Extreme events as a multi-point feature - Entropy production as a criterion for cascade process

Size: px

Start display at page:

Download "Extreme events as a multi-point feature - Entropy production as a criterion for cascade process"

Ashley Oswald McCormick
6 years ago
Views:

1 Extreme events as a multi-point feature - Entropy production as a criterion for cascade process J. P., Nico Reinke, Ali Hadjihosseni, Daniel Nickelsen and Andreas Engel Institute of Physics, Carl von Ossietzky University of Oldenburg,

2 - motivation wind energy rogue waves financial data - n-point statistics of turbulence - nonequilibrium thermodynamics and extrem

3 modern wind turbines WEC >5MW area = m2 energy resource Wind WPI Wien 2015 Center for Wind Energy Research

4 wind measurements and data analysis wind conditions after IEC measurement at hub height in front of a turbine measured time series Center for Wind Energy Research WPI Wien 2015

5 wind ressource reference wind speed annual mean wind speed V ref profile extreme events -gusts Center for Wind Energy Research WPI Wien 2015

6 ~ 10 m/s wind measurements and data analysis characterization after IEC norm - or what is a gust?? 10 min Mittelwerte Center for Wind Energy Research WPI Wien 2015

7 statistics of gusts P(u τ σ 1 ) τ = 4 s P rob(u ø > 6æ) º 10 4 º /day Boundary-Layer Meteorology 108 (2003) P rob(u ø > 6æ) º /3000 years Center for Wind Energy Research WPI Wien 2015

8 P WT = 1 2 c p( ) u 3 wind A dynamics of power conversion Center for Wind Energy Research WPI Wien 2015

9 increment statistics of power fluctuations highly intermittent and turbulent power dynamics from wind turbines and wind farms Wind τ= 1s Wind (farm) τ=10s, Wind (farm) τ=1000s, Wind (farm) Guassian PDF τ=1s Wind (single) τ=10s, Wind (single) τ=1000s, Wind (single) P(Xτ) (b) (b) X τ /σ τ P. MIlan et.al.prl 110, (2013) Center for Wind Energy Research WPI Wien 2015

10 finance scale dependent quantity for measuring the disorder x(t + τ ) Q(x,r) => r(t, τ ) = or R(t, τ ) = log r(t, τ ) x(t) p(r) [a.u.] return or log return for different time scales T h 4h 1h 15 min 4 min R/σ 0.5

rogue waves measurement Japan 8 6 Surface elevation [m] 4 2 0 2 4 6 44800 45000 45200 45400 45600 45800 data from stoch.

11 rogue waves measurement Japan 8 6 Surface elevation [m] data from stoch. Time [s] model based in multi point statistics Surface elevation [m] Time [s] Time [s]

12 turbulent cascade

13 turbulent cascade: - large vortices are generating smaller multi scale analysis u r := u(x + r) u(x) L <u n r > r n Z <u n r >= u n r p(u r ) du r 10 4 L r r r η u / σ η

14 n-point statistics - expressed by increment statistics n-point statistics n- point statistics expressed by increment statistics p(u(x 1 ),...,u(x n+1 )) = p(u,...,u,u(x r1 rn 1 )) u ri = u(x + r i ) u(x i ) = p(u r1,...,u rn u(x 1 )) p(u(x 1 ) x 1 x n+1 " # $

15 p(u(x 1 ),...,u(x n+1 )) n-point statistics = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) L r... η Markow prop & cascade with Fokker-Planck Equation j p(u rj u rk,u(x 1 )) rj D (1) (u rj,r j,u(x 1 2 j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 ))

16 n-point statistics... Markow prop & cascade with Fokker-Planck Equ j p(u rj u rk,u(x 1 )) rj D (1) (u rj,r j,u(x 1 2 j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 )) ' (#) #$ $ #$ %#%*%+%*%# +%,%%%$%%%-%" + &. +%,%" + %-%" + &. +%,%%%" + %-%" + &. " # $ # " %&%" # shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%' (#(-(.(-(#.(/((($(((0(". )1.(/(". (0(". )'.(/(((". (0(". )' " # $ # " ()(" # %no%u/dependence%of%diffusion%func-on% Stresing et.al. New Journal of Physics 12 (2010)

17 how are the scaling and the process approaches connected multi scale analysis u r := u(x + r) u(x) <u n r > r n L r 2 r 1 stochastic cascade process evolving in r - multi r u r p r (u r ) r η u / σ

18 cascade process as stochastic process -2- stochastic process - stationary stochastic process - jump process stochastic process - jump process

19 thermodynamics of cascade L new approach to non- equilibrium thermodynamics by Seifert PRL 95 (2005) r 2 r 1 the evolution along cascade as thermodynamical process entropy production along a single trajectory trajectory obeys an integral fluctuation theorem - Jarzynski relation PRL 78 (1997) η

20 thermodynamics of cascade 1st law : energy balance of a single trajectory U = W (u r ) Q(u r ) U equilibrium energy difference W (u r ) work done on trajectory Q(u r ) heat transferred to system entropy production for different paths u r (Seifert 2005) S(u r )= Q(u r) k B T

21 thermodynamics of cascade : entropy balance 2nd law : entropy balance entropy production in the system (Seifert 2005) the potential S m (u r )= Z r u u '(u r,r)dr is the stationary solution of the Fokker-Planck equ. contribution of fluctuations relaxing to the steady state '(u r )=lnd (2) (u r,r) Z ur 1 D (1) (u 0,r) D (2) (u 0,r) du0 entropy production along trajectory S = ln p r(u r ) p r0 (u 0 ) contribution of the path along the steady state

22 thermodynamics of cascade : entropy balance 2nd law : entropy balance entropy production in the system (Seifert 2005) S m (u r )= entropy production along trajectory Z r u u '(u r,r)dr S = ln p(u L,L) p(u, ) stoch process ' (#) #$ $ #$ %#%*%+%*%# +%,%%%$%%%-%" + &. +%,%" + %-%" + &. +%,%%%" + %-%" + &. " # $ # " %&%" # shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%' (#(-(.(-(#.(/((($(((0(". )1.(/(". (0(". )'.(/(((". (0(". )' " # $ # " ()(" # %no%u/dependence%of%diffusion%func-on%

23 thermodynamics of cascade : entropy balance 2nd law : entropy balance entropy production in the system (Seifert 2005) S m (u r )= entropy production along trajectory Z r u u '(u r,r)dr S = ln p(u L,L) p(u, ) stoch process experimental quantities r ' (#) #$ $ #$ %#%*%+%*%# +%,%%%$%%%-%" + &. +%,%" + %-%" + &. +%,%%%" + %-%" + &. " # $ # " %&%" # shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%' (#(-(.(-(#.(/((($(((0(". )1.(/(". (0(". )'.(/(((". (0(". )' " # $ # " ()(" # %no%u/dependence%of%diffusion%func-on% u / σ

24 thermodynamics of cascade : entropy balance 2nd law : entropy balance entropy production in the system (Seifert 2005) S m (u r )= entropy production along trajectory Z r u u '(u r,r)dr S = total entropy production ln p(u L,L) p(u, ) experimental quantities stoch process S tot (u r )=S m (u r )+ S hs tot (u r )i 0 he S tot(u r ) i =1 2nd law integral fluctuation theorem Seifert (2005)

25 test of validity of the fluctuation theorem Renner et al JFM (2001) 1.5 exp [ Stot] N he S tot(u r ) i N N Nickelsen Engel PRL (2013)

26 results for turbulent cascade: - 1st fulfillment of fluctuation theorem he S tot(u r ) i =1 This integral fluctuation theorem - a generalized Jarzynski s relation - is truly universal since it holds for - any kind of initial condition, - any time dependence of force and potential, with and without detailed balance, - any length of trajectory without the need for waiting for relaxation. Seifert (2005)

27 - 1st fulfillment of fluctuation theorem he S tot(u r ) i =1-2nd test of the validity of turbulence models S = ln p(u L,L) p(u, ) S m (u r )= Z r u u '(u r,r)dr => IFT holds only for non - scaling stochastic processes

28 summary - nonequi. thermodynamics generalized 2nd law - (entropy maximization) fulfilled for cascade - non scaling process fits to data (?? => new window for blow up??>

29 extreme events and its entropy production S tot (u r ) to fulfill the IFT - there must be paths with positive and negative entropy production

30 extreme events and its entropy production S tot (u(r)) to fulfill the IFT - there must be paths with positive and negative entropy production 6 4 DS.pro d d S Row Index

31 1.) positive entropy production S tot (u r ) > 0 event: large small u L with low probability p(u L ) u with high probability p(u ) u L S tot (u r ) u L u

32 2.) extreme and negative entropy production S tot (u r ) < 0 event small u L with high probability 10 1 large u with low probability 10 3 extreme event S tot (u r ) u L u

3.) extreme and normal entropy production S tot (u r ) 0 10 3 10 1 event 10 1 large u L

33 3.) extreme and normal entropy production S tot (u r ) event 10 1 large u L with low probability 10 3 large u with low probability S tot (u r ) u L u

34 End concept of stochastic cascade - multipoint statistics - non- equilibrium thermodynamics Thank you

35 references U. Seifert: Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, (2012) D. Nickelsen, A. Engel: Probing small-scale intermittency with a fluctuation theorem; Phys. Rev. Lett. (2012) R. Stresing and J. Peinke : Towards a stochastic multi-point description of turbulence, New Journal of Physics 12, (2010). M. Wächter, et al: The turbulent nature of the atmospheric boundary layer and its impact on the wind energy conversion process, Journal of Turbulence 13, 1-21 (2012) P. Milan, M. Wächter, and J. Peinke : Turbulent character of wind energy, Phys. Rev. Lett. 110, (2013) A. Hadjihosseini, J. Peinke, N. Hoffmann : Stochastic analysis of ocean wave states with and without rogue waves, New Journal of Physics 16, (2014) N. Reinke, D. Nickelsen, A. Engel, and J. Peinke: Application of an Integral Fluctuation Theorem to Turbulent Flows (iti 2015)

stochastic models for turbulence and turbulent driven systems

stochastic models for turbulence and turbulent driven systems Dank an Mitarbeiter J. Gottschall M. Hölling, M.R. Luhur St Lück P. MIlan A. Nawroth Ph. Rinn N. Reinke Ch. Renner R. Stresing M. Wächter Koopertionen