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Appendix A
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Model equations are given in the following sections. Model parameters are listed in Table I.
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Phytoplankton processes
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Light extinction 𝜎(m-1) is calculated as a function of water, particulates and dissolved semi-labile
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organic carbon present in the medium:
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𝐶
𝐶
) + (𝜎𝑃𝑂𝑀 ∙ 𝑅𝑃𝑂𝑀
)
𝜎 = 𝜎𝑏𝑔 + (𝜎𝑃 ∙ 𝑃𝐶 ) + (𝜎𝑠𝑙 ∙ 𝑅𝑠𝑙
(1a)
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10
𝐶
𝐶
where 𝑃𝐶 , 𝑅𝑠𝑙
and 𝑅𝑃𝑂𝑀
are the carbon concentration of algal biomass, semi-labile DOC and POC,
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respectively.
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The average light (𝐿𝑎𝑣 ) to which algal biomass is exposed to is calculated as follows:
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𝐿
𝐿𝑎𝑣 = 𝜎∙∆𝑧 (1 − 𝑒 (−𝜎∙∆𝑧) )
(2a)
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where 𝐿 is the environmental Photosynthetically Active Radiation (PAR) and ∆𝑧 is the thickness of the
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water layer, here assumed to be 0.15 m.
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The temperature response factor for phytoplankton ( ʄ𝑇 ) is calculated with a specific 𝑄10 function at
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any temperature value T:
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ʄ𝑇 = 𝑄10 [(𝑇−10)/10] − 𝑄10 [(𝑇−32)/3]
(3a)
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1
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The general equation for algal biomass (P) is the following:
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𝑑𝑃
𝑑𝑡
= 𝑃𝐻𝑂𝑇𝑂𝑆𝑌𝑁𝑇𝐻𝐸𝑆𝐼𝑆 − 𝑅𝐸𝑆𝑃𝐼𝑅𝐴𝑇𝐼𝑂𝑁 − 𝐿𝑌𝑆𝐼𝑆 − 𝐸𝑋𝑈𝐷𝐴𝑇𝐼𝑂𝑁
(4a)
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where:
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𝑃𝐻𝑂𝑇𝑂𝑆𝑌𝑁𝑇𝐻𝐸𝑆𝐼𝑆 = 𝑟𝑎𝑠𝑠 ∙ ʄ𝑇 ∙ 𝜑 ∙ 𝑃𝐶
(5a)
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𝜑 is a function describing the light limitation in phytoplankton and is given by:
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𝜑 = {1 − 𝑒 [−𝛼∙𝐼∙𝜃/(𝑟𝑎𝑠𝑠 ∙ʄ
𝑇 )]
} ∙ 𝑒 [−𝛽∙𝐼∙𝜃/(𝑟𝑎𝑠𝑠 ∙ʄ
𝑇 )]
(6a)
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where I is the PAR and 𝜃 the actual chlorophyll to carbon ratio.
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The proportion of net photosynthesis directed to chlorophyll synthesis ( 𝜌 ) is given by:
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𝜌 = (𝜃𝑚𝑎𝑥 ∙ 𝑟𝑎𝑠𝑠 ∙ ʄ𝑇 ∙ 𝜑)/(𝛼 ∙ 𝐼 ∙ 𝜃)
(7a)
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Algal mortality due to lysis is calculated as:
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𝐿𝑌𝑆𝐼𝑆 = 𝑟𝑙𝑦𝑠 ∙ 𝑃𝐶
(8a)
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Cellular carbon exudation is described by the sum of two distinct terms, activity exudation (𝐴. 𝐸𝑋𝑈)
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and nutrient stress-induced exudation (𝑆. 𝐸𝑋𝑈):
2
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𝐴. 𝐸𝑋𝑈 = 𝑃𝐻𝑂𝑇𝑂𝑆𝑌𝑁𝑇𝐻𝐸𝑆𝐼𝑆 ∙ 𝑝𝐴.𝑒𝑥𝑢
(9a)
𝑆. 𝐸𝑋𝑈 = 𝑃𝐻𝑂𝑇𝑂𝑆𝑌𝑁𝑇𝐻𝐸𝑆𝐼𝑆 ∙ (1 − 𝑁𝑆) ∙ (1 − 𝑝𝐴.𝑒𝑥𝑢 )
(10a)
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DOC derived by lysis and exudation is split into labile and semi-labile components using the
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parameter rdetr in Table I.
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Algal respiration is composed by a basal metabolism (B.RES see eq. 10) and an activity respiration
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term (A.RES):
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𝑅𝐸𝑆𝑃𝐼𝑅𝐴𝑇𝐼𝑂𝑁 = 𝐵. 𝑅𝐸𝑆 + 𝐴. 𝑅𝐸𝑆
(11a)
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where:
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𝐴. 𝑅𝐸𝑆 = 𝑟𝐴.𝑟𝑒𝑠 ∙ (𝑃𝐻𝑂𝑇𝑂𝑆𝑌𝑁𝑇𝐻𝐸𝑆𝐼𝑆 − 𝐸𝑋𝑈𝐷𝐴𝑇𝐼𝑂𝑁)
(12a)
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Chlorophyll loss terms due to rest respiration, exudation and lysis are modeled as for carbon.
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Nutrient losses due to lysis are channeled to the dissolved organic nitrogen (DON) and phosphorus
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(DOP) pool, according to the phytoplankton nutrient to carbon ratio.
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Phosphorus (𝑃𝑂4) uptake is given by:
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𝑈𝑃𝑇𝐴𝐾𝐸𝑃 = 𝑀𝐼𝑁[𝑈𝑃𝑇𝐴𝐾𝐸𝑃.𝑟𝑒𝑞 , ( 𝑎𝑃 ∙ 𝑃𝑂4 ∙ 𝑃𝐶 )]
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3
(13a)
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𝑈𝑃𝑇𝐴𝐾𝐸𝑃.𝑟𝑒𝑞 is composed of two terms. The first is proportional to the net photosynthesis while the
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second describes cellular nutrient accumulation up to a threshold value (luxury uptake):
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𝑈𝑃𝑇𝐴𝐾𝐸𝑃.𝑟𝑒𝑞 = (𝑃𝐻𝑂𝑇𝑂. −𝐸𝑋𝑈. −𝐴. 𝑅𝐸𝑆) ∙ 𝑄𝑃𝑚𝑎𝑥 + (𝑄𝑃𝑚𝑎𝑥 ∙ 𝑃𝐶 ) − 𝑃𝑃
(14a)
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where 𝑄𝑃𝑚𝑎𝑥 is the maximum phosphorus to carbon cellular ratio and 𝑃𝑃 is the cellular phosphorus
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content.
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Nitrogen uptake takes into account of both ammonium (𝑁𝐻4 ) and nitrate (𝑁𝑂3 ):
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𝑈𝑃𝑇𝐴𝐾𝐸𝑁 = 𝑀𝐼𝑁[𝑈𝑃𝑇𝐴𝐾𝐸𝑁.𝑟𝑒𝑞 , (𝑎𝑁𝑂3 ∙ 𝑁𝑂3 ∙ 𝑃𝐶 ) + (𝑎𝑁𝐻4 ∙ 𝑁𝐻4 ∙ 𝑃𝐶 )]
(15a)
𝑈𝑃𝑇𝐴𝐾𝐸𝑁.𝑟𝑒𝑞 = (𝑃𝑅𝑂𝐷𝑈𝐶𝑇𝐼𝑉𝐼𝑇𝑌 ∙ 𝑄𝑁𝑚𝑎𝑥 ) + (𝑄𝑁𝑚𝑎𝑥 ∙ 𝑃𝐶 ) − 𝑃𝑁
(16a)
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where 𝑄𝑁𝑚𝑎𝑥 is the maximum nitrogen to carbon cellular ratio and 𝑃𝑁 is the nitrogen cellular content.
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Bacteria processes
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The temperature response factor for bacteria (ʄ𝑇𝐵 ) is calculated with a specific 𝑄10 function:
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ʄ𝑇𝐵 = 𝑄10𝐵 [(𝑇−10)/10] − 𝑄10𝐵 [(𝑇−32)/3]
(17a)
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The general biomass equation is given by:
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𝑑𝐵
𝑑𝑡
= 𝑈𝑃𝑇𝐴𝐾𝐸𝐵 − 𝑅𝐸𝑆𝑃𝐼𝑅𝐴𝑇𝐼𝑂𝑁𝐵 − 𝑀𝑂𝑅𝑇𝐴𝐿𝐼𝑇𝑌
4
(18a)
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where DOC uptake is given by:
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𝐵
𝑈𝑃𝑇𝐴𝐾𝐸𝐵 = 𝑀𝐼𝑁[(𝑟𝑎𝑠𝑠
∙ ʄ𝑇𝐵 ∙ 𝑒𝑂2 ∙ 𝑁𝑆 𝐵 ∙ 𝐵 𝐶 ) , 𝐷𝑂𝐶]
(19a)
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where 𝐵 𝐶 is the bacterial carbon biomass. 𝑒𝑂2is given by:
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𝑒𝑂2 = 𝑂𝑟𝑒𝑙 /(𝑂𝑟𝑒𝑙 + ℎ𝑜𝑥 )
(20a)
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where 𝑂𝑟𝑒𝑙 is the relative oxygen saturation and ℎ𝑜𝑥 is the oxygen concentration at which the limiting
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factor equals 0.5. 𝑁𝑆 𝐵 is given by the expression:
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𝑁
𝑁𝐻4 +𝑅𝐷𝑂𝑀
𝑁𝑆 𝐵 = 𝑀𝐼𝑁 [𝑁𝐻
𝑁
4 +𝑅𝐷𝑂𝑀 +ℎ𝑁
,
𝑃
𝑃𝑂4 +𝑅𝐷𝑂𝑀
𝑃
𝑃𝑂4 +𝑅𝐷𝑂𝑀
+ℎ𝑃
]
(21a)
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𝑁,𝑃
where 𝑅𝐷𝑂𝑀
are the nitrogen and phosphorus contents of DOM and ℎ𝑁,𝑃 are the nitrogen and
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phosphorus concentrations at which the limiting factor equals 0.5.
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Respiration is composed of an activity (dependent on DOM uptake) and a basal (depending on
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biomass) term:
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𝐵
𝐵
𝐵
𝑅𝐸𝑆𝑃𝐼𝑅𝐴𝑇𝐼𝑂𝑁𝐵 = 𝑈𝑃𝑇𝐴𝐾𝐸𝐵 ∙ [𝑟𝐴.𝑟𝑒𝑠
∙ 𝑂𝑟𝑒𝑙 + 𝑟𝑟𝑒𝑠𝑂𝑋
∙ (1 − 𝑂𝑟𝑒𝑙 )] + (𝑟𝐵.𝑟𝑒𝑠
∙ ʄ𝑇𝐵 ∙ 𝐵 𝐶 )
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𝐵
where 𝑟𝐴.𝑟𝑒𝑠
is the respired fraction of uptake.
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Mortality is given by:
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(22a)
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𝐵
𝑀𝑂𝑅𝑇𝐴𝐿𝐼𝑇𝑌 = 𝑟𝑙𝑦𝑠
∙ ʄ𝑇𝐵 ∙ 𝐵 𝐶
(23a)
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DOC derived by mortality process is split into labile and semi-labile components using the parameter
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rdetr in Table I. Nitrogen and phosphorous cellular content are channeled into the DON and DOP pool,
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respectively.
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Nutrient excretion is only active when nutrient (𝑖 = N or P) are in excess and is given by:
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𝐿𝑂𝑆𝑆𝑖 = (𝑄𝑖𝐵 − 𝑄𝑖𝐵𝑚𝑎𝑥 ) ∙ 𝐵 𝐶
(24a)
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where 𝑄𝑖𝐵 is the actual nutrient to carbon ratio. In case of internal nutrient shortage, eq. 23a is replaced
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by the following:
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𝑖
𝑖
𝑈𝑃𝑇𝐴𝐾𝐸𝐵𝐴𝐶
= (𝑄𝑖𝐵 − 𝑄𝑖𝐵𝑚𝑎𝑥 ) ∙ 𝐵 𝐶 ∙ (𝑖+ℎ )
(25a)
𝑖
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𝐶
𝐶
The breakdown of semi-labile (𝑅𝑠𝑙𝐷𝑂𝑀
) to labile DOC (𝑅𝑙𝐷𝑂𝑀
) is described by:
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𝐶
𝐶
𝑅𝑙𝐷𝑂𝑀
= 𝑟𝑑𝑖𝑠 ∙ 𝑅𝑠𝑙𝐷𝑂𝑀
(26a)
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