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,
- motivation wind energy rogue waves financial data - n-point statistics of turbulence - nonequilibrium thermodynamics and extrem
modern wind turbines WEC >5MW area = 12469 m2 energy resource Wind WPI Wien 2015 Center for Wind Energy Research
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
wind ressource reference wind speed annual mean wind speed V ref profile extreme events -gusts Center for Wind Energy Research WPI Wien 2015
~ 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
statistics of gusts P(u τ σ 1 ) τ = 4 s P rob(u ø > 6æ) º 10 4 º 10 6 1/day Boundary-Layer Meteorology 108 (2003) P rob(u ø > 6æ) º 10 10 1/3000 years Center for Wind Energy Research WPI Wien 2015
P WT = 1 2 c p( ) u 3 wind A dynamics of power conversion Center for Wind Energy Research WPI Wien 2015
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) 30 20 10 0 10 20 30 X τ /σ τ P. MIlan et.al.prl 110, 138701 (2013) Center for Wind Energy Research WPI Wien 2015
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 10 5 10 3 10 1 10-1 10-3 10-5 T 10-7 12 h 4h 1h 15 min 4 min -0.5 0.0 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. Time [s] model based in multi point statistics 8 6 8 6 4 2 0 2 4 Surface elevation [m] 4 2 0 2 6 3.0992 10 6 3.0994 3.0996 3.0998 3.1000 10 6 Time [s] 4 2.702 10 5 2.704 2.706 2.708 2.710 10 5 Time [s]
turbulent cascade
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 2 10 2 r r 1 10 0 10-2 η 10-4 10-6 -4-2 0 2 4 u / σ η
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 " # $
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 r j @ @r j p(u rj u rk,u(x 1 )) = { @ @u rj D (1) (u rj,r j,u(x 1 )) + @2 @ 2 j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 ))
n-point statistics... Markow prop & cascade with Fokker-Planck Equ r j @ @r j p(u rj u rk,u(x 1 )) = { @ @u rj D (1) (u rj,r j,u(x 1 )) + @2 @ 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)
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 point @ r u r @ r p r (u r ) 10 4 10 2 10 0 r 10-2 10-4 η 10-6 -4-2 0 2 4 u / σ
cascade process as stochastic process -2- stochastic process - stationary stochastic process - jump process stochastic process - jump process
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
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
thermodynamics of cascade : entropy balance 2nd law : entropy balance entropy production in the system (Seifert 2005) the potential S m (u r )= Z L @ r u r @ 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
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 L @ r u r @ 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%
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 L @ r u r @ u '(u r,r)dr S = ln p(u L,L) p(u, ) stoch process experimental quantities 10 4 10 2 10 0 10-2 10-4 r ' (#) #$ $ #$ %#%*%+%*%# +%,%%%$%%%-%" + &. +%,%" + %-%" + &. +%,%%%" + %-%" + &. " # $ # " %&%" # shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%' (#(-(.(-(#.(/((($(((0(". )1.(/(". (0(". )'.(/(((". (0(". )' " # $ # " ()(" # %no%u/dependence%of%diffusion%func-on% 10-6 -4-2 0 2 4 u / σ
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 L @ r u r @ 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)
test of validity of the fluctuation theorem Renner et al JFM (2001) 1.5 exp [ Stot] N 1 0.5 0 he S tot(u r ) i N 10 0 10 1 10 2 10 3 10 4 10 5 N Nickelsen Engel PRL (2013)
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)
- 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 L @ r u r @ u '(u r,r)dr => IFT holds only for non - scaling stochastic processes
summary - nonequi. thermodynamics generalized 2nd law - (entropy maximization) fulfilled for cascade - non scaling process fits to data (?? => new window for blow up??>
extreme events and its entropy production S tot (u r ) to fulfill the IFT - there must be paths with positive and negative entropy production
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 2 0 2 7200 7600 8000 Row Index
1.) positive entropy production S tot (u r ) > 0 event: large small u L with low probability 10 3 10 1 10 1 10 3 p(u L ) u with high probability p(u ) u L 10 5-4 -2 0 2 4 S tot (u r ) u L u
2.) extreme and negative entropy production S tot (u r ) < 0 event 10 3 10 1 small u L with high probability 10 1 large u with low probability 10 3 extreme event 10 5-4 -2 0 2 4 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 with low probability 10 3 large u with low probability 10 5-4 -2 0 2 4 S tot (u r ) u L u
End concept of stochastic cascade - multipoint statistics - non- equilibrium thermodynamics Thank you
references U. Seifert: Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (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, 103046 (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, 138701 (2013) A. Hadjihosseini, J. Peinke, N. Hoffmann : Stochastic analysis of ocean wave states with and without rogue waves, New Journal of Physics 16, 053037 (2014) N. Reinke, D. Nickelsen, A. Engel, and J. Peinke: Application of an Integral Fluctuation Theorem to Turbulent Flows (iti 2015)