Computer Lab: Hydrodynamics of sinking and swimming


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1 Computer Lab: Hydrodynamics of sinking and swimming 1 Introduction Presented by the OCEAN 497 Class 1 October 19, 2009 Small suspended particles are central to the biological, chemical, and geological dynamics of the world s oceans. Living particles plankton are responsible for the vast majority of carbon fixation (i.e., photosynthesis by phytoplankton) and nutrient cycling, and are the basis of oceanic food webs on which we ourselves depend. Other types of particles supply key limiting nutrients (e.g. delivery of iron by dust particles) and drive the biological pump (e.g. sinking marine snow and other detritus), ultimately affecting global productivity and carbon fluxes. Particles from terrestrial inputs and volcanism shape geological deposits that cover much of the earth s surface. The distributions of particles in space and time, and their strong impacts on their environment, depend crucially on the fact that they are suspended in the water column. However, particles are seldom exactly the same density as ambient seawater. Most are heavier, a few are lighter. Differences in density mean that particles tend to move out of the water column, usually to the sediments (but, for the few buoyant particles, to the surface). This tendency raises fundamental questions about the type and number of oceanic particles we should expect to see: If suspended particles are constantly leaving the water column, why are there so many of them? Which mechanisms either prevent particle sinking or floating, or balance losses by creating and transporting new particles? Are sinking and floating selective, disproportionately removing particles with some characteristics while leaving other kinds of particles relatively unaffected? For biological particles, are size, shape and composition constrained by the need to remain in particular parts of the water column (e.g., the photic zone)? The answers to all of these questions depend on how sinking, floating and swimming rates vary quantitatively with particle characteristics. In this computer lab, we will use a hydrodynamic model explore the relationships between vertical movement and particle geometry, composition, and (in plankton) swimming propulsion. 1 Please direct questions or comments to Danny Grünbaum, 1
2 Figure 1: A screenshot of the particle sinking/swimming hydrodynamics model. This graphic represents a prolate ellisoidshaped cell, that is propelled upwards by cilia but is so dense that it nonetheless sinks. Flow patterns are visualized by a set of cyan tracers released immediately below the cell, and a set of red tracers released a couple of cell diameters lower in the water column. 2 The Model To quantify these relationships, we will use a Matlabbased model of low Reynolds number hydrodynamics. Low Reynolds number (Re 1) means that viscous forces are much stronger than inertial effects. This is true for small, slow particles, which includes many oceanic particles. Restricting our attention to this subset of particles enables us to devise a flexible and efficient computer model. This model addresses a family of hypothetical biotic and abiotic particles that are various types of ellipsoids. An ellipsoid is similar to a squashed sphere. It satisfies the equation x 2 a 2 x + y2 a 2 + z2 y a 2 z By adjusting a x, a y, and a z, we can define a prolate (cigarshaped) spheroid, an oblate (diskshaped) spheroid, and other shapes that are not exact representations of any specific particles but that approximate a wide variety of particle = 1 2
3 Table 1: Sinking/swimming model parameters. To see tooltips, hover mouse over the parameter input boxes in the graphics window. Particle characteristics a x,a y,a z Size (radius) of larva in X, Y and Z ciliary velocity Max. ciliary tip speed (zero if nonmotile) particle density Density of particle Water characteristics ν ρ Visualization parameters N part X part range, Y part range, Z part range Tmax part dt part dt plot Kinematic viscosity Sea water density Number of particles Initial position ranges of particles Duration of trajectories Timestep for trajectories Plotting interval for trajectories shapes found in nature (Figure 1). In addition, the ellipsoids could represent shapes not found in nature. To model motile plankton, the model assumes that the surface of the particle is covered in cilia, that are beating tangentially to the surface in a downward direction. The maximum ciliary tip speed is a parameter you set, called ciliary_velocity. The tip speed of these cilia depends on their orientation: Cilia that are on a horizontal surface do not beat at all; cilia that are on a vertical surface beat at speed ciliary_velocity in the downward (negative z) direction; and cilia on surfaces between vertical and horizontal have intermediate tip speeds. In the model, a particle with a zero value of ciliary_velocity is a nonmotile particle. The geometry you specify (with your choice of a x, a y, and a z ) also specifies the volume of the ellipsoid. You must also specify the interior density of the particle, and the density of the ambient sea water. These parameters define the particle s excess weight in water, and hence the force acting on it due to gravity and buoyancy. 3
4 3 Code and parameters To run this model, download or copy all the mfiles in the directory ellipsoid_swimming497. The code has two parts. Part 1 sets the geometry and calculates the flow. To run this, navigate to the ellipsoid_swimming497 directory in Matlab and execute the mfile ellipse_swim4.m. A graphics window will come up (as in Figure 1). At the top of that window are editable boxes, into which you can enter the particle characteristics that you would like to model. In addition, there are four buttons on the left of the window that enable you to perform various calculations. Using this model involves three basic steps: 1. Change a parameter and hit return. The model will plot the geometry that you specified, and report its volume and excess weight (that is, the net force after both gravity and buoyancy are accounted for). 2. Clicking on the button labeled Calculate Flow causes the model to (guess what?) calculate the flow. This takes a few seconds, so in some cases you may want to fine tune the particle geometry you are interested in before you do this calculation. When the calculation is complete, the model will display the vertical velocity of the particle and plot some velocity vectors on its surface. If you want to get back the threedimensional plot of the skin, click on the Plot 3D button. In general, the model will display the vertical velocity if it is known. That is, if you change a parameter, then the velocity is not known until you recalculate it. Accordingly, the model will not display the velocity until after you click the Calculate Flow button. 3. To vizualize the flow patterns, click on the button labeled Track Particles (after the Calculate Flow button!!). This will bring up a window in which you can modify parameters such as how many tracers, where they are released, their color, and for how long to track them. Tracking can be stopped at any point by clicking the Stop Tracking button. You can release a new set of particles, e.g. with a different color, to visualize different parts of the flow as in Figure 1. These parameters are highly adjustable to enable you to explore the consequences of different geometries and material properties (though it is certainly possible to break the code... ). The parameters and their interpretations are summarized in Table 1. NOTE: It is easy to copy and save snapshots of the graphics window into a Word or OpenOffice document. That is a quick and effective way of saving both input parameters and results. 4
5 4 Your Task When you first get this model running, experiment with the effects of different parameters to get an intuition about what different values do to the geometry of the particle, its sinking/floating rate or swimming performance, and the surrounding flow patterns. Then, ask yourself: From the standpoint of ocean geology, chemistry, and biology, what is it about these shapes that most critically affects particles abundance and impacts? Try to devise a strategy for using the model to gain oceanographic insight. Some examples of topics are listed below, but you are encouraged to generate your own ideas. 1. Effects of size. How does swimming and/or sinking change with increases in size? Does geometric similarity imply functional similarity? 2. Effects of weight. Do heavy particles create flows that are different from light particles? What are consequences for particle transport and encounter rates of being heavy or light? If there are differences, how heavy is heavy and how light is light for particles of various sizes? 3. Effects of shape. What are the consequences of particle shape? For a given particle volume and density, would you expect to see equal proportions of highly prolate, highly oblate, and approximately spherical particles in the water column? What about in the sediments? 4. Swimming metrics. How well do various hypothetical organism shapes swim? Are there shapes that appear to have better overall swimming performance? Are there tradeoffs, in which a cell shape that is good in one respect is not so good in other respects? Is there an upper limit to the size and/or density that you would expect to see in ciliated plankton? Finally, use a short series of runs (three or four snapshots of the graphics window) to formulate a hypothesis based on the model results: When we sample off the Thompson, the model results suggest we should see. 5
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