Estimation of probable maximum floods from the southern Alps, New Zealand

Transcription

1 Extreme Hydroloeical Events: Precipitation, Floods and Droughts (Proceedings of the Yokohama Symposium, July 1993). IAHSPubf.no. 213, Estimation of probable maximum floods from the southern Alps, New Zealand S. M. THOMPSON Freshwater Division, National Institute of Water and Atmospheric Research Ltd, PO Box 8602, Christchurch, New Zealand Abstract When the American procedure for estimating PMFs is applied to the Southern Alps, South Island, New Zealand, the results are unduly sensitive to parameters that cannot be directly measured. The reason for difficulty is a much greater orographic effect than in the USA. An alternative approach is proposed, tentative results presented, and further work that could firm up these results is suggested. THE SOUTHERN ALPS The Southern Alps in the South Island of New Zealand are an approximately straight range of mountains with many similar river catchments draining either side. The range lies along a line from north east to south west. Heavy rain occurs during moist northwesterly winds. Mean rainfall isohyets are parallel to the range, and storm rainfall isohyets are also parallel, at least over several catchments (Henderson, 1993). Thus when estimating probable maximum floods (PMFs) depth-area reduction of point rainfalls is negligible in the direction parallel to the range. However across the range the isohyets are close together. For example the heaviest recorded storm rainfall peaked at 1810 mm over three days three km from the Alpine Fault, but for the same storm, had fallen to 900, 430 and 150 mm at 17, 29 and 37 km from the Alpine fault respectively. Figure 1 is a map of the Southern Alps showing river catchments, with the scale across the range 2.5 times the scale along the range. Mean rainfall isohyets are also shown. The main divide is indicated by a double line across the centre of the map. northwest scale is 2.5 times northeast scale. Isohyets have been drawn strictly parallel to better show the general pattern of mean rainfall. In a few places there are enough raingauges to define a more complicated pattern which follows local topography. When estimating the flood flow caused by a particular pattern of storm rainfall, it is useful to distort the catchment shape in this way because the accuracy of the position of rainfall isohyets is relevant in only one direction, i.e. across the range. Estimates of PMF are required for the 12 catchments which already have dams downstream, and would be useful when planning riparian developments on all the other catchments. TIMES OF CONCENTRATION ON STEEP CATCHMENTS These catchments typically fall 2000 m in 30 km, so that rain runs off rapidly and the "time of concentration", i.e. the time between a change of rainfall intensity and the consequent change in flow rate, is not more than 12 hours on any catchment and on

2 300 S. M. Thompson Fig. 1 Map of Southern Alps showing the river catchments. most catchments is much less. Figure 2 shows hourly rain and the consequent flow, from an alpine catchment northwest of the main divide, during a major storm. The average catch of the three raingauges is close to the mean flow from the catchment. The storm flow total was 322 mm and the rain totals 524, 350 and 576 mm. There are three peak intensities in the rain record from three gauges, and corresponding peaks in the flow record two hours later indicating a "time of concentration" of two hours. The ratio of peak flows to the corresponding two hour rain intensities are 0.31, then 0.40 then 01-DEC DEC 03-DEC Hokitika at Colliers Ck 3600*(Flow l/s)/ Cropp at Cropp Hut Rain mm Hokitika at Rapid Ck Rain mm Hokitika at Prices Hat Rain mm Fig. 2 Storm rainfall intensity (mm h 4 ) and flood flow (mm h" 1 ) recorded on the upper 352 km 2 of the Hokitika catchment shown dashed on Fig. 1.

3 Estimation of probable maximum floods from the Southern Alps, New Zealand , increasing as the storm proceeds and the peak rain intensities increase. When the rain is heavier than this we can expect the peak flow to be even closer to the peak rain. Throughout the northwest side of the main divide, the largest storms will saturate the catchment before the peak intensity occurs and have a peak flow the same as the peak intensity averaged over the catchment and over the time of concentration. The catchments southeast of the main divide extend into a dry rain shadow where saturation probably never occurs. However if we consider only the part of the southeast catchments within 50 km of the Alpine Fault (the 1 m year" 1 ) isohyet on Fig. 1 is 47 km), the situation during the largest storms is probably the same as in the northwest catchments, i.e. saturation with the flood peak (mm h" 1 ) equal to the storms maximum rainfall intensity (mm h" 1 ) averaged over the area and time of concentration (2 to 12 h). Thus to calculate the peak flow corresponding to a given rainfall pattern, and in particular the PMF from the probable maximum precipitation (PMP) it is simply necessary to determine the time of concentration and to find the maximum rain intensity over that duration. REDUCTION OF PRECIPITATION INTENSITY WITH DURATION There is a relationship between maximum intensity and duration at every site which may be approximated as: where I max is the intensity of the precipitation observed over duration D hours, in mm h" 1, and X is a constant for the site, in mm h' 05. The largest recorded X from the Southern Alps (X = 270 mm h' 0-5 ) was at Alex Knob in March 1982 and is shown with other data on Fig. 3. The probability that a significantly high X will be observed at a particular raingauge during a year is small, but if we operate many gauges at locations with the same X for enough years we can determine X directly. However, there is really no prospect of defining the spatial variation of X by direct observation Duration hours Fig. 3 Large recorded rainfall intensities for various durations. Southern Alp data are solid symbols and some international data are open symbols.

4 302 S. M. Thompson TIMES OF CONCENTRATION OF THE LARGE LAKES There are seven catchments on Fig. 1 with large natural lakes, and the maximum outflow from these depends on the maximum rain intensity over a much longer period than 12 hours as will now be shown. For these large natural lakes the PMF requires sufficient rain to raise the lake to its maximum control level. Lake Hawea is omitted from this discussion because it does not have a spillway on the dam, only a sluice with capacity 313 m 3 s" 1, and this detail prevents application of the particular calculation described below to that lake. The times of concentration of lake outflow peaks can be calculated by maximising the level (L mm) to which the lake rises. A lake rises as long as the inflow (I mm h" 1 ) exceeds the outflow (Q mm h" 1 ). Both these flows are expressed as water yield per unit of catchment area. The outflow depends on the lake level and the stage-discharge relation, or rating, at the lake outlet, which has been represented in each case by the formula: Q = S(L-L 0 f (2) Each lake has a particular value of spill constant S (mm h)" 1. The value of Lo was chosen to obtain a good overall fit of a parabola to the current outflow rating, and is close to but not equal to the maximum control level prescribed in the rules for release of floods from each lake. The rate of rise of a lake caused by a particular inflow intensity (I mm h" 1 ) depends on the ratio of catchment area (A c km 2 ) to lake area (A! km 2 ), and for this calculation the catchment area upstream of the 1 m year" 1 isohyet on Fig. 1 was used.: ^ l = ^(7-0 = ai-as(l-l 0 f (3) dt A l Time t is expressed in hours and a = A c /A,. The actual rise is the integral of the rate of rise over the duration of the storm D hours: L L 0 - r-i tanh[(/5) - 5 fl >] ( 4 ) This integration was done assuming the inflow I.mm h" 1 was constant over the duration of the storm 0 < t < D. Consider now the situation where I is related to D according to equation (1): X L-L V 0-25 tanhffxs) 0^ ] (5) ~S The time of concentration is the value of D for which L is a maximum, that is dl/dd = 0. Thus: D" L25 tanh[(xs) 0 ' 5 az) 0 ' 75 ] 0.5 IT 0-25 sech 2 [(XS) 0-5 û2> 0-75 ] (0.75(XS) - 5 GZr 0-25 )

5 Estimation of probable maximum floods from the Southern Alps, New Zealand 303 Using the identity sinh(c) cosh(c) = 0.5 sinh(2 c) gives: 0.5 smh[2(xs) 0-5 ad ols ] = 3(XS) - 5 ad 015 We look up a table of sinh function to find the c for which sinh(c) = 3c ie. c = 2.84: D 1.42 (XS) 05 a (6) After this time the storm ends, the inflow I rapidly declines because the time of concentration of the catchment feeding the lake is short compared to the time of concentration of the lake. Thus at this time the outflow from the lake peaks at: Umax " S(L t =D A)) = SXS- l D- 5 tanh 2 (1.42) = 0.79XD~ 05 Eliminating D between equations (6) and (7) we get: X = 1.42 Q^S- - 2S a- 0-5 Eliminating X between equations (6) and (7) we get: (V) (8) 0.5 D = l.26q m -S c -0.5 fl -l (9) Adopting 2.2 times the 1% AEP outflow as Qmax, an initial estimate of the PMF, we estimate X and D. The 1% AEP outflow from each lake was obtained by fitting a Gumbel frequency distribution to the record of outflows up to commissioning of any lake control structure, and thereafter a record of simulated natural lake outflows. The inputs and results of this calculation are presented in Table 1. The rain parameters which could generate the flood that peaks at 2.2 times the 1 % AEP flood are derived. The 2.2 multiplier produces conventional PMF estimates at least two of the lakes. The catchments areas used are less than the total in each case, and are intended to represent only the northwest part of the catchment which becomes saturated. This has been estimated as the part within 50 km of the Alpine Fault, except for Lake Te Anau where a larger area is used. A shorter time of concentration D and correspondingly larger X would be calculated for the same input data if we assumed that the inflow I varies over the Table 1 Calculations for the outflows from six large lakes. Lake name Tekapo Pukaki Ohau Wanaka Wakatipu Te Anau Areas catch. Ac km A lake AI km A ratio a Ac/AI Outflow Spillway peak peak coef. coef. Qmax - S m A 3/s mm/h m/s (kmh)m level Lo m asl Flood rise Lmax-Lo m Intensity coef. X mm/h'\ Time of concentration D h days Rain intensity depth P X/D*.5 mm/h PD mm

6 304 S. M. Thompson duration, being smaller at the start and correspondingly larger towards the end to preserve the total volume. Incorporation of this variation into the calculation should be done to calculate the PMF. Nevertheless these simpler calculations are adequate to show the magnitude of the various factors, and in particular D and X. PROPOSED ATMOSPHERIC MODEL A mathematical model of rain from a moist northwest wind which rises to pass over the Southern Alps could define the spatial variation of X. The model could include the detailed topography but model the flow in only two dimensions, i.e. vertical and horizontal along a northwest - southeast line. This model would calculate the orographic rain, un-enhanced by any other process than that which is modelled, and would be calibrated using records of minimum rainfalls at the time of a particular incoming moisture flux. Possible mechanisms to explain an excess of observed rainfalls above the orographic rain predicted by the proposed model include: (a) a cold front; (b) circulation around a low pressure centre; (c) convective cell over a small area; (d) upwind (northwest) displacement of rainfall from its normal location due to (e) increased blocking in a more stratified atmosphere (Baines, 1987); downwind (southeast) displacement of rainfall from its normal location due to inefficient rainfall generation and consequent advection of more moisture to the downwind extremity of the storm. Displacement processes seem likely to be of practical importance. Unusually severe flood damage often seems to be the result of rain with moderate intensity at locations where such intensity has not been experienced for several decades, although rain of the same intensity is frequently experienced only a few kilometres away. This perception is obviously a consequence of the steep spatial gradients of rainfall. The standard PMP method estimates a flat land rain un-enhanced by orography, which is called convergence rain, then multiplies it by a T/C (total/convergence) ratio to include the effect of orography. There are sites in the Southern Alps where T/C is 8 which suggests that, at these sites, un-enhanced orographic rain will be 7/8ths of the total rain! This would mean the model was very useful. A model of this kind has been applied in the Austrian Alps (Haiden et al, 1990). The Southern Alps should be better suited to this type of model because the incoming wind on the Southern Alps is more often a simple geostrophic wind, warm and with a 100% humidity, and only these situations need be analyzed to understand the generation of large floods. The Southern Alps are an inconvenient location to obtain measurements of the incoming moisture flux, but I understand that the necessary information can be back calculated using radiosonde measurements along the coast which is only 30 to 50 km northwest of the main divide. CONCLUSIONS The Southern Alps are an extreme topography in a relatively simple atmospheric

7 Estimation of probable maximum floods from the Southern Alps, New Zealand 305 situation that gives rise to large storm rainfalls. The gradients of storm rainfall are steep normal to the mountains, but negligible parallel to the mountains on a catchment scale. For estimation of PMFs in the Southern Alps, it is the maximum intensity that is relevant, not the storm rain total, provided the maximum intensity occurs after sufficient rain to saturate the catchment. A two-dimensional atmospheric model could provide the means to model the processes giving rise to the PMP, to give more defensible estimates than the standard transposition techniques. REFERENCES Baines, P. G. (1987) Upstream blocking and airflow over mountains. Ann. Rev. Fluid Mech. 19, Haiden T., Kahlig, P. & Kerschbaum, M. (1990) On the influence of mountains on large-scale precipitation: a deterministic approach towards orographic PMP. Hydrol. Sci. J. 35(5), Henderson, R. D. (1993) Extreme storm rainfalls in the Southern Alps, New Zealand. This conference. Paulhus, J. L. H. (1965) Indian Ocean and Taiwan rainfalls set new records. Mon. Weath. Rev. 93(5),

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