Modelling windstorm losses in a climate model

Extratropical cyclones cause vast amounts of damage across Europe throughout the winter seasons. The damage from these cyclones mainly comes from the associated severe winds. The most intense cyclones have gusts of over 200 kilometres per hour, resulting in substantial damage to property and forestry, for example, the Great Storm of 1987 uprooted approximately 15 million trees in one night. The average loss from these storms is over $2 billion per year (Schwierz et al. 2010) and is second only to Atlantic Hurricanes globally in terms of insured losses from natural hazards. However, the most severe cyclones such as Lothar (26/12/1999) and Kyrill (18/1/2007) can cause losses in excess of$10 billion (Munich Re, 2016). One property of extratropical cyclones is that they have a tendency to cluster (to arrive in groups – see example in Figure 1), and in such cases these impacts can be greatly increased. For example Windstorm Lothar was followed just one day later by Windstorm Martin and the two storms combined caused losses of over \$15 billion. The large-scale atmospheric dynamics associated with clustering events have been discussed in a previous blog post and also in the scientific literature (Pinto et al., 2014; Priestley et al. 2017).

A large part of my PhD has involved investigating exactly how important the clustering of cyclones is on losses across Europe during the winter. In order to do this, I have used 918 years of high resolution coupled climate model data from HiGEM (Shaffrey et al., 2017) which provides a huge amount of winter seasons and cyclone events for analysis.

In order to understand how clustering affects losses, I first of all need to know how much loss/damage is associated with each individual cyclone. This is done using a measure called the Storm Severity Index (SSI – Leckebusch et al., 2008), which is a proxy for losses that is based on the 10-metre wind field of the cyclone events. The SSI is a good proxy for windstorm loss. Firstly, it scales the wind speed in any particular location by the 98th percentile of the wind speed climatology in that location. This scaling ensures that only the most severe winds at any one point are considered, as different locations have different perspectives on what would be classed as ‘damaging’. This exceedance above the 98th percentile is then raised to the power of 3 due to damage from wind being a highly non-linear function. Finally, we apply a population density weighting to our calculations. This weighting is required because a hypothetical gust of 40 m/s across London will cause considerably more damage than the same gust across far northern Scandinavia, and the population density is a good approximation for the density of insured property. An example of the SSI that has been calculated for Windstorm Lothar is shown in Figure 2.

From Figure 2b you can see how most of the damage from Windstorm Lothar was concentrated across central/northern France and also across southern Germany. This is because the winds here were most extreme relative to what is the climatology. Even though the winds are highest across the North Atlantic Ocean, the lack of insured property, and a much high climatological winter mean wind speed, means that we do not observe losses/damage from Windstorm Lothar in these locations.

I can apply the SSI to all of the individual cyclone events in HiGEM and therefore can construct a climatology of where windstorm losses occur. Figure 3 shows the average loss across all 918 years of HiGEM. You can see that the losses are concentrated in a band from southern UK towards Poland in an easterly direction. This mainly covers the countries of Great Britain, Belgium, The Netherlands, France, Germany, and Denmark.

This blog post introduces my methodology of calculating and investigating the losses associated with the winter season extratropical cyclones. Work in Priestley et al. (2018) uses this methodology to investigate the role of clustering on winter windstorm losses.

This work has been funded by the SCENARIO NERC DTP and also co-sponsored by Aon Benfield.

References

Leckebusch, G. C., Renggli, D., and Ulbrich, U. 2008. Development and application of an objective storm severity measure for the Northeast Atlantic region. Meteorologische Zeitschrift. https://doi.org/10.1127/0941-2948/2008/0323.

Munich Re. 2016. Loss events in Europe 1980 – 2015. 10 costliest winter storms ordered by overall losses. https://www.munichre.com/touch/naturalhazards/en/natcatservice/significant-natural-catastrophes/index.html

Pinto, J. G., Gómara, I., Masato, G., Dacre, H. F., Woollings, T., and Caballero, R. 2014. Large-scale dynamics associated with clustering of extratropical cyclones affecting Western Europe. Journal of Geophysical Research: Atmospheres. https://doi.org/10.1002/2014JD022305.

Priestley, M. D. K., Dacre, H. F., Shaffrey, L. C., Hodges, K. I., and Pinto, J. G. 2018. The role of European windstorm clustering for extreme seasonal losses as determined from a high resolution climate model, Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2018-165, in review.

Priestley, M. D. K., Pinto, J. G., Dacre, H. F., and Shaffrey, L. C. 2017. Rossby wave breaking, the upper level jet, and serial clustering of extratropical cyclones in western Europe. Geophysical Research Letters. https://doi.org/10.1002/2016GL071277.

Schwierz, C., Köllner-Heck, P., Zenklusen Mutter, E. et al. 2010. Modelling European winter wind storm losses in current and future climate. Climatic Change. https://doi.org/10.1007/s10584-009-9712-1.

Shaffrey, L. C., Hodson, D., Robson, J., Stevens, D., Hawkins, E., Polo, I., Stevens, I., Sutton, R. T., Lister, G., Iwi, A., et al. 2017. Decadal predictions with the HiGEM high resolution global coupled climate model: description and basic evaluation, Climate Dynamics, https://doi.org/10.1007/s00382-016-3075-x.

Why become a Royal Meteorological Society Student member?

This week the Royal Meteorological Society (RMetS) published their strategic plan for the period of 2018 to 2020, and here at Social Metwork HQ we thought it would be a splendid idea to reflect on the benefits of being a student member of the Royal Meteorological Society.

An important benefit in my opinion is that when becoming a member of RMetS you join a well-established community who hold enthusiasm about the weather and climate at its core. Members come from all corners of the world and at different stages of their career spanning the entire range: from the amateur weather enthusiasts to professionals.  As a student, being an RMetS member can lead to conversations that could develop your career and bring unexpected opportunities. This has been greatly enhanced with the RMetS mentoring scheme.

RMetS host many different types of meetings, including annual conferences, meetings hosted by regional centres, and national meetings. Additional gatherings are held by special interest groups, ranging from Weather Arts & Music to Dynamical Problems. Meetings on a regional and national scale provide a platform for discussion and learning amongst those in the field. For a student, the highlight in the RMetS calendar is the annual student conference. Every year, sixty to eighty students come together to present their work and develop professional relationships that continue for years to come. This year’s conference is hosted at the University of York on the 5th and 6th July 2018 (more information). After two student conferences under my belt (see previous blog post), I would highly recommend any early career research scientist attending this event. It serves as a platform to share their own work in a friendly atmosphere and be inspired by the wider student community.

Other benefits to becoming an RMetS student member include eligibility to the Legacies Fund, grants and fellowships, and receiving a monthly copy of Weather magazine. Most importantly though, through becoming a RMetS member you support a professional society who are committed to increasing awareness of the importance of weather and climate in policy and decision-making. Alongside this week’s publication of RMetS’ strategic plan, both the Met Office and NASA have published press releases stating that 2017 was the warmest year on record without El Niño. The atmosphere and oceans of our planet are changing at unprecedented rates: rising sea levels, reductions in Arctic sea-ice, and an increased frequency of extreme weather events to name but a few climate change impacts. Becoming an RMetS student member does not only benefit your career and knowledge, but also supports a society that is committed to promoting and raising awareness of weather and climate science.

New Forecast Model Provides First Global Scale Seasonal River Flow Forecasts

Over the past ~decade, extended-range forecasts of river flow have begun to emerge around the globe, combining meteorological forecasts with hydrological models to provide seasonal hydro-meteorological outlooks. Seasonal forecasts of river flow could be useful in providing early indications of potential floods and droughts; information that could be of benefit for disaster risk reduction, resilience and humanitarian aid, alongside applications in agriculture and water resource management.

While seasonal river flow forecasting systems exist for some regions around the world, such as the U.S., Australia, Africa and Europe, the forecasts are not always accessible, and forecasts in other regions and at the global scale are few and far between.  In order to gain a global overview of the upcoming hydrological situation, other information tends to be used – for example historical probabilities based on past conditions, or seasonal forecasts of precipitation. However, precipitation forecasts may not be the best indicator of floodiness, as the link between precipitation and floodiness is non-linear. A recent paper by Coughlan-de-Perez et al (2017), states:

“Ultimately, the most informative forecasts of flood hazard at the seasonal scale are streamflow forecasts using hydrological models calibrated for individual river basins. While this is more computationally and resource intensive, better forecasts of seasonal flood risk could be of immense use to the disaster preparedness community.”

Over the past months, researchers in the Water@Reading* research group have been working with the European Centre for Medium-Range Weather Forecasts (ECMWF), to set up a new global scale hydro-meteorological seasonal forecasting system. Last week, on 10th November 2017, the new forecasting system was officially launched as an addition to the Global Flood Awareness System (GloFAS). GloFAS is co-developed by ECMWF and the European Commission’s Joint Research Centre (JRC), as part of the Copernicus Emergency Management Services, and provides flood forecasts for the entire globe up to 30 days in advance. Now, GloFAS also provides seasonal river flow outlooks for the global river network, out to 4 months ahead – meaning that for the first time, operational seasonal river flow forecasts exist at the global scale – providing globally consistent forecasts, and forecasts for countries and regions where no other forecasts are available.

The new seasonal outlook is produced by forcing the Lisflood hydrological river routing model with surface and sub-surface runoff from SEAS5, the latest version of ECMWF’s seasonal forecasting system, (also launched last week), which consists of 51 ensemble members at ~35km horizontal resolution. Lisflood simulates the groundwater and routing processes, producing a probabilistic forecast of river flow at 0.1o horizontal resolution (~10km, the resolution of Lisflood) out to four months, initialised using the latest ERA-5 model reanalysis.

The seasonal outlook is displayed as three new layers in the GloFAS web interface, which is publicly (and freely) available at www.globalfloods.eu. The first of these gives a global overview of the maximum probability of unusually high or low river flow (defined as flow exceeding the 80th or falling below the 20th percentile of the model climatology), during the 4-month forecast horizon, in each of the 306 major world river basins used in GloFAS-Seasonal.

The second layer provides further sub-basin-scale detail, by displaying the global river network (all pixels with an upstream area >1500km2), again coloured according to the maximum probability of unusually high or low river flow during the 4-month forecast horizon. In the third layer, reporting points with global coverage are displayed, where more forecast information is available. At these points, an ensemble hydrograph is provided showing the 4-month forecast of river flow, with thresholds for comparison of the forecast to typical or extreme conditions based on the model climatology. Also displayed is a persistence diagram showing the weekly probability of exceedance for the current and previous three forecasts.

Over the coming months, an evaluation of the system will be completed – for now, users are advised to evaluate the forecasts for their particular application. We welcome any feedback on the forecast visualisations and skill – feel free to contact me at the email address below!

To find out more, you can see the University’s press release here, further information on SEAS5 here, and the user information on the seasonal outlook GloFAS layers here.

*Water@Reading is “a vibrant cross-faculty centre of research excellence at the University of Reading, delivering world class knowledge in water science, policy and societal impacts for the UK and internationally.”

Full list of collaborators:

Rebecca Emerton1,2, Ervin Zsoter1,2, Louise Arnal1,2, Prof. Hannah Cloke1, Dr. Liz Stephens1, Dr. Florian Pappenberger2, Prof. Christel Prudhomme2, Dr Peter Salamon3, Davide Muraro3, Gabriele Mantovani3

2 ECMWF
3 European Commission JRC

RMetS Impact of Science Conference 2017.

“We aim to help people make better decisions than they would if we weren’t here”

Rob Varley CEO of Met Office

This week PhD students from the University of Reading attended the Royal Meteorological Society Impact of Science Conference for Students and Early Career Scientists. Approximately eighty scientists from across the UK and beyond gathered at the UK Met Office to learn new science, share their own work, and develop new communication skills.

Across the two days students presented their work in either a poster or oral format. Jonathan Beverley, Lewis Blunn and I presented posters on our work, whilst Kaja Milczewska, Adam Bateson, Bethan Harris, Armenia Franco-Diaz and Sally Woodhouse gave oral presentations. Honourable mentions for their presentations were given to Bethan Harris and Sally Woodhouse who presented work on the energetics of atmospheric water vapour diffusion and the representation of mass transport over the Arctic in climate models (respectively). Both were invited to write an article for RMetS Weather Magazine (watch this space). Congratulations also to Jonathan Beverley for winning the conference’s photo competition!

Alongside student presentations, two keynote speaker sessions took place, with the latter of these sessions titled Science Communication: Lessons from the past, learning for future impact. Speakers in this session included Prof. Ellie Highwood (Professor of Climate Physics and Dean for Diversity and Inclusion at University of Reading), Chris Huhne (Co-chair of ET-index and former Secretary of State for Energy and Climate Change), Leo Hickman (editor for Carbon Brief) and Dr Amanda Maycock (NERC Independent Research Fellow and Associate Professor in Climate Dynamics, University of Leeds). Having a diverse range of speakers encouraged thought-provoking discussion and raised issues in science communication from many angles.

Prof. Ellie Highwood opened the session challenging us all to step beyond the typical methods of scientific communication. Try presenting your science without plots. Try presenting your work with no slides at all! You could step beyond the boundaries even more by creating interesting props (for example, the notorious climate change blanket). Next up Chris Huhne and Leo Hickman gave an overview of the political and media interactions with climate change science (respectively). The Brexit referendum, Trump’s withdrawal from the Paris Accord and the rise of the phrase “fake news” are some of the issues in a society “where trust in the experts is falling”. Finally, Dr Amanda Maycock presented a broad overview of influential science communicators from the past few centuries. Is science relying too heavily on celebrities for successful communication? Should the research community put more effort into scientific outreach?

Communication and collaboration became the two overarching themes of the conference, and conferences such as this one are a valuable way to develop these skills. Thank you to the Royal Meteorology Society and UK Met Office for hosting the conference and good luck to all the young scientists that we met over the two days.

#RMetSImpact

Also thank you to NCAS for funding my conference registration and to all those who provided photos for this post.

Can we really use El Niño to predict flooding?

R. Emerton, H. Cloke, E. Stephens, E. Zsoter, S. Woolnough, F. Pappenberger (2017). Complex picture for likelihood of ENSO-driven flood hazard. Nature Communications. doi: 10.1038/NCOMMS14796

When an El Niño is declared, or even forecast, we think back to memorable past El Niños (such as 1997/98), and begin to ask whether we will see the same impacts. Will California receive a lot of rainfall? Will we see droughts in tropical Asia and Australia? Will Peru experience the same devastating floods as in 1997/98, and 1982/83?

El Niño and La Niña, which see changes in the ocean temperatures in the tropical Pacific, are well known to affect weather, and indeed river flow and flooding, around the globe. But how well can we estimate the potential impacts of El Niño and La Niña, and how likely flooding is to occur?

This question is what some of us in the Water@Reading research group at the University of Reading have been looking to answer in our recent publication in Nature Communications. As part of our multi- and inter-disciplinary research, we work closely with the Red Cross / Red Crescent Climate Centre (RCCC), who are working on an initiative called Forecast-based Financing (FbF, Coughlan de Perez et al.). FbF aims to distribute aid (for example providing water purification tablets to prevent spread of disease, or digging trenches to divert flood water) ahead of a flood, based on forecasts. This approach helps to reduce the impact of the flood in the first place, rather than working to undo the damage once the flood has already occurred.

Photo credit: Red Cross / Red Crescent Climate Centre

In Peru, previous strong El Niños in 1982/83 and 1997/98 had resulted in devastating floods in several regions. As such, when forecasts in early 2015 began to indicate a very strong El Niño was developing, the RCCC and forecasters at the Peruvian national hydrological and meteorology agency (SENAMHI) began to look into the likelihood of flooding, and what FbF actions might need to be taken.

Typically, statistical products indicating the historical probability (likelihood [%] based on what happened during past El Niños) of extreme precipitation are used as a proxy for whether a region will experience flooding during an El Niño (or La Niña), such as these maps produced by the IRI (International Research Institute for Climate and Society). You may also have seen maps which circle regions of the globe that will be drier / warmer / wetter / cooler – we’ll come back to these shortly.

These rainfall maps show that Peru, alongside several other regions of the world, is likely to see more rainfall than usual during an El Niño. But does this necessarily mean there will be floods? And what products are out there indicating the effect of El Niño on rivers across the globe?

For organisations working at the global scale, such as the RCCC and other humanitarian aid agencies, global overviews of potential impacts are key in taking decisions on where to focus resources during an El Niño or La Niña. While these maps are useful for looking at the likely changes in precipitation, it has been shown that the link between precipitation and flood magnitude is nonlinear (Stephens et al.),  – more rain does not necessarily equal floods – so how does this transfer to the potential for flooding?

The motivation behind this work was to provide similar information, but taking into account the hydrology as well as the meteorology. We wanted to answer the question “what is the probability of flooding during El Niño?” not only for Peru, but for the global river network.

To do this, we have taken the new ECMWF ERA-20CM ensemble model reconstruction of the atmosphere, and run this through a hydrological model to produce the first 20th century global hydrological reconstruction of river flow. Using this new dataset, we have for the first time estimated the historical probability of increased or decreased flood hazard (defined as abnormally high or low river flow) during an El Niño (or La Niña), for the global river network.

The question – “what is the probability of flooding during El Niño?”, however, remains difficult to answer. We now have maps of the probability of abnormally high or low river flow (see Figure 1), and we see clear differences between the hydrological analysis and precipitation. It is also evident that the probabilities themselves are often lower, and much more uncertain, than might be useful – how do you make a decision on whether to provide aid to an area worried about flooding, when the probability of that flooding is 50%?

The likely impacts are much more complex than is often perceived and reported – going back to the afore-mentioned maps that circle regions of the globe and what their impact will be (warmer, drier, wetter?) – these maps portray these impacts as a certainty, not a probability, with the same impacts occurring across huge areas. For example, in Figure 2, we take one of the maps from our results, which indicates the probability of increased or decreased flood hazard in one month during an El Niño, and draw over this these oft-seen circles of potential impacts. In doing this, we remove all information on how likely (or unlikely) the impacts are, smaller scale changes within these circles (in some cases our flood hazard map even indicates a different impact), and a lot of the potential impacts outside of these circles – not to mention the likely impacts can change dramatically from one month to the next. For those organisations that take actions based on such information, it is important to be aware of the uncertainties surrounding the likely impacts of El Niño and La Niña.

“We conclude that while it may seem possible to use historical probabilities to evaluate regions across the globe that are more likely to be at risk of flooding during an El Niño / La Niña, and indeed circle large areas of the globe under one banner of wetter or drier, the reality is much more complex.”

PS. During the winter of 2015/16, our results estimated an ~80% likelihood of increased flood hazard in northern coastal Peru, with only ~10% uncertainty surrounding this. The RCCC took FbF actions to protect thousands of families from potentially devastating floods driven by one of the strongest El Niños on records. While flooding did occur, this was not as severe as expected based on the strength of the El Niño. More recently, during the past few months (January – March 2017), anomalously high sea surface temperatures (SSTs) in the far eastern Pacific (known as a “coastal El Niño” in Peru but not widely acknowledged as an El Niño because central Pacific SSTs are not anomalously warm) have led to devastating flooding in several regions and significant loss of life. And Peru wasn’t the only place that didn’t see the impacts it expected in 2015/16; other regions of the world, such as the US, also saw more rainfall than normal in places that were expected to be drier, and California didn’t receive the deluge they were perhaps hoping for. It’s important to remember that no two El Niños are the same, and El Niño will not be the only influence on the weather around the globe. While El Niño and La Niña can provide some added predictability to the atmosphere, the impacts are far from certain.

Full reference:

R. Emerton, H. Cloke, E. Stephens, E. Zsoter, S. Woolnough, F. Pappenberger (2017). Complex picture for likelihood of ENSO-driven flood hazard. Nature Communications. doi: 10.1038/NCOMMS14796

Press Release:

The Influence of the Weather on Bird Migration

As well as being a meteorologist, I am a bird watcher. This means I often combine meteorology and bird watching to see the impact of the weather on birds. Now that we are well into March my focus in bird watching turns to one thing – the migration.

March generally marks the time when the first summer migrants start arriving into the UK. Already this year we have had reports of Sand Martin, Wheatear, Garganey, Little Ringed Plover, White Wagtail, Osprey, Swallow, House Martin, Ring Ouzel and Whitethroat (up to 9 March), some of which are depicted below.

There are many people that consider the arrival dates of certain migratory species of birds and how this arrival date changes over many years. I do keep extensive records of the birds that I see (and thus arrival dates), but what interests me more are the odd days in the record, and the sightings of unusual birds and working out how they arrived at their destinations.

A good example of this can be found by looking at my first Swallow sighting of the year in Kent and East Sussex. Since I started bird watching in 2001 my first Swallow of the year has moved from around 10 April to between 26-March and 1 April. However in 2013 my first record was 15 April. Then in 2015 and 2016 I saw my first Swallow on 1 April and 27 March respectively (I was in Cheshire in 2014 in late March/early April).

So what happened; why were the Swallows late in Kent in 2013? Well, it all comes down to wind direction. The spring of 2013 was very chilly and along the east coast there were plenty of N/NE winds – this would have provided a head wind so the Swallows would preferentially not migrate up the east coast in those conditions but instead migrate up the west coast where there were southerlies.

So, the wind direction plays a key part in the migration of birds. If conditions are for a tailwind or very light winds the birds will migrate; otherwise they will stay put. However, headwinds can lead to some interesting phenomena associated with bird migration – ‘falls’.

A ‘fall’ occurs when there are a large number of migrants building up along the coastline at a departure point (so for the interest of UK bird watchers Northern France), as they cannot get to their destination. When the wind direction changes the birds will then migrate en masse and quite literally fall out of the sky.

It’s not all about the wind direction though; rain is also a key factor that bird watchers consider when looking at weather forecasts. Essentially, fronts and showers are great for bird watchers. On migration birds will often fly higher than they normally would. This means on a clear sunny day you could easily miss birds passing overhead as they are so high up. However, with the rain the birds will often fly lower, avoiding the in-cloud turbulence. For many of the summer migrants their food sources (insects) also fly lower in these conditions.

This means that a forecast of showers with a southerly wind is generally what I look for from mid-April onwards (particularly as an inland birder), as it means there is a good chance of migratory species turning up – also because then I can head out after work as the evenings are brighter. This is something that I did last year and ended up recording the first Sandwich Tern (photo below (not of the bird I saw)) of the year in Berkshire.

So in summary, it’s not as simple as just keeping an eye on the wind direction – there are other factors that can influence the birds’ migration and where they will end up. For more information about the impact of weather on bird sightings (considering both rare and common birds) check out my blog.

The advection process: simulating wind on computers

Email: js102@zepler.net   Web: datumedge.co.uk   Twitter: @hertzsprrrung

If we know which way the wind is blowing then we can predict a lot about the weather. We can easily observe the wind moving clouds across the sky, but the wind also moves air pollution and greenhouse gases. This process is called transport or advection. Accurately simulating the advection process is important for forecasting the weather and predicting climate change.

I am interested in simulating the advection process on computers by dividing the world into boxes and calculating the same equation in every box. There are many existing advection methods but many rely on these boxes having the correct shape and size, otherwise these existing methods can produce inaccurate simulations.

During my PhD, I’ve been developing a new advection method that produces accurate simulations regardless of cell shape or size. In this post I’ll explain how advection works and how we can simulate advection on computers. But, before I do, let’s talk about how we observe the weather from the ground.

In meteorology, we generally have an incomplete picture of the weather. For example, a weather station measures the local air temperature, but there are only a few hundred such stations dotted around the UK. The temperature at another location can be approximated by looking at the temperatures reported by nearby stations. In fact, we can approximate the temperature at any location by reconstructing a continuous temperature field using the weather station measurements.

So far we have only talked about temperatures varying geographically, but temperatures also vary over time. One reason that temperatures change over time is because the wind is blowing. For example, a wind blowing from the north transports, or advects, cold air from the arctic southwards over the UK. How fast the temperature changes depends on the wind speed, and the size of the temperature contrast between the arctic air and the air further south. We can write this as an equation. Let’s call the wind speed $v$ and assume that the wind speed and direction are always the same everywhere. We’ll label the temperature $T$, label time $t$, and label the south-to-north direction $y$, then we can write down the advection equation using partial derivative notation,

$\frac{\partial T}{\partial t} = - \frac{\partial T}{\partial y} \times v$

This equation tells us that the local temperature will vary over time ($\frac{\partial T}{\partial t}$), depending on the north-south temperature contrast ($- \frac{\partial T}{\partial y}$) multiplied by the wind speed $v$.

One way to solve the advection equation on a computer is to divide the world into boxes, called cells. The complete arrangement of cells is called a mesh. At a point at the centre of each cell we store meteorological information such as temperature, water vapour content or pollutant concentration. At the cell faces where two cells touch we store the wind speed and direction. The arrangement looks like this:

The above example of a mesh over the UK uses cube-shaped cells stacked in columns above the Earth, and arranged along latitude and longitude lines. But more recently, weather forecasting models are using different types of mesh. These models tesselate the globe with squares, hexagons or triangles.

Weather models must also rearrange cells in order to represent mountains, valleys, cliffs and other terrain. Once again, different models rearrange cells differently. One method, called the terrain-following method, shifts cells up or down to accommodate the terrain. Another method, called the cut-cell method, cuts cells where they intersect the terrain. Here’s what these methods look like when we use them to represent an idealised, wave-shaped mountain:

Once we’ve chosen a mesh and stored temperature at cell centres and the wind at cell faces, we can start calculating a solution to the advection equation which enables us to forecast how the temperature will vary over time. We can solve the advection equation for every cell separately by discretising the advection equation. Let’s consider a cell with a north face and a south face. We want to know how the temperature stored at the cell centre, $T_\mathrm{cell}$, will vary over time. We can calculate this by reconstructing a continuous temperature field and using this to approximate temperature values at the north and south faces of the cell, $T_\mathrm{north}$ and $T_\mathrm{south}$,

$\frac{\partial T_\mathrm{cell}}{\partial t} = - \frac{T_\mathrm{north} - T_\mathrm{south}}{\Delta y} \times v$

where $\Delta y$ is the distance between the north and south cell faces. This is the same reconstruction process that we described earlier, only, instead of approximating temperatures using nearby weather station measurements, we are approximating temperatures using nearby cell centre values.

There are many existing numerical methods for solving the advection equation but many do not cope well when meshes are distorted, such as terrain-following meshes, or when cells have very different sizes, such as those cells in cut-cell meshes. Inaccurate solutions to the advection equation lead to inaccuracies in the weather forecast. In extreme cases, very poor solutions can cause the model software to crash, and this is known as a numerical instability.

We can see a numerical instability growing in this idealised example. A blob is being advected from left to right over a range of steep, wave-shaped mountains. This example is using a simple advection method which cannot cope with the distorted cells in this mesh.

We’ve developed a new method for solving the advection equation with almost any type of mesh using cubes or hexagons, terrain-following or cut-cell methods. The advection method works by reconstructing a continuous field from data stored at cell centre points. A separate reconstruction is made for every face of every cell in the mesh using about twelve nearby cell centre values. Given that weather forecast models have millions of cells, this sounds like an awful lot of calculations. But it turns out that we can make most of these calculations just once, store them, and reuse them for all our simulations.

Here’s the same idealised simulation using our new advection method. The results are numerically stable and accurate.