How does a change in climate variability impact the Greenland ice-sheet surface mass balance?

Abstract. The future of the Greenland ice-sheet largely depends on the changing climate. When ice-sheet models are run for time periods that extend far beyond the observational record they are often forced by climatology instead of a transient climate. We investigate how this simplification impacts the surface mass balance using the Bergen Snow Simulator. The model was run for up to 500 years using the same atmospheric climatology, but different variability, as forcing. We achieve this by re-arranging the years in the ERA-interim reanalysis while leaving the intra-annual variations unchanged. This changes the surface mass 5 balance by less than 5 % over the entire Greenland ice sheet. However, using daily averages as forcing introduces large changes in intra-annual variability and thereby overestimates the Greenland-wide surface mass balance by 40 %. The biggest contributor is precipitation followed by temperature. The most important process is that small amounts of snow fall from the daily climatology overestimate the albedo, leading to an increased SMB. We propose a correction that distributes the monthly precipitation over a realistic intra-monthly variability. This approach 10 reduces the SMB overestimation to 15-25 %. We conclude that simulations of the Greenland surface mass and energy balance should be forced with a transient climate. Particular care must be taken if only climatological data is available for simulations with a model that was calibrated with transient data. If daily transient data cannot be used, at least the precipitation should follow a natural daily distribution.

intra-annual variability. While it is common practice to use a constant temperature index to interpolate between the coldest (Last Glacial Maximum) and the warmest (Present Day) state (e.q. Forsström and Greve, 2004;Alvarez Solas et al., 2018), it has not been studied what impact additional variability on short time scales would have. The effect of additional non-resolved variability may be an even larger issue as the most common temperature proxies used are ice cores, which in turn rather reflect the precipitation events than only climatological temperatures (Madsen et al., 2019). Proxies vary greatly in their temporal resolution, so we investigate the variability on multiple time scales (50 -500 years). Although the initial question arises from 30 proxy and climate reconstruction it is equally applicable to projections of the distant future of the Greenland ice sheet.
In this study, we perform simulations using the latest version of the BErgen Snow SImulator (BESSI) (Zolles and Born, 2021). Prior model parameter tuning was performed relative to the GRACE satellite data set and RACMO simulations (Noël et al., 2018;Fettweis et al., 2020a;Holube et al., 2021). The model is designed for the simulations of long time scales, leading to a trade off between complexity and computational efficiency. Therefore, we need a representative climate forcing for longer 35 time periods.
Input data to force BESSI is derived from the ERA-interim reanalysis data set, instead of using an artificial inter-annual variability or internal climate model variability (Semenov, 2008;Verdin et al., 2018) based on a climatology. Firstly, the rapidly increasing temperature over the last 50 years is a good example of a non-representative climatological average. Secondly, ERAinterim provides a reasonable natural variability and daily data is available over the entire Greenland Ice-sheet at a sufficiently 40 high spatial resolution . Potential climate model data for climate reconstructions and projections will be of a similar or lower resolution. Climate variability of different time scales is achieved by a reordering the individual years.
The ultimate test is weather a re-arranged forcing mimics reality by simulating the same SMB as the transient -real -forcing.
For a longer simulation duration the ERA-interim period is copied multiple times. We use ERA-interim as its resolution is of the same order of magnitude as most Global Circulation climate Models (GCM) and refrain from higher resolution models like 45 MAR (Fettweis et al., 2017) or RACMO (van Meijgaard et al., 2008) as those will not be available for the most of the past (last glacial) and are computationally demanding. We choose the current rapid climate change as it provides an upper uncertainty estimate for the entire glacial. Furthermore, the model sensitivity of the surface mass balance model has been evaluated prior for this time period (Zolles and Born, 2021). This leaves us with three goals of the study:

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-Quantify the uncertainty associated with inter-annual variability and climatological forcing -Identify the reasons for and potentially reduce this uncertainty -Find a procedure to create a representative climate forcing for the past based on temperature proxies In section 2 we will give a brief description of the surface mass balance model and the set-up of the climate ensemble used in this study. The results in section 3 are split into the uncertainty of inter-annual variability, individual forcing variables, and 55 precipitation and associated albedo impact. After that, we discuss our findings in section 4 and conclude in section 5. The study uses the Bergen Snow SImulator (BESSI), which calculates the mass and energy balance with a daily time step (Born et al., 2019). It compares well to other surface mass balance models over Greenland with a slight positive bias for melt regions 60 (Fettweis et al., 2020a). The latest model version is described in detail in Zolles and Born (2021) so that we will only provide an abridged description here. The model domain is based on a stereo-graphic projection of Greenland and uses an equidistant grid with a resolution of 10 km. The model uses a mass based vertical grid of 15 layers, with up to 500 kgm −2 . The model uses five input fields with a daily resolution: surface temperature, total precipitation, dew point, and down-welling long-and shortwave radiation. A full energy balance is calculated at the surface including diffusion of heat in the snow pack and latent contributions 65 from freezing and melting of water and liquid precipitation. Liquid water in the snow is explicitly represented. Mass changes due to melting, precipitation, or sublimation processes. The model parameters have been tuned using a multi-variate calibration towards RACMO (Noël et al., 2018) and the GRACE data set.

Atmospheric climate forcing
We use the daily ERA-interim reanalysis data from 1979-2017 (Uppala et al., 2011). The input variables of atmospheric 70 temperature, precipitation, dew point, and short and long-wave radiation are bi-linearly interpolated to a 10x10 km grid over Greenland. This initial forcing data of 39 years is then taken 12 times to represent longer time periods. We define the natural transient forcing as the ERA-interim forcing in the true historical order and then looping forward and backward (F-BWD).
This means the following order 1979-2017-1979-2017-1979-.. . We arrange the original transient forcing in four different ways: repeating the ERA-interim forcing in its original order 75 multiple times (forward, FWD), repeating the same data in reverse order (backward, BWD), alternating between FWD and BWD to avoid the abrupt transition between the forcing years 2017 and 1979 (forward-backward, F-BWD), and again the same in reverse (backward-forward, B-FWD). This already creates synthetic time series with different frequencies (Fig. 1).
However, to achieve even lower frequencies with the same data we also re-arrange the original transient forcing based on the Greenland ice-sheet wide average annual air temperature. This changes the order of the 39 years in the record. Note that this 80 does not break the consistency between the atmospheric variables, or add energy or mass to the atmospheric system relative to the original natural forcing. Temporal continuity is only broken at the year break with arguably negligible consequences.
The other time series are the temperature ordered forcing with different frequencies (rows) and sequential arrangement (columns, similar to the first row). All these time series have the same average forcing values, respectively the daily same climatology, but different temporal variability. They are obtained by ordering the 12 cycles of 39 years by the Greenland wide 85 temperature from the coldest of the series to the warmest. Afterwards depending on the chosen frequency we sample every n-th member of this series starting at the coldest/warmest year, where n is the frequency and once the end of the series is reached we start over at the 2nd member sampling every n th member thereafter, this is repeated in total n times for one time series.    with different reoccurring patterns (2017-1979-2017x6,1979-2017-1979x6,1979-2017x12,2017-1979x12). Rows three to six show the temperature ordered sequence with increasing frequencies, with row one starting cold (F-BWD) and row two starting warm (B-FWD). Instead of looping back and forth from cold to warm the last two rows (orange) only increase/decrease in temperature and once the maximum/minimum is reached it starts over with the coldest/warmest forcing year again. While the temporal order for the forcing years is of marginal influence (< 5% difference in SMB) over the entire ice-sheet, it is larger on a regional level. The variability mainly impacts the SMB around the equilibrium line, with a standard deviation of up to 500 kg m −2 yr −1 on the local scale ( fig. 3). The standard deviation is also quite high in the northeast.

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As proxy resolution is variable we also study additional simulation lengths of 78, 117, 156, and 234 years, corresponding to two, three, four, and six ERA-interim cycles. The general results are similar for shorter simulation periods (78, 117, 156, and 234 years instead of 468), though the difference between the simulations decreases, as with fewer ERA-interim cycles the duration of extended warm or cold periods decreases (not shown).
Climatological forcing / Intra-annual variability As the order of the inter-annual variability has a low impact, can we   The daily climatology leads to a drastic overestimation of the SMB by 40 % (274 kg m −2 yr −1 Fig. 4 a,b.). We further 120 investigate this overestimation by studying the impact of the individual forcing variables: using a transient forcing for all but one variable, which comprises of daily climatological averages (right), and the climatological forcing is mixed with one transient variable (left) ( fig. 4 c-l). The SMB of these simulations exceed the transient forcing (4 b), meaning that daily climatologies always lead to an SMB increase. This is no surprise due to the non-linearity of the SMB to energy input. There is a clear The small effect and low variability of the radiation components shows that using climatologies is justified in this case 130 ( fig. 4 g-j), as the inter-annual variability of Greenland wide radiation is relatively low anyway. Though it is still connected to a slight bias of 5% in the current climate. The turbulent latent heat flux has a relatively low impact on the Greenland wide SMB (Zolles and Born, 2021), which is in line with the low effect the dew point change has ( fig. 4 k, l). While the biggest differences between the previous simulations where found around the equilibrium line ( fig. 3), the largest difference between climatological and transient forced SMB simulations is found in the melting region of Greenland ( fig. 5). Temperature has 135 the second highest influence, which can be attributed mainly to the non-linearity of the SMB. However, the overestimation by climatological precipitation cannot be explained by the non-linearity, but the albedo. Using a daily climatology leads to small amounts of mostly snowfall every day leading to a surface albedo increase. The annual average albedo increase is up to 0.1 in the melt region of Greenland. The drastic effect of daily climatologies of precipitation can be attributed to this albedo overestimation.

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Can we emulate intra-annual variability of precipitation? We have shown that BESSI overestimates the SMB drastically if daily climatologies of precipitation are used. A daily climatology is unrealistic as it has small amounts of snow fall every day.
This does not agree with observations of highly event-based precipitation in the Atlantic region (Sodemann et al., 2008). We therefore calculate alternative temporal precipitation distributions by taking monthly averages with a sub-monthly distribution instead. Regular precipitation frequencies of 2, 4, 8, 15, and 30 days are tested as well as the sub-monthly distributions from 145 each of the 39 ERA-interim years. For the ERA-interim based distributions the original daily time series P day is scaled to have the same monthly average: with P t m as the monthly mean of the year t, and P m the monthly climatological precipitation amount. This correction can be compared to the Delta Method for precipitation (Beyer et al., 2019). We obtain 39 possible precipitation time series, each with q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 6. Sub-monthly precipitation distribution for March and April of different simulations. The same monthly precipitation is either distributed via daily climatologies (blue), monthly climatology with the sub-monthly distribution of, for example, 2014 (black) or with regular frequencies (green, orange, light blue). The red distribution is the true distribution for 2014 which is then adjusted to the climatological average (black, eq. 1), as can be seen April 2014 was wetter than the average April of the ERA-interim period.
wet month, so for the resulting forcing it is adjusted to be less but still has four days with precipitation of up and above 10 kg m −2 (fig. 6).
The simulated SMB depends on the chosen sub-monthly precipitation distribution ( fig. 7). For regular precipitation the 30 kg m −2 yr −1 , which is much closer to the "true" value of the transient forcing ( fig. 7a). The amplitude of this simulations SMB time series is rather low as the same amount of precipitation falls every year, it was investigated further. Instead of using different sub-monthly frequencies every year the distribution from each year ERA-interim year is taken as the forcing for the entire simulation period (as example 2009: fig. 7 f,g; the entire range is given in fig. 8). It spans from 224-253 kg m −2 yr −1 .
Using the sub-monthly precipitation distribution for the climatology reduces the SMB overestimation from 40% to 10-25%. A The decrease in SMB is due to the non-linearity effect of the SMB, as in dry years earlier ice exposure triggers a feedback.
Due to the non-linearity of the mass balance and albedo feedback, the range of these simulations is larger than the amplitude of the single simulation ( fig.7 d).

Discussion
We study the impact of inter-annual variability by a simple reordering. The SMB shows a low dependency of 5% over 468 years on the order of the forcing. In case of unknown inter-annual variability the use of a climatological forcing over estimates the SMB by 40 % due to the non-linearity of the SMB and albedo overestimation. We try to reduce this effect by instead using only monthly precipitation averages with a sub-monthly distribution. The overestimation is reduced but an uncertainty of 15% 175 based on the chosen distribution is introduced.
Climate model simulations of the same time period vary in their inter-annual variability, they can very well represent the climatology, but not the order. We show that the effect of the order of the inter-annual variability is less than 5 %. This indicates that the memory effect of the Greenland wide integrated SMB to multiple warm or cold years is low enough to be modeled with climate forcing which may not have a realistic temporal variability. Even multiple warmer years over Greenland after each other 180 do not significantly lead to strong feedback. The used ERA-interim period with its temperature trend (Hanna et al., 2021) as the study period can be considered an upper boundary for steady state climate. The simulation lengths were 78,117, 156, 234 and 468 years, and even the extreme case of 12 consecutive years with the warmest temperature the average SMB only decreased by 3.5 %. If the climatology is known and the amplitude of the variability of the forcing data, the order does not really matter, despite the high inter annual variability observed in line with Van den Broeke et al. (2011). In case of climate simulations based 185 on climatologies derived from proxies or other boundary conditions they likely are applicable for SMB simulations as long as the amplitude of the variability is good, even if there is a sub-resolution trend not visible in the proxy data. However, the effect is larger on a regional basis and around the equilibrium line the sensitivity towards this inter-annual variability increases. For the ERA-interim climate the northeast of Greenland with its sparse precipitation and large inter-annual variability in particular shows a standard deviation of up to 300 kgm −2 yr −1 .

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If the inter-annual variability is not known as is most often the case for the distant past or future, the forcing has to be based on climatologies. BESSI uses daily forcing data and is sensitive to daily precipitation. A small amount of snow-fall every day leads to an albedo overestimation as BESSI resolves albedo adjustments on a daily bases. A possible solution is to  parameterize the albedo routine differently for climatology and transient data. Alternatively, the precipitation climatology has to be calculated in a physical more reasonable way which we explore here. We show that monthly climatologies with a natural 195 sub-monthly distribution reduce the SMB overestimation. In practice, there are multiple ways how to define such a distribution: regular or stochastic frequencies for a region using normalized precipitation from reanalysis or climate simulation data. Either approach, may be prone to the sampling period and not invariant in time, and multiple solution may exist. The redistributing of the same precipitation amount at each grid point within a month can change the SMB by 15% (fig. 8). This is to be considered when selecting the fields for projections or reconstructions, purely based on scalar temperature and/or precipitation anomalies 200 of a given field. The precipitation is quite variable in Greenland (Mosley- Thompson et al., 2005), but not only the total amount is important but also its temporal distribution, in particular in the melt region. There is no clear best representative of the precipitation variability among the individual years of the ERA-interim period.
Based on our findings we suggest that in the absence of full climate simulations with natural variability, temperature and precipitation anomalies are applied to a related climatology with sub-monthly frequency in precipitation. Still using clima-205 tological forcing may be overestimating SMB, as it does for BESSI, due to the non-linearity of mass balance, which is in line with (Mikkelsen et al., 2018) who found a 13 % overestimation of the SMB if inter-annual temperature fluctuation is not considered. The choice of the representative precipitation distribution which is scaled may be accompanied by an uncertainty of up to 15%.
BESSI does not use sub-daily parameterizations for the daily cycle, which could reduce the effect of small amounts of snow 210 falling every day and the accompanied albedo overestimation while using climatological forcing if considered. Though small amounts of precipitation every day are physical not reasonable for the region and it has to be considered in the snow models.
BESSI showed a positive SMB bias in general relative to other snow-models, we cannot state how big the mentioned effects are for the other SMB models (Fettweis et al., 2020b).
We did not try to adjust climatological fields for temperature, or the other forcing variables. Due to the event based nature of 215 the precipitation this has the biggest impact, but daily climatologies overestimate the SMB also due to the other variables too.
The effect of the non-linearity alone has been previously studied with the model (Born et al., 2019). We furthermore did not study the impact of precipitation distributions on the point scale.

Conclusions
A surface mass and energy balance model was run for up to 500 years with different climate forcing. They all share the same 220 climatology in the five forcing variables, atmospheric temperature, precipitation, long and short-wave radiation, and humidity.
While different frequencies of climate variability have very little impact (< 5 %), using an average climate leads to a drastic overestimation (40 %) of the surface mass balance. This is mainly observed around the melt region of the Greenland ice sheet. The biggest contribution to this overestimation is the precipitation forcing (≈30 %), due to the resulting albedo increase.
Averaging multiple years to obtain a climatology produces a data set with frequent light precipitation, and a high surface albedo 225 due to the continuous presence of fresh snow. Small amounts of snowfall are not physically reasonable for a region with event based precipitation like Greenland.
To overcome the problem we calculated alternative precipitation climatologies to be used together with daily climatologies of the other variables. Monthly averages following a natural sub-monthly distribution lead to the smallest errors. Though, there is a dependency on the chosen distribution. Using a regular frequency is not feasible as there is a large spatial dependency and 230 empirical relations may change through time periods. We conclude that the surface mass balance model is best forced with transient climate. If daily climatologies with an altered precipitation forcing are used an overestimation of 15-25 % of the SMB should be assumed.
Code availability. The BESSI model code is available on git-hub (https://github.com/TobiasZo/BESSI) Author contributions. TZ conducted the model tuning and ensemble simulations, the data analysis and wrote the main part of the manuscript.