Sea-State-Dependent Sea Spray and Air–Sea Heat Fluxes in Tropical Cyclones: A New Parameterization for Fully Coupled Atmosphere–Wave–Ocean Models

Benjamin W. Barr aUniversity of Washington, Seattle, Washington

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Shuyi S. Chen aUniversity of Washington, Seattle, Washington

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Christopher W. Fairall bPhysical Sciences Laboratory, National Oceanic and Atmospheric Association, Boulder, Colorado

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Abstract

Air–sea exchange in high winds is one of the most important but poorly represented processes in tropical cyclone (TC) prediction models. Effects of sea spray on air–sea heat fluxes in TCs are particularly difficult to model due to complex sea states and lack of observations in extreme wind and wave conditions. This study introduces a new sea-state-dependent air–sea heat flux parameterization with spray, which is developed using the Unified Wave Interface–Coupled Model (UWIN-CM). Impacts of spray on air–sea heat fluxes are investigated across a wide range of winds, waves, and atmospheric and ocean conditions in five TCs of various sizes and intensities. Spray generation with variable size distribution is explicitly represented by surface wave properties such as wave dissipation, significant wave height, and dominant phase speed, which may be uncorrelated with local winds. The sea-state-dependent spray mass flux is substantially different than a wind-dependent flux, especially when wave shoaling occurs with enhanced wave dissipation near the coast during TC landfall. Spray increases the air–sea enthalpy flux near the radius of maximum wind (RMW) by approximately 5%–20% when mean 10-m wind speed at the RMW reaches 40–50 m s−1. These values can be amplified significantly by coastal wave shoaling. Spray latent heat fluxes may be dampened in the eyewall due to high saturation ratio, and they consistently produce a moistening and cooling effect outside the eyewall. Spray strongly modifies the total sensible heat flux and can cause either a warming or cooling effect at the RMW depending on eyewall saturation ratio.

Significance Statement

Fluxes of heat and moisture from the ocean to the atmosphere are important for hurricane intensification, but the impact of sea spray generated by breaking waves on these fluxes is not well understood. We develop a new model for heat fluxes with spray that accounts for how waves control spray, and we apply this model to a set of five simulated hurricanes to better understand the broad range of ways that spray impacts heat fluxes in high wind conditions. We find that spray significantly affects heat fluxes in hurricanes and that impacts are strongly controlled by waves, which are not always correlated to winds. This research improves our understanding of how spray affects heat fluxes in hurricanes and provides a foundation for future studies investigating sea spray and its impacts on high-impact weather systems.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Benjamin Barr, bwbarr@uw.edu

Abstract

Air–sea exchange in high winds is one of the most important but poorly represented processes in tropical cyclone (TC) prediction models. Effects of sea spray on air–sea heat fluxes in TCs are particularly difficult to model due to complex sea states and lack of observations in extreme wind and wave conditions. This study introduces a new sea-state-dependent air–sea heat flux parameterization with spray, which is developed using the Unified Wave Interface–Coupled Model (UWIN-CM). Impacts of spray on air–sea heat fluxes are investigated across a wide range of winds, waves, and atmospheric and ocean conditions in five TCs of various sizes and intensities. Spray generation with variable size distribution is explicitly represented by surface wave properties such as wave dissipation, significant wave height, and dominant phase speed, which may be uncorrelated with local winds. The sea-state-dependent spray mass flux is substantially different than a wind-dependent flux, especially when wave shoaling occurs with enhanced wave dissipation near the coast during TC landfall. Spray increases the air–sea enthalpy flux near the radius of maximum wind (RMW) by approximately 5%–20% when mean 10-m wind speed at the RMW reaches 40–50 m s−1. These values can be amplified significantly by coastal wave shoaling. Spray latent heat fluxes may be dampened in the eyewall due to high saturation ratio, and they consistently produce a moistening and cooling effect outside the eyewall. Spray strongly modifies the total sensible heat flux and can cause either a warming or cooling effect at the RMW depending on eyewall saturation ratio.

Significance Statement

Fluxes of heat and moisture from the ocean to the atmosphere are important for hurricane intensification, but the impact of sea spray generated by breaking waves on these fluxes is not well understood. We develop a new model for heat fluxes with spray that accounts for how waves control spray, and we apply this model to a set of five simulated hurricanes to better understand the broad range of ways that spray impacts heat fluxes in high wind conditions. We find that spray significantly affects heat fluxes in hurricanes and that impacts are strongly controlled by waves, which are not always correlated to winds. This research improves our understanding of how spray affects heat fluxes in hurricanes and provides a foundation for future studies investigating sea spray and its impacts on high-impact weather systems.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Benjamin Barr, bwbarr@uw.edu

1. Introduction

Air–sea heat fluxes are widely recognized to play a key role in tropical cyclone (TC) intensification (e.g., Shapiro and Willoughby 1982; Emanuel 1995), but they are difficult to observe in high wind conditions (i.e., 10-m wind speed U1020ms1) (Zhang et al. 2008; Drennan et al. 2007) and are not well-represented in models (Sroka and Emanuel 2021a). The existing bulk algorithms for air–sea sensible and latent heat fluxes (SHFs and LHFs) represent vertical transport of heat and moisture by turbulent eddies using Monin–Obukhov (MO) similarity theory (Monin and Obukhov 1954) but do not address the important contribution of sea spray (e.g., Fairall et al. 2003). In high winds, sea spray ejected from breaking waves introduces a new avenue for heat transfer, i.e., cooling and evaporation of droplets (Veron 2015; Richter and Veron 2016), which is thought to substantially modify air–sea heat fluxes (Andreas 1992; Fairall et al. 1994, hereafter F94; Andreas and Emanuel 2001; Zhao et al. 2006; Bao et al. 2011; C. W. Fairall et al. 2014, unpublished report; Mueller and Veron 2014b; Richter and Stern 2014; Andreas et al. 2015; Troitskaya et al. 2018b; He et al. 2018; Sroka and Emanuel 2021b) in ways that promote TC intensification (Kepert et al. 1999; Bao et al. 2000; Wang et al. 2001; Cheng et al. 2012; Zhang et al. 2017; Garg et al. 2018; Prakash et al. 2019; Xu et al. 2021b).

Understanding and modeling of spray heat transfer has progressed in recent decades, but the crucial control of spray generation and heat transfer in high winds by the physics of breaking waves and wave dissipation related to sea state remains largely unresolved. Pioneering work on spray production (e.g., Monahan et al. 1986), which largely did not address high winds, recognized the importance of waves, whitecaps, air entrainment, and bubble bursting in spray production, but these effects were generally parameterized in terms of U10. This work gave rise to the most commonly used model for spray generation (F94), which prescribes total spray mass flux using a U10-based whitecap fraction and defines a universal (i.e., not wind- or sea-state-dependent) droplet size distribution based on observations. Recent laboratory studies (Veron et al. 2012; Troitskaya et al. 2017, 2018a; Ortiz-Suslow et al. 2016, hereafter OS16; Bruch et al. 2021) have illuminated the complex dynamics of spray generation by breaking waves in high winds, demonstrating both control of spray generation by wave processes and the droplet distribution’s ability to change with conditions. A number of wave-based spray generation models have recently been proposed; some of these allow the droplet size distribution to vary (Fairall et al. 2009; Mueller and Veron 2009; Troitskaya et al. 2018a), while others do not (Zhao et al. 2006; Zhang et al. 2017; Xu et al. 2021a). These models have large uncertainties and stand to benefit from further development and calibration based on ever-improving observational datasets.

In addition to their direct role in spray generation, the manner in which dynamic, spatially varying wave fields in TCs modulate spray heat fluxes is essentially unknown. Sea states in TCs are generally understood as a superposition of local wind seas and swell propagating from the high-wind region of the storm. Wave fields in translating TCs are highly asymmetric, with resonant growth of swell aligned with winds on the right side producing the highest and longest waves in the front-right (FR) quadrant (Wright et al. 2001; Chen et al. 2013; Zhang and Oey 2019; Tamizi and Young 2020; Northern Hemisphere convention used throughout). On the left side of the storm, unaligned swell and wind-sea produce smaller, shorter waves (Chen and Curcic 2016). Open ocean patterns are disrupted during storm landfall by processes such as wave shoaling, refraction, and offshore flow over very short fetch (Chen and Curcic 2016). In deep water, waves break by “spilling” and release about one-third of their energy to dissipation; when waves shoal in shallow water, they break by “plunging” and dissipate energy much more quickly (Donelan et al. 2012).

The effects of sea state on spray generation and air–sea heat fluxes with spray have not been systematically investigated across a wide range of conditions in TCs. In fact, control of heat fluxes with spray by storm-scale surface thermodynamics is also largely unknown, since previous studies of air–sea heat fluxes with spray (e.g., Andreas 1992; Mueller and Veron 2014b; Andreas et al. 2015; Troitskaya et al. 2018b) typically evaluate a limited set of test cases. Parametric variations that have been performed suggest that spray effects depend strongly on near-surface thermodynamics. For example, Andreas et al. (2015) varied relative humidity from 75% to 100% at U10 = 25 m s−1 and found that the spray latent heat flux varied from over 350 W m−2 to below 0 W m−2. Without knowledge of storm-scale variations in surface conditions to put these values in context, it is difficult to evaluate the true impact of spray on air–sea heat fluxes.

This study aims to better understand and improve model representation of sea-state-dependent air–sea heat fluxes with spray by developing and testing a new parameterization across a wide range of TC conditions found in model output of fully coupled atmosphere–wave–ocean (AWO) TC simulations, allowing us to characterize control of heat fluxes with spray by sea state and surface thermodynamics. We focus herein on characterizing air–sea heat fluxes with spray in complex TC conditions by exercising our new parameterization on output from coupled AWO model simulations of five TCs with various intensities; effects of spray-mediated air–sea heat fluxes on TC structure and intensity in the fully coupled model will be addressed in a future study. The coupled model and new spray heat flux parameterization are described in sections 2 and 3, respectively. In section 4, we characterize sea-state-based spray generation and heat fluxes in TCs, starting first with a single simulation of a translating TC in the open ocean and then expanding our study to explore spray impacts on heat fluxes across a diverse set of five TC simulations. In section 5, we summarize our conclusions.

2. Coupled model

The coupled model used for the TC simulations in this study is the Unified Wave Interface–Coupled Model (UWIN-CM), a fully coupled AWO model (Chen et al. 2013; Chen and Curcic 2016). The atmospheric component of UWIN-CM is the Weather Research and Forecasting Model (WRF-ARW) (Skamarock et al. 2008), v3.9, which was run with nested domains of 12-, 4-, and 1.3-km horizontal resolution and 44 vertical levels. Inner nests were set as vortex-following where appropriate. The surface-layer scheme is based on MO theory with scalar roughness lengths per Garratt (1992), and the boundary layer parameterization is the Yonsei University scheme (Hong et al. 2006). The surface wave component of UWIN-CM is the University of Miami Wave Model (UMWM) (Donelan et al. 2012), v2.1, which was run with 4-km horizontal resolution, 32 or 36 directional bins, and 37 frequency bins from 0.0313 to 2.0 Hz. The ocean component of UWIN-CM is the Hybrid Coordinate Ocean Model (HYCOM) (Wallcraft et al. 2009), v2.2.99, which was run with 0.04° (∼4-km) horizontal resolution and 41 or 32 vertical levels.

We performed simulations for five TCs. These are Hurricanes Harvey (2017), Florence (2018), Michael (2018), and Dorian (2019) and Typhoon Fanapi (2010). The simulations were initialized and concluded at 0000 UTC 24 August and 1200 UTC 26 August 2017 for Harvey, 0000 UTC 10 September and 1200 UTC 14 September 2018 for Florence, 0600 UTC 8 October and 0000 UTC 11 October 2018 for Michael, 0000 UTC 28 August and 0000 UTC 6 September 2019 for Dorian, and 0000 UTC 15 September and 0000 UTC 20 September 2010 for Fanapi. Initial and boundary conditions (ICs and BCs) for WRF come from European Centre for Medium-Range Weather Forecasts ERA5 0.25° data (Michael, Dorian) and National Centers for Environmental Prediction Global Forecasting System 0.25° forecast fields (Harvey, Florence, Fanapi). ICs and BCs for HYCOM come from 1/12° Global Ocean Forecast System, version 3.0 and 3.1, global HYCOM model analysis (all TCs). Simulations for Florence, Michael, Dorian, and Fanapi were initialized using a relocated vortex (Lin et al. 2018). Because of the long duration of the Hurricane Dorian simulation, it was necessary to reinitialize the large-scale atmosphere at 1200 UTC 30 August 2019 to improve the environmental steering flow affecting the storm track along the U.S. Atlantic coast. The simulation of Hurricane Michael underestimated the observed rapid intensification prior to landfall, which is a well-known issue in TC forecasting (e.g. Cangialosi 2020). To mitigate potential error in the ocean initial condition, we decoupled feedback of sea surface temperature from the ocean to the atmosphere after 2100 UTC 8 October 2018, which produced a realistic intensification.

3. Air–sea heat flux parameterization with spray

The new air–sea heat flux parameterization presented in this study predicts the total SHF and LHF (HS,1 and HL,1, respectively) by modeling the physical processes of spray generation, spray transport and heat transfer to the near-surface flow, and feedback between spray heat fluxes and the near-surface environment (Fig. 1a). Spray generation is sea-state based and is adapted from Fairall et al. (2009); we update several physical assumptions and scalings of this earlier model and calibrate it using available field and laboratory observations. Spray heat fluxes are calculated using radius-specific time scales for droplet cooling, evaporation, and settling, as is commonly done (Andreas 1989, 1990, 1992, 2005; F94; Andreas and DeCosmo 1999). Near-surface feedback is addressed by modeling the vertical divergence of turbulent heat fluxes due to spray within a MO model of the surface layer, building on earlier work (Andreas et al. 1995; Andreas 2004; Bao et al. 2011; C. W. Fairall et al. 2014, unpublished report; Mueller and Veron 2014b).

Fig. 1.
Fig. 1.

(a) Fluxes and physical processes addressed by our new air–sea heat flux parameterization with spray. Fluxes HS,drop, Hwb,drop, and HR,drop are the droplet-specific SHF, heat flux due to temperature change from air temperature T to salt-adjusted wet-bulb temperature Twb, and heat flux due to size change, respectively. These heat fluxes are integrated over the sea spray generation function to obtain HS,spr, Hwb,spr, and HR,spr, which are defined in the text. HS,0 and HL,0 are the SHF and LHF at the surface in the presence of spray, and HS,1 and HL,1 are the total SHF and LHF with spray produced by the parameterization. (b) Fluxes and variables for sample grid cells in the lowest two layers of the WRF Model. Subscripts 1 and 2 on variables U, θ, q, and z indicate values at the cell centers (i.e., mass points) of the lowest and second-lowest layers of cells, respectively. Fluxes HS,12 and HL,12 are the vertical SHF and LHF from the lowest to second-lowest layers, and fluxes HS,23 and HL,23 are the vertical SHF and LHF from the second-lowest to third-lowest layers.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Inputs to the parameterization are the wave energy dissipation flux (ε), significant wave height (HS), dominant phase speed (Cp,d), mean squared wave slope (sm), and friction velocity (u), which come from the wave model; ocean surface temperature (T0), which comes from the ocean model; and surface pressure (P0) plus horizontal wind speed, potential temperature, specific humidity, and height at the lowest model mass level (LML) of the atmospheric model (U1, θ1, q1, and z1, respectively). In WRF-ARW, model mass points are at gridcell centers, so height z (from geopotential) and components of horizontal wind speed U must be vertically and horizontally unstaggered, respectively, to obtain values at mass points. With U1, θ1, q1, and z1 defined at the LML, the parameterization produces HS,1 and HL,1, which are applied at the bottom boundary of the lowest layer of cells and used to calculate the vertical flux divergence at the LML during the WRF Model integration (Fig. 1b).

a. Spray generation

We define generation of spray spume droplets using a sea-state-based sea spray generation function (SSGF) that assumes that droplets are formed by converting a small portion of the turbulent kinetic energy (TKE) within breaking wave crests into surface potential energy of spray droplets (Fairall et al. 2009). Note that we neglect film and jet droplets in our model as these are produced by a fundamentally different wave process (i.e., bursting of entrained bubbles) and are broadly considered to have minimal impact on spray mass and heat fluxes (e.g., Andreas 1992). After formation, a fraction of the produced droplet population is ejected from the wave crest and enters the surface flow, with the probability of ejection determined by the ballistics of the droplets at the wave crest. The SSGF in terms of mass, dm/dr0, is
dmdr0=C1fsρswε˜r0WSS3σsurfexp[32C2αk(πηkr0)4/3]×12[1+erf(Uh,relυgC3smC4σhC5)].
The factors on the right to the left and right of the × sign are the size distribution of droplets produced from TKE and the droplet ejection probability, respectively; C1 through C5 and fs are model coefficients, ρsw is the density of seawater, ε˜ is the volumetric kinematic dissipation rate under actively breaking whitecaps [see Brumer et al. (2017) for an observationally driven treatment of whitecaps in high winds], r0 is the droplet radius at formation, WSS is a sea-state-based estimate of the actively breaking whitecap fraction, σsurf is the ratio of surface tension to density for water, αk is the Kolmogorov constant, and ηk is the Kolmogorov microscale on the water side based on ε˜. υg is the droplet gravitational settling velocity, sm is the mean squared wave slope, σh is the standard deviation of the horizontal turbulent wind speed at a gust height hgust, and Uh,rel = Uh − 0.8Cp,d is the ejected droplet velocity relative to the moving wave crest, with Uh as the horizontal wind speed at hgust and the factor of 0.8 per Banner et al. (2014). We define ε˜ based on measurements and scaling from Sutherland and Melville (2015) as follows:
ε˜=CdissεHSρswWSS.
Here Cdiss is a nondimensional constant that we select as 102 based on Fig. 11 of their paper. Note that substituting (2) into (1) eliminates WSS from the multiplicative group preceding the exponential but that the sea-state-based SSGF still depends on WSS through ηk (see appendix).
We compare our sea-state-based SSGF to an updated wind-based model based on F94, defined as
dmdr0=fsWwi(dmdr0)F94.
Here (dm/dr0)F94 is the F94 droplet spectrum per unit whitecap [implemented according to Mueller and Veron (2014b), section 3c(2)] and Wwi is a wind-based whitecap fraction parameterization.

Additional variable definitions and calibration of model coefficients are described in the appendix.

b. Spray heat fluxes

1) Spray heat fluxes due to temperature and size change

Detailed microphysical models (Andreas 1989, 1990, 1992) have shown that temperature change is roughly three orders of magnitude faster than size change for spume droplets, so that droplet cooling and evaporation processes are effectively decoupled. Spray droplet heat transfer may thus be approximated as 1) cooling/warming at constant r0 from T0 to the salt-adjusted wet-bulb temperature Twb of the air, followed by 2) evaporation/condensation at constant Twb to the droplet’s equilibrium radius req. Andreas (1989, 1990, 2005) approximated these processes as
Tdrop,f=Twb+(T0Twb) exp(τfτT),
rf=req+(r0req) exp(τfτR),
where Tdrop,f and rf are the droplet reentry temperature and radius, and τT, τR, and τf are characteristic time scales for droplet cooling, evaporation, and settling, respectively. From this, the spray heat fluxes into the near-surface flow due to temperature change (HT,spr) and size change (HR,spr) for the entire droplet population are
HT,spr=cp,sw(T0Tdrop,f)dmdr0dr0,
HR,spr=Lυ[1(rfr0)3]dmdr0dr0,
where cp,sw and Lυ are the specific heat capacity and latent heat of vaporization of seawater.
Similar to Fairall et al. (1990), we define req as
reqr0=[xs(1+νΦsMwMs1s)]1/3.
Here ν is the number of ions into which NaCl dissociates, Φs is the practical osmotic coefficient, Mw and Ms are molecular weights of water and salt, respectively, and xs is the mass fraction of salt in seawater. sq/qsat,0(T) is the saturation ratio of air with specific humidity q, temperature T, and saturation specific humidity over a plane surface of pure water qsat,0(T).
We define the thermodynamic time scales τT and τR by scaling the classical equations for droplet temperature and radius change (e.g., Pruppacher and Klett 1997), obtaining
τT=ρswcp,swr023kafυ,
τR=ρswr02ρaDafυqsat,0(T)β(1+y0s).
Here ka is the thermal conductivity of air, fυ is the mean ventilation coefficient, ρa is the air density, Da is the diffusivity of water vapor in air, and β is the wet-bulb coefficient. The parameter y0 accounts for the effect of salinity on saturation vapor pressure and equals −0.021 for typical surface seawater, so that air is saturated with respect to ejected spray droplets when s = 0.979.
We define τf in the usual way (Andreas 1992; F94; Andreas et al. 2015) as
τf=δυg,
where δ is the spray-layer thickness. We choose δ = HS based on Mueller and Veron (2014a); this study used detailed numerical simulations to suggest that the model τf = HS/υg may be incorrect by a factor between 0.5 and 2 but that HS is nonetheless the correct scaling for δ. The value of δ is not allowed to be greater than z1.

Finally, we define radius-specific heights for extracting ambient conditions (T, q) for spray calculations (i.e., specific heights for each droplet size that are appropriate for its thermodynamic time scales) following Peng and Richter (2019), which allows us to account for reheating and regrowth by condensation of small droplets as they reenter the ocean. More details on droplet reheating and regrowth and additional variable definitions are given in the appendix.

2) Spray specific available energy and heat transfer efficiency

It is useful to regroup terms in HT,spr and HR,spr to clarify how conditions and processes govern fluxes. First, we define specific available energies aT and aR for HT,spr and HR,spr, respectively, as
aT=cp,sw(T0Twb,10N),
aR=Lυ{1[(reqr0)10N]3}.
Here the subscript 10N indicates calculation using equivalent neutral 10-m variables, and the prime indicates that there are no spray feedback effects in the calculations. The specific available energies estimate the maximum energy extractable per unit mass of spray in terms of local conditions.
Next, we define ET(r0) and ER(r0) as dimensionless, radius-dependent droplet heat transfer efficiencies for temperature and size change, respectively:
ET(r0)=1aTcp,sw(T0Tdrop,f),
ER(r0)=1aRLυ[1(rfr0)3].
Plugging (9) and (10) into (5) gives
HT,spr=aTETdmdr0dr0,
HR,spr=aRERdmdr0dr0;
ET and ER represent how well transport and thermodynamic processes transfer the available energy at each radius to the near-surface flow.
Next, we define Mspr, the generated spray mass flux, as
Mspr=dmdr0dr0.
Finally, we define dimensionless mean spray heat transfer efficiencies E¯T and E¯R as
E¯T=1MsprETdmdr0dr0,
E¯R=1MsprERdmdr0dr0;
E¯T and E¯R weight ET and ER by the droplet sizes within the SSGF, providing a single number characterizing the efficiency of heat transfer for the droplet population as a whole. Plugging (13) into (11) gives
HT,spr=aTE¯TMspr,
HR,spr=aRE¯RMspr.
Thus, spray heat fluxes are determined by the mass of spray generated (Mspr), the energy available within the spray (aT and aR), and how well the near-surface flow extracts that energy from the droplet sizes present (E¯T and E¯R).

3) Spray sensible heat, latent heat, and enthalpy fluxes

The spray SHF and LHF, HS,spr and HL,spr, respectively, are in general not equal to HT,spr and HR,spr, respectively. This is because if T is between T0 and Twb, then the droplet temperature change from T to Twb exchanges droplet sensible heat for latent heat, producing a latent heat flux Hwb,spr. The Hwb,spr portion of HT,spr must be repartitioned with HR,spr to calculate HS,spr and HL,spr.

The energy for evaporation (or, if s > 0.979, from condensation) in HR,spr is taken from (is rejected to) the sensible heat of the air and must be subtracted from (added to) HS,spr to determine the net spray SHF, HSN,spr. Finally, the spray enthalpy flux HK,spr is the sum of HSN,spr and HL,spr, which is HT,spr, in accordance with Andreas and Emanuel (2001). These relationships are summarized as follows:
HS,spr=HT,sprHwb,spr,
HL,spr=HR,spr+Hwb,spr,
HSN,spr=HS,sprHR,spr,
HK,spr=HSN,spr+HL,spr=HT,spr.

c. Air–sea heat fluxes with spray and near-surface feedback

We calculate total heat fluxes with spray, including feedback, using a simple model for sensible and latent heat transfer through a MO surface layer experiencing vertical divergence of turbulent heat fluxes due to spray. This model is derived in the appendix, and final expressions for heat fluxes and feedback are presented below.

Total SHF and LHF, including both spray and turbulent effects, are HS,1 and HL,1, respectively, defined as
HS,1=HS+γS(HS,sprHR,spr),
HL,1=HL+γLHL,spr,
γS=ln(δz0t)ΨH,δ1+φH,δln(z1z0t)ΨH,1,
γL=ln(δz0q)ΨH,δ1+φH,δln(z1z0q)ΨH,1.
Here HS and HL are the turbulent interfacial SHF and LHF without spray, respectively, which are
HS=ρacp,aκu*(θ0θ1)ln(z1z0t)ΨH,1,
HL=ρaLυκu*(q0q1)ln(z1z0q)ΨH,1;
z0t and z0q are thermal and moisture roughness lengths (Garratt 1992), ΨH is the traditional integrated stability function for heat, and φH is a new term that is the analog of ΨH for a layer with volumetric heating. cp,a is the specific heat capacity of air, κ is the von Kármán constant, and θ is potential temperature. Subscripts 0, δ, and 1 indicate that variables or functions are evaluated at the surface, height δ, and LML, respectively. γS and γL are feedback coefficients that address the “geometric” resistance to spray heat fluxes through the surface layer. For realistic TC conditions (i.e., ΨH,δ, ΨH,1, and φH,δ are small compared to the other terms), γS and γL are always less than 1.0.
In addition to the effects of γS and γL, feedback modifies HS,spr, HR,spr, and HL,spr directly by changing the vertical θ and q profiles within the spray layer. We account for this by iteratively calculating spray heat fluxes (15) and modified profiles [(A11a) in the appendix] until they converge. The feedback effects between spray heat fluxes and vertical profiles are expressed by feedback coefficients αS, βS, and βL as follows:
αS=HS,sprHS,spr,
βS=HR,sprHR,spr,
βL=HL,sprHL,spr.
Here primed and nonprimed fluxes are calculated without and with spray modifications to vertical profiles, respectively.
Finally, we define 10-m neutral transfer coefficients for sensible heat (Ch,10N), latent heat (Cq,10N), and enthalpy (Ck,10N) below. Note that these definitions are based on the total fluxes HS,1 and HL,1 and apply to cases both with and without spray (in the latter case, HS,1=HS and HL,1=HL):
Ch,10N=HS,1ρacp,aU10N(T0T10N),
Cq,10N=HL,1ρaLυU10N(q0q10N),
Ck,10N=HS,1+HL,1ρaU10N[cp,a(T0T10N)+Lυ(q0q10N)].

4. Sea-state-dependent heat fluxes with spray in tropical cyclones

a. Sample conditions from Hurricane Dorian (2019)

We first characterize sea-state-dependent air–sea heat fluxes with spray using sample conditions from a UWIN-CM simulation of Hurricane Dorian (Avila et al. 2020). We select for analysis a 12-h period from 0000 to 1200 UTC 1 September (called Dor-O) when the storm is moving westward in the open Atlantic Ocean. Advanced Scatterometer (ASCAT) 0.25° satellite swath (Figa-Saldaña et al. 2002) surface wind speed and direction during this period (Fig. 2a) do not resolve the TC eyewall structure but indicate stronger wind speeds on the right side of the storm, reflecting established theory (e.g., Shapiro 1983). Surface wind speed estimates at 1 Hz derived from stepped frequency microwave radiometer (SFMR) measurements (Uhlhorn and Black 2003; Uhlhorn et al. 2007) at approximately the same time capture the high-wind eyewall structure. Daily gridded 9-km horizontal resolution Optimally Interpolated Sea Surface Temperature (OISST; Remote Sensing Systems 2017) shows a cold wake (Price 1981; Chen et al. 2007; Lee and Chen 2014) forming behind the storm, favoring the rear-right (RR) quadrant.

Fig. 2.
Fig. 2.

(a) Storm-relative 0.25° Advanced Scatterometer (ASCAT) surface wind speed (colored field) and wind direction (unit vector field) from satellite overpass at approximately 0230 UTC 1 Sep 2019. Also shown is stepped-frequency microwave radiometer-estimated surface wind speed (colored line) from a flight segment from 0150 to 0236 UTC 1 Sep 2019. (b) Storm-relative 9-km Optimally Interpolated Sea Surface Temperature (OISST) on 1 Sep 2019. Observed storm centers at 0230 and 0600 UTC 1 Sep are used for (a) and (b), respectively.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Simulated surface conditions are presented in Fig. 3, with storm-relative position in Fig. 3g expressed as the ratio of distance to storm center (R) to the radius of maximum azimuthally averaged 10-m wind speed (RMW). Surface winds (Figs. 3a,g) are strongest on the right side of the storm, as also seen in Fig. 2a, and wave dissipation flux (Figs. 3b,h) peaks in the FR quadrant. HS (Figs. 3c,i) is highest in the FR quadrant, agreeing with established patterns (e.g., Wright et al. 2001). The 10-m neutral, sprayless (i.e., not modified by spray feedback) air–sea temperature difference [(T0T10N); Figs. 3d,j] increases with wind speed, and the 10-m neutral, sprayless air–sea specific humidity difference [(q0q10N); Figs. 3e,k] decreases and then remains roughly constant as wind speed increases. The 10-m neutral, sprayless, salt-adjusted (i.e., equal to 1.0 when air is saturated with respected to the ejected saline droplets) saturation ratio [(s10Ny0); Figs. 3f,l] increases as wind speed increases and is close to 1.0 at the highest wind speeds. A cold wake is present in the RR quadrant as observed and is a major source of variability in the thermodynamic variables. Note that, in Figs. 3h–l, correlation of a particular variable with U10 does not necessarily indicate causation. In the fully coupled AWO system, the variables in Fig. 3 are related through storm-scale air–sea–wave interactions that must be considered when determining causation among them. Finally, note that points in the eye disrupt trends for U1010ms1 in Fig. 3 and future figures.

Fig. 3.
Fig. 3.

Sea state and surface conditions for UWIN-CM simulation of Hurricane Dorian. (a)–(f) Maps of (a) 10-m wind speed, (b) wave energy dissipation flux, (c) significant wave height, (d) 10-m neutral, sprayless air–sea temperature difference T0T10N, (e) 10-m neutral, sprayless air–sea specific humidity difference q0q10N, and (f) 10-m neutral, sprayless, salt-adjusted saturation ratio s10Ny0 at 0600 UTC 1 Sep 2019. (g)–(l) Mean of grid points (solid lines) with ±1 standard deviation bands (shading) for the same fields as in (a)–(f), from 0000 to 1200 UTC 1 Sep 2019.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

b. Spray generation

The wind-based SSGF maintains the same shape at all wind speeds, whereas the sea-state-based (SS-based) SSGF favors production of larger droplets at higher wind speeds (Figs. 4a,b), reflecting the growing ability of turbulent surface gusts to entrain large droplets at higher wind speeds. The SS-based model predicts approximately one order of magnitude higher (lower) spray mass flux than the wind-based model at 50 (20) m s−1 (Figs. 4c,d), demonstrating that dissipation-based and whitecap-based models produce fundamentally different behavior. The SS-based results in Figs. 4c and 4d also have nonzero variance, reflecting the range of sea states that may occur at a given wind speed. The SS-based model predicts approximately one order of magnitude higher (lower) spray mass flux than the wind-based model at 1RMW (5RMW) (Figs. 4e,f).

Fig. 4.
Fig. 4.

Spray production for Hurricane Dorian simulation from 0000 to 1200 UTC 1 Sep 2019. Sea-state-based (SS-based) and wind-based (a) SSGFs at selected wind speeds, (b) radius of SSGF peak, and spray mass flux vs (c),(d) 10-m wind speed and (e),(f) R/RMW on linear and logarithmic scales. In (a), smooth curves are SS based and jointed curves are wind based. The plots in (b)–(f) show means of grid points (solid lines) with ± 1 standard deviation bands (shading).

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

c. Spray and air–sea heat fluxes

HT,spr and HR,spr are controlled by the spray mass flux, specific available energies, and heat transfer efficiencies. aT decreases with increasing wind speed, levels off, and finally increases at the highest wind speeds (Fig. 5a). For reference, the quantity cp,sw(T0T10N), which is aT without the wet-bulb depression contribution, is also plotted. At low wind speeds, the wet-bulb contribution is substantial, but it decreases to become negligible at the highest wind speeds where (s10Ny0) is high (Fig. 3l). aR (Fig. 5b), which is controlled by saturation ratio, is large at low wind speeds but plummets to near zero in the high-humidity eyewall.

Fig. 5.
Fig. 5.

Specific available energies and heat transfer efficiencies for Hurricane Dorian simulation from 0000 to 1200 UTC 1 Sep 2019. Mean (a) aT and (b) aR, with ±1 standard deviation bands. For reference, the quantity cp,sw(T0T10N), which is aT without the wet-bulb depression contribution, is also plotted in (a). (c) ET and (d) ER without near-surface feedback at selected wind speeds. Mean (e) E¯T and (f) E¯R, showing results with and without near-surface feedback using both SS- and wind-based SSGFs.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Radius-specific heat transfer efficiencies ET and ER are shown in Figs. 5c and 5d without accounting for near-surface feedback effects, which makes the results independent of SSGF. Peak efficiency for ET occurs near r0 = 300 μm for all wind speeds, with larger droplets settling back to the ocean quickly and smaller droplets reheating to the ejection temperature upon reentry. Peak efficiency for ER occurs near r0 = 20 μm, with larger droplets settling quickly and smaller droplets regrowing by condensation as they pass through the near-saturated air at the ocean surface. ER generally decreases with wind speed due to the strong effect of s on τR (7b). Thus, there is a double effect of high s in dampening HR,spr; high s limits both the potential for evaporation (through aR) and the rate of evaporation (through ER).

We note that our results in Figs. 5c and 5d are analogous to those of Mueller and Veron (2014a, their Figs. 9a and 10a). Our model reproduces many of the qualitative features of this much more detailed analysis, an encouraging indication that our model could be calibrated using such studies. Additionally, analysis not presented shows that neglecting regrowth of the smallest droplets by condensation as they reenter the ocean (i.e., the falloff of curves at the smallest radii in Fig. 5d) produces an error in the spray modifications to total heat fluxes that is generally below 2%, so the regrowth effect may be neglected.

The SSGF-weighted efficiencies E¯T and E¯R reveal that the SSGF shape has a significant impact on spray’s ability to transfer heat. Since the wind-based SSGF shape does not change with wind speed, wind-based E¯T (Fig. 5e) is mostly modulated by the ability of Hs to increase the droplet residence time τf (cf. to Fig. 3i). For SS-based E¯T, the changing shape of the SSGF is also important, with efficiency dropping away from its peak near 30 m s−1 as the SSGF favors smaller (larger) droplets at lower (higher) wind speeds. The peak of the wind-based SSGF (220 μm) is far from the peak efficiency in ER, causing mean wind-based E¯R (Fig. 5f) to be very low (<0.04) in all cases. Increasing s causes wind-based E¯R to decrease as wind speed increases. At the highest wind speeds, large droplets cause SS-based E¯R to be even lower than wind-based E¯R, but SS-based E¯R rises sharply as wind speed decreases and the SSGF favors smaller droplets. Feedback effects (discussed later) have a generally small effect on E¯T and E¯R. We emphasize that, in the absence of feedback, differences between wind-based and SS-based E¯T and E¯R are due to the shape of the SSGF alone [the influence of Mspr is removed by (13)].

We now examine the spray heat fluxes HSN,spr, HL,spr, and HK,spr. Analysis not presented shows that Hwb,spr is very small (10Wm2 under most conditions) due to effects of low Mspr at lower wind speeds (Fig. 4c) and low ΔTwb at higher wind speeds (Fig. 5a). So, for the purpose of interpretation, we may consider HS,sprHT,spr and HL,sprHR,spr, allowing us to interpret HSN,spr, HL,spr, and HK,spr in terms of available energies and efficiencies.

At low wind speeds, higher Mspr and E¯T for the wind-based SSGF produce larger HK,spr (equal to HT,spr) than when using the SS-based SSGF (Fig. 6c). At higher wind speeds, very large SS-based Mspr counteracts the deficit in E¯T, and the SS-based SSGF produces higher HK,spr. HL,spr (Fig. 6b) rises steadily with wind speed for both SSGFs until around 35 m s−1, after which point it decreases due to high s, which attenuates both aR and E¯R. The shapes of HL,spr curves depend strongly on the SSGF selected. Note that we force zero spray heat fluxes for U10 < 10 m s−1 because this is the approximate observed threshold for producing spume droplets (e.g., Veron 2015); the wind-based SSGF produces a sharp (nonphysical) jump in HL,spr at this threshold whereas the SS-based curves are smooth. HSN,spr mirrors features of both HK,spr and HL,spr (Fig. 6a). Below approximately 45 m s−1, HR,spr > HS,spr, and evaporative cooling of spray produces negative HSN,spr. For higher wind speeds, HS,spr > HR,spr and HSN,spr is positive, although it is much larger for the SS-based SSGF than for the wind-based SSGF. Replotting these results versus R/RMW (Figs. 6d–f), we find that the SS-based SSGF produces substantially higher (lower) HK,spr than does the wind-based SSGF within (outside) 2 to 3RMW, with peak HK,spr occurring at the RMW for both SSGFs. HL,spr does not peak at the RMW but rather between 1 and 2RMW, with the SS-based SSGF producing a stronger peak that is closer to the RMW. Finally, while both SSGFs produce negative HSN,spr outside the eyewall, the SS-based SSGF produces strong positive HSN,spr at the RMW whereas wind-based HSN,spr is near zero there. Feedback effects (discussed later) do not change the qualitative behavior of any spray heat fluxes.

Fig. 6.
Fig. 6.

Spray and total heat fluxes for Hurricane Dorian simulation from 0000 to 1200 UTC 1 Sep 2019. Mean spray (a),(d) net sensible heat, (b),(e) latent heat, and (c),(f) enthalpy fluxes vs (a)–(c) 10-m wind speed and (d)–(f) R/RMW. Mean interfacial and total (g) sensible heat, (h) latent heat, and (i) enthalpy fluxes vs R/RMW. Mean percentage change in (j) sensible heat, (k) latent heat, and (l) enthalpy fluxes due to spray vs R/RMW.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Turning now to spray’s impact on total air–sea heat fluxes (Figs. 6g–l), SS-based spray contributes strongly to the total eyewall enthalpy flux (over 12.5% mean increase at the peak), whereas the wind-based increase is much smaller (∼2.5%). SS-based HL,spr produces a mean increase of LHF over 12.5% near 1.5RMW which diminishes quickly with storm radius, whereas wind-based HL,spr produces a mean increase of nearly 10% between 2 and 5RMW. Both SSGFs reduce the SHF strongly outside the eyewall. Mean increase in SHF at the RMW is over 20% using the SS-based SSGF but is near zero for the wind-based SSGF.

With spray heat fluxes described, we now characterize their feedback effects on the near-surface layer. Spray-layer warming/cooling (Fig. 7a) mirrors HSN,spr (Fig. 6d), with cooling outside the eyewall for both SSGFs and warming at the RMW for the SS-based SSGF. Spray-layer moistening (Fig. 7c) mirrors HL,spr (Fig. 6e). Heating/cooling and moistening cause s to increase under all conditions for both SSGFs, except at the RMW using the SS-based SSGF, where s decreases due to the stronger impact of warming than of moistening (Fig. 7e).

Fig. 7.
Fig. 7.

Feedback modifications to thermodynamic variables at the middle of the spray layer (δ/2) and feedback coefficients for Hurricane Dorian simulation from 0000 to 1200 UTC 1 Sep 2019. Mean mid-spray-layer changes to (a) air temperature, (c) specific humidity, and (e) saturation ratio. Mean feedback coefficients (b) αS, (d) βL, and (f) γS and γL.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Cooling of the spray layer amplifies HS,spr for all conditions for both SSGFs, resulting in αS > 1.0, except at the RMW using the SS-based SSGF, where warming produces αS < 1.0 (Fig. 7b). Outside the eyewall, increased s reduces HL,spr, causing βL < 1.0 (Fig. 7d). At the RMW, decreased s for the SS-based SSGF increases HL,spr, causing βL > 1.0 (note that βL can be very large when HL,spr is near zero). Because HL,sprHR,spr, βLβS (βS is thus not shown). γS and γL do not depend on the SSGF but rather on the thickness of the spray layer. Both peak above 0.9 at the RMW where waves are highest and decrease with distance outward, remaining high (∼0.85) at 5RMW (Fig. 7f).

Overall, the impacts of feedback on spray heat fluxes are modest (feedback coefficients are generally close to 1.0), as reflected in the efficiencies (Figs. 5e,f) and heat fluxes (Figs. 6a–f). The feedback modification to the interfacial fluxes is also small, as seen by comparing interfacial fluxes without spray (black lines) and with SS-based spray feedback (gray lines) in Figs. 6g–i. The modest impact of near-surface feedback on surface-layer thermodynamic variables and heat fluxes can be understood as follows. By injecting heat and moisture into the spray layer away from the surface, spray bypasses the region of highest resistance to vertical turbulent transfer (i.e., the lowest eddy diffusivity). Away from the surface, only a small increase in the vertical gradients of θ and q is required to drive the additional spray heat fluxes upward, so the surface layer can accommodate the increase in total heat fluxes due to spray without large changes to the vertical thermodynamic profiles. We should also point out that the SS-based and wind-based SSGFs produce unique feedback behavior. For instance, the SS-based (wind-based) model produces large positive (near-zero) HSN,spr at the RMW (Fig. 6d), which causes large (near-zero) spray-layer warming (Fig. 7a) and a decrease (almost no change) in spray-layer s (Fig. 7e) at the RMW. Consequently, HL,spr at the RMW (Fig. 6e) is amplified by feedback for the SS-based model but is essentially unchanged for the wind-based model.

Finally, we examine spray’s impact on heat transfer coefficients (with near-surface feedback included). Both SSGFs substantially decrease Ch,10N for U10N45ms1, and the SS-based SSGF increases Ch,10N considerably above 45 m s−1 (Fig. 8a). Both SSGFs increase Cq,10N, but the SS-based model produces a stronger effect at higher wind speeds (Fig. 8b). Wind-based spray has almost no effect on Ck,10N, but SS-based spray produces Ck,10N that levels off and then increases for U10N30ms1 (Fig. 8c).

Fig. 8.
Fig. 8.

The 10-m neutral heat transfer coefficients for Hurricane Dorian simulation from 0000 to 1200 UTC 1 Sep 2019. Mean values (solid lines) with ±1 standard deviation bands (shading) for heat transfer coefficients for (a) sensible heat, (b) latent heat, and (c) enthalpy, showing results without spray and with SS-based and wind-based SSGFs. All spray results include near-surface feedback.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

d. Air–sea heat fluxes with spray in diverse and complex TC conditions

We now expand upon the previous sections’ findings by examining sea-state-dependent air–sea heat fluxes with spray across a diverse set of TC simulations covering a wide range of environments. Spray calculations in this section are made using the SS-based SSGF only, and near-surface feedback is always included. In addition to Dorian, we examine (Fig. 9) Hurricanes Harvey (2017), Michael (2018), and Florence (2018) and Typhoon Fanapi (2010). For each new storm, open ocean and coastal periods were selected. Twelve-hour open-ocean periods (Mic-O, Flo-O, Fan-O, Har-O) were selected when the storms were translating in the open ocean with little curvature, ensuring that the wave fields are not disrupted by changing storm direction. Coastal periods were defined to demonstrate the influence of seafloor interaction (i.e., wave shoaling) on heat fluxes. For storms making landfall over continental shelves, we select coastal periods (Mic-C, Flo-C, Har-C) where the product of dominant wavenumber and water depth averaged within 1RMW is less than π (this parallels the classical linear deep-water wave definition; e.g., Young 1999). For Fanapi, which is over deep water until landfall, we select a 9-h period preceding landfall (Fan-C) for comparison.

Fig. 9.
Fig. 9.

National Hurricane Center Best Track and UWIN-CM simulated (a)–(e) storm center position and (f)–(j) maximum surface wind speed for (a),(f) Hurricane Harvey (2017), (b),(g) Hurricane Michael (2018), (c),(h) Typhoon Fanapi (2010), (d),(i) Hurricane Dorian (2019), and (e),(j) Hurricane Florence (2018).

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Per (1), the spray mass flux is modulated by wave properties including ε (holding other properties constant, Mspr increases as TKE due to ε increases), Hs (the breaking penetration depth increases with HS; holding other properties constant, Mspr decreases as increasing Hs spreads dissipation vertically), and Cp,d (holding other properties constant, increasing Cp,d reduces Mspr by decreasing the wave-relative ejection velocity of droplets). Comparison of these properties and Mspr across storms (Fig. 10) shows how storm-scale wave behavior controls Mspr. Open ocean storms and Fan-C, which do not experience shoaling, have dissipation curves that group together, relatively high HS, and high Cp,d. Shoaling (Mic-C, Flo-C, Har-C) causes a strong increase in dissipation (due to increased breaking), a reduction in Hs (by removing energy from the spectrum), and a reduction in Cp,d. These differences in behavior create two groups of curves for Mspr (i.e., shoaling and nonshoaling), with shoaling cases experiencing higher Mspr than nonshoaling cases for the same wind speed. Note that even within nonshoaling cases, the influence of storm-specific differences in sea state can be seen, as nonshoaling Mspr curves are reordered relative to dissipation curves due to Hs and Cp,d. Note also that curves for Dor-O from sections 4a4c are reproduced identically in Fig. 10 and future figures.

Fig. 10.
Fig. 10.

Mean (a) wave dissipation flux, (b) significant wave height, (c) dominant phase speed, and (d) spray mass flux vs 10-m wind speed across selected periods of UWIN-CM simulations of five storms.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Behavior of aT (Fig. 11a) is storm specific, with mean values ranging between roughly 1 × 104 and 2 × 104 J kg−1. Analysis not presented shows that at high winds most of the variation in aT among cases arises from variation in the air–sea temperature difference rather than the wet-bulb depression. aR (Fig. 11b) varies according to storm-scale behavior of s10Ny0 (Fig. 11c). Although all cases experience a drop in aR with increasing wind speed, the magnitude of aR at the highest wind speeds is very sensitive to the corresponding value of s10Ny0, with aR varying between approximately 1.5 × 106 and 0 J kg−1 as s10Ny0 varies between 0.95 and 1.0. The relationship between s10Ny0 and q0q10N is storm specific (Fig. 11d), indicating that spray LHF cannot be scaled using the traditional turbulent thermodynamic scale q0q10N.

Fig. 11.
Fig. 11.

Mean (a) aT, (b) aR, (c) 10-m neutral, sprayless, salt-adjusted saturation ratio s10Ny0, (e) efficiency E¯T, and (f) efficiency E¯R vs 10-m wind speed across selected periods of UWIN-CM simulations of five storms. (d) 10-m neutral, sprayless air–sea specific humidity difference q0q10N vs s10Ny0 for the same periods.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

For all storm periods, E¯T follows the inverted U-shape caused by the SS-based SSGF’s transition from smaller to larger droplets as wind-wave processes intensify (Fig. 11e). E¯T shows variation among cases due to the impact of Hs on τf (cf. to Fig. 10b), with larger waves increasing efficiency by lengthening droplet residence times. E¯R is high at low wind speeds and plummets to near zero as wind speed increases, with case-to-case variation arising from storm-scale differences in Hs and s10Ny0. (Fig. 11f).

We now address spray heat fluxes across our diverse set of storm cases and their impacts on total heat fluxes. Variation in HK,spr (Fig. 12c) arises from shoaling impacts on Mspr (Fig. 10d), case-specific variation in aT (Fig. 11a), and modulation of E¯T by Hs (Fig. 11e). Shoaling cases produce consistently higher HK,spr than nonshoaling cases. Shoaling impacts on Mspr and control of both aR and E¯R by s10Ny0 (Figs. 11b,c,f) dominate variation in HL,spr (Fig. 12b). High-s cases (e.g., Dor-O, Mic-O) produce HL,spr that remains low or even declines at the highest wind speeds, whereas low-s cases (e.g., Mic-C, Fan-C) produce HL,spr that rises continuously with wind speed. All aforementioned influences apply also to HSN,spr and produce a wide range of behavior (Fig. 12a). When HK,spr is low and/or HL,spr is high (e.g., Fan-C, Flo-O), HSN,spr is near zero or negative at all but the highest wind speeds. When HK,spr is high and/or HL,spr is low (e.g., Dor-O), HSN,spr becomes very large at high wind speeds. In all cases, negative HSN,spr occurs between U10 of approximately 20 and 40 m s−1. In R/RMW coordinates (Figs. 12d–f), peak HK,spr occurs near the RMW, but peak HL,spr occurs between 1 and 1.5RMW. Some cases (i.e., Dor-O, Mic-C, Flo-C) have strong positive mean HSN,spr in the eyewall, but all other cases show negative HSN,spr there. All cases show negative HSN,spr beyond 1.5RMW.

Fig. 12.
Fig. 12.

Spray and total heat fluxes across selected periods of UWIN-CM simulations of five storms. Mean spray (a),(d) net sensible heat, (b),(e) latent heat, and (c),(f) enthalpy fluxes vs (a)–(c) 10-m wind speed and (d)–(f) R/RMW. Mean (g) sensible heat, (h) latent heat, and (i) enthalpy fluxes without spray vs R/RMW. Mean percent change in (j) sensible heat, (k) latent heat, and (l) enthalpy fluxes due to spray vs R/RMW.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Spray augments the turbulent heat fluxes (Figs. 12g–i) to different degrees (Figs. 12j–l) depending on storm-specific wave and thermodynamic characteristics. Strong shoaling cases (Mic-C, Flo-C) have the strongest spray contribution to enthalpy fluxes (35% mean increase in enthalpy flux at the RMW for Mic-C). Spray increases enthalpy fluxes most strongly at the RMW for all cases, with percentage increase declining smoothly at larger radii. Because spray and turbulent LHFs depend on different thermodynamic properties, spray augmentation of the turbulent LHF depends on the storm-specific relationship between s10Ny0 and q0q10N. In all but the weakest case (Har-O), the largest mean increase of LHF due to spray is over 10%. Spray contribution to total SHF varies widely due to storm-scale differences in wave properties and evaporative cooling rates. Spray may produce a mean change in SHF at the RMW above 30% (Mic-C; due to high HK,spr) or below −60% (Fan-C; due to high HL,spr). Depending on the magnitude of HL,spr, spray may produce a mean reduction of SHF at large radii of below 20% (Har-O) or nearly 100% (Mic-C).

Spray impacts on heat transfer coefficients are shown in Fig. 13. Heat transfer coefficients without spray collapse for all cases, but there is substantial spread in all three coefficients across the selected cases when spray is included. In extreme cases (Mic-C, Fan-C), spray evaporative cooling is so strong that it completely negates the turbulent SHF, producing Ch,10N below zero. In all cases, Ck,10N levels off and then rises for U10N40ms1 when spray is included.

Fig. 13.
Fig. 13.

Mean heat transfer coefficients for selected periods of UWIN-CM simulations of five storms. (a),(b) Ch,10N, (c),(d) Cq,10N, and (e),(f) Ck,10N, showing results (a),(c),(e) without spray and (b),(d),(f) with spray.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Finally, we summarize the behavior of heat fluxes with spray by tracking several metrics for spray through the full simulation time period for all storms. Spray mass flux at the RMW for all storms falls near a common curve when plotted versus peak azimuthal-mean 10-m wind speed, with notable deviation from this curve occurring for shoaling cases (Fig. 14a). Percent increase in enthalpy flux at the RMW shows a very similar pattern, reflecting the strong connection between spray mass and enthalpy fluxes (Fig. 14f). Mean percent increase in LHF outside the RMW due to spray increases with peak azimuthal-mean wind speed, but there is a large spread in results due to storm-specific patterns in waves and thermodynamics outside the eyewall (Fig. 14e). Due to the coupling of sensible and latent spray heat fluxes by evaporative cooling, mean percent change in SHF outside the RMW due to spray shows a similar but negative pattern to that for LHF, with larger magnitudes of change due to turbulent LHF being generally much larger than turbulent SHF (Fig. 14d). Spray mean change to SHF at the RMW (Fig. 14c) is generally negative below 35 m s−1 but may become either strongly negative or strongly positive at higher wind speeds. The tendency for eyewall SHF change to be positive or negative seems to correspond to the eyewall saturation ratio (Fig. 14b), with high-s cases (e.g., Dor-O) increasing SHF and low-s cases (e.g., Fan-C) reducing SHF. Shoaling cases (e.g., Mic-C, Flo-C) seem to have increasing SHF despite relatively low s10Ny0 due to shoaling-enhanced HK,spr.

Fig. 14.
Fig. 14.

Time traces of metrics for spray through the full simulation time period for all storms (calculations performed hourly). (a) Mean Mspr at the RMW, (b) mean s10Ny0 at the RMW, (c) mean percentage change in SHF due to spray at the RMW, (d) mean percentage change in SHF due to spray between 1.5 and 4RMW, (e) mean percentage change in LHF due to spray between 1 and 4RMW, and (f) mean percentage change in enthalpy flux due to spray at the RMW. All metrics are plotted vs the peak azimuthal-mean 10-m wind speed.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

Because this study has demonstrated that spray effects may be storm specific, it is reasonable to wonder if they can be predicted by storm characteristics such as size and speed. While a full assessment is beyond the scope of this work, we briefly address this idea in Fig. 15 by tracking several metrics for spray against storm size (which indicates the areal extent of the high-wind regime where spray is most active) and storm translation speed (which affects the asymmetry of the wind and wave fields that drive spray). Spray impact on eyewall enthalpy flux does not show a clear dependence on storm size [measured as the radius of 17 m s−1 (gale force) winds] (Fig. 15e). Spray impact on LHF outside the RMW generally increases with storm size (Fig. 15c). Spray tends to increase eyewall SHF for smaller storms and decrease it for larger storms (Fig. 15a). There is no clear dependence of any of the preceding three metrics on storm translation speed (Figs. 15b,d,f).

Fig. 15.
Fig. 15.

Time traces of metrics for spray through the full simulation time period for all storms (calculations performed hourly). (a),(b) Mean percentage change in SHF due to spray at the RMW, (c),(d) mean percentage change in LHF due to spray between 1 and 4RMW, and (e),(f) mean percentage change in enthalpy flux due to spray at the RMW. (a), (c), and (e) are plotted vs the radius of 17 m s−1 (gale force) azimuthal-mean 10-m wind speed, and (b), (d), and (f) are plotted vs storm translational speed. Line styles and colors are as in Fig. 14.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0126.1

5. Conclusions

In this study, we present a new sea-state-dependent air–sea heat flux parameterization with spray and apply it over a wide range of winds, waves, and atmospheric and oceanic conditions using UWIN-CM output from five fully coupled atmosphere–wave–ocean TC simulations. Our spray parameterization models the physical processes of wave-dependent spray generation, transport, heat transfer, and near-surface feedback, allowing physical connections between waves, thermodynamics, spray, and heat fluxes to be explored. Testing our parameterization using UWIN-CM provides access to physically consistent sets of surface conditions that cover a wide range of TC environments. We believe that this study represents the first time that a sea-state-based SSGF with variable shape has been exercised and examined broadly across a range of simulated TC environments to characterize spray’s impact on heat fluxes. It thus represents a valuable benchmark for understanding sea-state-based spray effects that may inform future work incorporating these processes into models.

Our sea-state-based SSGF produces droplet size distributions that favor larger droplets as wind speed increases, reflecting findings from a recent laboratory study (OS16). Sea-state-based spray mass flux differs substantially from traditional wind-based predictions. Sea-state-based spray mass flux is approximately one order of magnitude higher (lower) than traditional wind-based mass flux at U10 = 50 (20) m s−1 and is modulated by wave properties such as wave dissipation, significant wave height, and dominant phase speed, which may be uncorrelated to U10. Wave shoaling in shallow water greatly increases sea-state-based spray mass flux by increasing dissipation and decreasing wave height and phase speed.

Spray heat fluxes are controlled by the total mass of spray generated, the energy available for heat transfer within that spray, and the efficiency with which the near-surface flow extracts that energy from the droplet sizes present. High eyewall s is ubiquitous in TCs and severely reduces both the energy available and the efficiency of transfer for heat transfer due to size change. Heat transfers due to temperature and size change are most efficient in high winds for droplets with r0 near 300 and 20 μm, respectively, and they are extremely attenuated for droplets with r0 above 1000 and 100 μm, respectively. Because wind-based and sea-state-based SSGFs possess different proportions of droplet sizes, they can produce population-averaged heat transfer efficiencies that differ substantially. The wave field modulates heat transfer efficiency by controlling the ejection height, and therefore residence time, of droplets.

Sea-state-based spray enthalpy flux peaks at the RMW. Spray LHFs may be severely attenuated in the eyewall due to high s, confirming the strong control of spray effects by s found by Shpund et al. (2012), but spray LHFs consistently produce a moistening and cooling effect outside the eyewall. Net sea-state-based spray SHF can be either positive or negative at the RMW depending on the relative strengths of spray enthalpy and latent heat fluxes.

Sea-state-based spray produces a mean increase in enthalpy flux at the RMW over 35% for Hurricane Michael at landfall, which experiences strong shoaling effects, and generally produces mean increases of 5%–20% when peak azimuthal-mean 10-m wind speed is between 40 and 50 m s−1. Mean increase in LHF between 1 and 4RMW is generally 5%–20% for peak azimuthal-mean wind speed between 30 and 50 m s−1, with wide variation due to waves and surface thermodynamics. Spray decreases the SHF outside the eyewall in all tested scenarios, but spray may increase or decrease SHF at the RMW. Spray increase (decrease) of eyewall SHF corresponds to high (low) eyewall s and tends to occur in smaller (larger) storms. SHF decrease outside the eyewall could potentially increase boundary layer stability and enhance inflow to the inner core (Lee and Chen 2014), and changes of stability in the eyewall due to spray SHF could impact convection.

Traditional heat transfer coefficient scaling fails when spray is present because spray introduces physical interactions (at least 6) that cannot be characterized by traditional wind and thermodynamic variables (i.e., U10, T0T10N, and q0q10N). These interactions are 1) spray generation depends strongly on sea state, which may be uncorrelated to wind, especially in coastal regions, 2) droplet heat transfer efficiency depends on the droplet ejection height, which is governed by sea state, 3) the wet-bulb depression, in addition to T0T10N, governs HT,spr, 4) HR,spr depends on s10Ny0 rather than q0q10N, 5) droplet sensible and latent heat fluxes are coupled through evaporative cooling, and 6) all droplet heat fluxes are coupled through near-surface feedback. Interactions 1, 2, 4, 5, and 6 apply to Ch,10N. Interactions 1, 2, 3, 4, and 6 apply to Cq,10N. Interactions 1, 2, 3, and 6 apply to Ck,10N. Although results are case-dependent, our sea-state-based spray model consistently produces Ck,10N curves that level off and then rise at very high wind speeds.

It is important to note that heat fluxes with spray in this study were diagnosed from UWIN-CM output rather than calculated within the coupled simulations themselves. When spray heat fluxes are allowed to interact with the resolved-scale AWO system, they will modify resolved-scale properties, creating feedbacks that may either dampen or amplify the spray heat fluxes (e.g., large spray LHF will increase q and s at the LML, creating a feedback that diminishes the spray LHF). For simplicity, we have chosen to limit this study’s scope to diagnosing heat fluxes from coupled model output only; all considerations related to coupled interactions between spray and the AWO system, including feedbacks between spray and the resolved-scale environment at the air–sea interface, are left for future work.

The new parameterization contains a number of parameters that are poorly understood and that would benefit from additional observational and modeling work. Spray generation remains poorly observed, especially in high-wind field conditions, and experimental and modeling work targeting the dynamics of droplet generation at wave crests and entrainment of droplets into the turbulent near-surface flow could help calibrate our sea-state-based SSGF or determine if more suitable scalings exist than those currently used. Our model predicts that droplet cooling and evaporation are most efficient for droplets with r0 near 300 and 20 μm, respectively (Figs. 5c,d), suggesting that these droplet sizes should be targeted when designing future observational studies on spray generation. Mueller and Veron (2014a) have shown that the droplet residence time is more complex than as assumed in (8); additional numerical work (e.g., large-eddy simulations) studying droplet transport across complex seas (including those with swell and misaligned wind and waves) would be helpful in calibrating our model. Finally, our model’s assumption of uniform vertical heating within the spray layer when calculating near-surface feedback (A10a) is likely an oversimplification, and the model would benefit from numerical work investigating the vertical distribution of spray heat flux into the spray layer for a wide range of droplet sizes and sea states.

Despite the uncertainty in our new parameterization, we believe that this study’s results indicate strong control of spray effects by wave processes and demonstrate that explicit representation of sea-state-based spray generation and heat transfer in high-wind air–sea heat flux calculations is essential. The new parameterization has been implemented in UWIN-CM, and analysis is underway to examine the coupled interactive processes between waves, spray heat fluxes, the atmospheric surface and boundary layers, and the TC vortex, including 1) feedback between spray heat fluxes and the resolved-scale environment at the air–sea interface, 2) the impact of spray moistening/cooling effects on stability and inflow outside the eyewall, and 3) the impact of spray on eyewall convection.

We hope that this study and related work will facilitate inclusion of spray physics in operation hurricane forecast models. Toward this goal, we emphasize the importance of the SSGF in driving all spray effects and recognize the challenge of implementing sea-state-dependent spray physics in operational models, which requires coupling to a wave model with sufficient resolution to resolve the relevant wave field characteristics. We hope that continued development of parameterizations like the one presented herein and of computationally efficient, coupling-friendly wave models like the UMWM will help realize the goal of fully coupled AWO operational model forecasts of TC intensity and impacts.

Acknowledgments.

This research is supported in part by the NASA MAP Grant 80NSSC17K0421, the NSF Grant OCE-1756412, and the Gulf of Mexico Research Initiative Grant SA 18-14 (Subaward SPC-000581 AM 1). B. W. Barr is supported by NASA Future Investigators in NASA Earth and Space Science and Technology (FINESST) Award 80NSSC19K1336. The authors thank Brandon Kerns, Milan Curcic, Edoardo Mazza, and Dalton Sasaki for their contributions to the UWIN-CM simulations used in this study. The authors also thank three anonymous reviewers for their thoughtful and constructive review comments on the manuscript, which helped improve the manuscript significantly.

Data availability statement.

Coupled model output, spray calculation fields, and further information about the new parameterization and its implementation in AWO models can be obtained upon request to Benjamin Barr (bwbarr@uw.edu).

APPENDIX

Additional Details for Air–Sea Heat Flux Parameterization with Spray

a. Spray generation

Additional details defining variables in section 3a and a summary of the sea-state-based SSGF calibration process follow. We define WSS based on Deike et al. (2017) as follows:
WSS=(0.018sm1)Cp,du*2gHS,
where g is the acceleration due to gravity. We define Wwi using a simple wind speed fit to the formula given as Eq. (12) in Blomquist et al. (2017):
Wwi=(6.5×104s1.5m1.5)(U102ms1)1.5;
ηk is defined in the usual way as ηk=(νsw3/ε˜)1/4, with νsw as the kinematic viscosity of seawater. Note that ηk depends on WSS through ε˜.

The argument of the error function in (1) without tuning coefficients is (Uh,relυg/sm)/σh and represents a comparison between the ballistics of droplets at the wave crest (numerator) and the strength of the gusts that entrain them into the surface flow (denominator). The numerator relates the slope of a droplet’s wave-relative trajectory off the crest (−υg/Uh,rel, assuming constant horizontal and vertical speeds Uh,rel and υg, respectively) to the slope of the underlying wave face (−sm). We assume that droplets instantly accelerate to the local horizontal wind speed Uh as they exit the wave crest, so that Uh,rel = Uh − 0.8Cp,d. In this simple model, the wind properties Uh and σh should be extracted at a characteristic height for the ejection/entrainment process, which is the gust height hgust. We define hgust herein as 200z0, where z0 is the momentum roughness length and the factor of 200 is chosen based on additional analysis so that Uh,rel will be positive for U1025ms1 (i.e., when spray effects become prevalent). We note that the dynamics encapsulated in the error function argument are highly uncertain and would benefit from targeted experimental and numerical studies.

υg is defined per Pruppacher and Klett (1997). σh is not available from WRF and is parameterized as being proportional to U10 [similar to the code accompanying C. W. Fairall et al. (2014, unpublished report)], with the constant of proportionality absorbed into C4. All wind speeds and z0 are calculated per standard MO theory using u and U1.

Our sea-state-based SSGF was calibrated using the laboratory results of OS16 and the F94 SSGF droplet spectrum shape. The OS16 results [their Fig. 11a and Eq. (23)] cover 80r02000µm, which includes the SSGF peak and large droplet tail but not the small droplet tail. The F94 spectrum’s small droplet tail (2r030µm) is based on data from Miller (1987). Its large droplet tail and peak (30r0500µm), which does not move to higher r0 as U10 increases, is based on data from Wu et al. (1984).

We calibrated model coefficients C1 and C2, which control the spectrum magnitude and small droplet tail, to the Miller (1987) portion of the F94 spectrum. We calibrated C3, C4, and C5, which control the peak and large-droplet tail, to the OS16 results. The five coefficients were calibrated simultaneously by fitting one set of coefficients (e.g., C3, C4, C5) to its relevant dataset (e.g., OS16) using least squares minimization while holding the other set of coefficients (e.g., C1, C2) constant, then repeating the process with updated coefficients for the other dataset (e.g., F94), and switching back and forth until all coefficients converged. The converged values of the coefficients are C1 = 1.92, C2 = 0.1116, C3 = 0.719, C4 = 2.17, and C5 = 0.852.

We then used fs to calibrate the total mass flux for our sea-state-based and wind-based SSGFs using the F94 SSGF with its original whitecap formulation (Monahan and O’Muircheartaigh 1980). Based on unpublished spray observations from E. L. Andreas (1993, private communication) and the original Andreas (1992) model, the original F94 SSGF is considered to predict reasonable spray mass flux at U10 = 30 m s−1 when fs = 0.4. Tuning our updated wind-based SSGF to match this gives fs = 2.2. For simplicity, we chose to assign fs = 2.2 to our sea-state-based SSGF as well, which required an adjustment of C1 to 1.35. Note that C1 and fs could be combined in the sea-state-based model. fs is the “sourcestrength” parameter used in various codes (e.g., C. W. Fairall et al. 2014, unpublished report) to scale the spray mass flux and is retained for historical consistency. Forcing fs = 2.2 in both our sea-state-based and updated wind-based SSGFs simplifies future adjustment of both as more observations become available.

b. Spray heat fluxes

Additional details defining variables in section 3b follow. We define the salt-adjusted wet-bulb depression ΔTwb = TTwb and the wet-bulb coefficient β per classical thermodynamics (e.g., F94; Pruppacher and Klett 1997) as follows:
ΔTwb=(1s1+y0)(1β)γ
β=[1+Lυγ(1+y0)cp,aqsat,0(T)]1.
Here γ=[dqsat,0(T˜)/dT˜|T]/qsat,0(T).
The parameter y0 is defined as
y0=νΦsMwMsxs1xs.
Finally, it is common to extract ambient conditions (T, q) for spray calculations at a height related to spray-layer geometry (e.g., δ in F94). However, Mueller and Veron (2014a) and Peng and Richter (2019) showed that this scaling is not appropriate for small droplets, which respond quickly to local conditions and can reheat or regrow by condensation as they reenter the ocean. We account for small droplet reheating and regrowth following Peng and Richter (2019) by specifying radius-specific heights zT and zR for use in HT,spr and HR,spr calculations, respectively, as follows:
zT(r0)=0.5min(δ,υgτT),
zR(r0)=0.5min(δ,υgτR).

c. Air–sea heat fluxes with spray and near-surface feedback

In this section, we derive the model for heat transfer within a MO surface layer with spray heating discussed in section 3c.

Using first-order closure with a mixing length model for eddy diffusivity, the vertical turbulent sensible heat (HS) and latent heat (HL) fluxes as a function of height z are
HS(z)=ρacp,aκu*zϕH(ζ)dθdz,
HL(z)=ρaLυκu*zϕH(ζ)dqdz.
Here ϕH is the universal stability function for heat, which depends on the stability parameter ζ = z/L, with L the Obukhov stability length. When spray is absent, these turbulent fluxes are assumed constant with height, and (A6a) can be integrated to obtain the turbulent interfacial sensible and latent heat fluxes, HS and HL (17). We do not address spray impacts on stability herein and therefore define L in terms of heat fluxes without spray (i.e., HS and HL) as follows:
L=u*2κ(gθ1)(HSρacp,au*0.61θ1HLρaLυu*).
When spray is present, it carries part of the total heat fluxes, so that there is an apparent vertical divergence of the turbulent heat fluxes:
dHSdz=H^S(z),
dHLdz=H^L(z).
Here H^S and H^L are height-dependent volumetric sources of sensible and latent heat, respectively, due to heat transfer from spray droplets to the surface layer. Plugging (A6a) into (A8a) and integrating twice from z0t and z0q to an arbitrary height z gives
ρacp,aκu*(θ0θ)=HS,0[ln(zz0t)ΨH(ζ)]+z0tzϕH(z˜˜/L)z˜˜z0tz˜˜H^S(z˜)dz˜dz˜˜,
ρaLυκu*(q0q)=HL,0[ln(zz0q)ΨH(ζ)]+z0qzϕH(z˜˜/L)z˜˜z0qz˜˜H^L(z˜)dz˜dz˜˜.
Here HS,0 and HL,0 are the heat fluxes at the surface, including spray effects. In (A9a) and the remaining derivation, we neglect small terms involving z0t or z0q, such as ΨH(z0t/L).
Kepert et al. (1999) and Bianco et al. (2011) suggest that volumetric heating due to spray is negligible above the spray layer. Similar to Smith (1990), Andreas et al. (1995), and C. W. Fairall et al. (2014, unpublished report), we assume that spray heats the spray layer uniformly, so that
H^S(z)={1δHSN,sprzδ0z>δ,
H^L(z)={1δHL,sprzδ0z>δ.
Plugging (A10a) into (A9a) and integrating twice, we obtain the θ and q profiles within the spray layer:
ρacp,aκu*(θ0θ)=HS,0[ln(zz0t)ΨH(ζ)]+zδHSN,spr[1φH(ζ)],
ρaLυκu*(q0q)=HL,0[ln(zz0q)ΨH(ζ)]+zδHL,spr[1φH(ζ)];
φH(ζ) is the analog of ΨH(ζ) for a layer with volumetric heating. Using the standard prescription for ϕH (Paulson 1970; Webb 1970; Dyer 1974), φH(ζ) is
φH(ζ)=[(116ζ)1/21]16ζ2,ζ<0,
φH(ζ)=0,ζ=0,
φH(ζ)=2.5ζ,ζ>0.
We recognize in (A11a) the standard log layer and stability terms as well as new terms contributed by spray.
The region between z = δ and z1 is governed by the standard log law. Enforcing continuity of θ, q, and flux at z = δ, we can solve for HS,1 and HL,1, which are (16). We can also solve for the fluxes at the surface, HS,0 and HL,0, which are
HS,0=HS(1γS)(HS,sprHR,spr),
HL,0=HL(1γL)HL,spr.

Finally, we briefly compare the physics of our near-surface feedback model to three other prominent published approaches (Andreas et al. 2015; Bao et al. 2011; Mueller and Veron 2014b).

Andreas et al. (2015) address feedback by applying tuning coefficients to their spray heat fluxes. These coefficients are determined empirically based on numerous datasets and do not directly deal with physical processes in the surface layer.

Bao et al. (2011) address feedback in a physics-based way by estimating changes to temperature and humidity due to spray heat fluxes that would occur in a thermodynamically lumped column of air extending from the surface to the lowest atmospheric model level, accounting for conservation of enthalpy by evaporating spray droplets. These changes are then applied as perturbations to the temperature and humidity potentials used to calculate turbulent and spray heat fluxes. Our approach differs from this in that we directly calculate the full spray-modified temperature and humidity profiles in the surface layer, rather than lumped estimates of their changes, for use in heat flux calculations. Additionally, our calculation of full modified profiles directly yields the heat fluxes with spray applied to the atmospheric model, (16).

Mueller and Veron (2014b) calculate spray-modified profiles of temperature and humidity in the surface layer and use these to calculate heat fluxes with near-surface feedback, making their approach to feedback similar to ours, although with a different derivation and final form for the total heat flux expressions. However, their model and ours take different approaches on several important and challenging modeling issues, including specifying the vertical distribution of spray heat fluxes in the surface layer, determining droplet reentry temperature and size, and modeling sea-state-dependent spray generation.

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