A method for calculating thermal thickening and ablation of an ice cover under snow cover

Abstract

The application relates to a calculation method of thermal thickening and ablation of an ice cover under snow cover, which comprises the following steps: establishing a basic heat conduction equation of a snow cover and an ice cover; calculating the time-space change of water temperature under the ice cover; establishing a mathematical model of heat exchange between the snow cover and the atmosphere; vertically distributing the snow surface temperature, net heat flux and ice cover temperature; and numerically simulating the ice thickness at 0 DEG C. s <0℃ when the ice thickness; thermal ablation of the snow cover and the ice cover. The application firstly establishes the basic heat conduction equation of the snow cover and the ice cover, then calculates the heat exchange between the snow cover and the atmosphere, and vertically distributes the snow surface temperature and the ice cover temperature, and finally carries out critical condition and numerical simulation of the thermal thickening and ablation of the ice cover, so that the calculation precision of the thermal thickening and ablation of the ice cover under the snow cover is improved, and a scientific calculation basis is provided for ice flood prevention of rivers.

Description

Technical Field

[0001] This invention relates to a method for calculating the thermal thickening and melting of ice sheets under snow cover, and is a method for calculating multiple parameters of ice sheets. Background Technology

[0002] Determining the change in river ice thickness over time is a crucial aspect of ice engineering. When predicting the opening date, channel volume, and risk of ice dam floods in a river, variations in ice thickness are a primary determining factor. Furthermore, when calculating the forces exerted by ice on hydraulic engineering projects, ice thickness and temperature determine not only the ice sheet's load-bearing capacity but also the compressive and tensile forces on slopes. Current models of ice sheet thermal thickening and melting, as well as river ice processes, only consider air temperature as a meteorological factor. With advancements in ice condition prototyping technology and the accumulation of observational data, a pressing issue to address in order to improve the accuracy of ice condition forecasts is establishing functional relationships between various observed physical parameters, particularly atmospheric thermodynamic parameters. Summary of the Invention

[0003] To overcome the problems of existing technologies, this invention proposes a calculation method for the thermal thickening and melting of ice sheets under snow cover. The method establishes fundamental heat conduction equations and calculates heat exchange between the snow cover and the atmosphere, performing vertical distribution of snow surface and ice cover temperatures, as well as critical conditions and numerical simulations of ice sheet thermal thickening and melting. This provides a scientific calculation basis for river ice jam flood prevention.

[0004] The objective of this invention is achieved as follows: a method for calculating the thermal thickening and melting of ice sheets under snow cover, the steps of which are as follows:

[0005] Step 1, establish the basic equations for heat conduction in snow and ice caps:

[0006] I. The heat conduction equation of snow cover:

[0007] If we assume the snow cover is a homogeneous flat plate, meaning heat conduction only occurs in the vertical z-direction, then the one-dimensional heat conduction equation for the snow cover is:

[0008]

[0009] In the formula: ρ s C is the density of snow. pi ρ is the specific heat of ice; T is the temperature at point z on the snow cover; t is time; k s ρ is the thermal conductivity of snow; z is the distance from the snow cover surface; I vis h represents the internal heat per unit volume of visible solar radiation at point z on the snow cover. s The thickness of the snow cover;

[0010] II. Heat conduction equation of ice sheets:

[0011] If we assume the ice sheet is a homogeneous flat plate and heat conduction occurs only in the vertical direction, then the one-dimensional quasi-steady-state heat conduction equation for the ice sheet is:

[0012]

[0013] In the formula: k i ρ is the thermal conductivity of ice; T is the temperature at point y on the ice sheet; y is the distance from the surface of the ice sheet; h i For ice thickness; I vis Let y be the internal heat per unit volume of visible solar radiation at point y on the ice sheet;

[0014] Step 2, Calculation of the spatiotemporal variation of water temperature under the ice sheet: The river section is divided into m segments.

[0015]

[0016] b = -Bh wi -χh wbe ,

[0017] In the formula: the subscript "i" represents the river section number, i = 1, 2, ..., m; T wp,i For the exit time t of river segment i i Water temperature; T wc,i For the exit time t of river segment i i The equilibrium water temperature; B is the width of the water surface; The net amount of solar radiation transmitted into the water beneath the ice; h wi The heat exchange coefficient between water and ice sheet; T m χ represents the bottom surface temperature of the ice cap; h represents the wetted perimeter of the channel. wbe T is the equivalent heat exchange coefficient between the water body and the canal bed. be The equivalent ground temperature of the canal bed subgrade;

[0018] Step 3, Mathematical model of heat exchange between snow cover and atmosphere:

[0019] The heat exchange between the snow cover and the atmosphere consists of six parts: ① net solar radiation, ② long-wave radiation from the snow surface, ③ atmospheric long-wave back radiation, ④ snow surface evaporation, ⑤ convection, and ⑥ snowfall. The mathematical model is:

[0020]

[0021] In the formula: This refers to the net heat flux conducted from the atmosphere to the snow cover surface. Net solar radiation heat flux over snow surface; This refers to the atmospheric longwave reverse radiation heat flux; This refers to the long-wave radiation heat flux over the snow surface. For snow surface evaporation heat flux; The heat flux of air convection over the snow surface; The heat flux generated by snowfall;

[0022] Step 4, Vertical distribution of snow surface temperature, net heat flux, and ice cover temperature:

[0023] Snow surface temperature T s Calculation:

[0024]

[0025] In the formula: T u For visible light from solar radiation to pass through snow and ice caps; h sa The heat exchange coefficient between the snow surface and the atmosphere; The subscript "0" indicates the net heat flux conducted from the atmosphere to the snow cover surface at time t0. T a Temperature; h i For ice thickness; h s The thickness of the snow cover; k s The thermal conductivity of snow;

[0026] The critical temperature at which snow cover melts due to thermal ablation:

[0027]

[0028] In the formula: T acri The critical temperature for the thermal ablation of snow cover; Solar radiation heat flux;

[0029] a sa =70.7 + (6.04 + 2.95V) z (1-R) h c1>70.7, b sa =0.4 + (6.04 + 2.95V) z (1-R) h c2>0

[0030] coefficient:

[0031]

[0032] In the formula: V z R represents wind speed. h Relative humidity;

[0033] Step 5, snow surface temperature T s Numerical simulation of ice thickness at <0℃:

[0034] The ordinary differential equation for ice thickness changing with time:

[0035]

[0036] In the formula: The subscript "0" indicates the net heat flux conducted from the atmosphere to the snow cover surface at time t0. T w The average water temperature across the cross-section; ρ i L is the density of ice. i The latent heat of ice;

[0037] Equation for thermal thickening of ice sheets:

[0038]

[0039] Where: h i0 The subscript "0" indicates the ice thickness h at time t0. i ;

[0040] α=2k i △t / (ρ i L i ), S=(T m -T a ),

[0041]

[0042] In the formula: T w0 The subscript "0" indicates the average water temperature T of the cross-section at time t0. w ;I vis0 The subscript "0" indicates the internal heat per unit volume of visible light from solar radiation at point y on the ice sheet at time t0. vis ;k vi Δt represents the extinction coefficient or attenuation coefficient of the ice sheet; Δt represents the time step.

[0043] Step 6, thermal melting of snow and ice caps:

[0044]

[0045] In the formula: ρ i L is the density of ice. i It is the latent heat of ice.

[0046] The advantages and beneficial effects of this invention are as follows: First, this invention establishes the basic equations for heat conduction of snow cover and ice cover, then calculates the heat exchange between snow cover and atmosphere, and performs numerical simulations of the vertical distribution of snow surface temperature and ice cover temperature, as well as the critical conditions and numerical simulations of ice cover thermal thickening and melting. This improves the calculation accuracy of ice cover thermal thickening and melting under snow cover, and provides a scientific calculation basis for river ice flood control. Attached Figure Description

[0047] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0048] Figure 1 This is a flowchart of the method described in an embodiment of the present invention. Detailed Implementation

[0049] Example 1:

[0050] This embodiment describes a calculation method for the thermal thickening and ablation of ice sheets under snow cover. The calculation process described is as follows: Figure 1 As shown, it includes the following steps:

[0051] Step 1, establish the basic equations for heat conduction in snow and ice caps:

[0052] When the ice sheet is covered by snow, when the surface temperature of the snow sheet is T s When the temperature is less than 0, the thermal thickening and melting of the ice sheet only occur at the bottom surface of the ice, and the heat conduction process of the snow sheet and ice sheet can be described by the flat plate heat conduction theory.

[0053] I. The heat conduction equation of snow cover:

[0054] If we assume the snow cover is a homogeneous flat plate, meaning heat conduction only occurs in the vertical direction, then the one-dimensional heat conduction equation for the snow cover is:

[0055]

[0056] In the formula: ρ s The density of snow; the density of fresh snow cover ranges from 70 to 190 kg / m³. 3 Between; C pi ρ is the specific heat of ice, J / kg℃; T is the temperature at point z on the snow cover,℃; t is time, s; k s denoted by , where is the thermal conductivity of snow, with fresh snow exhibiting a thermal conductivity between 0.060 and 0.200 W / m℃; z is the distance from the snow cover surface, in meters; I vis The internal heat per unit volume of visible solar radiation at point z on the snow cover, in W / m 2 h s Let be the thickness of the snow cover, in meters (m).

[0057] When ignored Assuming the thermal conditions within the snow cover are quasi-steady, the one-dimensional heat conduction equation for the snow cover can be simplified to:

[0058]

[0059] The net heat flux conducted from the atmosphere to the snow cover surface, excluding the 70% visible light region of solar radiation, is completely absorbed by a very thin layer on the snow cover surface. Therefore, the boundary conditions of the snow cover are:

[0060]

[0061] z = h s T = T is (4)

[0062] In the formula: The net heat flux transferred from the atmosphere to the snow cover surface, W / m 2 ;T s Snow surface temperature, °C; T is The ice surface temperature, i.e., the temperature at the interface between the snow cover and the ice cover, is expressed in °C; c par Visible light heat flux and net solar radiation heat flux on the snow surface The ratio, c par = 0.443~0.483, depending on weather conditions; smaller values ​​are used for sunny days and larger values ​​for cloudy days. When there is no snow cover... T represents the net heat flux transferred from the atmosphere to the ice surface. s =T is .

[0063] The visible light heat flux of solar radiation transmitted through the ice sheet can be described as

[0064]

[0065] In the formula: I vis0 The heat flux of visible light from solar radiation passing through the snow cover, W / m 2 .

[0066] According to equation (5), for a distance z in the snow cover, the heat flux transmitted by visible light is:

[0067]

[0068] Substituting equation (6) into equation (2), and combining it with the boundary conditions (3) and (4), we obtain the following solution:

[0069]

[0070]

[0071]

[0072] In the formula: T u1 The increase in snow temperature (°C) as visible solar radiation passes through the snow cover:

[0073]

[0074] II. Heat conduction equation of ice sheets:

[0075] If we assume the ice sheet is a homogeneous flat plate and heat conduction occurs only in the vertical direction, then the one-dimensional quasi-steady-state heat conduction equation for the ice sheet is:

[0076]

[0077] In the formula: k i ρ is the thermal conductivity of ice, W / m℃, typically taken as 2.034 W / m℃; T is the temperature at point y on the ice sheet, ℃; y is the distance from the ice sheet surface, m; h i For ice thickness, m; I vis The internal heat per unit volume of visible solar radiation at point y on the ice sheet, in W / m². 2 .

[0078] Visible light transmission radiation I in ice sheets vis Decays according to the natural exponential law:

[0079] I vis =I vis0 exp(-k vi y) (12)

[0080] In the formula: k vi The extinction coefficient or attenuation coefficient of the ice sheet, m -1 k vi =0.46~3.54m -1 This is largely related to the amount of air bubbles and impurities contained in the ice sheet.

[0081] When the bottom surface of the ice sheet undergoes a phase change from water to ice under the influence of negative air temperature, or when the bottom surface of the ice sheet melts under the influence of water temperature, the boundary conditions of the ice sheet are:

[0082]

[0083]

[0084] In the formula: T m This refers to the temperature of the bottom surface of the ice sheet, i.e., the freezing point temperature, which is generally taken as T. m =0℃; The heat flux conducted from the water to the bottom of the ice, in W / m² 2 ;ρ i The density of ice is kg / m³. 3 Generally, 917 kg / m 3 L i dh is the latent heat of ice, in J / kg, typically taken as 333400.0 J / kg; i / dt represents the rate at which the ice surface thickens or melts due to thermal action, in m / s.

[0085] Substituting equation (12) into equation (11), and combining it with boundary condition equations (13) and (14), we obtain the formula for calculating the vertical temperature of the ice sheet:

[0086]

[0087] And the basic equations for thermal thickening and melting of the ice bottom surface:

[0088]

[0089] In the formula: The net heat flux transmitted through solar radiation into the water beneath the ice, in W / m². 2 .

[0090] According to equation (15), the following important conclusions can be drawn: 1) There is no solar radiation at night, I vis0 =0, ice temperature T is a linear function of distance y; during the day, affected by solar radiation, T is a non-linear function of y.

[0091] At the bottom of the ice, T = T m Therefore, from equation (15), we can obtain the functional relationship between ice surface temperature and ice thickness:

[0092]

[0093] In the formula: T u2 The increase in ice temperature, expressed in °C, as visible solar radiation passes through the ice sheet.

[0094] T u2 =I vis0 (1-exp(-k vi h i )) / (k i k vi )≥0 (19)

[0095] Solving equations (9) and (18) simultaneously yields the net heat flux over the snow surface:

[0096]

[0097] In the formula: T u The increase in ice temperature, expressed in °C, as visible solar radiation passes through snow and ice sheets.

[0098]

[0099] When the net heat flux of solar radiation on the snow surface When T is known u It can be calculated from the above formula.

[0100] It should be noted that the snow surface temperature T s It is an unknown quantity, and It is T s The function depends on the heat exchange between the snow surface and the atmosphere, in addition Depends on the water temperature under the ice T w .

[0101] Step 2, Calculation of the spatiotemporal variation of water temperature under the ice sheet:

[0102] Since the water temperature under the ice sheet depends on the heat exchange between the water and the ice sheet and the seabed, as well as the transmission of solar radiation, the one-dimensional thermal convection equation for the water under the ice sheet along the flow direction is:

[0103]

[0104] dx / dt=V (23)

[0105] In the formula: ρ is the density of water, kg / m³ 3 At room temperature, ρ≈1000kg / m 3 C p The specific heat of water is C at 0℃. p = 4217.7 J / kg℃; A is the cross-sectional area of ​​the water passage, m² 2 ;T w χ represents the average water temperature of the cross-section, in °C; B represents the width of the water surface, in m; and χ represents the wetted perimeter of the channel, in m. The equivalent heat flux for heat exchange between water and the riverbed, W / m 2 x is the distance along the flow direction, in meters; V is the average flow velocity across the cross section, in meters per second.

[0106] Heat flux from water to ice sheet:

[0107]

[0108]

[0109] Where: h wi The heat exchange coefficient between water and ice sheet, W / (m 2 ℃); R w Let R be the hydraulic radius, in meters (m). Under normal circumstances, when the flow velocity V > 0.3 m / s, for rivers with a significant width-to-depth ratio, the hydraulic radius R for water transport under the ice sheet is... w ≈0.5H, when the water depth H = 1-10m, h wi =619.1~448.7W / m 2 ℃.

[0110] For concrete-lined channels, the equivalent heat flux of heat exchange between water and the channel bed can be calculated using the multi-layer flat plate heat conduction theory:

[0111]

[0112] Where: h wbe The equivalent heat exchange coefficient between the water body and the canal bed, in W / m³. 2 ℃; T be Let be the equivalent ground temperature of the subgrade beneath the canal bed, in °C. For a river channel, when the channel is considered to consist of flat plates with different thermal conductivity and thickness, then... It can also be described by equation (26).

[0113] Equation (22) can be rewritten as:

[0114] dT w / dt=(bT w +c) / (ρC P A) (27)

[0115]

[0116] When the river channel is divided into m segments, and the parameters b, c, and A remain constant along the course of the river in each segment, the recursive formula for calculating the change of water temperature over time is obtained by integrating equation (26) along the characteristic line:

[0117] T wp,i =(-c i +(b i T wp,i-1 +c i )exp(b i △t i / (ρC P A i ))) / b i ,i=1,2,...,m (29)

[0118] In the formula: the subscript "i" represents the river segment number; m represents the river segment number; T wp,i-1 For the inlet time t of river segment i i-1 Water temperature, °C; T wp,i For the exit time t of river segment i i Water temperature, °C; △t i =t i -t i-1 ,s. Generally speaking, when T wp,i-1 When the water temperature is relatively high, for example, when the upstream section of river i is an open water surface and the air temperature is positive, then the water temperature T wp,i Decrease with time; when T wp,i-1 When the water temperature is low, for example, if there is a long, completely frozen section upstream of river segment i, then the water temperature T wp,i Increase over time.

[0119] Due to △t i =△x i / V i Equation (29) can be rewritten as:

[0120] T wp,i =(-c i +(b i T wp,i-1 +c i )exp(b i △x i / (ρCP Q i ))) / b i ,i=1,2,...,m (30)

[0121] Where: Flow rate Q i =V i A i m 3 / s. Due to coefficient b i <0 is always true, water temperature T wp,i With distance x i = x0 + Δx1 + Δx2 + ... + Δx i The increase in temperature follows an exponential pattern, gradually approaching an equilibrium temperature T. wc,i ,Right now:

[0122]

[0123] Substituting (30) into equation (24), consider h wi >>h wbe B≈χ and T m =0, then:

[0124]

[0125] When observation point x i = x0 + Δx1 + Δx2 + ... + Δx i If there is a long ice-covered section upstream, then from equation (32) we get:

[0126]

[0127] The above formula shows that when the observation point x i When there is a long ice-covered river channel upstream, the net heat flux transferred from the water to the ice surface. Equal to the heat flux of solar radiation transmitted into the water body With riverbed geothermal heat flux h wbe T be sum.

[0128] Step 3, Calculation of heat exchange between snow cover and atmosphere:

[0129] The heat exchange between the snow cover and the atmosphere consists of six parts: ① net solar radiation (shortwave radiation), ② longwave radiation from the snow surface, ③ atmospheric longwave back radiation, ④ snow surface evaporation, ⑤ convection, and ⑥ snowfall. The mathematical model is:

[0130]

[0131] In the formula: Net solar radiation heat flux over snow surface, W / m 2 ; The atmospheric longwave reverse radiation heat flux, W / m 2 ; The longwave radiation heat flux over the snow surface, W / m 2 ; The heat flux due to evaporation from the snow surface, W / m 2 ; Heat flux of air convection over snow surface, W / m 2 ; Heat flux generated by snowfall, W / m 2 Under normal circumstances, snowfall heat flux It is small and can be ignored.

[0132] Net solar radiation heat flux The relationship with snow surface albedo is:

[0133]

[0134] In the formula: This represents the heat flux of solar radiation incident on the snow surface, which is related to atmospheric transparency, atmospheric mass, and cloud cover, in W / m². 2 ;a si denoted as snow surface albedo. Equations (34) and (35) also apply to ice surfaces when there is no snow accumulation on the ice cover.

[0135] a si The parameters are related to snow thickness, snow surface temperature, ice cover albedo, and snow type. A general parameterized model for water ice and snow albedo can be used for calculation.

[0136]

[0137] In the formula: a s The albedo of ice; a ss The daily average albedo of dry snow ranges from 0.83 to 0.85. Snow is essentially granular ice, so its freezing point temperature T... m ≈0℃. When -1 <T s When T < 0, ΔT < 0, min(ΔT,0) = ΔT, indicating that the snow surface is close to melting and the albedo begins to decrease; when T s =T m When ΔT = -1.0 and min(ΔT,0) = -1.0, it indicates that the snow surface begins to melt, forming a muddy, rough mixture of water and snow on the ice surface, which reduces the albedo. When the temperature changes from negative to positive, the snow cover will melt within a few hours, forming a mixture of water and snow. Then, when the nighttime temperature drops below 0°C, white ice or gray ice forms, at which point the albedo a... si ≈0.38.

[0138] ice albedo a sIt is a function of the solar altitude angle α, and is related to geographical latitude, the Earth's rotation, and its rotation around the sun. For black ice, the following a... s Calculation with respect to the parameterized model of α:

[0139] a s =0.0564 / α, α≥0.105rad (37)

[0140] a s =0.537-4.408(α-0.105), 0≤α<0.105rad (38)

[0141] In its natural environment, the river is surrounded by mountains, which block solar radiation at sunrise and sunset.

[0142] When weather data is complete, including water surface temperature T s Temperature T a Cloud cover (C), relative humidity (R) h Wind speed V z Local atmospheric pressure p a If so, then it is determined. And in T s =T a Linearize equation (34):

[0143]

[0144] In the formula: Net heat flux of snow surface In T s =T a The value of h; sa The heat exchange coefficient between the snow surface and the atmosphere, W / (m²) 2 ℃). Because and Smaller, therefore:

[0145]

[0146] Because there is no solar radiation at night. If the temperature T a <0℃, Always true; daytime solar radiation Even during the winter ice age, daylight may exist. Therefore, under normal circumstances

[0147] In addition, for the period of river opening or ice melting, T s ≈0℃, |T s -T a If the value is large, then the secondary term (T) in convective heat exchange needs to be considered. s -Ta ) 2 Due to the influence of this, equation (39) can be modified as follows:

[0148]

[0149] Where: h sa2 =0.158×10 -3 p a .

[0150] To facilitate ice condition forecasting, based on historical weather data from typical years, h sa and Linear regression can be used as an approximation. Taking Mohe region in Heilongjiang Province as an example:

[0151] h sa = (7.05 + 0.02T) a (1.0+0.25V) z (42)

[0152] and:

[0153]

[0154] In the formula:

[0155] a sa =70.7 + (6.04 + 2.95V) z (1-R) h c1>70.7, b sa =0.4 + (6.04 + 2.95V) z (1-R) h c2>0(44)

[0156] coefficient:

[0157]

[0158] Step 4, Vertical distribution of snow surface temperature, net heat flux, and ice cover temperature:

[0159] Solving equations (20) and (39) simultaneously, we can obtain the snow surface temperature T. s The calculation formula is as follows:

[0160]

[0161] When calculating T s ≥T m =0℃ indicates that the snow surface will melt. In reality, when the snow melts, T... s ≡T m =0.

[0162] When T s=0, according to equations (43) and (45), the critical temperature for the thermal ablation of snow cover can be obtained.

[0163]

[0164] In the formula: T acri The critical temperature for the thermal ablation of snow cover, ℃. When temperature T... a ≥T acri At this time, the phenomenon of snow cover thermal ablation occurs. The net solar radiation heat flux at night... and T u =0, so T acri =a sa / [h sa (1-b sa / h sa )]>a sa / h sa >0 indicates that the temperature T a =0℃ is not the critical condition for the thermal ablation of snow cover.

[0165] When T is calculated by equation (45) s At <0℃, substituting equation (45) into equation (20) yields the net heat flux of the snow surface:

[0166]

[0167] in this case, It always holds true.

[0168] Given T s and Under the condition of, the ice surface temperature T in equation (9) is Substituting into equation (15), we can obtain the formula for calculating the vertical distribution of ice sheet temperature:

[0169]

[0170] Since ice sheet strength and elastic modulus are functions of ice temperature, the above formula can be used as a reference for calculating the strength and elastic modulus of ice sheets in winter.

[0171] When T is calculated by equation (45) s When ≥0℃ is established, the snow surface melts, causing T s =T m From equation (20), the net heat flux of the snow surface can be obtained:

[0172]

[0173] It should be noted that when there is no snow cover on the ice, only h needs to be taken. s =0, then T s and These represent the ice surface temperature and the net heat flux of the ice surface, respectively.

[0174] Step 5, snow surface temperature T s Numerical simulation of ice thickness at <0℃:

[0175] When the snow surface temperature T s Substituting equations (24) and (47) into equation (16) yields the ordinary differential equation for ice thickness as a function of time:

[0176]

[0177] In the formula: the first term on the right represents the rate of thermal thickening of the ice sheet due to atmospheric heat conduction; the second term on the right represents the rate of thermal thickening or melting of the ice sheet due to water heat conduction, when the water temperature T w When the temperature is above 0℃, the ice thickness decreases; conversely, when the water temperature is above 0℃, the ice thickness decreases. w At ≤0℃, ice thickness increases; the third term represents the rate of thermal thickening of the ice sheet caused by solar radiation transmission. A necessary and sufficient condition for thermal thickening of the ice sheet is that the right side of equation (50) is greater than 0. Clearly, the snow surface temperature T... s A value less than 0 is a necessary condition, but not a sufficient condition, for thermal thickening of the ice sheet. When T... s At ≥0℃, the snow cover and ice cover will undergo thermal ablation, and the relationship of equation (50) does not hold.

[0178] In season and T u When =0, equation (50) simplifies to the equation for the thermal thickening of the ice sheet:

[0179] dh i / dt=[(T m -T a ) / (h s / k s +h i / k i +1 / h sa )-h wi (T w -T m )] / (ρ i L i (51)

[0180] The above formula is currently widely used in river ice development models.

[0181] When h s =0、 T u =0 and 1 / h sa When =0, equation (50) simplifies to the equation for the thermal thickening of the ice sheet.

[0182] dh i / dt=(T m -Ta ) / [(h i / k i )(ρ i L i (52)

[0183] In summary, we can conclude that the existing equation for the thermal thickening of ice sheets is only a special case of equation (50).

[0184] Multiply both sides of equation (50) by (h) s k i / k s +h i +k i / h sa From this, we can obtain:

[0185]

[0186] Assuming the heat exchange coefficient h sa He Xuehouh s Under the condition that the time interval Δt remains unchanged, the right side of equation (53) is only the temperature T. a Alternatively, the equation can be a function of time, with the left side being a function of ice thickness. Integrating this equation yields the following approximate recursive formula:

[0187]

[0188] In the formula: the subscript "0" represents time t0; Δt = t - t0 is the time step, in seconds;

[0189] α=2k i △t / (ρ i L i ), S=(T m -T a ),

[0190]

[0191] It should be noted that during the period of thermal thickening of the ice sheet, the water temperature under the ice is close to 0°C. Due to the limitations of thermometer measurement accuracy, there is currently a general lack of measured data on water temperature under ice and riverbed temperature both domestically and internationally. The following study examines a special case.

[0192] Since the freezing period is the dry season and the flow velocity changes slowly, for river sections far from the ice sheet front, the heat exchange between the water and the ice sheet can be obtained according to equation (33). Mainly affected by solar radiation transmission and the ground temperature of the underlying layer of the riverbed, in this case, in equation (55)

[0193] W = (h s / k s +h i0 / ki +1 / h sa )h wbe T be (56)

[0194] Actual measurements of ground temperature along the South-to-North Water Diversion Project's central route show that at a certain depth above the surface, such as 80cm, ground temperature changes are relatively stable in winter. In the Daqing Oilfield of Heilongjiang Province, measured depths of the constant-temperature layer in winter reached 11.5m, with a temperature of T... be Approximately 5℃. Due to the excellent insulating properties of snow and ice caps, the temperature T of the underlying layer of the underwater riverbed is... be The range of variation between 0-10℃ is valid.

[0195] Step 6, thermal melting of snow and ice caps:

[0196] Once the snow surface temperature T s =T m =0℃ or When the snow cover undergoes thermal melting, it forms a mixture of snow, water, and air bubbles, commonly known as slush, and may even result in water glazing, where a film of water containing numerous air bubbles covers the ice surface. Due to the presence of penetrating cracks and the uneven surface of the ice cover, some meltwater seeps into river channels through these cracks, while the rest forms numerous small puddles on the ice surface, or even clear ditches or exposed water. As nighttime temperatures drop, the ice surface temperature T... s <T m When the remaining snow slurry refreezes, the ice is usually grayish-white because it contains a large number of air bubbles and snow crystals. At this time, the ice surface has a higher solar radiation albedo than black ice.

[0197] Referring to the CCSM3 Arctic climate system simulation model, the net heat flux transferred from the atmosphere to the snow cover surface, excluding the heat flux from the 70% visible solar radiation region, is completely absorbed by a very thin layer of the snow cover surface to melt ice; that is, the rate of thermal ablation of the snow cover surface:

[0198]

[0199] Because the density of snow cover is much smaller than that of ice cover, i.e., ρ s <<ρ i Especially when the snow is not very thick, the snow cover will melt in a very short time once the temperature turns from negative to positive.

[0200] Under natural conditions, not only is the surface of the ice sheet uneven and irregularly distributed, but the size and distribution of cracks on the ice sheet are also random, making theoretical calculations and analyses of water-ice accumulation phenomena very difficult. On the other hand, the thermal ablation process of snow covers is quite complex. Because snow covers are composed of ice particles filled with air, meltwater easily seeps onto the ice surface, causing simultaneous thermal ablation of the ice. To simplify the problem, in the following analysis of the thermal ablation process of the ice sheet, we assume: 1) When T... s = At 0℃, the snow cover melts in a very short time, so the effect of the snow cover melting process on the ice cover melting process can be ignored; 2) Water icing does not affect the melting process of the ice cover surface.

[0201] When T is calculated by equation (45) s When the temperature is above 0℃, the snow and ice caps begin to melt thermally, at which point T s ≡T m =0℃. Unlike the thermal thickening of ice sheets, which generally occurs at the bottom of the ice, the thermal melting of ice sheets can occur simultaneously at the surface, inside, and bottom of the ice sheet.

[0202] The net heat flux conducted from the atmosphere to the ice sheet surface, excluding the 70% visible light region of solar radiation, is completely absorbed by a very thin layer on the ice sheet surface to melt the ice. In other words, the rate of thermal ablation of the ice sheet surface can be described as follows:

[0203]

[0204] In the formula: dh iu / dt represents the rate of thermal melting of the ice sheet surface, in m / s.

[0205] Due to the transmission and scattering of visible light from solar radiation, the heat flux lost inside the ice sheet is... That is, the equivalent thermal ablation rate inside the ice sheet:

[0206]

[0207] In the formula: dh ie / dt represents the equivalent thermal ablation rate inside the ice sheet, in m / s. Because ice melting occurs inside the ice sheet, the vertical ice temperature distribution during thermal ablation becomes irregular. In other words, the heat conduction equation for the ice sheet in Section 2 is no longer applicable to the thermal ablation process.

[0208] The net heat flux transferred from the water body to the ice sheet is Therefore, the rate of thermal melting at the bottom of the ice is:

[0209]

[0210] From equation (58)+(59)+(60), the equivalent rate of thermal ablation of the ice sheet can be obtained, that is:

[0211]

[0212] In the formula: dh i / dt=dh iu / dt+dh ie / dt+dh ib / dt is the equivalent rate of thermal ablation of the ice sheet.

[0213]

[0214] It should be noted that equation (61a) is exactly the same as the basic equation (16) for thermal thickening and melting of the ice bottom surface. In other words, when the thermal thickening and melting of the ice sheet surface, interior and bottom are attributed to the ice bottom surface, equation (16) is the general mathematical model for thermal thickening and melting of the ice sheet.

[0215] If |T m -T a If the value is large, then the secondary term (T) in convective heat exchange needs to be considered. m -T a ) 2 The effect, from equation (39), is:

[0216]

[0217] because With solar radiation heat flux Proportional, therefore, from equation (61a), we can conclude that the rate of thermal melting of the ice sheet, dh i / dt and It is directly proportional. This is significantly different from the thermal thickening of the ice sheet, as can be seen from equation (50) that the thermal thickening rate dh of the ice sheet is... i / dt and Proportional. When the ice sheet is thicker, h s / k s +h i / k i +1 / h sa →1, while h sa >>1, therefore, the same solar radiation heat flux has a much greater effect on the thermal ablation of ice sheets than on the thermal thickening of ice sheets.

[0218] Integrating equation (61a) yields:

[0219]

[0220] For river sections far from the ice sheet front:

[0221]

[0222] At this point, equation (63) can be rewritten as:

[0223]

[0224] In the formula: ρ i L is the density of ice. i It is the latent heat of ice.

[0225] Finally, it should be noted that the above is only used to illustrate the technical solution of the present invention and not to limit it. Although the present invention has been described in detail with reference to preferred arrangements, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solution of the present invention (such as the applied river or channel, the application of various formulas, the order of steps, etc.) without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for calculating the thermal thickening and ablation of ice sheets under snow cover, characterized in that, The steps of the method are as follows: Step 1, establish the basic equations for heat conduction in snow and ice caps: I. The heat conduction equation of snow cover: If we assume the snow cover is a homogeneous flat plate, meaning heat conduction only occurs vertically... If the direction is as described, then the one-dimensional heat conduction equation of the snow cover is: In the formula: The density of snow; The specific heat of ice; For snow cover Temperature at that location; For time; The thermal conductivity of snow; The distance from the snow cover surface; For snow cover The internal heat per unit volume of visible solar radiation at a given location; The thickness of the snow cover; II. Heat conduction equation of ice sheets: If we assume the ice sheet is a homogeneous flat plate and heat conduction occurs only in the vertical direction, then the one-dimensional quasi-steady-state heat conduction equation for the ice sheet is: In the formula: is the thermal conductivity of ice; For ice cap Temperature at that location; The distance from the surface of the ice sheet; The ice is thick; For ice cap Internal heat per unit volume at which visible solar radiation is emitted; Step 2, Calculation of the spatiotemporal variation of water temperature under the ice sheet: The river section is divided into... part, ， In the formula: subscript " "Numbering the river sections," ; River section Exit time Water temperature; River section Exit time The equilibrium water temperature; The width of the water surface; The net effect of solar radiation penetrating into the water beneath the ice; The heat exchange coefficient between water and ice cap; Temperature of the bottom surface of the ice sheet; For the channel to be wet; The equivalent heat exchange coefficient between the water body and the canal bed; The equivalent ground temperature of the canal bed subgrade; Step 3, Mathematical model of heat exchange between snow cover and atmosphere: The heat exchange between the snow cover and the atmosphere consists of six parts: ① net solar radiation, ② long-wave radiation from the snow surface, ③ atmospheric long-wave back radiation, ④ snow surface evaporation, ⑤ convection, and ⑥ snowfall. The mathematical model is: In the formula: This refers to the net heat flux conducted from the atmosphere to the snow cover surface. Net solar radiation heat flux over snow surface; This refers to the atmospheric longwave reverse radiation heat flux; This refers to the long-wave radiation heat flux over the snow surface. For snow surface evaporation heat flux; The heat flux of air convection over the snow surface; The heat flux generated by snowfall; Step 4, Vertical distribution of snow surface temperature, net heat flux, and ice cover temperature: snow surface temperature Calculation: In the formula: This allows visible light from solar radiation to penetrate snow and ice caps. The heat exchange coefficient between the snow surface and the atmosphere; express The net heat flux transferred from the atmosphere to the snow cover surface at that time; Temperature; The ice is thick; The thickness of the snow cover; The thermal conductivity of snow; The critical temperature at which snow cover melts due to thermal ablation: In the formula: The critical temperature for the thermal ablation of snow cover; Solar radiation heat flux; 70.7， coefficient: ， In the formula: Wind speed; Relative humidity; Step 5, snow surface temperature T s Numerical simulation of ice thickness at <0℃: The ordinary differential equation for ice thickness changing with time: In the formula: express The net heat flux transferred from the atmosphere to the snow cover surface at that time; The average water temperature across the cross-section; The density of ice; The latent heat of ice; Equation for thermal thickening of ice sheets: ， In the formula: express The ice was thick at that time; ， ， ， ， In the formula: express The average water temperature at the cross-section of the water flow at that time; express ice cap Internal heat per unit volume at which visible solar radiation is emitted; This refers to the extinction coefficient or attenuation coefficient of the ice sheet. For time step; Step 6, thermal melting of snow and ice caps: In the formula: The density of ice; It is the latent heat of ice.

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