A Theoretical Model of Wind-Wave Growth Over an Ice-Covered Sea

Abstract

A wind-wave generation model over an ice-covered sea is proposed. The wind velocity over the ice upper surface is decomposed into the mean velocity profile of the boundary-layer flow and small perturbations, while the ice cover is modelled as a viscoelastic layer, with the water part modelled as an inviscid fluid. The present model is based on two-dimensional linear flow-instability theory, with no-slip boundary conditions at the air–ice interface, and both normal and shear stress boundary conditions matched on the air–ice interface. It is shown that the model converges to the field and experimental data for open-water cases. The ice elasticity is found to be the critical factor for generating wind waves, and the generation of flexural-gravity waves and elastic waves in ice is analyzed.

Appendix: Reduced Boundary-Condition Equations

The reduced boundary-condition equations in terms of the potential functions and stream-functions for the wind-wave generation model are

$$ - \frac{{\partial \phi_{1} }}{\partial x} - \frac{{\partial \psi_{1} }}{\partial z} + \frac{{\partial \psi_{0} }}{\partial z} + \frac{{U_{0}^{\prime } }}{c}_{0} = 0, z = h. $$

(31)

$$ - \frac{{\partial \phi_{1} }}{\partial z} + \frac{{\partial \psi_{1} }}{\partial x} - \frac{{\partial \psi_{0} }}{\partial x} = 0,z = h. $$

(32)

$$ \rho_{ice} \nu_{1} \left( { - 2\frac{{\partial^{2} \phi_{1} }}{\partial x\partial z} - \frac{{\partial^{2} \psi_{1} }}{{\partial z^{2} }} + \frac{{\partial^{2} \psi_{1} }}{{\partial x^{2} }}} \right) + \rho_{air} \nu_{air} \left( {\frac{{\partial^{2} \psi_{0} }}{{\partial z^{2} }} - \frac{{\partial^{2} \psi_{0} }}{{\partial x^{2} }}} \right) = 0,z = h. $$

(33)

$$ \rho_{ice} \left[ { - \frac{{\partial^{2} \phi_{1} }}{{\partial t^{2} }} + g\left( { - \frac{{\partial \phi_{1} }}{\partial z} + \frac{{\partial \psi_{1} }}{\partial x}} \right) + 2\nu_{1} \left( { - \frac{{\partial^{3} \phi_{1} }}{{\partial z^{2} \partial t}} + \frac{{\partial^{3} \psi_{1} }}{\partial x\partial z\partial t}} \right)} \right] + 2\rho_{air} \nu_{air} \left( {\frac{{U_{0}^{\prime } }}{c}\frac{{\partial \psi_{0} }}{\partial x} - \frac{{\partial^{2} \psi_{0} }}{\partial x\partial z}} \right) = 0,z = h. $$

(34)

$$ - \frac{{\partial^{2} \phi_{1} }}{{\partial t^{2} }} + 2\nu_{1} \left( { - \frac{{\partial^{3} \phi_{1} }}{{\partial z^{2} \partial t}} + \frac{{\partial^{3} \psi_{1} }}{\partial x\partial z\partial t}} \right) + \frac{i\omega }{{k\tanh k\left( {z + H} \right)}}\frac{{\partial w_{1} }}{\partial t} = 0,z = 0. $$

(35)

$$ - 2\frac{{\partial^{2} \phi_{1} }}{\partial x\partial z} - \frac{{\partial^{2} \psi_{1} }}{{\partial z^{2} }} + \frac{{\partial^{2} \psi_{1} }}{{\partial x^{2} }} = 0,z = 0. $$

(36)

With (31) and (32), we eliminate the streamfunction $\psi_{0}$ in (33) and (34). Finally, six boundary conditions are reduced to four equations for the coefficients A, B, C, and D, which enables solution the determinant of this 4 × 4 system to obtain the dispersion relation.