Transcript Slide 1

Lecture 5
Principles 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
Mass with =
Time
Sum of
Inputs
-
Sum of
Outputs
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
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
=
all elements
Fsediment
all elements
+
Fhydrothermal
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)
see Fig. 2 of Broecker (1971)
Quaternary Research
“A Kinetic Model of Seawater”
v is in m y-1
Flux = VmixCsurf = m yr-1 mol m-3 = mol m-2y-1
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 / V
fraction of total depth
volume
that is deep ocean
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.
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)
fraction of input
removed as B
because fB = VrCr
fraction of element removed to
sediment per visit to the surface
fxg
In his model Vr = 10 cm y-1
Vmix = 200 cm y-1
so
Vmix / Vr = 20
Here are some values:
g
f
N
0.95
0.01
P
0.95
0.01
C
0.20
0.02
Si
1.0
0.01
Ba
0.75
0.12
Ca
0.01
0.12
See Broecker (1971) Table 3
fxg
0.01
0.01
0.004
0.01
0.09
0.001
Q. Explain these values and
why they vary the way they do.
Why is this important for
chemical oceanography?
What controls ocean C, N, P?
The nutrient concentration of
the deep ocean will adjust so that
the fraction of B preserved in the
sediments equals river input!
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 > source)
B↓
CS
CD
if VMIX = m y-1 and C = mol m-3
flux = mol m-2 y-1
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
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)
1. Conservation of water
R = evap – precip
Vu = Vd = V
2. 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 - B
4. 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
Example: Global Carbon Cycle
CO2,atm = 590/130 = 4.5 y
C,biota = 3/50 = 0.06 y
C,export = 3/11 = 0.29 y
export/tbiota = 0.27/0.06 = 4.5 times recycled
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!