9.4: Application of Ekman Theory

Because steady winds blowing on the sea surface produce an Ekman layer that transports water at right angles to the wind direction, any spatial variability of the wind, or winds blowing along some coasts, can lead to upwelling. And upwelling is important:

1. Upwelling enhances biological productivity, which feeds fisheries.
2. Cold upwelled water alters local weather. Weather onshore of regions of upwelling tend to have fog, low stratus clouds, a stable stratified atmosphere, little convection, and little rain.
3. Spatial variability of transports in the open ocean leads to upwelling and downwelling, which leads to redistribution of mass in the ocean, which leads to wind-driven geostrophic currents via Ekman pumping.

Coastal Upwelling

To see how winds lead to upwelling, consider north winds blowing parallel to the California Coast (figure \(\PageIndex{1}\) left). The winds produce a mass transport away from the shore everywhere along the shore. The water pushed offshore can be replaced only by water from below the Ekman layer. This is upwelling (figure \(\PageIndex{1}\) right). Because the upwelled water is cold, the upwelling leads to a region of cold water at the surface along the coast. Figure \(10.8.4\) shows the distribution of cold water off the coast of California.

Figure \(\PageIndex{1}\): Sketch of Ekman transport along a coast leading to upwelling of cold water along the coast. Left: Plan view. North winds along a west coast in the northern hemisphere cause Ekman transports away from the shore. Right: Cross section. The water transported offshore must be replaced by water upwelling from below the mixed layer.

Upwelled water is colder than water normally found on the surface, and it is richer in nutrients. The nutrients fertilize phytoplankton in the mixed layer, which are eaten by zooplankton, which are eaten by small fish, which are eaten by larger fish and so on to infinity. As a result, upwelling regions are productive waters supporting the world’s major fisheries. The important regions are offshore of Peru, California, Somalia, Morocco, and Namibia.

Now I can answer the question I asked at the beginning of the chapter: Why is the climate of San Francisco so different from that of Norfolk, Virginia? Figures \(4.2.1\) or \(\PageIndex{1}\) show that wind along the California and Oregon coasts has a strong southward component. The wind causes upwelling along the coast, which leads to cold water close to shore. The shoreward component of the wind brings warmer air from far offshore over the colder water, which cools the incoming air close to the sea, leading to a thin, cool atmospheric boundary layer. As the air cools, fog forms along the coast. Finally, the cool layer of air is blown over San Francisco, cooling the city. The warmer air above the boundary layer, due to downward velocity of the Hadley circulation in the atmosphere (see figure \(4.2.2\)), inhibits vertical convection, and rain is rare. Rain forms only when winter storms coming ashore bring strong convection higher up in the atmosphere.

In addition to upwelling, other processes influence weather in California and Virginia.

1. The oceanic mixed layer tends to be thin on the eastern side of ocean, and upwelling can easily bring up cold water.
2. Currents along the eastern side of the ocean at mid-latitudes tend to bring colder water from higher latitudes.

All these processes are reversed offshore of east coasts, leading to warm water close to shore, thick atmospheric boundary layers, and frequent convective rain. Thus Norfolk is much different than San Francisco due to upwelling and the direction of the coastal currents.

Ekman Pumping

The horizontal variability of the wind blowing on the sea surface leads to horizontal variability of the Ekman transports. Because mass must be conserved, the spatial variability of the transports must lead to vertical velocities at the top of the Ekman layer. To calculate this velocity, we first integrate the continuity equation \((7.7.3)\) in the vertical: \[\begin{align*} \rho \int_{-d}^{0} \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) dz &= 0 \\ \frac{\partial}{\partial x} \int_{-d}^{0} \rho u \ dz + \frac{\partial}{\partial y} \int_{-d}^{0} \rho v \ dz &= -\rho \int_{-d}^{0} \frac{\partial w}{\partial z} dz \\ \frac{\partial M_{Ex}}{\partial x} + \frac{\partial M_{Ey}}{\partial y} &= -\rho [w(0) - w(-d)] \end{align*} \]

By definition, the Ekman velocities approach zero at the base of the Ekman layer, and the vertical velocity at the base of the layer \(w_{E}(-d)\), due to divergence of the Ekman flow, must be zero. Therefore: \[\frac{\partial M_{Ex}}{\partial x} + \frac{\partial M_{Ey}}{\partial y} = \rho w_{E}(0) \nonumber \]\[\boxed{\nabla_{H} \cdot \mathbf{M}_{E} = -\rho w_{E} (0)} \nonumber \]

where \(\mathbf{M}_{E}\) is the vector mass transport due to Ekman flow in the upper boundary layer of the ocean, and \(\nabla_{H}\) is the horizontal divergence operator. \((\PageIndex{1})\) states that the horizontal divergence of the Ekman transports leads to a vertical velocity in the upper boundary layer of the ocean, a process called Ekman Pumping.

If we use the Ekman mass transports \((9.3.3-4)\) in \((\PageIndex{1})\) we can relate Ekman pumping to the wind stress. \[\begin{align} w_{E}(0) &= -\frac{1}{\rho} \left[\frac{\partial}{\partial x} \left(\frac{T_{yz}(0)}{f}\right) - \frac{\partial}{\partial y} \left(\frac{T_{xz}(0)}{f}\right)\right] \\ w_{E}(0) &= -\text{curl}_{z} \left(\frac{\mathbf{T}}{\rho f}\right) \end{align} \nonumber \]

where \(\mathbf{T}\) is the vector wind stress and the subscript \(z\) indicates the vertical component of the curl.

The vertical velocity at the sea surface \(w(0)\) must be zero because the surface cannot rise into the air, so \(w_{E}(0)\) must be balanced by another vertical velocity. We will see in Chapter 12 that it is balanced by a geostrophic velocity \(w_{G}(0)\) at the top of the interior flow in the ocean.

Note that the derivation above follows Pedlosky (1996: 13), and it differs from the traditional approach that leads to a vertical velocity at the base of the Ekman layer. Pedlosky points out that if the Ekman layer is very thin compared with the depth of the ocean, it makes no difference whether the velocity is calculated at the top or bottom of the Ekman layer, but this is usually not true for the ocean. Hence, we must compute vertical velocity at the top of the layer.