# Ring

Corresponding Wikipedia article: Vortex ring

The vortex ring is the only possible form of a stable vortex, which may not consume the external energy. Although it can consume the energy.

The vortex geometry is determined by the environmental conditions. In any case, the vortex ring is a torus deformed in any way.

Hill's vortex cross section, where the warm colors correspond to the high flow velocities

The sphere is the natural vortex geometry in a homogeneous continuum in the absence of the external forces. The annular vortex tube surrounds the vortex axis. All parameters are symmetric with respect to this axis, and therefore they are considered in any section plane, which passes through this axis.

The Hill's vortex is, known in the theoretical hydrodynamics, the simple spherical vortex, which exists only in a limited volume. The rotation around its axis does not exist, and the streamlines are shaping a deformed torus.

The Hicks's vortex is a spherical solution of the wave Helmholtz equation for a helical flow ("Helix and spiral", 5). This is the only known spherical vortex, which has an infinite field in the shape of a wave. The solution in the cylindrical coordinates $$(r=R\sin\chi, \theta, z=R\cos\chi)$$ is[1]:
$v_r=Ck^2J_2(kR)\sin\chi\cos\chi$ $v_\theta=Ck^2J_1(kR)\sin\chi$ $v_z=2Ck\frac{J_1(kR)}{R}-Ck^2J_2(kR)\sin^2\chi$ where the constants $$C$$ and $$k$$ are determined by the conditions ((vortex energy etc.), and $$J_1(x)$$, $$J_2(x)$$ are the spherical Bessel functions: $J_1(x)=\frac{\sin x}{x^2}-\frac{\cos x}{x}$ $J_2(x)=\left(\frac{3}{x^2}-1\right)\frac{\sin x}{x}-\frac{3\cos x}{x^2}$ Obviously, at the small distances $$R$$, the Hicks's vortex is somewhat like a spherical wave with a wavenumber $$\approx k$$, which moves with an arbitrary velocity, including zero. Such wave is called the soliton.