lwjgl – Windblown fire particle effect

I want to create a fire effect and give it a variable "wind" to blow it aside. I do not want the particles to just bend to the side, but to flow semi-realistically through the air, much like the fire in this video.

Section of a linked video showing fire particles

I assume that this requires the use of trigonometric functions to achieve a "wavy" pattern.

My engine is completely new and built on Slick2D / LWJGL. I've already set up a particle system that I've implemented for simple spatter effects right down to projectiles.

Simple effects like "spray" or blood spatter are easy, but I do not know how to make something more complicated. I'm not really sure how to make it look, except for a random burst of small squares or poorly drawn fire sprites.