Category Archives: Paper Discussion

Summaries, interpretation and general discussion of journal articles.

Dilution of Convection

Publishing a paper as the sole author seems to be a rite of passage for any respectable scientist, and I’ve finally crossed the threshold!

Hannah, W. M., 2017: Entrainment vs. Dilution in Tropical Deep Convection. J. Atmos. Sci.743725–3747.

I started this work back in graduate school, but I had to put it on the back burner during my first postdoc. When I finally got back to working on this I realized that there were some big problems with my approach. I went through several rounds of review to work out all those kinks, but luckily the reviewers provided lots of good feedback. Part of this process involved rerunning the simulations and re-building the analysis code more times than I can remember. But I eventually ended up with a finished product I feel good about.

There were several things that got me thinking about this subject, but the main motivation came from reading papers about clouds and seeing the word “dilution” thrown around without any attempt to define or quantify it. We all have an intuitive sense of what “dilution” means, but can we measure it? Specifically, can we estimate the dilution rate of a cloud due to entrainment?

What is Entrainment?

in-ˈtrān-mənt \
verb – to draw in and transport by the organized current of a fluid

The word entrainment can also mean “to board a train”, which is a decent analogy for how it’s used in atmospheric science (the train in this case would be a cloudy updraft). Notice there is nothing in the definition of entrainment about dilution, but the words are often used interchangeably because the air that is being entrained is generally drier than the air inside the cloud, so we intuitively expect dilution. However, this is not always the case since the air being entrained can sometimes be relatively humid air that was previously detrained by a cloud.

What is Dilution?

\ diˈlo͞oSH(ə)n \
noun – the action of making something weaker in force, content, or value.

In the most general sense for our context of fluid mechanics, dilution is the process of reducing the concentration of a substance. An undilute substance is in it’s purist state (i.e. highest concentration). Conversely, a completely dilute substance is mixed with other substances such that it is unrecognizable from the undilute state (i.e. low concentration).

There are a few things I could find online about calculating a dilution rate, like this Wikipedia article, but these are too simplified to apply to a cloud. The term is also used in finance, but that isn’t very helpful here.

A measure of dilution rate needs to produce a rate of zero if identical substances are mixed together, and should also allow for negative dilution (i.e. concentration) if a more concentrated version of the substance is added. So to describe dilution we need to know about the “dilutee” (substance being diluted) as well as the “diluter” (process doing the dilution). Additionally, a measure of dilution rate needs to account for the mass of each substance, since increasing the amount of the “diluter” should also increase the rate of dilution.

For now, consider a simpler problem that’s slightly more complicated than diluting an ambiguous chemical with some inert water. Imagine you want to describe the dilution of a cup of coffee as you add milk to it. First we need to know about the properties of the coffee.

coffee diluted by milk

  • How is the concentration of coffee defined?
    • by the concentration of caffeine?
    • by the concentration of acids
    • by the concentration of other chemicals that define the “flavor” of the coffee?

Each of these properties may be diluted at different rates, and they all might be important. The dilution rate of each property may also change if we change the properties of the milk being added. For instance, whole milk with a higher fat content will neutralize more acid than skim milk, but have the same effect on the level of caffeine. Also, the amount of coffee we start with and the rate at which we add a non-coffee substance will affect the dilution rate as well.

Dilution of a Cloud

So how do we apply all this to estimate the dilution of a cloud by entrainment?

First we need to decide on a cloud property Φ to use for measuring dilution. Then we need to write down a basic equation that describes the behavior of this variable and manipulate it to get an equation that describes the change in concentration over the volume of the cloud. I’ll skip the algebra, but essentially we end up with a ratio between a number describing the effect of entrainment and the anomalous value of Φ. Both of these quantities are summed over the volume of the cloud in order to consider the effects on the whole cloud and how that volume might be changing. The resulting dilution rate calculation looks something like this:

The units of the dilution rate are seconds-1, so the dilution rate is like an exponential decay rate or an e-folding timescale. If we invert the dilution rate we get the time it takes to reduce the anomalous value of Φ by a large fraction. It’s important to use the anomalous value of Φ, so that the dilution rate is relative to an anomaly value of zero, which in this case is what we will consider the state of total dilution. Otherwise the dilution rate would describe the timescale to reach absolute zero, which for many quantities does not make physical sense, such as temperature.

An interesting aspect of the dilution rate is that it will go to zero if the air being entrained is the same as the air already in the cloud. This tends to make the dilution higher at upper levels in the atmosphere where the air tends to be drier, even though the entrainment tends to be weaker.

We can adapt the dilution rate to describe the dilution by any process. For example, we could estimate the dilution of cloud water by rainfall if we knew precisely how rain was falling out of a cloud. Unfortunately, these are all quantities that we can’t measure in the real world, so this technique is only really useful in high resolution cloud simulations. Luckily this tool can still help us learn more about entrainment and convection.

Dilution Results

I’ll just summarize a couple of the important findings from the paper, using total water (qt) for the dilution variable Φ. The results are from several simulations of individual clouds created by releasing a warm bubble in the SAM cloud model (Khairoutdinov and Randall, 2003).

The plot below shows the average profiles of dilution by different processes for the cloud core (i.e. strong updraft region), which I also refer to as “proportional tendencies“. The left panels shows the net dilution as the clouds are rising. The two middle panels show dilution by entrainment and detrainment, and the right panel shows the dilution by everything else, like precipitation fallout and turbulence.

There are a few interesting things about this figure. First, entrainment has a negative tendency, which means it dilutes the cloud as we expect. However, I was surprised to find that detrainment actually plays a large role in concentrating the cloud water, which counteracts the dilution by entrainment. I think the explanation for this is that drier air is preferentially detrained, which leaves the cloud more humid on average. Since entrainment and detrainment tend to balance each other out, then the main factor diluting the cloud turns out to be all the other stuff like rain falling to the ground! I was pretty surprised to find this.

Another interesting thing we can do with the dilution rate is to calculate a hypothetical dilution rate that would occur if the air being entrained were different. Normally, air is entrained near the edge of a cloud, which may or may not have been influenced by the cloud already. The air may have also been influenced by a previous cloud. However, in a typical cumulus parameterization it is assumed that the air being entrained is from the average environment, which is dry because it is mostly devoid of clouds.

In the figure below I’ve plotted the dilution rate in solid lines for the cloud core (red) and the entire cloud volume (blue). The dashed lines show what those dilution rates would be if the entrained air came from the average air at each level. These hypothetical dilution rates are much higher than the actual dilution rates, and this difference is especially dramatic for the cloud core. This shows that the cloud is entraining air that tends to be much more humid than the environment. This also suggests that the moist air being entrained into the core has previously been in the cloud core, so there appears to be a large amount of recycling of cloud air, consistent with Yeo and Romps (2013).

These results suggest that a moist convecting thermal might be more accurately described with a model that includes a shell of wet, non-buoyant air made up of air detrained by the thermal core, as shown in the cartoon below.

de Roode et al. (2012): Parameterization of the Vertical Velocity Equation for Shallow Cumulus Clouds

I’ve been stewing on an idea to build a convective parameterization for a couple years, but there are a lot of issues to work out so I’ve been trying to study up on how other parameterizations work. The biggest issue with a convective parameterization always seems to revolve around entrainment. In reading about recent parameterization developments, like Chikira and Sugiyama (2010), many people seem to like the idea of relating entrainment to the vertical velocity of the cloud updraft. This is not a new idea, but it has taken awhile to be utilized in weather and climate models. The paper discussed below provides a great discussion of this relationship, and I was surprised to learn that it is based on an inaccurate assumption about how important entrainment is for cloud momentum.

de Roode, S.R., A.P. Siebesma, H.J. Jonker, and Y. de Voogd, 2012: Parameterization of the Vertical Velocity Equation for Shallow Cumulus Clouds. Mon. Wea. Rev., 140, 2424–2436.

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Raymond and Jiang (1990): Long-Lived Mesoscale Convective Systems

I still struggle with using potential vorticity (PV) to think about atmospheric phenomena, but many extremely smart people find it very useful, so it’s a good thing to study. PV is perhaps most often used to discuss circulations on very large scales, but there has been a lot of work to make “PV thinking” work for mesoscale circulations as well. This paper describes such a theory to utilize PV to understand the maintenance of long-lived mesoscale systems in which convection plays a substantial role.

Raymond, D. J., and H. Jiang, 1990: A Theory for Long-Lived Mesoscale Convective Systems. J. Atmos. Sci., 47, 3067–3077.
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Convective Parameterization Reading list

At a recent conference I was talking to a colleague about research ideas, and we kept coming back to the conclusion that we don’t know as much as we should about how clouds are treated in the various models used around the world. There are many different schemes in use today and most of them are closely related to one another, but it’s hard to keep track of the subtle differences between the approaches. Sometimes it’s hard to find any information about the particular configuration of the convective schemes in certain models. I decided to compile a list of the different types of parameterization schemes, and gather some notes on the key papers that developed these schemes. Continue reading

Cook (2015): Role of Inertial Instability in the West African Monsoon Jump

I find the concept of inertial instability somewhat difficult to grasp. However, it seems to explain some interesting and important atmospheric phenomena, so I’m committed to wrapping my head around it. The current paper is an example of how inertial instability can be used to explain the seasonal behavior of the African monsoon system.

Cook, K. H., 2015: Role of inertial instability in the West African monsoon jump. J. Geophys. Res. Atmos., 120, 3085–3102.  Continue reading

Benestad (2016): A Mental Picture of the Greenhouse Effect

A recent post on touched on the question of what is the best simple mental model of the greenhouse effect that should be used to educate the public. Specifically, the author mentions that it is important to consider how convection affects the optical depth, in addition to the obvious role of radiative transfer. This led me to read the more thorough paper by Rasmus Benestad.

Benestad, R. E., 2016: A mental picture of the greenhouse effect. Theor. Appl. Climatol., doi:10.1007/s00704-016-1732-y.

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O’Gorman (2011): The Effective Static Stability Experienced by Eddies in a Moist Atmosphere

A lot of interesting weather phenomena are associated with various types of atmospheric waves. In most cases, we have a good understanding of wave dynamics when the atmosphere is dry and there’s no condensation or evaporation, but we lack a solid understanding of how things change when moist convection gets involved (i.e. clouds). A good example is how convectively coupled Kelvin waves move at a fraction of the speed compared to their dry counter parts (Kiladis 2009). I recently read the paper below by Paul O’Gorman that discusses a simple way to modify dry theory such that the effects of heating by convection can be implicitly included.

Paul A. O’Gorman, 2011: The Effective Static Stability Experienced by Eddies in a Moist Atmosphere. J. Atmos. Sci., 68, 75–90.

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Biasutti and Sobel (2009): Delayed Rainfall over the African Sahel in a Warmer Climate

The Sahel is a particularly sensitive region with respect to the effects of global warming. The Sahel has been plagued by decadal-ish periods of drought, such as in the 1980’s.  It’s generally understood that the variations in Sahel rainfall are mostly attributable to ocean variability (Giannini et al. 2003), but there are still many questions about the importance of other natural and anthropogenic factors. These questions are hard to answer with current climate models, because ocean temperature biases can lead to large differences in the distribution of rainfall. In spite of the lack of agreement between models, this paper was able to identify a robust result across a group of models from the CMIP3 data archive.

Biasutti, M., and A. H. Sobel, 2009: Delayed Sahel rainfall and global seasonal cycle in a warmer climate. Geophys. Res. Lett., 36, L23707.

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Thorncroft and Blackburn (1999): Maintenance of the African Easterly Jet

In a recent discussion with some colleagues someone asked the simple question, “Do you know why the African easterly jet occurs at 600mb?” and I realized didn’t actually know the whole story. The usual explanation is related thermal wind balance and the meridional temperature gradient at the surface. However, this explanation misses some subtle aspects of the vertical structure of wind over Africa. This paper appears to be the first to provide a thorough explanation of the processes which maintain the African easterly jet (AEJ).

Thorncroft, C. D., and M. Blackburn, 1999: Maintenance of the African Easterly Jet. Quart. J. Roy. Met. Soc., 125, 763–786.

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Skinner and Diffenbaugh (2014): Projected Changes in African Easterly Wave Intensity and Track in Response to Greenhouse Forcing

African easterly waves (AEW) have a large influence on weather in the African Sahel region, and are known to provide seed disturbances for many Atlantic tropical cyclones (TC). Because of all the factors that can affect the development of a TC (SST, shear, etc.), it is difficult to know how any given change in AEW activity will affect the climatology of Atlantic TCs. This paper is one the few that has tried to characterize future AEW activity from the model projections (CMIP5).

Skinner, C. B., and N. S. Diffenbaugh, 2014: Projected Changes in African Easterly Wave Intensity and Track in Response to Greenhouse Forcing. Proc. Natl. Acad. Sci., 102(50),17891–17896.

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