## Abstract

This paper formulates and presents opening solutions of a one-dimensional coastal polynya flux model in which frazil ice is characterized by its depth and concentration. In comparison with polynya flux models in which variable frazil ice concentration is absent, this model is found to predict a smaller heat flux to the atmosphere. Consequently the model in this paper exhibits a wider steady-state polynya and longer opening times when compared with models in which ice concentration is neglected. The aforementioned polynya flux model is then coupled to a lower-atmosphere boundary layer model, and it is demonstrated that the polynya opening time and the steady-state width are significantly altered in the coupled, as compared with the decoupled, system. In essence, the heating of the lower atmosphere above the evolving polynya in the coupled system reduces the sensible heat flux between the ocean and atmosphere, thereby reducing the frazil ice production rate and hence leading to longer polynya opening time and wider steady-state width. This phenomenon is particularly noticeable when the potential temperature of the atmosphere at the coast is only slightly below the freezing point. In addition, a cutoff atmospheric wind speed is shown to exist, above which a steady-state polynya can never be obtained. Solutions calculated by the two models, using parameters representative of the St. Lawrence Island polynya, show that the new models contain substantial predictive capability.

## 1. Introduction

Polynyas are areas of low sea ice concentration, or open leads, within regions of ice cover caused by the divergence of ice motion. This divergence may be due to either prevailing winds or oceanic currents. Within the polynya the water is normally at the freezing point and there is a large heat flux across the air–sea interface. Ice is continually formed within the polynya and is advected away from its formation site. Latent-heat polynyas (“latent heat” is used to emphasize the heat flux related to open water) are areas of high ice production rates and therefore are also regions of large saline input to the ocean. This generates large downslope oceanic flows of dense water in both the Arctic (Schauer and Fahrbach 1999; Winsor and Björk 2000) and the Antarctic (Baines and Condie 1998; Comiso and Gordon 1998). A comprehensive review of the post-1991 literature on the observations, modeling, and climactic role of coastal polynyas is given by Morales-Maqueda et al. (2004).

Within a latent-heat polynya frazil ice is formed and through advection it moves downwind, as shown schematically in Fig. 1. The polynya edge corresponds to the region where the frazil ice piles up against the receding consolidated ice pack. Frazil ice is converted into consolidated new ice as a result of the pile-up process. During opening, the evolving polynya edge is governed by the imbalance between the arriving frazil ice flux, normal to the edge, and the exported consolidated new ice flux in the same direction. If these fluxes balance, the polynya edge is steady.

In previous polynya flux models, such as Pease (1987), Ou (1988), Morales-Maqueda and Willmott (2000), Biggs et al. (2000), and Biggs and Willmott (2004), the frazil ice was characterized solely by its depth. However, polynyas are routinely identified as regions of low sea ice concentration. For example, Special Sensor Microwave Imager (SSM/I) satellite data provides a synoptic view of sea ice concentration, in which coastal polynyas can be identified (Lynch et al. 1997; Skogseth et al. 2004). Although satellite altimeter data allows, under certain circumstances, the depth of sea ice to be calculated (Wadhams et al. 1992), the technique fails in polynyas containing thin frazil ice. Therefore, at the present time it is not possible to validate polynya flux models that characterize frazil ice in terms of depth alone with remotely sensed sea ice data. Clearly, the development of a polynya flux model that characterizes frazil ice in terms of depth and concentration will facilitate comparison with satellite observations of coastal polynyas.

The purpose of this paper is to (i) develop a *polynya flux model with variable frazil ice concentration* (hereinafter abbreviated to VCPFM) and (ii) elucidate the feedback process when the VCPFM is coupled to anatmospheric lower boundary layer model. When sea ice concentration is neglected in polynya flux models, the prescribed frazil ice production rate corresponds to an “open water value.” When variable sea ice concentration is introduced, we must distinguish between the frazil ice production rates in ice-free and ice-covered waters. Consequently, the effective frazil ice production rate in the latter type of model is reduced when compared with the former class of models. We therefore anticipate that the inclusion of variable sea ice concentration will lead to an increased steady-state polynya width (to establish the required frazil ice flux to balance the consolidated ice flux) when compared with models in which the ice concentration is neglected.

In previous polynya flux models a number of parameterizations of the frazil ice pile up at the edge of the consolidated new ice have been used. These will be referred to as “collection depth” parameterizations in this paper. In many polynya flux models the collection depth has been parameterized as a named constant following Pease (1987). This allows for simple analytical solutions to be obtained, although the steady-state width and opening times are highly sensitive to the value assigned to the collection depth and this is a severe drawback with the parameterization. In examining the dense water production of the Arctic Ocean, Winsor and Björk (2000) used a linear relation between collection depth and wind speed, based upon the model results of Martin and Kauffman (1981). The drawback of this parameterization, which also applies to the constant collection depth parameterization, is that there is no guarantee that the ice arriving at the polynya edge is shallower than the consolidated ice depth at the polynya edge. If this condition is violated, the assumptions used to formulate the polynya width equation are violated and the model is invalid. Again, using the work of Martin and Kauffman (1981) and Bauer and Martin (1983), Biggs et al. (2000) developed a robust collection depth parameterization that depends upon the relative speed of the frazil ice to the pack ice normal to the polynya edge. The collection depth parameterization is further refined in Biggs and Willmott (2004). In this paper the affect of using these collection depth parameterizations will be examined. It must be noted in passing that there are no in situ field measurements to determine which collection depth parameterization is the most realistic.

Polynyas are regions of large heat flux to the atmosphere (see Morales-Maqueda et al. 2004). This input of heat into the atmosphere modifies the local air temperature, which in turn acts to reduce the sensible heat flux from the ocean to the atmosphere. To date this process has not been studied in a coupled polynya–atmosphere model. In the context of the Weddell Sea polynya, Timmermann et al. (1999) coupled a sea ice–mixed layer numerical model with a simple analytical model of the lower atmosphere. Using this model it was found that the perturbations to the wind speed, produced by the heat flux within the polynya, significantly altered the atmospheric circulation and hence the sea ice drift.

Coastal polynya–atmosphere interaction is discussed in a satellite observational study by Dethleff (1994). Dethleff shows that spectacular buoyancy-driven atmospheric plumes can originate above coastal polynyas in response to the ocean to atmosphere heat and moisture fluxes. Without doubt, air–sea interaction processes deserve consideration when modeling the evolution of coastal polynyas.

Following Renfrew and King (2000) a one-dimensional convective internal boundary layer (CIBL) model is used to represent the atmosphere above a polynya and is coupled, in this paper, to the VCPFM. This enables us to consider the impact of the simple atmospheric model on the steady-state width, opening time, and heat flux within the polynya.

The polynya flux models in this paper follow the development methodologies of previous polynya flux models. The inclusion of variable concentration provides an entirely new aspect to the model, which allows for more direct comparison with observations, especially those obtained from satellite or aerial platforms. The coupling of the polynya model to the simple atmospheric model is also new and provides, for the first time, some indication of the effects of the feedback mechanisms between the polynya and the atmosphere.

The paper initially develops, in section 2, a one-dimensional polynya flux model incorporating variable concentration. The results of this model are discussed in section 3. In section 4, the CIBL model is coupled to the polynya model, and the results of this model are analyzed in section 5. Conclusions and discussion are presented in section 6.

## 2. Polynya flux model formulation

Consider the opening of a 1D polynya, where sea ice is categorized by its depth and concentration. The coast is located at *x* = 0 and the polynya occupies the region 0 < *x* < *l*(*t*), where *l*(*t*) is the width of the polynya at time *t*, as shown in Fig. 1. The frazil ice moves offshore with speed *u* and the consolidated new ice with speed *U*, both assumed constant and *u* > *U* > 0. The frazil ice produced in the polynya will either be converted into consolidated new ice or remain within the polynya, and the mass balance is written as

where *F*(*x*, *t*) is the frazil ice production rate, *A* is the consolidated ice concentration, *H* is the consolidated ice depth at the polynya edge, and *R _{i}* is the total volume of frazil ice within the polynya given by

where *a*(*x*, *t*) is the concentration and *h*(*x*, *t*) is the depth of the frazil ice within the polynya. The volume of ice within the polynya is governed by the equation

where the subscript *c* in (5) denotes the value of *ah* at the polynya edge, *x* = *l*. Substituting (5) into (4) and using (3) to rewrite the time derivative of the frazil ice volume in terms of the production rate and the space derivative terms yields

where *a _{c}* ≡

*a*(

*l*,

*t*) and

*h*≡

_{c}*h*(

*l*,

*t*). Equation (6) is the generalization of the polynya equation developed by Ou (1988). Indeed, when

*A*=

*a*= 1, it reduces to the polynya width equation of Ou (1988).

_{c}Within the polynya the frazil ice is in free drift, and the governing equations for the ice concentration and depth are given by

where Γ and Δ are the constant ice production rates in open water and ice-covered regions, respectively, and *h*_{0} is the constant ice accumulation depth, following Lemke et al. (1990). The frazil ice production rate is assumed constant across the polynya in this model for simplicity; this assumption is not retained in the coupled model. The solutions to (7) and (8), under the condition that both the ice concentration and ice depth are zero at the coast, are

where ℋ denotes the Heaviside function. The consolidated pack moves with speed *U*, which is strictly less than the frazil ice speed *u*. Therefore, the polynya edge is behind the information wavefront, and thus within the polynya, (9) and (10) reduce to

Initially, consider the steady-state problem. The steady-state frazil ice depth and concentration and the corresponding polynya width, *l _{s}*, will be calculated. In the steady state (6) becomes

This can be solved for *l _{s}* using numerical methods, or an approximation can be derived based upon the relative size of the Δ term to the Γ term and unity. The ice production rate in open water is much greater than in ice-covered regions; thus, the term Δ/Γ is much smaller than unity. Second, in the coastal polynyas considered here, the width of the polynya will not be sufficient to make the Δ

*l*order 1. If the width of the polynya is great enough to make the term order one, then the frazil ice concentration in the polynya will be approximately unity, much too large for a realistic polynya value. Exploiting these approximations, (14) can be simplified to

_{s}To calculate *l _{s}* from (15), a parameterization for

*AH*must be specified. Two models of the collection volume are developed and compared. The first sets

*A*and

*H*as specified constants, comparable to the constant collection parameterization used in Pease (1987). The second follows Biggs et al. (2000) and parameterizes the conversion of frazil ice to consolidated ice at the polynya edge by

where *c* ≈ 0.665. The Biggs et al. collection depth parameterization was developed following the work of Martin and Kauffman (1981) on leads. In a frame of reference moving with the evolving polynya edge the rate of change with respect to time of the momentum flux of frazil ice is balanced by the pressure gradient associated with the consolidated ice edge.

Using the approximate steady-state width equation, (15), together with the constant collection volume assumption, the steady-state width, *l*, is given by

and the concentration and depth of frazil ice at the polynya edge are

Similarly, the approximate steady-state width using parameterization (16) yields

and the frazil ice concentration and depth at the polynya edge are

The accuracy and limitations of the approximate solutions (17)–(22) are examined in Tables 1 and 2. In these tables the relative sizes of the ice production terms are altered and the steady-state widths are compared with the exact solutions obtained by solving (14). The concentration at the polynya edge is also calculated in these tables. Table 1 shows that, when the ice production in ice-covered water is significantly less than in open water regions, the approximate solutions (17)–(19) compare favorably with the exact solutions. A similar conclusion is valid for the results in Table 2. In both of these tables the approximation is good, even when the ice production in open water regions is only an order of magnitude larger than in ice-covered regions. For the constant collection volume the error in the steady-state width is about 0.7% and in the nonconstant volume model the error is only 0.3%. The parameter values used in these calculations are stated in Table 3. The large differences in values for steady-state width, *l*, and maximum concentration, *a _{c}*, are due to the choice of parameterization for the collection volume.

The polynya opening time for the variable concentration model is obtained by solving (6) using (11) and (12). This again must be solved numerically to obtain exact opening times. However, we can again exploit the fact that Δ/Γ ≪ 1 to derive an asymptotic solution for the opening time. When Δ/Γ ≪ 1, (6) becomes

Equation (23) can be integrated analytically in the cases when *AH* is assumed constant and when the parameterization (16) is adopted. When *AH* is constant, we find that

where *a _{c}* is the concentration at the polynya edge given by (18).

On the other hand when (16) is adopted, we find that

where *a _{c}* is given by (21). Using (11) it is readily seen that the polynya opening time is unbounded. Following earlier studies, we define the opening time to be the time taken for the polynya to reach the fraction 1 −

*ε*of the steady-state width, where typically

*ε*= 0.05. The time to open to the fraction 1 −

*ε*of the steady-state width, for constant collection volume, is given by

where *a _{c}* is again given by (18). The analogous opening time for the nonconstant collection volume parameterization is given by

where *a _{c}* is given by (21).

## 3. Polynya model results

Following Pease (1987) and Biggs et al. (2000) the parameterization for the ice production rate Γ, given *u*_{10}, the wind speed, and *T _{a}*, the air temperature, both measured at 10 m above the ground, is introduced:

where *σ* = 5.67 × 10^{−8} W m^{−2} K^{−4} is the Stefan–Boltzmann constant, *e _{a}* = 0.95 is the emissivity of the air,

*Q*= 301 W m

_{lu}^{−2}is the upward longwave radiation,

*ρ*= 1.30 kg m

_{a}^{−3}is the air density,

*C*= 2.0 × 10

_{h}^{−3}is the sensible heat coefficient,

*= 1004 J °C*

_{Cp}^{−1}kg

^{−1}is the specific heat of air,

*T*= −1.8°C is the water temperature,

_{w}*ρ*= 0.95 × 10

_{i}^{3}kg m

^{−3}is the ice density, and

*L*3.34 × 10

_{f}=^{5}J kg

^{−1}is the latent heat of fusion. The consolidated new ice velocity is

*U*= 0.01

*u*

_{10}, which means that the drag coefficient between ice and water must be about 10 times the drag coefficient between air and water; the frazil ice velocity is

*u*=

*2U*, again following Pease (1987) and Biggs et al. (2000); and the ice-covered region ice production rate is chosen as Δ = Γ/100. The consolidated new ice velocity is one-third of the value used by Pease (1987); however, the paper of Drucker et al. (2003) shows that the consolidated new ice can move at speeds approaching 1% of the wind speed rather than the larger 3% adopted by Pease. It must be noted that the steady-state polynya width and opening times are sensitive to changes in this value and large consolidated new ice velocities will results in unrealistically wide polynyas.

To study the parameter sensitivity of each model, the change in steady-state width as a function of the wind speed and air temperature is examined. These variations will also facilitate comparisons to be made between the steady-state width, obtained numerically, and the corresponding asymptotic solutions. In Fig. 2 the steady-state widths of each model are shown as they vary with wind speed and air temperature. Solutions with the constant collection volume model are shown in Figs. 2a and 2c. Figure 2e shows the solution when *H* is constant, corresponding to the Pease (1987) model where ice concentration is neglected. Here the qualitative behavior of all models is seen to be the same. A critical wind speed is reached when the steady-state width becomes almost independent of the wind speed. Figures 2b and 2d show the exact solution of (45) and the asymptotic solution (A1), respectively, in the case when *AH* is parameterized by (16). Figure 2f shows contours of *l _{s}* when ice concentration is neglected for comparison with Figs. 2b and 2d, and corresponds to the model proposed by Biggs et al.. The asymptotic solutions perform well over the whole parameter space of wind speed and air temperature. Only at very high wind speeds or low air temperatures can discrepancies be seen between the exact and asymptotic solutions. Quantitatively, the differences between the values predicted for

*l*using models with variable ice concentration and those in which it is neglected are large.

_{s}The asymptotic solution (20) shown in Fig. 2d performs well over the entire parameter space (cf. with Fig. 2b). Quantitatively, the variable concentration model has a much larger steady-state width for the same wind speed and air temperature than in the model where concentration is neglected. Figures 2b and 2f show that the qualitative behavior of the models, when ice concentration is included and when it is neglected, is similar. The nonconstant collection volume solutions also have a significantly different qualitative behavior from the constant collection volume solutions. There is now no critical wind speed beyond which the steady-state width is almost independent of the wind speed. In these models the steady-state width is dependent upon both the wind speed and air temperature throughout the parameter space.

Using the values given in Table 3 a comparison is made of the opening times of the polynyas (defined to be 95% of the steady-state width corresponding to *ε* = 0.05) as the wind speed and air temperature varies. Figures 3a and 3e show that the constant collection volume parameterization yields qualitatively similar results with the model where concentration is neglected. The approximate opening times (Fig. 3c) are in good agreement with the numerically obtained solutions although, in general, they slightly underestimate the opening time. It can be clearly seen that there is a critical wind speed beyond which the opening time becomes weakly dependent on the wind speed. This is due to the form of parameterization of the ice formation rate in ice-free water regions, with quartic dependence on air temperature and linear dependence on wind speed. When the nonconstant collection parameterization is used, the asymptotic solutions are more accurate (cf. Figs. 3b and 3d). However, comparison between Figs. 3b and 3f shows that the new variable concentration model solutions have a different qualitative form to the model without variable concentration.

Using data obtained from extensive satellite imagery, Morales Maqueda and Willmott (2000) derived opening times and widths of the St. Lawrence Island polynya (SLIP). Using this data a comparison is made between the steady-state widths and opening times predicted by the VCPFM, adopting the nonconstant collection volume parameterization, and the observed polynya steady-state widths and opening times. The predicted steady-state widths and opening times obtained using the polynya flux model without variable concentration are also compared. In Table 4, *A ^{e}* and

*T*are the predicted area and opening time of the polynya as in Morales-Maqueda and Willmott (2000). The VCPFM results are denoted by the subscript

*ac*. The model can be seen to predict the opening widths in accordance with the 90% accuracy estimates of the observed widths. The new model predicts that the opening times are significantly longer than those calculated by Morales-Maqueda and Willmott (2000).

## 4. 1D VCPFM coupled to a convective internal boundary layer model

In this section we couple the VCPFM to a simple model of the atmospheric CIBL. Both components of the coupled model that we will develop are simple enough to allow substantial progress to be made analytically. The key questions to be address are whether the polynya opening time and steady-state width are substantially altered in a coupled system as compared with a polynya model simply forced by the atmosphere.

Consider the steady-state width of a 1D polynya, where sea ice is categorized by its depth and concentration, coupled to a simple model of the CIBL. Once again, the coast is located at *x* = 0 and the polynya occupies the region 0 < *x* < *l*, where *l* is the width of the polynya. The frazil ice moves with speed *u* and the consolidated new ice with speed *U*, both constant and *u* > *U* > 0. The CIBL is modeled as a 1D slab model following Renfrew and King (2000). The governing equations for conservation of heat and momentum in the CIBL model are, respectively,

and for the polynya flux model they are

The constants *A*, *B*, and *C* are defined as

and *u _{a}* is the constant wind speed in the CIBL;

*θ*is the potential temperature;

*h*is the height of the CIBL;

*a*is the frazil ice concentration;

*d*is the frazil ice depth;

*β*is the entrainment parameter;

*u*

_{10}is the wind speed at the height of 10 m and is related to

*u*by

_{a}*u*

_{10}=

*νu*, where

_{a}*ν*is a constant;

*u*is the speed of the frazil ice and is related to

*u*

_{10}by

*u*=

*ϕαu*

_{10};

*ϕ*is the ratio of the frazil ice velocity to the pack ice velocity;

*γ*is the upstream stratification;

_{θ}*ρ*is the air density;

_{a}*C*is the specific heat of air;

_{p}*ρ*is the ice density;

_{i}*L*is the latent heat of fusion; and

_{f}*d*is the depth at which new ice is formed. The terms

_{a}*Q*(

^{c}*x*) and

*Q*(

^{o}*x*) are the heat fluxes associated with ice-covered regions and ice-free regions, respectively.

First, (30) is rewritten as

from which we observe that the right-hand sides of (29) and (36) differ only by a constant, and *θ* and *h* can therefore be linked by the equation

Integrating (37) we obtain

where the subscript 0 on a variable denotes its value at the coast, *x* = 0.

Integrating with respect to *x* from the coast and using (38) to eliminate *h* yields

since *d* = 0 at the coast. The integral in (40) is straightforward to evaluate allowing the frazil ice depth to be written as a function of the atmospheric potential temperature:

To make further analytical progress we make the assumption that the heat flux in ice-covered regions, *Q ^{c}*(

*x*), is a small fraction,

*ε*, of the heat flux in ice-free water,

*Q*(

^{o}*x*). This assumption is the same as used in the VCPFM developed in section 2. Now the unknown ice concentration,

*a*, may be calculated. Using (31), the governing equation for the ice depth (32) may be written as

Integrating (42) with respect to *x* yields the ice depth as a function of ice concentration; namely,

Using (41) and (43) we write the potential temperature, *θ*, as a function of the ice concentration, *a*, as

where the positive square root is used because the potential temperature must be increasing with fetch as the polynya is warming the atmosphere.

We can now obtain a numerical solution for the steady-state width of the polynya, given a suitable formulation for the heat flux, *Q ^{o}*(

*x*). The steady-state width of a polynya, using the collection parameterization (16), is given by

where *U* is the pack ice velocity, given by *U* = *αu*_{10}, and subscript *c* denotes the value of the variable at the polynya edge. Writing (45) as an equation for the ice concentration, by using (43) to eliminate the ice depth, yields

Solving the transcendental Eq. (46) yields the ice concentration at the polynya edge. Using this together with (44) and the heat flux parameterization,

where *θ* is the sea surface temperature, allows (31) to be integrated between 0 and *a _{c}* to obtain the steady-state polynya width. In this parameterization of the heat flux the emmisivity constant for air and water are assumed to be equal for mathematical simplicity; removing this assumption does not affect any qualitative changes and simply introduces small quantitative changes.

It is possible to make analytical progress if the heat flux in ice-covered regions is neglected; that is, *ε* = 0 in (43) and (44). Then, the ice depth is given by

and the concentration and potential temperature at the steady-state polynya edge are

as the potential temperature is constrained to be greater than or equal to the initial temperature, *θ* ≥ *θ*_{0}. It is also required that the potential temperature be less than or equal to the sea surface temperature, *θ* ≤ *θ*, so that the heat flux remains from the ocean to the atmosphere. Thus, for a fixed initial temperature, *θ*_{0}, using (50) and (51) it can be shown that there is a maximum wind speed such that a steady-state polynya may be maintained, given by

Figure 4 shows the maximum wind speed, given an initial temperature for the parameter values used in this paper. Here it can be clearly seen that the maximum wind speed increases as the potential temperature decreases.

Now the steady-state polynya width *l _{s}* is given by integrating (29):

where *h*(*θ*) is given by (38) and

An expression for the steady-state polynya width is readily obtained, although it is rather complicated, and can be found in appendix A.

The opening time for the polynya can be determined by noting that the characteristic time scale for the atmosphere and ice characteristics, depth and concentration, are faster than the characteristic time scale for the polynya edge. Thus, the solutions for the ice depth as a function of ice concentration (43) and as a function of potential temperature (41) remain valid for the opening time problem. The evolution of the polynya edge is given by

where *Â* is the pack ice concentration, *D* is the pack ice depth, and *d _{c}* is the ice depth in the polynya at the polynya edge. For numerical progress the polynya opening evolution equation, (55), can be written in terms of the ice concentration, using (31), as

using (43) and (44) to eliminate the ice depth, *d*, and the potential temperature, *θ*. Parameterizing the collection volume *ÂD* using (16) allows (55) to be evaluated numerically. The opening time is defined as the time elapsed when 95% of the steady-state width has been achieved.

Analytical progress may be made if the assumption used previously, about the relative sizes of the heat fluxes in regions of ice-covered water and ice-free water, is retained. Using the collection volume parameterization (16), again, the polynya opening evolution (55) may be written in terms of the potential temperature, using (29), as

where again *h*(*θ*) is given by (38) and *a*(*θ*) is given by (54). Thus, the time taken to open the polynya to 95% of the steady-state width it given by

where *θ̂* is the potential temperature when the polynya has opened to 95% of the steady-state width. The solution of (58) is easily obtained, but is unwieldy, and is therefore given in appendix B.

## 5. Results

The parameter values used in the model runs shown below are listed in Table 5 together with those used previously and listed in Table 3. To compare the VCPFM with and without the CIBL the air temperature and wind speed at the coast are chosen to be identical. It can be clearly seen in Fig. 5 that the inclusion of the atmospheric boundary layer has a large impact on the quantitative behavior of the polynya. The steady-state width of the polynya greatly increases with the inclusion of the active atmospheric boundary layer. This is due to the decrease in heat flux across the polynya and, thus, a decrease in the ice production rates within the polynya. The largest impact on steady-state width is seen at high air temperatures. This is due to the warming of the atmosphere by the polynya, and at temperatures near the freezing point a small increase in air temperature has a dramatic effect on ice production. The demarcation line for the cutoff of steady-state polynyas is that shown in Fig. 4.

The polynya opening times are shown in Fig. 6. Here, again, the opening times are longer throughout the parameter space although the greatest difference, quantitatively and qualitatively, lies in regions of warmer atmospheric temperature.

We next examine how various fields in the coupled model vary with the offshore distance. First, we calculate the offshore distribution of the potential temperature, net ocean to atmosphere flux, frazil ice depth, and concentration at steady state. A model simulation was run with the air temperature at the coast −10°C and the wind speed 10 m s^{−1}. The steady-state width of the polynya is 20.9 km. In Fig. 7a, the potential air temperature is seen to increase by about 1.5°C with increasing offshore distance within the polynya. This results in a significant decrease, of almost 20%, in heat flux from the polynya to the atmosphere, as shown in Fig. 7b. Figures 7c and 7d show the ice concentration and ice depth within the polynya.

Using the SLIP data once again to examine the predictive capability of the coupled model, a comparison of opening times and steady-state widths is presented for comparison with the VCPFM of section 2. Table 6 shows that the coupled model can predict the opening width in accordance with the 90% accuracy estimates of the observed widths. The new model also predicts that the opening times are similar to the VCPFM. The prediction for the steady-state width in May is significantly closer to the observed width. This is due to the relatively warm air temperatures that would bring the polynya near the cutoff regime described above, resulting in large polynyas for relatively slow wind speeds. In calculating these results the initial CIBL height was altered so that the average heat flux from the polynya, that is, the integrated heat flux across the polynya width divided by the steady-state width, corresponds to that shown in the table. The initial CIBL height used in the six periods between January and December is 170, 200, 180, 200, 40, and 80 m, respectively.

## 6. Conclusions

A polynya model with variable frazil concentration has been developed that facilitates comparison with observed data obtained primarily through satellite observations (e.g., SSM/I). The introduction of variable concentration to characterize the frazil ice leads to a decrease of the net ocean to atmosphere heat flux within the polynya. This, in turn, leads to wider steady-state polynyas. The opening times are also increased through the inclusion of variable concentration. Approximate solutions, where the heat flux through ice-covered regions in the polynya is neglected in favor of the heat flux from ice-free water, provide excellent agreement with the exact numerical solutions over the entire wind speed–air temperature parameter space. The approximate solutions also provide analytical expressions of both the steady-state width and the opening time of the polynya.

Subsequently the coupling of this polynya model to an atmospheric boundary layer model (CIBL) allows, for the first time, an investigation of how the polynya opening time and steady-state width are influenced by a frazil ice production rate that is calculated within the model, rather than being prescribed. A feedback is demonstrated to exist between the polynya and the atmosphere, whereby frazil ice production decreases with offshore distance in response to warming of the lower atmosphere above the polynya. The inclusion of an atmospheric model introduces a new qualitative effect, namely that a maximum wind speed for a given coastal air temperature now exists above which a steady-state polynya cannot exist. This cutoff wind speed is seen most clearly when the air temperature is near the freezing point, when a relatively small amount of heating from the polynya creates a large relative change in the heat flux. The inclusion of the CIBL also results in wider steady-state polynyas, under the same parameter regime, due to the decrease in frazil ice production rates with offshore distance within the polynya, which is in turn due to the atmospheric warming by the polynya. The opening times also increase relative to a polynya model with no coupled atmosphere.

The development of more realistic process models for calculating the evolution of the polynya area allows for a more credible examination of the significance of polynyas as regions of dense water production. The polynyas in the Barents and Kara Seas are believed to contribute significantly to the total dense water produced in the Arctic. To test this hypothesis, polynya flux models can be employed in a manner similar to that described by Winsor and Björk (2000). Alternatively, numerical shelf sea models, possibly incorporating parameterizations of polynya dense water production based on flux models, can be used to investigate this hypothesis.

## Acknowledgments

This research was supported by the U.K. Natural Environment Research Council Thematic Programme RAPID, via Grant NER/T/S/2002/00979. We thank Victor Shrira for useful discussions and advice on this research.

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### APPENDIX A Expression for the Steady-State Polynya Width in the Coupled VCPFM–CIBL Model

The analytical solution of (53), after some algebra, gives the steady-state width as

where *μ _{i}*,

*i*= 1, . . ., 3 are the roots of the cubic equation

and Λ* _{j}*,

*j*= 1, 2, are given by

### APPENDIX B Expression for the Opening Time of the Coupled VCPFM–CIBL Model

## Footnotes

*Corresponding author address:* I. A. Walkington, Dept. of Mathematics, Keele University, Keele, Staffordshire ST5 5BG, United Kingdom. Email: i.a.walkington@keele.ac.uk