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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:
quente = hot, frio = cold
What I don't understand is why in practice the low slopes towards the cold air with height:
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.
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.
