Ekman Flow:

Welcome to the wonderful world of Ekman Flow. On this page we are going to address the following issues:
  1. Where does Ekman flow come from?
  2. What does Ekman flow do?
  3. Our Model
  4. Future Work

1. Where does Ekman flow come from?

Let's consider wind blowing over a deep region of the ocean, and what happens to the water below. As the wind blows over the ocean, the air interacts with the surface water, feeling friction from the relatively stationary water. The air wants to slow down (losing momentum) and the water is pushed faster by the air (gaining momentum). We can model this behavior by looking at the momentum of these interactions through the lens of Navier-Stokes:

$\frac{D\vec{u}}{Dt}=\sum _i\frac{F_i}{m_i} \ \ \Rightarrow \ \  \frac{\partial\vec{u}}{\partial t} + (\vec{u}\cdot \vec{\nabla})\vec{u} = \vec{g} +\nu\triangle\vec{u} - \frac{\vec{\nabla} P}{\rho_0}  $

But, if we have homogeneous fluids and steady conditions (no time derivatives), we can reduce this equation down to:

$\nu\frac{\partial^2\vec{u}}{\partial^2 z} = f \left( (u-\bar{u})\hat{u} - (v-\bar{v})\hat{v} \right)$

$f$ is the Coriolis parameter and $\nu$ is eddy viscosity. Let's further constrain that at our water surface, which we'll now define to be z=0, there is a no-slip condition. And finally as the water depth goes to infinity the affect of the wind must go to zero:

at z=0     : $\rho_0 \nu \frac{\partial \vec{u}}{\partial z} = \vec{\tau}$
at z=-$\infty$   : $\vec{u}=\vec{\bar{u}}$

Where $\tau$ is wind-shear, $\rho_0$ is mass density, and $\vec{\bar{u}}$ is the background flow. These boundary conditions applied to the momentum equations can then be shown to have the following solution:

$\vec{u} = \vec{\bar{u}} + \frac{\sqrt{2}}{\rho_0 f d} e^{\delta}
\left[ {\begin{array}{cc}
   \cos(\delta-\phi) & -\sin(\delta-\phi) \\
   \sin(\delta-\phi) & \cos(\delta-\phi) \\
  \end{array} } \right]  \cdot \vec{\tau}  \doteq \vec{\bar{u}} + A(\delta) \mathbf{R}(\delta)\cdot\vec{\tau} $

$d=\pi \sqrt{ \frac{2\nu}{|f|} }$    $\delta=\frac{z}{d}$    $\phi=\frac{\pi}{4}$    $A(\delta)=\frac{\sqrt{2}}{\rho_0 f d} e^{\delta}$    $\mathbf{R}(\delta)=
\left[ {\begin{array}{cc}
   \cos(\delta-\phi) & -\sin(\delta-\phi) \\
   \sin(\delta-\phi) & \cos(\delta-\phi) \\
  \end{array} } \right]$

Well, that's great, but what does it mean?



2. What does Ekman flow do?

The most obvious behavior can be seen at z=0. The solution at z=0 reduces $\mathbf{R}(\delta)$ to:

$\mathbf{R}(\delta=0) =
\left[ {\begin{array}{cc}
   \cos(\phi) & \sin(\phi) \\
   -\sin(\phi)  & \cos(\phi) \\
  \end{array} } \right] = \frac{\sqrt{2}}{2}
\left[ {\begin{array}{cc}
   1   & 1 \\
   -1  & 1 \\
  \end{array} } \right]$

This result shows a surface flow 45$^\circ$. With a positive (negative) signed f, the rotation will be to the right (left) in the Northern (Southern) hemisphere. Next we can look at the net transport across an entire Ekman layer. Integrating the velocity from the surface to the infinitely deep bottom yields the net transport to be:

$\vec{U}_T = \int_{-\infty}^0 \left(\vec{u} - \vec{\bar{u}} \right) dz = \int_{-\infty}^0 \left(  A(\delta)\mathbf{R}(\delta)\cdot\vec{\tau}  \right) dz = d \cdot \vec{\tau}\cdot \int_{-\infty}^0 \left(  A(\delta)\mathbf{R}(\delta)  \right) d\delta $

The result of this integration can be shown to be:

$\vec{U}_T =\frac{1}{\rho_0 f}
\left[ {\begin{array}{cc}
    0  & 1 \\
   -1  & 0 \\
  \end{array} } \right] \cdot \vec{\tau}$

The resultant rotation matrix is a 90$^\circ$ rotation to the right (left) of the wind direction given a Northern (Southern) hemisphere latitude and thus a positive (negative) sign of f. This behavior is plotted below for an Eastward wind:

You are unable to view images... think of the memes you've missed!
This view can, however, be misleading. A proper 3D view has been plotted at the link below corresponding to the same initial conditions:

Super Awesome 3D Plot of Ekman flow



3. Our Model


Our model consists of a wind field over the Pacific ocean in the Northern hemisphere. The grid is constrained to longitudes $\in\left[ 180^\circ W,120^\circ W \right]$ and latitudes $\in\left[ 10^\circ N,70^\circ N  \right]$ with a resolution of $0.5^\circ$ in both directions. We made a Gaussian style wind that exhibited both strong divergence and curl, along with varying speed, with $0.5\%$ Gaussian noise added. A 20$\frac{m}{s}$ max speed was applied to the Gaussian wind profile and time steps were incremented with 3600$s$ resolution. Ekman depth and f number were calculated at each grid element to provide more authentic values. Finally ten surface buoys were placed in the water and tracked for surface transport over time, they are represented as diamonds on the plots. The data product we output has four subplots: Upper Left (UL) showing wind and bulk transport, Upper Right (UR) showing wind and surface transport, Lower Left (LL) showing the vertical component of the curl of the wind, and Lower Right (LR) showing the horizontal divergence of the wind field.



In the video version of this data product below you can see the bulk transport (UL plot, calculated as above) is orthogonal to the wind, and the surface speed of the water is 45$^\circ$ to the right (UR plot, calculated as above).

Not Quite as Awesome Ekman Transport Video



4. Future Work





2017/12/3 John Elliott, University of Alaska Fairbanks: jelliott4@alaska.edu