The Ekman spiral for piecewise-constant eddy viscosity

ABSTRACT

We investigate the classical problem of wind-stress-induced non-equatorial steady ocean currents, considering a three-valued piecewise-constant eddy viscosity. This extends and generalizes recent results on piecewise-uniform eddy viscosity with only two layers. The aim is to determine how variations in the relative values of the eddy viscosity at different depths may account for the deviations from the surface current deflection of 45${}^{\circ }$, predicted in the classical constant eddy viscosity case. Such variations are observed in field data, and we investigate analytically the effect the presence of a third region makes compared to the two-layer case.

Keywords:

• Boundary-value problem
• viscous flow
• wind-driven currents
• deflection angle

AMS CLASSIFICATIONs:

• 34B05
• 34B40
• 35Q35
• 76U05

Acknowledgments

The author would like to thank the anonymous referee for the helpful comments and suggestions.

Data availability

The plots in Figure 1 are easily generated from Equation (21(21) $\begin{array}{rl}\tan \left({\theta }_0\right)& =\{{\phi }^2({\alpha }^2-{\text{β}}^2)+{\omega }^2({\gamma }^2-{\delta }^2)+2({\phi }^2+{\omega }^2)(1-{\ell }_1^2)\sin \left(2h_1\right)\\ & +4\left(1-\frac{{\ell }_2^2}{{\ell }_1^2}\right)[\left((1-{\ell }_1^2)\sinh \left(2h_1\right)+(1+{\ell }_1^2)\sin \left(2h_1\right)\right)\cos \left(2\frac{h_2-h_1}{{\ell }_1}\right)\\ & +2{\ell }_1\cos \left(2h_1\right)\sin \left(2\frac{h_2-h_1}{{\ell }_1}\right)]\}\\ & \times \{{\phi }^2({\alpha }^2-{\text{β}}^2)+{\omega }^2({\gamma }^2-{\delta }^2)-2({\phi }^2+{\omega }^2)(1-{\ell }_1^2)\sin \left(2h_1\right)\\ & +4\left(1-\frac{{\ell }_2^2}{{\ell }_1^2}\right)[\left((1-{\ell }_1^2)\sinh \left(2h_1\right)-(1+{\ell }_1^2)\sin \left(2h_1\right)\right)\cos \left(2\frac{h_2-h_1}{{\ell }_1}\right)\\ & -2{\ell }_1\cos \left(2h_1\right)\sin \left(2\frac{h_2-h_1}{{\ell }_1}\right)]{\}}^{-1}.\end{array}$(21) ).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Vienna Science and Technology Fund (WWTF) under grant MA16-009.