1. Principles of Mass Balance
Lecture 11
Principles of Mass Balance
Box Models:
The modern view about what controls
the composition of sea water

2. We have already seen examples of a few different types of box models, and have seen how they can be useful – simple models are tools to better understand complex natural systems.
Can determine fluxes, reservoir sizes, residence times, whether or not a system is in steady state, etc.
Can draw simple conclusions, identify gaps in knowledge, etc.

3. Change in Mass with Time
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
Two types of models:
Box Models
CTRM: box models that take into account transport and mixing processes and production/consumption of chemical species.
Either way, we can model the change in the amount of mass in the reservoir as a balance between the mass inputs and mass outputs.

4. Change in Mass with Time = 0
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 Outputs
At steady state the mass of the reservoir does not change with time.
The sum of the sources must equal the sum of the sinks.
Sum of Inputs
Sum of Outputs

5. How could we verify that this 1-Box Ocean is in steady state?
We have drawn a one box model of the ocean with the main fluxes indicated:
River input, glacial input
Atmospheric exchange, Hydrothermal exchange, sediment exchange, radioactive decay (internal source/sink) (notice these are shown as both inputs and outputs – these processes are sinks and sources, depending on what element you are looking at)
How could we determine whether this system is in steady state?
We could measure M over time/composition of species over time
We could try to measure all of the sources and sinks over time, to see if they are balanced.
Both of these things are hard to do. And there is a lot of uncertainty in both types of measurements.

6. 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 = Fsediment Fhydrothermal
all elements all elements source: Li, Rb, K, Ca, Fe, Mn
sink: Mg, SO4, alkalinity
For most elements, the main sources are atmospheric deposition and river input.
The main sink is deposition in ocean sediments. For some elements, hydrothermal vents are a source (+) and for some HT vents are a sink (-).
If we assume steady state (dM/dt = 0) and that atmospheric deposition is negligible, then we get an even simpler balanced equation.

7. Residence Time   = mass / input or output flux = M / Q = M / S
We have seen this figure before.
This is our very simple box model with 1 input (Q), 1 output (S), and the reservoir of our constituent M.
We know we can calculate the residence time with respect to either flux. This represents how long it would take to fill the reservoir (wrt Q) or drain the reservoir (wrt S) if the other flux were zero.
Q = input rate (e.g. moles y-1)
S = output rate (e.g. moles y-1)
[M] = total dissolved mass in the box (moles)

8. Source = Q = e.g. river input flux
d[M] / dt = Q – S
Source = Q = e.g. river input flux
= Zeroth Order flux (flux is not proportional to how much M is present in the ocean)
Sink = S = 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)
We could also look at the time rate of change of M: this is the difference between the source and the sink.
In most cases, the input is independent of how much M is in the box (zeroth order flux)
The removal flux, however, is often proportional to how much M is there. These are first order removal mechanisms.

9. 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
These fluxes can be described as k[M], or some first order removal rate constant times how much M is present.
So, we can rewrite our time rate of change equation for M in our reservoir.
At steady state Q = k[M] (input flux equals the removal rate constant times the total mass).
If we rearrange this equation, we see that this equation predicts an inverse relationship between k and tau. K can also be thought of as elemental reactivity (if reactivity is high, the removal rate constant is high). SO, if elemental reactivity is high, by this equation, residence time will be short. As k increases, tau decreases. Elements will have a short residence time if they are readily removed from the ocean via particle reactivity, biological uptake, etc.

10. Reactivity vs. Residence Time sw Elements with small k haveCl
Al, Fe
In an analysis by Whitfield and Turner (1979), we see a robust correlation between mean ocean residence time of an element and the partitioning of that element between seawater and crustal rock
*Note that log plots tend to hide a lot of scatter*
Y-axis = log residence time
X-axis = log of seawater/crust partitioning
(i.e. elements with short residence time are readily incorporated into particles that are removed from ocean water)
This is essentially a parameterization of particle reactivity (when the ratio is small, elements are mostly adsorbed onto particles.
Notice that Cl- is mostly in seawater and has a long residence time. Al and Fe are at the lower left – they are mostly in the crust and have a shorter residence time.
*When  < sw the element is not evenly mixed! Tau cannot be strictly interpreted as a residence time if tau of the element is less than the tau of seawater. This indicates that the element is not homogenously distributed in the ocean.
Elements with small k have
short residence times.
When  < sw the element is not evenly mixed!

11. 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 ”

12. Dynamic Box Models
M(t) = M1 – [(M1 – Mo)e-kt]
t =
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 = t) we find that:
Mt / M1 = (1 – e-1) = This is 37% reduced, t = 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.
E-folding time here is defined as the time interval over which is takes something to decrease by a factor of e. (the timescale for M to reach a new value that is 1/e its original value).
Walk through solution of Mo=0.
Remember e = 2.7
If the sink is proportional to the amount of your species in the reservoir ([M]), then the e-folding response time of that species will be equal to the residence time of that species.
If tau is longer, e-folding time will be longer.
It tau is shorter, e-folding time will be shorter.

13. Introducing the 2-Box Model
Introducing the two box model.
Same principles from a one box model can be applied. A two box model is often applied to the ocean in order to derive insight into how the ocean works.
Assume the ocean consists of two reservoirs – the surface ocean (~100 m deep) and a deep box (~3700m deep). Assume each box is well mixed and separated by a thermocline (a rapid temperature transition/steep T gradient).
Walk through mass balance and steady state equations.
Mass balance for surface box
Vs dCs/dCt = VrCr + VmCd – VmixCs – B
At steady state:
B = VrCr + VmixCd – VmixCs and fB= VrivCriv

14. Broecker (1971) defines some parameters for the 2-box model
Two important parameters are g and f:
g = 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
f = the fraction of particles that are buried
(the efficiency of ultimate removal from the water column)
= VrCr / B = VrCr / (VmixCd + VrCr - VmixCs)
g = efficiency of bioremoval from the surface as particles = bioremoval/inputs = B/(VrCr + VmixCdeep)
f = fraction of particles that are buried = the efficiency of ultimate removal
fB = VrCr
Walk through equation derivations.