## Introduction

Over the past decades, rapidly enhanced atmospheric warming has been observed in the Arctic1,2,3. The accelerated warming is pronounced in the lower troposphere during the cold season4,5,6. An accompanying drastic reduction of sea ice7,8 has pronounced implications for global climate changes by affecting energy exchange between ocean and atmosphere9, and is often referred to as a key factor for accelerated warming in the Arctic10,11,12. A particularly significant sea ice reduction can be found over the Barents and Kara Seas, which potentially influences cold winter extremes over the Eurasian continent13,14,15,16,17,18,19. Physically, sea ice loss involves a positive ice-atmosphere feedback, which leads to an enhanced warming signal in the Arctic region. This feature is generally referred to as Arctic amplification6,9,20. Previous studies have proposed the physical mechanisms of Arctic amplification, which involve the effect of atmospheric heat transport21,22, oceanic heat transport23,24,25,26, cloud and water vapor changes27,28,29,30,31,32, and/or diminishing sea ice cover5,6,33. The accurate physical process of the Arctic amplification, however, is subject to debate.

Due to the large seasonal variation of insolation, there exists pronounced seasonality in the air-sea interaction process over the Arctic Ocean. During summer, open water readily absorbs solar radiation, which results in increasing heat content in the oceanic mixed layer. This represents the so-called albedo feedback5,6,9,34,35, meaning that the Arctic Ocean is efficient in absorbing atmospheric heat during summer. The albedo feedback is also important during the snow and ice melt in spring and early summer even before the appearance of open sea. After the sun sets over the Arctic Ocean, the ice-albedo feedback is suppressed and the primary air-sea interaction mechanism becomes oceanic horizontal advection and vertical convection of heat36. The stored heat in the ocean mixed layer is released back to the colder atmosphere above, which will result in warming of the atmosphere. The decreased insulation effect36 due to the loss of sea ice also promotes further sea ice reduction. Thus, heat transfer between the ocean and atmosphere is generally considered as the fundamental mechanism of Arctic amplification, which is pronounced only during the cold season. On the other hand, increased cloud cover and water vapor27,28,29,30,31,32,37 can also contribute to an increase in downward longwave radiation.

Despite the general consensus that heat transfer between the ocean and atmosphere is a crucial element in the physical mechanism of Arctic amplification and sea ice reduction, a quantitative understanding of individual contributions of heat flux components is still controversial. Further, the role of upward and downward longwave radiation in Arctic amplification is vague and not fully understood. Accurately quantifying the contribution of these different mechanisms, therefore, is required for a complete understanding of the Arctic amplification.

In the previous study33, we showed that the temporal pattern of sea ice variation indeed differs significantly between the Barents–Kara Seas and the Laptev and Chukchi Seas. Sea ice refreezes and the sea surface exposed to air is closed up in late fall in the Laptev and Chukchi Seas. As a result, significant absorption of solar radiation in summer does not lead to increased turbulent heat flux in winter. However, sea surface does not freeze up completely in the Barents–Kara Seas. Consequently, we hypothesis that turbulent heat flux becomes available in winter in the Barents–Kara Seas for heating the atmospheric column, which in turn increases downward longwave radiation.

In the present study, a quantitative assessment of energy fluxes involved in the Arctic amplification is investigated in relation to the sea ice reduction over the Barents and Kara Seas. This is an extension of the previous study with a specific goal of delineating the feedback mechanism between sea surface and the atmosphere. In particular, we extract a physically meaningful warming signal in the Arctic region and investigate how sea ice loss and individual energy fluxes are linked in a quantitative manner. For this goal, cyclostationary empirical orthogonal function (CSEOF) analysis38,39,40 is carried out on surface and pressure-level variables derived from the ERA interim daily reanalysis data41 in winter (Dec. 1–Feb. 28, d = 90 days). It should be noted that our discussion is restricted to processes in the Arctic; forcing from lower latitudes can also be important in the process of Arctic amplification and sea ice reduction.

## Results and Discussion

Figure 1 shows the sea ice loss mode identified through CSEOF analysis. Since the loading vector (Fig. 1a; see also Figs S1 and S2 in the supplementary information) and the amplitude (PC) time series (Fig. 1g) describes the sea ice reduction, together with natural variability of sea ice concentration, this mode represents the loss of sea ice in the Barents and Kara Seas during the past 37 years and explains 24% of the total variability of the sea ice concentration in the Arctic Ocean. The pattern of sea ice reduction (Fig. 1a) is nearly identical with the trend pattern of sea ice concentration in the Arctic Ocean (see Fig. S1). As can be seen in Fig. 1h, the sea ice reduction trend in the Barents and Kara Seas (boxed area in Fig. 1a) is captured by this mode. In particular, the rate of sea ice loss has significantly increased since 2004–200542. In association with the sea ice loss, 2 m air temperature, 850 hPa temperature, specific humidity, upward longwave radiation, downward longwave radiation, and upward heat flux have increased significantly over the region of major sea ice loss [21°–79.5°E × 75°–79.5°N] (boxed area in Fig. 1a). Multiplying the amplitude time series (Fig. 1g) with the loading vector (Fig. S2) of the sea ice loss mode as in equation (7), actual sea ice concentration time series is obtained as in Fig. 1h. According to Fig. 1h, sea ice concentration has decreased by ~40% during the last 37 years (1979–2016).

As can be seen in Fig. 1a,c and e, the central areas of anomalous 2 m air temperature, upward longwave radiation and turbulent (sensible + latent) heat flux match well with the region of sea ice loss36. On the other hand, the centers of the downward longwave radiation and lower-tropospheric specific humidity match well with that of the 850 hPa air temperature (Fig. 1b,d and f).

Figure 2 shows the anomalous surface (2 m) air temperature, the lower tropospheric geopotential height and wind and the vertical cross section of anomalous temperature, geopotential height and wind along 60°E and 80°N associated with the sea ice reduction. A significant warming is seen in the lower troposphere3,4,12. Note that the anomalous temperature pattern is similar to the second EOF pattern in Graversen et al.21. The anomalous temperature and geopotential height are consistent according to the hydrostatic equation (see Fig. S3). Anomalous wind and geopotential height are consistent according to the thermal wind equation. As can be seen, an anticyclonic circulation is established over the region of sea ice loss. This anticyclonic circulation results in advection of warmer air over the Barents and Kara Seas and advection of colder air over the mid-latitude East Asia19.

The winter-averaged patterns of anomalous downward longwave radiation and specific humidity look fairly similar to that of 850 hPa air temperature (Figs 1 and S4). It appears that the increased downward longwave radiation is the result of the tropospheric warming (Fig. 2). Specific humidity also increases with the tropospheric warming. Note specifically that these changes are observed over or close to the region of sea ice reduction. The pattern of total cloud cover, however, differs significantly from that of sea ice reduction. Since cloud is a difficult variable to simulate accurately, we also examine total column liquid water and total column ice water, which are the key variables for the formation of clouds. The patterns of total column liquid water and total column ice water exhibit a strong response over the region of sea ice reduction although their centers of action are shifted toward the Greenland Sea (Fig. S4d). Therefore, we postulate that the increased downward longwave radiation is due to the increased 850 hPa air temperature and the greenhouse effect produced by the increased specific humidity and cloudiness to a lesser extent; this is consistent with several previous studies43,44. Further note that net (upward minus downward) longwave radiation is positive over the region of major sea ice reduction, whereas it is slightly negative over the surrounding areas (Fig. S4c). Thus, at the surface level, there is a net loss of longwave energy over the region of sea ice reduction, while there is a net gain of longwave radiation over the surrounding area.

A prominent source of energy available for heating the atmospheric column is the increased turbulent heat flux from the wider area of sea surface exposed to air due to sea ice reduction (Fig. 3). Figure 4 shows the winter daily variations of the regressed loading vectors in equation (12) (terms in curly braces) averaged over the region of sea ice reduction (21°–79.5°E × 75°–79.5°N); it may be interpreted as the atmospheric response to the sea ice reduction shown in Fig. S2. Although the total (area-weighted) magnitudes of sensible and latent heat fluxes are generally smaller than those of upward and downward longwave radiation (Fig. 4a), turbulent heat flux is locally more pronounced than longwave radiation (Fig. 3)35. Furthermore, the combined effect of turbulent heat flux is about 6 times larger than that of longwave radiation, since upward and downward longwave radiation tends to offset each other and the resulting net longwave radiation is comparatively smaller than the net upward turbulent heat flux (Fig. 4a). In the presence of turbulent heat flux, air temperature and, henceforth, downward longwave radiation can increase continually leading to further sea ice reduction.

Therefore, we propose a feedback mechanism as suggested in Fig. 5. Sea ice reduction in this area leads to an increase in upward heat flux, which is used to raise temperature in the lower troposphere. Warming in the lower troposphere increases downward longwave radiation. As a result, sea ice reduction is accelerated. This feedback process can be written mathematically as follow:Step 1:

$$\frac{dF{L}^{\uparrow }}{dt}=-\,\alpha \frac{dS}{dt},F{L}^{\uparrow }=S{W}^{\uparrow }-\,S{W}^{\downarrow }+L{W}^{\uparrow }-\,L{W}^{\downarrow }+S{F}^{\uparrow }+L{F}^{\uparrow },$$
(1)

Step 2:

$$\frac{dT}{dt}=\beta \frac{dF{L}^{\uparrow }}{dt},$$
(2)

Step 3:

$$\frac{dL{W}^{\downarrow }}{dt}=\gamma \frac{dT}{dt},$$
(3)

Step 4:

$$\frac{dS}{dt}=-\,\delta \frac{dL{W}^{\downarrow }}{dt},$$
(4)

This proposed feedback mechanism, in its present form, does not require any delayed action of increased absorption of insolation during summer in terms of albedo feedback. In winter, a significant amount of turbulent heat flux can be released from the ocean exposed to cold air without excessive energy stored in summer. Summer heating, on the other hand, may be a fortifying factor for this feedback loop by preventing sea ice from refreezing during fall and winter.

It should be noted that there are other processes, particularly forcing from lower latitudes, which are important for Arctic amplification and sea ice reduction. As can be seen in Fig. S6a and b, there are net convergence of moisture transport and heat transport over the region of sea ice reduction, although the center of action is over the Greenland Sea. Thus, moisture and heat transports from lower latitudes apparently affect the variation of sea ice concentration43,44. On the other hand, the horizontal transports of moisture and heat cannot explain one essential element of specific humidity anomaly and air temperature anomaly, respectively. As can be seen in Fig. S6c and d, moisture and heat transports contribute only about 30–40% of the mean value of anomalous specific humidity and air temperature, respectively. The remainder should derive from a vertical process. Therefore, vertical processes are an important mechanism for explaining winter sea ice reduction47.

According to the amplitude time series in Fig. 1g, the rate of sea ice reduction appears to be accelerating. A curve fit with an exponential function results in

$$pc(t)=a\exp (\lambda t)+b=a{({e}^{\lambda })}^{t}+b\approx a{(1+\lambda )}^{t}+b,$$
(5)

where pc(t) is the amplitude time series in Fig. 1g, and t is time in years since 1979. We obtained the fitting curve (dashed curve in Fig. 1g) with parameters a = 1.275 × 10−1, λ = 8.916 × 10−2, and b = −9.055 × 10−1. Equation (5) can be rewritten as

$$pc(t)-c=(pc(0)-c){(1+\lambda )}^{t}.$$
(6)

That is, the amplitude of sea ice reduction and atmospheric warming increases at the rate of ~8.9% every year.

## Methods

### Data

ECMWF Reanalysis (ERA) interim daily variables are used from 1979–201641. Both surface and pressure-level variables during winter (Dec. 1–Feb. 28) are analyzed over the Arctic region (north of 60° N) to understand the detailed physical mechanism of sea ice loss and Arctic amplification.

### CSEOF analysis and regression analysis in CSEOF space

Analysis tool used for this study is the CSEOF technique38,39,40. In CSEOF analysis individual physical processes in space-time data are decomposed as:

$$T(r,t)=\sum _{n}{B}_{n}(r,t){T}_{n}(t),\,{B}_{n}(r,t)={B}_{n}(r,t+d),$$
(7)

where Bn(r, t) depicts daily winter evolution of the nth physical process and Tn(t) describes how the amplitude of the evolution varies on a longer time scale, and r and t denote location and time, respectively. Since the nested period d = 90 days, each loading vector, Bn(r, t), consists of 90 spatial patterns which depict evolution of a variable throughout the winter. These winter evolution patterns, Bn(r, t), repeat every winter, but its amplitude varies from one year to another according to the corresponding PC time series. CSEOF loading vectors are mutually orthogonal to each other in space and time and represent distinct physical processes. The principal component (PC) time series, Tn(t) are uncorrelated with (and are often nearly independent of) each other. Each loading vector depicts a temporal evolution of spatial patterns seen in a physical process (such as El Niño or seasonal cycle), and corresponding PC time series describes a long-term modulation of the amplitude of the physical process. Thus, the CSEOF technique is suitable for extracting and depicting temporal evolution of (nearly independent) physical processes and often yields valuable insight that cannot be attained from single spatial pattern.

In order to make suitable physical interpretation of the analysis results, CSEOF analysis is conducted on a number of key variables. It is, then, extremely important to make CSEOF loading vectors derived from individual variables to be physically consistent with each other. For the purpose of generating physically consistent CSEOF loading vectors, regression analysis is carried out in CSEOF space40. A target variable is chosen such that its major CSEOF mode best depicts the physical process under investigation; target variable is sea ice concentration in the present study.

Once CSEOF analysis on the “target” variable is completed as in equation (7), physically consistent loading vectors of another variable, called the “predictor” variable, are obtained as follows:

Step 1: CSEOF analysis on a new variable

$$P(r,t)=\sum _{n}{C}_{n}(r,t){P}_{n}(t)$$
(8)

Step 2: regression analysis on a target PC time series

$${T}_{n}(t)=\,\sum _{m=1}^{M}{\alpha }_{m}^{(n)}{P}_{m}(t)$$
(9)

$${Z}_{n}(r,t)=\,\sum _{m=1}^{M}{\alpha }_{m}^{(n)}{C}_{m}(r,t)$$
(10)

Then, the target and predictor variables together can be written as

$$\{T(r,t),\,P(r,t)\}=\sum _{n}\{{B}_{n}(r,t),\,{Z}_{n}(r,t)\}{T}_{n}(t).$$
(11)

Namely, the loading vectors of the two variables, Bn(r, t) and Zn(r, t), share an identical PC time series, Tn(t), for each mode n. As a result, the evolution of a physical process manifested as Bn(r, t) and Zn(r, t) in two different variables is governed by a single amplitude time series. Otherwise, Bn(r, t) and Zn(r, t) do not represent the same physical process and henceforth are not physically consistent. This process can be repeated for as many predictor variables as needed. As a result of regression, then, entire data can be written in the form

$$Data(r,t)=\sum _{n}\{{B}_{n}(r,t),{Z}_{n}(r,t),{U}_{n}(r,t),\ldots \}{T}_{n}(t),$$
(12)

where the terms in curly braces denote physically consistent evolutions derived from various physical variables. A rigorous mathematical explanation of the regression analysis in CSEOF space can be found in Kim48.