# Analysis of Different Freezing/Thawing Parameterizations using the UTOPIA Model

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## Abstract

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## 1. Introduction

## 2. The UTOPIA Model

## 3. The Freezing Parameterization in UTOPIA

#### 3.1. Energy vs. Temperature

_{s}is the soil volumetric thermal capacity, Δz is the soil layer depth and T

_{0}is the freezing point of water (0 °C). The maximum possible change of liquid water (ice) Δη

_{w,max}(Δη

_{i,max}) is:

_{f}is the latent heat of fusion and ρ

_{w}is the water density of water (999.8 kg/m

^{3}). The ice content η

_{i,}depends, as shown by the formula (3), on the soil energy available for the phase change, also on the soil water content η

_{w}(if freezing is taking place) or on the soil ice content η

_{i}(if thawing is taking place):

_{i}is the actual change of ice content.

#### 3.2. Thermal Capacity vs. Temperature

_{T}is the soil thermal conductivity. The volumetric ice content is evaluated by:

_{v}is the vegetation cover, η

_{fc}is the soil water content at the field capacity, and f(T) is a function of temperature given by:

_{0}+ 1K e T

_{2}= T

_{0}- 3K, and T

_{0}is the freezing point of water. The behavior of f(T) is shown in Figure 1.

#### 3.3. Thermal Capacity vs. Temperature

_{i}is given by:

_{w,min}is the minimum quantity of water in soil (set to 0.01 m

^{3}/m

^{3}in order to avoid undesired numerical problem when soil water content tends to zero) and η = η

_{w }+ η

_{i}. Thus, in this case, the ice water content does not depend on the vegetation cover, and is limited by the actual liquid water content.

## 4. Sensitivity Experiments Using Synthetic Data

- Relative humidity (%): min = 61.1; max = 91.5.
- Wind velocity (m/s): min = 2.0; max = 5.0.
- Overcast conditions (W/m
^{2}): max radiation = 74.65; clear sky radiation max = 248.85 . - Pressure: min = 1022.17; max = 1023.73.

SYNTHETIC DATA – UTOPIA setting | ||
---|---|---|

Depth of the six soil layers (cm, top to bottom): 20, 20, 40, 80, 160, 320 | ||

t = 60 s | Initial
_{w} = 0.4 m^{3}/m^{3} | Initial T = 1 °C |

Soil type: silty clay | ||

Soil porosity
_{s}[m ^{3}/m^{3}] | Wilting point
_{wi}[m ^{3}/m^{3}] | Field capacity
_{fc}[m ^{3}/m^{3}] |

0.492 | 0.283 | 0.4 |

Vegetation type: short grass Vegetation cover _{v} = 0.8 |

## 5. Results

_{w}during the freezing period. The warming of the layer in which ice forms is proportional to the quantity of ice present in the soil: this fact is not surprising, as the heating is linked to the latent heat release. In this sense, V and C parameterizations simulate the warmest temperatures and the biggest ice quantity.

**Figure 5.**Simulated soil moisture in the 10 cm soil layer. The cyan line is the porosity η

_{s}, the pink line is the wilting point η

_{wi}.

#### 5..1 Numerical Instability during the Thawing Phase

_{wz}is the soil water flux.

_{w}

_{η}and from the hydraulic diffusivity of water vapor D

_{v}

_{η}. The second term is the water flux due to gravitational drainage, while the last one is the water vapor flux due to the temperature gradient.

^{3}/m

^{3}. For this reason, the water vapor hydraulic diffusivity in the equation of soil water vertical flux has been set to zero for η

_{w}> 0.06 m

^{3}/m

^{3}, leading to a noticeable reduction of the instability (Figure 6 and Figure 7).

**Figure 6.**Soil temperature in the 10 cm soil layer (as for Figure 4) for the simulation performed when the water vapor hydraulic diffusivity is set to zero.

**Figure 7.**Soil moisture in the 10 cm soil layer (as for Figure 5) for the simulation performed when the water vapor hydraulic diffusivity is set to zero

#### 5.2. The Problem of Water Overproduction during Soil Freezing

_{TOT}calculation, we considered the water density equal to 999.8 kg/m

^{3}and the ice density equal to 917.0 kg/m

^{3}.

_{TOT}values corresponding to the simulation referring to Figure 6 and Figure 7. It is evident that, during the period of the soil freezing, the increase of η

_{TOT}is not justified on the basis of considerations related to hydrological soil balance. In fact, in the freezing period, there is no precipitation, and the transpiration stops because the temperatures drop below 0 °C, thus minimizing also the evaporation. Considering also the drainage from the bottom layer, a decrease, and not an increase, of total water content may be considered realistic.

_{tot}is still present.

**Figure 9.**Representation of the moisture fluxes between the different soil layers and the external environment. During the sensitivity test discussed in Section 5.2, transpiration evaporation, infiltration and gravitational drainage have been set to zero.

**Figure 10.**η

_{ tot}trends evaluated in the sensitivity test summarized in Figure 9. The three simulations have been executed using three different time step (DTSEC) values. The numerical convergence toward a constant value is just slightly shown. The complete numerical convergence would have been reached shorter values of DTSEC, however this is not shown because it requires too large a computation time.

_{tot}(in units of volumetric water content). The first one, called η

_{tot;correct}, conserves the water mass during the freezing, while the second variable, called η

_{tot;effective}, does not conserve the water mass.

_{i}> 0.01 m

^{3}/m

^{3}). Thus η

_{tot}begin to increase when the total VSWC of the second layer, where the volumetric ice content is greater than 0.01 m

^{3}/m

^{3}, is considered.

**Figure 11.**η

_{tot}trend relative to the sensitivity test summarized in Figure 9, considering: the three deepest soil layers (top left), the four deepest soil layers (top right), the five deepest soil layers (bottom left), and all soil layers (bottom right).

**Figure 12.**η

_{i}(volumetric ice content) trends relative to the sensitivity test summarized in Figure 9, for the fourth layer (top left), the third layer (top right), second layer (bottom left) and first layer (bottom right).

_{i,tot}, i.e., the total volumetric ice content in all soil layers, is calculated. If η

_{i,tot}is greater than the minimum value 0.01 m

^{3}/m

^{3}(i.e., if there is a significant quantity of ice in soil), the variable η

_{tot,correct}is computed by setting to zero the hydraulic diffusivity (D

_{w}

_{η}= 0), in order to neglect the water fluxes among the soil layers due to the humidity gradient. In this way, the total VSWC (η’

_{w}+η’

_{i}) of each layer contributes to conserve the total VSWC. Thus, the variable η

_{tot;correct}is calculated as:

_{tot,effective}is calculated by accounting the true hydraulic diffusivity (D

_{w}

_{η}≠ 0) and it does not conserve the total VSWC:

_{w}is performed only in those soil layers in which the volumetric ice content is significant (with η

_{i}> 0.01 m

^{3}/m

^{3}):

_{tot,effective}is recalculated from the η

_{w,new}values in the previous iteration based on the following scheme:

_{tot,effective}, as shown in Figure 13.

**Figure 13.**Trends of η

_{tot;correct}for parameterization V without correction (black line) and of η

_{tot;effective}with corrections (other lines) and with different iterations numbers.

_{tot}is justified on the basis of consideration related to hydrological soil balance. The decrease of η

_{tot}is mainly due to drainage from the bottom layer. In fact, in the period with temperature below zero degrees, the transpiration is stopped and evaporation is reduced.

**Figure 14.**Trends of η

_{tot;correct}for simulations without correction (blue line) and of η

_{tot;effective}with correction (red line). In these simulations each soil layer has been initialized with a value of η

_{w }= 0.4 m

^{3}/m

^{3}.

**Figure 15.**Same as Figure 14 but for the parameterization V.

**Figure 16.**Same as Figure 14 but for the parameterization C.

**Figure 17.**Hydrological balance and its components with (a, b) and without (c, d) the correction for the water overproduction and relative to the parameterization V. Notice the different scales of the plots (10

^{-6}for the hydrological balance, 10

^{-7}for the components).

## 6. Conclusions

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**MDPI and ACS Style**

Bonanno, R.; Loglisci, N.; Cavalletto, S.; Cassardo, C.
Analysis of Different Freezing/Thawing Parameterizations using the UTOPIA Model. *Water* **2010**, *2*, 468-483.
https://doi.org/10.3390/w2030468

**AMA Style**

Bonanno R, Loglisci N, Cavalletto S, Cassardo C.
Analysis of Different Freezing/Thawing Parameterizations using the UTOPIA Model. *Water*. 2010; 2(3):468-483.
https://doi.org/10.3390/w2030468

**Chicago/Turabian Style**

Bonanno, Riccardo, Nicola Loglisci, Silvia Cavalletto, and Claudio Cassardo.
2010. "Analysis of Different Freezing/Thawing Parameterizations using the UTOPIA Model" *Water* 2, no. 3: 468-483.
https://doi.org/10.3390/w2030468