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Meander spectra of the Angabunga River
Journal of Hydrology 3 (1965) 1-15; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microlilm without written permission from the publisher
MEANDER
S P E C T R A OF T H E A N G A B U N G A
RIVER
J. G. SPEIGHT Division of Land Research and Regional Survey, CSIRO, Canberra, Australia
Abstract: Three sets of air photos spanning twenty-four years are used in an analysis of the dominant modes of meandering of the Angabunga River, in central Papua, using the method of the power spectrum. Spectra are constructed lbr successive reaches of the stream based on the auto-correlation of the direction of flow measured at equi-spaced points on the talweg of the channel, Such spectra display the meander intensity associated with each band of frequency, where frequency is the reciprocal of meander-wavelength, measured along the talweg. It is shown that the subjectively chosen characteristic meander wavelength used by previous workers is a poor indicator of the dominant frequencies of oscillation. All spectra show a number of stable peaks of meander intensity, of which the dominant one corresponds to a wavelength of the order of 90 times the root of the bankfull discharge, that is, about 50'>~,longer than the expected "characteristic" wavelength. The most prominent spectral peaks are more stable in a downstream direction than are other parameters, such as channel width. The relative intensities of the various peaks, however, are subject to rapid change associated with change in channel cross-sectional shape. Parts of the channel with a lower width-depth ratio have relatively high intensity associated with high-frequency peaks, a fact which strongly influences the subjective judgement of meander wavelength. 1. Introduction Since 1949 when the w o r k o f inglis ~) on the relation o f m e a n d e r wavelength of a river to d o m i n a n t discharge was published, there have been a n u m b e r o f papers z-6) on correlations between m e a n d e r wavelength a n d width, discharge, or v a l l e y - m e a n d e r wavelength. The wavelengths q u o t e d in these studies have generally been m e a s u r e d at favourable sites, t a k i n g twice the distance between successive points o f inflection in the channel. This m e t h o d has three s h o r t c o m i n g s : (i) " t y p i c a l " , well defined m e a n d e r s are chosen at the discretion o f the g e o m o r p h o l o g i s t ; (it) wavelengths are measured along a succession o f straight lines, the directions o f which m a y bear little relevance to the direction of flow o f the river (the angle between successive lines is frequently m o r e than 90 ~); (iii) the possibility of m o r e t h a n a single m o d a l wavelength is ignored. The a s s u m p t i o n implicit in the m e a s u r e m e n t of m e a n d e r wavelength at selected sites where the m e a n d e r is a simple S-shape is t h a t there is a single d o m i n a n t wavelength o f m e a n d e r i n g (;re) which is obscured at all other sites
2
J.G. SPEIGHT
by irregularities of a quasi-random nature. Only Hjulstr6m 7) and Schumm s) have suggested that two characteristic wavelengths may be present in the one stream at the same time. This investigation proceeds from a consideration of the total variability of direction of flow of the stream and the possibility of its analysis as a spectrum of oscillations in direction measured against distance downstream. The type of spectrum implied by most previous workers is a single intense peak of frequency v = 1/2L superimposed on a "noise" the intensity of which is widely distributed through the frequency range. 2. The Angabunga River The Angabunga River in the Territory of Papua was chosen for study because of its long unobstructed and undisturbed alluvial plains course and
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ANGABUNGA RIVER I, owcr Courses
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Fig. 1. The plains course and lower mountain course of the Angabunga River.
MEANDER SPECTRA OF 'FILE ANGABUNGA RIVER
3
its highly developed, rapidly changing meander pattern (Fig. I). The basic data for the analysis was derived from three sets of vertical air photos at a scale of 1:40 000 taken in 1938, 1957, and 1963. For comparative purposes flow characteristics, topography, channel width, shape, and roughness, and sediment type were also investigated; these aspects are reported fully in the following paper 9). The stream is of moderate size, with a catchment of 940 square miles under a 90 inch rainfall, and a mean flow of nearly 4000 ft:~/sec. The variability of daily flows is low, although the rise and fall during a fresh is very rapid. A monsoonal climate causes a moderate seasonality of high and low flows. Until very recently, the channel length of the plains course, which has no tributaries, was about 57 miles, but a crevasse which occurred in 1954 near the mid-point of the course resulted in a reduction of channel length to 39 miles. Although the new lower course was fully formed and was carrying all tile flow by 1963, in 1957 the main flow was still carried by the old lower course, and it was reasonable to ignore the effect of the crevasse in the 1957 analysis. The channel cross-section of tile plains course is of two types: a migrating wide triangular section dominated by point-bars, characteristic of tile upstream reaches which have a gravel bed and sandy banks, and a fixed narrow rectangular section dominated by levees, a characteristic of the downstream reaches with a sandy bed and silt-clay banks. In its lower mountain course the river is confined between rock walls on a valley floor seldom much wider than the unvegetated channel.
3. Spectral analysis On each set of photos a smoothly curving line was drawn to follow the talweg of the stream and a pair of dividers was used to make marks on this line at scaled intervals of 300 feet, that is, approximately the mean width of unvegetated channel. A precisely oriented arbitrary datum line was drawn on each photo and the direction of flow at each marked point on the talweg was recorded. Distributaiies, being both small and short, were ignored. This procedure was applied not only to the three plains courses, but also to about 25 miles of the lower mountain course and to four of the longest wellpreserved prior stream channels visible on the plain. For the purpose of the analysis the several river courses were broken into overlapping reaches usually 126 000 feet long at intervals of 31 500 feet downstream, so that in each the number of 300-foot points was 420, although there was some variation of length in special cases. When a trial showed that the points were too closely spaced for useful analysis the angles at successive points were summed in pairs, giving an effective point-spacing of 600 feet. The resulting angles in
4
J.G. SPEIGHT
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Mountain Course
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Fig. 2. Some examples of meander spectra for various reaches of the Angabunga River, with insets showing the trace of the talweg in the central part of the reach in each case.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
5
degrees, xa, where d = 1, 2 ..... n; n = 210, were then analysed by digital computer for auto-correlation using the formula:
cx(1
; l °-'
i
(1)
(n
f o r / = 0 , 1, 2 ..... m - 1, where m is the number of frequency bands, the value of m being taken as 40 in this case. The intensity X (k) in each frequency band k = 0, 1, 2 ..... m was then calculated by m-
1
m/
m 3_1
/=1
and spectra were constructed as in the examples of Fig. 2. The total area under the curve is proportional to the variance, while the ordinate X(k) is arbitrarily defined as meander intensity, the probable error of which was about + 30%. For a general discussion of power-spectral techniques the text by Blackman and Tukey 1°) should be consulted. The spectral envelopes clearly contain a number of coherent peaks, but in order to quantify these it is necessary to make assumptions about the shape of the peaks and about the character of the non-periodic components, or noise. There does not appear to be a standard procedure for this. The method adopted consists of replacing spectral peaks by curves of normal distribution. It was suggested by the excellent fit obtained in the case of the most isolated peaks, such as the peak labelled " F " on the spectrum illustrated for Prior Channel No. 2, and by the way more complex peaks may be completely represented as the sum of only two or three normal curves. As the figure shows, the total spectral envelope may be represented by quite a small number of such normal curves. It must be emphasised that this procedure is merely a device to obtain precise quantification of the spectral peaks and does not imply that all of the peaks are "real" in the sense of representing stable oscillations rather than noise. In effect the method causes the noise in the spectrum to be absorbed into the spread of values associated with the individual peaks. At the low-frequency end, in particular, some quite intense peaks have been analysed into normal curves which, while representing the simplest construction which will fit, may well be quite different from the peaks (or noise) which produce the spectrum. The 1938 Lower Plains Course illustrates such a doubtful case. The spectral intensity peaks produced by the analysis could now be represented by three parameters: the peak intensity X (p) in arbitrary units, the peak frequency v in cycles per (foot x 1057, and the band-width Av, measured at half the peak intensity. The variations in these parameters are plotted
6
J . G . SPEIGHT
against mean distance downstream for all spectra in Fig. 3, the height of each rectangle being the band-width, and the breadth being proportional to the peak intensity. Data from spectra for prior channels Nos. 1, 3 and 4 are not presented as the remnants of these courses are too short to allow of adequate resolution. Both the complex structure of the spectra and the persistence of individual peaks in successive reaches are remarkable and were completely unexpected on the basis of previous studies. For comparison the talweg wavelengths of typical, well-developed meanders near the mid-point of each reach were measured and their reciprocals plotted on the figure. The unreliability of the measure as an indicator of dominant mode of meandering is plainly shown. For convenience of discussion fifteen relatively persistent frequency peaks have been labelled alphabetically. It is their very persistence which allows them to be traced downstream and to be labelled in this manner. One may doubt the "reality" of some of them, either through low spectral intensity or through lack of resolution, as is common at the low frequency end, but the majority appear to represent stable frequencies of oscillation. In general lowfrequency peaks A, B and C dominate the spectrum, together with variable amounts of very low-frequency noise which is probably aperiodic. Because the C-peak is characteristically the shortest wavelength dominant peak the subjectively assessed "typical" meander-wavelength often corresponds to it, but any appreciable peaks at higher frequencies, even of quite low intensity, may cause a very short meander-wavelength to be judged "typical", as in the lowermost reach of 1938 (Fig. 3b). The B-peak is the most stable and persistent peak of all, dominating nearly all plains-course spectra and occurring in every course. The value of v at the peak lies in the neighbourhood of 10 (i.e. 2r= 10 000 feet), but tends to rise steadily from v = 8 in the upper plains to v = 12 in the lower plains, except that the increase is accelerated in the 1963 lower course. Prior channels Nos. 1,3, and 4 also are dominated by peaks in the vicinity of v = 10, so that if wavelength is strictly related to discharge, there are strong grounds for assuming a general uniformity of flow regime over the whole time represented. If one compares the upper parts of the 1938 and 1963 plains-courses, three of the spectra are found to be almost identical except for an increase in the intensity of the F-peak and other high-frequency peaks. This is surprising in view of the radically different aspect of the corresponding 1957 spectra, in which the C-peaks (v = 15) have been elided and replaced by intense D-peaks (v = 19), and the frequency of the A-peaks has been drastically reduced. This alternation of spectral shape is quite impossible to detect by visual examination of the river pattern and may represent a sensitive response to some short term variation in flow regime. The lower reaches of 1938 and 1957, which
7
MEANDER SPECTRA OF THE ANGABUNGA RIVER
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8
J.G. SPEIGHT ~Mountain
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Fig. 3d Fig. 3. Frequencies, band widths, and intensities of spectral peaks plotted against mean distance downstream.
9
MEANDERSPECTRAOF THE ANGABUNGARIVER
appear virtually identical on the air photos have similar, but not identical spectra. Although there are large A-peaks and some very-low-frequency noise, mainly due to the pair of right angle bends north of the delta, the characteristic appearance of the meander pattern is strongly influenced by numerous higher-frequency peaks, notably F, H, and J, which all show variable shifts toward lower frequency between 1938 and 1957. Errors of measurement account for part of this shift but one must also recognize a general lengthening of meander loops. The newly-formed lower 1963 course has developed spectra similar to those of the upper course except for a loss of intensity in the low-frequency peaks and a notable increase in the frequency of each peak. The development of a relatively strong H-peak shows an affinity with the older lower courses. 4. Relation of spectral peaks to other parameters In Fig. 4 wavelengths corresponding to the most stable and persistent 20 0001938
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150 (l:t x I03)
200
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10
J. G. SPEIGHT
spectral peaks are plotted on a logarithmic scale to show their changing relation to channel width with distance downstream. It is clear that violent fluctuations in tree-to-tree width are not reflected in the behaviour of individual peaks. It is more likely that there is a correlation between grass-tograss width and peak wavelength, but the relation does not appear to be strictly linear, and the peak wavelengths constitute rather more stable para-
20 0 0 0 1957
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100 -150
i
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150
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(Ft x 103)
Fig. 4b
meters than channel width. The conclusion of Leopold and Wolman ~) that "typical" meander wavelength and width were related by the formula 2L = 10.9W1.o~ cannot be compared with the present data as it stands, because they measured wavelength along straight lines (2L). However, as their data included the sinuosity of each meander considered, it was possible to compute the relation of talweg wavelength (2T) to width, establishing the regression 2T= 16.3Wt'°t with a slightly reduced variance. For the range of widths considered in the case of the Angabunga this relation may be approximated
MEANDER SPECTRA OF THE ANGABUNGA RIVER
ll
by 2T= 17W. Such a relation is approached most closely in the case of the C-peak which, as previously noted, is a peak likely to be chosen as representing "typical" meanders. Much more variable than the wavelengths of individual peaks is the spectral shape, and in particular the relative intensities of different peaks. There is a definite association of intensified F-, H-, and J-peaks with re-
20 000 1963
10 000
• o
6 000"
~+
4000"
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100 -160
~ - - - - - r
-100
-50
0
50
100
150
200
250
300
Distance downstream (Ft x l 0 3 )
Fig. 4c Fig. 4. Relations between river width and the wavelengths of certain stable and persistent spectral peaks. The wavelength predicted on the basis of discharge is also shown.
latively deep trench-shaped channels, and it seems likely that other details of spectral shape will prove to be dependent on the shape of the channel cross section. Dury 4) has collected data which indicate that the relation 2L=30Q °'5, where Q is the bankfull discharge, holds true for a number of flood-plain streams through a very wide range of discharges. In the present instance, if
12
J. G. SPEIGHT
one allows for a mean sinuosity of 2.0 (i.e.)-T ---- 2.0 ~'L) the expected relation is 2T=60X (14000)°'5=7100feet. This figure is plotted on Fig. 4c, which shows that, once again, there is a general agreement with the wavelength of the C-peak. When spectral methods are employed however it appears preferable to discuss the characteristic wavelength in terms of the dominant B-peak, and the validity of the relation 2T~B)= 90Q °'5 deserves investigation, but there is insufficient information on the variation of Q in the present case to test it. The influence of channel slope on the meander spectrum has not been identified. *Its effects appear to be completely masked by those of channel shape.
5. Variance and sinuosity The area under the spectral curve is proportional to the variance of the directions of flow used in the analysis. It is thus a precise measure of the sinuosity of the channel, if precautions are taken to eliminate very-low-
2000
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3
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Fig. 5. Relation between sinuosity and the variance of flow-direction, with a regression curve fitted by eye.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
13
frequency oscillations likely to be associated with topographic obstacles rather than with the phenomenon of meandering. To conform to the definition of sinuosity in the companion paperg), an arbitrary cut-off point was fixed at a frequency corresponding to a meander-wavelength 100 times the mean unvegetated channel width. The relation between variance and sinuosity is then as shown in Fig. 5. The isolated very low value is influenced by a lack of resolution between a very-low-frequency peak and an A-peak. The use of variance as a sinuosity measure carries the obvious advantage that the sinuosity component due to a given wavelength of meander can be isolated. 6. Meander migration
A cross-spectral analysis between analogous reaches on different dates was carried out to investigate the rate of migration of meanders. Orderly downstream migration of meanders, which would have produced a straight-line relationship between change of phase and frequency, could not be identified, and it was concluded that such migration does not occur in this case. 7. The river as an oscillatory system
The characteristic multi-peaked spectrum of oscillations of meandering suggests that the river, like a musical instrument, may be considered as a resonant oscillatory system, the most notable difference being that the oscillations of interest represent fluctuations relative to distance rather than to time. In each case one may consider three elements: a source of energy, a forcing vibration, and a resonant response, represented in the case of a brasswind instrument by air under pressure, a vibration of the player's lip, and the characteristic vibrations due to the dimensions and material of the instrument respectively. For a river the main source of energy is plainly the potential energy due to altitude above base-level, but the other elements require further discussion. The nature of the forcing vibration may not be the same for all streams. In small steep streams it might be identified with an alternation of subcritical and supercritical flow, but most streams have quite low Froude numbers (0.20 to 0.35 in the case of the Angabunga) and the forcing vibration must be sought under conditions of uniform or gradually varied flow. Friedkin 11) claims to have produced meanders in a model with uniform flow; on the other hand the perfection of meanders seen in tidal creeks argues for the effectiveness of gradually varied flow. Hsieh Wen Shen 12) produced a pattern of alternating scour holes in a straight laboratory flume at subcritical flow by making the banks very much rougher than the bed. He attributed the pattern
14
J. G. SPEIGHT
to secondary currents perpendicular to the main flow induced by the difference in shear stress between the rougher and smoother surfaces. Exposure of a greater area of rough bank in a scour hole generated a secondary current which tended to excavate another scour hole further downstream on the opposite side of the channel. Once a stable oscillation of channel shape has been initiated in a natural stream by this or some other mechanism bank-caving generally occurs, creating alternation of channel curvature, which in turn calls into play a complex of other factors which feed back oscillatory energy into the system. Where the stream flows in bedrock large-scale features of the geologic structure may also have an effect. Probably only the broader features of the meander spectrum are due to these primary mechanisms: the sharply defined stable peaks constitute a response of the system, in which the main determinants appear to be flow magnitude and variability and channel cross-sectional shape, the latter being ultimately controlled by the materials of the channel perimeter. 8. Remarks on the method
Since the character of a river meander pattern changes rapidly in a downstream direction and may be completely altered by the entry of a tributary stream, the length of reach available for spectral analysis is strictly limited, and is usually barely adequate to produce acceptable resolution in the spectrum. Parameters which influence the resolution and reliability of the spectrum are: (i) the spacing of points at which direction of flow is measured Ad, (ii) the number of points analysed n, and (iii) the number of frequencybands used in producing the spectrum, m. The spacing of points used in the analysis determines the shortest wavelength recorded in the spectrum, which will be 2ztd. However, spacing the points at closer intervals than necessary results in wasting frequency-bands on high frequencies exhibiting negligible meander intensity. In the present case Ad was set (in effect) at 600 feet; about twice the mean width of unvegetated channel. The longest wavelength resolved depends directly on the number of frequency-bands employed, and it was found necessary to use 40 bands to obtain adequate resolution of the major low-frequency peaks. For acceptably reliable estimates of spectral intensity it was then necessary to consider reaches of not less than 200 points, that is, reaches with lengths exceeding 120000 feet or approximately 400 channel-widths. Adequate definition of downstream changes in spectral shape was achieved by overlapping the reaches analysed so that their mid-points were spaced at intervals of approximately I00 channel-widths.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
15
Acknowledgements While t a k i n g responsibility for the i n t e r p r e t a t i o n o f spectra in terms o f n o r m a l curves, the a u t h o r wishes to a c k n o w l e d g e t h a t the spectral analysis was carried out by Professor E. J. H a n n a n o f the A u s t r a l i a n N a t i o n a l University, who first suggested how the m e t h o d might be a p p l i e d to the problem. T h a n k s are due to Professor G. H. Dury, Dr. S. A. Schumm, and the a u t h o r ' s colleagues in the C S I R O for reading drafts o f the manuscript. The study was p a r t o f the research p r o g r a m m e o f the Division o f Land Research a n d R e g i o n a l Survey, CS1RO.
References 1) C. C. lnglis, The Behaviour and Control of Rivers and Canals, Central Waterpower Irrigation and Navigation Research Station, Poona, India, 13 (1949) 2) L. B. Leopold and M. G. Wolman, U.S. Geol. Survey Prof. Paper 282-B (1957) 3) Geol. Soc. America Bulletin 71 (1960) 769 4) G. H. Dury, U.S. Geol. Survey Prof. Paper 452 (in press) 5) Geographical Review 50 (1960) 219 6) . . . . . Institute of British Geographers, Trans. and Papers (1963) 83 7) F. Hjulstr6m, Geografiska Annaler 31 (1949) 83 8) S. A. Schumm, Geol. Soc. America Bulletin 74 (1963) 1089 9) J. G. Speight, Journal of Hydrology 3 (1965) 16 10) R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1959) 11) J. F. Friedkin, A Laboratory Study of the Meandering of Alluvial Rivers, U.S. Waterways Experiment Station, Vicksburg, Missouri (1945) 12) Hsieh Wen Shen, A Study on Meandering and Other Bed Patterns in Straight Alluvial Channels, University of California, Berkeley (1961)
MEANDER
S P E C T R A OF T H E A N G A B U N G A
RIVER
J. G. SPEIGHT Division of Land Research and Regional Survey, CSIRO, Canberra, Australia
Abstract: Three sets of air photos spanning twenty-four years are used in an analysis of the dominant modes of meandering of the Angabunga River, in central Papua, using the method of the power spectrum. Spectra are constructed lbr successive reaches of the stream based on the auto-correlation of the direction of flow measured at equi-spaced points on the talweg of the channel, Such spectra display the meander intensity associated with each band of frequency, where frequency is the reciprocal of meander-wavelength, measured along the talweg. It is shown that the subjectively chosen characteristic meander wavelength used by previous workers is a poor indicator of the dominant frequencies of oscillation. All spectra show a number of stable peaks of meander intensity, of which the dominant one corresponds to a wavelength of the order of 90 times the root of the bankfull discharge, that is, about 50'>~,longer than the expected "characteristic" wavelength. The most prominent spectral peaks are more stable in a downstream direction than are other parameters, such as channel width. The relative intensities of the various peaks, however, are subject to rapid change associated with change in channel cross-sectional shape. Parts of the channel with a lower width-depth ratio have relatively high intensity associated with high-frequency peaks, a fact which strongly influences the subjective judgement of meander wavelength. 1. Introduction Since 1949 when the w o r k o f inglis ~) on the relation o f m e a n d e r wavelength of a river to d o m i n a n t discharge was published, there have been a n u m b e r o f papers z-6) on correlations between m e a n d e r wavelength a n d width, discharge, or v a l l e y - m e a n d e r wavelength. The wavelengths q u o t e d in these studies have generally been m e a s u r e d at favourable sites, t a k i n g twice the distance between successive points o f inflection in the channel. This m e t h o d has three s h o r t c o m i n g s : (i) " t y p i c a l " , well defined m e a n d e r s are chosen at the discretion o f the g e o m o r p h o l o g i s t ; (it) wavelengths are measured along a succession o f straight lines, the directions o f which m a y bear little relevance to the direction of flow o f the river (the angle between successive lines is frequently m o r e than 90 ~); (iii) the possibility of m o r e t h a n a single m o d a l wavelength is ignored. The a s s u m p t i o n implicit in the m e a s u r e m e n t of m e a n d e r wavelength at selected sites where the m e a n d e r is a simple S-shape is t h a t there is a single d o m i n a n t wavelength o f m e a n d e r i n g (;re) which is obscured at all other sites
2
J.G. SPEIGHT
by irregularities of a quasi-random nature. Only Hjulstr6m 7) and Schumm s) have suggested that two characteristic wavelengths may be present in the one stream at the same time. This investigation proceeds from a consideration of the total variability of direction of flow of the stream and the possibility of its analysis as a spectrum of oscillations in direction measured against distance downstream. The type of spectrum implied by most previous workers is a single intense peak of frequency v = 1/2L superimposed on a "noise" the intensity of which is widely distributed through the frequency range. 2. The Angabunga River The Angabunga River in the Territory of Papua was chosen for study because of its long unobstructed and undisturbed alluvial plains course and
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Fig. 1. The plains course and lower mountain course of the Angabunga River.
MEANDER SPECTRA OF 'FILE ANGABUNGA RIVER
3
its highly developed, rapidly changing meander pattern (Fig. I). The basic data for the analysis was derived from three sets of vertical air photos at a scale of 1:40 000 taken in 1938, 1957, and 1963. For comparative purposes flow characteristics, topography, channel width, shape, and roughness, and sediment type were also investigated; these aspects are reported fully in the following paper 9). The stream is of moderate size, with a catchment of 940 square miles under a 90 inch rainfall, and a mean flow of nearly 4000 ft:~/sec. The variability of daily flows is low, although the rise and fall during a fresh is very rapid. A monsoonal climate causes a moderate seasonality of high and low flows. Until very recently, the channel length of the plains course, which has no tributaries, was about 57 miles, but a crevasse which occurred in 1954 near the mid-point of the course resulted in a reduction of channel length to 39 miles. Although the new lower course was fully formed and was carrying all tile flow by 1963, in 1957 the main flow was still carried by the old lower course, and it was reasonable to ignore the effect of the crevasse in the 1957 analysis. The channel cross-section of tile plains course is of two types: a migrating wide triangular section dominated by point-bars, characteristic of tile upstream reaches which have a gravel bed and sandy banks, and a fixed narrow rectangular section dominated by levees, a characteristic of the downstream reaches with a sandy bed and silt-clay banks. In its lower mountain course the river is confined between rock walls on a valley floor seldom much wider than the unvegetated channel.
3. Spectral analysis On each set of photos a smoothly curving line was drawn to follow the talweg of the stream and a pair of dividers was used to make marks on this line at scaled intervals of 300 feet, that is, approximately the mean width of unvegetated channel. A precisely oriented arbitrary datum line was drawn on each photo and the direction of flow at each marked point on the talweg was recorded. Distributaiies, being both small and short, were ignored. This procedure was applied not only to the three plains courses, but also to about 25 miles of the lower mountain course and to four of the longest wellpreserved prior stream channels visible on the plain. For the purpose of the analysis the several river courses were broken into overlapping reaches usually 126 000 feet long at intervals of 31 500 feet downstream, so that in each the number of 300-foot points was 420, although there was some variation of length in special cases. When a trial showed that the points were too closely spaced for useful analysis the angles at successive points were summed in pairs, giving an effective point-spacing of 600 feet. The resulting angles in
4
J.G. SPEIGHT
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Mountain Course
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Fig. 2. Some examples of meander spectra for various reaches of the Angabunga River, with insets showing the trace of the talweg in the central part of the reach in each case.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
5
degrees, xa, where d = 1, 2 ..... n; n = 210, were then analysed by digital computer for auto-correlation using the formula:
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; l °-'
i
(1)
(n
f o r / = 0 , 1, 2 ..... m - 1, where m is the number of frequency bands, the value of m being taken as 40 in this case. The intensity X (k) in each frequency band k = 0, 1, 2 ..... m was then calculated by m-
1
m/
m 3_1
/=1
and spectra were constructed as in the examples of Fig. 2. The total area under the curve is proportional to the variance, while the ordinate X(k) is arbitrarily defined as meander intensity, the probable error of which was about + 30%. For a general discussion of power-spectral techniques the text by Blackman and Tukey 1°) should be consulted. The spectral envelopes clearly contain a number of coherent peaks, but in order to quantify these it is necessary to make assumptions about the shape of the peaks and about the character of the non-periodic components, or noise. There does not appear to be a standard procedure for this. The method adopted consists of replacing spectral peaks by curves of normal distribution. It was suggested by the excellent fit obtained in the case of the most isolated peaks, such as the peak labelled " F " on the spectrum illustrated for Prior Channel No. 2, and by the way more complex peaks may be completely represented as the sum of only two or three normal curves. As the figure shows, the total spectral envelope may be represented by quite a small number of such normal curves. It must be emphasised that this procedure is merely a device to obtain precise quantification of the spectral peaks and does not imply that all of the peaks are "real" in the sense of representing stable oscillations rather than noise. In effect the method causes the noise in the spectrum to be absorbed into the spread of values associated with the individual peaks. At the low-frequency end, in particular, some quite intense peaks have been analysed into normal curves which, while representing the simplest construction which will fit, may well be quite different from the peaks (or noise) which produce the spectrum. The 1938 Lower Plains Course illustrates such a doubtful case. The spectral intensity peaks produced by the analysis could now be represented by three parameters: the peak intensity X (p) in arbitrary units, the peak frequency v in cycles per (foot x 1057, and the band-width Av, measured at half the peak intensity. The variations in these parameters are plotted
6
J . G . SPEIGHT
against mean distance downstream for all spectra in Fig. 3, the height of each rectangle being the band-width, and the breadth being proportional to the peak intensity. Data from spectra for prior channels Nos. 1, 3 and 4 are not presented as the remnants of these courses are too short to allow of adequate resolution. Both the complex structure of the spectra and the persistence of individual peaks in successive reaches are remarkable and were completely unexpected on the basis of previous studies. For comparison the talweg wavelengths of typical, well-developed meanders near the mid-point of each reach were measured and their reciprocals plotted on the figure. The unreliability of the measure as an indicator of dominant mode of meandering is plainly shown. For convenience of discussion fifteen relatively persistent frequency peaks have been labelled alphabetically. It is their very persistence which allows them to be traced downstream and to be labelled in this manner. One may doubt the "reality" of some of them, either through low spectral intensity or through lack of resolution, as is common at the low frequency end, but the majority appear to represent stable frequencies of oscillation. In general lowfrequency peaks A, B and C dominate the spectrum, together with variable amounts of very low-frequency noise which is probably aperiodic. Because the C-peak is characteristically the shortest wavelength dominant peak the subjectively assessed "typical" meander-wavelength often corresponds to it, but any appreciable peaks at higher frequencies, even of quite low intensity, may cause a very short meander-wavelength to be judged "typical", as in the lowermost reach of 1938 (Fig. 3b). The B-peak is the most stable and persistent peak of all, dominating nearly all plains-course spectra and occurring in every course. The value of v at the peak lies in the neighbourhood of 10 (i.e. 2r= 10 000 feet), but tends to rise steadily from v = 8 in the upper plains to v = 12 in the lower plains, except that the increase is accelerated in the 1963 lower course. Prior channels Nos. 1,3, and 4 also are dominated by peaks in the vicinity of v = 10, so that if wavelength is strictly related to discharge, there are strong grounds for assuming a general uniformity of flow regime over the whole time represented. If one compares the upper parts of the 1938 and 1963 plains-courses, three of the spectra are found to be almost identical except for an increase in the intensity of the F-peak and other high-frequency peaks. This is surprising in view of the radically different aspect of the corresponding 1957 spectra, in which the C-peaks (v = 15) have been elided and replaced by intense D-peaks (v = 19), and the frequency of the A-peaks has been drastically reduced. This alternation of spectral shape is quite impossible to detect by visual examination of the river pattern and may represent a sensitive response to some short term variation in flow regime. The lower reaches of 1938 and 1957, which
7
MEANDER SPECTRA OF THE ANGABUNGA RIVER
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8
J.G. SPEIGHT ~Mountain
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Fig. 3d Fig. 3. Frequencies, band widths, and intensities of spectral peaks plotted against mean distance downstream.
9
MEANDERSPECTRAOF THE ANGABUNGARIVER
appear virtually identical on the air photos have similar, but not identical spectra. Although there are large A-peaks and some very-low-frequency noise, mainly due to the pair of right angle bends north of the delta, the characteristic appearance of the meander pattern is strongly influenced by numerous higher-frequency peaks, notably F, H, and J, which all show variable shifts toward lower frequency between 1938 and 1957. Errors of measurement account for part of this shift but one must also recognize a general lengthening of meander loops. The newly-formed lower 1963 course has developed spectra similar to those of the upper course except for a loss of intensity in the low-frequency peaks and a notable increase in the frequency of each peak. The development of a relatively strong H-peak shows an affinity with the older lower courses. 4. Relation of spectral peaks to other parameters In Fig. 4 wavelengths corresponding to the most stable and persistent 20 0001938
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J. G. SPEIGHT
spectral peaks are plotted on a logarithmic scale to show their changing relation to channel width with distance downstream. It is clear that violent fluctuations in tree-to-tree width are not reflected in the behaviour of individual peaks. It is more likely that there is a correlation between grass-tograss width and peak wavelength, but the relation does not appear to be strictly linear, and the peak wavelengths constitute rather more stable para-
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meters than channel width. The conclusion of Leopold and Wolman ~) that "typical" meander wavelength and width were related by the formula 2L = 10.9W1.o~ cannot be compared with the present data as it stands, because they measured wavelength along straight lines (2L). However, as their data included the sinuosity of each meander considered, it was possible to compute the relation of talweg wavelength (2T) to width, establishing the regression 2T= 16.3Wt'°t with a slightly reduced variance. For the range of widths considered in the case of the Angabunga this relation may be approximated
MEANDER SPECTRA OF THE ANGABUNGA RIVER
ll
by 2T= 17W. Such a relation is approached most closely in the case of the C-peak which, as previously noted, is a peak likely to be chosen as representing "typical" meanders. Much more variable than the wavelengths of individual peaks is the spectral shape, and in particular the relative intensities of different peaks. There is a definite association of intensified F-, H-, and J-peaks with re-
20 000 1963
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Fig. 4c Fig. 4. Relations between river width and the wavelengths of certain stable and persistent spectral peaks. The wavelength predicted on the basis of discharge is also shown.
latively deep trench-shaped channels, and it seems likely that other details of spectral shape will prove to be dependent on the shape of the channel cross section. Dury 4) has collected data which indicate that the relation 2L=30Q °'5, where Q is the bankfull discharge, holds true for a number of flood-plain streams through a very wide range of discharges. In the present instance, if
12
J. G. SPEIGHT
one allows for a mean sinuosity of 2.0 (i.e.)-T ---- 2.0 ~'L) the expected relation is 2T=60X (14000)°'5=7100feet. This figure is plotted on Fig. 4c, which shows that, once again, there is a general agreement with the wavelength of the C-peak. When spectral methods are employed however it appears preferable to discuss the characteristic wavelength in terms of the dominant B-peak, and the validity of the relation 2T~B)= 90Q °'5 deserves investigation, but there is insufficient information on the variation of Q in the present case to test it. The influence of channel slope on the meander spectrum has not been identified. *Its effects appear to be completely masked by those of channel shape.
5. Variance and sinuosity The area under the spectral curve is proportional to the variance of the directions of flow used in the analysis. It is thus a precise measure of the sinuosity of the channel, if precautions are taken to eliminate very-low-
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Fig. 5. Relation between sinuosity and the variance of flow-direction, with a regression curve fitted by eye.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
13
frequency oscillations likely to be associated with topographic obstacles rather than with the phenomenon of meandering. To conform to the definition of sinuosity in the companion paperg), an arbitrary cut-off point was fixed at a frequency corresponding to a meander-wavelength 100 times the mean unvegetated channel width. The relation between variance and sinuosity is then as shown in Fig. 5. The isolated very low value is influenced by a lack of resolution between a very-low-frequency peak and an A-peak. The use of variance as a sinuosity measure carries the obvious advantage that the sinuosity component due to a given wavelength of meander can be isolated. 6. Meander migration
A cross-spectral analysis between analogous reaches on different dates was carried out to investigate the rate of migration of meanders. Orderly downstream migration of meanders, which would have produced a straight-line relationship between change of phase and frequency, could not be identified, and it was concluded that such migration does not occur in this case. 7. The river as an oscillatory system
The characteristic multi-peaked spectrum of oscillations of meandering suggests that the river, like a musical instrument, may be considered as a resonant oscillatory system, the most notable difference being that the oscillations of interest represent fluctuations relative to distance rather than to time. In each case one may consider three elements: a source of energy, a forcing vibration, and a resonant response, represented in the case of a brasswind instrument by air under pressure, a vibration of the player's lip, and the characteristic vibrations due to the dimensions and material of the instrument respectively. For a river the main source of energy is plainly the potential energy due to altitude above base-level, but the other elements require further discussion. The nature of the forcing vibration may not be the same for all streams. In small steep streams it might be identified with an alternation of subcritical and supercritical flow, but most streams have quite low Froude numbers (0.20 to 0.35 in the case of the Angabunga) and the forcing vibration must be sought under conditions of uniform or gradually varied flow. Friedkin 11) claims to have produced meanders in a model with uniform flow; on the other hand the perfection of meanders seen in tidal creeks argues for the effectiveness of gradually varied flow. Hsieh Wen Shen 12) produced a pattern of alternating scour holes in a straight laboratory flume at subcritical flow by making the banks very much rougher than the bed. He attributed the pattern
14
J. G. SPEIGHT
to secondary currents perpendicular to the main flow induced by the difference in shear stress between the rougher and smoother surfaces. Exposure of a greater area of rough bank in a scour hole generated a secondary current which tended to excavate another scour hole further downstream on the opposite side of the channel. Once a stable oscillation of channel shape has been initiated in a natural stream by this or some other mechanism bank-caving generally occurs, creating alternation of channel curvature, which in turn calls into play a complex of other factors which feed back oscillatory energy into the system. Where the stream flows in bedrock large-scale features of the geologic structure may also have an effect. Probably only the broader features of the meander spectrum are due to these primary mechanisms: the sharply defined stable peaks constitute a response of the system, in which the main determinants appear to be flow magnitude and variability and channel cross-sectional shape, the latter being ultimately controlled by the materials of the channel perimeter. 8. Remarks on the method
Since the character of a river meander pattern changes rapidly in a downstream direction and may be completely altered by the entry of a tributary stream, the length of reach available for spectral analysis is strictly limited, and is usually barely adequate to produce acceptable resolution in the spectrum. Parameters which influence the resolution and reliability of the spectrum are: (i) the spacing of points at which direction of flow is measured Ad, (ii) the number of points analysed n, and (iii) the number of frequencybands used in producing the spectrum, m. The spacing of points used in the analysis determines the shortest wavelength recorded in the spectrum, which will be 2ztd. However, spacing the points at closer intervals than necessary results in wasting frequency-bands on high frequencies exhibiting negligible meander intensity. In the present case Ad was set (in effect) at 600 feet; about twice the mean width of unvegetated channel. The longest wavelength resolved depends directly on the number of frequency-bands employed, and it was found necessary to use 40 bands to obtain adequate resolution of the major low-frequency peaks. For acceptably reliable estimates of spectral intensity it was then necessary to consider reaches of not less than 200 points, that is, reaches with lengths exceeding 120000 feet or approximately 400 channel-widths. Adequate definition of downstream changes in spectral shape was achieved by overlapping the reaches analysed so that their mid-points were spaced at intervals of approximately I00 channel-widths.
MEANDER SPECTRA OF THE ANGABUNGA RIVER
15
Acknowledgements While t a k i n g responsibility for the i n t e r p r e t a t i o n o f spectra in terms o f n o r m a l curves, the a u t h o r wishes to a c k n o w l e d g e t h a t the spectral analysis was carried out by Professor E. J. H a n n a n o f the A u s t r a l i a n N a t i o n a l University, who first suggested how the m e t h o d might be a p p l i e d to the problem. T h a n k s are due to Professor G. H. Dury, Dr. S. A. Schumm, and the a u t h o r ' s colleagues in the C S I R O for reading drafts o f the manuscript. The study was p a r t o f the research p r o g r a m m e o f the Division o f Land Research a n d R e g i o n a l Survey, CS1RO.
References 1) C. C. lnglis, The Behaviour and Control of Rivers and Canals, Central Waterpower Irrigation and Navigation Research Station, Poona, India, 13 (1949) 2) L. B. Leopold and M. G. Wolman, U.S. Geol. Survey Prof. Paper 282-B (1957) 3) Geol. Soc. America Bulletin 71 (1960) 769 4) G. H. Dury, U.S. Geol. Survey Prof. Paper 452 (in press) 5) Geographical Review 50 (1960) 219 6) . . . . . Institute of British Geographers, Trans. and Papers (1963) 83 7) F. Hjulstr6m, Geografiska Annaler 31 (1949) 83 8) S. A. Schumm, Geol. Soc. America Bulletin 74 (1963) 1089 9) J. G. Speight, Journal of Hydrology 3 (1965) 16 10) R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1959) 11) J. F. Friedkin, A Laboratory Study of the Meandering of Alluvial Rivers, U.S. Waterways Experiment Station, Vicksburg, Missouri (1945) 12) Hsieh Wen Shen, A Study on Meandering and Other Bed Patterns in Straight Alluvial Channels, University of California, Berkeley (1961)