# barotropic component definition

In this 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?

• In what paper? You don't include the rference – arkaia Jan 24 '18 at 15:23
• @arkaia Sorry. I edited the question – shamalaia Jan 24 '18 at 23:13

The barotropic pressure gradient is also the integral result for the entire water column. A good illustration is the 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 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.