Ekman Layers

Near the upper and lower boundaries of the ocean (and lower boundary of the atmosphere), boundary layers form. These boundary layers, called Ekman layers, are typically modeled as geostrophic balance in the presence of an eddy viscosity. The model equation is

$$f\hat{k} \times \vec{u} = -\frac{\nabla p}{\rho_0} + \nu \frac{\partial^2 \vec{u}}{\partial z^2}$$

where \(\nu\) is the eddy viscosity, and we take \(\rho_0\) and \(f\) to be constant (and gravity is absorbed into the pressure term). We further assume that \(\vec{u}=0\) at the boundaries and that \(\vec{u}=\vec{u_g}\) in the interior, where

$$\vec{u_g} \equiv \frac{1}{f\rho_0}\hat{k} \times \nabla p$$

is the geostrophic velocity.

Now consider specifically a bottom boundary with a surface stress:

$$\frac{\vec{\tau}}{\rho_0} = \nu \frac{\partial \vec{u}}{\partial z}$$

  1. Show that the transport in the Ekman layer is$$\vec{U_e} \equiv \int\limits_{-H}^{-H+\delta} \vec{u_e}dz = \hat{k} \times \left(\frac{\vec{\tau}}{f\rho_0}\right)$$where \(\vec{u_e} = \vec{u}-\vec{u_g}\), \(H\) is the depth of the ocean, and \(\delta\) is the thickness of the layer. Remember that the ageostrophic velocity induced by the surface stress vanishes at the edge of the Ekman layer.
  2. Using the continuity equation and remembering that vertical velocity is identically zero at the bottom surface, show that the divergence of the total transport in the Ekman layer is$$\nabla \cdot \vec{U} = -w_e$$where \(w_e\) is the vertical velocity at the upper edge of the Ekman layer.
  3. Relate the vertical velocity to the divergence of the geostrophic transport in the Ekman layer and the applied stress.
  4. Give a scaling estimate for the depth \(\delta\) of the Ekman layer.
  5. It is well known that the sediment in the bottom of a cup of fluid congregates toward the center of the cup when the fluid is stirred, regardless of the direction of stirring. Qualitatively modify the above Ekman-layer theory to explain this phenomenon.
Answer

  1. Show that the transport in the Ekman layer is$$\vec{U_e} \equiv \int\limits_{-H}^{-H+\delta} \vec{u_e}dz = \hat{k} \times \left(\frac{\vec{\tau}}{f\rho_0}\right)$$where \(\vec{u_e} = \vec{u}-\vec{u_g}\), \(H\) is the depth of the ocean, and \(\delta\) is the thickness of the layer. Remember that the ageostrophic velocity induced by the surface stress vanishes at the edge of the Ekman layer.

By separating the geostrophic component of velocity from the total, we can rewrite the model equation in terms of ageostrophic velocities alone:

$$\begin{align*}
f\hat{k} \times \vec{u_e} &= \nu \frac{\partial^2\vec{u_e}}{\partial z^2} \\
f \vec{u_e} &= -\hat{k} \times \left(\nu \frac{\partial^2\vec{u_e}}{\partial z^2}\right)
\end{align*}$$

Integrating:
$$\begin{align*}
f\int\limits_{-H}^{-H+\delta} \vec{u_e}dz &= \int\limits_{-H}^{-H+\delta}-\hat{k} \times \left(\nu\frac{ \partial^2\vec{u_e}}{\partial z^2}\right)dz\\
&= -\hat{k} \times \int\limits_{-H}^{-H+\delta}\nu \frac{\partial^2\vec{u_e}}{\partial z^2}dz\\
&= -\hat{k} \times \int\limits_{-H}^{-H+\delta}\frac{\partial}{\partial z}\left(\frac{\vec{\tau}}{\rho_0}\right)dz\\
&= \hat{k} \times \left(\frac{\vec{\tau}|_{-H}}{\rho_0}\right)\\
\implies U_e &\equiv \hat{k} \times \left(\frac{\vec{\tau}}{f \rho_0}\right)
\end{align*}$$

since \(\tau\) disappears at the edge of the Ekman layer.

  1. Using the continuity equation and remembering that vertical velocity is identically zero at the bottom surface, show that the divergence of the total transport in the Ekman layer is$$\nabla \cdot \vec{U} = -w_e$$where \(w_e\) is the vertical velocity at the upper edge of the Ekman layer.

We know via continuity that the divergence of the three-dimensional velocity is zero, and thus so is its integral across the boundary layer. Separating the horizontal components from the vertical:

$$\nabla \cdot \vec{U} = -\frac{\partial W}{\partial z} = -\frac{\partial W_e}{\partial z}$$

since the geostrophic velocity has no vertical component. Using the definition of \(W_e\):

$$\begin{align*}
-\frac{\partial W_e}{\partial z} &= -\frac{\partial }{\partial z}\int\limits_{-H}^{-H+\delta} w_e dz\\
&= -w_e|_{-H+\delta} + 0 = -w_e
\end{align*}
$$

  1. Relate the vertical velocity to the divergence of the geostrophic transport in the Ekman layer and the applied stress.

We can use the result from part (2) of this problem, but separate out the geostrophic and ageostrophic components of horizontal transport, and use our result from part (1) to introduce the applied stress:

$$
\begin{align*}
w &= -\nabla \cdot \vec{U_g} - \nabla \cdot \vec{U_e}\\
&= -\nabla \cdot \vec{U_g} - \nabla \cdot \left(\hat{k} \times \left(\frac{\vec{\tau}}{f\rho_0}\right)\right)
\end{align*}
$$

  1. Give a scaling estimate for the depth \(\delta\) of the Ekman layer.

We can look at the momentum equations for the ageostrophic velocity:

$$
f\hat{k} \times \vec{u_e} = \nu \frac{\partial^2 \vec{u_e}}{\partial z^2}
$$

If we nondimensionalize this equation and let \(z\) scale as \(\delta\), we find that

$$
f \sim \frac{\nu}{\delta^2}
$$

which means that \(\delta\) scales like \(C\sqrt{\frac{\nu}{f}}\), where \(C\) is a constant.

  1. It is well known that the sediment in the bottom of a cup of fluid congregates toward the center of the cup when the fluid is stirred, regardless of the direction of stirring. Qualitatively modify the above Ekman-layer theory to explain this phenomenon.

This is called the tea leaf paradox. A secondary circulation arises when stirring the tea due to friction at the bottom and cup walls. This circulation is depicted in the image below.

  • Due to a reduction of angular momentum near the walls, gravity allows the lea leaves to sink there.
  • The angular momentum is also reduced at the ground, and the pressure gradient moves the leaves from the edges to the center (pressure is higher on the edges because the surface of the water is higher here).
  • Because the flow converges at the bottom, the vertical velocity in the Ekman layer should be positive. So the flow moves up the axis of rotation, but if the tea leaves are heavy they will collect near the bottom.
  • Because the vertical velocity approaches zero in the interior, \(\frac{\partial w}{\partial z}<0\), meaning there must be a divergence of horizontal velocities at the top of the circulation.

Thus, sediment will congregate towards the center, regardless of the direction of stirring.

From Wikipedia.


From Wikipedia.