How is density involved in natural events that happen everyday on Earth or in the atmosphere?

density is involved in natural mechanical phenomena via its gradients.

when a gradient in the density exists, in the presence of a local gravity field, less dense fluid rises in the field and more dense fluid sinks. This phenomenon is also known as natural convection.

If rotating frame/coriolis considerations are added to the picture, gradients in the density field can generate currents orthogonal to both the density gradient and the local rotation vector via the thermal wind mechanism.

density perturbations also play a prominent role in acoustics and the propagation of sound/compression waves.

Air is less dense in areas of low atmospheric pressure, and winds form to equalize the pressure as air moves in from areas of higher density and pressure. Ever since Earth was formed denser elements or materials tender to sink lower, whereas lighter materials formed most of the upper layer or crust.

One of the best sources .........of answer to this question!!

Go through this booklet >>

http://gfdrr.org/docs/WPS5161.pdf