## Cloud-Radiation Interactions and Their Contributions to Convective Self-Aggregation

Convective self-aggregation is the process by which initially randomly scattered convection becomes spontaneously clustered in space despite uniform initial conditions. This process was first identified in numerical models, however it is relevant to real world convection (Holloway et al., 2017). Tropical weather is dominated by convection, and the degree of convective aggregation has important consequences for weather and climate. A more organised regime is associated with reduced cloudiness, increased longwave emission to space (Bretherton et al., 2005), and a higher frequency of long-lasting extreme precipitation events (Bao and Sherwood, 2019).

Because of its relevance to weather and climate, self-aggregation has been the focus of many recent studies. However, there is still much debate as to the processes that cause aggregation. There is great variability in the rate and degree of aggregation between models, and there remains uncertainty as to how aggregation is affected by climate change (Wing et al., 2020). Previous studies have shown that feedbacks between convection and shortwave & longwave radiation are key drivers and maintainers of aggregation (e.g. Wing & Cronin 2016), and that interactive radiation in models is essential for aggregation to occur (Muller & Bony 2015).

This blog summarises results from the first paper from my PhD (Pope et al., 2021), where we develop and use a framework to analyse how radiative interactions with different cloud types contribute to aggregation. We analyse self-aggregation within a set of three idealised simulations of the UK Met Office Unified Model (UM). The simulations are configured in radiative-convective equilibrium over three fixed sea surface temperatures (SSTs) of 295, 300 and 305 K. They are convection permitting models that are 432 × 6048 km2 in size with a 3 km horizontal grid spacing. The simulations neglect the earth’s rotation, so they approximately represent convection over tropical oceans within a warming climate.

Our analysis framework is based on that used in Wing and Emanuel (2014) which uses the variance of vertically-integrated frozen moist static energy (FMSE) as a measure of aggregation. FMSE is a measure of the total energy an air parcel has if all the water (vapour and frozen) was converted to liquid, neglecting its velocity. Variations in vertically-integrated FMSE come from perturbations in temperature and humidity. As aggregation increases, moist regions get moister and dry regions get drier, so the variance of vertically-integrated FMSE increases.

The problem with using FMSE variance as an aggregation metric is that it is highly sensitive to SST. A warmer atmosphere can hold more water vapour via the Clausius-Clapeyron relationship. This means there is a greater difference in FMSE between the moist and dry regions for higher-SST simulations, so the variance of FMSE is typically much greater for higher SSTs. To account for this problem, we normalise FMSE between hypothetical upper and lower limits which are functions of SST. This gives a value of normalised FMSE between 0 and 1.

Wing and Emanuel (2014) derive a budget equation for the rate of change of FMSE variance which shows how different processes contribute to aggregation. By rederiving their equation for normalised FMSE , we get:

$\displaystyle \frac{1}{2}\frac{\partial\widehat{h'}_n^2}{\partial t} = \widehat{h'}_nLW'_n + \widehat{h'}_nSW'_n + \widehat{h'}_nSEF'_n - \widehat{h'}_n\nabla_h\cdot\widehat{\textbf{u}h_n}$

where $\widehat{h}$ is vertically-integrated FMSE, $LW$ and $SW$ are the net atmospheric column longwave and shortwave heating rates, $SEF$ is the surface enthalpy flux, made up of the surface latent and sensible heat fluxes, and $\nabla h \cdot \widehat{\textbf{u}h}$ is the horizontal divergence of the $\widehat{h}$ flux. Primes ($'$) indicate local anomalies from the instantaneous domain mean. The subscript ($_n$) denotes a normalised variable which is the original variable divided by the difference between the hypothetical upper and lower limits of $\widehat{h}$. The equation shows that the rate of change of $\widehat{h'}_n$ variance (left hand side term) is driven by interactions between $\widehat{h}_n$ anomalies and anomalies in normalised net longwave heating, shortwave heating, surface fluxes and advection.

We use the variance of $\widehat{h}_n$ as our aggregation metric. Hovmöller plots of $\widehat{h}_n$ are shown in Figure 1 for each of our SSTs. In these plots, $\widehat{h}_n$ is averaged along the short axis of our domains. The plots show how initially randomly-distributed convection organises into bands which expand until the point where there are 4 to 5 quasi-stationary bands of moist convective regions separated by dry subsiding regions. This demonstrates that once our domains become fully-aggregated, the degree of aggregation appears similar. Figure 2a shows time series of each of the variance of $\widehat{h}_n$, and shows that the variance of non-normalised $\widehat{h}_n$ is ~4 times greater for our 305 K simulations compared to our 295 K simulation. Figure 2b shows time series of the variance of $\widehat{h}_n$. From this, we can see the convection aggregates faster as SST increases, yet the degree of aggregation remains similar via this metric once the convection is fully aggregated. Values of $\widehat{h}_n$ variance around 10-4 or lower correspond to randomly scattered convection, whereas values greater that 10-3 are associated with strongly aggregated convection.

To understand the processes contributing to aggregation, we have to look to Equation 1. We mainly focus on the two radiative terms on the right hand side. The terms show that regions in which the radiative anomalies and the $\widehat{h}_n$ anomalies have the same sign contribute to aggregation. We can start to get an intuitive understanding of this concept by looking at maps of these variables. Figure 3b-d show maps of $\widehat{h'}_n$, $LW'$ and $SW'$. We can see $SW'$ and $\widehat{h'}$ are closely correlated since $SW'$ is mainly determined by the shortwave absorption by water vapour. Clouds have little effect on the shortwave heating rates, with ~90% of the shortwave heating rate in cloudy regions being due to absorption by water vapour. $LW'$ is closely linked to cloud condensed water path (Figure 3a). This is because the majority of our clouds are high-topped clouds which, due to their cold cloud tops, are able to prevent longwave radiation escaping to space, so they are associated with positive longwave heating anomalies.

The sensitivity of the budget terms to both aggregation and SST can be seen in Figure 4. This figure is made by creating 50 bins of $\widehat{h}_n$ variance and then averaging the budget terms in space and time for each bin and for each SST. Where the terms are positive, they are helping to increase aggregation. Where they are negative, the terms are opposing aggregation. The terms tend to increase in magnitude since every term has $\widehat{h'}_n$ as a factor, which increases with aggregation by definition.

In general, we find the longwave term is the dominant driver of aggregation, being insensitive to SST during the growth phase of aggregation. Once the aggregation is mature, the longwave term remains the dominant maintainer of aggregation, however its contribution to aggregation maintenance decreases with SST. The shortwave term is initially small at early times but becomes a key maintainer of aggregation within highly-aggregated environments. This is because humidity variations are initially small, so there is little variation in shortwave heating. Once the convection is aggregated, moist regions are very moist and dry regions are very dry, so there is a large difference in shortwave heating between moist and dry regions. The variations in shortwave heating remain very similar with SST, meaning shortwave heating anomalies contribute the same amount to non-normalised $\widehat{h}$ variance. Therefore, shortwave heating contributes less to aggregation at higher SSTs because they contribute to a smaller fraction of $\widehat{h}$ anomalies. The radiative terms are balanced by the surface flux term (negative because there is greater evaporation in dry regions) and the advection term (negative because circulations tend to smooth out $\widehat{h'}_n$ gradients). The decrease in the magnitude of the radiative terms with SST is balanced by the surface flux and advection terms becoming more positive with SST.

To understand the behaviour of the longwave term, we define different cloud types based on the vertical profile of cloud, assigning one cloud type per grid box in a similar way to Hill et al. (2018). We define a lower and upper level pressure threshold, assigning cloud below the lower threshold to a “Low” category, cloud above the upper threshold to a “High” category, and cloud in between to a “Mid” category. If cloud occurs in more than one of these layers, then it is assigned to a combined category. In total, there are eight cloud types: Clear, Low, Mid, Mid & Low, High, High & Low, High & Mid, and Deep. We can then find each cloud type’s contribution to the longwave term by multiplying the cloud’s mean [Equation] covariance by its domain fraction.

To see how the cloud type contributions change with aggregation, we define a Growth phase and Mature phase of aggregation. The Growth phase has $\widehat{h}_n$ variance between $3\times10^{-4}$ and $4\times10^{-4}$ and the Mature phase has $\widehat{h}$ variance between $1.5\times10^{-3}$ and $2\times 10^{-3}$. The contribution of longwave interactions with each cloud type to aggregation during these two phases is shown in Figure 5a, with their mean $LW'\times\widehat{h'}$ covariance and fraction shown in Figures 5b & c.

We find that longwave interactions with high-topped clouds and clear regions drive aggregation during the Growth phase (Figure 5a). This is because high clouds are abundant, have positive longwave heating anomalies and occur in moist, high $\widehat{h}$ environments. The clear regions are the most abundant category, have typically negative longwave heating anomalies and tend to occur in low $\widehat{h}$ regions, so their $LW'\times\widehat{h'}$ covariance is positive. During the Growth phase, there is little SST sensitivity within each category. During the Mature phase, longwave interactions with high-topped cloud remain the main maintainer of aggregation however their contribution decreases with SST. This sensitivity is mainly because there is a greater decrease in high-topped cloud fraction with aggregation as SST increases. This also has consequences for the $LW'\times\widehat{h'}$ covariance of the clear regions. As high-topped cloud fraction reduces, the domain-mean longwave cooling increases. This makes the radiative cooling of the clear regions less anomalous, resulting in an increasingly negative $LW'\times\widehat{h'}$ covariance during the Mature phase as SST increases.

There is great variability in the degrees of aggregation within numerical models, which has important consequences for weather and climate modelling (Wing et al. 2020). With cloud-radiation interactions being crucial for aggregation, understanding how these interactions vary between models may help to explain the differences in aggregation. This study provides a framework by which a comparison of cloud-radiation interactions and their contributions to convective self-aggregation between models and SSTs can be achieved.

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REFERENCES

Bao, J., & Sherwood, S. C. (2019). The role of convective self-aggregation in extreme instantaneous versus daily precipitation. Journal of Advances in Modeling Earth Systems11(1), 19– 33. https://doi.org/10.1029/2018MS001503

Bretherton, C. S., Blossey, P. N., & Khairoutdinov, M. (2005). An energy-balance analysis of deep convective self-aggregation above uniform SST. Journal of the Atmospheric Sciences62(12), 4273– 4292. https://doi.org/10.1175/JAS3614.1

Hill, P. G., Allan, R. P., Chiu, J. C., Bodas-Salcedo, A., & Knippertz, P. (2018). Quantifying the contribution of different cloud types to the radiation budget in Southern West Africa. Journal of Climate31(13), 5273– 5291. https://doi.org/10.1175/JCLI-D-17-0586.1

Holloway, C. E., Wing, A. A., Bony, S., Muller, C., Masunaga, H., L’Ecuyer, T. S., & Zuidema, P. (2017). Observing convective aggregation. Surveys in Geophysics38(6), 1199– 1236. https://doi.org/10.1007/s10712-017-9419-1

Muller, C., & Bony, S. (2015). What favors convective aggregation and why? Geophysical Research Letters42(13), 5626– 5634. https://doi.org/10.1002/2015GL064260

Pope, K. N., Holloway, C. E., Jones, T. R., & Stein, T. H. M. (2021). Cloud-radiation interactions and their contributions to convective self-aggregation. Journal of Advances in Modeling Earth Systems13, e2021MS002535. https://doi.org/10.1029/2021MS002535

Wing, A. A., & Cronin, T. W. (2016). Self-aggregation of convection in long channel geometry. Quarterly Journal of the Royal Meteorological Society142(694), 1– 15. https://doi.org/10.1002/qj.2628

Wing, A. A., & Emanuel, K. A. (2014). Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations. Journal of Advances in Modeling Earth Systems6(1), 59– 74. https://doi.org/10.1002/2013MS000269

Wing, A. A., Stauffer, C. L., Becker, T., Reed, K. A., Ahn, M.-S., Arnold, N., & Silvers, L. (2020). Clouds and convective self-aggregation in a multi-model ensemble of radiative-convective equilibrium simulations. Journal of Advances in Modeling Earth Systems12(9), e2020MS0021380. https://doi.org/10.1029/2020MS0021380

## Main challenges for extreme heat risk communication

For my PhD, I research heatwaves and heat stress, with a focus on the African continent. Here I show what the main challenges are for communicating heatwave impacts inspired by a presentation given by Roop Singh of the Red Cross Climate Center at Understanding Risk Forum 2020.

There is no universal definition of heatwaves

Having no agreed definition of a heatwave (also known as extreme heat events) is a huge challenge in communicating risk. However, there is a guideline definition by the World Meteorological Organisation and for the UK an agreed definition as of 2019. In simple terms a heatwave is:

“A period of above average temperatures of 3 or more days in a region’s warm season (i.e. all year in the tropics and in the summer season elsewhere)”

We then have heat stress which is an impact of heatwaves, and is the killer aspect of heat. Heat stress is:

“Build-up of body heat as a result of exertion or external environment”(McGregor, 2018)

Attention Deficit

Heatwaves receive low attention in comparison to other natural hazards I.e., Flooding, one of the easiest ways to appreciate this attention deficit is through Google search trends. If we compare ‘heat wave’ to ‘flood’ both designated as disaster search types, you can see that a larger proportion of searches over time are for ‘flood’ in comparison to ‘heat wave’.

On average flood has 28% search interest which is over 10 times the amount of interest for heat wave. And this is despite Heatwaves being named the deadliest hydro-meteorological hazard from 2015-2019 by the World Meteorological Organization. Attention is important if someone can remember an event and its impacts easily, they can associate this with the likelihood of it happening. This is known as the availability bias and plays a key role in risk perception.

Lack of Research and Funding

One impact of the attention deficit on extreme heat risk, is there is not ample research and funding on the topic – it’s very patchy. Let’s consider a keyword search of academic papers for ‘heatwave*’ and ‘flood*’ from Scopus an academic database.

Research on floods is over 100 times bigger in quantity than heatwaves. This is like what we find for google searches and the attention deficit, and reveals a research bias amongst these hydro-meteorological hazards. And is mirrored by what my research finds for the UK, much more research on floods in comparison to heatwaves (https://doi.org/10.1016/j.envsci.2020.10.021). Our paper is the first for the UK to assess the barriers, causes and solutions for providing adequate research and policy for heatwaves. The motivation behind the paper came from an assignment I did during my masters focusing on UK heatwave policy, where I began to realise how little we in the UK are prepared for these events, which links up nicely with my PhD. For more information you can see my article and press release on the same topic.

Heat is an invisible risk

Heatwaves are not something we can touch and like Climate Change, they are not ‘lickable’ or visible. This makes it incredibly difficult for us to perceive them as a risk. And this is compounded by the attention deficit; in the UK most people see heatwaves as a ‘BBQ summer’ or an opportunity to go wild swimming or go to the beach.

And that’s really nice, but someone’s granny could be experiencing hospitalising heat stress in a top floor flat as a result of overheating that could result in their death. Or for example signal failures on your railway line as a result of heat could prevent you from getting into work, meaning you lose out on pay. I even know someone who got air lifted from the Lake District in their youth as a result of heat stress.

A quote from a BBC one program on wild weather in 2020 sums up overheating in homes nicely:

“It is illegal to leave your dog in a car to overheat in these temperatures in the UK, why is it legal for people to overheat in homes at these temperatures

For Africa the perception amongst many is ‘Africa is hot’ so heatwaves are not a risk, because they are ‘used to exposure’ to high temperatures. First, not all of Africa is always hot, that is in the same realm of thinking as the lyrics of the 1984 Band Aid Single. Second, there is not a lot of evidence, with many global papers missing out Africa due to a lack of data. But, there is research on heatwaves and we have evidence they do raise death rates in Africa (research mostly for the West Sahel, for example Burkina Faso) amongst other impacts including decreased crop yields.

What’s the solution?

Talk about heatwaves and their impacts. This sounds really simple, but I’ve noticed a tendency of a proportion of climate scientists to talk about record breaking temperatures and never mention land heatwaves (For example the Royal Institute Christmas Lectures 2020). Some even make a wild leap from temperature straight to flooding, which is just painful for me as a heatwave researcher.

So let’s start by talking about heatwaves, heat stress and their impacts.

## The Scandinavia-Greenland Pattern: something to look out for this winter

The February-March 2018 European cold-wave, known widely as “The Beast from the East” occurred around 2 weeks after a major sudden stratospheric warming (SSW) event on February 12th. Major SSWs typically occur once every other winter, involving significant disruption to the stratospheric polar vortex (a planetary-scale cyclone which resides over the pole in winter). SSWs are important because their occurrence can influence the type and predictability of surface weather on longer timescales of between 2 weeks to 2 months. This is known as subseasonal-to-seasonal (S2S) predictability, and “bridges the gap” between typical weather forecasts and seasonal forecasts (Figure 1).

Figure 1: Schematic of medium-range, S2S and seasonal forecasts and their relative skill. [Figure 1 in White et al. (2017)]

In general, S2S forecasts suffer from relatively low skill. While medium-range forecasts are an initial value problem (depending largely on the initial conditions of the forecast) and seasonal forecasts are a boundary value problem (depending on slowly-varying constraints to the predictions, such as the El Niño-Southern Oscillation), S2S forecasts lie somewhere between the two. However, certain “windows of opportunity” can occur that have the potential to increase S2S skill – and a major SSW is one of them. Skilful S2S forecasts can be of particular benefit to public health planners, the transport sector, and energy demand management, among many others.

Following an SSW, the eddy-driven jet stream tends to weaken and shift equatorward. This is characteristic of the negative North Atlantic Oscillation (NAO) and negative Arctic Oscillation (AO), and during these patterns the risk of cold air outbreaks significantly increases in places like northwest Europe. So, by knowing this, S2S forecasts issued during the major SSW were able to highlight the increased risk of severely cold weather.

Given that we know that following an SSW certain weather types are more likely for several weeks, and forecasts may be more skilful, it might seem advantageous to know an SSW was coming at a long lead-time in order to really push the boundaries of S2S prediction. So, what about in 2018?

In the first paper from my PhD, published in July 2019 in JGR-Atmospheres, we explored the onset of predictions of the February 2018 SSW. We found that, until about 12 days beforehand, extended-range forecasts that contribute to the S2S database (an international collaboration of extended-range forecast data) did not accurately predict the event; in fact, most predictions indicated the vortex would remain unusually strong!

We diagnosed that anticyclonic wave breaking in the North Atlantic was a crucial synoptic-scale “trigger” event for perturbing the stratospheric vortex, by enhancing vertically propagating Rossby waves (which weaken the vortex when they break in the stratosphere). Forecasts struggled to predict this event far in advance, and thus struggled to predict the SSW. We called the pattern the “Scandinavia-Greenland (S-G) dipole” – characterised by an anticyclone over Scandinavia and a low over Greenland (Figure 2), and we found it was present before 35% of previous SSWs (1979-2018). The result agrees with several previous studies highlighting the role of blocking in the Scandinavia-Urals region, but was the first to suggest such a significant impact of a single tropospheric event.

Figure 2: Correlation between mean sea level pressure forecasts over 3-5 February 2018 and subsequent forecasts of 10 hPa 60°N zonal-mean zonal wind on 9-11 February, in (a) NCEP and (b) ECMWF ensembles launched between 29 January and 1 February 2018. White lines (dashed negative) indicate correlations exceeding +/- 0.7, while the black dashed lines indicate the nodes of the S-G dipole. [Figure 3 in Lee et al. (2019)]

So, we had established the S-G dipole was important in the predictability onset in 2018, and important in previous cases – but how well do S2S models generally capture the pattern?

That was the subject of our recent (open-access) paper, published in August in QJRMS. We define a more generalised pattern by performing empirical orthogonal function (EOF) analysis on mean sea-level pressure anomalies in a region of the northeast Atlantic during November-March in ERA5 reanalysis (Figure 3).  While the leading EOF (the “zonal pattern”) resembles the NAO, the 2nd EOF resembles the S-G dipole from our previous paper – so we call it the “S-G pattern”.

Figure 3: The first two leading EOFs of MSLP anomalies in the northeast Atlantic during November-March in ERA5, expressed as hPa per standard deviation of the principal component timeseries. The percentage of variance explained by the EOF is also shown. [Figure 1 in Lee et al. (2020)

We then establish, through lagged linear regression analysis, that the S-G pattern is associated with enhanced vertically propagating wave activity (measured by zonal-mean eddy heat flux) into the stratosphere, and a subsequently weakened stratospheric vortex for the next 2 months. Thus, it supports our earlier work, and motivates considering how the pattern is represented in S2S models. To do this, we look at hindcasts – forecasts initialised for dates in the past – from 10 different prediction systems from around the world.

We find that while all the S2S models represent the spatial pattern of these two EOFs very well, some have biases in the variance explained by the EOFs, particularly at weeks 3 and 4 (Figure 4). Broadly, all the models have more variance explained by their first EOF compared with ERA5, and less by the second EOF – but this bias is particularly large for the three models with the lowest horizontal resolution (BoM, CMA, and HMCR).

Figure 4: Weekly-mean ratio between the variance explained by the EOFs in each model and the ERA5 EOF. [Figure 6 in Lee et al. (2020)]

Additionally, we find that the deterministic prediction skill in the S-G pattern (measured by the ensemble-mean correlation) can be as small as 5-6 days for the BoM model – and only as high as 11 days in the higher resolution models. Extending this to probabilistic skill in weeks 3 and 4, we find models have only limited (if any) skill above climatology in weeks 3 and 4 (and much less than the skill in the leading EOF, the NAO-like pattern).

Furthermore, we find that the relationship between the S-G pattern and the enhanced heat flux in the stratosphere decays with lead-time in most S2S models, even in the higher-resolution models (Figure 5). Thus, this suggests that the dynamical link between the troposphere and stratosphere weakens with lead time in these models – so even a correct tropospheric prediction may not, in these cases, have a subsequently accurate extended-range stratospheric forecast.

Figure 5: Weekly mean regression coefficients between the S–G index and the corresponding eddy heat flux anomalies at (a) 300 hPa on the same day, (b) 100 hPa three days later, and (c) 50 hPa four days later. The lags correspond to days with maximum correlation in ERA5. Stippled bars indicate a significant difference from ERA5 at the 95% confidence level. [Figure 11 in Lee et al. (2020)]

So, when taking this all together, we have:

• The S-G pattern is the second-leading mode of MSLP variability in the northeast Atlantic during winter.
• It is associated with enhanced vertically propagating wave activity into the stratosphere and a weakened polar vortex in the following weeks to months.
• S2S models represent the spatial patterns of the two leading EOFs well.
• Most S2S models have a zonal variability bias, with relatively more variance explained by the leading EOF and correspondingly less in the second EOF.
• This bias is largest in the lowest-resolution models in weeks 3 and 4.
• Extended range skill in the S-G pattern is low, and lower than for the NAO-like zonal pattern.
• The linear relationship between the S-G pattern and eddy heat flux in the stratosphere decays with lead-time in most S2S models.

The zonal variance bias is consistent with S2S model biases in Rossby wave breaking and blocking, while these biases have been widely found to be largest in the lowest resolution models. The results suggest that the poor prediction of the S-G event in February 2018 is not unique to that case, but a more generic issue. Overall, the combination of the variability biases, the poor extended-range predictability, and the poor representation of its impact on the stratospheric vortex at longer lead-times likely contributes to limiting skill at predicting major SSWs on S2S timescales – which remains low, despite the stratosphere’s much longer timescales. Correcting the biases outlined here will likely contribute to improving this skill, and subsequently increasing how far we are able to predict real-world weather.

## How Important are Post-Tropical Cyclones to European Windstorm Risk?

To date, the 2020 North Atlantic hurricane season has been the most active on record, producing 20 named storms, 7 hurricanes, and a major hurricane which caused \$9 billion in damages across the southern United States. With the potential for such destructive storms, it is understandable that a large amount of attention is paid to the North Atlantic basin at this time of year. Whilst hurricanes have been known to cause devastation in the tropics for centuries, until recently there was little appreciation for the destructive potential of these systems across Europe.

As tropical cyclones such as hurricanes move poleward – away from the tropics and into regions of lower sea surface temperatures and higher vertical wind shear, they undergo a process called extratropical transition (Klein et al., 2000): Over a period of time, the cyclones change from symmetric, warm cored systems into asymmetric cold core systems fuelled by horizontal temperature gradients, as opposed to latent heat fluxes (Evans et al., 2017). These systems, so-called post-tropical cyclones (PTCs), often reintensify in the mid-latitude Atlantic with consequences for land masses downstream – often Europe. This was highlighted in 2017, when ex-hurricane Ophelia impacted Ireland, bringing with it the strongest winds Ireland had seen in 50 years (Stewart, 2018). 3 people were killed, and 360,000 homes were without power.

In a recent paper, we quantify the risk associated with PTCs across Europe relative to mid-latitude cyclones (MLCs) for the first time – in terms of both the absolute risk (i.e. what fraction of high impact wind events across Europe are caused by PTCs?) and also the relative risk (for a given PTC, how likely is it to be associated with high-impact winds, and how does this compare to a given MLC?). By tracking all cyclones impacting a European domain (36-70N, 10W-30E) in the ERA5 reanalysis (1979-2017) using a feature tracking algorithm (Hodges, 1994, 1995, 1999), we identify the post-tropical cyclones using spatiotemporal matching (Hodges et al., 2017) with the observational record, IBTrACS (Knapp et al., 2010).

Figure 1 shows the distributions of maximum intensity for PTCs and MLCs across the entire European domain (top), Northern Europe (48-70°N, 10°W-30°E; middle) and Southern Europe (36-48°N, 10°W-30°E; bottom), using all cyclone tracks all year round. The distribution of PTC maximum intensities is higher (in terms of both wind speed and MSLP) than MLCs, particularly across Northern Europe. The difference between the maximum intensity distributions of PTCs and MLCs across Northern Europe is statistically significant (99%). PTCs are also present in the highest of intensity bins, indicating that the strongest PTCs have intensities comparable to strong wintertime MLCs.

Whilst Figure 1 shows that PTCs are stronger than MLCs even when considering MLCs forming all year round (including the often much stronger wintertime MLCs), it is also useful to compare the risks posed by PTCs relative to MLCs forming at the same time of the year – during the North Atlantic hurricane season (June 1st-November 30th).

Figure 2 shows the fraction of all storms, binned by their maximum intensity in their respective domains, which are PTCs. For storms with weak-moderate maximum winds (first three bins in the figure), <1% of such events are caused by PTCs (with the remaining 99% caused by MLCs). For stronger storms, particularly those of storm force (>25 ms-1 on the Beaufort scale), this percentage is much higher. Despite less than 1% of all storms impacting Northern Europe during hurricane season being PTCs, almost 9% of all storms with storm-force winds which impact the region are PTCs, suggesting that a disproportionate fraction of high-impact windstorms are PTCs. 8.2% of all Northern Europe impacting PTCs which form during hurricane season impact the region with storm-force winds. This fraction is only 0.8% for MLCs, suggesting that the fraction of PTCs impacting Northern Europe with storm-force winds is approximately 10 times greater than MLCs.

Here we have shown that PTCs, at their maximum intensity over Northern Europe, are stronger than MLCs. However, the question still remains as to why this is the case. Warm-seclusion storms post-extratropical transition have been shown to have the fastest rates of reintensification (Kofron et al., 2010) and typically have the lowest pressures upon impacting Europe (Dekker et al., 2018). Given the climatological track that PTCs often take over the warm waters of the Gulf stream, along with the contribution of both baroclinic instability and latent heat release for warm-seclusion development (Baatsen et al., 2015), one hypothesis may be that PTCs are more likely to develop into warm seclusion storms than the broader class of mid-latitude cyclones, potentially explaining the disproportionate impacts they cause across Europe. This will be the topic of future work.

Despite PTCs disproportionately impacting Europe with high intensities, they are a relatively small component of the total cyclone risk in the current climate. However, only small changes are expected in MLC number and intensity under RCP 4.5 (Zappa et al., 2013). Conversely, the number of hurricane-force (>32.6 ms-1) storms impacting Norway, the North Sea and the Gulf of Biscay has been projected to increases by a factor of 6.5, virtually all of which originate in the tropics (Haarsma et al., 2013). Whilst the absolute contribution of PTCs to hurricane season windstorm risk is currently low, PTCs may make an increasingly significant contribution to European windstorm risk in a future climate.

Interested to read more? Read our paper, published in Geophysical Research Letters.

Sainsbury, E. M., R. K. H. Schiemann, K. I. Hodges, L. C. Shaffrey, A. J. Baker, K. T. Bhatia, 2020: How Important Are Post‐Tropical Cyclones for European Windstorm Risk? Geophysical Research Letters, 47(18), e2020GL089853, https://doi.org/10.1029/2020GL089853

References

Baatsen, M., Haarsma, R. J., Van Delden, A. J., & de Vries, H. (2015). Severe Autumn storms in future Western Europe with a warmer Atlantic Ocean. Climate Dynamics, 45, 949–964. https://doi.org/10.1007/s00382-014-2329-8

Dekker, M. M., Haarsma, R. J., Vries, H. de, Baatsen, M., & Delden, A. J. va. (2018). Characteristics and development of European cyclones with tropical origin in reanalysis data. Climate Dynamics, 50(1), 445–455. https://doi.org/10.1007/s00382-017-3619-8

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Hodges, K., Cobb, A., & Vidale, P. L. (2017). How well are tropical cyclones represented in reanalysis datasets? Journal of Climate, 30(14), 5243–5264. https://doi.org/10.1175/JCLI-D-16-0557.1

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## Exploring the impact of variable floe size on the Arctic sea ice

The Arctic sea ice cover is made up of discrete units of sea ice area called floes. The size of these floes has an impact on several sea ice processes including the volume of melt produced at floe edges, the momentum exchange between the sea ice, ocean, and atmosphere, and the mechanical response of the sea ice to stress. Models of the sea ice have traditionally assumed that floes adopt a uniform size, if floe size is explicitly represented at all in the model. Observations of floes show that floe size can span a huge range, from scales of metres to tens of kilometres. Generally, observations of the floe size distribution (FSD) are fitted to a power law or a combination of power laws (Stern et al., 2018a).

The Los Alamos sea ice model, hereafter referred to as CICE, usually assumes a fixed floe size of 300 m. We can impose a simple FSD model into CICE derived from a power law to explore the impact of variable floe size on the sea ice cover. Figure 1 is a diagram of the WIPoFSD model (Waves-in-Ice module and Power law Floe Size Distribution model), which assumes a power law with a fixed exponent, $\alpha$, between a lower floe size cut-off, $d_{min}$, and an upper floe size cut-off, $d_{max}$. The model also incorporates a floe size variable, $l_{var}$, to capture the effects of processes that can influence floe size. The processes represented are wave break-up of floes, melting at the floe edge, winter floe growth, and advection. The model includes a wave advection and attenuation scheme so that wave properties can be determined within the sea ice field to enable the identification of wave break-up events. Full details of the WIPoFSD model and its implementation into CICE are available in Bateson et al. (2020). For the WIPoFSD model setup considered here, we explore the impact of the FSD on the lateral melt rate, which is the melt rate at the edge surfaces of floes. It is useful to define a new FSD metric that can be used to characterise the impact of the FSD on lateral melt. To do this we note that the lateral melt volume produced by a floe is proportional to the perimeter of the floe. The effective floe size, $l_{eff}$, is defined as a fixed floe size that would produce the same lateral melt rate as a given FSD, for a fixed total sea ice area.

Here we will compare a CICE simulation incorporating the WIPoFSD model, hereafter referred to as stan-fsd, to a reference case, ref, using the CICE standard fixed floe size of 300 m. For the WIPoFSD model, $d_{min}$ = 10 m, $d_{max}$ = 30 km, and $\alpha$ = -2.5. These values have been selected as representative values from observations. The reference setup is initiated in 1990 and spun-up until 2005, when either continued as ref or the WIPoFSD model imposed for stan-fsd before being evaluated from 2006 – 2016. All figures in this post are given as a mean over 2007 – 2016, such that 2005 – 2006 is a period of spin-up for the incorporated WIPoFSD model.

In Figure 2, we show the percentage reduction in the Arctic sea ice extent and volume of stan-fsd relative to ref. The differences in both extent and volume over the pan-Arctic scale evolve over an annual cycle, with maximum differences of -1.0 % in August and -1.1 % in September respectively. The annual cycle corresponds to periods of melting and freeze-up and is a product of the nature of the imposed FSD. Lateral melt rates are a function of floe size, but freeze-up rates are not, hence model differences only increase during periods of melting and not during periods of freeze-up. The difference in sea ice extent reduces rapidly during freeze-up because this freeze-up is predominantly driven by ocean surface properties, which are strongly coupled to atmospheric conditions in areas of low sea ice extent. In comparison, whilst atmospheric conditions initiate the vertical sea ice growth, this atmosphere-ocean coupling is rapidly lost due to insulation of the warmer ocean from the cooler atmosphere once sea ice extends across the horizontal plane. Hence a residual difference in sea ice thickness and therefore volume propagates throughout the winter season. The interannual variability shows that the impact of the WIPoFSD model with standard parameters varies significantly depending on the year.

Although the pan-Arctic differences in extent and volume shown in Figure 2 are marginal, differences are larger when considering smaller spatial scales. Figure 3 shows the spatial distribution in the changes in sea ice concentration and thickness in March, June, and September for stan-fsd relative to ref in addition to the spatial distribution in $l_{eff}$ for stan-fsd for the same months. Reductions in the sea ice concentration and thickness of up to 0.1 and 50 cm observed respectively in the September marginal ice zone (MIZ). Within the pack ice, increases in the sea ice concentration of up to 0.05 and ice thickness of up to 10 cm can be seen. To understand the non-uniform spatial impacts of the FSD, it is useful to look at the behaviour of $l_{eff}$. Regions with an $l_{eff}$ greater than 300 m will experience less lateral melt than the equivalent location in ref (all other things being equal) whereas locations with an $l_{eff}$ below 300 m will experience more lateral melt. In Figure 3 we see the transition to values of $l_{eff}$ smaller than 300 m in the MIZ, hence most of the sea ice cover experiences less lateral melting for stan-fsd compared to ref.

For Figures 2-3, the parameters used to define the FSD have been set to fixed, standard values. However, these parameters vary significantly between different observed FSDs. It is therefore useful to explore the model sensitivity to these parameters. For α values of -2, -2.5, -3 and -3.5 have been selected to span the general range of values reported in observations (Stern et al., 2018a). For $d_{min}$ values of 1 m, 20 m and 50 m are selected to reflect the different behaviours reported in studies, with some showing power law behaviour extending to 1 m (Toyota et al., 2006) and others showing a tailing off at an order of 10 s of metres (Stern et al., 2018b). For the upper cut-off, $d_{max}$, values of 1000 m, 10,000 m, 30,000 m and 50,000 m are selected, again to represent the distributions reported in different studies. 50 km is taken as the largest value for $d_{max}$ as this serves as an upper limit to what can be resolved within an individual grid cell on a CICE 1$^{\circ}$ grid. A total of 19 sensitivity studies have been completed used different permutations of the stated values for the FSD model parameters. Figure 4 shows the change in mean September sea ice extent and volume relative to ref plotted against mean annual $l_{eff}$, averaged over the sea ice extent, for each of these sensitivity studies. The impacts range from a small increase in extent and volume to large reductions of -22 % and -55 % respectively, even within the parameter space defined by observations. Furthermore, there is almost a one-to-one mapping between mean $l_{eff}$ and extent and volume reduction. This suggests $l_{eff}$ is a useful diagnostic tool to predict the impact of a given set of floe size parameters. The system varies most in response to the changes in the α, but it is also particularly sensitive to $d_{min}$.

There are several advantages to the assumption of a fixed power law in modelling the sea ice floe size distribution. It provides a simple framework to explore the potential impact of an observed FSD on the sea ice mass balance, given observations of the FSD are generally fitted to a power law. In addition, the use of a simple model makes it easier to constrain the mechanism of how the model changes the sea ice cover. However, there are also significant disadvantages including the high model sensitivity to poorly constrained parameters, as shown in Figure 4. In addition, there is evidence both that the exponent evolves over an annual cycle and is not a fixed value (Stern et al., 2018b) and that the power law is not a statistically valid description of the FSD over all floe sizes (Horvat et al., 2019). An alternative approach to modelling the FSD is the prognostic model of Roach et al. (2018, 2019). The prognostic model avoids any assumptions about the shape of the distribution and instead assigns sea ice area to a set of adjacent floe size categories, with individual processes parameterised at floe scale. This approach carries its own set of challenges. If important physical processes are missing from the model it will not be possible to simulate a physically realistic distribution. In addition, the prognostic model has a significant computational cost. In practice, the choice of FSD modelling approach will depend on the application.

Bateson, A. W., Feltham, D. L., Schröder, D., Hosekova, L., Ridley, J. K. and Aksenov, Y.: Impact of sea ice floe size distribution on seasonal fragmentation and melt of Arctic sea ice, Cryosphere, 14, 403–428, https://doi.org/10.5194/tc-14-403-2020, 2020.

Horvat, C., Roach, L. A., Tilling, R., Bitz, C. M., Fox-Kemper, B., Guider, C., Hill, K., Ridout, A., and Shepherd, A.: Estimating the sea ice floe size distribution using satellite altimetry: theory, climatology, and model comparison, The Cryosphere, 13, 2869–2885, https://doi.org/10.5194/tc-13-2869-2019, 2019.

Stern, H. L., Schweiger, A. J., Zhang, J., and Steele, M.: On reconciling disparate studies of the sea-ice floe size distribution, Elem. Sci. Anth., 6, p. 49, https://doi.org/10.1525/elementa.304, 2018a.

Stern, H. L., Schweiger, A. J., Stark, M., Zhang, J., Steele, M., and Hwang, B.: Seasonal evolution of the sea-ice floe size distribution in the Beaufort and Chukchi seas, Elem. Sci. Anth., 6, p. 48, https://doi.org/10.1525/elementa.305, 2018b.

Roach, L. A., Horvat, C., Dean, S. M., and Bitz, C. M.: An Emergent Sea Ice Floe Size Distribution in a Global Coupled Ocean-Sea Ice Model, J. Geophys. Res.-Oceans, 123, 4322–4337, https://doi.org/10.1029/2017JC013692, 2018.

Roach, L. A., Bitz, C. M., Horvat, C. and Dean, S. M.: Advances in Modeling Interactions Between Sea Ice and Ocean Surface Waves, J. Adv. Model. Earth Syst., 11, 4167–4181, https://doi.org/10.1029/2019MS001836, 2019.

Toyota, T., Takatsuji, S., and Nakayama, M.: Characteristics of sea ice floe size distribution in the seasonal ice zone, Geophys. Res. Lett., 33, 2–5, https://doi.org/10.1029/2005GL024556, 2006.

## How do ocean and atmospheric heat transports affect sea-ice extent?

Downward trends in Arctic sea-ice extent in recent decades are a striking signal of our warming planet. Loss of sea ice has major implications for future climate because it strongly influences the Earth’s energy budget and plays a dynamic role in the atmosphere and ocean circulation.

Comprehensive numerical models are used to make long-term projections of the future climate state under different greenhouse gas emission scenarios. They estimate that the Arctic ocean will become seasonally ice free by the end of the 21st century, but there is a large uncertainty on the timing due to the spread of estimates across models (Fig. 1).

What causes this spread, and how might it be reduced to better constrain future projections? There are various factors (Notz et al. 2016), but of interest to our work is the large-scale forcing of the atmosphere and ocean. The mean atmospheric circulation transports about 3 PW of heat from lower latitudes into the Arctic, and the oceans transport about a tenth of that (e.g. Trenberth and Fasullo, 2017; 1 PW = 1015 W). Our goal is to understand the relative roles of Ocean and Atmospheric Heat Transports (OHT, AHT) on long timescales. Specifically, how sensitive is the sea-ice cover to deviations in OHT and AHT, and what underlying mechanisms determine the sensitivities?

We developed a highly simplified Energy-Balance Model (EBM) of the climate system (Fig. 2)—it has only latitudinal variations and is described by a few simple equations relating energy transfer between the atmosphere, ocean, and sea ice (Aylmer et al. 2020). The latitude of the sea-ice edge is an analogue for ice extent in the real world. The simplicity of the EBM allows us to isolate the basic physics of the problem, which would not be possible going directly with the complex output of a full climate model.

We generated a set of simulations in which OHT varies and checked the response of the ice edge. This is a measure of the effective sensitivity of the ice cover to OHT (Fig. 3a)—it is not the actual sensitivity because AHT decreases (Fig. 3b), and we are really seeing in Fig. 3a the net response of the ice edge to changes in both OHT and AHT.

This reduction in AHT with increasing OHT is called Bjerknes compensation, and it occurs in full climate models too (Outten et al. 2018). Here, it has a moderating effect on the true impact of increasing OHT. With further analysis, we determined the actual sensitivity to be about 1.5 times the effective sensitivity. The actual sensitivity of the ice edge to AHT turns out to be about half that to the OHT.

What sets the difference in OHT and AHT sensitivities? This is easily answered within the EBM framework. We derived a general expression for the ratio of (actual) ice-edge sensitivities to OHT (so) and AHT (sa):

A higher-order term has been neglected for simplicity here, but the basic point remains: the ratio of sensitivities mainly depends on the parameters BOLR and Bdown. These are bulk representations of atmospheric feedbacks and determine the efficiency of outgoing and downwelling longwave radiation, respectively. They are always positive, so the ice edge is always more sensitive to OHT than AHT.

The interpretation of this equation is simple. AHT converging over the ice pack can either be transferred to the underlying sea ice, or radiated to space, having no impact on the ice, and the partitioning is controlled by Bdown and BOLR. The same amount of OHT converging under the ice pack can only go through the ice and is thus the more efficient driver.

Climate models with larger OHTs tend to have less sea ice (Mahlstein and Knutti, 2011). We have also found strong correlations between OHT and the sea-ice edge in several of the models listed in Fig. 1 individually. Ice-edge sensitivities and B values can be determined per model, and our equation predicts how these should be related. Our work thus provides a way to investigate how much physical biases in OHT and AHT contribute to sea-ice-projection uncertainties.

## An inter-comparison of Arctic synoptic scale storms between four global reanalysis datasets

The Arctic has changed a lot over the last four decades. Arctic September sea ice extent has decreased rapidly from 1980-present by approximately 3.4 million square-kilometres (see Figure 1). This has made the Arctic more accessible for human activities such as shipping, oil exploration and tourism. As Arctic sea ice is expected to continue to decline in the future, human activity in the Arctic is expected to continue to increase. This will increase the exposure to hazardous weather conditions, such as high winds and high waves, which are associated with Arctic storms. However, the characteristics of Arctic storms are currently not well understood.

One way to investigate current Arctic storm characteristics is to analyse storms in global reanalysis datasets. Reanalysis datasets combine past observations with current weather models to produce spatially and temporally homogeneous datasets, that contain atmospheric data at grid-points around the world at constant time intervals (typically every 6-hours) per day from 1979-present (for the modern, satellite-era reanalyses). Typically, a storm tracking algorithm is used to efficiently process all of the 6-hour data in the reanalysis datasets from 1979 (60,088 time steps!) to identify all of the storms that may have occurred in the past. Storms can be identified in the mean sea level pressure (MSLP) field (as low pressure systems), or in the relative vorticity field (as large rotating systems). The relative vorticity field at 850 hPa (higher in the atmosphere than the atmospheric boundary layer) is typically used so that the field is less influenced by boundary layer processes that may produce areas of high relative vorticity.

At the moment, atmospheric scientists are spoilt for choice when it comes to choosing a reanalysis dataset to analyse. There are reanalysis datasets from multiple institutions; the European Centre for Medium Range Weather Forecasts (ECMWF), the Japanese Meteorological Agency (JMA), the National Aeronautics and Space Administration (NASA), and the National Centers for Environmental Prediction (NCEP). Each institution has created their reanalysis dataset in a slightly different way, by using their own numerical weather prediction model and data assimilation systems. Atmospheric scientists also have to choose whether to use the MSLP field or 850 hPa relative vorticity field when applying their storm tracking algorithm to the reanalysis datasets.

In my recent paper, I aimed to assess Arctic storm characteristics in the multiple reanalysis datasets currently available (ERA-Interim, JRA-55, MERRA-2 and NCEP-CFSR), using a storm tracking algorithm based on 850 hPa relative vorticity and MSLP fields. Below is a short summary of some of the results from the paper.

Despite the Arctic environment changing dramatically over the last four decades, we find that there has been no change in the frequency and intensity of Arctic storms in all the reanalysis datasets compared in this study. It was found in preceding, older versions of atmospheric reanalysis datasets that Arctic storm frequency had increased from 1949-2002 (Walsh. 2008 and Sepp & Jaagus. 2011). This is in contrast with results from the modern reanalysis datasets (from this study, and Simmonds et al. 2008, Serreze and Barrett. 2008 and Zahn et al. 2018) which show no increase in Arctic storm frequency.

Across all the reanalysis datasets, some robust characteristics of Arctic storms were found. For example, the spatial distribution of Arctic storms is found to be seasonally dependent. In winter (DJF), Arctic storm track density is highest over the Greenland, Norwegian and Barents Seas, whereas in summer (JJA), Arctic storm track density is highest over and north of the Eurasia coastline (a region known as the Arctic Frontal Zone (Reed & Kunkel. 1960)) (see Figure 2). The number of trans-Arctic ships in summer is much higher than in winter, and these ships typically use the Northern Sea Route to travel between Europe and Asia (along the coastline of Eurasia). Figure 2b shows that this in fact is where most of the summer Arctic storms occur. In addition, the reanalysis datasets show that ~50% of Arctic storms have genesis in mid-latitude regions (south of 65°N) and travel northwards into the Arctic (north of 65°N). This shows that storms are a significant mechanism for transporting air from low to high latitudes.

In general, there is less consistency in Arctic storm characteristics in winter than in summer. This may be because in winter, the occurrence of meteorological conditions such as low level cloud, stable boundary layers and polar night that are more frequent, which are more challenging to represent in numerical weather prediction models, and for the creation of reanalysis datasets. In addition, there is a low density of conventional observations in winter, and difficulties in identifying cloud and estimating emissivity over snow and ice limit the current use of infrared and microwave satellite data in the troposphere (Jung et al. 2016).

The differences between the reanalysis datasets in Arctic storm frequency per season in winter (DJF) and summer (JJA) (1980-2017) were found to be less than 6 storms per season. On the other hand, the differences in Arctic storm frequency per season between storms identified by a storm tracking algorithm based on 850 hPa relative vorticity and MSLP were found to be 55 storms per season in winter, and 33 storms per season in summer. This shows that the decision to use 850 hPa relative vorticity or MSLP for storm tracking can be more important that the choice of reanalysis dataset.

References:

National Snow & Ice Data Centre (2019) Sea ice index. https://nsidc.org. Accessed 4 Mar 2019.

Reed RJ, Kunkel BA (1960) The Arctic circulation in summer. J. Meteorol. 17(5):489–506.

Sepp M, Jaagus J (2011) Changes in the activity and tracks of Arctic cyclones. Clim. Change 105(3–4):577–595.

Simmonds I, Burke C, Keay K (2008) Arctic climate change as manifest in cyclone behavior. J. Clim. 21(22):5777–5796.

Serreze MC, Barrett AP (2008) The summer cyclone maximum over the central Arctic Ocean. J. Clim. 21(5):1048–1065.

Vessey, A.F., Hodges, K.I., Shaffrey, L.C., Day, J.J., (2020) An inter‑comparison of Arctic synoptic scale storms between four global reanalysis datasets. Clim. Dyn., https://doi.org/10.1007/s00382-020-05142-4

Walsh, J.E., Bromwich, D.H., Overland, J.E., Serreze, M.C. and Wood, K.R., 2018. 100 years of progress in polar meteorology. Meteorological Monographs, 59, pp.21-1.

Zahn M, Akperov M, Rinke A, Feser F, Mokhov I I (2018) Trends of cyclone characteristics in the Arctic and their patterns from different reanalysis data. J. Geophys. Res. Atmos., 123(5):2737–2751.

## The (real) butterfly effect: the impact of resolving the mesoscale range

What does the ‘butterfly effect’ exactly mean? Many people would attribute the butterfly effect to the famous 3-dimensional non-linear model of Lorenz (1963) whose attractor looks like a butterfly when viewed from a particular angle. While it serves as an important foundation to chaos theory (by establishing that 3 dimensions are not only necessary for chaos as mandated in the Poincaré-Bendixson Theorem, but are also sufficient), the term ‘butterfly effect’ was not coined until 1972 (Palmer et al. 2014) based on a scientific presentation that Lorenz gave on a more radical, more recent work (Lorenz 1969) on the predictability barrier in multi-scale fluid systems. In this work, Lorenz demonstrated that under certain conditions, small-scale errors grow faster than large-scale errors in such a way that the predictability horizon cannot be extended beyond an absolute limit by reducing the initial error (unless the initial error is perfectly zero). Such limited predictability, or the butterfly effect as understood in this context, has now become a ‘canon in dynamical meteorology’ (Rotunno and Snyder 2008). Recent studies with advanced numerical weather prediction (NWP) models estimate this predictability horizon to be on the order of 2 to 3 weeks (Buizza and Leutbecher 2015; Judt 2018), in agreement with Lorenz’s original result.

The predictability properties of a fluid system primarily depend on the energy spectrum, whereas the nature of the dynamics per se only plays a secondary role (Rotunno and Snyder 2008). It is well-known that a slope shallower than (equal to or steeper than) -3 in the energy spectrum is associated with limited (unlimited) predictability (Lorenz 1969; Rotunno and Snyder 2008), which could be understood through analysing the characteristics of the energy spectrum of the error field. As shown in Figure 1, the error appears to grow uniformly across scales when predictability is indefinite, and appears to ‘cascade’ upscale when predictability is limited. In the latter case, the error spectra peak at the small scale and the growth rate is faster there.

The Earth’s atmospheric energy spectrum consists of a -3 range in the synoptic scale and a $-\frac{5}{3}$ range in the mesoscale (Nastrom and Gage 1985). While the limited predictability of the atmosphere arises from mesoscale physical processes, it would be of interest to understand how errors grow under this hybrid spectrum, and to what extent do global numerical weather prediction (NWP) models, which are just beginning to resolve the mesoscale $-\frac{5}{3}$ range, demonstrate the fast error growth proper to the limited predictability associated with this range.

We use the Lorenz (1969) model at two different resolutions: $K_{max}=11$, corresponding to a maximal wavenumber of $2^{11}=2048$, and $K_{max}=21$. The former represents the approximate resolution of global NWP models (~ 20 km), and the latter represents a resolution about 1000 times finer so that the shallower mesoscale range is much better resolved. Figure 2 shows the growth of a small-scale, small-amplitude initial error under these model settings.

In the $K_{max}=11$ case where the $-\frac{5}{3}$ range is not so much resolved, the error growth remains more or less up-magnitude, and the upscale cascade is not visible. The error is still much influenced by the synoptic-scale -3 range. Such behaviour largely agrees with the results of a recent study using a full-physics global NWP model (Judt 2018). In contrast, with the higher resolution $K_{max}=21$, the upscale propagation of error in the mesoscale is clearly visible. As the error spreads to the synoptic scale, its growth becomes more up-magnitude.

To understand the dependence of the error growth rate on scales, we use the parametric model of Žagar et al. (2017) by fitting the error-versus-time curve for every wavenumber / scale to the equation $E\left ( t \right )=A\tanh\left ( at+b\right )+B$, so that the parameters $A, B, a$ and $b$ are functions of the wavenumber / scale. Among the parameters, a describes the rate of error growth, the larger the quicker. A dimensional argument suggests that $a \sim (k^3 E(k))^{1/2}$, so that $a$ should be constant for a $-3$ range $(E(k) \sim k^{-3})$, and should grow $10^{2/3}>4.5$-fold for every decade of wavenumbers in the case of a $-\frac{5}{3}$ range. These scalings are indeed observed in the model simulations, except that the sharp increase pertaining to the $-\frac{5}{3}$ range only kicks in at $K \sim 15$ (1 to 2 km), much smaller in scale than the transition between the $-3$ and $-\frac{5}{3}$ ranges at $K \sim 7$ (300 to 600 km). See Figure 3 for details.

This explains the absence of the upscale cascade in the $K_{max}=11$ simulation. As models go into very high resolution in the future, the strong predictability constraints proper to the mesoscale $-\frac{5}{3}$ range will emerge, but only when it is sufficiently resolved. Our idealised study with the Lorenz model shows that this will happen only if $K_{max} >15$. In other words, motions at 1 to 2 km have to be fully resolved in order for error growth in the small scales be correctly represented. This would mean a grid resolution of ~ 250 m after accounting for the need of a dissipation range in a numerical model (Skamarock 2004).

While this seems to be a pessimistic statement, we have observed that the sensitivity of the error growth behaviour to the model resolution is itself sensitive to the initial error profile. The results presented above are for an initial error confined to a single small scale. When the initial error distribution is changed, the qualitative picture of error growth may not present such a contrast between the two resolutions. Thus, we highlight the need of further research to assess the potential gains of resolving more scales in the mesoscale, especially for the case of a realistic distribution of error that initiates the integrations of operational NWP models.

A manuscript on this work has been submitted and is currently under review.

This work is supported by a PhD scholarship awarded by the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth, with additional funding support from the ERC Advanced Grant ‘Understanding the Atmospheric Circulation Response to Climate Change’ and the Deutsche Forschungsgemeinschaft (DFG) Grant ‘Scaling Cascades in Complex Systems’.

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Judt, F. (2018). Insights into atmospheric predictability through global convection-permitting model simulations. J. Atmos. Sci. 75, 1477—1497. https://doi.org/10.1175/JAS-D-17-0343.1

Leung, T. Y., Leutbecher, M., Reich, S. and Shepherd, T. G. (2019). Impact of the mesoscale range on error growth and the limits to atmospheric predictability. Submitted.

Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. J. Atmos. Sci. 20, 130—141. https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

Lorenz, E. N. (1969). The predictability of a flow which possesses many scales of motion. Tellus 21, 289—307. https://doi.org/10.3402/tellusa.v21i3.10086

Nastrom, G. D. and Gage, K. S. (1985). A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42, 950—960. https://doi.org/10.1175/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2

Palmer, T. N., Döring, A. and Seregin, G. (2014). The real butterfly effect. Nonlinearity 27, R123—R141. https://doi.org/10.1088/0951-7715/27/9/R123

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