•OURNALOF GEOPHYSICAL
RESEARCH
VOL. 72, NO. 22
NOVEMBER15, 1967
On River Discharge into the Northeastern Pacific Ocean and the
Bering Sea
GUNNAR I. RODEN
Department o• Oceanography, University o• Washington
Seattle, Washington 98105
The information obtained from monthly mean and extreme river dischargerecords and the
ioint variation of salinity and river dischargeat coastal stations are analyzed. The average
annual fresh water dischargeinto the Pacific between California and the Aleutian Islands
amountsto approximately21,000mS/sec.The fresh water dischargeinto the Bering Sea by
Alaskan and Siberian rivers occursat an average annual rate of at least 10,000 mS/sec.There
have been no significanttrends in natural streamflowduring the past half century. Prolonged
droughtsin the Far West, as those between 1928 and 1931,are causedby a persistentcold,
dry anticycloneover the plateau states.Extreme flooding,as in 1861-1862and 1964-1965,is
related to strong polar outbreaks and subsequentinvasion of warm moist air from the sub-
tropical Pacific.The spectraof river dischargeshowpeaksof meteorologicalorigin at annual
and semiannualfrequenciesand suggestnonlinear interaction. The probability distributions
of river dischargeare skew and limited on the left. For small and moderate values of skew-
hessthe probabilitiesof floodsand droughtscan be estimatedfrom theoriesof a nonlinear
randomvariable.There is goodagreementbetweenthe observedand the theoreticallyderived
probabilities
for the durationof floodsand droughts.
Salinityfluctuations
at coastalstations
reflectcloselythe salinity fluctuationsof river discharge.
The response
betweennonseasonal
salinityand river discharge
fluctuations
is largelyindependent
of frequencybut dependson
the size of the river and the distance of the salinity observing station from the river mouth.
Abnormal salinities due to floodsand droughtscan be estimatedto an accuracyof 1 to 2%o
Alaska,from the provinceof British Columbia,
INTRODUCTION
Riversplay an importantpart in the coastal
oceanographicenvironment. They transport
sediments an'd contribute to beach accretion
and erosion.They carry dissolvedchemical
compounds
to the seaandinfluence
the compositionof seawater. They give riseto characteristic marine environments that can support spe-
cific communities
of organisms.
They discharge
fresh water into the sea and lower the salinity
of the oceanic surface waters. Traces of sea
water dilutedby rivers can be foundseveral
hundredkilometersoffshore.A knowledgeof the
amount of fresh water dischargedinto the sea
seems,
thus,to be pertinentto severalbranches
of the marinesciences.
Yet, many detailedand
long-periodaspectsof river discharge
have remained largely unexplored.
Canada, and from the Chukhotsk an'd Kamchatka regionsof Siberia. River dischargefrom
the Pacific slopesof Mexico has been discussed
by Tamayo [1964] and will not be considered
here.
The river dischargeat any gagingstation is
the net amount of water passingthrough the
cross-sectionalarea of the river channel per
unit time. The volume of water carried past
the station depends,in general,on the season
and the particular weather conditions in the
water-shed area. The time variations. of stream-
flow consist,therefore,of a more or lessregular component upon which superimposedare
irregular or random fluctuations.The regular
component is of meteorologicalorigin an'd re-
flects the seasonalchange in rainfall, snowmelt, evaporation, and soil infiltration rate.
Thispaperdealswith riversdischarging
into The randomcomponentis alsostronglyaffected
the Pacific Oceanand the Bering Sea from the
statesof California, Oregon,Washington,and
by meteorologicalconditionsand may be regarded as reflecting the effects of indivi'dual
storms on streamflow.
The river dischargeis a random variable in
•Contribution 429 from the Department of
Oceanography,
Universityof Washington,Seattle. the sensethat its value at. a given time cannot
5613
5614
GUNNAR
I. RODEN
be predicted exactly; instead, a multitude of unnoticed displacementmay lead to spurious
values are possible at any time instant. The trends in the records. In the absence of land
number of these values is not unlimited, how- uplift or subsidence,the river level recorder
ever. In fact, the streamflowvaluesare bounded measures the fluctuations in water level due to
at the lower end by the con'ditionof no flow hydrologicaland meteorologicalcauses.These
and at the upper end by the capacity of the fluctuationsare likely to consistof a random
river channel.The capacity of the river chan- part superimposedon the more or less regular
nel may be extendedto 'potential river chan- seasonalan'ddiurnal changes.The random river
nel,' to includenormally dry areas along the level fluctuationsare mainly determinedby the
river banks, in anticipation of floods. Ultiturbulenceof streamflow,wind waves, and, in
mately, the dimensionsof the drainage basin caseof broad and sluggishlymovingrivers,pile
determine the upper limit of discharge.
up of water on the downwind bank. If gage
height measurements
are made frequentlyor if
I) ATA
continuouslyrecordinggagesare used,the efStreamflow
records in the
western
United
States date back to 1858, when the Oregon
Navigation Company started to keep a diary
of the annualmaximumstagesof the Columbia
River near The Dalles, Oregon. In 1878, the
U.S. Army Corps of Engineerstook over the
observationsand exten'dedthem to includedaily
river stages.Stream gagingby the U.S. Geological Survey in the region concernedstarted
in 1896 with a continuouslyexpandingstation
network. In Canada, the responsibility for
stream gagtriglies with the Department of Energy, Mines, and Resources.
Streamflowis commonlydeterminedby recordingthe elevationor stageof the water level
above an arbitrarily selected reference point
and by entering the recorded value into an
empirical stage-dischargerelationship, which
yields the dischargein the desired units. The
stage-dischargerelationshipis establishedby
measuringwith a current meter the velocity at
severalpoints of the river channelcrosssection
and by relating the calculatedvalue to the recorded river level. At most stream gaging stations in the United Statesthe speedof the river
is measured at only two depths: 0.2 and 0.8
times the distanceto the bottom. In very shallow water only one measurementis made at
0.6 times the actual depth. Details of theseand
other measurementsare discussedby Corbett
et al. [1945], Trestman [1964], and DeWtest
[1965].
The accuracy of the published streamflow
values obviously dependson the accuracy of
the stage-discharge
relationship.A river level
recorder measures the elevation
of the water
surfacetelative to a fixed referencepoint. If the
fixed point is embeddedin unstableground,its
fects due to turbulence and wind waves can be
reducedby suitable averagingtechniques,but
the effect of the pile up of water cannot be
eliminatedunlessmore than one gage is used.
The presence of ice seriously interferes with
river level measurements,and under such conditionsthe stage-discharge
relationshiplosesits
validity.
The determination of the average flow
through a cross-sectional
area of the river channel presupposes
an accurateknowledgeof the
profile as well as accurate velocity measurements in the presenceof turbulent e'ddies.The
profileitself may changewith time, particularly
in regionsof unconsolidatedsediments.Obtaining a representative velocity-depth structure
for the cross-sectionalarea presents considerable difficulties.The widespreadpractice of
measuringvelocity at only two points of the
vertical and assumingthat it is zero at the
bottom is equivalentto assuminga parabolic
velocity-depthrelationship.For turbulent flow
observedin most streamsthis is hardly a valid
supposition.The averaging time to eliminate
the turbulent velocity fluctuationsdependson
local conditionsand may be further affected
by the presenceof ice. Near the river estuary
and, in somerivers, for a considerabledistance
upstream tidal effectsare felt.
A stable and valid stage-dischargerelation-
ship can exist only in regionsin which the
above-mentioned
disturbinginfluences
are minimal. Thus, there cannotbe any meaningfulrelation between these two variables where tidal
influencesexist, where ice is present,or where
the profile changesrapidly with time.
The streamflow records published by the
U.S. GeologicalSurvey give a roughfirst ap-
RIVER
DISCHARGE--NE
PACIFIC
proximationto the actualstreamflow.
The daily
dischargevalues are consideredto be excellent
if the errorsare lessthan 5%, goodif they are
lessthan 10%, fair if they are lessthan 15%,
and poor if they exceed15%. For most of the
stationsconsidered
here, the quality of the records is goo'd,in the above-definedsense.The
AND BERING
Gaged
Drainage
river basins is shown in Tables 1 and 2. The
term 'gaged drainage area' refers to one for
which the surface runoff has been directly
measured.It differs from the actual drainage
area by not including any ungagedareas bet.ween the
last
downstream
station
and
the
river mouth. Hence, the gage'ddrainagearea is
always lessthan the actual one. The difference
between them is small when the downstream
gagingstation is closeto the river estuaryand
may becomelarge if any ungagedtributaries
enter the main stem after the last gagingstation. The listed dischargerates in Tables I and
2 are, therefore,minimum valuesof the actual
fresh water transport into the ocean.Although
in engineeringcirclesit is sometimesthe custom
to estimatethe rate of the remainingungaged
flow by making use of correctionfactors, this
procedurewill not be consideredhere because
of its low accuracy.
The largest river between Mexico and Canada is the Columbia. The average annual discharge near its mouth is approximately 7200
mS/sec.This is about twice the amount of fresh
water dischargefrom all other rivers in California, Oregon,and Washingtoncombined.The
influence of the Columbia River on the surface
waters of the ocean is felt several hundred
kilometersseawardby its effect on the salinity
[Barnes and Paquette, 1957; Duxbury et al.,
1966]. The secondlargest contributor of fresh
water is the SacramentoRiver, which, together
with the San Joaquin River, discharges840
mS/secinto San FranciscoBay on an annual
average.There the fresh water is mixed with
sea water and is carried seawardby tidal action. At the Golden Gate, salinities fluctuate
Measured
Mean Annual
Area,
km •-
ice, basedon direct measurements
throughice
holesis, however,poor.
The mean annual river dischargeinto the
Pacific Ocean and the Bering Sea for various
5615
TABLE 1. Mean Annual River Discharge(mS/see)into the
PacificOceanfrom California,Oregon,and Washington
Onlygaged
riverbasinshavinga meanannualdischarge
of at
least10 mS/secare considered.
Becauseof this,the valuesbelow
indicatea lower limit of the actual dischargeinto the ocean.
accuracy of the daily values for flow under
P•IVER DISCHA'RGE
SEA
Discharge,
ma/sec
California, OceanDrainage
Sacramento River basin
Klamath River basin
Eel River basin
31400
486
8950
234
San Joaquin River basin
40500
163
1670
106
Smith River basin
Russian River basin
Mad River basin
•attole
River basin
Redwood Creek basin
Navarro River basin
Gualala River basin
Garcia River basin
Salinas River basin
65100
677
3630
67
1260
44
622
38
720
31
785
15
417
12
256
11
10700
11
Total
1895
Oregon,OceanDrainage
Rogue River basin
Umpqua River basin
Nehalem River basin
CoquilleRiver basin
AlseaRiver basin
Siletz River basin
Wilson River basin
NestuccaRiver basin
Trask River basin
SiuslawRiver basin
12800
10100
1730
1960
924
524
417
236
376
882
Total
302
228
77
70
51
45
34
28
27
25
887
ColumbiaRiver, OceanDrainage
Columbia River at Vancouver,
Washington
624200
Willamette River basin, Oregon
28290
Cowlitz River basin, Washington
6160
Lewis River basin, Washington
2320
Kalama River basin, Washington
513
Grays River basin, Washington
184
ElochomanRiver basin, Washington
171
Total
5660
1063
273
161
35
19
11
7222
Washington, OceanDrainage
Chehalis.Riverbasin
QueetsRiver basin
Quinault River basin
Hoh River basin
Quillayute River basin
Humptulips River basin
North River basin
Willapa River basin
Naselle River basin
Dickey River basin
4700
1150
684
655
990
337
567
409
231
223
Total
220
117
79
71
62
37
27
24
15
14
666
Washington,Inside PassageDrainage
Skagit River basin
SnohomishRiver basin
Nooksack River basin
Stillaguamish River basin
8010
4500
1730
1170
467
300
106
95
5616
GUNNAR
TABLE
1 (continued)
Gaged
Drainage
Area,
km 2
Puyanup River basin
Elwha
River basin
Duwamish
River basin
Nisquany River basin
I. ROI)EN
Measured
Mean Annual
Discharge,
m3/sec
2450
95
697
42
1140
40
determinedaccurately,but a value of 400 mS/
secis sometimesquote'dby Canadianauthorities
(personal communication).If this estimate is
true, the mean annual dischargeof the Fraser
River is about half the dischargeof the CoTABLE 2. Mean Annual River Discharge into the Pacific
Ocean and the Bering Sea from British Columbia, Alaska,
and Siberia
1570
31
Skokomish River basin
Hamma-Hamma
River basin
588
189
31
15
Dosewanips River basin
244
13
Duckabush
175
12
414
11
132
11
Only gaged rivers and tributaries having a mean annual discharge of at least 100 m•/sec are considered. Because of the
existence of many ungaged tributaries between the gaging station and the river mouth, the values given below represent a
lower limit of the actual. fresh water discharge into the sea.
Rivers 'belonging to the same draiuage basin are indicated by
404
11
braces.
Deschutes
Hoko
River
River
River
basin
basin
basin
Dungeness River basin
Total
1280
mostly between26 and 32% and correlatewell
with the amountsof fresh water dischargedinto
San Francisco Bay by these two rivers. The
effectof the diluted San FranciscoBay water on
the salinitiesfarther offshoreis, however,small.
This is not surprisingin view of the relatively
small amountof freshwater adde'd(1/10 of the
Columbia) and of the fact that the salinitiesat
the oceanentranceare already high (29 instead
of 0%).
On a regional basis, very little fresh water
reaches the sea between the Mexican
border
and San FranciscoBay. The total mean annual
dischargefor the entire stretch of coastlineis
less than 50 m3/sec. Between San Francisco
Bay and the Oregon border, the annual dis-
chargeis about 1000 m3/sec.From the Oregon
border to the mouth of the Columbia River,
about 900 mS/secare dischargedannually.Between Cape Disappointmentand Cape Flattery
the mean annual dischargeis roughly 700
sec.Dischargeinto the inlandwaterwaysof the
state of Washingtonaveragesaround 1300 mS/
sec annually.The total fresh water discharge
into the Pacific Ocean from the states of Cali-
Gaged
Drainage
Area,
km2
British Columbia, Vancouver Island Drainage
Somass River near Alberni
River
1740
106
228
British Columbia, Mainland Drainage
Fraser
River
atHope
202800
Harrison
Riverat Harrison
Hot Springs
Skeena River
at Usk
Stikine
River
atTelegraph
Creek
Iskut River near Telegraph Creek
Nass River at Aiyansh
Vgannok
River
Homathko
near Rivers
River
Inlet
near Stuart
Island
Taku River near Tulsequah
Squamish River near Brackendale
Bella Coola River near Hagensborg
2700
8340
44O
38850
94O
29270
42O
9350
48O
18000
82O
4220
37O
5540
290
15540
27O
1980
24O
4066
120
Total
7090
Alaska, Ocean Drainage
Copper River near Chitina
Susitna River near Gold Creek
Chulitna River near Talkeetna
Knik River near Palmer
Skwentna River near Skwentna
Matanuska River at Palmer
Chakachatna River at Tyonek
53350
15950
6660
3060
5830
5360
2900
Total
1050
280
250
193
180
113
100
2166
Alaska, Bering Sea Drainage
from the Columbia River and San Francisco
Yukon River at Kaltag
Kuskokwin
chargeof the Fraser at Hope and of its most
important downstreamtributary, the Harrison,
is about 3100 mS/sec.The contributionof the
remainingdownstreamtributaries has not been
122
Total
Bay, amountsthus to approximately12,000
is the Fraser. The combined mean annual dis-
1310
Campben River below Quinsam
fornia,Oregon,and Washington,
includingthat
mS/sec.
In British Columbiathe most important river
Measured
Mean Annual
Discharge,
m3/sec
River at Crooked Creek
Nuyakuk River near Diningham
Wood River at Aleknagik
766600
6220
80550
95O
3860
2870
164
Total
140
7474
•iberia, Bering Sea Drainage
Anadyr River at Mouth
Kamchatka River at Mouth
Total
200000
55700
1660
1000
2660
RIVER
DISCHARGE--NE
PACIFIC
lumbia River. The fresh water carriedby the
Fraser has a profound effect on the hydrography of the Strait of Georgia,as discussed
by
AND BERING
SEA
5617
of the rainy season.Where both snowmeltand
rainfall control runoff, there are two maxima.
Almost all the large rivers in Alaska and Brit-
Waldichuk[1957]. The seeon'd
and third largest ish Columbia and Columbia River have their
Columbia are the Skeena and
highestdischargein summer.Mos• rivers origithe Stikine. The Skeenaat Usk discharges
about nating in the coastalmountainrangesof Cali940 mS/secannually. The Stikine at Telegraph fornia, Oregon,and Washington,however,have
rivers in British
Creek and its main downstreamtributary the
Iskut have a combinedmean annual discharge
of approximately 900 mS/see.Since many ungagedtributaries enter these rivers betweenthe
gagtrigstation an'dthe mouth, the above values
shouldbe regardedas a lower limit to the actual
fresh water dischargeinto the ocean.
In Alaska, only a few short streamflowrecordsexist.The largestriver discharginginto the
Pacific is the Copper.At Chitina, its mean annual dischargeis 1050mS/see.Many large rivers
flow into CookInlet, with a combinedmeanannual dischargeprobably equal to that of the
Copper River.
The most important rivers carrying fresh
water to the Bering Sea are the Yukon and the
Kuskokwim in Alaska, and the Anadyr and
Kamchatka in Siberia. The largest of these is
the Yukon, which at Kaltag has a mean annual
dischargeof 6220 mS/sec.Next in importance
is the Anadyr with a mean annual discharge'of
1660 mS/see,accordingto Sokolo.v[1964]. The
third largestriver is the Kuskokwim,which dischargesyearly 950 m•/sec at Crooked Creek.
The Kamchatka River at Kozyrevsk discharges
closeto 500 m•/seeannually [Lyubimova, 1961],
and the dischargeat its mouth is reported to
be closeto 1000 m•/see [Sokolov, 1964]. If we
consider the many ungaged rivers that carry
fresh water to the Bering Sea, it is reasonable
to assumethat the total mean annual discharge
their maximum in winter, the rainy season.
I)uring summer,many of the rivers in these
states carry only a negligibleamount of water.
In southernCalifornia,they tend to dry up entirely in their downstreamsections.The seasonal dischargevaries considerablyfrom one
year to the next, and the mean valuesshown
in Table 3 are generallyaccurateonly to 15%,
at the 95% confidence
level.
SECULARCHANGES OF RIVER I)ISCI-IARGE
Severalriver gagingstationsin the western
United
States and in British
Columbia have
been operated for more than half a century,
and it. is of interest to inquire into the long term
changesof river flow. Only the rivers where
activitiesby man have not markedly affected
natural runoff will be considered,To bring out
the salient features more clearly and to make
different stations comparableto each other, the
mean month-to-month
variation
was eliminated
from the records.The remainingmonthly
anomalieswere then divided by the monthly
standard
deviations.
The results are shown in
Figures 1 and 2. The outstandingfeature is the
asymmetryof the randomfluctuationsabout the
zero line. The positive peaks are generallymore
pronounced than the negative ones, and the
durations of above average river dischargeare
usually smaller than the durations of below
average discharge.The reason for this asyminto this sea exeee'ds that from the states of
merry is that runoff-producingrains are of
California, Oregon,and Washingtoncombined. shorter duratioi• than the intervening'dry peThe low surfacesalinitiesobservedin the Bering riods. The feature is most pronouncedfor CaliSea are largely due to the diluting effects of fornia rivers, where summerflow is negligible,
river runoff [Barneset al., 1935; Fleminq,1958; and least so for rivers in the Pacific northwest,
where summer rains and glacial snowmeltare
Uda, 1962].
The seasonal change of river discharge is more common.
summarizedin Table 3 for the most important
There is very little evidence that natural
rivers. The type of seasonalvariation depends streamflow has changed over the past half
on the meteorologicaland hydrographic con- century. Periods of above average runoff have
ditions in the drainage area. Where runoff is alternatedwith periodsof belowaveragerunoff,
controlledby snowand ice melt, the maximum but, taken over the entire available record
occursin summer.Where runoff is controlledby length, the fluctuations appear to have been
rainfall, the maximum is observedat the peak stationary.The mostextendedperio'dof drought
GUNNAR
5618
TABLE 3.
I. RODEN
Mean Monthly River Discharge (ms/see)into the Pacific Oceanand the Bering Sea
Jan. Feb. March
April
May
June
July
Aug.
304
164
9
67
299
103
4
28
Sept. Oct. Nov. Dec. Annual
California, Ocean Drainage
Sacramento River at Sacramento
Klamath River near Klamath
Eel River at Scotia
San Joaquin River at Vernalis
940 1183
1000 1217
465 581
132 185
1017
831
354
191
960
851
261
203
850
685
101
249
501
388
34
229
335
101
3
38
324
194
24
46
410
377
683
861
650
564
123 389
59 101
195
128
Oregon, Ocean Drainage
Rogue
RivernearAgness
312 352 260
229
i81
47
37
36
64 154 427 182
Umpqua River near Elkton
Illinois River near Agness
443
210
450
256
350
195
278
158
191
105
113
25
94
51
11
34
7
33
5
56
37
196 364
139 284
213
119
Nehalem River near Foss
170
182
120
78
39
1•
7
4
6
26
111 168
77
10260 13990 9646
5268
3468
2812 2794 2821
5510
173
199
377
915 1566
837
300
Columbia River, Ocean Drainage
Columbia River at The ])alles, Oregon
Willamette River at Wilsonville,
Oregon
Cowlitz River at Castle Rock,
Washington
Lewis River at Ariel, Washington
2748 3025
3592
5703
1628 1630
1186
942
742
455
234
370
352
299
324
344
289
150
76
64
123
415
259
200
198
169
165
159
114
56
35
48
85
170 214
134
Skagit River near Mount Vernon
Snohomish River near Monroe
Chehalis River at Porter
Queers River near Clearwater
446
410
279
214
432
359
250
171
592
294
16
45
323
156
11
25
269
138
13
39
387
178
39
98
482 507
305 364
176 239
170 224
467
300
120
116
Fraser River at Hope
905
839
773
1630
4818
7031
5605
3586
2418
1966 1553 1127
2688
Skeena River at Usk
205
185
146
328
1916
3010
1889
1016
751
880
Nass River at Aiyansh
Stikine River at Telegraph Creek
195
91
179
61
142
54
338
109
1128 212• 2017
681 1538 1071
1224
506
756
332
1055
279
Copper River near Chitinn
Susitna River near Gold Creek
Chulitna River near Talkeetna
Knik River near Palmer
160
42
38
23
130
34
31
19
2917
641
603
574
1501
425
383
342
595
174
150
114
Washington, Ocean Drainage
346
217
186
144
431
304
134
112
637
383
66
91
753
486
32
68
British Columbia, Ocean Drainage
624
344
941
464 227
182
91
821
416
Alaska, Pacific Ocean Drainage
124
29
27
13
147
37
32
19
871
388
243
56
2384
811
649
244
3310
709
730
814
282
75
62
63
200
52
44
31
1052
284
249
193
Bering Sea Drainage from Alaska and Siberia
Yukon River at Kaltag, Alaska
Kuskokwin River at Crooked Creek,
1321 1098
Alaska
915
930
7975 18329 13137 11767
9814
5560 2352 1492 6224
328
284
315
2080
2346
1302
325 250
300
350
385
Kamchatka River at Kozyrevsk,
450
2742
1100
2006
925
2403
600
500
400
599
457
1270
250 225
473
Siberia
1879
4
1881
1883
1885
1887
1889
1891
1893
1895
1897'
1899
1901
1903
0
-2
Fig. 1. Oldest record of river dischargeanomalies from long term monthly means in the
Far West. (• refers to the standard deviation.)
RIVER
DISCHARGE--NE
PACIFIC
AND BERING
SEA
5619
1905 07 09 II
13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65
I I I i i i i I i i I i I i i i i i i i i i i i I i i I i i i i i i I I i I I i I I I I I I I I i i I I I I' I I i I i i I_
6LARROYO
SECO
NEAR
SOLEDAD,
CALIF.
36ø16'50"N
121ø19'20"W
b
_,
,I
,
_•r•111
i i i i1 i i i i i I • • • • • i I ii i i i i I i i i • • i • • • • • i • i i i i i I i i
8
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I i I I I I i I I I I I I I I 1
EEL RIVER NEAR SCOTIA,CALIF. 40 ø 29' 30"N 124ø 05' 55"W
6
, -
DRAINAGEAREA 80,623 krn2
2
J•, .•
o
I
•,
. .,..I.,
Ju
,
.I,h,._
-2
,•b I0
i I i I I I I i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
•
UMPQUA
RIVER
NEAR
ELKTON,
ORE.43ø35'IO"N123
ø33'50"W
Z::
ß
'L DRAINAGE
AREA
9.õ$9
,m'
II
,• ,v I
I -.,]i
....__.:
._......,,.,,.,,,
l.,,...v.-.
:7
•
IIlil•
8
I I•11
jilli'll
I i i i i I i Iiiiii
liSlllill
Illill
•,,
i II
Ill
II I,
,•.,.
ill
. d.JLlJ,.
,,.,,,,,..,,.
,-,-,.
,,.,,,--,,,.,
i i i I i i i I i i I Ii
III
•
lllllllll
I iii
I I i i i i I i i i i i i I ill
I III
I1'l
COLUMBIARIVER AT THF' DALLES,ORE. ,4õø B6' IO"N I:::)1
ø I0' ,40" W
n•6--
[u
•
,
Ill
Sill
Sli
IJJ /
--I
DRAINAGE
AREA
613,830
km
2
A...•,l
,,,
k •,
n
i .,,
A
0
Z
-2
•
-4 I I I I I I I I '1I I I ! I I I I I I I I I I I I I I I I I I I I' I I I I I I I I I I I I I I I I I I I I I I I ! I I I/
ill
0 8 | I I I I !I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
a::
"•
'•_•
QUINAULT
RIVER AT QUINAULTLAKE,WASH. 47"27'30"N 123ø53'30" W
6--
DRAINAGE
AREA 684 km2
4
"'
.
>
•:
I I I I I I
I I
I
V
6 • I I I I , I , I I I , •
IIIIIII IIIIII
8
.
I I
I
iL
•
I
I
_,
I
I
I I
•
•
I
.k.
I • •
I
• • I • I I I
I I • • I
,
I
..•t
• I I I • I I I
ELWHA
RIVER
'NEAR'
PORT'ANGELES.
'WASHJ
;48ø03'20"N
' i23ø34
' 55"WI
III
II
II IIII
III
IIII
I
I
I
I
I
1
I
JI JI I I I I I t i I I I I I I I I I I I I I I I I I I I I I I I I
FRASERRIVER AT HOPE,B.C..CANADA49ø22' 50"N 121ø27'05"W
--i
• ! • I I I I I I I I I I I I I ! ! I I I I I I I I I I I I ! I I I I I
Fig. 2. River discharge
anomaliesfrom long term monthlymeansfor selectedstations.
was in the late 1920's and early 1930's,when
almost all rivers from California to Washington
carried less water than normal. Droughts of
The
absence of secular trends in natural
streamflowdoesnot necessarilyimply that the
mean dischargeof the river has remainedexactly constant,as the aboveexampleshows.In
the Pacific northwestand in the early 1960'sin fact, oscillationsof arbitrary period may be
California. Extended periodsof above average generatedby any variable that interactswith
streamflow (duration, not intensity) in the the environmentin a nonlinear way. This leads
Pacific northwest occurred in the mid-1890's
to the interestingconclusionthat long-period
and early 1950's,but in California they were fluctuations,or 'trends',may be generatedby
two distinctprocesses:
natural phenomenawith
principally in the late 1930'sand late 1950's.
lesser duration
occurred in the mid-1940's
in
5620
GUNNAR
I. RODEN
long time constants, or nonlinear interaction
with the medium [Munk, 1966].
The spectrum of river dischargeis seen to
consist of a few peaks superimposedon a
fluctuating level of random noise..The. most
prominent peak is centeredat a frequency of
i cpy. It is essentiallyof meteorologicalorigin
and reflectsthe annual occurrenceof the rainy
seasonand of snowmelt.Peaksof lessermagni-
SPECTRUM OF I•IVER DISCHA'RGE
The periodic and nonperiodicfluctuationsof
river dischargecan be studiedby meansof the
power spectrum, which representsthe distribu-
tion of meansquareamplitude,or energy,over
a rangeof frequencies.
The powerspectrumis
based on the notion of a stationaryrandom
variable that can be expressed
in terms of linear
superpositionof waves of independentamplitu'des and different frequencies.For such a
variable the power spectrumservesas a con-
tude occurfrequenfiyat 2 cpy and sometimes
at 3 cpy. The occurrenceof more than one
centrated measure of all the relevant informa-
peak in a spectrumis noteworthy,sincerainfall
and snowmelt,on the average,have only one
annualmaximumand minimum.Multiple peaks
can, however,be generatedby nonlinearinteractionamongfrequencies
of neighboring
spectral
ban'ds.The existenceof multiple peaks in the.
tion containedin the originaltime series.In the
spectrum can therefore be taken as an indication
more general case of a nonlinear or nonstation-
of the nonlinearityof the randomprocess.
The river dischargespectrum above 6. cpy
has not yet beenmeasured.In rivers receiving
their water from the meltingof snowand ice,
onemight expectto find a peak at a frequency
of I cycleper day, reflectingthe daytime thaw
and nighttime freeze,during the warm season.
ary random variable, the power spectrum
furnishesonly limited information.
The powerspectrafor the monthlymeanand
for the monthly extremeriver discharges
are
shown in Figures 3 and 4, respectively.The
computationswere made for the frequency
rangebetweenzeroand6 cyclesperyear (cpy),
with a resolutionof 0.125.cpy for stations
Large diurnal variations in streamflow due to
this melting' and freezing are known to occur
having'at least 15 years of continuousrecord at severalAlaskan stream gagingstations.Close
and 0.250 cpy for stationswith a shorter record to estuaries,streamflowis affectedby the ti'des,
length.
and undersuchconditions
thereshouldbe peaks
CALlFORNIA
OREGON
ARROYO SECO
WASHI NGTON
UMPQUA RIVER
NEARSOLEDAD 5
QUINAULT
NEAR
ELKTON
BRITISH
RIVER
I
'
ß !
'ø4E,
H
/I I I I I I •o
3 •
MERCED RIVER
NEARFOSS
,o
•
103
5
I
I
I
'1
I
/I
ELWHA RIVER
FRASER RIVER
3 NEAR
PORT
ANGELES
RIVER
RIVER
SKEENA
AT PUYALLUP
6
,o'
z
i
i
i
i
,-;,I'
,½ '•'
/I I I I I I
RIVER
YUKON
AT USK
RIVER
AT KALTAG
..
,½.:.
'?
,o"' ' ' , ' ' •'•,
•o
i
ATCROOKED
CREEK
.
'
I
KUSKOKWIM RIVER
ATHOPE
%•'
PUYALLUP
AT THEDALLES
I I I I I.I
'•
•o/I I I I I I •ø5 I I I
COLUMBIA
NEARSCOTIA
I
I0
loll I I I I I io••
RIVER
ALASKA
COPPER
RIVER
NEAR CHITINA
•øa• •
NEHALEM RIVER
NEAR
POHONO
BRIDGE4
EEL
!03
COLUMBIA
NANAIMO RIVER
NEAR EXTENSION
ATQUINAULT
LAKE
4
6
0
Z
4
6
,0
Z
4
6
0
2
4
6
FREeUE,CY, cpy
Fig. 3. Spectrum of monthly mean river discharge.The arrows refer to the 95% confidence
limits.
RIVER
DISCItARGE--NE
PACIFIC
CALIFORNIA
ARROYO
SECO
NEAR
SOLEDAD
UMPQUA
MINIMUM
6
RIVER
MAXIMUM
5621
ELKTON
QUINAULT
MINIMUM
5
RIVER
MAXIMUM
AT QUINAULT
3
LAKE
MINIMUM
i'o/;,i,,1 ,o,..... "ø1'.... I
102111111 ]
MAXIMUM
I
10511 I I I I I
4
COLUMBIA
MINIMUM
6
/I
0
I I I I I
2
4
RIVER
AT THE
MAXIMUM
I,o,[.]
2
,o.t.:,.
o411 I I I I I 102/• m • • •
EEL
RIVER
AT
SCOTIA
0
SEA
WASH INGTON
NEAR
4
ß
;7
AND BERING
OREGON
6
i0•/I
DALLES
I I I I I I0•1
ELWHA
MINIMUM
RIVER
MAXIMUM
NEAR
3
I I I I I
PORT
ANGELES
MINIMUM
4,l .... JIO
I II I,I1
,o.F,','
o
2
4
6
0
2
4
6
/I 2I I 4I I 6I I00/ I 2I "'14I I 6I
0
FREQUENCY.cpy
Fig. 4. Spectrum of monthly floods and monthly droughts. The arrows refer to the 95%
confidence limits.
corresponding
to the dominantti'dalfrequencies.
ß = i--1
Thus, the shapeof the river dischargespectrum
is likely to be influencedby meteorological,
radißattohal,and,in someplaces,astronomicaleventsß the a, are constantsand the x, are independent
random variables with zero mean Xoand variance
8•".As the numberof observations
N -• oo,and
8," -• 0, the distribution tends to a Gaussian
A knowledgeof the probabilitiesof river disdis:tributionwith variance 8' -- Z8•". Writing
charge extremesis of considerableimportance
f = Xo/8,we have
in the analysisof floodsand droughts.Suppose
we have a long seriesof daily observationsand
I•ROBABILITIES OF RIVER DISCHARaE EXTREMES
p(•) = (•,r•')-'"' exr (-«I •)
we are interested in extremes on a time scale of
(•)
I month. For each month of the record we may The above approach,which is linear in both a•
obtain the highestand lowest daily dischargesß and x,, leads to a symmetricalprobability disThis yields two new time series, one for the tribution, which is quite unsuitedfor studying
monthly floods and one for the monthly river dischargeextremes.
Representation1 can be generalizedto indroughts,the terms 'fioo'd'and 'drought' being
clu'denonlineareffectsand skewhess.
According
used here in a generalizedsenseto denote river
dischargemaxima and miniinc. Associatedwith to Longuet-Higgins [1963a], a random varias the sum of a large
these two sets of extremesare probability dis- able can be considered
number
of
independent
components
of the type
tributions,whichmay be characterized
in terms
of the moments or cumulants.
Examples of observed river discharge extremesare shownin Table 4 for stationshaving
at least 30 years of continuousrecord.The outstanding features are that the lowest values
frequently approach zero (no flow) and that
ß=
i-1
Z
i-1
...
where the a• and a•j are constantsand the x•
are in'dependentrandom variables with zero
mean and variance
The probability distributionscan be obtained
closerto the lowest than to the highest values. from a knowledgeof the cumulants of x. The
This suggeststhat the probability functionsare results are essentially Gaussian distributions
asymmetrican'dpositivelyskew.It is of interest multiplied by a sequenceof Hermite polynomto.inquireinto the nature of variablespossessing ials.
For the probability distributionwe have
•ch skewhess.
the mean
values of the extremes
tend
to. be.
As is well known, many random variablescan
be consi'dered
as the sum of a large number of
independentcomponents.
Thus, in
•(x) = (2•-•)-'/•' exp(-«#).[1 + •'XaHa
+ (•X?.o + -•,X•3 + '--],
(4)
5622
GUNNAR
I. RODEN
,
••o•o
•oo••
•oooooo
.•oo•o•
• .....
•oooooo
• ............
•
•
........
•oooooo
•
....
•oooooo
•oooooo
•
••'
•oooooo
.•
Too
'•
'••
..
oo
RIVER
DISCItARGE--NE
PACIFIC
AND
BERING
SEA
5623
and for the cumulativeprobability distribution
we have
P(x) = (2•r&)
-z/•
exp(--«•) d•
-- (2•r$)
-'/• exp(--«]"
+ (•x•
+ •x•)
+ ...]
(5)
In the above expressionsthe Hermite polynomialsare givenby
H• : I • -- 3I
H4:
]4_ 6i• + 3
(6)
H,:I
'-
10I•+1•I
H•:I
•-- 1•I•+4•I •-- 1•
and the constantsX• and X, expressthe skewhess
and kurtosis,respectively.• and X• are defined
by
x•:
•/•2
•
x•:
•/•J
(7)
whereK•, K•, and K• are the secon'd,
third, and
fourth cumulants of x.
The cumulative probability is a monotoni-
cally increasingfunctionof the argument• =
(x -- xo)/3 and, hence,can possesso•y one
maximum
and one minimum.
In the Gaussian
case,these extremesare not reachedfor •ite
values of •. To see what happens in the nonGaussiancase,we equate (4) to zero and find to
the first approximation
[• + {x•(r - al)]:
0
(s)
Solving this cubic equation, we obtain, for
•:Xo+•
-•+
•-I
-•--I3,3a
/• --1)1/211/3
}
The limiting value e is the argument at which
the cumulativeprobability is zero. In terms of
river dischargeextremes,this satisfiesthe condition of an empty river bed or no flow. We
have thus the interestingresult that the probability functionsof a nonlinearvariable of the
type (3) are limited on the left.
GUNNAR
5624
I. RODEN
A differentapproachto estimatethe probabilities of extreme values that are bounded was
DURATION OF •FLOODSAND DROUGHTS
The duration of abnormal streamflow is im-
presented
by Gumbel[1958]. Making cer•in portant in understanding
the salinityand nutriassumptions
with regardto the differentiability ent fluctuations in tidal estuaries and nearshore
of the cumulativeprobabilityin the vicinity of waters,as well as in many engineeringproblems.
the lowerlimit, he obtaine'd
In general, the duration of anomalousstream-
F(x) -- exp[--((x -- e*)/(v-- e*))•]
(10)
flow dependson the occurrenceof anomalous
weather con'ditionsin the drainage area. Both
eters,k, v, ande*. The parameter
k is relatedto
past and current meteorological
eventsare of
importance. Severe flooding in Pacific coast
the skewnessX, by
drainage basins can be expected when warm
The distribution is determinedby the param-
•k3 ---
r(• •- 3/k) -- 3r(1 •- 2/k)r(1 •- •/k) 4- 2r•(• •- •/k)
Jr(1 + 2/k) - r(1 + l/k)] •/2
and the characteristicvalue v is givenby
v = xo+ $[1 -- r(1 4- 1/k)]/[r(1 4- 2/k)
- r(1 4- l/k)] 1/•
(12)
and the limiting.value.e*by
e* = xo- air(1 +
+
-- r•(1 •- l/k)] 1/•
(13)
x• being the samplemean and 3 the sample
standard deviation.
It is of interestto comparethe limitingvalues
e and e* for fixed values of skewness.Assuming
(11)
moist air from subtropical latitudes invades
areaspreviouslycoveredby abundantsnowfall
from subarcticstorms.Frequently this occurs
in the wake of strongpolar outbreaksthat displace the track of cyclonicstorms several degreesof latitude southwardof their normal position. The two largest recordedCalifornia floods
(January 1862 [McGlashan and Briggs, 1937;
Roden, 1966a] and December1964 [Rantz and
Moore, 1965]) were of this origin. Under such
conditions,the river dischargeincreasesvery
rapidly with time, reachesa peak, and then
decreases
more slowly toward the prefioodrate.
The time interval during which the river flow
in (9) and (13) that samplemeanis zeroand remains above normal can be estimated from an
the samplestandarddeviationis unity,we ob- inspectionof dischargehydrographs;in caseof
tain the resultsgivenin Table 5. It is seenthat large floods, typical value lie between 1 and
the limiting values obtainedfrom the first 3 weeks.
approximation
of Longuet-Higgins'
distribution Droughts in Pacific coast drainage basins
(5) are alwaysslightlylowerthan the values frequently occur when a cold and dry anticyobtained from Gumbel's distribution (10).
The agreementbetweenthe observedand TABLE 5. Lower limit of the Variate f -•
theoreticalcumulativeprobabilitiesof river dis- (x - xo)/• at Which the Cumulative Probability
Distribution
Is Zero
chargeextremesare shownin Figure5. If the
sample
skewness
is small(X• _<1), eithertheory
The parameterX• refersto the sampleskewhess.
fits the observationswell. If the sample skew-
nessis moderate(1,5 • _<3), the two theories
followthe observations
closelyonly for small
valuesof the variate x/3. For large valuesof
x/3 Gumbel'sdistribution(10) agreesbetter
with the observedvaluesthan doesLonguet-Hig-
gins'distribution
(5). If the sampleskewness
is
large (• • 3) no goodagreement
is obtained
betweentheoryand observation.
The observed
limiting values of the variate x/3 for the
monthly droughtsseem to fall betweenthe
theoreticallyderivedcurves.
•
0.1
0.2
0.3
0.5
1.0
1.5
2.0
2.5
3.0
Longuet-Higgins
[1963a]
-4.18
-3.42
-3.08
-2.71
-2.35
-2.20
-2.10
--2.04
--2.00
Gumbel
[1958]
--2.94
--2.64
--2.44
--2.10
--1.53
--1.20
--1.00
--0.78
--0.76
RIVER DISCI-IARGE--NE
PACIFIC AND BERING
ARROYO SECO NEAR SOLEDAD, CALIE
•0.0
'•1 '• ' I • '1' '1)"'
''1'' ' I • •1' 'l'':J•10'0
6.0
FLOODS •l•x'/
DROUGHTS
•- 6.0
4.0
X'3.48y•
X=2.00
•-4.0
2.0
B •-•.o
1.0
0.6
•
0.4
-
'
-0.4
•
0.2
-o.•
L
•tl,d•
0.!
I , •1, ,I•
•
EEL RIVER
I0.0
'•'1'''
I'
•''1'''
A/I,,,
-
X= 2.
4.0
-01
AT SCOTIA, CALIF.
'1''1''•
FLOODS
6.0
, ,I
I'
-I0.0
'1''1'
DROUGHTS'
-6.0
X= 1.87
-
_
-4.0
.
-2.0
2.0
AB-
•.o
-I.0
0.6
-0.6
0.4
-0.4
illill
Iiillill!
!ii
0.2
0.!
COLUMBIA
m.o
.'•'1'''
6.0
-
0.1
-0.1
-
I' '1'•i''• -I0'0
DROUGHTS
X=
0.90
•-6.0
-4.0
I
-'
_
i
i
I
-0.4
I
ELWHA RIVER NEAR PORT ANGELES, WASH.
I0.0
6.0
4.0
2.0
IJ'l'•'
I'
- FLOODS
'1•'1 ''. A ,.•
''1'''
i j ,1•,1,,-I0.0
DROUGHTS
- -6.o
- • =1.21
•
•= 0.74
-
- -4.o
B-
_
B' -2.o
_
_
!.0
-I.0
0.6
-0.6
0.4
,
0.2
• •.• I,
0.1
0.9
West
occurred between
1928 and
1931,when almostall rivers from California to
meteorological
aspectsof the early part of this
period were discusse'd
by Bowie [1929]. Other
periods of extendeddrought occurredin 1889,
1940-1941,1943-1944in the Pacificnorthwest,
X109- ,•_o.6
_i
--
The observed extreme durations of abnormal
river dischargeare given in Table 6. Only stations that are not markedly affected by upstream diversionand regulation an'd that have
at least 50 years of continuousrecord are considered. The outstanding feature is •hat the
extreme durations of below average river dischargegenerallyexceedthoseof aboveaverage
discharge.This may be ascribedto the unequal
persistenceof cyclones and anticyclones, as
discussedabove. The most conspicuous
drought
Washington carried less water than usual. The
-I.0
0.6
0.4
''1'''
cipitation over extended periods an'd the absenceof snowmelt at higher elevationsreduce
the amountof water carriedby rivers and lead
to droughts.
in the Far
-2.0
hO -
0.2
J' '1''1'':
5625
-0.:>
AT THE DALLES, ORE.
FLOODSA•
4.0
•.o
RIVER
SEA
0.5
-0.4
-0.2
•,
0.1
I0-• 10-3 10'4 0.9
-0. I
0.5
0.1
i0-• 10-3 10'4
Fig. 5. Cumulative probabilities of monthly
floods and monthly droughts. The dots refer to
the observations; curve A to Gumbel's distribution; curve B to Longuet-Higgins' distribution
and in 1946-1947 and 1960-61 in California. Pro-
longedperiodsof above averagestreamflowin
the Pacific northwest were 1893-1894, 19201922, 1933-1934 and 1950-1951, and in California they were observedin 1917, 1937-1938,
1940-1941, 1955-1956, and 1957-1958.
It is of interest to investigatethe probability
distributions
for
the
durations
of
abnormal
streamflow.Ideally, long and continuoushydrograph recordswould be needed.Such records
of sufficientlength do not exist at present,however. Instead, it is necessaryto use discrete
records of observations and to draw conclusions
from them. Considera long record from which
the mean annual
variation
and the trend
have
been eliminated.The remaining fluctuationsof
above and below average river discharge can
then be regarded as consistingof a series of
(see text). • refers to the standard deviation and
alternating positiveand negativepulsesof variX refers to the skewness.
able duration, such as depicted in Figure 2.
Denoting the number of positive pulses by
clonesettlesover the plateaustates.If the anti- N (+), the numberof negativepulsesby N (--),
cycloneis of sufficientdimensions,
easterlywinds the duration of the individual positive pulses
are predominantin the entire region from Cali- by D(+)• and the duration of the individual
fornia to Washington,and the moistureladen negativepulsesby D(--)•, and the total record
oceanic storm systems are deflected toward length by T, we have the obviousrelations
southeasternAlaska. Contrary to flood-producing cyclonicstorms,which rarely last more than
i
i
a few days at a given place, anticyclonesmay
last for several weeks at a time. Lack of pre=
•. D(-•),
d-•. D(--),
= T (14)
5626
GUNNAR I. RODEN
TABLE 6.
ObservedExtreme Durations, in Months, of Above and Below Normal River Discharge
Onlyriversthat arenotmarkedly
affected
by regulations
andthat haveat least50yearsof continuous
recordareconsidered.
ß
Above Average
Extreme
Dates
Period
Columbia River at The Dalles, Oregon
20
Below Average
Extreme
Dates
1888-191S
May 1893 to Dec. 1894
16
Oct. 1888 to Jan. 1890
Period 191 •-1989
Arroyo Seco near Soledad, California
Merced River at Pohono Bridge, California
Eel River near Scotia, California
Umpqua River near Elkton, Oregon
Columbia River at The Dalles, Oregon
Clackareas River at Estacada, Oregon
Quinault River at Quinault Lake, Washington
Ehvha River near Port Angeles, Washington
Puyallup River at Puyallup, Washington
Fraser River at Hope, British Columbia
9
Feb. 1938 to Oct. 1938
52
Aug. 1927 to Novq 1931
12
9
Dec. 1937 to Nov. 1938
Feb. 1917 to Oct. 1917
44
21
June 1928 to Jan. 1932
Jan. 1930 to Sept. 1931
11
12
Aug. 1920 to June 1921
June 1933 to May 1934
24
44
March 1930 to Feb. 1932
July 1928 to Feb. 1932
13
Dec. 1920 to Dec. 1921
12
March 1930 to Feb. 1931
9
11
13
Mas; 1933 to Jan. 1934
June 1933 to April 1934
May 1933 to May 1934
10
20
13
Feb. 1923 to Nov. 1923
July 1928 to Jan. 1930
March 1930 to March 1931
20
June 1920 to Jan. 1922
17
(Aug. 1928 to Dec. 1929
(May 1930 to Sept. 1931
Period 19•0-1965
Arroyo Seconear Soledad,California
Merced River at Pohono Bridge, California
Eel River near Scotia, California
11
11
Dec. 1940 to Oct. 1941
Dec. 1955 to Oct. 1956
27 Dec.
1946
toFeb.
1949
27
21
Nov. 1959 to Jan. 1962
May 1960 to Jan. 1962
17
May 1957 to Sept. 1958
14
Jan. 1946 to Feb. 1947
Umpqua River near Elkton, Oregon
12
Nov. 1955 to Oct. 1956
17
April 1940 to Aug. 1941
Columbia River at The Dalles, Oregon
Clackareas River at Estacada, Oregon
12
June 1950 to May 1951
28
Sept. 1943 to Dec. 1945
13
9
13
17
13
Feb. 1950 to Feb. 1951
June 1950 to Feb. 1951
Feb. 1950 to Feb. 1951
Nov. 1949 to March 1951
June 1964 to June 1965
17
18
20
17
17
April
May
May
April
Dec.
Quinault River at Quinault Lake, Washington
Ehvha River near Port Angeles, Washington
Puyallup River at Puyallup, Washington
Fraser River at Hope, British Columbia
1940 to Aug. 1941
1943 to Oct. 1944
1943 to Dec. 1944
1940 to Aug. 1941
1944 to April 1946
We can now determine the mean durations of
extreme value consi'derationsto each of them.
the positiveand negativepulses
We also note that the durationsof the positive
and negativepulseshave a lower limit. For a
m(+) = •
continuousvariable the limit approacheszero,
'
m(--) = •
i
(15) and for a discretevariable it is equal to the
D(--),/N(--)
For a Gaussianvariable,m(+) -- m(--), and
sampling
interval.For a finiterecord,therealso
existsan upperlimit, equalto the total period
of observation. The information on the duration
the mean durationcanbe determinedby theories we can extract from a discrete and finite record
developed
by Rice [1958] and Longuet-Higgins is thus limited to time scales between the sam[1962], which fit the observations
rather well pling intervaland the total recordlength,in
[Roden,1966a,b]. For the slightlynon-Gaus- our casebetween1 month and roughly50 years.
sian case,Longuet-Higgins[1963b] presented Applying Gumbel's [1958] theory of a
a theory for estimatingthe duration of positive boundedvariable,we obtainfor the cumulative
and negative pulses taken together. In the probability distributionsof the durationsof
present case, it is more interestingto deal aboveand belowaverageriver di•scharge
separatelywith the durationsof posi'tiveand
negativepulses,so as to bring out more clearly
the persistence
of floodsand droughts.
Rememberingthat the originaltime seriesis
made up of an alternatingsequenceof positive
and negativepulses,we regardthe durationof
the positivepulsesas maxima and the duration respectively.The parametersk(+), v (+),
of negative pulsesas minima, and we apply •*(+) etc. are related to the samplemean,
F(+)
=exp
--'v(+)
--e*(+) (16)
[-- v(--)--- .*(-))"-']
F(--)= exp
RIVER DISCHARGE--NE
PACIFIC AND BERING SEA
samplevariance,and sampleskewness
of the
duration sequenceof positiveD(+),
5627
ARROYOSECO NEAR SOLEDAD, CALIF.
IOO
or nega-
60
40
tive D(--) pulses,and can be easilyobtained
2o
from an inspectionof (11), (12), and (13).
IO
The agreementbetweenthe observe'd
and
6
calculatedcumulativeprobabilitiesfor the du-
4
ration of abnormal streamflow is shown in
Figure6. The ordinategivesthe durationin
months,and the abscissa
denotesthe probabilityof exceedances.
In general,
thereisa pleasing agreement
betweentheoryandobservation.
Thisisparticularlytruefor largevaluesof duration, indicatingthat Gumbel'sdistributioncan
be used asymptoticallyto estimateduration
EEL RIVER AT SCOTIA, CALIF.
IO0•:
[ [,1•,, i ' 'i''1'':
),
40
20
''1'''
I ' 'i''1''• I00
-:
-
20
--.
I0
.
I0
probabilities.
I:[ELATIONBETWEENSALINITY AND
RIVE2 DISCHARGE FLUCTUATIONS
i/,/_.• • I,, , 1 , ,I, ,!,,
COLUMBIA
The fresh water that reachesthe seais gradu-
IOOl:'''l'''
ally mixedinto the surrounding
salinewaters.
•
3 36
'"'..............
1
aT THE DALLES, ORE.
RIVER
I:'''1'''
1' 'l''l'•
I' '1''1'': I00
)•= 1.45
--_60
--dO
The effectis most stronglyfelt at the surface,
20
becausefresh water is of lower d•nsity than
seawateris.It is,therefore,reasonable
to expect
*
--
I0
IO
•
lhat fluctuationsin river dischargewill be re-
4
flectedby corresponding
changes
of the surface
salinityin the vicinityof the river mouth.The
relationship
betweenthe variables
will obviously
dependonthe strengthof thefreshwatersource
(the magnitude
of the river) and the distance
of the salinityrecordingstationfrom the river
iIJ:,,IH,
ELWHA RIVER NEAR PORT ANGELES, WASH.
I00
60
40
6
4
2
I
FRASER RIV'ER AT HOPE, B.C.
IO0-,,,I,,,
I [ ,I,,I
60 - • =1.69
_
The frequencyresponsebetweenthe non-
,'-
]''1'''
I ' '1''1 ''-I00
)• =2.28
-
.
-
.
' 60
40-.
periodic
salinityandriverdischarge
fluctuations
bles.The coherence
spectrumis shownin Figure
7 for the frequencyrangebetween0 and 6 cpy.
It is seenthat significantcoherenceexistsbe-
IO
I0
of opposingcurrents.
whereK(f) is the response
as a functionof the
frequency
], E,,(f) is the powerspectraldensity of nonperiodic
salinityfluctuations,
E•(f)
is that of river discharge
fluctuations,
and
is the coherence
spectrumbetweenthesevaria-
'''1'''
I''i''1''
'l''l''
I''1''
i
20
hydrographic
and meteorological
conditions
in
the region.In addition,thebehavior
of thefresh
waterplumein the oceanwill be influenced
by
lhe deflectingforce of the earth'srotation.For
westward-flowing
rivers, the deflectionof the
plumewill be towardthe north,in the absence
K(J)= [E,,(J)/E,.,.(J)]'/2'C,,.(J)
(17)
!
,
mouth. It will also dependon the prevailing
can be calculated from
[
1, ,I,,I,L
.
.
I0
_-
I0
6
--
6
4
-
4
2
-
I
0.9
2
I
-
0.5
o.I
IO-2 I0-3 io-I 0.9
I
0,,5
o.I
10-210-310-1
Fig. 6. Cumulative probabilities for the duration of above and below normal monthly river
discharge.The dots refer to the observationsand
the solid curve was derived from theoretical
siderations (see .text). X refers to skewhess.
con-
5628
GUNNAR
I. RODEN
FRANCISCO
(FORT POINT) SALINITY-SACRAMENTO
RIVER DISCHARGEAT SACRAMENTO,
CAL.
I0
o+'
0.4
.....
0
I.Oj
......
I
I
I
I
I
I ' I
I
I
I
I
•
-- 270
i
I
190
,.ß.ß
,ßßßß
Iß
.......... diS0
I
I
I
04'CITY
I I NEAR
I ICRESCENT
I ICITY,
127o
CRESCENT
SALINITY-SMITH
iiER
DISCHARGE
CAL.
O6 ßßøßßßßøøøßßßßßßßßß
•
tween salinity and runoff at almostall frequencies. The phase of 180ø indicates an inverse
relationshipbetween the variables,as expected.
ResponsefactorsK(]) for a few representative station pairs are given in Table 7. Like the
coherence,the responseis largely independent
of frequency,for the time scalesconsidered
hereß
The responsefactorscan be usedto estimatethe
effect of floodsand droughts on the salinity at
o•
0
1
J
I
I
t-......................
J
/
I
I
I
I
•,eo
I
190
DEPARTURE
BAYSALINITY-NANAIMO
RIVER
DISCHARGE
NEAREXTENSION,
BC.
08
0.6''''*
04
o.•
0
ß
ß
ß
ß"•:
..
ß
j
*
2
i
I
i
i
i
j27o
coastal stations. The agreement between ob-
J-"
' 't ßßß..ßßß•- ß' ,"'" ' t ß*"1•eø
3
4
5
6
0I•
90
served and predicted salinity departuresfrom
FREQUENCY,
(cpy)
normal
due •to abnormal
runoff
is shown
in
Fig. 7. Coherence and phase between nonseasonal fluctuations of coastal surface salinity
Table 8. In most instances,the differencebe-
and river discharge.The dashed line refers to the
95% confidence limit.
than 1%o.
tween observed and estimated
salinities is less
TABLE 7. FrequencyResponsebetweenNonperiodicSalinity and River Discharge
Fluctuationsin per mille/m ssec-•
Fort Point Salinity
Sacramento
Frequency
River
CresentCity Salinity
at
Sacramento, Calif.
0
I
2
3
4
5
6
Mean Response
Departure Bay Salinity
Smith River near
CresentCity, Calif.
-0.0075
-0.0052
-0.0052
-0.0205
-0.0167
-0.0161
-0.0050
-0.0163
-0.0057
-0.0055
-0.0179
-0.0171
Nanaimo
River
near
Extension, B.C.
-0.0390
-0.0429
-0.0113
-0.0349
-0.0055
-0.0142
-0.0281
-0.0281
-0.0229
-0.0057
-0.0169
-0.0287
TABLE 8. Observedand PredictedHighestand LowestSalinity Anomaliesfrom Long
Term Monthly Means
The predictionsare basedon the highestand lowestobservedmonthly river dischargeanomalies,using
the mean responsefactors given in Table 7.
,
Fort Point Salinity
Sacramento
River
at
Sacramento, Calif.
CrescentCity Salinity
Departure Bay Salinity
Smith River near
Nanairno River near
CrescentCity, Calif.
Extension,B.C.
Period of joint records
Oct. 1948 to Sept. 1964 Oct. 1934 to Sept. 1947
Highest dischargeanomaly,
mS/sec
+1231
+276
Date
Observedsalinity anomaly,
•o
Predicted salinity anomaly,
•o
Lowest dischargeanomaly,
mS/sec
Date
Observedsalinity anomaly,
%0
Predicted salinity anomaly,
•oo
Dec. 1950
March
1938
Oct. 1948 to Septß1964
+97
Jan. 1953
-7.3
-4.7
-1.0
-7.1
-4.7
-2.8
--779
-- 168
Feb. 1949
Dec. 1936
-59
Jan. 1948
+3.5
+2.5
+2.4
+4.4
+2.8
+1.7
RIVER
Acknowledgments. I
DISCHARGE--NE
PACIFIC
am indebted to C. A.
Barnes,L. K. Coachman,R. t{. Fleming, M. G.
Gross, and M. Rattray for discussionand advice.
The river dischargerecordswere kindly furnished
by the Water ResourcesDivision of the U, S. Geo-
logical Survey in Alaska, California, Oregon, and
Washington, and by the Canadian Water Resources Branch of the Department of Energy,
Mines, and Resources.
The researchreported herein was supportedby
the
Office of
Naval
Research
under
contract
477(37), project Ni• 083 012.
Canada: Department o• Energy, Mines and Resources, Ottawa.
British Columbia. Daily dischargesare published in Surface Water Data ]or Pacific Drainage
since 1911. Monthly summaries occur in Water
Supply Papers 30, 51, 67, 80, and 94.
Geological Survey, Wash-
Alaska. Daily discharges
are publishedannually
in Surface Water Records ]or Alaska. Monthly
summaries of pre-1950 data occur in Water Supply Paper 1372 and of 1951-1960 data in Wat•er
Supply Paper 1740.
California. Daily dischargesare published an-
nually in Sur]ace Water Records Cali]ornia, parts
I and 2 (prior to 1960 parts 11A and 11B).
Monthly summaries of pre-1950 data occur in
Water Supply Palpers 1315A and 1315B and of
1951-1960 data in Water Supply Paper 1735.
Oregon. Daily discharges are published annually in Sur]ace Water Records Oregon (prior
to 1960 parts 12 and 14). Monthly summariesof
pre-1950 data occur in Water •pply
Papers
1316 a•d
1318 and of 1951-1960
SEA
5629
Duxburg, A. C., B. A. Morse, and N, McGary, The
Columbia
River
effluent
and its distribution
at
sea, Univ. Wash., Dept. Oceanog. Tech. Rept.
156, 105 pp., 1966.
Fleming, 1•. H., Notes concerning the halocline
in the northeastern Pacific Ocean, J. Ma•ine
Res., 17, 158-173, 1958.
Gumbel, E. J., Statistics o/ Extremes, 375 pp.,
Columbia University Press, New York, 1958.
Longuet-Higgins, M. S.., Intervals between zeros
of a random function, Phil. Trans. Roy.. Soc.,
London,A,.254,557-559,1962.
DATA RESOURCES
United States : U.S.
ington, D. C.
AND BERING
data
in
Water
Supply Papers 1736 and 1738.
Washington. Daily discharges are published
annually in Surface Water Records Washington
(prior to 1960 parts 12 and 14). Monthly summaries of pre-1950 data occur in Water Supply
Papers 1316 and 1318 and of 1951-1960 data in
Water Supply Papers 1736 and 1738.
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