Upper Ocean 1-D Seasonal Models
File last modified 16 November 1998
19.3a Adding Gases to the Model: Gas Exchange
What we have accomplished so far is a simple model which mimics the gross behavior of the water column on a seasonal basis. The result is that four important features are satisfactorily simulated:
We make no claims beyond this, but this should be sufficient to provide a happy home for our gases. The next step is then to set up the initial and boundary conditions for gas concentrations.
The initial conditions, we can deal with right away by stipulating that the water column be in solubility equilibrium with the atmosphere. We would then let the model run sufficient numbers of annual cycles to reach some kind of cyclical steady state. Our guess is that with gas exchange time scales of order of a month or so, a few years would be adequate for the upper ocean to reach some semblance of a steady cycle. However, there may be a long term drift driven by the much slower change of the deeper part of the water column (the part not directly exposed to the winter convection, say below 200m). How long might the lower part take? Well if it were left to vertical diffusion alone, then the characteristic time scale would be
You might think that we also have the help of vertical advection, which in our scheme is of the order of 10s of m/y near the surface. But by the 200m mark, however, it is down to about 10 m/y or less, so the time constant for advection is
No help there. So what we have then, is a deep water relaxation time scale uncomfortably close to the time length of our model runs. You'll see that in the time evolution of some of our early runs, just to let you know this is the case. It turns out not to greatly affect our conclusions here, but obviously it bears thinking about.
This, in fact, is a classical problem in numerical models, where the interesting region is "dragged along" by a much slower responding part of the system. Your first instinct might be to just let the model run for a much longer time, but then we are faced with the potential of a long term (unrealistic) drift in the physical part of the model (the temperature distribution). The way around that is to run the model for a few years, and then restart it with the same gas concentration anomalies, but the original/initial physical configuration. We won't get into this technique in detail here, but these problems of "spinup" are common in modeling.
Suffice it to say that our simple starting condition could be refined by redoing the model calculations starting with an initial (gas) state that more closely approximates, say the third March of a previous model run. Do this enough times, and the net result will not greatly depend on the original problem (ahem!).
Gas exchange between the ocean and the atmosphere is a complex and contentious topic. It is still a very active area of research, and we will gloss over the details and present you with a very simple model of gas exchange suitable for our calculations. If you really want to do state-of -the-art modeling in this area, be warned that the field is "rife with challenges" and not without some "unconstrained and wild models". Be that as it may, let's get on with things.
Our strategy will be an extremely pragmatic approach of choosing a "canonical" gas exchange formalism, and treat it as a "given" subject to a multiplicative scale factor. That is, we will use a commonly accepted gas exchange formulation, then reference our model runs to that gas exchange rate. We will then evaluate the sensitivity of our model results and how they compare with field data to changes in the gas exchange rate (for example doubling or halving the "canonical rate") and try to determine if our observations can be used to constrain the gas exchange rate. Our point will be to demonstrate that our model results and field observations can be used as a constraint to test/evaluate the "accepted" numbers. The canonical numbers, after all, are really estimates based on field observations coupled with even simpler models, so our approach actually represents a legitimate determination of gas exchange rates on seasonal time scales… one which is at least as good as theirs, and perhaps better in some ways.
The basic gas exchange model we will use is that of Liss and Merlivat (1986) , which consists of a trilinear fit of gas exchange rate as a function of wind speed. The model further stipulates that (for our purposes) the gas exchange rate, which is expressed as a velocity, further depends on the molecular diffusivity of the gas. This is customarily expressed in terms of the Schmidt number, defined by
that is, the ratio of kinematic viscosity of the water to the molecular diffusivity of the gas. In a stagnant film model, you might picture this as the thickness of the stagnant film being proportional to the viscosity of the water, and the impedance to gas transfer being inversely related to the molecular diffusivity. So you would expect the gas exchange rate to depend inversely on Sc. Experiments and theory point to a somewhat weaker dependence, where in fact for reasonable wind speeds you have
where "k" is usually used to refer to the gas exchange velocity. Thus for He, which has a molecular diffusivity more than three times that of oxygen, would have as much as 1.7 times greater gas exchange rate. As is the custom (seems to be a rather "custom filled" field, doesn't it?), the gas exchange rate is presented as a function of wind speed for a "reference gas" with Sc=600.
You'll note that there are two Y-scales on the plot. The left hand one indicates yet another quaint tradition in this business: transfer velocities in cm/h. We (and you should ) prefer the mks or some other rational scale, like the one shown on the right. But in keeping with the grand history of geochemistry, you may want to convert it some more obscure unit system (e.g. furlongs/fortnight). Another obvious aspect of this plot is the large scatter in data points. The dashed line labeled "Lab" corresponds to wind tunnel results. There are reasons to believe that they may be biased high due to uncharacteristic vortical motions that develop in wind tunnels which create high gas exchange rates. The other "points" correspond to different field techniques and experiments, and the range probably reflects a real variation in actual exchange rates: gas exchange does not depend solely on wind speed alone, but also on sea surface state (white caps, bubbles, wave height/length), the presence of surfactants, and possibly the Dow-Jones Industrial Average. Let's face it, it's a hopeless task. We simply hope to capture the gross behavior of gas exchange in our model, and that's about all. The other functionality will be buried in the annual cycle.
Below is a table of Sc numbers for different gases at different temperatures.
| Temp/Gas | Helium | Neon | Argon | Oxygen | Krypton | Xenon |
| 18 | 173 | 324 | 547 | 538 | 741 | 932 |
| 22 | 148 | 271 | 450 | 442 | 603 | 753 |
| 26 | 125 | 226 | 365 | 358 | 488 | 607 |
| 30 | 105 | 186 | 292 | 287 | 392 | 486 |
Which serves also to underline part of our strategy here. Note the similarity in Sc numbers between oxygen and argon. Since things tend to depend on the square root of Sc, then the difference in behavior between the two gases is less than 1%. Thus Ar looks like a good proxy for oxygen, i.e. an abiogenic analog of oxygen. Further, note the large range in Sc between He, Ar and Xe (more than a factor of four). This means that observation and successful modeling of Ar will do a good job of tracking abiotic effects in oxygen, and emulating the remainder of the gases will effectively constrain the small differences between Ar and O2.
So all we'll use for gas exchange in our model is for each time step to construct a flux of gas from (to) the ocean surface box driven by the gas exchange velocity multiplied times difference between the actual gas concentration and the equilibrium gas concentration. Thus we have
where k(Sc,WS) is the gas exchange rate in m/s as a function of Schmidt number (gas and temperature) and wind speed, C0 is the equilibrium concentration as a function of temperature and salinity, and A is the surface area of the surface box (=1m2). If we spread the resultant concentration change due to this flux over the time step (delta T) and over the mixed layer depth, we have
where H is the mixed layer depth in m. (The area A cancels out because the volume of the water column is the height times the area). Thus our budgeting amounts to accounting for a gas flux over each time step due to disequilibrium with the atmosphere.
Now we'll do one final, seemingly kinky thing. The original model was calculated in terms of the wind stress climatology. You'll recall that this approach is more accurate for our physical representation than using climatological wind speeds. To make our gas behavior consistent with the physical climatology, we'll back calculate the wind speed from the instantaneous wind stress. This has the attraction of taking into account what is normal for the oceans: wind bursts. The globally non-linear dependence of the gas exchange velocity on wind suggests that we need to take this into account.
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