# Dynamical impact of wind-waves on energy-containing eddies

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## A PHENOMENOLOGICAL SPECTRAL LINK

In the previous section, the description of wall-bounded turbulence has been related to statistical eddies, transporting momentum and energy both in physical space (the downward turbulent momentum flux) and in spectral space (the 3D isotropic cascade). In particular, the central role of « attached eddies », which interact with the mean wind shear at a given height, has been emphasized. Relying on this phenomenology, several authors have proposed a model, termed « spectral link », relating the averaged (or bulk) properties of turbulence to its spectral properties. The original model of Gioia et al. [2010] and its extension for a stratified flow by Katul et al. [2011] are first reviewed to introduce their use in the modeling of the mean velocity and MOST functions. They rely on the geometry of the eddies, without necessity of discussing spectral budgets. The latter are then discussed, along the lines of Katul et al. [2013] and Katul and Manes [2014].
BULK MODEL At the core of the concept of an attached eddy is the concept of a momentumtransporting eddy, i.e. which supports u′w′ at a given height. This was made explicit by Gioia et al. [2010] by expressing the momentum flux across a horizontal surface at a height z as the product of an energy-containing eddy turnover velocity ve, depending on the streamwise scale of the eddy sh, and a streamwise momentum difference transported by the eddy across a vertical distance sv. To first order, the momentum difference u(z + sv) − u(z − sv) reads 2∂U ∂z sv, which yields the following expression for the momentum flux − u′w′(z) ∝ ve[sh(z)] ∂U ∂z (z)sv(z).

### THE DYNAMICAL INTERACTION BETWEEN NEAR-SURFACE TURBULENCE AND WAVES

Understanding the properties of turbulent motions close to the sea surface is of uttermost importance for air-sea interactions, and is a complex problem due to the presence of ocean surface waves [see, e.g., the recent reviews on wind-wave interactions in Jones et al., 2001, Janssen, 2004, Sullivan and McWilliams, 2010, LeMone et al., 2019]. Field observations [e.g. Edson et al., 2013] indicate that, in the presence of surface waves and for sufficiently strong winds, the turbulent momentum flux on top of the so-called wave boundary layer (WBL), of height of about 10 m, is increased with respect to a flat surface, and has a one-to-one dependence on the mean wind speed (for averaging periods of about 30 minutes). Hence the disturbances generated by surface waves, whose amplitude is coupled to atmospheric motions, result in an overall change in the properties of turbulence in the WBL, for a prescribed mean 10m-wind [see experiments of Edson and Fairall, 1998, Sjöblom and Smedman, 2002]. In the following we review the dynamical properties of the WBL, with emphasis on theoretical (and, when possible, analytical) models for the interaction between atmospheric turbulence and waves. What follows is by no means an exhaustive or historical review, but rather one with a specific focus theoretical studies of the WBL.
Flow over surface waves, while sharing some similarities with flow with hills and wavy boundaries, exhibits some unique features due to the intrinsic properties of surface waves [see the review by Belcher and Hunt, 1998, where both flows are compared]. First, surface waves are moving undulations of the sea surface, which follow a dispersion relation. More precisely, the phase speed c = ω/k of a monochromatic wave depends on its wavenumber k and frequency ω, which are linked as ω2 = gk + Tswk3, where g is the gravity acceleration and Tsw is the dynamical surface water tension. The phase speed has a minimum for waves of wavelength (λ = 2π/k) of 1.6 cm, which marks the transition between capillary (smaller) and gravity (larger) waves. The phase speed of gravity waves increases with their size, as opposed to capillary waves. This first feature implies that the the impact of waves on atmospheric turbulence should depend not only on their geometry, but also on relative velocity of the wave with respect to the airflow, and hence on the scale of the wave [see for instance Kitaigorodskii, 1973, p. 27 to 36, where the surface is modelled as a linear superposition of moving roughness elements]. The relative velocity of the wave is termed wave age, c/u∗ , with u∗ the friction velocity (the square root of the momentum flux on top of the WBL).

#### MOMENTUM BALANCE IN THE WAVE BOUNDARY LAYER

As mentioned above, in the following we focus on the properties of the WBL, which is stationary and homogeneous, and above which the impact of waves on turbulent motions can be regarded as negligible. In this section we discuss its momentum balance. WAVE-INDUCED MOTIONS At the core of the coupling between wind and waves is the existence of perturbations of atmospheric quantities which are correlated to waves, and that can extract or lose energy either to the mean flow or to turbulent motions, leading to wave growth or decay. As described in Stewart [1961], as these motions carry energy down to the surface, their coherency must increase, in the sense that the different components of the motions become increasingly correlated. The existence of such motions is similar to that of internal waves in stably stratified turbulence [e.g. Zilitinkevich et al., 2008], or to flow over inhomogeneous surfaces, such as canopies [see Kaimal and Finnigan, 1994, p. 84]. In the presence of waves, the flow, for example the streamwise velocity, is linearly decomposed into a mean component hui = U, a wave-induced component u = uw, and a turbulent component u′: u = U + uw + u′.

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THEORETICAL MODELS OF WAVE-INDUCED STRESS

The previous section highlighted that the impact of waves on the WBL momentum balance occurs through wave-induced stress. As an energy balance reveals (see Sect. 1.3.3), waveinduced stress is related to the extraction of mean-flow energy by waves, and hence to wave growth. Historically, the determination of wave growth begins with the « separated sheltering » mechanism of Jeffreys [1925], according to which airflow separation on the leeside of the wave leads to a pressure drop and hence to wave growth. However, the growth rates predicted by this mechanism did not match measurements. Phillips [1957] and Miles [1957] proposed two mechanisms which aim at explaining the initial and later stage of wave growth, respectively.
The mechanism of Phillips [1957] considers the effect of turbulent pressure fluctuations on the surface, leading to a linear wave growth. The mechanism proposed by Miles [1957] considers the inviscid growth of wave-induced motions and waves in the context of stability analysis and neglects turbulent Reynolds stresses, leading to an exponential wave growth. Belcher and Hunt [1993] later argued that the Miles mechanism might not be valid for short waves, for which the effect of turbulence becomes important. This led to the « non-separated sheltering » mechanism, whereby, even without airflow separation (which requires waves to be steep), the presence of waves results in a pressure anomaly correlated to wave slope. In the following, we discuss in more details the Miles [1993] and Belcher and Hunt [1993] mechanisms, since our interest lies in the description of a stationary wind-over-waves system, and not in the initial stages of wave growth described by Phillips [1957].

WIND-OVER-WAVES MODELS AND THE EQUILIBRIUM RANGE

Several wind-over-waves models have been developed for the description of the local equilibrium between turbulence and and waves in the WBL. Within a one-dimensional vertical column, those models couple an atmospheric component, e.g. a momentum balance and TKE equation, to a wave component, i.e. a balance equation describing the evolution of the wave spectrum under the action of the wind. The coupling between those two components occurs through wave-induced stress, which characterizes both the changes in the atmospheric momentum balance (Eq. 1.24) and the momentum input from the wind to the waves (the impact of wave-induced stress on the TKE balance is described in Sect. 1.3.3.2 below). Assuming that both the wind and wave systems are stationary, the wind-over-waves models thus describe the equilibrium between turbulence and locally-generated wind-waves. More precisely, the range of gravity waves crucial to determine the stationary wind-overwaves system, described by the wave component of the wind-over-waves model, is the so-called equilibrium range. It covers wavelengths between the ten-meter scale and the centimeter scale, when surface tension starts to be important [Kitaigorodskii, 1983, Phillips, 1985]. In this range, wind input to the wave field is balanced by dissipation, mainly due to wave breaking.
This results in a wave spectrum with a shape as k−4 [see the theoretical works of Phillips, 1958, 1985, Belcher and Vassilicos, 1997]. More recently, the role of centimeter-to-millimeter waves in the determination of the wind-and-waves equilibrium has been emphasized, as their amplitude is very sensitive to mean wind speed [Yurovskaya et al., 2013], and those scales might support a significant fraction of wave-induced stress [Kudryavtsev et al., 1999, 2014].
Larger waves result mostly from an inverse energy cascade due to wave-wave interactions, and are not directly coupled to the local wind. Their amplitude depends on the history of the wave field (e.g. on fetch). It is out of the scope of this review to discuss the impact that these long gravity waves can have on the atmospheric momentum budget, or their impact on the local wind-over-waves equilibrium through a change in the properties of the overall wave spectrum, the latter being still an open question.

Remerciements
Résumé étendu en français
Contents
Introduction
1 Turbulence and waves: a literature review
1.1 Wall-bounded turbulence with stratification
1.1.1 General considerations for neutral conditions
1.1.2 Effect of stability: Monin-Obukhov Similarity Theory
1.2 A phenomenological spectral link
1.3 The dynamical interaction between near-surface turbulence and waves
1.3.1 Momentum balance in the wave boundary layer
1.3.2 Theoretical models of wave-induced stress
1.3.3 Turbulence in the wave boundary layer
1.3.4 Concluding remarks
1.4 Objectives of the present work
2 Towards a « spectral link » for the vertical velocity spectrum?
2.1 Introduction
2.2 Article: « Scalewise return-to-isotropy in stratified boundary layer flows »
2.3 Article: « Scaling laws for the Length Scale of Energy-containing Eddies in a
Sheared and Thermally Stratified Atmospheric Surface Layer »
2.4 Conclusion
3 Geometrical impact of wind-waves on energy-containing eddies
3.1 Introduction
3.2 Article: « On the impact of long wind-waves on near-surface turbulence and momentum fluxes »
3.3 Conclusion
4 Dynamical impact of wind-waves on energy-containing eddies
4.1 Introduction
4.2 Article: « Revisiting Beaufort scale: the dynamical coupling between turbulence and breaking waves »
4.2.1 Main text
4.2.2 Supplementary material
4.3 Conclusion
Conclusion
Appendices
Annex A: Some steps for the derivation of the spectral budget
Annex B: Comparison of several return-to-isotropy models
Annex C: Details on the preliminary numerical simulation

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