The motion of kelp blades and the surface renewal model

We consider how the flapping of kelp blades may enhance the flux of nutrients to a blade, by stripping away the diffusive sub-layer and renewing the fluid at the blade surface. The surface renewal model explains the degree of flux enhancement observed in previous studies under different flow and flapping conditions. We measured the motion of real kelp blades of Laminaria saccharina, Macrocystis pyrifera, and Nereocystis luetkeana under unidirectional current in a laboratory flume. Observed flapping frequencies coupled with the renewal model, suggest that the flapping of blades in the field has the potential to significantly enhance flux to the blade surface at low current speed, but has little effect on flux at high current speeds. Many species of kelp have blades with a flat morphology in regions of high wave and current action, called exposed sites, and blades with a ruffled morphology in regions of low wave and current action, called sheltered sites (Koehl et al. 2008). Researchers have suggested that this morphological shift between exposed and sheltered sites is a trade-off between the need to minimize drag and prevent breakage and the need to maximize photosynthesis (Gerard and Mann 1979; Koehl and Alberte 1988; Haring and Carpenter 2007). Under steady current, ruffled blades spread out and flap, tendencies that increase both light interception and drag (Koehl et al. 2008). Blade flapping has also been observed to enhance the rate of nutrient uptake (Koehl and Alberte 1988). In contrast, flat blades collapse into streamlined clumps under high flow, which reduces drag but also light interception (Koehl et al. 2008). Finally, previous research has suggested that at sheltered sites the flux of nutrients to a blade surface is limited by mass-transport to the blade surface (Gerard and Mann 1979; Wheeler 1980; Koch 1993). To summarize the above ideas, at exposed sites, the mean and wave-induced flow is consistently high enough that mass-flux limitation does not occur, so that drag reduction dominates the morphological choice, and a streamlined blade shape is produced. At sheltered sites, the mean currents are low enough that mass-transfer limitation is a greater threat than hydrodynamic drag, and a ruffled blade shape is produced, because this morphology promotes flapping, and flapping has been observed to enhance flux. In this paper we provide some new insight into this hypothesis by (1) demonstrating that the surface renewal model can explain previous observations of flux to flapping blades, (2) measuring the flapping frequencies of four different real blades, and (3) using the surface renewal model to describe the magnitude of flux enhancement expected from the observed range of flapping frequencies. How flapping enhances fluxes—the surface renewal model—Previously, the mass-flux to blade surfaces has been described using the thin-film model, which assumes that a static boundary layer exists on the surface of the blade (Wheeler 1980; Hurd et al. 1996). However, some authors have suggested that turbulence and wave-induced blade motion can periodically disturb or strip away the diffusive sub-layer and, thereby, enhance flux to the blade (Koch 1994; Stevens and Hurd 1997; Hurd 2000). Stevens and Hurd (1997) used the surface renewal model from Higbie (1935) to describe a mechanism of flux enhancement for kelp blades. The model proposes that the flux at a surface is enhanced by the periodic renewal of water at the surface. Each renewal, or disturbance, replaces the fluid in the diffusive sub-layer with fluid from outside this sub-layer, producing an instantaneously higher concentration gradient at the surface and, thus, higher flux. The subsequent evolution of the concentration profile is described below and depicted in Fig. 1. Let the surface of the blade be z 5 0, and z is positive upward. Next to the boundary there exists a fluid region, called the diffusive sub-layer, in which turbulent transport is negligible, and flux occurs only through molecular diffusion. Advection is very small within this layer, and can be neglected. The thickness of the diffusive sub-layer, dD, is related to the viscous sub-layer thickness, dn. For fully turbulent boundary layers, dn < 10n/u*, with n the molecular kinematic viscosity and u* the friction velocity. Because of the difference in magnitude between molecular diffusivity (D) and kinematic viscosity, the diffusive sub-layer is smaller than the viscous sub-layer. Specifically, dD 5 dn Sc21/3, with Schmidt number Sc 5 n/D (Boudreau and Jorgensen 2001). In water n 5 1026 m2 s21, and for most dissolved species D < 1029 m2 s21, so that in water, we generally find dD 5 0.1 dn. The diffusive sub-layer can control the uptake of nutrients by a blade, if the rate of diffusion across dD is slower than the rate of biological incorporation occurring at the surface. Under these conditions we can assume that the blade instantly takes up any chemical arriving at its surface, so that the concentration at the surface is zero, C(z 5 0) 5 0. The concentration at the top of the diffusive sublayer is Co. The steady-state concentration profile within the diffusive sub-layer is linear, and the flux is