1 Introduction

Large eddy simulations (LES) are considered a critical tool for atmospheric scientists in order to understand the atmospheric boundary layer (ABL) dynamics. This high accuracy approach has undergone multiple enhancements since it was initially implemented by Deardorff (1970). It explicitly resolves the turbulent structures carrying most of the kinetic energy, whilst smaller filtered structures are modeled. These smaller structures (subgrid scales) typically carry only a relatively small portion of the kinetic energy and mainly dissipate energy transferred from the larger scales.

Energy exchange at the marine atmospheric boundary layer (MABL) air–sea interface is one of the most important physical components of Earth's climate and its variability. Enormous amounts of heat, moisture and momentum are transported across this interface mostly by turbulence. This energy exchange affects from the maintenance of stratocumuli (Bretherton and Wyant 1997) to how hurricanes intensify (Andreas and Emanuel 2001; Black et al. 2007; Emanuel 1991; Wang et al. 2001).

On average, low-level clouds, particularly stratocumuli cover 23% of the ocean surface each year (Wood 2012). Thus, they constitute an important climatological component as the cloud cover can modify the solar radiation which reaches the Earth's surface. Turbulent fluxes affect the structure and evolution of the cloud-topped MABL. Furthermore, radiative cooling, evaporative cooling, and entrainment rates are all strongly connected processes which produce and are driven by turbulence and, thus, they influence the development and evolution of cloud layers as a whole (Mellado 2017; Rauterkus and Ansorge 2020). For instance, radiative cooling promotes convective instability, which is a primary mechanism of turbulence generation that enhances the entrainment of dry air from aloft. This dry air mixing in the MABL can change the near-surface moisture and, hence, the latent heat flux (Zheng et al. 2018). Therefore, when simulating the MABL with LES the impact of cloud microphysics and radiation processes must be considered, as these processes closely interact with one another and turbulence.

The interest in wind-wave interaction first emerged in the latter half of the twentieth century (Large and Pond 1981; Wu 1982; Yelland and Taylor 1996). The main goal of their research was to determine the surface total wind stress (or drag coefficient) using in-situ observations in most cases up to moderate wind conditions and parameterize it using the neutral wind speed at 10 m, \({U}_{10n}\), above the mean sea level (amsl). This simplification, however, implicitly includes the effects of various parameters of sea state and other processes, including wave breaking, the subsequent airflow separation, and formation of sea spray by bubble bursting or direct surface detachment at high winds conditions. Thus, the individual contribution of each process to the wind stress or heat fluxes could not be quantified.

The dependence of stress (momentum flux) on wind velocity was usually extrapolated to high wind conditions, yielding a parameterization of drag coefficient that grows linearly with wind speed. Recent research has challenged this approach, confirming that in hurricane-force winds the surface drag coefficient saturates or even decreases (Bell et al. 2012; Donelan 2004; Powell et al. 2003). The theoretical explanation for drag leveling-off includes the "sheltering effect" and the air-flow separation between consecutive wave crests, which results in the air-flow not feeling the actual sea roughness (Kudryavtsev and Makin 2007; Kukulka and Hara 2008; Kukulka et al. 2007; Mueller and Veron 2009). In addition, the surface drag can be affected by the development of a foamy layer smoother than the actual surface (Donelan 2004; Holthuijsen et al. 2012; Powell et al. 2003), and by changes in the near-surface wind speed due to redistribution of horizontal momentum by the falling sea-spray droplets (Kudryavtsev 2006; Makin 2005; Richter and Sullivan 2013).

Sea spray droplets lofted into the air increase the effective contact area between the ocean and the atmosphere, offering an additional mechanism in strong wind conditions for the exchange of heat fluxes, i.e. sensible and latent heat. Veron (2015) and Sroka and Emanuel (2021) present a thorough analysis of this topic while addressing the advancement of spray heat transfer modeling over the last few decades as a result of theoretical, observational, and numerical modeling approaches. Sea spray modeling employs bulk algorithms, Eulerian and/or Lagrangian approaches (Andreas et al. 2015; Bao et al. 2011; Kepert et al. 1999; Mueller and Veron 2014a, b; Peng and Richter 2017, 2019; Richter and Wainwright 2023; Shpund et al. 2012, 2014).

Two approaches are commonly used for the surface boundary conditions when simulating the MABL using LES. The first considers the air–sea interface to be a solid rough surface, with the waves being characterized as roughness elements. The Monin–Obukhov Similarity theory (MOST) framework is used and the surface shear stress is parameterized with a bulk aerodynamic method based on wind speed (Andreas et al. 2012; Foreman and Emeis 2010) or on information about the wave field. These wave data are either provided in the form of bulk variables like wave-age and wave-steepness (Drennan 2003; Edson et al. 2013; Taylor and Yelland 2001), or obtained from wave spectra (Hara and Belcher 2004; Janssen 1989; Makin and Kudryavtsev 1999; Moon 2004). The second approach couples the turbulent boundary-layer flow to the sea surface via a time-varying waving boundary condition (Hao and Shen 2019; Hara and Sullivan 2015; Sullivan and McWilliams 2010; Sullivan et al. 2000, 2008, 2014, 2018; Yang et al. 2013). This solid boundary is either moving in a prescribed fashion or in response to the airflow. However, these wave-resolving LES are highly computationally demanding, with requirements which exceed today’s typical high-performance computing resources, especially when microphysics and radiation are considered (Hao and Shen 2019).

Our main goal is to improve MABL representation by incorporating the air–sea interaction mechanism into an LES code which already includes the effects of water phase transition, cloud microphysics, and radiation processes. The MISU MIT Cloud and Aerosol (MIMICA) LES model for cloudy ABL (Savre et al. 2014) is employed, which is extended to include two bulk air–sea flux algorithms, namely COARE 3.5 (Edson et al. 2013; Fairall et al. 2003) and Andreas' latest version 4.0 (Andreas et al. 2015). Realistic data from the CBLAST-Hurricane experiment (Black et al. 2007) during hurricane Isabel on 12 September 2003 are used to initialize the model and validate our results, as it is one of the few datasets which include strong winds and turbulence measurements. We do not attempt to actually simulate a HBL, therefore the large-scale pressure gradient and meso-scale tendencies like advection and centrifugal force are not taken into consideration (Bryan et al. 2017; Worsnop et al. 2017). We analyze the impact of sea spray on the surface fluxes and, thus, on MABL structure and cloud macro-physical properties under severe wind conditions.

2 Methodology

Over the ocean, there is additional stress induced by the wave form in addition to the stress caused by viscous boundary effects and turbulence (Belcher and Hunt 1993; Edson and Fairall 1998; Hare et al. 1997; Janssen 1989; Makin et al. 1995). The form stress is caused by the interaction between air-flow and waves and can be further divided into non-breaking (regular waves) and breaking waves (inducing air-flow separation) (Kudryavtsev and Makin 2001). Various studies have also examined the potential impact of sea spray on stress (Andreas 2004; Kudryavtsev and Makin 2011; Makin 2005). These stress components vary with height with the total stress \(\tau\) considered constant in the surface layer. The bulk aerodynamic method is used for the parameterization of total stress, \(\tau ,\) and reads:

$$\tau =\rho {C}_{dz}{{U}_{z}}^{2}$$
(1)

where \(\rho\) is the air density and \({U}_{z}\) is the wind velocity at height \(z\); \({C}_{dz}\), is a non-dimensional drag coefficient which is related to the momentum roughness length, \({z}_{o}\), through the logarithmic profile (Stull 1988). For non-neutral atmospheric stability conditions, the drag coefficient at \(z=\) 10 \(\text{m}\) is given by:

$$\sqrt {{C_{d10} }} = \frac{{u_{{ \star }} }}{{U_{10} }} = \frac{\kappa }{{ln\left( {10/z_o } \right) - \psi_m (10/L)}}$$
(2)

where \(u_{{ \star }} = \sqrt {\tau /\rho }\) is the friction velocity, \(\kappa\) is the von Karman constant, \({\psi }_{m}\) is a dimensionless function which accounts for the effects of atmospheric stratification and \(L\) is the Monin–Obukhov length. Following Smith (1988), the momentum roughness length is expressed as:

$${z}_{o}= {z}_{o}^{smooth}+{z}_{o}^{rough}$$
(3)

where \({z}_{o}^{smooth}\) accounts for an aerodynamically smooth flow (for \({U}_{10}\)<2 \({\text{ms}}^{-1}\)) where viscous stress dominates, while \({z}_{o}^{rough}\) accounts for the rough flow (for \({U}_{10}\)>8 \({\text{ms}}^{-1}\)) where the momentum transferred by turbulent eddies interacting with the waves dominates. The term, \(z_o^{rough} = \alpha u_{{ \star }}^2 /g\) was introduced by Charnock (1955) where \(g\) is the gravitational acceleration and \(\alpha\) is the Charnock coefficient, which is usually expressed as a function of sea state in terms of wind speed, wave age or wave steepness or other wave-related parameters.

Similarly to momentum, the scalar transfer coefficients for sensible heat, \({C}_{t10}\), and latent heat, \({C}_{q10},\) are related to the scalar roughness length \({z}_{ox}\) (x stands for \(t\) or \(q\)) through:

$${C}_{x10}= \left[\frac{\kappa }{\mathit{ln}\left(10/{z}_{o} \right)-{\psi }_{m}(10/L)}\right]\cdot \left[\frac{\kappa }{\mathit{ln}\left(10/{z}_{ox} \right)-{\psi }_{x}(10/L)}\right]$$
(4)

In the Liu–Katsaros–Businger model (LKB; Liu et al. 1979), the scalar roughness length is parameterized in terms of the roughness Reynolds number, \({R}_{r}\), though:

$${z}_{ox}= \frac{\nu }{{u}_{\star }}{f}_{x}({R}_{r})$$
(5)

where \(\nu\) is the kinematic viscosity of air and \({f}_{x}\) is a polynomial function. LKB is an extension of Liu and Businger (1975) with eddies transporting heat between the bulk fluid and the sea surface through the interfacial layer (~ 1 \(\text{mm}\) depth) where molecular constraints on the transport are important.

The LKB method has been validated with field data for low wind speeds. Fairall et al. (2003) retained this interfacial scaling, yet they extended the parameterization for winds up to \(18\text{ m}{\text{s}}^{-1}\), based on a large amount of field data. In particular, they considered equal scalar roughness lengths, i.e. \({z}_{ot}={z}_{oq}=\) min \((1.1\times {10}^{-4}\), \(5.5\times {10}^{-5}{R}_{r}^{-0.6})\). Andreas et al. (2008) have argued that the interfacial scaling should not be expanded in higher wind conditions (\({U}_{10}>10\,{\text{ms}}^{-1}\)) where sea spray has a significant influence on scalar fluxes.

According to Andreas et al. (2008, 2015), the sensible and latent heat fluxes \({H}_{s}\) and \({H}_{L}\), respectively, are the sum of the interfacial and spray contributions (6) as the field measurements are taken above the droplet evaporation layer, which typically extends about one significant wave height amsl:

$$\begin{aligned} H_L & = H_{L,int} + H_{L,sp} \\ H_S & = H_{S,int} + H_{S,sp} \\ \end{aligned}$$
(6)

with the subscripts \(int\) and \(sp\) referring to interfacial and spray contribution, respectively. \({H}_{L,int}\) and \({H}_{S,int}\) are parameterized based on LKB through the temperature and humidity roughness length using (5). Furthermore, \({H}_{S,sp}\) depends on the difference between the equilibrium temperature of spray droplets and the sea surface temperature (\(SST\)) while \({H}_{L,sp}\) depends on the mass lost by evaporation. Andreas et al. (2008) hypothesized that the microphysical behavior of droplets with initial radii of 50 μm and 100 \(\upmu \text{m}\), which contribute most of the \({H}_{L,sp}\) and \({H}_{S,sp}\), respectively, are representative diameter values for making the spray flux estimation simpler and computational effective. It is important to note that Andreas’ algorithm does not account explicitly for the spray particle production. Instead, it parameterizes spray-mediated fluxes through the microphysical behavior of 50 \(\upmu \text{m}\) and 100 \(\upmu \text{m}\) radius droplets and friction velocity. Microphysical quantities, such as e-folding time of temperature and radius for spray flux calculations are based on Andreas (2005). However, due to the complexity of the system and the difficulty to obtain direct observations, the extent to which sea spray facilitates air–sea transfer remains uncertain. Peng and Richter (2017, 2019) have investigated the underlying assumptions of sea spray bulk models through idealized direct numerical simulations in a Lagrangian–Eulerian framework. They found that the spray flux algorithm of Andreas et al. (2015) probably overestimates spray-mediated heat flux.

2.1 MIMICA LES model extended

MIMICA LES model (Savre et al. 2014) solves a set of anelastic, non-hydrostatic governing equations (Clark 1979) including momentum, ice/liquid potential temperature and total water mixing ratio on a Cartesian staggered Arakawa-C grid. The 2nd order Runge–Kutta time integration scheme is employed and the integration time step is calculated continuously to satisfy the Courant–Friedrichs–Lewy criterion. Momentum advection is performed using a 3rd order equation in conservative form. A quadratic upstream interpolation for convective kinematics type finite volume scheme is used to calculate scalar advection. A first order closure based on the turbulent kinetic energy (TKE) equation (Stull 1988) is used to estimate the subgrid scale TKE, \(e\). Eddy diffusivity is modeled as \({K}_{m}={C}_{m}l\sqrt{e}\), where \({C}_{m}\) is a model constant and \(l\) is a characteristic turbulent length scale. The turbulent Prandtl number, \(Pr\), is used to relate the momentum to the scalar eddy diffusivity as \({K}_{h}={K}_{m}/Pr\). MIMICA takes into account a maximum of six classes of hydrometeors, and a two-moment bulk microphysics scheme (Seifert and Beheng 2001, 2006) is used to calculate the prognostic variables (mass mixing ratio, number concentration). A simple parameterization for cloud condensation nuclei (CCN) activation is applied (Khvorostyanov and Curry 2005), where the number of cloud droplets formed is a function of super-saturation and CCN concentration. In this study, grid points with condensed water \({q}_{c}\) exceeding 0.001 \({\text{g kg}}^{-1}\) are considered cloudy. The radiation solver of Fu and Liou (1993) considers the effects of the water vapor mixing ratio, cloud droplets and ice particles on radiation. The surface is treated as a solid wall with vertical velocity equal to zero with the following conventional surface boundary condition:

$${\tau }_{i3,s}=\langle {\tau }_{b}\rangle \frac{{\widetilde{u}}_{i}}{|{\widetilde{u}}_{r}|}$$
(7)

where \({\tau }_{i3,s}\) is the instantaneous local (i.e. over an LES grid element) surface stress at each grid point (\(i\) = 1, 2 denotes the stream-wise and span-wise components); \(\left|{\widetilde{u}}_{r}\right|=\sqrt{{{\widetilde{u}}_{1}}^{2}+{{\widetilde{u}}_{2}}^{2}}\) is the instantaneous resolved velocity magnitude at the first LES level (the \(\sim\) denotes LES resolved variable); \(\langle {\tau }_{b}\rangle\) is the total stress calculated by a bulk air–sea flux algorithm in our study.

MIMICA has been expanded to include two bulk air–sea flux algorithms, COARE 3.5 (Edson et al. 2013) and Andreas' version 4.0 (Andreas et al. 2015). MOST provides ensemble-mean flux-gradient relationships in the surface layer. Nevertheless, MOST is commonly used in the LES approach in order to relate surface turbulent fluxes to resolved-scale variables (Albertson and Parlange 1999; Moeng 1984; Stoll and Porté-Agel 2006). Similarly, in this study the spatially averaged resolved wind velocity, air temperature, humidity and other atmospheric parameters at the first LES level are used as input parameters to both algorithms in order to estimate surface fluxes at each time step.

COARE 3.5 is an air–sea flux algorithm which estimates the momentum flux through \({C}_{d10}\) by taking into consideration the momentum roughness length, gustiness, and atmospheric stability. The Charnock coefficient is obtained by the piecewise linear function of wind speed of Edson et al. (2013):

$$\alpha = {0.017U}_{10n}-0.005$$
(8)

Further, this algorithm applies the empirical fit for \({z}_{ox}\) of Fairall et al. (2003), as it was mentioned above.

Andreas’ algorithm explicitly treats the interfacial and the spray-mediated fluxes. Both total heat fluxes (6) can be used as bottom boundary conditions to account for spray without considering the evaporation layer transfer mechanisms. The drag coefficient follows the work of Foreman and Emeis (2010), similar to the left hand side of Eq. (2) but with a constant added. The new parameterization is \({u}_{\star }={C}_{m}U+b\), where \(\mathop{{\text{lim}}}\nolimits_{U\to \infty }{C}_{d}={C}_{m}^{2}\) and b is a constant with dimensions of meters per second. The benefit of this method is that it produces a natural limit to the drag coefficient as the wind speed increases. This modification is necessary in hurricane models, as a simple extrapolation of the \({C}_{d}\) from moderate to hurricane wind speeds underestimates the storm intensity.

2.2 Design of LES-CBLAST simulations

The simulated case is based on observations from the CBLAST-Hurricane experiment acquired during Hurricane Isabel on September 12, 2003. The N43RF research aircraft flew low-level stepwise descents within the cold wake between rain bands at a distance of 100–150 \(\text{km}\) east of the storms’ center. This aircraft collected HBL observations at 88–374 \(\text{m}\) amsl, which are suitable for estimation of heat and momentum flux as well as turbulence statistics (Drennan et al. 2007; French et al. 2007; Zhang 2007; Zhang et al. 2009; Zhang and Drennan 2012). In the vicinity of N43RF, the N42RF aircraft launched GPS dropsondes to measure the HBL vertical structure. The averaged profiles of wind speed, air temperature and humidity have been used for the model initialization (see Sect. 3.1.1).

In our study, the mesh resolution is chosen to be anisotropic and as fine as our computer resources permitted. Although higher resolution resolves smaller scales it showed little influence on the surface fluxes. The computational domain is a 512 \(\times\) 512 grid in the horizontal plane with resolution set to \(\delta x\) = \(\delta y\) = 10 \(\text{m}\). The vertical grid-spacing is \(\delta z\) = 5 \(\text{m}\) with a total of 300 layers. According to Berg et al. (2020), the LES grid-sensitivity issue persists down to a resolution of \(\delta x\) = \(\delta y\) = 2.5 \(\text{m}\) and \(\delta z\)= 1.75 \(\text{m}\). Highly anisotropic grid spacing with coarse resolution suffices to reproduce the bulk cloud parameters (cloud fraction, liquid water path), but turbulence statistics are better reproduced as mesh resolution increases (Pedersen et al. 2016; Mellado et al. 2018).

Periodic lateral boundary conditions are used, and a sponge layer is defined in the vertical direction at the top 20% of the domain (from 1200 \(\text{m}\) to 1500 \(\text{m}\)). A constant \(SST\) of 300.18 \(\text{K}\), and a surface pressure of 1002.9 \(\text{hPa}\), derived from the observations, are used. The background (large-scale) vertical velocity is estimated using a horizontal wind divergence \(D=-4\times {10}^{-6} {\text{s}}^{-1},\) as it was determined from the ERA5 Global Reanalysis (Hersbach et al. 2020). For all simulations, the solar zenith angle, and hence the incoming solar energy, are kept constant in time at the observations local time (11:00 am). A constant surface albedo of \(0.08\) is considered across the domain and a value of the Coriolis parameter corresponding to the latitude \(\varphi\)=\({21.61}^{\text{o}}\)N. Assuming that the sea-spray particles which potentially act as CCN are already included, the CCN concentration is equal to 100 \({\text{cm}}^{-3}\). This is a typical value for marine air masses with no anthropogenic or continental effect (Allan et al. 2008; Wex et al. 2016). CCN are passively advected within the model domain and removed during droplet activation.

We performed three simulations with the CBLAST conditions, in order to examine the impact of sea spray on sensible and latent heat fluxes and their impact on turbulence statistics, MABL development and cloud properties. The first simulation (C) employs the COARE algorithm, which does not take into account the effect of sea spray. The other two simulations use Andreas' algorithm; the original one (O) with the neutral drag coefficient estimated from Andreas' parameterization and a modified version (M) with a constant neutral drag coefficient (\({C}_{d10n}\) = 0.0015) primarily for sensitivity testing. In order to avoid inaccurate spray fluxes in the M simulation, we employed the same spray-mediated heat transfer parameterization as in the O simulation. The reason for this is that the spray's transfer coefficients are estimated using the stress parameterization based on eddy-covariance measurements. Hence, whenever the stress parameterization is changed they need to be readjusted.

The simulations are compared using time-averages over the large eddy turnover time scale \({T}_{e}\), which is calculated by the ratio of \({Z}_{i}\) (the boundary layer height) and the friction velocity scale \({u}_{\star }\text{ as}\) \({T}_{e}\equiv {Z}_{i}/{u}_{\star }\) (Berg et al. 2020; Moeng and Sullivan 1994). The boundary layer height \({Z}_{i}\) is estimated every 30 \(\text{s}\) using the maximum gradient method (Sullivan et al. 1998) applied to the instantaneous horizontal-averaged virtual potential temperature \({\theta }_{v}\), which accounts for both pressure and water vapor variations.

A non-nudging approach has been used for continuous turbulence development. The total simulation's duration is limited to ~ 5 h to avoid simulated profiles drifting further away from the initial CBLAST profiles. This would lead to a further reduction (> 26% for O simulation) of the near surface wind speed and the sea spray impact. Alternatively, a much higher gradient wind speed would be required to balance the frictional drag and prevent the surface wind from decreasing, which is unrealistic and computationally inefficient (Ma and Sun 2021). To address this issue and limit non-stationarity effects, a 2 \(\text{h}\) spin-up (~ 4 large eddy turnovers, i.e. ~ 22,000 time-steps) has been set with prescribed constant surface fluxes (\({u}_{\star }\) = 0.4 \({\text{ms}}^{-1}\), \({H}_{s}\) = 5 \({\text{Wm}}^{-2}\), \({H}_{L}\) = 50 \({\text{Wm}}^{-2}\)) lower than the prevailing ones. That allows turbulence and convection to develop within the domain before MIMICA starts to apply the surface fluxes through the bulk algorithms. The three simulations continued for ~ 3 h (~ 10 large-eddy turnovers, i.e. ~ 70,000 time-steps) after the spin-up and MABL reached a quasi-steady state. We define this quasi-steady state as the time when the variables of most interest in this study, i.e. the wind speed and momentum flux near surface, reach an equilibrium value.

The simulation results are averaged during the last large eddy turnover. In this study, the height of 10 \(\text{m}\) is used as reference height to estimate the LES surface turbulent fluxes, and the length scale \({z}_{o}\) and the Obukhov length \(L\) are used to reduces the transfer coefficients to neutral stability. Table 1 shows the results of the averaged LES output and CBLAST data with a focus on fluxes and key meteorological parameters. Further analysis is provided in 3.1.1 and 3.1.2. It should be noted that the study focuses on a small region with hurricane-like conditions to investigate the impact of sea-spray fluxes under strong winds. A validation of the LES output against the available CBLAST flight measurements is not the scope of this study.

Table 1 N43RF mean values (French et al. 2007; Zhang 2007) and LES time-averaged values over \(1T_e\) at the end of the simulation time
Full size table

2.3 Spectral calculations

The initial velocity magnitude profile from CBLAST is aligned in the x direction, which has consequences for one-dimensional spectral properties (Berg et al. 2020). In this study the two-dimensional power spectral density from the LES is calculated using the fast Fourier transform method. The two wind components have been treated as scalar quantities assuming axisymmetric isotropy in the plane according to Kelly and Wyngaard (2006). Following Durran et al. (2017), the spectral density at \({k}_{h}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\) is calculated using an integration around a ring of radius \({k}_{h}\) centered at the origin of the wavenumber plane. The power spectral density of a variable integrated over \({k}_{h}\) gives the variance. We use an ensemble average of the power spectral density derived by computing one spectrum per minute for 15 min and then temporal averaging.

3 Results

3.1 Observations and LES evaluation

3.1.1 Vertical structure

The CBLAST averaged profiles of wind speed \(U\), potential temperature \(\theta\), water vapor mixing ratio \(q\), virtual potential temperature \({\theta }_{v}\), equivalent potential temperature \({\theta }_{e}\), cloud water mixing ratio \({q}_{c}\), together with the available observed leg-averaged values and the corresponding LES profiles are shown at Fig. 1. The LES parameters are horizontally and temporally averaged for one large eddy turn-over interval (\({T}_{e}\) = 1425, 1487 and 1584 \(\text{s}\) for C, O and M, respectively).

Fig. 1
figure 1

Horizontally and temporally averaged (time averaging is done for \(1T_e\)) profile of a wind speed \(U\); b potential temperature \(\theta\); c water vapor mixing ratio \(q\); d virtual potential temperature \(\theta_v\); e equivalent potential temperature \(\theta_e\); f cloud water mixing ratio \(q_c\). The shaded profiles areas indicate the standard deviation of the profiles, blue line indicates the N42RF averaged CBLAST profile during Hurricane Isabel on September 12, 2003 (Zhang and Drennan 2012), closed blue circles represent flight leg-averaged values taken by N43RF at 374, 286, 279, 194, 191, 137, 126, 95, 88 \({\text{m}}\) height, and open circles are the corresponding near-surface values (French et al. 2007)

Full size image

Τhe potential temperature profile used for initialization is composed by a nearly constant \(\theta\) layer below 500 \(\text{m}\) with a stable layer above it (Fig. 1b) (Zhang and Drennan 2012). This HBL structure is attributed to the moist processes in the wider storm area (latent heat release in clouds and rainfall evaporation below), as well as variations of radial advection of \(\theta\) with height (Kepert et al. 2016). According to the dropsonde’s averaged measurements the near surface atmosphere is warmer (\({\theta }_{10n}\) = 300.82 \(\text{K}\)) than the \(SST\) (300.18 \(\text{K})\). These observations are taken in the right rear quadrant of the storm, where intense ocean mixing causes sea surface cooling (Nolan et al. 2009). Furthermore, the initial equivalent potential temperature, \({\theta }_{e}\), a conserved variable in dry and moist adiabatic processes without precipitation, indicates moist convective instability, as it decreases with height (Fig. 1e). The equivalent potential temperature has been calculated using Bolton (1980) formula; \({\theta }_{e}=T{\left(\frac{1000}{P}\right)}^{0.2854(1-0.28\times {10}^{-3}q)}exp\left[(\frac{3.376}{{T}_{L}}-0.00254)(1+0.81\times {10}^{-3}q)\right]\), where \(T\), \(P\), \(q\) are the absolute temperature, pressure and water vapor mixing ratio at the initial level, respectively, and \({T}_{L}\) is the absolute temperature at the lifting condensation level.

The LES profiles evolved as the simulations progressed, resulting in a much deeper and better mixed layer than the initial one as evidenced by the vertical distributions of \(u\), \(\theta\), \(q,\) \({\theta }_{v}\) and \({\theta }_{e}\) (Fig. 1). This is consistent with previous LES studies conducted under conditions of severe winds and moderate air–sea heat fluxes, in which the boundary layer grows beyond \(1 \text{km}\) depth as the inversion is eroded by the strong winds (Bryan et al. 2017; Chen et al. 2021; Green and Zhang 2015). According to Chen et al. (2021), if the term of rotation and associated centrifugal acceleration were included in the simulations, the boundary layers would have been shallower. The three simulations estimate average wind speed in the interval of 25.2–26.7 \({\text{ms}}^{-1}\) (Table 1) at the layer of aircraft measurements, albeit they are within the low-end range of the N43RF leg-averaged wind speed values (24.3–35.2 \({\text{ms}}^{-1}\), Fig. 1a). Correspondingly, the simulated \({U}_{10n}\) values (17.9–19.8 \({\text{ms}}^{-1}\)) are at the lower-end range of the observed near surface wind speed values (16.5–24.5 \({\text{ms}}^{-1}\)). It is also interesting to mention that the near surface LES wind speed profiles deviate from the logarithmic law by up to − 21%, − 18% and − 16% for C, O, and M, respectively. There is also a step-like decrease of the fluxes below 10 \(\text{m}\) (Fig. 2a–c), indicating that the near-surface flow is poorly resolved (Maronga et al. 2020). Simulation C forms the coldest, driest, and shallowest MABL (1012 \(\text{m}\)), while O and M develop warmer and more humid layers of 1109 \(\text{m}\) and 1120 \(\text{m}\) height, respectively. Regarding the temperature values, those from O and M simulations are found to be higher than the observed ones, whereas those from C are in the lower range (Fig. 1b). This is due to the heat flux spray-mediated contribution as well as the higher heat budget caused by the enhanced entrainment of warmer air from aloft. The entrainment rates \(w_e = dZ_i /dt + DZ_i\), which have been determined according to Bretherton and Wyant (1997) and averaged throughout the simulations, have values of 0.034, 0.043 and 0.044 \({\text{ms}}^{-1}\) for C, O and M, respectively.

Fig. 2
figure 2

Time-averaged (\(1T_e\)) profile of a stress covariance \(\left| {\tau /\rho } \right|^{1/2}\); b vertical velocity and potential temperature covariance \(w^{\prime} \theta ^{\prime}\); c vertical velocity and water vapor mixing ratio covariance \(w^{\prime} q^{\prime}\). The shaded profiles areas indicate the standard deviation of the profiles, and blue circles represent leg-averaged values taken by N43RF (French et al. 2007; Zhang et al 2008)

Full size image

The sea-surface water vapor mixing ratio is estimated to be ~ 23 \({\text{g kg}}^{-1}\)(with 98% saturation), resulting in a 4.25 \({\text{g kg}}^{-1}\) initial air–sea difference. As a result, all simulations produce a moister layer than the observations, with O and M accumulating more water vapor than C due to spray-mediated contribution to the moisture flux (Fig. 1c). Shpund et al. (2012) found that when the relative humidity (RH) is low (< 90%), the sea spray evaporation moistens and cools the boundary layer, which is consistent with our simulations where the near surface RH is less than 85%. The moist conditions, aided by the conditional instability (i.e. negative \({\theta }_{e}\) and positive θ gradient at the same time), contribute to the development of a cloud layer (no drizzle is detected) in all simulations, although the initial cloud water mixing ratio, \({q}_{c}\), is zero. The mean cloud liquid water path (LWP) for C, O and M, however, is small at 0.95, 2.4 and 3.3\({\text{gm}}^{-2}\), despite a cloud layer depth of 228, 453 and 561 \(\text{m}\), respectively (see Sect. 3.3). The difference in the amount of condensed water accumulated between simulations M and O is due to the larger contribution of the spray component to the moisture flux in simulation M (see Sect. 3.1.2), as the entrainment and \(\theta\) structure of these simulations is nearly identical.

3.1.2 Fluxes

The surface momentum flux changes abruptly when the algorithms are employed after spin up, with \({u}_{\star }\) more than doubling its prescribed value, which is analogous to airflow encountering a new roughness zone (from smooth to rough). Due to the strong feedback between the momentum flux and the near surface wind speed, the wind velocity decreases exponentially during the early stages of the simulations, with C and M having the highest and lowest reductions, respectively, due to the different stress parameterizations. During that time-frame, the vertically integrated TKE grows linearly with time in response to the change in surface fluxes before flattening (not shown). The reduction of \({u}_{\star }\) and \({U}_{10n}\) during the averaging \({T}_{e}\) is less than 2.3% and 2%, respectively. Accordingly, the vertical momentum and heat fluxes distribution initially exhibit a high rate of change with height, gradually decreasing as the boundary layer evolves toward a quasi-steady state (not shown).

Figure 2a–c displays the time-averaged profiles of stress \({\left|\tau /\rho \right|}^{1/2}\), kinematic sensible heat \(\langle {w}^{\prime}\theta ^{\prime}\rangle\) and moisture \(\langle {w}^{\prime}q^{\prime}\rangle\) flux at the final stage of the simulation period, as well as the N43RF observed flux values. The averaged fluxes show a linear change with height from their surface values to their respective entrainment values in agreement with theory (Wyngaard 1992). Furthermore, the LES Reynolds stress varies slightly between the simulations (0.60–0.63 \({\text{ms}}^{-1}\)) and is at the low end of the observed range (0.55–1.26 \({\text{ms}}^{-1}\), Fig. 2a). However, the scatter of the measurements, due to measurements errors (sampling errors, correlation time lag errors, deviations of the aircraft altitude etc.), prohibits an explicit conclusion about the stress behavior at or near hurricane wind speeds (French et al. 2007; Drennan et al. 2007). Richter and Wainwright (2023) also report minor differences of the surface stress between spray-laden and unladen simulations in an LES framework which explicitly took sea spray particles into account. According to our simulations, stress increases from the aircraft level (88 and 374 m) to the surface by 3% and 17%, while the height-based correction of the observed stress to the surface is 10%–15% (French et al. 2007).

Α small downward surface sensible heat flux, which increases with height up to the middle of the cloud layer, is apparent in all simulations (Fig. 2b). The heat flux profile derived from the best fit line of the observations shows the same trend (Zhang et al. 2008). The air–sea temperature difference in simulation C is the smallest of the three, whereas its surface downward \(\langle {w}^{\prime}\theta^ {\prime}\rangle\) is the highest (Table 1). Within the evaporation layer, the spray-mediated sensible heat flux, which is considered by the Andreas’ algorithm in simulations O and M, has positive values of 0.012 and 0.015 \({\text{Kms}}^{-1}\) and compensates for the negative interfacial heat flux (− 0.023 and − 0.026 \({\text{Kms}}^{-1}\), respectively). The spray-mediated sensible heat flux is caused by two processes: the cooling of the saline spray droplets to obtain thermal equilibrium with the environment releasing sensible heat (Andreas 1995), and the evaporation of sea spray droplets consuming sensible heat (Andreas et al. 2008). Thermal equilibrium is reached when the sea spray droplets attain temperatures close to the wet-bulb temperature (~ 298.4 \(\text{K}\)) which is much lower than the air temperature at 10 \(\text{m}\) height.

For all simulations, the moisture flux vertical distribution is roughly constant with height up to the cloud base (Fig. 2c). Likewise, the measured best fit moisture flux profile is nearly constant with height (Drennan et al. 2007). While O and M are found to be near the middle of the observed range (0.052–0.136 \({\text{g kg}}^{-1}{\text{ms}}^{-1}\)), the moisture flux of C is slightly below due to the absence of the spray-mediated contribution. According to Andreas et al. (2015), the large spray heat fluxes typically correspond to near-neutral conditions with strong winds (\(\left|z/L\right|\to\) 0), where \(z\) is the measurement height. Since \(L\) is proportional to \({u}_{\star }^{3}\). the spray contribution is expected to be considerable for these conditions (i.e. near-neutral stability, high air temperature and strong winds) as \({H}_{L,sp}\) scales according to \({u}_{*}^{2.39}\) and the droplet’s evaporation rate increases exponentially with ambient temperature. Within the evaporation layer, the spray-mediated component of the moisture flux is, 0.026 and 0.038 \({\text{g kg}}^{-1}{\text{ms}}^{-1}\) for simulation O and M as the RH is low, which is consistent with Shpund et al. (2012). The interfacial component is 0.052 and 0.049 \({\text{g kg}}^{-1}{\text{ms}}^{-1}\), respectively. The difference in wind velocity (\(\Delta {U}_{10n}\) = 1.3 \({\text{ms}}^{-1}\)) is the main factor behind the discrepancy in \({H}_{L,sp}\) between O and M rather than the difference in water vapor (\({\Delta q}_{10n}\) = 0.4 \({\text{g kg}}^{-1}\)) or RH. Evaporation of spray droplets is also greater for M, due to the droplets longer residence time in the air under higher by 0.6 \(\text{m}\) significant wave height conditions (which is proportional to \({U}_{10n}\), according to Andreas and Wang 2007).

The surface virtual potential temperature flux, which is defined as \(\langle {w}^{\prime}{\theta }_{v}^{\prime}\rangle = \langle {w}^{\prime}{\theta }^{\prime}\rangle \left(1+0.61q\right)+0.61\theta \langle {w}^{\prime}q^{\prime}\rangle\), is initially positive in the O and M simulations, but evolves into a downward flux due to the considerable reduction of the spray flux towards equilibrium. In contrast, it is always downward in the C simulation which excludes spray contribution. Richter and Wainwright (2023) have reported that sea spray fluxes can modify the total fluxes under particular conditions as the excess of spray heat fluxes saturates the nearby surface layer and brings temperature and humidity values back into equilibrium. As a result, during the averaging \({T}_{e}\) time period, the stability parameter \(-{Z}_{i}/L\) has similar values in all three simulations (− 0.45, − 0.35, and − 0.38 for C, O, and M, respectively), which indicates near neutral conditions with negative buoyancy flux. This is consistent with the observational study of Zhang et al. (2008) and Zhang (2010) who found slightly stable conditions above the cold wake of Hurricane Isabel in the right rear quadrant. It should be noted that Barr et al. (2023) reproduced the cold wake in a coupled numerical study, which was a major source of variability of near surface thermodynamics when compared to the other storm quadrants.

3.2 Turbulence

3.2.1 Flow visualization

Visualization of the instantaneous turbulent quantities \({x}^{\prime}=x-\overline{x }\), where \(x\) is the instantaneous resolved quantity and \(\overline{x }\) the instantaneous horizontal average at a specific level, are shown in Figs. 3, 4, 5 and 6. The selected flow fields of the residual resolved stream-wise velocity (x-axis) \({u}^{\prime}\), vertical velocity component \(w{\prime}\), potential temperature \({\theta }^{\prime}\) and water vapor mixing ratio \({q}^{\prime},\) in the \(x\)-\(y\) (z = 0.2 \({Z}_{i})\) and \(z\)-\(y\) (x = 2500 \(\text{m}\)) planes are instantaneous snapshots obtained in the final stage of the simulation.

Fig. 3
figure 3

Contours of the instantaneous residual stream-wise velocity \(u^{\prime}\); 1st row, at 0.2 \(Z_i\) in the x–y plane for simulation a C; b O; c M; 2nd row, the same as the 1st row but for the z-y plane (\(x\) = 2500 m) the height is normalized by \(Z_i\). The dashed lines in a, b and c figures indicate the position of the cross sections shown in d, e and f figures, respectively, and vice versa

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Fig. 4
figure 4

Same as in Fig. 3, but for the instantaneous residual vertical velocity \(w^{\prime}\)

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Fig. 5
figure 5

Same as in Fig. 3, but for the instantaneous residual potential temperature \(\theta ^{\prime}\)

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Fig. 6
figure 6

Same as in Fig. 3, but for the instantaneous residual water vapor mixing ratio \(q^{\prime}\)

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Coherent streaky structures are closely aligned with the mean flow direction in simulation C (Fig. 3a). These elongated high and low-speed agglomerations are visible in Fig. 3d in the bulk of the layer (\(z\)<0.9 \({Z}_{i}\)). However, since they are related to shear (Lee et al. 1990) they become more cellular, lose intensity, and rotate in relation to the mean flow with increasing height (Fig. 10). They are the primary coherent features of sheared boundary layers (Robinson 1991), which have been reported in both numerical and observational hurricane boundary layer studies (Lorsolo et al. 2008; Chen et al. 2021).

The small horizontal streaks found in C are arranged into larger scale structures in simulations O and M (Fig. 3b, c) as a result of the combined effects of shear and enhanced latent heat flux. Figure 3c shows a well-defined structure with its axis parallel to the mean flow, spanning the entire domain lengthwise with a high-speed region width of around 1.5 \({Z}_{i}\). This structure is not spatially stationary in time and it is shifting to the left in relation to the x-axis. The results from simulation O lie between the other two simulations; the streaks are wider than in simulation C and the \({u}^{\prime}\) structure is less organized than in simulation M (Fig. 3b). This is also demonstrated by the magnitudes of the spectral density at the larger resolvable scales (Fig. 11). The structures lose coherence as height increases, yet they can be identified up to ~ 0.5 \({Z}_{i}\) for O and ~ 0.7 \({Z}_{i}\) for M (Fig. 3e, f). A visual inspection of the \(u{\prime}\), \({\theta }^{\prime}\) and \({q}^{\prime}\) fields (Figs. 3, 5 and 6) shows that the lower-speed regions (\({u}^{\prime}\)<0) are strongly correlated with the moister (\({q}^{\prime}\)>0) and colder ones (\({\theta }^{\prime}\)<0) due to the upward moisture flux as well as the downward heat flux. An opposite correlation was found between \(u{\prime}\) and \(\theta^ {\prime}\) variations by Khanna and Brasseur (1998) under near-neutral conditions but with an upward temperature flux. The correlation coefficient, which is defined as \({\gamma }_{x,y}=\langle x{\prime}y{\prime}\rangle /\langle {x}^{{\prime}2}\rangle \langle {y}^{{\prime}2}\rangle\), is high near the surface (\({\gamma }_{u,\theta }\approx\)+0.85 and \({\gamma }_{u,q}\approx\)-0.85) for all simulations, and decreases rapidly with height up to the middle of the layer for simulation C. For simulation M it remains strong up to the middle of the layer, likely due to the stronger scalar fluxes and the presence of the coherent \({u}^{\prime}\) structure (Fig. 12). The correlation coefficient for simulation O is found between the other two. The increased correlation values just below \({Z}_{i}\) show that the temperature inversion plays a similar role in the organization of streaks as a boundary (lid) like a surface does. A similar effect has been seen in LES simulations (Moeng and Rotunno 1990) for the skewness of vertical velocity near \({Z}_{i}\), and it was attributed to change of eddy shape due to the presence of the inversion. However, this was generally not seen in the observations (Moyer and Young 1991).

Near the surface, streaky small plumes are found in all simulations with comparable structures, which is also evidenced by their spectra (not shown). These streaky small plumes merge with the nearby ones as they rise, and become organized into larger elongated updrafts (shifting the spectral peak towards lower frequencies) with increased vertical velocities (Fig. 4). The buoyancy forces within the large coherent \(u{\prime}\) structures, where the scalars (moisture and temperature) are concentrated, are negligible (− 0.4 < -\(Z_i /L\)<-0.3). Hence they do not greatly differentiate the vertical velocity structure of O and M simulations with regard to C. This structure is in contrast to the one reported by Khanna and Brasseur (1998) and Moeng and Sullivan (1994) on moderately convective sheared boundary layers, but buoyant enough to drive large coherent sheet-like updrafts. Despite the fact that the typical vertical structure is streaky throughout the MABL in all simulations, M appears to produce wider scale vertical motions above mid-MABL, as the elongated updrafts appear to be suppressed/enhanced above the high/low speed \((u^{\prime} )\) region (Fig. 13c). This is corroborated by the fact that the spectral density in M is higher than the other two simulations at large resolvable scales (not shown).

3.2.2 Variances

The time-averaged variance profiles of the resolved wind speed components, potential temperature and water vapor mixing ratio, normalized by the surface layer scaling parameters (\(u_{{ \star }} , \theta_{{ \star }}\) and \(q_{{ \star }}\)), are presented in Fig. 7.

Fig. 7
figure 7

Horizontally and temporally averaged (time averaging is done for \(1T_e\)) normalized variance profiles of the resolved a stream-wise; b span-wise; c vertical component of wind speed; d potential temperature and e water vapor mixing ratio

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The variance of the horizontal velocity components takes its maximum value near the surface (\(z\)< 0.05 \(Z_i\)) due to wind shear and decreases with height (Fig. 7a, b). Similar non-dimensional variance profiles have been reported by Chen et al. (2021) and Nakanishi and Niino (2012). The peak variance of the cross-stream velocity is lower than the value of the stream-wise component since the initial velocity profile is considered to align in the x direction. In particular, the peak values of \(u^{{\prime}2} /u_{{ \star }}^2\) and \(v^{{\prime}2} /u_{{ \star }}^2\) profiles in simulations C, O and M, are 16.5, 17.5 and 19 and 3.7, 2.7 and 2.5, respectively. Simulation C shows the highest cross-stream variance, due to increased wind rotation induced by the frictional drag (not shown). Further, simulation M has a higher stream-wise velocity variance than the other two simulations up to 0.6 \(Z_i\), which enhances the corresponding profile of turbulent kinetic energy (not shown). The vertical velocity variance \(w^{{\prime}2} /u_{{ \star }}^2\) profile peaks (~ 1.20) at a higher level (~ 0.15 \(Z_i\)) for all simulations (Fig. 7c), which is consistent with the Monin–Obukhov scaling functions \(w^{{\prime}2} /u_{{ \star }}^2 = 1.25(1 + 0.2z/L)\) for near-neutral conditions according to Högström (1988).

Because of entrainment processes, the most intense fluctuations, as demonstrated by the visualization of the turbulent flow fields \(\theta^{\prime}\) and \(q^{\prime}\) (Fig. 5 and 6), occur around \(Z_i\). Hence, both variance profiles peak at this height for all simulations (Fig. 7d, e). Although simulation C has the lowest \(q^{{\prime}2}\) (not shown), it has the highest \(q^{{\prime}2} /q_{{ \star }}^2\) peak value due to the lower surface moisture flux compared to O and M (Fig. 7e). In all simulations, the temperature and water vapor variances decrease with height and become nearly constant below 0.2 \(Z_i\) and 0.7 \(Z_i\), respectively, reaching their lowest values near the surface. It should be noted that in simulation M similarly to the variance of the stream-wise velocity, the variances of potential temperature and water vapor mixing ratio have their largest values in the mixed layer compared to the other simulations. This is due to the convergence of warmer/moister air within the higher/lower speed zones of \(u^{\prime}\) coherent structure in simulation M.

The vertical velocity skewness \(\langle w^{{\prime3}}\rangle /\langle w^{{\prime2}}\rangle^{3/2}\) represents the symmetry in the turbulence structure. The nearly steady positive value inside the MABL (Fig. 14a), similarly to observations in stratocumulus cloud (Lenshcow et al. 1980; Moyer and Young 1991), suggests that updrafts are narrower/stronger than downdrafts. At the upper portion of the temperature inversion, vertical velocity skewness turns negative, which indicates that downdrafts are narrower and more intense than the updrafts. The negative temperature and positive humidity skewness suggest that downdrafts are dominated by colder and moister air (Fig. 14b, c). These negatively buoyant downdrafts may be produced by a small radiative cooling at cloud top (Fig. 14d). However, part of these downdrafts may be due to the head of folding thermal plumes, which have reached the MABL top from below and then they descend, leading to engulfment of warmer and drier air.

3.2.3 Quadrant analysis

A quadrant analysis of momentum, heat, and water vapor fluxes (Mahrt and Paumier 1984; Sullivan et al. 1998) is carried out to determine the differences found between simulations C and M (Fig. 8).

Fig. 8
figure 8

a Temporally averaged vertical profiles of quadrant component momentum fluxes normalized by the surface flux for simulation C (solid lines) and M (dashed lines); b the same as (a) but for heat flux; c the same as (a) but for moisture flux

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The lack of an entrainment peak in the momentum fluxes indicates the negligible net momentum transfer from the free atmosphere to the boundary layer (Fig. 8a). Up to the height of 0.6 \(Z_i\) where the \(u^{\prime}\) structure loses coherence, each component of M is larger than its corresponding component of C in absolute magnitude due to the increased turbulent kinetic energy. Above this level, only the difference between their outward interactions (\(w^{\prime} > 0, u^{\prime} < 0\)) remains noteworthy. However, the profile of total flux in both simulations is similar and peaks at the surface due to drag (Fig. 8a). Furthermore, the ejections (\(w^{\prime} > 0, u^{\prime} > 0\)) and sweeps (\(w^{\prime} < 0, u^{\prime} < 0\)) are roughly equal for both simulations, whereas the outward interactions are stronger than the inward ones (\(w^{\prime} < 0, u^{\prime} > 0\)).

The maximum total downward heat flux for both simulations is located at a height of about 0.8 \(Z_i\), as it is shown in Fig. 8b. The peak magnitude of the individual quadrant fluxes is found near \(Z_i\), where the potential temperature gradient is steepest and the corresponding fluctuations are most pronounced. The quadrant heat fluxes are largely self-canceling around \(Z_i\), and even though air is exchanged between the boundary layer and the free atmosphere, as the large individual quadrant fluxes show, there is no considerable net vertical heat transfer. Maronga and Li (2022) argued that gravity waves at the top of the boundary layer are responsible for strong quadrant component values and at the same time for absence of a pronounced net heat transport. In both simulations, cooler air moving upward contributes the most to the total heat flux followed by warmer air moving downward, with the other two components (\(w^{\prime} \theta^{\prime} > 0\)) contributing similarly (Fig. 8b).

Since the moisture supply is from below, the net moisture flux is upward, and the maximum flux in both simulations is at the surface (Fig. 8c). The peak magnitude of the outward and inward interactions (\(w^{\prime} q^{\prime} < 0\); Fig. 8c) appears near \(Z_i\), for both simulations. The MABL of M simulation is well-mixed and, thus, the ejections and sweeps (\(w^{\prime} q^{\prime} > 0\)) are roughly constant with height. In contrast, there are two peaks in the C simulation, one stronger in the region of 0.8–0.9 \(Z_i\) and the other one near the surface. The dominant eddy type contributing to the total profile is moist air moving upwards, followed by dry air moving downward. Similar to momentum, the magnitude of each individual component in M, especially the upward ones, appears to be strengthened in comparison to C simulation below 0.6 \(Z_i\), which results in a slightly greater net moisture transport in this regime (Fig. 8c).

3.3 Clouds and radiation

In this section the thermodynamic impact of sea-spray on cloud formation is examined. Large eddies can transport spray droplets upward, changing the thermodynamic structure of the mixed layer and forming clouds by condensation and collisions (Shpund et al. 2012, 2014). Note that investigation of the role of sea-spray as CCN is beyond the scope of this study and, thus, all simulations are performed with the same CCN conditions (see Sect. 2).

Simulation C produces scattered cloud structures (Fig. 9a), while simulations O and M, which are characterized by larger surface moist fluxes (Fig. 2c), produce an extensive cloud deck throughout the domain (Fig. 9b, c). In all simulations, the modeled cloud tops are confined by the MABL top, which is a typical structure of marine stratocumulus (Fig. 15). Thus, the development of the MABL is critical to cloud formation, growth and accumulation of liquid water. However, in the present study the development of the MABL may not be realistic due to the lack of consideration for storm-scale dynamics.

Fig. 9
figure 9

1st row, contours of the instantaneous liquid water path for simulation a C; b O; c M; 2nd row, the same as the 1st row but for the surface long-wave net radiative flux; 3rd row, the same as the 1st row but for the surface short-wave net radiative flux

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LWP values are very low, usually below 5 \({\text{gm}}^{ - 2}\) (8 \({\text{gm}}^{ - 2} )\) in C (O and M) simulations. Local LWP maxima do not exceed 22 \({\text{gm}}^{ - 2}\) in any simulation, with the largest values found in M (Fig. 9c). These differences in cloud liquid properties result in differences in the surface net long-wave (LWN) and short-wave radiation fluxes (SWN). The near cloud-free condition in simulation C is generally characterized by LWN values between − 40 and − 56 \({\text{Wm}}^{ - 2}\) (Fig. 9d). As clouds develop in simulations M and O, they affect the surface LWN balance by emitting long-wave radiation downwards, which partially offsetts the surface long-wave cooling. LWN values correlate with differences in LWP, with higher liquid amount resulting in increased LWN. In particular, clouds with LWPs < 8 \({\text{gm}}^{ - 2}\) usually correspond to LWNs between − 45 and − 30 \({\text{Wm}}^{ - 2}\), while optically thicker cloud structures with LWPs > 10 \({\text{gm}}^{ - 2}\) (mostly found in M simulation) can increase surface LWN up to − 10 \({\text{Wm}}^{ - 2}\) (Fig. 9c, f). SWN in simulation C varies between 890 and 950 \({\text{Wm}}^{ - 2}\) (Fig. 9g). As clouds form in simulations O and M, they reflect the incoming solar radiation reducing SWN locally (Fig. 9h, i). When LWP exceeds 8 \({\text{gm}}^{ - 2}\), changes in the surface shortwave budget become very prominent with SWN values falling below 850 \({\text{Wm}}^{ - 2}\). Overall, the mean surface net radiative fluxes (LWN + SWN) for C, O, and M are 870, 855 and 840 \({\text{Wm}}^{ - 2}\), respectively, which indicates that changes in the air–sea flux treatment can alter the surface radiation budget by 30 \({\text{Wm}}^{ - 2}\) through modifications in the cloud field.

4 Conclusions

In this study, the air–sea interaction mechanism has been incorporated into MIMICA LES code to study the MABL structure and cloud macro-physical properties under severe wind conditions. The air–sea interaction mechanism is implemented into the LES code via two bulk air–sea flux algorithms: Andreas' algorithm which explicitly account for the interfacial and the spray-mediated fluxes and COARE algorithm. Three high resolution simulations have been performed using the COARE algorithm (simulation C), the original Andreas algorithm (O) and a modified version (M) which reduces the shear stress and increases the scalar fluxes for the same wind speed compared to O.

The profiles used for model initialization are from the CBLAST-Hurricane campaign, which acquired turbulence flux data from Hurricane Isabel within the storm's cold wake on September 12, 2003. Measurements demonstrate friction velocity values with substantial scatter (0.55–1.26 \({\text{ms}}^{ - 1}\)), strong moisture fluxes (0.052–0.136 \({\text{g kg}}^{ - 1} \,{\text{ms}}^{ - 1}\)), and the sensible heat fluxes in the stable to near neutral regime (− 0.05 to − 0.0023 Km s−1).

All simulations produce a deeper well mixed layer and a decrease in wind speed values compared to the initial profiles. The values of the near-surface wind speed and the momentum flux are at the low end of the observed range and they differ only slightly between simulations. Sea spray fluxes decrease as the simulations proceed, primarily due to a decrease in near-surface wind speed and, secondarily, due to changes in near-surface thermodynamic conditions. To sustain large spray fluxes, a mechanism should be incorporated such as advection, strong updrafts or water phase transition which could effectively remove moisture and heat from the surface layer and prevent saturation by the enhanced fluxes (Richter and Wainwright 2023).

Spray-mediated sensible heat flux is opposite to the interfacial flux and accounts for up to 60% of its magnitude. Thus, sea spray distinguishes the O and M simulations thermodynamically by increasing the potential temperature profiles compared to C. This is mainly due to the increased TKE production when spray is considered, which promotes MABL growth and entrainment of warmer air from aloft. In all simulations, the total sensible heat flux is downward, (but there is conditional convective instability) which is less than the observed mean but within the observed range of values. For the same simulations O and M, sea spray contributes considerably (up to 45%) to the total moisture flux and produces more humid layers than simulation C. When spray is included, the total moisture flux is in the middle of the observed range of values but it is slightly lower when it is not included.

Due to the spray-mediated latent heat flux, the O and M simulations develop large-scale structures of the residual stream-wise velocity found in ABLs driven by shear and buoyancy. On the other hand, simulation C shows streaky structures throughout the MABL which are usually observed in neutral shear-dominated layers. For all simulations, lower-speed regions are strongly correlated with moister and colder regions in the surface layer, but only simulation M with the increased scalar fluxes preserves this up to the mid-MABL. The vertical velocity structure below the mid-MABL is not greatly influenced by the scalars concentrations within the large-scale structures of the residual stream-wise velocity, because buoyancy forces are negligible. However, differences are observed above that level.

The horizontal velocities variance is maximum near the surface due to shear, while the vertical velocity variance peaks at a height of about 0.15 \(Z_i\). The most intense fluctuations of potential temperature and water vapor mixing ratio occur near \(Z_i\) due to the entrainment process. Turbulence intensity, and individual momentum and moisture quadrant fluxes are stronger in simulation M simulation than those of simulation C, up to the point where the large-scale structures of the residual stream-wise velocity loses coherence. At the same time, while the normalized total momentum flux of M is equivalent to that of C, its net moisture transfer efficiency is increased 5–10% above the surface layer.

Simulation C produces scattered cloud structures throughout the domain with extremely low LWP (< 5 \({\text{gm}}^{ - 2}\)), while simulations M and O produce more extensive cloud decks. The highest liquid content is produced in simulation M, which is also characterized by the largest surface moisture flux. Despite the fact that all simulations produce optically-thin clouds, the differences in cloud properties have an impact on surface radiation. The simulation with the optically-thicker clouds (M) results in increased downward longwave emission and reduced shortwave radiation reaching the surface. Thus, the net effect on total surface radiation is a decrease of 30 \({\text{Wm}}^{ - 2}\) (15 \({\text{Wm}}^{ - 2}\)) compared to simulation C (O). These results indicate that modeling the thermodynamic impact of spray-mediated fluxes can considerable affect cloud characteristics and radiation patterns, which has substantial implications for climate studies.

The simulations used a value of 100 \({\text{cm}}^{ - 3}\) for CCN concentration, which we consider to be a typical maritime value for the sea-spray particles that potentially act as CCN. According to Xu et al. (2022), the spray contribution to marine CCN is underestimated under strong wind conditions. A dynamic production of sea spray particles would be a more realistic contribution from this CCN source. However, this should be complemented by further research, including atmospheric chemistry in order to better understand the interactions of sea spray aerosol-derived CCN with other aerosols.