Lohmann, Gerrit

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Marine temperature trends  (last 6000 years)Alkenone-based temperature trendsAnnual mean sea surface temperature trends276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Lohmann et al., 2013 Response too low276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Marine temperature trends  (last 6000 years)Lohmann et al., 2013 Annual mean sea surface temperature trendsSeasonality of the recorder systemsMarine temperature variability                 276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Power spectrumLaepple and Huybers, 2014Current climate models seem to underestimate long-term variability6Climate variability and sensitivity are relatedPower spectrum: Holocene variability276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Noise Forcing276 CHAPTER 8. BROWNIAN MOTION, WE THER AND CLIMATEExercise 51 – Climate sensiti ity and variability in the Stochas ic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T Qnet + f(t) , (8.44)whe e the air- ea fluxes due to weather systems are represented by a white-noi e process withzero average < Qnet >= 0 and  -correlat d in time < et )Qnet(t + ⌧ ) >=  (⌧ ). Theunction f(t) is a tim dependent determin stic forcing. Assume furthermore that f(t) = c ·u(t)with (t) as unit step or th so-c lled Heaviside step functi n and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moment . Because < Qnet >=0, < T (t) > can be solved using th L place transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1 exp(  t)) (8.46)b caus we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The flu tuation can b characterized by the spect um (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium respon are clos ly coupled (fluctuation-dissipationtheorem).276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).276 CHAPTER 8. BR WNIAN MO ION, WEATHER AND CLIMATEExercise 51 – Cl mate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine th the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where th air-sea fluxes due to weather syst ms are represen d by a white-noise process withzero av rage < Qnet >= 0 an  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelation h of the dissipation (through   ) and the fluct ations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic pr cess, it has t be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >+  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As quilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can b characterized by the spectrum (exercise 50)S(!) < T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the quilibrium response are closely coupled (fluctuation-dissipationtheorem).Stochastic climate model (Hasselmann, 1976)276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-c lled Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibriu response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).276 CHAPTER 8. BR WNIAN MOTIO , WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, i agine that the temperature of th ocean mixed layer of epth h is governedbydTdt=   T + Qnet + f(t) , (8.44)wher the air-sea fluxes du to w athe ystems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step r the so-called Hea iside step function and solve (13.51). What is therelationship of the dissipation (thro gh   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved usi g the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because w hav < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characte ized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).ForcingResponse too lowDamping too high 276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withz ro averag < Qnet >= 0 and  -co related in tim < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q( ) is a sto h stic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   x (  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qn (t)Qnet(t ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Variance too lowLohmann, 2018(Fluctuation Dissipation Theorem)276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dep ndent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solv d using the Laplace transform:< T (t) > = L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).Holocene trends276 CHAPTER 8. BROWNIAN MOTION, WEATHER AND CLIMATEExercise 51 – Climate sensitivity and variability in the Stochastic Climate ModelAs in exercise 50, imagine that the temperature of the ocean mixed layer of depth h is governedbydTdt=   T + Qnet + f(t) , (8.44)where the air-sea fluxes due to weather systems are represented by a white-noise process withzero average < Qnet >= 0 and  -correlated in time < Qnet(t)Qnet(t + ⌧ ) >=  (⌧ ). Thefunction f(t) is a time dependent deterministic forcing. Assume furthermore that f(t) = c ·u(t)with u(t) as unit step or the so-called Heaviside step function and solve (13.51). What is therelationship of the dissipation (through   ) and the fluctuations (through the spectrum S(!)) ?SolutionSince Q(t) is a stochastic process, it has to be solved for the moments. Because < Qnet >=0, < T (t) > can be solved using the Laplace transform:< T (t) > L 1{F (s)}(t) = L 1⇢< T (0) >s +  +cs·1s +   (8.45)= T (0) · exp(  t) +c (1   exp(  t)) (8.46)because we have < T (0) >= T (0). As equilibrium response, we have T = limt!1< T (t) >=c . (8.47)The fluctuation can be characterized by the spectrum (exercise 50)S(!) =< T̂ T̂ ⇤ >=1 2 + !2. (8.48)and therefore, the spectrum and the equilibrium response are closely coupled (fluctuation-dissipationtheorem).And1mu, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=47864359Climate variability and sensitivity are relatedDamping DResponse Variation(Fluctuation Dissipation Theorem)D = spring constants = deflectionHookes Law: F = -D s

Holocene temperatures: Variability and Sensitivity are coupled

https://epic.awi.de/id/eprint/56735/1/FDT_Climate_Lohmann_2018.pdf

Holocene temperatures: Variability and Sensitivity are coupled

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