- Email: [email protected]
Vertical mixing influence on the compensation depth
Journal of Marine Systems 21 Ž1999. 169–177
Vertical mixing influence on the compensation depth Wojciech Szeligiewicz
)
Institute of Ecology PAS, Dziekanow ´ Lesny, ´ 05-092 Łomianki, Poland Received 14 November 1997; accepted 26 August 1998
Abstract Localization of compensation points of phytoplankton photosynthesis is simulated within cross-section of vertical circular eddies in the mixed water surface layer. The dynamic behaviour of net photosynthesis is considered. The euphotic zone may reveal complex structure and diminishes or disappear with the rotation. The area within which phytoplankton biomass growth during the circulation may depend on the speed of the rotation. The results are linked with the static compensation depth. q 1999 Elsevier Science B.V. All rights reserved. Keywords: compensation depth; euphotic zone; photosynthesis; vertical circular mixing
1. Introduction Light is one of the principal factors limiting development of aquatic plants Že.g., Kirk, 1983; Capblancq, 1995.. The euphotic zone according to definition Že.g., Dussart, 1992. represents the net-photosynthetic production area and it stimulates biological processes and geochemical cycles in a water body. In particular, production of phytoplankton, which is the dominant producer on most of the water impoundments, strongly depends on vertical distribution of phytoplankton cells in relation to the euphotic zone. A rule in aquatic biology is that the euphotic zone is the layer ranged from the water surface to the compensation depth Zc . The compensation depth is defined as the depth at which respiration of a plant cell L is balanced by its gross photosynthesis P Že.g., Parsons et al., 1984.. It is assumed in hydrobiological practice that Zc corresponds to that depth at )
Fax: q48-22-751-31-00; E-mail: [email protected]
which light intensity, i.e., compensation light, is equal to 1% of light reaching the water surface Že.g., Kirk, 1983; Tett, 1990.. It might imply an impression that the lower border of the euphotic zone would represent horizontal plane and the zone would be homogeneous if the light were evenly horizontally distributed. However, because compensation light depends on the function describing P and L it should be linked with internal state of plant cells rather than with external light. The former is, in particular, affected by previous light history of the cells Že.g., Pahl-Wostl and Imboden, 1990; PahlWostl, 1992; Beardall et al., 1994.. Hence, compensation depth may depend on hydrodynamic conditions. The aim of this paper is to show by simple mathematical formalism and by relatively simple assumptions about phytoplankton growth and water mixing that vertical circulation of phytoplankton cells in the surface layer of a water body may potentially produce spatial heterogeneous euphotic zone.
0924-7963r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 7 9 6 3 Ž 9 9 . 0 0 0 1 1 - 1
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
170
2. Model description I assume that dynamic behaviour of gross photosynthesis P is described by empirical models of Pahl-Wostl and Imboden Ž1990. and Pahl-Wostl Ž1992.. I will consider for simplicity only this part of the model that concerns uninhibited photosynthesis. Then, according to these authors: d Prdt s k Ž Peq y P .
Ž 1.
where
than the growth of individual phytoplanktonic organisms. All phytoplankton cells of a parcel have common light history. The life cycle of phytoplankton is not considered explicitly. Finally, applying the above assumptions to Eq. Ž1. produces the dynamic photosynthesis P PŽ t. s
1
t
H tanh t 0 r
Io
ž
Ik
ey ´
X
ž
=e Ž t yt .r t r d tX q tanh
k s 1rtr for P F Peq , k s ` for P ) Peq .
Fig. 1. The shape of the water circulation eddies.
ž
Io Ik
X
2P T
//
/
/
eyŽ ´ z 0 . eyt r t r
Ž 3.
Ž 2.
Peq is the equilibrium Žstatic. uninhibited rate of photosynthesis Ž Peq s tanhŽ IrI k ., where I is the light intensity, and I k is the characteristic light intensity, describing the equilibrium productivity curve., tr is the response time of P for increasing light. I assume that light intensity diminishes with depth and with extinction coefficient ´ according to Lambert–Beer’s law. L is constant model parameter. P, Peq and L are dimensionless rates as photosynthesis and respiration are calculated relative to the maximum equilibrium rate of P and Pmax . Vertical mixing in the upper part of the water is expressed by a set of concentric rotating circles Žwhich may be seen as a caricature of Langmuir eddies. ŽFig. 1.. Each circle has the same rotation period T. The maximal diameter of the circle is Zmax . Phytoplankton is treated as a passive component of the water motion. There is no phytoplankton exchange among the circles. The model follows phytoplankton growth in a chosen parcel of water rather
ž
z 0 qr cos t
at a point 2P
ž / ž /
x Ž t . s x 0 q r cos t z Ž t . s z 0 q r sin t
T
2P T
Ž 4.
for a cell circulating around the point Ž x 0 , z 0 . with radius r Ž0 F r F Zmaxr2.. P s Peq at angle s 0 Žangle s t 2 PrT .. In order to clarify the presentation, the model I have introduced has two quantities: a static or equilibrium compensation depth, static Zc , and a dynamic compensation depth, dynamic Zc . The former corresponds to the depth of the points along phytoplankton cell trajectories, for which Peq s L, where Peq is ‘static’ photosynthesis because it only depends on local light intensity, hence, it is unaffected by water circulation. The latter, on the contrary, corresponds to the depths of the points along phytoplankton cell trajectories, for which P s L, where P is ‘dynamic’ photosynthesis calculated according to Eq. Ž3., i.e., it potentially also depends on the ‘light history’ of the phytoplankton cells, which in turn, is related to phytoplankton circulation in the vertical light gradient. So, the dynamic Zc may decline from the static Zc due to water circulation. If there is no phytoplankton circulation, then static P s dynamic P and static Zc s dynamic Zc . Hence, water circulation differentiates these values. The size of the vortices, Zmax , may be treated as a mixing depth for simplicity Žalthough they may differ with one another., but the knowledge about the mixing depth is not necessary in the model.
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
Given the static Zc , i.e., the depth at which Peq s L the value of L is imposed implicitly Žand inversely. when keeping the same external growth conditions and considering the same algal species. So, it implies a solution for the dynamic compensation depth at which P s L Žand as a consequence P s Peq .. Therefore, the static Zc may be regarded as a supplementary or a vicarious parameter Žin place of L..
171
The parameter values are arbitrarily chosen. L does not depend on light and is assumed to be the same and constant for phytoplankton in each of the water parcel. Light extinction coefficient in the water ´ s 0.4 my1 . This value characterizes rather polluted or euphotic oceanic coastal waters and, in particular, is common for inland waters. In the open ocean, extinction coefficient is a few times smaller. However, it is assumed so high in order to the effects of
Fig. 2. Primary production P calculated for the set of 10 circles of diameters decreasing with the numeration, ŽA. T s 6 min and ŽB. T s 2 min. Zma x s 4 m, Io rI k s 0.6, ´ s 0.4 my1 , tr s 2 min. L Ždashed line. s 0.243 is chosen as an example.
172
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
phytoplankton circulation on the compensation depth be more apparent in numerical simulations. I assumed IorI k s 0.6 Žwhere Io is surface radiation. based on the work of Pahl-Wostl and Imboden Ž1990. where this ratio ranges from 0 to 1.65 according to experimental data reported by these authors from Marra Ž1978.. The model does not impose any limit to IorI k . However, the moderate instead of a high value was chosen, thereby justifying the respiration constancy assumption more. The tr value is constant within the range of 30 s–5 min taken also from the work of Pahl-Wostl and Imboden Ž1990.. Zmax is chosen to be within the range of surface mixed layer depth of medium size lakes. The length of rotation period T is in relation to Zmax and to the maximum downwelling velocity. The latter value is assumed to be the same as the maximum downwelling velocity of Langmuir circulation and is approximated as 1% of the wind speed Že.g., Li and Garret, 1993.. The wind varies between 3.0 and 10.5 m sy1 . The spatial and temporal scales of the circulation are of order of the values reported in literature Že.g., Zagarese et al., 1998.. 3. Results Two examples of dynamic photosynthesis calculated for cross-section of counter-clockwise rotated eddies are shown in Fig. 2. Arbitrarily chosen respiration value L is denoted by horizontal dashed line. Intersections of P and L lines represent compensation points. Localization of compensation depths over crosssection of the eddies are shown in Fig. 3. The shape of the eddies is the same as in Fig. 1. The dynamic and the static compensation depths were calculated for these eddies for various assumed T and L values Žcases: a, b, c, d, e, f, g and h.. The centres of small, bold and of the same dimension circles mark that points along the circular trajectories which correspond to the dynamic compensation of photosynthe-
sis. The coordinates Ž t,r . of these points are found numerically from the equality: P s L, where P is calculated according to Eq. Ž3.. Time t g w0,T x in these calculations Ži.e., model runs over one rotation period., because the solutions repeat itself for t q nT, where n is a natural. It is the consequence of the assumption expressed by Eq. Ž2., which leads to the conclusion that if angle s 2p n then P s Peq as it is, in particular, for angle s 0. The calculations starts with the angle s 0, i.e., with P s Peq . Therefore, P calculated from Eq. Ž3. gets the same set of values in the consecutive rotations because the field of light and the other parameters do not change in a given case. Increasing the L value decreases the static compensation depth Žcases: a, b, c, d or e, f, g, h, appropriately. because the static compensation depth, marked by a horizontal line, corresponds to that depth at which Peq s L and because Peq decreases with descent light intensity with depth. The shifting of the static compensation depth makes apparent changes in the dynamic compensation points spatial distribution ŽFig. 3. and in the euphotic zone structure ŽFig. 4. comprising a set of the described above counter-rotating eddies. The euphotic zone is the area Žor areas. where Pnet ) 0. The results are qualitatively different depending on the rotation period T Žcases: A and B in Figs. 3 and 4.. Compensation depth distributions of counter-rotating eddies are symmetrical each other relative to the vertical axes separating the eddies. The dynamic Zc is generally smaller than the static Zc in the model and may locally approach water surface. Thus, the euphotic zone diminishes due to the circulation. The localization of dynamic Zc is limited to the small area inside downwelling andror central part of the eddies, converged at the static compensation depth if static Zc is sufficiently shallow Žwhich may happen for high respiration rate or small surface light intensity or high light extinction coefficient in the water.. Fig. 3 suggests that the euphotic zone does not have to be a uniform layer but may reveal complex struc-
Fig. 3. Cross-section of the circular eddies rotating counter-clockwise. Centres of the circles: small bold circles mark positions of the simulated dynamic compensation depths. ‘Static Zc ’ corresponds to the depth at which Peq s L, i.e., to the compensation depth not affected by circulation. The difference between dynamic Zc and static Zc reflects the effect of rotation on the compensation depth at a given field of P values. Zma x s 4 m, IorI k s 0.6, ´ s 0.4 my1 , tr s 2 min and T s 6 min for Ža. L s 0.150; Žb. L s 0.243; Žc. L s 0.340; Žd. L s 0.390; and T s 2 min for Že. L s 0.150; Žf. L s 0.243; Žg. L s 0.248; Žh. L s 0.280.
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
173
174
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
Fig. 4. Simulated structure of the euphotic zone Ži.e., area of Pnet s P–L ) 0. for the same data as Fig. 3. It is assumed for simplicity that there is no vertical mixing outside the eddies.
ture due to the vertical mixing. The zone may be composed of a set of distinct subareas of Pnet ) 0
separated by the regions of Pnet - 0 and separated from the water surface. The changes of the structure
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
175
Fig. 5. Plots of integrated dynamic photosynthesis, static photosynthesis and respiration as a function of radius r Ž Psum s 1rTH0T Pd t, Peq sum s 1rTH0T Peq d t, Lsum s 1rTH0T Ld t s L.. The integration was performed along a circle of radius r.
are shown when L value increases. If static Zc is sufficiently shallow the dynamic euphotic zone may not occur at all. Then, phytoplankton cannot grow at any point within the eddies. The dynamic photosynthesis P along with equilibrium photosynthesis Peq and respiration L integrated over one cycle of rotation as a function of radius r of the circle are shown in Fig. 5. The integrated equilibrium photosynthesis Peq sum is relatively greater in the external part of the eddies while integrated dynamic photosynthesis Psum diminishes with radius of rotation and with rotation speed. For L s 0.243, for example ŽFig. 5., only the central area of the eddies, i.e., for r - 0.34 m approximately with rotation period T of 2 min and for r - 0.59 m with rotation period T s 6 min, enables phytoplankton cells to grow during rotation time equal or greater than T. For L ) 0.27, all the parts of the eddies disable from phytoplankton surviving, whereas for L - 0.21 Žwith T s 6 min. or for L - 0.17 Žwith T s 2 min. the whole eddies are productive over consecutive cycles.
4. Conclusions and discussion The above results are highly sensitive to the model parameters, which is particularly evident, e.g.,
for small light attenuation coefficient. Constancy of respiration is key assumption from the view point of these results. In essence, such an assumption may be found in other more sophisticated plankton models Že.g., Wolf and Woods, 1988; Woods and Barkmann, 1993a,b; Barkmann and Woods, 1996. if they were restricted to the conditions described above, i.e., when primary production was solely light-limited. It is worth to note that Woods and Barkmann Ž1993a. also simulated compensation depths in relation to turbulent mixing but they considered one-dimensional case only. Nevertheless, photorespiration and enhanced post-illumination respiration may change the above results significantly. The presented model could be improved in future as further details on the phytoplankton loss rate were accessible. Calculations including effect of photoinhibition will be presented elsewhere ŽSzeligiewicz, in preparation.. The diminishing of the euphotic zone due to the rotation is the consequence of the assumption of Pahl-Wostl and Imboden ŽEq. Ž2.. that the cells ‘memorize’ only relatively small ambient light intensities while moving to the water surface but do not memorize the relatively high light while they move down. Some of the above effects may emerge with other phytoplankton primary production models included
176
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
photoadaptation or photoresponse and convectiontype or Langmuir-type vertical mixing whenever the response time scale was comparable to the time scales of mixing while dynamic behaviour of P is the most pronounced Že.g., Pahl-Wostl, 1992.. It would be the case, for example, for the models of Kamykowski et al. Ž1994. or Barkmann and Woods Ž1996. Žif these models were developed to two-dimensional space. where response times scale ranged from 0.05 to 100 h or 1.7 to 7.1 h, correspondingly, whereas time scale of such vertical circulation is in the range of minutes to hours ŽImboden and Wuest, ¨ 1995.. The results show that if such structure occurred, it would be probably unstable as the influencing factors undergo in reality significant daily changes. Moreover, if the cells of a given phytoplankton ensemble were characterized by different model parameters it would lead to blurring the outlines of the resulting euphotic zone structure. The model presented above is entirely speculative, nevertheless, it reveals potential role of organized vertical mixing in the euphotic zone forming. The possibility of the complex spatial structure appearance of the zone would suggest that some of literature considerations based on static compensation depth concept referring to only one spatial dimension Ži.e., water column. need to be verified. For example, it may concerns the ratio of Zeu r Zmix , where Zeu is the euphotic zone depth Žs Zc . and Zmix is the mixed layer depth. The ratio up to now was regarded as an index of community photosynthetic balance Že.g., Capblancq, 1995., or as proportion of aphotic to euphotic zone volume Že.g., Reynolds, 1984. or euphotic zone stratification index Že.g., Capblancq, 1982.. Bearing in mind above presented results these meanings could be misleading. Likewise, considerations on net phytoplankton growth by means of ‘integrated water-column photosynthesis’ ŽFalkowski and Raven, 1997. or Sverdrup’s critical depth model Ž1953. or critical light models ŽHuisman and Weissing, 1994; Weissing and Huisman, 1994. may not be justified in case of organized water motion in the surface mixed layer. Furthermore, the presented above model emphasizes the ‘selective role of mixing in the competition between algal species’ ŽPahl-Wostl and Imboden, 1990., and the role of hydrodynamically stimulated
light climate that may promote or inhibit the phytoplankton development in the surface mixed layer. Moreover, it indicates that the further research must be focused on the dynamics of photosynthesis and respiration to enable modelling these processes more properly.
Acknowledgements This research was supported by KBN Grant No. 6P04G 003 09. I thank the suggestions of anonymous referees that improved the manuscript and for useful discussions during a meeting on the Mathematical Modelling of Plankton Population Dynamics at the Isaac Newton Institute for Mathematical Sciences, Cambridge.
References Barkmann, W., Woods, J.D., 1996. On using Lagrangian model to calibrate primary production determined from in vitro incubation measurements. J. Plankton Res. 18, 767–788. Beardall, J., Burger-Wiersma, T., Rijkeboer, M., Sukenik, A., Lemoalle, J., Dubinsky, Z., Fontvielle, D., 1994. Studies on enhanced post-illumination respiration in microalgae. J. Plankton Res. 16, 1411–1432. Capblancq, J., 1982. Phytoplancton et production primaire. In: Pourriot, R., Capblancq, J., Champ, P., Meyer, J.A. ŽEds.., ´ cologie du Plankton des Eaux Continentales. Masson, Paris, E pp. 1–48. Capblancq, J., 1995. Production primaire autotrophe. In: Pourriot, R., Meybeck, M. ŽEds.., Limnologie Generale. Collection ´ ´ ´ d’Ecologie 25, Masson, Paris, pp. 228–252. ´ Dussart, B., 1992. Limnologie. L’Etude des Eaux Continentales. Edition N. Boubee ´ & Cie, Paris. Falkowski, P.G., Raven, J.A., 1997. Aquatic Photosynthesis. Blackwell, Oxford. Huisman, J., Weissing, F.J., 1994. Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model. Ecology 75, 507–520. Imboden, D.M., Wuest, A., 1995. Mixing mechanisms in lakes. ¨ In: Lerman, A., Imboden, D.M., Gat, J.R. ŽEds.., Physics and Chemistry of Lakes. Springer, Berlin, pp. 83–138. Kamykowski, D., Yamazaki, H., Janowitz, G.S., 1994. A Lagrangian model of phytoplankton photosynthetic response in the upper mixed layer. J. Plankton Res. 16, 1059–1069. Kirk, J.T.O., 1983. Light and Photosynthesis in Aquatic Ecosystems. Cambridge Univ. Press, Cambridge. Li, M., Garret, C., 1993. Cell merging and the jetrdownwelling ration in Langmuir circulation. J. Mar. Res. 41, 737–769. Marra, J., 1978. Phytoplankton photosynthesis response to vertical movement in a mixed layer. Mar. Biol. 46, 203–208.
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177 Pahl-Wostl, C., 1992. Dynamic vs. static models for photosynthesis. Hydrobiologia 238, 189–196. Pahl-Wostl, C., Imboden, D.M., 1990. DYPHORA—a dynamic model for the rate of photosynthesis of algae. J. Plankton Res. 12, 1207–1221. Parsons, T.R., Takahashi, M., Hargrave, B., 1984. Biological Oceanographic Processes. Pergamon, Oxford. Reynolds, C.S., 1984. The Ecology of Freshwater Phytoplankton. Cambridge Univ. Press, Cambridge. Tett, P., 1990. The euphotic zone. In: Herring, P.J., Campbell, A.K., Whitfield, M., Maddock, L. ŽEds.., Light and Life in the Sea. Cambridge Univ. Press, Cambridge, pp. 59–87. Weissing, F.J., Huisman, J., 1994. Growth and competition in a light gradient. J. Theor. Biol. 168, 323–336. Wolf, K.U., Woods, J.D., 1988. Lagrangian simulation of primary
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production in the physical environment—the deep chlorophyll maximum and nutricline. In: Rotschild, B.J. ŽEd.., Towards a Theory on Biological–Physical Interactions in the Worlds Ocean. Kluwer, pp. 51–70. Woods, J., Barkmann, W., 1993a. Diatom demography in winter —simulated by the Lagrangian Ensemble method. Fisheries Oceanography 2, 202–222. Woods, J., Barkmann, W., 1993b. The plankton multiplier—positive feedback in the greenhouse. J. Plankton Res. 15, 1053– 1074. Zagarese, H.E., Cravero, W., Gonzales, P., Pedrozo, F., 1998. Copepod mortality induced by fluctuating levels of natural ultraviolet radiation simulating vertical water mixing. Limnol. Oceanogr. 43, 169–174.
Vertical mixing influence on the compensation depth Wojciech Szeligiewicz
)
Institute of Ecology PAS, Dziekanow ´ Lesny, ´ 05-092 Łomianki, Poland Received 14 November 1997; accepted 26 August 1998
Abstract Localization of compensation points of phytoplankton photosynthesis is simulated within cross-section of vertical circular eddies in the mixed water surface layer. The dynamic behaviour of net photosynthesis is considered. The euphotic zone may reveal complex structure and diminishes or disappear with the rotation. The area within which phytoplankton biomass growth during the circulation may depend on the speed of the rotation. The results are linked with the static compensation depth. q 1999 Elsevier Science B.V. All rights reserved. Keywords: compensation depth; euphotic zone; photosynthesis; vertical circular mixing
1. Introduction Light is one of the principal factors limiting development of aquatic plants Že.g., Kirk, 1983; Capblancq, 1995.. The euphotic zone according to definition Že.g., Dussart, 1992. represents the net-photosynthetic production area and it stimulates biological processes and geochemical cycles in a water body. In particular, production of phytoplankton, which is the dominant producer on most of the water impoundments, strongly depends on vertical distribution of phytoplankton cells in relation to the euphotic zone. A rule in aquatic biology is that the euphotic zone is the layer ranged from the water surface to the compensation depth Zc . The compensation depth is defined as the depth at which respiration of a plant cell L is balanced by its gross photosynthesis P Že.g., Parsons et al., 1984.. It is assumed in hydrobiological practice that Zc corresponds to that depth at )
Fax: q48-22-751-31-00; E-mail: [email protected]
which light intensity, i.e., compensation light, is equal to 1% of light reaching the water surface Že.g., Kirk, 1983; Tett, 1990.. It might imply an impression that the lower border of the euphotic zone would represent horizontal plane and the zone would be homogeneous if the light were evenly horizontally distributed. However, because compensation light depends on the function describing P and L it should be linked with internal state of plant cells rather than with external light. The former is, in particular, affected by previous light history of the cells Že.g., Pahl-Wostl and Imboden, 1990; PahlWostl, 1992; Beardall et al., 1994.. Hence, compensation depth may depend on hydrodynamic conditions. The aim of this paper is to show by simple mathematical formalism and by relatively simple assumptions about phytoplankton growth and water mixing that vertical circulation of phytoplankton cells in the surface layer of a water body may potentially produce spatial heterogeneous euphotic zone.
0924-7963r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 7 9 6 3 Ž 9 9 . 0 0 0 1 1 - 1
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
170
2. Model description I assume that dynamic behaviour of gross photosynthesis P is described by empirical models of Pahl-Wostl and Imboden Ž1990. and Pahl-Wostl Ž1992.. I will consider for simplicity only this part of the model that concerns uninhibited photosynthesis. Then, according to these authors: d Prdt s k Ž Peq y P .
Ž 1.
where
than the growth of individual phytoplanktonic organisms. All phytoplankton cells of a parcel have common light history. The life cycle of phytoplankton is not considered explicitly. Finally, applying the above assumptions to Eq. Ž1. produces the dynamic photosynthesis P PŽ t. s
1
t
H tanh t 0 r
Io
ž
Ik
ey ´
X
ž
=e Ž t yt .r t r d tX q tanh
k s 1rtr for P F Peq , k s ` for P ) Peq .
Fig. 1. The shape of the water circulation eddies.
ž
Io Ik
X
2P T
//
/
/
eyŽ ´ z 0 . eyt r t r
Ž 3.
Ž 2.
Peq is the equilibrium Žstatic. uninhibited rate of photosynthesis Ž Peq s tanhŽ IrI k ., where I is the light intensity, and I k is the characteristic light intensity, describing the equilibrium productivity curve., tr is the response time of P for increasing light. I assume that light intensity diminishes with depth and with extinction coefficient ´ according to Lambert–Beer’s law. L is constant model parameter. P, Peq and L are dimensionless rates as photosynthesis and respiration are calculated relative to the maximum equilibrium rate of P and Pmax . Vertical mixing in the upper part of the water is expressed by a set of concentric rotating circles Žwhich may be seen as a caricature of Langmuir eddies. ŽFig. 1.. Each circle has the same rotation period T. The maximal diameter of the circle is Zmax . Phytoplankton is treated as a passive component of the water motion. There is no phytoplankton exchange among the circles. The model follows phytoplankton growth in a chosen parcel of water rather
ž
z 0 qr cos t
at a point 2P
ž / ž /
x Ž t . s x 0 q r cos t z Ž t . s z 0 q r sin t
T
2P T
Ž 4.
for a cell circulating around the point Ž x 0 , z 0 . with radius r Ž0 F r F Zmaxr2.. P s Peq at angle s 0 Žangle s t 2 PrT .. In order to clarify the presentation, the model I have introduced has two quantities: a static or equilibrium compensation depth, static Zc , and a dynamic compensation depth, dynamic Zc . The former corresponds to the depth of the points along phytoplankton cell trajectories, for which Peq s L, where Peq is ‘static’ photosynthesis because it only depends on local light intensity, hence, it is unaffected by water circulation. The latter, on the contrary, corresponds to the depths of the points along phytoplankton cell trajectories, for which P s L, where P is ‘dynamic’ photosynthesis calculated according to Eq. Ž3., i.e., it potentially also depends on the ‘light history’ of the phytoplankton cells, which in turn, is related to phytoplankton circulation in the vertical light gradient. So, the dynamic Zc may decline from the static Zc due to water circulation. If there is no phytoplankton circulation, then static P s dynamic P and static Zc s dynamic Zc . Hence, water circulation differentiates these values. The size of the vortices, Zmax , may be treated as a mixing depth for simplicity Žalthough they may differ with one another., but the knowledge about the mixing depth is not necessary in the model.
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
Given the static Zc , i.e., the depth at which Peq s L the value of L is imposed implicitly Žand inversely. when keeping the same external growth conditions and considering the same algal species. So, it implies a solution for the dynamic compensation depth at which P s L Žand as a consequence P s Peq .. Therefore, the static Zc may be regarded as a supplementary or a vicarious parameter Žin place of L..
171
The parameter values are arbitrarily chosen. L does not depend on light and is assumed to be the same and constant for phytoplankton in each of the water parcel. Light extinction coefficient in the water ´ s 0.4 my1 . This value characterizes rather polluted or euphotic oceanic coastal waters and, in particular, is common for inland waters. In the open ocean, extinction coefficient is a few times smaller. However, it is assumed so high in order to the effects of
Fig. 2. Primary production P calculated for the set of 10 circles of diameters decreasing with the numeration, ŽA. T s 6 min and ŽB. T s 2 min. Zma x s 4 m, Io rI k s 0.6, ´ s 0.4 my1 , tr s 2 min. L Ždashed line. s 0.243 is chosen as an example.
172
W. Szeligiewiczr Journal of Marine Systems 21 (1999) 169–177
phytoplankton circulation on the compensation depth be more apparent in numerical simulations. I assumed IorI k s 0.6 Žwhere Io is surface radiation. based on the work of Pahl-Wostl and Imboden Ž1990. where this ratio ranges from 0 to 1.65 according to experimental data reported by these authors from Marra Ž1978.. The model does not impose any limit to IorI k . However, the moderate instead of a high value was chosen, thereby justifying the respiration constancy assumption more. The tr value is constant within the range of 30 s–5 min taken also from the work of Pahl-Wostl and Imboden Ž1990.. Zmax is chosen to be within the range of surface mixed layer depth of medium size lakes. The length of rotation period T is in relation to Zmax and to the maximum downwelling velocity. The latter value is assumed to be the same as the maximum downwelling velocity of Langmuir circulation and is approximated as 1% of the wind speed Že.g., Li and Garret, 1993.. The wind varies between 3.0 and 10.5 m sy1 . The spatial and temporal scales of the circulation are of order of the values reported in literature Že.g., Zagarese et al., 1998.. 3. Results Two examples of dynamic photosynthesis calculated for cross-section of counter-clockwise rotated eddies are shown in Fig. 2. Arbitrarily chosen respiration value L is denoted by horizontal dashed line. Intersections of P and L lines represent compensation points. Localization of compensation depths over crosssection of the eddies are shown in Fig. 3. The shape of the eddies is the same as in Fig. 1. The dynamic and the static compensation depths were calculated for these eddies for various assumed T and L values Žcases: a, b, c, d, e, f, g and h.. The centres of small, bold and of the same dimension circles mark that points along the circular trajectories which correspond to the dynamic compensation of photosynthe-
sis. The coordinates Ž t,r . of these points are found numerically from the equality: P s L, where P is calculated according to Eq. Ž3.. Time t g w0,T x in these calculations Ži.e., model runs over one rotation period., because the solutions repeat itself for t q nT, where n is a natural. It is the consequence of the assumption expressed by Eq. Ž2., which leads to the conclusion that if angle s 2p n then P s Peq as it is, in particular, for angle s 0. The calculations starts with the angle s 0, i.e., with P s Peq . Therefore, P calculated from Eq. Ž3. gets the same set of values in the consecutive rotations because the field of light and the other parameters do not change in a given case. Increasing the L value decreases the static compensation depth Žcases: a, b, c, d or e, f, g, h, appropriately. because the static compensation depth, marked by a horizontal line, corresponds to that depth at which Peq s L and because Peq decreases with descent light intensity with depth. The shifting of the static compensation depth makes apparent changes in the dynamic compensation points spatial distribution ŽFig. 3. and in the euphotic zone structure ŽFig. 4. comprising a set of the described above counter-rotating eddies. The euphotic zone is the area Žor areas. where Pnet ) 0. The results are qualitatively different depending on the rotation period T Žcases: A and B in Figs. 3 and 4.. Compensation depth distributions of counter-rotating eddies are symmetrical each other relative to the vertical axes separating the eddies. The dynamic Zc is generally smaller than the static Zc in the model and may locally approach water surface. Thus, the euphotic zone diminishes due to the circulation. The localization of dynamic Zc is limited to the small area inside downwelling andror central part of the eddies, converged at the static compensation depth if static Zc is sufficiently shallow Žwhich may happen for high respiration rate or small surface light intensity or high light extinction coefficient in the water.. Fig. 3 suggests that the euphotic zone does not have to be a uniform layer but may reveal complex struc-
Fig. 3. Cross-section of the circular eddies rotating counter-clockwise. Centres of the circles: small bold circles mark positions of the simulated dynamic compensation depths. ‘Static Zc ’ corresponds to the depth at which Peq s L, i.e., to the compensation depth not affected by circulation. The difference between dynamic Zc and static Zc reflects the effect of rotation on the compensation depth at a given field of P values. Zma x s 4 m, IorI k s 0.6, ´ s 0.4 my1 , tr s 2 min and T s 6 min for Ža. L s 0.150; Žb. L s 0.243; Žc. L s 0.340; Žd. L s 0.390; and T s 2 min for Že. L s 0.150; Žf. L s 0.243; Žg. L s 0.248; Žh. L s 0.280.
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Fig. 4. Simulated structure of the euphotic zone Ži.e., area of Pnet s P–L ) 0. for the same data as Fig. 3. It is assumed for simplicity that there is no vertical mixing outside the eddies.
ture due to the vertical mixing. The zone may be composed of a set of distinct subareas of Pnet ) 0
separated by the regions of Pnet - 0 and separated from the water surface. The changes of the structure
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Fig. 5. Plots of integrated dynamic photosynthesis, static photosynthesis and respiration as a function of radius r Ž Psum s 1rTH0T Pd t, Peq sum s 1rTH0T Peq d t, Lsum s 1rTH0T Ld t s L.. The integration was performed along a circle of radius r.
are shown when L value increases. If static Zc is sufficiently shallow the dynamic euphotic zone may not occur at all. Then, phytoplankton cannot grow at any point within the eddies. The dynamic photosynthesis P along with equilibrium photosynthesis Peq and respiration L integrated over one cycle of rotation as a function of radius r of the circle are shown in Fig. 5. The integrated equilibrium photosynthesis Peq sum is relatively greater in the external part of the eddies while integrated dynamic photosynthesis Psum diminishes with radius of rotation and with rotation speed. For L s 0.243, for example ŽFig. 5., only the central area of the eddies, i.e., for r - 0.34 m approximately with rotation period T of 2 min and for r - 0.59 m with rotation period T s 6 min, enables phytoplankton cells to grow during rotation time equal or greater than T. For L ) 0.27, all the parts of the eddies disable from phytoplankton surviving, whereas for L - 0.21 Žwith T s 6 min. or for L - 0.17 Žwith T s 2 min. the whole eddies are productive over consecutive cycles.
4. Conclusions and discussion The above results are highly sensitive to the model parameters, which is particularly evident, e.g.,
for small light attenuation coefficient. Constancy of respiration is key assumption from the view point of these results. In essence, such an assumption may be found in other more sophisticated plankton models Že.g., Wolf and Woods, 1988; Woods and Barkmann, 1993a,b; Barkmann and Woods, 1996. if they were restricted to the conditions described above, i.e., when primary production was solely light-limited. It is worth to note that Woods and Barkmann Ž1993a. also simulated compensation depths in relation to turbulent mixing but they considered one-dimensional case only. Nevertheless, photorespiration and enhanced post-illumination respiration may change the above results significantly. The presented model could be improved in future as further details on the phytoplankton loss rate were accessible. Calculations including effect of photoinhibition will be presented elsewhere ŽSzeligiewicz, in preparation.. The diminishing of the euphotic zone due to the rotation is the consequence of the assumption of Pahl-Wostl and Imboden ŽEq. Ž2.. that the cells ‘memorize’ only relatively small ambient light intensities while moving to the water surface but do not memorize the relatively high light while they move down. Some of the above effects may emerge with other phytoplankton primary production models included
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photoadaptation or photoresponse and convectiontype or Langmuir-type vertical mixing whenever the response time scale was comparable to the time scales of mixing while dynamic behaviour of P is the most pronounced Že.g., Pahl-Wostl, 1992.. It would be the case, for example, for the models of Kamykowski et al. Ž1994. or Barkmann and Woods Ž1996. Žif these models were developed to two-dimensional space. where response times scale ranged from 0.05 to 100 h or 1.7 to 7.1 h, correspondingly, whereas time scale of such vertical circulation is in the range of minutes to hours ŽImboden and Wuest, ¨ 1995.. The results show that if such structure occurred, it would be probably unstable as the influencing factors undergo in reality significant daily changes. Moreover, if the cells of a given phytoplankton ensemble were characterized by different model parameters it would lead to blurring the outlines of the resulting euphotic zone structure. The model presented above is entirely speculative, nevertheless, it reveals potential role of organized vertical mixing in the euphotic zone forming. The possibility of the complex spatial structure appearance of the zone would suggest that some of literature considerations based on static compensation depth concept referring to only one spatial dimension Ži.e., water column. need to be verified. For example, it may concerns the ratio of Zeu r Zmix , where Zeu is the euphotic zone depth Žs Zc . and Zmix is the mixed layer depth. The ratio up to now was regarded as an index of community photosynthetic balance Že.g., Capblancq, 1995., or as proportion of aphotic to euphotic zone volume Že.g., Reynolds, 1984. or euphotic zone stratification index Že.g., Capblancq, 1982.. Bearing in mind above presented results these meanings could be misleading. Likewise, considerations on net phytoplankton growth by means of ‘integrated water-column photosynthesis’ ŽFalkowski and Raven, 1997. or Sverdrup’s critical depth model Ž1953. or critical light models ŽHuisman and Weissing, 1994; Weissing and Huisman, 1994. may not be justified in case of organized water motion in the surface mixed layer. Furthermore, the presented above model emphasizes the ‘selective role of mixing in the competition between algal species’ ŽPahl-Wostl and Imboden, 1990., and the role of hydrodynamically stimulated
light climate that may promote or inhibit the phytoplankton development in the surface mixed layer. Moreover, it indicates that the further research must be focused on the dynamics of photosynthesis and respiration to enable modelling these processes more properly.
Acknowledgements This research was supported by KBN Grant No. 6P04G 003 09. I thank the suggestions of anonymous referees that improved the manuscript and for useful discussions during a meeting on the Mathematical Modelling of Plankton Population Dynamics at the Isaac Newton Institute for Mathematical Sciences, Cambridge.
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