On the Role of the Explosive Interaction of Three Surface Waves at the Initial Stage of Spray Generation in Strong Winds

Abstract

This work is devoted to a theoretical study of the hydrodynamic instability of the water–air interface. The development of this instability may result in “bag-breakup” fragmentation, which is one of the main sources of droplets in hurricane winds. A hypothesis according to which the initial water surface elevations subjected to fragmentation are formed by the hydrodynamic instability of disturbances of the wind drift current in the water is proposed. The weakly nonlinear stage of instability in the form of a resonant three-wave interaction is considered. It is found that the nonlinear resonant interaction of a triad of wind drift perturbations, one wave of which is directed along the flow and two other waves are directed at an angle to the flow, leads to an explosive increase in amplitudes. As part of the piecewise continuous model of the drift current profile, the characteristic time and spatial scales of disturbances are found and it is shown that their characteristic dependences on the air friction velocity agree with previously obtained experimental data.

APPENDIX

Let us consider the situation when the quantity Re_w is finite. Then, at $z = 0$, the boundary condition of zero shear stress must be satisfied:

$${{\left. {{{\partial }_{z}}u + ikw} \right|}_{{z = 0}}} = 0.$$

(A1)

It follows from the fluid incompressibility condition that

$$u = {{i{{\partial }_{z}}w} \mathord{\left/ {\vphantom {{i{{\partial }_{z}}w} k}} \right. \kern-0em} k}.$$

(A2)

We represent the vertical velocity perturbation in the form

$$w = {{w}_{{{\text{low}}}}}{{{\text{e}}}^{{kz}}} + {{w}_{{{\text{upp}}}}}{{{\text{e}}}^{{{z \mathord{\left/ {\vphantom {z {{{{{\delta }}}_{{\text{1}}}}}}} \right. \kern-0em} {{{{{\delta }}}_{{\text{1}}}}}}}}},$$

(A3)

where ${{\delta }_{1}} \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{w}}}}} }}} \right. \kern-0em} {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{w}}}}} }}$ is the characteristic scale of variation of the viscous addition.

Combining (A1)–(A3), we obtain an estimate for ${{w}_{{{\text{upp}}}}}$

$${{w}_{{{\text{upp}}}}} \propto \frac{{{{w}_{{{\text{low}}}}}}}{{{\text{R}}{{{\text{e}}}_{{\text{w}}}}}}.$$

(A4)

At the boundary $z = - 1$, the no-slip condition and the equality of shear stresses must be satisfied:

$${{\left. {{{u}_{1}}} \right|}_{{z = - 1}}} = {{\left. {{{u}_{2}}} \right|}_{{z = - 1}}},$$

(A5.1)

$$\frac{1}{{{\text{R}}{{{\text{e}}}_{{\text{w}}}}}}{{\left. {({{\partial }_{z}}{{u}_{1}} + ik{{w}_{1}})} \right|}_{{z = - 1}}} = \frac{1}{{{\text{R}}{{{\text{e}}}_{{\text{t}}}}}}{{\left. {({{\partial }_{z}}{{u}_{2}} + ik{{w}_{2}})} \right|}_{{z = - 1}}},$$

(A5.2)

where subscripts 1 and 2 correspond to the upper and lower regions. We again represent the velocity in the form

$$\begin{gathered} {{w}_{1}} = {{w}_{{{\text{1low}}}}}{{{\text{e}}}^{{k(z + 1)}}} + {{w}_{{{\text{1upp}}}}}{{{\text{e}}}^{{{{(z + 1)} \mathord{\left/ {\vphantom {{(z + 1)} {{{{{\delta }}}_{{\text{1}}}}}}} \right. \kern-0em} {{{{{\delta }}}_{{\text{1}}}}}}}}}, \\ {{u}_{1}} = {{u}_{{{\text{1low}}}}}{{{\text{e}}}^{{k(z + 1)}}} + {{u}_{{{\text{1upp}}}}}{{{\text{e}}}^{{{{(z + 1)} \mathord{\left/ {\vphantom {{(z + 1)} {{{{{\delta }}}_{{\text{1}}}}}}} \right. \kern-0em} {{{{{\delta }}}_{{\text{1}}}}}}}}}, \\ {{w}_{2}} = {{w}_{{{\text{2low}}}}}{{{\text{e}}}^{{k(z + 1)}}} + {{w}_{{{\text{2upp}}}}}{{{\text{e}}}^{{{{(z + 1)} \mathord{\left/ {\vphantom {{(z + 1)} 2}} \right. \kern-0em} 2}}}}, \\ {{u}_{2}} = {{u}_{{{\text{2low}}}}}{{{\text{e}}}^{{k(z + 1)}}} + {{u}_{{{\text{2upp}}}}}{{{\text{e}}}^{{{{(z + 1)} \mathord{\left/ {\vphantom {{(z + 1)} {{{{{\delta }}}_{{\text{2}}}}}}} \right. \kern-0em} {{{{{\delta }}}_{{\text{2}}}}}}}}}, \\ \end{gathered} $$

(A6)

where ${{\delta }_{1}} \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{w}}}}} }}} \right. \kern-0em} {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{w}}}}} }}$ and ${{\delta }_{2}} \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{t}}}}} }}} \right. \kern-0em} {\sqrt {{\text{R}}{{{\text{e}}}_{{\text{t}}}}} }}.$

At ${\text{R}}{{{\text{e}}}_{w}} \gg {\text{R}}{{{\text{e}}}_{t}} \gg 1$, it follows that

$${{w}_{{{\text{1low}}}}}\sim {{w}_{{{\text{2low}}}}}\sim {{u}_{{{\text{1low}}}}}\sim {{u}_{{{\text{2low}}}}}.$$

(A7)

Let us send ${\text{R}}{{{\text{e}}}_{{\text{w}}}} \to \infty ;$ then, from (A5.2), we obtain that ${{\left. {{{\partial }_{z}}{{u}_{2}} + ik{{w}_{2}}} \right|}_{{z = - 1}}} = 0.$ Whence, with allowance for (A2) and (A6), we find that

$${{w}_{{{\text{2low}}}}}\sim {\text{R}}{{{\text{e}}}_{{\text{t}}}}{{w}_{{{\text{2upp}}}}}.$$

(A8)

From (A5.1) and with allowance for (A2), we obtain the estimate

$${{w}_{{{\text{2low}}}}}\sim \sqrt {{\text{R}}{{{\text{e}}}_{{\text{w}}}}} {{w}_{{{\text{1upp}}}}}.$$

(A9)

Viscosity in the upper layer at $z = - 1$ may not be taken into account only if the shear stress in the lower layer is much more than the stress in the upper layer. To test the condition, it is sufficient to prove the smallness of ${{\left( {\frac{{{{\partial }_{z}}{{u}_{1}}}}{{{\text{R}}{{{\text{e}}}_{{\text{w}}}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{{{\partial }_{z}}{{u}_{1}}}}{{{\text{R}}{{{\text{e}}}_{{\text{w}}}}}}} \right)} {\left( {\frac{{{{w}_{2}}}}{{{\text{R}}{{{\text{e}}}_{{\text{t}}}}}}} \right)}}} \right. \kern-0em} {\left( {\frac{{{{w}_{2}}}}{{{\text{R}}{{{\text{e}}}_{{\text{t}}}}}}} \right)}}.$ Indeed, with allowance for (A6)–(A9), we have

Thus, at ${\text{R}}{{{\text{e}}}_{w}} \gg {\text{Re}}_{t}^{2} \gg 1$, the fluid in the upper layer $- 1 \leqslant z \leqslant 0$ can be considered nonviscous everywhere but in the critical layer.