Lecture 11
Principles of Mass Balance
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 = ΣdMi / dt = 0
(i.e. the sum of the sources must equal the sum of the
sinks at steady state)
Change in
Mass with
Time = 0
Sum of
Inputs
Sum of
Inputs
Sum of
Outputs
Sum of
Outputs
How could we verify that this 1-Box Ocean is in steady
state?
For most elements in the ocean:
(dM/dt)ocn = Fatm + Frivers - Fseds + Fhydrothermal
If we assume steady state, and assume atmospheric flux is
negligible (safe for most elements)… 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 output 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
Source =
Q =
=
Sink =
S
e.g. river input flux
Zeroth Order flux (flux is not proportional to how
much M is present in the ocean)
= many removal mechanisms are First Order (the
flux is proportional to how much M is there)
(e.g. radioactive decay, plankton uptake,
adsorption by particles)
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, we can rewrite d[M]/dt = Q – S, to include the first order
sink:
d[M] / dt = Q – k [M]
At steady state when d[M]/dt = 0, Q = k[M]
Rearrange:
[M]/Q = 1/k =  * and [M] = Q / k
*inverse relationship
between first order
removal constant and
residence time
Reactivity vs.
Residence Time
Cl
sw
Al, Fe
Elements with small k have
short residence times.
When  < sw the element is not evenly mixed!
Dynamic Box Models
In some instances, the source (Q) and sink (S) rates are not constant with time
OR they may have been constant, but suddenly change.
Examples: Glacial/Interglacial cycles, 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 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)e-kt]
“M increases from Mo to the new value of M1 (Q1/k) with a response time of
k-1 or ”
Dynamic Box Models
M(t) = M1 – [(M1 – Mo)e-kt]
=
This response time is defined as the time it takes to reduce the imbalance (M1 – Mo). to
e-1 (or 37%) of the initial imbalance ((1/e)*( 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
This is 37% reduced,  = e-folding time!
For a single box model with a 1st order sink, response time = residence time.
Elements with a short residence time will approach their new value faster
than elements with long residence times.
Introducing the 2-Box Model
Mass balance for surface box
Vs dCs/dCt = VrCr + VmCd – VmixCs – B
At steady state:
B = VrCr + VmixCd – VmixCs and fB= VrivCriv
Broecker (1971) defines some parameters for the 2-box model
Two important parameters are g and f:
g
=
=
=
f
=
=
the fraction of an element put in at the surface that is
removed as B (the efficiency of bioremoval of an
element from the surface – how efficiently it sinks as a
particle (as B flux) out of the surface ocean)
B / surface ocean input
(VmixCD + VrCr – VmixCs) / VmixCd + VrCr
the fraction of particles that are buried
(the efficiency of ultimate removal from the water
column)
VrCr / B = VrCr / (VmixCd + VrCr - VmixCs)