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

CC BY-SA 4.0
2020-10-28T21:25:52.993Z
what does ''m'' and ''b'' mean in your equation?
gulraiz safdar
CC BY-SA 4.0
2020-10-28T21:49:58.430Z
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
CC BY-SA 4.0
2020-11-01T04:46:15.087Z
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
CC BY-SA 4.0
2020-11-01T16:36:55.483Z
@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
CC BY-SA 4.0
2020-11-01T23:38:06.000Z
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
CC BY-SA 4.0
2020-10-27T21:09:27.500Z
2020-10-27T21:09:27.500Z
BarocliniCplusplus
<p>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?</p>

<meteorology><atmospheric-circulation><fluid-dynamics>
Vertical air speed
gulraiz safdar
# Vertical air speed

(Asked by: username_0 on 2020-10-24)

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?

**Tags:** meteorology, atmospheric-circulation, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2020-10-27. Score: 1

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.

> **Comments:**
> **username_0**: what does ''m'' and ''b'' mean in your equation?  
> **username_0**: 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.)  
> **username_2**: 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.  
> **username_1**: @username_2 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.  
> **username_2**: 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 :-/  

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