Assume we parametrize our surface by the solution of the Euler or NS equations over ,say, the torus. Then the peaks of the surface will be the high values of the solution.

For example, consider solution $\omega(t_{0},x)\in \mathbf{R}$ (eg.vorticity) for $x\in \mathbf{R}^{2}$ and fixed $t_{0}$. The graph of $\omega(t_{0},x)$ is a surface over $\mathbb{R}^{2}$.

I am curious of any papers/books on this approach i.e. of looking the solutions from a differential geometry view.

Here is a vorticity surface I have in mind from the paper "Coherent structures and turbulence in two-dimensional hydrodynamic":

Interestingly, 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.

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"): 

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?

Assume we parametrize our surface by the solution of the Euler or NS equations over ,say, the torus. Then the peaks of the surface will be the high values of the solution.

For example, consider solution $\omega(t_{0},x)\in \mathbf{R}$ (eg.vorticity) for $x\in \mathbf{R}^{2}$ and fixed $t_{0}$. The graph of $\omega(t_{0},x)$ is a surface over $\mathbb{R}^{2}$.

This is very common with minimal surfaces.

I am curious of any papers/books on this approach i.e. of looking the solutions from a differential geometry view. And there are interesting questions here: what is a natural connection for such surface? what is the relation with Arnold's formalism?

Here is a vorticity surface I have in mind from the paper "Coherent structures and turbulence in two-dimensional hydrodynamic":

Interestingly, 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.

Assume we parametrize our surface by the solution of the Euler or NS equations over ,say, the torus. Then the peaks of the surface will be the high values of the solution.

For example, consider solution $\omega(t_{0},x)\in \mathbf{R}$ (eg.vorticity) for $x\in \mathbf{R}^{2}$ and fixed $t_{0}$. The graph of $\omega(t_{0},x)$ is a surface over $\mathbb{R}^{2}$.

I am curious of any papers/books on this approach i.e. of looking the solutions from a differential geometry view.

Here is a vorticity surface I have in mind from the paper "Coherent structures and turbulence in two-dimensional hydrodynamic":

Interestingly, 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.

Assume we parametrize our surface by the solution of the Euler or NS equations over ,say, the torus. Then the peaks of the surface will be the high values of the solution.

For example, consider solution $\omega(t_{0},x)\in \mathbf{R}$ (eg.vorticity) for $x\in \mathbf{R}^{2}$ and fixed $t_{0}$. The graph of $\omega(t_{0},x)$ is a surface over $\mathbb{R}^{2}$.

This is very common with minimal surfaces.

I am curious of any papers/books on this approach i.e. of looking the solutions from a differential geometry view. And there are interesting questions here: what is a natural connection for such surface? what is the relation with Arnold's formalism?

Here is a vorticity surface I have in mind from the paper "Coherent structures and turbulence in two-dimensional hydrodynamic":

Assume we parametrize our surface by the solution of the Euler or NS equations over ,say, the torus. Then the peaks of the surface will be the high values of the solution.

For example, consider solution $\omega(t_{0},x)\in \mathbf{R}$ (eg.vorticity) for $x\in \mathbf{R}^{2}$ and fixed $t_{0}$. The graph of $\omega(t_{0},x)$ is a surface over $\mathbb{R}^{2}$.

This is very common with minimal surfaces.

I am curious of any papers/books on this approach i.e. of looking the solutions from a differential geometry view. And there are interesting questions here: what is a natural connection for such surface? what is the relation with Arnold's formalism?

Here is a vorticity surface I have in mind from the paper "Coherent structures and turbulence in two-dimensional hydrodynamic":

Interestingly, 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.