Lecture 11Principles of Mass Balance

Simple Box Models

The modern view about what controls the composition of sea water.

Four Main Themes

1.Global Carbon Cycle2.Are humans changing the chemistry of the ocean?3.What are chemical controls on biological production?4. What is the fate of organic matter made by biological production?

tCO2,atm = 590/130 = 4.5 ytC,biota = 3/50 = 0.06 y

tC,export = 3/11 = 0.29 y

texport/tbiota = 0.27/0.06 = 4.5 times recycled

Example: Global Carbon Cycle

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 ofMass with = Inputs - OutputsTime

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

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 = Fsediment + Fhydrothermal

all elements all elements 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 inflow equals outflow

Q = S

then

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.

Then:

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

inverse relationship

sw

Reactivity andResidence Time Cl

Al,Fe

A parameterization of particle reactivityWhen the ratio is small elements mostly on particles

Elements with small KY have short residence times.

When t < tsw not evenly mixed!

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 “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.

t =

e = Σ 1/n!

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 atmospheret = 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 riverst = 1.37 x 109 km3 / 37 x 103 km3 y-1 = 37,000 yrs