Chapter 4: Estuarine Mixing

4 Estuarine Mixing Initial concepts: tides and salinity Tide-resolving models Tidal-average models Tracers for model calibration Mixing diagrams Residence time Dual tracers What is an estuary? A semi-enclosed coastal body of water which has a free connection with the open sea and within which sea water is measurably diluted with fresh water derived from land drainage (Pritchard, 1952) Where the river meets the ocean Like a river but with tides and salinity gradients Tidal motion Tidal Channel Ocean η(t) 2ao Tt Head 2ξo ao = tidal amplitude 2ao = tidal range Mouth Tt = tidal period Gravitational and centrifugal acceleration (E with M & S) 2ξo = tidal excursion Ocean range ~ 0.5 m Coastal waters may have much larger ranges Equilibrium tide; moon only Low Water surface M High E High Low At any time: 2 high and 2 low tides; At any location: ~ 2 high and 2 low tides per day Combined sun and moon T=6.8d Lunar 29.5d month S 27.3d 13.6d 20.5d Sun and moon aligned (full and new moon) => spring tide; Sun and moon opposed (1st and 3rd quarters) => neap tide Because the earth revolves, period of spring-neap cycle = 365d/[(365/27.3)-1] = 29.5 days Number of full moon’s per year And because the moon revolves 24.8 E Lunar day 24 h M Lunar day = 29.5 d /(29.5 – 1) = 24.8 hours Dominant (lunar semi-diurnal tidal) period is 12.4 h Also a diurnal period “Side” View In general a number of tidal constituents are required to compose an accurate tidal signal L H L Because of the earth’s declination higher latitudes tend to experience a single (diurnal) cycle per rotation H “Top” View Full Moon Last Quarter New Moon Tidal displacement above reference Moon Phase First Quarter Los Angles, California (outer harbor) o o 33 43' N, 118 16' W 12.4 hr Mixed tide (with strong semi-diurnal component; lower latitude) 5 0 14.7d St. Michael, Alaska (on the being sea) o o 63 29' N, 162 02' W 24.8 hr Diurnal tide (higher latitude) Spring-neap cycle 5 0 5 10 15 20 25 30 Date in July 1963 Ippen, 1969 Figure by MIT OCW. Idealized (linear) tidal motion η(t) Qf η (t ) = a cos(ωt ) u (t ) = Qf A 2ξo + u max cos(ωt + φ ) ξ (t ) = u t + 2ξ o ( x) = ω = 2π/Tt u max Tt sin(ωt + φ ) 2π u max Tt Tidal excursion π Pu ( x) = Vu ,high ( x) − Vu ,lowh Upstream tidal prism Pu(x) x L High Tide Low Tide 2ξο(x) High Tide Vu(x) Low Tide 2ξ(x) Pu(x) 0 x L Now introduce salinity River Estuary Tidal, Freshwater Qf Ocean Salinity Intrusion + + + Mouth Head of Tide 10 S=0 20 30 S=35 psu PSU = practical salinity unit, an operational definition of salinity (mass fraction: ppt, o/oo or g/kg) Equation of State ρ = ρ (T ) + ∆ρ ( S ) + ∆ρ (TSS ) ⎡ ρ (T ) = 1000⎢1 − (Gill, 1982; ch 6) (Also pressure at deep depths) ⎤ T + 288.9414 (T − 3.9863) 2 ⎥ 508929.2(T + 68.12963) ⎦ ⎣ ∆ρ ( S ) = AS + BS 3 / 2 + CS 2 A = 0.824493 − 4.0899 x10 −3 T + 7.6438 x10 −5 T 2 − 8.2467 x10 −7 T 3 + 5.3875 x10 −9 T B = −5.72466 x10 −3 + 1.0227 x10 − 4 T − 1.6546 x10 −6 T 2 C = 4.8314 x10 − 4 1 ⎤ ⎡ −3 ∆ρ (TSS ) = TSS ⎢1 − 10 x ⎣ SG ⎥⎦ ρ = kg/m3, T in oC, S in PSU (g/kg), TSS in mg/L Seawater Density ( σ1 Units ) 44 -8 40 -6 σt = 1000*(ρ-1) -4 36 Temperature (oC) Fischer, et al. (1979) -10 48 -2 (ρ in g/cm3) 0 32 2 4 28 6 8 10 24 12 14 20 16 18 16 20 22 12 24 26 8 28 30 4 0 0 4 8 12 16 20 24 SALINITY (% ) 28 32 36 40 Figure by MIT OCW. Seawater Density ( σ1 Units ) 44 -8 40 -6 Salt water – Freshwater density difference ∆ρ o ρ -4 36 Temperature (oC) Example: -10 48 -2 0 32 Ocean salinity ~ 35 psu Freshwater salinity 0 psu 2 4 28 6 8 10 24 12 14 20 16 18 16 Temperatures 0 to 30C 20 22 12 ∆ρ o ρ = [28-0]/1000=0.028 24 26 8 28 (0C) 30 4 0 0 4 8 12 16 20 24 28 SALINITY (% ) Figure by MIT OCW. 32 36 40 = [22-(-4)]/1000=0.026 (30C) Estuary classification 10 S=0 20 30 S=35 psu Well mixed: isohaline lines approach vertical (Delaware R) Partially mixed: isohaline lines slant Vertically stratified (salt wedge): isohaline lines approach horizontal (Mississippi R.) Desire to classify to know what type of model/analysis to use; several options available; none is perfect Estuary classification, cont’d Densimetric Estuary number (Harleman & Abraham, 1966; Thatcher & Harleman, 1972) 2 Pt = tidal prism; Q f = freshwater flow rate; Pt Fd Ed = Q f Tt Fd = Tt = tidal period uo g (∆ρ o / ρ )h Fd is a densimetric Froude number u o = maximum tidal velocity; h = estuary depth; ∆ρ o /ρ = salt water − fresh water density difference Estuary classification, cont’d Estuary Richardson number (Fischer, 1972; 1979) R= ∆ρ o gQ f / W ~ Ed ρu t 3 −1 W = estuary width; u t = RMS tidal velocity ≅ 0.71u o R ~ potential energy rate/kinetic energy rate R < 0.08 well-mixed 0.08 < R < 0.8 partially stratified 0.8 < R vertically stratified (salt wedge) Example later Estuary classification, cont’d The definitions are related E d ~ R −1 ~ ut 3 u f ud 2 Each involves 3 velocities: u t = RMS tidal velocity Tends to mix estuary u f = fresh water velocity = Q f /A Tends to stratify estuary u d = density velocity = g(∆ρ o /ρ )h Tends to stratify estuary Hanson-Rattray (1966) 10 10 -1 3.3 x 10 -2 P = 10 P= 1 x „ 3. 3 x 10 -4 Increases w/ P, decreases w/ Fm = „ P Fm = 10 -3 10 = P -4 -3 10 x 3. 3 = P 10-2 10-3 1 10 us uf salinity stratification δS S o = ( S b − S s ) S Fm = 10 10 = P Fm = 10 -2 -2 -3 -1 Fm = 10 P = 3. 3 δS -1 S0 10 Fm = 1 < - Salinity stratification Semi-empirical Predicts -1 2 2 10 Velocity stratification -> Figure by MIT OCW. 10 3 Velocity stratification average surf vel / u s /u f =tidal tidal and depth aver vel Decreases w/ Fm uf uf ; Fm = P= ud ut Tide resolving models Well-mixed (1-D) estuary ∂c ∂c 1 ∂ ⎛ ∂c ⎞ q L (c L − c ) ′ + ∑ ri + ∑ re + u (t ) = ⎜ AE L (t ) ⎟ + A ∂t ∂x A ∂x ⎝ ∂x ⎠ Major difference between river and well-mixed estuary are 1) u is time-varying, 2) EL is constrained by reversing tide. Look at 2) first Characteristic dispersion time scales EL ~ Uc2Tc ~ u*2Tc (Fischer et al., 1979) For rivers, two possible time scales, Tc: „ Ttm ~ B2/ET and Tvm ~ h2/Ez „ Ttm >> Tvm => EL ~ u*2 Ttm (after transverse mixing) For estuaries, additional possibility: Tc = Tt/2 „ Ttm >> Tt/2 ~ Tvm => EL ~ u*2 Ttm or u*2 Tt/2 Previous example, B = 100 m, H = 5 m, u = 1 m/s Tvm = 750 s, Ttm = 34000 s, Tt/2= 22000 s (6.2 h) Dispersion in reversing flow “narrow” channel B t=0 Tc=Ttm Dispersion governed by Ttm, 0.5Tt 2 u* B 2 EL ~ ET Dispersion in reversing flow, cont’d “narrow” channel B t=0 Ttm Dispersion governed by Ttm, E L ~ 2 u* B ET 0.5Tt 2 “wide” channel B l ~ ET Tc t=0 Tc = 0.5Tt 2 2 u* ET Tt 2⎛ l ⎞ E ~ u T ~ ⎟ t Dispersion governed by Tt, * ⎜ L B B2 ⎝ ⎠ 2 Effects of reversing u(t) Mass continuously injected at x = 0 C land ocean High tide Low tide x 0 2ξo An actual simulation Harleman, 1971 1.0 0.5 N = 400.5 N = 400.0 L.W.S H.W.S N = 30.5 N = 30.0 0.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Figure by MIT OCW. 1.0 1.5 2.0 x / 2ξ o Continuous injection at x = 0; output after 30, 400 tidal periods (high slack) and 30.5 and 400.5 tidal cycles (low slack) Tidal-average models Perhaps we don’t care to resolve intratidal time-dependence Strong non-uniformities prevent resolution of intra-tidal variability Long term calculations more efficient with tidal-average time step However, averaging obscures physics Tidal-average models, cont’d Analogous, in principle, to time and cross-sectional averaging u = u + u ′′′ Triple bars imply tidal average c = c + c ′′′ Insert into GE and tidal-average ∂c ∂ c 1 ∂ ⎛⎜ ∂ c ⎞⎟ AE L ⎟ + ∑ ri + ∑ re +u = ⎜ ∂t ∂x A ∂x ⎜ ∂x ⎟ ⎠ ⎝ Tidal average Tidal average velocity disp coef Structurally similar to equation for river transport => similar solutions Tidal average dispersion Tidal pumping (shown) „ „ Ebb „ Asymmetric ebb (a) & flood (b) Tidal averaging => mean velocity (c) Trans mixing + trans velocity gradients => dispersion! Similar drivers Flood „ „ „ Net Tidal trapping Coriolis + density Depth-dependent tidal reversal EL ~ (2ξo)2/Tt General result A B C c Conservative Tracer; 3 injection locations x c Non-conservative tracer; middle location x c Non-conservative tracer; 3 locations x Comments For conservative tracer, c(x) „ „ Is independent of xd for x > xd Decrease with xd for x < xd If you must pollute, do it downstream (more discussion later) Several specific solutions in notes Conclusion applies loosely even if not 1-D Signell, MWRA (1999) One example m& q" = A Rectangular channel; no through flow 0= d ⎛ dc ⎞ ⎜ E L ⎟ − kc dx ⎝ dc ⎠ E L ~ ( 2ξ o ) 2 / Tt = αx 2 2 dc 2d c − kc 0 = 2αx + αx 2 dx dx 0 xd L Solution q ′′ ⎡ x −1 2−κ 2 x −1 2+κ 2 ⎤ c + ( x, x d ) − c L = ⎢ 1 2−κ 2 − κ 1 2−κ 2 ⎥ ακ ⎣ x d L xd ⎦ x > xd q " ⎡ x −1 2+κ 2 x −1 2+κ 2 ⎤ c _( x, x d ) − c L = ⎢ 1 2+κ 2 − κ 1 2−κ 2 ⎥ ακ ⎣ x d L xd ⎦ x < xd κ = 1 + 4k / α x WE4-1 Proposed relocation of Gillette’s Intake Proposal to shorten Fort Point Channel as part of the Big Dig threatened to limit Gillette’s cooling water source Details Boston Harbor 1700m 1700m 2ao=2.9 m; h=6 m; k=0.1 day-1 700 600 Discharge (xd) Intake (xi) 400 Qo = 1.4 m3/s; ∆To = 6C 0 “Existing” Channel “Modified” Channel Proposed remedies: move discharge and/or intake downstream. How far? Results of analysis 3 Temperature (C) 2.5 Existing Mod Chan 2 MC + Disch 1.5 1 0.5 xd 0 0 400 m 500 1000 1500 2000 Distance (m) Existing: Ti (x=600) ~ 0.8C; Modified: Ti ~ 2.4C Moving intake 400 m downstream (x=600) yields Ti ~ 0.8C Moving discharge 300 m downstream (x=900) also yields Ti ~ 0.8C Tidal Prism Method Treats whole channel as single well-mixed box High tide Low tide Qf Mass that leaves on ebb does not returns Except for harbors/short channels, this overestimates flushing; underestimates c. Hence common to “discount” P by defining the effective volume P’ of “clean” water. E.g., P’ = 0.5 P Formal ways to compute return factor using phase of circulation outside harbor Pf = Qf Tt Pctp Tt = m& m& Tt ctp = P P = total tidal prism f = “freshness” =(So-Sn)/So Modified Tidal Prism Method Divides channel into segments of length 2ξo Assumes EL = (2ξo)2/Tf <=> net ds transport during Tt is Pn 2ξo,n Pn Qf High tide Low tide Vn Pn f n = Qf Tt Pn cn = m& Tt cn = Vn+1=Vn+Pn m& Tt Pn fn = “freshness” =(So-Sn)/So mass injected continuously upstream of section n (behaves like freshwater) Comments Modified Tidal Prism Method has been modified and re-modified many times Ad-hoc assumption => not always agreement with data Non-conservative contaminates reduced in concentration by χ r 1 − (1 − r )e − kTt r = 2a / h χ= Salinity as tracer to measure EL Steady, tidal average flow d (u f AS ) = d ( AE L dS ) dx dx ds Integrate with S = dS/dx at head, x=0 EL = uf S dS / dx Example: Delaware R (WE 4-2) Measured salinity profiles 5.0 10.0 8.0 14.0 12.0 18.0 16.0 22.0 20.0 7.0 26.0 24.0 28.0 8.0 9.0 20.0 40.0 60.0 DISTANCE FROM BAY MOUTH (km) 100.0 120.0 12 4 5.0 6.0 6 10 18 7.0 3.0 22 4.0 26 14 8.0 9.0 60.0 40.0 0.0 20.0 DISTANCE FROM BAY MOUTH (km) November 1987 0.0 Should it be? What is EL? 22 18 14 6 24 10 20 9.0 12.0 140.0 0.0 18.8 Salinity profiles show river to be well-mixed. 26 4 8.0 11.0 80.0 20.0 12 7.0 11.0 100.0 2 6.0 10.0 120.0 40.0 16 5.0 10.0 12.0 140.0 24 DRBC RIVER MILES 55.9 37.3 74.6 2.0 24 20 12 60.0 80.0 C&D 16 0.0 18.8 Cheater C&D 8 2 26 April 1987 PRESSURE (db) PRESSURE (db) 4.0 Cheater 2.0 3.0 DRBC RIVER MILES 55.9 37.3 16 DISTANCE FROM BAY MOUTH (km) October 1986 74.6 8 9.0 12.0 140.0 0.0 22 8.0 11.0 80.0 18 7.0 11.0 100.0 14 10 0.0 18.8 20 4 6.0 10.0 120.0 6 2 5.0 10.0 12.0 140.0 Cheater 3.0 C&D 2.0 C&D 6.0 4.0 DRBC RIVER MILES 55.9 37.3 74.6 4.0 2.0 6.0 0.0 18.8 PRESSURE (db) PRESSURE (db) 4.0 Cheater 2.0 3.0 DRBC RIVER MILES 55.9 37.3 74.6 28.0 8 120.0 100.0 80.0 60.0 40.0 DISTANCE FROM BAY MOUTH (km) April 1988 20.0 0.0 Figure by MIT OCW. Kawabe et al. (1990) Å Head 3.0 PRESSURE (db) 4.0 Cheater 2.0 C&D 74.6 DRBC RIVER MILES 55.9 37.3 8 12 2 4 5.0 6.0 6 16 10 18 7.0 Mouth (ocean) 0.0 18.8 24 20 22 26 14 November 1987 8.0 9.0 10.0 11.0 ~ h (m) EL = uf S 12.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 Figure by MIT OCW. DISTANCE FROM BAY MOUTH (km) ≅ (Q f / A) S dS / dx ∆S / ∆x (260)(8) ≅ (1.5 x10 4 )(8) / 20000) ≅ 350 m 2 / s Qf = 260 m3/s; A = 1.5x104 m2; S = 8 psu (80 km); ∆S/∆x = (12-4) psu/20 km (70-90 km) Should river be well-mixed? ∆ρ R= ≅ ρ g ut Qf W 3 (0.025)(10)(260) / 4000 ≅ 0.02 < 0.08 3 1 Yes! Box models Qf Qf Qf Q12 c1 Qf Q23 c2 Q34 c3 c4 Q f + f 2 Q1, 2 = f 1 (Q1, 2 + Q f ) f1 (Q1, 2 + Q f ) + f 3 Q2,3 = f 2 (Q1, 2 + Q2,3 + Q f ) f 2 (Q2,3 + Q f ) + f 4 Q2,3 = f 3 (Q2,3 + Q3, 4 + Q f ) n equations in n unknowns; boxes dictated by geometry Salinity as direct measure of c Qf x 0 xd Use measured salinity distribution S(x) resulting from river discharge Qf entering at head (x=0) to infer concentration distribution c(x) of mass entering continuously at downstream location xd. L Qf x 0 xd L S/So 1 S/So 0 0 xd L Qf x 0 xd S/So, f f = (So-Sx)/So f 1 L Freshness: ND concentration of fresh water S/So 0 0 xd L Qf x 0 xd L Effective downestuary transport rate, Qeff Qeff = Qeff Qf fx Hypothetical flow rate necessary to transport freshness downstream by advection only (no tidal dispersion) df f = Q f = Q f f − EL A dx Qeff = Q f − E L A df f dx Qeff really accounts for both advection and dispersion Qf x 0 xd S/So, f Qeff = Qf/f L Qeff/Qf Qeff/Qf 4 f 1 S/So 2 0 0 0 xd L Qf x 0 xd L Downstream from xd, mass is transported like freshness Qeff c = m& Qeff f = Q f cx m& = fx Qf ⎛ So − S x c x = ⎜⎜ ⎝ So Concentration at xd ⎞ m& ⎟⎟ ⎠ Qf ⎛ So − Sd c d = ⎜⎜ ⎝ So ⎞ m& ⎟⎟ ⎠ Qf Qf x 0 xd S/So, f ⎛ S − Sx c x = ⎜⎜ o ⎝ So L ⎞ m& ⎟⎟ ⎠ Qf Qeff/Qf Qeff/Qf 4 c f 1 S/So 2 0 0 0 xd L Qf x 0 xd Upstream from xd, mass is transported like salinity cx S x = cd S d S o − S d m& S x cx = So Q f Sd cd L Qf x 0 xd L S/So, f c = S o − S d m& S x x So Qeff/Qf Qeff/Qf Q f Sd 4 c f 1 S/So 2 0 0 0 xd L (Conservative) Mixing Diagrams Concentration of conservative contaminant discharged at head (using freshness as tracer) c cmax ⎛ So − S x c x = ⎜⎜ ⎝ So x c x = a − bS x 0 0 ⎞ m& ⎟⎟ ⎠ Qf So m& = c max S a= Qf aka C-S (or T-S, etc.) diagram, or property-salinity diagram Uses for Property-S diagrams Determine end-member concentration & ) and loading (So, Qf known, but not m Identify extraneous sources (we think we know m& = Q f co but cmax > co) Distinguish different water masses Predict quality of mixed water masses Detect non-conservative behavior Determining end member c 1) Extrapolate to get cmax c 2) m& = Q f c max 3) If cmax > measured co, difference is extraneous source(s) 0 0 So Distinguishing water masses N-S diagram for Massachusetts Bay, Kelly (1993) Used to identify coastal water vs offshore waters Non-conservative behavior c cmax x 0 S 0 So Non-conservative behavior c cmax + x - 0 S 0 So Non-conservative behavior c cmax + x - 0 S 0 So Note that conservative mixing curve is only linear if conditions are steady and there is a single source Two conservative sources look like one NC source c 1 2 0 0 So Two conservative sources look like one NC source c 1 1+2 2 0 0 So Transient Conditions WE4-2 Nitrate-Salinity diagrams in Delaware R 200 X Fall 100 Ciufuentes, et al. (1990) 200 Summer Solid lines are predictions for conservative tracer & salinity at 4 times (not linear because river flow varies in space and time) Nitrate (µM) 100 200 Spring 100 Symbols are data for nitrate & salinity 200 Water 100 Why the discrepancy in fall, spring? 0 16 Salinity (% ) 32 Figure by MIT OCW. Residence times Why? Compare with k-1 „ „ tres >> k-1 => reactions are important tres << k-1 => reaction not important Also to determine if model has reached steady state Approaches „ „ Continuous tracer Instantaneous tracer Related time scales Continuous tracer release; c(x,y,z) monitored after steady state V t res = ∫ cdV 0 m& = M m& SS inventory over renewal rate; heuristic interpretation m& Types of Tracers Advantages and Disadvantages of each m& Deliberate tracer (e.g., dye) Tracer of opportunity (e.g. trace metals from WWTP) Freshwater inflow (freshwater fraction approach; residence time sometimes called flushing time) V t res = ∫ fdV 0 Qf WE 4-4 Trace metals to calculate residences times for Boston Harbor Residence Time in Boston Harbor 2500 3.4 Days 10 Days Zn 2000 Cu 1500 Cr 2 Days 1000 Ni 500 PCB 0 Pb Cd 0 Hg 20 cV t res = m& (2.5 x10 −6 kg / m 3 )(6.3 x10 2 m 3 ) = = 3.4d 5 (1.7 x10 kg / yr ) /(365d / yr ) PAH 40 60 80 100 120 140 160 180 Total Load to Harbor (kg/yr) (Thousands) Figure by MIT OCW. Shea and Kelly, 1992 Comments Ignore re-entries (by convention) If multiple sources, tres is average time weighted by mass inflow rate Assumes steady-state, but “fix-ups” applicable to transient loading Residence time reflects injection location; not property of water body… unless well mixed, in which case: c( x, y, z ) = c = const t res cV = m& Tres depends on discharge location A B C V t res = c x c x c x tres A ∫ cdV 0 m& > tres = M m& B > tres C Instantaneous Release; c(x,y,z,t) monitored over time Unit mass m& m* 1 1 0 t Rate of injection f * (t ) = − f* 0 t Mass remaining in system dm * dt ∞ ∫ f * (t )dt = 1 0 0 Mass leaving rate t Instantaneous release, cont’d f* is also distribution of residence times (mass leaving no longer resides). By definition, tres is mean (first temporal moment) of f* ∞ t res = ∫ 0 ∞ ∞ dm& ∞ f *tdt − ∫ tdt = −m * t 0 + ∫ m * (t )dt dt 0 0 1st moment of f* 0th moment of m* For mass of arbitrary loading Mo (not necessarily one) ∞ t res = ∞ ∫ f (t )tdt ∫ M (t )dt 0 Mo = 0 Mo M(t) = mass remaining f(t) is mass leaving rate Thus two more operational definitions of residence time: 1st temporal moment of f(t) and 0th temporal moment of M(t) WE 4-5 Residence time of CSO effluent in Fort Point Channel Rhodamine WT injected instantaneously at channel head on three dates; results for one survey: N Boston Inner Harbor 18 18 ft ft Northern Ave. Congress St. Summer St. 18 ft Gillette Dorchester Ave. Broadway Adams, et al. (1998) BOS 070 Figure by MIT OCW. Dye 100 Meters 0 500 Fort Point Channel dye release, cont’d Total Mass of Dye (Kg) 12 10 ∞ 8 6 t res = 4 2 0 0 20 40 60 80 100 ∫ M (t )dt 0 Mo ≅ 2.7 day 120 TIME (Hours after injection) Figure by MIT OCW. tres Adams, et al. (1998) Comments f(t) can be obtained from time rate of change of M(t); or from measurements of mass leaving (at mouth) Residence times for continuous and instantaneous releases are equivalent f(t) of f*(t) conveniently used to assess first order mass loss. ∞ F = ∫ f * (t )e − kt dt 0 F = total fraction of mass that leaves Mass loss from FPC (%/10hrs) WE 4-6 Residence time of bacteria in CSO effluent in Fort Point Channel (Adams et al., 1995) 40 1.6 35 1.4 30 1.2 f*(t) 1990 25 1 20 0.8 15 0.6 1 10 5 0.4 0.25 0.5 0.2 2 0 50 100 150 0 200 Time (h) ∞ F = ∫ f * (t )e − kt dt 0 1− F Residence time distributions f(t) determined from distributions of m(t). Indicator bacteria “disappear” (die or settle) at rates of 0.25 to 2 d-1 What fraction of bacteria would disappear for 1990 conditions? Figure by MIT OCW. Fraction (of viable bacteria) that leave Fraction that are removed within channel k=2.0 d-1 => F=0.15 (85% removed); k=0.25 d-1=> F=0.55 (45% removed) Relative advantages of 3 approaches? M (t) C(V) f (t) Mo t t V Instantaneous Instantaneous Continuous t1 = V ∞ ∞ ∫ M (t )dt 0 Mo t2 = ∫ f (t )tdt 0 Mo t3 = ∫ cdV 0 m& Amount of tracer (e.g., dye) required? Effort to dispense? Number of surveys and their spatial extent? Total duration of study? Other related time scales Flushing time use to describe decay of initial concentration distribution (convenient for numerical models); used by EPA for WQ in marinas (see example) Age of water (oceanography): time since tracer entered ocean or was last at surface (complement of tres) Concepts often used interchangeably, but in general different; be careful Dual Tracers Used to empirically distinguish fate from transport: introduce two tracers (one conservative; one reactive) instantaneously. Applies to any time of water body, but consider well mixed tidal channel dM c = −k f M c dt dM nc = −k f M nc − kM nc dt Mass of conservative tracer declines due to tidal flushing Mass of NC tracer declines due to tidal flushing and decay d ⎛ M nc ⎞ = −k ⎛ M nc ⎞ Ratio of masses declines due to ⎜ ⎟ ⎜ ⎟ decay M M c⎠ c⎠ ⎝ dt ⎝ M nc =⎡ M c ⎢⎣ M nc ⎤ e − kt M c ⎥⎦ o WE 4-7 Fort Point Channel again R 2 1.8 1.6 1.4 1.2 Best-fitted line 1 0.8 R k3 = 0.25 d-1 0.6 Fluorescent pigment particles (yellow DayGlo paint) were injected with dye. Pigment particles settle as well as flush. R = (Mp/Mpo)/(Md/Mdo) k = ksettle = 0.25 d-1 0.4 k = ws/h 0.2 0 20 40 Time (hr) Figure by MIT OCW. Adams, et al. (1998) 60 80 ws = kh = (0.25d-1)(6m) =1.5 m d-1 More in Chapter 9