On the effect of sea spray on the aerodynamic surface drag under severe winds

Abstract

We investigate the effect of the sea spray on the air-sea momentum exchange during the entire “life cycle” of a droplet, torn off the crest of a steep surface wave, and its fall down to the water, in the framework of a model covering the following aspects of the phenomenon: (1) motion of heavy particle in the driving air flow (equations of motion); (2) structure of the wind field (wind velocity, wave-induced disturbances, turbulent fluctuations); (3) generation of the sea spray; and (4) statistics of droplets (size distribution, wind speed dependence). It is demonstrated that the sea spray in strong winds leads to an increase in the surface drag up to 40 % on the assumption that the velocity profile is neutral.

Appendix. Derivation of the law of the momentum flux conservation

Here, we derive the conservation law for the momentum flux averaged over horizontal coordinate. Transforming Eq. (5) with account of Eq. (6) in Section 1 gives:

$$ \begin{array}{l}\frac{\partial {\rho}_a\left\langle u\right\rangle }{\partial {t}^{*}}+\frac{\partial }{\partial {z}^{*}}\left({\rho}_a\left\langle u\right\rangle \left(\left\langle v\right\rangle -\frac{\partial \eta }{\partial {t}^{*}}-\left\langle u\right\rangle \frac{\partial \eta }{\partial {x}^{*}}\right)\right)+\frac{\partial {\rho}_a{\left\langle u\right\rangle}^2}{\partial {x}^{*}}+\\ {}+\frac{\partial \left\langle p\right\rangle -{\rho}_a{\sigma}_{xx}}{\partial {x}^{*}}-\frac{\partial }{\partial {z}^{*}}\left(\left(\left\langle p\right\rangle -{\rho}_a{\sigma}_{xx}\right)\frac{\partial \eta }{\partial {x}^{*}}\right)=\frac{\partial {\rho}_a{\sigma}_{xz}}{\partial {z}^{*}}+f.\end{array} $$

(A1)

Averaging (A1) over the coordinate x* yields the conservation law for the momentum flux in MABL in curvilinear coordinates:

$$ \frac{\partial {\rho}_a{\overline{\left\langle u\right\rangle}}^{x*}}{\partial {t}^{*}}+\frac{\partial }{\partial {z}^{*}}\left({\rho}_a{\overline{\left\langle u\right\rangle \left(\left\langle v\right\rangle -\frac{\partial \eta }{\partial {t}^{*}}-\left\langle u\right\rangle \frac{\partial \eta }{\partial {x}^{*}}\right)}}^{x*}-{\overline{\left(\left\langle p\right\rangle -{\rho}_a{\sigma}_{xx}\right)\frac{\partial \eta }{\partial {x}^{*}}}}^{x*}\right)=\frac{\partial {\rho}_a\overset{\_\_\_x*}{\sigma_{xz}}}{\partial {z}^{*}}+\overset{\_\_x*}{f}. $$

(A2)

In stationary conditions (A1) yields:

$$ \frac{d}{d{z}^{*}}\left({\rho}_a\overset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_x*}{\left\langle u\right\rangle \left(\left\langle v\right\rangle -\frac{\partial \eta }{\partial {t}^{*}}-\left\langle u\right\rangle \frac{\partial \eta }{\partial {x}^{*}}\right)-\left(\left\langle p\right\rangle -{\rho}_a{\sigma}_{xx}\right)\frac{\partial \eta }{\partial {x}^{*}}}\right)=\frac{d{\rho}_a\overset{\_\_\_x*}{\sigma_{xz}}}{d{z}^{*}}+\overset{\_\_x*}{f}. $$

(A3)

Integrating (A3) over z* and taking into account the boundary condition $\overset{\_\_\_\_x*}{\sigma_{xz}}\to {u}_{*}^2$ when z* → ∞, yields the following expression for the conservation of momentum flux:

$$ {\rho}_a\overset{\_\_\_x*}{\sigma_{xz}}={\rho}_a{u}_{*}^2-{\rho}_a{\tau}_{\mathrm{wave}}\left(z*\right)-F\left(z*\right). $$

(A4)

Here

$$ {\tau}_{\mathrm{wave}}\left(z*\right)=-{\overline{\left\langle u\right\rangle \left(\left\langle v\right\rangle -\frac{\partial \eta }{\partial {t}^{*}}-\left\langle u\right\rangle \frac{\partial \eta }{\partial {x}^{*}}\right)+\left(\frac{\left\langle p\right\rangle }{\rho_a}-{\sigma}_{xx}\right)\frac{\partial \eta }{\partial {x}^{*}}}}^{x*} $$

(A5)

is a wave momentum flux caused by the form drag of the water surface (a function of z* decreasing with the distance from the surface).

$$ F\left(z\ast \right)=-{\displaystyle \underset{z\ast }{\overset{\infty }{\int }}{\overline{f}}^{x\ast}\left(z{\textstyle \hbox{'}}\right)dz}{\textstyle \hbox{'}} $$

(A6)

is the momentum delivered from the air flow to spray in the layer from current z* to infinity.