Modelling Interactions Between Weather and Climate

Transcription

1 Modelling Interactions Between Weather and Climate p. 1/33 Modelling Interactions Between Weather and Climate Adam Monahan & Joel Culina & School of Earth and Ocean Sciences, University of Victoria Victoria, BC, Canada

2 Modelling Interactions Between Weather and Climate p. 2/33 Introduction Earth s climate system displays variability on many scales

3 Modelling Interactions Between Weather and Climate p. 2/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m

4 Modelling Interactions Between Weather and Climate p. 2/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond)

5 Modelling Interactions Between Weather and Climate p. 2/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather

6 Introduction From von Storch and Zwiers, 1999 Modelling Interactions Between Weather and Climate p. 3/33

7 Modelling Interactions Between Weather and Climate p. 4/33 Introduction From von Storch and Zwiers, 1999

8 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales

9 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored

10 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored Classical deterministic parameterisations assume very large scale separation

11 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored Classical deterministic parameterisations assume very large scale separation Following Mitchell (1966), Hasselmann (1976) suggested stochastic parameterisations more appropriate when separation finite

12 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored Classical deterministic parameterisations assume very large scale separation Following Mitchell (1966), Hasselmann (1976) suggested stochastic parameterisations more appropriate when separation finite This talk will consider:

13 Modelling Interactions Between Weather and Climate p. 5/33 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored Classical deterministic parameterisations assume very large scale separation Following Mitchell (1966), Hasselmann (1976) suggested stochastic parameterisations more appropriate when separation finite This talk will consider: stochastic averaging as a tool for obtaining stochastic climate models from coupled weather/climate system

14 Introduction Earth s climate system displays variability on many scales space: 10 6 m m time: seconds years (and beyond) Loosely, slower variability called climate and faster called weather Dynamical nonlinearities coupling across space/time scales modelling climate, weather must be parameterised, not ignored Classical deterministic parameterisations assume very large scale separation Following Mitchell (1966), Hasselmann (1976) suggested stochastic parameterisations more appropriate when separation finite This talk will consider: stochastic averaging as a tool for obtaining stochastic climate models from coupled weather/climate system two examples: coupled atmosphere/ocean boundary layers, extratropical atmospheric low-frequency variability Modelling Interactions Between Weather and Climate p. 5/33

15 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules

16 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules Kinetic energy of individual molecules randomly distributed with Maxwell-Boltzmann distribution p(e)

17 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules Kinetic energy of individual molecules randomly distributed with Maxwell-Boltzmann distribution p(e) As N mean energy approaches sharp value E = 1 N N E i N E = E p(e) de i=1

18 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules Kinetic energy of individual molecules randomly distributed with Maxwell-Boltzmann distribution p(e) As N mean energy approaches sharp value E = 1 N N E i N E = E p(e) de i=1 For N large (but finite), E is random E = E + 1 N ζ such that (by central limit theorem) ζ is Gaussian

19 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules Kinetic energy of individual molecules randomly distributed with Maxwell-Boltzmann distribution p(e) As N mean energy approaches sharp value E = 1 N N E i N E = E p(e) de i=1 For N large (but finite), E is random E = E + 1 N ζ such that (by central limit theorem) ζ is Gaussian Very large scale separation deterministic dynamics

20 Modelling Interactions Between Weather and Climate p. 6/33 A statistical mechanical perspective Consider a box at temperature T containing N ideal gas molecules Kinetic energy of individual molecules randomly distributed with Maxwell-Boltzmann distribution p(e) As N mean energy approaches sharp value E = 1 N N E i N E = E p(e) de i=1 For N large (but finite), E is random E = E + 1 N ζ such that (by central limit theorem) ζ is Gaussian Very large scale separation deterministic dynamics Smaller (but still large) separation stochastic corrections needed

21 Modelling Interactions Between Weather and Climate p. 7/33 Hasselmann s ansatz: a stochastic climate model Observation: spectra of sea surface temperature (SST) variability are generically red

22 Modelling Interactions Between Weather and Climate p. 7/33 Hasselmann s ansatz: a stochastic climate model Observation: spectra of sea surface temperature (SST) variability are generically red Observations from North Atlantic weathership INDIA From Frankignoul & Hasselmann Tellus 1977

23 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed:

24 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed: local dynamics of small perturbations T linearised around climatological mean state

25 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed: local dynamics of small perturbations T linearised around climatological mean state fast atmospheric fluxes represented as white noise

26 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed: local dynamics of small perturbations T linearised around climatological mean state fast atmospheric fluxes represented as white noise simple Ornstein-Uhlenbeck process with spectrum dt dt = 1 τ T + γẇ E{T (ω) 2 } = γ 2 τ 2 2π(1 + ω 2 τ 2 )

27 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed: local dynamics of small perturbations T linearised around climatological mean state fast atmospheric fluxes represented as white noise simple Ornstein-Uhlenbeck process with spectrum Linear stochastic model dt dt = 1 τ T + γẇ E{T (ω) 2 } = γ 2 τ 2 2π(1 + ω 2 τ 2 ) simple null hypothesis for observed variability

28 Modelling Interactions Between Weather and Climate p. 8/33 Hasselmann s ansatz: a stochastic climate model Simple model of Frankignoul & Hasselmann (1977) assumed: local dynamics of small perturbations T linearised around climatological mean state fast atmospheric fluxes represented as white noise simple Ornstein-Uhlenbeck process with spectrum Linear stochastic model dt dt = 1 τ T + γẇ E{T (ω) 2 } = γ 2 τ 2 2π(1 + ω 2 τ 2 ) simple null hypothesis for observed variability We ll be coming back to this again later on...

29 Modelling Interactions Between Weather and Climate p. 9/33 Coupled fast-slow systems Starting point for stochastic averaging is assumption that state variable z can be written z = (x,y)

30 Modelling Interactions Between Weather and Climate p. 9/33 Coupled fast-slow systems Starting point for stochastic averaging is assumption that state variable z can be written z = (x,y) where x slow variable (climate) y fast variable (weather) d dt x = f(x,y) d dt y = g(x,y) such that

31 Modelling Interactions Between Weather and Climate p. 9/33 Coupled fast-slow systems Starting point for stochastic averaging is assumption that state variable z can be written z = (x,y) where x slow variable (climate) y fast variable (weather) d dt x = f(x,y) d dt y = g(x,y) such that where is small" τ f(x,y) g(x, y)

32 Modelling Interactions Between Weather and Climate p. 10/33 A simple example As a simple example of a coupled fast-slow system consider d dt x = x x3 + (Σ + x)y d dt y = 1 τ x y + 1 Ẇ τ

33 Modelling Interactions Between Weather and Climate p. 10/33 A simple example As a simple example of a coupled fast-slow system consider d dt x = x x3 + (Σ + x)y d dt y = 1 τ x y + 1 Ẇ τ The weather process y is Gaussian with stationary autocovariance C yy (s) = E {y(t)y(t + s)} = x ( 2 exp s ) τ x as τ 0 τx 2 δ(s)

34 Modelling Interactions Between Weather and Climate p. 10/33 A simple example As a simple example of a coupled fast-slow system consider d dt x = x x3 + (Σ + x)y d dt y = 1 τ x y + 1 Ẇ τ The weather process y is Gaussian with stationary autocovariance C yy (s) = E {y(t)y(t + s)} = x ( 2 exp s ) τ x as τ 0 τx 2 δ(s) Intuition suggests that as τ 0 d dt x x x3 + τ(σ + x) x Ẇ

35 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t)

36 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with d dt x = f(x)

37 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with where f(x) = d dt x = f(x) f(x,y)dµ x (y) = lim T 1 T T 0 f(x,y(t)) dt

38 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with where f(x) = d dt x = f(x) f(x,y)dµ x (y) = lim T 1 T T averages f(x,y) over weather for a fixed climate state 0 f(x,y(t)) dt

39 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with where f(x) = d dt x = f(x) f(x,y)dµ x (y) = lim T 1 T T averages f(x,y) over weather for a fixed climate state 0 f(x,y(t)) dt To leading order, climate follows deterministic averaged dynamics

40 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with where f(x) = d dt x = f(x) f(x,y)dµ x (y) = lim T 1 T T averages f(x,y) over weather for a fixed climate state 0 f(x,y(t)) dt To leading order, climate follows deterministic averaged dynamics For simple model d dt x = E y x { x x 3 + xy + Σy } = x x 3

41 Modelling Interactions Between Weather and Climate p. 11/33 Averaging approximation (A) As τ 0, over finite times we have x(t) x(t) with d dt x = f(x) where f(x) = f(x,y)dµ x (y) = lim T 1 T T 0 f(x,y(t)) dt averages f(x,y) over weather for a fixed climate state To leading order, climate follows deterministic averaged dynamics For simple model d dt x = E y x { x x 3 + xy + Σy } = x x 3 i.e. depending on x(0) system settles into bottom of one of two potential wells x = ±1

42 Modelling Interactions Between Weather and Climate p. 12/33 Analogy to Reynolds averaging Approximation (A) analogous to Reynolds averaging in fluid mechanics: split flow into mean and turbulent components u = u + u

43 Modelling Interactions Between Weather and Climate p. 12/33 Analogy to Reynolds averaging Approximation (A) analogous to Reynolds averaging in fluid mechanics: split flow into mean and turbulent components u = u + u where u(t) = 1 2T T T u(t + s) ds

44 Modelling Interactions Between Weather and Climate p. 12/33 Analogy to Reynolds averaging Approximation (A) analogous to Reynolds averaging in fluid mechanics: split flow into mean and turbulent components where u(t) = 1 2T u = u + u T T Full flow satisfies Navier-Stokes equations u(t + s) ds t u + u u = force

45 Modelling Interactions Between Weather and Climate p. 12/33 Analogy to Reynolds averaging Approximation (A) analogous to Reynolds averaging in fluid mechanics: split flow into mean and turbulent components where u(t) = 1 2T u = u + u T T Full flow satisfies Navier-Stokes equations u(t + s) ds t u + u u = force mean flow satisfies eddy averaged dynamics t u + u u = force u u

46 Analogy to Reynolds averaging Approximation (A) analogous to Reynolds averaging in fluid mechanics: split flow into mean and turbulent components where u(t) = 1 2T u = u + u T T Full flow satisfies Navier-Stokes equations u(t + s) ds t u + u u = force mean flow satisfies eddy averaged dynamics t u + u u = force u u Note: averaging is projection operator only if scale separation between u and u Modelling Interactions Between Weather and Climate p. 12/33

47 Modelling Interactions Between Weather and Climate p. 13/33 Central limit theorem approximation (L) For τ 0 averaged and full trajectories differ; this difference can be modelled as a random process

48 Modelling Interactions Between Weather and Climate p. 13/33 Central limit theorem approximation (L) For τ 0 averaged and full trajectories differ; this difference can be modelled as a random process Basic result: for τ small x(t) x(t) + ζ

49 Modelling Interactions Between Weather and Climate p. 13/33 Central limit theorem approximation (L) For τ 0 averaged and full trajectories differ; this difference can be modelled as a random process Basic result: for τ small x(t) x(t) + ζ where ζ is the multivariate Ornstein-Uhlenbeck process d ζ = Df(x) ζ + σ(x)ẇ dt

50 Modelling Interactions Between Weather and Climate p. 13/33 Central limit theorem approximation (L) For τ 0 averaged and full trajectories differ; this difference can be modelled as a random process Basic result: for τ small x(t) x(t) + ζ where ζ is the multivariate Ornstein-Uhlenbeck process d ζ = Df(x) ζ + σ(x)ẇ dt Diffusion matrix σ(x) defined by lag correlation integrals σ 2 (x) = E y x {f [x,y(t + s)]f [x,y(t)]} ds where f [x,y(t)] = f[x,y(t)] f[x]

51 Modelling Interactions Between Weather and Climate p. 13/33 Central limit theorem approximation (L) For τ 0 averaged and full trajectories differ; this difference can be modelled as a random process Basic result: for τ small x(t) x(t) + ζ where ζ is the multivariate Ornstein-Uhlenbeck process d ζ = Df(x) ζ + σ(x)ẇ dt Diffusion matrix σ(x) defined by lag correlation integrals σ 2 (x) = E y x {f [x,y(t + s)]f [x,y(t)]} ds where f [x,y(t)] = f[x,y(t)] f[x] Note that std(ζ) τ so corrections vanish in strict τ 0 limit

52 Modelling Interactions Between Weather and Climate p. 14/33 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected

53 Modelling Interactions Between Weather and Climate p. 14/33 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected feedback of weather on climate only through averaging; weather can t drive climate between metastable states

54 Modelling Interactions Between Weather and Climate p. 14/33 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected feedback of weather on climate only through averaging; weather can t drive climate between metastable states (L) approximation still has been very successful in many contexts (e.g. Linear Inverse Modelling)

55 Modelling Interactions Between Weather and Climate p. 14/33 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected feedback of weather on climate only through averaging; weather can t drive climate between metastable states (L) approximation still has been very successful in many contexts (e.g. Linear Inverse Modelling) For the simple example model: f (x, y) = (x x 3 + xy + Σy) (x x 3 ) = (x + Σ)y

56 Modelling Interactions Between Weather and Climate p. 14/33 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected feedback of weather on climate only through averaging; weather can t drive climate between metastable states (L) approximation still has been very successful in many contexts (e.g. Linear Inverse Modelling) For the simple example model: f (x, y) = (x x 3 + xy + Σy) (x x 3 ) = (x + Σ)y so σ 2 (x) = (x + Σ) 2 E y x {y(t + s)y(t)} ds = (x + Σ) 2 x 2

57 Central limit theorem approximation (L) Averaging solution x(t) augmented by Gaussian corrections such that x(t) itself not affected feedback of weather on climate only through averaging; weather can t drive climate between metastable states (L) approximation still has been very successful in many contexts (e.g. Linear Inverse Modelling) For the simple example model: f (x, y) = (x x 3 + xy + Σy) (x x 3 ) = (x + Σ)y so σ 2 (x) = (x + Σ) 2 E y x {y(t + s)y(t)} ds = (x + Σ) 2 x 2 x(t) 1 + ζ such that d dt ζ = 2ζ + (1 + Σ)2 Ẇ Modelling Interactions Between Weather and Climate p. 14/33

58 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate

59 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt

60 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states

61 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states Noise-induced drift further potential feedback of weather on climate

62 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states Noise-induced drift further potential feedback of weather on climate Simple model: d dt x = x x3 + (Σ + x) x Ẇ

63 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states Noise-induced drift further potential feedback of weather on climate Simple model: d dt x = x x3 + (Σ + x) x Ẇ Multiplicative noise introduced through:

64 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states Noise-induced drift further potential feedback of weather on climate Simple model: d dt x = x x3 + (Σ + x) x Ẇ Multiplicative noise introduced through: nonlinear coupling of x and y in slow dynamics

65 Modelling Interactions Between Weather and Climate p. 15/33 Hasselmann approximation (N) More complete approximation involves full two-way interaction between weather and climate x(t) solution of stochastic differential equation (SDE) d x = f(x) + σ(x) Ẇ dt Most accurate of all approximations on long timescales; can capture transitions between metastable states Noise-induced drift further potential feedback of weather on climate Simple model: d dt x = x x3 + (Σ + x) x Ẇ Multiplicative noise introduced through: nonlinear coupling of x and y in slow dynamics dependence of stationary distribution of y on x

66 Modelling Interactions Between Weather and Climate p. 16/33 When does this work? For the (A) approximation to hold all that is required is a timescale separation

67 Modelling Interactions Between Weather and Climate p. 16/33 When does this work? For the (A) approximation to hold all that is required is a timescale separation For (L) and (N) further require fast process is strongly mixing

68 Modelling Interactions Between Weather and Climate p. 16/33 When does this work? For the (A) approximation to hold all that is required is a timescale separation For (L) and (N) further require fast process is strongly mixing Informally, what is required is that the autocorrelation function of y decay sufficiently rapidly with lag for large timescale separations the delta-correlated white noise approximation is reasonable

69 Modelling Interactions Between Weather and Climate p. 16/33 When does this work? For the (A) approximation to hold all that is required is a timescale separation For (L) and (N) further require fast process is strongly mixing Informally, what is required is that the autocorrelation function of y decay sufficiently rapidly with lag for large timescale separations the delta-correlated white noise approximation is reasonable Crucially: none of this assumes that the fast process is stochastic. Effective SDEs (L) and (N) arise in limit τ 0 as approximation to fast deterministic or stochastic dynamics

70 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0

71 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0 MTV" theory considers averaging assuming that weather influence strengthens in this limit

72 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0 MTV" theory considers averaging assuming that weather influence strengthens in this limit An example: dx dt dy dt = x + a τ y = 1 x 1 τ τ y + b Ẇ τ

73 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0 MTV" theory considers averaging assuming that weather influence strengthens in this limit An example: dx dt dy dt = x + a τ y = E y x {y} = τx, std y x = b 2 /2 1 x 1 τ τ y + b Ẇ τ

74 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0 MTV" theory considers averaging assuming that weather influence strengthens in this limit An example: dx dt dy dt = x + a τ y = E y x {y} = τx, std y x = b 2 /2 1 x 1 τ τ y + b Ẇ τ Stochastic terms remain in the strict τ 0 limit dx dt = (1 a)x + abẇ

75 Modelling Interactions Between Weather and Climate p. 17/33 Relation to MTV Theory" Have assumed that influence of weather on climate is bounded as τ 0 MTV" theory considers averaging assuming that weather influence strengthens in this limit An example: dx dt dy dt = x + a τ y = E y x {y} = τx, std y x = b 2 /2 1 x 1 τ τ y + b Ẇ τ Stochastic terms remain in the strict τ 0 limit dx = (1 a)x + abẇ dt Present analysis will not make MTV ansatz of increasing weather influence as τ decreases; rather than a τ = 0 theory", it is a small τ theory"

76 Modelling Interactions Between Weather and Climate p. 18/33 Case study 1: Coupled atmosphere-ocean boundary layers Atmosphere and ocean interact through (generally turbulent) boundary layers, exchanging mass, momentum, and energy

77 Modelling Interactions Between Weather and Climate p. 18/33 Case study 1: Coupled atmosphere-ocean boundary layers Atmosphere and ocean interact through (generally turbulent) boundary layers, exchanging mass, momentum, and energy Idealised coupled model of atmospheric winds and air/sea temperatures: d dt u = Π u c d(w) h(t a, T o ) wu d dt v = c d(w) h(t a, T o ) wv d w e h(t a, T o ) u + σ uẇ1 w e h(t a, T o ) v + σ uẇ2 dt T a = β (T o T a )w β θ(w µ w ) λ a T a + Σ 11 Ẇ 3 + Σ 12 Ẇ 4 γ a γ a γ a d dt T o = β (T a T o )w + β θ(w µ w ) λ o T o + Σ 21 Ẇ 3 + Σ 22 Ẇ 4 γ o γ o γ o

78 Modelling Interactions Between Weather and Climate p. 19/33 Case study 1: Coupled atmosphere-ocean boundary layers 60 o N 54 o N 48 o N OSP 42 o N 36 o N 30 o N 165 o E 180 o W 165 o W 150 o W 135 o W 120 o W Ocean station P

79 Modelling Interactions Between Weather and Climate p. 20/33 Case study 1: Coupled atmosphere-ocean boundary layers Atmosphere and ocean interact through (generally turbulent) boundary layers, exchanging mass, momentum, and energy Idealised coupled model of atmospheric winds and air/sea temperatures: d dt u = Π u c d(w) h(t a, T o ) wu d dt v = c d(w) h(t a, T o ) wv d w e h(t a, T o ) u + σ uẇ1 w e h(t a, T o ) v + σ uẇ2 dt T a = β (T o T a )w β θ(w µ w ) λ a T a + Σ 11 Ẇ 3 + Σ 12 Ẇ 4 γ a γ a γ a d dt T o = β (T a T o )w + β θ(w µ w ) λ o T o + Σ 21 Ẇ 3 + Σ 22 Ẇ 4 γ o γ o γ o Atmospheric boundary layer thickness determined by surface stratification h(t a, T o ) = max [ h min, h(1 α(t a T o )) ]

80 Modelling Interactions Between Weather and Climate p. 21/33 Case study 1: Coupled atmosphere-ocean boundary layers h (m) T a T o (K) Boundary layer height and surface stability

81 Case study 1: Coupled atmosphere-ocean boundary layers correlation lag (days) T a (obs) T a (sim) T o (obs) T o (sim) T o (K) T a (K) Obs Sim 0.8 Obs Sim Obs Sim correlation p(w) lag (days) w (m/s) Observed and simulated statistics (Ocean Station P) Modelling Interactions Between Weather and Climate p. 22/33

82 Modelling Interactions Between Weather and Climate p. 23/33 Case study 1: Coupled atmosphere-ocean boundary layers Observed timescale separations: τ w τ Ta 0.7

83 Modelling Interactions Between Weather and Climate p. 23/33 Case study 1: Coupled atmosphere-ocean boundary layers Observed timescale separations: τ w τ Ta 0.7 Take x = T = (T a,t o ), y = (u,v) with

84 Case study 1: Coupled atmosphere-ocean boundary layers Observed timescale separations: τ w τ Ta 0.7 Take x = T = (T a,t o ), y = (u,v) with Averaging model (A) 5 f 1 ( K / s ) d T = f(t) + ΣẆ dt 5 f 2 ( K / s ) T o (K) T a (K) T a (K) x x 10 6 Modelling Interactions Between Weather and Climate p. 23/33

85 Modelling Interactions Between Weather and Climate p. 24/33 Case study 1: Coupled atmosphere-ocean boundary layers Analytic expression for σ(t a, T o ) σ(t a, T o ) = β γ 2 a + γ 2 o γ o γ a 1 1 γ a γ o ψ(t a T o )

86 Modelling Interactions Between Weather and Climate p. 24/33 Case study 1: Coupled atmosphere-ocean boundary layers Analytic expression for σ(t a, T o ) σ(t a, T o ) = β γ 2 a + γ 2 o γ o γ a 1 1 γ a γ o ψ(t a T o ) where ψ(t a T o ) = (T a T o + θ) E {w (t + s)w (t)} ds

87 Case study 1: Coupled atmosphere-ocean boundary layers Analytic expression for σ(t a, T o ) σ(t a, T o ) = β γ 2 a + γ 2 o γ o γ a 1 1 γ a γ o ψ(t a T o ) where ψ(t a T o ) = (T a T o + θ) E {w (t + s)w (t)} ds 0.5 ψ (K m s 1/ ) T a T o (K) 5 10 Modelling Interactions Between Weather and Climate p. 24/33

88 Modelling Interactions Between Weather and Climate p. 25/33 Case study 1: Coupled atmosphere-ocean boundary layers correlation correlation lag (days) T (reduced) a T (full) a T o (reduced) T (full) o lag (days) T (reduced) a T (full) a T o (reduced) T (full) o T o (K) T o (K) T a (K) T a (K) Reduced Full Reduced Full τ = 0.7 τ = 0.35

89 Modelling Interactions Between Weather and Climate p. 26/33 Case study 1: Coupled atmosphere-ocean boundary layers Sura and Newman (2008) fit observations at OSP to linear multiplicative SDE d T = AT + B(T) Ẇ dt

90 Modelling Interactions Between Weather and Climate p. 26/33 Case study 1: Coupled atmosphere-ocean boundary layers Sura and Newman (2008) fit observations at OSP to linear multiplicative SDE d T = AT + B(T) Ẇ dt Form of this SDE justified through simple substitution in full coupled dynamics w µ w + σẇ

91 Modelling Interactions Between Weather and Climate p. 26/33 Case study 1: Coupled atmosphere-ocean boundary layers Sura and Newman (2008) fit observations at OSP to linear multiplicative SDE d T = AT + B(T) Ẇ dt Form of this SDE justified through simple substitution in full coupled dynamics w µ w + σẇ Form of SDE did not account for feedback of stratification on h; physical origin of parameters in inverse model somewhat different than had been assumed

92 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV)

93 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV) Associated with hemispheric to global spatial scales

94 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV) Associated with hemispheric to global spatial scales Northern Annular Mode (NAM) From

95 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV) Associated with hemispheric to global spatial scales Northern Annular Mode (NAM) Basic structures characterised statistically; dynamical questions remain open From

96 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV) Associated with hemispheric to global spatial scales Northern Annular Mode (NAM) Basic structures characterised statistically; dynamical questions remain open Empirical stochastic models have been useful for studying LFV From

97 Modelling Interactions Between Weather and Climate p. 27/33 Case study 2: extratropical atmospheric LFV Variability of extratropical atmosphere on timescales of 10 days + descrbed as low-frequency variability (LFV) Associated with hemispheric to global spatial scales Northern Annular Mode (NAM) From Basic structures characterised statistically; dynamical questions remain open Empirical stochastic models have been useful for studying LFV Stochastic reduction techniques natural tool for investigating dynamics

98 Case study 2: extratropical atmospheric LFV Kravtsov et al. JAS 2005 Modelling Interactions Between Weather and Climate p. 28/33

99 Modelling Interactions Between Weather and Climate p. 29/33 Case study 2: extratropical atmospheric LFV Using PCA decomposition, identify 2 slow modes: propagating wavenumber 4, stationary zonal

100 Modelling Interactions Between Weather and Climate p. 29/33 Case study 2: extratropical atmospheric LFV Using PCA decomposition, identify 2 slow modes: propagating wavenumber 4, stationary zonal Kravtsov et al. (2005) concluded that regime behaviour resulted from nonlinear interaction of these modes

101 Modelling Interactions Between Weather and Climate p. 29/33 Case study 2: extratropical atmospheric LFV Using PCA decomposition, identify 2 slow modes: propagating wavenumber 4, stationary zonal Kravtsov et al. (2005) concluded that regime behaviour resulted from nonlinear interaction of these modes Consider reduction to 3-variable, 1-variable (stationary mode) models

102 Case study 2: extratropical atmospheric LFV Modelling Interactions Between Weather and Climate p. 30/33

103 Case study 2: extratropical atmospheric LFV Modelling Interactions Between Weather and Climate p. 31/33

104 Modelling Interactions Between Weather and Climate p. 31/33 Case study 2: extratropical atmospheric LFV Stationary mode potential shows prominent shoulder

105 Modelling Interactions Between Weather and Climate p. 31/33 Case study 2: extratropical atmospheric LFV Stationary mode potential shows prominent shoulder In this model, appears that multiple regimes produced by nonlinearity in effective dynamics of stationary mode

106 Modelling Interactions Between Weather and Climate p. 31/33 Case study 2: extratropical atmospheric LFV Stationary mode potential shows prominent shoulder In this model, appears that multiple regimes produced by nonlinearity in effective dynamics of stationary mode Non-gaussianity doesn t require state-dependent noise

107 Modelling Interactions Between Weather and Climate p. 32/33 Case study 2: extratropical atmospheric LFV MTV approximations with minimal regression tuning

108 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems

109 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems Reduction approach is straightforward, although implementation in high-dimensional systems challenging

110 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems Reduction approach is straightforward, although implementation in high-dimensional systems challenging This approach provides a more general approach than e.g. MTV theory, with

111 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems Reduction approach is straightforward, although implementation in high-dimensional systems challenging This approach provides a more general approach than e.g. MTV theory, with benefit of less restrictive assumptions

112 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems Reduction approach is straightforward, although implementation in high-dimensional systems challenging This approach provides a more general approach than e.g. MTV theory, with benefit of less restrictive assumptions drawbacks of more difficult implementation, lack of closed-form solution

113 Modelling Interactions Between Weather and Climate p. 33/33 Conclusions Stochastic reduction tools allow systematic derivation of low-order climate models from coupled weather/climate systems Reduction approach is straightforward, although implementation in high-dimensional systems challenging This approach provides a more general approach than e.g. MTV theory, with benefit of less restrictive assumptions drawbacks of more difficult implementation, lack of closed-form solution Real systems do not generally have a strong scale separation; an important outstanding problem is a more systematic approach to accounting for this