Lecture 11 Principles of Mass Balance. Simple Box Models The modern view about what controls the composition of sea water. Four Main Themes Global Carbon Cycle Are humans changing the chemistry of the ocean? 3.What are chemical controls on biological production?
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Principles of Mass Balance
Simple Box Models
The modern view about what controls
the composition of sea water.
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
No red export!
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
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 at steady state
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
= 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 inflow equals outflow
Q = S
d[M] / dt = 0 or steady state
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.
d[M] / dt = Q – k [M]
at steady state when d[M] / dt = 0 Q = k[M]
[M] / Q = 1/k = and [M] = Q / k
Elements with small KY have
short residence times.
When t < tsw not evenly mixed!
A parameterization of particle reactivity
When the ratio is small elements mostly on particles
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 M
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
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
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!
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)
“A Kinetic Model of Seawater”
Mass balance for surface box
VsdCs/dCt= VrCr + VmCd – VmixCs – B
At steady state:
B = VrCr + VmixCd– VmixCsand fB= VrivCriv
How large is the transport term:
If the residence time of the deep ocean is 1000 yrs (from 14C)
and t = Vold / Vmix
Vmix= (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 and returns as river flow.
fraction 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
fraction of input
to surface box
removed as B
Fraction of B flux
preserved in sediments
because fB = VrCr
fraction of element removed to
sediment per visit to the surface
Here are some values:
gff x g
Q. Explain these values and
why they vary the way they do.
See Broecker (1971) Table 3
Why is this important for
What controls ocean C, N, P?
assume g ≈ 1.0
Mass Balance for whole ocean:
C/ t = VRCR – f B
CS = 0; CD = CD
VU = VD = VMIX
Negative Feedback Control:
B ↑ (assumes g is constant!)
f B ↑ (assumes f will be constant!)
then CD↓ (because total ocean balance
VUCD↓ has changed; sink > source)
The nutrient concentration of
the deep ocean will adjust so that
the fraction of B preserved in the
sediments equals river input!
if VMIX = m y-1 and C = mol m-3
flux = mol m-2 y-1
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
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 y-1
Q. Is the water content of the Atmosphere at steady state?
Residence time of water in the ocean with respect to rivers
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!