Comparison of Eulerian and Lagrangian Approaches

The techniques for describing the statistical properties of the concentrations of marked particles, such as trace gases, in a turbulent fluid can be divided into two categories Eulerian and Lagrangian. The Eulerian methods attempt to formulate the concentration statistics in terms of the statistical properties of the Eulerian fluid velocities, that is, the velocities measured at fixed points in the fluid. A formulation of this type is very useful not only because the Eulerian statistics are readily measurable (as determined from continuous-time recordings of the wind velocities by a fixed network of instruments) but also because the mathematical expressions are directly applicable to situations in which chemical reactions are taking place. Unfortunately, the Eulerian approaches lead to a serious mathematical obstacle known as the closure problem, for which no generally valid solution has yet been found. 

By contrast, the Lagrangian techniques attempt to describe the concentration statistics in terms of the statistical properties of the displacements of groups of particles released in the fluid. The mathematics of this approach is more tractable than that of the Eulerian 

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