Conserved quantities (invariants) are important for analysing PDEs. In this talk, I will present two fluid problems where the conserved quantities and the structure of nonlinear terms leading to conserved quantities help us to understand the turbulence.
The first problem focuses on the energetics in geophysical turbulence, where different from classic homogeneous isotropic turbulence energy transfers to both large- and small-scales at the same time, however, the mechanism behind this bidirectional energy transfer is so-far not well understood. The geophysical turbulence is more complicated compared with the classic homogeneous isotropic turbulence by the presence of the rotation and stratification, which lead to inertial-gravity waves. Theories for both vortex turbulence and (weak) wave turbulence are proposed, but no present theory tells the turbulent mechanism when wave and vortex coexist. Using the Hamiltonian model capturing the interaction between near-inertial wave and vortex, we propose a heuristic argument that based on the three conserved quantities -- energy, potential enstrophy and wave action -- a turbulent state with bidirectional energy transfer must exist. We run numerical simulations to justify our prediction and show the existence of a phase transition between turbulent states with unidirectional and bidirectional energy transfer.
The second problem is the high-order structure functions in one-dimensional Burgers turbulence, which is treated as a toy model to study turbulence due to its nonlinear advection term. However, differing from the Navier--Stokes equation, the one-dimensionality makes the Burgers equation preserves infinite-many conserved quantities. For the Navier--Stokes turbulence, Kolmogorov proposed the exact theory for the third-order structure functions in the inertial range, which is the foundation for turbulence study. We show that due to the infinite-many conserved quantities exact results for infinite-many high-order structure functions can be calculated for the Burgers turbulence, and to make the theory more practically useful we extend the validity of the exact results beyond the inertial range. Numerical results from finite-volume simulations confirm our theoretical findings.