ALLUVIAL CHANNEL-FORM ADJUSTMENT AND THE VARIATIONAL PRINCIPLE OF LEAST ACTION

Huang and coworkers’

model*

Julien and Wargadalam’s

(1995) model**

Theoretical results

(Huang and Nanson, 2001)

Abstract: A mathematical analytical approach proposed by Huang and Nanson (2000) is examined against a wide range of flow conditions in order to provide a convincing explanation for the mechanisms governing alluvial channel-form adjustment. Here we show that the extremal hypotheses of MSTC (maximum sediment transporting capacity) and MSP (minimum stream power) are inherent in laws governing alluvial channel flow and that MSTC and MSP are the complementary expressions of a unifying principle – that of MFE (maximum flow efficiency). Importantly, conditions that limit the application of MFE are illustrated and it is shown that in most cases maximally efficient straight channels mathematically derived from basic flow relationships are highly consistent with ‘regime theory’ observations from stable canals and observed bankfull hydraulic geometry relations from natural channels. Finally, it is argued that MFE should be regarded as a general principle for understanding alluvial channel-form adjustment because it is the product of the widely applied variational principle of least action and the principle of energy conservation for sediment transport.

1 INTRODUCTION

Since the time of Lacey (1929), it has been known that alluvial channels tend to self adjust such that the imposed water and sediment loads can be transported through an extended reach of channel without progressive erosion or deposition. To explain this, numerous studies have focused on those conditions that determine the equilibrium condition, including erosion/deposition process-based analytical approaches (e.g. Parker, 1978; Vigilar and Diplas, 1997) and extremal hypotheses (e.g. Kirkby, 1977; Chang, 1979, 1980; Yang et al. 1981; White et al., 1982). However, such process-based analytical approaches have encountered considerable difficulty in solving proposed boundary shear distribution equations. In some cases advanced analyses involving singular perturbation and numerical techniques have been deployed to obtain approximate solutions. Alternatively, extremal hypotheses have been proposed without giving convincing theoretical explanations for their use and there have been conflicting opinions as to how appropriate such hypotheses are (Knighton, 1998).

Recently, a mathematical analysis approach for understanding the behavior of river channel-form adjustment has been developed, showing that a direct analysis of basic flow relationships can lead to further understanding of the mechanism governing alluvial channel-form adjustment (Huang and Nanson, 2000). Most importantly, this analytical approach reveals that the hypotheses of MSTC (maximum sediment transporting capacity) and MSP (minimum stream power) are complementary expressions of a unifying principle - that of MFE (maximum flow efficiency). The optimal channel geometries determined by using either MSTC or MSP or MFE are highly consistent with ‘regime theory’ for stable canals and with bankfull hydraulic geometry relations for natural streams. However, this analytical approach has been applied only to specific flow conditions using the Lacey flow resistance and the DuBoys sediment transport conditions). There are numerous flow resistance and sediment transport formulas available and it needs to be known whether MSP, MSTC and thus MFE are inherent in all combinations of those relationships. Furthermore, even if MSP, MSTC and MFE are demonstrated to occur, clarification is needed to reveal under what specific conditions theoretically derived optimal channel geometries are compatible with field observations. Finally, it would also be desirable to understand the physics behind the findings. For these purposes, this paper details the findings of our recently undertaken studies and illustrates the conditions that restrict the application of MFE. A physical explanation for the existence of MFE in alluvial channel flow is provided by showing how the widely-applied variational principle of least action and the derivative energy conservation principle work in sediment transport formulas that are based on various tractive theories.

2 MFE AS A UNIFYING CRITERION FOR ALLUVIAL CHANNEL-FORM ADJUSTMENT

Available flow resistance and sediment (bedload) transport relations normally have complicated expressions and are not suitable for providing a mathematical solution to stable channel forms. However, as proposed by Huang and Nanson (2000, 2001), incorporating a non-dimensional channel shape factor

, width/depth ratio, into the relationships of resistance and transport enables the following expression for sediment transport to be obtained:

where

is flow rate,

is sediment transport rate,

is a representative size of sediments, and

is energy gradient, which is equivalent to channel slope in steady, uniform channel flow.

For given

, and

, therefore, the condition of MSTC can be found from (1) when

is allowed to vary in a sufficiently wide range:

Furthermore, a detailed analysis of (1) reveals that for given

and

, the condition of minimum channel slope occurs at exactly the same points of

for a maximum

Moreover, for given

and

, conditions that rarely occur in natural fluvial systems but that are possibly useful for canal design, the condition of minimum flow discharge can be identified to occur in (1) at exactly the same points of

for a minimum

or a maximum

Letting

be total stream power, or

, (3) and (4) are then seen as the complementary expressions of a more generalized optimum flow condition – that of MSP:

Although most sediment transport formulas have been developed not directly based on the principle of energy conservation for sediment transport as interpreted by Bagnold (1966), they are transformable into the following (Huang and Nanson, 2001; Huang et al. 2001):

where

is a scaling factor and

;

is an efficiency operator for energy expenditure on sediment (bedload) transport and for various tractive theories, is determined by:

where

is average shear stress, or

(

is hydraulic radius), and

is critical shear stress for the incipient motion of sediments.

As a result, the following optimum flow condition, defined as MFE also occurs at exactly the same points of

for a minimum

or a maximum

, or a minimum

These results demonstrate the applicability of two concepts for understanding the behavior of river channel-form adjustment. The first is that the transport of a given amount of sediment load with the least energy loss (stream power) (the hypothesis of MSP), is equivalent to the transport of a maximum sediment load with the energy available (the hypothesis of MSTC). The second concept is that the condition of MFE is the general form of MSP and MSTC. In other words, the adjustment of alluvial channel form is governed by a single principle – that of MFE. Based on the above analyses, it is understandable why the use of MSTC and MSP has been found to produce consistent stable channel geometries (White et al. 1982).

For a very wide range of flow resistance and sediment transport relations, this mathematical analytical approach demonstrates that the optimal channel shape factor

for the achievement of MSTC or MSP or MFE varies in the range of (Huang and Nanson, 2000, 2001):

in which

and

are the exponents in the following generalized sediment transport formula:

where in (10) and (11),

is bedload discharge per unit channel width,

is a coefficient relating to sediment characteristics,

is channel cross-sectional area, and

is a coefficient.

The values of exponents

and

in (11) are determined by channel bed forms or flow regimes. For fixed-bed or flat-bed flow regime, the Manning-Strickler formula gives

and

values of

and

, respectively. According to the study of Brownlie (1983),

and

have respective values of 0.5293 and 0.3888 for the lower flow regime and 0.6005 and 0.4605 for the upper flow regime.

In line with field observations, the possible range of

can be assumed to vary from 2 to 30 at least. As a result, it is identified from (9) that the reasonable variation of

for all of the flow regimes concerned needs to be in the range of:

It is clear in (12) that the U.S. waterways Experimental Station (1935) formula largely falls into the range because it gives

and

. The Meyer-Peter-Muller (1948) formula assigns

and

, and the sum of

in the formula has difference with the lower value of

illustrated in (12) only in decimal figures. The Parker (1979) formula is also very close to the low value of

defined in (12) for it gives an ideal value of 1.5 (

and

). However, the DuBoys (1879) formula falls slightly outside the range for it provides a value of 2.0 (

). This may explain why among numerous sediment transport formulas, few of them assign a value of less than 1.5 to the term of

. This may also explain why the DuBoys formula is now regarded as a classic but out of date relationship whereas the Meyer-Peter-Muller and Parker formulas have gained substantially wider application.

Table 1 Comparison of averaged optimum channel geometry relations with 'regime theory'

Theoretical results

(Huang and Nanson, 2001)

‘Regime theory’

(e.g. Lacey, 1929)

Due to the complicated expressions in flow resistance and sediment transport formulas, optimal channel geometry relations determined by using either (2) or (3) or (4) are very complicated in form. Following the procedures proposed by Huang and Nanson (2000) for obtaining averaged power functions of optimal channel geometry, these derived relations can be reasonably averaged and compared with ‘regime theory’ and the hydraulic geometry relations from a wide range of natural streams (Tables 1 and 2). When sediment concentration

is maintained unchanged, Table 1 shows that optimal channel geometry relations are very close to ‘regime theory’ relations. For natural streams,

varies dramatically but observed hydraulic geometry relations have only minor differences with theoretically derived optimal channel geometry relations when, due to a shortage of field observations on

, channel slope

is used as an alternative to

Table 2 Comparison of averaged optimum channel geometry relations with observed bankfull hydraulic geometry relations

It needs to be pointed out, however, that when hydraulic radius

is replaced with channel depth

, all of the combinations of available flow resistance and sediment (bedload) transport formulas give

a fixed value, leading optimal channel geometry relations to be non-compatible with field observations. This is consistent with the findings of Griffiths (1984) who, based on the equivalence assumption, identified that none of the extremal hypotheses, including MSTC and MSP, is able to produce results compatible with field observations. However, the equivalence of hydraulic radius and channel depth represents only an approximation and can not be used to derive generally applicable results.

3 THE VARIATIONAL PRINCIPLE OF LEAST ACTION AS A FUNDAMENTAL LAW GOVERNING ALLUVIAL CHANNEL-FORM ADJUSTMENT

The least action principle (LAP) was originally formulated by mathematical physicists (Pierre – Louis Maupertuis, Leonhard Euler, Joseph Louis Lagrange, William Hamilton and Karl Jacobi) during the eighteenth and nineteenth centuries. It contains a curious and subtle twist on Newton’s laws for its variational formulation of motion does not use force but the physical quantities (work and energy) whose definition does not depend on any coordinate systems. As a result, it often shows structural analogies between various areas of physics and has been found useful in unifying subjects and consolidating theories in various branches of science (Lanczos, 1966). While LAP had been widely applied in classic mechanics during the 18^th and 19^th centuries, during the 20^th century even more widespread application has been seen. In the 1940s, Richard Feynman identified the applicability of LAP in quantum physics and since then physicists have found that LAP also underlies the fundamental gauge theories of particle physics, leading to the establishment of what is termed fundamental physics. LAP has also been applied outside of physics. A notable example is the study of George Zipf of Harvard University, who, within the context of LAP, tried to derive the power-law form of his law for understanding the behavior of humans on the basis of a principle of least effort (Zipf, 1949). Recently, Huang et al. (2001) examined the applicability of LAP for explaining why fluvial systems exhibit regular bankfull hydraulic geometry relations in very different geographical regions, and showed that for steady, uniform straight channel flow, LAP can be specifically written as the principle of minimum potential energy against the variation of channel shape factor

As alluvial-channel flow is maintained by an elevation difference

, the potential energy

has a form of:

If fluid mass

or flow discharge

is given within the fixed time scale of

, the variation of

(width-depth ratio) occurring on a fixed length of channel

leads (13) to be easily determined by solving:

According to the principle of energy conservation for sediment transport, the rate of sediment transport in alluvial channels reflects the rate of work done by the available stream power

. In other words, sediment discharge

measures the work done by the stream power available for sediment transport (e.g. Bagnold 1966). This notion appears generally applicable because the representative sediment transport formula in (10) can be written into the following form:

where

and

is a complicated function of

, and

. For different flow regimes as stated earlier, it is found in (16) that

has a value of between 0.8768 and 0.8772.

It is apparent that in a very simple way, the combination of (15) and (16) explains why a maximum

, a minimum

or a minimum

, and a maximum

can occur at exactly the same channel width-depth ratio

. Therefore, MFE, MSTC and MSP are the products of the widely applied variational principle of least action and the principle of energy conservation for sediment transport, and should therefore be regarded as general principles governing the adjustment of alluvial channel form.

4 DISCUSSION AND CONCLUSIONS

Our studies demonstrate that MFE (maximum flow efficiency), MSTC (maximum sediment transporting capacity) and MSP (minimum stream power) are the criteria that control the state of stable equilibrium in alluvial channel systems. The equilibrium channel sections that satisfy the physical relationships of flow continuity, resistance and sediment transport can actually be numerous, and only when the above criteria are satisfied does the equilibrium channel section provide a unique stable solution.

It is recognized that MFE, MSTC and MSP are specific forms of the widely applied variational principle of least action, and that they maintain stability of form in alluvial channel systems. This recognition of the applicability of the principle of least action is an advance over the thermodynamic analogies and the empirical formulas previously used to explain alluvial channel-form adjustment. We show clearly that among the many extremal hypotheses proposed, only MSTC and MSP are appropriate.

Although MFE, MSTC and MSP are indeed inherent in natural laws governing alluvial channel flow, all the alluvial channels are assumed to be straight and fully adjustable. If the conditions external to the assumed flow system are imposed, such as valley morphology and slope, the condition of MFE can not be fully satisfied and alluvial channels may develop into meandering, braiding or anabranching planforms. Further detailed study of this issue is presently underway.

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