# Ice Phenology and Thickness Modelling for Lake Ice Climatology

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Equations for Water Temperature

^{−3}°C

^{−2}is the fit parameter of the quadratic density equation in (2e). The dimension of ${\lambda}_{a},{\lambda}_{1}\mathrm{a}\mathrm{n}\mathrm{d}{\lambda}_{2}$ is inverse time, they are the inverse relaxation times in the system; and the dimension of ${f}_{a}\mathrm{a}\mathrm{n}\mathrm{d}{f}_{b}$ is temperature/time, they represent the surface and bottom heating rates. The surface heat flux parameters in general depend on time. ${K}_{0}$ is highly positive in the summer due to solar radiation while in the winter, it depends on latitude, and ${K}_{1}~10\u201320$ W m

^{−2}°C

^{−1}can be taken as fixed in the first approximation [17]. ${K}_{0}$ also depends on evaporation, and ${K}_{1}$ depends largely on wind speed.

#### 2.2. Ice–Water Model

## 3. Equilibrium

#### 3.1. Temperature Equilibrium

^{−2}°C

^{−1}represents the turbulent thermal conductivity.

^{−1}to keep the open surface state.

**Example**

**1.**

^{−2}, ${K}_{1}~20$ W m

^{−2}°C

^{−1}, ${K}_{0}~0$, and ${k}_{w}~50W{m}^{-2}{\mathbb{C}}^{-1}$. Then, the equilibrium is ${T}_{1}={T}_{a}+0.25\mathbb{C}$, ${T}_{2}={T}_{1}+0.1\mathbb{C}$. The lake freezes at ${T}_{a}<{T}_{f}-0.25\mathbb{C}.$ The relaxation time is ${\lambda}_{a}^{-1}\approx 12.5$ d.

#### 3.2. Ice Equilibrium

^{−2}${\mathbb{C}}^{-1}$, at the extreme, down to the molecular value ${k}_{w}=0.56$ W m

^{−2}${\mathbb{C}}^{-1}$. For ${k}_{w}~10$ W m

^{−2}${\mathbb{C}}^{-1}$ and ${Q}_{b},{\delta Q}_{s}~5\mathrm{W}{\mathrm{m}}^{-2}$, we have ${T}_{1}~{T}_{f}+1\mathbb{C},{T}_{2}~{T}_{1}+0.5\mathbb{C}$. If ${T}_{f}-{T}_{0}=10\mathbb{C}$, the equilibrium ice thickness is 2.1 m.

^{−2}to reach the equilibrium in 3 months. Thus, in typical conditions in boreal and tundra lakes, there is no climatic equilibrium state for ice thickness but only ice growth and melt periods. The growth period ends when the ice surface temperature settles to the melting point in late winter. In low-latitude, dry climate alpine lakes, a large fraction of sunlight penetrates the ice, making seasonal equilibrium possible [26].

**Example**

**2.**

^{−2}, we have $h~2.1$ m. The time scale of ice growth is then almost 2 years. Then, at $h~1$ m, the growth rate is 0.85 cm d

^{−1}. With low equilibrium thickness, the time scale is short, e.g., for ${{\delta Q}_{s}+Q}_{b}~100$ W m

^{−2}, the equilibrium thickness is 20 cm, and the time scale is 1 week.

^{−2}. In the case of sediment heat flux, if ${T}_{b}>8\mathbb{C}$ and ${k}_{b}>0$, the heating keeps on convection through all depths, resulting in an open-water situation.

## 4. Time Evolution

#### 4.1. The Two-Layer System

^{−1}divided by depth in meters, with ${\lambda}_{1}/{\lambda}_{2}={H}_{2}/{H}_{1}$.

**Example**

**3.**

^{−1}, and ${\left|{r}_{\mathrm{1,2}}\right|}^{-1}=$52 d, $7.6$ d. In a deep lake, ${H}_{1}=10$ m and ${H}_{2}=50$ m, and we have ${\left|{r}_{\mathrm{1,2}}\right|}^{-1}=210d,9.5$ d. Thus, there is a rapid first adjustment of the upper layer followed by a slow adjustment of the whole system.

^{−1}and ${H}_{max}=30$ m, ice extent increases by 6.7% per day and reaches 100% 15 days after first freezing at the shore. The solution is sensitive to the initial freezing date ${t}_{F0}$, which depends on the air temperature evolution, e.g., in Lake Ladoga, where the mean and maximum depths are 48 m and 210 m, it takes an average of 58 days to reach the complete ice coverage [14].

#### 4.2. Ice-Cover Thickness

^{−2}, we have ${h}_{e}=$ 2.1 m. The time scale is $\tau \approx 2$ years, and the thickness is half of the equilibrium or 1.05 m at $t=0.193\xb7\tau $ ≈ 140 days. For ${Q}_{w}=0,{h}_{e}=\infty $, and in 140 days, the ice would grow to 1.2 m.

^{−1}(a very high amplitude, corresponding to $\u2206{Q}_{w}\approx 35$ W m

^{−2}) and ${T}_{f}-{T}_{0}=10\mathbb{C}$, then ${h}_{e}=60$ cm and $\eta =2.16$ month

^{−1}. For a daily cycle, we have $\u2206e=0.11$ cm and $\eta =0.017$ rad, while for a monthly cycle, $\u2206e=3.4$ cm and $\eta =0.50$ rad. The phase shift is nearly $\pi /2$ rad or 6 h in the first case, and it is 1.49 rad or 7.1 d in the second case.

## 5. Discussion

^{−2}in the summer and ${K}_{0}~-50$ W m

^{−2}in the winter; while ${K}_{1}$ is more stable, ${K}_{1}~10$ W m

^{−2}°C

^{−1}is strictly positive in practice. The ratio ${K}_{0}/{K}_{1}$ adds a term in the solution of the surface temperature; at ${K}_{0}/{K}_{1}=0$, the upper-layer temperature equals a low-pass-filtered air temperature, while for large $\left|{K}_{0}/{K}_{1}\right|$, the situation is dominated by the radiation balance.

## 6. Concluding Remarks

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A sketch of the model structure. Red arrows are for heat transfer; yellow disc is the Sun.

**Figure 2.**A plot of dimensionless ice thickness vs. dimensionless time illustrating the influence of heat flux from the water body $\left({Q}_{w}\right)$ on ice growth. Ice thickness approaches the asymptotic equilibrium thickness $\left({h}_{e}\right)$ with time proportional to ${h}_{e}/{Q}_{w}$.

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Leppäranta, M.
Ice Phenology and Thickness Modelling for Lake Ice Climatology. *Water* **2023**, *15*, 2951.
https://doi.org/10.3390/w15162951

**AMA Style**

Leppäranta M.
Ice Phenology and Thickness Modelling for Lake Ice Climatology. *Water*. 2023; 15(16):2951.
https://doi.org/10.3390/w15162951

**Chicago/Turabian Style**

Leppäranta, Matti.
2023. "Ice Phenology and Thickness Modelling for Lake Ice Climatology" *Water* 15, no. 16: 2951.
https://doi.org/10.3390/w15162951