Limnol. Oceanogr., 44(7), 1999, 1781–1787

A turbulent diffusion model shows that there are two different mechanisms for the development of phytoplankton blooms. One of these mechanisms works in well-mixed environments and corresponds to the classical critical depth theory. The other mechanism is based on the rate of turbulent mixing. If turbulent mixing is less than a critical turbulence, phytoplankton growth rates exceed the vertical mixing rates, and a bloom develops irrespective of the depth of the upper water layer. These results demonstrate that phytoplankton blooms can develop in the absence of vertical water-column stratification. A common paradigm in aquatic ecology and oceanography is that phytoplankton blooms can develop only if the upper mixed-water layer is shallower than some critical depth (e.g., Sverdrup 1953; Smetacek and Passow 1990; Nelson and Smith 1991; Platt et al. 1991; Kirk 1994; Mann and Lazier 1996; Obata et al. 1996; Falkowski and Raven 1997). The idea is that once the upper water column is sufficiently shallow, the light conditions will be favorable for photosynthesis, so that depth-integrated photosynthesis exceeds the depth-integrated losses of the phytoplankton. Hence, the phytoplankton population increases. This critical depth concept was recently challenged by Townsend et al. (1992) and Eilertsen (1993). Townsend et al. (1992) reported that the phytoplankton spring bloom in the Gulf of Maine preceded the onset of vertical water-column stratification. Similarly, Eilertsen (1993) observed that in several Norwegian fjords with water depths of .200 m, phytoplankton blooms developed in the absence of vertical water-column stratification throughout. These observations were criticized by Platt et al. (1994) and Mann and Lazier (1996), basically for two reasons. First, it was not clear whether the observations of Townsend et al. and Eilertsen indeed violated the critical depth concept. It could be that the mixed-layer depth was misjudged, that the phytoplankton losses were unusually low, or that the production rates were unexpectedly high. Second, if the observations of Townsend et al. and Eilertsen did violate the critical depth concept, a proper theory to explain their observations was lacking. Here, we develop a new theory that provides a theoretical underpinning for the observations of Townsend et al. (1992) and Eilertsen (1993), as well as for many other observations. The mathematical models that underlie the critical depth concept assume that the phytoplankton population is homogeneously distributed over depth (Sverdrup 1953; Platt et al. 1991). We relax this crucial assumption, as it may not hold in many aquatic environments. Our strategy will be to let the dynamics of phytoplankton growth and turbulent mixing determine the phytoplankton distribution over depth. The analysis reveals that there are two different mechanisms for the development of phytoplankton blooms. A bloom can develop if the upper mixed-water layer is shallower than some critical depth or if turbulent mixing rates are less than some critical turbulence. The model—We consider a growth-diffusion model (Okubo 1980) in which the population dynamics of phytoplankton are governed by light-limited growth, local phytoplankton losses, and local transport of the phytoplankton by turbulent diffusion. Following Sverdrup (1953), the model is kept as simple as possible. Thus, the model neglects many additional complexities like nutrient limitation, photoinhibition, buoyancy regulation, and sinking, in order to focus on the most fundamental aspects of phytoplankton bloom development. Our diffusion equation forms the core of various more complicated models used in oceanography and ecosystems research (Cloern 1991; Slagstad and Støle-Hansen 1991; Koseff et al. 1993; Donaghay and Osborn 1997; Lucas et al. 1998) and has attracted considerable mathematical treatment (Shigesada and Okubo 1981; Ishii and Takagi 1982; Totaro 1989), but its features have never been fully analyzed. Phytoplankton dynamics: Let s denote the depth within the water column, where s runs from zero (top) to z (bottom). And let v(s, t) denote the phytoplankton population density (cells per unit volume) at depth s and time t. The changes in population density can be described by a partial differential equation. 2 ]v ] v (s, t) 5 [p(I(s, t)) 2 ,]v(s, t) 1 D (s, t). (1) 2 ]t ]s Here, p(I(s, t)) is the specific production rate as an increasing function of light intensity I(s, t), with p(0) 5 0; , is the specific loss rate, which incorporates all sources of phytoplankton loss; and D is the turbulent diffusion coefficient (also known as ‘‘vertical eddy diffusivity’’). For notational convenience, we introduce the net production rate, g(I) 5 p(I) 2 ,. We assume that the water column is closed, with no phytoplankton cells entering or leaving the column at the top or the bottom. This gives the following boundary conditions: ]v ]v (0, t) 5 (z, t) 5 0. (2) ]s ]s It is useful to keep track not only of the population density