# Part II : Residence time

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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 re-enter 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 five-dimensional 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 σ-levels free-surface, 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 : set-up mixed layer depth = h air-sea 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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