Q=r*DA

As you move down a watershed, the drainage area increases and the discharge increases

x4

x3

x2

x1

Since Q ↑ as DA↑ downstream

and Q=w*v*dThen d, w, and v all tend to increase downstream as WA increases.

x1x3

x4x2

Stream cross-section

w=nQa

v=pQb

d=qQc

n*p*q=1a+b+c=1

Q=wvd=(nQa)*(pQb)*(qQc)=npqQa+b+c

which can only be true if npq=a+b+c=1

rDA=wvd=(n(rDA)a)*(p(rDA)b)*(q(rDA)c)

So that w=nrDAa, v=prDAb, and d=qrDAc

We can also write w,v and d as functions of DA

since Q=rDA

These relationships can tell us how the width, velocity and depth of a river will change as its discharge increases or decreases.

Drainage Area (km2)

10

1.0

100

o

ooo

o

ooo

Slope =0.55Width (m)

1 10 100 10000

1

2

Log10 width

0 1 2 3

Log10 Drainage Area

Log w =0.23 + 0.55 Log DAw= 1.7DA0.55

How stream width increases with Drainage Area in the Upper Oldman R

W

DA

Drainage Area (km2)

oo

oo

Slope of depth line=0.25

o

o

o

o oSlope velocity line =0.2

Depth m

1.0

0.1

0.1

1.0

Velocity m/sec

o

o

o

o

o

o

o

1 10 100 1000

v=0.24 DA0.2

d=0.22 DA0.25The exponents for width, velocity and depth add up to 1

How stream velocity and depth increase with drainage area in the Upper Oldman R

Since w=nQa and v=pQb and d=qQc

We can writeLog w=Log (nQa)=Log n + Log Qa = Log n + aLogQOr since Q=rDALog w=Log (nrDAa)=Log n + Log r + Log DAa = Log n + Log r + aLogDA

And similarlyLog v=Log p + Log r + bLogDA, andLog d=Log q + Log r + qLogDA, andLog Q=Log r + LogDA

These relationships are useful since they allow us to plot the non-linear functions as linear graphs, and to establish exponent values using linear techniques.

0

0.2

0.40.6

0.8

1

1.2

1.41.6

1.8

2

0 500 1000 1500 2000 2500

Drainage Area (km2)

Mean velocity (m/s)

v= 0.24 DA 0.2

•The bell curves rising out of the plane of the graph depict the variability of the river’s flow regime at a series of points along the drainage—the bell curves get wider toward the right, illustrating the increasing range of variability downstream

frequency

velocity (m/s)

Cumulative Frequency (percentiles)

100%50%

About 90% of the river has velocity less than this value

100%

50% of the river has velocity less than this value

Only about 10% of the river has velocity less than this value

Two different ways of depicting variability

0

0.2

0.40.6

0.8

1

1.2

1.41.6

1.8

2

0 500 1000 1500 2000 2500

Mean velocity (m/s)

Drainage Area (km2)

The % of the river <1.2 m/s decreases downstream

At lower velocity (eg 0.6 m/s) the downstream decrease in % occurs more rapidly

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

0 500 1000 1500 2000 2500

Proportion< given velocity

1.2 m/s

1.0 m/s

0.8 m/s0.6 m/s

0.2m/s0.4m/s

The distance between these two lines represents the proportion of the river with velocity between 0.8 – 1.0 m/s at the point where DA = 1000 km2

Drainage area (km2)

10

20

30

50

40

500 15001000 2000 2500

oo

o

oo

o

o

o

o

o

o

oo

o

o

o

o

o

0.6-1m /sec adult trout0.2-0.6m/sec juvenile trout

0-0.2m/sec trout fry

% of habitat

Drainage area (km2)

o

o

o

o >1m/s very large adults

Low velocity habitats predominate in the upstream sections and medium and high velocity habitats become more predominant downstream

Fish and other aquatic biota that live in rivers and streams have to contend with the variability of the flow regime.

How variable is runoff/discharge?

From year to year?

From month to month

From day to day

Runoff is highly variable from year to year

Fig 5-14 from your text