Why do cold-core lows slope towards the cold air with heigth? How to show mathematically that wind intensifies with height in this case?
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
$$ \frac{\partial T}{\partial n}&gt; 0 $$
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 &lt; 0$ and $ \frac{\partial T}{\partial n} &gt; 0 $, we have $\frac{\partial  V_g}{\partial ln p} &lt; 0$, which implies $\frac{\partial  V_g}{\partial z} &gt; 0$, 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.