Session K.pdf - Clarkson University

hτιj⎡∂(hu∂ ⎤i ) (hu j )= ν t ⎢ + ⎥ .ρw⎢⎣∂ x j ∂ xi⎥⎦Here h – water depth; H=h+z B – the level of the free water surface; z B – the bottomlevel; u x , u y – the projections of the averaged (in depth) velocities on the axes x, y;Ω c = 2ω sinϕ – Coriolis’ parameter; ω – angular velocity of the Earth rotation; ϕ – thearea latitude; g = 9.81 m/s 2 – the acceleration of gravity; ρ w – the water density; ρ a – theair density; τ bx ,τ by – the projections of the friction forces on the axes x, y; ν t – turbulentviscosity coefficient; C – Chezy coefficient; n – roughness coefficient; С cur – thecoefficient of the bottom friction owing to the flow; τ wx , τ wy – the projections of thewind shear stress on the axes x, y; θ w – the angle between the wind direction and theaxis x; κ = 0,4 – Karman’s constant; W – the wind speed on the height z (compared withthe surface level); z 0 – the roughness coefficient of water surface; q(x, y) – the intensityof the mass source in the point with the coordinates (x, y). The viscosity coefficient ν tcan be determined on the basis of the Prandtle’s model−12 2ν = κ h , u = ρ ( τ + τtu* *The heat transfer equation in the context of the 2D (plan) model is derived on the basisof the averaging in depth 3D equation of transferring and has a formwbxby) .∂Th∂Thu+∂t∂xx∂Thu+∂yy=∂ ⎛ ∂T⎜ µTh∂x⎝ ∂x⎞⎟⎠∂ ⎛ ∂T+ ⎜ µTh∂y⎝ ∂y⎞⎟⎠1+ (ΦSρ wc w+ ΦB+ q ) . (4)THere T – averaged (in depth) water temperature; µ T , µ T – the total thermal diffusivitycoefficient with regard to dispersion; c w – specific heat of water; Φ S – the heat-fluxdensity on the water surface; Φ B – the heat-flux density on the bottom; q T – the intensityof the internal heat sources. The calculation of the heat-flux density Φ S is performed onthe basis of (Methodic Recommendation, 1976; Recommendation, 1979;Recommendation, 1986).The changing of the ice thickness on the water surface is determined according to therelationd hIρ I L = Φ SI – Φ IW . (5)d tHere ρ I – the ice density; L – the latent melting heat; h I – the ice thickness; Φ SI – theheat-flux density on the boundary of ice-air (from ice to air); Φ IW – the heat-flux densityon the boundary of ice-water. The relation (5) is used in two cases – when there is alreadyice cover (h I > 0) or when the calculated surface water temperature T SW gets lowerthan the temperature of water freezing (T SW