2.1 Kinetic description of the rain

The kinetic description employed here can be used to compute the force, and hence the pressure, exerted by drops falling on a surface (see Chapman & Cowling Reference Chapman and Cowling1990). For the sake of completeness, and since the reader might not be familiar with kinetic gas theory, we find it useful to briefly summarize below certain aspects of our approach.

A rain drop is a volume of fluid described by its density and velocity, which are in turn governed by the Navier–Stokes equations. At larger scales, the number of drops falling on a unit surface at the air–sea interface is large, and conversely, the radius of any drop is much smaller than the macroscopic length scales (e.g. the surface gravity waves). Thus, it is appropriate to use a statistical description of the drops. This description is similar to that used for spray modelling, as is commonly done in combustion engineering (e.g. see Marchiso & Fox Reference Marchiso and Fox2013).

We assume that each rain drop is made of a single fluid with constant density ${\it\rho}_{d}$ , and is spherical with radius $r$ and mass $m(r)={\it\rho}_{d}(4/3){\rm\pi}r^{3}$ . Each drop is then uniquely defined by:

1. (i) its position $\boldsymbol{x}\in \mathbb{R}^{3}$ ;

2. (ii) its velocity $\boldsymbol{v}\in \mathbb{R}^{3}$ , which is the mean velocity of the molecules inside the drop; and

3. (iii) its radius $r\in \, ]0,+\infty [\,$ .

One could also consider additional variables such as temperature or chemical species but this will not be done here.

The set of drops is then described by the distribution function $f(t,\boldsymbol{x},\boldsymbol{v},r)$ , which is the number density of drops at time $t$ in the phase space $\mathbb{R}^{3}\times \mathbb{R}^{3}\times \, ]0,+\infty [\,$ . In other words, $f(t,\boldsymbol{x},\boldsymbol{v},r)\,\text{d}\boldsymbol{x}\,\text{d}\boldsymbol{v}\,\text{d}r$ 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 $\boldsymbol{v}\pm \text{d}\boldsymbol{v}$ and with a radius $r\pm \text{d}r$ .

Averaged quantities can be easily obtained through the distribution $f(t,\boldsymbol{x},\boldsymbol{v},r)$ . For instance, the time-dependent radius distribution $n$ is

which is the number of drops of radius $r$ at time $t$ and position $\boldsymbol{x}$ , per unit volume and per radius increment.

2.2 Pressure induced by drops on a surface

To compute the surface stress exerted by rain drops, we first consider a surface $\text{d}S$ defined by a normal unit vector $\boldsymbol{n}$ and moving with a velocity $\boldsymbol{u}$ ; we trust the reader will not confuse the unit vector $\boldsymbol{n}$ with the drop size distribution $n$ (2.1). Here, we are interested in the drops with a given velocity $\boldsymbol{v}$ that will impact $\text{d}S$ . This simply implies that we only need to consider drops for which $(\boldsymbol{v}-\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{n}<0$ (see figure 1). Thus, summing over all possible radii and all incident velocities, the total number of drops impacting $\text{d}S$ during a short time $\text{d}t$ is given by:

In the frame of reference attached to the surface, the momentum from all the drops impacting $\text{d}S$ is

Consequently, the momentum flux ${\bf\varphi}$ (the momentum per unit of surface and per unit of time) across a surface of normal $\boldsymbol{n}$ is obtained by dividing $\boldsymbol{M}$ by $\text{d}S\,\text{d}t$ such that:

where $\otimes$ denotes the tensor product. Therefore, the force exerted by the rain drops on the surface $\text{d}S$ is

where ${\bf\varphi}$ is the momentum flux of the drops defined in (2.4). Here, we neglect the force exerted by the drops ejected from the surface. When rain falls on a free surface, there are indeed secondary drops generated at the surface. Drops pinch off from the stalks, and smaller so-called film droplets are ejected from crowns and craters resulting from rain-drop impacts. However, these secondary drops are either much slower than the rain in the case of stalks, or significantly smaller than rain drops in the case of film droplets (see Veron Reference Veron2015). In either case, the momentum carried by these secondary drops is considered negligible.

Figure 1.

Schematic, notation and sign convention for rain drops falling on the sea surface.

We finally consider that drops falling on the surface exert a stress on $\text{d}S$ such that the corresponding stress tensor ${\bf\sigma}_{d}$ is given by the relation $\boldsymbol{F}={\bf\sigma}_{d}\boldsymbol{n}\,\text{d}S$ . Using (2.4) and (2.5), this analysis then yields:

2.3 Accounting for falling rain in Navier–Stokes and Euler free-surface models

In this section, we incorporate the rain-induced stress ((2.6) above) into an incompressible Navier–Stokes framework which includes a free surface. The surface is described by the equation $z={\it\eta}(t,\boldsymbol{x}_{\bot })$ , where $t>0$ is the time, $\boldsymbol{x}_{\bot }=(x,y)$ is the two-dimensional horizontal position vector perpendicular to the $z$ direction and ${\it\eta}$ is the height of the free surface (see figure 1). The velocity $\boldsymbol{u}(t,\boldsymbol{x}_{\bot },z)=(\boldsymbol{u}_{\bot },w)$ of the water under the free surface is given by

where ${\bf\sigma}=-p\unicode[STIX]{x1D644}+{\it\mu}(\boldsymbol{{\rm\nabla}}\boldsymbol{u}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\text{T}})$ is the stress tensor, ${\it\rho}$ is the water density, $p$ is the pressure, ${\it\mu}$ is the dynamic viscosity of water and $\boldsymbol{g}$ is gravity. Here, $\unicode[STIX]{x1D644}$ is the unit tensor, $\boldsymbol{{\rm\nabla}}\boldsymbol{u}$ is the gradient velocity tensor and $(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\text{T}}$ its transpose. The boundary conditions at the free surface are the kinematic boundary condition

and the dynamic boundary condition

where ${\bf\sigma}_{a}$ is the stress tensor exerted by the air on the surface, ${\it\Gamma}$ is the surface tension and ${\it\kappa}$ is the curvature of the free surface. In the absence of wind and other shear stresses, the stress tensor reduces to ${\bf\sigma}_{a}=-p_{a}\unicode[STIX]{x1D644}$ , where $p_{a}$ is the atmospheric pressure.

Therefore, we can account for the effect of the rain drops on the free surface by simply incorporating the stress from the rain (2.6) into the dynamic boundary condition (2.10), which is then replaced by

Neglecting viscosity both in the fluid and at the surface, and neglecting surface tension (thus limiting our analysis to moderate surface curvatures), we obtain the following Euler model:

with the unchanged kinematic relation (2.9). Since viscosity is neglected, the dynamic boundary condition is limited to normal stresses only and reduces to:

where $p_{a}$ is the atmospheric pressure and $p_{d}=-{\bf\sigma}_{d}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{n}$ is the pressure exerted by the rain on the free surface expressed as:

3.1 Linear deep-water surface gravity waves with falling rain

Here, we use the standard linear water wave theory using the inviscid Euler model (2.12)–(2.13). For an irrotational flow, the linearized Euler system written in the potential form, gives

where ${\it\Phi}$ is the velocity potential such that $\boldsymbol{u}=\boldsymbol{{\rm\nabla}}{\it\Phi}$ is the velocity associated with the surface waves, and $C(t)$ depends only on time $t$ . The surface boundary conditions (2.9) and (2.14) give Finally, we consider that velocity perturbations from the surface waves are confined near the surface (deep-water waves) and add the kinematic boundary condition

Considering a surface-wave mode of elevation ${\it\eta}=a\text{e}^{\text{i}(\boldsymbol{k}_{\bot }\boldsymbol{\cdot }\boldsymbol{x}_{\bot }-{\it\omega}t)}$ with amplitude $a$ , angular frequency ${\it\omega}$ , and propagating along the horizontal wavenumber $\boldsymbol{k}_{\bot }$ , the Laplace equation (3.1) and accompanying linearized kinematic conditions (3.3) and (3.5) give ${\it\Phi}=-\text{i}((a{\it\omega})/k)\text{e}^{kz}\text{e}^{\text{i}(\boldsymbol{k}_{\bot }\boldsymbol{\cdot }\boldsymbol{x}_{\bot }-{\it\omega}t)}$ with $k=|\boldsymbol{k}_{\bot }|$ and the surface-wave wavelength ${\it\lambda}=k/(2{\rm\pi})$ . Note that the linear theory assumes that $ak$ is a small parameter. In turn, Bernoulli’s equation (3.2) and the dynamic condition (3.4) linearized about $z=0$ lead to the usual dispersion relationship ${\it\omega}=(gk)^{1/2}$ for deep-water surface gravity waves when $p_{d}=0$ and choosing $C(t)=p_{a}$ . Therefore, we can now examine how incorporating the rain-induced pressure into the dynamic boundary condition modifies the classical dispersion relationship.

In order to do so, the rain-induced pressure defined in (2.15) must be evaluated. To make the computation possible, we make several assumptions. First, we assume the rain-drop distribution is homogeneous in space and monokinetic, that is to say $f(t,\boldsymbol{x},\boldsymbol{v},r)=n(r){\it\delta}_{\boldsymbol{v}-\boldsymbol{v}_{d}(r)}$ , where ${\it\delta}$ is the delta-Dirac function centred on $\boldsymbol{v}_{d}$ so that ${\it\delta}_{\boldsymbol{v}-\boldsymbol{v}_{d}(r)}=0$ for $\boldsymbol{v}\neq \boldsymbol{v}_{d}$ . In other words, we assume the same drop radius distribution $n(r)$ at each time $t$ and position $\boldsymbol{x}$ , and all the drops of radius $r$ have the same velocity $\boldsymbol{v}_{d}(r)$ . Note that despite its simplicity, this choice allows for different drop velocities depending on the radius. Indeed, various analytic formulae of radius distribution and velocity can be used (see § 3.2). Second, we estimate the integral (2.15) assuming that $(\boldsymbol{v}_{d}(r)-\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{n}<0$ for all drop radii. With these assumptions, the rain-induced pressure (2.15) reduces to

This second assumption simply means that, in the estimate of the rain-induced pressure, we count all drop contributions including very slow falling drops for which $(\boldsymbol{v}_{d}(r)-\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{n}$ may not be negative. In other words, we include drops of very small radii that may locally fall more slowly than the wavy surface. We will show in the following section that these small droplets (as is the case with secondary rain-generated drops) carry a negligible fraction of the total rain-induced momentum flux. However, their inclusion here significantly simplifies the integration (3.6) and yields practical analytical expressions (see appendix A). Note here that $p_{d}$ is quadratic with the impact velocity, and cubic with the drop radius.

Without loss of generality, we choose a frame of reference aligned with $\boldsymbol{x}_{\bot }$ (i.e. such that $\boldsymbol{x}_{\bot }=(x,0)$ is the direction of propagation of the wave) and project $\boldsymbol{v}_{d}(r)$ with the horizontal and vertical components of the drop velocity given by $\boldsymbol{v}_{d}(r)=(u_{d}(r),w_{d}(r))$ respectively. Keeping only the linear terms in $ak$ and $a{\it\omega}$ , the rain-induced pressure $p_{d}$ from (3.6) reduces to $p_{d}=p_{d}^{(0)}+p_{d}^{(1)}$ with

and where $I$ and $J$ are constants defined by Here, $p^{(0)}$ is a constant pressure term (thus dynamically insignificant) arising from the vertical momentum flux due to the rain. Also, $p^{(1)}$ is a wave-coherent pressure component in which the term $Iak$ results from the surface-slope–drop-velocity correlation and the term $Ja{\it\omega}$ is caused by the vertical velocity of the interface. We note here that $J/(2{\it\rho}_{d})$ is the rain-induced volume flux per horizontal (i.e. perpendicular to $w_{d}$ ) unit surface per unit time. It has the dimension of a velocity and is commonly referred to as the rain rate $R$ . We also note that with our sign convention where the vertical $z$ -direction is positive upward, $w_{d}<0$ and $J/(2{\it\rho}_{d})=R$ where $R<0$ . Generally, rain rates are reported in $\text{mm}~\text{h}^{-1}$ and refer to $|R|$ .

We now re-introduce the dynamic condition (3.4) with $p_{d}=p_{d}^{(0)}+p_{d}^{(1)}$ given by (3.7)–(3.8) into Bernoulli’s equation (3.2) linearized about $z=0$ and obtain

where we have chosen the constant $C(t)$ such that $C(t)=p_{a}+p_{d}^{(0)}$ . Inserting the expressions of ${\it\Phi}$ , $p_{d}^{(1)}$ and ${\it\eta}$ into this relation gives the following equation for ${\it\omega}$ :

In this case, we find that the dispersion relationship is a quadratic equation with complex coefficients. We find that its solutions are couples of complex frequencies ${\it\omega}={\it\omega}_{R}+\text{i}{\it\omega}_{I}$ . The real part ${\it\omega}_{R}$ is given by

Without loss of generality, we choose the solution with ${\it\omega}_{R}>0$ (surface waves going in the $x>0$ direction), and we find that the imaginary part ${\it\omega}_{I}$ depends on the sign of $I$ :

where we consider that $\text{sign}(0)=0$ . Since ${\it\eta}=a\text{e}^{\text{i}(kx-{\it\omega}t)}=a\text{e}^{{\it\omega}_{I}t}\text{e}^{\text{i}(kx-{\it\omega}_{R}t)}$ , the falling rain induces wave attenuation if ${\it\omega}_{I}<0$ , or amplification if ${\it\omega}_{I}>0$ .

3.2 Rain size and velocity distributions

In order to assess the impact of rainfall on the surface waves, we must evaluate (3.11) and (3.12) and thus estimate $I$ and $J$ . To do so, we need to choose a number density function $n(r)$ and a fall velocity $\boldsymbol{v}_{d}(r)$ for the rain. There are number of different formulae for either $n(r)$ or $\boldsymbol{v}_{d}(r)$ in the literature but we note here that $I$ and $J$ depend on their integral values and are thus not dramatically influenced by the details of their formulations. For the drop size distribution, we choose the classical gamma function formulation proposed by Marshall & Palmer (Reference Marshall and Palmer1948):

where ${\it\Lambda}=344.34|R|^{-0.21}~\text{m}^{-1}$ with $R$ in $\text{m}~\text{s}^{-1}$ , and $\mathscr{N}$ is a normalization factor (see appendix A) that allows recovery of the rain rate, which is the variable generally measured and reported. The vertical fall velocity $w_{d}(r)$ is taken from Best (Reference Best1950):

with ${\it\Upsilon}_{w}=9.32~\text{m}~\text{s}^{-1}$ the maximum fall velocity for rain, and ${\it\alpha}_{w}=1296~\text{m}^{-1}$ . Equation (3.14) is considered valid for drop sizes $0.1~\text{mm}<r<3~\text{mm}$ . The horizontal rain velocity is usually ignored (Ho et al. Reference Ho, Asher, Bliven, Schlosser and Gordan2000; Harrison et al. Reference Harrison, Veron, Ho, Reid, Orton and McGillis2012) or considered to be independent of the radius and assumed to be approximately 85 % of $U_{10}$ , the wind speed at 10 m height (Caldwell & Elliott Reference Caldwell and Elliott1971, Reference Caldwell and Elliott1972). However, while it is clear that large rain drops falling through the near surface atmospheric boundary layer will rapidly reach the surface without having the time to decelerate significantly, small droplets take longer to fall and also have a smaller mass and lower Stokes number (inertial response time). Therefore, the small rain drops will impact the surface after having exchanged momentum with the air flow and thus decelerated substantially. Past studies have focused on the vertical terminal velocity of rain, and there are few published studies on the horizontal transport of rain drops. To the authors’ knowledge, there are no published radius-dependent horizontal drop velocity formulae. To remedy this paucity, we used results from a series of simulations based on the Lagrangian stochastic model of Mueller & Veron (Reference Mueller and Veron2009) for wind speeds ranging from 0 to $30~\text{m}~\text{s}^{-1}$ . In these simulations, rain drops of various sizes were left to fall freely through the turbulent atmospheric boundary layer above the wavy surface until they impacted the surface. Impact velocities were recorded and based on the results from these simulations (note that the simulations explore the free fall of water drops of several sizes through the stratified turbulent atmosphere near the wavy interface; they are very comprehensive and thus beyond the scope of this paper, but the results will be the subject of a subsequent publication), we propose the following formulation for rain-drop horizontal velocity at impact on the surface:

with ${\it\Upsilon}_{u}=0.85U_{10}$ and ${\it\alpha}_{u}\approx 5000~\text{m}^{-1}$ . Note that (3.15) implies that the wind and wave are travelling in the same $x$ direction. In the case of misaligned wind and waves, $u_{d}$ is simply the component of the horizontal drop velocity in the direction of propagation of the wave and ${\it\Upsilon}_{u}=0.85U_{10}\cos {\it\theta}$ , with ${\it\theta}$ the angle between the wind and the wave directions.

Figure 2.

(a) The rain-drop size distribution from Marshall & Palmer (Reference Marshall and Palmer1948) for multiple rain rates. (b) The vertical drop velocity $w_{d}$ and horizontal velocity $u_{d}$ normalized by $U_{10}$ (right axis) at impact on the surface, as a function of drop radius.

Figure 2(a) shows the drop size distribution of Marshall & Palmer (Reference Marshall and Palmer1948) for several rain rates. As expected the number density increases with increasing rain rate but the shape of the distribution remains. Figure 2(b) shows the rain-drop velocity at impact on the surface as a function of the radius. The vertical fall velocity is that of Best (Reference Best1950) given by (3.14). The horizontal drop impact velocity, shown in grey, is normalized by the 10 m wind speed $U_{10}$ . Large drops are found to impact at approximately 85 % of the 10 m wind speed in accord with Caldwell & Elliott (Reference Caldwell and Elliott1972).

3.3 Validity of assumptions

In this section, we discuss the validity of several assumptions made above and offer limits on the range of validity of the approach presented here. Overall, we show below that the model presented here is valid for linear deep-water gravity waves with wavelengths in the range $O(1{-}3)\;\text{m}<{\it\lambda}<O(250)\;\text{m}$ .