Part II : Residence time


 Bernard Lyons
 1 years ago
 Views:
Transcription
1 Part II : Residence time CART  the Constituent Oriented Age and Residence time Theory p. 42
2 Residence time Widely used concept in environmental studies : 9814 references found in the Elsevier Catalog (Science Direct  Environmental Science category) over the last 10 years! Very appealing concept to biologists and decision makers! CART  the Constituent Oriented Age and Residence time Theory p. 43
3 Lagrangian approach ω!!! Mixing!!! (t 0,x 0 ) (t out,x out ) distribution of residence times Define the control domain ω Introduce a particle at (t 0,x 0 ) Compute / observe the path of this particle and register its exit time (t out,x out ) Residence time at (t 0,x 0 ) = t out t 0 CART  the Constituent Oriented Age and Residence time Theory p. 44
4 Eulerian approach m(t 0 + τ) ω Define the control domain ω Introduce a unit discharge at (t 0,x 0 ) Follow the fate of the tracer and monitor the mass m(t 0 + τ) remaining in ω Cumulative distribution of RT τ CART  the Constituent Oriented Age and Residence time Theory p. 45
5 Mean residence time m(t 0 + τ) τ m(t 0 + τ) m(t 0 ) = Mean residence time θ = 1 m(t 0 ) Fraction of the initial release with a RT larger than τ Z 0 m(t 0 + τ)dτ CART  the Constituent Oriented Age and Residence time Theory p. 46
6 Forward Eulerian procedure Solve ] + v C = [ K C + Q c C t C(t 0,x) = δ(x x 0 ) and compute m(t;t 0,x 0 ) = ZZZ ω C dv But... Open boundary conditions? CART  the Constituent Oriented Age and Residence time Theory p. 47
7 BC  version 1 Residence time = time required to leave the control domain for the first time (Bolin and Rodhe, 1973; Takeoka, 1984) Lagrangian approach : discard particles when they leave the control domain (Lagrangian approach) Eulerian approach : put C = 0 at both inflow and outflow boundaries of the control domain (if diffusion 0) θ = 0 at the open boundary. (Delhez & Deleersnijder, Ocean Dynamics) CART  the Constituent Oriented Age and Residence time Theory p. 48
8 Asymmetric Random Walk ω = ( L,L) Z α β = 1 α p (n) i : probability distribution of particles at time t n. p (n+1) i = β p (n) i 1 + α p(n) i+1, i ( L,L) Z p (n+1) L = α p (n) L+1, Equivalent to p (n) L+1 = p(n) L 1 p(n+1) L = β p (n) L 1 = 0, i.e. C = 0 outside ω CART  the Constituent Oriented Age and Residence time Theory p. 49
9 Boundary layers of the RT (1) 1D infinite domain x (,+ ) ω = [ l,+l] Constant and uniform u and κ (and Pe = ul/κ) Pe = 100 Pe = θ u/l Pe = 10 Pe = 5 1 Advection time scale Pe = 1 Pe = x/l CART  the Constituent Oriented Age and Residence time Theory p. 50
10 Boundary layers of the RT (2) Idem with tidal (1 m/s) + residual (0.1 m/s) flow ωt 2π x/l Residence time CART  the Constituent Oriented Age and Residence time Theory p. 51
11 Exposure time Residence time = time required to leave the control domain for the first time Exposure time : allow particles to exit the domain and reenter at a later time No BC at the boundary of the control domain Appropriate BC at the boundary of the computational domain CART  the Constituent Oriented Age and Residence time Theory p. 52
12 Exposure time M(t 0 + τ) ω (t 0,x 0 ) Z Θ = 1 M(t 0 + τ)dτ M(t 0 ) 0 = Measure of the time concentration to which the control region is exposed to particles originating from (t 0,x 0 ). = Exposure time τ CART  the Constituent Oriented Age and Residence time Theory p. 53
13 ET : 1D example revisited θ, Θ κ = Const. u = Const.(> 0) ω M(t;t 0,x 0 ) m(t;t 0,x 0 ) Θ(t,x) θ(t,x) CART  the Constituent Oriented Age and Residence time Theory p. 54
14 Basin average RT & ET Compute 1 V ω ZZZ ω θ(x)dx or 1 V ω ZZZ ω Θ(x)dx or solve C ] + v C = [ K C + Q c t C(t 0,x) = 1 in ω + Approp. B.C. } < θ > = 1 Z m(t;t 0,x 0 )dt = 1 Z ZZZ C dv dt < Θ > V ω t 0 V ω ω t 0 CART  the Constituent Oriented Age and Residence time Theory p. 55
15 Direct Eulerian procedure Solve and compute C t ] + v C = [ K C C(t 0,x) = δ(x x 0 ) C(t,x) = 0 on ω m(t;t 0,x 0 ) = ZZZ ω C dv But... multiple runs are needed to compute θ(t 0,x 0 ). CART  the Constituent Oriented Age and Residence time Theory p. 56
16 Operator formulation m(t;t 0,x 0 ) = <A t,t0 δ(x x 0 ),δ ω (x) > where A t,t0 = forward operator such that C(t,x) =A t,t0 δ(x x 0 ) δ ω = characteristic function of control domain ω, < f,g > = δ ω (x) = ZZZ { 1 if x ω, 0 elsewhere R 3 f(x)g(x)dv CART  the Constituent Oriented Age and Residence time Theory p. 57
17 Operator formulation (2) m(t ;t 0,x 0 ) = <A T,t0 δ(x x 0 ),δ ω (x) > = < δ(x x 0 ),A T,t 0 δ ω (x) > wherea T,t 0 = adjoint operator ofa T,t0. C T v CT t = [K CT C T (T,x) = δ ω(x) C T (t,x) = 0 on ω Adjoint state C T (t 0,x 0 ) = m(t ;t 0,x 0 ) ] CART  the Constituent Oriented Age and Residence time Theory p. 58
18 Backward procedure C ] T v CT t = [K CT CT (T,x) = δ ω(x), (t,x) = 0 on ω C T Must be integrated backward in time. A single run of the adjoint model provides m(t ;t 0,x 0 ) for a range of (t 0,x 0 ). But m(t 0 + τ;t 0,x 0 ) is required for all τ > 0, solve the adjoint problem for a range of initial conditions Ct 0 +τ(t 0 + τ,x) = δ ω (x) (unless the hydrodynamics is constant) CART  the Constituent Oriented Age and Residence time Theory p. 59
19 Backward procedure (2) Definition : D(t,τ,x) = C t+τ(t,x) = m(t + τ;t,x) D t D D+ [ ] τ + v K D = 0 D(t,0,x) = δ ω (x), D(t,τ,x) = δ(τ) on ω to be solved in a fivedimensional space. Describes the spatial and temporal variations of the cumulative distribution function of residence times m(t + τ;t, x). CART  the Constituent Oriented Age and Residence time Theory p. 60
20 Mean residence time Z 0 θ(t,x) = Z 0 m(t + τ;t,x)dτ = Z 0 D(t,τ,x)dτ D t D D+ [ ] τ + v K D = 0 D(t,0,x) = δ ω (x), D(t,τ,x) = δ(τ) on ω...dτ, assuming lim τ D(t,τ,x) = 0 θ t + δ ω + v θ+ θ(t,x) = 0 on ω [ ] K θ = 0 CART  the Constituent Oriented Age and Residence time Theory p. 61
21 Boundary conditions or D t D D+ [ ] τ + v K D θ [ ] t + δ ω + v θ+ K θ = 0 = 0 At open boundaries of the control region ω prescribe D(t,τ,x) = δ(τ 0) or θ(t,x) = 0 Residence time in ω CART  the Constituent Oriented Age and Residence time Theory p. 62
22 1D Exemple  Residence time θ κ = Const. u = Const.(> 0) ω ( ) t = 0 δ ] L,L[ + u d θ dx + θ κd2 dx 2 = 0 θ(+l) = θ( L) = 0 CART  the Constituent Oriented Age and Residence time Theory p. 63
23 Initial conditions θ [ ] t + δ ω + v θ+ K θ = 0 Must be solved by backward integration from initial conditions given at some time T. θ(t,x) =? CART  the Constituent Oriented Age and Residence time Theory p. 64
24 Initial conditions (2) τ T D(t,τ,x) known? assume D = 0 θ(t,x) = 0 θ(t,x) = T T Z 0 D(t,τ,x)dτ = Z T 0 T D(t,τ,x)dτ only material with RT < T is taken into account t CART  the Constituent Oriented Age and Residence time Theory p. 65
25 Initial conditions (3) Concentration of material with RT < T is given by C T = 1 C T where C ] T v CT t = [K CT CT (T,x) = δ ω(x) CART  the Constituent Oriented Age and Residence time Theory p. 66
26 Summary θ [ ] t + δ ω + v θ+ K θ θ(t,x) = 0, θ(t,x) = 0 on ω = 0 mean residence time C ] T + v CT t + [K CT = 0 CT (T,x) = δ ω(x), CT (t,x) = 0 on ω convergence if C T = 1 C T 1 CART  the Constituent Oriented Age and Residence time Theory p. 67
27 NWECS model x = z = 10, 10 σlevels freesurface, baroclinic, k turbulence model 10 tidal constituents, NCEP Reanalysis met. data U.K. North Sea 51 Dover Strait Belgium 51 B English Channel 358 A 0 Control region 2 France CART  the Constituent Oriented Age and Residence time Theory p. 68
28 Residence time (days) U.K France 1 days Residence time Snapshot on 15/08/ Surface value CART  the Constituent Oriented Age and Residence time Theory p. 69
29 Exposure time (days) U.K France 1 days Exposure time Snapshot on 15/08/ Surface value CART  the Constituent Oriented Age and Residence time Theory p. 70
30 Variability 120 Convergence check Exposure time Residence time /1/1983 1/5/1983 1/9/1983 1/1/1984 1/5/1984 1/9/1984 Initialization Model start Basin average RT & ET (days) 0 CART  the Constituent Oriented Age and Residence time Theory p. 71
31 RT in the mixed layer Does turbulence increase or decrease the residence time in the surface mixed layer of settling particles? CART  the Constituent Oriented Age and Residence time Theory p. 72
32 RT in the mixed layer : setup mixed layer depth = h airsea interface κ(z) w = Cte. pycnocline wc+ κ C z = 0 κ θ z = 0 C t = ( wc+ κ(z) θ ) z z C(0,z) = δ(z z 0 ) ( d wθ κ(z) dθ ) = 1 dz dz κ C z = 0 wc κ θ z = 0 CART  the Constituent Oriented Age and Residence time Theory p. 73
33 RT in the mixed layer : results θ(z) = z w + 1 w z w No diffusion Z h z exp [ Z ζ w z ] dζ dζ κ(ζ) < θ(z) < h w Infinite mixing 1 2 h w < θ = 1 h Z h 0 Factor 2 only! θ(z)dz < h w Residence time increases with turbulence! CART  the Constituent Oriented Age and Residence time Theory p. 74
34 Conclusion of part II Useful diagnostic for numerical models Flexibility : residence / exposure time Can be generalized to tracers with linear dynamics. CART  the Constituent Oriented Age and Residence time Theory p. 75
An Overview of the HYSPLIT_4 Modelling System for Trajectories, Dispersion, and Deposition
An Overview of the HYSPLIT_4 Modelling System for Trajectories, Dispersion, and Deposition Roland R. Draxler NOAA Air Resources Laboratory Silver Spring, Maryland, U.S.A. and G.D. Hess Bureau of Meteorology
More informationOnedimensional modelling of a vascular network in spacetime variables
Onedimensional modelling of a vascular network in spacetime variables S.J. Sherwin, V. Franke, J. Peiró Department of Aeronautics, South Kensington Campus, Imperial College London, London, SW7 2AZ, UK
More informationONEDIMENSIONAL TRANSPORT WITH INFLOW AND STORAGE (OTIS): A SOLUTE TRANSPORT MODEL FOR STREAMS AND RIVERS
ONEDIMENSIONAL TRANSPORT WITH INFLOW AND STORAGE (OTIS): A SOLUTE TRANSPORT MODEL FOR STREAMS AND RIVERS U.S. GEOLOGICAL SURVEY WaterResources Investigations Report 98 4018 ONEDIMENSIONAL TRANSPORT
More informationInterfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification
J. Fluid Mech. (26), vol. 55, pp. 149 173. c 26 Cambridge University Press doi:1.117/s2211257998 Printed in the United Kingdom 149 Interfacial conditions between a pure fluid and a porous medium: implications
More informationSome open problems and research directions in the mathematical study of fluid dynamics.
Some open problems and research directions in the mathematical study of fluid dynamics. Peter Constantin Department of Mathematics The University of Chicago Abstract This is an essay in the literal sense:
More informationErgodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains
Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains Peter Balint 1, Kevin K. Lin 2, and LaiSang Young 3 Abstract. We consider systems of moving particles in 1dimensional
More informationHedging with options in models with jumps
Hedging with options in models with jumps ama Cont, Peter Tankov, Ekaterina Voltchkova Abel Symposium 25 on Stochastic analysis and applications in honor of Kiyosi Ito s 9th birthday. Abstract We consider
More informationOnedimensional wave turbulence
Physics Reports 398 (4) 65 www.elsevier.com/locate/physrep Onedimensional wave turbulence Vladimir Zakharov a;b,frederic Dias c;, Andrei Pushkarev d a Landau Institute for Theoretical Physics, Moscow,
More informationUnderstanding the FiniteDifference TimeDomain Method. John B. Schneider
Understanding the FiniteDifference TimeDomain Method John B. Schneider June, 015 ii Contents 1 Numeric Artifacts 7 1.1 Introduction...................................... 7 1. Finite Precision....................................
More informationThe Backpropagation Algorithm
7 The Backpropagation Algorithm 7. Learning as gradient descent We saw in the last chapter that multilayered networks are capable of computing a wider range of Boolean functions than networks with a single
More informationONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK
ONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK Definition 1. A random walk on the integers with step distribution F and initial state x is a sequence S n of random variables whose increments are independent,
More informationGraphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations
Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations Jure Leskovec Carnegie Mellon University jure@cs.cmu.edu Jon Kleinberg Cornell University kleinber@cs.cornell.edu Christos
More informationA PROCEDURE TO DEVELOP METRICS FOR CURRENCY AND ITS APPLICATION IN CRM
A PROCEDURE TO DEVELOP METRICS FOR CURRENCY AND ITS APPLICATION IN CRM 1 B. HEINRICH Department of Information Systems, University of Innsbruck, Austria, M. KAISER Department of Information Systems Engineering
More informationStorm Track Dynamics
2163 Storm Track Dynamics EDMUND K. M. CHANG* Department of Meteorology, The Florida State University, Tallahassee, Florida SUKYOUNG LEE Department of Meteorology, The Pennsylvania State University, University
More informationControllability and Observability of Partial Differential Equations: Some results and open problems
Controllability and Observability of Partial Differential Equations: Some results and open problems Enrique ZUAZUA Departamento de Matemáticas Universidad Autónoma 2849 Madrid. Spain. enrique.zuazua@uam.es
More informationStreamflow Depletion by Wells Understanding and Managing the Effects of Groundwater Pumping on Streamflow
Groundwater Resources Program Streamflow Depletion by Wells Understanding and Managing the Effects of Groundwater Pumping on Streamflow Circular 1376 U.S. Department of the Interior U.S. Geological Survey
More informationFast Greeks by algorithmic differentiation
The Journal of Computational Finance (3 35) Volume 14/Number 3, Spring 2011 Fast Greeks by algorithmic differentiation Luca Capriotti Quantitative Strategies, Investment Banking Division, Credit Suisse
More informationTHE PROBLEM OF finding localized energy solutions
600 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997 Sparse Signal Reconstruction from Limited Data Using FOCUSS: A Reweighted Minimum Norm Algorithm Irina F. Gorodnitsky, Member, IEEE,
More informationHow to Use Expert Advice
NICOLÒ CESABIANCHI Università di Milano, Milan, Italy YOAV FREUND AT&T Labs, Florham Park, New Jersey DAVID HAUSSLER AND DAVID P. HELMBOLD University of California, Santa Cruz, Santa Cruz, California
More informationSteering User Behavior with Badges
Steering User Behavior with Badges Ashton Anderson Daniel Huttenlocher Jon Kleinberg Jure Leskovec Stanford University Cornell University Cornell University Stanford University ashton@cs.stanford.edu {dph,
More informationTIME SERIES. Syllabus... Keywords...
TIME SERIES Contents Syllabus.................................. Books................................... Keywords................................. iii iii iv 1 Models for time series 1 1.1 Time series
More informationMEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationHow Do Scientists Develop and Use Scientific Software?
How Do Scientists Develop and Use Scientific Software? Jo Erskine Hannay Dept. of Software Engineering Simula Research Laboratory Dept. of Informatics, Univ. of Oslo johannay@simula.no Carolyn MacLeod
More informationQuantitative Strategies Research Notes
Quantitative Strategies Research Notes March 999 More Than You Ever Wanted To Know * About Volatility Swaps Kresimir Demeterfi Emanuel Derman Michael Kamal Joseph Zou * But Less Than Can Be Said Copyright
More informationCore Academic Skills for Educators: Mathematics
The Praxis Study Companion Core Academic Skills for Educators: Mathematics 5732 www.ets.org/praxis Welcome to the Praxis Study Companion Welcome to The Praxis Study Companion Prepare to Show What You Know
More informationchapter Geography, hydrography and climate
chapter 2 Geography, hydrography and climate 5 GEOGRAPHY 2.1 Introduction This chapter defines the principal geographical characteristics of the Greater North Sea. Its aim is to set the scene for the more
More informationThe Variance Gamma Process and Option Pricing
European Finance Review : 79 105, 1998. 1998 Kluwer Academic Publishers. Printed in the Netherlands. 79 The Variance Gamma Process and Option Pricing DILIP B. MADAN Robert H. Smith School of Business,
More informationBayesian Models of Graphs, Arrays and Other Exchangeable Random Structures
Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures Peter Orbanz and Daniel M. Roy Abstract. The natural habitat of most Bayesian methods is data represented by exchangeable sequences
More informationReduction of Wind and Swell Waves by Mangroves
Natural Coastal Protection Series ISSN 20507941 Reduction of Wind and Swell Waves by Mangroves Anna McIvor, Iris Möller, Tom Spencer and Mark Spalding Natural Coastal Protection Series: Report 1 Cambridge
More informationSpaceTime Approach to NonRelativistic Quantum Mechanics
R. P. Feynman, Rev. of Mod. Phys., 20, 367 1948 SpaceTime Approach to NonRelativistic Quantum Mechanics R.P. Feynman Cornell University, Ithaca, New York Reprinted in Quantum Electrodynamics, edited
More information