Inthispaper 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?

Inthispaper 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?