What is the relationship between vertical motion of air, air pressure, and storm

The relationship between the two is that air temperature changes the air pressure. For example, as the air warms up the molecules in the air become more active and they use up more individual space even though there is the same number of molecules. This causes an increase in the air pressure.

The same goes for cooler air. When the temperature cools, the molecules slow down and they do not move and bump into each other. This causes a decrease in air pressure.

An easy way to understand this concept is by comparing car tires in the winter and car tires in the summer. In the summer the air is a lot warmer so the molecules are moving around a lot. The tire expands and you not need as much air because the pressure in the tires is high enough. In the winter, when the air is cold, the molecules are moving very slowly. Many people need to add more air to their tires because there is not enough pressure.

Formation of a Storm Surge
When a hurricane is in deep ocean waters, the circulating wind pushes the ocean surface to create a vertically circulating column of water, where the surge is barely visible. However, as the storm moves closer to the shore, the water which is being pushed downwards by the wind cannot move any lower, so the water forces itself from the sides towards land, causing a storm surge wave. Although low pressure also contributes to the surge, its influence is very small, i.e., around 5%. An average storm surge can reach heights of around 15 feet, while a storm tide will reach around 17 feet. However, larger and smaller ones have also been observed and recorded. On the other hand, the width of these waves depends upon the shoreline length, with larger surges being recorded at longer beaches.

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What is the relationship between vertical motion of air, air pressure, and storm

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