(A) (B) Free Drift Linear Viscosity (C) (D) Ideal Plastic Viscous Plastic Collision Induced Rheology Figure 2.1: Schematic representation of the most commonly used rheologies includ- ing (A) free drift, (B) linear viscosity, (C) ideal and viscous plastic, and (D) collision induced. Modified from Washington and Parkinson (2005; Figure 3.24) 15 Figure 2.2: Schematic representation of the energy balance vertically through an ice pack. Modified from Washington and Parkinson (2005; Figure 3.21). is balanced along the air/snow, air/ice, snow/ice, and ice/ocean interfaces. The steady-state equation for the conservation of energy at the surface of ice covered water follows: 0 if T0 < Tf QH + QL + QLW + (1 α0)QSW I0 QLW + QG0 = (2.14) ↓ − ↓ − − ↑ Q if T = T M 0 f where I0 is the amount of solar radiation that penetratesthe snow/ice column, and Tf is the salinity dependent freezing point. The surface energy balance for the sea- ice zone will be equal to zero for surface temperatures below freezing (T0 < Tf ), otherwise melt will occur (Wadhams 2000, Washington and Parkinson 2005). It should be noted that for sea ice, the sensible and latent heat fluxes are positive downward ( ) (Washington and Parkinson 2005). ↓ The steady-state equation for the conservation of energy along the air/snow interface follows equation 2.14 for the snow surface and the values for emissivity, 17 albedo, and the conductive flux are specific to the snow surface ("s, αs, and QGs ). Snowmelt is dependent on surface temperature, which is that of the snow surface, and equals 0 for surface temperature below freezing. Once the T0 = Tf , snow will melt until the snow thickness, hs, equals 0, at which time the ice surface will begin to melt (Wadhams 2000, Washington and Parkinson 2005). In the case of bare ice, there is no snow cover to insulate the ice surface, therefore, warming will result in direct surface ice melt rather than snow melt. The energy balance for the air/ice interface can be expressed as QH + QL + QLW + (1 αi)QSW Ii QLW + QG Mi = 0 (2.15) ↓ − ↓ − − ↑ i − where "i and αi represent sea-ice emissivity and surface reflectance respectively, and (FG)i is the surface value of the upward conductive flux through the ice pack (Washington and Parkinson 2005). The conservation of energy along the snow/ice interface depends on the bal- ance between the conductive fluxes of snow and ice (QGs = QGi ) such that: δT δT k s = k i (2.16) s δz i δz ! "hs ! "hs In this case, hs indicates the depth to the snow/ice interface, which is equivalent to the snow thickness (Wadhams 2000, Hibler 2003, Washington and Parkinson 2005). The conservation of energy along the ice/water interface depends on the balance between the ice melt/growth term (QMi ) and the difference between the ocean long- wave radiation flux (QLWw ) and the conductive flux through the ice column (QGi ). The relative magnitudes of the ocean and conductive heat fluxes at the ice/water interface determines whether bottom, or basal, ice melt (+Q ) or growth ( Q ) Mi − Mi will occur following: δhi δTi Lf = QLW 0 ki (2.17) − δt ↓ − δz ! "hs+hi ! "hs+hi In this case, hs + hi indicates the depth to the ice/water interface, which is equal to the sum of the snow and ice thickness. If the ocean flux is positive and larger 18 in magnitude than the conductive flux, then the right-hand side of the equation is positive and sea-ice melt will occur ( δh /δt). If the ocean flux is negative or smaller − i in magnitude than the conductive flux, the flux difference is negative resulting in ice growth (δhi/δt) (Wadhams 2000, Washington and Parkinson 2005). 2.2.3 Thermal Structure of Sea Ice The transfer of energy through a snow-covered ice pack may also be used to calculate the temperature profile of ice and snow layers. Using a numerical approximation for heat conduction through ice and snow, the temperature profile for each can be estimated by: 2 δT δ T κz ρc = k + κI exp − (2.18) δt δz2 0 where κ is the bulk extinction coefficient for snow or ice (Wadhams 2000, Hibler 2003, Washington and Parkinson 2005). The second term on the righthand side of equation 2.17 allows solar radiation to penetrate the snow and ice layers using κ to approximate Beers extinction law (Maykut and Untersteiner 1971, Washington and Parkinson 2005). Although the specific heat and thermal conductivity of ice are often parame- terized and assumed constant in many cases, they are both functions of temperature and salinity (Wadhams 2000). The ice thermal conductivity can be approximated by: Si ki = ko + β (2.19) Ti where ko is the thermal conductivity of pure ice such that: 0.0057Ti ko = 9.282 exp− (2.20) 19 (Yen 1981), Si is the salinity of the ice in practical salinity units (psi), Ti is the 1 ice temperature in ◦C, and β = 0.13Wm− . Similarily, the specific heat can be approximated by: Si ci = co + aTi + b 2 (2.21) Ti 1 2 1 where Cpo is the specific heat of pure ice, a = 7.53 J kg− ◦C− , and b = 0.018 MJ ◦C kg− . In addition to approximating ki and Cpi , ρiCpi can be approximated by: Si ρiCpi = (ρCp)o + γ 2 (2.22) Ti 1 where (ρc)p is the product of the density and specific heat for pure ice and γ = 17.15 MJ kg− K (Wadhams 2000). 2.2.4 Ice Thickness Distribution An ice thickness distribution is the numerical approximation of the distribu- tion of sea-ice thickness within an ice pack and is a quantifiable way to define the character and state of sea-ice. Information on the distribution of sea-ice thickness is essential for understanding the exchange of energy between the ocean and the at- mosphere, degree of ice deformation, and strength of the sea ice. The ice thickness distribution may also be used to determine average thickness, which together with the ice velocity, is used to determine the mass flux (rate of transport) of sea ice (Wadhams 2000). A typical ice thickness distribution is shown in Figure 2.3a resulting from de- formation processes governed by sea-ice dynamics as well as thermodynamic growth and decay (Thorndike 1992, Wadhams 2000, Hibler 2003). Sea-ice thermodynamics lead to thinner ice by the ablation of thick ice and ridges, and thicker ice through thermodynamic growth of thin ice (Figure 2.3b; Hibler 2003). Sea-ice deformation causes ice to converge (Figure 2.3c) creating thicker ice through pressure ridging, and diverge (Figure 2.3d) producing areas of open water in which new ice forms. By creating areas of open water, where new ice forms, and forming ridges, sea-ice 20 dynamics affect the amount of ice that falls within the thinnest and thickest ice thickness categories, while sea-ice thermodynamics affect the amount of ice within the middle of the ice thickness distribution (Hibler 2003). ) Figure 2.3: Schematic representation of the sea-ice thickness distribution produced by (A) thermodynamic and dynamic processes, (B) only thermody- namic processes, (C) divergence, and (D) mechanical redistribution. Modified from Wadhams (2000, Figure 5.2) Numerical approximation of the ice thickness distribution involves solving a thickness distribution function for a given area using equations that represent both dynamic and thermodynamic processes (Thorndike et al. 1975). The ice thickness distribution is represented by an areal probability density function (g) of ice thickness (hi), which represents the proportion, or area (A), of ice within region 21 R(x, y, t) at time t with an ice thickness between h + (h + dh) given by: dA(h,h + dh) g(h)dh = (2.23) R (Thorndike 1992, Wadhams 2000). The right-hand side of equation 2.23 represents dynamic and thermodynamic processes and can be expanded such that: dg δ = g V (fg) + Ψ (2.24) dt − # • − δh where V is the horizontal velocity vector, f is the vertical growth rate, and Ψ is the mechanical redistribution function. The first term in equation 2.24 accounts for divergence within the ice pack, the second represents thermodynamic growth, and the redistribution function accounts for mechanical processes such as ridging and the formation of leads (Thorndike 1992). The thermodynamic term allows ice thickness to be redistributed between ice-thickness categories such that the areal distribution of ice within each thickness category varies as ice grows or melts (Hibler 2003). The representation of sea-ice thermodynamics in equation 2.24 only accounts for vertical growth of sea ice, however, the lateral growth and decay of sea ice also affects the distribution of sea-ice thickness. Therefore equation 2.24 expands to include an additional term, L(h,g), to account for the lateral growth and decay of sea ice (Hibler 1980). Equation 2.24 is now expressed as: dg δ = g v (fg) L(h,g) + Ψ (2.25) dt − # • − δh − 4 1 2 3 ()*+ in which term 1 represents(th)e*dy+nam(ic )r*edi+stri(bu)t*io+n of ice thickness due to di- vergence, terms 2 and 3 represent the vertical and horizontal redistribution of ice thickness due to thermodynamic growth and decay, and the fourth and final term represents the mechanical redistribution of sea-ice thickness due to ridging and raft- ing (Hibler 2003, Briegleb 2004). Horizontal divergence within the ice pack, as represented by the first term in equation 2.25, accounts for ice divergence and the creation of leads. Term 1 therefore 22 Thermodynamic sea-ice models improved as the understanding of sea-ice thermody- namics improved, leading to the development of multi-level numerical models (Wad- hams 2000, Washington and Parkinson 2005).