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Start with the original equation. Let's first write the Hydrostatic equation: $$\frac{\partial p}{\partial z}=-\rho g$$ So let's prove that $$-g=c_p\theta\frac{\partial \pi}{\partial z}$$ If we use the product rule, we observe $$-g=c_p(\frac{\partial \theta \pi}{\partial z}-\pi\frac{\partial \theta}{\partial z})$$ Since $\pi=\frac{T}{\theta}$, we can ...

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High pressures and low pressures at any height do not exist in isolation. Had to remind myself a lot of this (so anyone feel free to correct), but the first thing worth highlighting is that the Bermuda High is a warm core high pressure and so extends a great depth through the atmosphere. This image is a fairly typical cross section of the atmosphere at 35 ...

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After looking back into this I found my error to lie in the long equation $\dfrac{\partial \pi}{\partial z}=\left(\frac{1}{p_0}\right)^{\frac{R}{c_p}}\dfrac{\partial}{\partial z}\exp\left(\frac{R}{c_p}\log(p)\right)=\left(\frac{1}{p_0}\right)^{\frac{R}{c_p}}* \frac{T}{\theta}*{\frac{R}{c_p}}*\frac{1}{p}*\dfrac{\partial p}{\partial z}$. I was wrong, ...

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The storm and basics: Here is an animation I got from this NOAA Tweet of the lightning in Harvey hours before landfall: https://mashable.com/2017/08/31/harvey-weather-satellite-maps-lightning/, where I found the imagery, offers some basic thoughts, including the key concept that: As with most hurricanes, you'll notice that the most active area of lightning ...

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