Primitive Equations
du 1 ∂p − fv = − x-component momentum equation dt ρ ∂x dv 1 ∂p + fu = − y-component momentum equation dt ρ ∂y dp = −ρg hydrostatic equation dz ∂u ∂v ∂w 1 dρ + + = − continuity equation ∂x ∂y ∂z ρ dt dT dp c −α = Q thermodynamic energy equation p dt dt p = ρRT equation of state 6 equations with 6 dependent variables: u, v, w, p, ρ, T
Chaos and Numerical Weather Prediction
Why does it seem that the snowstorm in the 360-hour GFS forecast never happens? Predictability of the first kind: Predict the future based on initial conditions, with boundary conditions constant. This is limited by the chaotic nature of the atmosphere, which is a physical system with built-in instabilities, in vertical convection (e.g., thunderstorms) and horizontal motion (e.g., baroclinic instability - development of low pressure systems, such as hurricanes and Nor’easters).
(Source: Prof. Alan Robock, Rutgers University)
2 Xn+1 = a Xn - Xn
Consider a prediction using the above equation of the future state of the variable X, say the surface air temperature. The subscript n indicates the time, say the day, and a is a constant representing the physics of the climate system. X for any day is a times its value on the previous day minus X squared on the previous day.
With such a simple equation, it should be possible to predict X indefinitely into the future. Right?
(Source: Prof. Alan Robock, Rutgers University) 2 Xn+1 = a Xn - Xn
X0 2.200 2.200 2.210 2.20 a 3.930 3.940 3.930 3.93 Precision 3 3 3 2 (decimal places)
Let’s assume that a is exactly 3.930 and that a prediction with three decimal places is the exact solution. Then let’s consider three types of errors: imprecise knowledge of the physics of the climate system, imprecise initial conditions, and rounding due to limited computer resources. This example is from Edward Lorenz.
(Source: Prof. Alan Robock, Rutgers University)
2 Xn+1 = a Xn - Xn
X0 2.200 2.200 2.210 2.20 a 3.930 3.940 3.930 3.93 Precision 3 3 3 2 (decimal places)
Initial n (Time Step) Control Physics Rounding Conditions 0 2.200 2.200 2.210 2.20 1 3.806 3.828 3.801 3.81 2 0.472 0.429 0.490 0.46 3 1.632 1.506 1.686 1.60 4 3.750 3.666 3.783 3.73 5 0.675 1.004 0.556 0.75 6 2.197 2.948 1.876 2.39 7 3.807 2.924 3.853 3.68 8 0.468 2.971 0.297 0.92 9 1.620 2.879 1.079 2.77 10 3.742 3.055 3.076 3.21 11 0.703 2.704 2.627 2.31 12 2.269 3.342 3.423 3.74 13 3.769 1.999 1.735 0.71 14 0.607 3.880 3.808 2.29 15 2.017 0.233 0.465 3.76 16 3.859 0.864 1.611 0.64 17 0.274 2.658 3.736 2.11 18 1.002 3.408 0.725 3.84 19 2.934 1.813 2.324 0.35 20 2.922 3.856 3.732 1.25
(Source: Prof. Alan Robock, Rutgers University) 2 Xn+1 = a Xn - Xn
X0 2.200 2.200 2.210 2.20 a 3.930 3.940 3.930 3.93 Precision 3 3 3 2 (decimal places)
2 Xn+1 = a Xn - Xn
4.0 3.5 3.0 2.5
Xn 2.0 1.5 1.0 0.5 0.0 01234567891011121314151617181920 n
Control Physics Initial Conditions Rounding
(Source: Prof. Alan Robock, Rutgers University)