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8.7.3.19. Temperature dependence

Temperature functions are selected by the cpp options DARWIN_TEMP_VERSION and DARWIN_TEMP_RANGE. The cpp option DARWIN_NOTEMP turns off all temperature dependence. The temperature dependence of mortality and grazing can be turned off for individual plankton types by setting the traits tempMort, tempMort2 and tempGraz to 0.

Note that the temperature functions in all versions except 1 can be greater than 1.0.

8.7.3.19.1. DARWIN_TEMP_VERSION 1

\[\begin{split}\begin{gathered} f^{\text{phy}}_j(T) = \left[ c_j \left[ e_{1 j}^{T/1^\circ{\rm C}} {\mathrm{e}}^{-e_{2 j} {|T - T^{\text{opt}}_j|}^{p_j}} - n \right]_{\ge 10^{-10}} \right]^{\le 1} \\ f^{\text{up}}_j(T) = f^{\text{graz}}_z(T) = f^{\text{remin}}(T) = f^{\text{mort}}(T) = f^{\text{mort2}}(T) = 1 \end{gathered}\end{split}\]

where the exponential is only present with DARWIN_TEMP_RANGE.

8.7.3.19.2. DARWIN_TEMP_VERSION 2

\[\begin{split}\begin{gathered} f^{\text{phy}}_j(T) = c^{\text{Arr}} \left[ {\mathrm{e}}^{A^{\text{Arr}}_{\text{e}} \bigl( (T+273.15)^{-1} - {T^{\text{Arr}}_{\text{ref}}}^{-1} \bigr)} {\mathrm{e}}^{-e_{2 j} {|T - T^{\text{opt}}_j|}^{p_j}} \right]_{\ge 10^{-10}} \\ f^{\text{up}}_j(T) = f^{\text{graz}}_z(T) = f^{\text{remin}}(T) = f^{\text{mort}}(T) = f^{\text{mort2}}(T) = f_{\text{Arr}}(T) \\ f_{\text{Arr}}(T) = c^{\text{Arr}} \left[ {\mathrm{e}}^{A^{\text{Arr}}_{\text{e}} \bigl( (T+273.15)^{-1} - {T^{\text{Arr}}_{\text{ref}}}^{-1} \bigr)} \right]_{\ge 10^{-10}}\end{gathered}\end{split}\]

where the second exponential in \(f^{{{\text{phy}}}}\) again is only present with DARWIN_TEMP_RANGE.

8.7.3.19.3. DARWIN_TEMP_VERSION 3

\[\begin{split}f^{\text{phy}}_j(T) = f^{\text{up}}_j(T) = f^{\text{het}}_j(T) = f^{\text{graz}}_z(T) = f^{\text{mort}}(T) = f^{\text{mort2}}(T) = f^{\text{remin}}(T) = \\ = \left[ {\mathrm{e}}^{A_{\text{e}} (T - T_{\text{ref}})} \right]_{\ge 10^{-10}}\end{split}\]

where \(A_{{\mathrm{e}}}=0.05/{}^\circ{\rm C}\) 1 and \(T_{\text{ref}}=20\,^\circ{\rm C}\).

8.7.3.19.4. DARWIN_TEMP_VERSION 4

Temperature functions are exponential, with an optional restriction on their range,

\[ \begin{align}\begin{aligned}f^{\text{phy}}_j(T) &= \mathrm{e}^{A^{\text{phy}}_{\text{e} j} (T - T_{\text{ref}})} \mathrm{e}^{-e_{2 j} {|T - T^{\text{opt}}_j|}^{p_j}}\\f^{\text{het}}_j(T) &= \mathrm{e}^{A^{\text{het}}_{\text{e} j} (T - T_{\text{ref}})} \mathrm{e}^{-e^{\text{het}}_{2 j} {|T - T^{\text{opt het}}_j|}^{p^{\text{het}}_j}}\\f^{\text{graz}}_j(T) &= \text{e}^{A^{\text{zoo}}_{\text{e} j} (T - T_{\text{ref}})} \mathrm{e}^{-e^{\text{graz}}_{2 j} {|T - T^{\text{opt graz}}_j|}^{p^{\text{graz}}_j}}\\f^{\text{mort}}(T) &= \mathrm{e}^{A^{\text{mort}}_{\text{e}} (T - T_{\text{ref}})}\\f^{\text{mort2}}(T) &= \mathrm{e}^{A^{\text{mort2}}_{\text{e}} (T - T_{\text{ref}})}\\f^{\text{remin}}(T) &= \mathrm{e}^{A^{\text{remin}}_{\text{e}} (T - T_{\text{ref}})}\\f^{{{\text{up}}}}_j(T) &= \mathrm{e}^{A^{\text{uptake}}_{\text{e}} (T - T_{\text{ref}})}\end{aligned}\end{align} \]

where \(T_{\text{ref}}=20\,^\circ{\rm C}\) and the exponentials with \(e^{*}_{2j}\) are only present if DARWIN_TEMP_RANGE is defined. The main exponential temperature dependence corresponds to the use of a Q₁₀ temperature coefficient,

\[\mathrm{e}^{A_{\text{e}}(T-T_{\text{ref}})} = Q_{10}^{(T-T_{\text{ref}})/10\,^\circ\mathrm{C}}\]

where

\[Q_{10} = \mathrm{e}^{A_{\text{e}}\cdot 10\,^\circ\mathrm{C}} \quad \text{or} \quad A_{\text{e}} = \tfrac{1}{10\,^\circ\mathrm{C}} \ln Q_{10}\]

The default value corresponds to \(Q_{10}\approx1.55\).

Note that while the temperature dependences of versions 2, 3 and 4 are similar, versions 3 and 4 yield rates about 70% higher than version 2 with the default TempCoeffArr. The magnitude of version 1 functions is similar to version 2 ones. Default phytoplankton temperature functions for all DARWIN_TEMP_VERSION. shows the phytoplankton temperature functions with default parameters for the four versions.

Figure 8.16 Default phytoplankton temperature functions for all DARWIN_TEMP_VERSION.

The parameters of all temperature functions are summarized in Table 8.57.

Table 8.57 Temperature function parameters

Trait	Parameter	Default Value
for version 1:
phytoTempCoeff	a_phytoTempCoeff	\(c_j=1/3\)
phytoTempExp1	a_phytoTempExp1	\(e_{1j}=1.04\) 2
	tempnorm	\(n=0.3\)
for version 2:
	TempCoeffArr	\(c^{\text{Arr}}=0.5882\)
	TempAeArr	\(A^{\text{Arr}}_{{\text{e}}}=-4000\,{\rm K}\) 3
	TempRefArr	\(T^{\text{Arr}}_{\text{ref}}=293.15\,{\rm K}\)
for version 4:
phytoTempAe	a_phytoTempAe	\(A^{\text{phy}}_{\text{e}j}=0.0438/{}^\circ\mathrm{C}\) 4
hetTempAe	a_hetTempAe	\(A^{\text{het}}_{\text{e}j}=0.0438/{}^\circ\mathrm{C}\)
grazTempAe	a_grazTempAe	\(A^{\text{graz}}_{\text{e}j}=0.0438/{}^\circ\mathrm{C}\)
	reminTempAe	\(A^{\text{remin}}_{\text{e}}=0.0438/{}^\circ\mathrm{C}\)
	mortTempAe	\(A^{\text{mort}}_{\text{e}}=0.0438/{}^\circ\mathrm{C}\)
	mort2TempAe	\(A^{\text{mort2}}_{\text{e}}=0.0438/{}^\circ\mathrm{C}\)
	uptakeTempAe	\(A^{\text{uptake}}_{\text{e}}=0.0/{}^\circ\mathrm{C}\)
for TEMP_RANGE:
phytoTempExp2	a_phytoTempExp2	\(e_{2j}=0.001\)
phytoTempOptimum	a_phytoTempOptimum	\(T^{\text{opt}}_j=2.0\,^\circ{\rm C}\)
phytoDecayPower	a_phytoDecayPower	\(p_{j}=4.0\)
hetTempExp2	a_hetTempExp2	\(e^{\text{het}}_{2j}=0.001\)
hetTempOptimum	a_hetTempOptimum	\(T^{\text{opt het}}_j=2.0\,^\circ{\rm C}\)
hetDecayPower	a_hetDecayPower	\(p^{\text{het}}_{j}=4.0\)
grazTempExp2	a_grazTempExp2	\(e^{\text{graz}}_{2j}=0.001\)
grazTempOptimum	a_grazTempOptimum	\(T^{\text{opt graz}}_j=2.0\,^\circ{\rm C}\)
grazDecayPower	a_grazDecayPower	\(p^{\text{graz}}_{j}=4.0\)

With random trait generation, \(T^{\text{opt}}_j\) is drawn from a range [tempmax–temprange, tempmax].

1

Corresponds to an activation energy of 35.725 kJ mol^–1 at 20°C.

2

Corresponds to an activation energy of 28.023 kJ mol^–1 at 20°C.

3

Corresponds to an activation energy of 33.257 kJ mol^–1 at 20°C.

4

Corresponds to an activation energy of 31.314 kJ mol^–1 at 20°C.