The Arctic sea ice represents an important energy reservoir for the climate of the northern hemisphere. The shrinking of the polar ice in the past decades decreases the stored energy and raises serious concerns about future climate changes. [1][2][3][4] Model calculations of the present authors [5,6] suggest that half of the global warming during the past fifty years is directly related to the retreat of the sea ice, while the cause is not well understood, e.g. the role of surface pollution [7][8][9][10]. We have analysed the reported annual melting and freezing data of the northern sea ice in the years 1979 to 2018 [11] to gain some insight. Two features can be deduced from our simple model: (i) recent results [12,13] are confirmed that approximately 60 % of the loss of sea ice stems from energy transport to the arctic region. (ii) We find evidence that the remaining part of the ice retreat originates from an increasing surface absorption of solar radiation, obviously due to the rising surface pollution of the sea ice. While the phenomenon was previously considered by several authors in a qualitative way, our analysis contributes semi-quantitative information on the situation. We estimate that the relevant fallout of light absorbing aerosols onto the sea ice increased by 17 +/- 5 % during the past fifty years. A deposition of additional 3 +/- 1 % of solar radiation in the melting region results that accounts for the ice retreat. Recalling the important role of the ice loss for the terrestrial climate, [3,5,9] the precipitation of air pollution in the Arctic seems to be an important factor for the global warming. The relationship between Arctic warming and sea ice loss is not well understood. [14] Some authors point out that is partly due to natural reasons. [2,3,15,16] There is evidence for climate changes in past centuries evolving on time scales of hundreds of years, accompanied by distinct changes of the Arctic sea ice [3,[17][18][19] that may continue at the present. The loss of sea ice was also related to the [20][21][22] and the Atlantic Driver [23,24]. On the other hand, evidence was reported for a large contribution of summertime atmospheric circulation of 60 % to the September sea ice loss. [13]. Notz and Stroeve reported that the observed Arctic sea ice loss directly follows anthropogenic CO2 emissions, whereas climate model simulations for the sea ice retreat differ substantially. [25] The discussion presented below is based on the satellite observations of Ref. [11] for the daily sea ice extent and the data of Ref. [26] for the maximum and minimum sea ice area in recent decades. For the analysis of the experimental information an analytic model with several simplifications is used: -The data for the daily sea ice area are derived from the daily ice extent values, assuming that the ratio of the two quantities in each respective year is approximately constant. [26] -Following Ref. [27] and averaging over seasonal changes, the ice thickness in the melting region is taken to be 1.5 m and approximately constant in the years 1979 to 2018. As a result the daily energy consumed by ice melting in the summer period is just proportional to the daily loss of sea ice area. -For the melting energy of sea ice we only distinguish two sources: direct absorption of solar radiation of the ice surface and transport of heat to the melting zone by sea water and/or the atmosphere. -The energy transport that obviously contributes to the ice melting in the summer period is estimated from the time evolution of the freezing energy ( = negative melting energy) of ice at the beginning and end of the winter period when solar radiation is negligible (for details see below). -The solar radiation in the daily melting zone (i.e. difference of the southern edge of the melting region with maximum ice and the daily melting frontier) is estimated for a transmission of 0.49 of the atmosphere of the sun light and omitting geographical details of the Arctic region. In other words, the earth is treated as rotating sphere with known radius while the daily angular position of the sun is varying between the Tropic of Cancer (23.44 deg N) and Tropic of Capricorn (23.44 deg S). Geographical details of the ice are omitted. In terms of the model, the melting zone of the Arctic ice extends between the frontier line of the annual maximum of sea ice (i.e. minimum latitude at the end of the freezing period) and the daily melting frontier. The latter shifts to the north and reaches a maximum value at the end of the melting period (i.e. annual minimum of ice area). For example, for 1979 the latitude limits are 70.34 deg N and 78.00 deg N, respectively (71.37 deg N and 81.32 deg N in year 2017). Because of the various simplifications of the model the results reported below are of semi-quantitative nature, only. The analysis was carried out for nine years in the period 1979 to 2018. An example for 1979 is presented in Figs. 1 and 2. The reported data of Ref. [11] for the sea ice extent were smoothed to remove some of the experimental scatter; the data were converted to ice area values by the help of Ref. 26 for maximum and minimum ice area (dash-dotted black curve in Fig. 1), assuming that ice extent and ice area are approximately proportional [28]. The ice area is shown in the Fig. to rise from January 1 st (day = 1) to a maximum at March 6 th (day = 62, end of the freezing period). For subsequent days the area declines to a minimum at day = 255 (September 12 th , end of melting period). The daily loss of ice area is illustrated by the red curve in Fig. 1 that simply represents the daily change of the dash-dotted line.
We note that maximum daily melting occurred in July 1979 on day 199, notably later than the beginning of summer (day 172, see red curve). Comparing the reported data set for 1979 and 2018 (data not shown) the decrease of the annual area minima (factor 0.51) is more pronounced than that of the annual maxima (factor 0.90); i.e. the seasonal variation of sea ice area in individual years is rising in the period 1979 to 2018 (factor 1.13). Comparing the two curves we note that daily melting still increases after day 150 when sunshine in the region is already declining. Rising daily heat transport to the melting zone and / or increasing absorption of solar radiation by the ice surface is required to explain the delayed melting behaviour. For melting the sign of the melting energy is defined to be negative, while heat transport is chosen to be positive. In the summer period the heat transport from the south contributes to melting, while solar input accounts for melting only in part. In other words, heat transport changes sign. The dotted part of the blue curve in Fig. 2 is an estimate of this energy transport in the summer The lines are linear fits to the data for nine years in 1979 to 2018. The annual energy demand for melting of sea ice (absolute value) is found to rise from 3.8 x 10 /* J to 4.3 x 10 /* J (compare fitted red line). The rise by 13 +/- 1 % (red curve) is directly related to the increasing melting amounts of the sea ice, corresponding to the difference of the annual maxima and minima of sea ice area (and volume). The annual minima decrease more rapidly than the maxima (not shown in the figure) so that the difference is growing. The available solar energy for melting in the respective years is calculated to rise from 1.3 x 10 // J to 1.6 x 10 // J (increase of approximately 20%, see dashed green line). The positive slope may be compared with the growing annual ice retreat (1979: 9.3 x 10 */ m 2 ; 2018: 10.5 x 10 */ m 2 ). The required annual contribution of heat transport appears to be approximately constant with ( 2.2 +/- 0.7) x 10 /* J (see dotted blue line in Fig. 3). The latter finding may be compared with two opposing factors: the rising global warming on the one hand and the north shift of the melting region on the other one, hindering heat transport from the south. Air temperatures in the pole region are reported to rise much more than the global mean temperature (factor of 2 or more) [29] while the sea water directly beneath the permanent polar ice shield (i.e. water-ice interface) is obviously constant at ~272 K. An increasing absorption of sun light by the ice surface may be inferred from the data of Although different air pollutants contribute to the warming of the atmosphere [29,32] or cooling of the surface, several authors pointed to the important role of black carbon (from fuel combustion and biomass burning) and mineral dust for the darkening of the northern snow and ice surface. [29,32,33] In the light of these arguments it is suggested that light absorbing aerosols in the atmosphere should be considerably reduced to limit further global warming . and ice surface. [29,32,33] In the light of these arguments it is suggested that light absorbing aerosols in the atmosphere should be considerably reduced to limit further global warming .