The Sea Ice Model
2.1 Dynamics and Thermodynamics
2.1.3 Balance Equations
The sea ice cover is mainly modified by two processes: thermodynamic growth (or decay) and advection.
Thermodynamic growth describes the freezing of sea water and thus the for-mation of sea ice as well as the melting of sea ice (negative growth). It is the source and the sink of sea ice. Owing to the low angle of incidence of solar ra-diation polar regions usually show a negative rara-diation balance. This leads to a cooling of the atmosphere and further to a cooling of the surface. If a cold at-mosphere cools the oceanic boundary layer (the uppermost tens of metres) to the freezing point and continues withdrawing energy from the surface the re-sult is a phase transition of water and sea ice is formed. Melting of sea ice is the same thermodynamic process in reverse: a warmer atmosphere and ocean sur-face heats the ice layer up to freezing point from above and below. Continued warming results in the phase transition from solid to liquid, i.e. melting. The en-ergy balance at the ocean/atmosphere interface describes the thermodynamic ice growth.
The second process that modifies the regional distribution of the ice volume is the horizontal transport, also referred to as ice drift, or advection. Advection is responsible for the typical pattern of ice growth rates (positive and negative).
Ice drift velocity is calculated using a simplified momentum balance, which cor-responds to a force balance in which the inertia of sea ice is neglected. The temporal evolution of the prognostic variables - mean ice volume and mean ice
1The ice density is assumed to be constant here. Therefore, the ice volume can be calculated directly and more easily by applying the mass balance, as was done in the original formulations.
That is why the ice thickness is not treated as a variable itself rather than using the terms ice volume over ice concentrationh/Afor the thermodynamic calculations.
concentration - is described by the following balance equations:
Sh = ∂h
∂t +∇ ·(~uh) (2.3)
SA = ∂A
∂t +∇ ·(~uA) (2.4)
where h denotes the ice volume, A denotes the ice concentration and S rep-resents the thermodynamic source or sink of the prognostic variables. If these terms are zero, the prognostic variables are conservative quantities. On the right hand side the equation is a form of continuity equation. The first term is the lo-cal temporal evolution and the second term is the advection, the in- and outflow of ice by horizontal exchange between adjacent regions.
In contrast to the numerical sea ice model byHibler III.(1979) the balance equations (2.3) and (2.4) have no explicit diffusion. The diffusion in Hibler’s model does not describe a physical process, but only takes care of numerical stability. This artificial diffusion is needed for the numerical solution of the advection equation applying the central differences scheme. This scheme has the disadvantage of producing the numerical artifact of negative ice thicknesses (Fischer,1995). Here, a modified upstream scheme is implemented as a numer-ical method for calculating the advection (Smolarkiewicz,1983). This guaran-tees numerical stability without the need of including diffusion explicitly in the balance equations and does not produce negative values for quantities which may physically only take positive values.
2.1.4 Dynamics
The sea ice drift velocity~uin equations (2.3) and (2.4) is derived from the mo-mentum balance, which in Cartesian co-ordinates (Hibler III.,1979) is given by
mD~u
Dt =−mf~k×~u+τa+τw−mg∇H+F~ (2.5) whereD/Dt = ∂/∂t+~u· ∇is the total derivative in time,mis the ice mass per unit area,f is the Coriolis parameter,~k is the unit vector perpendicular to the surface,~uis the sea ice velocity,gis gravitational acceleration, andH is the sea surface topography. The momentum balance (2.5) comprises the Coriolis force
2(−mf~k×~u), the air stress(τa), the water stress(τw), the surface tilt(−mg∇H), and the force due to variation of internal ice stress(F~). The air and water stress as well as the internal stress are the dominant terms in equation (2.5) and are of
2Coriolis force is an apparent force named after the French mathematician Gustave Gaspard Coriolis (1792-1843). Any movement in the Northern hemisphere is diverted to the right, since the Earth is a rotating and therfore represents an accelerated system. The Coriolis parameter f = 2Ω sinϕis a function of the rotation frequencyΩ = 7.29·10−5s−1and of latitudeϕ.
approximately equal magnitude. The local acceleration term is generally small and is neglected here.
These stresses are determined from simple nonlinear boundary layer theo-ries with constant turning angle (McPhee,1979). For the atmosphere
τa =ρaca|U~a |[U~acosφ+~k×U~asinφ] (2.6) and for the ocean
τw =ρwcw |U~w−~u|[(U~w−~u) cosθ+~k×(U~w−~u) sinθ] (2.7) with ρa = air density
ρw = water density U~a = geostrophic wind
U~w = geostrophic ocean current ca = atmospheric drag coefficient cw = oceanic drag coefficient φ = air turning angle
θ = water turning angle
The non-dimensional air and water drag coefficientsca andcw are assumed to be constant in space and time. The turning angle for the geostrophic ocean cur-rent isθ= -25◦ . The atmospheric turning angle is set toφ= 0◦ and the surface wind is used as wind forcingU~a. Usually|U~a| |~u|, therefore the relative ve-locity between ice drift and wind speed can be represented by the wind speed alone.
Rheology
The sea ice model accounts for shear and compressive deformation, but shows no resistance against divergent ice drift. The bulk and shear viscosities as well as the ice pressure are non-linear functions of the ice volume and the ice concen-tration. For normal strain rates the ice behaves like a plastic material whereas it shows viscous behavior for very small strain rates. A detailed description of the rheology scheme is given in Harder(1996) andKreyscher (1998). In this real-ization of the numerical model a viscous-plastic rheology is implemented and Kreyscher et al.(2000) give a comparison of this rheology with other constitu-tive laws used in dynamic sea ice models. The internal forcesF~ are calculated as the divergence of a two dimensional stress tensor σ which is related to the ice velocity field by a viscous-plastic rheology afterHibler III.(1979). Table2.1 summarizes the main dynamic parameters and values.
Table 2.1: Dynamic parameters and values
Dynamic parameter Symbol Value
drag coefficient, atmosphere ca 2.2·10−3 drag coefficient, ocean cw 5.5·10−3 turning angle, atmosphere φ 0◦
turning angle, ocean θ -25◦