~ ...... ~ ~ - - -- .. ~~ -- ...... , - .. - ...... , .... IX. ADYNAMIC HESEHV01H SIMULATION MODEL-DYHESM:5 i\ 311 c. transverse and longitUdinal direction playa secondary role and only the variations) ." I in the vertical enter lhe first order balances of mass, momentum and energy. 1/ I Departures from this Stilte of horizontalisopyc'nalsare possible, but these \ tI l A DYNAMIC RESERVOIR SIl\olULATION MODEL enter only as isolated events or as \I/eak pe.!lurbatiQ.D.S. In both cases the.•net eJJ;cJ,J CI DYRESM: 5 is e~plured wi(h a parame!efizalion of their inp,ut (0 the vertical s(rUelure"iiild , ) I comparison of the model prediction and field data must thus be confined to ~ ~ .....of...... ,.calm when the structure is truly one-dimensional. lorg 1mberger and John C.. Pattetsun .. ~ ,. The constraints imposed by ~uch a one-dimer.:Jional model may best be University of Western Australia quantified by defining a series of non-dimensional llUmbers. The value of the Nedlands, Western Australia Wedderburn number :)
LV =.i.!!.. h (.J" I I '( 14.2 • L- '7 y(l .. n, (I) , \ ..,' I
/ 1. INTRODUCTION where g' is an effective reoufed gravity across the thermocline, h the depth of the mixed layer, L the basin scale, and u· the surface shear velocity, is a measure of """·".',j<}·,t-·;~·'",,,"~~,'ti The dynamic reservoir simulation model, DYRESM, is a one-dimensional the activity within the mixed layer. Spigel and Imberger (I980) have shown thah, numerical model for the prediction of temperature and salinity in small to medium for W > 00) the departure fmm one-dimensionality is minimal and for I ':I sized reservoirs and Jakes. The model was constructed toiorm a suitable basis for 0(11/ L) < W< 0(1) the departure is severe but may be successfully parameter- r 17 I ized. For W < O(h/L) the lake ove!1urns,- " • I, I a more general water quality model. The requirements imposed by this aim are J':~I made by Imberger (1978), who shows that for both substances with a fast tumo,ter Inflows do not lead to severe vertical motions in the bulk of the reservoir I lime such as phytoplankton and for conservative substances such as salt, the mix provided the internal Froude number ing mechanisms within the reservoir must be modeled accurately. Not only must a -- .mOdelbe.capable of predicting, the~v~ Y~rlieal tri!illifer--!!.t:.J!1.!!~.'0!~"~~ ~roi~ U <1 and the subsequent flow processes, but it must also be capable of F; = .Jg'H (2) Iresolving individual mixing regioos. - - -.----. This is a great departure from what has been assumed to date. Previously it I has been generally accepted that once a partkular reservoir model was capable of where U is the inflow velocity, g' the reduced gravity between the surface reser being tuned to correctly predict the vertical temperature profiles over a particular voir Water and the inflow and H the reservoir depth. study period, that this then constituted a reliable model for the purposes of water The outflow dynamics..is governed by a similar Froude number criterion I quality modeling. However, as discussed in Fischer et al., (1979) this is not the case since gif'f~r~I]t paraml;t~r~are dependent on different mixing Qro.9~sses fQ[ f~ = Q their. dis~ributio!l. g,I/2Hs/2 < 1 (3) The problem confronting the modeler is thus difficult. On the one hand the model should be widely applicable and cheap to run, and on the other it must cap' where Q is the outflow discharge and g' the reduced gravity between the surface ture the mixing induced by the small ·scales of motion. The marriage of these and bottom water. seemingly conflicting aims requires careful consideration of not only the concept I Laslly, the earth '5 rotation will not lead to vertical motions if J but also the architecture ora proposed--model. Although increasing availability oi' I . computing power has led to an increased potential for application of two or three- dimensional reservoir models, the required !]§.Ql!Jlio_1h. down to centimeters and g'8 minutes, is not currently possible within the context of long term simulations, and f 2B 2 < (4) I aor.e-dimensional model appears to be the only realistic option. 7 l F • I' -~ >Thelli.Qng stratification normally found in 2.m.all to lUe.t!ium sized reservoirs where g'is the effective reduced gravity over the depth 8, the scale bf the ~lucity A r may be used to advantage. strong stratiJicallon of the water inhibits vertical .toncell1ral.ious~ I ,IS the Coriolis parameter and B is the maximum width of the lake. - I~ c motions and reduces the turbulence to intermittent bursts. For such lakes the I~"··"'·:'I • ~ ~ (-t1I'IIll(hl InKI "l AIl"lt'U!IL'I'I~ ~ III'" • nllllWI '>lOll ::.~J ' 'lill \u ,I. I 1".I,f,I"'- 1Jt,lh\ JO.HI ri • •_,:, ...... ___ I' 4' ••:->4 J ""., '- ' L t t' . " -\~;:fi'l!f~.$$'- rt~~ i~lX~) 1l'~ '~~";. ~~"" ~~.L_~·~." ~"~'~.:...... 1' "'_ ...... '.. r ~~- >t-....-,...:.-.;,._-:.~,. _~~,:~~~-_.;_:~'>:~....--.-. _.;,.._~. ~~,._.~,,--, ~-::""· _-~---c-"i -~ ~"_., ~~. "-~. ~ ~;;.~r't."": .:.:~-r:_',:~" ·?>-,-<~~~':;4.14>4$i.,:j;;'~~,4:t.~;. tf"''''''. . __ ... __ ...... l .. , ...... -Q,'- "" - r, . . ,':'-",'-':' ,;;:/-',) . ~ ~
~ 312 JOHG IMBEHGEH AND JOHN C. PAT'l'E.\SON IX. A DYNAMIC HESEHVOm SIMULATION MODEL-DYRESM:5 313
Within these constraints DYRESM yields an llccurate descriplion of the 2. DY.R.E~M MAIN PROGRAM dynamics governing the verlical structure in a lake. It must, however, always be emphasized that superimposed on the structure predicted by DYRESM there are I 12[£'h&~ The model DYRESM is cocstructed as a main program with :mbroutines \ mixini which have only a second order influence on the vertical slruc which separately model each of the -e.hY~'!!..PJ~~se~ .of inflow, withdrawal, mixed !tures, but which are the first order mechanisms for transporting subslances hor- layer dynamics, and vertical transport in the hypolimnion. In addition, thm'e are a .~. I izontally. These find their origins in _~.!Wan~~~ .b~~timt al!..ne day and ) v' little advantage would be gained. . a ~ariable. syeb-d:illK. time st~PJorJ1!e_ mixing algorllhm. The length of this sub- .I 2lliL9y'ar~~r...baur I: I I I dally step IS determmed by the dynamiCs and ranges between i!nJ I' t -;lith these considerations in mind. the model DYRESM was dey'eloped by ~elye. ho.!:'~!Jf averaged data is used, This procedure allows small time steps Jl.-/, Imberf: "r et al. (1978) with a different approach to the task. Concentrating on 4' when the dynamics so require; in less critical periods, the time step expands "paraJTI.I:!~erizatioJLQL th~'physical proSS&~k~Lrather than numerical solution of the without loss in accuracy. appropriate differential eq'uations, DYRESM makes use of a layer concept, in which the reservoir is modeled as a ,.System..of !!.orizoo.taLJ
n " .. which incorporate the finest scales become a realistic proposition. Ij
, ,-- ~ L~, ••...•.•....••.• t' a.- ,- ,.,.-r----,J""\---.~ ~~ ~ ~ ~ ~ ,;" ...... ~ ~ .. ...~~- .~ I ''"! !>" ....:J...... '. """,,,.-""'Il ilL l ~_~, IL~•.••••••.•·..•....•.••.•••.. r...... ' ...... ~ I-L ,..-.....J •.J . "J I ~ - -~~ _~_.-.: ~I ~! ~J 1! - -" '- -JJ .• -.... '----I L-J e...-Jo ---..J/ --.1/ '- __ • _d '_. 0 ~. ~~ _~.-~ _---~----_ ~ ~. ---·-\\·.,~--_·_,.,.--_·· ·_-~_· ...... '...... -.. ..- ..._.•...... __.__ ...... - ..._- __···-··-··_·-...... --·__ __·-c·--·-'--'.. . l .. _ ._-_.-_ _ \\.., .- .-
" ..,~-> ~5 i· ete ,- j ~~ ~-- ~ ~ ...... , ...... , L....I L...... -J L ...... , 314 jOne IMBEHCEH ANDJOBN C. PATTEHSON IX. ADYNAMIC lU~SEHVOlH SiMULATION MODEL--DYHESM:5 315
with i/l increased ,by. the number of quarter hours in this time, and the algorithm I 1'11' proceeds. t~_l!l!xe~. J~.~~LgY'l'Ll1llg,~ Js modeled by a call to MIXER, which internally r calls the Ke!Yi!1~H~lm~9.l!~.RmmrinUQ!!U!1~ These subroutines are describ.;d " in Section 3.2. The eddy diffusivity described in Section 3.3 is then computed in ~ the main program and the diffusion routines FCT and DIFUSE (Section 3.4) called ~ . f in one quarter hour time steps for the sub-daily period for both .h~~d. sail. .. The 1. counter ill, incremented in HEATR, is checked against one day; if a day has not elapsed, control is looped back and the process repeated. ....w ~i ~ After complelion of the sub-daily loop, subroutine INFLOW is ca.l1ed for ~ 1 each inflowing river. This subroutine is described in Section 3.5. Subroutine ....~ OUTFLO (Section 3.6) models withdrawal from each submerged offtake, and if ~ necessary, flow over the crest. The model permits overflow only if the predicted "§• surface level is above bOlb tbe '!1easuredlevel and tbe crest level; tbe quantity of I '1 overflow permitted is the.ll which restores the predicted level without overflow to' • the measured level. This action is taken because of the unreliable nature of most -, --R,,,,,n••t1h (d,ef~"'''''''<'l tJ...· 1~J<7,/~') overflow volume data. I I updo-' 'H' n I -l' i • ,,0f t'."<- jtA ,,-da i {"} T/hl< j ""'-,!;J;.'"ilt.1o; '!I.• ',. i! ,f , ~tc p At this stage the predicted temperature and salinity structure is output, along • J I~ tho ~1,,;.,-Hel ...,hbI1d p,llc'''JA,3 ro,.f"•• f;.J/ will other information, and the daily loop is complete. If the simulation has hb 3. DYRESM SUBROUTINES \} : .1 ~ 4 ~ q{p ~{/ ('( .:' dr / 3.1 SUrface Heat and Mass Exchange 3.1.1 Theoretical background. Profile data of the air temperature,humiditx and ~ind..speedabove the lake surface are rarely available and the modeler must rely on single elevation data at a single station on or near the lake. p.,'Ofm wnhd,lW. '0' MCho"i.... endo..,tlo. This fact necessitates the Use of the following bulk aerodynamic formulae for CALL QUfflQ the assessment of the fluxt~s of momentum, rip, heat, iJ, and moistJ,Jre E ~=UUr 'Ir 'Ir = CUD 2 P (5) ;z:: ND iJ == -U'lr(}" "'"' -elfU (O-Os) pcp (6) I~I,' i Flowchart 2.1. DYRESM Mainline. I I I ~~..' ',··~ f I ~,' .•. '.'.! ,c .:,;./ ., ".1 '~1~: ~ ~."(~'-l_~~~~~LJj II ..... ~: ~-~~~.~;;t~. ~'~~~~-~~~~~~!i~~~~·~~~;P~1~~~~~~=vvv j' " . :;.. • :',•.. "', ",' ."", "-'- ~., ~-~- ,.-."".~_.".= ,_.~ , ,-, .•. '". .'0"'-.' - ,e..,., ".. - - , l·"" t . '. •. ',. , ..... ':. foet- -- . ,-" •.•.'. .::p - _. .' .. ;~ 316 IX. ADYNAMIC RESEHVOm SIMULATION MODEL-DYHESM:5 317 Ch>..Y.ds..emiUQ!J~~Y~ riiq! where T2 is the air temperature at 2 m height, and Rt), the water surface reflectivity ',I , ,', ' " ( 0.3,1. Field lesls show this fannulato be accurale 10 ;r,12 WJl\-2, 1 I ;: j e-<~r~, ,~ .... a--" ~ ~ JII8II ~ ~ ~ .-...... - - .. - -! ::; •i" i,;~,",:,',','/, "' ...... , -. _. • - - -- - _ "::' ...._...... ""'-i - ...... -- ., - ...... ~, "c, t.::; ~ ~. ~"~. ( ~• • ,1.,.,:1ib_.·~_~_.~_,~~~ • ~~~ L._~; ~ ~ ~ ~ ~ l-J~:"t ,J I_. L_.J L--J '---' '---' ••''''''' L...... i \-....II \-....II l-J ~ '-"-~"" -~- ~ (r'''''---'-~ ·:'¥"f:t,l!\'I\~'~~~i.~~·J ...' .·-···..,.,..,..rr ----":,,,,"-', ...,. "" '-'co - , .; 0'.; , () ",." f), J()He IMUEHCEB AND JOliN C. PATTEHSON 318 1.',-, .. Cooling at the' surface or evaporation affect only the s:'lrface layer, and the lI) Ii !: density profile may be unstable following all the heat exchange occurring in a time ... a step. HEATR makes no attempt to rectify this situation and merely passes the .:' profile. on to MIXER for stabilization. v The quarter hour time counter iJl is set to zero before the first call to 4i!-ii BEATR. It is ~suQ1ed that the short-wave radiation penetration occurs only for ttl ... I:~ I :t '"...III the second 12 hours of the day and thus is only applied if iii > 48. The ~ra.: I ... iii III tion lawLEq.O 1), is applied, and the heat input to layer K evaluated as II{ '0 ~I~ (3 C IoOIH ~ r I IX: ~~- ~. Q~ r!t_~ QK.~ (12) " ~ JiQK == AK-1B(QK) + (AK - AK-t)QK A, - 6 r: 'iii § c III it '0 'j where AJ.( is the area of the layer, QK the radiation intenSity arriving at the top of ~ f! the layer determined by Eq, (I I) and B(QK) = QK - QK-I ::l ,j ~ The .ewporative,. cQndw:.Uye ~ncLJQng.-wav.e neaLe1lchanges .occur fur all I:; & ~ E J values of i and aff~~t QUI)l. the.surfacelay~,1ID'e.r...NS. The heat input calculated ~ ~ H J from these effects is added to JiQNS' and the temperature increment for each layer 6 J M.I'\ evaluated for a single quarter hour time step. Since the me~Qr.ologild'J.._-. :5 Pr["~e~ponsible for .th~~~ .e1f.~£~~ !~ ltveraged pver .~ !t!L-it is assumed that the tempera Si ~ ture increment is a linear function of time and therefore that the 3°e change in 'u r.~.,,~4 .d.l'.Yt.J1Idt"" surface temperature will occur innl/ (T) == 3/1 AT I quarter hours, where ~ T is -uo I ;>- the surface change in one quarter hour, and nl/ is rounded down, with a minimum 'o value of one. Independently, the mean velocity criterion is given by III II" I I w III I I.~ q: I1I/(U) == 0.lh/u·2, where h is the mixed layer depth from the previous time step Z 1 ,, :::; ~ I I I and u* the current wind shear velocity; again nq has a minimum value of one. IS: w ~ '0 q > III The minimum of these two ",alues is then taken as the number of quarter hours n ~ ~ E t q c II. ::l for the current time step, ill is incremented by nl/'! cut off and n modified if the w ..l :< ~ new iN exceeds 48 or 96, and the total temperature increment fer nq quarter hours :2 Q. i llI:"' .cIII added into the temperature profile. ::> ! is '0- I C The salinitY... of the surface layer is adjusted for the effects of evaporation and .. <'II f I, t- ... III I rainfall, and finally a new density profile computed. Control is returned to the 111-' .;- '0 I I mainline ready for MIXER to be called. 'i I x: ,.. I ._.; ...... J ~,c"! ~ •I " :. I N :r ·,· :1!'\4 ::-;y:i..... , , • " '-"f'c, . L lJJ:.:'O-i~;,.~'"""" ",_,__ _ _.. ~, t *.. " .~nrvi-iiIi" ~ ';;,,;,,;,,;;j:- OHG 1MlHmGl':R AND JOHN C. PATTERSON 320 IX. ADYNAMIC HESEHVOIH SIMULATION MODEL-DYRESM;5 321 continually being mixed or stirred into the surface mixed layer. The turbulent 2 kinetic energy (T.K.EJ field will adjust to changes of the surface inputs in times of ~ P.E. =- .£ 1211 A dh + ~ d6 + '[h 2 + 8 ] ~ (A dl 2 P dt· 6 dt 12 dl P)1 (16) O(hlu"'), where his the mixed layer depth. Consider first the behavior of this layer in the absence of horizonlal pressure gradients Which would normally l,e set up by the presence of the brdndaries of the The above evaluations now allow an accounting of the turbulent kinetic lake. With this simplification the ..w.0mentum equation in any part of the profile energy budget expressed by theei]uation (see Denman, 1973) ---.....-...... - .. "".'----- reduces to (see Niiler and Kraus, 1977) -- o (f ~ C 'lA·.f) au __lL (u'w') (13) .1 aE =.: _ u'w' au _ -!-. 'Iw' IL + £1'_ g"j[Wi - E (17) at - az 2 at az az Po 2 r Integration of Eq. (1 J) from below the shear layer to any part of the profile, using the velocity distribution shown Fig. 3.1 yields the distribution of Reynolds stress where E/2 = (u,2 + v';! + W'2)/2 is TKE per unit mass, u'w'au/az is the shear sketched in Fig. 3.1. production of TKE by Reynolds stresses u'w' and velocity shear of the mean flow au/az, ala z[w'(P'lpq+E/2] is transport ofTKE by vertical velocity fluctuations \ The density profile shown in Fig. 3.1 allows a similar calculation of w'p' from , Wi, gp'w'/po = g(-aO'w' + pSIW ) is the work done locally against buoyancy 1 the mass equation forces by TKE lifting heavier fluid (a,p are compressibilities for heat and salt; 1 .. p' ,0',s' are density, temperature and salinity fluctuations, respectively), and E is the ~ _~-!-( 'W') _ aaq(z) (14) dissipation of TKE by viscosity. Equation (17) is valid at any point in the water €~8n. ~ at uz P cp CJz column and may be integrated from the base of the thermocline l =- to the ••".u..... II $'': d.(.plf" water surface z :t: H using ttid assumed profiles of Fig. 3.1. with a finite thermo 1 J { "t Jer/vl. cline thickness 6. where q is the radiation heat flux at a depth z. h }l1ia .... 1 ! I I 1 Making lise of the integrated equations for beat and mass as well, the " To obtain the mechanical energetics of the mean flow Eq. (13) may he multi III I I integrated TKE bUdget may be written as plied by the velocitYu and integrated over the mixed layer. The balance so obtained shows that during deepening (dh/dt > 0) or billowing (d6/ dt > 0) 2 events,a certain amount of mean kinetic energy is lost both in the surface shear 1 a {~ dh + ..£L .!!.- A + g8/ip d8} .. I 2" ii (£sh) + layer and in the base shear layer. As is shown in Lumley and Tennekes (972) 1 2po dt 24po dt P 12po dt turbul~nt 10 this energy change is not a loss, but merelY..J!transfer to kinetic energy 2 and therefore It m';st be accounted for in the TKE bUdget. v{ The total transfer of energy accomplished via this adjustment is given by 2 + {AU6 dliU + _1 AU2.!@. + .AU dh} c 3 6 dt 12 dt 2 dt + - u· ~+#2 H • 2 I J u'w' duo dz + 1. U'W,dudz = + AU6 dAu + ...!...(AU)2 d6 3 4 f-8/2 dz _y dz 6 dt 12 dt (AUF dh+ .3+ cu· dl1U (15) + 2 dt" cu YT-;Jt + aglliJ'" + hwSs ~ W'!L+Po 2E] .!3g _, g(h+8)"p';;i l. III 2poCp 2 2po -8/2 D where AU is the velocity increment across the base of the mixed layer and c is a i 5 6 7 fi constant. 8 " Similarly, the profile of w'p' shown in Fig. 3.1 leads to an adjustment ~f the gy mixe~ stru~tu. .Rl!~I·n~ d.e~p_enin._.g .~~bJl~Q'Y-~. •... m.ea.n.. p.ot.e.ntia.l.h~~rene.r .. belqJlJh~_J1l1~e;dof t. he. la.Ye.r!a'y~r ent~a!n~dreo . nto .1~e .b~seQf fl1l~~~._ ", .ing !he fluJg. IS l - ,tE s (h +6/'lJ - u'w'k_oI2AU r . layeLcau~_ an Adiustrn~nt of the profile shown m_.Ei~ With corresponding + w'lf. + ill -8/2 I~ raising. of the center .of gravity of the water column and a rate of chang~ oL- potentl~l, energy~ ~ 4 .E., gIVen by 9 10 II (I8) i" 0 ~~ ,~ ~ .~ ~ ! .... > ,'.I... . z'- r-' ]...... ,....1 _ _ ,...~....._...... -...... _ 1-<__"".-.._.~ ...... ---~- ~ ~- i.m-~ ...... -.'--'" ,,;;: ~. ~ ~ L...~ : • , ; I•I I I I tIL _j L __> L..- .'. L..... L...... L...... L...... L...i. L..J {¥'." l--J .~. " t'''7 , () MODEL~DYRESM:5 "[)- -l't" 1"' .n.~.'.n I; U A 11.11" 1f\J.ll\! (' I> A'I."I'E'lH··ON IX. ADYNAMIC HESEHVOIH SIMULATION 323 322 J" ... MOCllUJ.:I;.l\nl'IJJ" ...... _ ...... _ ... _ \u. ET __ ao'w' 1_ 2.!L Each term and its related scales are as follows: fJz Pu C" fJz (22) J L Rate of change of depth integrated TKE,as determined by the various at - sources and sinks of TKE on the right hand side of the equations. Es is the depth integrated TKE in th~ mixed layer. Since turbulence is generated by both wind If q (z) is a linear function of z, then aqlfJz= constant so that heating of the water and convective overturn, we introduce the scale ~. incorporating both these effects column {) TlfJt by solar radiation will be Uniform with depth, with no net stabiliz (see also Zeman and Tennekes, 1977) ing or destabilizing eftects produced. (Note that for q (z) linear, the solar radiation terms in Eq. (2l) sum to zero). As Rayner (1980) points out, the form of Eq. (19) (21) emphasizes the importance of correct modeling of :,:)Iar radiation absorption q*J == w..J + 'T}J u*:3 for mixed layer energetics. Finally, we note that term (6) can be a sink for TKE if tz- ir is negative (net heating). where now ,/7. TKE produced by convective overturn due to salinity increases caused by evaporation. Terms 2, 6 and 7 result from the integral (20) w*J = aghir +{3gh WSs 1/ Pocp f w'p'dz e-6/2 is modified to include the effects of evaporation at the surface, which leaves a resi due of heavier, saltier water at the surface. fr is net surface cooling corrected for ·~-;::.5~)':~~.I!,,",,*···:"~ the stabilizing effects of solar radiation absorption, as described below in conn~c j 8. Leakage of turbulent fluxes of heat and salt from the mixed layer into the tion with term 6. Hence, Es~q·2 and fJ(Eslz)/fJt~q*2dh/dt. hypolimnion "2. Rate of change of potential energy due to deepening and billowing. Bil- lowing is due to shear instability at the base of the mixed layei. I ~ == (-aO/w' + f3s-",,1W J ~-fi/2 i I;''1 -"l -. ..j 3. Shear production within the lower shear layer. Po -6/2 ,I (' J 4. Shear production within the surface shear layer. "5. Downward transport by vertical velocity lluctuations of TKE input at the is a sink for mixed layer TKE. free surface. Terms 4 and 5 both scale with u*) and are the sources of TKE for V 9. Downward transport by vertical velocity fluctuations of TKE into the I the wind stirring mechanism. hypolimnion. As with J6. TKE produced by convective overtu. n resulting from surface cooling, corrected for the effects of solar radiation. ir is defined by _I u'w' k-6/241 U ir __ .. 21/ (2l) --- = O'w'(H) + q(H) + q(g) - h f q(z)dz term 9 may act asa source for·internal waves in the hypolimnion and is a sink for Po~ ~ mixed layer TKE. v 10. Depth integrated dissipation in the mixed layer, ES _qeJlh. 0' Wi (8) is the net heat exchange that 0 . ms just at the water surface, including all turbulent and long-wave radiative tranSl\1.~. q (z) is the solar radiation flu~ in the v 1L Rate of working by the lower shear layer on the hypolimnion. water. The three solar radiation h.rms in Eq. (21) account for the stabilizing effect on the water column of the exponential attentuation of solar radiation with depth. Using the scales given above we introduce the following parameterization Whereas 9' w'{8} is a somce of TKE for net cooling, q (H) + q (g) - 2/b JI Jq (z) dz win be a sink for TKE if the q (z) profile is concave downward, as is usu- 1 d(Esh) _ C*2 dh 2" dt - rq dt (23) ally~ the case.• This may be better understood by considering the thermal energy t':i equation " /:. .~. ~;4 .. ;::. I' • • 'v.,:' ., .i~?:::"_':.' ~'. ~'ro;o. ll~~ . :~~ ~ fiilI'l'.t,,~ .... '" • ':; _. "-"<_'.__.__~"_~""_~,.",,,,,,,,,,,-.,,,,,,,,,",,,-"'_'"J ~''''''''''-i~_~~~.~.,_-,-_,..,,~~._. • ~_.;::;:.; -:,..::;,..~, ..._--_.-'...... ~,~~ ....._ .• --.----_._--.--_:~-~---".'-."--.!...,.....-.-....,....""".~--.... _.-.. ~,,'-'-"-~"""""""~'~'-.--.'.;-' L ,;:<,., "'. .. .) n 7i ,. . £'7 's -. ? • ....1I?IIIIIiiIi JOHG IMBEHGEH AND JOHN C. PATTIWSON IX. ADYNAMIC HESERVOIH SIMULATJON MODEL-OYHESM:5 325 324 summarized in Corcos (979) who suggests the time scale for billowing, collapse cu"') CK J (24) ,,~>O~ ..... _~-....~·w "~...~. ..."._.~~~~.~, ".~- ~~"~~.l ~.....~ ~~~~ ~.~ "'J~__ • ti 11;.;0,. .. _ f!'I:Im.... liIii!a6f.lIiII _," --- .. ~"" -- ""L...... Jij• j ~~ Lj.. It.. t .' 1 l . I l .1 ._~ ~.~._~ ~._~.~.l-~ ~_ ~ ~L--~_~ L..-i_.~1.-... ~ ~~ ~,,~.J ,~ __ __!__._.. .._"_I :__ ! __. i __ _L..-i , " . :J ~i ;.j 0 " . "::;"~'" ~ ~, ~~j"'ilJ'fj~\i.' F., ,"ifq:(>~'rJ's(flf:W(i~AA.. _.~ .. -.. _""." .. _~, ..".9Cl' IX. A DYNAMW HESEHVOlH SIMULATION MODEL-DYnESM:5 326 J()HG IMllEHGEH AND JOHN C. PATTEHSON 327 I 3.2.2 M/~\:ecl laver dYl:amicsalgorithm. The mixed layer dynamics are mulation of .1'''' Rlay be equated with the TKE form Eq. (20). If layers K, K +I, I \.:' ~j!:.'.~: simulated by subrouiines' 'MIXER and KH, which determine the OlixetlJ1!ye.L ---, NS have mixed, w'" is then given by 1 ,: deepening and the Kdvio-Helmbo1.1z...billo.w1llg...al the interface respectively. The r i:ci roUtines are based on Eqs. (27), (30) and (31). The algorithm acts \ j'e .d.e.nsilY.. ;oJ' J l.~ (~I,+d,-l) I tvt w· - 1'/,11 P,(d,-,I,>li profile .generated by heat transport subroutine HEATR. As this profile is the result ~~ of both surface heat exchanges and penetrative heating from radiation, there may l o be a density instability ,present at the surface. Stabilizatiotl of this profile is (dNS +__d.;.;...K---:1_) NS .j I equivalent to the calcula.tion of the buoyancy.t1ux and the total distributed heat L p, (d,-d'_I) (32) flux ir. Since §.Y.ilIlQra!h',Q l()§~e~jlre also included, w· may be directly calculated I-K from Eq. (20), or alternatively, froman..eguivalent.I]l~~n.J?~nf!mj!!l. ~.I1e.!,gyJormula l[' tion. If no surface,.&OO1iruLhas occurred, the profile win not require stabilization, where P, i3 the density and d, the height of the top of the i'll layer, and PI the the [email protected]! becomes the top model layer thickness, and w· = O. density of the mixed layers. The mixed layer depth becomes h == dNS - dK-l' I The simplest possible solution ofEq. (27) would be an explicit evaluation of Should the mixed layer deepen to the bottom the time counter is incremented and all of the available energies in a time step concerned with the mixini:. of a particu conlrol is returned to the main program with reinitialized velocity and residual lar layer, followed by a comparison with the required energy, with the process energy. I applied layer by layer until the available energy is exhausted. The difficulty here is S/ining. With the evaluation of w· from the stabilization section complete, tha.t in c.ases of~Qigb....wi!1d s!r~~s-and waH miX. ec!.lay~.r. denth, .anexcessively high the total energy made available for mixing by stirring in a single time step ~t is if. shear velocity is produced in the fixed time step At. In these cases, the time step calculated. To this is added any residual available energy from the previous time \I needs to be reduced below that set by HEATR. In the context of DYRESM, this step. Thus the lotal available energy from stirring mechanisms becomes is undesirable. 1 3 >"~'-l To avoid this problem, the balance described by Eq. (27) is applied in sec CKq !lt tions. Firstly, the deepening by stirring alone is applied layer by layer until there is Ea = ... +Eo (33) J insufficient energy available for further deepening. To the residual is added energy f I t J made available from shear production and billowing during this initial deepening after an adjustment of the shear velocity via Eq. (31). Deepening by shear pro wh re q is given by Eq. (19) i '1 ! ,] duction ...nd billowing is then applied to further layers by again comparing the The energy required to mix layer K -1 is given by :j available energy, inclUding the residual, to that required. At each incremental :l deepening, all of the variables are updated to their current values. ~n 6.h I 1 The resulting interface is then opened an amount corresponding to the shear E, = (=:- gh + CTq 2) T (34) " instability for the time step by subruutine KH, which ,evaluates the size of the I Kelvin..Helmhollz billows. If these meet certain size and time criteria, the sharp where 6.h == dK- 1 - dK-2 the thickness of layer K -1 and 6.p/p == 2(PK-l - Pl)/ interface is relaxed over the billow thickness. (p K-I + PI)' The layer is mixed if Eo ~ E" in which case a new Pf is calCUlated, . Thus the mixed layer dynamics.is modeled in four distinct sections; deepen- K decremented, Eo reduced by an amount 1:.."" and n~\l' values of6.p/p and 6.h I ing.' byc.onVective overturn, deepening by stirring, deepening. by shear production established. The process is repeated until Eo < E" at which stage the mixed layer .which includes continual readjustment of the shear velocity, and mixing of the has deepened by a further amount ,i. Thus the total mixed layer depth is now lI pycnocline by Kelvin-Helmholtz billows. The routin~ MIXER is called with the density profile stored in model layers 1 II h = h + Ii + d - d - (35) to NS, where layer NS is the surface layer, with the residual energy available for NS K 1 mixing from the previous step, and with the various parameters from the previous ! step required by the shear productinn algorithm. Again, if deepening. is vigorous enough for the mixed layer to reach the bottom, Convective overturn. The dep..sity of layer NS is checked against that of layer the algorithm is complete and control is returned after incrementing the time I counter and reinitializing the velocity and energy residual. NS-l; if an .in~tability exists, t.he-laYm. are..mixfill..,.a new density computed and compared with layer NS-2. The process is repeated until a stable,gradie.o.li.L Shear production. The deepening resulting from shear at the interface .qchiev~d. Eu:h such mixing re!easesa quantiiy of potential energy, and the requires the solution of Eq. (3 I) for the mean velocity shear 6. U. The precompu tat ion of mixing by the stirring mechanism ensures that in times of hi~h wind I corresponding turbulent velocity scale w· is calculated. This potential energy for- -) ? I I ~th~!llJMd..laye[ depth wHLaJ.SQ pe r~lany~!y.~ and the resultant shear i j I, · .... ~~,II .~~'"' • .I ....,- _j~~~--~~. , j---j~-->-----.,'',-~~--,- ~l!'l4_~-_._~_ ,,-~~-,", _----.~,.~,,-> I, .. U•••••••••• _... ",_...... 1-1"" ~ 't' .~'.' b- .. ! . $ 7 ; ... . '1 328 J('UC IMBEHCEll AND JOHN C. PATTEHSON IX. A DYNAMIC HESERVOlH SIMULATION MOOEL-DYHESM:5 r''19 ,) velocity remains ala sensible level. The procedure for determining the mean is reached. An average velocity for the time step is formed velocity is a complex one and is discussed fully by Spigel (1978). Briefly, the pro cedure is as foHows. 1 I ~ Equation (31) yields a linear growth in U with time for a fixed u'*, up 10 all Uuv = A f AU dt (40) effective cut nff time telf at which time AU falls to zero. The cut off lime ut 0 assumes use only of lhe energy produced by shear at the interface during the first wave period ~, and thus is defined by This velocity is used to evaluate the energy produced by shear and billowing dur ing the stirring phase, given by ~f. . 2 2 T1 T/4 2 2 u'* t . U'*t 1.... (--.) dt:::= f zPdt = 4 J (--) dt (36) o h 0 0 h E, ~ ~s lu.,', it + U.,',:(8) + U",8~(u",)] where ~. is calculated for an equivalent two layer profile and no damping is 2 +lg.!E- 8 ii - g~ 8.1. (8) 1 assumed. This yields p 24h p 12 (41) tdl"'" 1.59 ~/4 (37) where A(Uov ) is the change in Uuv from the previous time step, A(8) is the current billowing thickness change given by Spigel (1978) inserts a correCh-tll for damping to yield 0.6 UovA(Uav ) (42} .1.(8) =:= gf:.pjp ~I d ttll = 4 •0.59(1.. - sech (1jT - l)J + 1] (38) ! 1,1 To this energy is added any residual energy from the previous phase. The algorithm now considers the mixing of the next layer, layer K From I Where Ldis the damping time for the equivalent two layer system. Thus the shear -1. velocity iseifectivelygiven by shear production, additional energy \ fat ov av Qv AU- E..- + AU. t .~tell E = c.s IU 2 AI + U A(8) U 8A(U >] ~ h 0 a 2 av I 6 + 3 AU-O I > tell (39) 2 +Ig .Ae.. 8 Ah - .k 8 .1..(8)] p 24h g p 12 (43) sin~e where AVo is the last value of AU and, telf may extend over more than one day, u· may be a function of time. Once cut off has occurred, AU remains zero untit a change in wind stress occurs. is available, where /).h = dX-l - dX- 2. The energy required is again given by Thus the algorithm requires the last value of AU from the previous time step ~ and a time counter to determine the position of the current time relative to telf· Er = (g + C q2) b.h 2 p T 2 (44) The previous value of the slope of the velocity function u- / h is also required to determine a new start up timf1 The slope of the function may of course change within one period of telf as the wind stress changes from day to day. Layer K-1 is mixed if Ea ~ En in which case K is decremented, a new density With the procedure for establishing L\ U within a time step, the algorithm computed, Ea reduced by an amount En Uav adjusted for the deeper mixed layer, proceeds.in the following manner. The shear velocity computed in the previous and 8 and Apjp recalculated. The process is then continued, with A([lu~) and t; time step is adjusted for the new mixed layer depth from momentum considera· A(8) being the changes in Uav and 8 froIll the previous comparison. The loop tions. This adjusted value is used as an initial value and the layer accelerated with concludes when Ea < Er for some layer, or the deepening has reached the bot acceleration u·2/h over the time step At or, if cut off occurs before At, until tel; lorn, in which case control is passed back to the main program, after incrementing ...... , ...-J ,.... ,.... ~ ,.. ~ ~ ~ ~ ...... -.~_..... JIlIIIIIIIII ----... .,..... ~ ~ .... It... ,,, ~ ~ 13 ="'!... ~ ~ ;lI '----' L-J l.....l L --l L.-I L-J L---I L.-..J L--J l-.-J I..-..a L...... l.-..I ... --.,..L_.:w'r_~: ~,.. ,~ .J..f7~'"J ::;.;\ - - -- Q ---...... I L .,J ., .;, ~ ...--' ~ ~l~'''''''''j,!'''' I' ' I J()nG IMBEHGEHAND JOliN C. PATTEHSON 330 IX. ADYNAMIC RESERVOIR SIMULAT10N MODEL-DYRESM:5 331 the time counter and reinitializing the velocity and re$idual. Any residual E~is ~ stored for use in the following time step. A weND POWER INPUT l'w BiJlllWillg. The Interface resulting from the mixing phases above is unstable to shear if its thickness has been reduced below that given by Eq. (29). In this I case Kel:Wl-!l~!m.h..Qtt~ ..Qmo~~ will be formed. This, is accounted for by a call to subroutine KH, which evaluates a billow scale given by Eq. (29). If the billows are OSCILa.~TORY~llf~A~~~"~~~"p'~"/~~~"/;===~:;~;;~:;;~~~~~~~L~YE~1 ~ of sufficient thickness, at least six layers are formed over the shear zone, incor BOUNDA,RY ""'11;,,,,{ poratingor spliuingexisHl~g lUyersas required. These layers are then successively 'NSTABILlT"1 ~ ~ . mixed from the interface out, yielding an approximately linear density gradient I1 acroSs the billow thickness. If the billows are of insufficient size, the interface is K - tI BILLOWS unaffected. ! The final act of KH is to coalesce all the layers above the shear zone into a /. INTERN~L MIXlr.iC !'ROCESSES (z -2 ! single large layer to facilitate computation in the following stages of the main pro N gram. Control then returns to MIXER, which runs a final check on the density profile stability before returning control to the main program. B WIND POWER INPUT Pw I j 3.3 Vertical Diifdsion in the Hypolimnion SEICHING INDUCED BY I ~ PW~PS I I 3.3.1 Theoretical background. The vigorous mixing in the epilimnion has ~...... -... I been parameterized in Section 3.2, and this allows the calculation of the depth of the thermocline. The waters of the hypolimnion are protected by the thermocline I from distllrbances above. In addition, the densilygradient in the hypolimnion, I I,~-' I j ~although weak, has a slabilizing elfeCI, and II may be expected 'hal vertical mixing in the hypolimnion is small, perhaps only on a molecular level. HYPOL'MNETIC VERTICAL MIXING There have been many studies in which the evolution within the hypolimn BY BOUNDARY MIX'PfG ~l~ +l' ) ion of concentration distributiolls of natural or artificially introduced tracers have EZ-O.D•• - .J!..--!. been m(·~sured. Fischer, et ai., (1979) summarize the results of these studies and e s conclude that there is evidence or relatively Vigorous, but isolated, mixing in Ihe c hypolimnion. Th~ only explanation for such patchy mixing is that although there is not suffici.mt kinetic energy to cause overall mixing, there are areas in the lake II at any particular time where the energy density has been increased by some type of (', B concentrating mechanism allowing a local breakdown and mixing of the structure. ~ .--...../ -, \.. '- '- , ...... ---=:.---.. - The mixing mechanisms are most likely varied from one lake to the next, THERMOCLINE but it appears (see Garrett, 1979) that they faJl into three major classes. First, internal wave interaction leads, under the right conditions, to a growth of one VOLUME vI! AFFECTEO BY ,·INt:~C:~SEO!;'HEAR wavelength at the expense of the others until breaking occurs. These interactions ! AND WEAK may involve triads of internal waves, parametric instabilities or wave-wave shear .A TU.BU....." ....'NflOW'"",••All I interaction. Second, the local shear may be raised, by the combining of long and 1 0,081 P short internal waves, to such a level that Kelvin-Helmholtz billowing can take INTAtJSIGr~ VERTICAL MIXINQ f -,.z Z ttt 1 " place. Thirdly, gravitational overturning canbe~nduced by absorption of wave I I energy at critical layers. • o Regardless of the mechanism responsible, mixing in a stratified fluid is Figure 3.2. (a) Model of basin scale mixing in the hyplimnion of a lake. Wind and river inflow l characterized by an overturning motion with a scale of at most a few meters (see induce basin scale mol ions which in turn ind'l..:e internal as z"ell as boundary mixing. (b) Boundary Fig. 3.2a). The energetics of the mixing has so far not been explained in any mixing model. The vertical structure is actually adjusted via laminar horizontal intrusions which are I driven h!, boundary mixing, (c) Local river induced mixing. A river intrusion onen takes place near Ihe seas"nal thermucline, with little associate:J basin scale oscillation. Such intrusions do, however, r""'f-_ induce mixing immedi~tely below and above the intrusion. 3~4 • iIIIiIi ' ." f! , I r'w::w i~","il/lo!.''''lf!r """I' '~ M()DEL~DYHESM:5 1 JOHG IMBEHGEH AND JOHN C. PA'n'EHSON IX. A DYNAMIC nESEHVOm SIMULATION 333 l 332 detail, but ingenerallhe energy introduced by the wind at the lake surface or by Such a process sugget•.:) a diffusion coefficient formUlation as PI'oposed by ) the inflow in tumbling to a depth within the lake, induces mean motions and inter Ozmidov (965) and we may postulate a local. diffusion coefficient ,D nal waves with scales ranging from the basin size to a few centimeters. These I ij, non-mixing motions interact in the hypolimnion until locally one or the above E (46) r mechanisms leads to a local overturning event with subsequent mixing. The tur Ez = N2 bulent motions within the mixing patch lead to a readjustment of the potential I t energy,. The remainder of the mean kinetic energy is either dissipated or radiated J: away by internal waves gene led by the collapsing mixed patch. The local where € is the local dissipation. Once again if equilibrium flow is assumed an esti V V l:p' increase in ,potential energy caused by the mixing is in turn utilized to drive hor mate for € = a2Ps! s, where s is the volume of the reservoir effected by the ':;F J, izontalintrusions which are ultimately responsible for the adjustment of the mean stream intrusion. density profile in the hypolimnion. At present DYRESM uses the diffusion coefficient given Eq. (45) for values ; Garrett and Munk (1975) postulate a spectrum of internal waves truncated to of E < 1O~4m2s-1 and equal to 1O-4.n2s-1 for values greater than 1O-4m2s-1. For z allow the right amount of mixing for the particular e>lternal energy inputs. The the Wellington lake where most major inflows plunge to the bottom of the lake presence of the external lake boundaries prohibits the direct application of the this procedure works very well. However, simulations on Lake Argyle and I Garrett and Munk (1975) spectrum since, in lakes, one of the dominant mixing Kootenay Lake, British Columbia, have shown an intensification of mixing aronod mechanisms appears to be the dissipation of basin scale seiching by turbulent the inflows.. The algorithm is therefore being modified to include intensification of concefl~ the diffusion coeffic:ent by adding the effects of Eq. (45) and (46). M~'~h more t~ boundary layers at the boundary of the lake (Fig. 3.2b). Here, the of equilibdum mixing related to the local gradients via the Cox nllmher has only local work is required to justify the arbitrary cut-off of 1O-4m2s-1 si l1ificance. A simple dimensional argument may be used to derive an effective 6 diffusion coefficient resulting from the general interaction between localized mix 3.3.2 Turbulent diffusion algorithm. The effective turbulent mixing occurring I~"'~l' ing and subsequent gravitational adjustment of the mixed patches. durirtg a time step is simulated by submutine DIFUSE, which calculates ,.1, Let P be the power introduced by the wind at the surface of the lake,' Ps (separately) the redistribution of salt and heat governed by the diffusion equation. w 11 I the power InU oduced by the inflowing streams and E the potential energy of the The form of the diffusion equation solved is the constant flux model H;1 stratification in the whole lake. Then the global vertical diffusion coefficient Ez ill III may be written as \ oy, 1 0 I .ay, I i = 1, NS (47) = /I,A, A,p,E ~ at az az \'_. r H2(Pw+Ps) (45) Ez == at ES whe? Y, is the property (temperature or salinity), Pi the density, and A, the area of where "'I isa functio' depending o. the basin shape, the st heat and mass due to mixing induced by basin scale motions, the simulation of / Lake Argyle has indicated that the assumption that the power Ps is spread uni- I I NS NS I fotmlyover the volume of the lake is not alw.ys valid. River inflows which do nOI E = g /I.,:r. VA - :r. /I,. VA (49) plunge to. gr~t depth. but which intrude horizontallY just above or below the ther- I ,-I ,-I I ,1 moeline, do not induceseiching, but rather merely c.uselo0-41 turbulence and ! I ,:;- 11 shear within tho mtrusion itselr (Fig. 3.2c). ; p.~~~~~~~~~~_ ...... ~ ...,._ ... ,,,...... "'.- --.. ~ ~ w.w.~ ~.,..... i, ~~ ------.' - -- ~ ~ ...-. --- .....:..;.. ======~ . ,. I I. J I --, '---' '---' '---' L...J' ...... L..,-o . L...... ;: ..-.....-~"""'_~~ '"'_.,..-.~ __ "_".~" '",:"':=",,,,_,,__== " '' '~~_. __ ,,_. ~_,". . "-_ I-::: . I---.lI '--'.~~ ,";'"'-~----~-l..-..i L.....J l...... I L..... L-.J L.....J!i ''':- 334 J()HG IMBEHGEHAND JOliN C.PATTEHSON IX. A DYNAMIC HESEHVOlHSIMULATION MODEL-DYHESM:5 335 and Psund Pw are the ratcs of working ofthe inflows and wind respectively PLUNGE LINE OR ?OINT D + ~j .,if? Ps =g /::"P D Q (50) IT 4 for each stream, and " ? \1> • (51) Pw = CANS U3 , J with Pal' the average density~ ~ the volume and d, jhe height of the ithlayer, !lp, Dand Q "~~e density Jump, the level of neutral ~~oyancy, the river discharge (computed in this instance without entrainment), U the average wind speed,and ea parameter which incorporates the drag coefficient and a factor representing the degree of sheltering of the surface from wind. Figure 3.3.. Schematic of the various inl10w configurations parameterized in DYRESM. S9.broutiiie FCT is called with this value of TAt, the edGy diffusivityis com in which dis the non-dimensional density anomaly li =. (Pu - Pr)/P" Pu and plr puted, the molecular value for heat is added, and the remainder of the right hand the underflow and reservoir densities, u the mean velocity of the underflow,and side is evaluated for each layer. The redistribution of salt during one quarter h ur D!1/DI the variation jn A moving with the mean velocity u. Direct measurements period is evaluated by DIFUSE, which ensures Ihat no reversals in gradient occur, of tI,!land hyielded a value of B = 1.9x 10-4 for the Collie River Valley. I the lime counter checked against the current time step and the process repeated if l-~~!".b~~ necessary. The entire process is then repeated for temperature. By assuming that the discharge varies slowly in comparison to the internal l' • < adjustment time, and the valley section Was approximately triangular, Hebh-::rt et I Following these diffusion calculations, the service subroutine THICK is al. (I979) showed that the drag coefficient may be expressed as i ,I II call1ed to split the completely mixed upper layers inio a number ofsmaller layers in I, anticipation of the next action, either further heat transfer with HEATR or the I I inflow complJtations with INFLOW. ~).·I D sinaI5.in", _.i E 3 + (53) I C =·-5-. F,2 3 1 F,2 I 3,4 ltiflow Dynamics I where ~ is the internal Froude number, F, = u/(g!J.h)l/2, h is the hydraulic depth, and the angle between the river bed and the horizontal. For normal 1.4.1 Theoretical background. The inflow process ma; be divided into three flow, F, is constant. With the above value of E, tao. = 10-3, and F, := 0.24, Eq. stages as shown in Fig. 3.3 As the stream enters the lake it will push the stagnant (52) implies a value of CD = 0.015 for the Collie River Valley. lake \'ilter ahead of itself until buoyancy forces, due to any density differences which may be present, have become sufficient to arrest the inflow. At this point The corresponding Manning's n is 0.05, which is typical for such a natural stream. the flt w either floats over the reservoir surface if the inflow is lighter or it plunges beneaib the reservoir W&ter if it is heavier. If the entrance point isa well defined R\ewriting Eq. (53) for the entrainment E yields drowned river valley, as in most reservoirs, then the side of the valley will confine the flow and a plunge line will be visible across the reservoir at which point the river water submerges uniformly and travels down the channel in a one E = 1IStan • G'l ~. • • • • , ~ ---- -..;lOo " ~...... of f . ~" .. ~ .. , ¥ 'II .. ~». E~.~ '1' ''';(''5' .~ ,x. 336 J()UG IMBEHCEH AND JOHN C. PATTEHSON IX. A DYNAMIC RESEIWOm SIMULATION MODEL-DYRESM:5 337 induced by the bottom shear stress u·, where This formulation applies only for streams of small bed slope~ in the case of larger bed slopes, shear production and mixing at the plunge. point must also be ,0 considered. u· == CA'2 U ,~) {'; The entrainment given by Eq. (55) leads to a decrease in the density of the ,J Now since .F; is small, CT= 0, and w*== 0, Eg. (27) reduces to plunging Inflow until at some level, the inflow and reservoir densities balance and the inflow penetrates the reservoir. This intrusion into a stratified water body has been studied by Jmberger et al., (I976) and is characterized by the non 3 2 E= 1] Ck CJ/2F;2 (55) dimensional number R =0: FiGrl/3, where Gr is the Grashof number N L 2/€; and ,,..") F; is the Froude number ONL2/Q, both being determined at the point of insertion. The length of the intrusion e is given by t:' On first thought '1I3CK should be the same as found for the mixed layer e = 0.44 LR 1/2 (' dynamics. However, the experiments of Hebbert, et aL, (l979) and Elder and t' ~. R (59) Wunderlich (1972) suggest a considerably higher effidency of 3.2 for mixing in a downflow. The difference is most probably due to the very different distribution e= 0.51 LR2/3 (t)S/1i 516 of mean shear within the underflow. R< l' ~. Pr (60) Equating Egs. (54) and (55) leads to the result l 6 where I' = INIGr / • Pr is the Prandt! number of the fluid, N the Brunt-Vaisala frequency and L the reservoir length at the level of insertion. Equation (59) F,2:= sinatanlf>, (I - 0.85 CJl2 sinal (56) , CD describes an inertia.. buoyancy balance and Eq. (60) a viscous-buoyancy balance. For R ~ 1 Eq. (59) is valid for the whole intrusion process; on the other hand, the smaller the value of R. the larger the applicability ofEq. (60). In the context ~ I I I Hence, once CD is fixed F, and E, can be determined. of DYRESM, the process is split at R =0: 1. using Eq. (59) for R 1, and Eq. The entrainment E leads to an increase AQ in the underflow volume .Q (60) for R < 1, irrespective of the time of insertion. Given the two preceding II I t given by conservation of volume expressions for the length of the intrusion as a function of time it is possible to compute an entrance thickness 26 such that the inserted fluid is aiways in "static" I equilibrium with the fluid in the slabs which it pushes aheud of itself. Thus t (57) AQ" Qn~r-11 8-~{1-1J (61) The flow may thus be routed down the river channel of a stratified reservoir by assuming the flow to be gradually varied and by applying Eqs. (54), (56) and in which 0 is the widthai the entrance. The flow to be inserted during one day is <') (57) locally in each horizontal slab within which the density is assumed constant. then distributed over a thickness, 26. such that At the transition from one slab to the next ho is -redefined so that Qand F, are cominuolls across the transition. In this way the flow may be routed down the channel slope until neutral conditions are achieved, at which stage the horizontal ~ Un~,{cos ~z penetration of the slug is assumed to commence. u + IJ (62) To start the algorithm the initial ho is required from a plunge point analysis. f Fora triangular cross section, Hebbert et al., (1979) equated Eq. (56) to the ~~. entranceF.roude number and showed that the initial flowing depth hQ is given by 304.2 by/ow dynamics algorilhm. The dynamics of the river discharlv~senter 115 ing the main body of the reservoir are modeled by the subroutine INFLO\V. which (58) 2Q2 '1. is based on the equations described above. The subroutine is called separately for h ={ 2 (} . F,gAtan (j( each river entering the reservoir. The entn,ilce (if the river inflow into the reservoir is modeled by the , () 338 J()HG IMHEHCEH AND JOHN C. PATTEHSON lX. A DYNAMIC HESEHVOIH SIMULATION MODEL-DYHESM:5 insertion of .he volume into a number of existing Jayersal the appropriale height. 339 excessive~ !j !f the layer volumes become new layers are formed. The increased and the volume inserted into layer i, above the level of insertion is Y·)lume ·of these layers causes those above to move upwards~ decrcusing their thickness in accordance with the given volume-depth relationship, to accommodale the voltr, ne increase. The layers affected by inflow are those encompassed by the ~ dV, 7 q r (d,-d,,) - ("'-l-d )} d,-d,-l] (65) inflow layer thickness 2& centered at the level of neutral buoyancy. This level is [;'Is;n a: sin a: p + determined by a comparison of the inflow density, modified by the entrainment from the Jayers already passed, with the current layer density. The ~pporlionment of the total inflow volume (discharge + entrainment) is done in such a way thai A similar expression is obtained for Icwers below the level of insertion. the inflow velocity takes a bell shaped profile. In detail, the computational procedure is as follows. The inflow properties of 3.5 Out}low Dynamics volume Q, temperature T, salinity s, and density p are initialized to the ri ver values. The inflow density pis compared with that of tile top layer fJ N.~-: if P < PNS, Or NS == 1, the total volume is add~d ho' the top layer, a new surface 3.5.1 Theoretical background. The outflow structures of a reser ....oir vary level and properties are computed,and the control is returned to the main pro greatly and range from flow over a sharp crested spillway extending the whole I0 gram. If, however~ P > PNS. NS.~ 1, and underflow occurs, entrainment calcu width of the dam to single pipe inlets housed in an offtake tower situated in the central basin. Once again the single pipe outlets differ greatly in design from one 1 lations are necessary. af(~ bl!§i£.!!HYJw.Q~~!reme j ~~Y I r 1 <. .... • ~ ""J' ~ ':..J: .•... . l-l'.'-'-'~~'-'--"""~'--~'-'~~"-Ir--':-'-'-'--'~~ ~J Ii , . ,.' .$1.1 N{ ~i'<7::'f~' ~r ~ v••...... • ', > .'" ...... ,--.-=,15._4.""' -_._,_.__ '., » ... _:~ ..., .. - .. ,.' ~ .. ~''W~'i~:!,,,_:''',;,._,"._;..L__ "'"'~. ..;;~..~~~>.) .:'_.. - ,',,:'••• ',-, :,;s;· .. ~.1. - .' - C~:;· 340 JOne IMBEHCER AND JOHN C.PATTERSON IX. A DYNAMIC HEsrmVOIH SIMULATION MODEL-DYRESM~5 341 C,I vertical motion and the pycnocline acts as the free surface in the first case. The water in the epilimnion once again is that ina container with a falling bottom (Fig. 3.4c). Third, as the discharge is raised to a critical value,the pycnocline wHl also be drawn down (or up for an outlet in the epilimnion) and the water withdrawn will be a mixture of epilirnnion and hypolimnion water. The ratio of the mixture will tend increasing to that given by the potential flow SOlution (Fig. 3.4a) as the discharge is increased or the density difference across the pycnocline is decreased. In practice, the density profile is as shown in Fig. 3.5. There is usually a well POTENTIAL FLOW STREAMLINES mixed layer containing a diurnal thermocline, a strong pycnociine and a hypolimn ion stabilized with a definite density gradient. For weak outlet discharges, the I linear. stratificatio.n mOde.l .•is m.ost. applicable, but as the. discharge is inc.reas.e.d care. must. be taken that no contradiction arises between the applicction of this theory and the possibility of drawdownof the seasonal pycnocline. A complete review of the problems has recently been given by Imberger (I980) and we shall confine [} Cb) fREE SURfAC~ ourselves to a summary of the applicable formulae. .'~' z III- ; I' L -,,~ ZoN,'I .• • r---t----t-- r 7 All/III" II I' • ~tHlMmCL_ - r Z·+f I ~.. ,I ·1 ! ! :·"'t CIL r ! I! I z·, I ~ Of_TV f'Mf'LE lit' VlELOCITY 'ftOfILI CUlations.Figure J.5. Definition sketch of the denSity and velocity profiles for the selective withdrawal cal voirs. Consider first slot outlets, Spillway flow or point outlets in very narrow reser Figure 3.4. (a) Potential flow out through a strong discharge. (b) Selective withdrawal under Let q be the discharge per unit width, N the .average buoyancy frequency very stable condhions. (c) Influence of a strong pycnocline on the oUttlow streamlines. OVer the layer thickness, Ez the effective vertical diffusion coefficient, L the length of the reservoir and I i L..... -...... , -- - ~-...... ----,~.....-,.... '--' .. . /'" < - ~ • -!!"'~ ~·4.· • """.1 .. ." . oP :. '...., I • .- ~ I .j J . ¥ w" . ., ~ . .' :0... I I "',• • .,! .' ..'ot,J,~ • .. • .... • 0#. .. "" JI, .... " u. ,i1-~"¥('Y .tr!"i57" ~ ••:~~._m· ~itjlj~ •• .J> ,A" "'-----' ---.. -~ ----.. --.. ~ I...... iIIIIIIIIl ...... ~ ~ 1 J, ,,"" f ~V\ll 342 Jilll(: IMBEBCERAND JOHN C.!'ATTEIISON IX. ADYNAMIC HESEHVOlH SIMULATJON MODEL-DYHESM:5 ~ ,<'" I 343 ... \ Yv ((LI • ('-'/"&1 I I (]VL 2) 2 Y I)~" For the two results to agree as Ap -. 0 requires a =. l) :3. The correct procedure where FL ='CiJ' and Gr = N L4/E} then i,. /977 f"'l""t (F'IIF~'/ p ~ ;1./.. thus is to test if a change in formulue is warranted or not. F ;:...1- ttful I'IF In the case of RL < I and the close proximity of an interface is best handled Grl1/6~ ~'r" 8 = 5.5L R < 1 f e- r (67) with the average gradient method. In practice the discharge is rarely larger than L "'dlt 54..-.{' Jl:fit1;1 "" that b·ven by Eq. (70). OYRESM makes no allowance for this case other than the average gradient method. with 64% of the layer lying above the sink centerline and Second, consider the flow into single outlets of a dam wall or offtake struc lure where the lake is very wide and the outflow is essentially radial in the hor 8 = 4.0 L F1,'2; RL > 1 (68) izontal plane. We shall quantify this assumption after establishing the present for mulae. l I The set up time for these two steady state solutions is TL ::::: 0(N- Gr1'6} for For very small discharges~ where the withdrawal layer is governed by a 1-.f\\ 0 1 1/2 RL < 1 and TL = O(N- Fi: ) for RL > L viscous buoyancy balance there is no difference to that in the two~dimensional The velocity distribution associated with these flows is a complex integral of cast" ...;iiIOUgh Koh~s work (l966) indicates that the layers could be about 20% the equations of motion. However~ a good approximation is again given by the U>.cker. For Q large enough for the dynarnics to be a balance of buoyancy and I equation. inertia leads to a withdrawal layer thickness Umberger, 1980) of t I" 8 = 1.58 QI/J ! RJ•L > 1 (72) I u - u,." It + cos 2:'I (69) J NI/J where II This neglects the weillt. back flo~ ~ns at the boundaries of the withdrawal layers, but in terms of a water quality model this is insignificant as these flows are L I' internal and do not affect the outflow. 4 RJ,L = FJ,LGrj!l; FJ,L= N~3 and GrJ,L- N2; (73) Ez The withdrawal laYer will intercept the· seasonal pycnocline when l 8/2::::: O(g-t). The present versions of DYRESM retains the formulation given by Eqs. (67) and (68) and for~!~ny possible sharp density changes are r b Hence~ once again a critical discharge is reached when the layer intercepts the incorporated into the computation by using the average of N. This is not slrictly seasonal pycnocline. This will occur when I correct since once the discharge has reached the critical value where the layer §J!Q~nJiilY. ,gr.a..di~J1J~ 1 intersects the pycnocline, the there inhibit the develo'J 1 ment of th~ withdrawal to a far greater degree th';r. is accounted for by the average Q = Q(. = 1.00 ({ _ g)5/2 Up: g gradient. -!A '1 /2 (74) The layer intercepts the pycnocline when For the case of a sink situated in a dam wall (other cases can be considered) draw A I] 1/2 down occurs when q - 0.35 (, - elJ/2 I•• at. g (70) Q " Q, = 2.54 (, _ el'12 [de ::'dl!.' gj'" (75) yet drawdown does not occur until L Once again taking the limit Ap -. 0 requires a= 0.80. For Q much larger than q .2.6 (, - elJn dl' (71) = I ;'1" {" Q(. withdrawal will be dominated by the two layer flow and the ratio of epilimnion I I .. !) to hypolimnion draw can be obtained from the Jaboratory experiments of Lawrence i~ and Imberger (979). For value of Q-Q(. uncertainty still remains. Version 5 of DYRESM has previsions only for two dimensional withdrawal. t /"""- t .. '.': 1 I '~ .' ,',' ~I ~ (J' ·LJ···.·..·······'.. ·.·.··'.:.1 .\;"\4 {!>I ~''''''''------''';-----''''''''------. ~~.,~"._----- .--...... ,..--~-_.~----,-,~~--.~---.-~.~'~:~.,...... ~_ ~.,..,.,.'-._" ------...... '--'-'-...... il!"~_~.T--':!,_ ...':'_.~~~t1..~I; E...... :_~·'!'.:...... _~~·~.!r...::'..::~Ui ~ ;.I;~~~ mllttC~~;~ ltAAUM "I 3 . ._.'C... fie ••. .. .§!J__ ~" ·· 1:--' ~ -- ..~ ••,> "'o:::.-~ '!::: ! .•. ,;6 r~c,-l"'''ltwo~!~~ .~ --...... •¥'''''I'' "---' --." 1I0o--.I -- ...... (~ ...... , t ..... r I~\I" I I ," 342 / Y,c J()HG IMBEHCEH AND JOHN C. PNI"l'EHSON IX. A DYNAMIC HESEHVOLH SIMULATION MODEL-DVHESM:5 At , 343 t, Yr, B ~ ~ ./ correc~ (J"L 2) 2 t :; 1- /.. G"r I" I For the two results to agree as Ap --+ 0 requires a = 0.13. The procedure ~ where FL =r;:q/and Gr = N L4/E; then in /97i to< r- ~ ~ $:.:H F'II~ ;y/I thus is to test if a change in formulae is warranted or not. ' " . F== J-- ,.. fuJ In the case of RL < I and the d.ose proximity of an interface .is best handled ~ '" Ll, ~ lS = 5.5 L Gr[I/6; R 1 Gr (67) with the average gradient method. In practice the discharge is ral'ely larger than r.l L < ""iLl 54.",(' dtf;,,;-!'''' rr1' ~ that given by Eq. (70). DYRESM makes no allowance fol' this case o'>her than the average gradient method. 1'1 with 64% of the layer lying above the sink cenlerline and '1 i, Second, consider the flow into single outlets of a dam wall or offtake struc !O' ~. ture where the lake is very' wide and the outflow is essentially radial in the hor F1'2~ ~ 8 = 4.0 L RL > 1 (68) izontalplanc. We shall quantify this assumption after establishing the present for r; mulae. I I6 The s~t up lime for these two steady state so~utions is TL = 0(N- Gr,c ) for For very small discharges, where the withdrawal layer is governed by a RL < 1 and TL = 0(N-:;.--;1/2) for RL > 1. viscous buoyancy balance there is no difference to that in the two-dimensional The velocity distribution associated with these flows is a complex integral of case, although Koh's work (I966) indicates that the layers could be about 20% the equations of motion. However, a good approximation is again given by the thicker. For Q large enough for the dynamics to be a balance of buoyancy and equation inertia le~ds to a withdrawal layer thickness Umberger, 1980) of 8= L58~ +~os2;2] NII3 R3,L > 1 (72) U = urn" 11 (69) J I., M"l.~_~.~_.l~~., I where ~!!k. ~at I I I This neglects the back flow the boundaries of the withdrawal 1 0 layers, but in terms of a water quality model this is insignificant as these flows are 1'1 .:11 2 4 internal and do not affect the outflow. 1/2 -fL N L R 3,L = FJ•L Gr3,L; F3,L = NLf and GrJ,L- ~ (73) o The withdrawal layer will intercept the· seasonal pycnncline when Ez 8/2 = O(e-tl. The present versions of DYRESM retains the formulation given by 'Ii Eqs. (67) and (68) and forl~.s~J1i~£!!9n~~~ny possible sharp d~".;::y changes are \' Hence, once again a critical discharge is reached when the layer intercepts the "~' incorporated into the computation by using the average of N. This is not strictly seasonal pycnocline. This will occur when I correct since once the discharge has reached the critical value where the layer intersects the pycnocline, the str.QJ:!1Lill:I1~ity &r.a.dj~.n.!~ there inhibit the develop ment of the, withdrawal toa far greater degree than is accounted for by the average A, Q = Q(.- = LOO (t - g)S/2 1~~g ]1/2 gradient. (74) The layer intercepts the pycnocline. when For the case of a sink situated in a dam wall (other cases can be considered) draw (J AIp g ]112 down occurs when q = 0.35 ({ _ gp12 ..u (70) Pu c. I ~ Q Q, = 2.54 (, - $)5/21 Ap +p:'AP' f' (75) \" yet drawdown does not occur until Once again taking the limit Ap --+ 0 requires a = 0.80. For Q much larger than q ,. 2.6 <' - $)l/2[ AI';'i.- ern (70 Q(' withdrawal will be dominated by the two layer flow and the ratio of epilimnion o to hypolimnion draw can be obtained from the laboratory experiments of Lawrence Q-Q~. and Imberger (1979). For value of uncertainty still remains. Version 5 of Ii DYRESM has provisions only fol' two dimensional withdrawal. ,\ I I r. ; ~ ~ .,' f • ~~J I, - JILJ'", ) .Wl ~ ~ " " ' -~-~-"~~"-'-~~--~--_._---~,-,--,---- ..!_lJ'"r:.~._:'_::"~'-:''!.~~I'_ ".~7~_:,::",,!-s ~_~·IF0:!':',':~~fSi_f_ ~~ _._~£,~~ @1.~!.;'~~':":"'- 1 LJ' ---...----.--:-, sa £ _1)(.)\;" . ... I) ,J .- .• ~ ...... , .. . '.. ;...r. "," 'II A - ~., .. K; : •• • " • • ~,.'.: ~ t. ",' If'. t. I." . , ;,.. : .; , ' • ~ ,Il ~ -c.:' ~jiiVw~niP;Yfi!).,lnrpt ~.'~ IIlIIIMh " ~ >• ."iJ. i .. ~"'9Jlqffy;'(;;jM'_'~~'i' t~6~~ 1 jOne IMBEHCEUANDJOUN C. PATTE1\SON 344 IX. ADYNAMIC HESEHVOIH SIMULATION MODEL-DYRESM:5 345 ~ 3.6 Service Subroutines I f I \', - ~ ~ ~ 1. I; ,. ~ - . -i.; - ,---- L--..a ~. L.-...-a l-..-.a ~L-....i ~ l-.I! ...... -.. I", I ..... --. ~~ --,:-.:...." .._-----~----_._"'----- ~1; "':'t :. •• ." • '. ~.:..... _ ,. • ~ i ~, •• .. . .t ) 346 -..r""a r~'~'-"'OQ>h'-'" j()HG IMBEHGEH AND JOHN C. PATTEW:iON f' q.,,' IX. A DYNArvIlC RESEHVOIR SIMULATION MODEL-DYRESM:5 ~ ..' 'I The routine is calledafler any mixing of two layers, or any readjustment of a layer 347 ! property.' JIll \J deep:tt the dam wall, extends Some 24 km along the main Collie Ri,er Valley and , has a surface area of approximately 16 sq. km Cl:t the crest level. () ?';' SATVAP: In many cases the surface temperature is not available as daily f data. Thus the saturated vapor preSSure at the surface temperature, required by The catchment has been severely disturbed by clearing native forest for agrr ',--- -n'r HEATR, is not able to be precalculated. Subroutines SATVAP evaluates this vari M I able, using the predicted surface temperature T. culturaf development, This has resulted in substantial increases in the stream I salinities with the first winter flows possessing salinities as high as 3500 p.p.m. RESINT; Subroutine RESINT calculates the fayer Yolumes and sudiweareas Salinities generally (~ecrease during the late winter and: spring. These high salini corresponding to given layer heights or conversely, layer heights and surface areas I ties, although extremely detrimental to the usefulness of t:he storage, are valmlble corresponding 10 given volumes. The routine is caHed following any operation for the critical evaluation of the model. In this. lake the temperature and the con which affects either volumes or heights. ductivity of the water form two independent tracers with independent inputs. !> £) In either case the calculation is an interpolation on the given depth-volume I The seasonal variability of the various inputs to the resl..rvoir over the period area data for the reservoir, which is assumed to have the local functional form fromd.e Julian day 133 in 1975 to day 365 in 1977 are shown in Fig. 4.1a. Dep icted are the wind speeds, the short-wave solar radiaHon as computed from cloud l cover records, the salinity and temperature of the inflo'Wing water and the total rate v d. Ja Jt:~1 =.Id'~l· (76) of inflow from the Collie River which contributes approximately 85% of the total I Fig.inflow.4.1a.The remaining inflow is inclUded in the simulations, but is not shown in L 1'-' l f Figure5 4.1 band 4.1 c show the field temperature and salinity structuJies A d Jb I~~1 length averaged along the Collie River Valley as a function of time over the periOd I), A/ i = (77) I January 1975 to August 1978. The salinity data gathered between October 1977 L, and Jun.e .1978 is regarded as unreHab~e and is not shown. The broken Jines in Pig. ~'"IIf-~~':'i-!r;;_~ -~_ii1iq_ 1h ~ where Jt:,A J and d, are the i volume, area, and height data value given. 4.1 b indicate that no data Was taken in the period Covered and the thermal struc ture is interpreted from the structure before and after the period of interruption. An additional function of RESINT is to evaluate the model layers ! I I I corresponding to the levels of the various offtakes and the ,crest. This is per The yearly cycle is clearly evident in Figures 4.la and 4..1c. The cold salty each~'ayer inflows lodge in the base of the homogeneous reservoir in the months of June, I I I J formed by comparing height with the given h::ight of each offtake. I july and August. Summer stratification builds up until December, when surface I winds begin to mix the surface layers and a thermocline forms, protecting the I 4. DISCUSSION waters below. in early winter, the air temperature falls and the winds increase, f with the result that the reservoir is completely mixed before the following inflows arrive. The marked difference in the thermocline structure between 1976 and r' The developmem of DYRESM and the associated algorithms described in 1977 was caused by a change in the withdrawal policy. In 1976 all the water was Section 3 are part of a program designed to understand t.he physical mixing proces~es processf~S withdrawn from the offtakes at 15 m height, whereas in 1977 a large quantity of in a lake. The priority attached to the many operating in a lake Sany Water was Scoured through the offtake at the bottom of the dam wal/. is determined by the long standing aim to obtain a fuller 'understanding of the interaction between physical mixing processes and the bioI lOgical kinetics of the h is also clear from Fig. 4.1 b that the temperature regime of the resel''Voir is; lake. The initial construction and constant updating of DYRESM thus serves a determined by the inflows and the surface heating and Cooling. The bettom tem twofold purpose. First, it is an organizedwa} to evaluate thte influence of a partic perature of the reservoir for most of 'he year is determined by the temperature of ular process to the. overall mixing patterns in a lake, to validate a particular param the coldest inflows, whereas the surface temperature is determined by the eterization and to study the competition for available energy among the great host meteorOlogical forcing, This is typical of lakes in temperate j'egians, and is in con trast to tropical lakes, where the temperature regime is solely determined by the of mixing mechanisms. Second, DYRESM serves as a useful predictive tool for meteorological inputs. testing reservoir management strategies, for evaluating methods for the control of the lake stratification and for water quality simulations. The data set gathered at the Wellington Reservoir forms an ideal test for DYRESM or any other mOdeL The data was collected at reasonably regular inter DYRESM has been tested on a number of different lakes, but its major () evaluation and development has taken place with data from the Wellington Reser vaJs and a series of process oriented field expeditions yielded further data oa many voir. This storage is situated about 160 km south of Perth in Western Australia of the details of individual events. This work is continuing and it is t9 be expected 6 3 that further development of DYRESM will take place from these results. ,- and has a storage capacity of 185 x 10 m • The reservoir is approx;:nately 30 m I \i l, .~ ~. 1 il[ LJ'"' ~ o ... ~ .~... 17 f ,go, "" c ( ill-...... si. '" 348 J()RG Il\'fBERGER AND JOHN C.PATTERSON IX. ADYNAMIC RESERVOIR SIMULATION MODEL-DYRESM:5 349 12 :II: There are seven constants which must be specified by thl.~ user before apply e 9 ing DYRESM. Of these only one is trulyadjustable--the others are reJ~ted to well (5) w 1>a.. 1 II) 6 identified physical processes and are determined from experimental or field datil. en The constants are described below, together with experimentally determined z:: ?l.:s: .'.0 hw values. =u :D 0 1. CD is the drag coefficient for inflowing streams. CD was determined en :JC .:D,.. c... independently of DYRESM in a field study described in Hebbert et 01., (1978)1. ... ~. :x %... ~..- The value determined in that study, CD - 0.015, is used here. -< -0 2. 111 is an extinction coefficient for short-wave solar radiaUQl1penetrating % .:. :D n the water. It relates the solar radiation received at the water surface, to tha,t ,.. penetrating to a depth z. A single exponential decay formula was used as only "'ll ·0 'II - 28 :x limited field measurements were available. An average value of 0.35 is taken (I t ~24 5 from data presented by HutChinson 975) based on the fact that the WellingtQn IX ., is fairly clear in the summer months when surface heating is an important effect. ~20 ~. ~ 3. al is a constant occurring in the expression for the diffusivity calculated ~t6 o 3 0..r i: for the deep hypolimnetic mixing. It represents the efficiency with which the ~12 :x2 o power input from the surface wind and river inflows is converted to again in 8 J ~ 1 potential energy of the lake water due to vertical mixing. A value al - 0.0411 ...z: determined in earlier calibrations of DYRESM over the 100 day period from day -~ 1II~ -~." -~ ollllltllltld ....."•...t. q .. ..,.( 133 to 233 has proven satisfactory throughout. OJIF'"IAIIIJtJIAI510tUIDIJIFII1.AII1'J'JIA'SIOIN'OIJIFII1'A'I1I~'J'A'S'O'N'O' , 1975 I 1916 I t:l77 , 4. CK is the coefficient measuring the stitring efficiency of convective over.. I I turn. Experimental results summarized by Fischer et al., (1979) snggest an aver.. I "f/~,) age value of CK - 0.125. (w/..t ,If..t y,o,,)< 'Ill""" I 11~ wELLINGTON THERMAL STRUCTURE 5. in combination with CK as CK",3, is a coefficient measuring the stirrinil (b) 35 ,f It I n tRIP • efficiency of the wind. The value given by Wu (I973) is adopted here sinc~ it W&!I t shown by Spigel (1978) that in his experiments stirring effects dominated the: entranmentt with shear production, temporal etfects,and internal waveradiatioill i~~ ~ijf~~1 n- ~ lfr'l ~ { . '1~~'~ j' 1: j ;:,,~ ~ • J II \\\\\ \ I 'I }-r\\\ i1': . \ , t1 losses negligible. Wu's deepening laW is dh/dt - 0.23 u·/Rt giving CK113,andi '; IS '-.::"""", -'''''''--;~ ~ ~'{l, 1 I~' d· : \. / thus .,.,- 1.23. .... I Q I ' , , ,'I' I I> " I • .j '-. \" I, " I "'5 14 1'1 1 A "I I' • I 1 \.' ' I II s m~asuring o n-T""r ~ -1L-t-r-r~~' t r-ri.u.T.~I ..t-.r"T'...'.T.~7'.TT , . 6. C is the coefficient the efficiency of shear production for' DJ.f'H·R.I1·J'J.A,J;·O'N.Q'J'f·M'A'I'·J·.J'R·5·O'N>O·J·f 'M·H'I1·.J'J'A'l>·Q'N'/JQ,1 '!"'IH", J> • !l<,'i' " ,;. entrainment. Values rangh;tg from 0.2 to 1 are reported by Sherman el 01., (1978). . i91l:> • i9'JG • 197'1 !')"B, Cs - 0.2 was chosen as representing a good estima,te for most experimental results. A value of O.S was used before the energy released by billowing was !~) 35 ,fiELD TRIf' WELLINGTON NACL STRUCTURE separately inclUded• as is done• in version 5 of DYRESM. ~25_3D 7. Cr IS associated with temporal, or unsteady, non-equilibrium (:.ft"ects due to changes in surface wind stress or surface cooling. Cr is constructed by the ~2(, requirement that for a turbulent fron~ entraining into a homogeneous fluid, ~:: ~ dh/dl - 0.3u· (Zenman and Tennekes, 1911) giving a value of ~s ~ CT - CK .,/0.3 - 0.510. 1.) :1 0 'J'f'""'"'~~"""'N'O'J'f'""'"'j'J""'O'N'.'J'f'"'""'j¥"""'N'O'J'f'""'"'~'J"'S'O'N'O' These coefficient values were used for a 962 day simulation of the lake "\ ' I 5 I I 76 • I 7 , 1 76' dynamics starting with initial profile data on day 75133. The results from this J.I F'1gUre 4.1. (a) Seasonal variation of wind speed, short-wave radIation.•• sahOlty•• and temperat~re simulation are".shown in Figs. 4.2a and 4.2b. . ~ of inflow and discharge for the simulation period of the Wellington Reservoir. (b) Field data showma COmpaClSOn of Figs. 4.1 band 4.2a shows that the thermal structure IS repro- J 1h.1000pc,,m,,,Fie~d '''o<'u'. in 'b. Wem••'on R...rvoi, fo' 'h. duration of Ih. ,iOlulalioo pcriod, .(e) duced extremely well with .U thestratifyinS lind mixing regimes occurring at the '!.' data S.howina the salini'ty structure in the Wellineton Reservoir for. the duration of the simulation ,1 PCrlod' ,.... ,....,.... ~ ·,.,"',....~~,....lf:;I;.:~:;~~:.tnlf~;; "",~l'~ 9~t£~~~I~.~. ~ ,....-~-~- .lilI"J;"~"""....~ ~ > 1'..·.··.. .J.'.. _ _fl·%!..·.. •. •__ .,...... ,.....I"",______>.; ••• '" l!....-.l. i-.:_>...•.•.•....••..•...... •J. • ..l-...--l-...... J .0.-1 ...... J -- ...... , ....., ...... ~ ...... Io..i""'-.l ...... ~::; '-' '--' I:? .-...... ,-~--"';'l!:J, .. .it.,.~; ~~ i.(3t'':''.,. l<;'.. ..-...... --~ ..... • ...... ~.~'qVIW,i'_.liiolHI;Woo • --J1 f . ~ < ;r.~ IX. ADYNAMIC HESERVOlH SIMULATION MODEL-DYHESM:5 I ' 3tH .. > ~1 ;~. ' .', correct times and of the correct magnitUde. The slug of Ire water predicted Illy ) the model in August to September ~975 does not appear in the field data. This o " IllINOU~A313 llHNOUttA313 ~ OD'D' 00'1( 00'" DO". 00" 00'0 slug is derived from the previous inflow and its presence may be due to errors in 00'0' 00'IE 00'" 00'1\ 00" OD'D ,,,,,,I,I, lr the inflow temperature data. In any case, the predicted temperature gradient in the t I bottom at this time is smalJ,and the actual difference between field and predicted " temperatures is less than Ice. f I~1 B The predicted and observed salinity variations also compare well, the most , CJ) '""') .. 1\ '""') 11 significant anomaly being the mismatch of the 600 p.p,m.line at the period of "'") E I /,' '""') maximum inflow in 1976. The field data suggest somewhat more energetic surface ., ::c ~ :e , mixing which may be due to small errors in the wind data, Additionally, the I' CJ) i CC CC ... ::J I model does not reproduce the peak salinities of the inflows. This is due to the 1 ~(O ;:00 U ) L..e- 1J...e- ::J construction of the model .Iayers in terms of layer volumes, hence fixed by the , en t: -sen ?- III depth resolutiori in the upper parl. The bottom three or four meters always appealr .. ..-. ~ ( :5 as mixed, and any salinity slug oocupying less than this will be mixed with the ii layer above. Thus, there is no error in terms of the salt load, but one of display of ! -0 U the structure, and it was accepted in order to save computer time. If greater reso I" ai '3 lution were required, lhiscould be achieved by specifying smaller minimum slab -===t...:., E sizes. i Vi 1, '1 ...... j ,~ ~ Overall, DYRESM appears to faithfuHy reproduce even very severe changes ...... ~ b? in the reservoir structure caused by such diverse forcing as large saline inflows. f. "WAIlWt.'q.,.,_r• .-. ....~ active scouring, strong wind deepening and winter convective cooling.Perjlaps -.j. more importantly I the model correctly simulates two independent parameters~ ,I cil I salinity and temperature, toa resolution equal to that of the field program. Morel ·1· ii: Ii 'iI r• II B accurate field data is required if more stringent tests are to be applied. t, ..u ( 11 Profile datal as computed by the various COmponents of MIXER are repro. E duced in Fig. 4.3 for day 75138. The result of each computational step .is shown · ...... 8 starting with Fig. 4.3a which depicts the result of HEATR adding both the surface § r'""') u '""') .. heat losses and the subsurface heating due to soiar radiation penetrating to some ::J :e U · ::J depth. Are deficit due to surface COOling, Was quickly achieved, leaving a con.. CC t;;.. vectively unstable density profile, Fig. 4.3b, with a density anomaly of ncady ;:(0 CJ) J ...e- .. 0.5 kg m- . The potential energy stored in this unstable profile is released by , • en a ~) ?- ~ relaxing the density profile until it and the mixed layer have reached a depth of 5 8- ] \:., E meters. The total amount of potential energy released yiCilds the effective convec u tive velocity scale w· (Fig. 3.4c). The stirring energy C 12TJ3U·3 4f is then added "C- x U to this pool of Ti~E causing a further deepening of the mb:ed layer of 3.6 meters t ~~...... --- .9 ::J (Fig. 3.4d). The velocity in the mixed layer is reduced due to the deepening, but E r ~~ ~ ?=3~ Vi there is still sufficient mean kinetic energy to cause further 2 meters of deepening j ...... by shear production as is seen in Fig. 3.4e. At this stage the algorithm leaves a =;: ~ L structure characterized by a mixed layer and a sharp thermocline. The subroutine ! a CC t'i ,~ a 2 1 J' II: ...;. K-H smears this interface over a distance 0.3 du 1g which in this Case amounted , 'J ~ :elf.) ~ ~e- ~ to nearly 3 meters (Fig, 4.3f). This process is carried forward from one time step ,~~ ~ '0) n ~ to the next. ~~ ta ?- L&:;u 00'•• 00'11' 00'" 00'01 00'1. ..; oo'n OO'U llQ·l. 00" OO'w, 00" . 00:0:- - llUNOUtlU13 (UIIIOUtlA313 cit On day 75138 all the deepening processes were additive. This is not always I ii: the case as is illustrated by the deepening on day 75228. In Fig. 4.4a the net sur .\- face heattlux is plotted as a function of time and in Fig. 4.4b the corresponding /.r I AU (final) for the mixed layer. The plot in Fig. 4.4c shows the resulting mixed ,C"" 'I 'i.., . , . I I 'k ~ i _.,_. ._.__.~_~, :l!'\4 l~ ~-~ ""'""'- b ...... LJ -'~--"-<""""_~."'_""""'~""_'~--~"'.-'''''''''''',~~~.....,...... ".,.,..,,,,...._.~, ...-.....,...,--'0...... ,-,...... :...... ,'""".~ ...... ---_-...-.- ~ 'k<' U \.1 \) >9 .,....~' h\/-' . ". ,.-' ....' .. , ..... f. ' •••• '. '. ,, ' • "1 t '2&'1 '. Zf ' ftHP1t"'Ptiii:rttH'j£ b -P-'iii",." " b C 99S.5 998.9 j:.4__----.------T. ...,(O...,._C_) -,-__-, 998.9 27 +1_.-• .:lT---:----+__ _ ..J! ---- .... t' ~ r' ~ r' ~ ~ ~ ~ i ] t \ ~ J" \ r I I h ~ l [ ~,] I I. r ! ~ 20 I I I 1 I I [ \ I I I 1 I , I \ I l ) ",.., ",I I "1 [ I J i [ 13 113 Figure 4.3. Computed density profiles on day 75138 (7.00 to 12.00). (a) Temperature profile after HEATR. iI (b) Density profile after HEATR. (c) Density profile after convective overturn has stabilized profile. j I ~[ J ------.0)-I e f II [ Je::.9 998.9 99S.!J 21 27 .27 r------,----....----...... Po I Po" Po ) I 1 [ ] I I I I I I I I I I I h I I ,I u I -+ r h ...... u 20 20 I .:lp I I I 1 I hI! I I , tip ! ... I 1 ....-- t.p--+ I I I I I t I , o '-"'--1 I- ...... - .. I o 't~ . ... 13 I Figure 4.3. r:ontlnufOd. (cJ). Del ~ityprofiJe s.howing d.eepening cJue to surface energy inpu~s. (e) Deepening due to shear prlXfuction. (f) S;:","anng 01 the pycnocline due to K·H billowing. l [ . . I ... Iit_ t ----SJ- ----,~r:--.,-----+ p h 16 ....1------' 0/2 14 12 'I 10 E 8 ~h '== f- SHEAR ~ ... 6 h STIRFUNG f 4 . t . 1 2 [ o I 24 4 o 4 8 12 16 20 24 4 8 12 16 20 DAY 229 l DAY 228 r TIME IN HOURS ·1 F,guff 4.4. Deepening predictions on day 75228. (a) Thermal input at the surface. (b) Calculated mean speed of mixed !. layer. (c) Predicted deepening. Solid line is net depth of mixed layer. Thin line represents deepening only due to the stirring mechanism. Jumps at ecch lime step account fOT }(-H billowing. Notice the decrease in the depth during the 4th time step if ·1 I only stirring energy is accounted for. 11 I ())""l' I ! 200 - I 150 - I I N 100 - Ie 50 l- ~ ~I o I I I I • • I • -50 .. -100 ...... 150 .. II 0.3 I -I... 0.2 e u. :::I <:I 0.1 0.0---.-L.,.--..L...--.1----...L...-.J--...L-_-'--_...L- ..Ll_..L..._.J..._...L-_...J EX,PLICIT FORWARD DIFFERENCING o Figure 4.4, COntlnUfd. l,':f,i1"*f'" , ,,·'t ·.1i't'lIlY( '1M~ -- i' ". -,' , ' '" . i . •• ~.'>"""~'~<'~ ,'I ·,,"~~t"'ilIi!!!!¥, Ii Ii~~@)k_t('ii~?'n,~.•.. " ..-L o ..l.,;\,.\.j:",__ __. .' ':1 ,;-'. 356 JORG IMBEBr'ER AND JOHN C. PATTEHSON IX. A DYNAMIC HESEHVOm SIMULATION MODEL-DYRESM:5 357 layer behavior. The solid line is the total depth ~fter all the processes iIIustraled in Fig. 4.3 have acted. The thin line shows the deepening due to slirring only and 3 999.6 P Ikll m- 1 1000.2 10- the. jump at the end of each time period is ·the result of K-H billowing decreasing 7 e Izl /ro s' 2) 32/ II : II IIr-~~~~_ I 10-.e the layer depth. In the fourth time period. at the beginning of severe surface heat -, ',: ing the stirring mechanism actually predicted a decrease in layer thickness. This is 4 " explained simply by noting that the Monin-Obukhov length was decreased by the positive buoyancy flux due to the surface heating. The decreased depth however raised the velocity and greatly enhanced shear deepening. Each day of the simulation isa variation of these two extremes and the depth .of the mixed layer at the end of a season is merely the cumulative effect of indivi dual diurnal deepening events. The subroutine DIFUSE is called after the mixed layer dynamics calculations have been completed. Fig. 4.5 is a graph of the density profile of day 75138 and d lOll the associated vertical diffusion coefficient for heat. The value of 10-4m sec-I is the cut-off value introduced to simulate a fully turbulent condition. It is seen that 16 H as N increases €z decreases sharply leading to decreased mixing in stable regions. The great advantage of DYRESM is lhat each mixing process may be identified and each constant has a well-defined influence on the simUlated results. In Fig. 4.6b to 4.6dare shown three perturbations around the standar1 simulation reproduced in Fig. 4.6a. First, increasing the shear production efficiency greatly increases the deepen I t I ing during the wind events in January of 1976•. but otherwise introduces little J ..=~l,ge. The structure recovers quickly from the overdeepening indicating that the dynamics of the Wellington Reservoir are strongly forced with little long term ~'ringingH of the structure. This is also reflected in the short seiching periods. This is not the case for a recent simulation of Kootenay Lake in British Columbia. wa~ .J Here 1; close to 28 days and the structure tended to oscillate for long periods after a wind event. This necessitated a modification of MIXER routine which cap tured tbe wind direction as well as the wind speed. Figure 4.5. A plot ofdensity and vertical diffusivity on day 75138. The arbitrary cut-off of 10-4 m2 sec-1 in the diffusion coefficient has only a minor influel1.e on the temperature structure as is illustrated in Fig. 4.6c where the cut-off has been raised to 10-2 m2 sec-I. The salinity concentrations were however. appreciably altered with the increased diffusion reducing the peak salinity at the bottom. These simulations make clear that the diffusion in the hypolimnion s. CONCLUSrON r i, isa two parameter process; the second parameter being the appropriate diffusion!n I the absence of stratification. ~he dynamic simulation model DYRESM appears to successfully model the ! The importance of good input data is shown in Fig. 4.6d. A bad data point in dynam!cs of small and medium sized lakes. The model has only onecalibratable I u February of 1976 was not discovered until the recent introduction of the billowing coefficient and even tbe value of this is not expected to vary much from one lake I subroutine. Previous versions ofDYRESM were insensitive to isolated short dura to ~h: nex.t. The .model has been validate~in terms of temperature and salinitJfI tion. wind episodes. but as seen in Fig.4.6d a 12 m sec-I wind in February now vanatlorlS In 3 major lakes and two lakes with only poor data. It is now necessary I had a dramatic deepening influence. A check of the wind record revealed an error to collec~ data specific to certain. processes in the lake and valida~e {he details of I in transcript~on. !he algorithms MIXER and DIFUSE. Such a field program has been commenced I m Western Australia'j Certain changes to version 5 are presently in progress. These include the: 1(0 ~rovis.ion for drawdown at. the ~ffla~e slructure: localizing mixing around a river t., : . I lO~r~slon. lhe ~ffecl of wmd dl.reclwn. a specific para!lleterization of boundary .t.e • ...... JIIIiII'l ...... ~ ~ ... II1II\ ... lIIII\i p~.:]jferiz~., -.../III!IIi' l~ ~ ~ mlxmg and the mfluence~ of~rotation on the c:hp!oIr~pro."""-)n ~:)l'Il . . . ~.:.. .•1" •...... •.....•••.•..•- ,-~;_ r:-') .._ __ ...... - ...... ' ..- v , \,.<. J. L ~ '"'7 ~~""~ ---I~-' J~ J~ .J_(_J_~_. ~---,..--.-~.'--J---.~-.~~-~. ~ __ .. .._,,__ ..._..-'" ..._.. __...... , . . . \1 _~c~_. - ."'.~';.:;,~"~":" ~.~ ~ .~,~'.!·,<;,,~"'""'..;'''·~~lr~:;,':·~''\·~1>t/;f '·~,_1;)''::'.:':tf«M ~ , ."",,--*-. ,,' .. " ...... -""""""--,...... C\ ' ",",,,, .,~,..._.";("---...... -;-~,.:. .~~- .. .. 'Wib)QW&..grw••cftYC,,.nri**iI.1*+4- I I A B ~1 ITNfDAl'lD i g g :: ~ ~s r, sl!z .. ·z "0 \"-"""" ..c" ~g Bi ...... ~...... - f-... 8 ~ .. • g ,,....,...... ,J,...I.,II....,...... ,,,...... ,..J.,-,...... ,,-.....,,.....,.,.-,::.--,.,....-!J...... ,."'':['''-:t"'''''X....,.,,...J,,,.-...J,-l;~I.:l-:f-'L,+.J..,J.~~L...,,''"'"C,...... ,.M:r--"':''''''1,,.....~~"r'"'..,....,...... ,--:o:~:-;:r:-:-:''''':~.: I :u I!~ ;.n ~I B € :xl ..~ C,·O.5 ..~ ;;8 Ii: z'; ';z c".. "0.. '~g g~...... w-.... "'w g.. .i! I g .! ... 0 FigUre 4.6. (a)Predieted temperature structure. (b) Predicted temperature structure with Cs increased to 0.5 ~,c g ~ ~ :: MAXlllllJM DIFFUSlOfj COEFFICIENT. to-a... ..,a,-1 ~ ~ -0 1:0 :,.-..; o 0- ~r ...... -z ....~g c ~~ ...... I~.... ~d .~ ..~ ~-":""I.~~--;f-.-..!--"h:.~~ U a;"'7:'-n-~<;"".\.l;~-H-;I;-'cl....J..f"""'fr.""'~~~i"'"F""-rr-:n-.::..-.--;-'?\""';:~-..7'"' .~ • ...... r-,'TJ"F,r-,M"".Rr-,rrM.....J....."J" ?\"_ _ 1976 ~ c:n ~1 D = ~ ~ t ]I ..~ ,- FAUlTV !'lIND SPEED D...TA i! I'.' ;;8 :: z': c" gj; .-- ';z , C: a "0 ....>0 L.,~ ..I • I ... - 8~...... - ... I.. ~ J,I .. 'I :J. " l? ..... , ...... I• .. ·J'·M·R·M'J'J. .J. 'M'R.M'J'J' 0 1976 1978 Figure 4.6, Coni. (c) Predicted temperature structure -with diffusion coefficient cut-off raised to 10-2 m2 sec-I. (d) Predicted temperature structure showing the influence ofan erroneous wind llpeed of 12 m scc-l in February 1976. I ·f.I I ·-····c.:-. --_. rr() ~i" ...... ji-' . , c - .il j: ~.w' "W'C.t1*iJU 'P¥'/!\'ili)[lI',,*""P"l' "bll{'" '--,~ ,/:,,:,::.~/.. ;,'",. to '" ( , . -'-, , ..._-~~~lt ~'iJ !i"lHa:J$rl;;~llitt,~~]O;bW'WV'~l'A'~" -~----""""'~~:' Q ~.....,....~,- --"-",~",----".,"".,~"._,-.~:~.:.,,,,-,~'- <.....,,~ ""' --- --.... ::-.?!:~- ~ 360 JOBG iMBEHGEn AND JOHN C. PATTEHSON IX. A DYNAMIC RESEHVOJHSIMULATION MOOEL-DYRESM:5 361 Acknowledge.ments. DYRESM has been under development for over 6 years and numerous people have contributt:d to its constructkm. In particular the Imbcrgci, J., Thompson, R. and Fandry, C. (\976>. Selective withdrawal from a finite rectanglilar authors would like to thank Dr. Robert H. Spigel for has involvement in the lank. J. Flwd Mech. 78, 389-512. MIXER subroutine. Koh, R. C. Y. (196M. Viscous stratified flow towards a sink. J. Fluid Mech. 24, 555-575. Lawrence, G. A. and 1m berger, J. (1979). Selective Withdrawal through a Point Sink in a Continu The model WpS developed with funding from the Au!>tral1an Water Resources ously Stratified Fluid with a Pycnocline. Department of Civil Engineering, University of Western Council; the Water Resources Center of the University California, the Australian Australia, Report No. ED-79·oo2. Lumley, J. L. (965). On the interpretation of time spectra measured in high intensity shear flows. Research Grants Committee, the Public Works Department of Western Australia Phys. ofFlUids 8, 1056. and the Metropolitan Water Supply, Sewerage and Drainage Board of Western Niiler, P. P. and Kraus,E. B. (19771. One-dimehsional models of the upper ocean. In "Modelling and Australia. This assistance is gratefully acknowledged. Prediction of the Upper Layers of the Ocean," (Ed. E. B. Kraus). Jlergamon Press, New York. Ozmidov, R. V. (]965a). Some features of the energy spectrum of oceanic turbulence. Dokl. Acad. Nauk. SSSR Earth Sci. 160, 11-18. Ozmidov, R. V. (19650). On the turbulent exchange in a stably stratified ocean.Almos. Oceanic. Phys., Ser. I, 853-860. Pond, S., Fissel, D. B.and PaUlson, C. A. (]974). A note on bulk aerodnamic coefficients for sensible heat and moisture fluxes. Boundary-Layer Meleorol. 6, 333-;?;~ Rainer, K. N. (1980). Diurnal Energetics of Ii Reservoir Surface l.ayer. Joint Report: Environmental Dynamics, Department of Civil Engineering, University of Western Australia. Report EO.80-005, ~ R.EFERENCES and Department of Conservation and Environment, Western Australia, Bulletin No. 75. Sherman, F. S., 1mberger, J., and Cor,cos, G. M. (1978). Turbulence and mixing in stably stratified n waters. An. Rev. Fluid Meck 10. 267-288. 1/I, Carson, ,D. J. aud Richards, P. J. R. (1978>. Modelling surface turbulent fluxes in stable conditions. Spigel, R. H. (1978). Wind Mixing ,in Lakes. UnpUblished doc,oralthesis, University of California, '1 Baundary-Luyer MeteorG/. 14.67-81. Berkeley. Chen,C T. ami Millero, F. J 0977>' The use and misuse of pure water PVT properties for lake Spigel, R. H. andlmberger, J. H980). The classification of mixed layer dynamics in lakes of small to waters. Nalure 266. 707-708. medium size. J. Phys. Deeanogr. (jnpress). I Cor..:os, G. M. 0979). The Mixing Layer: Deterministic MOdels of a Turbulent Flow. College of Swinbank, W. C. (963). Longwave radiation from Clear skies. Quarl. J. Roy. Meleorol. Soc. 89, 339 c I ,I Engineering, University ofCalifornia, Berkeley, Report No. FM-79:2. . 348. Corcos, G. M.and Sherman, F. S. (976). Vorticity concentrations and the dynamiCS of unstable shear Thorpe, S. A.and Hall, A. J. (1977). Mixing in 8he lI.pper layer ofa lake during heating cycle. Nature I lavers. J.. .Fluid Mech 73, 241-264. 265,719-722. Deardorff, J. W. 0968). Degendence of air-sea tranercoefficients on bulk stability. J. Geophys. Res. Tennessee Valley Authority (972). Heat and Mass Transfer between a Water Surface and the Atmo 13, 2549-2557. sphere. Division of Water Resources Research, Tennessee VaHey Authority, Report No. 14. Denman. 1(. L. (973). Time dependent model of the Upper cone. J. Phys. Oceanogr. 3, No.2. 173 Wu, J. (1973). Wind induced entrainment across a stable density interface. J.Fluld Mech. '.,275- 184. 278. . 1 Oyer, K. R. (1974). The salt balance instratifiede;-;tuaries. Estuarme Coaslal Mar. Sci. 2.273-281. Zeman, O. and Tennekes, H. (977). Parameterisation of the turbulent energy budget at the lop of the t ~, Elder, R. A. and Wunderlich, W. O. (912). Inflow rtensity currents in TVA reservoirs. Proc. lnt. daytime atmospheric boundary layer. J. Almos. Sci. 34, 111-123. Symp. Stalrified Flow., Am. Soc. Civ. Eng. Ellison, T. H. and Turner, J. S. (959). TurbUlent entrainmentt in stratified flows. J. Fluid Meeh. 6, 423-448. FISCher, H. B., List, E. J•• Koh, R. Y. C." Imberger, J. and Brooks, N. H. (979). UMixingin Inland and Coastal Wal,ers." Academic Press, New York.. Garrett, C. (979). Mixing in the Ol:ean interior_ Dyn. Almos. Oceans 3,239-256. Garrett. C. 1. R. and Munk, W. H. (975). Space-rime scales of internal waves: a progress report. J. Geophys. Res. gO, 291-297. Hebbert, B., 1mberger, J, Loh. I. tlndPaUerson, J. (1979). Collie River underflow into the Wellington Reservoir. J. Hydraul. Dill. Proc. Am. SOC'- 0\1; Eng. ISO, HY5, 533-545. Hicks, B.B. 0972>' Some evaluation!. of drag and bulk transfer coefficients over water bodies of different sizes. Boundary-Layer Meteorol. 3, 201-213. Hicks, B. a. (975). A procedure for the formulation of bulk transfer coefficients over water. Boundary-Layer Meteorol. It 515-524. Huber, D. G. (960). Irrotational motion of two Jluid strata towanls a line sink. J. Eng. Meclt. Dll'. froe:. Am. Soc. Civ. Eng. 53, No.2. 329-349. Imbe{ger, J. (977). On the validity of water quality models for lakes and reservoirs. pf()('. ('ongr. Inl. Assoc. Hydraul. Res. 17th 6, 293-303. Imberger. J., Patterson, J., Bebbert. :a. and Loh. 1. (1978), Dynamics c.f reservoir of medium sitc. J. llydrau/lc. DJlI. Prtx:. Am. Soc. em. Eng. 104, No. HY5, 725-743. I Imberger, J. ana Splgel, R II (980). Billowing and lIs inlluence on mixed laycr dccpcning. Prot. k fi:vfI1J1. Sur}. Waler ltnpoundtTleJllS.• MUll," ("mi/!>. In J)rc~s. 1 :: 1<;-·-: ; :: ~' ~ : : :~ ~ I : I d~~; -~ ~~~. ;~ot~~r"6iJ L ~. ~ ·-'"···-.r~'~·-C-·~ .,~~ -~~~~~~~-.-.~. "- ~-- ~_~~~._•.. ~__. .".-..... _""~"~_ .... ~.,.-,. --~-' ,..-:- .... -."~".~_.-"'_._--'_.:-''-:"" .._....-_,_.".,__ ...-,.,. ".~ \," , -,_·.·.'~---i.. ·-. __,_., ._.'". ~ ..,,_' .. '_'" ._""_ ~.. ~. __'"_ ~'_·-_',:-:'-::;;_-~'-H: ,,,.. h·.~·.'~ •. _,"-·~ __""'~-'_·.._,,•. ~_._,, " ·· .. ~-·__~---·._'_ri~-.,..-'- ..'..:"---.----....-.-•.-~.~ ...... ,.., ,..- __ ~ ""'>'- __ ::\,"~_'_""'_'?" .~_~ ~, __, .__"--.....,...~_,..:'''' • 0': .'--<-_._.. __ l