Key Concepts and Fundamental Theorems of Atmospheric Science

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1 Key Concepts and Fundamental Theorems of Atmospheric Science Part I Global Thermodynamics John A. Dutton Meteo 485 Spring 2004

2 Goals and strategy Because skillful prediction requires understanding, we will try to Clarify links between cause and effect in large-scale atmospheric flow Illustrate how theory uses observations to create models and reach conclusions Inquire why the atmosphere behaves as it does Why are things the way they are?

3 Geometry of the Planet Area of a circle 2π r dr dθ = π R 2 0 R 0 r θ Surface area of a sphere π R 2 2π 2 0 π 2 cosϕ dϕ dλ = 4π R 2 R cosϕ ϕ

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5 Let there be light Sunlight of intensity I 0 falls on a disc of area πr 2 Earth emits radiation F = σt 4 as a sphere of area 4πR 2 F = σt 4 = I o / 4 I 0 = 1360 W/m 2, σ = W/m 2 K 4 ==> T 0 ~ 278 K Planetary albedo α = 0.32 J = (1 - α) I 0 /4 = (J/ σ) 1/4-273 = C

6 Let there be warmth Average surface temperature T s = K compared to T α = 253 K implies there must be another flux of energy G F s = σt s 4 = J + G µ = G/F s = (F s - J)/F s = J µf s F s (1-µ) F s

7 Let there be latitudinal variation α(ϕ)= a Global Albedo Global Albedo +b cos(ϕ / 2) a = b = Albedo (per cent) Approximation Observed Latitude α = π 2 α(ϕ)cos(ϕ)dϕ =

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9 Zenith Angles cos Z = sin ϕ sin δ + cos ϕ cos δ cos h at equinox, δ = 0; at sunrise and sunset, h = π/2 Z cos Z = cos ϕ cos h Averaging along a latitude circle 1 2π π /2 cos(h) dh = 1 π /2 π J(ϕ) = 1 π (1 α(ϕ))i 0 cos(ϕ)

10 Surface temperature estimate F = J + G = J + µf T s (ϕ) = [J(ϕ)/σ(1- µ)] 1/4 Surface Temperature (C) Latitude

11 Zonal Average Sea Surface Temperature Observation Temperature (C) Approximation Latitude aa=28.72 bb=0.564 Tocn = aa cos2(ϕ)/(1+bb ϕ2) Focn = σ Tocn4(ϕ) 100

12 Solar and Longwave Radiation 400 Solar J(ϕ) Watts/m^ Longwave F ocn (1-µ) Latitude

13 Solar and Longwave Radiation 400 Solar J(ϕ) 300 Watts/m^2 200 Longwave F ocn / (1-µ) Sin(Latitude)

14 Balancing the energy flow Poleward Heat Flux H(ϕ) = F ocean (ϕ)(1 µ) J(ϕ) 30 H p (ϕ) = ϕ π 2 H(ψ )cos(ψ )dψ H(ϕ) = F ocean (ϕ)(1 µ) J(ϕ) Heat Flux (W/m^2) H p (ϕ) = π 2 ϕ H(ψ )cos(ψ )dψ Latitude

15 Balancing the energy flow Solve for the µ that balances the energy flow H(ϕ) = F ocean (ϕ)(1 µ) J(ϕ) µ = 0 π 2 (F ocean (ϕ) J(ϕ))cos(ϕ)dϕ 0 π 2 F ocean (ϕ)cos(ϕ)dϕ = 0.412

16 Poleward Heat Flux 30 Heat Flux (W/m^2) Latitude

17 Latitudinal Heat Flux Estimates Houghton (1954) Oort & Vonder Haar (1976) Present Model (mu=0.399) Present Model (mu=0.411) Heat Flux (W/m^2) Latitude

18 So what do the thermodynamic gods ordain that the atmosphere and ocean must do? Transfer heat from the equator toward the poles, forevermore. As long as the Earth is round, the wind may never cease.

19 Let us turn this inescapable but qualitative conclusion into a definite statement a theorem, if you will.

20 T t c V ρ dt ρ t The Equations of Atmospheric Motion No motion: v(x,t) = v t = 0 x, t = t 0 dv dt = 1 ρ p gk 2Ω v + 1 µ v +... ρ dt + p v = R + k T + C+ v dρ dt + ρ v = 0 p = ρrt dq dt + q v = S + k q q d(g) dt = (g) t + v (g) q = C = 0

21 The Motionless State p = gρk z p = 0, p z z = p z z = g z ρ = 0 p z = gρ z p = Rρ z T + RT z ρ T z = 0 for Q = R, z Q = 0

22 The First Fundamental Theorem of Atmospheric Science If no motion z T = 0, then z T 0 motion. Theorem (Jeffreys, 1925) As long as there are horizontal variations in heating (or equivalently, temperature), then the atmosphere must remain in motion in the sense that the velocity and acceleration cannot both vanish identically.

23 But if the thermodynamic gods ordain that the atmosphere and ocean transfer heat from the equator toward the poles, why are there westerlies instead of northerlies or southerlies? Because the Earth rotates and the poleward temperature gradients are large.

24 why do the westerlies increase with height?

25 For Ro = V fl ~ 10 1 Re = VL υ ~ and slowly-varying flows,... and then v H v g = 1 ρ f k z p v g z g ft k z T v g p = R fp k p T v g θ = c p f θ k θ T

26 why do the westerlies increase with height? Because it is colder toward the poles and that s because the Earth is round.