HomeMy WebLinkAboutSUS218RESPONSE TO COtn1ENTS BY HARZA-EBASCO
SUSITNA JOINT VENTURE ON AEIDC'S REPORT
ENTITLED "STREAM FLOW AND TEl.fPERATURE
MODELING IN THE SUSITNA BASIN, ALASKA."
TK
1425
.S8
S9
no.218
Response to Comments by Harza-Ebasco Susitna Joint Venture
on AEIDC's Report Entitled "Stream Flow and Temperature
Modeling in the Susitna Basin, Alaska"
This document is numbered "SUS 218", and is the edition also containing the original
comments.
Alaska Resources Library and Information Services (ARLIS) is providing this table of contents.
Table of Contents
Part 1 Harza-Ebasco Susitna Joint Venture comments on AEIDC report "Stream flow
and temperature modeling in the Susitna Basin, Alaska".
Cover letter to William Wilson of AEIDC from John R. Bizer, Lead Aquatic
Ecologist at Harza-Ebasco.
Comments on AEIDC stream flow and temperature model.
o General comments.
o Specific comments.
Part 2 AEIDC's response to general comments.
Response to general comments.
Response to specific comments.
Attachment 1 - SNTEMP mathematical model description.
Attachment 2 - Heat flux components for average mainstem Susitna conditions.
Attachment 3 - Weather wizard data.
Attachment 4 - Daily Indian River temperatures versus Devil Canyon air temperatures.
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Dr. William Wilson
Au gust 9 , 1983
4.3 .1.4
Arctic Environmental Information & Data Center
707 A Street
Anchorage, Alaska 99501
Subject: Susitna Hydroelectric Project
Dear Bill:
Report on Stream Flow and Temperature
Modeling i n the Susitna Basin, Alaska.
\\' \\
Attached are our comments on the AEIDC report entitled "Stream Flow and
Temperature Modeling in the Susitna Uasin , Alaska." In general, we
fe l t tha t the report is well writ ten and provides a good documentatio n
of the mode l ing effort of AEIDC .
We have several specific comments t o which we wo uld like your r esponses.
If it is efH.cient, please revise the draft report where appr opriate in
response to t hese comments. However, many of the comments may be mo~e
appropriately addressed in a separate memo r a n dum .
When the repo rt is final , please s ubmit twenty five copies wh i ch we will
distribute to the appr opriate entities .
Sincerely,
~1 I
/:: / /$<"") .. :_'-
John R. Bizer, Pn:D .
Aquatic Ec o logist
JRB:baj
cc: G. Lawley, H-E
E. Marchigiani, APA
attachments
CmiMENTS ON AEIDC STREA.M FLOW AID TEMPERATURE MODEL
General Comme nts:
Generally we found the AEIDC stream fl ow and t empe ratur ~ model to be a
we !.l written docu men t. It provide s a thorough and .theoreti cal approa ch
to the determination of s tream t em p e r a tures . Howeve r, since the report
was wri t ten for a t echnical audience , we would have preferred a more
detailed description of the vario us submodels rather than a reference to
Theurer et al. 1983.
We question whether AEIDC's use of thre e meth ods to determine subba sin
flow contributions was wo rth the expenditure. \~il e we do not objec t to
assessing the relative differences a mongs t methods , we wonder why the
computations were made for all subbasins f o r each method. Th e time vari-
a t ion of these contributions may be importa nt and has not been consid ered .
We also questi o n whether the technique to d etermine t he distributed fl ow
tempera tures is mo re sophisticated than i s necessary . We would recommend
using availa ble h is t o rical tributary t empe rature data and perhap s corre-
lating this wi th a ir t empera tures t o gener at e a tribut ary t empe ra tu re
time series. We doubt tha t errors in estima t i n g tributary temperatures
will have a significant effect on ma i nstem t em pera tures.
We would like to see a daily p r e dic tio n of main s tem t emp e r a ture s, as we
feel month ly t empe r a ture s may be too coarse to proper ly assess project
impac ts. This will not o nly be n ecessar y for the instream ice s tudy b ut
has also been reques ted b y the r esource agencies . We will need to exam-
ine the effect of high, medium a n d low flows o n s tream temperatures.
Water years 1981, 1982 and 1 9 74 have tentatively been selected for this
purpose. We would like t o see sensitivity t ests using vario u s me teoro-
logical sequen ces with each of the fl ow conditio ns.
We do c o11plime nt AEIDC for incorporaLing a shading factor and ac counting
for tribu ta ry inflow in t h e model. These a re two significant i mprove ment s
over the HEATSIM model.
p.l Par. 2
p .S Par.l &
p.ll Par. 2
p.9 Fig. 3
p.lO,Bo ttom
p. 18, Par. 1
p.l9,Bottom
p.20
p .21
p.24,Par 4 ·
SPECIFIC COMMENTS
Note that ADF&G and USB~S have undertaken studies of
temperature effects on salmonid egg incubation .
Since subjectiveness is involved in aieal
w~ighting (method 2), is using this me tho d
1~ .. -t~a? using· the drain~ge_ area method?
precipitation
more approp~ia~e ~-
Since Met h od (2 ) yields a higher Watana discharge, we
r ~commend this method not be used at this time. The
nigh dis charg e implies additional economic benefits .
For ec(',JOmic runs, we need to be conservative. However, a
final decision -on the selected m~·thod will be reade by
H/E in the near future. -...
Mean annual water yield for several subbasins appears
to be greater than the mean annual precipitation (Tsusena,
Fog . Devil, Chin-Chee, Portage).
Calculated Cw for Method (1) is 0.5104. Acres used 0.515.
Why is there a difference? \vere areas replanimetered?
We s u gges t using s'olar radiation measurements when avail-
able r a ther than calculated values. We would also like
to see daily comparis6ns of observed versus computed
solar radiation. Please provide descriptions t•f the six
SNTEMP s ubmodels.
More discussion on heat flux would be helpful. Statements
regarding the relative importance of h ea t inputs and out-
puts should be made. Please provide all heat sources and
sin ~s considered.
In Eq. (9), how was Te(Equilibrium t e mperature) es time t ed?
What are the parameter values of K1 and K2?
There are potential problems with using temJ=.erature lapse
rates at Fairbanks and Anchorage . Both sites are subj e c t
to temperature inversions because of topography. This may
not occur along the Susitna River . l''e recommend that the
existing Weather Wizard data be reviewed.
How have we demonstrated that topographic shading has an
impo rtant influence on the Susitna River? While we do not
dispute this, we woul d like to see this verlfied with a
sens'tivity run.
p.27,Par.2
p.29, Par.l
Fig. 12
p.39, Par.3
p.40, Par.2
p.41, Bottom
p.44
p.45,46
Stream surface area is necessary to compute heat flux .
According to Figure 26, we are considering only ten (10)
reaches. How repre sentative a r e these reaches for
determining stream width and hence surface areas for
the river segment between Watana and Sunshine? While
App e ndix B illustrates the representativeness of t h e ten
(10) reaches,i t appears that we may have los t some of
the r efinement of the Acres model with its approximately
sixty (60) reaches.
To compute daily m1n1mum and ma ximum temperatures, we
suggest the use of HEC-2 velocities rather than obtain-
ing Manning's nvalues t o compll te Stt"eam velocities .
To reduce client costs, we must be conscious of the
information th e t is available and not redo computations
where they are n o t warran ted .
This fig ure is excellent. It shou ld probably be ex-
panded to include the months of Ma y a nd October.
We s u ggest tha t AEIDC discontinu e its literature search
f o r techniques to i mprove the re~olution of the (grgu nd
temperature) model.
ls the Talkeetna climate station representative of
conditions further north in the basin? Presumably Fig.
19 is a comparison of mo nthly observed versus predicted
wnich appea rs to be a good c o mparison. However, Fig. 19
does not show the comparison of Talkeetna tempe ratures
with o ther basin t emperatures. Thus, if Talkeetna data
are to be used in the model, are the y represent a tive of
basin conditions ?
Since monthly average wind speeds a re us e d in the mod el,
we fa il to s ee the ju ~tification for obtaining wind speeds
directly over the wa t er surface . We could understand this
for a lake ,Lut for a river?
Top figure. Is the value (9 .3°C pre dicted, 2°C observed)
for Watana correct?
There appears to be s omething seriously wrong h e re. We
believe more work is nece s sary to under s tand what the
problem is. F or example, how do the ob se rved relati'.re
humidities a t the stations compare with o ne another?
p.Sl-54
p.SS Future
Applications
The predicted temperatures in Appendix C g enerally
indicate increasing temperature with distance downstream
except for the Chulitna confluence. We are not convinced
that the observed data show t~ls. Thus, c a n we say the
model is calibrated? To apply the model to postproject
conditions may not be valid.
1) Normal and ex treme flow regimes for the 32-year reco rd
should be defined in coordination with H-E. (See
general comments).
2) Please explain what is meant by "This will identify
the area facing possible hydrologic/hydraulic impacts?''
3) Good, but do in coordination with H-E,as this is neces-
sary for other models.
8) Techniques for imp,roving the g r oundwater t emperature
should not be pursued at thi3 tirne.
Pclr+ 9 ;
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RESPONSE TO GENERAL COMMENTS
We feel that. although the AEIDC report ent:f.tled "Stream Flow and
Temperature Modeling in the Susitna Basin. Alaska" is written for a technical
audience. a detailed description of the SNTEMP model would be unnecessary
since the temperature model description is available frnm the Instream Flow
Group. U.S. Fish and Wildlife Service (the reference The1rer et al. 1983 in
the draft report). The description is lengthy and its inclusion in the A ~IDC
report would detract from the purpose of the report: a descri tion of the
modifications of the stream temperature model, the tecnniques used for data
genesis, and the methods employed for validation and calibration.
Attachment 1 of this memo is a copy of the mathematical model description from
a draft of the Theurer et al. 1983 paper which we hope will be useful in
providing background to the AEIDC report.
The decision to investigate other methods of determining subbasin flow
contributions was made at a March 15, 1983, meeting between Harza-Ebasco and
AEIDC personnel. We agreed then t examine more sophisticated approaches
which included the effects of precipitation distribution, and to respond in a
letter report to Dr. B.K. Lee in April.
The decision to test tht! three weighting methods using a large set of
subbasins rather than one or two individual subbasins was based on a number o f
reasons. The resoluti n of the precipitation and water yield distribution
maps used to determine weighting coefficients are low enough to allow
substantial miscalculation of coefficients for any single subbasin. By
testing on a composite set of subbasins, higher basinwide ac..:uracy would be
expected. Additionally the largest set of flow data available to test these
coefficients was on the mainstem river rather than on individual tributaries.
-1-
This is important as the weighting coefficients were derived from maps
representing average trends; anomalous runoff events on small subbasins could
easily lead to unrepresentative short-term flow records. Finally, delineation
and planimetry of all subbasins was necessary for watershed area weighting.
Once this and the additional work transferring precipitation and water yield
isopleths onto the base map was done, little extra time was required to
calculate water yield and precipitation coefficients for all subbasins.
As described later in this memo, alternate techniques could be u~ed in
predicth:t; tributary temperatures. The technique chosen should be physically
based to insure reasonable predictions when the model is used to extrapolate
t~ibutary temperatures. We have discovered that ~he tributaries have a major
influence on the mainstem c emperature i n simulations of postproject
cond i tions. We also feel that accurate tributary temperature predictions may
be necessary to address thermal shock effects on spawners traveling from the
mainstem into the tributaries .
We are presently organiz ing the data necessary to simulate daily stream
temperatures . Our initial •!ffort will be validation of the stream temperature
model predictions using 1982 data. A coordinated approach will be necessary
for determining which periods should be simulated and for de f ining the purpose
of daily simulations.
p. 1, para. 2
RESPONSE TO SPECIFIC COMMENTS
Note that ADF&G and USFWf have undet .-:aken stt:.hes of
temperc:<ture effects on salmonid egg incu: ~~·i 'J n.
-2-
The introduction to this temperature report paper was not intended to be
all inclusive concerning the literature on temperature effects on the various
fish life stages. We are aware of the studies being done by ADF&G and USFWS.
Their respective reports are due out during the month of August 1983 and we
will utilize the information as it becomes available.
p. o, Par. 1 and
p. 11, Par. 2
Since subjectiveness is involved in areal precipitation
weighting (method 2), is using this method more
appropriate than using the drainage area method?
Since Method (2) yields a higher Watana discharge, we
recommend this method not be used at this time. The
high discharge implies additional economic benefits.
For economic runs, we need to be conservative. However,
a final decision on the selected method '.Jill be made by
H/E in the nea r future.
The subjectiveness of the precipitation weighting coefficients is due
both to the methods used to arrive at those coefficients from the
precipitation distribution map, and to the inherent "art" involved in
developing that isohyetal map from the pauc ity of data available for the
Susitna basin. Method 2 was chosen s o lely on the merit of its better
agreement in predicting Watana streal!lflows than the other two methods. We
think this method has merit and could be improved by refining the basin
isohyetal map with the additional data that is being collected.
However, in the short 'term, we agree that the simpler drainage area
method can be n sed. It should be clarif ied, though , that no matter which
method is used, we have been running SNTEMP usi ng the available monthly data
sets provided in Ex hibit E (ACRES 1983) (with the exception of the Sunshine
data set). Flows a t Watana (or at Devil Canyon for the two-dam scenario) and
at Gold Creek are input to the water balance program, and are thus consistent
with those used by ACRES and Harza-Ebasco. It is only the apportionment of
water between gage sites that differs between these methods.
-3-
p. 9, Fig. 3 Mean annual water yield for several subbasins appears
to be greater than the mean annual precipitation
(Tsusena, Fog, Devil, Chin-Chee, Portage).
This is true. Mean annual precipitation values were developed using the
map of Wise (1977), and mean annual water-yield values using the map of Evan
Merril of the Soil Conservation Service (Jq82). These numbers are clearly in
dispute. This figure was included to demonstrate the differences between
tnose weighting methods.
p~ 10, Bottom Calculated C for Method (1) is 0.5104. ACRES ~sed
0. 515. lfuy '!s there a d ifference? Were these areas
replanimetered?
The basin between Cantwell and Gold Creek was divided into ten subbasins
(Clarence through Indian, Figure 4 of the draft report), four upstream from
the Watana dam site, and six downstream. The area of each subbasin was found
by planimetry; the areas of the basin above and the basin below Watana were
arrived at by sununing the appropriate subbasin areas. Discrepancies in basin
area measurements are expected when those basins a re delineated and
planimetered independently. Moreover, our procedure incorporates p o ssible
errors from a number of individual planimetry measurements, and compounding
errors can occur. However, the agreement of these two figures is to less than
one-half percent (0.0046) of the area between Cantwell and Gold Creek. This
2 difference c o rres ponds to an area l e ss than 9 mi in a watershed (defined at
2 Watana) larger than 5000 mi •
-4-
Once again and most importantly, these coefficients are defined for the
Cantwell to Gold Creek basin. When running SNTEMP, only the flow
apportionment between basin sites havi.ng input data is affected. Thus
mainstem flows at Watana, Gold Creek and Susitna Station are consistent with
those flows used by other groups.
p. 18, Par. 1 We suggest using solar radiation measurements when
available rather than calculated values. We would
also like to see daily comparisons of observed versus
computed solar radiation. Please provide descriptions
of the six SNTEMP submodels.
We have decided to use predicted solar radiation rather than observed
values so that we would be able to simulate water temperatures for periods
when there was no data collected. This is useful for predicting average and
extreme conditions which did not necessarily occur during the 1980 to 1982
periods. We h:r.re made an effort to calibrate the solar model to observed
solar radiation data to make our predictions as r~presentative as possible.
As Figure 22 indica tee predicted solar radiation values are
representative f basin for monthly average conditions. This figure
demonstrates a tendency to overpredict Watana and underpredict Devil Canyon
insolations. Thus, the solar model is predicting an average basin insolation.
Since the current implementation of SNTEMP allows for only one meteorological
data station, basin average solar rad~ations would have to be estimated from
alternative means or area weighted averages. The solar model essentially
averages conditions for us.
Calculated solar radiation is also necessary for simulating topographic
shade effects. The solar model tracks the sun during the day and accounts for
the time the stream surface is in shade due to the adjacent topography.
-5-
We will produce a plot similar to Figure 22 but with daily values if it
becomes necessary to predict daily water temperatures.
Attachment 1 contains pertinent pages from the paper by Theurer et al.
(1983) which describes the six SNTEHP submodels. These pages will be useful
in clarifying some of the comments to other sections of AEIDC's draft flow and
temperature report.
P• 19, Bottom More discussion on heat flux ~~uld be helpful. ~tatements
regarding the relative importanc _ of heat inputs and
outputs should be made. Please rovide all he t sources
and sinks considered.
Attachment 1 discussed in the previous response should clarify how the
beat flux components (atmospheric, topographic, and vegetative radiation;
solar r adiation; evaporation; free· and forced convection; stream friction;
stream bed conduction; and water back radiation) are simulated by SNTEMP. We
are working on a graphic presentation to demonstrate the values of the
individual heat flux components for average monthly conditions but do not feel
it will be available f or the final version ~f this report. Preliminary plots
of the beat flux components are presented in Attachment 2. The relatively
high friction heat i nput is interesting and will p Lo bably be a maj or influence
in fall and winter simulations .
p. 20 In Eq. (9), how wasT (Equilibrium temperature)
estimated? What are ihe parameter values of K1 and K2 ?
The values of the equilibrium tP.mperature (Te) and 1st (K 1) and 2nd
(K 2) thermal exchange c.oefficients are computed within SNTEMP. To visualize
-6-
the technique used, it is necessary to realize that the net heat flux (EH) is
an analytical but nonlinear fun cti on of the stream temperature (due to the
back radiation, evaporation, and convection heat components); i.e. EH •
f(T ) where T is stream temperature. w w When stream temperature equals
equilibrium temperature, the net heat flux is zero ( l:H = f (T -=T ) -= 0). w e
Newton's method is used to iterate to the equilibrium temperature with the air
temperature being the initial estimate of Te. The values for K1 and K2
follow since the first and second derivations of the heat flux are also
analytical functions and :
d(EH) dfK JfK
1 2 = Kl ~ ~ dT T = T w w w w e
d 2 ( EH) d 2 f d 2 f
= K2 K2 = K2 ---
dT 2 dT 2 dT 2 T T w w w w e
Average values of Te, K1 , and K2 will be presente d in a subsequent
report which will include 1983 data/SNTEMP simulation validation.
p.21 There are potential problems with using temperatur ~ lapse
rates at Fairbanks and Anchorage. Both sites are
subject to temperature inversions because of topography.
This may not occur along the Susitna River. We
recommend that the existing Weather Wizard data be
reviewed.
-7-
No long term upper air data are a~ailable for Talkeetna. Anchorage and
Fairbanks vertic al temperature (and humidi ty) data a v eraged over a six-year
period (1968, 1969, 1970, 1980, 1981. a nd 1982) are felt to be the best
a vdilable repre sentation of vertical air temperature profiles for the Susitna
River basin. Examination of numerous winter daily synoptic weather maps for
surface, 850 mb, and 500 mb levels verifies the assumption that inversion
strength and thickness in th~ Susitna River basin are roughly halfway between
those observed in An c horage and Fairbanks.
The Susitna basin is surrounded by mount ains on the north, east and west.
To the south i t is open t ~ the Cook I n l et and Gulf of Ala ska . In winter, the
Alaska range blocks most l ow level interio r air from reaching and influencing
the Susitna basin and Anchorage. However, radiative processes in concert with
topography are responsible for producing a strong, well documented low level
inversion in the Susitna valley (Comiskey, pers. comm.). This inversion is
not as severe as in Fairbanks, but more severe tha n in Anchorage. Data from
both station s a r e retained since upper air tempera tures for a ll three regions
are relatively uniform.
Topographic variability will introduce local systematic error in the
vertical profiles . Cold a ir flows downhill where r a diative cooling in the
valleys further reduces air temperatures. Weather Wizard data gathered at
stations within the b asin may reflect highly localized weather activ ity.
Within the mountain walls vertical and lateral air mass extent and movement is
limited compared to that of the synoptic scale e vents governing the major air
mass properties. Loc:al topographic effects cannot be reliably incorporated
into the larg er scale vertical lapse =ate regime.
-8-'
1 _is strong inversion is not ju.st a statewide phenomena, but occurs
throughout the high latitudes in winter. Due to the small heat capacity of
the land surface its temperature is highly dependent upon absorption of solar
radiation. Minimal radiation is absorbed in Alaska (i.e., the Susitna River
basin) :f.n win.ter for the following four reasons: (l) a high albedo, (2) short
hours of daylight, (3) the oblique angle of the sun's rays, and (4) screening
by clouds of ultraviolet rays . Consequently, a wana maritime air mass flowing
fro• the North Pacific or Bering Sea over Alaska will be strongly cooled at
the earth's surface. When subsequent air masses move onshore they are forced
to flow ciloft by the previously cooled, dense stable surface layer. Daytiae
beating at the earth's surface is usually not strong enough to destroy the
inversion. Over a 24-hour cycle no well-defined mixed L tyer remains and
fluxes of latent and sensible beat are very small. The i .nversion' s longevity
is enhanced when the wind speeds are low and corresponding momentum transfer
is weak. Talkeetna is typified by comparatively low average wind speeds, on
the order of 5 mph during the winter months. A singll! strong wind event can
disperse the inversion temporarily; however, it will occur frequently each
winter and is considered a semi-permanent feature.
Translocating average t e mperature profiles from Anchorage and Fairbanks
in the spring, summer, and fal to the Susitna Ri· er basin is well within
acceptable limits. The temperature profiles generated by this method fall
precisely within the moist adiabatic lapse rate, as predicted by standard
theory. The temperature data gathered from upper air National Weather Service
radiosonde instruments is highly correlated with temperatures measured in the
basin by the Weather W~zard. This argument further substantiates use of large
scale data to predlct local temperature patterns.
-9-
p.24, Par. 4 How have we demonstrated that topographic shading has an
important influence on the Susitna River? While we do not
dispute this, we would like to see this verified with a
sensitivi:.t run.
Our statement is in error since we have not demonstrated that topographic
shading has an important influence on Susitna stream temperatures. Initial
sensitivity simulations without topographic shade have shown that the
corresponding increase in solar radiation has only a small effect on the
stream temperatures. The significance Df the shade effects has only been
tested for average natural June through September conditions where an increase
C'!f less than 0.2 C was simula tfl'd without shade from Cantwell to Sunshine.
Based on the solar path plots in Appendix A of the draft report, we would
expect that the shading effects in other months would be greater but still
relatively small. The wording of tbis paragraph will be changed to reflect
the new knowledge gained from this sensitivity study.
p. 27, Par. 2 ~tream surface area is necessary to compute heat flux.
According to Figure 26, we are considering only ten (10)
reaches. How representative are these reaches for
determining stream width and hence surface areas for the
river segment between Watana and Sunshine? While Appendix B
illustrates t he representativeness of the ten (10) reac11es,
it appears that we may have lost some o f "the refinement
of the Acres model with its approximateJ y sixty (60) reaches.
We feel that increasing the number of simulated reaches would improve the
representativeness of the stream temperature model as would any increase in
data detail. Based en our familiarity with SNTEMP, we did not originally feel
that this many reaches were necessary. Nevertheless, we can increase the
number of reaches for simulation purposes; the data is already available and
the only increase in the client's costs will be the manpower to add them to
SNTEMP data files and the increased computational time.
-10-
We are not familiar with the ACRES st ~eam temperature model and do not
know the model's stream width or hydra'Ulic data req~iremer.t ts.
p. 29, Par. 1 To compute daily minimum and maximum tmperatures,. .,,e
suggest the use of HEC-Z velocities rather than
obtaining Manning's n values to compute stream velocities.
To reduce client costs, we must be cons~ious of the
information that is available and not t"edo <!amputations
where they are not warranted.
There would be two objections •_o us.ing HEC-2 velocities as input to
SNTEMP: (1) HEC-2 simulati<.ns ·..,ould be required for all water temperature
simulations where the minimum and maximum water temperatures were desired; and
(2) SNTEMP would have to be modified to accept velocities.
Velocity input is noc currently necessary to run SNTEMP for minimum and
maximum temperatures since it is computed internally. This allows us to use
SNTEMP for simulating any ice-free period from 1968 to 1982 (or later, when
the required data are received). Thus, we can determine the extreme
meteorological/flow periods for simulating tnaximum and minimum average daily
temperatures and the diurnal variation around these extreme daily
temperatures. If the HEC-2 velocity estimates are required, this flexibility
would be lost. If the Susitna Aquatic Impact Study Team could agree on the
periods for minimum and maximum temperature predictions, this first rroblem
could be eliminated .
Modifying SNTEMP to accept velocities, however, would be a major
undertaking. The explanation for this would be lengthy; we would prefer to
discuss this potential modification at a technical meeting to explain the
amount of work necessary and t ~ help decide if SNTEHP should be moditied or
alternate techniques used.
-11-
Figure 12 This figure is exceller,t. It should proba.'>ly be expanded
to include the months of May and Octobe ~.
We agree that Figure 12 is both useful and usable and should be expanded
to include Hay and October data as well as 1983 Jata. However, due to
budgetary and time constraints, we will not be able to revise this figure
until after the October 14 report.
p. 39, Par 3 We suggest that AEIDC discontinue its literature search
for techniques to improve the resolution of the (ground
tempera~ure) model.
This is not an intensive literature search. We are limiting our search
to the journals and reports we normally read within the course of our
professional maintenance and to conv'ersations with other professionals who may
have experience and knowledge of lateral flows and temperature in gen ~ral and
Susitna conditions specifically. The last sentence of this paragraph will be
replaced with "AEIDC believes this model current:ly provides the best available
approximation of the physical conditions existing in the Susitna basin and
will be applied without validat i on until b etter estimates of existing
conditions are obtained."
p. 40, Par 2 Is the Talkeetna elimate station representative of
conditions further north in the basin? Presumably Fig. 19
is a comparison of monthly observed versus precicted
which appears to be a good comparison. However, Fig. 19
does not show the comparison of Talkeetna temperatures
with other basin temperatures. 7hus, if Talkeetna data
are to be used in the model, are they representative of
basin conditions7
-12-
Talkeetna climate data would not be representative of conditions wi ~hin
the basin if applied without adjustment. The last two sentences of this
paragraph will be changed to "This period of record allows stream temperature
siaulations under extreme and normal meteorology once these data are adjusted
to better represent conditions throughout the Susitna basin. We used
meteorologic data collected specifically for the Susitna study to validate
this meteo"Lologic adjustment and the solar model predictions." We hope this
wil clarify that we are not blindly applying Talkeetna data without
adjustment.
Apparently Figure 19 bas been misunderstood . The predicted temperatures
are based on observed temperatures at Talkeetna and the lapse rates which we
have developed (Figure 7 in the report). Given the observed temperature at
the Talkeetna elevation, the lapse rate equations are used to predict
temperatures at any elevation. The air temperatures predicted for the
elevations of the Sherman, Devil Canyon, Watana, and Ko s na Weather Wizards
weT"e compared to the air temperatures observed by R&M (Figure 19 in the
report).
p . 41, Bottom Since month ly average wind speeds are used in the model,
we fail to see the justification for obtaining wind speeds
directly over the water surface. We could understand this
for a lake, but for a river?
As Fig ure 21 suggests, t he wind speed data collected at Talkeetna
represents average basin winds as collected at the four R&M sites (at least
the data at Talkeetna is not extremely different). What these wind speed data
represent, however, is not fully und erstood. The evaporative and convective
heat flux is driven by local (2 m above the water surface ) ~nd speeds. The
Watana, Devil Canyon, and Koslna stations are located high above the water
-13-
surface (as we understand, we have not visited the sites). This implies that
the data collected do not meet the model's requirements; however, we agree
that it is not necessary to collect additional data if this would be very
expensive. In our initial conversation with Jeff Coffin of R&M Consultan.ts,
we inquired if it would be possible to obtain this data easily as part of
their existing collection effort;. Be felt it would be possible. A return
call from St <!ve Bredthauer informed us that equipment necessary to collect
this data was not available and ~ould hav e to be purchased. Our response was
that this data would improve our understanding of · in-canyon winds but would
not be necessary at the expense envisioned. We have replaced this last
sentence on Page 41 with "Since it appears to be impractical to collect wind
speed data within the canyons below the existing meteorological data site s
(Bredthauer 1983), the wind speed data collected at talkeetna will be used as
representative of average ba ~·in winds."
p. 44 Top figure . Is the value (q.3o C predicted, 2° C observed)
for Watana correct?
SNTEMP did predict an air temperature of 9. 3 C and an average air
temperature of 2 C was observed for August 1981 at the Watana weather station.
The observed Watana data is obviously in error (e.g., a temperature of -30.9 C
was recorded for 15 August 1981) and probably should not have been included
for validation of the air temperature lapse model in this plo~. As stated in
the report, none of the Weather Wizard data were used in the water temperature
simulations but are presented 88 8 validation of the adjustment of the
observed Talkeetna data. Careful review of the Wea~her Wizard data
(especially humidities) would be necessary if these data were to be used in
-14-\
water temperature simulations. This data point will be removed from the plot
in ~he final draft.
p. 45, 46 There appears to be something seriously wrong here. We
believe more work is necessary to understand what the
rroblem is. For example, how do the observed rela~ive
humidities at the stations compare with one another?
The large variability in observed Weather Wizard data gives rise to
doubts of its reliability. Data which are smoothed b ; monthly averaging are
not expected to exhibit the year to year range of humidities which was
observed at the Weather Wizard stations. The entire data set is characterized
by irregular large annual changes in average relative humidities on the order
of 30% to 40%. Talkeetna relative humidity values, measured by the National
Weather Service, are consistently greater by approximately 20% throughout the
data. Talkeetna values are in agreement with the large scale picture
generated by averaged Anchorage and Fairbanks data. For this reason, and
those enumerated on Page 41 in the draft report, AEIDC maintains that the
predictive scheme derived for input into the stream temperature model is the
best representation of relative humidity with height for input in the surface
flux calculations.
Five sample figures from the R&M raw data are presented for inspection
(Attachment 3). Figures 1 and 2 present summer (June 1981) and winter
(November 1980) situations where the correlation between Weather Wizard data
at two stations is illustrated. In both instances the relative humidity data
is in good agree.ment from one station to another. These were chosen as
exemplary months; they are not, however, typical. Figure 3 indicates two
C01111DOn errors, missing days of data and an unvarying upper limit. Another
c01111110n error discussed in the report is illustrated by Figure 4. Erratic
-15-
·;.
daily swings from zero to 100 percent exist throughout the data. Figure 5
illustrat~s simultaneous comparison of Watana Weather Wizard data and surface
relative humidities measured at Talkeetna by the National Weather Service.
The correlation between the two is poor.
Attempts to explain the erratic swings in the data (daily, monthly and
annually) as highly localized topographic or microscale weather events is also
unsatisfactory. Over time, monthly averaging would smooth anomalies.
However, a three-year average for each month still retains a high variability
with elevation (see Figure 6, Attachment 3). From year to year topography
requires that highly localized atmospheric events be fairly consistent,
thereby giving rise to identifiable trends in the data. Such is not the case .
AEIDC meteorologists concur that instrument calibration problems are the
probable explanation for the high variability in the data.
The best way to verify these conclusions regarding the reliability of the
relative humidity data collected in the Susitna basin would be to perform a
spot calibration of the Weather Wizards. A wet btdb-dry dry bulb sling
psychrometer could be carried to the remote wea~her stations where the
relative humidities measured by each method can be compared.
p. 51-54 The predicted temperatures in Appendix C generally
indicate increasing temperature with dis tance downstream
except for the Chulitna confluence. We are not convinced
that the observed da t a show this. Thus, can we say the
model is r.alibrated? To apply the model to postproject
conditions may not be valid.
We have some problP.ms in believing the observed data, especially the
variation in downstream temperatures observed in August !981, September 1981,
and August 1982. We do not understand what would cause the types of
variations indicated unless there were tributar; impacts which we were not
-16-
·considering. We feel, &1owever, that we have ma <!e a thorough attempt at
modeling tributary flows and temperatures.
We are not thoroughly familiar with the techniques used by ADF&G to
verify and calibrate their thermographs. Their techniques are not publishe~
in any Susitna reports.
We recommend that data verification be performed. Wayne Dyok, H-E, has
collected some longitudinal temperature data which tends to support the
downstream increase in temperature which we have predicted. Wayne's effort
was helpful but does not identify which thermographs or data sets may be in
error. Until faulty data sets are identified (if any) we do not feel we
should attempt to increase the degree of fit of the model.
As to applying the model to postproject conditions, we feel that, at the
very least, it is necessary that some initial estimates of project impacts be
made at this time. It may be necessary to label these simulations as
preliminary results until temperature data is verified.
p. 55, Future
Applications
1) Norma l and extreme flow regimes for the 32-year record
should be defined in coordination with H-E. (See
general comments).
Our intent here is to identify the natural range of flow regimes in the
Susitna basin, not to necessarily "define" representative flow years for more
detailed stuJy. We agree that identifying such years should be done by AEIDC
and R-E together, insuring the most thorough results for the efforts of each.
p. 55 2) Please explain what is meant by "This will identify
the area facing possible hydrologic/hydraulic im.;:>acts?"
-17-
If possible, we will determine the location downstret.t m from the proJect
where operationa ~ flows become statistically indistinguishable from natural
flows. This will vary on a month-by-month basis. If project flows downstream
from a given location are insignificantly different from natural flows, we
reason that flow-related impacts must also be indistinguishable , and,
therefore, need not be examined further.
p. 55 3) Good, but do in coordination with H-E, as this is
necessary for other models.
We have met with Wayne Dyok of Harza-Ebasco and discussed our approach in
simulating normal and extreme stream temperature changes. The periods we
selected were not the same as the periods selected by Harza-Ebasco . Sin ~e we
had a deadline to meet in producing a stream temperature effects paper, there
was insufficient time fo r: a mo re coordinated approach. We feel that more
coordination will be of mutual benefit in the future.
p. 55 8) Techniques for improving the groundwater temperature
should not be pursued at this time.
We have found that the influence of the tributaries on the mainstem is
significant, especially in postproject simulations. The distributed flow
temperature model was developed to improve the tributary temperature
predictions with a physically reasonable model. There are other approaches to
predicting tributary temperatures but the technique used will have to meet
several requirements: (1) it must h e general enough to apply to June-September
periods without observed tributary temperatures, (2) it must be applicable to
winter conditions for future ice simulations, and {3) any technique used
cannot depend on more data than is available. The technique which you have
-18-
suggested (relating tributary temperatures to air temperatures) may be
possible when the 1983 field data becomes available, although we would
recommend a regression model based on computed equilibrium temperatures.
There is not enough monthly tributary data currently available for any
regression approach. Daily air temperature and tributary temperature data
suggests a correlation (Attachment 4 is a scattergram of recorded Indian River
temperatures versus air temperatures) but we believe that a regression model
based on daily data would result in a tributary temperature model which would
not be as capable as the distributed flow temperature model.
As you request, we will not pursue techniques for improving the
distributed flow temperature model at this time. This model will be used as
is for all simulations until the 1983 tributary temperature data becomes
available. When the 1983 data are available, we will look at possible
regression models for predicting tributary temperatures. We will then select
the best approach. Harza-Ebasco's involvement in this selection process would
be appreciated.
-19-,\
Attachment 1
S~TT-EMP MATHEMATICAL MODEL DESCRIPTION
INTRODUCTION
This part is to explain each of the physical processes affecting instream
'"ater temperatures and their ms.thel'llatical descriptions so that thP. responsible
engineer/scientist can understand the behavior of the model. It will enable
the responsible engineer/scientist to determine the applicab il ity ~~ the
mode 1 , the utility of 1 inking the mode 1 with other mo,de 1 s, and the va 1 i di ty of
results.
The instream 'tfat~r temperature model incor ., ates: (1) a complete solar
model including b;th topographic and riparian vegetat ion shade; (2) an
adiabatic meteor"-lo~ical !:orrection model to account for the change fn air
temperature, rElative humidity, and atmospheric pressure as a function of
elevation; (3) a complete set of heat flux-cc .. ponents to account for all
significa~t heat sources; (4) a heat transport mod 1 to determine ongitudinal
water te~per4ture changes; (5) regression models to mooth or complete known
water temperature data sets at measured points for st1rting or interior
vaTidatic~/calibration temperatures; 6) a f~ow mixing model at tributary
s
junt:tion$; and (7) calibration models to eliminate bia~ and/or reduce the
probable errors at interior calibration nodes.
SOLAR RADIATION
The solar rad i at i on model has four parts: (1) ex t ra-terrestri al radia-
tion, (2) correction for atmosp~eric conditions, (3) correction for cloud
cover, and ( 4) correction for reflection from water surface. The extra-
terrestrial radiation, when corrEcted for both the atmosphere and cloud cover,
predicts the average daily solar radiation received at the ground on ~ ho~i
zontal surface of unit area :-Therefore, it is the total amount of solar
energy per unit area that projects onto a lev~l surface in a 24-hour period.
It is expressed as a constant rate of heat energy flux over a 24-hour perioc
even though there is no sunsh i ne at night and the actual solar radiation
var1 es f'rom zero at sunrise and sunset to a maximum i nt.ensi ty at solar noon.
EXTRA-TERRESTRIAL RADIATION
The extra-terrestrial ra~iation at a site is a f unct i on of the latitude,
general topographic features, and time of year. The general topographic
features affect the actual time of sunrise and sunset at a site. Therefore,
the effect of solar shading due to hills and canyon wa l ls can be measured .
The time of year directly predicts the angle of the sun above or below the
equator (declinat i on) and the distance between the earth and the sun (orb i tal
position). The latitude is a measure of the angle between horizontal surfaces
along the same longi~uoe at the equator and the site .
The extra-terrestrial solar radiation equ ?tion is
H . = (qs/~) {[(l + e cosa 1• :/(l-e 2 )]} s ~, 1
( ) .
where:
{[h .(sin .. sin6.)] + [sinh .(co s9 cos6i)]} S ~1 1 S ,1
q 5 -solar constant= 1377, J/m 2 /sec.
e -orbital eccentricity = 0.0167238, dimensionless.
ai -earth orbit position abcut the sun, radians.
f: site latitude for day i, radians.
51 -sun declination for day i, radians.
:sunrise/sunset hour angle for day i, radians.
average daily extra-terrestrial solar radi htion for day 1,
J/m%/sac.
The extra-terrestrial solar radiation may be averaged over any time
pertod according to
where:
N = ( I
i=n
H .]/(N-n + 1] sx, l
H . -extra-terrestrial solar radiation for day i, J /m2 /sec. sx,,
N-last day in time period, Julian days.
n -first day in time period, Julian days.
-day counter, Julian days.
extra-terrestrial solar radiation averaged over time
period n to N, J/m 2 /sec.
( )
~ The earth oroit position and sun declinatio n as a funct i on of the day of year 71 .
are
ei = [(2~/365) (0 1-2)] ( )
61 = 0 .40928 cos [(2~/365) (172-0;)] ( )
where : D; -day of year, Julian days ~ 01=1 fo -January 1 and 01=365
for December 3 1. .... ,...---~
9; = earth orbit po~ition fo·r day ;(.Julian d~ys . \, ('V ,....__ _ __...
6i = sun declination for day f, Juliin days .
The sunrise/sunset hour angle is a measure of time, expressed as an angle,
between solar noon and sunrise/sunset. Solar noon is when the sun is at its
zenith. The t i me from sunrise to noon is equal to the time from noon to
sunset only for symeter i cal topographic situations . Howevel!', for simplicity,
this mode l w111 assume that an ave·rage of the -sola r attitudes "t sunr ise/
sunset is used . Therefore, the sunr i s~/s unset hour angle is
h . =arccos {[sina -(si n; sin6 .)]/[cos~ co s 6 .]} ( ) S,l S 1 1
where:
N
iis = [ t
i=n
h .]/[N-n + 1] s. 1
; -site latitude, radians.
61 -sun declination for day i , radians .
(
a
5
-average solar a ltitude at sunrise/sunset, radian ~; a = 0
for flat t errian, as> 0 for hilly or canyon terrian~
)
,,
9-
~
0
0
)
,,
LEVEL PLANE ON
EARTH'S SURFACE
N
s
Figure 2.1. Solar angular Measurements .
LATITUDE
EQUATOR
h :sunrise/sunset hour angle for day 1, radians s,i
~•eraqe sunrise/sun s et hour angle over the t i me period n to
\~, radians.
n -first day of time period, Julian days.
N -last day Gf time period, Julian days.
i =day counter, Julian days.
It is possible for the sun to be completely shaded during winter months
at some sites. This is why snow melts last on the north slopes of hillsides.
Therefore, certain restrictions are imposed on as; i.e., as~ (w /2)-; + 61 .
The average solar attitu d~ at sunrise/sunset is a measure of the obstruc-
tion of topographic features . It is det e ~ined by measuring the average angle
from the horizon to the point where ·the sun rises and sets . Therefore, the
resulting prediction of extra-terrestrial solar radiation includes only the
solar radiat i on between the est i mated actual hours of sunrise and sunset.
SUNRISE TO SUNSET DURATION
The sunrise to sunset duration at a specific s i te is a function of
latitude, time of year, and topographic features . It can be computed directly
from the sunrise/sunset hour angle hs;· The average sunrise to sunset duration
. .-.-\ \.1'.:
over the t i me period n toN is
0 ( )
where: average sunrise to sunset duration at the specific
site over the time period n to N, hours.
average sunrise/sunset hour angle over the time
period n to N, radians.
ATMOSPHERIC CORRECTION
The extra-terrestrial solar radiation is attenuated on its path through
the atmosphere by scattering and absorbtion when encoun t ering gas molecules,
water vapor, and dust particles. Furthermore, radiation is reflected from the
ground back into the sky where it is again scattered and reflected back again
to the ground.
The attenuation of solar radiation due to the atmosphere can be approxi-
mated by Beer's law
where:
( )
Hsx -average daily extra-terrestrial solar radiation; J /m%/sec.
-average daily solar radiation corrected for atmosphere
only, Jfm%/sec.
~ = absorbt i on coefficient, 1/m.
z -path len gth, m.
While Beer's law is valid only for monochr omatic radiation, it i s useful
to predict the form of and significant var i ables for the atmospheric correction
equation. Repeated use of Beer's law and recognition of the im portance of the
~-opt'ical air mass (path length), atmospheric moisture content (water vapor),
dust particles , and ground refl ectivity results in a useful emperica l atmos-
'
pher ic correction approximation.
where:
v :--
e-,z = [a11 + (1-a'-d)/2]/(1-R (1-a'+d)/2] ( ) g
a' -mean atmospheric transmission coeffici ent for dust free
moist air after scattering only, dimensionless.
a11 -mean distance transmission coefficient for dust free moist
air after scattering and absorbiton; dimensionless.
d -total depletion coefficient of the direct solar radiation
by scattering and absorbtion due to dust, dimension l ess.
R9 -total reflectivity of the ground in the vicinity of the
s ite, dimensionless.
The two transmission coefficients may be calculated by
a' = exp {-[0.465 + 0.134 w] (0.129 + 0.171 exp (-0.880 m ,.,
p ' ( )
a 11 = exp {-[0.465 + 0 .134 w] (0 .179 + 0.421 exp (-0.721 mp)] mp} ( )
where: w -precipitab i e water content, em.
mp -opt i cal air mass, dimens i onless .
The precipitable water content, w, of the atmosphere can be obtained
using the following pair of formulas.
T
(1 .0640 d)/(Td+273.16)
T = (Rhl .0640 a)/(Ta+273 .16) ( )
w = 0.85 exp (0.110 ~ 0 .0614 Td) l )
!a ... where : T -average daily air temp@rature, c. a
Rh -rela :ive humidity, d im ensionl f!ss.
Td -mean dew po i nt, c.
w -precipitable water content, em.
The optical air mass is the measure of both the path length and absorb-
tion coeffic i ent of a dust-free dry atmosph e re ~ It is a funct i on of the site
elev3t i on and instantaneous solar altitude. The solar altitude varies accord-
ing to the latitude of the site , time of year, and t 'ime of day . For practical
appl i cation, the optical air mass can be t i me-averaged over the same time
period as the extra-terr estrial so l ar radiation . The so l ar alt i tude function
is
where :
(1 . = 1 arcsin {(sin; sin6 .] +
1
; .
[cos~ (cos• cos6 i )]}
N h -{ ! [( 1 s , i dh)/h .]}/[N-n + 1] Cl = Cli 0 s , 1 i =n
~ -site lat i tude, radians .
6 1 -sun decl i na t i on on day i, radians .
h -in stanta neous hour ang l e, radians .
h . -sunrise/sunset hour angle for day i, radians. s . 1
n -first day in time period, Ju l ian days .
N -last day in time per i od, Jul i an days .
i -day counter, J u li an days .
cs 1 -i nstanta neous so l ar a l t i tude during day i , r a dia ~s.
( )
( )
cs -ave r age so l ar a lt i t ude over ti me per i od n to N, radians.
.... ..
Equation A14 can be solved by numerical integration to obtain a precise
so 1 uti on . However, i f the time periods do not exceed a month, a r ea so nab 1 e
approximation to the solut i on is
N -
a = [i!n ai]/[N-~ + 1]
where: ai =average solar altitude during day i, radians .
remaining parameters as previously defined.
The corresponding optical a i r mass i s
\<fhere:
mp = {[(288-0 .006SZ)/288]5 ·256 }/{s i n ~
+ 0.15[(180/~) ~ + 3.885]-1·253 }
Z -site e l evat i on abo ve mean sea l evel, m.
a -average s olar alt i tude for time period n to N, radians .
mp -average optical air mass, dimensionless.
( )
( )
The dust coeff i ci e nt d and the ground reflectiv ity R9 may be estimated
from Tables Al and A2 respectively or they can be calibrated to published
solar radiat i on d ata (Cinquemani et. a1, 1978) after cloud cover corrections
have been made.
Table Al. Dust coeff i cient d.1
Season Washington, DC Madison, Wisconsin
m =1 p m =2 p m =1 p m =2 p
Winter 0.13 0.08
Sprfng 0.09 0 .13 0.06 0 .10
Summer 0.08 0.10 0 .05 0.07
Fall 0 .06 0.11 0 .0'7 .0.08
1 Tennessee Valley Authority 1972, page 2 .15.
Table A2. Ground re fl ect ivi~y
Ground condition
Meadows and fields
leave and need l e fo rest
Dark., extended mix ed f~re5t
Heath
Flat ground, gra~s covered
Flat ground, rock.
Sand
Veget at ion ea rl y summer leaves
high water content
Vegetat i on late summery;;eaves
low water content ~
Fresh snow
Old snow
wi th
w i ~h
1 Tennesee Valley Author ity 19 72 , page 2 .1 5 .
Rg
Lincoln, Nebrask.a
m =1 p m =? p -
0.06
0 .05 0 .08
n 03 0.04
0 .04 0.06
1
R~
0.14
0.07 -0.09
0 .045
0.10
0 .25 -0.33
0.12 -0.15
0 .18
0.19
0 .29
0 .83
0 .42 -0 .70
Seasona l vari at i ons app uar to occur i n both d and R9 . Such seasonal
ve~i;ti ons can be pred icted result ing i n reasonab l e est i mates of gro und so l ar
r ad i ation .
The dust coeffi c i errt d of the atmosphere can be seasonally distributed by
the following empirical relationship .
d = d 1 + {[d2 -d 1] sin [(Zw/365) (D 1-213)]} ( )
~·here: d1 -mi'limum dust coeffic i ent occurring i n late July -earl y
August, d im ens i on l ess .
dt -maximum dust coe ffici ent occurr i ng in late January -earl y
February, dimens i onless .
D; :day of y~3r, Ju li an days ; D.=1 for January 1 and D 1 ~365
for December 31 . 1
The ground re f lect i v ity Rg can be seasonally d i stributed by th'e following
empirical relationship.
where : R 9 1
R 92
D.
1
Rg = R + {[R - R l s i n [(2w /365) (D.-244)} g1 92 91 1
-minimum ground reflectivity occurring in mid -Sept ember,
dimensionless .
-maximum ground reflectivity occurring in mid-March,
d i mens i onless .
-day of ye ar, Julian days; D1=1 for January 1 aod Di=365
for December 31 .
( )
The average min imum-maximum value f or both the dust coefficient and
ground reflec~ivities can be ca li brated to actual recorded solar radiation
data. Summaries of recorded so l ar radiat1on can oe fo und i n Cinquemani,
et a 1 . 1978.
i
CLOUD COVER CORR ECTION
Cloud cover signif i cantly reduces d i rect sc l ar r ad i a ti on and somewhat
reduces diffused so 1 ar radiation . The preferr ed measure of the effect of
cloud cover is the "percent poss i ble sunshine" recorded va l ue (S /S0 ) as
published t y NOAA. It is a direct measurement of solar radiation durat i on.
( )
where : Hsg -daily solar radiat1on at ground l e ve l.
H -solar r ad i ation corrected for at:nosphere O'l l y . sa
s -actual sunshine duration on a cloudy day .
so -sunrise to sunset durati on at the spec i f i c site.
If direct S/S0 va lues are not ava i lable, t hen S/S0 can be obta i ned from
estimates of c loud cover C1 .
SI S = 1-C S/J
0 l. ( )
where : ct -cloud cover , d i mens i onless.
DIURNAL SOLAR RA DIA TI ON
Obv i ousl y , the solar radiation intensity varies throughout the 2J.-hour
daily pe ri od. It i s zero at ni ght, incre ases from zero at s unrise to a max f mum
ij at noon, and decreases to zero at sunset. This diurna l vari at io n can be
approximated by:
wh ere:
Hnite = 0
Hnite -average nighttime solar radiation, J/m:/sec.
Hday -average dayt im e solar radiation, J/m 1 /sec .
H sg -average daily solar radiation at ground level,
h5 : average sunr ise/sunset hour angle over ~he tim e
period n to N, rJdians.
SOLAR RADIATION ~ENETRATING WATER
( )
( )
J /m :/sec .
Solar or shortwave radiat i on can be reflected from a water sur ~ace. The
relat i ve amoun~ of so l ar radiat i on ref l ectad (Rt) is a function of the solar
angle and the proport i on of direct to diffused short:wave radiati on. The
a 1eragc: solar angle a i !: a meas ure of the ang l e and the percent possible
sunshine S/S0 reflects the d irect-diffused proportions .
B(S /S ) R~ = A(S/S ) [~{180/~)J 0 0 s Rt s 0 .99 ... 0 )
where: Rt -solar-water reflect.ivity coeffi cient, dimensionless.
a -average so 1 a r a 1 t itude , radians .
A(S /S0 ) -coeff i cient as a function of S/S
0
.
B(S /50 ) -coefficient as a fu nc tion of S/50 .
S/S
0 -percent possible sunshine , dimensionless .
Both A(S /S0 ) and 8(S/S 0 ) are based on values given i n Table 2.4 Tennessee
Valley Authority , 1972. The following average high and low cloud values wer~
selected fro~ this table to fit the curves.
where:
ct
0
0 .2
1
S/S0
1
0 .932
0
A
1.18
2.20
0.33
A' = dA/dC and 8' = dB/dC t 1
A'
0
8
-0 .77
-0.97
-OAS
8'
0
The resulting curves are:
A(S/S 0 ) = [a 0 + a 1 (S/S0 ) + a 2 (S/$0 )2]/[1 + aJ(S/S0 )] ( )
B(S/S 0 ) = [bo + b 1 (S/S 0 )_+ b 2 (S /S0 )1 ]/[l + bt (S /S0 )] ( )
where: a = 0 0.3300 b = -0.4500 0
al = 1.8343 bl = -0.1593
a2 ·--2.1528 b2 = 0.59E6
a) = -0.9902 bl = -0 .9862
The amount of so l ar radiation actually penetrat i ng an unshaded water
surface is:
( )
where: H -SW daily so l ar radiat i on entering water, J /m:/sec
R. -solar-water reflectivity, dimensionless
I.
H -ca i 1y so 1 ar radiation a t ground 1 eve 1 , J /m :/sec sg
SOLAR SHADE
The so 1 ar shade factor is a combi 1at ion of topographic and riparian
vegetation shading. It is a modifaction and extension of Quigley's (1981)
work.. It distinguishes between topographic and riparian vegetation shading,
and does so for each side of the stream . It was modified to include the
intensity of the s o lar radiation throughout the entire day and is completely
con s istent with th E: heat flux components used with the water temperature
model.
Topographic shade dominates the shading effects because i t determines the
local time of sunrise and sunset. Rip~r i an vegetation is important for shading
between lo~al sunrise and sunset only if it cast ~ a shadow on the water
surface.
Topographic shade is a function of the : (1) time o f year , (2) stream
reach latitutde , (3) gen eral stream reach azimuth, and (4) topographic a l titude
angle. The riparian vegetatio n is a function of the topographic shade plus
the riparian vegetati on parameters of: (1) height of vegetation, (2) crown
measurement, (3) vegetation o ffset, and (4) vegetation density. The model
al lows for different condit ions on opposite sides of the stream.
The time of the year (Di) and stream reach latitude (~) parameters were
explained as a part of the solar rad i ation section . The remain in g shade
parameters are pecul i ar to determi nation of the shading effects .
The general stream reach azimuth (Ar) is a measure of the average depar-
ture ang 1 e of the stream reach from a north-south ( N-S) reference 1 i ne when
looking south . For streams oriented N-S, the azimuth i s 0°; streams oriented
NW-SE, the azimuth is less than 0°; and streams oriented NE-SW, the azimuth is
gre-:. .. er than 0°. Therefore, a 11 stream reach az i muth angles are bounded
between -90° and +90°.
The east siae of the stream is always on the left-hand side because the
azimuth is always measured looking south for streams located in the north
latitudes . Note that an E-W oriented stream dict ates the east or left-hand
side by whether the az i muth is a -90° (left-hand is the north side) or +90°
(left-hand is the south side).
The topographic al ti tude angle (at) is the vertical angle from a level
line at the streambank to the general top of the local terrian when l ooki ~g 90°
from the genera l stream reach az i muth . There are two a l titude ang l es --one
for for the left-hand and one for the right-hand sides . The alt i tude is 0 for
level plain topography; at> 0 for hilly or canyon terrian . The altitudes for
opposite sides of the stream are not necessarily identical. Sometimes streams
tend to one s ~de of a valley or may be flowing past a bluff line .
The he i ght of vegetation (Vh) is the average max i mum existing or proposed
height of the overstory riparian vegetation above the water surface . If the
height of vegetat i on changes dramatically--e.g., due to a change in type of
vegetation --then sudividing the reach i nto smaller sub~eaches may be
warranted.
---
At
J..-..-SOUTH -------------------
------------------
Ffgure 2 .2 . local solar and stream orientation angular measurements.
Crown measurement (V ) is a function of the crown diameter and accounts c
for overhang. Crown meast:rement for hardwoods is the crown diameter, soft-
woods is the crown radius.
Vegetation offset (V 0 ) is the average distance of the ~ree trunks from
the waters edge. Together with crown measurement, the net overhang is deter-
mined. This net overhang, (Vc /2) -(0 , must always be equal to or greater
than zero.
Vegetation density (Vd) is a measure of the screening of sunlight that
would oterhwise pass thru the shaded area determined by the riparian vegeta-
tion. It accounts for both the continuity of riparian vegetation along the
t stream bank and the filtering effect.of leaves and stands of trees along the
stream. For example, if only 50% of the left side of the stream has riparian
vegetation (trees) and if those trees actually screen on l y 50% of the sunlight,
then the vegetation density for the left-hand (east side) is 0 .25. vd must
always be between 0 and 1.
The solar shade model allows for separate topographic alt :tudes and
riparian vegetation parameters for both the east (left-hand) and west (right-
hand) sides of the stream.
The solar shade model is calculated in two steps. First the topographic
shade is deter:ni ned according to the 1 oca 1 sunrise and sun ! ~t t irnes for the
specified time of year . Then the riparian shade is calculated between the
local sunrise and sunset times .
...... ""-r~--·:>·Y :· .. :· :;·. ' .. :·. :~:.=::_{./·,._....,........., _____ _ ..
Vc = diameter, hardwoods
= radius, softwoods
Vd = ratio of shortwave
radiation eliminated
to Incoming over entire
reach shaded area
Figure 2 .3 . R1par1an vegetation shade parameters.
~ Topographic shade is defined as the ratio of that portion of solar radia-
~ ..
tion excluded between level-plain and local sunrise/sunset to the solar rad;a-
tion between level-plain sunrise and sunset.
Riparian vegetation shade is defined as the ratio for that portion of the
solar radiation over the water surface intercepted by the vegetation between
local sunrise and sunset to the solar radiation between level-plain sunrise
and sunset.
The following math models are based upon the previous rationals. There
are five groupings of these models: (1) level-plain sunrise/sunset hour angle
and azimuth (h and A ), (2) local sunrise/sunset altitude (1%s and l%.s), s so r ~
(3) topographic shade (St)' (4) ripa~ian vegetation shade (Sv), and (5) total
solar shade (Sh). The order is suggested for direct solutions .
Indicator function notation, I[•], is used. If the relationship shown
within the brackets are true. the value of the indicator function is 1; if
false. the value is 0. Definitions for each variable is given after the last
groupting of math mod~ls.
The global condit i ons of latitude and time of year determine the relative
movements of the sun which affect all subsequent calculations . They were
explained i n the solar radiat ~on section. The time of year directly determines
the solar declination, which is the starting point for the following math
models.
LEVEL-PLAIN SUNRISE /SUNSET HOUR ANGLE AND AZIMUTH
The level-pla i n sunrise/sunset group of math models are to determine the
hour angle and corresponding solar az i muth at sunrise and sunset. The solar
movements are symetrical about solar noon; i.e., the absolute values of the
sunrise and sunset parameters are identical, they differ only in sign . The
math mode 1 i s :
6 = 0.40928 cos((2T/365) (172-Di)]
hs = arccos [-(sin ~ sin 6)/(cos ; cos 6)]
Aso =)a rcsin (cos 6 sin hs)
LIt" -o.n:. Z.\.,.... ( (.o ~ & c ;"" '-" :.)
The level-pla i n sunrise hour ang l e i s equa l to -hs; the sunset hour angle
is hs. The hour angles are referenced to solar noon (h = 0). Therefore, the
duration from sunrise to solar noon is the same as from so l ar noon to sunset .
One hour of t i me is equa l to 15° of hour angle .
The solar az i muth at sunrise is -As 0 ; the sunset azimuth is Aso· Azimuths
are referenced from the north-south line looking south for streams located in
the north lat i tudes .
LOCAL SUNRISE/SUNSET ALTITUDES
Loca l s~nrise and sunset i s a funct i on of the loca l topography as we ll as
th~ global conditions . Fur~hermcre, the 1ocal terrain may not be i dentical on
opposite sides of :he stream . Also, some streams are ori~nted such that the
II 1
J
~ sun may ~i se and set on the same side of the stream dur i ng part or even all of
the year. The fo 11 owing 1 oca 1 sunri se/sunset mode 1 s properly account for the
relative location of the sun with respect to each side of the stream .
The model for the local sunrise is:
~tr = ~te I[-Aso s Ar] + ~tw I[Aso > Ar]
hsr = -arccos {(sin asr -{sin ~ sin 6)]/~cos ; cos 6]}
.J .
Asr = -arcsin [cos 6 s~n hsr)/[cos ~sr)]
~sr =arctan [(tan atr) {sin i Asr-Ar l )]
but, sin asr s (sin ~ sin 6) + (cos ~ cos 6)
The model for the local sunset is:
~ts = ate I[Aso s Ar] + a I[A > A ] tw so r
h = arccos {[sin a -(sin ~ sin o)]/(cos ; cos 6]} ss ss ~; '
A = arcsin [cos 6 sin hss~/[ccs ass)] ss
a = arctan [(tan a ) (sin i Ass-Ar i )J ss ss
but, sin ass S (sin ~ sin 6) + (cos ~ cos 6)
The reason for the restriction on the sin asr and sin ass is that the sun
never raises higher in the sky than indicated for that latitude and time of
year regardless of the actual topographic altitude. For example, anE-W
oriented stream in :he middle latitudes could be flowing through a deep canyon
which is casting con tinuous shade for a portion of the winter months.
'
~-
TO POGRAPHIC SHA D ~
Once the leve l -p lain and local sunsrise and sunset times are k.n01~n . the
topographic shade can be computed d i rectly i n closed form . The def i nit i on f or
topograp hic shade 1eads to the follow i ng:
s. = 1
'"
{h -h ) (s~n 9
' SS S!""
h ss .
h s r
S in ~ ,_i\ ..1 • ]
I r. s s i n dn I ::
I -h s
(co s¢ c o s 5) J / iz [<ns sino ;in 6 ) +(s i n h 5 oos o cos 5)11
RIPARIA N VEGE TATION SHADE
The r i par i a n vegetatio n shade r e quires keeping t r ack of the shadows cast
throug~ou t the sun l:ght time because only that port io n ove r the wa t e r sur f ace
is o f interest. The model mus·t acc o un t for sun si de of the stream and the
length o f t he shadow cJst ov er the wa t er . The model is :
V = V I[A 5 s A ] + Vd IrA > A ] c ce r w ~ s r
;
but,
s = 'I f
:,
,"\
' = "
_,.._
• ..I -
! ,..
)
I
vd = v . I(A ~ Ar] + vdw I[A > A J ae s s r
vh = vhe I(A ~ A~] + v Ir A > A J s I hw -s r
v = v I[A ~ A J + v I(A > A ]
0 oe s r ow s r
a ·= s i n-1 [~sin~ sin 6) +(cos~ cos 6 cos h)]
A = sin-1 [(cos 6 si n h) I (cos a)] s
0 ~ 8 s ~ 8
;, l I . J hs ,:; <::"" -~ 't; ~:;-; -. -,.. s ;n \. ·c: -. ~ J-·· ,_
h 5 ) I -n --s ...
-) .... "" -·-·t
-... ---~ .:. •••= ..,. ,_ ...... I :> -~,-·~·-~,-.! . -......... --. CpprOXi~C.:iC :1 i s:
:l:i
...
~:-
( 1/-3 5 1 ' jl -' • . -3 -J :..:1 { _ _,
J \ I l
sin ~· Si:1 o) -(sin h s .::s 9 c:s
E~u a:io ns __ th:--ough __ are used to determine t he jth value of Vd'
3s, and a f or h . = h + j~h. J sr Sixteen intervals, or ~h = (h -h )/16 will ss sr '
gi ve better :han !: ~rec i sion when us ing the trapezoida l rule an d better tha n
.01~ ~r~c ~s ~c n wh e~ us~~g Sim~s o n's rule for functions witho ut discont inuiti es .
44
However, t !"l e fur~C':.i on will have a d iscont in u i t y if t he st ream bec om e s fuli y
shaded due :o r ip ar i a n vegeta t i on a f ter s u nr is e o r be for e su nset .
SOL~R SHADE FACiOR
ihe solar shade fac t or is s i mply the sum of the topographic and r i par i an
vegetation shades . I t i s :
s. + s .. v
S~nce the so l ar decl i n i tion and subseque nt solar r e l ated parameters
depend upon the time of year, i t will be necessa ry to ca l c u late the var i ous
shade fac:ors for each day of t1e time per i od to obtain the average factor for
the time :;~eriods . This wi l l result in shade f ac t o r s completely compa t ib i e
wi t h the heat flux components . This is do n e by :
< st . + • 1
DEF!NITICNS
The 7ol lowi ng de fi ni t ion s pertain to all t he var i ables used in thi s s o l ar
s hace sec-:.io n :
c: -s olJ~ al ~i t u ce , ~ad i ans
a sr -l oc3 1 sunr :se so l ar alt i tude, rad i ans
local sunset solar altitude, radians
eastside topographic altitude, rad i ans
sunrise side topographic alti tude, rad i ans
sunset side topographic altitude, radians
atw -westside topographic altitude, radians
Ar -stream reach azimuth, radians
As -local azimuth at time h, radians
A50 -level-plain sunset azimuth, radians
Asr -local sunrise solar azimuth, radians
A5 s -local sunset solar azimuth, radians
8 -· average stream width , meters
n
N
stream solar shade width, meters
time of year, Julian day
solar declination, radians
solar hour angle, radians
level-plain hour sunset hour angle , radians
local sunrise hour angle , radians
local sunset hour angle, radians
day counter, Julian days
first day in time peri od, Julian days
last day in time period, Julian days
stream reach latitude, radians
total solar shade, decimal
topod r aph ic shade, dec i mal
riparian vegetation shade, decimal
ri ~ar i an vegetation crown factor, meters ; crown d i ameter for
hardwoods, crown radius for soft~oods
; vee -eastside crown factor, meters
vcw = westside crown factor , meters
vd -riparian vegetation density factor, decimal
vde -eastside density, decimal
vct.t = westside density, decimal
vh -riparian vegetation height above water surface, meters
vhe -eastside height, meters
vhw = westside height, meters
vo -riparian vegetation waterline offset distance, meters
voc a eastside offset, meters
vow : westside offset, meters
47
METEOROLOGY
There are five meteorological parameters usee in the instream water
temperature model: (1) air temperature, (2) humidity, (3) sunshine ratio/c l oud
cover, (4) wind speed, and (5) atmospheric pressure. The first four are
expected as input data for a specific elevation in the basin. The meteroology
model assumes adiabatic conditions to transpose the air temperature and
humidity vertically throughout the basin. Atmosphe~ic pressure is calculated
directly from reach elevations. Sunshine ratio/cloud cover and wi nd speed is
assumed constant throughout the basin.
ADIABATIC CORRECTION MODEL
The atmospheric pressure for each reach can be computed with sufficient
accuracy directly from the respective reach elevations ~ The form ula is:
' ;.~. ·~ ' , .. ,\ _..~
P = 1013[(288-0.0065Z)/288]5 •256 ( )
where: P -atmo sphe ric pressure at elevation Z, mb.
Z -average reach elevation, m.
Ai r temperatures gen~rally decrease 2°F for every 1000 ft . increase in
elevation. Therefore, correcting for the meteric system, the following formula
is used :
I
p
I
(O:OO•G.S"b ~ ')-( )5.'251.. 1. D f:>c.j 'J..6 __ £,_-_0~0 6 5 -t
"'2.-~5
J • • ..:
I
I t>o •.·
0 ·-----;--------------
0
I
fJ
I
I ''"'"" ~:~~ ...... .........
XXJ.
''"""' S:88
I -c-o
-"""" """""" ---""" C'OC'OC'O
I~~
I
I
••
I
I
I
I
I
,
(
e o~.~ =
--........ ~ \
--~ y
f =
CP--
qh q -=
r-.; i
1 .
..,.
F~
\(~-~-0 ·
<..
_.r -~-fZ.;, T c. <..
\..o,.,~_,_,,
·~~-,...._. __
V,.J p ... ..
4 1,: :
=
c: :' --
-r:--
~ b . b 0 . I. (i~ ~ 1<:\c
~) (\ -0 -~1 8
?o or
(1 -o .: 7 8 ~a ~)
p, ....
-l . t ••r
r_._.-
Vt~, I ti v ~ .. ; , , \:., (.p-n,_.:\
I' ..,-:-·-' \\ ,;.~ "'"~ \c..:..,
·". ~ r ••....:\.. •ro\ ......__ '-J
. ("'\
t o:_ c~ ;,-. . '/ ~r;-,_
---·--" '-" ....,.. ___ ....._A} -------
f t-(. e ,., .... Gz.:. ~<6 --"-
pi -'7"'-o . ~ ... r ,. (..-
/, i ' c..( ...-(:,{{_ e.-e-
e ---"' .... ,9 . --·-· 't-•
~;f ') r ' ---~. \-' ;
::.
---r:;~ b ob O • 1 ,0 {_.,'-i
-------r-, .. ( .. ::-n · I , Of. ._, ' •
f-rn , -l,q '\
Of..Y
,...,
P~
't:..
r~ ~&~ (-;.S ;~-o.oo '-S" l -1 /z.e>s)
( ~ -:" --0 . a o 6 s-i ~--/ l.S 8) ~. st.
I
I ''"'"" ........... ......... .........
%%% .........
000
111100
I -"
-""' .,.,.,
••
I
I
~t,.·';::
R\,"
C..o.."./E, . _, ;:. ----~Cl o(C-::._o
'<A; e-:.-:
---:-;:.
Cr,r e"--·
r
<2:. <' \<.. ~ 2.7 ~ -e-:.~ ~ • 2-7:.
e :..c-l :c-~-2 7 '':.
j!q -,---
•C> ~-'2 7 ~
..
I
~~ I
I
I "'''"'' ......... ......... .........
:Z:%:Z:
"'"'"' 000
"'00
I -«"4
-('4· .. .,
~r.r. ""('4
I
I
I
• 1
·)
1-oe •'
IOC"'I
-#to t i~, -te e: c...,. = c '.)0 (, ';(..
" __ J'' ·~: '.,
r: ,··-~ \
l_ __ _ ---+------------------
t:> •• 0./ O.<: . . , I .
~ T = T -C (Z-Z ) a o T o ; ( )
-
where: T a -air temperature at elevation E, C
To -air temperature at elevation ~o' c
z -average elevation of reach, m
Z0 -elevation of station, m
CT -adiabatic temperature correction coefficient = 0.00656 C/m
Both the mean annual· air temperatures and the actual air temperature for
the desired time period must be corrected.
The relative hum7dity can also be corrected for elevation assuming that
the total moisture content is the same over the basin and the station. There-
fore, the formula is a fun ction oft~ original re1ative humidity and the two
different air temperatures. It is based upon the ideal gas law .
(T -T )
R = R {(1.0640 ° a ] ((T +273 .16)/(T +273.16)]} ( ) h o a o
where : Rh -relative humidity for temperature Ta, dimensionless .
R
0 -relative humidity at station, dimensionless.
Ta -air temperature of reach, c.
T
0 -air temperature at station, c.
0 ~ Rh s 1.0
The sunshine factor is assumed to be the same over the entire basin as
over the station. There is no known way to correct the windspeed for trans f er
D to the basin . Certainly local topographic features will influence the wind-
~. speed over the water . However, the stat io n windspeed is, at l east, an
indicator of the basin windspeed . Since the windspeed affects only the con-
j
vection and evaporation heat flux components and these components have the
least reliable coefficients in these models, the windspeed can be used as an
important calibration parameter when actual water temperature data is avail-
able.
AVERAGE AFTERNOON MEiEOROLOGICAL CONDITIONS
The average afternoon air temperature is greater than the daily air
temperature because the maximum air temperature usually occurs during the
middle of the afternoon . This model a~sumes that
( )
where: fax average daytime air temperature between noon /sunse~. ,. -...
T ax -maximum air temperature during the 24-hour period, c.
fa -average da il y air temperature during the 24 -hour period, c.
A regression model was se lected to incorporate the significailt daily
meteorological parameters to estimate the incremental in crease of the average
daytime air tempera ture above the daily . The resulting average daytime air
temperature model is
( )
where: T ax -maximum a i r temperature, c.
Ta -da il y air temperature, c.
Hsx -extra-terresterial solar radiation, J /m1 /sec .
Rh -relative humidity, decima 1.
S/S 0 -percent possible sunshine, decimal.
ao thru a, -regression coeffi cients.
Some regression coefficients were determined for the 11 norma1 11 meteor-
ological conditions at 16 selected weather stations . These coeffici ents and
their respective coefficient of multiple correlations R, standard deviation of
maximum air temperatures S .Tax' and probable differences 5 are given in
Table 81.
The corresponding afternoon average relati ve humidity is
(1' -1' )
Rhx = Rh [1 .0640 a ax J[(Tax+273.16)/(Ta+273.16)] ( )
where: Rhx -average afternoon relative hum idi ty, dimensionless.
Rh -average daily relative humidity, di mensionless .
Ta -daily air temperature, C.
Tax -average afternoon air temperature, C.
$.) Table 81
~.
c c
Regression coefficients
Station name R S.Tax 6 ao al a2 a,
Phoenix, AZ .936 0.737 0.194 11.21 -.00581 - 9 .55 3.72
Santa Maria, CA .916 0.813 0.243 18.90 -.00334 -18 .85 3.18
Grand Junction, co .987 0 .965 0.170 3.82 -. 00147 -2 .70 5.57
Washington, DC .763 0.455 0 .219 6.64 -.00109 -7.72 4 .85
Miami, FL .934 0.526 0.140 29 .13 -.00626 -24.23 -7.45
Dodge City, KA .888 0 .313 0.107 7 .25 -. 00115 -5.24 4.40
Caribou, ME .903 0 .708 0.226 0.87 .00313 0 .09 7 .86
Columbia, MO .616 0.486 0 .286 4.95 -.00163 -2 .49 4 .54
Great Fa 11 s, MT .963 1.220 0.244 9.89 .00274 -9.56 1.71
Omaha (North), NE .857 0.487 0.187 9.62 -.00279 - 9 .49 6.32
Bismark, NO .918 1 .120 0.332 11.39 -.00052 -13.03 5 .97
Charleston, SC .934 0.637 0 .170 9.06 -.00325 -8.79 7 .42
Nashville, TN .963 0.581 0.117 5.12 -.00418 -4.55 9.47
Brownsvi 11 e, TX .968 0.263 0.049 9.34 -.00443 -4.28 0.72
Seattle, WA .985 1.180 0.153 -9 .16 .00824 12.79 3 .86
Madison, WI .954 0 .650 0.145 1.11 .00219 1.80 3.96
ALL .867 1.276 0.431 6.64 -.00088 - 5 .27 4 .86
t . HEAT FLUX
THERMAL PROCESSES
There are five basic thermal processes recognized by the heat flux rela-
tionships: (1) radiation, (2) evaporation, (3) convection, (4) conduction,
and (S) the conversion from other energy forms to heat.
THERMAL SOURCES
The various relationships for th~ individual heat fluxes will be discussed
here. Each f s considered mutua 11 y exc 1 us i ve and when added together account
for the heat budget for a single column of water. A heat budget analysis
would be applicable for a stationary tank of continuously mixed body of water.
However, the transport model is necessary to account for the spatial location
of the column of water at any point i n time.
RADIATION
Radiation is an electomagnetic mechanism, which allows energy to be
transported at the speed of light through regions of space that are devoid of
matter. The pnys ical ~henomena caus ing radiation is sufficiently well-
understood to pr~vice very dependable source-component models. Radiation
models have been :.heoretically derived from both thermodynamics and quantum
,\
ALSO : ( 1) liE AT LOSS DUE TO
EVAPOnATIOU
(2) IlEA T GAIN DUE TO
FLUID FniCTIOf~
(3) IlEAl EXCIIANGE DUE TO
I
ATMOSPtiERIC RADIATION
STREAMBED CONDUCTION
Aln ClnCULATION (COUVECTION)
Figure 2 .4. Heat flux sources.
~. physics and have been experimentally verified with a high degree of precision
and reliability. It provides the most dependable components of the heat flux
submodel and, fortunately, is also the most important source of heat exchange.
Solar, back. radiation from the water, atmospheric, riparian vegetation, and
topographic features are the major sources of radiation heat flux. There is
an inter-action between these various sources; e.g., riparian vegetation
screens both solar and atmospheric radiation while replacing it with its own.
SOLAR RADIATION CORRECTED FOR SHADING
The solar radiation penetrating the water must be further modified by the
local shading due to riparian vegetation, etc. The resulting model is:
( )
•
where: sh -solar shade factor, decimal.
Hsw -average daily solar radiation entering unshaded water, J /m~/sec .
H s -average daily solar radiation entering shaded water, J/m=/sec.
ATMOSPHERIC RADIATION
The atmosphere emits longwave radiation (heat). There are five factors
affecting t he amount of longwave radiation enteri ng the water: (1) the air
temperature i s the pr i mary factor; (2) the atmospheric vapor pressure aff ects
ij the emissiv i ty; (3) the cloud cover converts the shortwave so1ar radiation
~ into additional longwave radiation , sort of "hot spots 11 i n the atmosphere ;
(4) the reflection of longwave radiation at the water-air interf ace; and
(5) the interception of longwave radiat i on by vegetative canopy cover or
shading . An equation which approximates longwave atmospheric radiation enter-
ing the water is :
where: c.t = [1-(S/$0 )~/5 -cloud cover, decima 1
S/S 0 -sunshine ratio, decimal
k. -type of cloud cover factor, 0.04 s k s 0 .24
r:a -atmospheric emissivity, decimal
sa -atmospheric shade factor, decimal
r.t -l ongwave radiation reflection , decimal
Ta -air temperature, c
a = 5 .672•10-1 J /m 2 /sec/K~ -Stefen-Bol tzm an constant .
The preferred est i mate of ta is :
ta = a+b lea, decimal
a = 0.61
b = 0.05
--·-1 ------T
I ea = vapor pressure= Rh (6 .60(L 06 40) a],
I --------------------------
( t_oUD , ~-\vt\ 1 !.,. -, (\. \ K.. \
:: <: . .' .. ' )
HuM ''-" =-1-!.J~.,orJ • ( 1. Ob 'l ··"DI) • ( \ --pT I (-r~ 1 K. ¢ t-'27?. !::--))
~\2-: TA 1~¢ -1>1
\)I o.o~ bS"S. ( ~1-E.\l -€.LE.V,t5)
o .riL
3\. ')
An alternate estimate of e:a is:
The preferred estimate accounts for water vapor which also absorbs solar
radiation which, in turn , is converted into longwave radiation . If the
absorbtion of solar is overpredicted, then some of the overprediction is
returned as longwave and vice versa. Therefore, errprs in one (solar) tend to
be compensated by the other (atmospheric). The alternate form is mentioned in
the literature as a simpler model and possibly a better predictor of longwave
radiation alone. However, for purpose of predicting water temperatures, it
ultimately makes l i ttle difference as to the form of radiation (short or
longwave) as long as the total heat exchange fs accurately predicted. The
alternate form is only used when the solution technique requires simple steps.
Ass ·Jming k = 0.17, rt = 0.03, and using the preferred estimate of e:a ,
this equation reduces to :
The atmospheric shade f actor (S ) is assumed to be identical to the solar a
shade factor (Sh).
TOPOGRAPHIC FEAT URES RADIATION
Currently, the radiation from topographic features is assumed to be
included as a part of the riparian vegetation radiation. Therefore, no
separate component model is used.
RIPARIAN VEGETATION RADIATION
The riparian vegetation intercepts all other forms of radiation and
radiates its own. Essentially it totally eliminates the estimated shade
amount of solar, but replaces the other longwave sources with its own longwave
source. The difference is mostly in the emissivity between the differ 1 ~nt
longwave sources. The model is:
( )
where: tv -vegetation emissivity= 0 .9526 decimal
o -Stefan-Bo l tzman constant = 5 .672•10-, J /m:/sec /K~
H v -riparian vegetation radiat i on, J /m2 sec
s v -r i pari an vegetat i on shade factor , decimal
T -r i par i an vegetat i on temperature, assumed to be the ambient a air temperature, c
The riparian vegetation shade factor (S ) is assumed to be identical to the v
solar shade factor (Sh).
WATER RADIATION
The water emits radiation and this is the major balancing heat flux which
prevents the water temperature from increasing without bounds. The model is:
A
Hw = two(Tw+273.16)' ( )
where:
, ...
radiation, J/~2 /sec Hw -water
T -water temperature, c w
t w -water emissivity= 0.9526 decimal
o -Stefan-Boltzman constant = 5.672•10-1 J /m1 /sec/K'
A first-order approximation to equat~on A36 with less than ± 1.8% error
of predicted radiation for OC ~ T ~ 40C is: w
" Hw = 300 + 5.500 Tw ( )
" where: Hw -approximate water radiation, J/m 1 /sec
Tw -water temperature, C
STREAM EVAPORATION
Evapora t ion, and its counterpart condensation, requires an exchange of
heat . The isothermal (same temperature) conversion of liquid water to vapor
requires a known fixed amount of heat energy called the heat of vaporization.
Conversely, condensation releases the same amount of heat. The rate of evapora-
tion --the amount of liquid water converted to vapor--is a function of both
58
the circulation and vapor pressure (relative humidity) of the surrounding air .
If the surrounding air were at 100% relative ·humidity, no evaporation would
occur. If there were no circulation of air, then the air immediately above
the water surface would quickly become saturated and no further net evaporation
would occur .
Evaporation, while second in importance to radiation, is a significant
form of heat exchange. Most available models are derived from lake environ-
ments and are probably the least reliable of the thermal processes modeled.
However, one mode 1 was derived from a sing 1 e set of open channe 1 flow data.
Both model types are offered . They differ only in the wind function used.
The wind function for the flow-type model was adjusted by approximately 3/4 to
better match recorded field data.
Two evaporation models are available. They differ only in the wind
function assumed. The first is the simplest. It was obtained largely from
lake data, and is used only for small hand held calculator solutions tech-
niques . The second is the preferred. It was obtained from open channel flow
data, and is us ~d for all but the simplest solutions technique.
The lake-type model is:
T T
He= (26 .0Wa)(Rh(l.0640) a -(1.0640) w] ( )
~ The flow-type model is:
T T
He = (40 .0 + 15.0Wa)[Rh(l.0640) a -(1 .0640) w] ( )
where: He : evaporation heat flux, J /m.z /sec
wa -wind speed, m/sec
Rh -relative humidity, decimal
Ta -air temperature, c
Tw -water temperature, c
CONVECTION
Convection can be an important source of heat exchange at the air-water
interface . Air is a poor conductor, but the ability of the surround i ng air to
circulate , either under forced cond i tions from winds or freely due to temper-
ature differences, constantl y exchanges the air at the a i r-water interface.
Convection affects the rate of evaporat i on and, therefore, the models are
related. But the actua l heat exchange due to the two di f ferent sources are
mutua ll y exc lusive . Convection is not quite as im portan t as evaporation as a
source of heat flux but is still s i gnificant. The ava il able models suffer
from the same defects since both use the same circulation model .
The heat exc hange at the air-water interface i s due mai n l y to convection
of the air . Air is a poor conductor, but the abi 1 i ty of the atmosphere to
convect freely constantly exchanges the air at the air-wate r i nterf ace. The
c~rrent models ar~ largely based upon lake models but wi ll be used here . The
convection model is based upon the evaporation model using what is called the
Bowen ratio; i.e.
Bowen ratio= Bf P(Tw-Ta)/(es-ea) ( )
where: p -atmospheric pressure, mb
T -water temperature, C w
Ta -air temperature, C :
e s -saturation vapor pressure, mb
ea -air vapor pressure, mb
Bf -Bowen ratio factor
Air convection heat exchange is approximated by the product of the Bowen
ratio and the evaporation heat exchange:
where: He -air convection heat flux, J /m2 /sec
R -Bowen ratio, decimal
He -evaporated heat flux, J/m 2 /sec
( )
Since the air convection heat flux is a function of the evaporation heat
flux, two models are offered. The first. the simplest, is a lake-type model.
The second, the preferred, is a flow-type model.
The lake-type model is:
( )
61 '\
The flow-type model is :
where: He -air convection heat flux, J /m2 /sec
wa -wind speed, m/sec
p -atmospheric pressure, mb
Tw -water temperature, c
Ta -air temperature, C
STREAMBED CONDUCTION
Conduction occurs when a temperature gradient a temperature difference
between two points --exists in a ma~erial medium in which there is molecular
contact. The only important conduction heat flux component is through the
streambed. The thermal processes are reasonably well-understood although some
of the necessary data may not be easily obtained without certain assumptions.
However, the importance of this component, while not negilible, does allow for
some liberties and suitable predictions can be made for most applications.
Streambed conduction is a function of the difference in temperature of
the streambed at the water-streambed interface and the streambed at an equilib-
rium ground temperature at some depth below the streambed elevation , this
equilibrium depth, and the thermal conductiv i ty of the streambed material.
The equation is:
( )
62
where: Hd = conduct i on heat flux, J/m 2 /sec
Kg-thermal conductivity of the streambed material, J/m /sec/C
Tg -
Tw -
streambed equilibrium temperature, C
streambed temperature at the water-streambed interface,
assumed to to be the water temperature, C
Alg -equil i brium depth from the water-streambed interface, m
Kg = 1.65 J/m/sec/C for water-saturated sands and gravel
mixtures (Plukowski ~ 1970)
STREAM FRICTION
Heat is generated by fluid friction, either as ·work done on the boundaries
or as internal fluid shear, as the water flows downstream. That portion of
the potential energy (elevation) of the flowing water that is not converted to
other uses (e .g., hydroelectric generation) is converted to heat. When ambient
conditions are below freezing and the water in a stream is still flowing, part
of the reason may be due to this generation of heat due to friction. The
available model is straight-forward, simple to use, and solidly justified by
basic physics . However, fluid friction is the least significant source of
heat flux, but it can be noticeable for steep mountain streams.
The stream friction model is :
where: Hf -fluid f riction heat flux , J /m2 /sec
Sf -rate of heat energy conversion, generally the stream
gradient, m/m.
63
( )
Q = discharge, ems.
B = average top width, m
NET HEAT FLUX
The various heat flux components, when added together, form the net heat
flux equation, i.e.,
H = H + H + Hd + H + H + H - H ( ) n a c e ·s v w
where: Ha• etc . are as previously defined
Hn -net heat flux
When the equations for the separate components are substituted into
equation 01, it can be reduced to:
T
H = A(T +273 .16)~ + BT + C (1 .0640) w-0 ( ) n w w
where: A = 5 .40•10-1
B = (C • C P) + (K /~Zg) r e g
C = (40.0 + 15 .0Wa)
0 = H + Hf + H + H + (C • Ce PTa) + a s v r
C = a + bW + c 1-w-e a a
64
The equilibrium water temperature Te i s defined to be the water tempera-
ture when the net heat flux is zero for a constant set of input parameters;
i .e.,
T
A(Te+273.16)• + BTe + C (1.0640) e -0 = 0
where: A, B, C, and 0 are as defined above.
( )
The solution of equation 03 forTe, given A, B, C, and 0, is the equilib-
rium water temp t>rature of the stream for a fixed set of metero 1 ogi c, hydro-
logic, and stream geometry conditions . A physical analology is that as a
constant discharge of water flows downstream in a prismatic stream reach under
a constant set of meterologic conditions, then the water temperature will
asymptotically approach the equilibrium water temperature regardless of the
initial water tempe r ature.
The first order thermal exchange coefficient K1 is the first derivative
of equation 02 taken at T . e
T
K1 = 4A(Te+273.16)l + B + [Cln (1 .0640)] (1.0640) e ( )
where: Te, A, 8, and C are as defined above .
The second order therma 1 exchange coefficient is the coefficient for a
second order term that collocates the actual heat flux at the initial water
temperature (T 0 ) with a first-order Taylor series expansion about Te .
T
K2 = {(A(T 0 +273.16)• + BT 0 + C(l .0640) 0
-D)-(K 1 (T
0
-Te)]l/((T0 -Te)2 ] ( )
65
; where: A, 8, C, 0, K1 , T0 , and Teare as defined before.
Therefore, a first-order approximation of equation 02 with respect to the
equilibrium temperature is
( )
And a second order approximation of equation 02 with respect to the
equilibrium temperature is
Hn = K1 (T - T ) + K: (T -T ): ( ) e w e w
HEAT TRANSPORT
The heat transport model is based upon the dynamic temperature -steady
flow equation. This equation, when expressed as an ordi11ary differential
equation, is identical in form to the less general steady-state equation.
However, it is different in how the input data is defined and in that the
dynamic equation requires tracking the mass movement of water downstream. The
simultaneous use of the two identical equations with different sets of input
is acceptable s~n-<= the actual water temperature passes through the average
daily water temperature twice each day --once at night and then again during
the day . The steldy-state equation assumes that the input parameters are
constant for each 24-hour period. Therefore, the so l ar radiation, metero-
logical, and hydrology parameters are 24-hour averages . It follows, then,
that the predicted water temperatures are also 24-hour averages. Hence, the
term "average daily" means 24-hour averages --from midnight to midnight for
each parameter .
The dynamic model a ll ows the 24-hour period to be div ided into night and
day times. While the solar radiation and metero logical parameters are
different between night and day, they are still considered constant during the
cooler nighttime period and diffe rent, but still constant, during the warmer
daytime period. Since it is a steady flow model, the discharges are constant
over the 24-hour period.
It can be visualized that the wat~r temperature would be at a minim um at
sunrise, continua lly ri s e during the day so that the average daily water
c-
temperature would occur near noon and be maximum at sunset, and begin to cool
so that average daily would again occur near midnight and return to a minimum
just before sunrise where the cycle would repeat itself .
The steady-state equation, with input based upon 24-hour averages, can be
used to predict the average daily water temperatures throughout the entire
stream system network. Since these average daily values actually occur near
mid-night and mid-day, the dynamic model can be used to track the column of
water between mid-night and sunrise and between noon and sunset to determine
the minimum nighttime and maximum daytime water temperature respectively. Of
course, the proper solar radiation and meterological parameters reflecting
night and daytime conditions must be used for the dynamic model.
ihe minimum/maximum simulation requires that the ups~ream average daily
water temperature stations at mid-night/mid-day for the respective sunrise/
sunset stations be simulated. This step is a simple hydraulic procedure
requiring only a means to estimate the average flow depth.
DYNAMIC TEMPERATURE -STEADY FLOW
A control volume for the dynamic temperature -steady flow equation is
shown in Figur'! Al. It allows for lateral flow. To satisfy the fundamental
laws of physics regarding conservation of mass and energy, the heat energy in
the incoming waters less the heat energy in the outgoing water plus the net
heat fl~x across the control volume boundaries must equal t he change in heat
,1
10 c
"'1
fl)
N
U1
0
'<
:l
Cl>
3
~.
n
Ill
:l
11)
"'1
10
'<
n
0
:l
ct
"'1
0
<
0
c
3
11)
B
= PCp(OT)I t
pcp(aOT/ax)t\x
energy of the water with i n t he contro l vo l ume . The mat hemat i ca l e xpress i on
; s:
where:
((BtH) 6x]}6t = {(pcp(a(AT)/at)]6t}6x
p = water density, M/L 1
cp -specific heat of water , E/M/T
Q -discharge~ L1 /t
T -water temperature, T
ql -lateral flow, L1 /t
Tl-lateral flow temperature, T
x -distance, L
t -time, t
A -flow area, L1
i -inflow index
o -outf l ow index
8 -stream top wi dth, L
tH = net heat flux across control vo l ume, E/L=/t
note: units are
M -mas s
T -temperature
L -length
t -time
E -heat energy
( )
Equation A38 reduces to :
( )
Assuming steady flow (aA/at=O), letting Hn = B!H, recogn i :i ng q 1 -aQ /ax, and
dividing through by Q, leads t o:
( :.. )
1
-<<_...;;d~y....;;n..;:;.;am;.;.;..i;..;c;__> I <--...:s:..::t;..;:e.;::;a..;:;.;dy"---s:..t;.;:a;.;:t;..;:e:,_;;,ea~u:::..;a:..t:...:i...;;o;,;,;n ___ >
term
__ d;;;;.,jy..,;_n...:a;,..;.m;,..;.i .;::;c_t.;;..e:..m;.;.;;o;..;;e;..;.r...;;a..;;_t..;:;.;u.;_re;;;._-__;;s...:t...;;e..;:;.;a d.;;;.yc___.;f;....l...;;o...;..;w......;;;..e a;;..u;;;..;a:..t:..i...;;o;,..;.n_>
If the dynamic temperature term is neg l ected (aT /at= 0), then the steady-
state equation is left. Since the steady-state equation contains only a
single independent variable x, it converts directly i nto ln ordinary differ-
ential equation with no mathematical restrictions:
( )
If the dynamic temperature term i s no t neglected (aT, 3t ~ 0), then eq ua-
tion A40 can still be solved using the classica l mathemat ical technique known
as the "Method of Characteristics". If, for notional purposes only, we
substitute
( )
into equation A40 and use the definition of the total derivative for the
dependent var i ab l e T , a res ul t i ng pa i r of depe ndent s i mu l ~aneous f i rst-order
part i al diff erentia l equat i ons emerge
(A/Q) (aT/at) + (1) (aT /ax) = + ( )
(dt) (aT /at) + (dx) (aT/ax) = dT ( )
Since the equations are dependent, the solution of the coe f ficient matrix is
zero; f . e. ,
(
(A/Q)
dt
--
1
] = 0
dx
which leads to the characteristic line equation,
dx = (Q /A)dt
For the same reason, the so l ut i on matrix is also ze r o; i .e .,
which leads to the charac~eristic integ r al equation,
when t is replaced by its orig i nal terms of equation A42.
( )
( )
Equation A46 is identical i n form to equation A41 , and is valid for
dynamic temperature conditions when solved along the c haracteristic line
equation (equation A45). This presents no spec i al problem since equat i on A45
simply tracts a c~lumn of water downstream--an easily s i mu l Jted task .
Clo sed-form so lutions f or the ordinary different i al equation forms
(equations A41 and A46) of the dynamic temperature-steady flow equations are
poss i ble with two important assumptions : (1) uniform flow exists, and
(2) first and /or second order approx im ations of the heat flux versus water
temperature relationships are valid.
FIRST-ORDER SOLUTIONS
First-order solut i ons are possible for all three cases of q1 : Case 1,
q1 >0; Case 2, q 1 <0; and Case 3, q1=0.
The ordinary differential equation with the first-order substitution is:
( )
Since Q = Q
0
+ q1 x, equation 08 becomes
~
[Q 0 + q 1 x] dT /dx = ([q 1 T1 ] + [(K 1 8)/(pcp)]Te} -(q 1 + [(K1 W)/(pcp)]}T ( )
Then 09 becomes
(Q 0 + q~x) dT /dx = a -bT
Using separation of variables,
and the solution is
Case 2, q1 < 0:
dx
Q + q X 0 f.
If q1 < 0, then T1 = T and equat io n 08 becomes
The soluti on is
Case 3, q1 = 0 :
If q 1 = 0, t he n Q ; Q(x) and equation 08 becomes
( )
( )
( )
( )
( )
The solution is
( )
SECOND-ORDER SOLUTI ONS
A second-order so lution for case 3 i s as follow s .
Let q1 = 0 and using equation A48 results in
( )
The solution is
(Te -T0 ) exp [-(K 1 Bx 0 )/(pc 0 Q)] T = T -___ ____;;....__.;:_ ______ ~----"------
w e ( )
1 + (K 1 /K 1 ) (Te -T0 ) {1 -exp [-(K 1 Bx 0 )/(pcPQ)]}
Using the first-order solution and making second-order co rrections according
to the form suggested by equation 018 resu lts in
Case 1. q>O :
I
Te = a/b
Case 2 . q<O :
I
T = T e e
Case 3 . q=O :
I
T = T e e
w
0:
::> r-
<{
a: w
a.
~ w ....
~
<{
w
0: r-cn
EQUILIBRIUM TEMPERATURE ----------------------
~ INITIAL WATER TEMPERATURE
0
LONGITUDINAL OIST ANCE
figure 2.6 . Typical longitudinal water temperature profile
predicted by heat transport equation.
TIME PERIODS
The basic math model for the overall basin network. i s a steady-state
model because it assumes that the input is a constant over an indefinite
period of time. Conceptually it assumes that the input conditions exist
sufficiently long for the steady-state results to reach the lowest point in
the network.. If the travel time from the upstream most point to the down-
stream end of the network. becomes significant compared to the time period,
then the results become less reliable.
If the travel time to the lowes~ point is 30 days, it should be
recognized that the water passing this point on the first day of the 30 day
period originated upstream 30 days prior. Therefore, the meterological condi-
tions that determine downstream daily water temperatures on the first day are
not included in the t i me period averages . In fact, only the last day's water
column was infl uenced ent i rely by the metero l ogic data use d in the input for
the time period.
One way to overcome t~is problem is to redefine the time periods to
smaller increments (as small as a day if necessary) and track each day's water
column movement using the previous day's results as the i nit i a l c onditions for
the current day .
DIURNAL FLUCTUATIONS
The following relationships can be solved explicitly at any study site or
point of interest to determine the maximum temperature rise of the water above
the average. It is based upon the fact that the water temperature passes
through the average values twice each day. That the average water temperature
occurs approximately half way through the day. That the remainder of the day
the water temperature increases steadily to a maximum close to sunset. The
same logic is used for determining the minimum water temperature by substitu-
ting nighttime conditions in lieu of daytime.
d = {[(Q /B)n]/[IS ]}315
e ( )
( )
( )
( )
where: d -average flow depth, m.
n -Manning•s n-value .
Q -discharge, ems.
8 -average top width, m.
s -energy gradient, m/m. e
t -X
travel time from noon to sunset, sec.
s -0
durat ion of possible sunshine from sunrise to sunset, hours.
Ted -equilibr ium temperature for average daily condi:i ons, c.
T ex -equil i o riOJm temperature for average daytime con di tions, c.
0
T ox
Twx
-average daily water temperatu re (at solar noon) at point of
interest, C.
-average dai l y water temperature at travel t im e d i stance upst~eam
from point of interest, C.
-average max i mum daytime water temp erature (at su nset) at pcint
of interest, C.
first order thermal exchange coeffici ent for dai ly conditions,
J/m 1 /sec/C.
Kx -first order thermal exchange coeffici ent for da yti me condi ti ons,
J/m1 /sec/C .
p =density of water= 1000 kg /m2 •
cp -specific heat of water= 4182 J /kg/C.
Because of the symmetery assumed for the daytime conditions, it is only
necessary to calculate the difference between the maximum daytime and average
daily water temperatures to obta i n the mi nimum water temperature.
where : T wn
Twx
T = T (T wn wd -wx -T ) wd (
-average m1n1mum night im e ,wa t er t emperature (at sunrise) at
point of interest, C.
-average maximum daytime water temperature (at sunset) at
point of interest , C.
-average da il y water temperature (at solar noon) at point of
interest, C.
.·
.,,.
)
FLOW MIXING
The equation for determining the final downstream water temperature when
flows of different temperatures and disc harges met at junctions , etc. is:
where: water temperature below junction
water temperature above junction on the mainstem
(branch node)
water temperature above junction on the tri butary
(terminal node of the t~ibutary)
Q8 -disc harge above junction on the mainstem (branch node)
discharge above junct i on on the tr i butary (~erminal
node on the tr i butary)
( )
REGRESSION MODELS
Regression modesl are commonly used to smooth data and /or fill-in missing
data. They are used as a part of the instream water temperature model:
first, to provide ini tial water temperatures at headwaters or point sources to
start the transport mode; and second, as an independent prediction of water
temperatures at interior network points for purposes of validation and calibra-
tion. Obviously, regression models are only useful at the points of analysis
and cannot be used in lieu of longitudinal transport. Two regression models
are included in the instream water temperature model package : (1) a standard
regression model, and (2) a transformed regression mode l . Each requires
measured or known water temperatures as the dependent variable along with
associated meteorological, hydrological, and stream geometry independent
parameters. However, the standard regression model requires less detail than
the transformed. The standard model i s satisfactory for most appl i cations,
but the transformed version has a b~tter physica l bas i s. Th e cho i ce becomes a
matter of judgement by the responsible engineer/sc i entist.
STANDARD REGRESSION MODEL
IFG studies during the model development have shown that the following
simple linear multiple regression model provides a high de~ree of correlat i on
for natural condit i ons . The model is:
1\ T = a, + a 1 T + a 1 W + a 1 Rh + a~ (S I S ) + a, H + a, Q w a a · o sx
81
A
where: T -estimate of water tem perature, c w
a,-a, -regress i on coefficients
Ta -air temperature , c
w a -wind speed, mps
Rh -relative humidity , dec i mal
S/5 0 -sunshine rat i o, decimal
Hsx -extra terrestri al solar radiat i on_, J /m%/sec
Q -discharge, ems
It is rec omme nded tha t the meterological parameters an d the solar r adiation at
the l"!eterological stati on be used for eac h regress i on anl l ysis. Obvious l y ,
the discharge, Q, a nd the dependent var i ab l e water temoeratures must be
obtained at t he po int of analys i s .
These six independant var iab l es are read il y obta i nab l e and are a l so
neces ~ary for the transport model . A mini mum of seven data sets are necessary
t o obta i n a solut i on. Howe ver, a great er number i s des irable for s t atist i cal
va l i dity . Also, it needs to be emphas ized that the result in g regress i on model
is only val i d at the point of analysis and only if upst r eam hydrologic cond i -
tions do not c hange . For .examp l e, if a reservo i r has been con structed upstream
subsequent to the data set, t he mode 1 is not 1 ike l y to be va 1 i d because t.he
rel e ase temperatures have been affected .
TRANSFORMED REGRESSION MODEL
The best regression model would be one that not only uses the same
parameters as the best physical-process models; but has the same, or nearly
the same, mathematical form. That is, the regression model equation uses
physical-process transformed parameters as the independent variables. This
transformed regression model uses all of the input parameters used in the
transport model except for stream distance and initial water temperatures.
The first-order approximation of the constant-di sch~:>q e heat transport
model was chosen as the basis for the phys i ca]-prccess regression model.
Water temperature and discharge data at t ne specif i ed locat ion together with
the corresponding time period metero l ogic data fr-om a nearby station are
needed. The meteorologic data is used to determine the equilibrium tempera-
ture (Te) and first-order thermal exchange coeff;c i ent (K1 ). The Te and K1
are combined with the corresponding time period di scharg€:s as independent
variables to determine the regression coeffici.ants for est i mati ti g the corre-
sponding time period water temperature dependent variab l e. An estimate of the
average stream width 'II above the site location is necessary as an arbitrary
constant in the regression. The resulting regression coefficients are tant-
amount to synthetically determining an upstream source water temperature as a
function of time and the distance to the source.
The constant discharge neat transport model is:
( )
where: T -eo ui{ib r ium water temperature, C e
T, -i nit i a l water temperature, C
Tw-water t emperat.ure at x0 , C
K1 -first-order therma i exchange coeffic i ent, J /m2 /sec/C
B -average stream width , m
Xo -distance from T,, m
p = water density = 1000 k.g/m,
c -specific heat of water = 4182 J/kg p
Q -discharge, ems
X
The definition of exp (x) = e is
( )
If T0 is a function of the time period on ly, then it can be approx im ated
as
Ta =fa + 6T 0 cos((2rr/365) (D.-213)]
l
( )
where : To -dVerage initial water temperature over al l time periods ; c
6T 0 -ha 1f in it i a 1 t emperat ure range over all t im e peri ods ; c
D. average Julian day fo r .th time per iod; Ja nuary 1 = 1 and -1
l
December 31 = 365..
Let, Z1 = - (K1 B)/(pcPQ) ( )
Zz = -J e ( )
Z, = cos [(21T/365) ( D; -213)] ( )
84
If equations C2 through CS are substituted into equation Cl and the terms
rearranged, then Tw can be expressed as:
( )
If the converging power series is truncat~d after the final fourth-order term
and the following substitutions are made, then a possible multiple linear
regression model results.
Let , a, = 1',
al = ~T, xl = Z,
az = T,x, X:~ = zl
a, = ~T.x, X, = zlz,
a., = :<, x~ = z1z:
a, = T,x,2 /2 Xs = zl 2
a~ = AT,x,2 /2 X, = zl:z,
a, = Xo 2 /2 X,.= Zl 2 Z1
41 , :.'! T,x,1 /2 X, = z\ l
a, = AT,x,)/6 X, = Z 1
1 Z,
a111 = Xo 1 /6 xlO = Z l ,Zz
• T,x ,"/2 X11 zl .. , a 11 = =
SSt
,
If the resulting independent transformed variables X1 , through Xll are
regressed on the dependent variable T , then the following regression equation w
results
( )
The best estimates of the synethic physical-process parameters are
f. = a, ( )
~To = al ( )
'Xo = a .. ( )
86
Attachment 2
HEAT FLUX COMPONENTS FOR AVERAGE
HAINSTEM SUSITNA CONDITIONS
19:'0
300
tzZT/Z/A I 00
.nso
8
rZ%2 !
-lOO
-288
-380
-490
ATI'IOSPHCP.IC
4 (11)
1970
300
rz-'l.V~·\a
1~77 ~CJO
fZ2/zzza 101)
11£'0
f-p::::;:-r-
1';
/ " ,.
1-, '
V, / / ':.I , rr. /
f-I' ~ /
/
0
E / //I
-INJ f-
-Z90
-:aN
_ .. ,.,
(, v'/ \-.· \ ( r. , • • ' ':..;) r. r , .. (· f
6 ~
SUSITNR RIVER HE~T FL II~:-:1
-I I
SOLAR
JUliE
F"R ICTIOII , COHDUCTIO!t EV~~ilRATIOII
COHP OHEHT
SUSITNR RIVER HE~T FL II~:-~ -I I
JUL'o'
/
[T :~ r;: .. · . . •
,~Pln .. / .' ..
t1l..J
-
JACK P.AII
l :1 . . /I ' / ' ·: . ~j ··· . ~
~ l '-'--
-
)
"":. ;:)/./,-:-c:: /=' r.
SUSI TNR RIVER HERT FLU X
~(1 0
1970
f>&;X&"$&1
300
1977 200
nso • tZZZI
-100
-298
.~
-308
ATI1 0 SPH£P.IC SOLAR r R !CTIOH -COU DUC TI OH EV ~PO RATIOII JAC K RA D
C011?0H £tlT
SUSITN~ RIVER HERT FLU X
SE PT£11BE P
~1)0 > -:
1979 ':
300
f;x.~~1
1977 200
fV/@_2 100
1939
0
E / //I
-1 00
r
~vr-... r~
f-._-: v" . ~~ . ;.:v_ ~v. r:J flR nr1 n / .. · .. --.._ ~~ . L...J .,
)( ~ ,·
'·.
-2(10 ~ :.. .. ,
~:
-:,oo .. ; -~..:..__
-~ •"> f P; 0:. T I u ll
O:O II P I)II(III
Attachment 3
WEATHER WIZARD DATA
H4.CJ P .
1111/1 U I
lltt'fi<MIIK
J{ll 1'\JIIII
llLC. Cl
,, ~~ ()I". I
UfliU ~Hl D
R\H C006U. T~T, Itt:. SUSJTNA HYDAOEL(CTRJC ~J(CT
~~~?-~T-T-~~~~~~~~~~,_,-,-,-,-,-~~~-r-r-r-r-r-r-r-,t'l
n
ta SQ.IIl
-4a RADII-ITIW
~ ftiU'CHlJ •
--.J::.~ ___ :._ _________________________ :=~~---Traced from R&N
Processed
Climatic Data ~ VUfv~~ ~tj~~~~i~Y.uVMJ~t~:;;~ Rll..ATIV£ Vol. 6 ·
•u11DITY Devil Canyon
lXI Station
-----------------------------··-------------------------------------
• -·-·-·-·-·-·-·-·-·-·--1-·-·-·-·-~-r ··-·"T•-r ·-·-·_..,.-·-·-·-·-·r-·~.-·~· !. :: lr ! ; ~~ ~J •: I··~ 1 1 ; • ~ ; I • • : :: --:: :! J. I :: : ;: .: ;; : ~ :;! ~~:: l ~~~ ·~ . :: ~ ii 0 0 to j ;. ~~ :j : ; ~:!.}·~ :-:. )'t : -r-·.:,.·,..-··t,. tt : :::,:;~ ·>·· ·• ~-•'\,...;":..:· ::t.r :...:l.t, :~ .,. ....... ....,'·.,: .r': .... ~·-~·''-'' · "'! =,-J ::;•~: t6 Ult '" · .... · .. · "' · ........ : .. §.-.. · ··· , a·1. , .•• •' ... -• • ••••• , :,• w ::::::.a :: ~:~: :.~: :~ .;· :; < : .:,;~ :s : · ::...-: : ::~ ::1 ·:: :; f :::! •t ::: 1 ; :··1 .: : i· ~ : :~ l : :f :~, : ::-. ::; 1! 144 Dtf<L:CfiOI
• • • • • 0 ,.. • • ... • t I .... I ' "' .. f '. .. • .• :, · ! : ,. • ·.;, : :; i 1 ! t : ~ I ~ t; :: 72 f U:G I
. .' ... : .. ~ ............................ :.! . .!. .•.•. 1 ...... ! ....... · ...... :.: .. , .... 1]: ............ t ...... : ........ · .... 1 ... ) ....... l ..... -~ 1
ltlr.i I -4
~~~~~~~~~~~~~~~-4~~-L-C~~~~~~~~~~~~~
5 11 25
~TtviA LI:AT~ STAT 1()4
D"•TA STrllTI It .1\.1£ • l!Jlt
.!
From R&H Proc essed ClimAtic Data, Vol. 5, Watana Station
· Fi ~ure 1
l[)i•ftlfoTIF(
IJ U I~Jltl l
II.LC. (I
11 (•' ·~'J'
Ill liD ':.liT U
1 rv:; 1
RVt C().ISU. T~T, rue. SuSll~ H'l'l'J\GCL(ClHIC 1-'HOJ(CT
r--~~-r-T-,--r-~,--r-r~-,r-r-~,--r-r-o~r-r-,--r-r~-,--r-~,--r-r-o-,r-11~~
---------·-·O"T·--------------------·-·-·-·-------------l·---·1·-·n·----: :.: :I : . ; ~ : ~ I : : . :: -:-: : . : . ·: .. ... • • • : f .. -. . . . '.
. •. ~1 • ~~ 1 J·. . . .. . ~ . . ~ f: . J ( . ;, •. t:•;:'-'·"'"''·'··~··.:1:-~~..; ~ r·.;,.:v.... _: • .,-.-.·.-~;·, ... : ~"r.'j :;!': ~-·. f""'{ 1: ..... ,. •. v . ~.) .. ~ ...... :.~ t}·' .. :;:;'._. ~J :.._ .•.• :::"1 ..... ~ .. : .. ::.!: .'!~-.. • ...... ! .... :::·:: .......... : ... : ........ : ....... : .. ~--...... 7: ... : .. :.: .. •. : .... :.:::.:': ... .!. ;) ........... .
ll
MTC\ SUe1TI Cl t()V(I()(;t , t!}U
Fro m R&H Processed Climatic Data, Vol. 5 , {"a tana Station
Figur e 2
u
u sa.r .. 1
-4~ n:,DIATrc.;l
2~ lhl/00)
e
I
R£LAT IV[
1-t.J II D IT Y
ll( I
Ultl.lJ
·'
Traced from
R&M Processed
Clima tic Da t a
Vol. 6
D ~vi l Ca nyon
Station
1
I
l~tJP,
lrtV\kl
lft11.1~fl.H:
l.(UKJIIII
I LCG C I
1111' ur->1
Ulti.l ~'(TD
ltv~ I
---------------.. -----------
lfU1 C() 6U. T f-1 IT, II.C. !>USITt~ ttYu;OCl[CTRlC PI\OJ[CT
-:;~
-~r-;;:-·-~·"1··-·-·-·-·-:-·-·-··-;-:·;·------T.·-·;-·-;··7·-----·-=·-~--~---
( ~: :: .: .. I : • t : :: : • :; t 1' ..
.,. •,• •• .,. " • r J ., • :• ~ ~ '• • ' •• )' 4 • ~ ~ ·.:~.~ ,.,,....,_;..~-,:: .:"":. ::.:.~-..1~;.~.-.:·-:>-:' ~.· ,,_ ·•.: •. ,_ •• _ ;>·i·, :·: 1 J·: , l' .: -:•:'l-\:.. ,.,..~.. ..1. •
• • I • • •• • I . • I . • • • • • ' • • • ' •• I I ·' ' ••• I · :: • :: : :: : =: : ~ ·! :': ·v:: : = ; · : = : ~ ·-/ r ;:1 : i i ; :: · ;_;: ::; ; ·l L~ J · . · i P ..
r ~ ~~~·. '!! :~.:··:~/ ::· f '. -~1 J ·'1 20: ........................................... 1 ................................. -...... : ...................... · ......................... .
t £.
·~t
I .':.. ~ \( : • : ,:-, .... ·, . ! ;... ' L .. ' • :.. ' •
:-•
•
•
•
lll I . • • • ) I . m:vM?J . ·:!!·~ . .;~~ . .1'\.t:j I ':FI .... J.; •• • ;,( ... : ;• • : ~ · vv~ L.n~ril.~.&J~t_....·;;_· ..a.;::"--~w~-· ........ · --~.-~..:....=~.;.t.......:JI-.L.__,
5
DATA STAAT: Ql
l'l
Jt.l.'f
W\Ttl~ I,.CAni:R STATlGJ
19i~
From R&~1 Processed Climatic Data, Vo l. 5, tvatana Station
Figure 3
OWCit21
R£L.AT l'r£
.. U11DIT't
lXI
( U.:G I
Traced from
R&H Processed
Climatic Data
Vol. 6
Devil Canyon
Station
v
'
R~H CONSULTANTS, INC.
SUSITNR HYDRO ELECTRIC PROJECT
WRTANA WEATHER STATION
Augu st , 1981
From R&M Processed Climatic Data, Vol. 5, l~atana St ation
· Figure 4
K.C(II>.
I I V VI~I
l()i'flb'tTl F£
U ~ HJitH
I {.(.(J (I
l'flj: (j(f.,;f
UIIIO ':i{T IJ
IIVS I
-------~ ~~ .... ~ ---
, .. ,
J ..
I C:~
~
(.O -· .J ..
"
MTc. Slr..RT a 11 tDVEI«R • ssaa
From R&M Processed Clima tic Da ta, Vo l. 5, \~a t an a Sta tion
Fi gure 5
Figure 6. Monthly averaged observed relative and absolute humidity data
from R&.M Weather Wizzards in Susitoa basin.
JUNE
10 5 JliT.Y
X 105 AUG
X 10 5 SEPT
Rh p X Rh pv Rh pv Rh p X v v
(decimal) 3 (kg/m ) (decimal) 3 (kg/m ) 3 (decimal) (kg/m ) (decimal)
Talkeetna 1
105 m
1980 .785 8.2 .810 10.0 .833 9.0 .813 6.7
1981 • 713 7.7 .805 9.4 .835 9.1 .785 6.7
1982 .755 8 .6 .790 9.4 .820 9.4 .903 7.0
3-y~ar average .751 8.2 .802 9.6 .829 9.2 .834 6.8
Sherman
198.0 m
1980
1981
1982 .40 4.0 .44 4.9 .. 22 1.8 .35 2 .8
3-year average .40 4.0 .44 4 .9 .22 1.8 .35 2.8
Devil Canyon
457.0 m
1980 .65 7 .6 .54 6 .0
1981 .67 6.4 .78 7 .1 .82 7.6 .66 4 .2
1982 .37 3.5 .43 4 .2 .35 3 .5 .52 3.9
3-year average .52 5.0 .62 6.3 .57 5.7 .59 2 .7
Watana
671.0 m
1980 .so 4.S .47 5.0 .71 s.o
1981 .29 2.7 .37 3.4 .26 1.6 .30 2 .0
1982
3-ye.a r average .40 3.6 .42 4.2 .26 1.6 .so 3 .5
Kosina Creek
792.5 m
1980 .66 5.2 .10 0.6
1981 • 51 4.3 .65 6.1 .56 5.0 .46 2.7
1982 .29 2 .5 .35 3.4 .26 2 .3 .53 3.6
3-year average .40 3.4 .so 4.8 .49 4.2 .36 2.3
1 Data from National Weather Service Local Climatological Data Summary
10 5
3 (kg/m )
Figure 7. Monthly averaged observed tempera ture (o C)
from R&M Wea ther Wizza r d.
JUNE JULY AUG SEPT
Talkeetna 1
105.0 m
1980 11.9 14.7 12 . t 7 .7
1981 12 .2 13 .5 12 .4 7 .7
1982 11.7 13 .7 13 .2 7.8
3-ye ar average 11 .9 14.0 12 .6 7.7
Sherman
198 .0 m
1980
1981
1982 10.7 12 .8 11.6 7 .1
3-year average 10 .7 12.8 11.6 7 .1
Devil Canyon
457 .0 m
1980 13.7 12.5
1981 10.0 9 .3 9.2 3.3
1982 9.9 11.7 10 .8 6 .0
3-yea r average 10 .0 11.6 10.8 4.7
Watana
671.0 m
.... ~ ...
1980 9.1 11.9 4.8
198 1 9.3 9.3 2.0 4.0
1982 8.6 10.8 10.0 5.0
3-year average 9 .0 10 .7 6.0 4.6
Kosina Creek
792.5 m
1980 6 .8 3 .1
1981 8.0 9 .7 9 .0 2.9
1982 8.4 10 .4 9. 1 4.4
3-y ear av erage 8 .2 10 .l 8.3 3 .5
1 Data from Na tiona l \~eather Service Lo cal Clima tological Data Summary
'
Attachment 4
DAILY INDIAN RIVER TEMPERATURES VERSUS
DEVIL CANYON AIR TEMPERATURES
--
If 1
L
I
J ... .,
" t .
._ -~.-
. .. . .
~~
-. -..