Available global records reveal regions of increasing fog occurrence over the ocean over long timescales and decreasing occurrence over land. connect a reduction in dense fog to a decrease in European aerosol loading during 1976–2006. Marine visual observations for 1950–2007 reveal a positive summertime trend in at least two parts of the world where fog is most frequent: expansive regions of the western North Pacific and North Atlantic, centered on the Kuril Islands and the Grand Banks, respectively . A relative paucity of observations in the Arctic prohibits similar historical analyses, but observed and expected increases in moisture availability and temperature motivate an exploration of both warm and cold fog occurrence (relative to 0 ∘C) in this region as well .

This study seeks to identify baseline long-term changes in warm fog and visibility that might be expected in the Arctic and North Atlantic marine regions. We assert that a growing number of climate simulations (e.g., ) permit an initial multiyear prediction. Our pragmatic assumption is that the processes resolved by different climate models, explicitly including sub-synoptic and larger scales, provide a basis for estimating the future trend in fog occurrence. Contemporary studies, both observed and modelled , highlight that fog formation and dissipation are indeed sensitive to these resolved scales. In an attempt to approach the fog-process scales, we apply a regional model downscaling of Hadley Center Earth System model (HadGEM2) global climate simulations to provide an enhanced representation of radiative processes and moist energy transports. As in situ observations provide a complementary measure of such processes, this study can be seen as an extension (by simple statistical modelling) of physical model downscaling.

It is convenient to estimate visibility for various climate scenarios, insomuch as parameterizations are available that employ common climate model variables like surface temperature or relative humidity (e.g., ). With multiple combinations of models and parameterizations, however, there are at least two related challenges to identifying a reasonably consistent baseline change. First, it is well known that different models may seek to represent or support a variable like surface temperature in subtly different ways. Bayesian data assimilation, for example, acknowledges a separate target analysis for each model grid . Another familiar inconsistency relates to spatiotemporal representations of a variable by models versus observations (e.g., ). Based on a prior averaging of cloud observations, conclude that input data representation is critical for tuned parameterizations of fog and visibility. Of course, differences in representation by separate models, for example, are prevalent even at a common resolution. That is, any aspect of a model may ultimately determine its representation, from initialization and core numerics to physical parameterizations and a final vertical extrapolation to the surface. We may recognize a need to map between two representations, but given sometimes subtle differences in the target of distinct observations, models, and analyses, as well as the challenges of mapping (e.g., ), representation differences are to be expected nonetheless.

A related challenge to identifying a baseline change in relative humidity and visibility is that rudimentary and advanced estimates alike can be sensitive to a representation of the variables used to calculate them. Observational support is given by their Fig. 10, where visibility varies during the course of a day between 50 km and a few meters within a relative humidity range of 92 %–98 %. By convention, fog is defined as visibility of less than 1 km and is often identified with a relative humidity of greater than 95 %. note that the lack of a consistent representation of surface relative humidity has led to numerous proposed visibility parameterizations, perhaps applicable to one model each. In our search for a baseline change among multiple models, because the calculation of relative humidity and visibility are sensitive to a representation of their input variables, it is instructive to explore whether systematic differences among models can be adjusted to yield so-called homogeneous datasets. In other words, we seek a baseline that nominally employs a common reference of visibility to which all our estimates can be calibrated.

We begin with a parameterization of visibility based on in situ observations of relative humidity with respect to water. Because marine observations (e.g., ships) generally include air and dew point temperature (and thus relative humidity), they provide a proxy representation for the climate models of interest. Following , however, a parameterization of warm fog can be more accurately expressed in terms of condensation nuclei, liquid water content, and drop size distribution (among other variables). By comparison, rudimentary parameterizations are sometimes described as yielding a form of “relative humidity mist”. We also acknowledge that a model approximation of warm fog can cover a domain (i.e., a grid box) that is larger than an entire marine fog event. Our chosen parameterization is expected to capture neither the intensity of warm fog events nor processes that govern cold fog below 0 ∘C nor ice fog below about -20 ∘C . Finally, in selecting an in situ based parameterization, we consider the corresponding in situ representation (i.e., of the input variables air and dew point temperature) to be our target or true representation. Although any estimate of visibility involves a parameterization that is at least partly empirical, and almost as challenging to relate to prognostic model variables as fog itself , this in situ representation is not necessarily optimal. However, it is a separate ongoing challenge that, even for a precise instrument, measurement error ranging from 10 % to 50 % (and more in Arctic winter conditions) is common .

A somewhat broader accommodation of targets for truth and error is explored in the context of more than one visibility parameterization by . This parallel exercise points to relative humidity as a key source of information and allows us to focus on an in situ based, midlatitude parameterization. To establish consistent baseline 21st century trends for either the Arctic or North Atlantic, we focus on a homogeneous calibration of gridded historical estimates of surface air and dew point temperature. Such a calibration is called homogeneous because it seeks to make one dataset as consistent as possible with another. In accordance with our assumption of an error-free ICOADS reference, we adopt ordinary linear regression as our statistical calibration model (a justification is given in the Appendix). Ship and fixed platform marine observations are then taken as a large-scale reference for all model simulations, either directly or indirectly via surface variables of the ERA-Interim reanalysis . Individual datasets that nominally overlap with each other during the 1979–2004 historical period are described in Sect. 2. A visibility parameterization and two exploratory methods of linearly reconciling climate model simulations to observations are discussed in Sect. 3. The resulting high relative humidity and low visibility estimates are then used in Sect. 4 to indicate possible 21st century trends in fog. Conclusions are provided in Sect. 5.

A linear calibration of collocated air or dew point temperature is capable of matching the mean and variance of HadGEM2, WRF, and ERA distributions to those of ICOADS. In spite of a preexisting conformity of ERA surface variables to in situ observations , the assumption of no ICOADS error yields, unsurprisingly, a slight adjustment of ERA temperature and dew point (Table ). Also as expected, adjustment is smaller (i.e., the additive component is closer to zero and the multiplicative component is closer to one) when the ICOADS and ERA collocation data are separately ranked (single-step) than when they are unranked (two-step, where the first step preserves synoptic pairing). The single-step adjustment of HadGEM2 and WRF generally involves non-negligible additive and multiplicative components, whereas in two-step adjustment (to ERA), the multiplicative component is negligible (equal to one) and the additive component is similar in the Arctic and North Atlantic (cf. Fig. ). For brevity in the remainder of this section, single-step calibration is discussed in the context of our two-step results.

Figure a–d reveal good similarity in ERA (red) and ICOADS (green) distributions of air and dew point temperature, both before (dashed) and after (solid) calibration. As in Fig. , WRF is a nested model whose representation of temperature differs from the HadGEM2 driving model, and both differ somewhat from the ERA and ICOADS distributions. The calibration of HadGEM2 is more apparent than for WRF, with a shift of about 2 ∘C in all temperature distributions (Table ). Essentially by design, however, differences in the shape of each distribution are unchanged after both the single-step and two-step calibrations. Figure e, f reveal greater variation in the representations of relative humidity and less similarity with ICOADS. The contrast with WRF is most apparent, but in general, ICOADS observations sample more evenly a range in relative humidity from 45 % to 50 % to a limit of just over 95 %. In turn, a linear calibration of temperature has the desired impact of shifting all gridded relative humidity distributions towards drier conditions, but because uncalibrated WRF relative humidity is quite frequently high, linear calibration shifts the peak of its distribution by as much as 5 % to 10 %. This occurs following both the single-step and two-step calibrations and highlights the challenge of conserving relationships between variables in univariate (marginal) adjustments .

Comparison of the single-step and two-step calibrations suggests that, in this study, it is more appropriate to employ an additive calibration. As noted above, if ERA-ICOADS differences are ignored, then the single-step calibration is multiplicative, whereas the two-step calibration is additive (Table ). Both are applied locally in time and space, but unfortunately, the annual and areal averages of a single-step calibration (not shown) do not yield the degree of similarity shown in Fig. . Both before (Fig. ) and after (Fig. ) two-step calibration, we find that divergence of the RCP4.5 and RCP8.5 trends occurs more so in the Arctic than in the North Atlantic, after 2050, and for temperature, dew point, and specific humidity. The RCP4.5 and RCP8.5 relative humidity and visibility trends are quite similar throughout these simulations. To the extent that relative humidity increases and visibility decreases, this is predominantly found in the Arctic. We estimate the 21st century visibility decrease (in kilometers) to be in the range of 8 %–12 % in the Arctic and 0 %–5 % in the North Atlantic.

Marine environments of the Arctic and North Atlantic can be characterized by an observed peak in relative humidity that is near saturation, followed by an abrupt limit beyond this value that represents the physical process of condensation, which often includes the fog formation process of interest (Fig. a, b). Thus, although our large-scale calibration aligns each distribution with an ICOADS distribution in some mean sense (e.g., with a slight shift in ERA and HadGEM2 relative humidity), accompanying our linear calibration of WRF is the following interpretive burden: there is a notable impact on the frequency of relative humidity near 100 % (i.e., a decrease of about an order of magnitude) and on peak relative humidity (a drying of 5 %–10 %) in the Arctic and North Atlantic. The occurrence of fog is often associated with relative humidity above 95 % , and while this is somewhat unchanged in the HadGEM2 calibrated distributions (Fig. ), its frequency is considerably reduced for WRF.

Figure  depicts the impact of linear calibration on the WRF and HadGEM2 RCP4.5 distributions and their 21st century trends in relative humidity (i.e., as a fraction of marine coverage in early and late 30-year periods). The RCP8.5 distributions are nearly the same (not shown). The number of gridded marine samples is 2 to 4 orders of magnitude larger than the number of ICOADS collocations, but as expected, two-step calibration has the same impact as on the historical distributions (Fig. ). In other words, if we allow that condensational processes are shifted from near 100 % to just greater than 90 % relative humidity in the calibrated WRF RCP distributions, then differences in the early and late 30-year distributions are otherwise the same before and after our simple linear calibration. By accommodating this shift, we find the same signal in Fig.  that we find at large scales in Fig. . That is, both WRF and HadGEM2 reveal an increase in the frequency of fog-producing (high) relative humidity. The Arctic enhancement in each model (blue shading) is over 10 % of the peak frequency and is roughly double that of the North Atlantic.

The basic premise of this study is that models, observations, and analyses target distinct representations of the marine surface, with similarly distinct forms of bias relative to an unknown, fully supported representation (i.e., the “broad” definition of truth in ). Because we recognize different target representations, there is not necessarily a need to calibrate them.

Homogeneous linear calibration to a singular in situ reference is prompted by our choice of visibility parameterization, but the symmetry recognized by , , , and others invites us to consider the reverse calibration of in situ air and dew point temperature using a numerical model or gridded analysis as a reference. Moreover, this accommodation of symmetry may facilitate a less restrictive form of linear calibration.