# Vertical air speed

Assuming that the divergence varies linearly with the pressure between the surface (1000 hPa) and the tropopause (200 hPa). Can someone tell me how do we calculate the vertical speed on the pressure surface of 700 hPa if the maximum vertical speed in this column of air is 1 Pa/s?

Well, we start with making assumptions. Let's write the continuity equation with the Boussinesq approximation: $$\frac{\partial \omega}{\partial P}=-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=D\tag{1}$$. If we assume a priori that the divergence varies linearly with pressure, then we can write divergence in point-slope form: $$D(P)=\frac{D(1000 \text{ hPa})-D(200 \text{ hPa})}{1000 \text{ hPa}-200 \text{ hPa}}\left(P-200 \text{ hPa}\right)+D(200\text{ hPa})=mP+b \tag{2}$$
Let's combine (1) and (2): $$\frac{\partial \omega}{\partial P}=mP+b$$. Integrating over P yields $$\omega(P)=\frac{1}{2}mP^2+bP+P_0+\omega(P_0)$$. I'll let you plug in your numbers.