Hydrodynamics of Copepods: a Review

HYDRODYNAMICS OF COPEPODS: A REVIEW

HOUSHUO JIANG1 and THOMAS R. OSBORN2,3 1Department of Applied Ocean Physics and Engineering, MS 12 Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA E-mail:[email protected] 2Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA 3Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, Baltimore, MD 21218, USA

(Received 1 October 2002; accepted 19 September 2003)

Abstract. This paper reviews the hydrodynamics of copepods, guided by results obtained from recent theoretical and numerical studies of this topic to highlight the key concepts. First, we briefly summarize observational studies of the water flows (e.g., the feeding currents) created by copepods at their body scale. It is noticed that the water flows at individual copepod scale not only determine the net currents going around and through a copepod’s hair-bearing appendages but also set up a laminar flow field around the copepod. This laminar flow field interacts constantly with environmental background flows. Theoreti- cally, we explain the creation of the laminar flow field in terms of the fact that a free-swimming copepod is a self-propelled body. This explanation is able to relate the various flow fields created by copepods to their complex swimming behaviors, and relevant results obtained from numerical simulations are summarized. Finally, we review the role of hydrodynamics in facilitating chemoreception and mechanoreception in copepods. As a conclu- sion, both past and current research suggests that the fluid mechanical phenomena occurring at copepod body scale play an important role in copepod feeding, sensing, swarming, mating, and predator avoidance.

Keywords: chemoreception, copepod, feeding current, hydrodynamics, mechanoreception, numerical simulation, self-propelled

1. Introduction

Calanoid copepods live in water environments. Water ﬂows are created, whenever the copepods feed and/or swim by rapid beating of their antennae, mandibular palps, maxillules and maxillae (i.e., the cephalic appendages), elicit an escape reaction resulting from the combined actions of the antennules and swimming legs, or even sink freely through the water column by stopping all the activities of the appendages. (This is because most of them are negatively buoyant.) The water ﬂows so created are crucial for the survival of the copepods, as they have to engage in various survival tasks of feeding, predator avoidance, and mating in the three-dimensional water environments (e.g., Yen and Strickler, 1996; Yen, 2000).

Surveys in Geophysics 25: 339–370, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands. 340 HOUSHUO JIANG AND THOMAS R. OSBORN

The study of the water flows created by copepods and associated interactions with environmental background flows forms the scope of the hydrodynamics of copepods. Generally, the studies of the hydrodynamics of copepods may be divided into three research directions: (1) the water flows at a copepod’s appendage scale, i.e., the water flows around and through a copepod’s hair-bearing appendages, (2) the water flows at the scale size of an individual copepod (e.g., the feeding current created by a copepod hovering in the water column, the flow field around a swimming copepod, and the vortical flow structure shed in the wake of a copepod eliciting an escape reaction), and (3) the interaction between the water flows created by a copepod and the environmental background flows surrounding it (e.g., the manner in which the small-scale turbulence around a copepod erodes the laminar feeding current created by the copepod). Important questions are the following: (i) How does a copepod alter its feeding current by adjusting its body orientation and the forcing which it applies to the adjacent water, in response to a turbulent eddy or turbulence-induced shear of a scale comparable with the size of the copepod? (ii) In general, how does a copepod change its swimming behavior and therefore change the flow field around its body in response to the small-scale turbulence? (iii) How does the interaction between the flows created by a copepod and the environmental background flows affect the copepod’s sensory mechanisms such as mechanoreception and chemoreception? The study of the water flows at a copepod’s appendage scale has been reviewed by Jørgensen (1983), LaBarbera (1984) and Shimeta and Jumars (1991), mainly focusing on the mechanisms of suspension feeding. In addition, Childress et al. (1987) utilized a number of highly simplified models of appendage motion, such as the movement of Stokeslets, spheres or stalks, to set up an average scanning current in Stokes flow in a suitable far-field formulation; they discussed the possible applications of these models in understanding the feeding efficiency and strategies of small organisms such as copepods. Some more recent studies have discovered that different aspects of morphology and behavior are important in determining the leak- iness of a hair-bearing appendage at different Reynolds numbers, and have provided insights about the function of arrays of hair-like olfactory antennae (e.g., Koehl, 1992, 1995, 1996; Loudon et al., 1994). As to the study of the interaction between the water flows created by a copepod and the environmental background flows surrounding it, experimental studies have documented the behavioral responses of copepods to laboratory-generated turbulence (e.g., Costello et al., 1990; Marrase´ et al., 1990; Saiz and Alcaraz, 1992; Hwang et al., 1994; Caparroy et al., 1998). Marrase´ et al. (1990) revealed the difference in the flow field around a tethered copepod under two different background flow conditions: a non-turbulent condition and a turbulent condition. Kiørboe and Saiz (1995) HYDRODYNAMICS OF COPEPODS: A REVIEW 341 provided a simple analysis of the erosion of copepod feeding current by the small-scale turbulence. Osborn (1996) proposed a conceptual model of the interaction between a copepod’s feeding current and the surrounding small-scale turbulence, in which copepod feeding is considered as turbulent diffusion of the food in towards the region where the feeding current serves to capture the food well before it is identified. Although there is a large literature on the effect of the small-scale turbulence upon the encounter rate between predators and prey (see the reviews in Dower et al. (1997), and Lewis and Pedley (2000)), only some speculations about the interactions between the copepod-created water flows and the environmental background flows can be found from the up-to-date literature (e.g., Strickler, 1985; Granata and Dickey, 1991; Yamazaki, 1993; Yamazaki and Squires, 1996; Strickler et al., 1997). Studies of the interactions in a more dynamic way are still needed. That is, future studies are needed to investigate the spatial and temporal variations of the flow field around a copepod under the influence of environmental background flows at suitable spatial and temporal scales. Since one of these two research directions has been reviewed extensively by other researchers and the other has not yet been investigated extensively, we choose not to include them in this review. The main purpose of this paper is to review (from the viewpoint of hydrodynamics) the study of the water flows at individual copepod scale. The content includes (1) descriptions of the flow fields obtained from observational studies, (2) understanding the creation of the flow fields based on the evidence from both observations and theoretical analyses, (3) numerical simulations of the flow fields, and (4) effects of the flow fields on copepod sensory mechanisms. As has been pointed out at the very beginning, the water flows at individual copepod scale play an important role in copepod feeding, predator avoidance, and mating. It is noteworthy that the water flows at individual copepod scale not only determine the net water flows going around and through a copepod’s hair-bearing appendages but also set up a laminar flow field around the copepod; the created laminar flow field is constantly under the influence of environmental background flows.

2. Observations of the ﬂow ﬁeld at individual copepod scale

More than twenty years ago, a high-speed microcinematographic technique based on a Schlieren optical pathway was used to observe the world of zooplankton such as copepods (Strickler, 1977, 1985; Alcaraz et al., 1980). In some earlier applications of this technique (e.g., Strickler and Bal, 1973; Strickler, 1975a, b, 1977; Kerfoot et al., 1980), the ‘‘footprints’’ created by free-swimming copepods were registered on film. In fact, these ‘‘footprints’’ 342 HOUSHUO JIANG AND THOMAS R. OSBORN have visualized the hydrodynamic disturbances created by the copepods at their body scale. The idea underlying this technique is direct visual observation, which now has been widely used by researchers to study the feeding, swimming, breeding, and predator–prey interactions of zooplankton, and their interactions with environmental conditions. On the other hand, this technique directly contributed to an important finding – many calanoid copepods create feeding currents – and made possible the quantification of the feeding currents. By analyzing the images taken on the film or later on the digital videotapes, researchers were able to measure the flow field (e.g., the feeding currents) created by copepods.

2.1. For a tethered copepod

Many laboratory experiments used tethered copepods and measured the feeding currents created by them (e.g., Koehl and Strickler, 1981; Vander- ploeg and Paffenho¨ fer, 1985; Paffenho¨ fer and Lewis, 1990; Yen and Fields, 1992; Fields and Yen, 1993; Yen and Strickler, 1996; Fields and Yen, 1997; van Duren et al., 1998). Obviously, ‘‘tethering’’ made the data gath- ering much easier. Here, we adopt the descriptions by Fields and Yen (1993) of their measurement of the flow field around a tethered Pleuro- mamma xiphias (3.5 mm prosome length) to show the three-dimensional flow structure of the feeding current created by a tethered copepod (Figure 1). A maximum velocity of 38 mm s)1 occurred at the base of the downward swing of the second antennae just lateral to the sides of the body. Lateral symmetry was found in the flow field with water velocity decreasing rapidly from the head to the distal tips of the antennules. Asym- metry in the flow field was present between the dorsal and ventral side of the copepod with the 1 mm s)1 velocity isoline approximately 1.5 times further from the body ventrally than dorsally. The hydrodynamic disturbance defined by the 1 mm s)1 velocity isoline was located as far as 4.1 mm above the head, 4.6 mm lateral, 5.6 mm ventral and 3.6 mm dorsal to the copepod. The lower extent of the 1 mm s)1 velocity isoline could not be identified; however, the lower portion of the 7 mm s)1 velocity isoline was found 7.5 mm directly below the head.

2.2. For a free-swimming copepod

However, the experimental technology has also allowed free-swimming copepods to be followed so that the ﬂow ﬁeld around the free-swimming copepods can be obtained (e.g., Strickler, 1982, 1985; Greene, 1988; Tise- lius and Jonnson, 1990; Yen et al., 1991; Bundy and Paffenho¨ fer, 1996). HYDRODYNAMICS OF COPEPODS: A REVIEW 343

Figure 1. Feeding current of a tethered Pleuromamma xiphias. Velocity contour plots from (a) a dorsal view; (c) a lateral view. The labels in (a) and (c) are in mm s)1. Particle trajectories from (b) a dorsal view; (d) a lateral view. (From Fields and Yen, 1993.) 344 HOUSHUO JIANG AND THOMAS R. OSBORN

Strickler (1982) reported his measurement of the flow field around free-swimming Eucalanus crassus. During feeding bouts of 10–30 s, the structure of the flow field (i.e., the feeding current) around E. crassus was constant. The flow field in front of the mouthparts showed a double shear field, one extending laterally from the median plane and another parallel to the median plane. Once an alga was entrained into this flow field, its path through it was determined (indicating the laminar property of the flow field). Strickler (1982) also pointed out the flow difference between the feeding mode and the cruising mode of E. crassus, i.e., no anterior double shear field was created during cruising. Greene (1988) illustrated the structure of the feeding current created by a hovering Neocalanus cristatus; the feeding current structure looks visually similar to that due to a point force in an infinite domain [see Figure 6.4.1a in Pozrikidis (1997)]. Tiselius and Jonnson (1990) measured the flow field created by copepods among three different feeding strategies: (1) slow-swimming or stationary suspension feeding (Temora longicornis, Pseudocalanus elongatus, and Paracalanus par- vus), (2) fast-swimming interrupted by sinking periods (Centropages typicus and Centropages hamatus), and (3) motionless sinking combined with short jumps (Acartia clausi). It was found that flow fields were similar for all suspension-feeding species, but the anterior velocity gradient moved closer to the copepod in fast-swimming species. By tracing the paths of entrained algae, Yen et al. (1991) reconstructed the velocity field of the feeding current created by a hovering Euchaeta rimana. The velocity field was actually a two-dimensional velocity vector field on the dorsal–ventral plane nearly perpendicular to the stretched antennules and also parallel to the body axis. Based on this velocity field, some characteristics of the velocity gradient field such as vorticity, shear, and squared rates of strain were calculated. Furthermore, by assuming axisymmetry of the feeding current (not necessarily a good approximation here), Yen et al. (1991) were able to calculate the viscous energy dissipation per E. rimana feeding current as ) ) 9.3 · 10 10 W individual 1. Bundy and Paffenho¨ fer (1996) utilized high-resolution video observations of free-swimming adult female copepods to characterize the flow fields created by Centropages velificatus (an omnivore with strong tenden- cies toward carnivory), and Paracalanus aculeatus (a herbivore). For the purpose of comparison, the flow fields around tethered copepods were also measured. The velocity vectors were calculated on the dorsal–ventral plane, so that the obtained flow fields were two-dimensional. Large differences in flow geometry were found between tethered copepods and free-swimming copepods. This was attributed to the fact that, for free-swimming copepods, flow field velocity and geometry are controlled by the balance of forces (drag, negative buoyancy, and forces exerted on the water by the appendages) (Strickler, 1982), while, for tethered copepods, tethering will HYDRODYNAMICS OF COPEPODS: A REVIEW 345

Figure 2. Lateral view of the feeding current of a free-swimming Paracalanus aculeatus female. (a) Flow velocity contour plot. The contour interval is 1 mm s)1. (b) Flow velocity vector plot. Single large arrow represents the copepod’s swimming direction. Note that the frame of reference is fixed on the copepod. (From Bundy and Paffenho¨ fer, 1996.) alter the balance of these forces. Since different species may have different configurations of the balance of forces, it is not surprising that large differences in the geometry of flow fields were found between species. They also observed that a free-swimming copepod creates a flow field with areas of high velocity (larger than its swimming velocity) located in a short distance away from the body surface, and in most situations ventrally or anterior- ventrally to the body surface (Figure 2).

3. Creation of the flow field at individual copepod scale: the significance of ‘being self-propelled’

How does a copepod create a quasi-steady flow field at its body scale? To answer this question, first, the characteristics of beating movements of the cephalic appendages of copepods must be taken into account, as many observations have shown that the creation of the flow field is closely related to the beating movements (e.g., Koehl and Strickler, 1981; Paffenho¨ fer et al., 1982; Cowles and Strickler, 1983; Strickler, 1984; Price and Paffenho¨ fer, 1986a). After revisiting some observational evidence obtained by other researchers, Jiang et al. (2002b) have generalized that two characteristics of the beating movements are responsible for the creation of a quasi-steady flow field at individual copepod scale. (1) The beating movements are rapid, i.e., the beating movements occur at a high frequency, ranging from 20 to 80 Hz (e.g., Storch and Pfisterer, 1925; Can- non, 1928; Lowndes, 1935; Koehl and Strickler, 1981; Price et al., 1983; 346 HOUSHUO JIANG AND THOMAS R. OSBORN

Price and Paffenho¨ fer, 1986a, b; Gill, 1987). (2) The cephalic appendages usually operate in specific motion patterns during swimming and feeding, displaying asymmetry in the propulsive stroke and the recovery stroke (e.g., Gauld, 1966; Strickler, 1984). A high frequency of the beating movements ensures that the unsteadiness of the flow field can onlypffiffiffiffiffiffiffiffi exist within a short distance characterized by the viscous length scale v=x (where m is the kinematic viscosity of the fluid, and x ¼ 2pf, where f is the beating frequency of the appendages) away from the surface of the appendages. Therefore, at a scale of copepod body length, which is usually much larger than the viscous length scale, the flow field is quasi-steady. On the other hand, copepods usually live at low Reynolds numbers (e.g., Zaret, 1980; Koehl and Strickler, 1981; Strickler, 1984; Naganuma, 1996); because of the reversibility of the low Reynolds number flow, the asymmetry in motion patterns of the appendages is of ultimate importance for copepods to create a non-zero mean flow field at their body scale. Next, factors that control the magnitude and geometry of the non-zero mean flow field at a copepod’s body scale must be determined. These factors include the copepod’s excess weight, swimming behavior, and morphology. Stricker (1982) first pointed out that a copepod’s excess weight (i.e., negative buoyancy) is important for the copepod to create a strong feeding current. Intuitively, he suggested that the configuration of forces acting on a free-swimming copepod determines the copepod’s body orientation and swimming velocity; further, he drew diagrams of different configurations of forces for several different copepod species. Along the same line, Emlet and Strathmann (1985) argued that the drag on the main body of a copepod also plays an important role in setting up the flow field around the copepod. In fact, their argument emphasizes the role of the copepod’s swimming behavior and morphology, since the drag is determined by the swimming behavior (including the body orientation, and swimming direction and speed) and morphology (including the morphology of the main body, and the morphology and motion pattern of the cephalic appendages). Childress et al. (1987) indicated that the far-field flow associated with the main body resistance decays much faster than the far-field associated with counterbalancing the excess weight. Indeed, the far-field flow associated with the net force exerted by the copepod on the fluid (a force mono- pole) falls off as 1=r and is independent of the morphology. The far-field flow associated with the main body resistance (a force dipole) falls off like 1=r2 and depends on the morphology and the details of how the copepod is propelling itself (e.g., Childress, 1981; Pedley, 1997; Jiang et al., 2002b). (Here, r is the distance from an origin inside the body.) The reason is the following. When a neutrally buoyant copepod swims, the copepod applies both drag (through body resistance) and thrust (through the beating move- HYDRODYNAMICS OF COPEPODS: A REVIEW 347 ment of appendages) on the water, and the first-order effect due to one is cancelled in the far-field by the first-order effect due to the other. However, the intense currents associated with the details of how the copepod is propelling itself will be found in the immediate vicinity of the ‘‘scanning machine’’ (a name given by Childress et al. (1987) to the beating appendages). Based on this, it is worth pointing out that if the ‘‘scanning machine’’ is well separated from the body parts where most of the resistance originates, the currents around it will be very strong and well- extended, resembling ‘‘first-order scanning’’, their name for the motion associated with overcoming the excess weight. For example, a few calanoids such as those in the family Calocalanidae have a long and huge tail-like appendage attaching to their urosome (see Figure 2.2.1B in Huys and Boxshall (1991)). A small swimming speed of the copepod’s body will result in a large drag originating from the tail-like appendage. That force has to be balanced by the force exerted on the fluid by the beating movement of the cephalic appendages (the ‘‘scanning machine’’). Therefore, a strong, well-extended, first-order scanning-like feeding current may be created around a relatively large volume around the cephalic appendages (J.R. Strickler, pers. comm.). On the other hand, the effect of tethering on the flow field around a copepod is discussed by Childress et al. (1987). The effect of the tethering is twofold. First, the tethering may restrain the normal swimming motion of the copepod and therefore alter the drag. Second, if the tether force is not zero, the tethering will change the thrust that the copepod applies to the water through the beating movement of the cephalic appendages, and therefore modify the flow that one interprets as the first-order scanning component. The above descriptions about the effects of excess weight and main body drag on the creation of the flow can be unified by the concept of being self-propelled, i.e., a free-swimming copepod must beat its cephalic appendages in a certain way and therefore gain propulsion (equal in magnitude but opposite in direction to the thrust) from the surrounding water in order to counterbalance the drag force by the water and its excess weight. Based on this concept, a self-propelled Stokes-flow model has been developed to calculate the flow field around a free-swimming, negatively buoyant, spherical copepod in steady motion through properly coupling the Stokes equations, which govern the flow field around the spherical copepod under the Stokes approximation, with the dynamic equation of the copepod’s spherical body (Jiang et al., 2002b). A point force is applied on the water at a given position outside the spherical body, representing the net effect of the beating movement of the copepod’s cephalic appendages. The magnitude and direction of the point force (the thrust) is adjusted accord- ing to the copepod’s swimming behavior through the coupling between the fluid dynamics and the body dynamics, so that the propulsion balances the 348 HOUSHUO JIANG AND THOMAS R. OSBORN drag force by the water and the copepod’s excess weight. Although the model is highly abstract, it reflects the key concept of being self-propelled. Above all, analytical solutions of the model can be obtained for arbitrary steady motion, such as hovering, sinking, or steady swimming at a given speed along a given direction. The results obtained from the self-propelled Stokes-flow model are con- sistent with intuition and observational evidence. Intuitively, the net force exerted by a free-swimming copepod in steady motion on its surrounding water must be equal to its excess weight in spite of the swimming behavior, because the copepod is self-propelled. Concerning the decay of the velocity field around a negatively buoyant copepod, this indicates that the velocity field should decay in the far-field to the velocity field generated by a point force of magnitude of the copepod’s excess weight in an infinite domain (termed the ‘‘point force’’ model). Fortunately, the self-propelled Stokes- flow model is able to reproduce this important property in velocity decay. It is clearly shown that the velocity magnitudes for different swimming behaviors (e.g., hovering, forward swimming fast or slowly) decay to the velocity field generated by the point force model (Figure 3). On the other hand, observations have shown that the geometry of the flow field around a free-swimming copepod varies significantly with different swimming behaviors. (A review of copepod swimming behaviors as well as the associated flow fields is given in Jiang et al. (2002b).) The observations have been confirmed by the modeling study, in which the streamtube through the capture area of the copepod is used to visualize the flow geometry. Specifically, the streamtube associated with a copepod swimming slowly (i.e., swimming at a speed at least several times smaller

point force model velocity magnitude standing-still model forward swimming model (u=4.4 mm/s) forward swimming model (u=1.1 mm/s) (normalized by terminal velocity)

X (normalized by sphere radius)

Figure 3. Velocity decay for diﬀerent swimming behaviors. The velocity magnitudes have been normalized by the terminal velocity of the spherical copepod (4.4 mm s)1 for the present case). (From Jiang et al., 2002b.) HYDRODYNAMICS OF COPEPODS: A REVIEW 349

(a) (b) 6.0 0.0 t=0.0s t=–0.1s t=–0.5s

4.0 t=–2.5s -2.0 =4.4 mm/s

t=–1.2s sinking t=–1.2s V 2.0 t=–0.5s -4.0 Z (mm) Z (mm) t=–0.1s 0.0 t=0.0s -6.0

-2.0 -8.0 t=–2.5s

-4.0 -2.0 0.0 2.0 4.0 6.0 -4.0 -2.0 0.0 2.0 4.0 X (mm) X (mm)

5.0 t=–1.2s t=–0.5s

4.0 t=–2.5s t=–0.1s t=–1.2s 2.0 0.0 t=0.0s

Z (mm) t=–0.5s Z (mm) t=–0.1s Vswimming=4.4 mm/s 0.0 t=0.0s

-5.0 V =1.1 mm/s -2.0 swimming

-2.0 0.0 2.0 4.0 6.0 0.0 5.0 10.0 X (mm) X (mm)

Figure 4. Lateral view of the streamtube through the capture area of a spherical model copepod (a) hovering (like a helicopter) in the water, (b) sinking freely at its terminal velocity (4.4 mm s)1), (c) swimming forward (in positive x-direction) at a speed of 1.1 mm s)1, and (d) swimming forward (in positive x-direction) at a speed of 4.4 mm s)1. Note that the frame of reference is fixed on the copepod. (From Jiang et al., 2002b.) than the copepod’s terminal velocity of sinking, termed the slow-swimming behavior) resembles the streamtube of a copepod hovering in the water. In both situations, the cone-shaped and wide streamtube transports water to the capture area of the copepod, and a feeding current is created (Figure 4a and c). Conversely, when a copepod swims at a speed equal to or greater than the terminal velocity (termed the fast-swimming behavior), the streamtube is cylindrical, long and narrow and the corresponding flow field is not a feeding current (Figure 4d). In addition, when a copepod sinks freely, the flow comes from below relative to the copepod and the streamtube is much narrower and longer than hovering and swimming 350 HOUSHUO JIANG AND THOMAS R. OSBORN slowly, but shorter than swimming fast (see Figure 4b). Again, the flow field around a free-sinking copepod is not like a feeding current. The differences in the flow geometry with the different swimming behaviors are due to the relative importance between the two factors in generating the flow field: the copepod’s swimming motion and the requirement to counterbalance the copepod’s excess weight. Here, comparing the actual swimming velocity with the terminal velocity of sinking distinguishes between slow- swimming and fast-swimming among species, and also measures the relative importance between the two factors in generating the flow field. Modeling free-swimming copepods as self-propelled bodies not only serves as an imperative to obtaining a correct spatial decay of the flow field and understanding the relationship between swimming behavior and flow geometry, but also plays a crucial role in reproducing the key aspects of the flow field, which have been revealed by observational studies. These key aspects include the double shear field (Strickler, 1982), the spatial configuration of the velocity gradient field such as vorticity, shear, and squared rates of strain (Yen et al., 1991), and the spatial configuration of the velocity field such as the locations of velocity maximums relative to the body surface (Bundy and Paffenho¨ fer, 1996) and asymmetry (e.g., ventral–dorsal asymmetry). For instance, the self-propelled Stokes-flow model is able to reproduce an important flow characteristic reported by Bundy and Paffenho¨ fer (1996), i.e., an area of high flow velocities (larger than its swimming velocity) is located a short distance away from the body surface, and in most situations ventrally or anterior-ventrally to the body surface (Figure 5b and d). On the contrary, the Stokes-flow field due to the translating motion of a solid sphere, termed the towed body model, cannot reproduce this flow characteristic; the velocity maximum (equal to the translating velocity) is reached at the body surface when using a stationary frame of reference (Figure 5a and c). In addition, the self-propelled body model creates a flow field with a ventral-dorsal asymmetry (Figure 5b and d), while the flow field created by the towed body model is axisymmetric with respect to the body axis along the direction of the translating motion (Figure 5a and c). Some studies, although very heuristic for understanding the biological aspects of copepod feeding and/or sensing, used the towed body model to calculate the flow field created by a free-swimming copepod, not consider- ing the free-swimming copepod as a self-propelled body. Tiselius and Jons- son (1990) used the towed body Stokes solution to model the flow field around a swimming copepod and tried to understand the efficiency of hovering behavior compared with swimming behavior. In their study, the clearance of a swimming copepod was simply calculated as the cross sec- tion of the copepod times swimming velocity. However, the most important contribution to the clearance by the beating movement of the copepod’s cephalic appendages was neglected. In addition, for the same HYDRODYNAMICS OF COPEPODS: A REVIEW 351

Figure 5. Contour plots of velocity magnitudes of the flow field calculated from (a) and (c) the towed body model, and (b) and (d) the self-propelled body model. The velocities have been calculated using a stationary frame of reference and the velocity magnitudes normalized by 4.4 mm s)1 (the terminal velocity of the spherical copepod). In (a), the sphere (a towed body) translates in the negative x-direction at a speed of 1.1 mm s)1. In (b), the spherical copepod (a self-propelled body) swims backward in the negative x-direction at a speed of 1.1 mm s)1. In (c), the sphere (a towed body) translates in the negative x-direction at a speed of 4.4 mm s)1. In (d), the spherical copepod (a self-propelled body) swims backward in the negative x-direction at a speed of 4.4 mm s)1. (From Jiang et al., 2002b.) swimming velocity, the drag calculated from the towed body model is generally different from that calculated from the self-propelled body model (Ji- ang et al., 2002b). Similarly, Kiørboe and Visser (1999) applied the Stokes solution of the flow around a fixed sphere to model the flow around a moving plankter. The numerical model of a two-dimensional flow field around two circu- lar cylinders (which is actually a towed body model) was used to model the situation of a free-swimming copepod approaching an inert particle (Bundy et al., 1998). Since these studies did not consider a free-swimming copepod as a self-propelled body, their quantitative results may need to be re-exam- 352 HOUSHUO JIANG AND THOMAS R. OSBORN ined. A so-called ‘‘spherical pump’’ solution (the Stokes flow for a translating sphere with a frame of reference fixed with respect to the far-field fluid) was used to model the flow field of a copepod feeding current (e.g., Kiør- boe and Visser, 1999). Although the velocity decay happens to be correct, the spatial configurations of velocity field, deformation rate field and vorticity field (around the solid sphere) calculated from the model are apparently not comparable with available observational results around a copepod (e.g., Yen et al., 1991; Bundy and Paffenho¨ fer, 1996). Visser (2001) argued that, for the spherical pump model, the thrust is distributed over a finite spherical volume of space. If this is the case, we can only understand that the solid body in the spherical pump model is not the copepod main body since a hovering negatively buoyant copepod only applies thrust to a small volume of water where its cephalic appendages are located, with its main body retarding the created feeding current. Thus, the ‘‘spherical pump’’ model does not separate the main body resistance from the thrusting effect of the beating movement of the cephalic appendages, which is, however, a requirement for a self-propelled body model. Therefore, the model is still not a self-propelled body model. It is worth noting that a specific form of the separation reflects a specific way in which the copepod propels itself, which then can be used to reproduce the specific ‘‘footprint’’ left by the copepod.

4. Numerical simulations of the ﬂow ﬁeld at individual copepod scale

The equations governing the ﬂow velocity vector ﬁeld u(x) around a copepod in steady motion are approximately the steady Navier–Stokes equations and the continuity equation:

2 qu ru ¼rp þ lr u þ fa ð1Þ

ru ¼ 0 ð2Þ where q is the density of the seawater, l is its dynamic viscosity and p is the flow pressure field. fa represents the force field (force per unit volume) that models the net effect of the beating movement of the copepod’s cephalic appendages, and the volume integral of fa is the thrust (Jiang et al., 2002b). The boundary conditions of Eqs.(1) and (2) are the no-slip boundary condition on the surface of the main body (denoted as X mb, i.e., the body excluding the beating appendages):

u ¼ Vswimming; at X mb ð3Þ HYDRODYNAMICS OF COPEPODS: A REVIEW 353 and the boundary condition at inﬁnity:

u ! 0; at infinity ð4Þ

Here, Vswimming is the swimming velocity of the copepod. That a copepod is in steady motion means that it is hovering at a position (Vswimming=0), swimming at a constant velocity (Vswimming=constant), sinking at its terminal velocity of sinking (Vswimming=Vsinking and fa=0), or tethered at a position (Vswimming=0). The dynamic equation of a copepod in steady motion can be written as Z

Wexcess þ F faðxÞdx þ Te ¼ 0 ð5Þ x where Wexcess is the excess weight of the copepod, F is the drag force exerted by the flow field on the copepod’s main body, and Te is the force acting on the copepod by the tethering, if any. Eqs (1)–(5) are a set of equations that describe the dynamic coupling between a copepod’s swimming motion and the water flows surrounding the copepod. Apparently, along with the main body motion (i.e., the velocity and orientation of the main body), which determines the boundary condition at the body–fluid interface, the beating movement of the cephalic appendages (represented here by fa) plays a key role in the dynamic coupling. The input parameters of the set of equations are the morphology, swimming behavior, and excess weight of the copepod. Here, the morphology includes the main body morphology and the spatial distribution of the cephalic appendages relative to the main body (represented here by the spatial distribution of fa). The swimming behavior includes the swimming speed and direction. The excess weight is determined from the excess density and body volume of the copepod. The mass density of some copepod species is available (see Gross and Raymont, 1942; Lowndes, 1942; Green- law, 1977; Køgeler et al., 1987; Visser and Jo´ nasdo´ ttir, 1999; Knutsen et al., 2001). Information on the density contrast of copepods with respect to their natural seawater habitat can be found in Greenlaw and Johnson (1982) and Knutsen et al. (2001). Numerical simulation involving computational fluid dynamics (CFD) is needed for a full solution of Eqs. (1)–(5). The output of the numerical simulation is a three-dimensional flow velocity vector field around the copepod, which is detailed enough for further uses, such as visualization of the flow geometry, estimation of the feeding rate, and quantification of the sensory field, etc.. Till now, an accurate measurement of the three-dimensional flow velocity vector field still challenges experimental biologists. One of the virtues of numerical simulations is that parametric studies can be 354 HOUSHUO JIANG AND THOMAS R. OSBORN performed. That is, we can vary the input parameters, such as the morphology, swimming behavior, and excess weight of the copepod, and exam- ine their effects systematically. Some problems (e.g., the relationship between swimming behaviors and feeding efficiency, the relationship between swimming behaviors and sensory mechanisms, and the size effect) may be difficult to study experimentally. However, they can be attacked by performing the numerical simulation.

4.1. For a tethered copepod

Eq. (5) is neglected and only Eqs. (1)–(4) are solved numerically in the simulation of the feeding current created by a tethered copepod. The magnitude and distribution of the force field fa is adjusted so that the flow velocity field output from the simulation is reasonably matched to available observational data of the feeding current around a tethered copepod. From this flow field, the drag force F acting on the copepod’s main body by the water is calculated, and then the tethering force Te can be evaluated from Eq. (5). Following the above-described simulation process, a numerical feeding current has been generated (Jiang et al., 1999). The magnitude and spatial configuration of the simulated feeding current (Figure 6) are in good agreement with those of the feeding current around a tethered Eucha- eta norvegica observed by Yen and Strickler (1996). The geometry of the entrainment region, as visualized by tracking particles in the feeding current to construct a streamtube through the capture area of the copepod, is cone-shaped and wide. By calculating the viscous dissipation rates around the copepod, the influence field of the simulated feeding current is shown to be anisotropic, similar to the observational results obtained by Fields and Yen (1993). In addition, by varying the distribution of the force field fa, it is shown that a distributed force dissipates less energy, but results in a higher entrainment rate than a concentrated force does. Therefore, for a given amount of force, applying a more distributed force to the water is energetically more efficient. This interesting finding may be used to account for the evolution of the complex and delicate feeding appendages.

4.2. For a free-swimming copepod

To simulate the flow field created by a free-swimming copepod in steady motion, Eqs. (1)–(5) have to be solved together such that the model is a self-propelled body model. Given the main body morphology, spatial distribution of the force field fa (the spatial distribution resembles that of a real copepod’s cephalic appendages), and excess density of the copepod, HYDRODYNAMICS OF COPEPODS: A REVIEW 355

1.5 mm/s 1.5 mm/s 2 mm

Figure 6. Contours of velocity magnitudes of the simulated feeding current around a tethered copepod. The contour plot is drawn along a plane 0.3 mm ventral to the copepod. The contour levels are (1.50, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 18.5) mm s)1. the magnitude of the force field fa and the body orientation of the copepod are determined for different swimming behaviors (characterized by their swimming speed and direction). Then, the flow field around the copepod is determined. The above-described simulation process has been performed by Jiang et al. (2002a). The numerical results confirm the conclusions drawn from the theoretical analysis using Stokes-flow models by Jiang et al. (2002b) about the relationship between swimming behaviors and flow geometry, i.e., the geometry of the flow field around a free-swimming copepod varies significantly with different swimming behaviors (Figure 7). Above all, the simulated feeding current is comparable with available observational data, especially in the spatial configuration of the feeding current: by comparing a simulated velocity vector field around a forward swimming copepod (Figure 7b) with an observed velocity vector field also around a forward swimming copepod (Figure 2b), it is shown that the numerical simulation is able to reproduce the key features of an observed feeding current, such as the location of velocity maximum and the spatial configuration of the shear layer around the copepod. Here, the high veloc- 356 HOUSHUO JIANG AND THOMAS R. OSBORN ity area extends for up to a body length ventrally/anterior-ventrally away from the body surface. The spatial configuration of the shear layer around a copepod is shown to be different for different swimming behaviors (e.g., hovering/slow-swimming, fast-swimming, or sinking freely, see Figure 7). Each swimming behavior is associated with a specific three-dimensional flow structure. For example, for a hovering copepod (Figure 8), the streamtube through the area above the antennules does not overlap with that through the capture area. The flow above the antennules is decelerated when approaching the antennules, because it has to satisfy the no-slip boundary condition (i.e., zero velocity) at the surface of the antennules. In contrast, the flow going through the capture area is accelerated when approaching the capture area, since here the no-slip boundary condition is the beating movement of the cephalic appendages, the net effect of which is an accelerated flow. Further parametric studies reveal that the behavior of hovering or swimming slowly is energetically more efficient in terms of relative capture volume per energy expended than the behavior of swimming fast. That is, for the same amount of energy expended a hovering or slow-swimming copepod is able to scan more water than a fast-swimming one does. It is also found that the behavior of hovering or swimming slowly is hydrodynamically quieter than the behavior of swimming fast or sinking freely. That is, the influence field is smaller for the flow field created by a hovering/slow-swimming copepod than by a fast-swimming/free-sinking copepod, provided that the excess weight of the copepod is the same.

4.3. For two copepods in close proximity

When two copepods are in close proximity or approaching each other, the flow field created by the movement of one copepod is transmitted through the fluid medium and affects the flow field around, as well as the hydrodynamic force and torque on the other copepod. This is the hydrodynamic interaction between two copepods, which was first studied by Jiang et al. (2002c) using a numerical simulation method. Their results show that, when two copepods are in close proximity, the hydrodynamic interaction between them distorts the geometry of the flow field around each copepod (Figure 9a) and changes the hydrodynamic force on each copepod. The hydrodynamic interaction also results in a hydrodynamic signal, which can be measured as a distribution of flow velocity difference along each copepod’s antennules (Figure 9b). Parametric studies show that the hydrodynamic interaction as well as its resulting hydrodynamic signal is a function of the separation distance between the two copepods, their relative body positions and orientations, and their relative swimming HYDRODYNAMICS OF COPEPODS: A REVIEW 357

(a) (b)

Vswimming=1.047 mm/s

(d)

Vswimming=4.187 mm/s

Figure 7. Velocity vector plots along the median plane of a model copepod (a) hovering in the water; (b) swimming forward at a speed of 1.047 mm s)1; (c) swimming forward at a speed of 4.187 mm s)1; (d) sinking freely, with the anterior pointing upward, at its terminal velocity (4.187 mm s)1 and along its body axis for this case). The frame of reference is ﬁxed on the copepod. velocities. The numerical method may have further usage in understanding the swarming behavior of zooplankton, nearest neighbor distances (NND) in a zooplankton swarm, and the energetic advantage of maintaining swarm integrity. 358 HOUSHUO JIANG AND THOMAS R. OSBORN

Figure 8. Streamtubes for a model copepod hovering in the water. Two streamtubes are drawn: the streamtube through an area right above the copepod’s antennules and the streamtube through the copepod’s capture area. The dashed line connecting the stars is the streamline passing through the center of the capture area. Note that the copepod’s body has been made transparent in order to show the portion of the streamtube ventral to the copepod. (From Jiang et al., 2002d.)

5. Hydrodynamics and copepod sensory mechanisms

Behavioral, morphological and physiological studies have revealed that copepods are able to perceive food particles, prey, predators and con- speciﬁcs (including mates) via mechanoreception and/or chemoreception (for reviews see Atema, 1988; Lonsdale et al., 1998; Mauchline, 1998; Visser, 2001; see also Doall et al., 1998; Kiørboe and Visser, 1999; Moore et al., 1999; Paffenho¨ fer and Loyd, 2000; Jiang et al., 2002d, and references therein). Since the water environment surrounds the copepods HYDRODYNAMICS OF COPEPODS: A REVIEW 359

(a)

t=– 12.0s t=– 12.0s t=– 7.0s t=– 7.0s 2 t=– 3.0s t=– 3.0s t=– 1.0s t=– 1.0s

z (mm) 0 t=0.0s t=0.0s

– 2

––4 2 02 4 y (mm)

(b)

The paired dotted lines represent the velocity magnitude of the solitary copepod 3. +/- 20 µm/s. Any velocity magnitude outside 43m 4.8 m these dotted lines can be detected, based on a 8 m m 4 detection threshold of 20 µm/s velocity difference. .10 mm

Figure 9. (a) Ventral view of the two streamtubes respectively through the capture areas of the two copepods that are side-by-side, stationary and separated by a distance of 3.43 mm. (b) Velocity magnitudes along the line right above the antennules for the two side-by-side stationary copepods. The labels in mm indicate the separation distance between the two copepods. (From Jiang et al., 2002c.) as well as their items of interest, the ﬂow ﬁeld created due to the exis- tence of both parties will affect the generation and transmission of water-borne signals, whether the signals are mechanoreceptional or che- moreceptional. 360 HOUSHUO JIANG AND THOMAS R. OSBORN 5.1. Chemoreception

The role of the copepod feeding current in copepod chemoreception has been noticed for a long time (e.g., Alcaraz et al., 1980; Strickler, 1982). The shear configuration of the feeding current created by a copepod hovering or slightly drifting in the water column will elongate the active space of the chemicals around an entrained alga; this will enable the copepod to use chemoreception to detect the presence, as well as the trajectory, of the alga (Strickler, 1982). Strickler’s hypothesis was confirmed by Andrews (1983), who developed a two-dimensional numerical model to calculate the deformation of the active space surrounding an alga in the low Reynolds number feeding current created by a copepod. The experiments done by Moore et al. (1999) also confirm the hypothesis, further, they show that the deformation of the chemical signal can be different when entrained into feeding currents of different shear configurations. Copepod chemoreception capability has been compared among copepods of different swimming behaviors by using a three-dimensional alga- tracking, chemical advection–diffusion model to calculate the deformation of the active space surrounding an entrained alga (Jiang et al., 2002d). It is shown that an initially spherical active space will be more elongated when entrained by the flow field around a hovering or slow-swimming copepod, and that the copepod can have a few hundred milliseconds to respond to the approaching alga (Figure 10a and b). Thus, a copepod in slow-swimming behaviors (including the behavior of hovering) is capable of using chemoreception to detect individual algae entrained by the flow field. In contrast, the advance warning time for a fast-swimming copepod is much shorter (Figure 10c), so that the copepod is not able to rely on chemoreception for remote detection. The distinction is because the flow fields associated with different swimming behaviors have different velocity and shear configurations. It is emphasized that not only the shear configuration but also the velocity configuration of the flow field affect the advance warning time. The former determines the deformation of the active space; the latter determines the traveling time as well as trajectory of the alga. For a hovering or slow-swimming copepod, the flow streamlines are concentrated from a large region toward a small region around the capture area, and therefore there are larger differences in speed between adjacent streamlines and in speed as well as acceleration along streamlines. For a fast-swimming copepod, since the flow geometry is cylindrical, narrow and long, the differences in speed between adjacent streamlines and in speed as well as acceleration along streamlines are expected to be small. Thus, the deformation of the active space is more significant for a hovering or slow-swimming copepod than for a fast-swimming copepod. On the other hand, the flow velocity magnitudes around a fast-swimming copepod are generally HYDRODYNAMICS OF COPEPODS: A REVIEW 361

(a) 5 mm/s (b) 5 mm/s

Vswimming=1.047 mm/s

t=0.000 s

t=1.800 s t=0.000 s t=1.920 s

t=3.494 s t=3.550 s

t=3.998 s t=3.998 s

t=3.744 s

m t=3.996 s t=3.904 s t=3.600 s t=2.700 s t=1.800 s t=0.900 s t=0.000 s t=1.740 s

7 t=1.500 s

4 t=1.000 s

i t=0.500 s

t=0.000 s Vswimming=4.187 mm/s

5 mm/s

Figure 10. Deformation of the active space surrounding an alga entrained by the flow field around a model copepod (a) hovering in the water; (b) swimming forward at a speed of 1.047 mm s)1; (c) swimming forward at a speed of 4.187 mm s)1; (d) sinking freely, with the anterior pointing upward, at its terminal velocity (4.187 mm s)1 and along its body axis for this case). The alga is on the median plane of the copepod and to be entrained into the copepod’s capture area. The streamline is drawn through the centers of the alga. The velocity vectors of the flow field relative to the copepod are drawn on the median plane. (From Jiang et al., 2002d.) much greater than those around a hovering or slow-swimming copepod (the frame of reference is fixed on the copepod). Thus, the alga travels much faster when entrained into the flow field created by a fast-swimming copepod than by a hovering or slow-swimming copepod. For these two reasons, the advance warning time for a fast-swimming copepod is much shorter than for a hovering or slow-swimming copepod. The velocity and shear configurations for a free-sinking copepod seem to favor its chemoreception of an encountered alga (Figure 10d). It is also 362 HOUSHUO JIANG AND THOMAS R. OSBORN shown that advection by fluid motion dominates over diffusion during the transport of the chemical signals inside the active space surrounding an alga to the location of a copepod’s chemoreceptors.

5.2. Mechanoreception

Tremendous efforts have been made in order to identify and quantify the hydrodynamic signals detected by copepods, and to determine signal threshold strengths and the associated detection distances (e.g., Haury et al., 1980; Fields and Yen, 1996, 1997; Viitasalo et al., 1998; Kiørboe and Visser, 1999; Kiørboe et al., 1999; Fields et al., 2002). Kiørboe and Visser (1999) have argued that prey detection relies on the magnitude of the fluid velocity created by the prey, while predator detection depends on the magnitude of one or several of the components of the fluid velocity gradients (deformation rate, vorticity, acceleration) created by the predator. Most importantly, the hydrodynamic signals potentially detectable by copepods may be the velocity differences caused by all these components. Many studies have quantified the signal threshold strengths in terms of the fluid deformation (or shear) rate at the copepod’s body scale. However, the threshold deformation rate obtained ranges from 0.5 to 5 s)1 (Kiørboe et al., 1999), which is probably too broad. Our speculation on this is that the hydrodynamic signals might be better measured at the place where copepod mechanoreceptors are located or distributed, such as along the antennules. This is because essentially the bending of setae on copepod sensory organs enables the detection, and the bending is essentially due to the change of the flow field around the setae (Jiang et al., 2002c). However, a fairly accurate measure of the flow field around a copepod seta or even along the antennules is difficult. On the other hand, 20 lms)1 in the velocity difference between the tip and base of a seta is enough to elicit a neural response from the antennules of copepods (Yen et al., 1992). Using this threshold in velocity difference and a numerical simulation method, Jiang et al. (2002c) have quantified the hydrodynamic signals, resulting from an approaching copepod of comparable size, along a line right above the antennules of a copepod. Generally, the hydrodynamic disturbances (the signals) between two copepods (of comparable size) in close proximity are different from those between a copepod and its prey (of a much smaller size) or predator (of a much larger size). Since its size is much smaller than the copepod, the prey has less influence on the flow field around the copepod (Figure 11a). The hydrodynamic signals generated by the prey can only bend a very limited number of setae on the copepod’s antennules, and hence the array of setae along the copepod’s antennules detects the prey as localized velocity distur- HYDRODYNAMICS OF COPEPODS: A REVIEW 363 bances. In contrast, an approaching predator of a relatively large size may totally destroy the organized flow field around the copepod (Figure 11b). The copepod detects the predator when the spatially varying and temporally fluctuating velocity disturbances at the copepod’s whole body scale stimu- late the full array of mechanoreceptors on the copepod. Hwang and Stric- kler (2001) have suggested that copepods may use a simple form of pattern recognition to distinguish between sources of signals, prey or predators. Along the same line, pattern recognition may also be used by copepods to detect conspecies of comparable size, where the velocity disturbances at their partial body scale are detected (Figure 11c). As such, different deflec- tion patterns of the array of mechanoreceptors in response to different signal sources enable a copepod to distinguish among a small prey, a giant predator or a copepod of comparable size. Finally, it is worth noticing that the signals between two copepods of comparable size may not necessarily be symmetric, i.e., one copepod can detect the presence of the other copepod, while the latter cannot detect the former (see Figure 11d in Jiang et al., 2002c).

5.3. Effects of morphology

The effects of morphology can be divided into the effects of the main body and the effects of the appendages. Of the effects of the main body, the effect associated with size is the dominant one. Given the size of the main body, changes in the body shape have limited effects on the flow field around the body in the low Reynolds number regime (Panton, 1996). On the other hand, the local flow field around the body is largely dependent on the distribution and moving pattern of the appendages relative to the main body. Thus, it is not surprising that the results from the theoretical analyses using Stokes-flow models of a sphere with a point force (Jiang et al., 2002b) are so close to the results obtained from the numerical simulations using a realistic copepod body shape (Jiang et al., 2002a). The sphere and the realistic copepod body shape are of similar size, and the magnitude and distribution (relative to the main body) of the forces, which represent the net effect of the appendages, are also similar. Furthermore, the distribution and moving pattern of the appendages may reflect a copepod’s sensory requirements, in that they create flows going across the sensors on the copepod (e.g., Yen and Stricker, 1996). Correspondingly, the sensors may be distributed to fit for monitoring the changes in the key aspects of the flow field. Observational evidence of the locations of copepod chemoreceptors and mechanoreceptors supports the interplay of flow structure and sensor morphology (e.g., Fields and Yen, 1993; Lenz and Yen, 1993; Moore et al., 1999). The experiments done by 364 HOUSHUO JIANG AND THOMAS R. OSBORN

Figure 11. Illustrations of the three perspectives of copepod mechanoreception as predator, prey and conspecies, respectively. (a) Detection of the local flow disturbance due to the jumping motion of a small prey; the streamlines are drawn using a frame of reference fixed on the copepod’s body. (b) Detection of the flow disturbance at the copepod’s whole body scale due to the swimming motion of an approaching predator (the fish); the streamlines are drawn using a frame of reference fixed on the fish’s body. Note that the streamlines shown in (b) are just for the purpose of illustration; there are some situations in which the streamlines are different from here, such as the situation that a fish gener- ates suction to draw a copepod into its open mouth. [(a) and (b) are modified from Figure 1 in Jumars (2000); copepod drawing from Williamson (1987) and fish drawing from Drucker and Lauder (1999).] (c) Detection of the flow disturbance at partial body scale due to the presence of a nearby conspecific copepod (of comparable size); the streamlines are drawn using a frame of reference fixed on one of the two copepods. HYDRODYNAMICS OF COPEPODS: A REVIEW 365

Fields et al. (2002) further show that each seta, whether long or short, responds to only a portion of the overall range of ﬂow velocity in the copepod’s habitat, suggesting that the ensemble of setae of different morpho- logies and lengths may function as a unit to decode the intensity and directionality of complex hydrodynamic signals.

6. Summary

(1) Associated with their complex swimming behaviors, body morphology, and negative buoyancy, water flows are created at the body scale of calanoid copepods. The created water flows not only determine the net currents going around and through the hair-bearing appendages of the copepods but also set up a laminar flow field around them. The laminar flow field interacts constantly with the environmental background flows. (2) Physically, the creation of the water flows at copepod body scale can be explained in terms of the fact that free-swimming copepods are self- propelled bodies. That is, a free-swimming copepod must gain propulsion from the surrounding water in order to counterbalance its excess weight and the drag force exerted by the water. This also explains the observations that the flow field around a free-swimming copepod varies significantly with the different swimming behaviors. (3) Understanding the hydrodynamics at copepod body scale is helpful to understand many aspects of copepod feeding, swimming, sensing and swarming behaviors. (4) Numerical simulation is a useful tool to study the small-scale biological–physical interactions (i.e., the organism–organism and organism– environment interactions at the scale of the individual). (5) Future research directions of the hydrodynamics of copepods may include: The interaction between the water flows created by a copepod and its surrounding environmental background flows, such as small-scale turbulence. The effect of small-scale turbulence on copepod sensory mechanisms. The effect of the hydrodynamics on the mate locating processes. The hydrodynamics of copepod nauplii.

Acknowledgements

H.J. gratefully acknowledges the Postdoctoral Scholar Program at the Woods Hole Oceanographic Institution (WHOI), with funding provided by 366 HOUSHUO JIANG AND THOMAS R. OSBORN the Dr. George D. Grice Postdoctoral Scholarship Fund. T.R.O. is sup- ported by the Ofﬁce of Naval Research. The authors thank Dr. Mark Gro- senbaugh and an anonymous reviewer for valuable advice and comments. Thanks are due to Dr. David Fields for his kind help in preparing Figure 1 from his original work. H.J. also gratefully acknowledges the support from the Penzance Endowed Fund in Support of Assistant Scientists at WHOI. This is Contribution Number 10721 from WHOI.

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