Lecture 5Principles of Mass Balance

Simple Box Models

The modern view about what controls the composition of sea water.

Two main types of models used in chemical oceanography.

-Box (or reservoir) Models

-Continuous Transport-reaction Models

In both cases:

Change in Sum of Sum ofMass with = Inputs - OutputsTime

At steady state the dissolved concentration (Mi) does not change with time:

(dM/dt)ocn = dMi / dt = 0

Sum of sources must equal sum of sinks

Box Models

How would you verify that this 1-Box Ocean is at steady state?

For most elements in the ocean:

(dM/dt)ocn = Fatm + Frivers - Fseds + Fhydrothermal

The main balance is even simpler:

Frivers = Fsediment + Fhydrothermal

all elements all elements source: Li, Rb, K, Ca, Fe, Mn sink: Mg, SO4, alkalinity

Residence Time

= mass / input or removal flux = M / Q = M / S

Q = input rate (e.g. moles y-1)S = output rate (e.g. moles y-1)[M] = total dissolved mass in the box (moles)

d[M] / dt = Q – S

input = Q = Zeroth Order flux (e.g. river input) not proportional to how much is in the ocean

sink = S = many are First Order (e.g. Radioactive decay, plankton uptake, adsorption by particles)

If steady state, then inflow equals outflow

Q = S

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

[M] / Q = 1/k = and [M] = Q / k

Dynamic Box Models

If the source (Q) and sink (S) rates are not constant with time or they may have been constant and suddenly change.

Examples: Glacial/Interglacial; Anthropogenic Inputs to Ocean

Assume that the initial amount of M at t = 0 is Mo. The initial mass balance equation is:

dM/dt = Qo – So = Qo – k Mo

The input increases to a new value Q1.

The new balance at the new steady state is:dM/dt = Q1 – k M1

and 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 see Emerson and Hedges: Appendix 2.2

Dynamic Box Models

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 =) 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.

=

Broecker two-box model (Broecker, 1971)

v is in m y-1

Flux = VmixCsurf = m yr-1 mol m-3 = mol m-2y-1see Fig. 2 of Broecker (1971)Quaternary Research“A Kinetic Model of Seawater”

Vs dCs/dCt = VrCr + VmCd – VmixCs – B

B = VrCr + VmixCd - VmixCs

How large is the transport term:If the residence time of the deep ocean is 1000 yrs (from 14C)and = Vold / Vthen:

V = (3700m/3800m)(1.37 x 1018 m3) / 1000 y = 1.3 x 1015 m3 y-1

If River Inflow = 3.7 x 1013 m3 y-1

Then River Inflow / Deep Box Exchange = 3.7 x 1013/1.3 x 1015

= 1 / 38

This means water circulates on average about 40 times through the ocean (surface to deep exchange) before itevaporates.

volumefraction of total depth that is deep ocean

Broecker (1971) defines some parameters for the 2-box model

g = B / input = (VmixCD + VrCr – VmixCs) / VmixCd + VrCr

f = VrCr / B = VrCr / (VmixCd + VrCr - VmixCs)

f x g

In his model Vr = 10 cm y-1

Vmix = 200 cm y-1

so Vmix / Vr = 20

Here are some values:g f f x g

N 0.95 0.01 0.01P 0.95 0.01 0.01C 0.20 0.02 0.004Si 1.0 0.01 0.01Ba 0.75 0.12 0.09Ca 0.01 0.12 0.001

Q. Explain these values andwhy they vary the way they do.

because fB = VrCr

fraction of element removed to sediment per visit to the surface

fraction of inputremoved as B

See Broecker (1971) Table 3

Why is this important for chemical oceanography?What controls ocean C, N, P?g ≈ 1.0

Mass Balance for whole ocean:C/ t = VRCR – f B

CS = 0; CD = CD

VU = VD = VMIX

Negative Feedback Control:if VMIX ↑VUCD ↑B ↑f B ↑ (assumes f will be constant!)assume VRCR then CD ↓ (because total ocean balanceVUCD ↓ has changed; sink > source)B ↓

CS

CD

if VMIX = m y-1 and C = mol m-3

flux = mol m-2 y-1

The nutrient concentration of the deep ocean will adjust so thatthe fraction of B preserved in thesediments equals river input!

sw

Reactivity andResidence Time Cl

Al,Fe

A parameterization of particle reactivityWhen the ratio is small elements mostly on particles

Elements with small KY have short residence times.

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)

1. Conservation of water R = evap – precip Vu = Vd = V2. Surface Box mass balance (units of mol t-1) Vols dCs/dt = R[CR] + V [Cd] – V ([Cs]) - P

Vols dCs/dt = R[CR] – V ([Cs] – [Cd]) - P 3. Deep Box mass balance Vold d[Cd] / dt = V [Cs] – V[Cd] + P - B

Vold d[Cd] / dt = V ([Cs] – [Cd]) + P - B4. At steady state d[Ct] / dt = 0 and R [CR] = B

Example: Global Water Cycle

103 km3

103 km3 y-1

Q. 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 = 0.026 yr = 9.6 d

Residence time of water in the ocean with respect to rivers = 1.37 x 109 km3 / 37 x 103 km3 y-1 = 37,000 yrs

CO2,atm = 590/130 = 4.5 yC,biota = 3/50 = 0.06 y

C,export = 3/11 = 0.29 y

export/tbiota = 0.27/0.06 = 4.5 times recycled

Example: Global Carbon Cycle

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!