<p>The top expression is basically the depth-averaged velocity. Meanwhile the second expression only applies to geostrophic flows. The depth-averaged velocity can be caused by many mechanisms, with the geostrophic balance being one of them. The most common ones are the wind and the tides (but also differences in atmospheric pressure, sloping bottoms, ...). Both tides and wind create a barotropic pressure gradient that has to be compensated by other forces in the Navier-Stokes equations, with friction being the most common one. The <a href="https://en.wikipedia.org/wiki/Geostrophic_current" rel="nofollow noreferrer">geostrophic balance</a> is a specific solution of the NS equations that neglects the effects of friction and any other forces. <a href="https://marine.rutgers.edu/dmcs/ms501/2004/Notes/Wilkin20041104.htm" rel="nofollow noreferrer">John Wilkin (Rutgers)</a> has some good notes on how to estimate these currents from observations.</p>

<p>The barotropic pressure gradient is also the integral result for the entire water column. A good illustration is the <a href="http://en.wikipedia.org/wiki/Thermal_wind" rel="nofollow noreferrer">thermal wind</a> example:  You need a difference in slope between isobaric (pressure) and isopycnal (density) surfaces to give you a baroclinic gradient. In theory in a <a href="http://en.wikipedia.org/wiki/Barotropic_fluid" rel="nofollow noreferrer">barotropic fluid</a> the flow is the same at all depths: $u(z)=u_{bt}$. Clearly, in practice, any fluid will have boundary layers in which that is not the case. </p>

CC BY-SA 3.0
2018-01-25T05:48:50.700Z
Thank you for the reference. However, I still do not understand why the depth-averaged velocity is influenced only by barotropic pressure gradients. If there is some baroclinic disturbance, this would affect the velocity and, as a consequence, also its depth-averaged value. Or not?
shamalaia
CC BY-SA 3.0
2018-01-25T12:01:39.493Z
The barotropic pressure gradient is also the integral result for the entire water column. You can look at the thermal wind example: https://en.wikipedia.org/wiki/Thermal_wind You need a difference in slope between isobaric (pressure) and isopycnal (density) surfaces to give you a baroclinic gradient. In theory in a barotropic fluid (https://en.wikipedia.org/wiki/Barotropic_fluid) the flow is the same at all depths: u(z)=u_bt. Clearly, in practice, any fluid will have boundary layers in which that is not the case.
arkaia
CC BY-SA 3.0
2018-01-25T02:02:24.717Z
2018-01-26T15:13:55.047Z
2018-01-26T15:13:55.047Z
arkaia
<p>In <a href="https://www.sciencedirect.com/science/article/pii/S0967063709001344?via%3Dihub" rel="nofollow noreferrer">this</a> paper the author calculated the barotropic component $u_{bt}$ of the current $u$ as: </p>

<blockquote>
  <p>the vertical mean velocity weigthed with the depth differences</p>
</blockquote>

<p>If I am not wrong, in a continuous case, this translates to:</p>

<p>$$ u_{bt} = \frac{1}{d}\int_{-d}^{0}u(z)~dz$$</p>

<p>However, supposing the current to be geostrophic, the following expression can be found for the barotropic component:</p>

<p>$$ u_{bt} = -\frac{g}{f}\frac{\partial \eta}{\partial y} $$</p>

<p>where $g$ is the gravity acceleration, $f$ the Coriolis parameter and $\eta$ the SSH anomaly.</p>

<p>Are these two expressions equivalent? If yes, why? If not, which are the differences between the hypothesis at the base of the two expressions?</p>

CC BY-SA 3.0
2018-01-24T15:23:02.680Z
In what paper? You don't include the rference
arkaia
CC BY-SA 3.0
2018-01-24T23:13:34.067Z
@arkaia Sorry.  I edited the question
shamalaia
<oceanography>
barotropic component definition
shamalaia
# barotropic component definition

(Asked by: username_0 on 2018-01-24)

In [this](https://www.sciencedirect.com/science/article/pii/S0967063709001344?via%3Dihub) paper the author calculated the barotropic component $u\_{bt}$ of the current $u$ as:

> the vertical mean velocity weigthed with the depth differences

If I am not wrong, in a continuous case, this translates to:

$$ u\_{bt} = \frac{1}{d}\int\_{-d}^{0}u(z)~dz$$

However, supposing the current to be geostrophic, the following expression can be found for the barotropic component:

$$ u\_{bt} = -\frac{g}{f}\frac{\partial \eta}{\partial y} $$

where $g$ is the gravity acceleration, $f$ the Coriolis parameter and $\eta$ the SSH anomaly.

Are these two expressions equivalent? If yes, why? If not, which are the differences between the hypothesis at the base of the two expressions?

**Tags:** oceanography

> **Comments:**
> **username_1**: In what paper? You don't include the rference  
> **username_0**: @username_1 Sorry.  I edited the question  

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## 1 answers

### Answer by username_1 on 2018-01-25. Score: 2 | Accepted answer

The top expression is basically the depth-averaged velocity. Meanwhile the second expression only applies to geostrophic flows. The depth-averaged velocity can be caused by many mechanisms, with the geostrophic balance being one of them. The most common ones are the wind and the tides (but also differences in atmospheric pressure, sloping bottoms, ...). Both tides and wind create a barotropic pressure gradient that has to be compensated by other forces in the Navier-Stokes equations, with friction being the most common one. The [geostrophic balance](https://en.wikipedia.org/wiki/Geostrophic_current) is a specific solution of the NS equations that neglects the effects of friction and any other forces. [John Wilkin (Rutgers)](https://marine.rutgers.edu/dmcs/ms501/2004/Notes/Wilkin20041104.htm) has some good notes on how to estimate these currents from observations.

The barotropic pressure gradient is also the integral result for the entire water column. A good illustration is the [thermal wind](http://en.wikipedia.org/wiki/Thermal_wind) example: You need a difference in slope between isobaric (pressure) and isopycnal (density) surfaces to give you a baroclinic gradient. In theory in a [barotropic fluid](http://en.wikipedia.org/wiki/Barotropic_fluid) the flow is the same at all depths: $u(z)=u\_{bt}$. Clearly, in practice, any fluid will have boundary layers in which that is not the case.

> **Comments:**
> **username_0**: Thank you for the reference. However, I still do not understand why the depth-averaged velocity is influenced only by barotropic pressure gradients. If there is some baroclinic disturbance, this would affect the velocity and, as a consequence, also its depth-averaged value. Or not?  
> **username_1**: The barotropic pressure gradient is also the integral result for the entire water column. You can look at the thermal wind example: https://en.wikipedia.org/wiki/Thermal_wind You need a difference in slope between isobaric (pressure) and isopycnal (density) surfaces to give you a baroclinic gradient. In theory in a barotropic fluid (https://en.wikipedia.org/wiki/Barotropic_fluid) the flow is the same at all depths: u(z)=u_bt. Clearly, in practice, any fluid will have boundary layers in which that is not the case.  

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