Stirring by swimming bodies
We consider the stirring of an inviscid fluid caused by the locomotion of bodies through it. The swimmers are approximated by non-interacting cylinders or spheres moving steadily along straight lines. We find the displacement of fluid particles caused by the nearby passage of a swimmer as a function of an impact parameter. We use this to compute the effective diffusion coefficient from the random walk of a fluid particle under the influence of a distribution of swimming bodies. We compare with the results of simulations. For typical sizes, densities and swimming velocities of schools of krill, the effective diffusivity in this model is five times the thermal diffusivity. However, we estimate that viscosity increases this value by two orders of magnitude.
Munk (1966) was the first to ask whether biology has an important impact on mixing in the oceans. Since mixing affects the global circulation and stratification of the oceans, it is of great interest to physical oceanographers to settle this question. Dewar et al. (2006) have proposed that the mechanical energy delivered by the swimming motions of the marine biosphere could amount to almost , a figure comparable to the energy delivered by the winds and tides. This suggests a biological origin of about of the mixing in the oceans, an enormous figure. By assuming that this energy is delivered to the top three kilometers of the oceans, they estimate an effective diffusivity produced by swimmers to be approximately , or about 100 times the molecular value for heat. Kunze et al. (2006) have measured elevated levels of ocean turbulence due to swimming krill, though some have questioned whether this turbulence can efficiently overturn a stratified medium Visser (2007); Gregg and Horne (2009). Katija and Dabiri (2009) suggest that the displacement of fluid particles by swimming bodies, which viscous effects can lengthen, is more relevant to stirring than the scale of turbulence they produce. This is the viewpoint adopted in this Letter.
Huntley and Zhou (2004) considered the energy produced by representative species of schooling animals, from krill to whales, and found that regardless of size the energy input to the ocean per unit mass from the swimming of these animals was roughly a constant of order . If the average the biomass of the oceans has a volume fraction over a volume in cubic meters, we can thus (assuming biological materials have the density of water) arrive at a total energy input of . The area of the oceans is about , and if we assume the biomass is evenly distributed to a depth meters, we get an input of W. This suggests that for a typical depth of one kilometer the volume density of biomass should be of order –, and thus that the organisms form a dilute suspension, even if their distribution is patchy and organized into schools.
The object of the present paper is to determine the effective diffusivity and the statistics of the concentration field of a passive scalar that can result from the fluid motions caused by such dilute arrangements of swimming animals. The scalar could be for example heat, salt, or nutrients, and is assumed to have negligible feedback on the flow (at least at small scales). The animals considered by Dewar et al. (2006) and Huntley and Zhou (2004) have large Reynolds number, typically –. Our focus is therefore distinct from the mixing that can occur from dense, tightly interacting suspensions of Stokesian swimmers, as in Pedley and Kessler (1992); Wu and Libchaber (2000); Hernandez-Ortiz et al. (2006); Saintillian and Shelley (2007); Underhill et al. (2008); Leptos et al. (2009), though it shares many common features such as linearity in the density of swimmers. Real swimmers present a wide variety of motions and wakes, but these have generic forms in the far field Lighthill (1991). For example, the far potential field of a neutrally buoyant fish in steady unaccelerated swimming decays at least like a quadrapole. Following Katija and Dabiri (2009), we will model the swimmers by identical cylinders or spheres moving in potential flow, but these are merely stand-in examples. We emphasize that though the motivating application comes from oceanography, the simple model we introduce can be applied to a range of systems, such as the mixing caused by vortices or suspensions of solid particles.
Dilute suspensions of swimmers. As an extremely simple model of stirring by swimmers, we consider a fluid particle, called the target particle, which is influenced by the occasional passing of swimming bodies. We assume that every swimmer moves at a fixed speed , over distances large compared to the “range of influence.” This range will typically be a few body lengths normal to the swimming path, and represents the distance from a target particle where interaction with the swimmer becomes significant. We assume that encounters of our target particle with swimmers are occasional, in that at each encounter the particle moves by a distance , where the impact parameter is the perpendicular distance of the initially unperturbed particle from the extended line of motion of the approaching swimmer. Since we are assuming that is large compared to the distance of significant interaction, we may assume that by an “encounter” we mean that the may be computed from motion of the swimmer along a doubly-infinite line. Each encounter, the th say, moves the particle a distance , in the direction of the unit vector . Thus after encounters the position of a target particle initially at is given by
Assuming infrequent encounters, we regard the as independent and identically distributed random variables. We then follow Einstein’s derivation Einstein (1956) for computing the displacement,
where we assume spatial homogeneity to set , and the angle brackets denote ensemble averaging. The are isotropically distributed, so the second term above vanishes.
To evaluate , consider a swimmer at a large
distance from the target particle at the origin (see Fig. 1). Imagine a “target disk” (or sphere in 3D) of radius , with , where is a typical length scale for the swimmer. The fraction of swimmers that will hit the target disk (sphere) is ( in 3D). We can use the number density and the volume of the ‘shell’ at a distance to find that the number of swimmers hitting the target from a distance is ( in 3D), where is the number density of swimmers. We now integrate from to , since swimmers further than cannot hit the target, to find the number of swimmers that will hit the interaction disk (sphere) in time : we find ( in 3D). The expression for the squared displacement (2) is now
since the largest value of the impact parameter is . Here is the probability distribution of impact parameters; since the swimmers are assumed to arrive from far away, this is ( in 3D). Combining these results and taking , we find
assuming that the integrals converge. Since the effective diffusivity is defined by , with the spatial dimension, Eq. (3) can be used to determine .
Displacement due to a moving body. To find the displacement of a target particle due to a swimming body coming from infinitely far away and swimming in the direction, we need to integrate the equations for the position of a target particle at ,
for ranging from to . The impact parameter, , appears as the initial vertical position of the target particle. This classical ‘drift’ problem has been treated in great detail by many authors; see for example Maxwell (1869); Darwin (1953); Eames et al. (1994); Eames and Bush (1999).
As a working example of the displacement due to a moving body, we shall first treat the cylinder. We consider the displacement in two-dimensional flow due to the passage of a cylinder of radius moving at speed in potential flow. Equation (4) can be integrated numerically for a given to obtain , plotted in Fig. 2. The characteristic ‘ribbon’ shape is evident.
Also shown are the asymptotic forms Darwin (1953)
There is an integrable logarithmic singularity for small . For large , the trajectories are almost circles. By combining these asymptotic limits with numerical integration, we find the integral from Eq. (3) is . But note that : the integral is completely dominated by “head-on” collisions ( of the integral).
For a sphere in three-dimensional potential flow, the displacement appears superficially much as for the cylinder in Fig. 2, but drops off more rapidly for large :
As for the cylinder, there is an integrable logarithmic singularity for small , and the integral in Eq. (3) is . The small logarithmic singularity is mollified by the extra factor of in the integral, but still : as it was for for the cylinder, the integral is completely dominated by “head-on” collisions ( of the integral), due to the rapid decay of the displacement with impact parameter. For both the cylinders and spheres, the logarithmic singularity is the dominant contribution to the integral. The coefficient of the logarithm in Eqs. (5) and (6) is given by the linearized flow near the stagnation points at the front and rear of the cylinder or sphere, suggesting that the integral is easy to approximate for more complicated swimmers. Putting all the numerical factors together, we find the effective diffusivity
We can justify these formulas dimensionally by observing that the frequency of collisions is linear in both and , and since we are assuming the path length is infinite the only other length scale is the swimmer size .
Direct simulation of dilute suspensions. To validate our theoretical predictions, we consider the encounters of 2D swimmers moving in straight lines within the square : . We assume each swimmer is a cylinder of radius , with . We initially place swimmers at random positions within , which subsequently move with unit speed in a random direction. (Figure 3 shows a typical initial configuration.) Positions are subsequently computed mod in both directions, maintaining the number density in . Diluteness requires that . A target particle initially at the origin moves under the potential flow created by all of the cylinders. Since the cylinders are typically well-separated, we compute their net velocity field by linear superposition. We show an example of this computation in figure 3, for the values , . The larger “ribbons” caused by drift are easily identified, suggesting that in this dilute limit the approximation of encounters as being independent will hold.
Figure 4 shows the mean-squared displacement of a target particle over trials (realizations) for cylinders, again with unit radius and speed. The solid line confirms that grows linearly with time, and the dashed line shows the 2D theoretical prediction (7) for cylinders. The discrepancy is due to Eq. (7) only being valid in the limit of infinite dilution, i.e. as . We have verified that as the dilution is increased the slope approaches the theoretical prediction.
The mean is dominated by a few trajectories with large displacements, corresponding to small impact parameter . Analogous remarks apply to a suspension of spheres in three dimensions: the theoretical predictions are verified there as well.
Typical numerical values. We will use values for typical krill as in Visser (2007). We consider spheres of radius , swimming speed , and number density . Equation (7) then gives an effective diffusivity of , about five times the thermal molecular value , and five hundred times the molecular value for salt. This implies a considerable enhancement to the molecular diffusion, but we emphasize that these values apply within a school of krill: the distribution and size of the schools themselves is a more complicated matter Huntley and Zhou (2004). Note also that a small change in the swimmer size has a huge impact: for a radius of , the effective diffusivity is , an order of magnitude smaller than for . If we use mean densities as discussed in the introduction, the effective diffusivity decreases by a factor of to .
Effect of viscosity. We expect viscosity to greatly enhance . This will be the focus of future investigation, but for now we present a rough estimate of the impact of viscous no-slip boundary conditions at the surface of the swimmer. For inviscid flow, the displacement function has a logarithmic singularity near the axis of swimming; for viscous flow near a no-slip boundary, the displacement function has the stronger singularity
where for a sphere of radius Young and Jones (1991); Eames et al. (2003). This implies that the 3D squared-displacement integral in (3) diverges as . The divergence of the second moment is often associated with Levy flights, but here we are interested in scales that are much larger than the typical correlation length of swimming, i.e., the typical length for which a swimmer travels roughly in a straight line before changing direction. We thus expect the overall long-time transport to remain diffusive, and we can cap-off the displacement function at a maximum value . In other words, a particle which is directly in the path of a swimmer cannot travel further than the swimmer itself: this regularizes the integral (3) to give
where the lower bound of the integral is the value at which the displacement (8) achieves its maximum allowable value, . We introduce a transition length scale where we switch from the boundary-layer form to the inviscid form derived earlier, and find again that the dominant contribution to the integral arises from small , as was the case for potential flow, yielding
For spheres, this is . Inserting the same numerical values for krill as before, with a path length , we find , about times the molecular value. Thus, including the effect of viscosity and finite path length has increased the effective diffusivity by a factor of over the inviscid flow case. We emphasize that this is a rough estimate. The path length (or swimming correlation length) is a measure of how much a swimmer tends to move in one direction before turning. Our chosen value of is not based on any evidence, but has only a weak logarithmic dependence on . Assuming a value only raises from to .
Any conclusion regarding the importance of biomixing in the oceans must be carefully qualified: at the densities inside of schools, the inclusion of viscous effects suggests a rather large enhanced diffusivity, comparable with other processes Munk (1966), while outside of schools the effect is much weaker. However, our viscous estimate is rough and more effects will need to be included to form a complete theory: boundary layers, more realistic shape distributions for the swimming bodies, wakes, spatial correlations between the swimmers, patchiness and schooling, finite correlation length of swimming, distribution of velocities, and buoyancy and stratification effects. This last item is probably the most important: stratification can cause fluid parcels to return to their initial height after being displaced if they can’t equilibrate their density with their surroundings. A mechanism such as enhanced diffusion due to small-scale turbulence might assist this equilibration.
The simplicity of our model means that prefactors and scalings can be computed accurately. The numerical constants we obtained depend mostly on the flow near the stagnation points around the swimming body. Our simple model can serve as a platform on which to build complexity, or could be applied to other fluid-dynamical systems where a collection of objects causes mixing, such as in sedimentation.
Acknowledgements.The authors are grateful to W. Dewar, R. Ferrari, M. Graham, Z. G. Lin, C. Ortiz–Duenas, Y.-K. Tsang, and W. Young for helpful discussions, as well as to the hospitality of the 2008 Summer Program in Geophysical Fluid Dynamics (supported by NSF and ONR) at WHOI, where this work began, and the Institute for Mathematics and its Applications (supported by NSF). SC was supported by NSF under grant DMS-0507615, J-LT under grant DMS-0806821.
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