-length approximation for $A_{1}=0$
but is negative for Townsend’s values of $a_{1}$ and $A_{1}$,
$\textstyle\frac{1}{6}$ and $-0.03$.

##  3 Solution of boundary-value problem

Miles-Sajjadi theory of wave generation by turbulent wind (Miles 1996, Sajjadi
1998) reduces to the solution of the Orr-Sommerfeld-like equation

$\displaystyle(\nu_{e}\Phi^{\prime\prime})^{\prime\prime}=ik[(U-c)(\Phi^{\prime\prime}-k^{2}\Phi)-U^{\prime\prime}\Phi]\equiv{\mathscr{T}}^{\prime\prime}$
(47)

subject to the boundary conditions

$\displaystyle\Phi=ac,\hskip 14.22636pt\Phi^{\prime}=a(kc-U^{\prime})\hskip
14.22636pt\mbox{on $\eta=0$}$ (48) $\displaystyle\Phi,\hskip
14.22636pt\nu_{e}\Phi\rightarrow 0\hskip 14.22636pt\mbox{as
$k\eta\rightarrow\infty$}$ (49)

where $\Phi=\Phi(\eta)$, $U=U(\eta)$ and $(\,\,\,)^{\prime}\equiv d/d\eta$. In
equ (47) $\nu_{e}$ is a complex eddy viscosity given by

$\displaystyle\nu_{e}=2U_{*}^{2}(U^{\prime}+ikV)^{-1},\hskip
14.22636ptV\equiv(U-c)/a_{1}$ (50)

where $a_{1}$ ($\simeq\kappa^{2};\kappa=0.4$ is Kármán’s constant) is
Townsend’s boundary-layer constant, and $a,c$ and $k$ are the amplitude, speed
and wave number of the surface wave. The velocity profile has the logarithmic
asymptote

$\displaystyle U\sim U_{1}\ln(\eta/z_{0}),\hskip 14.22636pt(\eta\gg z_{0})$
(51)

where

$\displaystyle U_{1}=U_{*}/\kappa,\hskip 14.22636ptz_{0}=\Omega U_{1}^{2}/g$
(52)

$U_{*}$ is the wind friction velocity and $\Omega$ is Charnock’s constant. We
seek the impedance

$\displaystyle\alpha+i\beta=({\mathscr{P}}_{0}+i{\mathscr{T}}_{0})/(kaU_{1}^{2})$
(53)

where

$\displaystyle{\mathscr{P}}_{0}=kac^{2}+(ik)^{-1}(\nu_{e}\Phi^{\prime\prime})^{\prime}\hskip
7.11317pt{\rm and}\hskip
7.11317pt{\mathscr{T}}_{0}=\nu_{e}\Phi^{\prime\prime},\hskip 7.11317pt\mbox{on
$\eta=0$}$ (54)

are complex amplitudes of the wind-induced perturbation pressure and shear
stress action on the wave.

###  3.1 Reduction to second-order differential equations

It is convenient to reduce the fourth-order differential equation (47) to the
pair of second-order equations

$\displaystyle{\mathscr{T}}^{\prime\prime}=(ik/\nu_{e})(U-c){\mathscr{T}}-ik[U^{\prime\prime}+k^{2}(U-c)]\Phi$
(55) $\displaystyle\Phi^{\prime\prime}={\mathscr{T}}/\nu_{e}$ (56)

for which the respective boundary conditions are

$\displaystyle{\mathscr{T}}={\mathscr{T}}_{0},\hskip
14.22636pt{\mathscr{T}}^{\prime}=ik({\mathscr{P}}_{0}-kac^{2})\hskip
7.11317pt\mbox{on $\eta=0$}$ (57)

and (48). Note that, ${\mathscr{T}}_{0}$ and ${\mathscr{P}}_{0}$ are
implicitly determined by the null conditions (49).

We render the formulation dimensionless by referring to $\eta$ to $k^{-1}$,
$c,U$ and $V$ to $U_{1}$, $\Phi$ to $aU_{1}$, ${\mathscr{P}}$ and
${\mathscr{T}}$ to $kaU_{1}^{2}$, thereby reducing (50), (55)–(57) and (48) to

$\displaystyle{\mathscr{T}}^{\prime\prime}=\textstyle\frac{1}{2}iV(U^{\prime}+iV){\mathscr{T}}-i(U^{\prime\prime}+U-c)\Phi$
(58)
$\displaystyle\Phi^{\prime\prime}=(2\kappa^{2})^{-1}(U^{\prime}+iV){\mathscr{T}}$
(59) $\displaystyle{\mathscr{T}}={\mathscr{T}}_{0},\hskip
7.11317pt{\mathscr{T}}^{\prime}=i({\mathscr{P}}_{0}-c^{2}),\hskip
7.11317pt\Phi=c,\hskip 7.11317pt\Phi^{\prime}=c-U^{\prime},\hskip
7.11317pt\mbox{on $\eta=0$}.$ (60)

###  3.2 Inner expansion for surface layer

A constant-stress interpolation between the logarithmic profile (51) and a
viscous sublayer of vanishing thickness is given by (Rotta 1950)

$\displaystyle\displaystyle\frac{U}{U_{1}}=\log(\zeta+\sqrt{\zeta^{2}+1})-\displaystyle\frac{\zeta}{1+\sqrt{\zeta^{2}+1}}\equiv\hat{U}(\zeta),\hskip
14.22636pt\zeta=\displaystyle\frac{1}{2}e\displaystyle\frac{\eta}{z_{0}}$ (61)

in which

$\displaystyle\displaystyle\frac{dU}{d\zeta}=\displaystyle\frac{U_{1}}{1+\sqrt{1+\zeta^{2}}}$

Note that $U^{\prime\prime}={\cal O}(U_{1}/z_{0}^{2})$ for $\zeta={\cal O}(1)$
and therefore, in contrast to the conventional Orr-Sommerfeld problem, is not
negligible near the boundary, although it does vanish at $\zeta=0$.

The inner and outer length scales are $z_{0}$ (or, more conveniently,
$\hat{z}_{0}\equiv 2z_{0}/e$) and $k^{-1}$, and the introduction of the inner
variable $\zeta$ and the small parameter

$\displaystyle\varepsilon\equiv k\hat{z}_{0}=(2\Omega/e)c^{-2}\ll 1$ (62)

in (58–59) and (60) leads to the inner expansions

$\displaystyle\Phi=c+\left(\varepsilon
c-\textstyle\frac{1}{2}\right)\zeta+(\varepsilon/2\kappa^{2})\left[{\mathscr{T}}_{0}\int_{0}^{\zeta}U\,d\zeta+i{\mathscr{G}}(\zeta)\right]+{\cal
O}(\varepsilon^{2}),$ (63)

where

$\displaystyle{\mathscr{G}}(\zeta)=\int_{0}^{\zeta}d\zeta_{1}\int_{0}^{\zeta_{1}}U^{\prime}(\zeta_{2})\,d\zeta_{2}\int_{0}^{\zeta_{2}}\left[\textstyle\frac{1}{2}(c-U+\zeta
U^{\prime})-cU^{\prime}\right]\,d\zeta_{3},$ (64)

and

$\displaystyle{\mathscr{T}}={\mathscr{T}}_{0}+i\left[\textstyle\frac{1}{2}(c+U)\zeta-
cU-\displaystyle\int_{0}^{\zeta}U\,d\zeta\right]+{\cal O}(\varepsilon)$ (65)

Letting $\zeta\rightarrow\infty$ in (63) and (64) and invoking
$\zeta=\eta/\varepsilon$, we obtain

$\displaystyle\Phi\sim
c+\left(c-\textstyle\frac{1}{2}\varepsilon^{-1}\right)\eta+\textstyle\frac{1}{2}\kappa^{-1}\left[{\mathscr{T}}_{0}\eta(U-1)+\textstyle\frac{1}{4}i\varepsilon^{-1}\eta^{2}\left(c-U+\textstyle\frac{7}{2}\right)\right].$
(66)

Hence, the problem is reduced in determining ${\mathscr{T}}$ and
${\mathscr{T}}_{0}$. The form given by (65) is not very convenient for this
task. A better approach is to determine ${\mathscr{T}}$ in the inner via
shear-stress-layer approximation.

In this region, the complex amplitude of the wind-induced perturbation shear
stress may be expressed as (Sajjadi 1998)

$\displaystyle{\mathscr{T}}=\nu_{e}[{\mathscr{U}}(\Phi/{\mathscr{U}})^{\prime\prime}-2U^{\prime}(\Phi/{\mathscr{U}})^{\prime}],\hskip
14.22636pt{\mathscr{U}}\equiv U-c$ (67)

Rearranging this equation, we obtain

$\displaystyle\displaystyle\frac{{\mathscr{U}}{\mathscr{T}}}{\nu_{e}}={\mathscr{U}}\Phi^{\prime\prime}-U^{\prime\prime}\Phi.$
(68)

Under the assumption that $k\eta\ll 1$ in this region, we may neglect
$k^{2}{\mathscr{U}}\Phi$ in (47) and upon combining the result with (68) we
arrive at the shear-stress-layer approximation

$\displaystyle{\mathscr{T}}^{\prime\prime}-(ik{\mathscr{U}}/\nu_{e}){\mathscr{T}}=0;\hskip
14.22636pt(k\eta\ll 1)$ (69)

whose asymptotic solution for $k\eta\downarrow 0$, which may be expressed in
terms of of modified Bessel function of the first kind (Sajjadi 2010), is

$\mathscr{T}(\eta)=-2\left\\{1+4\eta{\rm
K}^{\prime}_{0}(\eta)\right\\}+O(\varepsilon),$

(cf. Appendix A of BH)222The asymptotic solution given here reduces to that
given by BH in the limit as $c_{i}\downarrow 0$., whence for growing waves we
obtain

$\tau=-2e^{kc_{i}t}\mathscr{T}(\eta)e^{ik(x-c_{r}t)}.$

The real part of the complex amplitude of shear stress as a function of non-
dimensional height is shown in figure 3. This figure clear that the shear
stress becomes negative after decending from a maxima and then rising again.
The latter part is not depicted in this figure

Figure 3: The real part of the complex amplitude of $\mathscr{T}$ as a
function of non-dimensional height $k\eta$.

However, for the purpose of calculating the energy-transfer parameter, from
wind to wave $\beta$, we multiply (69) by ${\mathscr{T}}$ and integrate by
parts over $0<\eta<\infty$ to obtain

$\displaystyle({\mathscr{T}}{\mathscr{T}}^{\prime})_{0}=-\int_{0}^{\infty}\left[{\mathscr{T}}^{\prime
2}+(ik\mathscr{U}/\nu_{e}){\mathscr{T}}^{2}\right]\,d\eta$ (70)

The integral (70) is stationary with respect to first-order variations in
${\mathscr{T}}$ about the true solution (69).

Substituting the trial function

$\displaystyle{\mathscr{T}}={\mathscr{T}}_{0}e^{-k\eta/\delta}$ (71)

where $\delta$ is a (complex) free parameter to be determined, and $\nu_{e}$
given by (50), with $a_{1}=\kappa^{2}$, into (70) and invoking the condition
$\partial({\mathscr{T}}^{\prime}_{0}/{\mathscr{T}_{0}})/\partial\delta=0$ we
obtain

$\displaystyle\displaystyle\frac{{\mathscr{T}}^{\prime}_{0}}{{\mathscr{T}}_{0}}=-\displaystyle\frac{ik}{4\kappa^{2}}\left[(1+\delta_{*})\left(L_{\delta}^{2}+\textstyle\frac{1}{6}\pi^{2}\right)+2\delta^{-1}_{*}-\hat{c}^{2}\right]$
(72)
$\left(L_{\delta}^{2}+2L_{\delta}+\textstyle\frac{1}{6}\pi^{2}\right)\delta_{*}^{2}+2L_{\delta}\delta_{*}-2=0,\hskip
14.22636ptL_{\delta}\equiv L_{0}+\ln\delta,\hskip 14.22636pt\delta_{*}\equiv
i\kappa^{-2}\delta$ $None$

where $L_{0}\equiv\Lambda^{-1}=-\gamma-\ln 2{\sf k}$, ${\sf
k}=\Omega\hat{c}^{-2}e^{\hat{c}}$, $\hat{c}=c/U_{1}$, $\gamma=0.5772$ is the
Euler’s number and $\Omega$ is the Charnock’s constant.

Solving (3.27a) as a quadratic in $\delta_{*}L_{\delta}$ and letting
$\delta_{*}\rightarrow 0$ we get

$\displaystyle\frac{i\delta L_{\delta}}{\kappa^{2}}=\sqrt{3}-1+{\cal
O}(\delta)$ $None$

##  4 Outer approximation above inertial critical-layer

In contrast to the inner region, the solution in the outer region is very
straightforward. In the outer domain $k\eta\gg 1$, $U$ may be approximated by
(51), and $\Phi$ admits the Green-Liouville approximation (Olver 1974, chap.
6)

$\displaystyle\Phi\sim{\mathscr{F}}(\mathscr{Z})\exp\left[\displaystyle\int_{0}^{\eta}{\mathscr{W}}(\mathscr{Z})\,d\eta\right],\hskip
14.22636pt{\mathscr{Z}}(\eta)\sim(1/\kappa^{2})\ln(\eta/\eta_{c})$ (73)

Substituting (73) into the dimensionless counterpart of (47), we obtain

$\displaystyle{\mathscr{W}}^{4}+\textstyle\frac{1}{2}{\mathscr{Z}}^{2}({\mathscr{W}}^{2}-1)=0$
(74)

and

$\displaystyle\displaystyle\frac{{\mathscr{W}}({\mathscr{W}}^{2}+{\mathscr{Z}}^{2})}{{\mathscr{F}}}\displaystyle\frac{d{\mathscr{F}}}{d{\mathscr{Z}}}+\left(6{\mathscr{W}}^{2}+\textstyle\frac{1}{2}{\mathscr{Z}}^{2}\right)\displaystyle\frac{d{\mathscr{W}}}{d{\mathscr{Z}}}+\displaystyle\frac{(i\kappa^{2}{\mathscr{W}}-2){\mathscr{W}^{3}}}{\mathscr{Z}}=0$
(75)

from which it follows that

$\displaystyle{\mathscr{W}}^{2}=\textstyle\frac{1}{4}\left[-{\mathscr{Z}}^{2}\pm({\mathscr{Z}}^{4}+8{\mathscr{Z}}^{2})^{1/2}\right],$
(76)
$\displaystyle\displaystyle\frac{d{\mathscr{W}}}{d{\mathscr{Z}}}=\displaystyle\frac{-{\mathscr{Z}}({\mathscr{W}}^{2}-1)}{{\mathscr{W}}(4{\mathscr{W}}^{2}+{\mathscr{Z}}^{2})}=\displaystyle\frac{2{\mathscr{W}}^{3}}{{\mathscr{Z}}(4{\mathscr{W}}^{2}+{\mathscr{Z}}^{2})}=\displaystyle\frac{\pm
2{\mathscr{W}}^{3}}{{\mathscr{Z}}({\mathscr{Z}}^{4}+8{\mathscr{Z}}^{2})^{1/2}}$
(77)

and

$\displaystyle\displaystyle\frac{d\ln{\mathscr{F}}}{d{\mathscr{Z}}}=-\displaystyle\frac{{\mathscr{W}}^{2}(4{\mathscr{W}}^{2}-{\mathscr{Z}}^{2})}{{\mathscr{Z}}(4{\mathscr{W}}^{2}+{\mathscr{Z}}^{2})^{2}}-\displaystyle\frac{i\kappa^{2}{\mathscr{W}}^{3}}{{\mathscr{Z}}(4{\mathscr{W}}^{2}+{\mathscr{Z}}^{2})}.$
(78)

We restrict further consideration to the asymptotic limit
$V\rightarrow\infty$, for which the admissible roots of (76) may be
approximated by

$\displaystyle{\mathscr{W}}\sim-1,\hskip 14.22636pt-i{\mathscr{Z}}/\sqrt{2}$
(79)

(the roots $+1$ and $+i{\mathscr{Z}}/\sqrt{2}$ are ruled out by the null
condition at ${\mathscr{Z}}=\infty$). The corresponding approximations to (78)
are

$\displaystyle\displaystyle\frac{d\ln{\mathscr{F}}}{d{\mathscr{Z}}}\sim\displaystyle\frac{1+i\kappa^{2}}{{\mathscr{Z}}^{3}},\hskip
14.22636pt-\textstyle\frac{3}{2}{\mathscr{Z}}^{-1}-2^{-3/2}\kappa^{2},$ (80)

the integration of which leads, through (73), to

$\displaystyle\Phi\sim
C_{1}e^{-\eta}+C_{2}{\mathscr{Z}}^{-3/2}\exp\left[-\displaystyle\frac{i\eta({\mathscr{Z}}-\kappa^{-2})}{\sqrt{2}}-\displaystyle\frac{\kappa^{2}{\mathscr{Z}}}{2\sqrt{2}}\right],\hskip
7.11317pt\mbox{for ${\mathscr{Z}}\rightarrow\infty$}$ (81)

where $C_{1}$ and $C_{2}$ are constants. [the term
$-\textstyle\frac{1}{2}{\mathscr{Z}}^{-2}$, derived from the real part of
(80), has been neglected in the exponent in (81) since it is dominated by the
error implicit in the approximation (79).] The corresponding approximation to
${\mathscr{T}}$, obtained through substitution of (81) into (59) is

$\displaystyle{\mathscr{T}}\sim
2i\kappa^{2}\left[C_{1}{\mathscr{Z}}^{-1}e^{-\eta}-\textstyle\frac{1}{2}C_{2}{\mathscr{Z}}^{-1/2}\exp\left\\{-\displaystyle\frac{i\eta({\mathscr{Z}}-\kappa^{-2})}{\sqrt{2}}-\displaystyle\frac{\kappa^{2}{\mathscr{Z}}}{2\sqrt{2}}\right\\}\right].$
(82)

Note that, using the standard exponential substitution of the Liouville-Green
method for the asymptotic solution of (47), we obtain the following expansion
of the phase function

$\displaystyle\Phi$ $\displaystyle\sim$
$\displaystyle\hat{\eta}^{\gamma_{1}}[\log(\hat{\eta}/z_{0})]^{\gamma_{2}}\exp[\gamma_{3}/\log(\hat{\eta}/z_{0})]...\exp\\{i[\vartheta_{1}\hat{\eta}\log(\eta/z_{0})$
(83) $\displaystyle+$ $\displaystyle\vartheta_{2}\hat{\eta}+\vartheta_{3}{\rm
Ei}(1,-\log(\hat{\eta}/z_{0}))+\vartheta_{4}\hat{\eta}/\log(\hat{\eta}/z_{0})+...]\\}$

where $\vartheta_{k}$ and $\gamma_{k}$ are real, $\hat{\eta}\approx k\eta$ and
Ei is the exponential integral. Formally there is an infinite number of terms
with coefficients $\vartheta_{k}$ that precede the determination of the
$\gamma_{k}$. However, as only the first of these, namely $\vartheta_{1}$,
enters into the formula for $\gamma_{1}$, the first two into $\gamma_{2}$
(although the $\vartheta$ contributions happen to cancel), the first four into
$\gamma_{3}$, and so on. The result is depicted in figure 4 where we see that
an initial expoential decay follows by an algebraic tail. It should be noted
that this is agreement with numerical simulation of Ierley and Miles (2001),
and the expression (83) is originally found by them. However, the results of
figure 4 is drawn from the analytical expression given by the equation (83).

Figure 4: The variation of the log of moduls square for perturbation stream
function $\Phi$ against the non-dimentional distance $k\eta$ showing the
initial exponential decay follows by an algebraic tail. The lower figure is a
close of the upper figure in the range $6\leq k\eta\leq 20$.

Since the governing equation is fourth order, we find $\vartheta$ by solving
for the roots of the fourth degree polynomial

$\displaystyle\vartheta^{2}_{1}\left[(2\kappa^{4}+A_{1})\vartheta_{1}^{2}+1\right]=0$

Two roots are zero and the other two constitute an imaginary pair. The double
zero leads to a particularly simple result at next order:
$\vartheta^{2}_{2}-1=0$, thus one pair of solutions is approximately
$\exp(\pm\eta)$. The pure imaginary root pair exhibits weak algebraic growth
(or decay), as reflected in $\gamma_{1}$. A suitable boundary condition for
(47) is to match the decaying exponential and the decaying algebraic
solutions. For this purpose, we employ the following expressions

$\displaystyle\vartheta_{1}=-(A_{1}+2\kappa^{4})^{-1/2}$
$\displaystyle\vartheta_{2}=(1+\kappa^{3})(A_{1}+2\kappa^{4})^{-1/2}-\displaystyle\frac{A_{3}+B_{1}}{2(2\kappa^{4}+A_{1})}-\displaystyle\frac{b(4\kappa^{4}+A_{1})}{2(2\kappa^{4}+A_{1})}$
$\displaystyle\gamma_{1}=-\displaystyle\frac{1}{2}\kappa^{2}(A_{1}+2\kappa^{4})^{-1/2};\,\,\,\,\,\,\,\,\gamma_{2}=-\displaystyle\frac{3}{2}$

after selecting $\tilde{a}_{1}=\kappa^{2}$.

##  5 Effect of the inertial critical-layer

###  5.1 Comparison with BHC

In the case of slow waves, where $c\ll 1$, the perturbation shear stress in
(47) can be neglected in the outer region and thus $\Phi$ will satisfies the
Rayleigh equation

$\displaystyle(U-c)(\Phi^{\prime\prime}-k^{2}\Phi)-U^{\prime\prime}\Phi=0$
(84)

where now we shall assume $c=c_{r}$ and $c_{i}=0$. The corresponding
expressions for unsteady waves will be given in the next subsection.

As was shown by Sajjadi (1988), the leading order solution to (84) is

$\displaystyle\Phi=(U-c)e^{-k\eta}\left[{\sf
A}+\Phi_{c}U_{c}^{\prime}e^{k\eta_{c}}\int_{0}^{\infty}\left\\{\displaystyle\frac{1}{(U-c)^{2}}-1\right\\}\,d\eta\right]$
(85)

The integral is (85) is regular at the critical height and hence, by indenting
the path of integration in (85) under the singularity $\eta=\eta_{c}$, we
obtain

$\displaystyle\Phi=(U-c)e^{-k\eta}\left[{\sf
A}+\Phi_{c}U_{c}^{\prime}e^{k\eta_{c}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{\infty}\left\\{\displaystyle\frac{1}{(U-c)^{2}}-1\right\\}\,d\eta-I\right)\right]$
(86)

where A is constant which can be determined from the boundary conditions and

$\displaystyle I=\lim_{\varpi\rightarrow
0}\int_{\eta_{c}-\varpi}^{\eta_{c}+\varpi}\left\\{\displaystyle\frac{1}{(U-c)^{2}}-1\right\\}\,d\eta$
(87)

Expanding $U(\eta)$ in a Taylor expansion in the vicinity of the critical
point, setting $\eta=\eta_{c}\varpi e^{i\theta}$, where $\varpi\equiv
c_{i}/U_{*}\ll 1$, and

$\displaystyle\tan\theta=-c_{i}/U_{c}^{\prime}(\eta-\eta_{c})$ (88)

then (87) becomes

$\displaystyle I$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{U_{c}^{\prime
2}}\left\\{\lim_{\varpi\rightarrow
0}\int_{\eta_{c}-\varpi}^{\eta_{c}+\varpi}\displaystyle\frac{d\eta}{(\eta-\eta_{c})^{2}}+i\pi\displaystyle\frac{U_{c}^{\prime\prime}}{U_{c}^{\prime}}\right\\}$
(89) $\displaystyle=$ $\displaystyle\displaystyle\frac{i\pi
U_{c}^{\prime\prime}}{U_{c}^{\prime 3}}$

which agrees with the result obtained by Belcher et al. (1999).

As also pointed out by Belcher et al. (1999), for a logarithmic mean velocity
profile (88) yields $\tan\theta=\varpi\eta_{c}/(\eta-\eta_{c})$. Hence
$\theta$ varies between $0$ and $\pi$ as $(\eta-\eta_{c})/\ell_{c}$ tends to
$\pm\infty$, respectively. Note that, the transition between these limiting
values occurs across the layer of thickness $\ell_{c}=\varpi\eta_{c}$. Note
also, the significance of the term $iU_{c}^{\prime\prime}/U_{c}^{\prime 3}$ in
the solution for $I$ is that it leads to an out of phase contribution to the
wave induced vertical velocity that gives rise to the same contribution to the
wave growth from Miles (1957) critical-layer mechanism.

The result of the present analysis confirms the earlier finding of Belcher et
al. (1999) in that Miles (1957) solution is only valid when the waves grow
significantly slowly such that

$\displaystyle c_{i}\ll U_{c}^{\prime}\eta_{c}\sim U_{*}$ (90)

As in Belcher et al. (1999), our analysis also shows that when inertial
effects controls the behaviour around the critical layer, there is a smooth
behaviour around the critical layer of thickness

$\displaystyle\ell_{c}\sim c_{i}/U_{c}^{\prime}\sim\eta_{c}c_{i}/U_{*}$ (91)

Hence this proves the effects of critical layer, as calculated by Miles
(1957), are only valid in the limit $c_{i}/U_{*}\downarrow 0$.

###  5.2 Steady monochromatic waves

For comparison with the waves that are unsteady, we calculate the energy-
transfer parameter due to critical layer, $\beta_{c}$, for steady
monochromatic waves. Thus, we let $\Phi=-\mathscr{UM}$. Thus, (47) becomes

$\displaystyle[\nu_{e}({\mathscr{U}}{\mathscr{M}}^{\prime\prime}+2U^{\prime}{\mathscr{M}}^{\prime}+U^{\prime\prime}{\mathscr{M}})]^{\prime\prime}=ik[({\mathscr{U}}^{2}{\mathscr{M}}^{\prime})^{\prime}-k^{2}{\mathscr{U}}^{2}{\mathscr{M}}]$
(92)

In quasi-laminar limit the left-hand side of (92) is negligible and thus (92)
reduces to

$\displaystyle({\mathscr{U}}^{2}{\mathscr{M}}^{\prime})^{\prime}-k^{2}{\mathscr{U}}^{2}{\mathscr{M}}=0$
(93)

Multiplying (93) by ${\mathscr{M}}$, integrating by parts over
$0<\eta<\infty$, and invoking the inner limits ${\mathscr{M}}\rightarrow a$
and ${\mathscr{U}}^{2}{\mathscr{M}}^{\prime}\rightarrow{\mathscr{P}}_{0}$ and
a null condition at $\eta=\infty$, we obtain

$\displaystyle
a{\mathscr{P}}_{0}=-\int_{0}^{\infty}{\mathscr{U}}^{2}({\mathscr{M}}^{\prime
2}+k^{2}{\mathscr{M}}^{2})\,d\eta$ (94)

Using the simplest admissible trial function for the variational integral
(94), i.e.

$\displaystyle{\mathscr{M}}=ae^{-k\eta/\varsigma}$ (95)

where $\varsigma$ is a free parameter. Substituting (95) into (94) together
with the approximation ${\mathscr{U}}\approx U_{1}\ln(\eta/\eta_{c})$ we get

$\displaystyle\hat{{\mathscr{P}}}_{0}\equiv{\mathscr{P}}/kaU_{1}^{2}$
$\displaystyle=$
$\displaystyle-k(\varsigma^{-2}+1)\int_{0}^{\infty}e^{-2k\eta/\varsigma}\ln^{2}(\eta/\eta_{c})\,d\eta$
(96) $\displaystyle=$
$\displaystyle-\xi_{c}(\varsigma^{-2}+1)\int_{0}^{\infty}e^{-st}\ln^{2}t\,dt$
$\displaystyle=$
$\displaystyle-\displaystyle\frac{\varsigma+\varsigma^{-1}}{2}\left\\{\displaystyle\frac{\pi^{2}}{6}+\ln^{2}\left(\displaystyle\frac{2\gamma\xi_{c}}{\varsigma}\right)\right\\}$

where $\xi_{c}\equiv k\eta_{c}$. It then follows from the variational
condition $\partial\hat{{\mathscr{P}}}_{0}/\partial\varsigma=0$ that

$\displaystyle\varsigma^{2}=\displaystyle\frac{L_{\varsigma}^{2}-2L_{\varsigma}+\pi^{2}/6}{L_{\varsigma}^{2}+2L_{\varsigma}+\pi^{2}/6}$
(97)

where $L_{\varsigma}\equiv 2\gamma\xi_{c}/\varsigma$, and $\varsigma=O(1)$.

The corresponding, quasi-laminar approximation to the energy-transfer
parameter may be calculated from (86), which implies
$\Phi_{c}={\mathscr{P}}_{c}/U_{c}^{\prime}\approx{\mathscr{P}}_{0}/U_{c}^{\prime}$,
and (89), which yields

$\displaystyle\beta_{c}=\pi\xi_{c}|\Phi_{c}/U_{1}a|^{2}$ $\displaystyle=$
$\displaystyle\pi\xi^{3}_{c}|\hat{\mathscr{P}}_{0}|^{2}=\textstyle\frac{1}{4}\pi(\varsigma+\varsigma^{-1})^{2}\left(L_{\varsigma}+\textstyle\frac{1}{6}\pi^{2}\right)^{2}$
(98) $\displaystyle=$
$\displaystyle\pi\xi_{c}^{3}L_{0}^{4}\left[1-\left(4-\textstyle\frac{1}{3}\pi^{2}\right)\Lambda^{2}+{\cal
O}(\Lambda^{3})\right]$ (99)

where $\Lambda=L_{0}^{-1}=-\gamma-\ln(2{\sf k}),L_{0}=L_{\delta}-\ln\delta$,
and $i\delta L_{\delta}/\kappa^{2}=(\sqrt{3}-1)+O(\delta)$.

To obtain the corresponding expression for the component of the energy-
transfer parameter, $\beta_{T}$, due to turbulence, we multiply (47) by
$-{\mathscr{M}}$, integrating over $0<z<\infty$, invoking the conditions

${\mathscr{M}}=a,\hskip 7.11317pt{\mathscr{M}}^{\prime}=ka,\hskip
7.11317pt{\mathscr{T}}^{\prime}=ik[{\mathscr{P}}_{0}-kac^{2}]$

on $z=0$ and the null condition for $z\rightarrow 0$, we obtain

$\displaystyle\int_{0}^{\infty}{\mathscr{M}}{\mathscr{T}}^{\prime\prime}\,dz$
$\displaystyle=$ $\displaystyle
ka[{\mathscr{T}}_{0}-i{\mathscr{P}}_{0}]+i(kac_{r})^{2}+\int_{0}^{\infty}{\mathscr{M}}^{\prime\prime}{\mathscr{T}}\,dz$
$\displaystyle=$ $\displaystyle
i(kac_{r})^{2}+ik\int_{0}^{\infty}{\mathscr{V}}^{2}\left({\mathscr{M}}^{\prime
2}+k^{2}{\mathscr{M}}^{2}\right)\,dz,$

with $c=c_{r}+ic_{i}$. Then, in the limit as $s\equiv\rho_{a}/\rho_{w}$, where
$\rho_{a}$ and $\rho_{w}$ are densities of the air and water, respectively, we
obtain from

$\displaystyle\alpha+i\beta\equiv(c^{2}-c_{w}^{2})/sU_{1}^{2}=(\mathscr{P}_{0}+i\mathscr{T}_{0})/kaU_{1}^{2}\equiv(\hat{\mathscr{P}}_{0}+i\hat{\mathscr{T}}_{0}),$
(100)

where $c$ is the complex wave speed, $c_{w}=\sqrt{g/k}$ is the speed of water
waves in the absence of the airflow above it and the suffix zero denotes
evaluation at $z=0$,

$\displaystyle\alpha_{T}+i\beta_{T}=(kaU_{1})^{-2}\int_{0}^{\infty}\left\\{i\nu_{e}\left[{\mathscr{V}\mathscr{M}}^{\prime\prime
2}+2U^{\prime}{\mathscr{M}}/{\mathscr{M}}^{\prime\prime}+U^{\prime\prime}{\mathscr{M}}{\mathscr{M}}^{\prime\prime}\right]\right.$
$\displaystyle\left.-k{\mathscr{V}}^{2}\left({\mathscr{M}}^{\prime
2}+k^{2}{\mathscr{M}}^{2}\right)\right\\}\,dz.$ (101)

The above integral can be evaluated asymptotically333The detail evaluations
may be obtained found in the appendix of the paper by Sajjadi (2007). whose
imaginary part yields

$\displaystyle\beta_{T}=5\kappa^{2}L_{0}^{-1}.$ (102)

Therefore, in summary, the energy-transfer parameter from wind to surface
waves for steady monochromatic waves may be given by the following formulae.

$\displaystyle\beta=\beta_{T}+\beta_{c},\hskip
28.45274pt\beta_{T}=\displaystyle\frac{5\kappa^{2}}{\Lambda},\hskip
28.45274pt\beta_{c}=\displaystyle\frac{5}{2}\pi\hat{W}L_{0}^{4}\left[1-\left(4-\displaystyle\frac{\pi^{2}}{3}\right)\varepsilon^{2}\right]$
$\displaystyle\Lambda=L_{0}^{-1},\hskip
14.22636ptL_{0}=-\gamma-\log\hat{W},\hskip
14.22636pt\hat{W}=kz_{0}e^{c_{r}/U_{\lambda}}(U_{\lambda}/c_{r})^{2},\hskip
14.22636ptU_{\lambda}=2U_{*}$

where $\kappa=0.4$ is von Kármán’s constant and $\gamma=0.5772$ is Euler’s
constant.

###  5.3 Unsteady waves

The generalization of the results just obtained above follows a similar
development, but with the exception that $c_{i}\neq 0$. Here we shall present
results for Stokes waves being a sum of two harmonics $(n=1,2)$. Hence, we
consider the surface wave expressed as

$z=\mbox{Re}\\{ae^{ik(x-ct)}+\textstyle\frac{1}{2}ka^{2}e^{2ki(x-ct)}\\},\qquad
c=c_{r}+ic_{i}$

Note that, results for monochromatic waves follows immediately from what will
be developed by setting $n=1$ and ignoring the second harmonic.

Therefore, we begin by using the expression (5.11), but now we take

$\displaystyle\mathscr{U}=U-c_{r}-ic_{i}$ (103)

Thus, the expression (5.11) now reads

$\displaystyle
a\mathscr{P}_{n0}=-\int_{0}^{\infty}[(U-c_{r})^{2}-2ic_{i}(U-c_{r})-c_{i}^{2}](\mathscr{M}^{\prime
2}_{n}+k_{n}^{2}\mathscr{M}_{n})\,d\eta$ (104)

where $k_{1}\equiv k$ and $k_{2}\equiv 2k$.

Once again using a similar admissible trial function as that given by (95),
namely

$\mathscr{M}_{n}=ae^{k_{n}\eta/\varsigma_{n}}\qquad(n=1,2)$

the variational integral (104) becomes

$\displaystyle a\mathscr{P}_{n0}$ $\displaystyle=$ $\displaystyle
a^{2}k_{n}^{2}(\varsigma_{n}^{-2}+1)\left\\{\int_{0}^{\infty}U_{1}^{2}\ln^{2}(\eta/\eta_{c})e^{-2k_{n}\eta/\varsigma_{n}}\right.\,d\eta$
$\displaystyle+$ $\displaystyle
2ic_{i}\int_{0}^{\infty}U_{1}\ln(\eta/\eta_{c})e^{-2k_{n}\eta/\varsigma_{n}}\,d\eta+\left.c_{i}^{2}\int_{0}^{\infty}e^{-2k_{n}\eta/\varsigma_{n}}\,d\eta\right\\}$

Evaluating the integrals, we obtain

$\displaystyle\hat{\mathscr{P}}_{n0}\equiv\mathscr{P}_{n0}/k_{n}a_{n}U_{1}^{2}=\displaystyle\frac{1}{2}\left(\varsigma_{n}+\varsigma_{n}^{-1}\right)\left\\{\left[\displaystyle\frac{\pi^{2}}{6}+\log^{2}\left(\displaystyle\frac{2k_{n}\gamma\eta_{c}}{\varsigma_{n}}\right)\right]-2i\hat{c}_{i}\log\left(\displaystyle\frac{2k_{n}\gamma\eta_{c}}{\varsigma_{n}}\right)+\hat{c}_{i}^{2}\right\\}$
(105)

where $\hat{c}_{i}=c_{i}/U_{1}$.

As before, applying the variational condition
$\partial\hat{\mathscr{P}}_{n0}/\partial\varsigma_{n}=0$ yields

$\displaystyle L_{n\varsigma}^{2}$ $\displaystyle-$ $\displaystyle
2(1+i\hat{c_{i}})L_{n\varsigma}+\left(\hat{c}_{i}^{2}+2i\hat{c}_{i}+\displaystyle\frac{\pi^{2}}{6}\right)$
$\displaystyle=$
$\displaystyle\varsigma_{n}^{2}\left[L_{n\varsigma}^{2}+2(1+i\hat{c_{i}})L_{n\varsigma}+\left(\hat{c}_{i}^{2}-2i\hat{c}_{i}+\displaystyle\frac{\pi^{2}}{6}\right)\right]$

whence

$\displaystyle\varsigma_{n}^{2}=\displaystyle\frac{L_{n\varsigma}^{2}-2\mathfrak{C}_{i}L_{n\varsigma}+\left(\hat{c}_{i}^{2}+2i\hat{c}_{i}+\displaystyle\frac{\pi^{2}}{6}\right)}{L_{n\varsigma}^{2}+2\mathfrak{C}_{i}^{*}L_{n\varsigma}+\left(\hat{c}_{i}^{2}-2i\hat{c}_{i}+\displaystyle\frac{\pi^{2}}{6}\right)}$
(106)

where $\mathfrak{C}_{i}=1+\hat{c}_{i}$, the superscript * denotes the complex
conjugate, and $L_{n\varsigma}\equiv 2\gamma k_{n}\eta_{c}/\varsigma_{n}$
where $L_{n\varsigma}=-\gamma-\log(2k_{n}\eta_{c})$.

The expression (106) may be approximated to $O(\Lambda_{n}^{2})$, to give

$\varsigma_{n}=\displaystyle\frac{1-\mathfrak{C}_{i}\Lambda_{n}}{1+\mathfrak{C}_{i}^{*}\Lambda_{n}}+O(\Lambda_{n}^{2})$

where $\Lambda_{n}\equiv L_{n0}^{-1}$, and therefore we obtain the following
expression for the energy-transfer parameter to the two $(n=1,2)$ harmonics of
the wave

$\displaystyle\beta_{nc}=\pi\left(\displaystyle\frac{\varsigma_{n}L_{n\varsigma}}{2\gamma}\right)^{3}L_{n0}^{4}\left[1-\left(4-\displaystyle\frac{\pi^{2}}{3}+10\hat{c}_{i}^{2}\right)\Lambda_{n}^{2}+O(\Lambda_{n}^{3})\right]$
(107)

Note that, for a steady wave ($c_{i}=0$) and a monochromatic wave ($n=1$ and
ignoring the second harmonic of the Stoke wave) the expression (107) reduces
to (5.16). We remark that for a steady Stokes wave we only need to assume
$c_{i}=0$.

In a similar manner, for an unsteady wave, we adopt the complex amplitude of
surface shear stress is given by (3.24) but with $\mathscr{U}$ given by (103)
and the following modification for the expression for the eddy viscosity,
given by (3.4), namely

$\displaystyle\nu_{ne}=2U_{*}^{2}\\{U_{1}[\eta^{-1}+(ik_{n}/\kappa^{2})\ln(\eta/\eta_{c})]+k_{n}c_{i}/\kappa^{2}\\}^{-1}$
(108)

Thus, upon substituting (108), and using an equivalent trial function to that
given by (3.25), namely

$\displaystyle\mathscr{T}_{n}=\mathscr{T}_{n0}e^{-k_{n}\eta/\delta_{n}}$ (109)

the ‘unsteady’ version of the integral (3.24) becomes

$\displaystyle(\mathscr{T}_{n}\mathscr{T}_{n}^{\prime})_{0}=-k_{n}\int_{0}^{\infty}[k_{n}/\delta_{n}^{2}+U_{1}^{2}(i\mathscr{U}^{\prime
2}-2k_{n}\mathscr{U}^{2})]\mathscr{T}_{n0}e^{-2k_{n}\eta/\delta_{n}}\,d\eta$
(110)

Invoking the condition
$\partial(\mathscr{T}_{n}/\mathscr{T}_{n})_{0}/\partial\delta_{n}=0$, after
substituting $\mathscr{U}$ and (109), and evaluating the integral (110) we
obtain

$\displaystyle(\mathscr{T}_{n}/\mathscr{T}_{n})_{0}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2i\delta_{n}}-\displaystyle\frac{(c_{r}+ic_{i})^{2}}{4U_{*}^{2}}+\displaystyle\frac{\delta_{n}}{4\kappa^{2}}\left(\displaystyle\frac{1}{\delta_{n}}-\displaystyle\frac{ik_{n}}{4\kappa^{2}}\right)\left(\displaystyle\frac{\pi^{2}}{6}+L_{\delta_{n}}^{2}\right)$
(111) $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{4\kappa^{2}}\left[\displaystyle\frac{2}{\delta_{n}}-\displaystyle\frac{(c_{r}+ic_{i})^{2}}{U_{1}^{2}}+(1+\hat{\delta_{n}})\left(\displaystyle\frac{\pi^{2}}{6}+L_{\delta_{n}}^{2}\right)\right]$

where $\hat{\delta_{n}}=i\delta_{n}/\kappa^{2}$,

$\displaystyle\left(L_{\delta_{n}}^{2}+2L_{\delta_{n}}+\displaystyle\frac{\pi^{2}}{6}\right)\hat{\delta}_{n}^{2}+L_{\delta_{n}}\hat{\delta}_{n}-2=0$
(112)

and $L_{\delta_{n}}=L_{n0}+\ln\delta_{n}$.

Solving the equation (112) for $\hat{\delta}_{n}$ we find that

$\displaystyle\hat{\delta_{n}}=(\sqrt{3}-1)(L_{n0}+\ln\delta_{n})^{-1}$ (113)

Note that, $\delta_{n}$ may be complex and $|\delta_{n}|\ll 1$ (but strictly
not equal to zero), we may use the expansion for $\ln\delta_{n}$, given by

$\ln\delta_{n}=2\displaystyle\sum_{m=1}^{\infty}\displaystyle\frac{1}{2m-1}\left(\displaystyle\frac{\delta_{n}-1}{\delta_{n}+1}\right)^{2m-1}\doteqdot
2\displaystyle\frac{\delta_{n}-1}{\delta_{n}+1}+O\left[\left(\displaystyle\frac{\delta_{n}-1}{\delta_{n}+1}\right)^{3}\right],\qquad
0<|\delta_{n}|\leq 2$

we may cast (113) as

$\displaystyle\hat{\delta}_{n}=(\sqrt{3}-1)\left[L_{n0}+2\left(\displaystyle\frac{\delta_{n}-1}{\delta_{n}+1}\right)\right]^{-1}$
(114)

Hence, the asymptotic evaluation of integrals in (110) yields to the following
expression (for further details see the appendix of Sajjadi (2007))

$\displaystyle\beta_{n}=2\kappa^{2}\left[1+(\sqrt{3}+1)\left(1-\mathscr{E}_{\delta_{n}}\right)L_{|\delta_{n}|}-(\sqrt{3}-1)\left(L_{|\delta_{n}|}+4\ln
2\right)\right.$
$\displaystyle-\left.4\left(\mathscr{E}_{\delta_{n}}-\mathscr{E}_{2\delta_{n}}\right)\right]-4\delta_{ni}L_{n0}^{2}$
(115)

where $\delta_{ni}=\mathscr{I}\\{\delta_{n}\\}$,

$\displaystyle\mathscr{E}_{p}=\mathscr{E}\left(\displaystyle\frac{2a_{1}}{ipL_{p}}\right)\approx
1-\displaystyle\frac{ipL_{p}}{2a_{1}},\qquad
L_{p}=-\gamma-\ln\left(2\sf{k}{p}\right)=L_{0}+\ln p$

and

$\mathscr{E}(\Theta_{n})=\Theta_{n}e^{\Theta_{n}}\int_{0}^{\infty}t^{-1}e^{-t}\,dt=\Theta_{n}e^{\Theta_{n}}{\sf
E}_{1}(\Theta_{n})$

From these expressions we obtain (see Sajjadi (2007)) we obtain

$\beta_{nc}=\displaystyle\frac{1}{2}(\Lambda_{n}+\Lambda_{n}^{-1})\left\\{L_{\varsigma_{n}}^{2}+\displaystyle\frac{\pi^{2}}{6}\right\\},\qquad\beta_{nT}=\displaystyle\frac{4\kappa^{2}}{\Lambda_{n}}$

which are in reasonable agreement with (5.19) and (5.24).

##  6 Results and conclusions

In figure 5, we show comparison of the energy-transfer rate, $\beta$, between
the present result for a monochromatic unsteady (growing) wave, both
analytically and numerically, and those calculated by Miles (1957) and Janssen
(1991) for the steady wave counterpart. Miles and Janssen both assume that the
drag $C_{D}$, and thence $\beta$, is dominated by the limiting inviscid wave
growth mechanism, thus their formulation is independent of $c_{i}$. In
contrast, the present calculation is for a viscous unsteady (growing) wave,
where $c_{i}/U_{*}=0.01$ and $kz_{0}=10^{-4}$.

We emphasize that the various models, such as those by Belcher & Hunt (1993),
Mastenbroek et al. (1996), Cohen (1997) etc., all generally agree with our
numerical simulations performed using Launder, Reece & Rodi (1975) Reynolds-
stress closure scheme for the energy transfer parameter, $\beta$, shown in
figure 5. This shows consistency between these models and the unimportance of
very small $c_{i}$ for which viscous processes are significant.

Figure 6 shows comparison of $\beta_{c}$ as a function of wave age
$c_{r}/U_{1}$, calculated according to (Conte & Miles 1959) for numerical
solution of inviscid Orr-Sommerfeld equation, against the numerical solution
of equation (1.6) for $c_{i}/U_{*}=0.01$, $kz_{0}=10^{-4}$ and $\nu_{e}\neq
0$. We remark that increasing $c_{i}/U_{*}$ from 0.01 to 0.1 (not shown here)
makes no significant difference in the magnitude of $\beta_{c}$. We conclude
therefore for a finite value of $\nu_{e}$ the right-hand side of equation
(1.6) is dominant and therefore the magnitude of $\beta_{c}$, calculated from
the solution of (1.6), is practically zero over a wide range of the wave age,
in particular for a ‘young’ wave, where $c_{r}/U_{1}<2.$ We thus conclude that
the critical-layer mechanism plays an insignificant role for $c_{r}/U_{1}<9$,
and very little effect for $9\leq c_{r}/U_{1}\leq 10.5$.

Figure 5: Total energy transfer parameter, $\beta$, using different models for
critical layer and sheltering mechanisms for unsteady waves (where $c_{i}\ll
U_{*}$) as a function of the wave age $c_{r}/U_{1}$. +++++, Miles (1957)
calculation ($c_{i}=0,\nu_{e}=0$) from his formula:
$\beta=\pi\eta_{c}\left\\{\textstyle\frac{1}{6}\pi^{2}+\log^{2}(\gamma\eta_{c})+2\sum_{n=1}^{\infty}\textstyle\frac{(-1)^{n}\eta_{c}^{n}}{n!n^{2}}\right\\}^{2}$,
where $\eta_{c}=kz_{c}$ is the critical height
$\eta_{c}=\Omega(U_{1}/c_{r})^{2}e^{c_{r}/U_{1}}$ and
$\Omega=gz_{0}/U_{1}^{2}$ is the Charnock’s (1955) constant. Thick solid line,
parameterization of Miles (1957) formula, for $c_{i}=0,\nu_{e}=0$, given by
Janssen (1991): $\beta=1.2\kappa^{-2}\eta_{c}\log^{4}\eta_{c}$, where
$\eta_{c}=\min\left\\{1,kz_{0}e^{[\kappa/(U_{*}/c+0.011)]}\right\\}$. Thin
solid line, present formulation: ($\beta_{T}+\beta_{c}$) for $c_{i}\neq
0,\nu_{e}\neq 0$. $\circ$, Numerical simulation using Launder, Reece & Rodi
(1975) Reynolds-stress closure model for $c_{i}\neq 0,\nu_{e}\neq 0$. Note
that, $\beta$ for Miles and Janssen is equivalent to $\beta_{c}$ in our
notation. Taken from Sajjadi et al. (2014).

We remark that these parameterizations have been incorporated and tested in a
spectral wave models, WaveWatch and WindWave, which shows superior results
when compared with field data (see Fitzpatrick et al. 2002 and Sajjadi et al.
2002).

Figure 6: Component of energy transfer parameter, $\beta_{c}$, by different
models of critical layer mechanisms for unsteady waves (where $c_{i}\ll
U_{*}$) as a function of the wave age $c_{r}/U_{1}$. $\bullet$, numerical
solution of inviscid Orr-Sommerfeld equation by Conte & Miles (1959) for
$c_{i}=0$ and $\nu_{e}=0$ using the singular critical layer approach; $\circ$
numerical solution of equation (1) for $c_{i}\neq 0$ and $\nu_{e}\neq 0$.
Taken from Sajjadi et al. (2014).

In conclusion we adopted an asymptotic multi-deck solution for turbulent shear
flows over unsteady surface waves, in the limits of low turbulent stresses and
small wave amplitude. The structure of the flow is defined, using an eddy-
viscosity turbulence model, in terms of asymptotically-matched thin-layers,
namely the surface layer and a critical layer. Solutions for both inner and
outer regions are constructed through an interpolation between an inner,
mixing-length and an outer, rapid-distortion approximations. The results
particularly demonstrate the physical importance of the singular flow features
and physical implications of the elevated critical layer in the limit of the
unsteadiness tending to zero. These agree with the variational mathematical
solution of Miles (1957) for small but finite growth rate. However, the
results obtained here, are not consistent physically or mathematically with
his analysis in the limit of growth rate tending to zero. In the present study
it is shown that in the limit of zero growth rate the effect of the elevated
critical layer is eliminated by finite turbulent diffusivity, so that the
perturbed flow and the drag force on determined by the asymmetric and
sheltering flow in the surface shear layer and its matched interaction with
the upper region, as physically demonstrated by Sajjadi, Hunt and Drullion
(2014). The results for an unsteady monchromatic waves is also extended to
those growing Stokes waves. Thus, estimation can be made as to what precentage
of total energy transfer from wind goes to each harmonic of a Stokes wave.

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