Last month, from the 21st July until the 3rd August 2019, I was in Ghana attending the African SWIFT and YESS International Summer School. What a catchy name you are probably thinking. SWIFT, or Science for Weather Information and Forecasting Techniques, is a programme of research and capability building, led by the National Centre for Atmospheric Science (NCAS), and funded by the UK Research and Innovation Global Challenges Research Fund. The project aims to improve African weather forecasting, especially on seasonal timescales, as well as build capability in related research. It’s worth a quick Google search at some point, and there are several people involved in the project at the University of Reading. YESS, the Young Earth System Scientists community, is an international, multidisciplinary network of early career researchers. Catchy!
Anyway, all this contributed toward a really remarkable summer school in tropical West Africa, with people from many different institutions and nations across Europe and Africa attending the summer school and science meeting alongside. It was also another chance to get consistently barraged with Brexit questions by a baffled international audience.
The days were long but engaging, with lectures, practical sessions and workshops on a huge variety of topics in tropical meteorology – from Rossby waves, to the monsoon, to remote sensing applications. It was quickly evident how tricky tropical African weather is to forecast. It is largely driven by convection, which is very difficult to forecast accurately on a small spatio-temporal scale, unlike nice, large mid-latitude weather systems. Furthermore, several different atmospheric features are at play. This is where we were introduced to the wonderful West African synoptic analysis/forecast charts (see below for an example of mine). We also had a chance to present our posters, with many of those from the science meeting – experts in their fields – coming round to look, and this was a fantastic networking opportunity. It was really beneficial being around other early career scientists in the same specific field as me, from different places around the world. It cannot be said enough how important this is for PhD students, who for the most part live quite an isolated existence where when you switch from your native English language to your ‘PhD language’, only your supervisors and a few select others can understand you!
For me, it is in the people attending where the strength of the summer school really lies. The people in Kumasi, Ghana were amazing people. They not only keep you going through 2 weeks of long days, 3 dozen lectures, and 400 meals of rice, but they reminded me what it meant to be a scientist. I found, to my discredit, that most of the students there were far more studious than I was, not because they were any less clever or anything like that, but because they simply loved knowledge, and loved applying it (meteorology is great for quickly being able to see how what we know manifests itself in the real world). On reflection, I think they are more aware of the fact millions of people (moreso in Africa than any other continent) simply do not have access to such knowledge, but in Kumasi we were learning about African meteorology from world experts. They did not take it for granted, in fact, it was clearly what drove them. Science wasn’t just an occupation for them, it had tangible importance, which came across in the way they spoke about their science, but also their future ambitions, hopes and plans.
Further to this, meeting people across universities, countries, and continents also brings a different perspective on your work and where it fits into the wider collection of research in the area. One sad point was learning how hard it was for African students to get PhDs. Not only do they typically have to travel much further (i.e. typically to Europe or the US) in order to get one, but they also rely on getting funding, which is often the final obstacle even after they have found the right PhD project. It’s a real shame.
So after 2 long weeks (and a very hot football game on a gravelly pitch with no shoes) I came back physically exhausted, but academically I was refreshed with lots of new ideas floating round, but even more importantly newfound inspiration. In the now famous words of the provost of the college during the closing ceremony, “let your research be SWIFT and YESS.”
From the 1st – 12th of July 2019, I was fortunate enough to be able to attend the Fluid Dynamics of Sustainability and the Environment (FDSE) summer school held at Ecole Polytechnique on the southern outskirts of Paris. Although it was held at Ecole Polytechnique this year, it alternates with the University of Cambridge, where it will be held in 2020.
As hinted at in the title, the summer school explores the fluid dynamical aspects of planet Earth, including, but not limited to: the atmosphere, the ocean, the cryosphere and the solid Earth, and was of particular relevance to me because I study clear-air turbulence (a fluid dynamical phenomenon) and its impact on aviation. To get a better sense of the summer school, have a watch of this 3-minute promotion video: https://www.youtube.com/watch?v=TGoF0L8gqXw
It was a busy, action-packed 2 weeks. The days consisted of: 4 hours of lectures held each morning (coffee was provided), followed by either lab or numerical practical sessions in the afternoons and something social (wine was provided) such as a poster session, barbecue, and an environmentally-themed film night followed by a discussion of the film’s (The Day After Tomorrow) fluid dynamical accuracy (or not, as the case may be!). During the mid-programme weekend, we were put up in a hostel in central Paris, treated to an evening on a moored boat on the Seine (champagne was provided) and then left to our own devices to explore Paris.
The other students were great,
with all sorts of backgrounds/PhD projects that linked in one way or another to
the FDSE theme. Many interesting and diverse conversations were had, as well as
a great deal of fun and laughter! No doubt many of the people who met here both
this year and others will collaborate scientifically in the future.
Not having come from a maths/physics background, I found a lot of the mathematical content quite challenging, but I made copious notes and my interest in and appreciation for the subject greatly increased. As I progress throughout my PhD (I am currently still in my first year), I feel many of the concepts that I encountered here are likely to resurface in a slow-burn fashion and I can see myself returning to the lecture material as and when I meet related concepts.
In particular, gaining an understanding of what an instability is and studying the different types was eye-opening, and seeing Kelvin-Helmholtz instabilities — which cause the shear that generates the clear-air turbulence I study in my PhD — form in a tube of dyed fluid was a particularly memorable moment for me.
Apart from being very
interesting theoretically, fluid dynamics also has many practical applications.
For example, insufficient understanding and modelling of the behaviour of
plumes at the Fukushima nuclear reactor led to hydrogen gas concentrations
exceeding 8%, resulting in dangerous explosions. Many other such examples could
The summer school was well-organised and many of the lecturers and guest speakers were both highly entertaining and informative, and really bought the subject to life with their enthusiasm for it. I highly recommend it to anyone with a related PhD!
Gristey, J.J., J.C. Chiu, R.J. Gurney, K.P. Shine, S. Havemann, J. Thelen, and P.G. Hill, 2019: Shortwave Spectral Radiative Signatures and Their Physical Controls. J. Climate, 32, 4805–4828, https://doi.org/10.1175/JCLI-D-18-0815.1
Sunlight reaching the Earth is comprised of many different colours, or wavelengths. Some of these wavelengths cannot be detected by the human eye, such as the ultraviolet (UV) wavelengths which famously cause sunburn. Fortunately for us, the most intense sunlight is found at harmless visible wavelengths and reaches the surface with relative ease, allowing us to see during the daytime. Sometimes nature aligns to dramatically separate these wavelengths, producing beautiful optical phenomena such as rainbows. More often, however, the properties of the atmosphere and surface lead to intricate differences in the wavelengths of sunlight that get reflected back to space (Fig. 1).
Satellites have observed specific
wavelengths of reflected sunlight to infer the properties and evolution of our
climate system for decades. Satellites have also independently measured the
total amount of reflected sunlight across all wavelengths to track energy flows
into and out of the Earth system. It has been less common to make spectrally
resolved measurements at many contiguous wavelengths throughout the solar
spectrum. In theory, these measurements would simultaneously provide the total energy
flow – by integrating over the wavelengths – and the “spectral signature” associated with all atmospheric and
surface properties that determined this energy flow. Our recent study puts this
theory to the test.
Almost 100,000 spectra of reflected sunlight were computed at the top-of-atmosphere under a diverse variety of conditions. Applying a clustering technique to the computed spectra (which identifies “clusters” in a dataset with similar characteristics) revealed distinct spectral signatures. When we examined the atmospheric and surface properties that were used to compute the spectra belonging to each spectral signature, a remarkable separation of physical properties was found (Fig. 2).
Surprisingly, the separation of physical properties by distinct spectral signatures, as shown in Fig. 2, was found to be robust up to the largest spatial scales tested of 240 km. This is similar to the footprint size of one of the only previous satellite instruments to measure contiguous spectrally resolved reflected sunlight, the SCIAMACHY**, providing an exciting opportunity to investigate spectral signature variability in real observations. We found that the frequency of spectral signatures in real SCIAMACHY observations followed the expected behaviour during the West African monsoon very closely (Fig. 3).
Overall, the separation of physical
properties by distinct spectral signatures demonstrates great promise for
monitoring evolution of the Earth system directly from spectral reflected
sunlight in the future.
Funding acknowledgement: This work was supported by
the Natural Environment Research Council (NERC) SCience of the Environment:
Natural and Anthropogenic pRocesses, Impacts and Opportunities (SCENARIO)
Doctoral Training Partnership (DTP), Grant NE/L002566/ 1, and from the European
Union 7th Framework Programme under Grant Agreement 603502 [EU project
Dynamics–Aerosol–Chemistry–Cloud Interactions in West Africa (DACCIWA)]
*Note several key simplifications in Fig. 1 for the purposes of
visual effect: atmospheric properties are separated, but often occur
simultaneously and throughout the atmosphere; the depicted path of sunlight is
one option, but sunlight emerging at the top of the atmosphere will come from
many different paths; sunlight reflected by the surface will need to travel
back through the same gases (and likely other properties) on its way back to
the top of the atmosphere, which is not shown. The spectra in Fig. 1 are
generated with SBDART
using a set of arbitrary but realistic atmospheric and surface properties.
** SCIAMACHY = Scanning Imaging Absorption Spectrometer for Atmospheric Chartography.
Jake completed his PhD at Reading in 2018 and now works at the NOAA Earth System Research Laboratory (ESRL) in Boulder, Colorado.
Earlier this month (9th – 17th July, 2019), Elena Saggioro and I from the Mathematics of Planet Earth Centre of Doctoral Training (MPE CDT) were in Montréal for the General Assembly of the IUGG, a quadrennial gathering of nearly 4000 geoscientists from all over the world sharing their latest scientific advances.
The IUGG, which celebrates its centenary this year, is an international organisation ‘dedicated to advancing, promoting, and communicating knowledge of the Earth system, its space environment, and the dynamical processes causing change’ (from the Mission Statement on its website). The IUGG consists of eight constituent associations, among which the International Association of Meteorology and Atmospheric Sciences (IAMAS) and the International Association for the Physical Sciences of the Oceans (IAPSO) are of the most relevance to meteorology students here in Reading. Other fields under the IUGG umbrella include hydrology, cryospheric sciences, seismology, volcanology, geodesy and geomagnetism.
In the General Assembly I presented a poster on my own PhD research, revisiting and proposing a new argument for the finite-time barrier of weather predictability. The poster turned out to be popular, with a good number of scientists visiting and discussing in depth. It is great to know these people, especially those who work in the relatively small field of predictability. Earlier that day, Elena gave an interesting talk on studying southern-hemisphere stratosphere-troposphere coupling using casual network. A member in the audience came to her after the talk for a follow-up chat which lasted for hours! In addition, our supervisor Ted Shepherd gave a solicited talk advocating his storylines approach to the construction of regional climate-change information.
For the variety of subjects covered, the General Assembly was also an excellent opportunity for us to interact with geoscientists of other fields and to get an idea of their research. I did this primarily through the poster sessions, as there’s already so much going on in the oral-presentation sessions of the IAMAS symposia (just a matter of fact: the IAMAS, at 21%, was by far the association with the most attendees), and because it’s easier for a beginner to learn through interacting with a poster presenter than listening to short talks that usually presume some background knowledge in the field. The outcome of visiting posters in such an international conference could be somewhat unexpected. This time, I gave a little more focus on posters from remote parts of the world and learnt how research is being done in these places. To give an example, I saw how hydrologists in French Polynesia use analogue techniques to forecast rainfall and flood on the island of Tahiti which has a complex geography of drainage basins (poster by Lydie Sichoix, University of French Polynesia). This is a very challenging problem, and I think their commitment to protecting the public’s safety during floods is clear, yet there’s only so much they can do as they don’t have the money to buy even a single RADAR instrument for nowcasting. The situation in underprivileged places like this definitely deserves more attention.
Aside from the scientific programme, Elena and I spent some time as a tourist in Montréal. We are delighted to learn how committed Montréal is to sustainability and climate-change adaptation. The Biosphère Museum of the Environment nicely outlines the resilient city’s master plan 50 years ahead: new space reserved for nature in the city centre, green alleyways throughout the city, and harvesting storm and rain water are just a few examples in their long-term plan.
Montréal is also rich in history, culture and diversity. Churches and museums are everywhere. There were also a multi-cultural festival and a series of fireworks depicting different national themes during our stay, and we went to some of them. Situated along St Lawrence’s River, the city is also home to a range of water sports, including white-water rafting which was a fun experience. Before coming home, Elena and I went up to Mount Royal for an exhilarating view of Montréal, a city that we much enjoyed!
Between the 1st and 12th July 2019, I attended the 2nd International Centre for Theoretical Physics (ICTP) Summer School in Hierarchical Modelling of Climate Dynamics at the ICTP guesthouse in Trieste, Italy. The focus of this summer school was on convective organisation and climate sensitivity, which is incredibly relevant to my PhD topic: Interactions between Radiation and Convective Organisation. So, I felt I had to attend this summer school (and not just because my lead supervisor, Chris Holloway, was one of the lead directors).
This was an international conference with staff and students coming together from all corners of the globe. In total there were 111 people attending the school, made up of 84 participants, 20 speakers and 7 directors. Without knowing anyone else going to this school (except my supervisor), I was initially a little apprehensive as I didn’t know what to expect but as soon as I met some of the other students I was put at ease. It was amazing to meet other people working on very similar projects to me, especially since my supervisor was the only other person I previously knew working on this convective organisation topic. So, it was great to not only make new friends but also meet potential future colleagues.
As expected, the schedule was pretty intense, with most days working from 9am until 6pm except for lunch and a couple of coffee breaks. The mornings consisted of a couple of lectures given by some of the leading experts in the field including Kerry Emanuel, Bjorn Stevens and Sandrine Bony, then in the afternoons we would do some group project work. In our groups of 4 or 5, we analysed some numerical model data, to study how convection organises within our model. I was surprised to find that our group tasks were very similar to what I’ve been doing for my first year, so I was a bit worried that we’d manage to do what I’ve been working on this past year within a couple of weeks! But actually, it ended up giving me almost too many new ideas for my own research! In the second week, each group then had to give a quick presentation on their work.
Each day, after the lectures and the group work, we were free to do what we wanted for the rest of the evening. With the venue being right on the coast, and with temperatures consistently between 26 – 32C in the day, it was perfect to relax by the sea or go for a swim. Or, if we were bored with the relentless supply of pasta in the canteen then we’d often go into town in search of pizza and of course gelato!
At the start of the second week, there was a poster session in which a lot of the participants brought posters to showcase their projects. This was the first time I’d presented my research at an event like this, so it was great to show what I’ve been working on in front of so many people. It was exciting to see so many people genuinely interested in my work and I got lots of useful feedback and ideas.
So overall, this summer school far surpassed my expectations
and I would strongly recommend attending a summer school if you get the chance.
I learned so much through the lectures, the group work, through chatting to the
professors and students and through presenting my work. I now have far too many
ideas to explore with my research, probably more than I can realistically
achieve! Perhaps the most valuable aspect of the school was being able to meet
so many people working in this field. Since this topic is very niche, I have
been very lucky to meet a very large proportion of the people working in the
topic so I’m sure some of our paths will cross in the future and we will be
able to collaborate on future projects.
This year at the University of Birmingham, from the 2nd to the 5th of July, the Royal Meteorological Society (RMetS) held two national conferences. The first, the Atmospheric Science Conference, was well attended by staff and post-docs. The second, the Student and Early Career Scientists Conference, was attended by PhD students, including some of us from Reading. It proved a great opportunity to share research and best practices as well as network with both old and new colleagues from other institutions.
The Student and Early Career conference is open to all students and researchers just embarking upon their science career. It aims to give those in the field the opportunity to meet and present work before going on to attend more specialized conferences. For some of the Reading delegates this was the first opportunity to present work outside of the department to a wider audience who they weren’t already familiar with, or in quite the same field as. Presentations from Reading students ranged from topics such as thermal updrafts to atmosphere and ocean model coupling (summaries below). There were also keynote sessions that discussed important topics in atmospheric sciences as well as addressing the impact and reach that social media can give research.
There was also time to socialize, with an ice-breaker event on the Wednesday before the conference and a conference dinner on the Thursday evening. Keen to give the participants an opportunity to maximise their networking time, on Wednesday several scientists who had attended the Atmospheric Science Conference that day volunteered to stay behind, share their experiences during their careers and chat to the Early Career conference delegates over a few drinks.
Having also attended events through other institutions (such as the doctoral training partnership SCENARIO) there were also many friendly faces from outside Reading in attendance, and it was a great opportunity to catch up and share progress on our work. One of the delegates was even an old friend from when I was an undergraduate, so you never know what familiar faces you might find!
The student conference is organised by a committee of students and early career scientists (usually but not always attendees from previous conferences) from around the UK. Being a member of the committee is a fantastic opportunity to hone one’s organizational and planning skills, as well as getting invaluable practice for things like chairing sessions. If you’re interested in helping organise next year’s conference please do get in touch with Catherine Bicknell at RMetS (firstname.lastname@example.org) or if you’re thinking about attending then you can start by joining the society where you’ll hear about all the other great events they host.
Highlights of the work presented by Reading students:
Kris Boykin presented work on clustering ensemble members in high resolution forecasts in order to extract likely scenarios and assign probabilities to each one.
Sally Woodhouse presented a study of the effect of resolution of atmospheric models on heat transport into the Arctic using a coupled ocean-atmosphere climate model.
Emanuele Gentile presented a poster on his work determining how coupled models can improve extreme surface wind predictions using storm Helene as a case study.
Jake Bland presented a poster on the humidity biases in the stratosphere in the Met Office operational model assessed relative to experimental radiosonde data gathered during the North Atlantic Waveguide and Downstream impacts EXperiment (NAWDEX) field campaign.
It is widely known that clouds pose a lot of difficulties for both weather and climate modelling, particularly when ice is present. The ice water content (IWC) of a cloud is defined as the mass of ice per unit volume of air. The integral of this quantity over a column is referred to as the ice water path (IWP) and is considered one of the essential climate variables by the World Meteorological Organisation. Currently there are large inconsistencies in the IWP retrieved from different satellites, and there is also a large spread in the amount produced by different climate models (Eliasson et al., 2011). A major part of the problem is the lack of reliable global measurements of cloud ice. For this reason, the Ice Cloud Imager (ICI) will be launched in 2022. ICI will be the first instrument in space specifically designed to measure cloud ice, with channels ranging from 183 to 664 GHz. It is expected that the combination of frequencies available will allow for more accurate estimations of IWP and particle size. A radiometer called ISMAR has been developed by the UK Met Office and ESA as an airborne demonstrator for ICI, flying on the FAAM BAe-146 research aircraft shown in Fig. 1.
As radiation passes through cloud, it is scattered in all directions. Remote sensing instruments measure the scattered field in some way; either by detecting some of the scattered waves, or by detecting how much radiation has been removed from the incident field as a result of scattering. The retrieval of cloud ice properties therefore relies on accurate scattering models. A variety of numerical methods currently exist to simulate scattering by ice particles with complex geometries. In a very broad sense, these can be divided into 2 categories –
1: Methods that are accurate but computationally expensive
2: Methods that are computationally efficient but inaccurate
My PhD has involved developing a new approximation for aggregates which falls somewhere in between the two extremes. The method is called the Independent Monomer Approximation (IMA). So far, tests have shown that it performs well for small particle sizes, with particularly impressive results for aggregates of dendritic monomers.
Radiometers such as ICI and ISMAR convert measured radiation into brightness temperatures (Tb), i.e. the temperature of a theoretical blackbody that would emit an equivalent amount of radiation. Lower values of Tb correspond to more ice in the clouds, as a greater amount of radiation from the lower atmosphere is scattered on its way to the instrument’s detector (i.e. a brightness temperature “depression” is observed over thick ice cloud). Generally, the interpretation of measurements from remote-sensing instruments requires many assumptions to be made about the shapes and distributions of particles within the cloud. However, by comparing Tb at orthogonal horizontal (H) and vertical (V) polarisations, we can gain some information about the size, shape, and orientation of ice particles within the cloud. If large V-H polarimetric differences are measured, it is indicative of horizontally oriented particles, whereas random orientation produces less of a difference in signal. According to Gong and Wu (2017), neglecting the polarimetric signal could result in errors of up to 30% in IWP retrievals. Examples of Tb depressions and the corresponding V-H polarimetric differences can be seen in Fig. 2. In the work shown here, we explore this particular case further.
Figure 2: (a) ISMAR measured brightness temperatures, showing a depression (decrease in Tb) caused by thick cloud; (b) Polarimetric V-H brightness temperature difference, with significant values reaching almost 10 K.
Using the ISMAR instrument, we can test scattering models that could be used within retrieval algorithms for ICI. We want to find out whether the IMA method is capable of reproducing realistic brightness temperature depressions, and whether it captures the polarimetric signal. To do this, we look at a case study that was part of the NAWDEX (North Atlantic Waveguide and Downstream Impact Experiment) campaign of flying. The observations from the ISMAR radiometer were collected on 14 October 2016 off the North-West Coast of Scotland, over a frontal ice cloud. Three different aircraft took measurements from above the cloud during this case, which means that we have coincident data from ISMAR and two different radar frequencies of 35 GHz and 95 GHz. This particular case saw large V-H polarimetric differences reaching almost 10 K, as seen in Fig. 2(b). We will look at the applicability of the IMA method to simulating the polarisation signal measured from ISMAR, using the Atmospheric Radiative Transfer Simulator (ARTS).
For this study, we need to construct a model of the atmosphere to be used in the radiative transfer simulations. The nice thing about this case is that the FAAM aircraft also flew through the cloud, meaning we have measurements from both in-situ and remote-sensing instruments. Consequently, we can design our model cloud using realistic assumptions. We try to match the atmospheric state at the time of the in-situ observations by deriving mass-size relationships specific to this case, and generating particles to follow the derived relationship for each layer. The particles were generated using the aggregation model of Westbrook (2004).
Due to the depth of the cloud, it would not be possible to obtain an adequate representation of the atmospheric conditions using a single averaged layer. Hence, we modelled our atmosphere based on the aircraft profiles, using 7 different layers of ice with depths of approximately 1 km each. These layers are located between altitudes of 2 km and 9 km. Below 2 km, the Marshall-Palmer drop size distribution was used to represent rain, with an estimated rain rate of 1-2mm/hr taken from the Met Office radar. The general structure of our model atmosphere can be seen in Fig. 3, along with some of the particles used in each layer. Note that this is a crude representation and the figure shows only a few examples; in the simulations we use between 46 and 62 different aggregate realisations in each layer.
To test our model atmosphere, we simulated the radar reflectivities at 35 GHz and 95 GHz using the particle models generated for this case. This allowed us to refine our model until sufficient accuracy was achieved. Then we used the IMA method to calculate the scattering quantities required by the ARTS radiative transfer model. These were implemented into ARTS in order to simulate the ISMAR polarisation observations. Fig. 4 shows the simulated brightness temperatures using different layers of our modelled atmosphere, i.e. starting with the clear-sky case and gradually increasing the cloud amount. The simulations using the IMA scattering method in the ARTS model were compared to the measurements from ISMAR shown in Fig. 2. Looking at the solid lines in Fig. 4, it can be seen that the aggregates of columns and dendrites simulate the brightness temperature depression well, but do not reproduce the V-H polarization signal. Thus we decided to include some horizontally aligned single dendrites which were not included in our original atmospheric model. The reason we chose these particles is that they tend to have a greater polarization signal compared to aggregates, and there was evidence in the cloud particle imagery that they were present in the cloud during the time of interest. We placed these particles at the cloud base, without changing the ice water content of the model. The results from that experiment are shown by the diagonal crosses in Fig. 4. It is clear that adding single dendrites allow us to simulate a considerably larger polarimetric signal, closely matching the ISMAR measurements. Using only aggregates of columns and dendrites gives a V-H polarimetric difference of 1.8K, whereas the inclusion of dendritic particles increases this value to 8.4K.
To conclude, we have used our new light scattering approximation (IMA) along with the ARTS radiative transfer model to simulate brightness temperature measurements from the ISMAR radiometer. Although the measured brightness temperature depressions can generally be reproduced using the IMA scattering method, the polarisation difference is very sensitive to the assumed particle shape for a given ice water path. Therefore, to obtain good retrievals from ICI, it is important to represent the cloud as accurately as possible. Utilising the polarisation information available from the instrument could provide a way to infer realistic particle shapes, thereby reducing the need to make unrealistic assumptions.
Eliasson, S., S. A. Buehler, M. Milz, P. Eriksson, and V. O. John, 2011: Assessing observed and modelled spatial distributions of ice water path using satellite data. Atmos. Chem. Phys., 11, 375-391.
Gong, J., and D. L. Wu, 2017: Microphysical properties of frozen particles inferred from Global Precipitation Measurement (GPM) Microwave Imager (GMI) polarimetric measurements. Atmos. Chem. Phys., 17, 2741-2757.
Westbrook, C. D., R. C. Ball, P. R. Field, and A. J. Heymsfield, 2004: A theory of growth by differential sedimentation with application to snowflake formation. Phys. Rev. E, 70, 021403.