Presentation on theme: "Principles of Mass Balance"— Presentation transcript:
1 Principles of Mass Balance Lecture 11Principles of Mass BalanceSimple Box ModelsThe modern view about what controlsthe composition of sea water.
2 Four Main ThemesGlobal Carbon CycleAre humans changing thechemistry of the ocean?3.What are chemical controlson biological production?4. What is the fate of organic mattermade by biological production?
3 texport/tbiota = 0.27/0.06 = 4.5 times recycled Example: Global Carbon CycletC,biota = 3/50 = 0.06 ytC,export = 3/11 = 0.29 ytexport/tbiota = 0.27/0.06 = 4.5 times recycledtCO2,atm = 590/130 = 4.5 yNo red export!
4 -Box (or reservoir) Models -Continuous Transport-reaction Models Two main types of models used in chemical oceanography.-Box (or reservoir) Models-Continuous Transport-reaction ModelsIn both cases:Change in Sum of Sum ofMass with = Inputs OutputsTime
5 At steady state the dissolved concentration (Mi) does not change with time:(dM/dt)ocn = SdMi / dt = 0Sum of sources must equal sum of sinks at steady state
6 Box ModelsHow would you verify that this 1-Box Ocean is at steady state?
7 For most elements in the ocean: (dM/dt)ocn = Fatm + Frivers - Fseds + FhydrothermalThe main balance is even simpler:Frivers = Fsediment Fhydrothermalall elements all elements source: Li, Rb, K, Ca, Fe, Mnsink: Mg, SO4, alkalinity
8 = mass / input or removal flux = M / Q = M / S Residence Time = mass / input or removal flux = M / Q = M / SQ = input rate (e.g. moles y-1)S = output rate (e.g. moles y-1)[M] = total dissolved mass in the box (moles)
9 d[M] / dt = Q – Sinput = Q = Zeroth Order flux (e.g. river input)not proportional to how much is in the oceansink = S = many are First Order (e.g. Radioactive decay,plankton uptake, adsorption by particles)If inflow equals outflowQ = Sthend[M] / dt = 0 or steady state
10 First order removal is proportional to how much is there. S = k [M] where k (sometimes ) is the first order removal rate constant (t-1)and [M] is the total mass.Then:d[M] / dt = Q – k [M]at steady state when d[M] / dt = Q = k[M][M] / Q = 1/k = and [M] = Q / kinverse relationship
11 sw Reactivity and Residence Time Cl Al,Fe Elements with small KY have short residence times.When t < tsw not evenly mixed!A parameterization of particle reactivityWhen the ratio is small elements mostly on particles
12 Dynamic Box ModelsIf the source (Q) and sink (S) rates are not constant with timeor they may have been constant and suddenly change.Examples: Glacial/Interglacial; Anthropogenic Inputs to OceanAssume that the initial amount of M at t = 0 is Mo.The initial mass balance equation is:dM/dt = Qo – So = Qo – k MoThe input increases to a new value Q1.The new balance at the new steady state is:dM/dt = Q1 – k Mand the solution for the approach to the new equilibrium state is:M(t) = M1 – (M1 – Mo) exp ( -k t )M increases from Mo to the new value of M1 (= Q1 / k) with a response time of k-1 or
13 Dynamic Box Modelst =The response time is defined as the time it takes to reduce the imbalance to e-1 or 37% of the initial imbalance (e.g. M1 – Mo). This response time-scale is referred to as the“e-folding time”.If we assume Mo = 0, after one residence time (t = t) we find that: Mt / M1 = (1 – e-1) = 0.63 (Remember that e = 2.7.). Thus, for a single box with a sink proportional to its content, the response time equals the residence time.Elements with a short residence time will approach their new value faster than elements with long residence times.e = Σ 1/n!
15 Broecker two-box model (Broecker, 1971) v is in m y-1Flux = VmixCsurf = m yr-1 mol m-3 = mol m-2y-1see Fig. 2 of Broecker (1971)Quaternary Research“A Kinetic Model of Seawater”
16 Mass balance for surface box Vs dCs/dCt = VrCr + VmCd – VmixCs – BAt steady state:B = VrCr + VmixCd – VmixCs and fB= VrivCriv
17 How large is the transport term: If the residence time of the deep ocean is 1000 yrs (from 14C)and t = Vold / Vmixthen:Vmix = (3700m/3800m)(1.37 x 1018 m3) / 1000 y= 1.3 x m3 y-1If River Inflow = 3.7 x 1013 m3 y-1Then River Inflow / Deep Box Exchange = 3.7 x 1013/1.3 x 1015= 1 / 38This means water circulates on average about 40 timesthrough the ocean (surface to deep exchange) before itevaporates and returns as river flow.fraction of total depththat is deep oceanvolume
18 Broecker (1971) defines some parameters for the 2-box model g = B / input = (VmixCD + VrCr – VmixCs) / VmixCd + VrCrf = VrCr / B = VrCr / (VmixCd + VrCr - VmixCs)f x gIn his model Vr = 10 cm y-1Vmix = 200 cm y-1so Vmix / Vr = 20fraction of inputto surface boxremoved as BFraction of B fluxpreserved in sedimentsbecause fB = VrCrfraction of element removed tosediment per visit to the surfaceHere are some values:g f f x gNPCSiBaCaQ. Explain these values andwhy they vary the way they do.See Broecker (1971) Table 3
20 Why is this important for chemical oceanography? What controls ocean C, N, P?assume g ≈ 1.0Mass Balance for whole ocean:C/ t = VRCR – f BCS = 0; CD = CDVU = VD = VMIXNegative Feedback Control:ifVMIX ↑VUCD ↑B ↑ (assumes g is constant!)f B ↑ (assumes f will be constant!)assume VRCR then CD ↓ (because total ocean balanceVUCD ↓ has changed; sink > source)B ↓The nutrient concentration ofthe deep ocean will adjust so thatthe fraction of B preserved in thesediments equals river input!CSCDif VMIX = m y-1 and C = mol m-3flux = mol m-2 y-1
21 Multi-Box Models Vt – total ocean volume (m3) Vs = surface ocean volumeVu,Vd = water exchange (m3 y-1)R = river inflow (m3 y-1)C = concentration (mol m-3)P = particulate flux fromsurface box to deep box (mol y-1)B = burial flux from deep box(mol y-1)
22 1. Conservation of waterR = evap – precipVu = Vd = V2. Surface Box mass balance (units of mol t-1)Vols dCs/dt = R[CR] + V [Cd] – V ([Cs]) - PVols dCs/dt = R[CR] – V ([Cs] – [Cd]) - P3. Deep Box mass balanceVold d[Cd] / dt = V [Cs] – V[Cd] + P - BVold d[Cd] / dt = V ([Cs] – [Cd]) + P - B4. At steady stated[Ct] / dt = 0 and R [CR] = B
23 Example: Global Water Cycle 103 km3103 km3 y-1Q. Is the water content of the Atmosphere at steady state?Residence time of water in the atmosphere= 13 x 103 km3 / 495 x 103 km3 y-1 = yr = 9.6 dResidence time of water in the ocean with respect to rivers= x 109 km3 / 37 x 103 km3 y-1 = 37,000 yrs
24 Summary Salinity of seawater is determined by the major elements. Early ideas were that the major composition was controlled by equilibrium chemistry.Modern view is of a kinetic ocean controlled by sources and sinks.River water is main source – composition from weathering reactions.Evaporation of river water does not make seawater.Reverse weathering was proposed – but the evidence is weak.Sediments are a major sink. Hydrothermal reactions are a major sink.Still difficult to quantify!