# A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem

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## Abstract

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## 1. Introduction

## 2. Derivation of the Wind-to-Current Transfer Function

#### 2.1. Transfer Function Fundamentals

#### 2.2. Equations of Motion for the Wind-Driven Flow

#### 2.3. Transformation to the Modified Bessel’s Equation

## 3. Behavior of the Transfer Function

#### 3.1. Essential Behavior of the Transfer Function

#### 3.2. Comparison with the Impulse Response Function

#### 3.3. A Second Self-Similarity of the Transfer Function

#### 3.4. Variability of Transfer Function Structure

## 4. Asymptotic Behavior of the Transfer Function

#### 4.1. Regimes of the Transfer Function

#### 4.2. Transfer Function Expressions

**Table 1.**Parameter space behavior of the general no-slip transfer function in the limit of large boundary layer depth h as a function of the $\delta /{z}_{o}$ ratio and $z/{z}_{o}$ ratio, numbered I–IX. Here ${\mathcal{K}}_{\eta}\left(x\right)$ and ${\mathcal{I}}_{\eta}\left(x\right)$ are decaying modified Bessel functions of order $\eta $ of the first and second kind, respectively, the Madsen depth $\mu (\delta ,{z}_{o})\equiv {\delta}^{2}/{z}_{o}$ is regarded as a function of the Ekman depth $\delta $ and roughness length ${z}_{o}$, and $s=s(\omega ,f)\equiv \mathrm{sgn}\left(f\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{sgn}(1+\omega /f)$ is a sign function. The functions ${\varphi}_{z}\left(\omega \right)\equiv {\zeta}_{0}\left(\omega \right)\left[1+\frac{1}{2}\frac{z}{{z}_{o}}\right]$, ${\zeta}_{z}\left(\omega \right)\equiv 2\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\frac{{z}_{o}}{\delta}\phantom{\rule{0.166667em}{0ex}}\sqrt{\left(1+\frac{z}{{z}_{o}}\right)\left|1+\frac{\omega}{f}\right|\phantom{\rule{0.166667em}{0ex}}}$, and ${\varsigma}_{z}\left(\omega \right)\equiv 2\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{z}{\mu}\left|1+\frac{\omega}{f}\right|\phantom{\rule{0.166667em}{0ex}}}$ are three different versions of the arguments to the Bessel functions that appear in the upper, middle, and lower rows, respectively; note ${\zeta}_{0}\left(\omega \right)={\varphi}_{0}\left(\omega \right)$. The boxed terms were previously presented by Elipot and Gille [12], with “EG” refers to the numbering nomenclature of those authors. The transfer functions for the Ekman [4] solution and Madsen [8] solutions are in the upper right-hand corner and lower left-hand corners.

Strong Gradient/Near-Inertial | Arbitrary Gradient/Any Frequency | Weak Gradient/Far-Inertial | |
---|---|---|---|

${\mathit{z}}_{\mathit{o}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\mathit{\delta}$ or $\mathit{\omega}\to -\mathit{f}$ | — | ($\mathit{\delta}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{\mathit{z}}_{\mathit{o}}$ and $\mathit{\omega}\ne -\mathit{f}$) or $\left|1+\mathit{\omega}/\mathit{f}\right|\to \mathit{\infty}$ | |

I | II | III, Ekman, EG-1a | |

$z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | $z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<(h,\delta )$ | $z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | $(z,\delta )\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ |

∞ | $\frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\frac{{\mathcal{K}}_{0}\left({\varphi}_{z}\left(\omega \right)\right)}{{\mathcal{K}}_{1}\left({\varphi}_{0}\left(\omega \right)\right)}$ | $\begin{array}{|c|}\hline \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{|1+\omega /f|}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{-(1+s\mathrm{i})(z/\delta )\sqrt{|1+\omega /f|}}\\ \hline\end{array}$ | |

IV | V, Mixed, EG-3a | VI | |

${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<(h,\delta )$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | $\delta \leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ |

$\frac{4}{\rho \left|f\right|\mu}{\mathcal{K}}_{0}\left({\zeta}_{z}\left(\omega \right)\right)$ | $\begin{array}{|c|}\hline \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{|1+\omega /f|}}\frac{{\mathcal{K}}_{0}\left({\zeta}_{z}\left(\omega \right)\right)}{{\mathcal{K}}_{1}\left({\zeta}_{0}\left(\omega \right)\right)}\\ \hline\end{array}$ | $\frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{|1+\omega /f|}}\frac{{\mathrm{e}}^{{\zeta}_{0}\left(\omega \right)-{\zeta}_{z}\left(\omega \right)}}{{\left(1+z/{z}_{o}\right)}^{1/4}}$ | |

VII, Madsen, EG-2a | VIII | IX | |

${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<(z,\delta )\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ | $\delta \leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<h$ |

$\begin{array}{|c|}\hline \frac{4}{\rho \left|f\right|\mu}{\mathcal{K}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right)\\ \hline\end{array}$ | $\frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{|1+\omega /f|}}\frac{{\mathcal{K}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right)}{{\mathcal{K}}_{1}\left({\zeta}_{0}\left(\omega \right)\right)}$ | 0 |

**Table 2.**As with Table 1, the parameter space behavior of the general no-slip transfer function $G(\omega ,z)$, but now including the effects of the finite boundary layer depth h and numbered I-h–IX-h. Again the functions ${\varphi}_{z}\left(\omega \right)\equiv {\zeta}_{0}\left(\omega \right)\left[1+\frac{1}{2}\frac{z}{{z}_{o}}\right]$, ${\zeta}_{z}\left(\omega \right)\equiv 2\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\frac{{z}_{o}}{\delta}\phantom{\rule{0.166667em}{0ex}}\sqrt{\left(1+\frac{z}{{z}_{o}}\right)\left|1+\frac{\omega}{f}\right|\phantom{\rule{0.166667em}{0ex}}}$, and ${\varsigma}_{z}\left(\omega \right)\equiv 2\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{z}{\mu}\left|1+\frac{\omega}{f}\right|}\phantom{\rule{0.166667em}{0ex}}$ are versions of the arguments to the Bessel functions appearing in the upper, middle, and lower rows. As before, $\mu (\delta ,{z}_{o})\equiv {\delta}^{2}/{z}_{o}$ is regarded as a function of $\delta $ and ${z}_{o}$, and $s=s(\omega ,f)\equiv \mathrm{sgn}\left(f\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{sgn}(1+\omega /f)$ is a sign function. The boxed terms were previously presented by Elipot and Gille [12]. Note that $z<h$ in all regimes, but only in the upper row, where $h\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}$, is there a required assumption for the size of h versus the other parameters.

Strong Gradient/Near-Inertial | Arbitrary Gradient/Any Frequency | Weak Gradient/Far-Inertial | |
---|---|---|---|

${\mathit{z}}_{\mathit{o}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\mathit{\delta}$ or $\mathit{\omega}\to -\mathit{f}$ | — | ($\mathit{\delta}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{\mathit{z}}_{\mathit{o}}$ and $\mathit{\omega}\ne -\mathit{f}$) or $\left|1+\mathit{\omega}/\mathit{f}\right|\to \mathit{\infty}$ | |

I-h | II-h | III-h, Depth-modified Ekman, EG-1b | |

$z<h\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}$ | $z<h\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\delta $ | $z<h\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}$ | $(z<h,\delta )\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}$ |

$\frac{2}{\rho \left|f\right|\delta}\frac{h-z}{\delta}$ | $\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{{\mathcal{I}}_{0}\left({\varphi}_{h}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varphi}_{z}\left(\omega \right)\right)-{\mathcal{I}}_{0}\left({\varphi}_{z}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varphi}_{h}\left(\omega \right)\right)}{{\mathcal{I}}_{0}\left({\varphi}_{h}\left(\omega \right)\right){\mathcal{K}}_{1}\left({\varphi}_{0}\left(\omega \right)\right)+{\mathcal{I}}_{1}\left({\varphi}_{0}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varphi}_{h}\left(\omega \right)\right)}}\hfill \end{array}$ | $\begin{array}{|c|}\hline \begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{sinh\left(\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\frac{h-z}{\delta}\sqrt{\left|1+\frac{\omega}{f}\right|}\right)}{cosh\left(\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{s\mathrm{i}\pi /4}\frac{h}{\delta}\sqrt{\left|1+\frac{\omega}{f}\right|}\right)}}\hfill \end{array}\\ \hline\end{array}$ | |

IV-h | V-h, Depth-modified mixed, EG-3b | VI-h | |

$z<h$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\delta $ | — | $\delta \leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}$ |

$\begin{array}{c}{\displaystyle \frac{4}{\rho \left|f\right|\mu}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \left[{\mathcal{K}}_{0}\left({\zeta}_{z}\left(\omega \right)\right)-\frac{{\mathcal{K}}_{0}\left({\zeta}_{h}\left(\omega \right)\right)}{{\mathcal{I}}_{0}\left({\zeta}_{h}\left(\omega \right)\right)}{\mathcal{I}}_{0}\left({\zeta}_{z}\left(\omega \right)\right)\right]}\hfill \end{array}$ | $\begin{array}{|c|}\hline \begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{{\mathcal{I}}_{0}\left({\zeta}_{h}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\zeta}_{z}\left(\omega \right)\right)-{\mathcal{I}}_{0}\left({\zeta}_{z}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\zeta}_{h}\left(\omega \right)\right)}{{\mathcal{I}}_{0}\left({\zeta}_{h}\left(\omega \right)\right){\mathcal{K}}_{1}\left({\zeta}_{0}\left(\omega \right)\right)+{\mathcal{I}}_{1}\left({\zeta}_{0}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\zeta}_{h}\left(\omega \right)\right)}}\hfill \end{array}\\ \hline\end{array}$ | $\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{1}{{\left(1+z/{z}_{o}\right)}^{1/4}}\frac{sinh\left({\zeta}_{h}\left(\omega \right)-{\zeta}_{z}\left(\omega \right)\right)}{cosh\left({\zeta}_{h}\left(\omega \right)-{\zeta}_{0}\left(\omega \right)\right)}}\hfill \end{array}$ | |

VII-h, Depth-modified Madsen, EG-2b | VIII-h | IX-h | |

${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z<h$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<(z<h,\delta )$ | ${z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z<h$ | $\delta \leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<{z}_{o}\leftarrow \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<z<h$ |

$\begin{array}{|c|}\hline \begin{array}{c}{\displaystyle \frac{4}{\rho \left|f\right|\mu}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \left[{\mathcal{K}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right)-\frac{{\mathcal{K}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right)}{{\mathcal{I}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right)}{\mathcal{I}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right)\right]}\hfill \end{array}\\ \hline\end{array}$ | $\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{{\mathcal{I}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right)-{\mathcal{I}}_{0}\left({\varsigma}_{z}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right)}{{\mathcal{I}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right){\mathcal{K}}_{1}\left({\zeta}_{0}\left(\omega \right)\right)+{\mathcal{I}}_{1}\left({\zeta}_{0}\left(\omega \right)\right){\mathcal{K}}_{0}\left({\varsigma}_{h}\left(\omega \right)\right)}}\hfill \end{array}$ | $\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{\rho \left|f\right|\delta}\frac{{\mathrm{e}}^{-s\mathrm{i}\pi /4}}{\sqrt{\left|1+\omega /f\right|}}\times \phantom{\rule{3.61371pt}{0ex}}}\hfill \\ {\displaystyle \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\left(\frac{{z}_{o}}{z}\right)}^{1/4}\frac{sinh\left({\varsigma}_{h}\left(\omega \right)-{\varsigma}_{z}\left(\omega \right)\right)}{cosh\left({\varsigma}_{h}\left(\omega \right)-{\zeta}_{0}\left(\omega \right)\right)}}\hfill \\ \u27f60\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{as}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{z}_{o}/z\u27f60\hfill \end{array}$ |

#### 4.3. Survey of Asymptotic Behavior

#### 4.4. A Depth/Frequency Interpretation of Regimes

#### 4.5. Impulse Response Functions

## 5. Discussion

## 6. Materials and Methods

`jLab`, is available for download from https://github.com/jonathanlilly/jLab with installation instructions and detailed online documentation available at http://www.jmlilly.net/software.html. The primary function related to this paper is called

`windtrans`. This implements the general no-slip transfer function of Equation (23), as well all of the boxed forms in Table 1 and Table 2. The default formulation uses the tilde-function approach developed in Appendix E to avoid numerical overflow. All figures are created with the script

`makefigs_transfer`.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Transfer Function Relation

## Appendix B. Derivation of the Modified Bessel’s Equation

## Appendix C. Verification of the Boundary Conditions

## Appendix D. The Free-Slip Transfer Function

## Appendix E. Numerical Computation of the Transfer Function

**Figure A1.**An illustration of overflow problems in computing the transfer function, and their solution. The transfer function $G(\omega ,z)$ at frequency $\omega =2f$ and depth $z=15$ m is computed on the $\delta $ vs. $\mu $ plane with $h=100$ m using (

**a**) Equation (23) directly or (

**b**) using the tilde-function formation of Equation (A20) for Bessel function ratio. In the latter case, 30 terms are used to compute ${\tilde{\mathcal{I}}}_{\eta}\left(z\right)$ and ${\tilde{\mathcal{K}}}_{\eta}\left(z\right)$ via Equations (A21) and (A22). The same color scale is used in both (

**a**,

**b**). Panel (

**c**) is a comparison of the asymptotic or one-term expansions of Equations (A16) and (A17) versus the 30-term expansions. The quantity shown in panel (

**c**) is the fractional error, defined as the magnitude of the asymptotic version minus the 30-term version, all divided by the 30-term version. The gray diagonal line marks the location on the $\delta $ vs. $\mu $ plane where $2\sqrt{2}({z}_{o}/\delta )\sqrt{|1+\omega /f|}=2\sqrt{2}\sqrt{3}(\delta /\mu )={10}^{2.9}$, and is the location where ${\mathcal{I}}_{0}\left({\zeta}_{0}\left(2f\right)\right)$ begins to overflow.

`besseli`function. The latter will be taken as the true value. Using 30 terms in Equation (A21), the approximation minus the true value, all divided by the true value, has a magnitude less than ${10}^{-14}$ with $x>23$ and for $\eta =0$ or 1. The same applies for ${\tilde{\mathcal{K}}}_{\eta}\left(x\sqrt{\pm \mathrm{i}}\right)$ in Equation (22) compared with

`besselk`with $x>15$. Thus, these series offer a high degree of numerical precision after 30 terms for even relatively small values of the argument. Because the ${\tilde{\mathcal{I}}}_{\eta}\left(x\right)$ and ${\tilde{\mathcal{K}}}_{\eta}\left(x\right)$ functions can be represented accurately, while the exponential growth terms have been arranged to cancel, we can now evaluate the transfer function for large values of ${z}_{o}/\delta $ with very high accuracy.

`besseli`and

`besselk`, and therefore we do not wish to use it for all parameter values. However, as the overflow due to large ${\zeta}_{h}\left(\omega \right)$ is already handled simply by rewriting the Bessel function ratio as in Equation (A18), the tilde-function approach only need be used when the second-largest Bessel function argument, ${\zeta}_{z}\left(\omega \right)$, also leads to overflow. In implementation, we switch to computing the Bessel function ratio using Equation (A20) whenever the magnitude of ${\zeta}_{z}\left(\omega \right)$ exceeds ${10}^{2.9}$, a threshold that is slightly below where ${\mathcal{I}}_{\eta}\left({\zeta}_{z}\left(\omega \right)\right)$ begins to overflow. In Figure A1b, the computation of the transfer function with the same parameter values as in panel (a) is accomplished by switching to the tilde-function version in this way. The transfer function can now be accurately computed over a wide parameter space.

## Appendix F. Derivation of the Asymptotic Forms

## Appendix G. The Ekman and Madsen Impulse Response Functions

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**Figure 1.**Examples of the transfer function for (

**a**) infinite boundary layer depth h and (

**b**) $h=50$ m. In both panels, the magnitude of the rescaled transfer $\tilde{G}(\omega ,z)\equiv \rho \left|f\right|G(\omega ,z)$ is shown. The parameter choices, chosen for display purposes rather than for realism, are $\delta =20$ m and ${z}_{o}=20$ m. The depth $z=20$ m is indicated in both panels by a dotted black line, while the curving white lines are curves of constant $|{\zeta}_{z}\left(\omega \right)|$. Roman numerals, shown here for future reference, refer to the regimes enumerated later in (

**a**) Table 1 and (

**b**) Table 2. Areas where the various forms from Table 1 and Table 2 apply are schematically indicated by the regions separated by white lines together with the gray horizontal lines. See Section 4.4 for further details regarding these regimes.

**Figure 2.**The rescaled transfer functions $\tilde{G}(\omega ,z)$ from Figure 1, plotted on the complex plane. Curves are plotted every 5 m from the surface to 45 m for (

**a**) the $h=\infty $ or (

**b**) $h=50$ m cases. In each panel, the transfer function at the surface is plotted with the heavy solid line, while the transfer function at 45 m is plotted with the heavy dashed line. Note the difference in axis limits between the two panels.

**Figure 3.**The impulse response or Green’s function $g(t,z)$ corresponding to the transfer function $\tilde{G}(\omega ,z)$ shown in Figure 1b and Figure 2b. For display purposes, the impulse response function is divided by its own maximum over all times and depths. Panel (

**a**) shows the magnitude of $g(t,z)$ as a function of time and depth with a logarithmic color scale, and with the white contours indicating marking $log\left(g(t,z)/\mathrm{max}\left\{g\right\}\right)={10}^{-n}$ for n being a non-negative integer. Panels (

**b**,

**c**) respectively show the real and imaginary parts of the transfer function at 5 m depth intervals, the same depths used in Figure 2. Line styles are also as in that plot. The horizontal lines in (

**a**) mark the depths of the corresponding curves plotted in (

**b**,

**c**). Vertical dotted lines mark locations of the zero-crossings of the real part of $g(t,z)$, which we observe to occur at $(1/4+n/2)$ inertial periods at all depths.

**Figure 4.**The rescaled inertial amplitude $A\equiv \rho \left|f\right|G(-f,z)$ on the $\delta $ vs. $\mu $ plane for $z=15$ m and $h=1000$ m; note that this quantity is independent of the choice of Coriolis frequency f. Black contours mark locations where $A={10}^{n}$ m${}^{-1}$ for integer n, while white lines are lines of constant ${z}_{o}$ with ${z}_{o}={\delta}^{2}/\mu ={10}^{n}$ m. Note that ${z}_{o}$ increases towards the lower right-hand corner. The heavy black line is the $A=1$ m${}^{-1}$ contour, a curve that will be referred to in subsequent figures, while the heavy white line is the ${z}_{o}=1$ m contour. Black dots mark intersections of the ${z}_{o}={10}^{n}$ m lines with the $A=1$ m${}^{-1}$ contour. As discussed subsequently, the limits of high and low values of ${z}_{o}$ correspond respectively to purely Madsen-like and purely Ekman-like transfer functions, modified by the finite value of the boundary layer depth h; this tendency is reflected with the “M” and “E” labels.

**Figure 5.**Transfer functions at 15 m depth on the complex plane with $A=1$ m${}^{-1}$ and with $h=16$ m, 100 m, 1000 m, 10${}^{4}$ m, 10${}^{5}$ m, and 10${}^{6}$ m in panels (

**a**–

**f**) respectively. The transfer functions in panel (

**c**), with $h=1000$ m, correspond to the dots on the $\delta $ vs. $\mu $ plane in Figure 4. In each panel, transfer functions are drawn for $\delta ={10}^{n}$ with n taking on all integer values from −12 and 12. The heavy black solid line is for the largest value of ${z}_{o}$, the point that is farthest into the Ekman-like regime along the $A=1$ m${}^{-1}$ contour, while the heavy black dashed line is for smallest value of ${z}_{o}$, i.e., the farthest point into the Madsen regime. Not all lines are visible because some almost exactly overlap others. A thin white line and thin black line show the asymptotic forms for the depth-modified Ekman and depth-modified Madsen solutions presented later in Table 2 as forms III-h and VII-h; these nearly exactly overlie the heavy solid and heavy dashed lines, respectively. According to the self-similarity established in Section 3.3, for each choice of h the transfer functions for a different A value but the same values of ${z}_{o}$ would appear identical to those shown here apart from an overall amplitude scaling; thus this figure essentially reflects the entire range of possible behaviors of the transfer function at depth $z=15$ m.

**Figure 6.**As in Figure 5c, the transfer function with $h=1000$ m, but for an observation depth of the surface, $z=0$, rather than $z=15$ m. All other parameter settings are as in that plot. The Madsen-like solution at ${z}_{o}=0$, plotted as a thin black line overlaying the heavy dashed line in Figure 5c, collapses to a single value at all frequencies at the surface, as indicated by the black dot.

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Lilly, J.M.; Elipot, S. A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem. *Fluids* **2021**, *6*, 85.
https://doi.org/10.3390/fluids6020085

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Lilly JM, Elipot S. A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem. *Fluids*. 2021; 6(2):85.
https://doi.org/10.3390/fluids6020085

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Lilly, Jonathan M., and Shane Elipot. 2021. "A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem" *Fluids* 6, no. 2: 85.
https://doi.org/10.3390/fluids6020085