The Ekman spiral for piecewise-constant eddy viscosity

Figures & data

Figure 1. Plots of $\tan \left({\theta }_0\right)$ as a function of ${\ell }_2$ for ${\ell }_1=2$, $h_1=0.7$, and two different values of $h_2$, namely $h_2=1.6$ (left) and $h_2=3.4$ (right). The dashed horizontal line represents the solution with no bottom layer, formula (23(23) $\tan \left({\theta }_0\right)=\frac{(1+{\ell }_1^2)\sinh \left(2h_1\right)+2{\ell }_1\cosh \left(2h_1\right)+(1-{\ell }_1^2)\sin \left(2h_1\right)}{(1+{\ell }_1^2)\sinh \left(2h_1\right)+2{\ell }_1\cosh \left(2h_1\right)-(1-{\ell }_1^2)\sin \left(2h_1\right)},$(23) ), independent of ${\ell }_2$.

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) ).