3
$\begingroup$

In "Pressure Field, Vorticity Field, and Coherent Structures in Two-Dimensional Incompressible Turbulent Flows" Larchev$\hat{e}$que computes the Gaussian curvature of the surface of the streamfunction $\psi$ to obtain criteria for coherent structures.

Specifically, he considers streamfunction $\psi(x,t_{0})\in \mathbf{R}$ (the streamfunction) for $x\in \mathbf{R}^{2}$ for fixed $t_{0}$. The graph of $\psi(x,t_{0})$ is a surface over $\mathbb{R}^{2}$. The following figures are for the vorticity $\omega(x,t)$ as time increases. For example, an initial data of interest is discrete point vortices: $$\omega(x,0):=\sum \Gamma_{k} \delta(x_{k}).$$

Ignoring the first image (a), the rest are a good example of the evolution of discrete point vortices (cf. "Coherent structures and turbulence in two-dimensional hydrodynamic"): "Coherent structures and turbulence in two-dimensional hydrodynamic"

Questions

1)Are there any rigourous mathematical papers on coherent structures and their relation with Gaussian curvature?

2)Have there been any studies of the vorticity surface and its evolution with initial data the point vortices?

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ Any function on a surface can be an initial condition for the height of a fluid, via the wave equation, so there is no differential geometric condition. You want to keep track of the fluid velocity for the Navier Stokes, so a tangent vector field, not a single function, so this doesn't make sense. $\endgroup$
    – Ben McKay
    Commented Oct 28, 2016 at 13:05
  • $\begingroup$ can you clarify: "Any function on a surface can be an initial condition for the height of a fluid, via the wave equation, so there is no differential geometric condition." I was thinking more along the line of minimal surfaces. $\endgroup$
    – Thomas Kojar
    Commented Oct 31, 2016 at 12:42
  • $\begingroup$ @BenMcKay The surface here is the graph of vorticity $\omega$, a scalar field (curl of velocity, or laplacian of the stream function $\psi$) that determines the divergence-less fluid velocity (up to a constant). It seems reasonable to look at this surface, or at the graph of $\psi$ and its curvature, to define coherent structures, a prominent feature of 2D turbulence. $\endgroup$
    – Jean Duchon
    Commented Nov 2, 2016 at 8:43
  • $\begingroup$ @Wille Wong Is it better now? $\endgroup$
    – Thomas Kojar
    Commented Nov 7, 2016 at 0:39
  • $\begingroup$ I've cast a vote to re-open since there are actually questions formulated. However, it may help for you to include a definition (or discussion) of what "coherent structure" means in this context. $\endgroup$
    – Willie Wong
    Commented Nov 7, 2016 at 4:11

0

Reset to default

You must log in to answer this question.