How does gravity affect the formation of vortices?

Question

Context: I understand the axis of vortices can form in any direction (like underwater used by fishes). However, in large vortices in water bodies, for example, in a closed water bottle, the centre of water often creates a hole that droops down into the water body, and the vortex forms an upside-down cone shape.

Problem: How is this shape formed, does gravity affect the individual fluid elements differently, or is it caused by small geometrical irregularities present in the forming of the vortex? I've only seen theoretical vortex formation equations assuming 0 density and viscosity, and don't know how to calulate the influence on gravity on vortex formation or movement inside the bottle.

Answer 1

This is what I think: when flow velocity is aligned in the same direction with strain, the vorticity increases. This is because the vorticity is the cross product of the directional derivative with flow velocity, so the rate of growth of vorticity is equal to the corresponding strain rate eigenvalue, meaning it will grow exponentially with strain rate. Each coefficient grows or decays exponentially at a rate given by the corresponding strain rate eigenvalue. \omega will approach the direction of greatest extensional strain, the largest positive eigenvalue. herefore, when local strain is uneven, local vorticity is uneven. We illustrate this through three examples:

Draining water in sink is stretched by weight acting on falling water Liquid in bottom of container is strained altitudinally by hydrostatic pressure Tornado is stretched by lift of rising air

Returning to the example of fluids in a cylindrical container, on Earth, hydrostatic pressure is uneven throughout the continuum, it is larger lower down in fluid, when h is larger, creating strain in the direction of gravity. For reference, hydrostatic pressure is defined p=\rho hg

As established in Scenario 1, when vorticity is aligned in the same direction with strain, the vorticity grows exponentially, and when local strain is uneven, local vorticity is uneven.

We will return to the example raised, of liquid in the bottom of the container being strained altitudinally by hydrostatic pressure on Earth. For Equation 2.2.7 in microgravity, gravitational acceleration is negligible, so difference in hydrostatic pressure along fluid elements, p=\rho hg is negligible.

In microgravity, altitudinal strain on Earth, in other words the stress arising from difference in hydrostatic pressure, is negligible. The uneven strain is hence gone everywhere, alongside the unevenness in local vorticity. Thus, in a microgravity environment, a cylindrical vortex will be formed.

Sources: Shames, I. H. (2003). Mechanics of fluids. McGraw-Hill. Smyth, W. (2019). All Things Flow: Fluid Mechanics for the Natural Sciences. Lulu.com. https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%253A_All_Things_Flow_-_Fluid_Mechanics_for_the_Natural_Sciences_(Smyth)