In Ensemble Forecasting I wrote a short section on parameterization using the example of latent heat transfer and said:

Are we sure that over Connecticut the parameter C_{DE} = 0.004, or should it be 0.0035? In fact, parameters like this are usually calculated from the average of a number of experiments. They conceal as much as they reveal. The correct value probably depends on other parameters. In so far as it represents a real physical property it will vary depending on the time of day, seasons and other factors. It might even be, “on average”, wrong. Because “on average” over the set of experiments was an imperfect sample. And “on average” over all climate conditions is a different sample.

Interestingly, a new paper has just shown up in JGR (“accepted for publication” and on their website in the pre-publishing format): *Seasonal changes in physical processes controlling evaporation over an inland water*, Qianyu Zhang & Heping Liu.

They carried out detailed measurements over a large reservoir (134 km² and 4-8m deep) in Mississippi for the winter and summer months of 2008. What were they trying to do?

Understanding physical processes that control turbulent fluxes of energy, heat, water vapor, and trace gases over inland water surfaces is critical in quantifying their influences on local, regional, and global climate. Since direct measurements of turbulent fluxes of sensible heat (H) and latent heat (LE) over inland waters with eddy covariance systems are still rare, process-based understanding of water-atmosphere interactions remains very limited..

..Many numerical weather prediction and climate models use the bulk transfer relations to estimate H and LE over water surfaces. Given substantial biases in modeling results against observations, process-based analysis and model validations are essential in improving parameterizations of water-atmosphere exchange processes..

Before we get into their paper, here is a relevant quote on parameterization from a different discipline. This is from *Turbulent dispersion in the ocean*, Garrett (2006):

Including the effects of processes that are unresolved in models is one of the central problems in oceanography.

In particular, for temperature, salinity, or some other scalar, one seeks to parameterize the eddy flux in terms of quantities that are resolved by the models. This has been much discussed, with determinations of the correct parameterization relying on a combination of deductions from the large-scale models, observations of the eddy fluxes or associated quantities, and the development of an understanding of the processes responsible for the fluxes.

The key remark to make is that it is only through process studies that we can reach an understanding leading to formulae that are valid in changing conditions, **rather than just having numerical values which may only be valid in present conditions**.

[Emphasis added]

### Background

Latent heat transfer is the primary mechanism globally for transferring the solar radiation that is absorbed at the surface up into the atmosphere. Sensible heat is a lot smaller by comparison with latent heat. Both are “convection” in a broad term – the movement of heat by the bulk movement of air. But one is carrying the “extra heat” of evaporated water. When the evaporated water condenses (usually higher up in the atmosphere) it releases this stored heat.

Let’s take a look at the standard **parameterization** in use (adopting their notation) for latent heat:

LE = ρ_{a}LC_{E}U(q_{w} −q_{a})

LE = latent heat transfer, ρ_{a} = air density, L = latent heat of vaporization (2.5×10^{6} J kg^{–1}), C_{E} = bulk transfer coefficient for moisture, U = wind speed, q_{w} & q_{a} are the respective specific humidity in the water-atmosphere interface and the over-water atmosphere

The values ρ_{a} and L are a fundamental values. The formula says that the key parameters are:

- wind speed (horizontal)
- the
**difference** between the humidity at the water surface (this is the saturated value which varies strongly with temperature) and the humidity in the air above

We would expect the differential of humidity to be important – if the air above is saturated then latent heat transfer will be zero, because there is no way to move any **more** water vapor into the air above. At the other extreme, if the air above is completely dry then we have maximized the potential for moving water vapor into the atmosphere.

The product of wind speed and humidity difference indicate how much mixing is going on due to air flow. There is a lot of theory and experiment behind the ideas, going back into the 1950s or further, but in the end it is an over-simplification.

That’s what all parameterizations are – over-simplifications.

The **real formula** is much simpler:

LE = ρ_{a}L<w’q’>, where the brackets denote averages,w’q’ = the turbulent moisture flux

w is the upwards velocity, q is moisture; and the ‘ denoting eddies

*Note to commenters, if you write < or > in the comment it gets dropped because WordPress treats it like a html tag. You need to write < or >*

The key part of this equation just says “how much moisture is being carried upwards by turbulent flow”. That’s the real value so why don’t we measure that instead?

Here’s a graph of horizontal wind over a short time period from Stull (1988):

From Stull 1988

*Figure 1*

And any given location the wind varies across every timescale. Pick another location and the results are different. This is the problem of turbulence.

And to get accurate measurements for the paper we are looking at now, they had quite a setup:

Figure 2

Here’s the description of the instrumentation:

An eddy covariance system at a height of 4 m above the water surface consisted of a three-dimensional sonic anemometer (model CSAT3, Campbell Scientific, Inc.) and an open path CO2/H2O infrared gas analyzer (IRGA; Model LI-7500, LI-COR, Inc.).

A datalogger (model CR5000, Campbell Scientific, Inc.) recorded three-dimensional wind velocity components and sonic virtual temperature from the sonic anemometer and densities of carbon dioxide and water vapor from the IRGA at a frequency of 10 Hz.

Other microclimate variables were also measured, including Rn at 1.2 m (model Q-7.1, Radiation and Energy Balance Systems, Campbell Scientific, Inc.), air temperature (Ta) and relative humidity (RH) (model HMP45C, Vaisala, Inc.) at approximately 1.9, 3.0, 4.0, and 5.5 m, wind speeds (U) and wind direction (WD) (model 03001, RM Young, Inc.) at 5.5 m.

An infrared temperature sensor (model IRR-P, Apogee, Inc.) was deployed to measure water skin temperature (Tw).

Vapor pressure (ew) in the water-air interface was equivalent to saturation vapor pressure at Tw [Buck, 1981].

The same datalogger recorded signals from all the above microclimate sensors at 30-min intervals. Six deep cycling marine batteries charged by two solar panels (model SP65, 65 Watt Solar Panel, Campbell Scientific, Inc.) powered all instruments. A monthly visit to the tower was scheduled to provide maintenance and download the 10-Hz time-series data.

I don’t know the price tag but I don’t think the equipment is cheap. So this kind of setup can be used for research, but we can’t put one each every 1km across a country or an ocean and collect continuous data.

That’s why we need parameterizations if we want to get some climatological data. Of course, these need verifying, and that’s what this paper (and many others) are about.

### Results

When we look back at the parameterized equation for latent heat it’s clear that latent heat should vary linearly with the product of wind speed and humidity differential. The top graph is sensible heat which we won’t focus on, the bottom graph is latent heat. Δe is humidity, expressed as partial pressure rather than g/kg. We see that the correlation between LE and wind speed x humidity differential is very different in summer and winter:

From Zhang & Liu 2014

*Figure 2*

The scatterplots showing the same information:

From Zhang & Liu 2014

*Figure 3*

The authors looked at the diurnal cycle – averaging the result for the time of day over the period of the results, separated into winter and summer.

Our results also suggest that the influences of U on LE may not be captured simply by the product of U and Δe [humidity differential] on short timescales, especially in summer. This situation became more serious when the ASL (atmospheric surface layer, see note 1) became more unstable, as reflected by our summer cases (i.e., more unstable) versus the winter cases.

They selected one period to review in detail. First the winter results:

From Zhang & Liu 2014

*Figure 4*

On March 18, Δe was small (i.e., 0 ~ 0.2 kPa) and it experienced little diurnal variations, leading to limited water vapor supply (Fig. 5a).

The ASL (see note 1) during this period was slightly stable (Fig. 5b), which suppressed turbulent exchange of LE. As a result, LE approached zero and even became negative, though strong wind speeds of approximately around 10 ms^{–1} were present, indicating a strong mechanical turbulent mixing in the ASL.

On March 19, with an increased Δe up to approximately 1.0 kPa, LE closely followed Δe and increased from zero to more than 200 Wm^{–2}. Meanwhile, the ASL experienced a transition from stable to unstable conditions (Fig. 5b), coinciding with an increase in LE.

On March 20, however, the continuous increase of Δe did not lead to an increase in LE. Instead, LE decreased gradually from 200 Wm^{–2} to about zero, which was closely associated with the steady decrease in U from 10 ms^{–1} to nearly zero and with the decreased instability.

These results suggest that LE was strongly limited by Δe, instead of U when Δe was low; and LE was jointly regulated by variations in Δe and U once a moderate Δe level was reached and maintained, indicating a nonlinear response of LE to U and Δe induced by ASL stability. The ASL stability largely contributed to variations in LE in winter.

Then the summer results:

From Zhang & Liu 2014

*Figure 5*

In summer (i.e., July 23 – 25 in Fig. 6), Δe was large with a magnitude of 1.5 ~ 3.0 kPa, providing adequate water vapor supply for evaporation, and had strong diurnal variations (Fig. 6a).

U exhibited diurnal variations from about 0 to 8 ms^{–1}. LE was regulated by both Δe and U, as reflected by the fact that LE variations on the July 24 afternoon did not follow solely either the variations of U or the variations of Δe. When the diurnal variations of Δe and U were small in July 25, LE was also regulated by both U and Δe or largely by U when the change in U was apparent.

Note that during this period, the ASL was strongly unstable in the morning and weakly unstable in the afternoon and evening (Fig. 6b), negatively corresponding to diurnal variations in LE. This result indicates that the ASL stability had minor impacts on diurnal variations in LE during this period.

Another way to see the data is by plotting the results to see how valid the parameterized equation appears. Here we should have a straight line between LE/U and Δe as the caption explains:

From Zhang & Liu 2014

*Figure 6*

One method to determine the bulk transfer coefficients is to use the mass transfer relations (Eqs. 1, 2) by quantifying the slopes of the linear regression of LE against UΔe. Our results suggest that using this approach to determine the bulk transfer coefficient may cause large bias, given the fact that one UΔe value may correspond to largely different LE values.

They conclude:

Our results suggest that these highly nonlinear responses of LE to environmental variables may not be represented in the bulk transfer relations in an appropriate manner, which requires further studies and discussion.

### Conclusion

Parameterizations are inevitable. Understanding their limitations is very difficult. A series of studies might indicate that there is a “linear” relationship with some scatter, but that might just be disguising or ignoring a variable that never appears in the parameterization.

As Garrett commented “..having numerical values which may only be valid in present conditions”. That is, if the mean state of another climate variable shifts the parameterization will be invalid, or less accurate.

Alternatively, given the non-linear nature of climate process, changes don’t “average out”. So the mean state of another climate variable may not shift, the mean state might be constant, but its variation with time or another variable may introduce a change in the real process that results in an overall shift in climate.

There are other problems with calculating latent heat transfer – even accepting the parameterization as the best version of “the truth” – there are large observational gaps in the parameters we need to measure (wind speed and humidity above the ocean) even at the resolution of current climate models. This is one reason why there is a need for reanalysis products.

I found it interesting to see how complicated latent heat variations were over a water surface.

### References

*Seasonal changes in physical processes controlling evaporation over an inland water*, Qianyu Zhang & Heping Liu, JGR (2014)

*Turbulent dispersion in the ocean*, Chris Garrett, *Progress in Oceanography* (2006)

### Notes

Note 1: The ASL (atmospheric surface layer) stability is described by the Obukhov stability parameter:

ζ = z/L_{0}

where z is the height above ground level and L_{0} is the Obukhov parameter.

L_{0} = −θ_{v}u*^{3}/[kg(w’θ_{v}‘)s ]

where θ_{v} is virtual potential temperature (K), u* is frictional velocity by the eddy covariance system (ms^{–1}), k is Von Karman constant (0.4), g is acceleration due to gravity (9.8 ms^{–2}), w is vertical velocity (m s^{–1}), and (w’θ_{v}‘)s is the flux of virtual potential temperature by the eddy covariance system

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