Principles of Mass Balance

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Lecture 5 Principles of Mass Balance
Simple Box Models The modern view about what
controls the composition of sea water.
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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 of Mass with Inputs -
Outputs Time
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At steady state the dissolved concentration (Mi)
does not change with time (dM/dt)ocn SdMi /
dt 0
Sum of sources must equal sum of sinks
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Box Models
How would you verify that this 1-Box Ocean is at
steady state?
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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
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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)
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dM / 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
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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 dM / dt Q k
M at steady state M / Q 1/k ? and
M Q / k
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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
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Dynamic Box Models
t
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.
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Broecker two-box model (Broecker, 1971)
v is in m y-1 Flux VmixCsurf m yr-1 mol m-3
mol m-2y-1
see Fig. 2 of Broecker (1971) Quaternary
Research A Kinetic Model of Seawater
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Vs dCs/dCt VrCr VmCd VmixCs B B VrCr
VmixCd - VmixCs
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How large is the transport term If the residence
time of the deep ocean is 1000 yrs (from 14C) and
t Vold / V then 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
it evaporates.
fraction of total depth that is deep ocean
volume
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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
fraction of input removed as B
because fB VrCr
fraction of element removed to sediment per
visit to the surface
Here are some values g f f x g N 0.95 0.01 0.01
P 0.95 0.01 0.01 C 0.20 0.02 0.004 Si 1.0 0.01 0.0
1 Ba 0.75 0.12 0.09 Ca 0.01 0.12 0.001
Q. Explain these values and why they vary the way
they do.
See Broecker (1971) Table 3
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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 balance VUCD ? has
changed sink gt source) B ?
The nutrient concentration of the deep ocean
will adjust so that the fraction of B preserved
in the sediments equals river input!
CS
CD
if VMIX m y-1 and C mol m-3 flux mol m-2 y-1
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Reactivity and Residence Time
Cl
?sw
Al,Fe
Elements with small KY have short residence
times.
A parameterization of particle reactivity When
the ratio is small elements mostly on particles
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Multi-Box Models
Vt total ocean volume (m3) Vs surface ocean
volume Vu,Vd water exchange (m3 y-1) R river
inflow (m3 y-1) C concentration (mol m-3) P
particulate flux from surface box to deep box
(mol y-1) B burial flux from deep box
(mol y-1)
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1. Conservation of water R evap precip
Vu Vd V 2. Surface Box mass balance (units
of mol t-1) Vols dCs/dt RCR V Cd V
(Cs) - P Vols dCs/dt RCR V (Cs
Cd) - P 3. Deep Box mass balance Vold
dCd / dt V Cs VCd P - B Vold
dCd / dt V (Cs Cd) P - B 4. At
steady state dCt / dt 0 and R CR B
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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

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Example Global Carbon Cycle
tC,biota 3/50 0.06 y tC,export 3/11 0.29
y texport/tbiota 0.27/0.06 4.5 times recycled
tCO2,atm 590/130 4.5 y
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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!