STUDY ON THE FREEZE-UP FORECAST MATHEMATICAL MODEL FOR THE INNER MONGOLIA REACH IN YELLOW RIVER UPSTREAM

Sujuan Ke,  Xianghui Yang  and  Jiuyu Lu

Bureau of Hydrology, YRCC, Zhengzhou, China,450004,fax: 86-371- 6352574,

E-mail:ksuj@371.net

Abstract: In this paper, river ice Hydraulics, freeze-up mechanism and the factors influencing freeze-up were analyzed. Based on this, the freeze-up forecast mathematical model was established. In the end of the paper, the function form of freeze-up forecast mathematical model was defined and tested by taking Inner Mongolia reach for example, the results show that the model can be used easily and its precision is high.

Keywords: freeze-up mechanism, forecast, mathematical model

1  STUDY ON THE FREEZE-UP MECHANISM

1.1  Flowing ice distribution

Rivers are usually in turbulent situation, so ice flower will be formed in whole water body when the water is super cooled. With the affection of water mixing, one part of suspended-ice float on water surface, others are moved to river bed, stick to the bodies of river bed and forming anchor-ice when suspended-ice are formed. So ice flower in water body can be divided into 3 layers: anchor-ice, suspended-ice and surface-ice. But there isn't a clear dividing line among 3 layers which in a continuous transformation. Anchor-ice is always produced in the rivers of smaller depth, but can be neglected in the rivers of bigger depth such as Yellow River and Songhuajiang River in the northern of China. The formation of anchor-ice will be neglected.

1.2  Freeze-up mechanism

It can be seen from above section that flowing ice can be divided into suspended-ice and surface-ice, ice flowers can be transformed between the two layers, moreover, flowing ice discharge will get bigger with continuously losing heat of water body. According to ice Hydraulics, the stress acting on ice flower are mainly friction with other ice flowers, freezing force, drag force of flow and wind force. In these forces, friction and freezing force prevent ice flower from flowing ahead, so they are called frazil resistance. Flow drag force acting on frazil ice makes frazil ice moved. Wind force becomes drag force when the direction of wind is as same as that of flow, if opposite, then resistance. The frazil ice resistance surely gets bigger when air temperature continues falling and ice volume increasing. The river begins to block up when frazil ice resistance is bigger than drag force.

From above analysis we know, frazil ice is proportional with ice discharge and negative temperature, drag force relates to factors such as flow discharge, surface slope, width ratio of up and down cross-section, and river channel curvature, etc., that is:

F_s=f(Q_i,T_a)

F_t=f(Q,J,B,S)

Where F_s—frazil ice resistance, Q_i—flowing ice discharge, T_a—daily average temperature(), F_t—drag force, Q—flow discharge(m³/s), J—surface slop, A－river width ratio of up and down cross-section(m), S—river channel curvature.

                                  when  F_s> F_t+F_w  ，freeze-up  begins                         

  F_w is wind force, it can be neglected as wind velocity is small.

2  ANALYSIS OF FACTORS INFLUENCING FREEZE-UP

From freeze-up mechanism we know, 3 kinds factors influencing freeze-up mainly are air temperature conditions, hydraulic conditions and channel boundary conditions.

2.1  Thermal conditions

The main thermal condition is air temperature. The lower the air temperature, the bigger the flowing ice discharge and the frazil ice resistance, the shorter the flowing ice period, the quicker the freeze-up. Sanhuhekou hydrologic station, which is in Inner Mongolia reach of Yellow River upstream, is taken for example here. The period from flowing ice to freeze up is called flowing ice period. We analyzed the relations between 3 years flowing ice period from 1992-1995 and averaged air temperature, look at following table1.

Table1  Freeze-up date and daily average air temperature statistics

The above table clearly shown , the lower the air temperature , the shorter the flowing ice period  and the quicker the freeze-up .In 1993-1994 years, as the air temperature was only -11,the river reach had been frozen up after 3 days since the beginning of flowing ice, but in 1992-1993 years ,because the air temperature was more than -5 ,it had spent one month from flowing ice to freeze the river up.

2.2  Hydraulic conditions

As ice flower is moved by flow drag force, so main factors influencing flow ice is flow velocity. When other conditions are fixed, the bigger the flow discharge and velocity, the bigger the capacity of ice transportation, freeze-up will be difficult. Otherwise, will be easy.

2.3  River channel conditions

The main river channel conditions influencing freeze-up are river channel slope, curvature and width ratio of up and down cross-section. If river channel slope is bigger, the freeze-up will be difficult. If curvature is bigger, the flow ice will move slowly, then the freeze-up will be easy. The bigger the ratio of up and down cross-section, the more narrow the width of down cross-section, then the easier the freeze-up.

For example, among the three hydrologic stations as Bayangele, Sanhuhekou, and Toudaoguai in Inner Mongolia reach of Yellow River upstream, the time of freeze-up in Sanhuhekou is earliest, in Bayangaole is latest. The reason for this, one hand is air temperature low, the other important hand are unfavorable channel conditions. Here we use R to express comprehensive river channel conditions(R=curvature ×width ratio/slope). The bigger the R is, the easier the freeze-up is. Table 2 shows each station 1991-1995 years average freeze-up date and river channel conditions. We can seen from the table that Sanhuhekou station’s R is the biggest, the freeze-up date is the earliest, while Bayangaole station’s R is the smallest, the freeze-up date is the latest.

Table 2  Each station in Inner Mongolia years average freeze-up date(1991-1995) and  comparative channel conditions

3  THE FORMATION OF FREEZE-UP FORECASTING MODEL

3.1  The principle of formation of the model

To form the freeze-up forecasting model, the factors influencing freeze-up and the freeze-up mechanism will be based on.

Some of the factors influencing freeze-up such as channel curvature, width ratio of up and down cross-section, negative temperature are helpful of freeze-up, some of the factors such as channel slope, flow discharge are unfavorable of freeze-up (wind force neglected). According to the freeze-up mechanism, the frazil ice resistance is proportional with negative temperature and flow ice discharge, that is F_s∝f(ta_-,c_i). Drag force relates to flow discharge, channel slope and curvature, width ratio of up and down cross-section, that is F_t∽f(Q,J,B,S)，when F_s> F_t，freeze-up begins.

The terms in above formula, J、B、S have known, Q can be gained from flowing ice discharge calculation(this kind of calculation method has matured). T_a- can be forecasted by air temperature forecasting. Now the key is the calculation of flow ice discharge C_i.  We can use above formula to forecast the freeze-up date when C_i has been calculated.

3.2  Flow ice discharge calculation

3.2.1  The method of calculation 

Flow ice discharge is the index to indicate ice volume, generally is expressed by concentration of flowing-ice(the percentage of flowing-ice area and water surface area). Flowing-ice is produced by water body losing heat and forming supercooling water after air temperature falls 0℃ below because of air-water heat exchange，so the concentration of flowing-ice can be calculated   by the one-dimensional convection-diffusion equation as following:

         (1)

in which _i—density of ice(917kg/m³),L_i—specific heat of ice(80cal/g), C_i—concentration of flowing-ice, A－cross-sectional area of river, B－channel width, Q－flow discharge, t－time, x－distance along the river, _T— net heat flux from the river per unit surface area including heat exchanges at the free surface and the river bad(w/m².), Ex－longitudinal dispersion coefficient.

  (2)

Because the longitudinal mixing terms  and  are very small, by neglecting them we can simplify above formula to the following form:

                  (3)

Generally, there are two methods to calculate Ф_T, one is to use  metrological materials such as radiation and convection, another is to only consider the difference of water temperature and  air temperature, that is:

                                              (4)

in which K_wa—energy exchange coefficient(30 w/m² ),  T_a—air temperature(),α—experience coefficient，to those rivers which air temperature varies steady, =1, like in American rivers, which air temperature varies severely, >1, like in Inner Mongolia reach of Yellow River upstream，=1.5.

The first method needs too perfect metrological materials to be satisfied, so the second method usually is used. 

By bring (4) into (3)，we can obtain the following as：

                                (5)

By simplifying above formula to limited difference form, flowing-ice concentration can be calculated.

3.2.2  Calculation of flowing-ice concentration in Inner Mongolia reach of Yellow river upstream

Inner Mongolia reach of Yellow river upstream is in the north end of the Yellow River basin, (shown as Figure 1), the air temperature during ice flood period is low, the average daily air temperature during ice flood period of Toudaoguai Station is shown as Figure 2. The disasters were serious at this reach. The flowing-ice period is five months long from November to March of next year. Generally, river begins to flowing ice and freeze up on November, break up on March, several years on the first ten days of April. Here we calculate the flowing-ice concentration of the two hydrological stations at this reach, Bayangele and Sanhuhekou. The distance from Bayangele to Sanhuhekou is 221km. In this reach, because of factors such as geographic location, channel conditions etc., generally the time of freeze-up in Byangaole is the earliest and in Sanhuhekou is the latest. So the freeze-up forecast model making by this two station’s materials is representative.

Fig. 1  The plane view of Inner Mongolia reach of Yellow River

Fig. 2  Average daily air temperature during ice flood period from 1955 to 2000 of Toudaoguai Station

Let t select one day when the flowing-ice concentration is calculated. Generally, flowing-ice may be formed when air temperature falls to -3 below, so we take the first date of the temperature falling -3 below as the beginning date of calculation. We assume the flowing-ice concentration to be 0 at the day before the beginning date. The rest may be deduced by analogy, analyzing the every year’s observed datum. Both the latest freeze-up date of the two stations are on December, so the time of the calculation is on till 31st, December.

When the flowing-ice begins, T_W=0()

known =1.5

                                  (6)

 where t=1day，when the flowing-ice concentration is C_ii ，the surface area of heat-transfer with the atmosphere will decrease to 1-C_ii ( the percentage of surface area of heat-transfer and water surface area. By taking the value of _i、L_i and K_wa to (6), the following 

form can be given：

                                        (7)

 From above formula (7) we can obtain every day’s flowing-ice concentration.

3.3  The formation of the freeze-up forecast model

3.4 According to the principles of freeze-up forecast model, when , F_s>F_t freeze-up will begins, F_s is directly proportional to T_a-and C_I, F_t is directly proportional to Q、J，is inversely proportional to B、S. Here  J、B、S are channel conditions. Let R=B*S/J express the comprehensive channel condition, then the freeze-up condition is：

                F_s/F_t = f (T_a-*C_i*R/Q)>1                                 (8)

As the rest items known, we can forecast the freeze-up date by equation (8) as soon as the flowing-ice concentration has been calculated. For example, use Bayangaole and Sanhuhekou two hydrological stations total ten years observed datum from 1991 to 1995 to determine the form of f(T_a-*C_i*R/Q)  as following：

             F_s/F_t= f (T_a-*C_i*R / Q) = 50* T_a-*C_i*R^{^0.2}/Q                         (9)

That is the freeze-up concentration in Inner Mongolia reach is：

                 50* T_a-*C_i*R^{^0.2}/Q≥1                                   (10)

As the hydrological datum in Yellow River basin have been arranged to 1997, then the datum in the calculation of this paper was before 1997. We applied above freeze-up condition to forecast the freeze-up of the two hydrological stations from 1996 to 1997, results show that the precision is high. The simulated results from 1991 to 1995 are given as following table 3.

Table 3  Comparative of freeze-up date for forecast and observation

The freeze-up condition formula(10)indicates that the factors influencing freeze-up are air temperature, flowing ice concentration and channel conditions. The front three terms are single power, the last term is only 0.2 power. This shows that the front three terms are important, while the influence of channel condition is small relatively. That is to say, if both thermal factors and hydraulic factors couldn’t satisfy the freeze-up condition, the freeze-up wouldn’t happen even if channel conditions were very bad. Conversely, river would freeze up when thermal and hydraulic factors satisfy the freeze-up condition even if channel conditions are very well. This is as the same as the fact.

 4  THE CHARACTERISTICS OF THE FREEZE-UP FORECAST MODEL

The freeze-up forecast model has following characteristics：

(1) The influence of channel conditions are first considered in this model, then the freeze-up discrimination can be applied by all of the hydrological stations.

(2) The calculation of flowing ice for one station only using itself datum from the beginning of flowing ice to freeze up, instead of calculating station by station in order from upstream to downstream, then the spread time needn’t be considered. The spread time is not a constant because of the influence of the variation of flow discharge and flowing ice discharge. The model escapes making new errors for the sake of the error of spread time. Moreover, it is used more convenient.

(3) The precision of the model is high. According to “The criterion of error evaluating”, the foreseen period is 10 days, allowed error is 4 days. The results using this model to forecast in the Inner Mongolia reach of Yellow River upstream on 1st,November shows that the foreseen periods are all beyond 12 days, the maximum error is only 2 days.

References

[1]  Hung Tao Shen. River Ice Processes-State of Research ,IAHR Ice Symposium 96,Beijing.

[2]  Sujuan Ke, Guangqi lu, Zhiyuan Ren. Study on the ice-blocking mechanism in Bayangaole reach of Yellow River. Hydraulic journal，2000(7).