# 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?

That's... a pretty specific question.

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.

• what does ''m'' and ''b'' mean in your equation? – gulraiz safdar Oct 28 '20 at 21:25
• I also have another related question if you could answer please? ( Now suppose that the air rises at a rate of 2 cm / s at 700 hPa. What will be the vertical speed omega if the temperature at this pressure surface is -10 ° C.) – gulraiz safdar Oct 28 '20 at 21:49
• Perhaps it would be better to only point to equations without doing the derivations when it appears to be a homework question? Especially when the asker hasn't made any attempt. – JeopardyTempest Nov 1 '20 at 4:46
• @JeopardyTempest In spirit, I agree with you. However, I noticed that the assumptions and data the asker was making were unreasonable, so I gave an equivalently unreasonable answer. For example, the linear profile of divergence is out of touch of even approximations of reality (like the bow-string profile of omega actually varies linearly with height not pressure). I agree, I won't answer the follow-up question and presume the asker will fail the exam. And I'll try to be a bit more guarded with my answers. – BarocliniCplusplus Nov 1 '20 at 16:36
• Hey, fair enough. And I'm not against helping people struggling, I want them to understand... just encourage everyone to be patient with it. I've certainly been guilty of fully answering some homework questions in my day :-/ – JeopardyTempest Nov 1 '20 at 23:38