<p>The link is the conservation of mass equation: $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0. $$
Briefly, Ekman pumping is the horizontal divergence of the Ekman transport. </p>

<p>If we integrate this expression over the Ekman layer,
$$ \int_{-D}^0 \frac{\partial u}{\partial x} dz + \int_{-D}^0 \frac{\partial v}{\partial y} dz + \int_{-D}^0 \frac{\partial w}{\partial z} dz = 0$$ 
where $D$ is the height of the Ekman layer we take $z=0$ to be the surface.  This gives
$$ \frac{\partial}{\partial x} \int_{-D}^0  u dz +  \frac{\partial }{\partial y} \int_{-D}^0 v dz + w_{z=0} - w_{z=-D} = 0$$ or,
$$ \frac{\partial}{\partial x} U +  \frac{\partial }{\partial y} V + w_{z=0} - w_{z=-D} = 0$$ 
where $U$ and $V$ are the horizontal volume (Ekman) transports.  As a boundary condition we take $w_{z=0} = 0$ leaving
$$ w_{z=-D} = \frac{\partial}{\partial x} U +  \frac{\partial }{\partial y} V. $$
So, the Ekman pumping velocity is the horizontal divergence of the Ekman transport (and thus also proportional to the curl of the wind stress).</p>

CC BY-SA 3.0
2016-05-29T23:14:32.670Z
2016-05-29T23:14:32.670Z
jmunroe
<p><a href="https://en.wikipedia.org/wiki/Ekman_transport" rel="nofollow">Ekman transport</a> is the integrated horizontal flow in the boundary layer resulting from the balance between frictional forces (at the surface or bottom) and Coriolis. Meanwhile, <a href="http://oceanworld.tamu.edu/resources/ocng_textbook/chapter12/chapter12_03.htm" rel="nofollow">Ekman pumping</a> is a vertical motion at the base of the boundary layer resulting from the curl of the wind. </p>

<p><strong>What is the relationship between Ekman transport and Ekman pumping?</strong></p>

<ocean><oceanography><ocean-currents>
What is the relationship between Ekman transport and Ekman pumping?
arkaia
# What is the relationship between Ekman transport and Ekman pumping?

(Asked by: username_0 on 2016-05-12)

[Ekman transport](https://en.wikipedia.org/wiki/Ekman_transport) is the integrated horizontal flow in the boundary layer resulting from the balance between frictional forces (at the surface or bottom) and Coriolis. Meanwhile, [Ekman pumping](http://oceanworld.tamu.edu/resources/ocng_textbook/chapter12/chapter12_03.htm) is a vertical motion at the base of the boundary layer resulting from the curl of the wind.

**What is the relationship between Ekman transport and Ekman pumping?**

**Tags:** ocean, oceanography, ocean-currents


---

## 1 answers

### Answer by username_1 on 2016-05-29. Score: 6 | Accepted answer

The link is the conservation of mass equation: $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0. $$
Briefly, Ekman pumping is the horizontal divergence of the Ekman transport.

If we integrate this expression over the Ekman layer,
$$ \int\_{-D}^0 \frac{\partial u}{\partial x} dz + \int\_{-D}^0 \frac{\partial v}{\partial y} dz + \int\_{-D}^0 \frac{\partial w}{\partial z} dz = 0$$
where $D$ is the height of the Ekman layer we take $z=0$ to be the surface. This gives
$$ \frac{\partial}{\partial x} \int\_{-D}^0 u dz + \frac{\partial }{\partial y} \int\_{-D}^0 v dz + w\_{z=0} - w\_{z=-D} = 0$$ or,
$$ \frac{\partial}{\partial x} U + \frac{\partial }{\partial y} V + w\_{z=0} - w\_{z=-D} = 0$$
where $U$ and $V$ are the horizontal volume (Ekman) transports. As a boundary condition we take $w\_{z=0} = 0$ leaving
$$ w\_{z=-D} = \frac{\partial}{\partial x} U + \frac{\partial }{\partial y} V. $$
So, the Ekman pumping velocity is the horizontal divergence of the Ekman transport (and thus also proportional to the curl of the wind stress).


---

