Thresholds and the spatial variability of flood power during extreme floods

Geomorphology, 5 (1992) 373-390 Elsevier Science Publishers B.V., Amsterdam

373

Thresholds and the spatial variability of flood power during extreme floods Francis J. Magilligan Department of Geography, Dartmouth College, 6017 FairchiM Building, Hanover, NH 03755, USA (Received March 10, 1992; revised April 3, 1992; accepted April 6, 1992 )

ABSTRACT Magilligan, F.J., 1992. Thresholds and the spatial variability of flood power during extreme floods. In: J.D. Phillips and W.H. Renwick (Editors), Geomorphic Systems. Geomorphology, 5: 373-390. Analysis of the effects of extreme events on overall landscape form has shifted geomorphic attention away from a focus on the magnitude and frequency relationship of discharge to a focus on flood power. This paper analyzes the variation in flood power both at-a-station and downstream in a basin of differing lithologies, and discusses the geologic and geomorphic controls on flood power. Design floods ranging from the 2-year discharge to the 500-year discharge are routed through a drainage basin, and channel boundary shear stress and unit stream power are calculated using HEC-2 along 21 cross valley transects. Because of the strong lithologic control on valley width and channel slope, an irregular downstream pattern of flood power emerges which reflects primarily the pattern in valley width. Narrow valleys correspond to a greater percent of channel conveyance during large magnitude floods and increasing rates of depth increases - thus maximizing flood power in these zones. The opposite occurs in anomalously wide valleys where flood power is minimized. A downstream spatial pattern emerges reflecting both local and general controls. For a given probability event, flood power varies at least threefold within the basin with a maximum occurring in a specific downstream section. Holding flood power constant produces an extremely wide range in flow probabilities. In order to establish a link between critical flood power magnitude and flow frequency, minimum flood powers are developed from an analysis of previous work on "catastrophic" flooding for humid, alluvial channels. These minimum thresholds tentatively correspond to 100 N / m 2 or 300 W / m 2 for shear stress and unit stream power, respectively. Discharges required to attain these critical flood powers vary from events roughly twice the 100-year flood to a maximum of roughly 18 times the discharge of the 100-year flood.

Introduction

Although no single subject matter connects all the sciences, many fields share common themes - each discipline with its respective methodology, perspective, or data source. One topic (and its associated extensions) linking geomorphology to the other physical and earth sciences is the analysis of equilibrium, with its focus on the relationship between input forces and overall system response. Because of its Correspondence to: F.J. Magilligan, Department of Geography, Dartmouth College, 6017 Fairchild Building, Hanover, N H 03755, USA.

combined attention to m o d e m processes and landscape evolution, theoretical geomorphology has been instrumental in the analysis and description of natural systems equilibrium (Chorley, 1962 ). Although the persistent and dominant question in geomorphology is the search for "what makes the landscape look the way it does", geomorphological foci have evolved over the past 20-30 years to one of its current themes of evaluating the role of catastrophic events in molding the landscape. This paper examines the hydrologic, geologic, and geomorphic controls on large magnitude (catastrophic?) floods in humid alluvial channels. The first part represents a combined hydrol-

0169-555X/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

374

ogic model (HEC-2) and field approach to determine the m a x i m u m shear stress and unit stream power for various probability floods in a Wisconsin watershed, and presents the spatial pattern of these flood power indices. The second part is based on a literature search of extreme floods to determine if some m i n i m u m threshold of instability typifies large magnitude events, and then generates their associated flow probabilities. One of the goals is to embed the analysis of critical thresholds of flood power within the broader theoretical framework of magnitudefrequency relationships. In particular this paper addresses the local (geologic and geomorphic) and general (e.g., downstream changes in the longitudinal profile) effects on the at-a-station and downstream variation in flood power. Floodplain width influences these local and general controls, and will directly and indirectly control flood power within the drainage network through its control on the rates of change of several hydraulic variables during discharge increases.

General background The attention to formative events in landscape formation can be traced to Wolman and Miller's (1960) claim that landscapes must be evaluated by both the magnitude and the frequency of resulting forces. Although considerable evidence accumulated to support the principle that large events may have minor impacts on the environment (Wolman and Eiler, 1958; Dury, 1973), several discrepancies needed to be reconciled. Large floods sometimes produce devastating impacts, and other times minor changes occur. Therefore, when does a large flood become catastrophic? The pursuit of a correct definition of the term "catastrophic" has become an almost Sisyphean task, yet it has initiated new ways to evaluate landscape and landform development. What became necessary in the analysis of formative events was a shift toward the magnitude of the

F.J. MAGILL1GAN

event; however, the focus on magnitude was not strictly absolute (i.e., peak flood discharge), but was relative to thresholds of the fluvial system (Schumm, 1973 ). The presence of these intrinsic or extrinsic thresholds explained why some extreme hydro-climatological events had minor effects on channel and valley morphology and why lesser precipitation events correspond with major fluvial adjustments. Current thought on landscape development and catastrophic events incorporates the salient features of these previous approaches, but also includes a more hydraulic foundation. Magnitude-frequency relationships, thresholds, and geomorphic effectiveness are incorporated into a hydraulic expression of critical flood power (Baker and Costa, 1987). Simply stated, destabilization of fluvial systems occurs when input forces (expressed as flood power) exceed resistance thresholds. The effectiveness of geomorphic events in molding the landscape is thus strongly tied to channel boundary shear stress and stream power, which are controlled by flood magnitude, channel morphology, and stream gradient (Baker and Costa, 1987). Resistance forces are largely controlled by lithology, sediment and soil type, and vegetation. The magnitude of the destabilizing event is important, but rather than focusing on discharge, the accurate expression of the hydrogeomorphic input is flood power (Baker and Costa, 1987). The focus on energy conditions and flood power represents one of the major advancements of the Baker-Costa principle as it provides a more accurate and encompassing expression of flow conditions and sediment transport capabilities. For this paper I am using the term flood power in a general sense to express the entrainment capabilities of large floods. Although geomorphologists differ on the appropriate expression of flood power, the two dominant expressions are shear stress and unit stream power which are both some function of discharge and interrelated. Shear stress

375

VARIABILITYOF FLOOD POWER DURING EXTREME FLOODS

(z) is the tangential boundary shear acting on the channel bed. Total stream power expresses power per unit length (Bagnold, 1966) yet is often expressed relative to stream width and designated as unit stream power (Baker and Costa, 1987 ) which is the power per unit wetted area of a specific reach (Rhoads, 1987). These variables are described as follows:

r=rRS

(1)

co=TQS/w

(2)

co = ?(Q/w)S= ~,D VS

(3)

where z is shear stress (N/m2), 7 is specific weight of water ( N / m 3), D is mean flow depth (m), R is hydraulic radius (m), Q is discharge ( m a / s ) , 09 is unit stream power (Watts/m2), S is slope, w is width (m), and V is mean velocity. In wide channels, where width-to-depth ratios generally exceed 20, mean flow depth (D) is commonly substituted for hydraulic radius (Chow, 1959). Thus: z = ~,DS

(4 )

co=TV

(5)

Initially, geomorphic focus concentrated on the interrelationship of magnitude and frequency (Wolman and Miller, 1960), but now the magnitude of the critical flood power becomes the focal variable. Flood power is directly linked to discharge, therefore the discharge necessary to engender the critical flood power can be calculated. If "catastrophic" floods can be described as those exceeding thresholds of instability, is it possible to determine a priori the particular zones along a drainage system where these thresholds may be attained? Once the critical discharge can be calculated, the probability of achieving that critical discharge can ultimately be determined. The link between flow probabilities and both critical flood power and discharge represents a second main theme of this paper.

Spatial variability of flood power According to the tenets of hydraulic geometry, the downstream rates of change of hydraulic variables are: D=cQ c

(6)

s=ta z

(7)

The exponent for depth changes in the downstream is commonly 0.4 (Leopold and Maddock, 1953; Williams, 1979) and the exponent for slope for humid alluvial channels is c o m m o n l y - 1 . 0 7 (Wolman, 1955). If shear stress is the product of depth and slope, then its downstream rate of change is the sum of these exponents. Thus, shear stress, theoretically, should decrease downstream to the - 0.67 power of discharge. Spatially, particular zones exist within watersheds where shear stress is maximized. Steep slopes occur in the headwaters, but this zone frequently does not possess sufficient drainage area to produce sufficiently high flood depths. Downstream, just the opposite occurs. In humid alluvial channels, with drainage areas on the order of 103 to 105 km a, downstream slopes are frequently 10- 3 to 10- 5. Therefore, these are zones where shear stress is minimal as slopes are too gentle to generate the requisite critical shear stress. Therefore, as Graf (1983a,b) and Baker and Costa (1987) point out, shear stress is maximized in moderatesized basins of approximately 101 to 10 2 km 2 possessing steep enough slopes and generating adequate flood depths for a given flood. The following section examines the maximization of flood power throughout a watershed in southwest Wisconsin of variable geology in determining the downstream variation of shear stress and unit stream power. Watershed characteristics

The Galena River is located in the Driftless Area of southwestern Wisconsin and north-

376

F,J. MAGILLIGAN

5 km

ville-Galena cuesta, and the basin consists entirely of a lower Paleozoic sequence of sedimentary rocks (Mullens, 1963 ). Because of the variation between the gradual structural dip and the stream's concave longitudinal profile, valley side lithology varies. The resistant Galena dolomite underlies the headwater portions of the basin. In the middle section of the basin, the stream has eroded through the Galena dolomite and exposed the less resistant Platteville limestone, Decorah shale, and St. Peter sandstone. In the downstream sections of the basin, where the structural dip exceeds the gentle longitudinal profile, the resistant Galena dolomite re-emerges as the dominant valley side lithology. Thus, unusually wide valleys with gentle gradients dominate in the middle sections of the basin, but unusually narrow and steep valleys occur in the lower portions of the watershed.

Methodology 7 45~i

)

'LL

GALENA RIVER

Fig. 1. The Galena River Basin.

western Illinois and flows into the Mississippi River near the town of Galena, Illinois (Fig. 1 ). Local relief is approximately 50 to 120 m with moderate-to-steep valley side slopes, which are especially steep near its confluence with the Mississippi River where deep dissection has occurred in response to major base level changes of the Mississippi River. Valleys in southwest Wisconsin are characterized by deep alluvial fills of approximately 15 m in downstream sections and less than 3 m in headwaters. The Galena River flows down the southwesterly-dipping backslope of the Platte-

The primary methodology consists of using a hydraulic flow model, HEC-2, for generating shear stress and unit stream power. HEC-2, a gradually-varied flow model developed by the U.S. Army Corps of Engineers, iterates to a best fit solution for hydraulic variables based upon input variables of slope, discharge, roughness, and valley and channel morphology. HEC-2 is a widely used flow model for generating water surface profiles (O'Connor et al., 1986; Baker and Pickup, 1987; Partridge and Baker, 1987 ) and its mechanics and documentation are well discussed by Feldman (1981). Floods for a given estimated recurrence probability were routed through the Galena River watershed to calculate flood power. Twenty-nine channel and valley cross-sections (herein referred to as Range Lines) were surveyed relatively evenly spaced throughout this 526 km 2 watershed, although only 21 contained adequate information to be subsequently used for HEC-2 analyses. Discharges for these design floods were then routed through these valley and channel

VARIABILITY

OF FLOOD

POWER

DURING

EXTREME

37 7

FLOODS

relationship is subsequently developed between the 2-year flood and drainage area. Thus, the 2-year flood can be estimated for any stream channel cross-section within the region. In order to determine the discharge of higher magnitude events, the 2-year flood is multiplied by its respective flood ratio as established previously. The combination of gauges used for this analysis passed the homogeneity test (Dalrymple, 1960) indicating that they typify the region and adequately represent the calculated recurrence probabilities. One of the gauges used for the regional flood frequency came from the Galena River basin, and the calculated and estimated discharges for the respective flow probabilities closely correspond (Table 2). The discharges for these design floods had been initially determined using a log-Pearson Type III calculation and were based on regional skew coefficients utilized by Conger ( 1981 ). HEC-2 calculates various hydraulic vari-

cross-sections. The discharges for these design floods (2-year, 10-year, 20-year, 50-year, 100year, 200-year, 500-year) were calculated by first determining a regional flood frequency relationship, and then establishing the relationship between drainage area and the discharge of the particular design flood. Southwest Wisconsin possesses a fairly dense stream gauging network, with some records extending as far back as the 1930s. Fourteen gauges (Table 1 ) were selected that typified the similar physiographic and hydrologic characteristics of the Galena River basin. The regional flood frequency analysis followed the procedure as outlined by Dalrymple (1960). The procedure consists of determining the discharges for these design floods for each gauge and expressing them as a ratio to the two-year discharge. The ratios are ranked, and the median ratio to the 2-year flood is selected for the range of gauges. The 2-year floods are plotted against their respective drainage areas, and a TABLE 1

Gauges used from southwestern Wisconsin for the regional flood frequency Gaging Area Station So. Km. 1. Bear Branch 7.05 5414200 2. Bishops Branch 18.83 5408800 3. Crooked Creek 33.41 5407200 4. E. Fk. Galena R. 45.40 5415500 5. Galena River 323.75 5415000 6. Grant River 696.71 5413500 7. Kickapoo R. 688.94 5408000 8. Knapp Creek 21.85 5408500 9. Morris Creek 11.88 54074O0 10. Pats Creek 21.98 5414900 11. Pigeon Creek 17.94 5413400 12. Platte River 367.78 5414000 13. Rock Branch 12.51 5432300 14. Yellowstone R. 73.81 5433500

Years

1940

1950

1960

1970

1980

1990

I......................................................

I ...................

I

1959-69

I

I..................................................... I ...................................................

I

I.............................................................................................

I

1937-87

I

1935-87 1939-87

=---I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I ........................

1959-87 1940-69

]

] .........................................................................................

I

of Rec~d 1958-87

1955-69

I

I.....................................

1960-80

I

I..................................................

1960-87

I

1960-87 I.............................................................................................

I................................... ] ..........................................................

I

1935-87 1959-78

I I

1955-87

378

vJ. MAGILLIGAN

TABLE 2 Discharges predicted from the regional flood frequency compared to discharges calculated using log Pearson Type III analyses. Log-Pearson Type III analysis is based on the Buncombe Gauge along the Galena River (USGS Gauge #5415000)

Q2

Qlo

Q2o

Qso

Qloo

Q2oo

Qsoo

Predicted discharge (m3/s) (using regional flood frequency analysis)

88

244

323

442

545

660

833

Computed discharge (m3/s) (using log-Pearson type III )

125

282

364

485

592

713

896

ables, and permits both lateral and vertical variation in input parameters. I tried to make the model as dynamic as possible and varied the roughness values for increasing flow magnitudes. Roughness values for the stream channel varied from a maximum of 0.033 for the 2-year flood to a minimum of 0.028 for the 500-year flood, and are based on an empirical study of the area (Magilligan, 1988). Lateral variation of roughness can also be incorporated into HEC-2, and various roughness values were assigned for discrete boundary areas across the valley cross-section, and roughness values for these overbank areas were also varied with increasing stage. The floodplains are primarily in pasture, and sometimes in corn, and overbank roughness values were estimated using Chow ( 1959 ) and Arcement and Schneider (1989) for these vegetation types. These selected roughness values also correspond with USGS slope-area calculations for post-flood indirect discharge estimates for the region. Thus, estimates of discharge from the regional flood frequency are entered into HEC2 utilizing the field-surveyed cross-sections and roughness estimates. HEC-2 provided calculations of flow depth, channel boundary shear stress, unit stream power, energy grade line slope, percent of total flow in channel, velocity head, and Froude number.

should decrease downstream because slope decreases faster downstream than depth increases. However, downstream changes in either shear stress or unit stream power do not follow this general pattern. Figures 2 and 3 are plots of the downstream changes in both shear stress and unit stream power, respectively, for

Qlofl 80-

10ft

200

300

400

500

DRAINAGE AREA (km2) Fig. 2. Downstream changes in shear stress for Q2, Qloo, and Qsoo for the Galena River watershed.

~. 2oo

Downstream changes in flood power DRAINAGE

If shear stress varied according to average downstream hydraulic geometry values, it

A R E A (kin 2

Fig. 3. Downstream changes in unit stream power for Q2, Q~oo,and Qsoofor the Galena River watershed.

VARIABILITYOF FLOODPOWERDURINGEXTREMEFLOODS

3 probability events: Q2, Gl0o and Qsoo. Shear stress and unit stream power both increase slightly downstream, but because of the high variability none of the relationships are statistically significant. Shear stress and stream power will be maximized where the depth-slope product is maximized - i.e., in zones where either slope or depth increase. Slope decreases downstream, but the rate of change of slope depends upon local geologic and geomorphic controls (e.g., coarse bed material). Channel depth increases downstream, but it may increase rapidly where either channel or valley conditions locally influence depth variation. For example, for a similar drainage area, channel morphology, and channel cross-sectional area, the rate of change of depth and shear stress at-a-station should be greater for a narrow valley than it is for a wide valley (Fig. 4). Thus, as drainage area increases, the depth-slope product decreases through normal downstream slope changes (Hack, 1957) and also through low depth values associated with wider valleys downstream. Narrow valleys a n d / o r conditions minimizing the width-depth ratio typically enhance the rate of change of depth downstream. The downstream pattern in either shear

379

stress or unit stream power reflects this lithologic control. Unusually wide valleys occur throughout the middle section of the watershed where the Platteville limestone and Decorah shale dominate the valley side lithology. Anomalously narrow valleys occur throughout the downstream sections where the resistant Galena dolomite re-emerges as the dominant lithology. This can be seen in Fig. 5 which expresses the plot of the standardized residuals of the regression of valley width versus drainage area. High negative residuals (i.e., narrow valleys) occur in the upper watershed and in the lower sections of watershed. High positive residuals (i.e., unusually wide valleys) occur in the less resistant limestones and shales. The geologic control of valley width strongly influences the variation in stream power and shear stress and manifests itself in two dominant ways. Resistant rocks contribute to steep slopes and enhance the rate of change of shear stress with increasing discharge - thus stream power will be maximized in those zones. Just the opposite occurs with erodible lithologies - gentle channel slopes and reduced rates of stage changes with increasing discharge. The pattern of downstream rates of change of stream power and shear stress evidence this lithologic control. A visual c0mpar-

A.

B.

Fig. 4. Theoretical control of valley width on flow depth with increasing discharge. Different stages (and thus shear stress ) of the 100-year flood for a narrow valley (A) and a wide valley (B).

380

F.J.MAGILLIGAN oa i°Pl 2.0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 !.0

/4,,0

-

~

-

i

lllJ i l l J

__

t If

16

.

op-Platteville limestone

':\1 13~1

~2~0

0

"~3 24

~

/

'('\

/

/

/

/

"

14

"Y~

Osp- SI. Peter sandstone

I ~

1'4

!

Dom,n n,.a a,v ,ey.ib-.

OgI-G~enad°l°mite -1 O 7!- H - Od - Decorah shale

" I I/

Od

i

.7 denotes range number

•

100

200

300

400

500

Drainage Area (km 2) Fig. 5. S t a n d a r d i z e d residuals o f t h e regression o f f l o o d p l a i n w i d t h v e r s u s d r a i n a g e area p l o t t e d against d r a i n a g e area. Bars i n d i c a t e z o n e s o f p a r t i c u l a r lithology. N u m b e r s refer to range n u m b e r .

ison of Fig. 5 with Figs. 2 and 3 demonstrates this phenomenon as they are inverted images of each other: high shear stress values (Fig. 2 ) and stream power (Fig. 3) correspond with unusually narrow valleys (Fig. 5 ). These downstream plots of either shear stress or unit stream power also document the nonlinear rates of increase of these flood power measure's with increasing discharge. Large differences exist between the flood power (expressed either as z or o9) for Q2 and Qloo (representing a fivefold increase in discharge), but only minor differences exist in energy exerted on the channel bed between Qloo and Qsoo (also associated with a fivefold increase in discharge). Because the floodplain stores large flow volumes, discharge increases during large magnitude floods result in nonlinear increases in flood power. This is typical of humid alluvial channels possessing gentle gradients and large floodplain storage. The implications of these zones will be discussed later.

At-a-station changes in flood power In addition to controlling the downstream rates of change of these hydraulic variables, geologic and geomorphic controls influence local conditions as well. In essence, the downstream rate of change reflects the continuum of the local conditions. Valley side lithology enhances flood power variation through its control on both slope and the rate of change of depth. However, certain channel geomorphic controls also influence flood power within a cross-section. As Graf (1983a) indicates, channel morphology greatly influences the flood power variation. He analyzes downstream changes in stream-power in semi-arid watersheds and discovers that stream power in pre-historical watersheds typically decreased downstream ( e x p o n e n t = - 0 . 2 4 ) , while it increased downstream following a period of arroyo cutting (exponent = 0.65 ). The arroyo incision increased the flow conveyance within the channel ultimately increasing stream power

VARIABILITY

OF FLOOD

POWER

DURING

EXTREME

381

FLOODS 100

~.~1 225

O0

80

.~0

70

175

j-f

60

R,*,'~{;E29

LI.

1.25

50 40

30 Y = 64.3

20

- 9.65(X)

r = - 0.61 p < 0.01

QII/Q2 I0

Fig. 6. Rates of change of shear stress (in N / m 2) with increasing discharge for Range 20 and Range 29. Both shear stress and discharge are expressed as the ratio of a given design flood "n" (Q, o through Qhoo) to their respective 2-year events to permit standardization.

-I

0

1

Residuals of Floodplain Width vs. Drainage Area Fig. 8. Standardized residuals of the regression of floodplain width versus drainage area plotted against the percent flow in channel during the 500-year discharge for each valley cross-section.

2.4 ¸

•~

1.2

Q,~/Q2 Fig. 7. Rates of change of shear stress (in N / m 2) with increasing discharge for Range 13 and Range 10. Both shear stress and discharge are expressed as the ratio of a given design flood "n" (Qlo through Qhoo) to their respective 2-year events to permit standardization.

dov~nstream. For the Galena River watershed, the accelerated floodplain sedimentation rates due to agricultural disturbance (Magilligan, 1985 ) and enlarged meander plains due to recent lateral migration (Knox, 1987; Magilligan, 1992) have combined to increase the channel capacity. Channel morphology will also influence the rate of change of flood power in a cross-section through the variation in the

rate of change in channel shape with increasing discharge. In order to assess the local channel effect on flood power, the percent of flow conveyed in the channel was calculated by HEC-2, as well as the rate of change of shear stress for each cross-section. Shear stress is standardized at each cross-section by expressing it as the ratio of the shear stress of a given design discharge (e.g. 10-year flood, 20-year flood, etc.) to the shear stress of the 2-year discharge. This standardization accounts for the difference between small upstream channels and larger downstream channels, which permits comparison. The effect of channel shape on rates of change of flood power can be seen in Figs. 6 and 7. The upper steep line in each plot corresponds to channel morphologies enhancing flood power increases with increasing discharge while the lower gently sloping line corresponds to a channel shape inhibiting increases in stream power with discharge increases. Channel shape can only accomplish so much, however, in controlling the total magnitude of flood power. Figure 6 demonstrates that Range 20 has a

382

F.J, M A G I L L I G A N

.<

>. 0

/-

Y =-4.7~c-5-().47(X~ l = -0.47 p < 0.(

-

I

0

I

2

Residuals: Valley Width vs. Drainage Area Fig. 9. Standardizedresidualsof the regressionof shear stress of Q~ooversusdrainage area plotted againstthe standardized residuals of floodplainwidth versus drainage area. much faster rate of change of shear stress with discharge than Range 29 does. However, this local channel control of Range 20 cannot overcome the other dominant control of slope within this section. Although the shear stress for the 500-year flood is 2.37 times greater than the 2-year flood, this only represents an increase from 18 to 42 N / m 2. Range 29 only experiences a 1.9 increase in shear stress for the 500-year flood compared to the 2-year event. Because Range 29 possesses a steeper slope and a higher stage for Q2, it subsequently has a very high shear stress for the 2-year event, and its channel and valley morphology do not greatly enhance shear stress increases with increasing stage. The other extreme scenario is portrayed in Fig. 7. The diagram shows how channel morphology enhances rates of change of shear stress for already high 2-year shear stresses, or dampening already low values. Range l 0 represents a cross-section flowing in the highly erodible

Platteville limestone, and its valley slope is one of the flattest in the watershed (0.0012) resulting in one of the lowest shear stress values (21.2 N / m 2) for Q2. Its channel and valley configuration ultimately d a m p e n shear stress increases with increasing discharge as the shear stress for Qsoo is only 1.78 times greater. Just the opposite occurs for Range 13. The combination of steep slopes in this section (0.0025) and narrow channels generates high shear stress values for Q2 (29.8 N / m 2) which increase by a factor of 2.6 to one of the highest values for shear stress for Qsoo (79 N / m 2 ) . Channel morphology directly controls local effects on flood power, but it is also responding to valley width controls during large magnitude events. During large magnitude events for especially wide valleys, the channel conveys progressively smaller increments of the total flow, whereas the opposite occurs with narrow valleys. Certainly channel morphology, independent of valley width influences,

383

VARIABILITYOF FLOOD POWER DURING EXTREME FLOODS

ultimately determines channel capacity. However, the percent flow conveyed within the channel also reflects these lithologic controls, as a correlation exists between these residuals of valley width and percent flow contained in the channel (Fig. 8). This figure shows that narrow valleys (high negative residuals) concentrate a greater percentage of total volume within the channel, while the opposite occurs with wide valleys (high positive residuals). Because of the control of valley width on both the rates of change of depth and percent flow in the channel, it partially explains the pattern of shear stress variation (Fig. 9 ). Narrow valleys (high negative residuals ) correspond with high positive residuals of shear stress, and wide valleys (high positive residuals) correspond with high negative residuals of shear stress ( r = -0.48, p<0.05).

Implications of spatial variability Minimum and maximum flood power occur in discrete sections along the Galena River. Irrespective of which probability event is used, the zone draining between approximately 200 and 290 km 2 experiences the largest magnitude of flood power, while the zone immediately upstream (80-150 km 2) experiences minimum values of shear stress (Fig. 2) and unit stream power (Fig. 3 ). Thus, a nonlinear downstream pattern results where particular zones reflect the lithologic controls. The probability of achieving a particular flood power varies widely throughout a basin depending on an array of geologic and geomorphic controls. For some sections of the basin, the probability of attaining a large flood power is especially "rare" while for other sections of a basin, that same magnitude of flood power is attained more "frequently." For example, because of the particular geologic and geomorphic controls for Range 17, the shear stress for the 2-year flood corresponds to the shear stress for the 500-year flood for Range 20, only several km downstream! This wide range of probabilities for

flood power partially explains why little correlation exists in the literature between major morphologic adjustments and flood magnitude, or its associated flow probability. Results from the Galena River demonstrate that for a specific probability event (e.g., 100-year flood), calculated values for either stream power or shear stress vary widely through a basin. Conversely, the probability of attaining a specific flood power (e.g., 100 W / m 2) may be associated with vastly different flow probabilities. The results presented here demonstrate the spatial and temporal (in a probabilistic sense) variability of achieving a specific flood power magnitude. This variability influences the transport, removal, and/or storage of sediments within watersheds. In a study of sediment removal on a southwestern watershed, Graf (1983b) notes that shear stress for the 10year discharge varies widely throughout the trunk stream (by a factor of 5 ). Graflimits his study to an analysis using only Q~o, yet the pattern of shear stress reflects lithologic and structural controls and determines the pattern of sediment evacuation. Flow competency for the 10-year flood decreases downstream contributing to sediment storage of coarse bedload in the downstream sections (Graf, 1983b). This variation in flood power throughout a basin demonstrates the episodic movement of sediment as certain sections within the Galena basin require exceedingly more infrequent events to generate a similar stream power or shear stress than within other zones. These zones operate as sediment sinks and will require longer time intervals to transport that material downstream. Thresholds of flood power in humid alluvial channels

Baker and Costa's ( 1987 ) primary goal is to establish the requisite flow conditions generating a catastrophic flood, which ultimately relates flood power to thresholds of instability.

384

The ratio of input forces to resistance is a widely used expression in geomorphology and engineering to explain and predict sediment transport and channel erosion (Hjulstrom, 1935; Shields, 1936; Sundborg, 1956; Baker and Ritter, 1975; Bull, 1979, 1988, 1990). However, in the case of catastrophic flooding its shortcoming pertains to determining resistance thresholds before the actual occurrence of a flood. Flood power can be calculated, but resistance cannot be determined until after the occurrence of a destabilizing flood. This section examines the possibility of determining when a catastrophic flood will occur (in a probabilistic sense). Implicit within this analysis is some notion of what actually constitutes a "catastrophic" event. In a geomorphological sense, catastrophic is usually associated with major morphologic adjustments: major erosion, deposition or channel re-alignment. Is there a c o m m o n magnitude of shear stress or unit stream power necessary to cross this threshold of resistance for gentle gradient alluvial channels? In order to determine if a m i n i m u m threshold exists for major morphological change, it becomes necessary to establish what constitutes a major morphological change. Certainly it corresponds with the transport of unusually large particles, or with channel scour/erosion on an event scale normally associated with longer time scales (Holocene?). I have adopted an approach similar to Williams (1983) who establishes m i n i m u m thresholds of velocity, shear stress, or unit stream power necessary to transport particles of a given diameter. Williams incorporates data from previously published work that provided information on particle size and flow conditions relative to whether a particle moved or remained stationary. The upper limits of flood power associated with extreme events has been previously evaluated. Baker and Costa (1987) analyze discharge data compiled from flow records from the United States Geological Survey (USGS),

F.J. MAGILLIGAN

and convert them into hydraulic units and plot these data as a function of drainage area (Fig. 10). They contend that an envelope curve adequately encloses this relationship, so that for a given drainage area the m a x i m u m value of shear stress or unit stream power can be identified. I suggest, instead, that a m i n i m u m curve can be identified as well as a m a x i m u m curve. Baker and Costa include data from the upper limits of USGS floods, and I have tried to include data from as many sources as possible where a researcher indicated major fluvial adjustments. I have restricted myself to alluvial channels in primarily humid to sub-humid environments, which severely limits the available sources. Several large floods were not included because of incomplete information (lacking slope, flood depth, or stream channel information). I relied mainly on the work of Wolman and Eiler (1958), Gupta and Fox (1974), Osterkamp and Costa (1987), Kochel (1988) and Miller (1990). Kochel's review paper on extreme floods (Kochel, 1988 ) provided data where he separated large floods culled from the literature into three major categories: major modification, moderate modification, and minor modification. I used those sections that had major modifications. Data

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from these and other sources are plotted against drainage area for both shear stress (Fig. 11) and unit stream power (Fig. 12) and a line is drawn at the base. In general, this line may approximate the minimum level of shear stress and unit stream power associated with causing major morphological adjustments. For gentle gradient alluvial channels in humid to sub-huIlk100 -

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mid environments, this minimum threshold appears to correspond with a shear stress of approximately 100 N / m 2 or to a unit stream power of 300 W / m 2. Is it possible to link flow probabilities to requisite flood power values necessary to generate a "catastrophic" flood? In other words, is it possible to attach a flow probability to the minimum thresholds of flood power expressed in Figs. 11 and 12? These minimum values of 100 N / m 2 for shear stress and 300 W / m : for unit stream power are not rigidly absolute thresholds, but they closely approximate the minimum thresholds. For example, in his analysis of major erosion and deposition in the Valley and Ridge section of Virginia following a series of major floods in 1985, Miller (1990) concludes that most of the major floodplain and channel scour occurs in zones where unit stream power exceeded 300 W / m 2. O'Connor et al. (1986) examine the sedimentological characteristics of pool-riffle sequences in relation to stream power. They identify boulder deposits accumulating where stream power is less than 270 W / m 2, roughly corresponding to the minimum in Fig. 12. Also, the lowest value for shear stress of the largest floods compiled by Baker and Costa (1987) roughly corresponds with 300 W / m 2 (Fig. 10). These threshold values suggested here are not absolute and do not necessarily represent a universal threshold as local variation in vegetation, channel bed armoring, or bank cohesiveness will act to enhance or impede flow resistance. However, given that they are approximately correct, they can be used as a gauge of threshold variation. In order to determine the flow probabilities of reaching these critical values of shear stress ( 100 N / m 2) and unit stream power (300 W / m2), increasingly larger discharges were input into HEC-2 until these threshold values of shear stress and stream power were reached. A problem surfaces, however, in trying to express the probabilities of these enormous discharges. One way would be to extrapolate the

386

F.J. MAGILLIGAN

regional flood frequency curve and determine the flow probability for that specific discharge. However, this approach strains the statistical reliability of flow frequency analysis. Instead of expressing these discharges as specific flow probabilities, they are presented as a ratio to the 100-year flood discharge to circumvent these statistical limitations. This ratio captures the sense of probability while also being standardized. If one accepts that these threshold values (Figs. 11 and 12 ) are realistic, then the probabilities of achieving them can be illustrated in Figs. 13 and 14. These plots reflect the local effects of geology and channel shape, but also demonstrate the improbability of achieving a "catastrophic" flood in humid alluvial channels. For some sections of the Galena River basin, the minimum flood power threshold will be reached with a discharge less than twice the 100-year event, and for other sections of the basin, it requires enormous discharges exceeding 15 times the magnitude of the 100-year flood. In a probabilistic sense, this roughly translates into events of roughly 200-500 year

probabilities for the lower ratios, and significantly greater than 1000-year recurrence probabilities for other sections. These frequency values and their spatial context correspond to field validated occurrences of erosion and sedimentation within the basin. The Galena River basin experienced an extreme event during June, 1969 approximating the 200-to-400 year flood, and floodplain scouring occurred in certain sections of the basin while minimal geomorphic effects occurred in other parts (Magilligan, 1985). Thus, for a similar low frequency event, widely different geomorphic effects occurred throughout the basin, as geomorphic controls either enhanced or diminished the flood power magnitude for the similar low frequency event. In a similar study of stream power distribution, Graf (1982) demonstrated that stream power for the 10-year discharge varies spatially through a watershed and responds to intrinsic controls of flow depth and stream gradient, and to structural and external controls. Results from the Galena River extend this earlier work of Graf by analyzing the spatial variability of flood power for other

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probability events, which reinforces that the spatial characteristics of these intrinsic and external controls on flood power continue to operate for even larger magnitude events.

Summary and conclusions As geomorphologists we are simultaneously confounded and enticed by the tenuous link between hydrology (flood magnitude, flow frequency, etc. ) and fluvial characteristics (sediment transport, channel avulsion, etc.). This paper attempts to link the magnitude-frequency relationship to thresholds of critical flood power, and to explain why events of similar magnitude or frequency produce dissimilar geomorphic effects. Much of this discussion has centered on morphological adjustments to "catastrophic" floods, yet no definitive explanation currently exists to determine what actually constitutes "catastrophic". Dury (1975) argued that geomorphic attention too often focuses on the continuous and on the central tendency of processes rather than on discontinuities, and he further suggested that catastrophic pertains to

"... a breakdown of systems behaviour induced by an overload of input or positive feedback." (Dury, 1975, p. 137). Therefore, we should be focusing on disequilibrium rather than on equilibrium conditions. Analysis of channelforming discharges has generated considerable geomorphic research over the years associating channel pattern and form to modal flow conditions (Leopold and Maddock, 1953; Wolman and Miller, 1960; Dury, 1973; Andrews, 1980). This analysis of the Galena River redirects the attention to the probabilities of channel de-forming discharges. Perhaps no unifying probability exists as evident in channel forming discharges (e.g., Q2.33), but these results show the range in this probability and its spatial variation. These results further demonstrate that at-astation and downstream flood power magnitudes respond to general controls of the fluvial system, but also to localized effects. Downstream changes in flood power do not necessarily follow a simple log-linear progression as identified by hydraulic geometry, but other controls exist to exacerbate these general patterns. Flow constrictions due to lithologic re-

388

sistance contribute to increases in flood power at-a-station because of increases in energy slope and flow depth. It will also concentrate a greater percentage of flow within the channel which enhances flood power, thus flood power within these sections are considerably greater than what would be anticipated simply based upon its position within the network. The effect of these normal downstream changes in hydraulic variables and localized lithologic effects combine to produce an irregular, but distinct, pattern of flood power within the Galena River Basin. Hydrologically, these results demonstrate that the flow probabilities of reaching a critical flood power also vary spatially. Whether these minimum values of 100 N / m 2 or 300 W / m 2 are indeed the thresholds is certainly arguable, but irrespective of what flood power value is used, the probability of achieving it varies irregularly through a basin due to both the localized lithologic effects and overall general patterns. Also, if these threshold flood power values define the critical threshold, they demonstrate the tremendous improbability of achieving a channel "de-forming" flood in certain valley reaches in gentle gradient alluvial channels. In some sections of the basin, critical threshold values will require discharge events greater than several thousand years recurrence probabilities. What effect does this spatial and temporal pattern exert on overall geomorphic characteristics? The effects can be separated into either overall geomorphic form or the processes generating these forms. The dominant control on process relates to sediment transport abilities. Low flood power values for even extreme events would therefore necessarily translate into decreased abilities to transport sediment ultimately encouraging sediment storage (O'Connor et al., 1986). Just the opposite would occur in narrow valleys where high flood power values are generated for progressively higher probability events. In a previous study, floodplain storage of historical alluvium was -

F.J. MAGILLIGAN

shown to be strongly controlled by variations in floodplain width (Magilligan, 1985). Resuits presented here suggest that floodplain width is an indirect control: the direct control is associated with flood power but indirectly controlled by floodplain width. Geomorphologically, this pattern manifests in various ways. As Nanson ( 1986 ) points out, floodplain morphology is strongly linked to specific valley and fluvial characteristics. Floodplain scouring, which is certainly a manifestation of a rare flood, occurs on a relatively frequent basis in his study of highly confined rivers in New South Wales, Australia. Floodplain scouring occurs during events < 100-year probability in narrow valley sections where slopes are locally steepened and flows confined; whereas overbank deposition occurred within the same basin during the same event, in non-confined sections (Nanson, 1986). The presence of these critical flood power thresholds may also control the vertical and horizontal channel position. Bull ( 1979 ) contends that channel lateral migration will occur in sections where channel energy is near thresholds of resistance, and downcutting will occur when these thresholds of resistance are crossed. Results from the Galena River demonstrate that certain sections along the longitudinal profile are considerably distanced from this threshold, even during infrequent events (e.g., Qsoo), while other sections can attain this threshold at lower magnitude discharges. Perhaps variations in lateral migration evidence this spatial control as well. Broadly speaking, these results also link the foundations of the magnitude-frequency relationship established by Wolman and Miller (1960) 30 years ago with the current analysis of flood power magnitudes. Wolman and Miller emphasized that geomorphic "work" (e.g., sediment yield) should be evaluated with reference to both the magnitude and the frequency of an event. Baker and Costa (1987) argue that "formative" events needed to be evaluated by flood power magnitudes (rela-

VARIABILITYOF FLOOD POWER DURING EXTREME FLOODS

tive to in h er en t thresholds o f instability). I have tried to e m b e d the Baker and Costa principle within the larger c ont e xt o f the W o l m a n a n d Miller principle by establishing the flow probabilities o f these critical thresholds. Thus, m a j o r mo d if icatio n s will oc c ur in distinct geom o r p h i c settings where flood pow e r is maximized. These results perhaps also help explain why the link between hydrologic inputs a nd g e o m o r p h i c effects has been diffuse.

Acknowledgements This work greatly benef i t e d f r o m the research assistance o f Brian Tracy. Brian helped t h r o u g h o u t all stages o f the project, a nd his advice an d counsel c o n t r i b u t e d significantly to the o u t c o m e o f this project. Melissa St a m p c o n t r i b u t e d to data collection during the emb r y o n ic stages o f the project, and N a n c y Fenton helped with the cartography. Bob Brakenridge an d R i c h a r d M a h o n read an earlier draft o f the m a n u s c r i p t a nd t hei r editorial advice i m p r o v e d the overall quality. I would also like to t h a n k Will G r a f for his review a nd comments on this paper, and especially t h a n k the editors, Bill Renwick and J ona t ha n Phillips for their support an d c o m m e n t s . Acknowledgem e n t is m a d e to the donor s o f the P e t r o l e u m Research Fund, a d m i n i s t e r e d by the A m e r i c a n Chemical Society, for partial support o f this research ( P R F ~ 2 4 8 9 5 - G B 8 ) . This work was also partially f u n d e d by the N a t i o n a l Science F o u n d a t i o n G e o g r a p h y and Regional Science P r o gr am (SES 9 l- 126 5 8 ).

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390 Hjulstrom, F., 1935. Studies of the morphological activity of rivers as illustrated by the river Fryis. Bull. Geol. Inst. Upsula, 25: 221-527. Knox, J.C., 1987. Historical valley floor sedimentation in the Upper Mississippi valley. Ann. Assoc. Am. Geogr., 77: 224-244. Kochel, R.G., 1988. Geomorphic impact of large floods: review and new perspectives on magnitude and frequency. In: V.R. Baker, R.G. Kochel and P.C. Patton (Editors), Flood Geomorphology. Wiley, New York, pp. 169-187. Leopold, L.B. and Maddock, T., 1953. The hydraulic geometry of stream channels and some physiographic implications. U.S. Geol. Surv. Prof. Pap., 252:57 pp. Magilligan, F.J., 1985. Historical floodplain sedimentation in the Galena River basin, Wisconsin and Illinois. Ann. Assoc. Am. Geogr., 75: 583-594. Magilligan, F.J., 1988. Variations in slope components during large magnitude floods, Wisconsin. Ann. Assoc. Am. Geogr., 78: 520-533. Magilligan, F.J., 1992. Sedimentology of a fine-grained aggrading floodplain. Geomorphology, 4: 393-408. Miller, A.J., 1990. Flood hydrology and geomorphic effectiveness in the central Appalachians. Earth Surf. Proc. Landforms, 15:119-134. Mullens, T.E., 1963. Geology of the Cuba City, New Diggings, and Schullsburg Quadrangles, Wisconsin and lllinois. U.S. Geol. Surv. Bull., 1123-H: 437-531. Nanson, G.C., 1986. Episodes of vertical accretion and catastrophic stripping: a model of disequilibrium floodplain development. Geol. Soc. Am. Bull., 97: 14671475. O'Connor, J.E., Webb, R.H. and Baker, V.R., 1986. The relationship of pool and riffle pattern development to

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large magnitude flow hydraulics within a canyonland stream. Geol. Soc. Am. Bull., 97: 410-420. Osterkamp, W.R. and Costa, J.E., 1987. Changes accompanying an extraordinary flood on a sand-bed stream. In: L. Mayer and D. Nash (Editors), Catastrophic Flooding. Allen and Unwin, Boston, pp. 201-224. Partridge, J.B. and Baker, V.R., 1987. Paleoflood hydrology of the Salt River, central Arizona. Earth Surf. Proc. Landforms, 12: 109-125. Rhoads, B.L., 1987. Stream power terminology. Prof. Geogr., 39: 189-195. Schumm, S.A., 1973. Geomorphic thresholds and complex response of drainage systems. In: M. Morisawa (Editor), Fluvial Geomorphology. George Allen and Unwin, London, pp. 299-310. Shields, A., 1936. Application of similarity principles and turbulence research to bed load movement. U.S. Dept. Agriculture, Soil Conservation Service Cooperative Lab, California Institute of Technology. Sundborg, A., 1956. The River Klaraven: a study of fluvial processes. Geogr. Ann., 38:125-316. Williams, G.P., 1979. Hydraulic geometry of river crosssections-theory of minimum variance. U.S. Geol. Surv. Prof. Pap., 1029:47 pp. Williams, G.P., 1983. Paleohydrologic methods and some examples from Swedish fluvial environments: I. Cobble and boulder deposits. Geogr. Ann., 65A: 227-242. Wolman, M.G., 1955. The natural channel of Brandywine Creek. U.S. Geol. Surv. Prof. Pap., 271:56 pp. Wolman, M.G. and Eiler, J.P., 1958. Reconnaissance study of erosion and deposition by the flood of August 1955. Am. Geophys. Union Trans., 39: 1-14. Wolman, M.G. and Miller, J.P., 1960. Magnitude and frequency of forces in geomorphic processes. J. Geol., 68: 54-74.