Assignment. Find connections between your own research and intraseasonal variability in the tropics

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3 Assignment Find connections between your own research and intraseasonal variability in the tropics a 15-minute oral summary with 5-minutes additional to answer questions and A 1-5 page written summary Please confirm your topic selection with me by March 29.

4 Internet Directory roundy/atm523 Contains the syllabus and will include pdfs of these notes

5 Background Animation

6 Dissect the Data Seasonal cycle Cycle is not sinusoidal, so Harmonics Includes periods of 365, 365/2, 365/3, and 365/4 days (first 4 harmonics)

7 Seasonal Cycle Low level flow Upper level flow Vertical structure at equator and 10N Winds Relative Humidity

8 Time longitude diagram of CLAUS Tb (2.5S 7.5N), January April 1987

9 Intraseasonal Mode Vocabulary Madden Julian Oscillation Kelvin wave (oceanic or atmospheric) Equatorial Rossby wave Mixed Rossby Gravity wave Inertio Gravity wave Easterly Wave

10 Intraseasonal Mode ITCZ/SPCZ/SACZ Monsoon trough Trade wind trough Gill Model Westerly Wind Burst Trade surge Baroclinic modes Vocabulary Rossby wave response

11 OBSERVATIONS OF KELVIN WAVES AND THE MJO Time longitude diagram of CLAUS Tb (2.5S 7.5N), January April 1987 MJO (5 m s-1) Kelvin waves (15 m s-1)

12 Simple Models of Wave Motion in the Tropics Full Equations of Motion and Scaling Equatorial Beta Plane Shallow Water (Following Matsuno (1966), JMSJ Boussinesq Model (Following Roundy and Janiga 2011)

13 u' t 2Ωv'sin(φ) + 2Ωw'cos(ϕ) = 1 p' ρ x

14 u' t 2Ωv'sin(φ) + 2Ωw'cos(ϕ) = 1 p' ρ x v' t + 2Ωu'sin(φ) = 1 p' ρ y

15 u' p' t 2Ωv'sin(φ) + 2Ωw'cos(ϕ) = 1 ρ x v' p' t + 2Ωu'sin(φ) = 1 ρ y w' t 2Ωu'cos(φ) + 1 ρ 0 p' z = s'

16 u' p' t 2Ωv'sin(φ) + 2Ωw'cos(ϕ) = 1 ρ x v' p' t + 2Ωu'sin(φ) = 1 ρ y w' t 2Ωu'cos(φ) + 1 ρ 0 p' z = 0

17 u' t 2Ωv'sin(φ) + 2Ωw'cos(ϕ) = 1 p' ρ x v' t + 2Ωu'sin(φ) = 1 p' ρ y w' t 2Ωu'cos(φ) + 1 ρ 0 p' z = 0 1 p' ρ t + gh e u' x + v' = 0 y

18 Shallow Water Model Assume a constant density shallow fluid of mean depth h e

19 f βy u t βyv'= 1 p' ρ x v' t + βyu'= 1 p' ρ y 1 p' ρ t + gh u' e x + v' = 0 y u' v' 1 ρ p' = u ˆ (y) v ˆ (y) p ˆ (y) exp i(kx νt) [ ]

20 f βy u' t βyv'= 1 p' ρ x v' t + βyu'= 1 p' ρ y 1 p' ρ t + gh u' e x + v' = 0 y u' u ˆ (y) v' = ˆ v (y) 1 ρ p' p ˆ (y) exp[ i(kx νt) ]

21 f βy u' t βyv'= 1 p' ρ x v' t + βyu'= 1 p' ρ y 1 p' ρ t + gh u' e x + v' = 0 y u' u ˆ (y) v' = ˆ v (y) exp i(kx νt) 1 ρ p' p ˆ (y) [ ] iνˆ u βy ˆ v = ikˆ p iνv ˆ + βyˆ u = ˆ p y iνˆ p + gh e ikˆ u + v ˆ = 0 y See Holton Chapter 11 (tropical dynamics) for solutions

22 Mixed Rossby-Gravity Wave Theoretical Structure Wind, Pressure (contours), Divergence, red negative

23 Kelvin Wave Theoretical Structure Wind, Pressure (contours), Divergence, blue negative

24 Boussinesq Model N(z) = g dρ ρ 0 dz 1/ 2 s = g ( ρ ρ) 0 ρ 0

25 u' t βyv'+2ωw'+ 1 ρ 0 p x = 0 (1)

26 u' t βyv'+2ωw'+ 1 p ρ 0 x = 0 (1) v' t + βyu'+ 1 p ρ 0 y = 0 (2)

27 u' t βyv'+2ωw'+ 1 p ρ 0 x = 0 (1) v' t + βyu'+ 1 p ρ 0 y = 0 (2) w' 2Ωu'+ 1 p' t ρ 0 z = s (3)

28 u' t βyv'+2ωw'+ 1 p ρ 0 x = 0 (1) v' t + βyu'+ 1 p ρ 0 y = 0 (2) w' 2Ωu'+ 1 p' t ρ 0 z = s (3) u' x + v' y + w' z = 0 (4)

29 u' t βyv'+2ωw'+ 1 p ρ 0 x = 0 (1) v' t + βyu'+ 1 p ρ 0 y = 0 (2) w' 2Ωu'+ 1 p' t ρ 0 z = s (3) u' x + v' y + w' z = 0 (4) s' t + N 2 w'= 0 (5)

30 (6) ( ( )) (6) ( u',v',w',ρ 1 0 p',s' ) = ( u ˆ, v ˆ, w ˆ, p ˆ, s ˆ )exp i kx + lz νt

31 iνˆ u βyv ˆ + 2Ωw ˆ + ikˆ p = 0 (7) iνˆ v + βyˆ u + ˆ p y = 0 (8) iνw ˆ 2Ωˆ u + ilˆ p = s ˆ (9) ikˆ u + v ˆ y + il w ˆ = 0 (10) iνˆ s + N 2 w ˆ = 0 (11)

32 2 v ˆ y + 2 l 4iβylΩ ˆ v N 2 + 4Ω 2 ν 2 y + 2 ν 2 k 2 ( N 2 ν 2 ) kβ ( ν N 2 ν 2 ) N 2 + 4Ω 2 ν 2 v ˆ + 2iΩlβ N 2 + 4Ω 2 ν 2 ˆ v l 2 β 2 y 2 N 2 + 4Ω 2 ν 2 ˆ v = 0

33 v ˆ = η(y)exp i 2 Γy 2 Γ 2βlΩ N 2 + 4Ω 2 ν 2

34 2 η l 2 ν 2 + k 2 ( N 2 ν 2 ) + kβ y ν N 2 ν 2 2 N 2 + 4Ω 2 ν 2 ( ) + y 2 β 2 l 2 ( N 2 ν 2 ) ( N 2 + 4Ω 2 ν 2 ) 2 η = 0

35 ( N 2 ν 2 ) 1/ 2 βl k 2 kβ + l2 ν 2 ν N 2 ν 2 = 2n + 1

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38 How Do Equatorial Waves Actually Behave? Analysis of observations

39 Morphology of a Tropical Mesoscale Convective Complex in the eastern Atlantic during GATE (from Zipser et al. 1981) Storm Motion

40 Observed Kelvin wave morphology (from Straub and Kiladis 2003) Wave Motion

41 Two day (WIG) wave cloud morphology (from Takayabu et al. 1996)

42 from Morita et al., 2006

43 Basic Fourier Analysis Assume some time series x(t) that includes various quasi-periodic processes How can we compare activity at different frequencies or filter activity in a specific frequency band from the rest of the signal?

44 Fourier Analysis Used to find out how much activity in a given frequency is found in a time series, and to filter for that activity Basic underlying principle: sin(2πft + ϕ) = asin2πft + bcos2πft

45 Fourier Analysis Used to find out how much activity in a given frequency is found in a time series, and to filter for that activity Basic underlying principle: sin(2πft + ϕ) = asin2πft + bcos2πft sin(2πft + ϕ) = sin2πft cosϕ + cos2πft sinϕ

46 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x

47 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x X(t) = 1 T T µ=1 ( )( t 1) /T X(µ) * exp( i * 2π µ 1 )...1 t T Inverse Transform

48 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x i = 1 r = x 2 + y 2 Polar Coordinates tanθ = y / x

49 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x i = 1 r = x 2 + y 2 Polar Coordinates tanθ = y / x z = rcosθ + irsinθ = r(cosθ + isinθ)

50 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x i = 1 r = x 2 + y 2 Polar Coordinates tanθ = y / x z = rcosθ + irsinθ = r(cosθ + isinθ) cosθ + isinθ = e iθ Euler s Formula

51 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x i = 1 r = x 2 + y 2 Polar Coordinates tanθ = y / x z = rcosθ + irsinθ = r(cosθ + isinθ) cosθ + isinθ = e iθ Euler s Formula

52 X(µ) = 1 T T x(t)exp i 2π(µ 1)(t 1) t=1 ( ) Fourier Transform of x The power spectrum is just X(µ)X(µ)* Often a signal is buried in noise. To emphasize significant signals, smoothing the spectrum is appropriate. This can include breaking the dataset up into many subsets, calculating the spectra of each subset, and averaging the results.

53 Pure sinusoidal signals Square wave example Harmonics Gibbs ringing phenomenon Aliasing Signals in observed data Seasonal Cycle

54 From Mathworld.wolfram.com

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56 Red noise White noise Blue noise How to distinguish noise from signal?

57 Some Definitions DFT, FFT, power, complex conjugate Frequency angular Wavenumber

58 Resolution of a Spectrum: Assume a time series length T days Maximum period resolved is length T Minimum period resolved is 2 days Periods resolved are T, T/2, T/3,, 2

59 Tropical Convection Tropical convection varies in space and time, so a spectrum characterizing it should include both zonal wavenumber and frequency. Break the dataset up into many overlapping time segments Take dft for a segment in space, then transform the result again in time. Average power over all segments

60 Symmetry of a Power Spectrum A power spectrum is redundant Second half reflects the first half Lowest Frequencies occur at the beginning and end, with highest at center Wavenumber-frequency spectra actually have four quadrants

61 Add full spectrum

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72 0.5 Spectral Bands for OLR and PW Filtering TD-type 3 days Frequency (CPD) MRG Kelvin 6 days Period (Days) 0.1 h=50m ER 30 days MJO ISOe Westward Zonal Wavenumber Eastward h=12m *Adapted from Wheeler and Kiladis, 1999

73 Variance Variance is the mean square difference between observed values and the mean value var = ((x mean(x)) 2 ) Knowing the mean climate is necessary but not sufficient to describing weather in a region Variance of OLR also has a seasonal cycle