Why is the mean meridional mass stream function multiplied by g/2π?

I was reading this material on the mean atmospheric meridional circulation and the definition of the meridional mass streamfunction: http://www.geo.cornell.edu/geology/faculty/Cook/Hadley_book.pdf (Pag 3 and 4).

The continuity equation in spherical coordinates and using pressure as the vertical coordinate reads:

$$\frac{1}{a \cos\phi}\frac{\partial u}{\partial \lambda}+\frac{1}{a \cos\phi}\frac{\partial (v \cos\phi)}{\partial \phi}+\frac{\partial \omega}{\partial p}=0$$

Where $$a$$ is the Earth radius, $$\phi$$ is the latitude, $$\lambda$$ is the longitude, $$u,v$$ the horizontal wind components on a pressure level, $$\omega$$ is the vertical pseudovelocity in pressure coordinates. My first doubt is in fact when they state that the zonal (longitudinal) mean of this equation is

$$\frac{1}{a}\frac{\partial[v]}{\partial \phi}+\frac{\partial[\omega]}{\partial p}=0$$

where [ ] indicates the zonal mean. I get why the first term $$\frac{1}{a \cos\phi}\frac{\partial u}{\partial \lambda}$$ vanishes; I don't get though why $$\frac{1}{a \cos\phi}\frac{\partial (v \cos\phi)}{\partial \phi}$$ reduces to $$\frac{1}{a}\frac{\partial[v]}{\partial \phi}$$. By the product differentiation rule, I'd expect it to look like

$$\frac{1}{a}\frac{\partial[v]}{\partial \phi}-\frac{\tan\phi}{a}[v]$$

especially because they then define the meridional streamfunction $$\Psi$$ such that

$$[v] = \frac{g}{2\pi a \cos\phi}\frac{\partial\Psi}{\partial p}$$

and

$$[\omega] = -\frac{g}{2\pi a^{2} \cos\phi}\frac{\partial\Psi}{\partial \phi}$$

so that when we do $$\frac{1}{a}\frac{\partial[v]}{\partial \phi}+\frac{\partial[\omega]}{\partial p}$$ using this definition it actually results in $$\frac{\tan\phi}{a}[v]$$, not $$0$$.

I wonder if there is a mistake there or is there something I'm not following?

My second doubt is, why multiply by the factor $$\frac{g}{2\pi}$$ the definition of the streamfunction, since it will be cancelled out anyway when replacing this definition into the continuity equation? I mean, it doesn't actually interfere in this definition satisfying the continuity equation or not.