Reconstruction of the Rheological Parameters in a Sea Ice Model with Viscoplastic Rheology

a. Viscoplastic sea ice model

1) Formulation

In the present study, we employed the sea ice model based on the formulation of Lemieux et al. (2008) with added basal stress parameterization Lemieux et al. (2015, 2016) and generalized Hibler (1979) yield curve. Equations of the model describe VP ice physics coupled with sea ice dynamics which is forced by the stresses τ exerted on ice through its interaction with the bottom τ_b, atmosphere τ_a, and the ocean τ_w:

$\rho hA({\partial }_t+f\mathbf{k}\times )\mathbf{u}=\text{div}\mathit{\sigma }+{\mathit{\tau }}_a+{\mathit{\tau }}_w,$(1)

${\partial }_{t}h=\text{div}(h\mathbf{u}),$(2)

${\partial }_{t}A=\text{div}(A\mathbf{u}).$(3)

Here, u = {u, υ}, h, and A are the 2D fields of sea ice velocities, ice effective thickness, and concentration, div is the divergence operator; and k is the vertical unit vector. With a VP formulation (Hibler 1979), σ is the 2D field of ice stress tensor defined through the deformation rate tensor $\dot{\text{є}}$:

$\mathit{\sigma }=\left|\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{xy}& {\sigma }_{yy}\end{array}\right|,\hspace{0.17em}{\sigma }_{ij}=2\eta {\dot{\text{є}}}_{ij}+\mathsf{E}[(\zeta -\eta )\text{Tr}(\dot{\text{є}})-P/2],$(4)

$\dot{\text{є}}=\frac{1}{2}\left|\begin{array}{cc}2u_x& {\upsilon }_x+u_y\\ {\upsilon }_x+u_y& 2{\upsilon }_y\end{array}\right|,$(5)

where $\mathsf{E}$ is the identity matrix and the pressure term P is parameterized as

$P=\hspace{0.17em}{P}^{\text{*}}hA\text{exp}\hspace{0.17em}[-\alpha (1-A)],$(6)

where $P^{\text{*}}$ and α are, respectively, the ice compressive strength and the compactness strength parameter. The typical values of the ice strength $P^{\text{*}}$ and α are listed in Table 1. According to Hibler (1979) and König Beatty and Holland (2010), ζ and η are the bulk and shear viscosities

$\zeta =\frac{P}{2\mathrm{\Delta }},\hspace{0.17em}\eta =\frac{\zeta }{e^2},$(7)

$\mathrm{\Delta }(\dot{\text{є}})=\frac{1}{e}{[(e^2-1){(\text{tr}\dot{\text{є}})}^2+2\text{tr}({\dot{\text{є}}}^2)]}^{-1/2}\hspace{0.17em},$(8)

where e is the ratio of the axes of the elliptical yield curve and k_T is the empirical parameter controlling the tensile strength. To avoid numerical singularities at $\dot{\text{є}}=0$, the values of $\mathrm{\Delta }(\dot{\text{є}})$ are limited from below by the additional parameter ${\mathrm{\Delta }}^{*}={10}^{-10}$, so that $\mathrm{\Delta }(\dot{\text{є}})=\max ({\mathrm{\Delta }}^{*},\mathrm{\Delta })$. The atmospheric, ocean, and bottom stresses in Eq. (1) were parameterized in accordance with Lemieux et al. (2015, 2016) and Hunke and Lipscomb (2010):

${\mathit{\tau }}_a=-C_a{\rho }_{a}A\left|{\mathbf{u}}_a^g\right|{\mathsf{R}}_{{\theta }_a}({\mathbf{u}}_a^g),$(9)

${\mathit{\tau }}_w=-C_w{\rho }_{w}A\left|\mathbf{u}-{\mathbf{u}}_w\right|{\mathsf{R}}_{{\theta }_w}(\mathbf{u}-{\mathbf{u}}_w),$(10)

where θ is the Heaviside step function, h_b is the ocean depth, ${\mathsf{R}}_{{\theta }_a}$ and ${\mathsf{R}}_{{\theta }_w}$ are 2 × 2 matrices rotating the velocity vector by the turning angles θ_a and θ_w counterclockwise, and ${\mathbf{u}}_a^g$ and u_w are the atmospheric wind and water velocities, respectively. The values of other parameters C_a, C_w, ρ_a, and ρ_w are listed in Table 1. Similar to the recent study by Panteleev et al. (2020), we focus on the retrieval of the spatial variability of $P^{\text{*}}$ and e from synthetic observations of the sea ice state in the 375 km × 280 km domain with horizontal resolutions of 7.5 and 7 km, respectively. The dimension of the model grid was 50 × 40 elements. The sea ice was forced by spatially and temporally varying wind, ocean currents, and sea ice inflow (or outflow) through the open boundaries.

Table 1

Model and assimilation system configuration parameters.

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b. Variational DA with VP sea ice model

a. OSSE descriptions

Conducting the OSSEs is a first step to analyze the performance and skill of a DA system. These experiments should be performed before applying the DA algorithm to the real observations.

The conventional OSSE methodology (e.g., Nitta 1975; Houtekamer and Mitchell 1998; Francis et al. 2018) includes several steps (see Fig. 1). First, “true” solutions of the sea ice model are generated with a given atmospheric forcing, initial-state conditions, and spatially varying RP’s distributions. For the experiments with zonal winds (see below) we specified smooth initial sea ice concentration and thickness distributions and define the initial velocity conditions by a 10–time step (100-min) forward model integration starting from a rest state with all other initial variables and parameters the same. For the experiments with cyclonic winds (see below) all initial conditions and forcing were derived from the model run configured for larger domain.

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Schematic of an OSSE. The general process of an OSSE is to generate noisy observations from an a priori natural solution, to assimilate/not assimilate those data into an inaccurately initialized model, and to compare the resulting solutions with the true one. Dashed black arrows correspond to noisy processes, and hollow arrows correspond to application of the forward model. Solid black arrows correspond to system input/output, and the blue arrows identify the quantities compared in the experiment validation. FG, OPT designate the first guess, optimized control, respectively. DA = data assimilation.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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As a second step, we generated synthetic data by contaminating this true, known solution with spatially correlated noise whose amplitude and scale depends on the type of observation the data are intended to represent. The level of contamination and spatial/temporal decorrelation scales are discussed below. On the third step, the sea ice model is reinitialized with inaccurate first guess (FG in figures) initial conditions and homogeneous distribution of some of the RPs distributions (e.g., $P^{*}$ or $P^{*}$ and e) to mimic the conventional approach with constant rheological parameters $P^{*}$ and e.

On the fourth step, the variational assimilation scheme presented in section 2 is applied with perfect model assumptions to determine the optimal control C_opt by assimilating observations during the data assimilation window of 1 day. The optimal model state trajectory and parameters (OPT in figures) result from initialization at the start of day 0.0. Finally, the optimized RP’s and sea ice state is compared with “true” sea ice model solution. The latter allows to define the performance and skill of the developed data assimilation algorithm

The major goal of the conducted OSSEs is to evaluate the feasibility of reconstructing the RPs through assimilation of the sea ice velocity, thickness, and concentration observations in the central part of the Arctic Ocean where sea ice concentration is close to 100%

Note also, in pack ice conditions (sea ice concentration > 97%), viscous-plastic rheological forcing in Eq. (1) plays a more important role when sea ice converges [div(u) < 0], because sea ice divergence will eventually result in the decrease of the sea ice concentration and the exponential decrease of the sea ice pressure P [Eq. (6)], which controls the magnitude of the rheological term in the momentum balance. Note that the impact of the sea ice divergence on the rheological term may be not so straightforward if another rheological hypothesis are applied.

Because of this we consider two types of OSSEs. The first OSSE simulated a rather strong convergence in the entire domain, which was achieved by setting convergent wind from opposite directions and intense inflow of the sea ice into the domain through the western boundary. In the second experiment the domain was forced by the gyre-shaped winds, ocean currents, and open boundary forcing extracted from a larger domain (80 × 70). As a result, ice convergence was much weaker and changed sign to divergence in some areas. In both OSSEs, we analyze the feasibility of the recovering of the spatially varying $P^{*}$ with a given distribution of e and, in addition, explore the case when spatial distributions of both $P^{*}$ and e are unknown and need to be reconstructed. The impact of the higher inaccuracy of the sea ice thickness observations for both OSSEs was also analyzed.

Sea ice pressure term P is also controlled by compactness strength parameter α in the Eq. (6). Note that impact of parameter α is essential only in the regions where sea ice concentration less than 100%. At the same time, any reasonable value (∼15–30) of the compactness strength parameter will significantly decrease sea the ice pressure [Eq. (6)] in the regions with sea ice concentration below 90%. That will decrease the impact of the rheological term in the momentum Eq. (1) and make the inversion of RPs inaccurate. Because of that, in order to evaluate the possibility of the inversion of both $P^{*}$ and α we conducted two additional OSSEs with initial and boundary sea ice concentration multiplied by 0.95. For simplicity, we also suggested that α does not vary in space, i.e., α was an unknown constant, which should be defined together with spatially varying $P^{*}$.

A list of the OSSEs with short descriptions is given in Table 2. The maximum number of control variables associated with the initial conditions (the number of ice model grid points occupied by the sea ice thickness, concentration and velocity fields) was about 8000. As it was mentioned above, the RP control fields were defined on a coarser grid (Δx = 10δx) and bilinearly interpolated on the model grid of the respective OSSEs. Thus, the maximum dimension of the RP control vector never exceeded (50/10 + 1) × (40/10 + 1)2 = 60 elements. Here, 50 and 40 are the grid dimensions. In all the experiments, we assumed that sea ice thickness, concentration and velocity observations were available at all the space–time grid points of the model domain. The atmospheric and oceanic forcing fields were contaminated by the large-scale noise, with decorrelation scales of ∼150 km. The magnitudes of the respective error fields were ∼10% for the ocean and atmosphere.

Table 2

List of the performed experiments.

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b. Synthetic data

In all OSSE experiments, we used three types of simulated sea ice observations, trying to keep the magnitude of respective errors close to realistic values, which could be derived from the MOSAiC or similar experiment.

In particular, we adopted rather accurate (0.005 m s⁻¹) sea ice velocity observations for the entire region because the new methods of sequential SAR image comparison can retrieve ice velocities with an accuracy of 0.005 m s⁻¹ (Komarov and Barber 2014). Combining these remotely sensed sea ice velocities with the direct observations from sea ice buoys and SAR velocities will likely provide even higher accuracy of the sea ice velocity observations. Thus, in the OSSEs reported below, inaccuracy of sea ice velocities is set to 0.005 m s⁻¹, which corresponds to ∼5%–7% of the relative sea ice velocity errors. Similarly, we adopted the same relative error (∼5%) of the sea ice concentration observations that are very well observed from multiple satellites, SAR images, and radars. Note that sea ice concentration in the central part of the Arctic Ocean is very close to 100% and because of that, in the most of our experiments, sea ice concentration ranges between 97% and 100% and only in two experiments we reduced sea ice concentration by 5%. Panteleev et al. (2020) showed that the sea ice thickness observations are very important for the proper recovering of the spatially varying rheological parameters. Taking into account that MOSAiC field experiments provide very accurate in situ sea ice thickness observations, we set the sea ice thickness observational error at ∼0.1 m, having in mind that multiple buoys and acoustic observations may provide this accuracy for the entire region. We also assumed that all observations are available in all the space–time grid points of the model domain.

a. Optimization of the sea ice strength

The impact of the sea ice strength on the sea ice state (i.e., velocity, thickness, and concentration) is significantly stronger than the impact of the yield ellipse axes ratio e. To evaluate the feasibility of optimizing $P^{\text{*}}$, the VP model was configured in a 350 km × 290 km rectangular domain with true values of sea ice thickness at t = 0 and 1 day shown in Figs. 2c and 2d. The true values of $P^{\text{*}}$ and e varied as shown in Figs. 2e and 2f with spatial variation in the following ranges: $17.5\le P^{\text{*}}\le 37.5\text{kN}{\mathrm{m}}^{-2}$ and 1.1 ≤ e ≤ 2.9. These ranges were adopted from various studies (e.g., Hibler and Walsh 1982; Kreyscher et al. 1997, 2000; Tremblay and Hakakian 2006; Lemieux et al. 2016). The true wind forcing was blowing from opposite directions with stationary position and strength, gradually increasing 1.2 times during the 1-day assimilation window. The resulting wind at t = 1 day is shown in Fig. 2b and has a maximum magnitude of 20 m s⁻¹. The stationary ocean currents were specified as shown in Fig. 2b with a maximum amplitude of 0.1 m s⁻¹, which is a reasonable amplitude for the central part of the Beaufort Sea according to our experiments with CICE6 sea ice model. The zonal open boundary sea ice velocity of 0.2 and 0 m s⁻¹ were specified along the western and eastern boundaries, respectively, without any variation in the meridional direction. Along the northern and southern open boundaries, the zonal velocity was interpolated linearly, while the meridional velocity was set to zero at all open boundaries. Note that Dirichlet open boundary conditions are different from Neuman type open boundary conditions utilized by Lemieux et al. (2008). That is because 4Dvar data assimilation minimize the difference between sea ice state and sea ice observations and updates open boundary conditions on each iteration to provide an optimal fit to the data inside the domain. Initial sea ice velocity conditions were determined by a 100-min model integration starting from rest, with all other initial variables and parameters being the same.

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Parameters of the true solution in the experiments CONV_P and CONV_P_e: (a) div(u) (×10⁻⁶ s⁻¹) for the time 0.5 days. White contour corresponds to div(u) = 0. (b) Sea ice velocity (blue arrows), wind (red arrows), and ocean currents (black arrows) at t = 1 day; (c),(d) sea ice thickness at t = 0 and t = 1 day; (e),(f) spatial distributions of $P^{*}$ and e.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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This type of forcing and sea ice rheology causes a strong convergence in the entire region with a maximum amplitude of −3.8 × 10⁻⁶ s⁻¹ (experiment CONV_P in Table 2). The meridional region of strongest convergence along the 120–180 km coincides with minimum in the $P^{*}$ distribution (Figs. 2a,e) resulting in a substantial (0.25 m) increase of the sea ice thickness in this region from 1.25 m up to 1.5 m in a single day. Sea ice thickness changes in other parts of the domain are less profound, which is probably due to higher $P^{*}$ and weaker wind forcing. The changes in the sea ice concentration are very moderate because the initial sea ice concentration distribution is close to 100% everywhere. Because of this, the impact of $P^{\text{*}}$ and e optimization on sea ice concentration is not discussed below, although we note that, in all OSSEs, a certain improvement of sea ice concentration distributions was observed.

Noisy sea ice concentration, thickness and velocity observations were generated by adding spatially decorrelated noise (with decorrelation scales of 50 km) to each of the state variables at the beginning and at the end of the assimilation window and linearly interpolated in time.

The simulated data mimic realistic observations and errors such as those obtained from sources discussed above. In all the experiments, we did not introduce any bias to ice observations and suggested that sea ice model is also nonbiased. The bias-free hypothesis are common assumptions in most of the existing DA systems. The initial conditions for the first-guess solution were generated in a way similar to the noisy observations, with slightly larger decorrelation length scales for sea ice concentration, thickness, and velocity and spatially uniform values of $P^{*}=29.167{\text{kN}\hspace{0.17em}\mathrm{m}}^{-2}$. The wind and ocean forcing were also contaminated, as discussed above. Figure 3a shows that the first-guess sea ice thickness distribution differ significantly from the true sea ice thickness (Fig. 2d) with a relative error of 0.55 estimated as

${\text{err}}_h={\left[{\displaystyle \sum {(h_{\text{fg}}-h_{\text{true}})}^2}/{\displaystyle \sum {(h_{\text{true}}-{\overline{h}}_{\text{true}})}^2}\right]}^{0.5},$(14)

where the overbar denotes the space–time averaging operator. The region of the most profound difference between h_fg and h_true coincides with the minimum in the true $P^{*}$ suggesting that the incorrect uniform $P^{*}$ in the first-guess solution strongly affects the sea ice thickness distribution even for a short (1-day) period of time.

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Results of the CONV_P experiment: (a) sea ice thickness at t = 1 day in the first-guess solution; (b) optimized $P^{*}$; (c) optimized sea ice thickness at t = 1 day. (a),(c) The relative error err_h values; (d) difference between optimized and true $P^{*}$ Yellow contours show the div(u) for the time 0.5 days (see Fig. 2a).

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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Similarly to Stroh et al. (2019), the optimization was conducted in three steps. First, we optimized initial and open boundary sea ice velocity, thickness, and concentration conditions [C₀, C_OBC]. Then we sequentially optimized rheological components of the control vector C_p and finally conducted an additional optimization of the full control vector C = [C₀, C_OBC, C_p]. Note that the simulated sea ice velocity, thickness, and concentration observations efficiently constrain the respective initial conditions and, thus, provide a more accurate first guess for the final optimization of the entire control vector.

Figures 3b and 3d show the optimized $P^{*}$ distribution which is very close to the true $P^{*}$ in Fig. 2e. Note that our optimization succeeded in a good reconstruction of the minimum in $P^{*}$ along the 150-km meridian, while the shape of the maximum along the 300-km meridian was reconstructed with higher errors. This is due to weaker sea ice convergence in these regions and less impact of the rheological term on the local ice dynamics (Fig. 3d). The optimization of the spatial distribution of $P^{*}$ results in a significant improvement in the sea ice thickness distribution and a reduction of relative errors of up to 0.24 (Fig. 3c).

Interestingly, that maxima of h_opt (1.5 m) is slightly smaller than maximum in h_true distributions (1.6 m) due to wider area with minimum $P^{*}$ along 150-km meridian. There is a slight (0.05-m) difference between h_opt and h_true in another area of the model domain. Again, this is due to the better reconstruction of the minimum in $P^{*}$. There is also a significant improvement in the optimized velocity field ${\overline{[{(u_{\text{opt}}-u_{\text{true}})}^2+{({\upsilon }_{\text{opt}}-{\upsilon }_{\text{true}})}^2]}}^{0.5}=0.01\mathrm{m}{\mathrm{s}}^{-1}$ with respect to the first-guess velocity fields ${\overline{[{(u_{\text{fg}}-u_{\text{true}})}^2+{({\upsilon }_{\text{fg}}-{\upsilon }_{\text{true}})}^2]}}^{0.5}=0.03\mathrm{m}{\mathrm{s}}^{-1}$.

The opposite wind forcing over the 350-km domain cannot be considered as a typical environmental condition in the central Arctic, so, we conducted a GYRE_P experiment with a more realistic gyre-like cyclonic wind and open boundary conditions, extracted from the sea ice model solution, configured for a larger domain. The initial/open boundary sea ice concentration was set to 1 everywhere, while the initial/open boundary sea ice thickness, ocean currents, $P^{*}$, and e were the same as in CONV_P experiment (Fig. 2). The gyre-like wind has a maximum magnitude of 18 m s⁻¹ at the beginning of the assimilation window and gradually increases to 22 m s⁻¹ by the end of model integration (Figs. 4c,d). Under such forcing, sea ice rotates counterclockwise, with a maximum speed of 0.13 m s⁻¹ at the beginning and 0.21 m s⁻¹ at the end of model integration period (Figs. 4c,d). Similarly to the CONV_P experiment, sea ice converged in most parts of the domain, with a maximum convergence of ∼−2 × 10⁻⁶ s⁻¹, which correlates well with the minimum in the $P^{*}$ distribution. However, there were also substantial areas where the sea ice weakly diverged. The resulting sea ice thickness distribution indicates an increase by 0.15 m, in the center, due to sea ice convergence (Fig. 4b).

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True solution in the experiments GYRE_P and GYRE_P_e: (a) div (u) (10⁻⁶ s⁻¹) for t = 0.5 days. White contour corresponds to div(u) = 0. (b) Sea ice thickness at t = 1 day; (c),(d) blue arrows—sea ice velocities at t = 0 and t = 1 day. Red and black arrows show the wind and ocean currents.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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The observations and the first-guess control vector were set in the same way as in the CONV_P experiment; i.e., $P_{\text{fg}}^{*}$ was set to 29.167 kN m⁻². The first-guess sea ice thickness, by the end of the period of integration, is relatively smooth (Fig. 5a) and does not have the maximum along the 150 km meridian, which is clearly seen in the true solution (Fig. 4b). This is due to a uniform $P_{\text{fg}}^{*}$. Optimization was conducted in a similar manner as described above and Fig. 5b shows the optimized $P_{\text{opt}}^{*}$ distribution. A comparison with the true $P_{\text{true}}^{*}$ (Figs. 5d, 2e) reveals some difference along the northern and southern boundaries, but, in the central part of the domain, the $P_{\text{opt}}^{*}$ $P_{\text{opt}}^{*}$ has the same configuration of maxima and minima and their absolute values as $P_{\text{true}}^{*}$ distribution. The optimization of $P_{\text{opt}}^{*}$ also provides significantly more accurate optimized sea ice thickness distribution (Fig. 5c) which now has several maxima whose position coincides with the positions of the maxima in the true sea ice thickness field (Fig. 4b).

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Results of the GYRE_P experiment: (a) sea ice thickness at t = 1 day in the first-guess solution; (b) optimized $P^{*}$ ; (c) optimized sea ice thickness at t = 1 day. (a),(c) The relative error magnitudes err_h; (d) difference between optimized and true $P^{*}$ Yellow contours show the div(u) (see Fig. 4a).

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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To evaluate the impact of the inaccuracy of the sea ice thickness observations, we conducted a series of CONV_P and GYRE_P experiments, gradually increasing the relative noise in the sea ice thickness observations and evaluating the difference between $P_{\text{opt}}^{*}$ and $P_{\text{true}}^{*}$, $d_p^{\text{opt}}={\overline{{(P_{\text{opt}}^{*}-P_{\text{true}}^{*})}^2}}^{0.5}$. The results in Table 3 show that decreasing the accuracy of the sea ice thickness observations results in significant degradation of the $P_{\text{opt}}^{*}$ distribution. However, taking into account that spatial variability of $P_{\text{true}}^{*}$ is $d_p^{\text{true}}=\text{std}(P_{\text{true}}^{*})={\overline{(P_{\text{true}}^{*}-\overline{P_{\text{true}}^{*}}){\hspace{0.17em}}^2}}^{0.5}=5.77\text{kN}{\mathrm{m}}^{-2}$, even relatively high sea ice thickness (∼0.2 m) errors allow satisfactory reconstruction of the $P_{\text{opt}}^{*}$, even in the case of the realistic GYRE_P experiment with cyclonic wind. Naturally, the reconstruction errors are nearly 2 times smaller than those obtained in the experiment with strong convergence (CONV_P).

Table 3

$d_p^{\text{opt}}$ (N m⁻²) as a function of std(h_opt − h_true) in the sea ice thickness observations. $d_p^{\text{opt}}$ is shown in the last column.

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b. Optimization of the sea ice strength and yield curve axes ratio

The yield curve axes e is another important rheological parameter, which can formally be reconstructed through 4Dvar. However, the impact of this parameter on the rheological term in the momentum Eq. (1) is much weaker than the impact of $P^{\text{*}}$, because σ_ij is directly proportional to $P^{\text{*}}$ [see Eqs. (4), (6), and (8)]. To evaluate the feasibility of optimizing spatially varying $P^{\text{*}}$ and e, we conducted two additional OSSEs (CONV_P_e and GYRE_P_e), and set $P^{\text{*}}$ and e to be spatially homogeneous in the first-guess solution. Observational errors, contamination of the first-guess solution with large-scale noise and inaccuracy in the ocean currents and atmospheric forcings were set in a similar manner as in the experiments CONV_P and GYRE_P, discussed above. Optimization of two unknown distributions was conducted in a sequential manner; i.e., we initially optimized spatially varying $P^{\text{*}}$, then spatially varying e, and finally conducted the optimization of the entire control vector C = [C₀, C_OBC, C_p].

The major results of the CONV_P_e experiment are shown in Fig. 6. The optimization clearly provided a reasonable estimate of $P^{\text{*}}$ (Fig. 6a), with $d_p^{\text{opt}}=2.4\text{kN}{\mathrm{m}}^{-2}$. The difference between $P_{\text{opt}}^{*}-P_{\text{true}}^{*}$ is shown in Fig. 6c. This accuracy is comparable with the results of the CONV_P experiment, with twice less accurate sea ice thickness observations (Table 3), but still much smaller than the standard deviation $\text{std}(p_{\text{true}\hspace{0.17em}}^{*})=5.77\text{kN}{\mathrm{m}}^{-2}$. The quality of the reconstruction of e is not so successful (Figs. 6b,d). The $d_e^{\text{opt}}$ between optimized and true e is about 0.89, which is larger than $\text{std}(e_{\text{true}}^{})={\overline{{(e_{\text{true}}-\overline{e_{\text{true}}})}^2}}^{0.5}=0.52$. Note, however, that the major inaccuracies in e_opt are in the eastern and western parts of the domain (Fig. 6d), while the strong meridional maximum in the vicinity of 150-km meridian agrees well with the true e_true distribution. This is due to strong ice convergence in this region of the true solution (Fig. 2a) which results in a stronger impact of the rheological term in the momentum balance [Eq. (1)].

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Results of the CONV_P_e experiment. (top) Optimized (a) $P^{*}$ and (b) e distributions. $d_p^{\text{opt}}={10}^{-3}{\overline{{(P_{\text{opt}}^{*}-P_{\text{true}}^{*})}^2}}^{0.5}$ and $d_e^{\text{opt}}={\overline{{(e_{\text{opt}}-e_{\text{true}})}^2}}^{0.5}$ are shown on each panel. (middle) Difference between optimized and true (c) $P^{*}$ and (d) e. Yellow contours show the div(u) for the time 0.5 days (see Fig. 2a). (bottom) Sea ice thickness at t = 1 day in the (e) first-guess and (f) optimized solutions. The relative error magnitudes err_h are shown in (e) and (f).

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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Despite the inaccuracies in the reconstructed $P_{\text{opt}}^{*}$ and especially e_opt, the optimized sea ice thickness distribution (Fig. 6f) demonstrates a strong resemblance to the true sea ice thickness (Fig. 2d), with a relative error of 0.33, which is about 1.5 larger than the relative errors in the CONV_P experiment. This is due to the stronger impact of $P^{*}$ on the sea ice dynamics and the relatively accurate reconstruction of $P_{\text{opt}}^{*}$.

The major results of the GYRE_P_e experiment are shown in Fig. 7. Due to the weaker sea ice convergence, the reconstruction of $P^{*}$ is not as accurate as in the CONV_P_e experiment; however, the $d_p^{\text{opt}}=25.3\text{kN}{\mathrm{m}}^{-2}$ is still smaller: $\text{std}(p_{\text{true}\hspace{0.17em}}^{*})=5.8\text{kN}{\mathrm{m}}^{-2}$, and there is a clear similarity with the true solution in the locations at the maxima and minima along the 150 and 300 km meridians, respectively (Fig. 2e). This similarity agrees well with the minima in $(P_{\text{opt}}^{*}-P_{\text{true}}^{*})$ distribution shown in Fig. 7c.

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Results of the GYRE_P_e experiment. (top) Optimized (a) $P^{*}$ and (b) e distributions. $d_p^{\text{opt}}={10}^{-3}{\overline{{(P_{\text{opt}}^{*}-P_{\text{true}}^{*})}^2}}^{0.5}$ and $d_e^{\text{opt}}={\overline{{(e_{\text{opt}}-e_{\text{true}})}^2}}^{0.5}$ are shown. (middle) Difference between the optimized and true (c) $P^{*}$ and (d) e. Yellow contours show the div(u) for the time 0.5 days (see Fig. 4a). (bottom) Sea ice thickness at t = 1 day of the (e) first-guess and (f) optimized solutions. The relative error magnitudes err_h are shown in (e) and (f).

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0158.1

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The $d_e^{\text{opt}}$ between the optimized and true e is about 0.4, which is slightly smaller than std (e_true) = 0.52. There is also some portion of similarity between optimized e_opt and true e_true distributions: for example, in locations of the minima along the western and eastern boundaries and of the maximum in the central part of the domain. This similarity agrees well with the minima in (e_opt − e_true) distribution shown in Fig. 7d.

Due to partial optimization of $\hspace{0.17em}{P}_{\text{opt}}^{*}\hspace{0.17em}$ and e_opt, the optimized sea ice thickness distribution (Fig. 7d) also demonstrates some improvements with respect to the first-guess sea ice thickness and the reduction of the relative error from 0.6 up to 0.4 (Figs. 7c,d). Overall, the results of both experiments CONV_P_e and GYRE_P_e indicate the feasibility of reconstructing the spatially varying $P^{*}$ distribution and a somewhat less accurate reconstruction of the spatially varying e distribution.

c. Optimization of the sea ice strength and compactness strength parameter

The compactness strength parameter α is the second parameter which defines the sea ice pressure in Eq. (6). In the regions entirely covered by sea ice, the impact of the compactness strength parameter is negligible, but it can be comparable or even increase the impact of the $P^{*}$ in the regions where sea ice concentration varies between 90% and 100%. With smaller sea ice concentration the impact of the rheological term in Eq. (1) significantly decreases. In many sea ice model applications α is equal to 20, but some studies suggest that it could be smaller and may significantly impact the sea ice dynamics in the models (e.g., Ungermann and Losch 2018).

Because of that we conducted two OSSEs with strong convergence (CONV_p_α) and cyclonic wind (GYRE_p_α) and evaluated the possibility to define both $P^{*}$ and α assuming that e is known. Taking into account that α is the dimensionless parameter, which defines the impact of the sea ice concentration on the sea ice pressure we assumed that α is unknown and its value does not vary in space. The true $P^{*}$ distribution of sea ice velocity and sea ice thickness were the same as in other OSSEs, while initial sea ice concentration was multiplied by 0.95. In the first-guess solution we specified α_fg = 25 and uniform $P^{*}$. Forcing was the same as in other OSSEs. Optimization was conducted in 3 steps as described above. We also found that optimization of α during the first step accelerates convergence providing more accurate reconstruction. That is probably due to small dimension of the C_α control, but also could be due to strong impact of the α in the region with 95% of the sea ice concentration. Results of the optimization are shown in Fig. 8.