2.1 Thickness measurements

The test channels (TCh01 and TCh02) were investigated in two winters and were positioned at the same location, ~200 m from the shore. These channels were formed by breaking an initial layer of level ice with an approximate thickness of 15 cm. The main channel, on the other hand, was only studied during the second winter and was positioned parallel to the test channel, ~400 m from the shore, as indicated in Figure 1c. This navigation route is used all year round, and the brash ice channel was formed in early winter from navigation through thin ice. The development of level ice at the shoreside of the test channels was examined in both years. The test channels were drilled every metre along a selected cross-section. At each location, measurements were taken for the overall brash ice thickness, side ridges, and the level ice in the channel's vicinity. Additionally, the dimensions of water-filled or slush-filled macro pores, the freeboard, as well as the thicknesses of snow and slush were measured. Cross-section measurements were carried out after the brash ice was partly consolidated.

The first channel was navigated 9 times, and 18 cross-sections were investigated in total. The second test channel was navigated 10 times, and 9 cross-sections were measured. The main channel was navigated 116 times until the final measurement, and the channel's cross-section was measured three times. The specific dates and times of the breaking events and corresponding measurements are detailed in an earlier study (Zhaka and others, Reference Zhaka, Bridges, Riska, Nilimaa and Cwirzen2023a, Reference Zhaka, Bridges, Riska and Cwirzen2023c). For each cross-section that was measured, several parameters were calculated. These include the average thickness of level ice at the vicinity of the channel, the total equivalent thickness of brash ice which assumes that all the brash ice remains in a ship channel, the average brash ice thickness remaining in the channel, and the equivalent thickness of the side ridges. Illustrative examples of cross-section profiles for TCh02 after the second and tenth breaking events are presented in Figure 3. These cross-sections demonstrate the progression in the geometry of TCh02. For example, the thickness, width and macroporosity increased from the second to the tenth breaking event.

Figure 3. Cross-section profiles of the second test channel (TCh02): (a) after the 2nd breaking event (BE) and (b) after the 10th breaking event. The thickness of brash ice blocks above and below the WL is represented by dark grey bars, while the thickness of snow is illustrated with lighter grey bars. Macropores' dimensions are depicted using blue bars. The snow top is delineated by a solid cyan line, while the freeboard and the bottom of the brash ice are outlined with solid black lines.

On the shoreside of both test channels, measurements were taken of various level ice parameters, including level ice, snow ice, congelation ice, snow and slush thicknesses, as well as the freeboard. During the first winter, level ice measurements were conducted six times, within a surface region specified by a grid pattern with 30 m spacing along the channel and 20 m perpendicular to the shore. In the second winter, level ice measurements were carried out 12 times, covering a distance of 60 m along the channel and 40 m towards the shore.

4.1 Brash ice accumulation

The snow–slush transformation does not contribute to the overall thickness of the brash ice after a breaking event. Instead, the slush fills the wet macropores between the submerged ice pieces. As a result, the equivalent brash ice thickness along the ship channel after the breaking event is determined by the mass conservation of the ice before and after the ship passage (Riska and others, Reference Riska2019). The equivalent brash ice thickness (H _B) after the breaking incident can be expressed as follows:

(9)$$H_{\rm B} = H_{{\rm D}( {\,j-1} ) } + \displaystyle{{H_{{\rm C}( {\,j-1} ) }} \over {( {1-p_0} ) }} + H_{{\rm W}( {\,j-1} ) }\displaystyle{{( {1-p_{( {\,j-1} ) }} ) } \over {( {1-p_0} ) }}.$$

Assuming that the average temperature of brash ice (T _av) is reached instantly, it can be calculated using the mass and heat conservation equations before and after the breaking event, taking into account the contributions from all four layers. This average temperature (T _av) can be expressed as:

(10)$$T_{{\rm av}} = \displaystyle{ { \eqalign{& { ( {T_{{\rm SA}} \! + \! T_{{\rm DS}}/2} ) ( {1 \!- \!p_{\rm s}} ) H_{{\rm S}( {\,j-1} ) }} { + ( {T_{{\rm SI}} \! + \! T_{{\rm DS}}/2} ) ( {1-p_0} ) H_{{\rm D}( {\,j-1} ) }} \cr& {+ ( {T_{\rm F} + T_{{\rm SI}}/2} ) H_{{\rm C}( {\,j-1} ) } + T_{{\rm F\;}}( {1-p_{\,j-1}} ) H_{{\rm W}( {\,j-1} ) }} }} \over {( {1-p_0} ) H_{{\rm D}( {\,j-1} ) } + H_{{\rm C}( {\,j-1} ) } + ( {1-p_{\,j-1}} ) H_{{\rm W}( {\,j-1} ) }}}, \;$$

where T _SA, T _DS and T _SI are the snow/air, dry brash ice/snow and dry brash ice/consolidated brash ice interface temperatures.

In the second step of the calculation, the initial porosity (p ₀) of wet brash ice is assumed to instantaneously change to a new porosity value p _i. The reduction in porosity occurs due to the freezing of the water fraction in the slush-filled voids or the water-filled voids, driven by the temperature gradient between the water and ice. If no snow is present, as described by Riska and others (Reference Riska2019), then the porosity can be expressed as:

(11)$$p_j = p_{0\;}-\displaystyle{{c_{\rm i}\Delta T} \over {L_{\rm i}}}( {1-p_{0}} ) ,$$

where ΔT is the temperature difference until equilibrium (T _F − T _av), and c _i is the specific heat capacity of ice, see Table 1 in Appendix.

If snow is present, and the slush formed after the ship passage completely fills the voids within the entire brash ice layer, then the energy and mass balance equations between the slush-filled pores and brash ice pieces, as well as the final porosity equation, can be expressed as:

(12)$$H_{\rm W}\rho _{\rm i}L_{\rm i}( {v_{\rm w}p_{0\;}-p_{\rm j}} ) = \rho _{\rm i}c_{\rm i}\Delta TH_{\rm W}( {1-p_{0}} ) ,$$

(13)$$p_j = v_{\rm w}p_{0\;}-\displaystyle{{c_{\rm i}\Delta T} \over {L_{\rm i}}}( {1-p_{0}} ).$$

If the voids within the brash ice layer are partially filled with slush and partially with water, then the thickness of brash ice with slush-filled voids is H _BSL = H _S/p ₀. The energy and mass conservation equation between the slush-filled and water-filled voids with the brash ice pieces, along with the derived porosity equation, are as follows:

(14)$$\begin{align}\displaystyle{{H_{\rm S}} \over {\,p_{0\;}}}&\rho _{\rm i}L_{\rm i}( {v_{\rm w}p_{0\;}-p_{\rm j}} ) + \left({H_{\rm W}-\displaystyle{{H_{\rm S}} \over {\,p_{0\;}}}} \right)\rho _{\rm i}L_{\rm i}( {\,p_{0\;}-p_{\rm j}} ) \cr & = \rho _{\rm i}c_{\rm i}\Delta TH_{\rm W}( {1-p_{0}} ) , \;\end{align}$$

(15)$$p_{\rm j} = \displaystyle{{H_{\rm S}( {v_{\rm w}-1} ) + H_{\rm W}( {\,p_{0\;}-c_{\rm i}\Delta T( 1-p_{0\;}) /L_{\rm i}} ) } \over {H_{\rm W}}}.$$

4.2 Brash ice consolidation

The third step of the model calculates the brash ice consolidation between two ship passages. Initially the model tests the presence of slush and estimates the thickness of brash ice occupied by slush (H _BSL = H _S/p ₀). If slush is present, the brash ice consolidation will occur only in the slush filled brash ice macropores and in particular in the water fraction of the slush.

If the entire wet brash ice column is filled with slush, then the freezing process will occur within the water fraction v _w of the slush-filled macropores until the wet brash ice fully consolidates, thereafter only water freezing is considered. In case that brash ice is filled partly with slush and partly with water, it is assumed that consolidation occurs initially in the H _BSL layer and then in the rest of wet brash ice.

The snow thickness that covers the ship channels between breaking incidents is calculated by the incoming snow thickness measured by the SMHI meterological station. The brash ice growth equation is derived by substituting the interface temperatures obtained from the first three continuous heat flux equations and integrating the fourth equation. These equations are illustrated in Figure 5. The brash ice growth equation is:

(16)$$H_{\rm C} = \sqrt {k_{{\rm bi}}^2 h_{\rm e}^2 -\displaystyle{{2k_{\rm bi}} \over {\rho _{{\rm bi}}p_jv_{\rm w}L_{\rm i}}}\theta } -k_{{\rm bi}}h_{\rm e}.$$

where p_jv _w accounts for the freezing of slush within the slush-filled macropores, ρ _bi is the brash ice density, v _w is the not considered for the part of wet brash ice that is solely filled with water. Additionally, p _j is assumed equal to one if the brash ice is fully consolidated and h _e is given as:

(17)$$h_{\rm e} = \displaystyle{{H_{\rm D}} \over {k_{\rm d}}}\;+\;\displaystyle{{H_{\rm S}} \over {k_{\rm S}}}\;+\;\displaystyle{1 \over {h_{\rm a}}}, \;$$

where k _d and k _s are the heat conductivities of the dry brash ice and snow, h _a is the surface convective heat transfer coefficient.

4.3 Expelling coefficient

The estimated brash ice equivalent thickness (H _B) can be compared with the brash ice equivalent thickness (H _eq) observed from all the measured cross-sections of the ship channel's (TCh01, TCh02, MCh). This equivalent brash ice thickness (H _eq) can be estimated as the total surface area of the ice accumulated on the channel (A _BI) and both side ridges (A _R) divided by the width of the vessel as:

(18)$$H_{{\rm eq}} = \displaystyle{{A_{{\rm BI}} + A_{\rm R}} \over {W_{{\rm BICh}}}}$$

A schematic illustration of the channels' cross-section and the surface areas of brash ice and side ridges, as well as a schematic illustration of the brash ice equivalent thickness are provided in Figures 6a, b. The channel's width (W _BICh) was considered to be 12 m for the TChs, which represents the beam of tug Viscaria. The main channel (MCh) was navigated by several vessels with beams varying between 12 and 31 m. Tug Viscaria would often navigate in the sides of this channel and not at the centre. Thus, the expelling coefficient for all three cross-section profiles of the MCh was calculated considering a constant width of 26 m representing the main navigation lane in the centre line. Also, the actual channel widths measured in each cross-section were used, these widths were 36, 30 and 42 m after 14, 73 and 116 BE, respectively.

Figure 6. (a) Typical cross-section of a ship channel assuming a width equal to the vessel's beam (W _BICh). The surface area covered by brash ice and the areas occupied by the side ridges are denoted as A _BI and A _R. (b) The equivalent brash ice thickness. A scheme of the first brash ice growth model (BIGM1) that does not consider the loss of ice sideways. (c) The average brash ice thickness that remains in the ship channel and the equivalent side ridge thickness. A scheme of the brash ice growth model that considers the ice loss sideways (BIGM2).

The trapezoid method was used to calculate the cross-sectional areas of brash ice and ridges (excluding the level ice thickness) for each cross-section measured in all three channels. If the measured thicknesses at each hole are H _i and the distance between these w _i then the surface area is equal to:

(19)$$A_{{\rm R/BI}} = \mathop \sum \nolimits^{} 0.5\;( {H_{\rm i} + H_{{\rm i\ + \ 1}}} ) w_{\rm i}.$$

H _i represents the thicknesses measured in each drilled hole along the channels' cross-section, while w _i is the distance between the drilled holes. In the case of the test channel, the thicknesses were measured every metre while in the main channel distances of 2, 2.5 and 5 m were used.

At each vessel passage, part of the brash ice is expelled sideways and piled under the level ice layer, forming side ridges on both sides of the channel, see Figure 6. The fraction of ice that has been moved sideways is calculated for each measured cross-section in all three ship channels assuming the following procedure. Initially, the average thickness of brash ice that remains in the channel is calculated as the surface area of the ice cross-section inside the channel divided by the channel's width, which is assumed to be equal to the vessel's beam. Thereafter, the equivalent thickness of the ice that is expelled sideways (H _Req) is estimated as the ratio between the surface area of ice cross-section in both side ridges with the channel's width. Then the cumulative expelling coefficient (χ_j) is calculated for each measured cross-section as:

(20)$$\chi _j = \displaystyle{{H_{{\rm Req}}} \over {H_{{\rm eq}}}}.$$

This cumulative expelling coefficient χ_j represents the total fraction of ice that is expelled sideways from the first ship passage until the jth ship passage when the cross-section measurements were conducted. The difference χ = χ_j − χ_j−₁ will give the fraction of ice that is expelled from the channel at each ship passage. It should be noted that the calculations of χ_j are based on the measured brash ice cross-sections, which were taken after the brash ice had undergone partial consolidation. As a result, the potential influence of the ice formation around the blocks on the value of χ_j at the time of measurement is neglected. χ_j can be included in the brash ice growth model discussed in the earlier section. The average thickness of the brash ice (H _BI) that remains in the channel and the side ridge equivalent thickness (H _R) can be estimated as follows:

(21)$$H_{{\rm BI}} = ( {1-\chi_j} ) H_{\rm B}, \;\;$$

(22)$$H_{\rm R} = \chi _jH_{\rm B}.$$

A schematic illustration of this concept is given in Figure 6c. For modelling purposes, the ridge porosities are assumed equal to the brash ice porosities. The analysis of the measured cross-sections showed that the maximum initial porosity, considering both brash ice and side ridges, was found to be about 25%, which is the value used in this study (see Zhaka and others, Reference Zhaka, Bridges, Riska and Cwirzen2023c).

The brash ice and ridge thicknesses estimated by the model (H _BI and H _R) can be validated with the results obtained from all three ship channels. For the test channels, it is assumed that the equivalent ridge thickness represents the amount of ice that is expelled from the channel and accumulates in a width equal to the vessels beam (12 m). However, similarly to the equivalent brash ice thickness that represents the total volume of brash ice, the equivalent ridge thickness calculated from the model and the cross-sections of all three channels represent the volume of ice expelled from the channel.

In addition, to using the cumulative expelling coefficient χ_j to estimate the brash ice and side ridge thicknesses, the fraction of brash ice that is expelled at each ship passage (χ) is assumed constant and is incorporated in the model to continuously estimate the thickness of the brash ice that remain in the channel and the equivalent ridge thickness at each ship passage.

5.1 Level ice adjacent to test channels

Figure 7a illustrates the measured and estimated thicknesses of snow, level ice and snow ice on the shoreside of TCh01. The snow–ice thickness adjacent to TCh01 was measured four times, while the thicknesses used to validate the model calculations were measured during each cross-section profile assessment. Figure 7b illustrates the measured and estimated thicknesses of snow, level ice and snow ice on the shoreside of TCh02. The level ice model best predicted the thicknesses for a surface heat convective coefficient of 10 W m⁻¹ °C⁻² in the first winter and 20 W m⁻¹ °C⁻² in the second winter. The air temperatures and snow thicknesses used as model input are given in Figure 2.

Figure 7. Measured and estimated thicknesses of level ice (H _I), snow ice (H _SI), and snow (H _S) adjacent to (a) the first test channel (2020–21) and (b) the second test channel (2021–22). The solid and dashed lines depict the model estimations where the snow–slush transformation was calculated using Eqns (4) and (5), respectively.

In the first winter, the predicted H _SI and H _S were in good agreement with the measured values for both approaches used to estimate the slush formation, the empirical linear regression and the mass conservation principles. A better agreement was achieved for the H _I when the slush thickness was calculated based on the mass conservation (Eqn 4), while using the empirical correlation between the H _SL and WL overestimated the level ice thickness from the 66th day, corresponding to 15 of February and thereafter.

During the second winter, the measured thicknesses of snow ice, and snow, similarly to the first winter were well predicted using both approaches, while the total level ice thickness was best predicted when H_SL was calculated from the empirical regression Eqn 5. On the other hand, when the slush thickness was estimated using Eqn 4, it led to an underestimation of the total ice thickness.

Using the empirical regression function (Eqn 5) resulted in slightly higher snow–ice thickness compared to the mass conservation approach. This difference started with the initial snowfall that occurred at the beginning of winter when the ice was thin. For example, in the second winter, the first 4 cm of incoming snow that covered the thin ice, based on the mass conservation approach, formed a slush layer of 1.6 mm. In contrast, the empirical regression function assumes that, on average, 4 cm of snow transforms into slush due to snow capillarity. Therefore, in this approach, all the initial snow thickness transforms into slush and snow ice. The difference in this initial snow–ice thickness between model approaches caused a difference in the total level ice thickness, which, in turn, affected the difference in snow–ice thickness between the two approaches, thereafter. With a higher level ice thickness and lower snow thickness (see the dashed lines in Fig. 7), there is a lower imbalance between the snow and ice layers, and therefore, the snow–ice thickness difference between the two approaches reduced over time.

5.2 Brash ice growth

The equivalent brash ice thickness was simulated using three different models: the brash ice growth model (BIGM) including two effects of snow, the model that considers only the snow insulative effect, and the original model which omits the snow effects or assumes a zero snow thickness. The brash ice thickness was simulated for a total of 150 days, using three levels of constant air temperatures, −5, −10 and −20°C, corresponding to a θ of 750, 1500 and 3000°C days. The total number of breaking events was 38 passages with a navigation frequency of one passage every 4 d. The snow thickness was assumed to increase linearly with time until it reached a total thickness of 0.75 m after 150 days. The effect that various ship frequencies have on the BIGM was assessed for a total of 298, 149, 75 and 38 ship passages. Finally, the effect that snow–slush transformation fraction and slush water content have on the brash ice equivalent thickness were assessed for a snow–slush transformation fraction varying from 0.3 to 1.0 and water contents in slush varying between 0.3 and 0.73.

5.2.1 Effects of snow

The equivalent brash ice thickness (H _B) obtained from the model that considers the snow effects for a total snow thickness of 0.75 m, is compared with the model results when only the insulative effect is considered for the same snow thickness input, and with the model that neglects the snow thickness, see Figure 8. The convective heat transfer coefficient used in all three models is equal to 10 W m⁻¹ °C⁻¹. The brash ice thickness was estimated for 38 ship passages and three different cumulative freezing air temperatures.

Figure 8. (a) Brash ice equivalent thickness (H _B) estimated from the BIGM which includes both effects of snow (H _SL), only the snow insulative effect (H _S), and the original model for a zero-snow thickness (H _Z). (b) The brash ice macroporosity (p _i) change simulated from the model that considers both effects of snow (P _SL), and the one considering only the snow insulative effect (P _S).

The slower growth rate caused by snow insulation is offset by slush freezing once the snow is submerged in water. The insulation effect of snow is more prominent than the increase in thickness due to the quicker growth of slush-filled pores compared to water-filled pores. The maximum difference in the total simulated brash ice thickness between bare ice and a scenario with a total snow thickness of 0.75 m is 10 cm for 750°C days, 24 cm for 1500°C days, and 45 cm for 3000°C days.

The brash ice thickness is lower when only the insulative effect is considered in the BIGM, compared to when both effects of snow are considered. The difference in H _B between the two models increases as the cumulative freezing air temperatures (θ) decreases. This indicates that in milder winters with relatively high snow thickness, neglecting the slush–snow–ice transformation but including the snow insulative effect leads to higher errors in estimating the brash ice thickness. The maximum differences in H _B between models were 20, 17 and 11 cm for 750, 1500 and 3000°C days, respectively.

A higher initial reduction in macroporosity is obtained when the snow effects are considered (see Fig. 8b), as the slush freezing till temperature equilibrium is quicker than freezing water for the same initial porosity (Zhaka and others, Reference Zhaka, Bridges, Riska, Hagermann and Cwirzen2023b). The porosity reduction increases as the cumulative freezing air temperatures increases when only the insulative effect is considered but decreases when the slush–snow ice transformation is also included. When only the snow insulative effect is considered, the porosity is higher for lower θ due to a lower growth rate. However, when the slush–snow–ice transformation is considered, the same slush content occupied a larger fraction of macropores for lower H _B compared to H _B caused by higher θ. As a result, there is a higher porosity reduction with lower θ scenarios, when the column of brash ice has a greater proportion of slush.

5.2.2 Frequency of navigation, water fraction in slush, snow–slush transformation fraction

The snow effects BIGM is assessed for different frequencies of ship navigation, various slush water fractions and snow–slush transformation fractions. A θ of 1500°C days and a total snow thickness of 0.75 m is considered in all the simulations. Figure 9a shows the maximum brash ice thickness reached after 150 days when a channel is navigated twice a day (2BEs/day), once a day (1BE/day), once in two days (1BE/2 days) and once in four days (1BE/4 days). All the input parameters were consistent, with the only variation being the number of ship passages. The increase in the number of breaking events has resulted in a higher amount of ice accumulating in the channel, as brash ice consolidates more rapidly when it is frequently broken and mixed. Moreover, the frequent submergence of snow, forming slush and subsequently snow ice may have influenced the overall process. According to the model estimates, the snow accumulation on the ship channel between two passages ranges from 0.005 m to 0.02 m for a navigation frequency between 2BEs/day and 1BE/4 days. The brash ice thickness was 1.5 m higher when the ship channel was navigated twice a day compared to when it was navigated once every four days.

Figure 9. (a) Brash ice equivalent thickness (H _B) and (b) macroporosity variation for different frequencies of navigation. (c) The equivalent brash ice thickness (H _B) was estimated assuming various water content (v _w) in the slush (0.3–0.73) and different snow–slush transformation rates, e.g. H _SL = 0.7*H _S.

Figure 9b illustrates that the initial porosity reduction is the same for all the breaking frequencies, as the first passage occurs at the same time, and the submerged snow thickness is equal. However, with an increasing number of ship passages, the reduction in porosity until thermal equilibrium decreases. This is because the cumulative freezing air temperatures between ship passages decrease, and the temperature gradient between water and ice decreases as well. The blue solid line (2BE/day) in this figure consists of similar porosity reduction steps to the other lines given in the same figure. However, due to a high number of breaking events, these steps are closer, and thus, the line gives the impression of a filled surface area. Additionally, the snow accumulated between ship passages decreases with the increase in the number of breaking events (Figure 9b). This figure demonstrates that as the number of breaking events increases, less ice forms due to the temperature difference between the ice and water until thermal equilibrium is reached.

Figure 9c presents the maximum equivalent brash ice thickness after 150 days under varying water contents in slush (0.3, 0.4, 0.5, 0.6 and 0.73, shown as blue bars). In this scenario, a 100% snow–slush transformation is assumed, meaning that all submerged snow transforms into slush. The figure also includes model results using different snow–slush transformation fractions 0.3, 0.5 and 0.7, meaning that 30, 50 and 70%, of the submerged snow transformed to slush (represented by the orange bars). The maximum difference in equivalent brash ice thickness (H _B) between water fraction inputs of 0.3 and 0.73 was 31 cm. While comparing the scenario where H _SL is assumed equal to H _S and when it is assumed equal to 30% of the submerged H _S, the maximum difference in the brash ice equivalent thickness between the two cases is 15 cm.

5.3 Model validation

The brash ice equivalent thickness was estimated using three models: the one considering the two effects of snow, the one considering only the insulative effect of snow and the original model (Riska and others, Reference Riska2019) that does not include the snow layer. These results were validated using data obtained from three ship channels located in the Bay of Bothnia, Luleå, Sweden. The same convective heat coefficient, equal to 10 W m⁻¹ °C⁻¹, was used in all models, except for the estimation of the equivalent brash ice thickness in the main channel, where the models performed better for a coefficient of 15 W m⁻¹ °C⁻¹.

5.3.1 Test channel 2020–2021

The equivalent brash ice thickness was calculated for each measured cross-section assuming a channel width equal to the vessel's beam, 12 m (Sandkvist, Reference Sandkvist1980; Reference Sandkvist1986). The predicted and observed equivalent thicknesses of brash ice in TCh01 are shown in Figure 10a.

Figure 10. (a) Observed and estimated equivalent brash ice thickness (H _B). (b) The brash ice macroporosity estimated with all three models and the snow thickness (H _S) accumulated on the channel between two ship passages.

The original model gave the highest brash ice thickness, while the snow's insulative effect model gave the lowest thickness. The value of H _B estimated with the model that considers the two effects of snow lies between the other two model results, as the quicker slush freezing compared to water freezing compensates for the reduction of the growth rate due to snow insulation. The maximum H _B estimated by the model considering the effects of snow was 26 cm lower than the original model and 10 cm greater than the snow insulative effect model. The initial amount of snow that was submerged at the first ship passage was 37 cm, and the corresponding reduction in macroporosity was 15%, as shown in Figure 10b. The macroporosity change was lower in the other two models, where snow–slush–snow ice transformation was neglected.

5.3.2 Test and main channels 2021–2022

The measured and estimated equivalent brash ice thicknesses for the second test channel (TCh02) and main channel (MCH) are illustrated in Figure 11. In TCh02, like TCh01, the results from the model which considered the two effects of the snow situated between the original model and snow insulative effect model predictions, with maximum differences of 30 and 14 cm. The predicted maximum H _B were 1.25 m for the two snow effects models, 1.55 m for the original model and 1.11 m for the snow insulative effect model.

Figure 11. Observed and predicted equivalent brash ice thicknesses (a) in the second test channel (TCh02) and (b) in the main channel (MCh).

However, this was not the case for the MCh, where H _B estimated from the two effects snow model was higher than those given by the other two models. This implies that in the main channel, the snow–slush–snow ice transformation had a greater impact on the equivalent brash ice thickness than the insulating effect of snow. This conclusion aligns with the observation that incoming snowfall in frequently navigated channels submerges quickly and does not insulate the ice growth for a long time, as seen in Figure 12. Omitting the snow's insulative effect only leads to a difference of 6 cm in the maximum thickness. The maximum difference in H _B between the snow effects model and the snow insulative effect model is 19 cm.

Figure 12. Brash ice macroporosity (p_j) and snow thickness (H _S) accumulated on the channels between two ship passages in the second test channel (a) and the main channel (b).

The equivalent brash ice thickness in the MCh was calculated considering a constant width, which was concluded to be equal to the centre track of the main channel (MCh cte). Also, the thickness was calculated considering the actual width of the channel (MCh ac) as measured. A wider channel results in a lower equivalent brash ice thickness. However, the H _B was better represented by the model for a constant width, equal to the central track of the channel.

The maximum porosity reduction in the test and main channels exceeds 8%, as shown in Figure 12. The porosity reduction estimated by the snow effects BIGM was higher than in the other two models for both channels. The maximum amount of snow submerged in TCh02 was 20 cm, while in MCh was 16 cm. This suggests that even in frequently navigated channels, the amount of snowfall that submerges at each passage can be significant.

5.4 BIGM including expelling coefficient

In this section, the cumulative expelling coefficient (χ_j) which represents the fraction of brash ice expelled sideways from the first ship passage until the jth ship passage is calculated for each measured cross-section of all three ship channels. An asymptotic envelope function was introduced to fit and represent the measurements. The asymptotic function was incorporated in the snow effects model. In addition, various levels of the ice fraction that is expelled from a channel at every ship passage (χ) were included in the BIGM model. Thus, the model's performance is assessed considering χ_j and various assumed constant values of χ. Finally, the estimated average thickness of brash ice (H _BI) that remains in the channel and the equivalent thickness of ridges (H _R) were validated against the results obtained from all three channels.

5.4.1 Expelling coefficient

The cumulative expelling coefficient (χ_j), calculated by Eqn 20 for all ship channels, and its progression with the number of breaking events is given in Figure 13. For TCh01, χ_j demonstrated a linear increase with the rising number of breaking events. However, for TCh02, the linear regression displayed a weak correlation. This can be attributed to the fact that the thickness of brash ice and ridges along the ship channel exhibit significant variability.

Figure 13. Calculated cumulative brash ice expelling coefficient (χ_j) for the test and main channels (TCh01, TCh02, MCh). The blue solid line illustrates the asymptotic envelope function (Eqn 23).

For the main channel, χ_j was calculated considering a constant width, representing the track at the centre of the channel, but was also calculated considering the actual channel's width. The results show that χ_j decreases with the increase of the channel width. This implies that in a wide channel, the fraction of ice that remains in the channel can be higher compared to a narrower channel. However, the value of χ_j will simultaneously depend on the position that the ship navigates along the channel and the ship's beam relative to the channel width. For example, if the ship navigates in the centre, the brash ice will be pushed to the sides but remain within the channel, but if the ship navigates on one side of the channel, the fraction of ice that is expelled out of the channel will increase.

Furthermore, an asymptotic envelope function was fitted to encompass all the calculated χ_j for each measured cross-section across all three ship channels. The data fit best with the asymptotic function, which can be expressed as:

(23)$$\chi _j = 0.58-0.6\;{\rm e}^{{-}0.3j\;}.$$

This function assumes that χ_j initially increases with the number of breaking events and subsequently levels off to a constant value of 0.58. This coefficient estimates the cumulative amount of ice expelled from the first breaking event (j = 1) to a certain number of BE (jth ship passage). From the cumulative expelling coefficient (χ_j), the expelling coefficient (χ) at each passage can be calculated as the difference in the cumulative expelling coefficient between two consecutive passages. The average expelling coefficient for the first 10 breaking events (BE) was ~0.09. Note that negative values of χ, which were obtained as a consequence of thickness variabilities along the channel, were neglected in these calculations.

5.4.2 Model estimations

The cumulative expelling coefficient (Eqn 23) for four navigation frequencies, and six various constant expelling coefficients (χ) from 5 to 50% were incorporated in the brash ice growth model that considers the two effects of snow (snow insulation and snow–slush–snow ice transformation), see Figure 14. According to the asymptotic function when χ_j approaches a constant value, the H _BI will account for 42% of the overall equivalent ice thickness, while the side ridges will constitute 58%. When the frequency of navigation increases from 1BE/4 days (38 BEs) to 2BEs/day (298 BEs), both the H _BI and H _R increase by a factor of 1.6. The model's results for χ ranging from 5% to 50% indicate that if the expelled ice after each ship passage is between 30 and 50%, H _BI after 5 BE remains nearly constant, ranging from 13 to 20 cm, while the ridge thicknesses continue to increase.

Figure 14. (a) Estimated average brash ice thickness (H _BI) and (b) ridge equivalent thickness (H _R) using the snow effects model that considered ice expelled sideways using Eqn (23), for four different navigation frequencies (i.e. H _ex38BE). Also using six different constant expelling coefficients (χ) varying from 5% to 50% (i.e. H ₁₀).

5.4.3 Model validation

The measured and estimated thicknesses of the brash ice that remains in the channel and the equivalent thicknesses of side ridges for all three channels are given in Figures 15 and 16. In the test channels, the corresponding thicknesses are estimated using the snow effects model for two constant expelling coefficients (χ) 0.1 and 0.2, and the cumulative expelling coefficient (χ_j), while for the main channel, the constant values of χ used are 0.01, 0.012 and 0.015. A convective heat transfer coefficient of 10 W m⁻¹ °C⁻² was employed for all three channels, as the model considering the snow effects performed best at this value.

Figure 15. Measured and estimated average thicknesses of brash ice (H _BI) and the equivalent ridge thicknesses (H _R) for the first test channel (TCh01). Two constant expelling coefficients of 10% and 20%, also the cumulative expelling coefficient (e.g. H _BIex) determined by Eqn (23) were used as input.

Figure 16. (a) Measured and estimated average thicknesses of brash ice (H _BI) and (b) the equivalent ridge thicknesses (H _R) for the second test channel (TCh02) and the main channel (MCh). The constant expelling coefficients (χ) used were 10 and 20% for the TCh01, also 1.0, 1.2 and 1.5% for the MCh. The cumulative expelling coefficient (χ _j) determined by Eqn 23 was also used as input (i.e. Tch_ex).

The BIGM predicts best the H _BI and H _R in both test channels for a constant expelling coefficient of 0.1, which is similar to χ calculated for the first 10BE of the test channels. In a previous study focused on brash ice formation in model-scale ship channels, the expelling coefficient calculated after each vessel passage was generally observed to be 0.1 or lower (Ettema and Huang, Reference Ettema and Huang1990). Employing the exponential function to account for the sideways movement of brash ice underestimated the average brash ice thickness and overestimated the amount of ice expelled sideways.

The brash ice thickness that remained in the main channel was not significantly affected by the channel width. This observation was made when calculating the brash ice thickness using two approaches: one considering a constant channel width (MCh_cte), representing the mid-lane of the channel, and the other considering the actual channels (MCh_ac). Refer to Figure 16a for details. This is because the increase in the amount of brash ice remaining in the channel was offset by the increase in the channel's width. The calculated equivalent ridge thickness was affected significantly when the channel width increased as the amount of ice expelled sideways decreased, see Figure 16b.

For the main channel, the BIGM performed well in predicting the average brash ice thickness that remains in the main channel using a expelling coefficient of 0.012. This constant coefficient implies that only 1.2% of the brash ice was expelled sideways at each ship passage. On the other hand, the exponential function underestimated the average brash ice thickness but was in good agreement with the ridge thickness, when assuming a constant channel width. However, it is important to note that this evaluation is limited to three measured cross-section profiles, which may not fully represent the actual development of the main channel. Also, the H_R calculated from the actual channel width are not well represented by the model.

The estimated brash ice thickness that was expelled from the test channels at each ship passage was about 10%, whereas in the main channel, ~1.2%. This difference between the two channel types is likely related to their width. While both test channels had a width comparable to the vessel's beam, the main channel had a width up to 42 m. As a result, in each ship passage in the test channel, the ice piled under the level ice, forming the side ridges. In contrast, in the main channel, ice was pushed to the sides but still remained within the channel, forming piles of ridged ice within the channel. Also, the main channel was navigated about 11 times more than the test channels till the last measurement.