Illustration of the GLM theoryΒΆ

The following might not be an illustration of the most general case but it may answer some questions relative to the meaning of the GLM theory.

We take a parcel, the trajectory of which makes one circle in 1000 units of time, while making smaller circles with a period of 10 units (light lines in Figs. 1 and 2). The X and Y position of the parcel is thus just the sum of two cosines with periods of T0 = 1000 and T = 10 units and an amplitude of 1 and 0.05, respectively. X and Y are 90 deg. out of phase to allow the trajectory to form circles.

The corresponding velocity field U = (U,V) is given by the time derivative of the vector position X = (X,Y) and is also the sum of two cosines, with the adequate phase change between U and V. Thus, the velocity field is uniform and the present illustration may not be an illustration for the most general case.

We then choose a time average Tav over which the position and the velocity field are averaged out. The first average gives the mean trajectory, the second the Lagrangian mean velocity.

In the first case, Tav = T (Fig. 1). The trajectory is the larger circle only (dashed thick lines) and the Lagrangian mean velocity is parallel to that trajectory and the larger circle. This sounds all as expected.

In the second case, Tav = 30T (Fig. 2). The trajectory is again a circle but of smaller radius. Indeed, it makes sense: With longer and longer period of averaging, the trajectory should collapse to one point. The velocity field is weaker (not shown) than in the first case, but still parallel to the mean trajectory.

I was surprised so I also plotted: the line (red) that joins two points separated by Tav and the mean position of the trajectory during that period (red star). The mean position is basically “the center of mass” of the trajectory during the period of averaging and the Lagrangian mean velocity is parallel to the line, that is to the direction that links the starting and ending positions of the parcel during the period of averaging.

In the mechanical analogy of Andrews and McIntyre (1978; p. 617), the mean position is also the center of mass of the rod and the Lagrangian mean velocity is the velocity of that center of mass. Thus, as in mechanics where we can replace any complex solid of mass m by one single point of mass m located at the center of mass of the solid, we can replace a complex trajectory by one parcel located at “the center of mass” of the trajectory and moving with the Lagrangian-mean velocity.

It would be worthy to test if these results hold in the case where the velocity field is not uniform anymore; but I suspect they will.