# Wind-wave growth over a viscous liquid

J. Zhang1,2 A. Hector1 M. Rabaud1 F. Moisy1 frederic.moisy@universite-paris-
saclay.fr 1Université Paris-Saclay, CNRS, FAST, F-91405 Orsay, France.
2PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université
Paris-Cité, F-75005, Paris, France.

###### Abstract

Experimental and theoretical studies on wind-wave generation have focused
primarily on the air-water interface, where viscous effects are small. Here we
characterize the influence of the liquid viscosity on the growth of
mechanically generated waves. In our experiment, wind is blowing over a layer
of silicon oil, of viscosity 20 and 50 times that of water, and waves of small
amplitude are excited by an immersed wave-maker. We measure the spatial
evolution of the wave slope envelope using Free-Surface Synthetic Schlieren, a
refraction-based optical method. Through spatiotemporal band-pass filtering of
the surface slope, we selectively determine the spatial growth rate for each
forcing frequency, even when the forced wave is damped and coexists with
naturally amplified waves at other frequencies. Systematic measurements of the
growth rate for various wind velocities and wave frequencies are obtained,
enabling precise determination of the marginal stability curve and the onset
of wave growth. We show that Miles’ model, which is commonly applied to water
waves, offers a reasonable description of the growth rate for more viscous
liquids. We finally discuss the scaling of the growth rate of the most
amplified wave and the critical friction velocity with the liquid viscosity.

##  I Introduction

The generation of waves by the wind, whether on a puddle, a pond, or at sea,
remains a fascinating phenomenon that has challenged researchers for decades
[1, 2, 3, 4, 5]. Gaining a comprehensive understanding of this fundamental
problem is of first importance for wave forecasting, assessing heat and mass
transfers in the ocean, and in engineering applications involving liquid and
gas transport in pipes or heat exchangers [6].

Since the pioneering works of Phillips [7] and Miles [8], several decades of
research have led to the following two-stage scenario. In the first stage,
incoherent wrinkles of very low amplitude are excited by the turbulent
pressure fluctuations of the wind (Phillips mechanism), resulting in a linear
increase in wave energy over time [7, 9, 10], with a saturation to a finite
amplitude governed by the liquid viscosity for moderate wind [11, 12, 13]. In
the second stage, for sufficient wind, the air-water shear flow becomes
unstable, leading to a subsequent exponential increase of the wave energy
(Miles mechanism) [8, 14, 15]. However, this highly idealized scenario still
requires experimental and numerical confirmations. In particular, there is no
consensus on the nature of the transition between the Phillips’ and Miles’
regimes and its dependence on the various physical parameters of the problem
[12, 10]. In this context, accurate laboratory measurements of growth rates
under controlled conditions, for physical parameters distinct from the usual
air-water configuration, are highly valuable.

The difficulty in modeling the wind-wave generation problem has several
origins. First, the flow on the air side is turbulent, and both the
characteristics of the mean velocity profile and the statistical properties of
the turbulent stress fluctuations play significant roles. Second, because of
the transport of wave energy, the temporal growth of waves translates into
spatial growth [14], but a quantitative relationship between the two requires
steady forcing conditions over long distances and durations, which are rarely
satisfied in open sea conditions. Third, the wind forces not only waves but
also currents, which may reach saturation on a timescale distinct from the
growth time of the waves [9]. Fourth, surface contamination, which is
inevitable in open sea environments and large water tanks, can significantly
impact the onset of wave generation [16]. Lastly, even in controlled
laboratory experiments, the presence of walls introduces additional damping
effects and undesirable wave reflections [17, 14, 18].

In this paper, we consider the problem of wind-wave generation on viscous
liquids (silicon oils of viscosity 20 and 50 times greater than that of
water), with the aim of circumventing some of the previous difficulties: the
flow in the liquid is purely laminar, enabling accurate analytical modeling;
the surface drift remains moderate, simplifying the air-liquid energy transfer
problem; the impact of surface contamination is reduced by the use of a liquid
of low surface tension; and interference with reflected waves is reduced by
the strong viscous damping.

As long as the liquid viscosity is not too high (typically less than 100 mm2
s-1), the two-stage scenario of the air-water configuration, with linearly
increasing wrinkles at small wind (Phillips mechanism) and exponentially
growing waves at large wind (Miles mechanism), is still relevant; larger
liquid viscosities are not considered here, as they exhibit a very distinct
behavior governed by the Kelvin-Helmholtz instability [19, 20, 21, 3, 22].
While previous studies using viscous liquids focused on waves naturally
generated by the wind [23, 11], here we are interested in waves triggered by a
small mechanical disturbance. Provided that their initial amplitude is
sufficient, this forcing bypasses the first stage of Phillips [7], and the
system directly enters the second stage of Miles [8]. This shortcut in the
two-stage wave generation problem has the advantage of directly assessing the
growth rate of each wave number, at the cost of disregarding the realistic
conditions required for the “true” wave generation process starting from an
initially undisturbed surface. As a result, different critical wind velocities
may be expected for natural and mechanically generated waves.

In the exponential growth regime, Miles demonstrated that the normalized
temporal growth rate of a given wave number $k$ is proportional to the
normalized wind shear stress $(u^{*}/c)^{2}$, where $u^{*}$ is the friction
velocity and $c(k)$ the phase velocity, with a coefficient $\beta$ that
depends on the curvature of the mean velocity profile at the ($k$-dependent)
critical height where the mean flow velocity matches the phase velocity. This
growth is mitigated by the dissipation in the liquid, either laminar or
turbulent, which defines in principle a critical friction velocity for the
growth of this wave number $k$. However, unlike standard hydrodynamic
instabilities arising from the infinitesimal disturbance of a base state, the
two-stage transition here also depends on the incoherent base state from which
the waves grow, which itself depends on the liquid viscosity and the wind
forcing conditions (intensity, duration) [12]. This complexity likely
contributes to the persistent challenge in defining a clear critical wind
velocity for the onset of wave generation under realistic conditions.

The growth rate of wind-generated waves has been the subject of a large number
of experimental works, either from their temporal growth at a fixed point [24,
25, 26], or from their spatial growth in steady conditions [17, 27, 28, 29,
30, 31] (the equivalence between these two approaches is discussed in Ref.
[14]), but only in the air-water configuration. These measurements, initially
limited to one-point probes, have made significant progress with the advent of
optical methods resolved in space and time [32, 23, 11, 33, 34, 35] and direct
numerical simulations [36, 37, 10, 38, 39, 40], revisiting this old problem
with new, high quality data.

In our experiments, the spatial evolution of mechanically generated waves are
measured using Free-Surface Synthetic Schlieren [32], a refraction-based
optical method offering a wave slope resolution of $3\times 10^{-3}$. To
isolate the spatial evolution of the wave at each forcing frequency, we use a
dedicated spatiotemporal band-pass filtering. This method enables us to
measure the growth or damping rate for each forcing frequency, even for cases
above wind onset where natural waves are also present, from which we determine
the marginal stability curve of the instability with unprecedented resolution.
We show that Miles’ model, which is commonly applied to water waves, offers a
reasonable description of the growth rate for more viscous liquids, and we
propose a scaling for the growth rate of the most amplified wave with the
liquid viscosity and friction velocity. We finally discuss the difference
between the critical friction velocity that triggers wave growth in the case
of mechanically generated waves and natural waves.

##  II Wave growth and attenuation mechanisms

The spatial evolution of wind-generated waves results from a combination of
energy transferred from the air flow and dissipated in the liquid. We first
recall the modeling of these two contributions. We start from a simple
harmonic surface deformation,
$\zeta\left(x,t\right)=\zeta_{0}e^{i\left(kx-\omega t\right)}$, of wave number
$k$ and angular frequency $\omega$, propagating over a liquid layer of depth
$h$, density $\rho_{\ell}$, and surface tension $\sigma$. In the absence of
dissipation and forcing, $\omega$ and $k$ are real and are related by the
dispersion relation

$\omega^{2}=\left(gk+\frac{\sigma
k^{3}}{\rho_{\ell}}\right)\tanh\left(kh\right)=gk\left(1+\frac{k^{2}}{k_{cap}^{2}}\right)\tanh\left(kh\right),$
(1)

where $k_{cap}=2\pi/\lambda_{cap}=\sqrt{{\rho_{\ell}g}/{\sigma}}$ is the
capillary wave number. The energy density per unit surface of the wave writes
$E=\rho_{\ell}g\zeta_{0}^{2}[1+(k/k_{cap})^{2}]$. Neglecting nonlinear
interactions and surface currents, the transport and growth of the wave energy
are governed by [41, 42, 43]

$\left(\frac{\partial}{\partial t}+c_{g}\frac{\partial}{\partial
x}\right)E=P-D,$ (2)

with $c_{g}=\partial\omega/\partial k$ the group velocity, $P$ the local power
injected by the wind and $D$ the local power dissipated in the liquid by
viscosity.

The injected power is the work per unit time of the wind stress (pressure and
viscous shear stress) applied on the liquid surface, and can be modeled in two
ways, depending on the wave amplitude. For very low wave amplitude, typically
smaller than a fraction of the viscous sublayer thickness, the wavy shape of
the liquid surface does not modify the flow in the air. In this case, only the
turbulent pressure fluctuations in the air deform the interface, resulting in
an injected power $P$ independent of the wave energy $E$, and hence a linear
growth of $E$: this is the starting hypothesis of the Phillips model [7]. On
the other hand, for larger wave amplitude, a feedback of the wave on the air
flow takes place, and the wind stress becomes modulated by the wave profile.
The resulting injected power $P$ is now proportional to $E$, yielding an
exponential growth of the wave energy: this is the Miles model [8].

In the present paper, we are interested in mechanically generated waves of
initial amplitude larger than the viscous sublayer thickness, which places us
directly in the second stage of wave growth. In this regime, both $P$ and $D$
are proportional to $E$, so the right-hand-side of Eq. (2) can be written in
the form $P-D=\gamma E$, with $\gamma$ the temporal growth rate. In this
framework, the onset of wave growth is defined by $\gamma=0$, when the
injected power overcomes the viscous dissipation.

We first focus on the injected power $P$. Since the flow in the air is
necessarily turbulent in our problem, the wave-induced wind stress variations
(pressure and viscous shear stress) are conveniently scaled with the applied
mean shear stress $\tau=\rho_{a}u^{*2}$, with $u^{*}$ the friction velocity
and $\rho_{a}$ the air density. In a developing boundary layer, this friction
velocity $u^{*}$ depends on the free-stream wind $U_{a}$ and the fetch $x$
(distance along which the stress is applied) [44].

To first order in wave slope, since the velocity of the surface is essentially
vertical, the work per unit time of the horizontal component of the stress
(shear stress) can be neglected, and only the vertical component (pressure)
contributes to the injected power. This contribution writes $P=\Re\langle
p^{\dagger}\dot{\zeta}\rangle$, where $p^{\dagger}$ is the complex conjugate
of the oscillating component of the pressure along the surface and
$\dot{\zeta}$ the vertical velocity of the surface [8, 1]. It therefore
depends on the phase relation between the surface elevation and the wave-
induced pressure oscillation. Following Miles [8], we can write the
oscillating pressure in the form $p=(\alpha+i\beta)\rho_{a}u^{*2}k\zeta$ to
first order in wave slope, with $\alpha+i\beta$ a nondimensional parameter
coding for this phase shift [39]. Only the pressure contribution in phase with
the wave slope ($\beta\neq 0$) provides work, yielding a growth rate
$\gamma=P/E$ in the form [8, 45]

$\gamma=\beta\omega\frac{\rho_{a}}{\rho_{\ell}}\left(\frac{u^{*}}{c}\right)^{2}.$
(3)

The normalized phase velocity $c/u^{*}$ is often referred to as the “wave
age”, because young waves are short and have small phase velocity $c$. The key
result of Miles theory [8, 46] is that $\beta$ is proportional to the
curvature $-U^{\prime\prime}(z_{c})$, with $z_{c}$ the elevation of the
critical layer where the wind velocity $U(z_{c})$ matches the phase velocity
of the considered wave. This indicates that too small or too large
wavelengths, for which $z_{c}$ is such that $U^{\prime\prime}(z_{c})$ vanishes
(in the viscous sublayer or outside the boundary layer), cannot be amplified
by this mechanism. The main difficulty in the determination of $\beta$
therefore relies in the detailed knowledge of the velocity profile in the air.

Since the original work of Miles, the experimental verification of the scaling
(3), and the question of the extent to which the parameter $\beta$ can be
considered a constant, have been the subject of numerous studies. For
naturally growing waves, $\beta$ corresponds to the most amplified wave
number, but it is possible to examine its dependence with $k$ using
mechanically generated waves. Compilation of field and laboratory experiments
suggests that the scaling $(u^{*}/c)^{2}$ approximately holds for the air-
water interface over a significant range of wave ages $(1<c/u^{*}<20)$, with
an average value $\beta\simeq 32\pm{16}$ [45, 1]. This large variability can
be ascribed to a number of parameters, such as the dependence with wave
number, the presence of currents, the incorrect modeling of wave dissipation
for nonlaminar flow in the liquid, nonlinear effects (dependence with wave
slope [43]), etc.

We now describe the dissipation in the liquid, focusing on the laminar
dissipation mechanisms, which are relevant for the viscous liquids considered
here. Viscosity introduces a negative imaginary part in the dispersion
relation, $\omega=\omega_{r}+i\omega_{i}$, yielding a temporal damping rate
$\gamma=-D/E=2\omega_{i}<0$ of the wave energy. The full dispersion relation
for arbitrary viscosity (without side effects) is given by Lamb [47] in
infinite depth, later improved by LeBlond and Mainardi [48] for arbitrary
depth. For moderate viscosity (including that considered in the present work),
the real part $\omega_{r}$ remains close to the inviscid dispersion relation
(1), i.e. the phase velocity $c=\omega_{r}/k$ is marginally reduced by
viscosity. The damping rate $\gamma=2\omega_{i}$ can be approximately split in
two contributions: bulk dissipation due to the internal shear stress,

$\gamma_{\rm bulk}=-4\nu_{\ell}k^{2},$ (4)

and dissipation at the bottom wall due to the shear stress in the Stokes
boundary layer of thickness $\delta=\sqrt{\nu_{\ell}/\omega_{r}}$ (provided
that $\delta$ is much smaller than $h$) [49],

$\gamma_{\rm BW}=-\frac{\sqrt{2\nu_{\ell}\omega_{r}}\,k}{\sinh{2kh}}.$ (5)

The bulk dissipation is dominant at small wavelength, while the bottom-wall
dissipation is dominant for wavelength much larger than liquid depth.

Finally, for waves in a channel of finite width $W$, a third contribution
arises from the friction with the side walls [50, 30, 51],

$\gamma_{\rm SW}=-\frac{\sqrt{2\nu_{\ell}\omega_{r}}}{W}.$ (6)

In the following, we use the numerically resolved LeBlond-Mainardi solution,
noted $\gamma_{\rm LM}$, which has the bulk and bottom-wall approximations (4)
and (5) as asymptotic solutions for large and small $k$, respectively, to
which we add the side-wall contribution $\gamma_{\rm SW}$ (6). The relative
magnitude of these contributions, for the specific geometry of our
experimental setup, is discussed in the Appendix.

To summarize, the net temporal growth rate $\gamma$ accessible to experiments
is the sum of the positive contribution (3) and a combination of negative
contributions (4) and (6). In most practical configurations (including the
present paper), waves are forced under permanent conditions and develop
spatially, so the energy transport equation (2) reduces to

$\frac{d}{dx}E=\gamma_{s}E,$

with $\gamma_{s}=\gamma/c_{g}$ the spatial growth rate [52]. Assuming
homogeneous forcing and dissipation (independent of $x$), this yields an
exponential variation of the energy profile, $E(x)=E_{0}e^{\gamma_{s}x}$, with
$\gamma_{s}>0$ for growing waves and $\gamma_{s}<0$ for damped waves. In the
following, we perform measurements of $\gamma_{s}$ by fitting $E(x)$ in the
range of exponential growth or decay for various $k$ and friction velocities
$u^{*}$, and discuss the relevance of the various growth and damping rate
contributions in the case of viscous liquids.

##  III Experiments and measurements

###  III.1 Experimental setup

Figure 1: Sketch of the experimental setup. (a) Side view; (b) cross sectional
view. $L=1500~{}\mathrm{mm}$, $h=35~{}\mathrm{mm}$, $H=105~{}\mathrm{mm}$,
$W=296~{}\mathrm{mm}$. The wave maker is a cylinder of 5 mm diameter, located
at $L_{wm}=125~{}\mathrm{mm}$ from the beginning of the tank and $-14$ mm
below the liquid surface.

The experimental setup is composed of a rectangular tank of length $L=1.5$ m
and width $W=296$ mm, located at the bottom of a wind tunnel; see Fig. 1 and
Refs. [23, 11, 21] for details. The wind velocity $U_{a}$, measured in the
center of the cross-section, ranges from $0$ to $7~{}\mathrm{m~{}s^{-1}}$. The
tank, of depth $h=35~{}\mathrm{mm}$, is filled with silicon oil (Rhodorsil®
oil 47 V20 and V50). Two kinematic viscosities are used:
$\nu_{\ell}=20.5~{}\mathrm{mm}^{2}~{}s^{-1}$ and $\nu_{\ell}=50.1\pm
0.1~{}\mathrm{mm}^{2}~{}s^{-1}$. The experiments are carried out at
temperature of $24.0\pm 1.0^{\circ}$C, giving an oil density
$\rho_{\ell}=956~{}\mathrm{kg~{}m^{-3}}$ and surface tension
$\sigma=20.7~{}\mathrm{mN~{}m^{-1}}$. The corresponding capillary wavelength
and capillary-gravity frequency are $\lambda_{cap}=9.33$ mm and $f_{cap}=18.3$
Hz, and the minimum phase velocity is $c_{min}=(4g\gamma/\rho)^{1/4}=0.171$
m/s.

Waves are generated using an immersed wave maker, made of a vertically
oscillating stainless steel cylinder of diameter $5~{}\mathrm{mm}$ and length
matching the width of the tank. The wave maker is located $125$ mm after the
beginning of the tank and defines the origin $x=0$ of our measurements. It is
hung by a rectangular frame and oscillated using an electromagnetic shaker
(Sinocera JZK-20) via a crank rod system [Fig. 1(b)]. The upper part of the
cylinder is at $z(t)=z_{0}+A\cos(\omega t)$, with $z_{0}=-14$ mm the mean
depth and $A$ the stroke amplitude. In order to minimize wave reflection, two
inclined perforated planes of length $85~{}\mathrm{mm}$ and slope $30^{\circ}$
are added at both ends of the tank to absorb the wave energy.

The forcing frequency $f=\omega/2\pi$ ranges from $3$ to $12~{}\mathrm{Hz}$.
This range is below the capillary-gravity crossover frequency $f_{cap}$,
indicating that excited waves are in the gravity regime. The corresponding
wavelengths range from 150 mm to 15 mm. The frequency range is limited at low
frequency by the fact that the small diameter of the cylinder cannot excite
efficiently large wavelengths, and at large frequency by the fact that the
wavelength and the attenuation length become too small to be measurable. The
Reynolds numbers $Re=\lambda_{cap}c_{min}/\nu_{\ell}$, which characterize
gravito-capillary waves of wavelength $\lambda_{cap}$ propagating at the
minimum phase velocity $c_{min}$, are $Re=78$ and 32 for
$\nu_{\ell}=20.5~{}\mathrm{mm}^{2}~{}s^{-1}$ and
$\nu_{\ell}=50.1~{}\mathrm{mm}^{2}~{}s^{-1}$, respectively. Since the
wavelengths considered in the following are significantly larger than
$\lambda_{cap}$, these values represent lower bounds for the actual Reynolds
numbers, confirming that the waves are in the weakly damped regime [48].

The stroke amplitude of the wave maker is kept fixed to $A=1~{}\mathrm{mm}$ in
all this study. This forcing induces waves of initial amplitude $\zeta_{0}$
between 0.1 and 0.4 mm depending on the forcing frequency. This initial
amplitude is small enough to satisfy the linear wave approximation (the
maximum wave slope near the wave maker remains below $k\zeta_{0}\simeq 0.02$).
On the other hand, this initial amplitude is large enough to be directly in
the Miles’ exponential growth regime: the characteristic surface deformation
corresponding to the transition from the Phillips’ incoherent wrinkles to the
Miles’ regular waves is estimated to $\zeta\simeq(0.11-0.14)\delta_{\nu}$ in
Refs. [12, 10], with $\delta_{\nu}=\nu_{a}/u^{*}$ the thickness of the viscous
sublayer, which lies in the range $40-100~{}\mu$m.

###  III.2 Friction velocity

In order to check the scaling of the growth rate (3), it is necessary to
determine accurately the friction velocity $u^{*}$ for a given wind velocity
$U_{a}$. We deduce $u^{*}$ from the stationary current $U_{s}$ that develops
at the liquid surface because of stress continuity
$\rho_{a}u^{*2}=\rho_{\ell}\nu_{\ell}\partial u_{x}/\partial z|_{z=0}$. The
flow in the liquid being laminar (the maximum Reynolds number
$U_{s}h/\nu_{\ell}$ is less than 70), a stationary Couette-Poiseuille flow of
zero flow rate develops, with a surface current $U_{s}$ given by [23]

$U_{s}=\frac{\rho_{a}u^{*2}h}{4\rho_{\ell}\nu_{\ell}}.$ (7)

Close to the onset of wave growth, this surface current is of the order of
$0.02~{}\mathrm{m~{}s^{-1}}$, which is less than 10% of the typical phase
velocity. In Aulnette et al. [21], we show that the friction velocity $u^{*}$
deduced from $U_{s}$ at various $x$ locations using Eq. (7) is in good
agreement with the empirical relation for a developing turbulent boundary
layer – see, e.g., Eq. (21.12) in Schlichting [44]

$\frac{u^{*2}(x)}{{U_{a}}^{2}}\simeq
C\left(\frac{(x+x_{0})U_{a}}{\nu_{a}}\right)^{-0.2},$ (8)

with $x_{0}=475$ mm the distance between the beginning of the flat plate at
the end of the wind-tunnel convergent and the wave maker, and $C=0.029$. This
decrease of $u^{*}$ along $x$ is related to the development of the boundary
layer along the tank, with a thickness that increases as $\delta(x)\simeq
x^{0.8}$ [22]. In the following, we use Eq. (8) to compute the friction
velocity $u^{*}$ from the prescribed wind velocity $U_{a}$ close to the wave-
maker, in the range $x\simeq 100-300$ mm where the growth rate is measured.
The uncertainty on $u^{*}$ is of the order of 5% [21].

###  III.3 Wave measurement

The surface deformations are determined using Free-Surface Synthetic Schlieren
(FS-SS) [32]. This method is based on the analysis of the refracted image of a
random-dot pattern through the interface. The dots are $1~{}\mathrm{mm}$ in
diameter (corresponding to 2.3 pixels) with a density of $60$ dots per
$\mathrm{cm^{2}}$. The pattern is printed on a transparent film located at the
bottom of the liquid tank and illuminated from below by a LED panel. Images of
the pattern are taken through the transparent upper-wall of the wind tunnel.
To reduce parallax errors, images are acquired through a 45o mirror located
above the wave tank by a camera located $3~{}\mathrm{m}$ from the liquid
surface [Fig. 1(b)]. Images are acquired using a high-speed camera (Photron
FASTCAM Mini WX50), fitted with a Canon $85~{}\mathrm{mm}~{}f/1.8$ macro lens.
Images of dimension of $890\times 320~{}\mathrm{mm^{2}}$ are recorded at a
frame rate at least $10$ times larger than the wave maker frequency $f$, for a
recording duration of at least 20 wave periods.

The surface slope field is obtained by computing the apparent displacement
field $\delta{\bf r}$ of the dot pattern induced by the surface deformation
using an image correlation algorithm [32]. In the limit of weak slopes and
within the paraxial approximation, the displacement field is proportional to
the surface height gradient,

$\delta{\bf r}=-h^{*}\nabla\zeta,$

with $h^{*}=14$ mm an effective distance that includes the surface-pattern
distance ($h_{p}=48.2~{}\mathrm{mm}$) and the refraction indices of the liquid
and intermediate layers between the surface and the pattern [32]. Although the
surface height field $\zeta({\bf r},t)$ can be reconstructed by inverting the
$\nabla$ operator, as done in Refs. [23, 11], here we work directly with the
refraction-induced displacement field $\delta{\bf r}$: for a quasi-
monochromatic wave propagating in the $x$ direction,
$\zeta(x,t)=\zeta_{0}(x)\cos(kx-\omega t)$, where $\zeta_{0}(x)$ is a slowly
varying amplitude, the displacement is simply proportional to the wave height,

$\delta x=-h^{*}\frac{\partial\zeta}{\partial x}\simeq
h^{*}k\zeta_{0}(x)\sin(kx-\omega t),$

so the spatial growth rate of the wave amplitude $\zeta_{0}(x)$ can be
obtained directly from that of the displacement $\delta x$.

The image correlation algorithm is based on interrogation windows of size 12
pixels (5.2 mm) with 50% overlap, yielding a spatial resolution of
$2.6~{}\mathrm{mm}$. This is approximately 6 times smaller than the smallest
wavelength considered here, $\lambda\simeq 15~{}\mathrm{mm}$ for
$f=12~{}\mathrm{Hz}$. The resolution in the apparent displacement $\delta x$
is 0.1 pixel (0.04 mm), yielding a resolution in surface slope $k\zeta\simeq
0.003$.

###  III.4 Data processing

Figure 2: Spatial evolution of the instantaneous and mean-square wave slope at
wind velocity $U_{a}=5~{}\mathrm{m~{}s^{-1}}$ and oil viscosity
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$. Left column:
$f=5~{}\mathrm{Hz}$, for which the wave is amplified; right column:
$f=9.5~{}\mathrm{Hz}$, for which the wave is damped. (a,b) Snapshots of the
raw surface gradient $|\nabla\zeta|$. (c,d) Snapshots of the filtered surface
gradient $|\nabla\bar{\zeta}|$ using the spatiotemporal filter (10). (e,f)
Wave slope profiles $\partial\zeta/\partial x$ (red dashed curves: raw data;
black continuous curves: filtered data). (g,h) Mean square envelopes
$\langle(\partial\zeta/\partial x)^{2}\rangle$ (raw and filtered). The two
vertical lines show the range used for the exponential fit (11).

Figures 2(a) and (b) show two snapshots of the magnitude of the wave gradient
field $\nabla\zeta=-\delta{\bf r}/h^{*}$ measured at a wind velocity
$U_{a}=5.0$ m s-1 in the case of an amplified wave ($f=5$ Hz, left) and damped
wave ($f=9.5$ Hz, right), for the oil of viscosity
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$. The slight tilt observed in the
wave crests originates from the slightly nonhomogeneous wind in the $y$
direction, of the order of 3% [53]. From the wave slope in the $x$ direction,
we compute the mean-square envelope, which writes for a quasi-monochromatic
wave as

$\langle(\partial\zeta/\partial
x)^{2}\rangle\simeq\frac{1}{2}k^{2}\zeta_{0}^{2}(x),$ (9)

with $\zeta_{0}(x)$ the slowly varying wave amplitude and
$\left\langle\cdot\right\rangle$ the average over time and $y$. Examples of
wave slope profiles $\partial\zeta/\partial x$ at fixed $y$ and their
corresponding mean-square envelopes are shown as red dashed lines in Figs.
2(e)- 2(h). In the amplified case, the envelope shows a well defined
exponential growth at moderate fetch, followed by a saturation at large fetch
[Fig. 2(g)]. This saturation, which results from a combination of nonlinear
interactions with other waves and the small decrease of the wind shear stress
along the developing boundary layer, is not considered in the following. The
damped case is more complex [Fig. 2(h)]: after a short range of exponential
decay, the envelope saturates or even slightly increases at large fetch,
because of the presence of other waves of frequencies different from the
forcing frequency. These unforced waves are the amplified waves naturally
present for a wind velocity above the critical threshold, and may hide the
underlying decay of the forced wave, which makes it difficult to accurately
measure the damping rate.

To extract the forced waves from the sea of ambient waves present in the
field, we perform a spatiotemporal band-pass filtering of the displacement
field $\delta x=-h^{*}\partial\zeta/\partial x$. Figure 3(a) shows the power
spectrum $|\delta\hat{x}|^{2}(k,\omega)$ (averaged in the $y$ direction) in
the case of a damped wave. The displacement field being real, its Fourier
transform is symmetric under $(k,\omega)\rightarrow(-k,-\omega)$. The
quadrants ($+,+$) and ($-,-$) correspond to waves propagating in the $x>0$
direction, while the quadrants ($+,-$) and ($-,+$) correspond to waves
propagating in the $x<0$ direction. In addition to the injected energy around
the forcing $\pm(k_{0},\omega_{0})$ (red ellipses), energy is also visible in
the reflected waves $\pm(-k_{0},\omega_{0})$, in harmonics, and along the
viscous dispersion relation (white dashed lines). Energy along the dispersion
relation is systematically found for a wind velocity above the onset, and is
due to the natural waves growing from initial disturbances and amplified by
the wind.

To filter out the undesired wave components, we compute for each $y$ the
space-time Fourier transform $\delta\hat{x}(k,\omega)$, convolute it with a
Butterworth filter kernel

$G\left(k,\omega\right)=\frac{H(k\omega)}{1+\left[\left(\frac{k-k_{0}}{\Delta
k}\right)^{2}+\left(\frac{\omega-\omega_{0}}{\Delta\omega}\right)^{2}\right]^{N/2}},$
(10)

and compute the inverse Fourier transform to reconstruct the filtered
displacement field $\delta\bar{x}(x,t)$, and hence the filtered wave slope
$\partial\bar{\zeta}/\partial x$. We choose a filter kernel centered around
the forcing $\pm(k_{0},\omega_{0})$, with a spectral bandwidth ($\Delta k$,
$\Delta\omega$) and a decay characterized by the exponent $N$, chosen here
equal to 6. A Heaviside operator $H(k\omega)$ is included to remove the
reflected waves ($k\omega<0$). A narrow temporal bandwidth $\Delta\omega$ is
chosen to select precisely the forced waves, but a wider spatial bandwidth
$\Delta k$ is chosen to correctly measure the spatial decay of the wave. The
temporal bandwidth is chosen as $\pm 3$ spectral points $\pi/T_{acq}$ (with
$T_{acq}$ the acquisition duration) surrounding the forcing frequency
$\omega_{0}$. The spatial bandwidth $\Delta k$ is chosen equal to $k_{0}$, so
the smallest measurable damping length is of the order of the wavelength. The
resulting filtered spectrum $|G(k,\omega)\delta\hat{x}|^{2}(k,\omega)$ in Fig.
3(b) shows that the energy of the forced wave is correctly selected by these
parameters.

Figure 3: (a) Power spectrum $|\delta\hat{x}|^{2}(k,\omega)$ of the
displacement field in logarithmic scale for $U_{a}=4.2$ m s-1,
$f=7.3~{}\mathrm{Hz}$ and oil viscosity
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$. The white dashed line shows the
viscous dispersion relation [48]. The energy of the forced wave
$\pm(k_{0},\omega_{0})$ is selected by the spatiotemporal filter (10) (the red
ellipses show the isovalue $G=0.05$). (b) Resulting filtered spectrum
$|G(k,\omega)\delta\hat{x}|^{2}(k,\omega)$.

The snapshots for the filtered instantaneous wave gradient
$\nabla\bar{\zeta}=-\delta\bar{\bf r}/h^{*}$, shown in Figs. 2(c)- 2(d), are
very close to the raw data in the amplified case, but are significantly
modified in the damped case, indicating that the naturally amplified waves are
efficiently filtered out by our procedure. The corresponding wave slope
profiles and envelopes are also shown as continuous black lines in Figs.
2(e)-2(h).

To validate our procedure over the full range of wind velocity, we show in
Fig. 4 a series of mean-square filtered wave slopes at a fixed frequency
($f=4.1$ Hz) for increasing $U_{a}$, illustrating the gradual transition from
damped to amplified waves. Close to the wave maker, the growth or decay is
well described by an exponential function, from which we fit the spatial
growth rate $\gamma_{s}$,

$\langle(\partial\bar{\zeta}/\partial x)^{2}\rangle\propto e^{\gamma_{s}x}.$
(11)

We choose the range $x\in[1.8,4]\lambda$ for the fits [shown by the two
vertical lines in Figs. 2(g) and 2(h) and 4], for which a robust exponential
fit can be performed at all $f$ and $U_{a}$. The overall uncertainty on
$\gamma_{s}$ is less than 8%.

Figure 4: Mean square envelopes $\langle(\partial\bar{\zeta}/\partial
x)^{2}\rangle$ (filtered data) at various wind velocities $U_{a}$, for
$\nu_{\ell}=50$ mm2 s-1 and $f=4.1$ Hz ($k=69$ mm-1). The vertical lines show
the range used to fit the spatial growth rate $\gamma_{s}$ using Eq. (11).

##  IV Results

###  IV.1 Phase velocity

To compare the spatial growth rates $\gamma_{s}$ determined experimentally to
the temporal growth rates $\gamma$ predicted theoretically, it is necessary to
convert the measured spatial variations of the wave energy to the
corresponding temporal variations. In the limit of small viscosity, the ratio
between the two growth rates is given by the group velocity [52], which cannot
be directly measured in our system. We must therefore use the group velocity
obtained from the dispersion relation of the free waves in a viscous liquid,
i.e. including viscous effects but without wind, as discussed in Sec. II. To
check to what extent we can rely on this viscous free-wave dispersion
relation, we first measure here the phase velocity of the waves and compare it
with this prediction.

Figure 5: Phase velocity as a function of the wave number for various wind
velocities and for the two liquid viscosities, (a)
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$ and (b)
$\nu_{\ell}=50.1~{}\mathrm{mm^{2}~{}s^{-1}}$. Error bars correspond to the
standard deviation of the data. The black lines show the phase velocity for
free waves, either inviscid (solid line) or with viscosity (dashed line).
(c,d) Difference between the phase velocity with and without wind. Solid lines
are the predictions including the (positive) surface drift correction (12) and
the (negative) aerodynamic pressure correction (13), with uncertainties shown
as color shaded areas.

Figures 5(a) and 5(b) show the phase velocity $c_{exp}=\omega/k$ for the two
liquid viscosities as a function of the wave number for various wind
velocities $U_{a}$ (here $k$ is simply measured from the distance between wave
crests, ensuring high accuracy). The data are compared to the inviscid
prediction (1) in solid line, and to the numerically computed viscous
prediction of LeBlond and Mainardi [48] including depth effects in dashed
line. These predictions are very close in our range of frequencies ($3-12$
Hz), within 2%, so we will simply consider the inviscid prediction in the
following. At low frequency, the inviscid phase velocity tends to the
nondispersive shallow water limit $\sqrt{gh}\simeq 0.58$ m/s, while the
viscous prediction falls to 0 when the thickness of the Stokes boundary layer
becomes of the order of the liquid depth (this cutoff is well below the
frequencies relevant to our study).

For our range of frequencies, the measured phase velocity without wind,
$c_{0-exp}$, is almost indistinguishable from the prediction (either inviscid
or viscous). In the presence of wind, it still remains close to the
prediction, to within 3% at $\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$ and
8% at $\nu_{\ell}=50.1~{}\mathrm{mm^{2}~{}s^{-1}}$. We can therefore use with
reasonable accuracy the predicted group velocity computed from the inviscid
dispersion relation (1) to convert spatial growth rates to temporal ones in
the following.

Before further proceeding, it is interesting to analyze in more detail the
small differences between the measured phase velocity with and without wind
($c_{exp}$ and $c_{0-exp}$, respectively). The difference $c_{exp}-c_{0-exp}$
is plotted in Fig. 5(c) and 5(d), with uncertainties shown as shaded areas. It
can be explained by a combination of two antagonistic effects [54], the
importance of which depending on the liquid viscosity. The first one
corresponds to an increase of the phase velocity by the surface current
induced by the wind shear stress, which dominates at small viscosity, and the
second one corresponds to a decrease of the phase velocity due to the wave-
induced aerodynamic pressure, which dominates at large viscosity.

The increase in phase velocity by surface current is simply given by $U_{s}$
for short wavelengths [see Eq. (7), with $U_{s}\simeq 0.02$ m s-1 close to the
wave onset], but is less pronounced for long wavelengths because the
perturbation they induce penetrates deeper into the liquid, where the current
is lower. This transport effect therefore depends on the mean velocity profile
in the liquid. For the Couette-Poiseuille profile considered here [23], the
modification in the phase velocity has been computed by Lilly, and can be
found in the appendix of the paper by Hidy and Plate [17],

$\Delta
c_{w}(k)=U_{s}\left(1-\frac{1+2\cosh{2kh}}{kh\sinh{2kh}}+\frac{3}{2k^{2}h^{2}}\right).$
(12)

The second effect is the slowdown of the phase velocity by the variations of
the aerodynamic pressure over the wave profile, and can be explained
qualitatively as follows. The phase velocity of gravity waves is governed by
gravity, i.e. by the weight of the liquid deformation. In the presence of
wind, a fraction of this weight is carried by the wave-induced aerodynamic
pressure (Bernoulli suction over the wave crests and over-pressure on the wave
troughs), of the order of $\rho_{a}U_{a}^{2}k\zeta$. The surface deformation
therefore experiences an effective reduced gravity, yielding a reduced phase
velocity, which can be written as [55]

$c(k)=\sqrt{c_{0}(k)^{2}-\frac{\rho_{a}}{\rho_{\ell}}U(z_{k})^{2}},$ (13)

with $U(z_{k})$ the wind velocity evaluated at a characteristic elevation that
depends on the wave number $k$. We evaluate this effect by considering a
classical logarithmic profile $U(z)=u^{*}[\kappa^{-1}\log(zu^{*}/\nu_{a})+B]$
(with $\kappa=0.41$ and $B=3.5$) computed at the elevation $z_{k}=\epsilon/k$
with $\epsilon\simeq 0.23$ [22].

The correction to the phase velocity, combining the surface current effect
(12) and the aerodynamic pressure effect (13), is plotted in Figs. 5(c) and
5(d). It provides a reasonable description of the data: At low viscosity and
large wave numbers, the waves are essentially transported by the surface
current, yielding a phase velocity larger than the inviscid prediction. On the
other hand, waves at low wave numbers are systematically slower than the free-
wave celerity, as a result of the dominant aerodynamic pressure effect.

To summarize, the phase velocity of the waves in presence of wind can be
correctly described by the finite-depth inviscid phase velocity, by including
the corrections due to the surface drift and the aerodynamic pressure.
However, in the range of liquid viscosities and wind velocities considered
here, these corrections remain small: for a wind velocity close to the wave
onset, the measured phase velocity matches the free-wave inviscid prediction
to within 10%. By extension, we can consider that the group velocity is also
only marginally affected by the surface current and aerodynamic pressure,
which allows us to use the free-wave inviscid prediction $c_{g}$ derived from
Eq. (1) to infer the temporal growth rate $\gamma$ from the measured spatial
growth rate $\gamma_{s}$.

###  IV.2 Spatial growth rates

Figure 6: (a,b) Spatial growth rate $\gamma_{s}$ as a function of the wave
number for various wind velocities, for (a)
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$ and (b)
$\nu_{\ell}=50.1~{}\mathrm{mm^{2}~{}s^{-1}}$. Black solid line: bulk
dissipation ($-4\nu_{\ell}k^{2}/c_{g}$); black dashed line: numerically solved
LeBlond-Mainardi dissipation (bulk and bottom-wall dissipation) combined with
side-wall dissipation. Color lines: local polynomial regression (“loess”
smoothing). (c,d) Marginal stability curve (green circles) and most amplified
wave number (blue triangles). Thin dashed lines are guides for the eyes.

Figures 6(a) and 6(b) shows the measured spatial growth rate $\gamma_{s}$ as a
function of the measured wave number, for different wind velocities. As
expected, without wind $\gamma_{s}$ is negative for all $k$. The corresponding
damping rate is compared to the bulk prediction
$\gamma_{s}=-4\nu_{\ell}k^{2}/c_{g}$ (solid line) and to the numerically
solved LeBlond-Mainardi prediction [48] (including bulk and bottom-wall
dissipation) completed with side-wall dissipation (6) (dashed line). The large
wave numbers ($k>70$ m-1) are equally well described by the two models, but
the smaller $k$ are much better described by the refined model, especially at
large viscosity, highlighting the significant effect of the bottom dissipation
at small $k$ (see Appendix Appendix: Relative importance of the various
contributions to the wave dissipation).

Table 1: Critical parameters at the onset of wave generation for the two liquid viscosities. $\nu_{\ell}~{}(\mathrm{mm^{2}~{}s^{-1}})$ |  $U_{ac}~{}(\mathrm{m~{}s^{-1}})$ |  $u_{c}^{*}~{}(\mathrm{m~{}s^{-1}})$ |  $k_{c}$ (m-1) |  $f_{c}$ (Hz)
---|---|---|---|---
$20.5$ |  $3.18~{}\pm 0.04$ |  $0.173~{}\pm 0.005$ |  $80\pm 5$ |  $4.35~{}\pm 0.1$
$50.1$ |  $5.92~{}\pm 0.04$ |  $0.303~{}\pm 0.005$ |  $100\pm 6$ |  $4.60~{}\pm 0.1$

As the wind velocity is increased, the waves become gradually less damped and,
above a critical wind $u^{*}_{c}$, the growth rate becomes positive in a
finite range of wave numbers. We recall that the friction velocity $u^{*}$ is
deduced from the free-stream velocity $U_{a}$ by using Eq. (8) for $x$ close
to the wave maker. From the set of growth rate curves $\gamma_{s}(k)$ at
various velocities, we can compute the marginal stability curve $u^{*}(k)$,
defined as the minimum friction velocity required for a given wave number $k$
to become unstable. It is plotted in Figs. 6(c) and 6(d) for the two liquid
viscosities. The lowest point $(k_{c},u^{*}_{c})$ of the marginal stability
curve defines the critical wave number and critical friction velocity, which
are summarized in Table 1 for the two liquid viscosities. The critical wave
numbers correspond to wavelengths $\lambda_{c}\simeq 60-80$ mm, much larger
than the capillary wavelength $\lambda_{cap}=9.33$ mm.

As the liquid viscosity is increased, the critical friction velocity strongly
increases, but the critical wave number is marginally modified. Increasing
$u^{*}$ shifts the most amplified wave number (shown as blue triangles) to
larger values, and rapidly widens the domain of unstable wave numbers. The
low-$k$ branch of the stability curve, for which the bottom-wall and the bulk
dissipation are of comparable importance (see the Appendix), shows a weak
dependence with $u^{*}$. On the other hand, the large-$k$ branch, which is
essentially governed by the bulk dissipation, strongly varies with $u^{*}$. We
note that this higher branch is defined with high accuracy, because
$\gamma_{s}$ sharply crosses 0, but that the lower branch is more difficult to
measure, because $\gamma_{s}$ shows little variations with $u^{*}$ for small
$k$.

###  IV.3 Comparison with Miles’ scaling

We now investigate to what extent Miles’ model, usually applied to the air-
water interface, also applies to the case of the more viscous liquids
considered here. Following Eq. (3), we plot in Fig. 7(a) and 7(b) the temporal
growth rate $\gamma=\gamma_{s}c_{g}$ normalized by the wave frequency $\omega$
as a function of the inverse wave age $u^{*}/c$ (only amplified waves are
considered here). In this traditional representation, “young” waves (high $k$,
low $c$) are on the right and “old” waves (small $k$, large $c$) are on the
left. However we note that, strictly speaking, there is wave aging in our
experiments: each data point corresponds to a single wave excited at a given
frequency.

Figure 7: Dimensionless growth rate $\gamma/\omega$ as a function of the
inverse wave age $u^{*}/c$, for (a,c) $\nu_{\ell}=20.5$, and (b,d) 50.1
$\mathrm{mm^{2}~{}s^{-1}}$. The solid line shows Eq. (3) for $\beta=45$. (a,
b) Raw growth rates, without viscous correction. (c, d) Growth rates corrected
by the viscous contribution $\gamma_{0}$ measured without wind. The shaded
area depicts the range $\beta=23-75$. Figure 8: Dimensionless net growth rate
$(\gamma-\gamma_{0})/f$ (with $\gamma_{0}$ the damping rate without wind) vs
the inverse wave age, showing our data (red circles:
$\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$, blue triangles:
$\nu_{\ell}=50.1~{}\mathrm{mm^{2}~{}s^{-1}}$) superimposed to data obtained in
water and compiled by Plant [45] (see also [56, 1, 57]). Open symbols: field
data; filled symbols and crosses: laboratory data; solid line: Miles’s
prediction [58].

Although individual curves for each $u^{*}$ fail to follow the $(u^{*}/c)^{2}$
scaling, the upper envelope of the set of curves is compatible with this
scaling. The strong departure at small and large wave age results from a
combination of dissipation effects (dominated by the bulk dissipation for
large $u^{*}/c$ and bottom-wall dissipation for small $u^{*}/c$) and the
selective amplification of the Miles’ mechanism: wave numbers $k$ far from the
most amplified wave numbers correspond to a critical height where the
curvature of the mean velocity profile $U^{\prime\prime}(z)$, and consequently
the growth rate, becomes zero.

To further test the $(u^{*}/c)^{2}$ scaling, it is therefore necessary to
subtract the contribution from the viscous dissipation. In order not to rely
on a specific dissipation model, we use the measured damping rate $\gamma_{0}$
without wind as the best estimate for the global dissipation. We therefore
plot the normalized net growth rate $(\gamma-\gamma_{0})/\omega$ in Figs. 7(c)
and 7(d), which is now consistent with the scaling $(u^{*}/c)^{2}$, at least
for large wind velocity and large wave numbers (large $u^{*}/c$). The best fit
with Eq. (3) gives the average value $\bar{\beta}\simeq 45$ with an
uncertainty range $\beta\simeq 23-75$ (shown in the shaded area).

The significant scatter in the dimensionless growth rates is not specific to
our experiments performed with viscous oils, and is also present in
experiments in the air-water case. In Fig. 8 we superimpose our normalized net
growth rates to the classical Plant’s compilation of laboratory and field
measurements [45], reproduced from Janssen [1]. The correct agreement between
our data in viscous oils and the data in water supports the robustness of the
scaling $(u^{*}/c)^{2}$ at large $u^{*}/c$, with a comparable scatter. The
solid line shows the prediction of Miles [58], which gives the asymptotic
scaling $(u^{*}/c)^{2}$ at large $u^{*}/c$, relevant to our “young” (slow)
waves, but a lower growth rate for “older” (faster) waves when the phase
velocity becomes of the order of the wind velocity.

###  IV.4 Maximum growth rate

We now turn to the maximum growth rate $\gamma_{max}$ (growth rate of the most
amplified wave number) and its dependence on the liquid viscosity. Figure 9(a)
shows $\gamma_{max}$ as a function of the friction velocity $u^{*}$ for the
two viscosities. By definition $\gamma_{max}$ increases with $u^{*}$ and
crosses 0 at the critical friction velocity $u^{*}_{c}$ (given in Table 1). We
note that negative values of $\gamma_{max}$ can be defined for
$u^{*}<u^{*}_{c}$ because slightly below $u^{*}_{c}$ the growth rate shows a
local (negative) maximum, corresponding to the least damped wave number.

Figure 9: (a) Maximum temporal growth rate $\gamma_{max}$ as a function of the
friction velocity $u^{*}$ for $\nu_{\ell}=20.5~{}\mathrm{mm^{2}~{}s^{-1}}$
(red) and $\nu_{\ell}=50.1~{}\mathrm{mm^{2}~{}s^{-1}}$ (blue). Vertical dash-
dotted lines show the critical friction velocities $u_{c}^{*}$ (Table 1). (b)
Viscosity-weighted maximum temporal growth rate $\gamma_{max}\nu_{\ell}^{3}$
as a function of the friction velocity $u^{*}$. The line corresponds to Eq.
(15) with $\bar{\beta}=45$, and the shaded area shows the range $\beta=23-75$.

Given the various sources of dissipation, with distinct scaling laws with
respect to $\nu_{\ell}$ (bulk dissipation in $\nu_{\ell}$ and wall dissipation
in $\nu_{\ell}^{1/2}$), a unique scaling for $\gamma_{max}$ with $\nu_{\ell}$
cannot be derived in general. However, an approximate trend can be proposed,
at the cost of a number of approximations: (1) we ignore the $k$ dependence of
the parameter $\beta$ in Eq. (3), which is acceptable close to the most
amplified $k$, and consider the average value $\bar{\beta}=45$; (2) we
consider that waves are in the deep-water gravity regime, $\omega=\sqrt{gk}$,
which again is acceptable because $k_{c}/k_{cap}<0.15$ and
$\tanh(k_{c}h)\simeq 0.99$; and (3) we consider the bulk dissipation
$\gamma_{\rm bulk}=-4\nu_{\ell}k^{2}$ as the only source of dissipation. This
third approximation is the most questionable in our experiments, with a bulk-
bottom transition at $k\simeq 70$ m-1 (see the Appendix), which is close to
the critical wave number $k_{c}$. Under these approximations, the temporal
growth rate can be modeled as

$\gamma\simeq\bar{\beta}s\frac{u^{*2}k^{3/2}}{g^{1/2}}-4\nu_{\ell}k^{2},$ (14)

with $s=\rho_{a}/\rho_{\ell}$ the density ratio. Solving for
$\partial\gamma/\partial k=0$ yields the maximum growth rate

$\gamma_{max}=\frac{3^{3}}{2^{14}}\bar{\beta}^{4}s^{4}\frac{u^{*8}}{g^{2}\nu_{\ell}^{3}}.$
(15)

To verify this scaling, we plot in Fig. 9(b) the combination
$\gamma_{max}\nu_{\ell}^{3}$ as a function of $u^{*}$. The data for the two
viscosities approximately gather along the power law $u^{*8}$, with variations
of the same order as the uncertainty on the parameter $\beta$ found previously
(the black line gives the average value $\bar{\beta}=45$, and the shaded area
shows the same range of $\beta$ as in Fig. 7). Although the data for the two
viscosities do not overlap, because of the very limited range of accessible
$u^{*}$ (at most twice the critical value $u^{*}_{c}$), the high-$\nu_{\ell}$
curve reasonably extends the low-$\nu_{\ell}$ curve, which is consistent with
the scaling law (15).

The variation of $\gamma_{max}$ with $u^{*}$ has been the subject of many
studies, but only in the air-water case [24, 25, 59, 60, 61]. Approximate
power laws $\gamma_{max}\sim u^{*n}$ are reported, but with significantly
smaller exponents $n$, in the range 1.5–3.5. The asymptotic scaling $n=8$ may
be difficult to access experimentally in water experiments because nonlinear
effects (wave saturation, frequency downshift, wave breaking) rapidly appear
as $u^{*}$ is increased, even close to the onset $u^{*}_{c}$.

###  IV.5 Critical friction velocity

We finally consider the influence of the liquid viscosity on the critical
friction velocity $u^{*}_{c}$ for the growth of mechanically generated waves.
This critical friction velocity cannot be directly inferred from Eq. (14),
based on the sole bulk dissipation, because it predicts a range of unstable
$k$ starting from $k=0$ as soon as $u^{*}>0$. This directly follows from the
approximation $\beta(k)\simeq\bar{\beta}$ and the omission of other
dissipation sources. In our experiments, the dissipation being given by a
combination of bulk and bottom-wall contributions (see the Appendix), we can
modify Eq. (14) by still assuming a constant coefficient $\beta$, while now
including the Leblond-Mainardi damping rate

$\gamma\simeq\bar{\beta}s\frac{u^{*2}k^{3/2}}{g^{1/2}}+\gamma_{\rm LM}(k).$
(16)

The critical friction velocity $u_{c}^{*}$, obtained by solving numerically
$\gamma=0$, is plotted in Fig. 10 as a function of $\nu_{\ell}$. The numerical
solution, shown again for the average value $\bar{\beta}=45$ (black line) and
its uncertainty range (shaded area), is in good agreement with the measured
$u_{c}^{*}$ for the two values of $\nu_{\ell}$.

Interestingly, the measured $u^{*}_{c}$ and the numerical solution are well
described by the approximate power law $u^{*}_{c}\sim\nu_{\ell}^{1/2}$.
Although this power law cannot be directly derived from Eq. (16), it is
compatible with the fact that the most unstable wave number $k_{c}$ shows
little variation with $\nu_{\ell}$. Considering $k_{c}$ constant in Eq. (14)
and solving for $\gamma=0$ indeed yields

$u^{*}_{c}\simeq\sqrt{\frac{4\nu_{\ell}\omega_{c}}{\bar{\beta}s}},$ (17)

with $\omega_{c}=ck_{c}$ the frequency of the critical wavenumber. This is
equivalent to the relation proposed by Wu and Deike [38], $u^{*}/c\simeq
a\sqrt{\nu_{l}k^{2}T}$, with $T=2\pi/\omega$ the wave period and $a$ a
numerical factor. The best fit to our data using this relation yields $a\simeq
3.0\pm 0.2$, which is a factor 2 below the value $a\simeq 6.76$ found in the
numerical simulations of Ref. [38]. In spite of this difference, probably
related to the difference in the velocity profile (which is linear in the
numerics), this suggests that the scaling (17) for the critical friction
velocity in the case of mechanically generated waves is robust.

We close this discussion by comparing this critical friction velocity and the
one obtained for natural waves, without mechanical forcing. In Fig. 10 we add
$u^{*}_{c}$ for natural waves as measured in the same setup in Refs. [11, 22],
using silicon oils and water-glycerol mixtures. Here $u^{*}_{c}$ is defined as
the transition from the small-amplitude wrinkles (Phillips regime) to the
exponentially growing waves (Miles regime). This natural-wave threshold is
larger than the forced-wave threshold, but shows a shallower dependence with
the liquid viscosity, $u^{*}_{c}\sim\nu_{\ell}^{0.2}$, with a very similar
numerical prefactor (within 5%) for the two types of fluids.

The two-stage scenario provides a natural framework to interpret this
difference: Mechanically-generated waves grow from a disturbance of initial
amplitude sufficient to disturb the air flow and trigger the Miles instability
mechanism. On the other hand, natural waves grow from a noisy base state
defined by the low-amplitude incoherent wrinkles excited by the turbulent
pressure fluctuations in the air, so they require a larger wind velocity for
the wrinkle amplitude to disturb the flow and trigger the instability. More
specifically, for $u^{*}<u^{*}_{c}$, the wrinkle amplitude results from the
balance between the power injected by the turbulent pressure fluctuations and
the viscous dissipation in the liquid, yielding $\zeta_{w}\simeq
s\delta(u^{*3}/g\nu_{\ell})^{1/2}$, with $\delta$ the boundary layer
thickness. Assuming a Phillips-Miles transition when the wrinkle amplitude
$\zeta_{w}$ becomes of the order of the viscous sublayer thickness
$\delta_{\nu}=\nu_{a}/u^{*}$ [12, 10], we obtain
$u^{*}_{c}\sim\nu_{\ell}^{1/5}$, in good agreement with the data in Fig. 10.
The strong difference between these two scalings,
$u^{*}_{c}\sim\nu_{\ell}^{1/2}$ for forced waves and
$u^{*}_{c}\sim\nu_{\ell}^{1/5}$ for natural waves, emphasizes the difficulty
in defining a unique onset for the wave generation problem, and the interest
of varying the liquid viscosity to gain more insight into this problem.

Figure 10: Comparison between the critical friction velocity $u^{*}_{c}$ for
natural and forced waves. $\circ$, Natural waves in water-glycerol mixtures
[11]. $\triangle$, Natural waves in silicon oils [22]. $\square$, Forced waves
in silicon oils (present data). The black line shows the numerical solution of
$\gamma=0$ using Eq. (16) for $\bar{\beta}=45$, and the shaded area shows the
range $\beta=23-75$.

##  V Conclusion

In this paper we investigated the wind-induced growth of mechanically
generated waves over liquids of viscosity 20 and 50 times larger than that of
water, with the aim to gain insight into the scaling of the growth rate in the
exponential regime. This growth rate has been extensively studied in the
literature but only on the air-water configuration. Varying the liquid
viscosity offers an interesting opportunity to test existing theories and shed
light on this problem.

Using Free-Surface Synthetic Schlieren with a spatio-temporal filtering
provides wave slope measurements with a very good resolution, and the
possibility to access the growth and damping rates of the forced waves even
when masked by other naturally growing waves. From these measurements we
reconstruct the marginal stability curves with unprecedented accuracy. Once
the viscous dissipation is subtracted, our measurements show correct overall
agreement with Miles’ scaling, with a significant scatter comparable to that
of experiments performed in water. Our analysis suggests a maximum growth rate
(growth rate of the most amplified wave number) that scales as
$u^{*8}\nu_{\ell}^{-3}$, in reasonable agreement with our data.

Our work raises the question of the origin of the critical friction velocity
$u^{*}_{c}$ for wave growth, and its dependence with liquid viscosity
$\nu_{\ell}$. For mechanically generated waves, provided that the initial wave
amplitude is not too small, the system is directly in the second stage of wave
growth (exponential amplification of the wave energy). In this case, the
threshold is governed by the balance between the power injected by the wind
and the dissipation in the liquid. On the other hand, for natural waves, a
larger threshold $u^{*}_{c}$ is found, because the wave instability is
triggered from the low-amplitude incoherent wrinkles excited by the turbulent
pressure fluctuations in the air [12, 10]. Previous works [11, 22] suggested a
dependence $u^{*}_{c}\sim\nu_{\ell}^{1/5}$ for natural waves, while the
present work, available for two liquid viscosities only, suggests a dependence
$u^{*}_{c}\sim\nu_{\ell}^{1/2}$ for mechanically generated waves, in
qualitative agreement with recent numerical results [38]. Extending these
results to arbitrary liquid depth and to a larger range of viscosities,
including the case of water, would be valuable to clarify this difference.

###### Acknowledgements.

We are grateful to F. Charru, J. Magnaudet, R. Mathis, F. Burdairon, M.
Aulnette and S. Perrard for fruitful discussions, and to P. Balondrade and J.
Sant-Anna for assistance with measurement and data processing. We thank A.
Aubertin, L. Auffray, J. Amarni and R. Pidoux for experimental help. This work
was supported by the project “ViscousWindWaves” (ANR-18-CE30-0003) of the
French National Research Agency.

## Appendix: Relative importance of the various contributions to the wave
dissipation

The damping rate of waves at the surface of a viscous liquid in a container,
provided that the thickness of the boundary layers is much smaller than the
container dimension, can be split into three contributions: bulk ($\gamma_{\rm
bulk}$), bottom-wall ($\gamma_{\rm BW}$) and side-wall ($\gamma_{\rm SW}$),
given by Eqs. (4), (5), and (6), respectively. The first two contributions are
contained in the finite-depth damping rate $\gamma_{\rm LM}$ for arbitrary
viscosity derived by LeBlond and Mainardi [48], which has $\gamma_{\rm bulk}$
and $\gamma_{\rm BW}$ as asymptotic solutions at low viscosity for large and
small $k$ respectively.

Figure 11: Various contributions to the spatial damping rates
$-\gamma_{s}=-\gamma/c_{g}>0$ for (a) $\nu_{\ell}=20.5$ and (b) 50.1
$\mathrm{mm^{2}~{}s^{-1}}$. LM indicates the numerically resolved Leblond-
Mainardi solution, which includes the bulk dissipation and the bottom-wall
dissipation. The gray horizontal bar shows the experimentally accessible range
of wave numbers $k$.

To estimate which contributions dominate in our experiments (tank depth $h=35$
mm, width $W=296$ mm), we plot in Fig. 11 the spatial damping rates
$-\gamma_{s}=-\gamma/c_{g}$ from the numerically resolved Leblond-Mainardi
dispersion relation ($\gamma_{\rm LM}$) and from the three separate
contributions ($\gamma_{\rm bulk}$, $\gamma_{\rm BW}$, $\gamma_{\rm SW}$) for
the two liquid viscosities $\nu_{\ell}=20.5$ and 50.1
$\mathrm{mm^{2}~{}s^{-1}}$. In the range of accessible wave numbers, shown as
gray bars, the bulk dissipation $\gamma_{\rm bulk}$ (4) is the dominant
contribution for $k>70$ m-1, while the bottom-wall contribution $\gamma_{\rm
BW}$ becomes significant for lower $k$. The side-wall contribution
$\gamma_{\rm SW}$, shown by the dashed blue line, is never dominant, but it
contributes approximately 20% to the wall dissipation at low $k$.

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