Chlorophyll can absorb and utilize photons of near ultraviolet and visible wavelengths; therefore, the relevant measure of the underwater light field is the PAR, which is a broadband quantity integrated from the wavelength of 350 nm to that of 700 nm (Morel, 1978). In Case 1 water, the concentration of phytoplankton is high compared with nonbiogenic particles (Morel and Prieur, 1977). Once the light energy is absorbed by the chlorophyll pigments of phytoplankton or aquatic macrophytes, it is used to generate useful chemical energy in the form of carbohydrate. Since the aquatic plants also contain lipids and nucleic acids, none of which conform closely in their overall composition to glucose, the biosynthesis of these substances requires additional photosynthetically generated reducing power and chemical energy, and so requires additional light quanta per carbon dioxide incorporated (Kirk, 1994). This is why the entropy argument for living phytoplankton is essential in terms of macroscopic heat energy transfer processes between the living system and its environment (Peixoto and Oort, 1992; Nakamoto et al., 2002).

We propose a simple model to provide the essence of Nakamoto et al. (2001) numerical model experiments: The model must contain the radiative transfer mechanism to generate geostrophic currents as seen in the OGCM experiment (Nakamoto et al., 2001). In order to capture the chlorophyll pigment-generated geostrophic current, annually averaged horizontal distribution of chlorophyll pigment concentration is expressed by the following function:

where the coordinate frame designated by x, y, z is assigned to eastward, northward, and upward, respectively. Bmax is the maximum chlorophyll pigment concentration and Lx and Ly are the zonal and meridional length scales for chlorophyll pigment distribution, respectively. The above function B(x, y) (Fig. 1) resembles a typical chlorophyll pigment distribution observed in remotely sensed sea-viewing wide field-of-view sensor (SeaWiFS) data, or a CZCS data (Feldman, 1989) used in the numerical experiments by Nakamoto et al. (2000, 2001).

Given the above surface chlorophyll pigment concentration distribution, how can an attenuation depth of the solar radiation be estimated in the ocean? In general, complex OGCMs employ numerous adjustable parameters that are not directly observable. These models are not falsifiable so that the opportunity to learn from a wrong prediction is short circuited.

The approach we employ here is based on the fact that phytoplankton increase the rapidity of attenuation of light with depth, and in productive waters may do so to such an extent that they become a significant factor limiting their own population growth. The contribution of phytoplankton to the vertical attenuation coefficient of PAR must therefore be taken into

account in any consideration of the extent to which light availability limits primary production in the aquatic biosphere (Kirk, 1994).

Prieur and Sathyendranath (1981) developed a pioneering bio-optical formalism for the spectral absorption coefficient. Their formalism was statistically derived from 90 sets of spectral absorption data taken in various Case 1 waters, and included absorption by phytoplankton pigments, by nonpigmented organic particles derived from deceased phytoplankton, and yellow matter derived from decayed phytoplankton. The contribution of phytoplankton to the total absorption was parameterized in terms of the chlorophyll concentration. The essence of the Prieur and Sathyendranath (1981) formalism is contained in a more recent and simpler variant given by Morel (1991). Statistically derived absorption coefficient values for pure water and for nondimensional chlorophyll-specific absorption coefficient are listed in Table 3.7 of Mobley (1994).

For the averaged vertical attenuation coefficient throughout the eupho-tic zone, it is assumed that the total vertical attenuation coefficient at a given depth can be regarded as the sum of a set of partial attenuation coefficients, each corresponding to a different component of medium. We make use of a further assumption, namely that the contribution of any component of the medium to the averaged attenuation coefficient is linearly related to the concentration of that components (Kirk, 1994).

A diffuse attenuation coefficient k(x, y) that varies with chlorophyll concentration is thus expressed as a sum of components related to pure water and that due to chlorophyll biomass (Apel, 1987; Kirk, 1994; Mobley, 1994):

where k(x, y) is the biomass dependent vertically average diffuse attenuation coefficient, kw is the attenuation coefficient in pure water, kc is the specific attenuation coefficient due to unit chlorophyll mass, I0 — I (x, y, 0) and I(x, y, 0) are the irradiance just below the surface, respectively.

It is noted that the above formalism is not exactly the same as that used in the numerical OGCM experiment conduced by Nakamoto et al. (2001); however, the physics employed in the above two equations (2) and (3) is simple enough to extract the essence of the radiative transfer mechanism perturbed by phytoplankton concentration, i.e., the solar energy penetration depth becomes smaller with increasing biomass concentration. This means that oceanic phytoplankton traps solar irradiance toward the ocean surface. The implication of the above formalism is that the upper ocean with phyto-plankton is optically less transparent than without phytoplankton. The above formalism is also consistent with recent measurements of the absorption spectrum for solar irradiance in the ocean, which shows significant diminishing of light energy under the existence of phytoplankton (Sasaki et al., 2001): Sasaki's observation of underwater solar radiation represents that living phytoplankton releases more heat energy toward the surrounding waters than the detritus of dead organism does.

In mathematical terms, phytoplankton in the upper part of the ocean localizes the heat source of the heat equation at habitat depth of phyto-plankton because living phytoplankton absorb visible energy and convert a small portion of it into glucose during photosynthesis, while releasing the rest of the energy toward the external environment as heat (Nakamoto et al., 2002). These are the physics exhibited in OGCM experiments with the effect of living phytoplankton, i.e., the life activity expressed in radiative energy budget in the upper ocean environment.

In order to describe the heating mechanism of the seawater through radiative transfer process perturbed by living phytoplankton, we employ the following anomalous temperature equation with phytoplankton-induced anomalous heat source:

@t pcp @z where 6(x, y, z, t) is the anomalous temperature of the water at a fixed point due to the irradiance perturbation caused by living phytoplankton, p is seawater density, and cp is heat capacity of seawater.

Integrating the above heat equation with respect to time leads,

Note that the above solution is valid for a vertically averaged attenuation coefficient, k(x, y), which is a function of chlorophyll concentration distribution specified in the equation (1) in this study. As the consequence of the above assumed heat equation, Equation (4), the anomalous temperature increases linearly in time: the temperature perturbation due to phytoplankton biomass is not large enough to induce heat advection influencing the mean heat equation. This model is simple enough to serve as "Fermi approach'' as suggested by Harte (2002): The analytical solution for the anomalous temperature rise due to living phytoplankton can be verified or rejected by careful laboratory measurements as well as numerical model experiments. Recently, Ohta and Nakagawa (2003) conducted laboratory experiment to measure photosynthesis efficiency with phytoplankton. They reported that the measured water temperature increased linearly in time, and that 82% of the incoming solar energy was released to the surrounding waters as heat.

Our approach is thus based on the observational fact that most of the solar energy in the visible spectral range is trapped by and released from living phytoplankton to the surrounding waters in the ocean. Thus, less solar radiation penetrates to deeper waters when a high density of phytoplankton exists in the ocean.

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