1. Introduction

A vortex is not a spiral drawn in a fluid. It is a flow field in organized rotation around an axis. The trajectories of fluid particles can be circular, helical or spiral depending on conditions, but the vortex itself is a three-dimensional dynamic structure.

2. The Navier-Stokes Equations

Vortices are solutions of the Navier-Stokes equations, which describe the motion of viscous fluids. Vorticity ω = ∇ × v measures the local rotation of the fluid. A vortex corresponds to a region of concentrated vorticity.

ω = ∇ × v(Vorticité)

Kelvin's theorem states that in a perfect fluid, the circulation around a material contour is conserved. This explains the persistence of vortices: once formed, they tend to maintain their structure.

3. Types of Vortex

The Rankine vortex combines a solid-rotation core (velocity proportional to radius) and an outer potential-flow region (velocity inversely proportional to radius). It is the most widely used model for tornadoes and bathtub vortices.

  • Free vortex (potential): v = Γ/(2πr), irrotational outside core
  • Forced vortex (solid): v = ωr, rotation like a rigid body
  • Rankine vortex: combination of both
  • Burgers vortex: with axial stretching, tornado model

4. Cyclones and the Coriolis Force

Tropical cyclones are large-scale atmospheric vortices. Their rotation — counterclockwise in the northern hemisphere, clockwise in the southern — is imposed by the Coriolis force, which results from Earth's rotation. It is not the bathtub that spins in a given direction depending on the hemisphere: at that scale, Coriolis is negligible.

5. Conservation of Angular Momentum

When a fluid converges toward an axis, its angular momentum is conserved: L = mvr = constant. If r decreases, v increases. This is why a skater who pulls in her arms spins faster, and why wind accelerates at the core of a cyclone. This acceleration is the dynamic signature of the converging spiral.

6. Conclusion

A vortex is a structured solution of the equations of fluid mechanics, not a simple visual spiral. Its apparent shape depends on the type of vortex, viscosity, scale and boundary conditions. Conservation of angular momentum is its central organizing principle.

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