The dynamics of vortices in the solar atmosphere

Sometimes, it can get pretty mathematical in solar physics too. Opposite formula describes the dynamical evolution of a vortical or swirling motion, of which there exists various types in the solar atmosphere. A vortex or swirl can be intuitively described as the rotation of fluid parcels around a common axis. Despite this simple concept, a rigorous mathematical definition is still an open issue in fluid mechanics. An effective physical quantity for characterising swirls is the swirling strength for which we have now derived the corresponding dynamical equation.

Evolution equation for the swirling strength λ. ℘ is the matrix composed of the eigenvectors of the Jacobian matrix of the velocity, ∂_j v_i, and p_g and p_m are the gas pressure and the magnetic pressure, respectively, Φ is a potential of conservative forces, and D/Dt is the material derivative. The terms between curly brackets are matrices, of which only the (2,2) components are needed. The symbol ⊗ denotes the tensor product between two vectors.

Commonly, one uses the vorticity of which the evolution equation is known. It consists in the case of magnetohydrodynamics of four terms describing different physical processes that generate vorticity. Unfortunately, vorticity does not differentiate between vortical motions and shear flows. A better physical quantity describing vortical flows is the swirling strength, a generalisation of the vorticity, which derives form an eigenanalysis of the tensor of velocity gradients, that is, the Jacobi-Matrix of the velocity vector. However, for this quantity no evolution equation was known so far. We have now derived the corresponding equation (Canivete Cuissa & Steiner, 2020). It consists of five terms describing the generation of swirling motion as arising from stretching (T1), hydrodynamic and magnetohydrodynamic baroclinicity (T2 and T3, respectively), magnetic tension (T4), and from conservative forces (T5). This equation serves us now to explore the origins of vortical flows in the solar atmosphere.

José R. Canivete Cuissa & Oskar Steiner: 2020, “Vortices evolution in the solar atmosphere: A dynamical equation for the swirling strength”, A&A 639, A118