ESTIMATION OF THE HYDRAULIC ROUGHNESS OF RIVER ICE USING DATA ASSIMILATION

Transcription

1 ESTIMATION OF THE HYDRAULIC ROUGHNESS OF RIVER ICE USING DATA ASSIMILATION Steven F. Daly 1 and Carrie Vuyovich 1 1 ERDC Cold Regions Research and Engineering Laboratory Hanover, NH ABSTRACT Ice covers increase river stages via an additional rough boundary, which increases channel wetted perimeter, reduces channel hydraulic radius, and increases effective channel roughness. The hydraulic roughness of the river ice cover can vary throughout the winter and yearly, which can affect simulation models used to forecast near term conditions or estimate rivers flows. This paper presents a hydraulic model that applies a Kalman filter, using observed stages and the position of the upstream leading edge of the ice cover, to efficiently estimate optimal river ice and hydraulic conditions in the St. Clair River, an uncontrolled connecting channel between Lake Huron and Lake St. Clair. Estimating ice conditions in the St Clair River is complicated because ice can form in the river itself or flow out of Lake Huron, and there is no direct observation of discharge in the river. Hydraulic roughness is estimated for several different approaches over one winter season. Key Words: river ice, St Clair River, Great Lakes, data assimilation, Kalman filter INTRODUCTION The St. Clair River is an important component of the Great Lakes system of lakes and connecting channels, linking Lakes Huron and St. Clair (Fig. 1). A portion of the international boundary between the United States and Canada follows near the centerline of the channel. The flow in the St. Clair River is an important factor controlling the water levels of Lakes Michigan, Huron, St. Clair, and the lower Great Lakes. The upstream and downstream lake levels and the hydraulic conditions of the channels themselves determine the flow in the St. Clair Rivers. The difference between the upstream and downstream lake levels can be thought of as the driving force for the flow. The hydraulic conditions of the channels the geometry and roughness provide the resistance that balances the driving force. Other factors, such as the momentum of the flow itself and wind stress can also be important, although relatively short-term factors. Information on the water levels and knowledge of the hydraulics can be used to estimate the flows in the connecting channels. The water levels in the lakes and channels are constantly monitored. The water level -1-

2 elevation is measured at a series of gages installed along the St. Clair River (Fig. 1). The channel geometries are known. Problems arise in determining the channel roughness because of the presence of aquatic plants and especially ice. Aquatic plants cause a slow and continuous increase in the hydraulic roughness of the channels throughout the warmer months. Ice, on the other hand, can cause sudden and dramatic changes in the water levels and have dramatic effects on the channel flows. The presence of ice increases the channel roughness and makes a portion of the channel unavailable to flow. Ice can be generated in the channels, through heat transfer to the atmosphere or be transported from Lake Huron into the upper end of the St. Clair River at Port Huron, Michigan. Large and extensive ice jams have occurred on the St. Clair River in the past (Derecki and Quinn, 1986) that have significantly affected St. Clair River flows and the levels of the Great Lakes themselves. Given knowledge of the ice cover thickness and roughness, the hydraulics of ice-covered channels can be simulated and the flow in the St Clair River estimated throughout the winter. The difficulties are that the ice-cover thickness can only be measured by expensive and timeconsuming measurements from icebreakers and the roughness cannot be measured directly at all. Both can vary throughout the winter and from year to year. The hydraulic roughness can be highly variable and reflects the thickness of the ice cover, the type of ice that formed the cover, and the evolution of the cover with time. This highly variable hydraulic roughness is one reason that operational simulation of river ice covers has proven difficult to undertake. In this study a state-space model with data assimilation, well described elsewhere (Daly, 2002, 2003), is used to simulate the rapidly changing stage, discharges, and ice conditions in the St Clair River. The three components of the simulation are a hydraulic model, which estimates the water surface elevations and flows in the river system; a thermal and ice transport model, which estimates the river water temperatures, the frazil ice and surface ice concentrations, and the surface ice thickness; and an ice progression model, which estimates the extent and thickness of any stationary ice covers. The model assimilates observations of the water levels and the river ice cover extent to update estimates of the channel conveyance. These automatic updates allow the model to update all the components of the state vector that includes flows, stages, ice extents, and conveyance factors. Errors in the estimated ice extent and ice roughness are compensated for through updating the conveyance factors. In this paper a brief background is presented on the St. Clair River. The state-space model is briefly described, with special attention to the development of the conveyance factor. The model is then applied to the St Clair River for the winter of Observed water levels and ice cover extents are assimilated into the model. Three difference approaches are used. The results are then described. BACKGROUND Ice in the St Clair River The St. Clair River is approximately 63 km long, with a total fall of about 1.5 m. Flow averages about 5330 cms, which varies from a winter low monthly flow of 4360 cms in February, to a summer high of 5800 cms in July (Quinn and Kelly, 1983; Hunter and Croley, 1993). The all time record monthly flow was 7715 cms in June The all time record low monthly -2-

3 flow was 3000 cms in February of Larger extremes can occur over shorter periods of time. The St. Clair River can be divided into two sections: a single channel river and a delta. The single stem St. Clair channel is about 45 km long and contains the majority of the river fall, about 1.4 m. The delta region, known as the St. Clair flats, forms the lower reach, which extends the remaining 18 km to Lake St. Clair. In this reach the river falls less than 0.2 m, and is characterized by marshy flats with multiple channels. Figure 1. Location of the St. Clair River and gages. The ice conditions in the St. Clair River have received relatively little attention over the years. The Great Lakes Ice Atlas, for example (Assel, 2003), has extensive information on the surface ice conditions of the Great Lakes for 1973 to the present, but no information on the ice conditions in the St Clair River. Derecki and Quinn (1986) described the record ice jam of April of This ice jam disrupted navigation and remained in place for nearly 24 days. It was estimated to have reduced the monthly average flow in the St. Clair by about 50%. Daly and Arcone (1989) investigated the brash ice cover on the lower St. Clair using airborne radar. -3-

4 They found that they could discern the top of the ice and the water surface elevations from the radar returns. This information could then be used to estimate the overall ice cover thickness if the ice was assumed to be floating at the local hydrostatic equilibrium. Daly (1991) investigated the influence of ship passage between the St. Clair River and Lake Huron on the quantity of ice entering the St. Clair River. The study was based on time-lapse video of the entrance of the St Clair River. He noted that the quantity of ice entering the river was highly variable and that it was difficult to find a direct relationship between ship passages and ice quantity. Daly (1992) also investigated the ice passage from Lake Huron into the St. Clair River over the winters of through Data on ice passage had been collected during this period to support studies of the Navigation Season Extension Program (Calkins et al., 1982, Sodhi et al., 1982). Daly quantified the ice observed to enter the St. Clair River each month of the winter season. He found that ice was less likely to enter the St. Clair River when an ice arch existed in Lake Huron immediately above the upper end of the St. Clair River. He found that an ice arch could be expected to form when the 5-day moving average air temperature dropped below 7.5C and that the arch would cease to exist when the moving 5-day air temperature rose above 3.9C. Detroit District personnel made a series of 10 observations of St Clair River ice conditions during the winter (R. Zomparelli, personal communication). The location of the various types of ice observed was recorded. Simulation Model The hydraulic model is used to estimate the flow stages and discharges for a channel whose ice cover changes dynamically. The basic continuity and momentum equations describing one-dimensional, unsteady flow in channels with a floodplain were developed for both openwater and ice-covered flow. The continuity and momentum equations are solved using the four-point, implicit finite-difference scheme. As is usual, the channel geometric properties were described at only a finite number of discrete cross sections that segments the river into a series of sub-reaches. The energy losses of the flow are estimated using Manning s equation for an equivalent steady flow. At any section, the energy slope, S f, can be estimated as S f = Q K 2 2 where K = conveyance of the channel; and Q = the discharge. The open-water conveyance, K o, and the ice-influenced conveyance, K i, can be estimated using Manning s equation: K K = 1 A R 23 o o o no = 1 AR 23 i i i nc where n o = hydraulic roughness of the channel in open water and n c = composite roughness including ice (found using the Belokon-Sabaneev formula); A o and A i = open-water and icecovered flow area respectively; and R o and R i = open-water and ice-influenced hydraulic radius respectively. The values of the components of the open-water conveyance, A o, R o, and n o, are assumed to be functions of the water level only. The value of n i, the hydraulic roughness of the ice cover, is assumed to be fixed and constant with time. The values of A i and R i are (1) (2) -4-

5 assumed to be functions of the ice thickness and water level only, and will vary in time only with variations in the ice thickness and water level. The impact of ice on the conveyance is estimated by defining an effective channel conveyance combining both the open-water conveyance and the ice-influenced conveyance. As the ice conditions may differ upstream and downstream of any section, two conveyance factors need to be developed for each section, one for the upstream side and one for the downstream side. The effective conveyance upstream of a section can be estimated as (an analogous effective conveyance is also found for downstream of a section) ( ) K = C us K us K i v Σ +Σ i oi i ii where Σus i = the fraction of the upstream reach associated with the cross section covered with ice (0 < = Σus i < =0.5, for details see Daly 2003); K oi and K ii are the open-water and ice-cover channel conveyances, respectively; and C v = a conveyance factor. The conveyance factor allows the effective conveyance to be adjusted to account for errors in the parameters that form the open-water and ice-influenced conveyances, and errors in the estimated ice-cover extent. This allows the effective conveyance to be updated, along with other elements of the state vector as described below, as observations are assimilated into the model. The updating of the conveyance factor accounts for errors in the estimates of any of the parameters that form the effective conveyance. If the values of these parameters are estimated accurately, the value of C v should be equal to one. However, during updates, the value can be adjusted to if the water elevations estimated by the model do not agree with the observations. As the effective channel roughness decreases, for example through smoothing of the river ice cover, C v increases. APPLICATION TO THE ST CLAIR RIVER The state-space model was used to estimate discharges in the St. Clair River for 1 December 2002 through 31 March The downstream limit of the model was located at the Algonac Gage and the upstream limit at the Ft. Gratiot (Fig. 1). The model was calibrated during June It reproduced the daily discharges estimated based on previously developed stage-fall equations. Observations indicated that a stationary ice cover existed in the St. Clair River upstream of the Algonac Gage for some periods during the period of interest. Figure 2 show observations of the distance of the upstream edge of the stationary ice cover from the Algonac Gage. There were 10 observations of the ice cover over the winter. The water levels recorded by the gages along the river are shown in Figure 3. The Algonac and Ft. Gratiot gages provide the boundary conditions to the model. Wind set-up, seiches, and short-term lake level fluctuations impact the Ft. Gratiot Gage, as can be seen. The intermediate gages are used for data assimilation. Elevation differences between the gages vary throughout the winter as a result of the ice cover impacts on the channel conveyances, as discussed above. The recorded hourly water temperatures at the Dunn Paper Gage, which fluctuated considerably (Fig. 4), were used as the upstream water temperature boundary condition. The water temperatures measurements reflect an unknown accuracy. In addition, the degree to which the temperature fluctuations are caused by the incomplete vertical mixing of the stratified Lake Huron water temperature is not known. The air temperature records were recorded at the Selfridge Air National Guard base, approximately 50 km from the lower end of the St. Clair River. These records are probably adequate as there is little horizontal variation in air temperature throughout the region of the St. Clair River. (3) -5-

6 20 15 KM Figure 2. Upstream leading edge of the ice cover Stage (METERS) Figure 3. Observed hourly water levels along the St. Clair River. -6-

7 Temp () Figure 4. Observed hourly water temperatures recorded at the Dunn Paper Gage and recorded air temperatures. As discussed in Daly (2003), a fundamental question that must be addressed is the number of conveyance factors. Theoretically, the number of conveyance factors could range from the number of cross sections (if a separate conveyance factor was used for each cross section) to one (if a single conveyance factor is applied at all cross sections). The approach used here was to follow something analogous to the standard procedure for calibrating hydraulic model roughness. In that procedure, conveyance factors are selected on the basis of the number of gages along the channel that measure the water surface elevation. A separate conveyance factor can be used for each river reach that has a reliable gage at its upstream and downstream end. In this case, three conveyance factors were used. Conveyance section 1 extended from the St. Clair State Police Gage to the Dry Dock Gage; conveyance section 2 from the Roberts Landing Gage to the St. Clair State Police Gage; and conveyance section 3 from the Algonac Gage to the Roberts Landing Gage. The upstream most section of the river, from the Dry Dock Gage to the Ft. Gratiot Gage, was not included because experiments indicated that adjusting the conveyance at the upstream boundary of the model led to numerical instability, and stationary ice was rarely, if ever, observed that far upstream. The hourly flow in the St. Clair River was estimated using the Algonac and Ft. Gratiot gages as the downstream and upstream boundary conditions for the hydraulic model, the water temperature recorded at Dunn Paper as the upstream air temperature, and the air temperature recorded at the Selfridge Air National Guard as the air temperature. The difference between the water and air temperature was used to estimate the heat transfer between the water and the atmosphere. Three different approaches were used to estimate the hourly discharge: -7-

8 1. A time series of hourly ice observations was developed for 1 December 2002 through 31 March 2003 in which the extent of the St Clair River ice cover was set consistently to zero. These observations were assimilated into the model to ensure that the no ice cover existed during the entire simulation. In this approach, the roughness of the ice cover was not relevant, as the model assumed that open water existed at all times. 2. A time series of hourly ice observations was developed for 1 December 2002 through 31 March 2003 by linearly interpolating between the ten ice observations that were made over the winter period. The time series developed is represented by the solid line connecting the observations in Figure 2. The Manning s roughness coefficient of the ice cover was set to 0.018, a moderate to low roughness value. 3. In the third approach, a time series of hourly ice observations was developed for 1 December 2002 through 31 March 2003, as in approach 2, by linearly interpolating between the 10 ice observations. The Manning s roughness coefficient of the ice cover was set to 0.036, a relatively high roughness value. RESULTS Water Levels Given that the model assimilated the observed water levels at the three intermediate gages, the model reproduced these observed water levels nearly exactly. Ice Cover Extent Given that the model assimilated the time series of the ice cover extent that are described above, the model reproduced these observed time series exactly in Approaches 2 and 3. Conveyance Factors The results for the conveyance factors are shown in Figure 5. Recall that changes in the conveyance factors result from assimilating the observed water levels. Whenever data from one of the assimilated water level was missing, the conveyance factor is automatically set to a value of one, and remains at one until the data stream resumes. The following general results can be seen. In Approach 1, no ice cover was included in the simulation model, meaning rather large modifications to the conveyance factor are required when an ice cover was in place during the winter, to account for the resistance of the ice cover to flow. This can be dramatically seen in conveyance section 3 and 2 (Figures 5a, b). In conveyance section 1, the most upstream section (Figure 5c), the conveyance factor is near its open water value throughout the winter. This indicates that no ice cover existed in this section during the winter, except for an unexpected period during mid-december. In all three conveyance sections, open water can be assumed whenever the conveyance factor is at its open water value. -8-

9 NONE CONV. SECTION 3 IAHR 1 CONV. ADJ CONV. SECTION 3 IAHR 2 CONV. ADJ CONV. SECTION 3 IAHR 3 CONV. ADJ 2.2 a NONE CONV. SECTION 2 IAHR 1 CONV. ADJ CONV. SECTION 2 IAHR 2 CONV. ADJ CONV. SECTION 2 IAHR 3 CONV. ADJ b. -9-

10 NONE CONV. SECTION 1 IAHR 1 CONV. ADJ CONV. SECTION 1 IAHR 2 CONV. ADJ CONV. SECTION 1 IAHR 3 CONV. ADJ c. Figure 5. Estimated conveyance factors. Circle: no ice (Approach 1); square: interpolated ice covers with relatively low hydraulic roughness (Approach 2); triangle: interpolated ice covers with relatively high hydraulic roughness (Approach 3). In Approach 2, the ice cover extent, based on a linear interpolation between the 10 ice observations, was assimilated into the model and a moderate ice roughness was assumed. The modifications to the conveyance section generally follow that of Approach 1, but the modifications are smaller than Approach 1 when the conveyance factor is less than one. The presence of the ice cover in the simulation model provides additional resistance to the flow, so that the correction to the conveyance factor does not have to be as large. There are some periods when the ice cover was included in the simulation model but probably did not exist in the conveyance section of the St Clair River. This indicates the inaccuracies introduced by linear interpolation of so few observations. During these periods, the conveyance factor of Approach 1 returns to its open water values, but the conveyance factor of Approach 2 must become larger than one to counteract the presence of ice in the simulation model. In Approach 3, the ice cover extent, based on a linear interpolation between the 10 ice observations, was assimilated into the model and a large ice roughness was assumed. The modifications to the conveyance section generally follow that of Approach 2, but the modifications are smaller than Approach 2, when the conveyance factor is less than one. This indicates that the actual roughness of the ice cover is closer to that assumed in Approach 3 than 2. During periods when the ice cover was included in the simulation model, but probably did not exist in the conveyance section of the St Clair River, the conveyance factor of Approach 3 must become larger than one to counteract the presence of ice in the simulation model. In this -10-

11 case, the conveyance factor is also larger than conveyance factor of Approach 2, because the modifications of the conveyance factor must counteract the large ice roughness. Flows The estimated hourly flows are shown in Figure 6. It can be seen that the impact of ice on the estimated flows can be dramatic, reducing the discharge in the channel to less near 50% of the open water value at its maximum reduction. All three approaches give the same results. Assimilating the observed water levels and modifying the conveyance factor can account for any problems that result from inaccuracies in estimating the ice extent and ice roughness Flow (cms) ALGONAC IAHR 1 FLOW ALGONAC IAHR 2 FLOW ALGONAC IAHR 3 FLOW Figure 6. Estimated hourly discharges at the Algonac Gage. Circle: no ice; square: interpolated ice covers with relatively low hydraulic roughness; triangle: interpolated ice covers with relatively high hydraulic roughness. ACKNOWLEDGEMENTS This work was supported by the Detroit District of the Corps of Engineers. The attention and contribution of John Koschik and Scott Thieme, of the District are gratefully acknowledged. REFERENCES Assel, R.A. (2003), An electronic atlas of Great Lakes ice cover, NOAA Great Lakes Ice Atlas, Great Lakes Environmental Research Laboratory, Ann Arbor, Michigan

12 Calkins, D. J., Deck, D., and Sodhi, D. S. (1982), Hydraulic model study of Port Huron ice control structure, USA Cold Regions Research and Engineering Laboratory, CRREL Report CR Daly, S. F., and Arcone, S. A. (1989), Airborne radar survey of a brash ice jam in the St. Clair River, USA Cold Regions Research and Engineering Laboratory, CRREL Report Daly, S. F. (1991), Effects of ship transits on the ice entering the St. Clair River from Lake Huron, USA Cold Regions Research and Engineering Laboratory Report submitted to the Detroit District, Corps of Engineers, CENCE-IA Daly, S. F. (1992), Observed ice passage from Lake Huron into the St. Clair River, Journal of Great Lakes Research, 18(1): Daly, S. F. (2002), Data assimilation in river ice forecasting, in Proceedings of the 16th International Symposium on Ice, Dunedin, New Zealand. 2 6 December 2002, Vol. I, pp Daly, S. F. (2003), A state space model for ice forecasting, U.S Army Engineering Research and Development Center, Cold Regions Research and Engineering Laboratory, ERDC/CRREL TR Derecki, J. A., and Quinn, F. H. (1986), Record St. Clair River ice jam of 1984, Journal of Hydraulic Engineering 112(12): Hunter, T. S., and Croley, II, T. E. (1993), Great Lakes monthly hydrologic data, NOAA Data Report ERL GLERL, National Technical Information Service, Springfield, Virginia. Quinn, F. H., and Kelley, R. N. (1983) Great Lakes monthly hydrologic data, NOAA Data Report ERL GLERL-26, National Technical Information Service, Springfield, Virginia. Sodhi, D. S., Calkins, D. J., and Deck, D. (1982), Model study of Port Huron ice control structure: wind stress simulation, USA Cold Regions Research and Engineering Laboratory, CRREL Report CR