NUMERICAL. SIMULATION OF LAGRANGIAN QUANTITIES
143
bulence (Charney, 1971). With the advent of satellites and constant level
balloons Lagrangian measurements in the " quasi two-dimensional " region
of the atmosphere are a reality. The EOLE experiment tracked 480 balloons
at the 200-mb level (Morel and Bandeen. 1973) and forthcoming experiments under GARP promise further expansion of this technique. In the light
of these experiments in the stratosphere and in view of the aforementioned
ability to treat a two-dimensional simulation without subgrid scale modeling, such simulation seems advisable. In particular, such numerical experiments can eliminate many of the features of real experiments such as
anisotropy and nonstationarity. In the present study an Eulerian twodimensional numerical turbulent field was used to study Lagrangian statistics. Particle velocity correlations. particle mean-square displacement, and
Eulerian-Lagrangian relations were examined. Results were limited to
single-particle analysis ; however, computer-generated movies allowed a
qualitative study of cluster dispersion. (Quantitative study of the multiparticle statistics was restricted by grid size limitations.)
The results in general verify the classical theories of diffusion (Taylor,
1921) and the Corrsin hypothesis (Corrsin, 1959) which related the Lagrangian autocorrelation to the Eulerian space-time correlation. Some of the
observed behavior of clusters was similar to that observed in the EOLE
experiments.
2.1. Eulerian Flow F i d I Simulutiori
The technique used in this study was to track particles, that is, fluid points
on a two-dimensional numerical sirnulation of a turbulent flow field. The
Eulerian field was that developed by Lilly (1969, 1972). A review and discussion of recent two-dimensional flow fields numerical simulations can be
found in the paper by Fox (1972). While the present study used physical
space representation, recent work by Orszag (this volume, p. 225) indicates
that considerable improvement in accuracy can be obtained by k space
techniques, that is Fourier space simulation techniques. However, these do
have the disadvantage of imposing considerable programming difficulty,
particularly when Lagrangian information is desired.
2.1 . I . Two-Dimriuiond Flow Field Equurions. The simulated twodimensional Eulerian field was generated by solving the basic equations of
vorticity and stream function on a 64 x 64 grid and utilizing periodic boundary conditions to maintain isotropy. The equations solved are as follows:
(2.1 )
ai/dt + V V i = F + v V2i - K i
(2.2)
V = - i dt,b/Jv + j ilt,b/dx
(2.3)
i = v2*
143
bulence (Charney, 1971). With the advent of satellites and constant level
balloons Lagrangian measurements in the " quasi two-dimensional " region
of the atmosphere are a reality. The EOLE experiment tracked 480 balloons
at the 200-mb level (Morel and Bandeen. 1973) and forthcoming experiments under GARP promise further expansion of this technique. In the light
of these experiments in the stratosphere and in view of the aforementioned
ability to treat a two-dimensional simulation without subgrid scale modeling, such simulation seems advisable. In particular, such numerical experiments can eliminate many of the features of real experiments such as
anisotropy and nonstationarity. In the present study an Eulerian twodimensional numerical turbulent field was used to study Lagrangian statistics. Particle velocity correlations. particle mean-square displacement, and
Eulerian-Lagrangian relations were examined. Results were limited to
single-particle analysis ; however, computer-generated movies allowed a
qualitative study of cluster dispersion. (Quantitative study of the multiparticle statistics was restricted by grid size limitations.)
The results in general verify the classical theories of diffusion (Taylor,
1921) and the Corrsin hypothesis (Corrsin, 1959) which related the Lagrangian autocorrelation to the Eulerian space-time correlation. Some of the
observed behavior of clusters was similar to that observed in the EOLE
experiments.
2.1. Eulerian Flow F i d I Simulutiori
The technique used in this study was to track particles, that is, fluid points
on a two-dimensional numerical sirnulation of a turbulent flow field. The
Eulerian field was that developed by Lilly (1969, 1972). A review and discussion of recent two-dimensional flow fields numerical simulations can be
found in the paper by Fox (1972). While the present study used physical
space representation, recent work by Orszag (this volume, p. 225) indicates
that considerable improvement in accuracy can be obtained by k space
techniques, that is Fourier space simulation techniques. However, these do
have the disadvantage of imposing considerable programming difficulty,
particularly when Lagrangian information is desired.
2.1 . I . Two-Dimriuiond Flow Field Equurions. The simulated twodimensional Eulerian field was generated by solving the basic equations of
vorticity and stream function on a 64 x 64 grid and utilizing periodic boundary conditions to maintain isotropy. The equations solved are as follows:
(2.1 )
ai/dt + V V i = F + v V2i - K i
(2.2)
V = - i dt,b/Jv + j ilt,b/dx
(2.3)
i = v2*