## Supply of nutrients

Based on the example of carbon, the well-developed nutrient harvesting capabilities of algae have already been indicated (see, especially, Section 3.4.2). However, it is not simply a matter of engineering a high affinity for the carbon dioxide (or, indeed, other nutrient in the adjacent medium) as the mechanisms can only be effective over a short distance beyond the cell. The operational benefits are really restricted to within the boundary layer adjacent to the cell.

Here, the movement of solutes are subject to Fick-ian laws of diffusion (cf. Eq. 3.19). The renewal, or replenishment, of nutrients in this immediate microenvironment of the cell can also be critical and, hence, so is any attribute of the organism that enhances the rate of entry of essential solutes into that boundary layer. Such adaptations in this direction may raise directly the effectiveness of nutrient gathering by the cell.

The importance of the movement of water relative to the phytoplankter (or, as we now recognise, to the phytoplankter plus its boundary layer) was famously considered by Munk and Riley (1952). They were among the first to point out that the effect of motion - either active 'swimming' or passive sinking or flotation - is to increase the solute fluxes to the cell above those that would be experienced by one that is non-motile with respect to the adjacent medium. This seemingly axiomatic statement was verified through the experiments of Pasciak and Gavis (1974, 1975) and the interpolation of the results to the benefits to nutrient uptake kinetics of a diatom of sinking through nutrient-depleted water. In consideration of these data, Walsby and Reynolds (1980) determined the trade-offs between sinking and uptake rates in sinking diatoms: there was always a positive benefit in material delivery but at the ambient external concentrations critical to sufficiency, the compensatory sinking rates become unrealistically large. In other words, motion relative to the medium undoubtedly assists the renewal and delivery of nutrients to the immediate vicinity of the plankter but, ultimately, is no guarantee of satisfaction of the plankter's requirements at low concentrations.

A modern, empirical perspective on this topic has been pursued in the work of Wolf-Gladrow and Riebesell (1997; see also the review of Riebe-sell and Wolf-Gladrow, 2002). Starting from the perspective of the single spherical algal cell with an adjacent boundary layer of thickness a, the concentration (C) of a given nutrient in the immediate microenvironment is subject only to diffusive change, in conformity with the equation:

where t is time, m is the coefficient of molecular diffusion of the solute (as in Eq. 3.19) and AN = (S/Sx, S/Sy, S/Sz) is an integral of the gradients in the x, y and z planes. Supposing steady state in a symmetrical sphere, this will reduce to:

where rb is the radial distance from the centre of the sphere. It may be solved for the space to the edge of the boundary, Csurface = C(rb = a), and beyond, Cbulk = C(rb ^ to), so that:

The flux (Fa) of the solute to the cell is calculable

If the live cell now retains the inwardly diffusing solute molecules, Csurface diminishes to zero and the flux increases towards a maximum:

The effect of the motion of the cell, sinking, floating or 'swimming' through water is to increase the flux to the diffusive boundary layer at the same time as compressing its thickness (Lazier and Mann, 1989). The distribution of a nutrient solute next to the cell is modified with respect to Eq. (4.1) by the advection owing to the hydrody-namic flow velocity, u:

The advection-diffusion equation is not easily soluble. The approach of Riebesell and Wolf-Gladrow (2002) was to rewrite the problem in dimensionless Navier-Stokes terms, using particle Reynolds (Section 2.3.4 and Eq. 2.13), Péclet and Sherwood numbers. The Péclet number (Pe) compares the momentum of a moving particle to diffusive transport. For a phytoplankton cell whose movement satisfies the condition of nonturbulent, laminar flow (Re < 0.1; Section 2.4.1),

where us is the intrinsic velocity of a spherical cell of diameter d. In the present context, where the particle is introduced into the flow field u i

Figure 4.2

The Sherwood number as a function of the Peclet numbers for steady, uniform flow past an algal cell (solid line) and turbulent shear (dashed line). Figure redrawn with permission from an original in Riebesell and Wolf-Gladrow (2002).

### Figure 4.2

The Sherwood number as a function of the Peclet numbers for steady, uniform flow past an algal cell (solid line) and turbulent shear (dashed line). Figure redrawn with permission from an original in Riebesell and Wolf-Gladrow (2002).

Sh = 2 + 2(1 + 2Pe)a influence of the turbulent shear rate (Karp-Boss et al., 1996). They showed:

when the Peclet number was derived from:

(x, y, z, Re), the Peclet number also expresses the ratio of the scales of advective (u AN C) and diffusive solute transport (m(AN)2C). The Sherwood number (Sh) is the ratio between the total flux of a nutrient solute arriving at the surface of a cell in motion and the wholly diffusive flux. Riebesell and Wolf-Gladrow (2002) showed that, for particles in motion with very low Reynolds numbers, Sherwood numbers are non-linearly related to Peclet numbers but, in the range 0.01 < Pe < 10 (which embraces the sinking motions of algae from Chlorella to Stephanodiscus; see Section 2.4.1), the relationship is adequately described by:

The relationship (sketched in Fig. 4.2) shows that, for small cells embedded deeply in the turbulence spectrum (Pe < 1), the benefit in terms of nutrient supply is marginal (Sh ~1). For larger units and motile forms generating Re > 0.001 and Pe > 1, the dependence on turbulence for the delivery of nutrients becomes increasingly significant (Sh > 1).

The conclusion fits comfortably with the demonstration of a direct relationship of algal size to Sherwood scaling, mediated through the in which equation, d is the diameter of a spherical cell, E is the turbulent dissipation rate, in m2 s-3, v is the kinematic viscosity of the water (in m2 s-1) and m is the coefficient of molecular diffusion of the solute (for further details, see Section 2.3.3).

A further deduction and reinterpretation of the comment of Walsby and Reynolds (1980; see above) is that relative motion does not in itself overcome rarefied nutrient resources. However, chronic and extensive resource deficiencies must exact a greater dependence of larger algae on turbulence to fulfil their absolute resources requirements to sustain growth requirements than they do of smaller ones. Conversely, smaller cells are less dependent upon turbulent diffusivity to deliver their nutrient requirements than are larger ones.

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