Key Concepts and Fundamental Theorems of Atmospheric Science

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1 Key Concepts and Fundamental Theorems of Atmospheric Science Part II Global Motion Systems John A. Dutton Meteo 485 Spring 2004

2 From simple waves to predictability In two lectures a walk through two centuries of mathematical crisis and progress, to two fundamental theorems about atmospheric flow, and, then, a critical question or two.

3 Long waves in the westerlies the Rossby theory For horizontal, incompressible flow, density varying only in the vertical, the perturbation equations are u t v t +U u x fv + x +U v x + fu + y u x + v y = 0 p ρ = 0 p ρ = 0 ς = v x u y = ( 2 x 2 + = H 2 ψ ( t +U )ζ + βv = 0 x 2 y 2 )ψ ( t +U x ) H 2 ψ + βψ x = 0 ψ x = ψ x

4 The wave equation ( t +U x ) H 2 ψ + βψ x = 0 i(ωt +κ x) ψ (x, y,t) = ψ ˆ ( y)e (ω +κu )[ ˆ ψ yy κ 2 ˆ ψ ] + βκ ˆ ψ = 0 ψˆ yy + λψ ˆ = 0, λ = βκ ω +κu κ 2

5 The simplest solution ψ yy + λψ = 0 βκ 2 λ= κ ω + κu ψ yy = 0 λ = 0 ω β c = =U 2 κ κ 2 βl =U 2 4π J. Namias F. W. Reichelderfer C.G. Rossby 1955 Copyright John A. Dutton 2004

6 The modern era begins βl c =U 2 4π 2

7 Rossby-Haurwitz waves in a channel of width D The eigenvalue problem ψˆ yy + λψ ˆ = 0, λ = βκ ω +κu κ 2 ˆ ψ ( y) = Asin( λ y) + Bcos( λ y) At y= ± D/2: v = ψ x = 0 ˆ ψ (±D / 2) = 0 y = 0 : A = 0 v y = 0 ˆ ψ y =0 λ D / 2 = πn,n = 1,3,... 2 β ω k,n +U 2πk / L (2πk L )2 = (πn / D) 2

8 The solution orthogonality relations ψ (x, y,t) = (2n +1)π cos( y)[a D n,k sin(ω n,k t + 2πkx / L) k =1 n=1 +B n,k cos(ω n,k t + 2πkx / L)] D/2 cos( D/2 (2n +1)π D (2m +1)π y)cos( y) = D 0 m n π m = n 2 2π 0 2π sin( jξ)sin(kξ)dξ = 2π 2π 0 cos( jξ)cos(kξ)dξ = cos( jξ)sin(kξ)dξ = 0 k j 0 sin 2 (kξ)dξ = cos 2 (kξ)dξ = π 0 2π 0

9 The solution specifying the coefficients ψ (x, y,t) = (2n +1)π cos( y)[a D n,k sin(ω n,k t + 2πkx / L) k =1 n=1 +B n,k cos(ω n,k t + 2πkx / L)] A n,k D L = 2 ψ (x, y,0)cos( B 0 n,k D 2 (2n +1)π D ) sin(2πkx / L) cos(2πkx / L) dy dx We appear to have a solution valid for all time, perfect predictability, and infinite resolution of detail depending only on the initial and assumed boundary conditions. Is this too good to be true?

10 Crisis in mathematics The crisis struck four days before Christmas The edifice of calculus was shaken to its foundations. the difficulties had been building for decades it would take fifty years before the full impact of the event was understood. The crisis was precipitated by the deposition at the Institute de France in Paris of a manuscript, Theory of the Propagation of Heat in Solid Bodies, by the 39-year old prefect of the department of L Isère, J.B. Fourier A Radical Approach to Real Analysis David Bressoud, 1994

11 0 X ( f n (x))dx = n=1 n=1 0 X f n (x) dx lim n 0 X f n (x)dx = 0 X lim n f n (x) dx G. B. Riemann The 19th century would see ever expanding investigations into the assumptions of calculus, an inspection and refitting of the structure from the footings to the pinnacle, so thorough a reconstruction that calculus was given a new name: Analysis A. L Cauchy Karl Weierstrass

12 Resolution of the crisis We use the Lebesgue integral convergence of partial sums of the series mean square convergence Grace C. Young Henri Lebesgue And in some cases we talk about equality except on a set of measure zero

13 But enough of this diversion into the world of misbehaving functions, recalcitrant limits, and divergent sequences. Let us return to the physics where all still seems to be tidy and nice

14 Energy conservation u t v t +U u x fv + x +U v x + fu + y u x + v y = 0 p ρ = 0 p ρ = 0 k = 1 2 (u2 + v 2 ) ( t +U x )ρk = H pu t ρk(x, y,t)dxdy = 0 A

15 A surprise in the westerlies ( t +U x )ζ + βψ x = ( t +U x )ζ 2 + β( 2 H ψ )ψ x = 0 ( H 2 ψ )ψ x = H (ψ x H ψ ) H ψ H ψ x = H (ψ x H ψ ) 1 2 ( H ψ )2 / x t ρζ 2 (x, y,t) dx dy = 0 A

16 Summing up so far We have a linear wave theory with flow constrained to conserve kinetic energy and mean square vorticity--enstrophy. But the symmetric Rossby-Haurwitz waves cannot transfer heat or momentum poleward. So we must have a nonlinear theory that preserves the features of the long waves in the westerlies and yet is simple enough to reach rigorous conclusions. Through careful, laborious scale analysis we arrive at the quasi-geostrophic equations.

17 The quasi-geostrophic equations ( t + u g H )u g + Ro(1 + λ)w u g z = k u' π ' z θ ' = 0 ( t + u )θ '+ w'σ = Q' g H H u'+ 1 + λ ρ 0 z (ρ w') βv = 0 0 g u g = k H π ' Jule Charney Dimensionless equations for large-scale, moderately heated flow.

18 A nonlinear wave equation Quasi-geostrophic equation (some constants absorbed) ( t + k H ψ H )L(ψ ) + βψ x = 0 L(ψ ) = H 2 ψ + 1 ρ 0 z ( ρ 0 σ ψ z = θ ' z ψ ) Rossby-Haurwitz equation ( t +U x ) H 2 ψ + βψ x = 0 (k H ψ H )L(ψ ) = ψ x L(ψ ) y ψ y L(ψ ) x t L(ψ ) + J(ψ, L(ψ )) + βψ x = 0

19 Quasi-geostrophic eigensystem L(ψ ) = H 2 ψ + 1 ρ 0 z ( ρ 0 σ L(ψ ) = λψ z ψ ) leads to eigenvalues and orthonormal eigenfunctions 0 < λ 1 < λ 2 <...λ n <... ψ 1 (x, y,z), ψ 2,...ψ n... λ 0 = 0,ψ 0 = const and if we put a n = V ρ 0 ψ n ψ dv then 2 a n = ρ 0 ψ 2 dv n=1 V lim N V ρ 0 N n=0 ψ a n ψ n 2 dv = 0

20 Spectral equation for quasi-geostrophic motion t L(ψ ) + J(ψ, L(ψ )) + βψ = 1 x ρ 0 z ( ρ 0 σ N and ψ = a k ψ k k =1 leads to Q') + µl L(ψ ) (ad hoc) λ n da n dt + a j a k ρ 0 ψ n λ k J(ψ j, ψ k ) dv β a j ρ 0 ψ n (ψ j ) x dv j,k j V = ˆq n µλ n 2 a n n = 1,2,...N V A set of N ordinary differential equations describing the evolution of the nonlinear quasi-geostrophic flow.

21 to an N-dimensional trajectory in phase space 1 From a flow in physical space N 2

22 The spectral energy equation The ordinary differential equations λ n da n dt + a j a k ρ 0 ψ n λ k J(ψ j, ψ k ) dv β a j ρ 0 ψ n (ψ j ) x dv j,k j V = ˆq n µλ n 2 a n n = 1,2,...N V imply with the aid of the boundary conditions that d dt d dt n n n λ n a n 2 = a n ˆq n µ λ n 2 a n 2 n n n λ n 2 a n 2 = λ n a n ˆq n µ λ n 3 a n 2

23 A global theorem d dt n n n λ n a n 2 = &E N = a n ˆq n µ λ n 2 a n 2 a N = µ (λ n a n ˆq n ) n 2µλ n 4µ n ( ˆq n λ n ) 2 &E < 0 ` &E > 0 E a 1 &E < 0

24 a N The trapping theorem (Lorenz,1963) &E > 0 &E < 0 &E < 0 E a 1 For finite heating and for N <, every trajectory φ t (y) starting with finite energy proceeds asymptotically to the closed ball A N with exterior boundary E*, and so A N is a global attractor for any bounded part of phase space. This is the second fundamental theorem of atmospheric science. Corollary 1. The trajectory φ t (y) has at least one fixed point, or equivalently, the quasi geostrophic system has at least one steady solution. Corollary 2. Every trajectory has at least one limit point, and hence there is at least one point for which the trajectory is recurrent. For proofs, and a version applicable to a more general set of equations, see Chapter 15, Ceaseless Wind,or Dynamics of Atmospheric Motion

25 The solutions to the nonlinear, quasi-geostrophic flow exist for every N < and can be computed to any degree of accuracy given a sufficiently powerful computer. But can we actually follow them? Are they periodic or apparently random? How could we find and describe the attractor? What happens if we change an initial condition by ε in quadratic norm? Another mathematical crisis loss of predictability and the topic of the next lecture.

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