Editorial Type:
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Article Type:
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Polydisperse Sea Spray Effect on the Vertical Momentum Transport in Hurricanes

Yevgenii Rastigejev aDepartment of Mathematics and Statistics and Applied Science and Technology Ph.D. Program, North Carolina A&T State University, Greensboro, North Carolina

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Sergey A. Suslov bDepartment of Mathematics, Swinburne University of Technology, Hawthorn, Victoria, Australia

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Online Publication:
09 Jul 2024
Print Publication:
01 Jul 2024
DOI:
https://doi.org/10.1175/JAS-D-23-0195.1
Page(s):
1127–1146
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Abstract

This study focuses on the influence of the sea spray polydispersity on the vertical transport of momentum in a turbulent marine atmospheric boundary layer in high-wind conditions of a hurricane. The Eulerian multifluid model treating air and spray droplets of different sizes as interacting interpenetrating continua is developed and its numerical solutions are analyzed. Several droplet size distribution spectra and correlation laws relating wind speed and spray production intensity are considered. Polydisperse model solutions have confirmed the difference between the roles small and large spray droplets play in modifying the turbulent momentum transport that have been previously identified by monodisperse spray models. The obtained results have also provided a physical explanation for the previously unreported phenomenon of the formation of thin low-eddy-viscosity “sliding” layers in strongly turbulent boundary layer flows laden with predominantly fine spray.

Significance Statement

Achieving better accuracy in hurricane forecasts requires an in-depth understanding and accurate modeling of the ocean spray effect on the vertical fluxes of momentum and heat in a hurricane boundary layer. It has been shown that this effect depends on the size distribution of spray droplets, also known as spray polydispersity. This study aims to investigate the influence of a polydisperse spray on the vertical momentum transport within hurricane boundary layers by employing a modern theory of turbulent disperse multiphase flows.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yevgenii Rastigejev, ye_rast@yahoo.com

Abstract

This study focuses on the influence of the sea spray polydispersity on the vertical transport of momentum in a turbulent marine atmospheric boundary layer in high-wind conditions of a hurricane. The Eulerian multifluid model treating air and spray droplets of different sizes as interacting interpenetrating continua is developed and its numerical solutions are analyzed. Several droplet size distribution spectra and correlation laws relating wind speed and spray production intensity are considered. Polydisperse model solutions have confirmed the difference between the roles small and large spray droplets play in modifying the turbulent momentum transport that have been previously identified by monodisperse spray models. The obtained results have also provided a physical explanation for the previously unreported phenomenon of the formation of thin low-eddy-viscosity “sliding” layers in strongly turbulent boundary layer flows laden with predominantly fine spray.

Significance Statement

Achieving better accuracy in hurricane forecasts requires an in-depth understanding and accurate modeling of the ocean spray effect on the vertical fluxes of momentum and heat in a hurricane boundary layer. It has been shown that this effect depends on the size distribution of spray droplets, also known as spray polydispersity. This study aims to investigate the influence of a polydisperse spray on the vertical momentum transport within hurricane boundary layers by employing a modern theory of turbulent disperse multiphase flows.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yevgenii Rastigejev, ye_rast@yahoo.com
Keywords: Turbulence; Air–sea interaction; Hurricanes/typhoons; Momentum; Surface fluxes; Numerical analysis/modeling

1. Introduction

The study of hurricanes and their dynamics has long been a topic of great importance, in particular, due to significant property damage and loss of life they can cause. To minimize them, it is important to accurately forecast hurricane track and intensity. There is observational and theoretical evidence (Fairall et al. 1994; Mestayer et al. 1996; Van Eijk et al. 2001; Donelan et al. 2004; Makin 2005; Wu et al. 2015; Druzhinin et al. 2017; Tang et al. 2017; Garg et al. 2018; He et al. 2018; Rastigejev and Suslov 2019; Xu et al. 2021; Huang et al. 2022; Rastigejev and Suslov 2022; Richter and Wainwright 2023; Troitskaya et al. 2023; Xu et al. 2023) that ocean spray plays an important role in modifying the vertical enthalpy and momentum fluxes in the marine atmospheric boundary layer (MABL) of a hurricane, thereby affecting its dynamics both thermodynamically and mechanically. However, the present understanding of the role of ocean spray in the air–sea turbulent exchange under high-wind conditions of hurricanes remains limited. This knowledge gap has hindered progress in the accurate prediction of hurricane intensity motivating further in-depth studies of sea spray that are also relevant to many other contexts such as understanding the influence of spray on ice accumulation on vessels at high latitudes (Panov 1978; Line et al. 2022), gas exchange between ocean and atmosphere (Staniec et al. 2021; Gutiérrez-Loza et al. 2022), formation of aerosols (Burrows et al. 2022; Su et al. 2022), and climate dynamics (Song et al. 2022; Bruch et al. 2023).

There are two distinct ways in which sea spray can affect the MABL: thermodynamic and mechanical (Rastigejev and Suslov 2016). The thermodynamic effect is due to moisture and heat introduced or removed by spray droplets. It leads to variations in the temperature, humidity, and the vertical fluxes of sensible and latent heat within the hurricane MABL (Van Eijk et al. 2001; Rastigejev and Suslov 2019). The mechanical effect of ocean spray on the MABL is related to its capacity to modify the mechanical characteristics of the airflow via air–droplet momentum transfer by several competing physical mechanisms: weakening of the wind due to spray inertia and its acceleration due to the spray-induced suppression of turbulence (Barenblatt et al. 2005; Rastigejev et al. 2011; Rastigejev and Suslov 2022). The spray-induced inertial air deceleration is caused by the air momentum transfer to the accelerating sea spray ripped off wave crests and entering an airflow with initial velocity that is much smaller than that of the wind and increasing the density of the air–spray mixture near the ocean surface. The competing wind acceleration effect is due to two different mechanisms responsible for reducing the turbulence intensity of the air: the turbulence attenuation due to air–droplet slip (DS) (Kulick et al. 1994; Li et al. 2016) and the gravity lubrication (GL) (Barenblatt and Golitsyn 1974; Bertsch et al. 2015). The DS turbulence attenuation is caused by turbulent energy dissipation due to air–droplet friction, while GL results from the reduction in the turbulent kinetic energy (TKE) due to the vertical transport of spray against gravity. The TKE suppression by both DS and GL mechanisms reduces the effective vertical turbulent transport coefficients such as eddy viscosity. Subsequently, to maintain a constant momentum flux from the MABL to the sea, the wind velocity profile develops a steeper gradient, leading to an increase in the airflow velocity above wave crests (Barenblatt et al. 2005; Kudryavtsev 2006; Rastigejev et al. 2011; Rastigejev and Suslov 2014, 2022).

Our previous studies have demonstrated that the mechanical suppression of turbulence and the subsequent reduction of turbulent vertical exchange strongly affect heat fluxes in the MABL (Rastigejev and Suslov 2016, 2019) and can induce variations in the sea surface temperature, ocean mixed layer depth, and currents and other phenomena (Feng et al. 2021; Z. Sun et al. 2021). At the same time, the thermodynamic influence on mechanical transport in the MABL is found to be much weaker, thereby justifying decoupling their studies as is done in the current work.

The conventional relationships for the ocean surface roughness length Charnock (1955), Large and Pond (1981) state that the drag coefficient increases monotonically with the wind speed. However, the data collected in recent field observations and laboratory experiments (Powell et al. 2003; Donelan et al. 2004; Jarosz et al. 2007; Black et al. 2007; French et al. 2007; Haus et al. 2010; Takagaki et al. 2012; Potter et al. 2015; Zhao and Li 2019; Zhang et al. 2023) demonstrate that the drag coefficient increases with wind speed only when it is sufficiently small, but it reaches maximum at ≳30 m s−1 and then starts falling off the values calculated for the classical logarithmic velocity profile. Multiple authors (Makin 2005; Barenblatt et al. 2005; Rastigejev et al. 2011) suggested that the observed reduction in the drag coefficient may be due to the presence of spray. In our previous paper discussing mechanical effects of a monodisperse spray (Rastigejev and Suslov 2022), we indeed demonstrated such a behavior using a consistent physical model.

Our previous papers (Rastigejev et al. 2011; Rastigejev and Suslov 2014, 2016, 2019, 2022) also contained a comprehensive investigation of the influence of a monodisperse spray on the vertical fluxes within the MABL. In agreement with studies undertaken by other authors (e.g., Peng and Richter 2020), they have shown that both thermodynamic and mechanical influences of ocean spray on hurricane dynamics sensitively depend on the spray droplet size. However, realistic sprays are polydisperse mixtures of droplets of different sizes. Therefore, in the current study, we aim to determine how the variation in droplet size spectra affects various mechanical characteristics of the MABL. We also discuss the influence of the ocean surface roughness length on the airflow characteristics in the presence of polydisperse ocean spray.

We focus on the mechanical effect of a polydisperse spray. We employ the modern theory of turbulent multiphase flows (Drew and Passman 2006; Brennen 2005) and adopt the multifluid approach (Reeks 1992; Massot 2007) that treats air and droplets of different sizes as separate interacting and interpenetrating turbulent continua. This approach, which has recently gained popularity in numerical simulations of multiphase flows in various fields (Andreini et al. 2016; Kartushinsky et al. 2016; Senapati and Dash 2020; Li et al. 2021; Lian et al. 2022), is also known as the Eulerian–Eulerian method. In our analysis, each continuum represents a distinct entity with its own velocity and TKE and obeys mass and momentum conservation laws. A multifluid E–ϵ epsilon (also known as k–ϵ) model (Jakobsen 2008) is used as a turbulence closure. This choice is motivated by the fact that presently, it is one of the most extensively validated and widely used turbulence models for multiphase flow simulations across diverse applications (Torno et al. 2020; Pouraria et al. 2021; Song et al. 2021; Che et al. 2022). It has also found widespread application in simulating the atmospheric boundary layer (Chalikov and Rainchik 2011; van der Laan et al. 2017; Walsh et al. 2017).

Spray droplet size distributions characterized by two spectral shape functions (SSFs) derived from volumetric spray generation functions (VSGFs) proposed by Andreas (1998), hereafter A98, and Ortiz-Suslow et al. (2016), referred to as OS16 below, are considered. The key distinction between these spray distributions important in the context of our study is that OS16 SSF contains a higher proportion of larger droplets compared with the A98 counterpart (see Figs. 1c and 1d). Additionally, we examine estimations for the aerodynamic roughness length: the classical expression suggested by Charnock (1955) and the empirical formulation proposed by Large and Pond (1981) that is frequently used by atmospheric research and meteorological communities.

Fig. 1.
Fig. 1.

The SSFs for VSGFs given in (a) A98 and (b) OS16. (c),(d) The corresponding number density SSFs. Distributions (b) and (d) contain larger droplets.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

The paper is organized as follows. Section 2 introduces the main governing equations for polydisperse spray. The results of numerical simulation and their discussion are presented in section 4. It includes a comparative analysis of the spray polydispersity effects for A98 and OS16 SSFs for two different choices of aerodynamic roughness length. Section 5 presents a qualitative explanation of the mechanism behind the formation of a sliding layer and the associated turbulence crisis. Section 6 provides a concise overview of the main results of the paper. Details of spray distributions and spectral shape functions are given in the appendixes.

2. Problem formulation and mathematical model

The developed mathematical model of the spray-laden MABL is designed to capture the complexity of air–spray interaction and consistently quantify the main mechanical effect of spray on the airflow. To achieve this in a tractable way, we assume that turbulent incompressible airflow is statistically steady and horizontally homogeneous. The air is laden with polydisperse sea spray generated at the average wave crest level. Therefore, all variables depend only on the elevation above the mean sea level. The spray is modeled as thermodynamically inert media, which implies no evaporation or heat exchange with the surrounding air. The droplets are idealized as spheres of various radii subject to drag and gravity forces; droplet breakup and coalescence are neglected. The forces acting on spray droplets due to the perturbations they cause to the undisturbed flow, the so-called Faxen correction, are estimated to be at least three orders of magnitude smaller than Stokes’ drag (Maxey and Riley 1983) and so is the Basset memory force (Armenio and Fiorotto 2001; Daitche 2015; Liang and Michaelides 1992). Therefore, they are also neglected so that coupling between the turbulent airflow and droplets is assumed to have two components: airflow carrying the slipping droplets and droplets influencing the drag force experienced by the airflow.

The spray droplet radius spectrum is partitioned into N bins (classes). Because it falls off rapidly at large sizes, to maintain a better representation it could be beneficial to consider bins of nonequal width, but for algebraic simplicity in the current model, we opted for 10 equally spaced bins. Comparison of the so-obtained numerical results for several representative regimes with their counterparts for 5 and 15 bins did not reveal any qualitative discrepancies.

The proposed mathematical model adopts a multifluid approach treating droplets of various sizes and the air as distinct interacting turbulent media. As a result, the air and droplets from different bins are governed by their respective sets of coupled equations for mass, momentum, and turbulent kinetic energy. Each droplet class is characterized by its own velocity, turbulent kinetic energy, and transport coefficients. The model flow uses a 1.5-order E–ϵ multiphase turbulence closure. The Boussinesq approximation is employed to represent turbulent transport in both air and water phases.

The nondimensional equations used to model the vertical transport in the MABL describe the variation with the altitude z from the mean sea level scaled by the average wave crest height zw of the average horizontal air ua and spray ui velocities scaled by the friction velocity u⋆, and the volumetric spray densities si (also referred to as spray volume fraction in the literature) scaled by the spray source intensity s0, which is defined in appendix A. The air ea and spray ei TKEs and the correlation between air and droplet turbulent fluctuating velocities qai are scaled by u⋆2 and the dissipation rate ϵa of the TKE of air by u⋆3/zw. The equations read
ddz(kaduadz)=π12π1∑i=1Nsi(ua−ui)τdi,
ddz(kisiduidz)=si(ui−ua)π17τdi,
2π173ddz(siei)+daiτdidsidz=−π1si,
αeddz(kadeadz)=ϵa−pa−qae,
αϵddz(kadϵadz)=daϵ−paϵ−qaϵ,
ei=c1iea ,andqai=2c1iea,
where i = 1, …, N. Equations (1) and (2) describe the horizontal momentum balance for the air and spray in the ith class, respectively. Terms quantifying the turbulent transport and air–droplet drag force appear on the left- and right-hand sides of equations, respectively. Equation (3), which is derived in our previous publication (Rastigejev and Suslov 2022), describes the conservation of mass in the ith spray class. The turbophoretic tendency of droplets to migrate toward regions with weaker turbulence and the turbulent spray drift are represented by the first and second terms on the left-hand side of the equation, respectively. They are balanced by the downward gravitational pull given by the term on the right-hand side. Equations (3) and (4) represent the E–ϵ model for the air, while (5) and (6) describe the turbulent energy of spray and fluctuating air–droplet velocity covariance, respectively. The turbulent transport coefficients ka, dai, ki scaled by u⋆zw, the source terms qae, pa, paϵ, daϵ, qaϵ, the air–spray mixture density ρ scaled by the mass density ρa = 1.2 kg m−3 of spray-free air, and the coefficient σ are defined as
ka=23eaτat,dai=13qaiτait,ki=13(τaitqai+π17τdiei),qae=π12π1∑i=1Nsiτdi(qai−2ea+uir⋅uid),pa=pau+pag,pau=ka(duadz)2,pag=−π121ρa∑i=1Nsiwid,paϵ=Cϵ,1ϵapaea,daϵ=Cϵ,2ϵa2ea,qaϵ=Cϵ,3ϵaqaeea,ρ=1+σs0∑i=1Nsi,σ=ρwρa,
where ρw = 1020 kg m−3 is the mass density of water. Similar equations for monodisperse spray have been derived in Rastigejev and Suslov (2022), while the general theoretical framework underlining these equations is discussed in Jakobsen (2008). Here, the source term pau quantifies the rate of shear production of TKE, while pag and qae represent the rate of TKE destruction due to the vertical transport of spray against GL and DS as defined in section 1, respectively. Note that the wake arising behind large droplets contributes to the generation of TKE. However, quantitatively, this effect remains negligible because the concentration of such droplets in spume spray is low and their sizes that define the scale of vortices generated in their wakes are at least three orders of magnitude smaller than the most energetic turbulent eddies present in the MABL (Gore and Crowe 1989). Thus, the effect of turbulence enhancement by spray is not included in the presented model.
The local spray drift uid and relative air–droplet uir velocities for droplets of the ith size are
uid=−dai∇sisi=(0,wid)=(0,−daisidsidz) ,uir=(ui−ua,π1wi−wid).
The characteristic times of turbulent eddies (τat) and of their interaction with droplets of the ith size (τait) are scaled by zw/u⋆, while the relaxation time of such droplets (τdi) is scaled by a0/g. They are given by
τat=3α22eaϵa ,τait=c0iτat,τdi=ai,
where a0 is the characteristic terminal speed of droplets given by (A6) in appendix A and g = 9.8 m s−2 is the gravity acceleration. The expressions for the characteristics of droplets of the ith size, the terminal speed ai in the turbulent airflow scaled by a0, the air–droplet drag coefficient cd(u) (Mikhailov and Freire Silva 2013), and the root-mean-square of the local relative air–droplet velocity ⟨|uir|⟩ scaled by u⋆ are given by
ai=8σgr0ri3u⋆a0cd (⟨|uir|⟩)⟨|uir|⟩,cd(u)=3.17×108+6.69×107Ru+4.47×105Ru21.31×107Ru+9.86×105Ru2−Ru3,⟨|uir|⟩=(ui−ua)2+(π1wi−wid)2+2(1−c1i)ea,
where Ru=2r˜iu⋆u/ν is the Reynolds number for a droplet of radius r˜i moving with speed u⋆u relative to air and ν is the kinematic viscosity of air. Expression (9) for the air–droplet friction coefficient cd that is used to calculate the droplet terminal velocity is valid for a wide range of droplet radii 1 µm≲r≲3 cm. Coefficients c0i and c1i quantifying the cross-trajectories effects (Yudine 1959) and the capacity of air to transfer turbulent kinetic energy to droplets (Hinze 1975) and coefficient Cβ are given by
c0i=1σ0t1+Cβξr2,c1i=τaitπ17τdi+τait,Cβ=0.45+1.35cos2θ ,ξr=⟨|uir|⟩32ea,cos⁡θ=|ui|2−ua⋅ui|ui−ua||ui|.
The vertical components of the average air and mass-weighted average spray velocities are wa = wi = 0 above the wave crest level in the considered flow. Therefore, the expressions for qae, ⟨|uir|⟩, and Cβ given by (7), (9), and (10), respectively, are simplified to read
Cβ=1.8 ,qae=π12π1∑i=1Nsiτdi[qai−2ea−(wid)2],⟨|uir|⟩=(ui−ua)2+(wid)2+2ea(1−c1i).
The standard values of coefficients have been used for the multiphase E–ϵ turbulence model (Stull 1988; Simonin and Viollet 1990):
α=0.3,αe=1,σ0t=0.67,αϵ=0.77,Cϵ,1=1.44,Cϵ,2=1.92,Cϵ,3=1.
The nondimensional parameters entering the governing equations are defined as follows:
π1=a0u⋆,π12=gσs0zwu⋆2,π17=a0u⋆gzw.
The numbering of parameters reflects that introduced in our previous publications Rastigejev et al. (2011), Rastigejev and Suslov (2014, 2016, 2019, 2022).
The nondimensional boundary conditions complementing the model equations (1)–(5) are
ua=1kpln⁡1z0,ea=1α,ϵa=1kp,ui=ua+π12π17τdi3kpσ0tc0i(U−ua),si=piai,
at the wave crest level z = 1 and
kaduadz=1,ui=ua,deadz=0,dϵadz=−ϵaz∞,
at the upper edge of the boundary layer z = z∞. Here, pi is the nondimensional discrete SSF defined by (B3) in appendix B, U is the horizontal droplet injection speed, and kp=(Cϵ,2−Cϵ,1)α/αϵ≈0.43 is the von Kármán constant. The boundary conditions [(12)] for ui and si have been derived in Rastigejev and Suslov (2022). In the present study, we compare the numerical results obtained using the classical expression for the aerodynamic roughness length z0 given by Charnock (1955)
z0,Ch=0.015u⋆2g,
and the one obtained from an empirical formulation suggested by Large and Pond (1981) assuming logarithmic dependence of the horizontal air velocity on the vertical coordinate,
z0,LP=z10exp⁡(−kpua10),
where the nondimensional airspeed ua10 at the reference altitude z10 (10 m above the mean sea level) is calculated by solving the following cubic equation:
0.065u⋆ua103+0.49ua102−103=0,
valid for ua10 ≥ 11 m s−1.

To assess the sensitivity of model predictions to the choice of spray generation correlations, we employ two SSFs p˜(r˜), A98 and OS16, derived from the VSGFs suggested in (A98, Fig. 4) and [OS16, (22)], respectively. These SSFs describe the distribution of the droplet radius r˜ (here and below tildes denote dimensional variables). The p˜ and p˜N SSFs (see appendix A) and their partitioning into discrete bins used in our calculations are shown in Fig. 1. The main difference between the two spray distributions is that the OS16 SSF obtained using laboratory data is skewed toward large droplets.

The model presented above enables us to compute vertical distributions of various physical quantities characterizing the MABL as well as particular characteristics that are of conventional interest in the field of hurricane dynamics. One of such quantities is the drag coefficient at elevation z. It is defined as
Cd(z)=1ua2+σs0∑i=1Nsiui2.
Other quantities of interest include the total volumetric spray concentration st, the total volumetric spray concentration sw scaled by s0, the droplet size-averaged spray TKE per unit mass of air ew, factors c0 and c1, the characteristic time τawt of a turbulent droplet–air interaction, and the droplet relaxation time τd defined as
st=s0sw,sw=∑i=1Nsi,ew=1sw∑i=1Nsiei,c0=sw(∑i=1Nsic0i)−1,c1=1sw∑i=1Nsic1i,τawt=sw(∑i=1Nsiτait)−1,andτd=sw(∑i=1Nsiτdi+τait/π17)−1−τawtπ17,
respectively. Note that τd∼τd¯a and ∼τd¯h, assuming c0i ∼ c0, for τdi≪τait/π17 and τdi≫τait/π17, respectively, where
τd¯a=1sw∑i=1Nsiτdi,τd¯h=sw∑i=1Nsiτdi,
are the arithmetically (τd¯a) and harmonically (τd¯h) size-averaged relaxation times, respectively. Given that the total volumetric fraction of spray is small, st < 10−3, we take the volumetric density of air to be 1 − st ≈ 1.

The variable parameters entering model (1)–(13) and the discretized SSFs pi in (12) depend on four physical quantities: the friction velocity u⋆, the spray injection velocity U, the average wave crest height zw, and the spray production intensity s0. In the reported computations, we fix the value of zw to 5 m, which corresponds to the typical crest-to-trough wave height of ∼10 m observed in hurricane conditions. Our computations show that model output is almost insensitive to the value of the spray injection speed up to at least 10 m s−1. Therefore, we report all major numerical results for U = 0 and only consider nonzero values of the droplet injection speed to highlight the tendencies its variation can trigger that may be of interest in principle from a physical point of view despite being quantitatively very weak. The remaining quantities s0 and u⋆ are varied between 10−7 and 2.5 × 10−4 and between 1.5 and 4 m s−1, respectively.

3. Mechanical effect of sea spray

Ocean spray affects the vertical air–sea momentum exchange in the MABL in two ways: by its inertia, which decelerates the airflow, and by its ability to suppress the air turbulence, which accelerates the flow. The interplay of these two effects causes a slight overall deceleration of the flow near wave crests and its acceleration at higher altitudes. To demonstrate that, we integrate (1) with respect to z and apply the boundary condition (13) for ua, which yields
kaduadz=1−π12π1∑i=1N∫z∞si(ua−ui)τdi dz′≡1−t2.
Here, term t2 quantifies the airflow deceleration caused by spray droplets that are torn off the wave crests by the wind. In its dimensional form, (19) reads
du˜adz˜=u⋆2k˜a(1−t2),
where k˜a, u˜a, and z˜ are the dimensional turbulent eddy viscosity, the average horizontal air velocity, and the vertical coordinate, respectively. Throughout the domain, the eddy viscosity k˜a decreases compared with the corresponding reference values for the spray-free atmosphere due to the suppression of turbulence by the DS and GL mechanisms. On the other hand, the spray inertia effect is much more confined. It reaches its maximum strength at the wave crest level, where spray droplets are ripped off the waves, and gradually decays with the distance from it becoming negligible (t2 ≈ 0) several meters above zw. As a consequence, the spray inertia reduces the vertical airflow gradient, and hence its velocity, only in the proximity of wave crests. Note that such a velocity reduction is insignificant because the inertia factor t2 is relatively small. However, several meters above wave crests, the suppression of turbulence by the spray dominates and leads to lowering k˜a. This increases the vertical gradient of the wind velocity and the wind speed.

4. Results of numerical simulation

The governing equations introduced in section 2 were solved numerically using a standard MATLAB (MATLAB 2021) routine bvp5c using at least 2000 spatial discretization points distributed over the interval between z = 1 and z = 100 (from the wave crest level up to the altitude of 500 m). This ensures that all reported numerical results are accurate up to at least three decimal places.

a. Suppression of TKE by polydisperse spray and subsequent wind acceleration

Typical vertical distributions of characteristics of the spray-free reference boundary layer and of that laden with polydisperse spray are presented in Fig. 2. Despite that the shown quantities have been computed for specific values of the main independent parameters u⋆ and s0, their plots are representative of the main qualitative features of several hundred other distributions found for wide ranges of u⋆ and s0 to construct comprehensive model output maps that will be discussed in section 4b. As seen from Fig. 2a, the wind speed decreases slightly just above the wave crest level due to spray inertia. However, the turbulence suppression by spray overpowers its inertia, resulting in a noticeable wind speed increase a few meters above wave crests (see the solid and dashed lines) compared to the spray-free atmosphere (the dotted line). This acceleration is the consequence of the TKE suppression by spray shown in Fig. 2b. Such suppression is more pronounced for the A98 droplet size distribution that has a higher proportion of fine spray than OS16 (see Figs. 1c,d). Small droplets attenuate turbulence more efficiently than large ones in a hurricane boundary layer because of their ability to propagate to greater heights in the turbulent flow (Rastigejev and Suslov 2022) (see Fig. 2e). As a result, the airflow containing the A98-distributed spray is characterized by a lower turbulent viscosity as seen from Fig. 2d. Figure 2f shows that the vertical distribution of the total spray TKE ew is much closer to that of air TKE ea shown in Fig. 2b, for spray with A98 SSF than with OS16 SSF. This is due to a greater proportion of small droplets, which follow the air turbulence fluctuations closer, in A98 SSF. Finally, Fig. 2c demonstrates that at low values of spray concentrations s0≲10−4, the vertical distribution of the TKE dissipation rate is not sensitive to the choice of the SSF and remains inversely proportional to elevation z above the mean sea level. Combined with the fact that the system’s TKE is approximately constant for z > 5, this explains the essentially linear behavior of the turbulent viscosity coefficient ka seen in Fig. 2d and confirms that the wind speed profile above about 25 m to a good accuracy remains logarithmic with, however, a significantly larger magnitude than that of its counterpart in the spray-free atmosphere.

Fig. 2.
Fig. 2.

Vertical distributions of (a) airspeed, (b) air TKE, (c) dissipation rate of TKE, (d) turbulent viscosity in the airflow, (e) total spray concentration, and (f) total TKE of spray in the MABL for u⋆=3 m s−1, s0 = 10−4, and the roughness length given by Charnock relationship [(14)]. The thick dotted, dashed, and solid lines correspond to spray-free air and air laden with spray with the OS16 and A98 droplet size distributions, respectively. The corresponding thin curves show similar distributions computed for monodisperse spray.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

For comparison, in Fig. 2, we also present the computational results obtained for a monodisperse spray with the same concentration s0 = 10−4 and the droplet radius req corresponding to the terminal droplet velocity in quiescent air given by (A6) (shown by thin curves). Physically, this choice ensures the equivalence of the total volumetric spray injection fluxes in the cases of poly- and monodisperse sprays with identical values of the total spray volumetric fraction s0 in a quiescent air. We found that for A98 and OS16 SSFs, (a0, req) ≈ (0.86 m s−1, 122 μm) and (a0, req) ≈ (2.56 m s−1, 304 μm). The main observations are that the monodisperse spray assumption leads to a quantitative underprediction of the wind velocity and of the total spray concentration at higher altitudes (see Figs. 2a and 2c) and overprediction of the eddy viscosity (see Fig. 2d) that are stronger pronounced for the spectrum containing larger droplets (OS16, dashed lines). The distribution of the TKE dissipation is almost insensitive to whether poly- or monodisperse spray is considered. Importantly, the difference between the air TKE distributions computed using poly- and monodisperse models depends on the adopted SSF: The monodisperse model predicts larger (smaller) air TKE values for spectra favoring larger (smaller) droplets. Such nonmonotonicity of variations between results obtained adopting mono- and polydisperse assumptions demonstrates that the model explicitly and consistently accounting for the natural spray polydispersity is a nontrivial extension of the previously suggested models (Rastigejev and Suslov 2014, 2022) effectively considering a single-sized spray.

b. Dependence of the air–sea drag coefficient on a spray production rate

In view of large uncertainty currently existing in the literature regarding the spray concentration data at the wave crest level (A98) that is required as the input to the model developed in section 2, here, we vary it over a wide interval producing a comprehensive parametric maps that could also be viewed as sensitivity maps of the model outputs.

The distributions of spray volumetric density s˜(zw) at the wave crest level zw and the 10-m air–sea drag coefficient Cd defined by (17) as functions of the main independent parameters, the spray source intensity s0 and the friction velocity u⋆, are shown by labeled contours in panels a and b of Figs. 3–5. Three different cases are presented. Figures 3 and 4 exhibit the data computed for Charnock’s aerodynamic roughness length [(14)] and for A98 and OS16 SSFs, respectively, whereas Fig. 5 shows the data obtained for the aerodynamic roughness length [(15)] and the A98 spray distribution.

Fig. 3.
Fig. 3.

Distributions of (a) the volumetric spray density s˜(zw) at the wave crest level and (c) the air–sea drag coefficient Cd at the 10-m level as functions of friction velocity u⋆ and spray source intensity s0 for A98 SSF and Charnock roughness length [(14)]; (b),(d) the same quantities as functions of wind speed ua10 at the 10-m level assuming the power or exponential law correlations between s˜(z) and u⋆ or ua10 mapped as shown by the thick solid, dashed, and dash–dotted lines in (a) and (c). The dotted line shows the dependence of Cd on ua10 for the reference spray-free atmosphere.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for the OS16 SSF.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for the aerodynamic roughness length given by (15) and (16).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

The figures show that the air–sea drag coefficient Cd decreases with the spray source intensity s0 and grows with u⋆ qualitatively similarly for both A98 and OS16 SSFs and both z0,Ch and z0,LP roughness lengths when the spray source intensity s0≲10−4. The reduction in the drag coefficient is somewhat stronger for A98 SSF because of the larger proportion of small droplets, which suppress turbulence more efficiently due to their better ability to propagate to higher altitudes (Rastigejev and Suslov 2022), in the mixture. However, A98 and OS16 SSFs result in noticeably different behaviors of Cd for s0≳10−4 and u⋆≳3 ms−1 (see the top-right corner in Figs. 3c and 5c). The drag coefficient computed using the A98 SSF experiences a very sharp decline evident in the region containing closely spaced contour lines in Figs. 3c and 5c. We will discuss the reasons for such a crisis of the drag coefficient in section 5.

The comparison of Figs. 3 and 5 reveals that the values of Cd are lower when they are computed for the roughness length z0,LP rather than z0,Ch for the same SSF. This is expected since z0,LP < z0,Ch for the considered wind speed range so that the predicted wind speed is higher when the air moves over a smoother surface.

So far, we discussed the model results obtained treating the spray production intensity s0 and the friction velocity u⋆ (and thus the 10-m wind speed ua10) as independent parameters. It is intuitively clear that this is not so in physical reality: More spray is expected to be ripped off the waves by faster moving and more turbulent air. Unfortunately, to this day, there is no well-accepted consensus on the correlation between these two parameters (Fairall et al. 2003; Zhao et al. 2006; Fairall et al. 2009; Ma et al. 2020; Bruch et al. 2021). Therefore, similar to our previous studies (Rastigejev and Suslov 2014, 2016, 2019, 2022), we consider several spray generation functions reported in the literature. Specifically, we employ exponential laws suggested by Monahan (1986) and Wu (1993) and power laws suggested by Fairall et al. (1994) and Andreas (2002). They are given by
s˜(zw)=Aexp⁡[δ(u˜a10−ur)],
where A = 2 × 10−7, δ = 0.2–0.6 s m−1, and ur = 22 m s−1 (Rastigejev and Suslov 2022), and
s˜(zw)=Anu⋆n,
for which we consider two sets of parameters: 1) n = 5.37 and An = 2.32 × 10−7 sn m−n and 2) n = 8 and An = 4.2 × 10−8 sn m−n that are estimated from data presented in A98 and OS16, respectively. The spray production functions presented in these two studies are based on the field and wind tunnel measurements, respectively. Strictly speaking, the results of experiments conducted in a wind tunnel must be extrapolated to the actual field conditions prior to them being used in simulations. However, such an extrapolation is a nontrivial problem that does not have an apparent solution at present. It has been observed through Ma et al. (2020) that the spray production rates measured in field conditions are approximately two orders of magnitude greater than those obtained in wind tunnel studies for the same wind speeds. In view of these observations, we increased the spray concentrations estimated from OS16 by a factor of 100 before feeding them into our model.

The thick solid, dashed, and dash–dotted lines in Figs. 3a,c, 4a,c, and 5a,c map the wind speed/spray concentration correlations (21) and (22) for n = 8 and n = 5.37 and δ = 0.3, respectively, to the (s0,u⋆) plane. The corresponding dependencies of s˜(zw) and Cd on the 10-m wind speed u˜a10 along these curves are shown in Figs. 3b,d, 4b,d and 5b,d. The values of u˜a10 were extracted from full wind velocity profiles computed numerically by solving the governing equations for various u⋆ and s0 and then establishing a one-to-one relationship between the u⋆ and u˜a10 values for a given spray generation function. Remarkably, the exponential (δ = 0.3) and power (n = 8) laws, while derived from data obtained using different methods, yield very similar dependencies of the ocean spray concentration and the drag coefficient on the wind speed for both SSFs and roughness lengths. On the other hand, the power law with n = 5.37 results in lower spray concentrations and higher drag coefficients. However, the qualitative behaviors of both the spray concentration and the drag coefficient are the same for all tested correlations showing the initial rapid increase in the spray density in the air with its subsequent saturation at high wind speeds and the existence of a characteristic maximum of the drag coefficient beyond which its value quickly falls off the straight line corresponding to reference spray-free atmosphere. Such behavior of the drag coefficient has been observed in both field and laboratory studies (Powell et al. 2003; Jarosz et al. 2007; D. Sun et al. 2021). The rate at which the spray density increases with the wind speed determines the location of the maximum of Cd. The spray volumetric densities at the wave crest level corresponding to these maxima are found to be 7.4 × 10−5–1.0 × 10−4 and 1.4 × 10−4–2.0 × 10−4 for the A98 and OS16 SSFs, respectively. It is found that these values are weakly dependent on the type of used correlation linking the spray density at the wave crest level and the wind speed, and insensitive to the choice of the roughness length.

The mechanical effect of ocean spray cannot be consistently characterized by the behavior of the drag coefficient alone even though its reduction can be explained by the spray-induced decrease in the turbulent intensity. It is more appropriate to evaluate the mechanical influence of spray by assessing the extent to which the spray attenuates the TKE in the MABL. This is because the influence of the spray extends beyond a thin layer above the ocean surface, where it is produced, and can be substantial up to several hundred meters above it. Furthermore, the spray indirectly affects the MABL dynamics by mechanically suppressing turbulence and subsequently reducing turbulent vertical exchange, which strongly affects heat fluxes in the MABL. Nonetheless, we can compare our results with those presented in the literature by observing the drag coefficient behavior in a spray-laden MABL.

c. Weak influence of the spray injection velocity

As we established previously (Rastigejev and Suslov 2022), the outputs of the models based on E–ϵ turbulence closure are almost insensitive to the speed with which spray droplets enter the airflow. Since in hurricane conditions, most of spray originates from spume that is torn off the wave crests by a wind shear, it is expected that the initial speed of such droplets would be approximately equal to the ocean wave propagation speed. Even for large ocean waves created by hurricane winds, it does not exceed around 60 km h−1. Thus, below we consider two droplet injection speed values: typical 6 m s−1 (U = 2) and near-maximum 15 m s−1 (U = 5).

As seen from Fig. 6a, the nonzero droplet injection speed leads to a very minor change in the overall wind velocity (well under 0.1%). There are several physical reasons for that. First is that even the maximum expected droplet injection speed is at least 3–5 times slower than that of the hurricane winds at the wave crest level. The second is that in a statistically steady regime, the spray droplets that have just been injected with a nonzero speed constitute a relatively small proportion of the total droplet population. The majority of droplets found at the wave crest level at any moment in time are those that have been ejected previously, then carried across the MABL by turbulent diffusion, and then brought back by the gravity pull. Since such droplets have been suspended in the air for a sufficiently long time, their mean speed relaxes to the local flow conditions with their initial injection momentum well dissipated. This scenario is illustrated in Figs. 7a and 7c, where the vertical distributions of ratios of the droplet velocities (uiU) injected with speed U ≠ 0 to those (ui0) of droplet injected with U = 0 (thick curves corresponding to the positive values). The figure demonstrates that while larger droplets retain the information about their initial speed over a longer distance due to their inertia, regardless of the injection speed in the physically expected range, even they “forget it” by the time they reach the altitude of at most 1.5 wave crest heights (i.e., by traveling less than 2.5 m up from the 5-m wave). After that, it takes about another half-wave height for droplets injected with any initial speed to assume the local wind speed (the thin lines corresponding to the negative values). Therefore, the influence of the injection speed dispersion is limited to a thin [exponential as shown in Rastigejev and Suslov (2022)] region just above the waves.

Fig. 6.
Fig. 6.

Relative variation in the airflow characteristics with droplet injection speed U: (a) horizontal airspeed ua and (b) turbulent eddy viscosity ka for u⋆=3 m s−1, s0 = 10−4, and roughness length given by (14).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

Fig. 7.
Fig. 7.

Relative variation in spray characteristics with droplet injection speed U: (a),(c) horizontal droplet speed ui and (b),(d) spray concentration si for various droplet radii in a polydisperse spray with the A98 spectrum and other parameters as in Fig. 6.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

Another noteworthy observation following from Fig. 6a is that while spray injection with positive speed accelerates the local airflow in a thin region near the wave crest level, it leads to the overall wind reduction, which may appear counterintuitive at the first glance. The physical explanation for such a phenomenon becomes evident from Fig. 6b. The steep air velocity gradient induced by the spray injection leads to a more intense turbulence production [pau term in (4) and (7)], which in turn leads to the enhancement of turbulent viscosity ka illustrated in Fig. 6b and the related reduction of the wind speed. Such an increase in the turbulent transport coefficient also intensifies spray flux across the MABL, which results in a decrease in the spray concentration compared with the zero-injection-speed case that is more pronounced for large droplets (see Figs. 7a,b).

Having observed these trends, in principle, we note that quantitatively, they are very weak, and thus, we do not pursue this aspect of the model further.

5. Turbulence crisis in a thin near-surface layer

Our numerical simulations reported in section 4b revealed that when the spray production intensity s0 exceeds the critical value, the computed drag coefficients decrease quickly (see the top right corners of Figs. 3c and 5c). Typical flow quantity distributions presented in Fig. 8 indicated by solid lines (corresponding to A98 SSF) for such regimes shed light on the physical reasons for such a phenomenon. They demonstrate that a thin layer with a small turbulent eddy viscosity (see Fig. 8c) is formed a few meters above wave crests, under which a large portion of the injected spray is confined (see Fig. 8e). We refer to this layer as “a sliding layer” in the subsequent discussion. The presence of a large amount of spray drastically increases the local rate of turbulence destruction due to the friction between droplets slipping relative to the surrounding air and due to the vertical transport of spray against gravity. In statistically steady regimes, such TKE destruction must be compensated by the increasing rate of shear production, which implies the steepening of the vertical gradient of wind speed (see Fig. 8a). The only way this can be achieved is through a drastic local reduction in the turbulent eddy viscosity k˜a, which is indeed observed in Fig. 8d. Since k˜a=l˜e˜a, where l˜ and e˜a are the dimensional local vertical turbulence scale and TKE of air, respectively, the decrease in k˜a requires reductions in l˜ and/or e˜a. We demonstrate in appendix D that e˜a does not decrease noticeably in the vicinity of the minimum of k˜a. Hence, l˜, which can be interpreted as the thickness of the sliding layer, decreases. This reduction leads to the observed increase in ϵ˜a for A98 SSF in Fig. 8c, consistent with the relationship ϵ˜a=α2e˜a3/2/l˜. This local decrease in the eddy viscosity hinders the vertical turbulent transport, and the bulk of MABL becomes isolated from the rough sea surface and effectively slides over it. The integral effect of such physical processes is the crisis of the drag coefficient that is evident from Figs. 3c,d and 5c,d.

Fig. 8.
Fig. 8.

As in Fig. 2, but for u⋆=3.5 m s−1 and s0 = 2.3 × 10−4. The thin solid and dashed vertical lines show the locations of the minimum TKE fluxes fe = −kadea/dz for A98 and OS16 SSFs, respectively.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

To clarify the influence that the spectral composition of spray mixture has on the processes discussed above, in Fig. 8, we also show computational results obtained for the same physical parameters but the OS16 SSF (see the dashed lines). Such a spray with spectrum shifted toward larger droplets does not induce the formation of a thin layer with strong turbulence suppression. This is because the TKE balance in this case can be achieved without its formation. Therefore, no evidence of the drag coefficient crisis is found in Fig. 4 computed for this spectrum.

Small droplets lead to stronger turbulence destruction near the wave crest level, which can be seen from the dimensional expression for its rate,
d˜e∼stρwτ˜d(q˜aw−2e˜a),
where τ˜d, e˜a, and q˜aw are the dimensional characteristic droplet relaxation time, the turbulent kinetic energy of a droplet, and the air–droplet fluctuating velocity covariance, respectively. The above expression neglects the effects of the spray GL and of the vertical spray drift [(C2)] as they are much weaker than that of DS in the vicinity of the wave crest level (see appendix C). The analytical expression for the air–droplet fluctuating velocity covariance developed in Tchen’s theory of discrete particle dispersion (Hinze 1975) is given by
q˜aw∼2e˜aτ˜awtτ˜d+τ˜awt,
where τ˜awt is the characteristic turbulent eddy–droplet interaction time. Substituting (24) into (23), we obtain
d˜e∼2e˜astρwτ˜d+τ˜awt.
From (25), it follows that the turbulence destruction rate d˜e decreases with the droplet relaxation time τ˜d in the vicinity of the wave crest level and increases with volumetric spray density st. Therefore, for the same spray production intensity in the vicinity of wave crests, fine spray characterized by smaller droplet relaxation time and higher concentration than those of large-droplet spray destroys turbulence more efficiently, which explains why sliding layers may occur with small but not large droplets (see appendix D for more detailed discussion and estimates).

The TKE crisis has a profound influence on the distribution of spray in the upper part of the MABL. When the crisis develops, the amount of spray reaching higher altitudes drastically decreases. Indeed, as seen from Fig. 2e, in the absence of the crisis, the total spray concentration for OS16 SSF favoring larger droplets is significantly lower than that for the lighter AS98 spray. However, when the latter gets affected by the crisis, its concentration distribution above the sliding layer becomes very close to that of the heavier OS16 spray that remains unaffected by the crisis (see Fig. 8e). This leads to another noteworthy effect: the degree of the air TKE recovery in higher atmosphere is larger when the drag crisis develops preventing spray from traveling to high altitudes. Since the crisis is more likely to develop for a fine rather than large droplet spray, their dominant roles in the suppression of turbulence swap at high altitudes: According to relationships (C8), at equal concentrations, heavy droplets suppress turbulence more efficiently (see Figs. 2b and 8b).

Overall, Fig. 8 demonstrates that the spectral composition of spray has a highly nonlinear influence on the practically important MABL characteristics such as the wind speed and the air–sea drag coefficient. The monodisperse spray model results, which are shown for comparison by thin lines in this figure, demonstrate that while they also exhibit the tendency toward forming a thin sliding layer when a sufficiently large amount of fine spray is injected in the air, it is much weaker in a uniform spray. This is the main reason why this process has not been reported previously. As a consequence of that, a monodisperse model results may noticeably underpredict the wind acceleration in a spray-laden atmosphere. Note that the appearance of sliding is controlled by the TKE balance, and in the present model, it is considered in the average sense. Ocean waves can vary this balance and thus affect the characteristics of such layers. To quantify such an influence, an accurate description of the average action of ocean waves has to be included in the model, which is left for future studies given the current lack of relevant field data.

6. Conclusions

This study has been concerned with the Eulerian modeling of a turbulent statistically steady MABL laden with polydisperse ocean spray, where droplets of different sizes have been treated as interacting and interpenetrating continua with individual transport properties. Two spray size spectra, A98 with the large number of fine droplets and OS16 favoring big droplets, have been considered. Just above the wave crest level, spray causes a slight wind speed decrease due to its inertia. However, several meters higher, the spray-induced turbulence suppression overcomes the effect of inertia. This leads to a noticeable increase in the wind speed and the corresponding decrease in the drag coefficient for both spectra and roughness lengths compared to the reference spray-free atmosphere. The turbulence suppression and the wind acceleration and the air–sea drag reduction that it causes are stronger for the A98 droplet size distribution than that for OS16. This is due to the larger proportion of fine spray in A98, which has a superior ability to propagate upward in turbulent flows, leading to a higher volumetric spray density above wave crests.

We have tested several correlation laws that relate the wind speed and spray production intensity and observed that the drag coefficient gradually deviates from its reference value in the spray-free atmosphere reaching its maximum at a spray volumetric fraction smax ∼ 10−4 as the wind speed increases. The specific value of smax depends sensitively on the chosen SSF but only weakly on the correlation law or the aerodynamic roughness length. However, for a fixed correlation law, the value of smax is approximately twice as high for OS16 SSF than for A98. In contrast, the wind speed at which the drag coefficient reaches its maximum is primarily determined by the correlation law and is almost insensitive to the choice of SSF and the roughness length.

Our numerical calculations have also shown the possibility of formation of a thin sliding layer, which is characterized by a low turbulent eddy viscosity, a few meters above wave crests, where the presence of large amount of spray strongly enhances turbulence destruction by the air–droplet friction. Numerical experiments have demonstrated that such layers occur in an MABL laden with A98 SSF spray with volumetric densities ∼2 × 10−4. However, sliding layers have not been observed for OS16 SSF containing a larger proportion of heavy droplets at the same volumetric spray production rate. The intense TKE destruction in a sliding layer is balanced by the increased shear, leading to a steepening of the vertical gradient of the wind speed profile, which implies a local reduction in the turbulent eddy viscosity. It is shown that this reduction can be achieved only via a substantial increase in the TKE dissipation. This also implies a significant reduction in the local turbulence length scale. The sliding layer with a strongly suppressed vertical turbulent transport caused by the low turbulent eddy viscosity effectively isolates the bulk of the boundary layer from a rough sea surface, ultimately leading to the crisis of the drag coefficient.

The comparison of the current numerical results with those obtained using the previously suggested monodisperse spray models has shown that the latter may noticeably underestimate the spray-induced wind acceleration. It is also concluded that due to the strong nonlinearity of the considered phenomena, the quantitative and qualitative differences between mono- and polydisperse model predictions of the air TKE depend on the specifics of the actual spray spectra. This means that more realistic polydisperse spray results cannot be generally expected to be qualitatively similar to those obtained using a simpler monodisperse spray approach. Therefore, the model explicitly taking into account the spray droplet size variation is a nontrivial extension of the single-size models reported in the literature previously.

Finally, we note that the k–ϵ multifluid turbulence model was chosen here and in our previous studies because of its strong track record in both fluid-particle flows and atmospheric boundary layer simulations. Even though it has its limitations, this well-validated model has been widely used for numerical modeling of diverse industrial and environmental multiphase flows. Notably, it enables one to simulate the energy exchange between continuous and disperse phases. However, the model may not fully capture air–droplet energy exchange across the spectrum of turbulent eddy sizes. Addressing this issue presents an opportunity for a future model refinement. One promising avenue lies in employing multiple time-scale (MTS) models (Hanjalic et al. 1980). It is expected that the MTS approach would account for the air–droplet energy exchange within diverse eddy scales through separate consideration of different spectral parts more consistently than a standard k–ϵ model. The incorporation of the MTS framework would be a natural extension of the current mathematical model.

Acknowledgments.

The authors acknowledge the support from the U.S. National Science Foundation via the Grant Awards 1832089 and 2302221.

Data availability statement.

The MATLAB code used for numerical simulations of the mathematical model will be made available at https://github.com.

APPENDIX A

Spray Generation Function

Various researchers (Fairall et al. 1994; A98; OS16) suggested that volumetric spray generation function (VSGF) f˜υ(uc,r˜) can be given in a separable form by the product of two functions h˜(uc) and p˜(r˜) that depend only on the characteristic flow speed uc and the spray droplet radius r˜, respectively,
f˜υ(uc,r˜)=h˜(uc)p˜(r˜).
Reflecting their physical meaning, p˜(r˜) and h˜(uc) are referred to as the spectral shape function (SSF) and the intensity of spray source functions, respectively. Typically, either the friction velocity u⋆ or the wind speed u˜a10 at the 10-m level are taken as the characteristic speed. It is easy to see that functions h˜ and p˜ are defined up to a nonzero multiplicative constant. To fix this ambiguity, we normalize p˜(r˜) so that
∫0∞[p˜(r˜)/a˜0r]dr˜=1,
where a˜0r is the terminal speed of a droplet of radius r˜ in quiescent air. After multiplying (A1) by a˜0r−1 and integrating the left- and right-hand sides of the obtained equation over the complete spectrum, we obtain
h˜(uc)=∫0∞s˜0rdr˜=s0(uc),
where s0 is the total volumetric fraction of spray in a quiescent air and s˜0r=f˜υ(uc,r˜)a˜0r−1 is the droplet volume fraction distribution over the radius in quiescent air. Hence, (A1) takes the following form:
f˜υ(uc,r˜)=s0(uc)p˜(r˜).
Note that (A3) implies that s0 can also be interpreted as the spray source intensity. Function p˜(r˜) can be expressed in terms of f˜υ(uc,r˜) using (A3) and (A4) as
p˜(r˜)=f˜υ(uc,r˜)∫0∞f˜υ(uc,r˜)/a˜0r dr˜.
The size-averaged droplet terminal speed is then given by
a0=1s0∫0∞f˜υdr˜=∫0∞p˜(r˜) dr˜.
It follows from expression (A6) that monodisperse spray with droplet radius req and the corresponding terminal speed a0 is characterized by the volumetric injection flux s0a0 equivalent to that of polydisperse spray ∫0∞f˜υdr˜ in quiescent air. The number density spray generation function (NSGF) can be given in terms of VSGF as
f˜N(uc,r˜)=3f˜υ(uc,r˜)4πr˜3.
By substituting (A4) into (A7), we obtain
f˜N(uc,r˜)=s0(uc)p˜N(r˜),
where the spectral shape function p˜N(r˜) for the NSGF is defined as
p˜N(r˜)=3p˜υ(r˜)4πr˜3.

APPENDIX B

Discrete Spray Generation Function

The considered droplet radius range r˜min⁡<r˜≤r˜max⁡ is partitioned into N bins (also referred to as classes):
r˜i<r˜≤r˜i+1,i=1,…,N,r˜1=r˜min⁡,r˜N+1=r˜max⁡.
The average discrete spray characteristics p˜i, f˜i, s˜0i, a˜0i and nondimensional pi associated with the ith bin are defined as
p˜i=∫r˜ir˜i+1p˜ dr˜,f˜υi=∫r˜ir˜i+1f˜υdr˜=s0p˜i,
s˜0i=∫r˜ir˜i+1s˜0r dr˜,a˜0i=s0p˜is˜0i,pi=p˜ia0.

APPENDIX C

Asymptotic Expressions for Turbulence Destruction Rate

Here, we obtain the asymptotic approximations for turbulence destruction rates de and dϵ defined by (D9) in the vicinity of the wave crest level z ∼ 1 and in the upper part of MABL z ≫ 1. By substituting the expressions
qai−2ea=−2π17τdieaπ17τdi+τait ,
uir⋅uid=−(wid)2,
into (7) and taking into account that wid≈−π1τdi for π17 ≪ 1, we obtain
qae≈q1+q2,q1=−π1π12swτd¯a ,q2=−2π12π17sweaπ1(π17τd+τawt),pag≈−π1π12ρswτd¯a,
where τd and τawt are defined by (18). Since ρ ∼ 1, using (C3), we obtain
pagq2≈q1q2=3a0α2c0gτd¯azw4ϵau⋆3+a02τdτd¯a2eau⋆2.

a. Rate of turbulence destruction at z ∼ 1

The ratio (C4) does not exceed 0.1 for typical values τd¯a∼τd=0.5, c0 = 1, ea = 2, u⋆ =3.5 m s−1, and ϵa≳1 in the vicinity of the wave crest level z ∼ 1. This implies that pag and q1 are negligible compared to q2. Therefore, neglecting pag and q1 in (D9), we obtain
de≈dϵ≈−q2=4σs0sweaϵa2π17τdϵa+3α2c0ea.

b. Rate of turbulence destruction at z ≫ 1

Given that ϵa ∼ 1/(kpz) for z ≫ 1, from (8), we obtain τat∼3kpα2eaz/2. Then, it follows that τawt=c0τat∼3kpα2c0eaz/2≫π17τd. Therefore, q2 in (C3) is simplified to
q2≈−2π17π12eaπ1τawt=−4σs0sw3kpα2c0z.
Substituting (C6) into (C3) and (C3) into (D9), we obtain
de≈4σs0sw3kpα2c0z+σs0swgzwa0τd¯au⋆3(1+1ρ),
dϵ≈4σs0sw3kpα2c0z+σs0swgzwa0τd¯au⋆3(1+Cϵ,1′ρ).
Note that the first term in (C8) dominates over the second term for large values of u⋆ typical of hurricanes, but it decreases with altitude at z ∼ 1 as ∼1/z so that the two terms become comparable at z ∼ 10.

APPENDIX D

Mathematical Analysis of Physical Mechanisms Causing the Appearance of Sliding Layers

Figure D1 shows that the vertical distributions of the nondimensional air TKE flux fe = −kadea/dz has a minimum at some location z = zmin in the lower part of MABL. We will demonstrate that the TKE budget balance at this location requires a large TKE dissipation rate ϵa ≫ 1 whenever spray concentration is high. Therefore, the turbulent eddy viscosity ka = (αea)2/ϵa achieves its minimum and, subsequently, sliding layers occur near the minimum of the TKE flux.

Fig. D1.
Fig. D1.

The vertical distributions of the TKE flux fe = −kadea/dz for the same parameters and A98 SSF as in Fig. 8. The vertical dashed, solid, and dash–dotted lines correspond to locations of the minimum of ea (and, thus, fe = 0), the minimum of ka, and the maximum of ϵa, respectively, all clustered in the close vicinity of the minimum of fe.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0195.1

Substituting the following expression for ua − ui (see Rastigejev and Suslov 2022),
ua−ui≈π1π17τdi12(ua1−U)2Δiexp⁡(1−zΔi),
into (19), we obtain the estimate
t2∼12π12π17(ua1−U)∑i=1Nτdi1siexp⁡(1−zΔi),
where τdi1 = τdi(1), c0i1 = c0i(1), ua1 = ua(1), and Δi=kpπ17τdi1σ0tc0i1. The magnitude of Δi is estimated to be in the range of 0.1–0.2 for typical values of the problem parameters. As a consequence, t2 rapidly decreases with z becoming negligible at z−1≳0.1 and leading to
kaduadz≈1.
From here, the rate of turbulence production due to shear pau is expressed as
pau=ka(duadz)2=1ka.
Since ka = (αea)2/ϵa,
pau=ϵa(αea)2.
After substituting (D5) into (4) and (5), we obtain
χeea=de+ϵa−ϵa(αea)2,
χϵϵa=dϵ+Cϵ,2′ϵa−Cϵ,1′ϵa(αea)2,
where
χe=αeeaddz(kadeadz) ,χϵ=αϵ′eaϵa2ddz(kadϵadz), andαϵ′=αϵCϵ,3 ,Cϵ,1′=Cϵ,1Cϵ,3,Cϵ,2′=Cϵ,2Cϵ,3,
de=−(pag+qae) ,dϵ=−(Cϵ,1′pag+qae)
are the rates of destruction of turbulent energy and its dissipation rate due to the spray GL and DS effects, which are represented by qae and pag, respectively. It is shown in appendix C that the effect of spray GL and of the vertical drift is negligible compared to that of DS in the vicinity of wave crest level z ∼ 1; hence,
de≈dϵ.
Note that it follows from (D8) that
χe(zmin⁡)=0.
Next, we derive the expression for the TKE dissipation rate ϵa at z = zmin. By substituting (C5) into (D6) and taking into account (D11), we obtain the TKE balance equation as
4σs0sweaϵa2π17τdϵa+3α2c0ea+ϵa−ϵa(αea)2=0
at z = zmin. Solving (D12) for ϵa yields
ϵa=4Xσs0sw−3α2c02π17τdea ,X=(αea)21−(αea)2.
It follows from (D6), (D7), and (D10) that
χϵ=χeeaϵa+Cϵ,2′−1+1−Cϵ,1′(αea)2
and, subsequently, after taking into account (D11), we obtain from (D14)
ea=1αCϵ,1′−1Cϵ,2′−1−χϵ≈1αCϵ,1′−1Cϵ,2′−1≈2.3.
The factor χϵ was ignored in (D15) since it is difficult to estimate analytically for the complete range of boundary layer parameters. However, (D15) produces the value consistent with ea ≈ 2.06 and ea ≈ 1.83 found numerically for OS16 and A98 SSFs, respectively, for u⋆ =3.5 m s−1 and s0 = 2.3 × 10−4 (see Fig. 8b), indicating that χϵ is indeed small for large values of s0 and u⋆.

The sliding layer starts developing when the TKE dissipation rate becomes larger than its value ϵa = 1/(kpz) in the reference spray-free atmosphere. This happens when s0 exceeds a certain critical value s0,c. We show that s0,c increases with the average terminal speed of spray a0 but decreases with the friction velocity u⋆ next.

Substituting expression (11) for π17 and the estimate sw=z−a0τd/(kpc0u⋆)/τd for the volumetric spray density (Rastigejev and Suslov 2022) into (D13), we evaluate the dissipation rate as
ϵa=[4Xσs0z−a0τd/(kpc0u⋆)−3α2c0τd]eagzw2τd2a0u⋆.
We express s0,c from (D16) assuming ϵa = 1/(kpz) as
s0,c=za0τd/(c0kpu⋆)τd(3gα2c0eakpzwz+2a0u⋆τd)4gXσeakpzwz.
Relationship (D17) shows that s0,c increases with a0.
It follows from (D17) that
∂s0,c∂u⋆<0
for
u⋆>ln⁡(z)a0τd2kpc0[1+1+6gα2c02eakp2zzwln⁡(z)a02τd2] .
For A98 SSF, the right-hand side of (D19) is approximately 3 m s−1 for the typical values c0 = 1, ea = 2, τd = 0.5, and z = 1.5. Therefore, it follows from (D18) that the value of s0,c decreases with u⋆ for u⋆ ≳3 m s−1. Our calculations show that a sliding layer forms for 3ms−1≤u⋆≤4ms−1 for A98 SSF across the considered values of the main independent parameters. For example, we found that s0,c ≈ 2.5 × 10−4 and 1.7 × 10−4 when the spray volumetric concentrations at the wave crest level are s˜(zw)≈4×10−4 and 3 × 10−4 at u⋆ ≳3 m s−1 and 4 m s−1, respectively (see Figs. 3c and 5c).

Furthermore, we employ (D13) to estimate ϵa(zmin) for A98 and OS16 SSFs for which the average terminal speeds are a0 = 0.86 m s−1 and a0 = 2.56 m s−1, respectively, for the same parameter values as in Fig. 8. By substituting ea = 1.83 and sw = 0.9 for A98 SSF and ea = 2.06 and sw = 0.57 for OS16 SSF obtained for typical values τd = 0.2 and c0 = 0.9, we obtain ϵa ≈ 4.6 and ϵa ≈ 0.74 for A98 and OS16, respectively. These values of the TKE dissipation rate are close to ϵa ≈ 5.5 and 0.78 obtained numerically. Therefore, to achieve the TKE balance, ϵa(zmin) has to be large for A98 when both u⋆ and s0 are large. This implies the presence of a sliding layer. In contrast, for OS16 SSF, the balance between the turbulence energy production and dissipation can be achieved for ϵa ∼ 1 at least for s0≲2.5×10−4 and u⋆ ≲4 m s. The ϵa(zmin): Hence, in this case, sliding layers do not occur.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The SSFs for VSGFs given in (a) A98 and (b) OS16. (c),(d) The corresponding number density SSFs. Distributions (b) and (d) contain larger droplets.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Vertical distributions of (a) airspeed, (b) air TKE, (c) dissipation rate of TKE, (d) turbulent viscosity in the airflow, (e) total spray concentration, and (f) total TKE of spray in the MABL for u⋆=3 m s−1, s0 = 10−4, and the roughness length given by Charnock relationship [(14)]. The thick dotted, dashed, and solid lines correspond to spray-free air and air laden with spray with the OS16 and A98 droplet size distributions, respectively. The corresponding thin curves show similar distributions computed for monodisperse spray.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Distributions of (a) the volumetric spray density s˜(zw) at the wave crest level and (c) the air–sea drag coefficient Cd at the 10-m level as functions of friction velocity u⋆ and spray source intensity s0 for A98 SSF and Charnock roughness length [(14)]; (b),(d) the same quantities as functions of wind speed ua10 at the 10-m level assuming the power or exponential law correlations between s˜(z) and u⋆ or ua10 mapped as shown by the thick solid, dashed, and dash–dotted lines in (a) and (c). The dotted line shows the dependence of Cd on ua10 for the reference spray-free atmosphere.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    As in Fig. 3, but for the OS16 SSF.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    As in Fig. 3, but for the aerodynamic roughness length given by (15) and (16).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Relative variation in the airflow characteristics with droplet injection speed U: (a) horizontal airspeed ua and (b) turbulent eddy viscosity ka for u⋆=3 m s−1, s0 = 10−4, and roughness length given by (14).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Relative variation in spray characteristics with droplet injection speed U: (a),(c) horizontal droplet speed ui and (b),(d) spray concentration si for various droplet radii in a polydisperse spray with the A98 spectrum and other parameters as in Fig. 6.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    As in Fig. 2, but for u⋆=3.5 m s−1 and s0 = 2.3 × 10−4. The thin solid and dashed vertical lines show the locations of the minimum TKE fluxes fe = −kadea/dz for A98 and OS16 SSFs, respectively.

  • <sc>Fig</sc>. D1.
    View in gallery
    Fig. D1.

    The vertical distributions of the TKE flux fe = −kadea/dz for the same parameters and A98 SSF as in Fig. 8. The vertical dashed, solid, and dash–dotted lines correspond to locations of the minimum of ea (and, thus, fe = 0), the minimum of ka, and the maximum of ϵa, respectively, all clustered in the close vicinity of the minimum of fe.

Fig. 1.

The SSFs for VSGFs given in (a) A98 and (b) OS16. (c),(d) The corresponding number density SSFs. Distributions (b) and (d) contain larger droplets.
Fig. 2.

Vertical distributions of (a) airspeed, (b) air TKE, (c) dissipation rate of TKE, (d) turbulent viscosity in the airflow, (e) total spray concentration, and (f) total TKE of spray in the MABL for u⋆=3 m s−1, s0 = 10−4, and the roughness length given by Charnock relationship [(14)]. The thick dotted, dashed, and solid lines correspond to spray-free air and air laden with spray with the OS16 and A98 droplet size distributions, respectively. The corresponding thin curves show similar distributions computed for monodisperse spray.
Fig. 3.

Distributions of (a) the volumetric spray density s˜(zw) at the wave crest level and (c) the air–sea drag coefficient Cd at the 10-m level as functions of friction velocity u⋆ and spray source intensity s0 for A98 SSF and Charnock roughness length [(14)]; (b),(d) the same quantities as functions of wind speed ua10 at the 10-m level assuming the power or exponential law correlations between s˜(z) and u⋆ or ua10 mapped as shown by the thick solid, dashed, and dash–dotted lines in (a) and (c). The dotted line shows the dependence of Cd on ua10 for the reference spray-free atmosphere.
Fig. 4.

As in Fig. 3, but for the OS16 SSF.
Fig. 5.

As in Fig. 3, but for the aerodynamic roughness length given by (15) and (16).
Fig. 6.

Relative variation in the airflow characteristics with droplet injection speed U: (a) horizontal airspeed ua and (b) turbulent eddy viscosity ka for u⋆=3 m s−1, s0 = 10−4, and roughness length given by (14).
Fig. 7.

Relative variation in spray characteristics with droplet injection speed U: (a),(c) horizontal droplet speed ui and (b),(d) spray concentration si for various droplet radii in a polydisperse spray with the A98 spectrum and other parameters as in Fig. 6.
Fig. 8.

As in Fig. 2, but for u⋆=3.5 m s−1 and s0 = 2.3 × 10−4. The thin solid and dashed vertical lines show the locations of the minimum TKE fluxes fe = −kadea/dz for A98 and OS16 SSFs, respectively.
Fig. D1.

The vertical distributions of the TKE flux fe = −kadea/dz for the same parameters and A98 SSF as in Fig. 8. The vertical dashed, solid, and dash–dotted lines correspond to locations of the minimum of ea (and, thus, fe = 0), the minimum of ka, and the maximum of ϵa, respectively, all clustered in the close vicinity of the minimum of fe.
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