Eigenmodes
We study the advection diffusion equation in two spatial dimensions. This relatively simple, linear partial differential equation governs the behavior of a passive scalar field (temperature, pollutant concentration, chemical species) under the combined action of (1) a prescribed vector velocity field which advects the scalar about and (2) Laplacian diffusion of the scalar. The partial differential equation is of great interest in own right, with applications ranging from the optimal design of micro-mixing devices to estimating planetary scale transport in the atmosphere and ocean. Mathematically, the form and structure of the linear advection-diffusion equation may also serve as a possible starting point for understanding the behavior of the fully non-linear partial differential governing equations of fluid mechanics.
Our concern here is an attempt to determine the 'modes of mixing' in the special case where the advecting velocities are time periodic, but non-integrable. In other words, the prescribed velocity field produces chaotic motion of passive fluid parcels. This strong stirring stretches and folds the concentration field, amplifying initial gradients and rapidly creating small scale whorls and tendrils. The diffusion term acts to counter this process, smoothing out scalar gradients and homogenizing the field. Although it is easy to prove that the diffusion always wins at very long times, at intermediate times the scalar field often evolves into a state where the two processes exist in delicate balance - what R.T. Pierrehumbert called 'A Strange Eigenmode' - where the advection and diffusion result in intricate patterns of scalar concentration which reproduce themselves periodically.
The pictures show our attempt to numerically determine just how periodic the advection-diffusion operator is. We start by evolving a large number ( ~ 2562) independent initial profiles (Fourier modes) of the scalar forward over one or two periods of the velocity field. Each one of these initial profiles evolves onto all others. Since these evolutions are independent, we can make efficient use of parallel algortithms and the hardware available on the CUNY High Performance Computational Cluster. The evolution over two or four periods of selected modes are shown. Once all the modes have been constructed, we can form a numerical Green's Function for the one or two period advection diffusion operator. This operator takes the form of a large, (2562 ×2562), non-symmetric matrix. The eigenspectrum of this matrix - a rather enormous computational task to determine - then fixes the rate of decay of the corresponding eigenvectors - these are the 'modes of mixing'. One expects that those modes with the slowest decay rates are the building blocks of the periodic patterns observed.
How periodic is the advection-diffusion equation? The figure below (above?) shows a comparison of the numerically determined eigenspectrum of the two period operator to that of the square of the one-period operator. One sees very good comparison for those modes with the largest eigenvalues, espeically those that lie on the real axis. This indicates that much of the dynamics of the scalar over two periods of oscillation can be captured by iterating the one period operator.
This work is supported by the Office of Naval Research, several PSC-CUNY Research Grants and involves Profs Vukadinovich, Schaefer and Poje. Undergraduate and Graduate Students interested in research opportunities in mathematics, applied mathematics and computational science are encouraged to contact us. We thank the CUNY High Performance Computing Facility for access and support.