<p>I've managed to show why wind speed increases with height in a cold core low, as follows. Considering low is in the Southern Hemisphere, the wind blows clockwise around the low. In natural coordinates, the normal vector points outwards. Since the center is colder than the surroundings, we have</p>
<p><span class="math-container">$$ \frac{\partial T}{\partial n}&gt; 0 $$</span></p>
<p>from the thermal wind relationship,</p>
<p><span class="math-container">$$\frac{\partial  V_g}{\partial ln p}\boldsymbol{\widehat{t}}= \frac{-R}{f}\boldsymbol{\widehat{k}} \times\nabla_p{T}=\frac{-R}{f}\boldsymbol{\widehat{k}} \times\frac{\partial T}{\partial n}\boldsymbol{\widehat{n}}=\frac{R}{f}\frac{\partial T}{\partial n}\boldsymbol{\widehat{t}} $$</span></p>
<p>Where <span class="math-container">$\boldsymbol{\widehat{t}}$</span>,<span class="math-container">$\boldsymbol{\widehat{n}}$</span> and <span class="math-container">$ \boldsymbol{\widehat{k}}$</span> are the tangent, normal (in the horizontal plane) and vertical unit vectors.</p>
<p>Since <span class="math-container">$ f &lt; 0$</span> and <span class="math-container">$ \frac{\partial T}{\partial n} &gt; 0 $</span>, we have <span class="math-container">$\frac{\partial  V_g}{\partial ln p} &lt; 0$</span>, which implies <span class="math-container">$\frac{\partial  V_g}{\partial z} &gt; 0$</span>, i.e., wind intensifies with height in a cold low.</p>
<p>I still have not understood why the low is displaced towards cold air with height. Since the wind intensifies with height, I would expect the low to be further ahead at higher levels than at the surface.</p>

CC BY-SA 4.0
2020-10-12T15:38:17.063Z
2020-10-12T15:38:17.063Z
paulo
<p>I understand that from the hypsometric equation, in a cold core low the pressure gradient must increase with height, because in the center the isobars will be closer together:</p>
<p><a href="https://i.sstatic.net/5pKlW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5pKlW.png" alt="enter image description here" /></a><br />
<sub><sup>quente = hot, frio = cold</sup></sub></p>
<p>What I don't understand is why in practice the low slopes towards the cold air with height:</p>
<p><a href="https://i.sstatic.net/4Meyz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4Meyz.png" alt="enter image description here" /></a></p>
<p>Also, I wonder how I can show mathematically (I guess from the thermal wind relationship + the hypsometric equation?), that in a cold core low, the wind must increase with height.</p>

CC BY-SA 4.0
2020-10-12T11:03:33.930Z
frio means cold and quente means hot :-)
JeopardyTempest
CC BY-SA 4.0
2020-10-12T11:08:45.943Z
Perhaps the geostrophic wind equation is another option towards showing why the wind increases?
JeopardyTempest
CC BY-SA 4.0
2020-10-12T11:19:25.437Z
Off the top of my head at this late hour, I can't reason out the slope towards cold.  Well, to be honest, you've really just about entirely explained why it slopes towards the cold (just consider more what you said at the start)... but why it slopes at all (or in other words: why it slopes towards warm air at low levels) is less clear to me at this moment.  My guess is it may have something to do with that thermal wind relationship you mentioned.  Hopefully people can give good hints (without full answers if this is a homework question?) :-)
JeopardyTempest
CC BY-SA 4.0
2020-10-12T22:24:27.100Z
Not sure that really answers why anymore than what I said?  I can give advanced reasons why the surface low develops where it does (vorticity advection and such)... but think that's beyond the complexity of this question.  Just a "this generally happens" seems not to the level this question is asked from.
JeopardyTempest
CC BY-SA 4.0
2020-12-18T07:59:16.293Z
Um...shouldnt a cold core low have low pressure at high altitude and high pressure at low altitude?
R. Emery
<meteorology>
Why do cold-core lows slope towards the cold air with heigth? How to show mathematically that wind intensifies with height in this case?
paulo
# Why do cold-core lows slope towards the cold air with heigth? How to show mathematically that wind intensifies with height in this case?

(Asked by: username_0 on 2020-10-11)

I understand that from the hypsometric equation, in a cold core low the pressure gradient must increase with height, because in the center the isobars will be closer together:

[![enter image description here](https://i.sstatic.net/5pKlW.png)](https://i.sstatic.net/5pKlW.png)  
quente = hot, frio = cold

What I don't understand is why in practice the low slopes towards the cold air with height:

[![enter image description here](https://i.sstatic.net/4Meyz.png)](https://i.sstatic.net/4Meyz.png)

Also, I wonder how I can show mathematically (I guess from the thermal wind relationship + the hypsometric equation?), that in a cold core low, the wind must increase with height.

**Tags:** meteorology

> **Comments:**
> **username_1**: frio means cold and quente means hot :-)  
> **username_1**: Perhaps the geostrophic wind equation is another option towards showing why the wind increases?  
> **username_1**: Off the top of my head at this late hour, I can't reason out the slope towards cold.  Well, to be honest, you've really just about entirely explained why it slopes towards the cold (just consider more what you said at the start)... but why it slopes at all (or in other words: why it slopes towards warm air at low levels) is less clear to me at this moment.  My guess is it may have something to do with that thermal wind relationship you mentioned.  Hopefully people can give good hints (without full answers if this is a homework question?) :-)  
> **username_1**: Not sure that really answers why anymore than what I said?  I can give advanced reasons why the surface low develops where it does (vorticity advection and such)... but think that's beyond the complexity of this question.  Just a "this generally happens" seems not to the level this question is asked from.  
> **username_2**: Um...shouldnt a cold core low have low pressure at high altitude and high pressure at low altitude?  

---

## 1 answers

### Answer by username_0 on 2020-10-12. Score: 1

I've managed to show why wind speed increases with height in a cold core low, as follows. Considering low is in the Southern Hemisphere, the wind blows clockwise around the low. In natural coordinates, the normal vector points outwards. Since the center is colder than the surroundings, we have

$$ \frac{\partial T}{\partial n}> 0 $$

from the thermal wind relationship,

$$\frac{\partial V\_g}{\partial ln p}\boldsymbol{\widehat{t}}= \frac{-R}{f}\boldsymbol{\widehat{k}} \times\nabla\_p{T}=\frac{-R}{f}\boldsymbol{\widehat{k}} \times\frac{\partial T}{\partial n}\boldsymbol{\widehat{n}}=\frac{R}{f}\frac{\partial T}{\partial n}\boldsymbol{\widehat{t}} $$

Where $\boldsymbol{\widehat{t}}$,$\boldsymbol{\widehat{n}}$ and $ \boldsymbol{\widehat{k}}$ are the tangent, normal (in the horizontal plane) and vertical unit vectors.

Since $ f < 0$ and $ \frac{\partial T}{\partial n} > 0 $, we have $\frac{\partial V\_g}{\partial ln p} < 0$, which implies $\frac{\partial V\_g}{\partial z} > 0$, i.e., wind intensifies with height in a cold low.

I still have not understood why the low is displaced towards cold air with height. Since the wind intensifies with height, I would expect the low to be further ahead at higher levels than at the surface.


---

