Modified parameterization of the vertical water temperature profile in the FLake model

Figures & data

Fig. 1. The scheme of the vertical temperature distribution in the system ‘snow – ice – water – bottom sediments’ realized in the model FLake: (a) open water case; (b) ice-covered period.

Table 1. Parameterizations suggested by different authors to describe the vertical distribution of temperature in water basins.

Authors	Parameterization		Application
Anderson and Munk 1947 (cited from Mironov 2008)	Verbal mention of the ‘almost constant shape’ of the temperature profile beneath the upper mixed layer		Lake water column
Kitaigorodskii and Miropolsky (1970)	$T^{\ast }(\zeta )=\frac{8}{3}·\zeta -2·{\zeta }^2+\frac{1}{3}·{\zeta }^4$	(8)	Oceanic active layer
Arsenyev and Felzenbaum (1977)	T*(ζ) = 1 – (1 – ζ)^3	(9)	Oceanic active layer
Malkki and Tamsalu (1985)	$T^{\ast }(\zeta )=\left\{\begin{array}{ccc}1-{(1-\zeta )}^3\hfill & \text{at}\hfill & \text{d}h/\text{d}t>0\hfill \\ 1-4·{(1-\zeta )}^3+3·{(1-\zeta )}^4\hfill & \text{at}\hfill & \text{d}h/\text{d}t\le 0\hfill \end{array}\right.$	(10)	Baltic Sea active layer, water column of lakes
Golosov et al. (2003)	T*(ζ) = (2 – B(t)) ⋅ ζ + (B(t) – 1) ⋅ ζ^2	(11)	Water column of lakes at ice cover period
Mironov (2008)	T*(ζ) = ζ	(12)	Water column of lakes at ice cover period, snow and ice cover
Golosov et al. (1998); Golosov and Kirillin (2010)	$\left\{\begin{array}{ccc}{T}_1^{\ast }({\xi }_1)=2·{\xi }_1-{\xi }_1^2\hfill & \text{at}\hfill & D\le z\le H\hfill \\ T_2^{\ast }({\xi }_2)=3·{\xi }_2^2-2·{\xi }_2^3\hfill & \text{at}\hfill & H\le z\le L\hfill \end{array}\right.$	(13)	Bottom sediments of lakes

Fig. 2. The scheme of linear temperature profile (Table of symbols in Fig. 2 are the same as those in Fig. 1), T(z) is temperature at depth z, A, B, C, D, E are the vertices of a right triangles.

Fig. 3. Allocation scheme of temperature sensors at the thermistor chain.

Fig. 4. Relationship between different parameterizations. Here empty circles mark ‘winter’ data, black squares stand for data, obtained during the open water period; upper solid line coincides with parameterization (16) at A = 3 and B = 0, the lower solid line coincides a new parameterization (16) at A = 0 and B = 3. Intermediate thin lines correspond to parameterizations (8)–(13).

Table 2. Relationship between different parameterizations.

Authors	Parameterization		Values of A(t) and B(t) in new parameterization
Kitaigorodskii and Miropolsky (1970)	$T^{\ast }(\zeta )=\frac{8}{3}·\zeta -2·{\zeta }^2+\frac{1}{3}·{\zeta }^4$	(8)	$\begin{array}{c}A=2.85\hfill \\ B=0\hfill \end{array}$
Arsenyev and Felzenbaum (1977)	T*(ζ) = 1 – (1 – ζ)^3	(9)	$A=3,B=0$
Malkki and Tamsalu (1985)	$T^{\ast }(\zeta )=\left\{\begin{array}{ccc}1-{(1-\zeta )}^3\hfill & \text{at}\hfill & \text{d}h/\text{d}t>0\hfill \\ 1-4·{(1-\zeta )}^3+3·{(1-\zeta )}^4\hfill & \text{at}\hfill & \text{d}h/\text{d}t\le 0\hfill \end{array}\right.$	(10)	$\begin{array}{cc}A=3,\hfill & B=0\hfill \\ A=?,\hfill & B=?\hfill \end{array}$
Golosov et al. (2003)	T*(ζ) = (2 - B(t)) ⋅ ζ + (B(t) – 1) ⋅ ζ^2	(11)	A(t) ∊ [0, 2], B(t) ∊ [0, 2]
Mironov (2008)	T*(ζ) = ζ	(12)	A = 1, B = 1
Golosov et al. (1998); Golosov and Kirillin (2010)	$\left\{\begin{array}{ccc}{T}_1^{\ast }({\xi }_1)=2·{\xi }_1-{\xi }_1^2\hfill & \text{at}\hfill & D\le z\le H\hfill \\ T_2^{\ast }({\xi }_2)=3·{\xi }_2^2-2·{\xi }_2^3\hfill & \text{at}\hfill & H\le z\le L\hfill \end{array}\right.$	(13)	$\begin{array}{cc}A=2,\hfill & B=0\hfill \\ A=0,\hfill & B=0\hfill \end{array}$

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