Curious Symmetry in a Simple Flow (and a Bottle of Champagne)
|The evolution of an initial point-like pulse of tracer at the origin (position="0"). Panels (a), (b), (c), and (d) show the shape of the pulse (tracer concentration versus position) at four different time intervals. The vertical dashed lines at positions +1 and -1 indicate the location of tracer concentration recording devises. Panels (e) and (f) show the time variation recorded at (e) position=+1 and (f) position=-1. Although the upstream magnitude is 10 times smaller than the downstream — note the different vertical scales in (e) and (f) — the shape of the time variation is identical.|
In this essay, I discuss a curious feature of transport in a simple fluid undergoing bulk flow and uniform mixing: while most tracer is swept downstream from a source point, the average speed of the small amount carried upstream by mixing is the same as the downstream tracer speed. This is counterintuitive, as one expects it to be "harder" to move against the flow than with the flow.
In the earth sciences one often wants to quantify the transport of a trace constituent, or "tracer". One useful diagnostic is the elapsed time, called the "tracer age," since the tracer was released into the flow. Generally, there is no single age, as a sample of fluid will contain many tracer molecules, which took a variety of times and paths to travel from source to sample. One must consider a statistical "tracer age distribution" and its mean, the "mean tracer age."
Often transport can be decomposed into large scale bulk motion and small scale dispersive motion. For example, a pollutant released into a river is carried en masse downstream by the current and is simultaneously dispersed into a growing "cloud" by eddies and molecular diffusion. An useful idealized model for these systems is "advection-diffusion." Advection is the bulk component of the motion (e.g., bricks carried along on a conveyer belt), and diffusion is the dispersive component (e.g., perfume spreading from a point in still air). In the very simplest such model, advection and diffusion occur in one spatial dimension (1-D) at constant and uniform rates, and any bounderies for the transport are far from the source of the constituent.
Despite centuries of study in numerous different contexts, the properties of advection-diffusion are often subtle and surprising results can still be found. The surprise here in this simple 1-D application is that the distribution of tracer age has the same shape in fluid samples taken at equally distant points upstream and downstream from the source. Consequently, the "mean tracer age" (the mean age of the distribution) is identical upstream and downstream; that is, it is symmetric. Similarly, the mean speed of tracer motion (distance divided by mean tracer age) is symmetric.
This tracer age symmetry, noted mathematically by Deleersnijder et al. (2000), is counterintuitive. The flow is strongly asymmetric (it has a direction), as is the resulting tracer concentration (i.e., more tracer is carried downstream than upstream), yet the tracer age manages to remain symmetric. Tom Haine at Johns Hopkins University and I have put forth a physical explanation for this symmetry, thereby winning the bottle of Champagne offered as a prize by Eric Deleersnijder for the first such explanation. The explanation invokes statistical properties of the random particle motions that underlie diffusive transport.
The essential elements are as follows: (1) Every random trajectory (a sequence of random steps) resulting in downstream displacement has an upstream "mirror image" counterpart. (2) All trajectories from the source point to a sample point have the same net displacement. Element (2) means that the trajectories differ from one another only in the number of "back-and-forth" motions, which strongly affect the travel time along the trajectory, but result in no net displacement. Each trajectory can be imagined as rearranged to form a direct trajectory from source to sampliing point, followed by the left over "back-and-forth" steps. The probability of a particle following one of these zero-net displacement "sub-trajectories" is independent of position — that is, it is the same upstream and downstream — since the diffusive component of the flow that causes "back-and-forth" motion is assumed to be uniform.
Combining these elements, one sees that the tracer release point is connected to equally distant upstream and downstream fluid samples with an equal but opposite set of trajectories. There is net transport downstream because an individual random particle step is more probable in one direction than the other. Each upstream trajectory is less likely to be followed by a particle than its downstream counterpart, and consequently there are fewer tracer particles upstream. But for the minority of particles that do follow upstream trajectories, the travel time along a trajectory is the same as its downstream counterpart. Moreover, the difference in likelihood of following an upstream trajectory compared to its downstream counterpart does not depend on the trajectory's travel time, which is largely determined by the "back-and-forth" motions that are independent of position.
In summary, upstream and downstream water samples are reached by equal and opposite sets of trajectories. The degree to which a tracer particle is less likely to move upstream than downstream is independent of travel time. Consequently, the tracer age distribution has the same shape, albeit different magnitude, upstream and down, and all "moments" of the distribution (e.g., the mean tracer age and the tracer age standard deviation) are symmetrical.
Deleernijder, E., E.J.M. Delhez, M. Crucifix, and J.-.M. Beckers 2001. On the symmetry of the age field of a passive tracer released into a one-dimensional fluid flow by a point-source. Proc. Royal Soc. London A 61, 1526-1544.
Hall, T.M., and T.W.N. Haine 2004. Tracer-age symmetry in advective-diffusive flows. J. Marine Systems 48, 51-59, doi:10.1016/j.jmarsys.2003.01.001.
Please address all inquiries about this research to Dr. Timothy Hall.