Parametrization of sub-mesoscale processes related to spatially heterogeneous land/water surface remains one of the key challenges in atmosphere and climate modeling. In polar regions, sea ice surface properties vary over a wide range of scales, from centimeters to tens of kilometers, influencing the oceanic and atmospheric boundary layers through many complex interactions. The quality of weather and climate predictions depends heavily on our ability to accurately represent the effects of those interactions in numerical weather prediction (NWP) models. As most of currently used climate models have horizontal grid spacing of tens of kilometers and are therefore unable to directly reproduce processes related to smaller-scale sea ice features, their effects must be parametrized. However, the scarcity of observations of atmospheric properties over fragmented sea ice and the resulting incomplete understanding of the sub-mesoscale processes hampers the development of parametrizations. Consequently, sub-grid interactions between the surface and the atmosphere, which may account for the inaccuracies present in model predictions [1
], are poorly represented in NWP models.
The influence of sea ice heterogeneity on the atmospheric boundary layer (ABL) has long been the subject of both modeling [3
] and observational studies [7
]. The general aspects of ABL’s response to the presence of open water areas in the form of singular or several leads in an otherwise continuous ice cover are relatively well studied. These effects include changes of the ABL stability over and downstream of open leads [5
], sensitivity to prevailing wind conditions [10
], formation of steam fog and cloud plumes [12
], or initiation of gravity waves [4
]. Similarly, the influence of leads width on the strength of the atmospheric response has been studied as well [13
], and it is generally assumed that for a given total surface area of leads, numerous narrow leads tend to induce stronger area-averaged turbulent exchange of heat and moisture than a few wide ones. However, recent satellite data studies indicate that such conclusion might be flawed [15
] and suggest that the larger contribution from small leads to the total surface heat flux may not be associated with their efficiency, but rather number density, length and frequency of their occurrence in the Arctic. Other studies point out that not only the average lead width is important, but also their spatial arrangement and the shape of their size distribution [16
]. According to modeling results [17
], even very small open water areas might result in the development of small convective plumes which penetrate the ABL and modify its circulation and properties. The areas of updraft air motion can be recognized in the winter Arctic from the satellites due to the presence of steam fog. Steam fog occurs because of cold (
C) air advection over water which is only slightly warmer than its freezing point. The fog is characterized by either columnar or banded structure and extends in height from 1 to 1500 meters with the liquid water content in the range of 0.01–0.5 g/m
]. This fog type can be recognized in satellite images as linear features, organized into bands of different width, interspersed with clear air areas [20
], or as plume-like streaks, with a typical spacing of a few kilometers [21
]. Numerical studies indicate that such features may have significant impact on atmospheric processes not only locally, but also regionally [5
Obviously, large scale models do not directly resolve heterogeneity at scales smaller than the model grid cell. To account for the larger-scale effects of sub-grid properties of the atmosphere and the underlying surface, NWP models must rely on estimation methods and simplified parametrizations. Computation of surface turbulent heat fluxes (STHF)—the main subject of the present paper—provides a very good example of problems associated with constructing such parametrizations. Locally, STHF is a function of several variables characterizing the surface and the lower atmosphere (temperature, wind speed, etc.). Due to the nonlinear character of the relationship between those variables and spatial correlations between them, the area-averaged STHF cannot be directly computed from the area-averaged variables, available in numerical models. The necessary correction depends on the characteristics of the surface (size and spatial arrangement of “patches” of different surface types; in the case of ice-covered sea surface analyzed here: spatial distribution of sea ice and open water) and the associated local response of the lower atmosphere. Several approaches to this problem have been developed and are used in various NWP models.
In the standard, most basic approach, a single, dominant type of the surface in a grid cell is used as representative for that cell. Thus, the cell-averaged surface fluxes are calculated from cell-averaged atmospheric and surface properties, like, e.g., in the standard version of COSMO-CLM or CCLm model [23
]. Such averaging of atmospheric and surface properties is termed an “aggregation” process. An important disadvantage of this approach is a possibility that the aggregation will produce biased results due to the above-mentioned nonlinear relationship between the variables involved. The problem of biased STHF estimates has long been recognized [14
] and several improved methods of flux calculation have been proposed. One of the most popular ones is a so-called mosaic method developed by Avissar and Pielke [27
]. In this approach, the sub-grid variability is accounted for by independently coupling each surface type within a grid cell to the overlying atmosphere. Surface-specific fluxes are thus calculated from area-averaged atmospheric properties (as previously), but with the consideration of the parameters specific for each surface type. Grid cell-averaged fluxes are then obtained by summing the fluxes from different surfaces, weighted by the fractional area they cover. This method has been adopted, sometimes with slight modifications, in various modeling [26
] and observational [30
] studies. The mosaic approach, to some extent, includes the effects of surface heterogeneity, but due to very complex relationship between the ABL and the underlying surface, it represents only a rough estimation of surface–atmosphere interactions, completely disregarding spatial structure of those interactions. As we demonstrate in this work, in the case of ABL over fragmented sea ice it is that spatial structure that is responsible for large deviations between area-averaged fluxes computed with different methods. It is very likely that in the case of winter sea ice fragmentation, the averaging of atmospheric variables is behind the largest errors in NWP predictions of STHF. This is because differences between local ABL properties over sea ice and open water are often very large (e.g., can reach even tens of degrees in the case of air temperature).
The methods of STHF calculation described above rely solely on sea ice concentration in a grid cell and do not take into account any information about spatial arrangement of ice and water within that cell (size of ice floes, presence and orientation of leads and fractures, etc.). This limitation seems particularly serious in view of the present climate change in the polar regions, associated with a rapid transition of the sea ice cover, with decreasing extent of multiyear sea ice and expanding proportion of seasonal ice [31
]. Due to the fact that seasonal, first-year sea ice is more prone to breaking and deformation due to external forcing, heterogeneous sea ice is going to appear more frequently [32
]. Since the sea ice cover moderates energy transfer between ocean and atmosphere and is highly sensitive to weather patterns changes, the atmospheric response to sea ice fragmentation becomes a particularly important research topic. A specific aspect related to sea ice fragmentation that attracted a lot of scientific attention throughout the years is the floe size distribution (FSD), defined as the number of floes in different size categories in a given region, divided by the area of that region [33
]. It is a key parameter in describing the state of the marginal ice zone (MIZ) and the evolution of sea ice cover. The information about floe size is essential to determine the rate of lateral sea ice melt [34
], to assess the sea ice dynamics and internal stresses [35
], and to resolve sea ice interactions with oceanic [36
] and atmospheric boundary layers [17
]. The importance of FSD drives the effort to include its representation into sea ice and climate models. Equations describing the evolution of the joint distribution of sea ice thickness and floe size were formulated and implemented in a continuum sea ice model by Horvat and Tziperman [37
]. The advantage of this model, further developed by Roach et al. [38
], is that it does not involve any assumptions about the form of that distribution, so that it is allowed to evolve in response to forcing from the ocean and atmosphere. Another approach, alternative to continuum models, is based on discrete-element methods (DEM), as, e.g., by Herman [39
] whose joint-particle DEM allows direct simulations of floe formation due to dynamic processes. However, due to extremely high computational costs of DEMs, larger-scale sea ice simulations must be based on continuum models (coupled with atmospheric, oceanic and possibly other components). The new possibility of including FSD in those models, described above, provides a wide range of possibilities in terms of implementing new physical processes, both influencing and influenced by the FSD. On the other hand, however, this can be done only if those processes are sufficiently well understood. In the case of the response of the ABL to sea ice fragmentation, this is at present not the case. Moreover, observations that could contribute to improving our knowledge are very rare if not non-existent. Therefore, in order to implement FSD-related effects into NWP models we should first focus on better understanding of its impact on the atmospheric and oceanic boundary layers and the physical processes involved, in parallel with the development of parametrizations that would make the models less expensive computationally.
Several aspects of the ABL response to sea ice heterogeneity have been studied thoroughly in the recent numerical study of Wenta and Herman [17
]. They examined the influence of small-scale sea ice surface variability (different floe size distributions or different lead widths and orientation) on domain-averaged STHF and properties of the ABL (e.g., the total moisture content). In agreement with earlier studies, mentioned above, the model predicted formation of convective plumes, rapid release of heat and moisture, strong local wind speed increase and intensification of vertical air motions. Crucially for the present analysis, at the same ice concentration, different size and spatial arrangement of ice floes resulted in changes of the domain-averaged ABL properties. This indicates that the above-described methods of computing STHF, based solely on sea ice concentration, might produce biased results, with inaccuracy dependent on the properties of FSD. Furthermore, significant correlation is observed not only between the local surface and atmospheric variables, but also within the ABL itself (i.e., between ABL properties at different heights), indicating that the transfer coefficients used in the STHF algorithms should take into account local variations of temperature and wind speed.
In this study, the influence of FSD on the ABL is studied further. The modeling results obtained by Wenta and Herman [17
] are examined, with the focus on the convective structures within the ABL and the surface turbulent moisture heat flux (the second component of STHF, the sensible heat flux, will be analyzed in a subsequent study). The goal of the first part of the analysis (Section 3.1
) is to explain how the spatial arrangement and size of convective structures depends on the floe size distribution, and to describe the properties of those structures found in the modeling results. (Anticipating Section 3.1
, it is important to stress here that convection cells in this case do not have characteristic, regular shapes typical for Rayleigh–Benard convection, and therefore they are referred to as convective structures rather than convective cells in the rest of this paper.) The convection analysis is also used to explain the variability of domain-averaged STHF and total cloud liquid water content found in Wenta and Herman [17
The goal of the research described in the second part of the paper (Section 3.2
) is to develop a formula that would allow incorporating the FSD-related effects in the calculation of surface turbulent moisture heat flux (further referred as surface moisture heat flux, or SMHF). More specifically, the goal is to propose a correction factor, dependent on floe size, which might be used in combination with the existing, ice-concentration-based flux formulae in order to improve their accuracy over fragmented sea ice. The proposed correction takes a form of a coefficient, denoted with
, which describes the ratio between the surface moisture flux
calculated with two different methods:
. In the first method, the effects of FSD are properly accounted for as the surface and atmospheric quantities on the sub-grid scale are taken directly from high-resolution numerical simulations. In the second method, the flux is determined with one of the algorithms described earlier, i.e., it is based on area-averaged properties of the surface and the atmosphere. Obviously, the computation of
depends on the availability of surface and atmospheric properties present in a given atmospheric model. As described in detail in Section 3.2
can be formulated as a function of area-averaged wind speed, ice concentration and floe size, so that, for given ice and wind conditions,
can be computed in a traditional way and corrected for FSD effects by multiplying it with
It must be stressed that the proposed correction is based on idealized modeling results and must be validated with observations and further modeling in the future. Therefore, it can be understood as the first step towards a parametrization suitable for implementation in NWP models.
We have shown in this study that the floe-size distribution plays an important role in sea ice–atmosphere interactions in the polar regions. It controls the spatial arrangement of sea ice and open water areas together with the associated convective structures in the ABL. Convective updrafts occur in every one of our simulations, but their intensity differs with varying FSD due to the changing extent and strength of breeze-like circulation. Our results are similar to the study of Esau [22
], who showed that secondary circulations like the ones found in our modeling results might be of large importance in the studies of ABL convection. Furthermore, his results do not entirely agree with the assumption that narrow leads produce more intensive fluxes, claiming that the associated processes are much more complicated and still not fully understood. Furthermore, in their study of atmospheric response to various configurations of polynya, Dare and Atkinson [16
] point out that the spatial arrangement of polynyas influences the magnitude and spatial distribution of vertical heat transfer in the atmosphere. The spaces between the floes in our results vary from a few meters to several kilometers, but in order to fully understand the atmospheric processes above them we also have to examine the size of adjacent floes and the extent of the associated breeze-like circulations. Therefore, our results agree with the suggestions of [16
] that, when studying the atmospheric response to sea ice fragmentation, the effects related to the spatial arrangement and size distribution of leads and floes must be considered.
Large local differences in atmospheric properties over ice with different floe-size distributions, associated with convective structures, are disregarded when the atmospheric quantities like air temperature or humidity are averaged over the whole model domain. Furthermore, Wenta and Herman [17
] pointed out that even when all other initial conditions remain identical, different results of area-averaged fluxes are obtained for the same ice concentration, but different FSDs—which further indicates that the lack of FSD information in NWP models may be at least partially responsible for the observed errors in the simulations over polar regions [1
]. Based on our WRF modeling results, we propose a simple correction method in the form of an
coefficient that might be used in computation of the surface moisture flux from area-averaged surface and atmospheric variables. The value of
depends on the mean wind speed, sea ice concentration and the median floe radius from particular FSD. In view of the fact that models do not include FSD or the size of the floes in particular grid cell and that presented results have not been validated with observational data, the proposed coefficient is not yet applicable for the GCMs. However, it indicates that the processes associated with the FSD and atmosphere interactions can be parametrized in a simple, computationally efficient way. With the development of models like Horvat et al. [36
], Roach et al. [38
] such straightforward solution, which includes the effects of different spatial arrangement and sizes of the floes, may soon become applicable in GCMs.
In general, the present research complements that of Wenta and Herman [17
] and provides an explanation of the effects described there. The differences in area-averaged values of turbulent fluxes, water vapor and liquid water content in simulations with different FSDs found in [17
] are explained based on the fact that FSD determines the spatial arrangement and intensity convection, which in turn controls the exchange of heat and moisture. Furthermore, the problem of high variability of local atmospheric conditions due to surface heterogeneity is addressed further and the first attempt to develop a coefficient integrating the effects of FSD into the calculation of surface moisture flux is made. The development of parametrization in the form of coefficient
will be continued in the subsequent research based on field observations and modeling with the full version of WRF model. Nonetheless, the results presented here provide new insights into the complex influence of sea ice fragmentation on the atmospheric boundary layer. In our opinion, the effects of FSD on the ABL can and should be included in the regional and global weather and climate models, as it is very likely that they are responsible for some of the known inaccuracies present in models results. Furthermore, it is expected that due to the ongoing warming of the Arctic region, sea ice fragmentation in winter is going to become more prevalent, thus further increasing the importance of properly taking into account sea ice heterogeneity in numerical models. Undoubtedly, more field campaigns are needed to improve our understanding of specific processes in situations with various sizes, shapes and orientations of ice floes and open water areas.
It must be noted that present study is based on an idealized model setup and that the results have not been validated with observations. It is a serious limitation. However, as already discussed in the previous paper [17
], the crucial aspects of the simulated processes in our results are realistic and have been observed in other modeling and observational studies in the polar regions (e.g., [8
]). Furthermore, the described analysis provides an invaluable support in planning of our further research which will include observations and modeling of atmospheric response to sea ice fragmentation in the Bay of Bothnia (Baltic Sea). The aim in this follow-up study is to confirm, based on the field data that FSD does affect the ABL properties and circulation, and to determine to which degree it controls the spatial arrangement and intensity of convective structures within the ABL.
Finally, it is worth noticing that the correction method presented here, based on a coefficient defined as the ratio of the “true” and biased value of the moisture flux, is not applicable to the second component of the total turbulent heat flux, namely the sensible heat flux. Whereas the values of the latent heat flux are always positive (with very few exceptions of extremely small negative values), the sensible heat flux can be both positive and negative, and our previous results [17
] showed that different estimation methods—analogous to Equations (5
) and (9
)—might produce results with a different sign, making a correction based on their ratio meaningless. Thus, a different approach will be necessary for an analogous parametrization of the sensible heat flux.