Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References A Hydrodynamic Model for Biogenic Mixing Zhi Lin 1 Jean-Luc Thiffeault 2 Steve Childress 3 1 Institute for Mathematics and its Applications University of Minnesota – Twin Cities 2 Department of Mathematics University of Wisconsin – Madison 3 Courant Institute of Mathematical Sciences New York University International Conference on Interdisciplinary, Applied and Computational Mathematics June 17 2011 1 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Outline • Motivation and History • Dilute Theory • Simulations • Squirmers • Open Problems and Summary 2 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Biomixing A controversial proposition: • There are many regions of the ocean that are relatively quiescent, especially in the depths (1 hairdryer/ km 3 ); • Yet mixing occurs: nutrients eventually get dredged up to the surface somehow; • What if organisms swimming through the ocean made a significant contribution to this? • There could be a local impact, especially with respect to feeding and schooling; • Also relevant in suspensions of microorganisms (Viscous Stokes regime). 3 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Bioturbation The earliest case studied of animals ‘stirring’ their en- vironment is the subject of Darwin’s last book. This was suggested by his uncle and future father-in- law Josiah Wedgwood II, son of the famous potter. “I was thus led to conclude that all the vegetable mould over the whole country has passed many times through, and will again pass many times through, the intestinal canals of worms.” 4 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Munk’s Idea Though it had been mentioned earlier, the first to seriously consider the role of ocean biomixing was Walter Munk (1966): “. . . I have attempted, without much success, to interpret [the eddy diffusivity] from a variety of viewpoints: from mixing along the ocean boundaries, from thermodynamic and biological processes, and from internal tides.” 5 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Basic claims The idea lay dormant for almost 40 years; then • Huntley & Zhou (2004) analyzed the swimming of 100 (!) species, ranging from bacteria to blue whales. Turbulent energy production is ∼ 10 − 5 W kg − 1 for 11 representative species. • Total is comparable to energy dissipation by major storms. • Another estimate comes from the solar energy captured: 63 TeraW , something like 1% of which ends up as mechanical energy (Dewar et al. , 2006). • Kunze et al. (2006) find that turbulence levels during the day in an inlet were 2 to 3 orders of magnitude greater than at night, due to swimming krill. 6 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Counterargument: Mixing efficiency Visser (2007) counterargument: The mixing efficiency is defined as Γ = change in potential energy work done Γ depends strongly on L / B , where L is the turbulence scale and B is the Ozmidov scale. ∗ For krill L = 1 . 5 cm , B = 3 to 10 m , so L / B = . 005 to . 0015. Γ = 10 − 4 to 10 − 3 : little turbulent energy goes into mixing. (from Visser (2007)) ∗ Vertical scale at which buoyancy force is comparable to inertial forces.) 7 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References But it’s not over. . . Katija & Dabiri (2009) looked at jellyfish: [movie 1] (Palau’s Jellyfish Lake.) 8 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Displacement by a moving body Maxwell (1869); Darwin (1953); Eames et al. (1994); Eames & Bush (1999) 9 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References A sequence of kicks Inspired by Einstein’s theory of diffusion (Einstein, 1905) : a test par- target particle ticle initially at x (0) = 0 under- � goes N encounters with an axially- a symmetric swimming body: swimmer b N � x ( t ) = ∆ L ( a k , b k )ˆ r k L k =1 ∆ L ( a , b ) is the displacement, a k , b k are impact parameters, and ˆ r k ( a > 0, but b can have is a direction vector. either sign.) 10 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References After squaring and averaging, assuming isotropy: | x | 2 � ∆ 2 � � � = N L ( a , b ) where a and b are treated as random variables with densities d A / V = 2 d a d b / V (2D) or 2 π a d a d b / V (3D) Replace average by integral: = N � | x | 2 � ∆ 2 � L ( a , b ) d A V Writing n = 1 / V for the number density (there is only one swimmer) and N = Ut / L ( L / U is the time between steps): = Unt � | x | 2 � ∆ 2 � L ( a , b ) d A L 11 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Effective diffusivity Putting this together, = 2 Unt � | x | 2 � ∆ 2 � L ( a , b ) d a d b = 4 κ t , 2D L = 2 π Unt � | x | 2 � ∆ 2 � L ( a , b ) a d a d b = 6 κ t , 3D L which defines the effective diffusivity κ . If the number density is low ( nL d ≪ 1), then encounters are rare and we can use this formula for a collection of particles. 12 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Simplifying assumption � L ( a , b ) a 2 d (log a ) d ( b / L ) κ = π ∆ 2 3D 3 Un Notice ∆ L ( a , b ) is nonzero for 0 < b < L ; otherwise independent of b and L . L ( a , b ) a 2 (sphere) ∆ 2 ∆ 2 L ( a , b ) a (cylinder) 13 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References We can make the simplification (for large L ) � ∆( a ) , 0 ≤ b ≤ L ; ∆ L ( a , b ) = 0 , otherwise , that is, the displacement vanishes if the swimmer is moving away from the particle, or if the particle doesn’t reach the swimmer. In that case we can do the b integral: � ∞ κ = Un ∆ 2 ( a ) d a , 2D 2 0 � ∞ κ = π Un ∆ 2 ( a ) a d a , 3D 3 0 There is no path length dependence. 14 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Displacement for cylinders 2 Small a : ∆ ∼ − log a 0 log a = − 5 Large a : ∆ ∼ a − 3 −2 log a = 2 −4 log ∆ (Darwin, 1953) −6 � 1 log a = 0 0 ∆ 2 ( a ) da ≃ 2 . 31 −8 −10 � ∞ 1 ∆ 2 ( a ) da ≃ . 06 −12 −8 −6 −4 −2 0 2 4 log a = ⇒ 97% dominated by “head-on” collisions (similar for spheres) 15 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Numerical simulation • Validate theory using simple simple simulations; • Large periodic box; • N swimmers (cylinders of radius 1), initially at random positions, swimming in random direction with constant speed U = 1; • Target particle initially at origin advected by the swimmers; • Since dilute, superimpose velocities; • Integrate for some time, compute | x ( t ) | 2 , repeat for a large number N real of realizations, and average. 16 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References A ‘gas’ of swimmers 1 L/2 0.5 start 0 −0.5 −1 y 0 y −1.5 end −2 −2.5 −L/2 −3 −L/2 0 L/2 −4 −3 −2 −1 0 x x [movie 2] N = 100 cylinders, box size = 1000 17 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References How well does the dilute theory work? n = 10 −3 n = 5 × 10 −4 200 n = 10 −4 theory �| x | 2 � / 2 nUℓ 3 150 100 50 0 0 20 40 60 80 100 t 18 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Cloud of particles 19 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Cloud dispersion proceeds by steps 12 10 N = 30 n = 7.5e−04 8 �| x | 2 � 6 4 2 0 0 1000 2000 3000 4000 5000 t 20 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Squirmers Considerable literature on transport due to microorganisms: Wu & Libchaber (2000); Hernandez-Ortiz et al. (2006); Saintillian & Shelley (2007); Ishikawa & Pedley (2007); Underhill et al. (2008); Ishikawa (2009); Leptos et al. (2009) Lighthill (1952), Blake (1971), and more recently Ishikawa et al. (2006) have considered squirmers: • Sphere in Stokes flow; • Steady velocity specified at surface, to mimic cilia; • Steady swimming condition imposed (no net force on (Drescher et al. , 2009) fluid). (Ishikawa et al. , 2006) 21 / 33
Biomixing Dilute theory Simulations Squirmers No-slip boundary Conclusions References Typical squirmer 3D axisymmetric streamfunction for a typical squirmer, in cylindrical coordi- nates ( ρ, z ): � 1 2 ρ 2 + 1 2 r 3 ρ 2 + 3 β � 4 r 3 ρ 2 z ψ = − 1 r 2 − 1 ρ 2 + z 2 , U = 1, radius of � where r = squirmer = 1. Note that β = 0 is the sphere in potential flow. We will use β = 5 for most of the re- mainder. 22 / 33
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