A new sea ice state dependent parameterization for the free drift of sea ice

Free drift estimates of sea ice motion are necessary to produce a seamless observational record combining buoy and satellite-derived sea ice motion vectors. We develop a new parameterization for the free drift of sea ice based on wind forcing, wind turning angle, sea ice state variables (concentration and thickness ) and ocean current (as a residual). Building on :::::::: thickness ::: and :::::::::::: concentration) :::: and :::::::: estimates :: of ::: the :::::: ocean ::::::: currents. :::::: Given : the fact that the spatially varying standard wind-ice transfer coefficient (considering only surface wind stress) has a structure as ::::: spatial :::::::::: distribution :: of ::: the :::::::::::: wind-ice-ocean ::::::: transfer ::::::::: coefficient 5 ::: has : a :::::: similar :::::::: structure :: to :::: that :: of : the spatial distribution of sea ice thickness, we introduce a wind-ice ::: take :::: the ::::::: standard :::: free ::: drift :::::::: equation ::: and :::::::: introduce :: a ::::::::::::: wind-ice-ocean transfer coefficient that scales linearly with ::: ice thickness. Results show a mean ::: bias : error of -0.5 cm/s (low-speed bias) and a root-mean-square error of 5.1 cm/s, considering daily buoy drift data as truth. This represents a 31 :: 35% reduction of the error on drift speed compared to the free drift estimates used in the Polar Pathfinder dataset (Tschudi et al., 2019b). The thickness-dependent wind transfer coefficient provides an improved seasonality and long10 term trend of the sea ice drift speed, with a minimum (maximum) drift speed in May (October), compared to July (January) for the constant wind transfer coefficient parameterizations which simply follow the peak in mean surface wind stresses. The :::: Over ::: the ::::::::: 1979-2019 ::::::: period, ::: the trend in sea ice drift in this new model is +0.45 cm/s decade−1 compared with +0.39 cm/s decade−1 from the buoy observations, whereas there is essentially no trend in the standard : a free drift parameterization (-0.01 :::: with : a ::::::: constant ::::::: transfer ::::::::: coefficient ::::: (-0.09 : cm/s decade−1) or the Polar Pathfinder free drifts (-0.03 :::: drift :::: input :::: data ::::: (-0.01 : cm/s 15 decade−1). The :::::: optimal : wind turning angle that minimize the cost function is equal of ::::::: obtained :::: from :: a ::::::::::: least-squares ::::: fitting :: is 25Â°, with a mean and root-mean square ::::::: resulting :: in : a ::::: mean ::::: error ::: and :: a :::::::::::::: root-mean-square : error of +2.6Â :: 3Â° and 51Â :::: 42Â° on the direction of the drift, respectively. The residual :::: ocean ::::::: current :::::::: estimates ::::::: obtained : from the minimization procedure (i.e. the ocean currents) resolves :::::: resolve key large scale features such as the Beaufort Gyre and Transpolar Drift Stream, and is ::: are in good agreement with ocean state estimates from the ECCO, GLORYS and PIOMAS ice-ocean reanalyses, and geostrophic 20 currents from dynamical ocean topography, with a root-mean-square difference of 2.4, 2.9, 2.6 and 3.8 cm/s, respectively. Finally, a repeat of the analysis on a two sub-section ::: two :::::::::: sub-sections : of the time series (preand post-2000) clearly shows the acceleration of the Beaufort Gyre (particularly along the Alaskan coastline) and an expansion of the gyre in the post-2000 ::::::::: post-2000s, : concurrent with a thinning of the sea ice cover and observations :: the :::::::: observed : acceleration of the ice drift speed and ocean current ::::::: currents. This new dataset is publicly available for complementing merged observations-based sea ice drift 25 datasets that includes :::::: include : satellite and buoy drift records.

. The offset between the seasonal cycles of wind speed and ice drift speed indicates that other sea ice state parameters , are essential for describing the seasonally varying drift of ice motion. This can also be understood in terms of an energy balance for sea ice, where the power input from the surface wind stress is mainly dissipated by the water drag as well as the internal ice stresses in compact ice regions, when ice interactions are important (Bouchat and Tremblay, 2014). Mapping the distribution of the wind-ice :::::::::::: wind-ice-ocean : transfer coefficient (based on passive microwave-derived ice 100 drift and geostrophic winds over the mid-1990s), Kimura and Wakatsuchi (2000) report a sharp contrast between seasonal ice zones, such as the Bering, Barents and Okhotsk Seas, where the value can reach 2%, and the Arctic interior where the value drops to 0.8% and below; a spatial pattern which, they hypothesize, relates to stresses internal to the ice pack, and therefore to ice thickness and concentration. The same spatial pattern is observed over the 2003-2017 period by Maeda et al. (2020), who additionally report on the seasonal cycle of the wind-ice transfer coefficient (minimum around 0.8% in March and maximum 105 around 1.1% in October), and on positive long-term trends across the whole Arctic. Interestingly, Maeda et al. (2020) note that the upward trend of the wind-ice transfer coefficient stops after 2010, particularly in regions where multi-year ice used to be prevalent. The marked differences in the seasonality of the surface wind stress and ice drift calls for a parameterization of the wind :::::::::::: wind-ice-ocean transfer coefficient that also considers ice state variables. The goal of this work is to develop such a parameterization for the wind-ice transfer coefficient (α). To the best of our knowledge, this has not been attempted before.

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The current study builds on the work of Lu et al. (2016) -who only considered a range of sea-ice concentration between 0-80 : 0 ::: and ::: 80 %, limiting the applications of their parameterization to the marginal ice zone. In contrast with these previous studies, we propose a simpler approach, in which the wind-ice transfer coefficient α is a function of the ice state, in the formulation for the free drift of sea ice. A state-dependent α can be conceptually understood as an integrated metric taking into account the spatial and temporal variability of both the atmosphere and ocean drag coefficients.

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Another challenge in estimating free drift ice motion vectors arises from the poorly constrained Arctic ocean surface currents under sea ice. Existing observational approaches to estimate surface currents in the Arctic Ocean include i) the use of ocean dynamic height derived from satellite altimetry, from which geostrophic currents can be derived (Armitage et al., 2017), ii) direct current measurement from ice-tethered acoustic Doppler current profiler (ADCP, McPhee, 2013), iii) using wind stress data and depth-integrated vorticity balance (Nøst and Isachsen, 2003) and iv) deriving the mean surface ocean circulation from 130 time-averaged sea ice drift data (Thorndike and Colony, 1982;Kimura and Wakatsuchi, 2000;Kwok et al., 2013). We expand on the approach of Thorndike and Colony (1982) for estimating the surface oceanic currents, using drifting buoys and wind data to produce updated estimates of the surface oceanic circulation in the Arctic.
One key contribution of this study is to produce a free drift product with documented errors (both spatially and temporally) that spans the shoulder season between spring and fall when satellite-based drift estimates are sparse. The hope is that this will 135 encourage a wider range of independent seamless ice motion datasets, also covering the summer period which is oftentimes avoided because of larger error in drift estimates from passive microwave. The need for a continuous dataset for ice tracking is clear (e.g. Pfirman et al. 2004;Krumpen et al. 2016;Williams et al. 2016;Newton et al. 2017;Mahoney et al. 2019;Belter et al. 2020).
Such a dataset is also included in the Sea Ice Tracking Utility, made publicly available recently on the National Snow and Ice Data Center website (SITU, Campbell et al. 2020), or the Alfred Wegener Institute ICETrack tool (Krumpen, 2018) -for 140 educational, scientific and field expedition planning purposes.
This paper is structured as follows. Section 2 introduces the buoy, sea ice, atmospheric and oceanic datasets. Section 3 describes the methodology for parameterizing of the wind-ice ::::::::::::: wind-ice-ocean transfer coefficient α. Section 4 presents the new estimates of free drift sea ice motion and quantifies the error with respect to buoy data. Section 5 summarizes the main findings presented in the paper.

Grid
We use a 25 km Equal Area Scalable Earth grid (EASE grid, Brodzik and Knowles, 2002) as a common grid for all datasets.

Sea ice thickness
Daily sea ice thickness is taken from the Pan-Arctic Ice Ocean Modeling and Assimilation System (PIOMAS) ice volume reanalysis (Zhang and Rothrock, 2003). PIOMAS is a coupled ice-ocean model, forced with atmospheric fields from the NCEP/NCAR reanalysis, and that assimilates sea ice concentration from the near real-time NSIDC product , based on the NASA Team algorithm (Lindsay and Zhang, 2006). PIOMAS correctly represents ::::::: captures the large scale structures of ice thickness in the Arctic; while being biased thick in thin ice regions and biased thin in thick ice regions (Schweiger et al., 200 2011). PIOMAS data is available in near-real time :::::::: year-round : for the full Arcticand year-round, and is stored on a generalized curvilinear coordinate system grid configuration with the pole shifted over Greenland.
The full convergence of the solution is reached within 5 iterations. Using this iterative procedure reduces the size of the matrix passed to the least-squares solver, and therefore greatly reduces the memory requirement. The initial guess for the iterative procedure is a value of one (unit-value) for all free parameters. The final solution (i.e. the minimum in the error function) is independent of the initial guess and the resulting wind-ice transfer coefficients ranges from 1-2.5% (in line with previous estimates) depending on whether α is constant or sea ice state dependent.

Results and discussion
Our goal is to derive a free drift parameterization that is bias-corrected with respect to the buoy drift data from IABP -and that is sea ice state-dependent in order to take into account the seasonality and long-term trends in the sea ice drift. This seasonality 300 is directly related to seasonal changes in sea ice thickness. The long-term changes in sea ice thickness associated with global warming are of the same order of magnitude as the seasonal change (Rothrock et al., 2008;Rampal et al., 2009), which implies that there will be a trend in α from the parameterization as well. For clarity, we use subscripts for denoting the wind-ice transfer coefficients and wind turning angle in the different parameterizations of free drift: α p , θ p : Polar Pathfinder parameterization (referred to as "standard" in the following Thorndike and Colony, 1982); α 0 , θ 0 : constant wind-ice transfer coefficient, no ocean 305 currents; α w , θ w : constant wind-ice transfer coefficient, including ocean currents; α h , θ h : thickness-dependent parameterization; as well as an additional free parameter β h for the thickness parameterization (see explanation below).

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A simple attempt at representing the dependence of the drift angle on sea ice thickness by a linear relationship was tested (see Appendix), but did not contribute to reducing the error on the drift direction. Leppäranta 2011 (eq. 6.7b) do a formal derivation for wind-ice turning angle in a free drift regime, showing that it is : it :: to :: be : a function of Coriolis forcing divided by the ocean drag. While this might help reduce the error, the introduction of the ice drift speed in the parameterization of the wind-ice turning angle removes the elegance of the simple linear free drift parameterization. Other interesting approaches 525 include Hongwei et al. (2020) and Park and Stewart (2016) who observe a cubic polynomial relationship between the wind turning angle and 10m wind speed, based on ice-tethered buoys deployed in the central Arctic in 2012. Future work will include testing parameterizations for the wind-turning angle in free drift models.

Surface ocean current estimates
A caveat of our approach is the use of a fixed ocean current climatology to calculate the ice motion estimates. The variations of 530 the sea ice drift speed have a significant seasonal signature on top of a long-term trend (Fig. 5a,b); and due to a strong coupling of the ice-ocean system in the Arctic, these variations of ice drift speed also drive variations of the surface currents. As explored by Meneghello et al. (2018) in model studies, the slowdown of the ice speed in the winter can lead to an inversion of the surface ice-ocean stresses; modulating the surface currents seasonally. With respect to the long-term trends, Armitage et al. (2017) report increased oceanic surface current speeds over the 2003-2014 period, concurrent with a thinner, weaker and looser sea ice cover.

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The inclusion of these two features of the oceanic currents variability in our free drift parameterization would help improve the representation of the seasonal cycle and the long-term trends in ice drift speed.
(2014) observe a stabilization of the freshwater content in the Beaufort Gyre in recent years. In a recent study, Armitage et al.

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Additionally, we present estimates of the ocean currents U w for the high and low sea ice drift speed seasons , :::: (Fig. :::::: 10a,b), respectively defined as July to December , :::::::::: (n=239,515), : and January to June :::::::::: (n=218,400). We refer to these as the 'summer' and 'winter' seasons, from the sea ice drift point of view. The currents tend to be faster in the summer, specifically :::::::: primarily in the Southern branch of the Beaufort Gyre, consistent with a seasonal reduction in ice thickness ( Figure ::: Fig. : 10c,e). We also note that the surface currents in Fram Strait and east of Greenland appear to be slightly slower in the summer, potentially explained 585 by the minimum in the wind speed seasonal cycle. Interestingly, the general winter/summer contrast in estimated ocean currents speed resembles the pre/post-2000s difference, which in both cases can be related to differences in ice thickness. The long term trend in sea ice thickness reduction affects the ice-ocean system in a way that is analogous :: to the seasonal cycle: the presence of perennial ice in the pre-2000s was favorable to drift conditions that were similar :: to the winter climatology, whereas the modern transition towards a seasonal pack ice constitutes an ice-ocean system that resembles the summer climatology.
In the proposed minimization procedure, the ocean currents appear as a residual (i.e. assumed to be :: the : part of the signal in sea ice drift speed that is not explained by surface winds ) Thorndike and Colony 1982) ::::::::::::::::::::::::: (Thorndike and Colony, 1982). The climatology of these surface ocean current estimates captures the general features of the Arctic Ocean general circulation, including the Beaufort Gyre, the Transpolar Drift Stream, and a fast outflow current along the eastern coast of Greenland. We explore climatological changes in the ocean state by retrieving the ocean currents for the pre-and post-2000s :::::::: post-2000 periods.

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Code and data availability. Code and data are available at https://web.meteo.mcgill.ca/~charles/freedrift/