Modelling of autumn plankton bloom dynamics

Transcription

1 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 Modelling of autumn plankton bloom dynamics HELEN S. FINDLAY,, ANDREW YOOL *, MARIANNA NODALE 3 AND JONATHAN W. PITCHFORD BIOLOGY DEPARTMENT, UNIVERSITY OF YORK, HESLINGTON, YORK YO 5DD, UK, NATIONAL OCEANOGRAPHY CENTRE, SOUTHAMPTON, EUROPEAN WAY, SOUTHAMPTON SO4 3ZH, UK AND 3 THE MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD 4-9 ST GILES, OXFORD OX3LB, UK *CORRESPONDING AUTHOR: axy@noc.soton.ac.uk Received August 5, 5; accepted in principle October 5, 5; accepted for publication December 6, 5; published online December 5, 5 Communicating editor: K. J. Flynn A simple system of parametrically forced ordinary differential equations is used to model autumn phytoplankton blooms in temperate oceans by a mechanism involving deepening of the upper mixed layer. Blooms are triggered provided the increase in nutrients in the mixed layer is rapid within the first few days of deepening and provided light-limited phytoplankton growth rate is relatively high. Blooms exist as transient trajectories between quasi-equilibrium states, rather than as bifurcations of steady states; therefore very gradual deepening cannot trigger blooms. Very rapid deepening also prevents blooms due to the deleterious effect on phytoplankton growth rate. The mechanisms identified by this simple model are vindicated by considering alternative grazing and deepening regimes and by comparison with a more ecologically complex model (Fasham, 993, in The Global Carbon Cycle, Springer-Verlag). Modelled estimates of primary productivity from both the simple model and the complex model parameterized for Ocean Weather Station India are around.5 g C m day during the autumn bloom, therefore comprising a significant component of annual production in temperate areas. INTRODUCTION Phytoplankton populations in most temperate and subpolar oceanic regions undergo strong seasonal cycles, with prominent blooms occurring particularly in the spring but also to a lesser extent in the autumn. Understanding the mechanisms causing such blooms, and quantifying their associated primary production (and the fate of this primary production), is of fundamental importance in assessing global carbon budgets and modelling scenarios of global climate change. Spring blooms have been studied for many decades (Sverdrup, 953; Evans and Parslow, 985; Lochte et al., 993; Truscott, 995; Bury et al., ; Dutkiewicz et al., ), and a good theoretical understanding has been achieved. It is generally accepted that they are caused by a combination of vertical stratification of the water column and increasing solar radiation after the replenishment of nutrients by convectional mixing over the winter months. Once initiated, spring blooms are controlled by increasing zooplankton levels and decreasing nutrient concentration within the upper mixed layer (Fasham, 993). By contrast, autumn blooms are less well studied, although their occurrence is widely documented (Raymont, 963; Hardy, 97; Ross, 977; Steele and Henderson, 978; Lalli and Parsons, 997; Herring, ). The classic hypothesis suggests that phytoplankton blooms in autumn are caused by the following mechanism. Increased vertical mixing and the breakdown of stratification in autumn causes an influx of nutrients into the upper layers of the ocean. Meanwhile, light levels remain high enough so as not to limit photosynthesis. Grazing by zooplankton will also be affected by increased vertical mixing this acts to dilute the grazer population and may further promote phytoplankton growth. This combination of factors in principle allows phytoplankton populations to increase their growth rate and initiate a bloom. However, this increased vertical mixing also has negative consequences for phytoplankton since it mixes them to greater depth and thus decreases the light levels they experience. Indeed, if phytoplankton are drawn below Sverdrup s critical depth (Sverdrup, 953), then, as a whole, population growth is impossible. doi:.93/plankt/fbi4, available online at Ó The Author 5. Published by Oxford University Press. All rights reserved. For Permissions, please journals.permissions@oxfordjournals.org Downloaded from on 9 May 8

2 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 Thus, the rate and timing of the breakdown in stratification, modelled here as a deepening in the mixed layer, is predicted to be crucial in controlling the magnitude and duration of an autumn bloom. Autumn blooms have been documented in the literature (Zingone et al., 995; Edwards et al., ; Aiken et al., 4), and the seasonal cycles have been quantified (i.e. phytoplankton concentration and chlorophyll levels measured), but no mechanism has been analysed with the important factors being quantified. This investigation tests the autumn bloom hypothesis by formulating a simple mixed layer ecosystem model, which is parametrically forced by deepening of the upper mixed layer. The mechanisms elucidated in this simple model are then examined in the context of the more complicated ecosystem model of Fasham (Fasham, 993) to determine their general applicability. METHODS Asimplemodeloftheopenoceanecosystemisused to test the theory of autumn blooms (Fig. ). The model is formulated as a bilayer system consisting of an upper, actively mixed layer (down to the seasonal thermocline) containing phytoplankton and a limited M Light Mortality loss Uptake Deepening of mixed layer and diffusion N Upper mixed layer Lower layer amount of nutrients (nitrate here), and a lower layer containing no phytoplankton (because of assumed low light) but an excess of nutrients, N (Evans and Parslow, 985). The model uses three coupled ordinary differential equations (ODEs) showing the change in mixed layer depth (MLD) M, nutrient concentration N, and phytoplankton concentration P in the upper mixed layer over time t measured in days (these variables are summarized in Table I). dn dt dp dt ¼ ¼ N k þ N N m þ hðtþ rðmþp gp k þ N M P ðþ rðmþp þ m þ hðtþ M ðn N Þ ðþ dm dt ¼ hðtþ ð3þ The model is adapted from the classic model of Evans and Parslow (Evans and Parslow, 985), which aimed to model annual plankton cycles and included a fourth equation describing herbivore dynamics. The herbivore equation is here simplified to a constant, g, reflecting phytoplankton mortality. If this mortality is assumed to be primarily due to herbivory, then it may be more appropriate to treat g as a dynamic quantity as described in detail below. Seasonality in day length is regarded as negligible over the time scale of an autumn bloom. Stochastic fluctuations, e.g. in cloud cover, were not incorporated for the sake of model tractability; this is reasonable in relation to short time scale (hours) fluctuations about a seasonal mean (Truscott, 995; James et al., 3) but may be unrealistic for variability over longer time scales (days). In short, equations () (3) represent a temporal snapshot of the annual cycle described by Evans and Parslow (Evans and Parslow, 985). Evans and Parslow (Evans and Parslow, 985) describe an asymmetry in the response of the ecosystem state variables to positive and negative changes in the MLD the simple ecosystem model in this study focuses solely on the deepening of the mixed layers over a relatively short, typically -day, time period. Table I: Model state variables Variable Symbol Unit Fig.. Schematic diagram of the model. Nutrients (N) and phytoplankton (P) are active in the upper mixed layer, which has depth (M); lower layer nutrients (N ) are introduced into the upper layer by diffusion and mixing. Phytoplankton can be lost from the upper mixed layer by diffusion and by dilution when the mixed layer is deepening. Mixed layer depth M m Nutrient concentration N mmol m 3 Phytoplankton concentration P mmol m 3 Downloaded from on 9 May 8

3 H. S. FINDLAY ETAL. j MODELLING AUTUMN BLOOMS The asymmetry in Evans and Parslow s study was generated by the annual variation in deepening and shallowing of the mixed layer, which is not relevant here. Phytoplankton [equation ()] Uptake by phytoplankton is modelled with a simple Michaelis Menten hyperbola N/(k+N) multiplied by the light-limited specific growth rate and the phytoplankton concentration. It is assumed that nutrient is limited by a saturation maximum and is immediately converted to growth (i.e. there is no modelling of intracellular pools of nutrient). Light-limited specific growth rate is a function of depth, r(m)=a/m (where a is a constant, M is depth). This approach is taken from Boushaba and Pascual (Boushaba and Pascual, 5), who derive this simplified functional form for photosynthesis within the mixed layer by implicitly considering irradiance, plankton self-shading and attenuation by water itself. This functional form is simpler than that used by Fasham (Fasham, 993), but the general agreement with simulations using this more complex model (see below) further justifies its use. Phytoplankton mortality occurs at a constant per capita rate, g. This simplifying assumption is used in other models (Fasham et al., 99) but is inappropriate where mortality is dominated by grazing. An alternative assumption, where a fixed grazer population is able to remain with the mixed layer, is explored below. Within the context of this simple model, the parameter g is chosen so as to produce realistic unforced equilibria for the (N, P) system; the fact that more complicated models involving dynamic and nonlinear grazing effects show the same bloom behaviour as the simple model justifies this expedient choice. Phytoplankton concentrations are also affected by cross-thermocline mixing, m, and by entrainment of water, h(t), from the lower layer. In both instances, because it is assumed that there are no phytoplankton in the lower layer, these processes decrease upper layer concentrations. Nutrient [equation ()] Nutrient concentrations are decreased due to uptake by phytoplankton, following the Michaelis Menten relationship described above. Nutrient addition into the mixed layer is first modelled as a diffusive process, such that nutrient concentration in the upper mixed layer, N, tends toward that in the lower layer, N, at a rate (m/m)(n N ). Nutrients can also be added to the mixed layer directly through entrainment due to deepening. Assuming that the upper layer is always perfectly mixed, this entrainment occurs at a rate (h(t)/m)(n N ). Note that phytoplankton losses are not returned as nutrient in this model. They are assumed either to enter (unmodelled) higher trophic levels, to quickly sink out of the modelled system as detrital particles, or to be regenerated in a time frame beyond that simulated. Forcing [equation (3)] The dynamic forcing in this model is represented by the deepening of the mixed layer with deepening rate h(t). For simplicity, h(t) can be assumed to be constant (y) for the -day period that the autumn bloom is being simulated. In most simulations presented here, depth is increased from 5 to 5 m in days, giving a deepening rate of.5 m day. This is a plausible autumn deepening regime according to Ryabchenko et al. (Ryabchenko et al., 997), who suggest it is these abrupt changes in MLD that cause the oscillatory phytoplankton behaviour found in their model and observed data. The effects of temporally varying deepening regimes, simulating storms, are described below. Calculation Numerical integration of the simple model [equations () (3)], using an explicit fourth-order Runge Kutta algorithm with fixed time step (Press et al., 99), allows a large range of parameter values to be investigated. Table II summarizes default parameter values used, based on those of Evans and Parslow (Evans and Parslow, 985), with the implicit phytoplankton mortality parameter Table II: Model parameters Parameters Symbol Unit Value Source Deep nutrient N mmol m 3 Evans and Parslow (985) Uptake half saturation k mmol m 3.5 Evans and Parslow (985) Diffusion rate m mday 3 Evans and Parslow (985) Light-limited specific growth rate r (M) day (max) Evans and Parslow (985) Phytoplankton mortality rate g day.8 This article Downloaded from on 9 May 8

4 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 calculated so as to produce a realistic equilibrium phytoplankton population based on Batten et al. (Batten et al., 3) before the deepening of the mixed layer. Comparison with a seven-component ecosystem model The Fasham (Fasham, 993) plankton ecosystem model consists of seven state variables linked together to form a more complete description of the ocean s nitrogen cycle than the model used here. Concentrations of phytoplankton, zooplankton, bacteria, detritus, nitrate, ammonium and dissolved organic nitrogen are simulated by forcing changes in MLD and light over an annual cycle (for a given latitude and longitude location). An implementation of this model by Yool (Yool, 998) (hereafter referred to as the YFDM model) was adapted here to reproduce the MLD deepening used in the simpler two-component model (i.e. the maximum depth reaches 5 m in winter, 5 m in summer, with shallowing occurring in spring and deepening occurring in late summer/early autumn). Phase plane analysis and equilibria of the simple model It is useful briefly to consider the model [equations () (3)] in the absence of the forcing induced by the deepening of the mixed layer, that is with h(t) set to zero. This simpler system of two ODEs, modelling nutrient and phytoplankton dynamics within a mixed layer of constant depth, can be shown to have two physically meaningful equilibria provided and r l> N > lk r l ; where l=g+dand d = m/m represents the per capita loss rate for phytoplankton. These criteria are readily satisfied by the parameter values considered here (Table II). If they are not satisfied then one is essentially considering a species or assemblage of phytoplankton which cannot persist, even at minute levels, in the given nutrient regime; such situations are not considered here. One equilibrium is at (N, ) and can be shown always to be a (mathematically unstable) saddle point. The second, more interesting, equilibrium may be denoted ðn ; P Þ¼ lk r l ; dn l dk r l and this equilibrium can be shown always to be linearly stable, according to the Routh Hurwitz criteria or simple phase plane analysis (Murray, ). Thus, in the absence of forcing, the system will simply equilibrate to (N *, P *) from any physically realistic initial condition. Simple algebra reveals that P * decreases as M (depth, regarded as a fixed parameter) increases but that the stability of (N *, P *) does not change. At first glance, this suggests that blooms can never be produced by a deepening of the mixed layer (i.e. a dynamic increase in M); as the mixed layer deepens the system is attracted towards a smaller equilibrium value of P. This is, however, misleading; the trajectory followed by the solution to the system can involve the P population increasing significantly before settling down to its new equilibrium value. This will be discussed in more detail, but here it is sufficient to note that autumn blooms, as described here, are a purely transient phenomenon; there is no bifurcation of steady states. RESULTS The results of simulations using the basic parameter values (Table II) are shown in Fig. a in comparison with the equilibrium values [the (N *, P *) values of section.4] calculated at each time step. The system was allowed to reach its equilibrium value at a constant MLD of 5 m. The mixed layer was then deepened at a rate of.5 m day, reaching a maximum depth of 5 m after days. In this and subsequent graphs, time in days is measured from the day on which the mixed layer begins to deepen. The results show that a phytoplankton bloom is produced, despite the fact that the equilibrium phytoplankton value decreases monotonically during the forcing period. This is because, despite there being no bifurcation of steady states, the trajectory of the dynamical system is temporarily attracted towards a quasiequilibrium state (Truscott, 995) with increased phytoplankton during the forcing period. The simulation in Fig. b uses the same deepening regime, but with a different nutrient equilibrium concentration, phytoplankton equilibrium concentration and mortality constant [(N *, P *, g) = (,.8,.)]. This illustrates that autumn blooms are not a necessary consequence of the model. Here, under conditions where phytoplankton mortality is high or nutrients are not depleted during the summer, the model system will not show a bloom in response to autumn deepening of the mixed layer. A more thorough investigation of parameter space elucidates which conditions are sufficient to trigger Downloaded from on 9 May 8

5 H. S. FINDLAY ETAL. j MODELLING AUTUMN BLOOMS a Concentration (mmol N m 3 ) Concentration (mmol N m 3 ) b Concentration (mmol N m 3 ) Concentration (mmol N m 3 ) Fig.. Modelled nutrient, N (top graph solid line) and phytoplankton, P (bottom graph solid line) values over a -day period of deepening (deepening rate.5 m day, initial depth 5m) and then a 5-day recovery period (depth maintained at 5m), shown together with equilibrium values (N*, P*) for nutrient (dots) and phytoplankton (dots) at each depth. Panel a has parameter values taken from Table II, illustrating conditions sufficient for a bloom [(N*, P*, g) = (.4,.3,.8)] and panel b demonstrates parameter values which, under identical forcing, do not trigger a bloom [(N*, P*, g) = (,.8,.)]. 3 Downloaded from on 9 May 8

6 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 autumn blooms. Investigations were carried out for the main variables, including varying the initial condition (N *, P *), to map out the range of conditions that would cause a temporary bloom in phytoplankton. This then allowed analysis of specific parameters to determine the model s sensitivity to each. The important results from this numerical investigation are summarized below and in Fig. 3. (i) Increasing N enhances the bloom, as might be expected because of increased resource availability (e.g. N = mmol m 3, phytoplankton concentration increases by 5%; N = mmol m 3, phytoplankton concentration increases by 65%) (Fig. 3a). This change is not pro rata since phytoplankton become more limited by light than nutrient availability at larger values at N. (ii) As r(m) is decreased, the magnitude of the bloom is also decreased (Fig. 3b). However, at these lower levels of light-limited specific growth rate, the phytoplankton population is sustained at a lower concentration for a longer period. Therefore, when averaged over the -day forcing period, there is still a significant increase in phytoplankton concentration above its equilibrium. (iii) As Fig. 3c shows, the model is sensitive to changes in the mortality parameter, g. Under decreased a r (M ) = r (M ) = r (M ) =.6 mortality, phytoplankton more easily escape temporarily onto a bloom trajectory. (iv) There is an optimum for deepening rate, y, which allows a maximum bloom to be produced with each set of parameters. As y tends to infinity, the maximum phytoplankton concentration decreases due to the rapid decrease in light-limited specific growth rate and the direct effect of dilution. As y tends to zero, the phytoplankton population is able to follow the equilibrium concentration for each time step, and no bloom occurs (Fig. 4). Primary production can be calculated by integrating the uptake of carbon over the MLD (assuming a fixed Redfield C:N ratio, consistent with no nutrient limitation). Figure 5 illustrates the primary production produced from the autumn bloom depicted in Fig. (N *, P *, g) = (.4,.3,.8) in comparison to primary production produced at the corresponding equilibrium concentrations and primary production when the growth rate r(m ) is lowered. Interestingly, when computed over the -day deepening period, total primary production is found to be quite insensitive to the absolute value of phytoplankton growth rate: setting a = 5 m day gives total primary production over the days of 575 mg C m, while a = m day gives production of 5968 mg C m. This result arises primarily from the differing patterns of production across the bloom period, but also from the initial conditions (N *, P *) going into the bloom (which are products of the summer equilibrium). Phytoplankton populations in our model with a high growth rate characteristically undergo a rapid increase followed by a rapid decline. In contrast, while P concentration (mmol m 3 ) b c N = 3 N = N = g =.5 g =.8 g = P concentration (mmol m 3 ) θ = θ = 5 θ =.5 Fig. 3. Phytoplankton concentrations plotted against time with changes in the following parameters: (a) light-limited specific growth rate, r(m) (day ); (b) deep layer nutrient concentration, N (mmol m 3 ); (c) mortality parameter, g (day ). Except for the parameter being varied in each graph, all other parameter values and the deepening regime are identical to those of the simulations in Fig. a Fig. 4. Phytoplankton concentrations plotted against time with a range of deepening rates, y (m. day ). There is an optimum rate between.5 and 5 m day that produces the greatest bloom. 4 Downloaded from on 9 May 8

7 H. S. FINDLAY ETAL. j MODELLING AUTUMN BLOOMS Primary production (mg C m ) Concentration (mmol N m 3 ) Concentration (mmol N m 3 ) Fig. 5. Primary production associated with the bloom of Fig. a [(N*, P*, g) = (.4,.3,.8)] (solid line); the bloom where r(m) =.8 day (instead of day ) (dashed line) and primary production for the corresponding equilibrium concentrations (dotted line). populations with a lower growth rate bloom more slowly, they reach a peak concentration nearly as great but also decline more slowly. Overall, this more gradual bloom evolution has similar production, despite phytoplankton concentrations seemingly experiencing a smaller bloom. Sensitivity analysis Alternative grazing functions The assumption of a constant per capita phytoplankton mortality rate is not appropriate to situations where mortality is primarily driven by grazing; an alternative formulation is more appropriate. Assuming a constant zooplankton grazer population, ĝ, over the short duration of a bloom leads to a new equation dp dt ¼ N k þ N rðmþp ^gp m þ hðtþ M M P ð4þ where ĝ/m parameterizes the reduced concentration of grazers as the mixed layer deepens [note that, neglecting diffusion, this is the same process of decreasing concentration due to entrainment from the lower layer as the final term in the original phytoplankton equation ()]. It should be noted that this assumption of constant grazer population is better suited to meso-, rather than, microzooplankton. The mathematical analysis for this revised system reveals that again the deepening of the mixed layer cannot cause any bifurcation of steady states; in this instance the equilibrium (N*, P*) is invariant to changes in M. Once more, equilibrium-based analysis would suggest that deepening of the mixing depth cannot induce blooms. Figure 6 shows that this is not the case; Fig. 6. Modelled nutrient (top graph solid line) and phytoplankton (bottom graph solid line) values with new grazing function (equation 4). Forcing occurs over a -day period of deepening (deepening rate.5 m day, initial depth 5m) and then a 5-day recovery period (depth maintained at 5m), shown together with equilibrium values (N*, P*) for nutrient (dots) and phytoplankton (dots) at each depth. with the same deepening rate as in Fig. a a phytoplankton bloom is triggered although the equilibrium P and N values remain unchanged. The general picture is the same as in Fig.. The only major difference lies in the response of the model to deepening rate y; there is no intermediate deepening rate associated with maximum bloom size, rather an increasing deepening rate always increases bloom size (see Discussion). Comparison with a seven-component ecosystem model A potentially serious caveat to the results expressed so far is that the study makes use of a simple plankton model that excludes much of the complexity of the ocean ecosystem. For instance, the model includes no grazers and does not directly consider the remineralization of organic material. To address this, the behaviour of a model (Fasham, 993) that includes many of the processes missing here is additionally examined. Simulations were carried out to assess the potential occurrence of autumn blooms. Using the model s default parameters for Ocean Weather Station India (located in the Atlantic Ocean, 598N 98W), a small autumn bloom can be triggered as soon as the MLD begins to deepen. The forcing can be controlled to show precisely that it is this method of MLD entrainment that causes the increase in upper layer nutrients (Fig. 7). By using a seasonally constant MLD, the YFDM model supports the simple model s demonstration of the importance of mixing (Fig. 8). In this latter example, while a small spring bloom is triggered by rising irradiance, the lack 5 Downloaded from on 9 May 8

8 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 (W m ) 4 a Irradiance (W m ) 4 a Irradiance (m) (mmol m 3 ) (g C m d ) 4 6 (mmol m 3 ).5 4 b c d Mixed layer depth Phytoplankton (solid) and zooplankton (dashed) Nitrate Primary production (solid) and export (dashed).6 e Fig. 7. YFDM model output showing (a) seasonal PAR; (b) mixed layer depth (MLD) shallowing from 5 to 5 m then deepening to 5 m again towards the end of summer. Shallowing and deepening each take 3 days; (c) phytoplankton and zooplankton increase in spring, decline over summer then bloom again in autumn; (d) the upper layer nutrient concentration alters with MLD, while the lower layer nutrient concentration remains constant at mmol m 3 ; and (e) primary production and export are greatest during periods of high phytoplankton concentrations. If the deepening rate is increased to.5 m day, the bloom increases in magnitude to >.5 mmol m 3. Deepening to greater depths does not increase bloom as long as the rate is the same. Growth rate (V p ) =.3 day. of a nutrient pulse in autumn prevents any phytoplankton response. Further to these baseline simulations, Fig. 9 illustrates the YFDM model response to parameter manipulations that approximate to those performed with the two-component model. The YFDM model parameters chosen are: g max, the maximum specific zooplankton grazing rate; N, the subthermocline nitrate concentration; k w, the attenuation coefficient of seawater; and y, the autumn deepening rate. In each instance, values of these parameters either side of the baseline value were used to explore the response of the model (see the figure caption for further details). The model responses closely follow those expected from the analysis of the two-component model. Increased zooplankton grazing decreases the autumn bloom size. Increased subthermocline nitrate increases the autumn nutrient pulse and increases the bloom size. (m) (mmol m 3 ) (mmol m 3 ) (g C m d ) b c d e Mixed layer depth Phytoplankton (solid) and zooplankton (dashed) Nitrate Primary production (solid) and export (dashed) Fig. 8. As Fig. 7, but with no change in MLD over the annual cycle. Increased seawater attenuation [approximating to decreased r(m)] decreases available light and stunts the autumn bloom. Furthermore, an increased rate of deepening more rapidly entrains nutrient and increases the bloom size (although excessive deepening decreases the bloom through its effect on light availability). There is an interesting deviation from the expected results in one instance. In Fig. 9b, increasing N results in a suite of curves with consecutively greater autumn blooms until the most extreme case, however, where the bloom is slightly decreased in size. Examination of this simulation in more detail reveals that, although the autumnal nutrient pulse fuels production (i.e. the mixed layer enters the autumn denuded of nutrient), a higher standing stock of zooplankton (more than double that of the closest simulation) is able to quickly respond to the blooming phytoplankton and control their numbers. The effect of explicit zooplankton grazing has an additional secondary effect. In Fig. 9a, increasing zooplankton grazing rate decreases the size of the autumn bloom in the most extreme simulation, the bloom is completely absent. Examination of these simulations finds that, as well as directly preventing the bloom by grazing down the phytoplankton, the zooplankton also decrease nutrient consumption during the summer. In the simulations in Fig. 8a, this means that nutrients are not depleted during the summer, which decreases the 6 Downloaded from on 9 May 8

9 H. S. FINDLAY ETAL. j MODELLING AUTUMN BLOOMS P (mmol m 3 ) P (mmol m 3 ) P (mmol m 3 ) P (mmol m 3 ) a b c d Maximum specific grazing rate (g max ) Subthermocline nitrate (N ) Attenuation coefficient (k w ) Deepening rate (θ) Fig. 9. YFDM model output showing phytoplankton concentrations during autumn blooms across ranges of four model parameters: (a) maximum specific zooplankton grazing rate, g max ;(b) subthermocline nitrate concentration, N ;(c) attenuation coefficient of seawater, k w ;(d) autumn mixed layer deepening rate, y. In each case, the intermediate thickness line represents the default value for the parameter, with progressively thinner lines for lower values (/ and /4 times baseline respectively) and progressively thicker lines for high values (two and four times baseline respectively). impact of nutrient entrainment during autumn mixing. These results are analogous to those of Fasham (995) for the sub-arctic Pacific station Papa. In contrast to the Atlantic station India, at this station phytoplankton concentrations only rarely show a bloom (spring or autumn) and nutrient concentrations are never limiting. Zooplankton data and the analysis of Fasham (995) suggest that these conditions are caused by zooplankton control through much of the annual cycle (although the micronutrient iron is also implicated). Aside from these interesting deviations, the YFDM model supports the two-component model s description of autumn bloom dynamics. Variable deepening rates and storms The previous results all describe the mixed layer increasing in depth at a constant rate. This may be an oversimplification; it has been argued that introducing a sudden increase in depth into a gradual deepening in the autumn, induced by transient storm events, may be the best way to simulate the observed variability (interannual and spatial) (Steele, 958). Breakdown of vertical stratification at the end of summer is due to the predictable decline in solar heating and the less predictable impact of mixing by weather events or storms. To examine this, Fig. shows storm events simulated within the model [equations () (3)] by allowing a sudden increase in deepening rate to occur for day, in the middle of a slower deepening trend. To enable comparison, all simulations are constrained to have the mixed layer deepening from 5 to 5 m over days, with storms of increasing severity occurring on day 5. There is a decrease in bloom size as the magnitude of the storm event increases, due mainly to the sharp decrease in light availability with depth, which can entirely counteract the large increase in nutrient availability caused by the rapid deepening. Simulations of the YFDM model under identical deepening regimes exhibit similar, although less abrupt, changes in bloom behaviour (results not shown). These results emphasize the necessity of including, where possible, a high level of temporal resolution in models involving the interplay between physics and biology, particularly where accurate estimates of primary production are sought. DISCUSSION This study shows that autumn blooms can be produced by the mixing of nutrients into a deepening mixed layer and reveals the key conditions required to trigger a bloom; these include surface and deep nutrient concentrations, phytoplankton mortality and the rate of mixed layer deepening. With these factors in mind, it is useful to explore the general dynamics of the parametrically forced system [equations () (3)] to try to understand the conditions under which autumn blooms are triggered. Examining the quasi-equilibria to which the system is attracted during forcing is helpful. When dp/dt = and dn/dt = with y =, forcing does not occur and the system is at equilibrium (see Fig. ; the null clines are represented by dotted lines in this figure). Switching on the forcing (i.e. setting y to be a positive constant) creates, instantaneously, a new N null cline which appears above its previous location [solid line in Fig. ]. Whilst the forcing persists, the solution trajectory is temporarily attracted to a quasi-equilibrium value on this new null cline. When the forcing ceases, the system is attracted to its new equilibrium (N *, P *) value, at the intersection of the original (unforced) N null cline and the new P null cline appropriate to the final MLD. A bloom occurs only when the attraction towards the quasi-equilibrium state causes the P population to exceed its original value, as 7 Downloaded from on 9 May 8

10 JOURNAL OF PLANKTON RESEARCH j VOLUME 8 j NUMBER j PAGES 9 j 6 Concentration N (dashed), P(solid) (mmol N m 3 ) Depth (m) Fig.. Phytoplankton and nutrient concentrations over a forcing period with a storm event interrupting the standard deepening event. Storm events occur in order, from top graph to bottom graph: m day, 5 m day, 5 m day, m day and 5 m day. P 3 days (5m) kl/(r-l) N Forcing on (Forcing off) Fig.. Null clines and example solution trajectory for equations () (3) for (N*, P*) = (.4,.3). Null clines are shown: dp/dt = null cline, solid line; dn/dt = with y =, dotted line; dn/dt = with y =.5, solid line. Forcing is turned on at day (5 m) and can either be turned off again on day (5 m) thick line or deepening can continue to some extended depth or time (e.g. m dashed line). In both cases, a bloom is triggered; the solution trajectory begins at its unforced equilibrium (direction of movement indicated by arrows), but is attracted temporarily towards a quasiequilibrium state with increased N and P before returning to its new stable equilibrium value with a higher N but lower P value. illustrated in Fig.. If either N * is too low or P * is too large, any increase in the nutrients will cause the quasiequilibrium to be at a lower P level, and hence phytoplankton will always decrease. If N * is close to N,orP * is too small, then although the quasi-equilibrium will be above the equilibrium state, the difference between the two states is only slight and any increase in nutrients will be unable to stimulate a bloom. 8 Downloaded from on 9 May 8

11 H. S. FINDLAY ETAL. j MODELLING AUTUMN BLOOMS It is interesting to note the lack of symmetry between the spring bloom and the autumn bloom. As Figs 7 and 8 illustrate, spring blooms can be driven solely by changes in the light regime, even in the absence of MLD changes. By contrast, as Fig. illustrates, autumn bloom development is highly dependent on mixing events, and their evolution is controlled by the decrease of light with both depth and time. This helps to explain the interannual variability of autumn blooms by underscoring the importance of the initial (summer) conditions in addition to the autumnal mixing conditions. The initial conditions are intrinsically affected by phytoplankton zooplankton dynamics over the spring and summer. A future study, making use of this simple model and the understanding of autumn blooms brought about by this paper, might be to assess the importance of changes in climate, storm frequency and warming on an annual cycle. Several coupled atmosphere ocean models have shown global warming to be accompanied by an increase in vertical stratification (Houghton et al., ). Such a change would decrease the rate of vertical mixing in the ocean, with the primary effect of a reduction in nutrient mixing into the upper layers. As demonstrated in both the simple model and the YFDM model, such a decrease in mixing rate or timing will have a large impact on the magnitude and period of an autumn bloom. The results of systematically changing y in the simple model reveal an optimum rate that produces blooms of greatest magnitude (Fig. 4). Although there is no optimum y in the YFDM model, it does demonstrate (and echo) the effects of decreased mixing on primary production. On a longer time scale, the full effects of global warming are still unknown. Aside from its local effects on mixing, ocean warming may also alter the major circulation patterns, which in turn could affect large-scale nutrient distributions (horizontal and vertical). Again, parameter investigation has shown that any change in nutrient parameterization (e.g. N ) can affect autumn blooming. Further, although the major oceanic pathway for carbon dioxide uptake is through the physicochemical pathway, biological productivity can play an important role. In the temperate and subpolar regions to which the work here is most relevant, Herring (Herring, ) estimates that total primary productivity of between 7 and g C m year. During the autumn blooms simulated here, the modelled primary production was approximately.5 g C m, with autumn blooms therefore contributing significantly to annual production in these regions (Fig. 7e). Both models suggest that the primary effect of global warming will be to decrease the magnitude of autumn blooms. The general principles of autumn bloom development outlined by this modelling study can be applied to ocean datasets to examine their validity. Although in situ sampling may miss autumn blooms because of their relatively limited duration, synoptic measurements of ocean colour (e.g. SeaWIFS) and sea surface temperature (as a proxy for ocean stratification) offer an avenue to test the hypotheses framed by this study. This will form the basis of a future paper. The mechanism explored in this investigation may also be used to explain bloom phenomena in different situations. In conclusion, the classic mechanism that was long thought to explain autumn bloom phenomena can be described by relatively simple equations, which when validated with plausible parameters, is a useful tool in determining the significance of an autumn bloom. The simple model shows that blooms develop provided that : (i) The increase in nutrients into the upper layer is rapid within the first few days of the forcing period, thus allowing there to be enough nutrients and light available to trigger a population bloom. This factor is dependent on the values of the upper and lower layer nutrient concentrations, the upper layer becoming depleted of nutrients over the summer while nutrients remain at relatively high concentrations in the lower layer. (ii) The light-limited specific growth rate is relatively high, thus allowing phytoplankton growth to reestablish before becoming drawn to low depths, and hence low light levels. (iii) The mixed layer deepens at an appropriate rate. When the rate is too slow, the change in conditions for the phytoplankton is not rapid enough to perturb them from the steady state. When the rate is too large, the phytoplankton are moved to depths with low light levels before they have a chance to utilize the increased nutrients. (The deepening of the mixed layer thereby provides the link between the above two factors.) More complex models of annual plankton cycles show the same mechanism at work and can be used in conjunction with the simple model to assess the impact of phytoplankton populations on various ecosystem features. The contribution of autumn blooms to primary production can be significant. ACKNOWLEDGEMENTS Simple autumn blooms models were initially developed at a workshop held at the University of California, Berkeley (ONR Grant ONR_URIP N4-9-J-57) attended by JWP together with Rob Armstrong, John Brindley, Peter Franks, Hariloas Loukos, Hal Smith and John Steele. We thank Khalid Boushaba for useful advice 9 Downloaded from on 9 May 8

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