Integrating morphological and process variation: unit stream power power

Variations in both morphological parameters and the prevailing process regime determine, and are determined by, variations in unit stream power and unit stream power expenditure. If variations in channel length are considered in terms of channel gradient, then length, width, and channel discharge determine unit stream power ( ), a measure of the total energy of flow per unit bed area, and hence, the total energy available for hydraulic and geomorphological processes.

= ( Q s ) / w (4.1)

where: = bulk unit weight of water (constant) Q = discharge

s = channel slope (inversely proportional to channel length) w = channel width

The relationship between channel length and channel width in the stream power equation is considered in Figure 4.21, assuming a constant valley slope and no long term trend in discharge. Unit stream power can remain constant if channel length and channel width vary by exactly inverse proportions. For example, if the length of a meandering channel through a reach halves following dramatic cutoff development, this corresponds to an effective doubling of channel slope. Hence, for unit stream power to remain at unity, channel width must also double. This is shown by the line of constant unit stream power in Figure 4.21. Unity can also be achieved when an increase in channel length is effectively balanced by a proportional decrease in channel width. This concept can be related to the variation of channel length and width of the Lower Mississippi River shown in Figure 4.22.

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Shorter and wider channel.

Figure 4.21 The implications of channel length and width variability for unit stream power.

Figure 4.22 Plots of average length against width in the period 1830-1975 for: a) the Lower Mississippi River from Cairo to Old River (reaches 1-5) and; b) the individual planform reaches identified in Figure 4.11 (with reaches 1 and 2 amalgamated).

Figure 4.22a displays the reach-averaged changes in channel length against corresponding changes in channel width. If no long term trend in discharge is assumed once again, there has been a general decrease in unit stream power in the 1830-1975 period because, whilst width has more than doubled, total channel length has shortened by only ten percent. However, this observation assumes reliability of the top-bank width for 1830, reported by Winkley (1977). If this is removed from the analysis and variation in channel length and width are considered from 1887, net lengthening in combination with slight net widening is suggestive of a decline in unit stream power in the period 1887-1930. In the latter 1930-1975 period however (incorporating the period of artificial cutoff construction), a decline of channel length by approximately ten percent, in combination with an increase in channel width of approximately ten percent, is suggestive of very little or no change in unit stream power. This latter finding suggests that the assertion made by Biedenharn et al.

(2000), that the contemporary, post-cutoff, Lower Mississippi River has a much larger stream power than prior to the cutoffs, may be misleading because it is based solely on stream power per unit length of channel, not unit stream power and thus, does not consider simultaneous variations in channel width.

In Figure 4.22b, the regional-scale changes in channel length and channel width observed in Figure 4.22a are decomposed to the reach-scale. Observed regional-scale changes mask the considerable inter-reach variation highlighted in sections 4.4 and 4.5. Most evidently, the upstream reaches 1 and 2 demonstrate considerably different behaviour in the period 1887-1975 to the artificial cutoff reaches (3-5).

This reveals that it may be misleading to consider changes in unit stream power at only the regional-scale.

If the variation in the process regime noted in section 4.6 is factored into the above discussion, the geomorphological significance of the artificial cutoff programme is further reinforced. Immediately prior to the artificial cutoff programme in 1930, the Lower Mississippi River had lengthened and widened in all reaches between Cairo and Old River over the previous 43 years. Although discharge may have slightly increased over the same period, unit stream power is likely to have been considerably lower in 1930 than in 1887. Published records indicate that measured suspended sediment loads were high in 1930 and hence, there was most likely, an imbalance

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between available stream power per unit width and available sediment load. In the 1930-1975 period, the artificial cutoff programme dramatically reduced channel length although simultaneous increases in channel width indicate that stream power per unit width may have remained approximately constant. However, bank stabilisation, together with improved land use practices, reduced rates of sediment supply and consequently, measured suspended sediment load. Biedenharn et al.

(2000) propose that excess stream power alongside declining suspended sediment loads in the post-cutoff period must be balanced by an increase in bed material transport. However, stream power per unit width cannot be simply related to rates of sediment transport because energy is first lost in overcoming flow resistance. Spatial and temporal variations of flow resistance therefore require investigation before variations in unit stream power can be related to sediment transport processes.

Resistance to flow is comprised of several different components (Leopold et al., 1964; see section 1.5). At the regional-scale, bend size and shape is a major source frictional resistance (Richards, 1982). This induces internal energy losses within eddies and vortices within the flow field and therefore exerts a strong control on total flow resistance. Flow resistance increases as bend tightness increases. However, the relationship between channel curvature and flow resistance is more complicated because tighter meander bends are associated with a longer channel length (lower channel slope) and therefore, the rate of energy loss per unit length may be reduced, particularly as energy is dissipated more efficiently in a meandering channel (Davies and Sutherland, 1980; Yang, 1971). In the following two sections, two complementary techniques are together used to inform variations in flow resistance:

spatio-temporal trends in radius of curvature is first examined to highlight changes in bend tightness; and then bend wavelength and amplitude are examined to highlight changes in bend size and shape.