2.1 Motion of individual spray drops

The motion of individual spray drops is given by

(2.1)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\boldsymbol{x}_{p}(t)}{\text{d}t}=\boldsymbol{v}_{p}(t), & \displaystyle\end{eqnarray}$$

(2.2)$$\begin{eqnarray}\displaystyle & \displaystyle m_{p}(t)\frac{\text{d}\boldsymbol{v}_{p}(t)}{\text{d}t}=\boldsymbol{{\mathcal{F}}}(t). & \displaystyle\end{eqnarray}$$

The first equation relates the location of the drop $\boldsymbol{x}_{p}(t)$ to its velocity $\boldsymbol{v}_{p}(t)$. The second equation is Newton’s second law where $\boldsymbol{{\mathcal{F}}}$ accounts for gravity and air frictional effects,

(2.3)$$\begin{eqnarray}\boldsymbol{{\mathcal{F}}}=m_{p}\boldsymbol{g}+\boldsymbol{F},\end{eqnarray}$$

but does not include higher-order effects such as the Basset history and the added mass terms, or Faxen, Saffman and Magnus effects. These equations follow individual drops and are therefore Lagrangian in nature. Gravity is denoted by $\boldsymbol{g}$ and $\unicode[STIX]{x1D70C}$ is the density of the (moist) air. The drag force exerted by the fluid on the drop is described with a general drag law:

(2.4)$$\begin{eqnarray}\boldsymbol{F}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}C_{Dp}A(\boldsymbol{u}-\boldsymbol{v}_{p})|\boldsymbol{v}_{s}|,\end{eqnarray}$$

where $\boldsymbol{u}$ is the velocity of the air (at the location of the drop as if it were absent), $C_{Dp}$ is the drag coefficient of the drop and $A=\unicode[STIX]{x03C0}r_{p}^{2}$ its cross-sectional area. This drag coefficient generally depends on the drop Reynolds number $Re_{p}=(2r_{p}|\boldsymbol{v}_{s}|)/\unicode[STIX]{x1D708}$, with $\boldsymbol{v}_{s}=\boldsymbol{v}_{p}-\boldsymbol{u}$, usually referred to as the slip velocity, and $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}$ the kinematic viscosity of air. It is convenient to express the drag coefficient as $C_{Dp}=(24/Re_{p})\unicode[STIX]{x1D6EF}$, where $\unicode[STIX]{x1D6EF}$ is an empirical correction factor that accounts for departures from the Stokes flow as $Re_{p}$ increases and the flow around drops becomes turbulent (e.g. Clift & Gauvin Reference Clift and Gauvin1970).

Equivalently, one can express the viscous drag force in a more compact form:

(2.5)$$\begin{eqnarray}\boldsymbol{F}=\frac{m_{p}}{\unicode[STIX]{x1D70F}_{D}}(\boldsymbol{u}-\boldsymbol{v}_{p}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{D}$ is the drop response time to the drag force. In this case, just like the drag coefficient, $\unicode[STIX]{x1D70F}_{D}$ can be expressed as $\unicode[STIX]{x1D70F}_{D}=\unicode[STIX]{x1D70F}_{p}/\unicode[STIX]{x1D6EF}$, where $\unicode[STIX]{x1D70F}_{p}=(2r_{p}^{2}\unicode[STIX]{x1D70C}_{p})/9\unicode[STIX]{x1D707}$ is the drop Stokes time scale. Naturally, in Stokes flow with $Re_{p}\ll 1$, $\unicode[STIX]{x1D6EF}\rightarrow 1$, $\unicode[STIX]{x1D70F}_{D}\rightarrow \unicode[STIX]{x1D70F}_{p}$, $C_{Dp}\rightarrow 24/Re_{p}$ and the viscous drag $\boldsymbol{F}\rightarrow (m_{p}/\unicode[STIX]{x1D70F}_{p})(\boldsymbol{u}-\boldsymbol{v}_{p})=6\unicode[STIX]{x03C0}r_{p}\unicode[STIX]{x1D707}(\boldsymbol{u}-\boldsymbol{v}_{p})$. Non-continuum effects can also be accounted for in the correction factor $\unicode[STIX]{x1D6EF}$ through the use of a Cunningham slip coefficient (Davies Reference Davies1945), but these non-continuum effects do not have a significant influence on the drag of drops with radii larger than approximately $0.3{-}0.5~\unicode[STIX]{x03BC}\text{m}$.

2.2 Thermodynamics of individual spray drops

In addition to the drop position and momentum, it is clear from (2.2) that we need to determine the mass of the drop and thus its thermodynamical state. As sea spray drops evaporate, their mass changes according to (Pruppacher & Klett (Reference Pruppacher and Klett1996), their equation (13.9))

(2.6)$$\begin{eqnarray}\frac{\text{d}m_{p}(t)}{\text{d}t}=4\unicode[STIX]{x03C0}r_{p}D_{v}^{\ast }(\unicode[STIX]{x1D70C}_{v}-\unicode[STIX]{x1D70C}_{v,p}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{v}$ is the water vapour density and $D_{v}^{\ast }$ the molecular diffusivity for water vapour modified to include non-continuum effects. While the formulation of Pruppacher & Klett (Reference Pruppacher and Klett1996) was originally derived for stationary drops in a motionless atmosphere, we also include ventilation effects using a correction coefficient based on the drop Reynolds number (see below). In this model, the mass transfer results from the difference between the ambient vapour density and that directly adjacent to the surface of the drop, denoted $\unicode[STIX]{x1D70C}_{v,p}$. To estimate $\unicode[STIX]{x1D70C}_{v,p}$ properly, it is generally necessary to account for the effects of drop curvature (surface tension) and salinity. With these effects explicitly written out, the evolution equation for the mass of a saline sea spray drop takes the following form (Pruppacher & Klett (Reference Pruppacher and Klett1996), equation (13.26)):

(2.7)$$\begin{eqnarray}\displaystyle \frac{\text{d}m_{p}(t)}{\text{d}t} & = & \displaystyle \frac{4\unicode[STIX]{x03C0}r_{p}D_{v}^{\ast }M_{w}p_{v}^{sat}(T_{air})}{RT_{air}}\nonumber\\ \displaystyle & & \displaystyle \times \left[Q_{RH}-\frac{T_{air}}{T_{p}}\exp \left(\frac{L_{v}M_{w}}{R}\left(\frac{1}{T_{air}}-\frac{1}{T_{p}}\right)+\frac{2M_{w}\unicode[STIX]{x1D6E4}_{p}}{R\unicode[STIX]{x1D70C}_{w}r_{p}T_{p}}-\frac{I\unicode[STIX]{x1D6F7}_{s}m_{s}(M_{w}/M_{s})}{m_{p}-m_{s}}\right)\right]\nonumber\\ \displaystyle & = & \displaystyle {\mathcal{M}}(t),\end{eqnarray}$$

where it is noted that the rate of variation ${\mathcal{M}}$ depends both on the fields of the carrier fluid and on the Lagrangian state of the drop, i.e. ${\mathcal{M}}={\mathcal{M}}(\unicode[STIX]{x1D70C},\boldsymbol{u},T_{air};\boldsymbol{v}_{p},r_{p},m_{p},T_{p})$. In the equation above, $Q_{RH}$ is the fractional relative humidity; $R$ is the universal gas constant; $M_{w}$ and $M_{s}$ are the molecular weight of water and salt, respectively; $p_{v}^{sat}(T_{air})$ is the saturation water vapour pressure at the air temperature $T_{air}$; $T_{p}$ is drop temperature assumed uniform inside the drop; $L_{v}$ is the latent heat of vaporization; $\unicode[STIX]{x1D6E4}_{p}$ is the surface tension for a flat surface at the interface of the drop; $m_{s}$ is the mass of salt in the drop; $I$ is the number of ions into which salt dissociates $(I=2)$; and $\unicode[STIX]{x1D6F7}_{s}$ is the osmotic coefficient. In (2.7), the term in square brackets simply represents the difference between the ambient humidity $Q_{RH}$ surrounding the drop and the humidity at the surface of the drop which results from three distinct effects that are reflected in the three terms inside the $\exp (\ldots )$ expression. The first term accounts for the drop temperature which differs from the ambient air temperature. The second term reflects curvature and surface tension effects. The last term accounts for salinity effects in the spray drop.

Noting that the mass of salt inside a spray drop remains constant (i.e. evaporation or condensation transfers only pure water), at a given time

(2.8)$$\begin{eqnarray}m_{p}(t)={\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{sw}r_{p0}^{3}+{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{w}(r_{p}(t)^{3}-r_{p0}^{3}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{sw}$ is the density of sea water, $\unicode[STIX]{x1D70C}_{w}$ is the density of (pure) water and $r_{p0}$ is the radius of a drop at formation ($t=0$). In the equation above, the first term is the mass of the drop at formation and the second term represents the mass lost due to the evaporation of pure water. Incidentally, this computation shows that the mass of salt in the drop (required in (2.7)) is $m_{s}=m_{p}(t)-\frac{4}{3}\unicode[STIX]{x03C0}r_{p}(t)^{3}\unicode[STIX]{x1D70C}_{w}$. Thus, assuming that $\unicode[STIX]{x1D70C}_{w}$ does not change substantially in time (because of temperature changes for example),

(2.9)$$\begin{eqnarray}\displaystyle \frac{\text{d}m_{p}(t)}{\text{d}t} & = & \displaystyle 4\unicode[STIX]{x03C0}r_{p}^{2}\unicode[STIX]{x1D70C}_{w}\frac{\text{d}r_{p}(t)}{\text{d}t}\nonumber\\ \displaystyle & = & \displaystyle 4\unicode[STIX]{x03C0}r_{p}^{2}\unicode[STIX]{x1D70C}_{w}{\mathcal{R}}(t)\nonumber\\ \displaystyle & = & \displaystyle {\mathcal{M}}(t),\end{eqnarray}$$

which readily gives the evolution equation for the drop radius.

Similarly, the temporal evolution of the drop temperature $T_{p}$ is given in Pruppacher & Klett (Reference Pruppacher and Klett1996) (their equation (13.65)):

(2.10)$$\begin{eqnarray}m_{p}c_{ps}\frac{\text{d}T_{p}(t)}{\text{d}t}=4\unicode[STIX]{x03C0}r_{p}k_{a}^{\ast }(T_{air}-T_{p})+L_{v}\frac{\text{d}m_{p}(t)}{\text{d}t},\end{eqnarray}$$

which yields

(2.11)$$\begin{eqnarray}\displaystyle \frac{\text{d}T_{p}(t)}{\text{d}t} & = & \displaystyle \frac{4\unicode[STIX]{x03C0}r_{p}}{m_{p}c_{ps}}(k_{a}^{\ast }(T_{air}-T_{p})+L_{v}D_{v}^{\ast }(\unicode[STIX]{x1D70C}_{v}-\unicode[STIX]{x1D70C}_{v,p}))\nonumber\\ \displaystyle & = & \displaystyle {\mathcal{T}}(t),\end{eqnarray}$$

where the rate of variation ${\mathcal{T}}={\mathcal{T}}(\unicode[STIX]{x1D70C},\boldsymbol{u},T_{air};\boldsymbol{v}_{p},r_{p},m_{p},T_{p})$ also depends on the macroscopic fields of the carrier fluid and on the Lagrangian variables of the drop. In the expression for ${\mathcal{T}}$, we neglect high-order effects related to temporal and spatial gradients in the air temperature (Michaelides & Feng Reference Michaelides and Feng1994). In the equation above, $c_{ps}$ is the specific heat of salty water and $k_{a}^{\ast }$ the modified thermal conductivity of air. The two terms in ${\mathcal{T}}(t)$ represent the changes to the drop temperature from the sensible and latent heat flux exchanges.

The modified diffusivity for water vapour, $D_{v}^{\ast }$, and the modified thermal conductivity of air, $k_{a}^{\ast }$, include ventilation and non-continuum effects and are, respectively,

(2.12)$$\begin{eqnarray}\displaystyle & \displaystyle D_{v}^{\ast }=\frac{f_{v}D_{v}}{\displaystyle \frac{r_{p}}{r_{p}+\unicode[STIX]{x1D6E5}_{v}}+\frac{D_{v}}{r_{p}\unicode[STIX]{x1D6FC}_{c}}\left(\frac{2\unicode[STIX]{x03C0}M_{w}}{RT_{air}}\right)^{1/2}}, & \displaystyle\end{eqnarray}$$

(2.13)$$\begin{eqnarray}\displaystyle & \displaystyle k_{a}^{\ast }=\frac{f_{h}k_{a}}{\displaystyle \frac{r_{p}}{r_{p}+\unicode[STIX]{x1D6E5}_{T}}+\frac{k_{a}}{r_{p}\unicode[STIX]{x1D6FC}_{T}\unicode[STIX]{x1D70C}_{a}c_{p,a}}\left(\frac{2\unicode[STIX]{x03C0}M_{a}}{RT_{air}}\right)^{1/2}}, & \displaystyle\end{eqnarray}$$

where $D_{v}$ and $k_{a}$ are the molecular diffusivity for water vapour and thermal conductivity of air, respectively. Also, $M_{a}$ is the molecular weight of dry air; $c_{p,a}$ and $\unicode[STIX]{x1D70C}_{a}$ are, respectively, the specific heat (at constant pressure) and density of dry air, with constants $\unicode[STIX]{x1D6E5}_{v}=8\times 10^{-8}$, $\unicode[STIX]{x1D6FC}_{c}=0.06$, $\unicode[STIX]{x1D6E5}_{T}=2.16\times 10^{-7}$ and $\unicode[STIX]{x1D6FC}_{T}=0.7$. We also include ventilation coefficients $f_{v}=f_{h}=1+\sqrt{Re_{p}}/4$, which are corrections to the heat and water vapour diffusivity for large instantaneous drop Reynolds numbers, i.e. $Re_{p}\gg 1$ (Edson & Fairall Reference Edson and Fairall1994). These are analogous to the mean Sherwood and Nusselt numbers corrected using Ranz–Marshall correlations (Ranz & Marshall Reference Ranz and Marshall1952).

In short, in this Lagrangian approach, the system that governs the evolution of sea spray drops is

(2.14)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\text{d}\boldsymbol{x}_{p}}{\text{d}t}=\boldsymbol{v}_{p},\quad \frac{\text{d}\boldsymbol{v}_{p}}{\text{d}t}=\frac{1}{m_{p}}\boldsymbol{{\mathcal{F}}},\\ \displaystyle \frac{\text{d}r_{p}}{\text{d}t}={\mathcal{R}},\quad \frac{\text{d}m_{p}}{\text{d}t}={\mathcal{M}},\quad \frac{\text{d}T_{p}}{\text{d}t}={\mathcal{T}}.\end{array}\right\}\end{eqnarray}$$

Figure 1 shows the time evolution of a water drop’s temperature (blue), radius (red) and velocity (black) when placed at time $t=0$ in air with a wind speed of $U_{10}=15~\text{m}~\text{s}^{-1}$, a temperature of $T_{air}=18\,^{\circ }\text{C}$ and a relative humidity of 75 %. The drop is initially at rest and with an initial radius of $r_{p}=100~\unicode[STIX]{x03BC}\text{m}$ and initial temperature of $T_{p}=20\,^{\circ }\text{C}$. From figure 1, it is apparent that the drop responds quickly to the drag forces imposed on it. It also transfers sensible heat to the air and rapidly cools down. The latent heat transfer between the air and the drop is slower than both momentum and sensible heat transfers. Indeed, the radius of the drop starts to diminish because of evaporation only after having been exposed to ambient atmospheric conditions for several seconds. The separation between latent and sensible heat transfers from a spray drop was first noted by Andreas (Reference Andreas1990) who subsequently explored the implication of this phenomenon for air–sea spray-mediated heat fluxes (e.g. Andreas Reference Andreas1990, Reference Andreas1992; Andreas et al. Reference Andreas, Edson, Monahan, Rouault and Smith1995; Andreas & DeCosmo Reference Andreas, DeCosmo and Geernaert1999; Andreas & Emanuel Reference Andreas and Emanuel2001). As such, the results of figure 1 are not new but they will inform results in this paper.

Figure 1. Time evolution of a water drop’s temperature (blue), radius (red) and velocity (black). The drop is placed at time $t=0$ in air with a wind speed of $U_{10}=15~\text{m}~\text{s}^{-1}$, a temperature of $T_{air}=18\,^{\circ }\text{C}$ and a relative humidity of $75\,\%$. The drop has an initial radius of $r_{p}=100~\unicode[STIX]{x03BC}\text{m}$ and initial temperature of $T_{p}=20\,^{\circ }\text{C}$.

2.3 Drop conservation and kinetic equation

Let us now consider a set of drops described by the distribution function $f(t,\boldsymbol{x},\boldsymbol{v},r,m,T)$, which is the number density of drops at time $t$ in the phase space $\mathbb{R}_{\boldsymbol{x}}^{3}\times \mathbb{R}_{\boldsymbol{v}}^{3}\times [\!0,+\infty \![_{r}\times [\!0,+\infty \![_{m}\times [\!0,+\infty \![_{T}$. It is the number of drops per $\text{d}\boldsymbol{x}$, per $\text{d}\boldsymbol{v}$, per $\text{d}r$, per $\text{d}m$, per $\text{d}T$ and has units of [(number of drops) $(\text{m})^{-3}~(\text{m}~\text{s}^{-1})^{-3}~(\text{m}^{-1})~(\text{kg}^{-1})(\text{K}^{-1})$]. In other words, $f(t,\boldsymbol{x},\boldsymbol{v},r,m,T)\,\text{d}\boldsymbol{x}\,\text{d}\boldsymbol{v}\,\text{d}r\,\text{d}m\,\text{d}T$ is the number of drops that at time $t$ are at position $\boldsymbol{x}\pm \text{d}\boldsymbol{x}$ (where $\text{d}\boldsymbol{x}=\text{d}x\,\text{d}y\,\text{d}z$) with velocity $\boldsymbol{v}\pm \,\text{d}\boldsymbol{v}$, with a radius $r\pm \text{d}r$, with mass $m\pm \text{d}m$ and at temperature $T\pm \text{d}T$.

The governing (conservation) equation for this distribution function is

(2.15)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}f+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{v}f)+\unicode[STIX]{x1D735}_{\boldsymbol{v}}\boldsymbol{\cdot }\left(\frac{\boldsymbol{{\mathcal{F}}}}{m}f\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}({\mathcal{R}}f)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}m}({\mathcal{M}}f)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}({\mathcal{T}}f)=0.\end{eqnarray}$$

This is known as the Williams–Boltzmann equation (Marchiso & Fox Reference Marchiso and Fox2013) and can be understood as a conservation law for the number of drops in the multi-dimensional phase space that represents physical space $\boldsymbol{x}$, but also velocity space $\boldsymbol{v}$, radius $r$, mass $m$ and temperature $T$ domains. Indeed, the first two terms of this equation are recognized as the classical conservation law involving the flux divergence. Likewise, $\unicode[STIX]{x1D735}_{\boldsymbol{v}}$ denotes differentiation with respect to the velocity vector components. Note that the Eulerian rates of variation of velocity, radius, mass and temperature used in (2.15) are defined through their Lagrangian versions (2.2), (2.7), (2.11), (2.9), in which the Lagrangian variables $\boldsymbol{x}_{p}$, $\boldsymbol{v}_{p}$, $r_{p}$, $m_{p}$ and $T_{p}$ are replaced by the Eulerian kinetic variables $\boldsymbol{x}$, $\boldsymbol{v}$, $r$, $m$ and $T$. If the drops do not interact, this change of variables is straightforward. In the case of spray drops, there is a weak interaction because the drops modify the airflow, which in turn can influence the drop transport (see § 3). In that case, equation (2.15) is nonetheless recovered using so-called mean-field techniques (see Desvillettes, Golse & Ricci (Reference Desvillettes, Golse and Ricci2008) for a rigorous derivation of a simple example). For particles with strong interactions, such as charged particles submitted to a long-range interaction force induced by all the other particles, the derivation of (2.15) can be substantially more complex and the Eulerian rates of variation require some averaging process. This is not necessary here in the case of sea spray.

Note also that collisions, breakup, coalescence and nucleation of drops are normally taken into account by source (and sink) terms on the right-hand side of (2.15). However, since the volume fraction of sea spray is expected to be very small (see the introduction), collision and coalescence are expected to be negligible. Nucleation processes are important in cloud formation but are neglected for sea spray applications. Finally, breakup is expected to be a substantial mechanism of spray formation per se, particularly for large spume drops (e.g. Mueller & Veron Reference Mueller and Veron2009b), but presumably only for short times after the initial ejection of the spray. Furthermore, the details of generation of large spume drops at the ocean surface remain poorly resolved. Thus, in this analysis, we consider the spray distribution after the spray is formed and once the drops are no longer fragmenting in the airflow after their ejection from the interface. Overall, source and sink terms are expected to provide minimal contributions to the overall flux balances and can be neglected.

The source of drops can therefore be naturally taken into account by boundary conditions. Indeed, equation (2.15) is set in a moving domain where one boundary is the free sea surface. At this boundary, where drops are injected into the airflow, the distribution function of incoming drops has to be defined. To that end, let $\boldsymbol{x}$ be a point on this free surface, $\boldsymbol{u}_{s}(t,\boldsymbol{x})$ be the velocity of the free surface at this point and $\boldsymbol{n}(t,\boldsymbol{x})$ the unit vector normal to this free surface, directed towards the airflow. Then, the source of drops coming from $\boldsymbol{x}$ is simply $f(t,\boldsymbol{x},\boldsymbol{v},r,m,T)$ for every $\boldsymbol{v}$ such that $(\boldsymbol{v}-\boldsymbol{u}_{s}(t,\boldsymbol{x}))\boldsymbol{\cdot }\boldsymbol{n}(t,\boldsymbol{x})>0$ (i.e. drops whose velocities are directed towards the airflow). This value, denoted by $f_{gen}$, thus needs to be known.

Fortunately, because of the interest in assessing the impacts of spray on Earth’s climate and weather, some efforts have been made in attempting to evaluate the air–sea flux of spray. In practice, so-called sea spray generation functions, $S(t,\boldsymbol{x},r)$, are commonly utilized. They give the number of drops generated at the surface per unit surface area, per unit time and per radius increment. We note, however, that $S(t,\boldsymbol{x},r)$ is not trivial to obtain and nearly all spray generation estimates derive from measurements of drop concentration, $N(t,\boldsymbol{x},r)$, at some elevation above the surface. Drop concentration $N(t,\boldsymbol{x},r)$ is then the number of drops per unit volume of air per radius increment, and is a function of space, time and drop radius. The difference between the flux and the concentration is typically estimated assuming steady state resulting from a balance between gravitational deposition and turbulent suspension. Under these assumptions, the source flux $S(\boldsymbol{x},r)$ can then be inferred from concentration measurements (e.g. Andreas et al. Reference Andreas, Jones and Fairall2010; Veron Reference Veron2015) and $S(\boldsymbol{x},r)=v_{o}(r)N(\boldsymbol{x},r)$, where $v_{o}(r)$ is a vertical suspension (or deposition) velocity (Slinn & Slinn Reference Slinn and Slinn1980; Andreas et al. Reference Andreas, Jones and Fairall2010). However, suspension velocities are likely to be a function of near-surface turbulence and the details of the drop formation process (Mueller & Veron Reference Mueller and Veron2009b; Veron et al. Reference Veron, Hopkins, Harrison and Mueller2012; Troitskaya et al. Reference Troitskaya, Kandaurov, Ermakova, Kozlov, Sergeev and Zilitinkevich2018).

For the purpose of the model, we assume that the (known) radius-dependent ejection velocity is $\boldsymbol{v}_{gen}(r)$. (Note that the ejection velocity can take a more general form in order to include, for example, distributions of ejection angles. Here, $v_{o}$ can be understood as the vertical component of $\boldsymbol{v}_{gen}$.) We also note that the drops generated at the surface have the same density as sea water, and therefore their mass $m_{gen}$ is given by $m_{gen}(r)=\frac{4}{3}\unicode[STIX]{x03C0}r^{3}\unicode[STIX]{x1D70C}_{sw}$. Likewise, we can also assume that all the drops are ejected with the same sea temperature $T_{sw}(t,\boldsymbol{x})$. Then, dimensional arguments show that the number of drops generated at the surface per unit surface area, per unit time, per velocity, radius, mass and temperature increment is

(2.16)$$\begin{eqnarray}(\boldsymbol{v}-\boldsymbol{u}_{s}(t,\boldsymbol{x}))\boldsymbol{\cdot }\boldsymbol{n}(t,\boldsymbol{x})f_{gen}(t,\boldsymbol{x},\boldsymbol{v},r,m,T)=S\unicode[STIX]{x1D6FF}_{\boldsymbol{v}-\boldsymbol{v}_{gen}(r)}\unicode[STIX]{x1D6FF}_{m-m_{gen}(r)}\unicode[STIX]{x1D6FF}_{T-T_{sw}(t,\boldsymbol{x})},\end{eqnarray}$$

where $(\boldsymbol{v}-\boldsymbol{u}_{s}(t,\boldsymbol{x}))\boldsymbol{\cdot }\boldsymbol{n}(t,\boldsymbol{x})>0$ and where the ‘$\unicode[STIX]{x1D6FF}$’ symbols denote Dirac delta functions centred in $\boldsymbol{v}_{gen}(r)$, $m_{gen}(r)$ and $T_{sw}(t,\boldsymbol{x})$ for the velocity, mass and temperature, respectively. Once $S$ and $\boldsymbol{v}_{gen}(r)$ are known, this relation fully defines the boundary value $f_{gen}$ for (2.15).

Finally, note that the Williams–Boltzmann equation (2.15) needs to be solved concurrently with the equations for the atmospheric airflow, i.e. the (Eulerian) conservation of mass, momentum, and energy, respectively:

(2.17)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.18)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u}\otimes \boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}+\unicode[STIX]{x1D70C}\boldsymbol{g}, & \displaystyle\end{eqnarray}$$

(2.19)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(E\boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{Q}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D749}\boldsymbol{u})+\unicode[STIX]{x1D70C}\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the air density, $\boldsymbol{u}$ the air velocity, $\unicode[STIX]{x1D749}$ the atmospheric stress including pressure, $E$ the total energy density of the air and $\boldsymbol{Q}$ the heat flux.

3.1 Conservation of mass

For example, since ${\mathcal{M}}\,\text{d}t$ is the mass lost (gained) from a drop during a small time interval $\text{d}t$, the mass gained (lost) by the fluid during $\text{d}t$, from all the drops at position $\boldsymbol{x}\pm \text{d}\boldsymbol{x}$, is

(3.1)$$\begin{eqnarray}-\!\text{d}\boldsymbol{x}\,\text{d}t\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{\mathbb{R}_{\boldsymbol{v}}^{3}}{\mathcal{M}}f(t,\boldsymbol{x},\boldsymbol{v},r,m,T)\,\text{d}\boldsymbol{v}\,\text{d}r\,\text{d}m\,\text{d}T.\end{eqnarray}$$

Note that the domain considered here now comprises all possible values in the entire phase space $\mathbb{R}_{\boldsymbol{v}}^{3}\times [0,\infty ]_{r}\times [0,\infty ]_{m}\times [0,\infty ]_{T}$. To lighten the notation, we denote

(3.2)$$\begin{eqnarray}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{\mathbb{R}_{\boldsymbol{v}}^{3}}{\mathcal{A}}(t,\boldsymbol{x},\boldsymbol{v},r,m,T)\,\text{d}\boldsymbol{v}\,\text{d}r\,\text{d}m\,\text{d}T=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{A}}\,\text{d}\boldsymbol{\mathfrak{X}},\end{eqnarray}$$

where $\boldsymbol{\mathfrak{X}}=(\boldsymbol{v},r,m,T)$ and $\unicode[STIX]{x1D734}_{\mathfrak{X}}=\mathbb{R}_{\boldsymbol{v}}^{3}\times [\!0,+\infty \![_{r}\times [\!0,+\infty \![_{m}\times [\!0,+\infty \![_{T}$.

So, conservation of mass in the air, accounting for the mass exchange between the air and the drops, reads

(3.3)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})=-\unicode[STIX]{x03C0}_{m},\end{eqnarray}$$

where

(3.4)$$\begin{eqnarray}\unicode[STIX]{x03C0}_{m}=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}\end{eqnarray}$$

is the mass production term due to spray drops. In other words, the water vapour transferred from (evaporation) or to (condensation) the spray drops will change the density of the air. In fact, while equations (2.17), (2.18) and (2.19) were pertinent for dry air, it is now necessary to consider moist air, i.e. the (binary) mixture of dry air and water vapour. As such, $\unicode[STIX]{x1D70C}$ in (3.3) is the density of moist air and $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{v}+\unicode[STIX]{x1D70C}_{a}$, the sum of the water vapour density and the density of dry air. In the remainder of the paper, unless explicitly noted as ‘dry’, the air is considered to be moist air.

Furthermore, we need an equation for the water vapour density which is used to estimate the rate of variation of both the mass and the temperature of the drops (see (2.6) and (2.11)). Water vapour advects with the same velocity as the moist air, plus a small molecular motion $\unicode[STIX]{x1D74A}_{v}$ induced by the gradients in water vapour. The equation for the water vapour density is then

(3.5)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}_{v}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{v}(\boldsymbol{u}+\unicode[STIX]{x1D74A}_{v}))=-\unicode[STIX]{x03C0}_{m}.\end{eqnarray}$$

The diffusion from molecular motion can further be modelled with a Fick diffusion law $\unicode[STIX]{x1D74A}_{v}=(-1/q)D_{v}\unicode[STIX]{x1D735}q$ with diffusivity $D_{v}$, and where $q=\unicode[STIX]{x1D70C}_{v}/\unicode[STIX]{x1D70C}$ is the specific humidity of air. Thus, the conservation of water vapour reads

(3.6)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}_{v}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{v}\boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}D_{v}\unicode[STIX]{x1D735}q)-\unicode[STIX]{x03C0}_{m}.\end{eqnarray}$$

For completeness, equations (3.6) and (3.3) yield the mass conservation equation for dry air:

(3.7)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}_{a}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{a}\boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}D_{v}\unicode[STIX]{x1D735}(1-q)),\end{eqnarray}$$

where $\unicode[STIX]{x1D74A}_{a}=(-1/(1-q))D_{v}\unicode[STIX]{x1D735}(1-q)$. Incidentally, $\unicode[STIX]{x1D70C}_{a}\unicode[STIX]{x1D74A}_{a}+\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x1D74A}_{v}=0$ since we consider here a binary mixture with $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{v}+\unicode[STIX]{x1D70C}_{a}$.

3.2 Conservation of momentum

Equivalently, if $\boldsymbol{F}$ is the force from the fluid on the drop, the force on the fluid from all drops at position $\boldsymbol{x}\pm \text{d}\boldsymbol{x}$ is $-\text{d}\boldsymbol{x}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{F}f\,\text{d}\boldsymbol{\mathfrak{X}}$. The change of momentum of the air is also due to an increase (decrease) in mass of water vapour coming from evaporating (condensing) drops. The corresponding rate of variation of momentum is $-\text{d}\boldsymbol{x}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{v}{\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}$. Then conservation of momentum in the air, accounting for the momentum exchange between the air and the drops, reads

(3.8)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u}\otimes \boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}+\unicode[STIX]{x1D70C}\boldsymbol{g}-\unicode[STIX]{x1D745}_{\boldsymbol{u}},\end{eqnarray}$$

where

(3.9)$$\begin{eqnarray}\unicode[STIX]{x1D745}_{\boldsymbol{u}}=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{F}f\,\text{d}\boldsymbol{\mathfrak{X}}+\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{v}{\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}\end{eqnarray}$$

is the momentum production term due to the drops. The corresponding equation for the air velocity is

(3.10)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}\right)+\unicode[STIX]{x1D735}p & = & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}+\unicode[STIX]{x1D70C}\boldsymbol{g}-\unicode[STIX]{x1D745}_{\boldsymbol{u}}+\unicode[STIX]{x03C0}_{m}\boldsymbol{u}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}+\unicode[STIX]{x1D70C}\boldsymbol{g}-\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{F}f\,\text{d}\boldsymbol{\mathfrak{X}}-\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}(\boldsymbol{v}-\boldsymbol{u}){\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}},\qquad\end{eqnarray}$$

where the stress tensor has been decomposed into its atmospheric pressure and viscous components as $\unicode[STIX]{x1D749}=-p\boldsymbol{I}+\unicode[STIX]{x1D748}$. Also, the viscous stress tensor is easily obtained from the gradient velocity tensor and its transpose using $\unicode[STIX]{x1D748}=\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-(\frac{2}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\boldsymbol{I})$. In developing (3.8) and (3.10), strictly speaking, one needs to first consider separate momentum equations for dry air and water vapour. These can easily be combined by neglecting the terms involving molecular diffusive motions for dry air and water vapour (see equations (3.6) and (3.7)) of $O(\unicode[STIX]{x1D74A}_{a}^{2})$ or $O(\unicode[STIX]{x1D74A}_{v}^{2})$.

3.3 Conservation of energy

Finally, we seek to estimate the energy exchanged between the drops and the air, which we denote $\unicode[STIX]{x03C0}_{e}$. The energy of a single drop is the sum of its internal and kinetic energy:

(3.11)$$\begin{eqnarray}E_{p}(t)=m_{p}(t)e_{p}(t)+{\textstyle \frac{1}{2}}m_{p}(t)\boldsymbol{v}_{p}^{2}(t),\end{eqnarray}$$

where $e_{p}$ denotes the drop specific internal energy which is a function of $T_{p}(t)$. So, the rate of change of a spray drop energy is

(3.12)$$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}E_{p}(t) & = & \displaystyle \left(e_{p}(t)+\frac{1}{2}\boldsymbol{v}_{p}^{2}(t)\right)\frac{\text{d}}{\text{d}t}m_{p}(t)+m_{p}(t)\left(\frac{\text{d}}{\text{d}t}e_{p}(t)+\boldsymbol{v}_{p}(t)\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{v}_{p}(t)}{\text{d}t}\right)\nonumber\\ \displaystyle & = & \displaystyle \left(e_{p}(t)+\frac{1}{2}\boldsymbol{v}_{p}^{2}\right){\mathcal{M}}+m_{p}\frac{\text{d}}{\text{d}t}e_{p}(t)+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}\nonumber\\ \displaystyle & = & \displaystyle {\mathcal{E}}(t).\end{eqnarray}$$

However, gravity is a force that is external to the closed air–drops system, and its work should not be accounted for in the energy exchange between the drops and the air. Consequently, to define the rate of variation of energy of the drop, the work of $\boldsymbol{{\mathcal{F}}}$ is replaced by the work of the drag force $\boldsymbol{F}$ in the previous relation. Furthermore, the specific internal energy of the drop is simply the specific internal energy of salty water at the temperature of the drop, so $e_{p}(t)=e_{sw}(T_{p}(t))$. Then, from the definition of $c_{ps}$, we obtain $\text{d}e_{p}(t)/\text{d}t=c_{ps}(\text{d}T_{p}(t)/\text{d}t)=c_{ps}{\mathcal{T}}$. In fact, equation (2.10) can now be understood as the rate of change of a drop’s internal energy. Finally,

(3.13)$$\begin{eqnarray}{\mathcal{E}}(t)=e_{sw}(T_{p}){\mathcal{M}}+m_{p}c_{ps}{\mathcal{T}}+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{F}+{\textstyle \frac{1}{2}}{\mathcal{M}}\boldsymbol{v}_{p}^{2}.\end{eqnarray}$$

In the equation above, one can see that changes in internal energy of a drop may arise from both a change of mass and a change of temperature, while the change of kinetic energy results from the work of friction forces and a change in drop mass. As with the mass and momentum conservation equations above, the energy production term from all the drops is

(3.14)$$\begin{eqnarray}\unicode[STIX]{x03C0}_{e}=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{E}}f\,\text{d}\boldsymbol{\mathfrak{X}}\end{eqnarray}$$

in which ${\mathcal{E}}$ has to be understood as an Eulerian quantity derived from its Lagrangian definition (3.13). In other words, equation (3.14) reads

(3.15)$$\begin{eqnarray}\unicode[STIX]{x03C0}_{e}=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}e_{sw}(T){\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}+\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}mc_{ps}{\mathcal{T}}f\,\text{d}\boldsymbol{\mathfrak{X}}+\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{F}f\,\text{d}\boldsymbol{\mathfrak{X}}+\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\textstyle \frac{1}{2}}\boldsymbol{v}^{2}{\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}.\end{eqnarray}$$

We can now develop the energy conservation for the air, accounting for the input from the spray. However, in contrast to the derivation of the momentum equation above, we chose here to keep the moist air energy density $E$ explicitly separated into the energy density for dry air $E_{a}$ and that for water vapour $E_{v}$. With spray effects included, equation (2.19) reads

(3.16)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(E_{a}+E_{v})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }((E_{a}+E_{v})\boldsymbol{u})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(E_{a}\unicode[STIX]{x1D74A}_{a}+E_{v}\unicode[STIX]{x1D74A}_{v})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{Q}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D748}\boldsymbol{u})-\unicode[STIX]{x1D735}\boldsymbol{\cdot }((\boldsymbol{u}+\unicode[STIX]{x1D74A}_{a})p_{a}+(\boldsymbol{u}+\unicode[STIX]{x1D74A}_{v})p_{v})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70C}\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}-\unicode[STIX]{x03C0}_{e},\end{eqnarray}$$

where $p_{a}$ is the partial pressure for dry air and $p_{v}$ the water vapour partial pressure, and where terms of $O(\unicode[STIX]{x1D74A}_{a}^{2})$ or $O(\unicode[STIX]{x1D74A}_{v}^{2})$ are once more neglected. The separation between water vapour and dry air is useful here because we wish to keep the rate of work done by the respective partial pressures. We note here that neglecting $\unicode[STIX]{x1D74A}_{v}p_{v}$ and $\unicode[STIX]{x1D74A}_{a}p_{a}$ would allow us to simply use $p=p_{v}+p_{a}$ and incorporate the pressure in the stress tensor of the (moist) air as in (3.10). However, it is convenient to keep the partial pressures explicitly out of the stress tensor so that expressions for the enthalpy of the dry air and water vapour can be uncovered. For example, as done above for the drop, the energy density of the water vapour can be decomposed into internal and kinetic energy:

(3.17)$$\begin{eqnarray}E_{v}=\unicode[STIX]{x1D70C}_{v}e_{v}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{v}|\boldsymbol{u}|^{2},\end{eqnarray}$$

where $e_{v}$ is the specific internal energy for water vapour, and where the contribution of the diffusion velocity to the kinetic energy can be neglected. By definition, the specific enthalpy for the water vapour is $h_{v}=e_{v}+p_{v}/\unicode[STIX]{x1D70C}_{v}$, and thus

(3.18)$$\begin{eqnarray}E_{v}+p_{v}=\unicode[STIX]{x1D70C}_{v}h_{v}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{v}|\boldsymbol{u}|^{2}.\end{eqnarray}$$

Similarly,

(3.19)$$\begin{eqnarray}E_{a}+p_{a}=\unicode[STIX]{x1D70C}_{a}h_{a}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{a}|\boldsymbol{u}|^{2},\end{eqnarray}$$

in which $h_{a}=e_{a}+p_{a}/\unicode[STIX]{x1D70C}_{a}$ is the specific enthalpy of dry air and $e_{a}$ the specific internal energy for dry air. Therefore, equation (3.16) can be written as

(3.20)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}_{a}e_{a}+\unicode[STIX]{x1D70C}_{v}e_{v})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }((\unicode[STIX]{x1D70C}_{a}h_{a}+\unicode[STIX]{x1D70C}_{v}h_{v})\boldsymbol{u})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{a}h_{a}\unicode[STIX]{x1D74A}_{a}+\unicode[STIX]{x1D70C}_{v}h_{v}\unicode[STIX]{x1D74A}_{v})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{1}{2}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}\right)+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{1}{2}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}\boldsymbol{u}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{Q}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D748}\boldsymbol{u})+\unicode[STIX]{x1D70C}\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}-\unicode[STIX]{x03C0}_{e}.\end{eqnarray}$$

Using (3.6), (3.8) and (3.20), we find the equation governing $e$, the specific internal energy of moist air:

(3.21)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}e)+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}e\boldsymbol{u}) & = & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{Q}+\unicode[STIX]{x1D748}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}-p\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D745}_{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{u}-\frac{1}{2}|\boldsymbol{u}|^{2}\unicode[STIX]{x03C0}_{m}-\unicode[STIX]{x03C0}_{e},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}e=\unicode[STIX]{x1D70C}_{a}e_{a}+\unicode[STIX]{x1D70C}_{v}e_{v}$ and

(3.22)$$\begin{eqnarray}\displaystyle \boldsymbol{J} & = & \displaystyle \unicode[STIX]{x1D70C}_{a}h_{a}\unicode[STIX]{x1D74A}_{a}+\unicode[STIX]{x1D70C}_{v}h_{v}\unicode[STIX]{x1D74A}_{v}\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D70C}D_{v}(h_{a}\unicode[STIX]{x1D735}(1-q)+h_{v}\unicode[STIX]{x1D735}q)\end{eqnarray}$$

is a flux of enthalpy due to small molecular motions.

For modelling purposes, it is generally practical to look at the temperature of the carrier flow $T_{air}$. With tedious but simple algebra, the energy conservation equation (3.21) can be written as

(3.23)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}c_{v}\left(\frac{\unicode[STIX]{x2202}T_{air}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T_{air}\right) & = & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}T_{air})+\unicode[STIX]{x1D748}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}-p\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}+e_{a}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{a}\unicode[STIX]{x1D74A}_{a})+e_{v}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x1D74A}_{v})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D745}_{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{u}-\frac{1}{2}|\boldsymbol{u}|^{2}\unicode[STIX]{x03C0}_{m}-\unicode[STIX]{x03C0}_{e}+e_{v}\unicode[STIX]{x03C0}_{m},\end{eqnarray}$$

where we have taken the heat flux as given by the Fourier law $\boldsymbol{Q}=\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}T_{air}$ and where $c_{v}$ is the specific heat of air at constant volume, derived from the corresponding specific heat of water vapour and dry air using

(3.24)$$\begin{eqnarray}c_{v}=qc_{v,v}+(1-q)c_{v,a}.\end{eqnarray}$$

Also, in (3.23), $e_{v}$ and $e_{a}$ are taken at the air temperature $T_{air}$.

Expanding the spray production terms leads to

(3.25)

In the equation above, the spray feedback terms are on the right-hand side and evidently contain $f$ and its integrals. The first three feedback terms represent the heat gained by the air due to the work of the frictional drag force exerted by the drops on the air, a loss of kinetic energy of the drops (because of a change of mass as the drop evaporate) and the exchange of sensible heat between the air and the drops due to the variation of the temperature of the drops. The last feedback term represents a transfer in internal energy from the drops to water vapour as drops evaporate.

Also, we develop the term resulting from molecular diffusion:

(3.26)$$\begin{eqnarray}\displaystyle {\mathcal{D}} & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}+e_{a}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{a}\unicode[STIX]{x1D74A}_{a})+e_{v}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x1D74A}_{v})\nonumber\\ \displaystyle & = & \displaystyle RT_{air}\left(\frac{1}{M_{w}}-\frac{1}{M_{a}}\right)\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}D_{v}\unicode[STIX]{x1D735}q)\nonumber\\ \displaystyle & & \displaystyle \quad +\,(c_{p,v}-c_{p,a})\unicode[STIX]{x1D70C}D_{v}\unicode[STIX]{x1D735}q\boldsymbol{\cdot }\unicode[STIX]{x1D735}T_{air},\end{eqnarray}$$

where $c_{p,v}$ and $c_{p,a}$ are, respectively, the specific heat at constant pressure for water vapour and dry air. Finally,

(3.27)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle e_{sw}(T)=c_{ps}(T-T_{ref})+e_{sw}(T_{ref})\quad \text{and}\\ \displaystyle e_{v}(T_{air})=c_{v,v}(T_{air}-T_{ref})-\frac{R}{M_{w}}T_{ref}+L_{v},\end{array}\right\}\end{eqnarray}$$

where $T_{ref}$ is a reference temperature, generally taken as $T_{ref}=0\,^{\circ }\text{C}$, such that the latent heat of vaporization, $L_{v}$, is the enthalpy difference between water vapour and liquid water at $T_{ref}$. To simplify the notation, we write the last feedback term in (3.25) as

(3.28)$$\begin{eqnarray}\unicode[STIX]{x1D6F6}=\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}(e_{v}(T_{air})-e_{sw}(T)){\mathcal{M}}f\,\text{d}\boldsymbol{\mathfrak{X}}.\end{eqnarray}$$

4.1 Characteristic quantities and dimensionless variables

Here, we take $r_{o}$ as a characteristic radius of a population of drops. Their corresponding vertical settling velocity is denoted by $v_{o}$. The space scale $x_{o}$ is a typical transport scale for these drops which we choose as the significant wave height (Andreas Reference Andreas2004). Thus, the time scale $t_{o}=x_{o}/v_{o}$ roughly corresponds to a typical average time for a drop to fall from the significant wave height to the mean water level. Andreas (Reference Andreas2004) considers $t_{o}$ as an adequate proxy for the drop time of flight, but we emphasize here that the details of the ejection mechanism such as the initial velocity and angle of ejection of the drops are not accounted for. There is some evidence that large drops ejected near the wave crest may fall back near the crest and not at the mean water level (Mueller & Veron Reference Mueller and Veron2014a). And the ejection velocity, particularly for small, bubble-generated drops, can be significant (Blanchard Reference Blanchard1989; Spiel Reference Spiel1995, Reference Spiel1997, Reference Spiel1998). Nevertheless, for the purpose of the non-dimensional scaling, $t_{o}$ is a suitable estimate for the drop time of flight. The characteristic temperature of the drops is naturally taken as the sea temperature and the characteristic drop mass is therefore $m_{o}=\unicode[STIX]{x1D70C}_{sw}\frac{4}{3}\unicode[STIX]{x03C0}r_{o}^{3}$. Finally, the corresponding characteristic values ${\mathcal{F}}_{o}$, ${\mathcal{R}}_{o}$, ${\mathcal{M}}_{o}$ and ${\mathcal{T}}_{o}$ of the the rates of variation $\boldsymbol{{\mathcal{F}}}$, ${\mathcal{R}}$, ${\mathcal{M}}$ and ${\mathcal{T}}$ of the drop momentum, radius, mass and temperature, respectively, are obtained using formulas (2.3), (2.7), (2.11) and (2.9). Because sea spray is made of spray drops that are disperse in size (ranging here from $O(1)~\unicode[STIX]{x03BC}\text{m}$ to $O(1)~\text{mm}$), defining the characteristic value $f_{o}$ of the distribution function $f$ is delicate. However, we shall see later that $f_{o}$ appears in all spray feedback terms and thus naturally falls out of a term-by-term dimensional comparison.

For the airflow, we take the same space and time scales. However, since airflow and spray drops can have very different velocities, especially for large spume drops with substantial mass and inertia, we use a velocity scale $u_{o}$ for the airflow as a typical wind speed, possibly different from $v_{o}$. An obvious choice for $u_{o}$ is the mean wind speed measured at a height of 10 m above the surface, $U_{10}$. The characteristic air mass density is $\unicode[STIX]{x1D70C}_{o}$, and its characteristic temperature is $T_{o}$ which is the same as the characteristic sea temperature. Derived characteristic values of viscosity $\unicode[STIX]{x1D707}_{o}$ and heat transfer $\unicode[STIX]{x1D705}_{o}$ coefficients can be easily obtained.

In the following, the corresponding dimensionless quantities are denoted with a prime symbol $^{\prime }$, and defined by $[.]^{\prime }=[.]/[.]_{o}$; for instance $t^{\prime }=t/t_{o}$ for the time variable.

4.2 Dimensionless Williams–Boltzmann equation

The Williams–Boltzmann equation (2.15) can be non-dimensionalized to

(4.1)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}f^{\prime }+\unicode[STIX]{x1D735}^{\prime }\boldsymbol{\cdot }(f^{\prime }\boldsymbol{v}^{\prime })+\frac{t_{o}}{\unicode[STIX]{x1D70F}_{v}}\unicode[STIX]{x1D735}_{\boldsymbol{v}^{\prime }}\boldsymbol{\cdot }\left(\frac{f^{\prime }}{m^{\prime }}\boldsymbol{{\mathcal{F}}}^{\prime }\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{t_{o}}{\unicode[STIX]{x1D70F}_{R}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\prime }}({\mathcal{R}}^{\prime }f^{\prime })+\frac{t_{o}}{\unicode[STIX]{x1D70F}_{m}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}m^{\prime }}({\mathcal{M}}^{\prime }f^{\prime })+\frac{t_{o}}{\unicode[STIX]{x1D70F}_{T}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T^{\prime }}({\mathcal{T}}^{\prime }f^{\prime })=0,\end{eqnarray}$$

where the relaxation times $\unicode[STIX]{x1D70F}_{v}$, $\unicode[STIX]{x1D70F}_{R}$, $\unicode[STIX]{x1D70F}_{m}$ and $\unicode[STIX]{x1D70F}_{T}$ are characteristic times for the variation of the momentum, radius, mass and temperature of the drops, defined by

(4.2a-d)$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{v}=\frac{m_{o}v_{o}}{{\mathcal{F}}_{o}},\quad \unicode[STIX]{x1D70F}_{R}=\frac{r_{o}}{{\mathcal{R}}_{o}},\quad \unicode[STIX]{x1D70F}_{m}=\frac{m_{o}}{{\mathcal{M}}_{o}},\quad \unicode[STIX]{x1D70F}_{T}=\frac{T_{o}}{{\mathcal{T}}_{o}}.\end{eqnarray}$$

We note that the momentum term can further be decomposed into its gravitational and drag components such that

(4.3)$$\begin{eqnarray}\frac{t_{o}}{\unicode[STIX]{x1D70F}_{v}}\unicode[STIX]{x1D735}_{\boldsymbol{v}^{\prime }}\boldsymbol{\cdot }\left(\frac{f^{\prime }}{m^{\prime }}\boldsymbol{{\mathcal{F}}}^{\prime }\right)=\unicode[STIX]{x1D735}_{\boldsymbol{ v}^{\prime }}\boldsymbol{\cdot }\left(\frac{t_{o}}{\unicode[STIX]{x1D70F}_{g}}\boldsymbol{g}^{\prime }f^{\prime }+\frac{t_{o}}{\unicode[STIX]{x1D70F}_{Do}}\frac{1}{\unicode[STIX]{x1D70F}_{D}^{\prime }}\left(\frac{1}{\unicode[STIX]{x1D702}}\boldsymbol{u}^{\prime }-\boldsymbol{v}^{\prime }\right)f^{\prime }\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D702}=v_{o}/u_{o}$ is the ratio of the characteristic drop and airflow velocities, and where the dimensionless gravity vector is $\boldsymbol{g}^{\prime }=\boldsymbol{g}/g=-e_{z}$, with $g=|\boldsymbol{g}|$ the gravity magnitude. Then, $\unicode[STIX]{x1D70F}_{g}=v_{o}/g$ is the characteristic drop time scale due to gravity and $\unicode[STIX]{x1D70F}_{Do}=2r_{o}^{2}\unicode[STIX]{x1D70C}_{sw}/(9\unicode[STIX]{x1D707}_{o}\unicode[STIX]{x1D6EF}_{o})$ is the characteristic drop response time to the drag force, i.e. $\unicode[STIX]{x1D70F}_{D}=\unicode[STIX]{x1D70F}_{Do}\unicode[STIX]{x1D70F}_{D}^{\prime }$ with $\unicode[STIX]{x1D70F}_{D}^{\prime }={r^{\prime }}^{2}/(\unicode[STIX]{x1D707}^{\prime }\unicode[STIX]{x1D6EF}^{\prime })$.

Figure 2. Characteristic times of variation of the drag force $\unicode[STIX]{x1D70F}_{Do}$, radius $\unicode[STIX]{x1D70F}_{R}$, mass $\unicode[STIX]{x1D70F}_{m}$ and temperature $\unicode[STIX]{x1D70F}_{T}$ of the drop as a function of the characteristic radius $r_{0}$ and for air temperature of $T_{air}=18\,^{\circ }\text{C}$, $T_{p}=20\,^{\circ }\text{C}$ and $75\,\%$ relative humidity. Also, the characteristic times $\unicode[STIX]{x1D70F}_{g}$ and $t_{o}$ are shown for wind speeds of $U_{10}=5~\text{m}~\text{s}^{-1}$ with a significant wave height of $x_{o}=0.5~\text{m}$ (solid lines) and $U_{10}=40~\text{m}~\text{s}^{-1}$ with a significant wave height of $x_{o}=6.4~\text{m}$ (dashed lines).

Figure 2 shows the characteristic times of variation of the drag force $\unicode[STIX]{x1D70F}_{Do}$, radius $\unicode[STIX]{x1D70F}_{R}$, mass $\unicode[STIX]{x1D70F}_{m}$ and temperature $\unicode[STIX]{x1D70F}_{T}$ of the drop as a function of the characteristic radius $r_{0}$. These time scales are plotted here for specific ambient conditions, but they do vary (albeit weakly), in particular with relative humidity (see Andreas Reference Andreas1990). They are, however, strong functions of the drop radius. Indeed, $\unicode[STIX]{x1D70F}_{Do}$ is proportional to the drag on the drop, i.e. its cross-sectional area, and $\unicode[STIX]{x1D70F}_{R}$, $\unicode[STIX]{x1D70F}_{m}$ and $\unicode[STIX]{x1D70F}_{T}$ are all controlled by molecular exchanges through the drop surface area. Thus, all these time scales are proportional to $r_{o}^{2}$. From figure 2, one can easily see that small drops transfer sensible heat faster than larger ones. Also, $\unicode[STIX]{x1D70F}_{T}\ll \unicode[STIX]{x1D70F}_{R}$ and the temperature of a drop changes and adapts to ambient atmospheric conditions much faster (approximately 1000 times) than its radius.

We note here that in the absence of wind or other sources of airflow fluctuations and turbulence, the terminal fall velocity of a drop, $v_{t}(r_{o})$, is such that $v_{t}/g=\unicode[STIX]{x1D70F}_{Do}$, and $v_{t}(r_{o})$ is a function of the drop radius only. However, when the airflow is turbulent, the terminal velocity $v_{t}$ and the settling (or deposition) velocity $v_{o}$, of small drops in particular, can be substantially different. Indeed, small drops are affected by the turbulence in the airflow such that $v_{t}$ does not adequately represent the actual velocity at which these drops effectively settle (e.g. Slinn & Slinn Reference Slinn and Slinn1980; Andreas et al. Reference Andreas, Jones and Fairall2010). Here, we adopt a formulation found in Smith, Park & Consterdine (Reference Smith, Park and Consterdine1993) and originally proposed by Carruthers & Choularton (Reference Carruthers and Choularton1986):

(4.4)$$\begin{eqnarray}v_{o}=\frac{v_{t}}{\displaystyle 1-\text{exp}\left(\frac{v_{t}}{C_{D}U_{10}}\right)},\end{eqnarray}$$

where $C_{D}\sim O(10^{-3})$ is the air–sea aerodynamic drag coefficient. Thus $\unicode[STIX]{x1D70F}_{g}$ is a function of both the drop radius and the wind speed. It is plotted in figure 2 for $U_{10}=5~\text{m}~\text{s}^{-1}$ with a significant wave height estimated at $x_{o}=0.5~\text{m}$ (solid line) and for $U_{10}=40~\text{m}~\text{s}^{-1}$ with a significant wave height estimated at $x_{o}=6.4~\text{m}$ (dashed line).

It is worthwhile to compare the relaxation times with $t_{o}$, the atmospheric residence time of spray drops. For convenience, we have plotted $t_{o}$ in figure 2, and we note that $t_{o}$ also depends on the drop settling velocity and thus on the wind speed. For the remainder of the analysis, it is useful to introduce the following dimensionless numbers:

(4.5a-e)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{g}=\frac{t_{o}}{\unicode[STIX]{x1D70F}_{g}},\quad \unicode[STIX]{x1D6FC}_{D}=\frac{t_{o}}{\unicode[STIX]{x1D70F}_{Do}},\quad \unicode[STIX]{x1D6FC}_{R}=\frac{t_{o}}{\unicode[STIX]{x1D70F}_{R}},\quad \unicode[STIX]{x1D6FC}_{m}=\frac{t_{o}}{\unicode[STIX]{x1D70F}_{m}},\quad \unicode[STIX]{x1D6FC}_{T}=\frac{t_{o}}{\unicode[STIX]{x1D70F}_{T}},\end{eqnarray}$$

such that the non-dimensionalized Williams–Boltzmann equation can be written as

(4.6)

Figure 3. Non-dimensional relaxation times for the drag force on the drop $\unicode[STIX]{x1D6FC}_{D}$, the drop temperature $\unicode[STIX]{x1D6FC}_{T}$, the drop mass $\unicode[STIX]{x1D6FC}_{m}$ and the drop radius $\unicode[STIX]{x1D6FC}_{R}$. Solid lines indicate $U_{10}=5~\text{m}~\text{s}^{-1}$ with a significant wave height estimated at $x_{o}=0.5$ m and dashed lines indicate $U_{10}=40~\text{m}~\text{s}^{-1}$ with a significant wave height estimated at $x_{o}=6.4~\text{m}$.

Figure 3 shows these non-dimensional $\unicode[STIX]{x1D6FC}$ parameters which are the ratios of the drop residence time $t_{o}$ to the relaxation times $\unicode[STIX]{x1D70F}_{v}$, $\unicode[STIX]{x1D70F}_{R}$, $\unicode[STIX]{x1D70F}_{m}$ and $\unicode[STIX]{x1D70F}_{T}$. They are plotted as a function of $r_{o}$ and for a wind speed of $U_{10}=5~\text{m}~\text{s}^{-1}$ (solid lines) and $U_{10}=40~\text{m}~\text{s}^{-1}$ (dashed lines). When these ratios are above 1, this simply indicates that the drops remain suspended in the airflow long enough to exchange their momentum, sensible heat or water (vapour) mass with the surrounding atmosphere. For example, from figure 3, we deduce that at $U_{10}=5~\text{m}~\text{s}^{-1}$, drops with $r_{o}<150~\unicode[STIX]{x03BC}\text{m}$ will exchange momentum and sensible heat with the atmosphere, while only drops with $r_{o}<30~\unicode[STIX]{x03BC}\text{m}$ will have the opportunity to exchange latent heat with the air, thereby transferring water vapour to the atmosphere.

4.3 Dimensionless equations for the airflow including feedbacks

In this section, we non-dimensionalize the equations for the airflow including spray feedback effects.

4.3.1 Conservation of mass

The dimensionless mass conservation equation for the air is

(4.7)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}\unicode[STIX]{x1D70C}^{\prime }+\frac{1}{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D735}^{\prime }\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime })=-\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}^{\prime }}}{\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime },\end{eqnarray}$$

where $\unicode[STIX]{x1D712}=m_{o}f_{o}v_{o}^{3}r_{o}T_{o}m_{o}/\unicode[STIX]{x1D70C}_{o}$. Physically, $\unicode[STIX]{x1D712}$ is a mass fraction, i.e. the mass of the drops represented by $f_{o}$ divided by the mass of a characteristic volume of air.

4.3.2 Airflow velocity

In non-dimensional variables, equation (3.10) reduces to

(4.8)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}^{\prime }\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}^{\prime }}{\unicode[STIX]{x2202}t^{\prime }}+\frac{1}{\unicode[STIX]{x1D702}}(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}^{\prime })\boldsymbol{u}^{\prime }\right)+\frac{1}{\unicode[STIX]{x1D702}}\frac{1}{\unicode[STIX]{x1D6FE}Ma^{2}}\unicode[STIX]{x1D735}^{\prime }p^{\prime } & = & \displaystyle \frac{1}{\unicode[STIX]{x1D702}}\frac{1}{Re}\unicode[STIX]{x1D735}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{\unicode[STIX]{x1D702}}\frac{1}{Fr^{2}}\unicode[STIX]{x1D70C}^{\prime }\boldsymbol{g}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6FC}_{D}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}^{\prime }}}\frac{m^{\prime }}{\unicode[STIX]{x1D70F}_{D}^{\prime }}(\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D702}\boldsymbol{v}^{\prime })f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}^{\prime }}}(\unicode[STIX]{x1D702}\boldsymbol{v}^{\prime }-\boldsymbol{u}^{\prime }){\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime },\end{eqnarray}$$

where the non-dimensional number in front of the pressure gradient contains the Mach number of the airflow defined by $Ma=u_{o}/c_{o}$, with $c_{o}=\sqrt{\unicode[STIX]{x1D6FE}p_{o}/\unicode[STIX]{x1D70C}_{o}}$ the sound speed in air, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ the ratio of specific heats of air and $p_{o}=\unicode[STIX]{x1D70C}_{o}RT_{o}$ the characteristic value for the atmospheric pressure. The non-dimensional number in front of the shear stress is the inverse of the Reynolds number of the airflow $Re=\unicode[STIX]{x1D70C}_{o}u_{o}x_{o}/\unicode[STIX]{x1D707}_{o}$, where $\unicode[STIX]{x1D707}_{o}=\unicode[STIX]{x1D707}(T_{o})$ is a characteristic value for the viscosity of air. The non-dimensional number in the gravity term contains the Froude number $Fr=u_{o}/\sqrt{x_{o}g}$. The other dimensionless numbers have been defined in previous sections.

It is possible to substantially simplify the above equation. Indeed, we can see from figure 3 that $\unicode[STIX]{x1D6FC}_{m}\ll \unicode[STIX]{x1D6FC}_{D}$ for all drop radii. Therefore, among the spray feedback terms, only that related to the drag force on the fluid from the drops remains.

4.3.3 Airflow temperature

The same procedure leads to the non-dimensional form of the temperature equation (3.25):

(4.9)

where the dimensionless number in front of the heat flux contains the Peclet number for heat $Pe_{T}=c_{p}x_{o}u_{o}\unicode[STIX]{x1D70C}_{o}/\unicode[STIX]{x1D705}_{o}$, with $c_{p}$ the specific heat of air at constant pressure.

The non-dimensional diffusion term reads

(4.10)$$\begin{eqnarray}\displaystyle {\mathcal{D}}^{\prime } & = & \displaystyle \frac{1}{\unicode[STIX]{x1D702}}\frac{1}{Pe_{v}}\frac{R}{c_{v}}q_{o}T_{air}^{\prime }\left(\frac{1}{M_{w}}-\frac{1}{M_{a}}\right)\unicode[STIX]{x1D735}^{\prime }\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}^{\prime }q^{\prime })\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{\unicode[STIX]{x1D702}}\frac{1}{Pe_{v}}q_{o}\frac{c_{p,v}-c_{p,a}}{c_{v}}\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}^{\prime }q^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}^{\prime }T_{air}^{\prime },\end{eqnarray}$$

where $Pe_{v}$ is the Peclet number for mass transfer of water vapour $Pe_{v}=x_{o}u_{o}/D_{v}$ and $q_{o}$ is a characteristic specific humidity. Finally, the non-dimensional internal energy term reads

(4.11)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}^{\prime } & = & \displaystyle \frac{c_{v,v}}{c_{v}}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\left(T_{air}^{\prime }-\frac{T_{ref}}{T_{o}}\right){\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\,\frac{L_{v}}{T_{o}c_{v}}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,\frac{RT_{ref}}{M_{w}T_{o}c_{v}}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,\frac{c_{ps}}{c_{v}}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}\left(T^{\prime }-\frac{T_{ref}}{T_{o}}\right){\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,\frac{e_{sw}(T_{ref})}{T_{o}}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}\int _{\unicode[STIX]{x1D734}_{\mathfrak{X}}}{\mathcal{M}}^{\prime }f^{\prime }\,\text{d}\boldsymbol{\mathfrak{X}}^{\prime }.\end{eqnarray}$$

Again, using figure 3, it is possible to see that several feedback terms can be neglected. With the typical values for $c_{ps}$, $c_{v}$, $c_{p}$, $c_{p,v}$, $c_{p,a}$ and $\unicode[STIX]{x1D6FE}$, and considering a low Mach number (see below) and values of $T$ and $T_{o}$ between 0 and about $40\,^{\circ }\text{C}$, only two feedback terms remain. They are the spray feedback due to sensible heat variations from the drop (second to last term in (4.9)) and the latent heat transfer from the drops to the water vapour (second term on the right-hand side of (4.11)). We also keep the molecular diffusion term (${\mathcal{D}}^{\prime }$ in (4.9)).