Singularities and extreme events in turbulent flows

Mathematical studies of turbulent flows are mostly based on the incompressible Navier-Stokes equations

$\frac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v}\cdot\nabla ) \mathbf{v} = - \frac{\nabla p}{\rho } + \nu\Delta \mathbf{v}$

which are standardly used in physics and engineering. Since the main dimensionless parameter, the Reynolds number, is high in turbulent flows, we are also lead to consider solutions of the Navier-Stokes equations in the limiting case of vanishing viscosity, which formally corresponds to the Euler equations

$\frac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v}\cdot\nabla ) \mathbf{v} = - \frac{\nabla p}{\rho }$

with a divergenceless velocity field.

In spite of substantial progress in the mathematical analysis of these two equations (see e.g. [1,2]), a number of fundamental problems remain unsolved. For example, it is unknown, for the Euler equations as well as for the Navier-Stokes equations, whether a three-dimensional flow with smooth initial conditions can become singular in a finite time. Actually, the regularity of the solutions of the Navier-Stokes equations is one of the seven Millennium Problems selected by the Clay Mathematical Institute.

One of the most prominent dynamic effects in inviscid flows or in flows with small viscosity is depletion, i.e. the tendency of the flow to suppress nonlinearities and to organize itself into structures that are almost stationary. It seems that the failure to prove the global regularity in the three-dimensional case is intimately related to this phenomenon. Depletion can be studied most effectively on the example of the Euler equations where it is not obscured by the presence of the viscous term. In the cases in which the existence and regularity of solutions is assured, such as the two-dimensional Euler and Navier-Stokes equations, it plays an important part in determining the long-time dynamics of the solutions.

Another interesting problem which arises in this context is the study of small-scales of turbulent flows, described by the Navier-Stokes or some similar equations such as the hydrostatic primitive equations. Even in the cases where the smoothness of the flows is assured, as in the case of the hydrostatic primitive equations [3], the velocity field (or some other fields like the temperature field) can still exhibit very violent, quasi-singular behavior.

To address these issues we use the following approach: for real analytic initial conditions solutions of the Euler and the Navier-Stokes equations are analytically continued to complex values of the spatial variables, in which case the hydrodynamic fields also assume complex values. It is known that any real finite-time singularities are necessarily preceded by complex ones. Therefore, numerically one can detect the possible appearance of real finite-time singularities by monitoring the dynamics of the complex singularities. One possibility to study the structure and the temporal behavior of the complex singularities consists in analyzing the Fourier coefficients of the solutions, see [4,5,6,7].

Contact: Eberhard Bodenschatz

[1] К. Бардос, Э.С. Тити, Уравнения Эйлера идеальной несжимаемой жидкости, УМН, 62, 5-46 (2007), English translation: C. Bardos and E.S. Titi, Euler equations for incompressible ideal fluids, Russ. Math. Surveys, 62, 409-451 (2007) [Link]
[2] C. Bardos, A basic example of non linear equations: the Navier-Stokes equations, [Link]
[4] C. Cao and E.S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Annals of Mathematics, 166(1), 245-267 (2008) [Link]
[4] U. Frisch, T. Matsumoto and J. Bec, Singularities of Euler flow? Not out of the blue!, J. Stat. Phys., 113, 761-781 (2003) [Link]
[6] T. Matsumoto, J. Bec and U. Frisch, The analytic structure of 2D flows at short times, Fluid Dyn. Res., 36, 221-237 (2005) [Link]
[6] W. Pauls, T. Matsumoto, U. Frisch and J. Bec, Nature of complex singularities for the 2D Euler equation, Physica D, 219, 40-59 (2006) [Link]
[7] W. Pauls, On complex singularities of the 2D Euler equation at short times, submitted to Physica D [Link]