$$ f \tilde{u} = A_v \frac{d^2 \tilde{v}}{dz^2}, \qquad -f \tilde{v} = A_v \frac{d^2 \tilde{u}}{dz^2} $$

$$ f \frac{d^2 \tilde{u}}{dz^2} = A_v \frac{d^4 \tilde{v}}{dz^4} \Rightarrow \frac{d^2 \tilde{u}}{dz^2} = -\frac{f}{A_v}\tilde{v} \Rightarrow \frac{d^4 \tilde{u}}{dz^4} + \frac{f^2}{A_v^2}\tilde{u} = 0 $$

The characteristic equation is $$ r^4 + \frac{f^2}{A_v^2} = 0 \Rightarrow r^2 = \pm i \frac{f}{A_v} \Rightarrow r = \pm \sqrt{\frac{f}{A_v}} e^{\pm i \pi/4} \Rightarrow r = \pm (1 \pm i)\sqrt{\frac{f}{2A_v}} \Rightarrow r = \pm (1 \pm i)\frac{1}{\delta_E} $$ where the Ekman layer thickness is $$ \delta_E = \left(\frac{A_v}{f/2}\right)^{1/2} $$ Thus, the general solution is $$ \tilde{u} = C_1 e^{(1+i)z/\delta_E} + C_2 e^{(1-i)z/\delta_E} + C_3 e^{-(1+i)z/\delta_E} + C_4 e^{-(1-i)z/\delta_E} $$

As $z \to \infty$, only the decaying exponentials are allowed. Set \(C_1 = C_2 = 0\) $$ \tilde{u} = C_3 e^{-(1+i)z/\delta_E} + C_4 e^{-(1-i)z/\delta_E} \Rightarrow \tilde{v} = -i C_3 e^{-(1+i)z/\delta_E} + i C_4 e^{-(1-i)z/\delta_E} $$

At the wall $z=0$ \[\tilde{u}(0) = -U, \quad \tilde{v}(0) = 0 \Rightarrow \tilde{u}(0) = -U, \quad \tilde{v}(0) = 0\] With constants substituted \[\tilde{v} = U e^{-z/\delta_E}\sin(z/\delta_E), \tilde{u} = -U e^{-z/\delta_E}\cos(z/\delta_E)\]

Since $u = U + \tilde{u}, \; v = \tilde{v}$, in terms of the total velocity $$ u = U\left[1 - e^{-z/\delta_E}\cos(z/\delta_E)\right], \quad v = U e^{-z/\delta_E}\sin(z/\delta_E)\sin(z/\delta_E) $$ These are the Ekman spiral solutions