I have wanted to run this story down since I saw the reference in Lamb’s *Hydrodynamics* to a paper by G. I. Taylor that contains a description of what oceanic and atmospheric scientists call “Ekman layers.” Physical oceanographers learn early in their careers that the Norwegian oceanographer Fritjof Nansen, on the **Fram expedition** of 1893-1896 noted that ice floes tend to drift to the right of the wind, and suggested to his colleague Vilhelm Bjerknes that the problem be assigned to a student. Bjerknes chose Ekman, who came up with the result associated with his name. Ekman’s result can be found in lots of places, e.g., the Wikipedia page on **Ekman layers** or just about any text on physical oceanography, e.g. Knauss, *Introduction to Physical Oceanography*. Ekman explained the crosswind transport of ice floes by assuming a balance of Coriolis acceleration and viscous drag. The bare bones of the derivation go like this:

Steady flow governed by a balance of Coriolis force and viscous drag looks like this:

\begin{align*}

-fv &= \nu u_{zz}\\

fu &= \nu v_{zz}\\

f &= 2\Omega \sin \phi

\end{align*}

where $(u,v)$ are the horizontal velocity components, $\Omega = 2\pi/86400{\_}$ sec is the angular rotation rate of the earth, $\phi$ is the latitude and $\nu$ is the viscosity, about which more later. For the flow near the ocean surface, the vector wind stress $(\tau^{(x)},\tau^{(y)})$ enters as a boundary condition $(\tau^{(x)},\tau^{(y)}) = \nu (u,v)_z$. The trick is to divide by $\nu$ on both sides, multiply the first equation by $i$ and add it to (and subtract it from) the second, yielding the complex conjugate scalar ODEs

\begin{align*}

&(u+iv)_{zz}-i(f/\nu)(u+iv) = 0\\

&(u-iv)_{zz}+i(f/\nu)(u-iv) = 0

\end{align*}

These equations are readily solved to find

\begin{align*}

u &= \frac{\surd 2}{fd}e^{z/d}\left ( \tau^{(x)}\cos(z/d -\pi /4) -\tau^{(y)}\sin(z/d – \pi /4)\right)\\

v &= \frac{\surd 2}{fd}e^{z/d}\left ( \tau^{(x)}\sin(z/d -\pi /4) +\tau^{(y)}\cos(z/d – \pi /4) \right)

\end{align*}

where $d=(2\nu /f)^{1/2}$. Integrating over the water column from $z=-\infty$ to $z=0$ yields the result that transport is to the right of the wind. If you draw the two-component current vectors at each depth as arrows, with the tails on the $z$-axis, you will see that the heads trace out a nice spiral. There are pictures in lots of places, e.g. the Wikipedia page on **Ekman layers**.

Oceanographers talk casually about Ekman layers and Ekman transports, but, to be precise, Ekman layers do not appear in nature. The real atmosphere and ocean are turbulent. Vertical momentum transfers do not occur by simple diffusion, and characterization of penetration of surface stress into the interior fluid by a simple scalar diffusion coefficient is only the crudest approximation. Ekman knew this. He noted that if he used the measured viscosity of sea water for $\nu$ and typical wind stress magnitudes for $\tau^{(x,y)}$ the surface layer would be less than a meter thick. He did not refer to the work of **Reynolds** on turbulence, and much of Ekman’s dissertation is concerned with trying to model flow in the real ocean in terms of the mathematical tools available to him.

G. I Taylor was apparently unaware of Ekman’s work when he applied the same analytical machinery to form an expression for the surface velocity profile in the atmospheric surface boundary layer. (Taylor, 1915: Eddy motion in the atmosphere, *Phil. Trans. R. Soc. Lond. A.*, **215**). Taylor was interested in quantifying the transport of physical properties by macroscopic eddies, i.e., he wanted to estimate what we now recognize as eddy diffusivity and eddy viscosity. Taylor compared the solutions of his equations to observations taken from balloons and backed out estimates of kinematic viscosities varying over an order of magnitude, from about $3\cdot 10^4$ and $6 \cdot 10^4 \mathrm{cm}^2 / \mathrm{sec}$ over land, and from $7.7\cdot 10^2$ to $6.9\cdot 10^3 \mathrm{cm}^2 / \mathrm{sec}$ over water. The molecular kinematic viscosity of air, by comparison, is more like $10^{-3} \mathrm{cm}^2 / \mathrm{sec}$. The sixth edition of Lamb’s *Hydrodynamics* contains discussions of both Ekman’s and Taylor’s work. It’s just as well that we refer to “Ekman layers” rather than “Taylor layers.” Ekman was first, after all.

Richardson, in his (1922) book {\it Weather Prediction by Numerical Process}, laid out specific methodology for numerical weather prediction. Richardson was a man far before his time. He could not have conceived of an electronic computer — his book was published decades before the work of Turing and von Neumann. He imagined that numerical weather prediction would be done by great halls full of people working the mechanical calculators of the day. Richardson understood that he would need to specify the viscosity of air, and, after examining the results available to him, including Taylor’s work, he concluded that air is slightly more viscous than **Lyle’s Golden Syrup** and slightly less viscous than shoe polish.

Ninety years after the publication of Richardson’t book, we have come a very long way but we still don’t know how to deal with turbulent transports in models of the ocean and atmosphere.

Robert Miller

College of Earth, Ocean, and Atmospheric Sciences

Oregon State University

miller@coas.oregonstate.edu