Lect-04-P4 - M.Sc in Computational Science Fundamentals of...

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M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met ´ Eireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January–April, 2004. Lecture 4 The Shallow Water Equations 2 Physical Laws of the Atmosphere NEWTON’S LAWS OF MOTION Describe how the change of velocity is determined by the pressure gradient, Coriolis force and friction GAS LAW, or EQUATION OF STATE Relates the pressure, temperature and density CONSERVATION OF MASS Continuity Equation: air neither created nor distroyed CONSERVATION OF WATER Continuity Equation for water (liquid, solid and gas) CONSERVATION OF ENERGY Thermodynamic Equation determines changes of tempera- ture due to heating, compression, etc. Seven equations; seven variables ( u, v, w, ρ, p, T, q ) . 3 The Primitive Equations du dt - f + u tan φ a v + 1 ρ ∂p ∂x + F x = 0 dv dt + f + u tan φ a u + 1 ρ ∂p ∂y + F y = 0 ∂p ∂z + = 0 p = RρT ∂ρ ∂t + ∇ · ρ V = 0 ∂ρ w ∂t + ∇ · ρ w V = [ Sources - Sinks ] dT dt + ( γ - 1) T ∇ · V = ˙ Q c p These equations are suitable for a forecast or climate model. For understanding the dynamics , we need to simplify them. Check: Look at the above equations. How many have we got so far? 4
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Equations in Component Form The equations of motion in the rotating frame are d V dt + 2 Ω × V + 1 ρ p - g = 0 . Next we split the equation of motion into components. We introduce local cartesian coordinates ( x, y, z ) . V = ( u, v, w ) d V /dt = ( du/dt, dv/dt, dw/dt ) g = (0 , 0 , - g ) p = ( ∂p/∂x, ∂p/∂y, ∂p/∂z ) Ω = (0 , Ω cos φ, Ω sin φ ) 2 Ω × V = ( 2 w Ω cos φ - 2 v Ω sin φ, 2 u Ω sin φ, - 2 u Ω cos φ ) Note: Certain trigonometric terms have been omitted from the acceleration. We assume w is much smaller than u and v , and we can neglect the term 2 w Ω cos φ in the Coriolis force. 5 The variables x and y are distances eastward and northward on the globe . We will ignore the effects of sphericity except in the Coriolis term (see below). Then ( x, y, z ) are equivalent to Cartesian coordinates . Coordinate System Components of Ω 6 The horizontal components of the equation of motion may now be written: du dt - fv + 1 ρ ∂p ∂x = 0 dv dt + fu + 1 ρ ∂p ∂y = 0 where f = 2Ω sin φ is called the Coriolis parameter . The vertical component of the equation of motion is dw dt - u cos φ + 1 ρ ∂p ∂z + g = 0 . In the absence of motion, this reduces to the hydrostatic equation, ∂p ∂z + ρg = 0 , expressing a balance between the vertical pressure gradient and gravity. Note that horizontal component of Ω no longer enters the equations, and we write 2 Ω = f k.
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