Data assimilation of sea ice concentration into a global ocean–sea ice model with corrections for atmospheric forcing and ocean temperature fields

Abstract

A multivariate data assimilation experiment was conducted in order to improve the global representation of both the ocean and sea ice fields through the inclusion of sea ice concentration (SIC) data. Our method corrects the surface forcing and ocean temperature fields (as well as the SIC field) through the use of three-dimensional variational analysis. The adjustments to surface air temperatures resulting from the SIC assimilation are estimated on the basis of two constraints. First, we assume that the interfacial temperature difference between the surface air and the average value at the “top” of the grid (which represents a weighted mean according to the relative coverage of sea ice to open water within the grid) is maintained at the pre-assimilation value. Similarly, the vertical temperature structure for each of the five sea ice categories considered here remains unchanged throughout the assimilation. In making the necessary adjustments to upper-layer ocean temperatures, we again adopt a weighting procedure based on the condition that ice-free water temperature must remain the same. Thus, areas containing sea ice are allotted the freezing-point temperature such that the weighted mean value across the grid can be derived. The reproduction of the SIC field in both hemispheres is improved by incorporating the resulting corrections to the surface forcing and ocean temperature values, indicating that these boundary conditions produce results that are more consistent with the corrected SIC field in the sea ice model. The enhanced ocean–sea ice fields provide initial conditions that are better suited for coupled atmosphere–ocean–sea ice prediction experiments.

Appendix: Experiment with corrections for atmospheric forcing and ocean temperature fields

As described in Sect. 3, the SAT (specifically, the 2-m air temperature) of the forcing field is modified by assuming that the top temperature of the sea ice in each category does not change, and that the differences between the SAT and the grid-averaged sea ice top-ocean surface temperature are conserved. The correction for SAT ($\Delta {\text{SAT}}^{1}$) is then calculated as follows:

$$\Delta \text{SAT}^{1} = \mathop \sum \limits_{n = 1}^{5} \left[ {\frac{{\text{SIC}_{n}^{f} }}{{\mathop \sum \nolimits_{{n^{'} = 1}}^{5} \text{SIC}_{{n^{'} }}^{f} }} \times \left( {\text{SIC}^{a} - \mathop \sum \limits_{{n^{'} = 1}}^{5} \text{SIC}_{{n^{'} }}^{f} } \right) \times { \hbox{min} }\left( {\text{TIT}_{n}^{f} - \text{TO}^{f} ,0} \right)} \right]\quad \text{for}\quad \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{f} \ge \text{SIC}_{{\text{small}}} ,$$

(3)

where SIC^f, TIT^f, and TO^f represent the first guesses of the SIC and the sea ice top and oceanic surface temperatures, respectively. The quantity SIC^a is the SIC analysis value obtained by 3DVAR, and n and nʹ are the category numbers. The function “min”, which returns the smallest value of the elements, is used because of the difference in the relationship between SAT and SIC when TIT^f > TO^f. SIC_small is introduced to avoid singularities. Note that TIT^f adopts the snow-top temperature if snow exists on the ice. If the first guess of SIC is smaller than SIC_small, then

$$\Delta \text{SAT}^{1} = { \hbox{min} }\left[ {\text{TF}\left( {\text{SO}^{f} } \right) - \text{TO}^{f} ,0} \right] \times \left( {\text{SIC}^{a} - \mathop \sum \limits_{n' = 1}^{5} \text{SIC}_{n'}^{f} } \right),$$

(4)

where TF represents the freezing-point temperature, which is dependent on the salinity, and SO^f is the first guess of the ocean surface salinity. The correction for atmospheric specific humidity ($\Delta q^{1}$) is calculated as follows:

$$\Delta q^{1} = \Delta \text{SAT}^{1} \times \frac{{\text{d}q}}{{\text{d}T}}|_{{T = \text{TO}^{f} }} ,$$

(5)

where the differential of saturated specific humidity with respect to temperature at the surface, $\frac{{\text{d}q}}{{\text{d}T}}$, is calculated following Gill (1982).

The atmospheric corrections described above are imposed on the forcing field as constant offsets during the whole assimilation window. This is called the “intermediate assimilation” run. Because the SIC increments are redistributed among the categories according to the dynamical processes in the sea ice model, the simulated SIC field can deviate from the 3DVAR analysis field, although it should be closer than the first-guess field. To achieve an SIC field that is closer to the 3DVAR analysis field, the corrections for the forcing field are modified as follows:

$$\Delta \text{SAT}^{2} = \Delta \text{SAT}^{1} + \mathop \sum \limits_{n = 1}^{5} \left[ {\frac{{\text{SIC}_{n}^{i} }}{{\mathop \sum \nolimits_{{n^{{\prime }} = 1}}^{5} \text{SIC}_{{n^{{\prime }} }}^{i} }} \times \left( {\text{SIC}^{a} - \mathop \sum \limits_{{n^{{\prime }} = 1}}^{5} \text{SIC}_{{n^{{\prime }} }}^{i} } \right) \times { \hbox{min} }\left( {\text{TIT}_{n}^{i} - \text{TO}^{i} ,0} \right)} \right]\quad \text{for}\quad \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} \ge \text{SIC}_{{\text{small}}} ,$$

(6)

$$\Delta \text{SAT}^{2} = \Delta \text{SAT}^{1} + \hbox{min} \left[ {\text{TF}\left( {\text{SO}^{i} } \right) - \text{TO}^{i} ,0} \right] \times \left( {\text{SIC}^{a} - \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} } \right)\quad \text{for}\quad \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} < \text{SIC}_{{\text{small}}} ,$$

(7)

$$\Delta q^{2} = \Delta q^{1} + \left( {\Delta \text{SAT}^{2} - \Delta \text{SAT}^{1} } \right) \times \left. {\frac{{\text{d}q}}{{\text{d}T}}} \right|_{{T = \text{TO}^{i} }} ,$$

(8)

where the modified (previous) correction values are denoted by the superscript “2” (“1”), and superscript “i” represents the intermediate assimilation result described above instead of the first guess (superscript “f”).

In addition to the second correction for the forcing field (Eqs. (6–8)), a correction for the ocean temperature field is introduced here. Only the former is introduced in the intermediate assimilation run in order to ensure that the effects of corrections for the atmospheric forcing and ocean temperature fields on the simulated SIC field are not duplicated. Note that the differences between the current and previous correction values for the forcing field are generally much smaller in amplitude than the previous correction values. We assume that the modeled ocean temperature at a grid point is the average temperature of the freezing-point temperature under sea ice and the ice-free ocean temperature within the grid. The ocean temperature correction from the SIC change is then calculated as follows:

$$\Delta \text{TO}_{1} = \hbox{min} \left[ {\text{TF}\left( {\text{SO}^{i} } \right) - \text{TO}^{i} ,0} \right] \times \frac{{\text{SIC}^{a} - \mathop \sum \nolimits_{n = 1}^{5} \text{SIC}_{n}^{i} }}{{1 - \mathop \sum \nolimits_{n = 1}^{5} \text{SIC}_{n}^{i} }}\quad\quad \text{for}\quad \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} \le 0.9.$$

(9)

A threshold value of 0.9 is used in Eq. (9), as this correction is assumed to work primarily in the marginal ice zones. In addition, we consider the effect of salinity changes on the ocean surface layer due to sea ice melting on the freezing-point temperature, as follows:

$$\Delta \text{TO}_{2} = \left[ {\text{TF}\left( {\frac{{H_{s} \times \text{SO}^{i} + H_{{\text{melt}}}^{i} \times S_{{\text{ice}}} }}{{H_{s} + H_{{\text{melt}}}^{i} }}} \right) - \text{TF}\left( {\text{SO}^{i} } \right)} \right] \times \text{SIC}^{a} \quad\quad \text{for}\quad \text{SIC}^{a} - \mathop \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} < 0,$$

(10)

where S _ice is sea ice salinity of 4 psu, and the 10-m-thick ocean surface layer is taken to be H _s = 10 m. The grid-averaged change in water thickness due to sea ice melting in the SIC assimilation is

$$H_{{\text{melt}}}^{i} = - \frac{{\rho_{i} }}{\rho_{w} } \times \sum \limits_{n = 1}^{5} \left[ {H_{n}^{i} \times \frac{{\text{SIC}_{n}^{i} }}{{\sum \nolimits_{{n^{{\prime }} = 1}}^{5} \text{SIC}_{{n^{{\prime }} }}^{i} }} \times \left( {\text{SIC}^{a} - \sum \limits_{n = 1}^{5} \text{SIC}_{n}^{i} } \right)} \right],$$

(11)

where $H_{n}^{i}$ is the modeled sea ice thickness (in meters) of category n, and ρ _i and ρ_w represent the density of sea ice and seawater, respectively. $\Delta \text{TO}_{2}$ is not generated when the increment ($\text{SIC}^{a} - \mathop \sum \nolimits_{n = 1}^{5} \text{SIC}_{n}^{i}$) is positive, because of the successful use of subgrid-scale parameterization to distribute the brine rejection associated with sea ice formation at the base of the mixed layer (e.g., Duffy et al. 1999; Matsumura and Hasumi 2008). The total ocean temperature correction is calculated as follows:

$$\Delta \text{TO} = \Delta \text{TO}_{1} + \Delta \text{TO}_{2}$$

(12)

and is inserted into the modeled ocean temperature field in the surface mixed layer using the IAU method, in addition to the original temperature update derived using the T–S EOF modes. Specifically, a value $\Delta \text{TO}/N$ is constantly inserted into the model at each time step during the assimilation window, where N is the number of time steps in the assimilation window. The surface mixed layer depth is defined here as the depth at which temperature and salinity values differ from the surface values by 0.5 °C or 0.2 psu, respectively, and is limited to a maximum value of 100 m. This definition basically corresponds to the top of the strong halocline in the Arctic Ocean (e.g., Steele and Boyd 1998), even in the coarse-resolution model used in this study. Note that the values of 0.5 °C and 0.2 psu are roughly equivalent in terms of their effect on potential density.

The integration with the SIC assimilation through the use of the corrections for both the atmospheric forcing and ocean temperature fields as described above is the final step in an assimilation window, and gives the initial condition for the next window.

The complexity of the present scheme is the result of sea ice characteristics. The updates of the correction for the atmospheric forcing field (ΔSAT² and Δq ²), as well as additional correction for the temperature field (ΔTO₂), are needed because of the nonlinearity of the sea ice process, which is adjusted iteratively in the four-dimensional variational method. Our use of criteria (SIC_small and SIC_large) for separating cases (e.g., Eqs. (3) and (4)) to avoid singularity is due to the discontinuous distribution of the sea ice field (e.g., 0 ≤ SIC ≤ 1). Note that sea ice models usually set a minimum thickness for a similar purpose. The details described in this appendix can facilitate an assimilation of sea ice data in other studies.