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I am puzzled by the dimensions of the formulas given here for the baroclinic component of ocean currents (first terms on the right-hand side of formulas 5 and 6).

If I am not wrong:

[f]=1/s

[$\rho$]=kg/m3

[$\frac{\partial}{\partial y}$]=1/m

[dz]=m

therefore: $$[\frac{1}{f\rho}\frac{\partial}{\partial y}\int_{-z}^0\rho(z)dz]=s\frac{m^3}{kg}\frac{1}{m}\frac{kg}{m^3}m = s$$

What am I missing?

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They are missing $g$ in equations 4,5 and 6. When they go from the hydrostatic equation to equation 4, they forget to include gravity in their equations. All the dimensions except from $g$ and $f$ compensate each other and then $g/f$ produces the right velocity dimensions $m/s$.

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