Part II : Residence time


 Bernard Lyons
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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
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