Part II : Residence time

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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 !

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Lagrangian approach

ω (t_out,x_out) b (t₀,x₀) !!! Mixing !!! ⇒ distribution of residence times

• _{Define the control domain} ω • _{Introduce a particle at} (_t0,_x0)

• _{Compute / observe the path of this particle and}

register its exit time (t_out,x_out)

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Eulerian approach

ω τ m(t₀ +τ)

• _{Define the control domain} ω

• _{Introduce a unit discharge at} (_t0,_x0)

• _{Follow the fate of the tracer and monitor the mass}

m(t₀ + τ) remaining in ω

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Mean residence time

τ m(t₀ + τ)

m(t₀ + τ)

m(t₀) =

Fraction of the initial release with a RT larger than τ

Mean residence time ¯θ = 1 m(t₀)

Z ∞

0

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Forward Eulerian procedure

Solve    ∂C ∂t + v · ∇C = ∇ · h K _· ∇Ci ₊ Qc C(t₀,x) = δ(x − x₀) and compute m(t;t₀, x₀) = ZZZ ωC dV

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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 6= 0)

⇒ θ¯ = 0 at the open boundary.

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Asymmetric Random Walk

b b b b b b b b b b b

α β = 1 − α ω = (₋L,L) ∩ _Z

p(_in) : probability distribution of particles at time t_n.

p(_in+1) = β p(_i−n)₁ + α p(_i+n)₁, i ∈ (−L,L) ∩ _Z p(₋n_L+1) ₌ α p(₋n_L)_+1, p(_Ln+1) ₌ β p(_Ln₋)₁

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Boundary layers of the RT (1)

• _{1D infinite domain x} ∈ (−∞,+∞)

• ω = [−ℓ,+ℓ]

• _{Constant and uniform u and} κ _{(and Pe} = _uℓ/κ₎

Advection time scale

-1 -0.5 0.5 1 0.5 1 1.5 Pe = 0.2 Pe= 1 Pe =5 Pe = 10 Pe = 20 Pe = 100 θ¯ u/ℓ x/ℓ

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Boundary layers of the RT (2)

Idem with tidal (1 m/s) + residual (0.1 m/s) flow

0 0.25 0.5 0.75 1 1.25 1.5 1.75 Residence time -1 -0.5 0 0.5 1 0 0.5 1 1.5 2 x/ℓ ωt 2π

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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}

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Exposure time

ω (t₀,x₀) τ M(t₀ + τ) ¯ Θ = 1 M(t₀) Z ∞ 0 M(t₀ + τ)dτ

= Measure of the time × concentration to which the control region is exposed to particles originating from (t₀, x₀).

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ET : 1D example revisited

u = Const.(> 0) κ = Const. ω ¯ θ,Θ¯ M(t;t₀,x₀) ≥ m(t;t₀, x₀) ¯ Θ(t,x) ≥ θ¯(t,x)

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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 ∂t + v · ∇C = ∇ · h K · ∇C i + Qc C(t₀,x) = 1 in ω + Approp. B.C. < θ¯ > < Θ¯ > ) = 1 V_ω Z ∞ t₀ m(t;t₀,x₀)dt = 1 V_ω Z ∞ t₀ ZZZ ωC dV dt

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Direct Eulerian procedure

Solve            ∂C ∂t + v · ∇C = ∇ · h K · ∇C i C(t₀,x) = δ(x − x₀) C(t,x) = 0 on ∂ω and compute m(t;t₀, x₀) = ZZZ ωC dV

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Operator formulation

m(t;t₀,x₀) = <

A

_t,t 0δ(x − x0), δω(x) > where •

A

_t,t

0 = forward operator such that C(t,x) =

A

_t,t

0δ(x − x0)

• δ_{ω = characteristic function of control domain} ω_,

δ_ω(x) = 1 if x _∈ ω_, 0 elsewhere • < _f ,_g > = ZZZ R3 f (x)g(x)dV

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Operator formulation (2)

m(T ;t₀,x₀) = <

A

_T,t 0δ(x − x0), δω(x) > = < δ(x − x₀),

A

∗ T,t0δω(x) > where

A

∗ T,t₀ = adjoint operator of

A

T,t0.            −∂_∂CT∗ t − v · ∇C ∗ T = ∇ · h K · ∇C_T∗ i C_T∗(T,x) = δ_ω(x) C_T∗(t,x) = 0 on ∂ω Adjoint state C_T∗(t₀, x₀) = m(T ;t₀,x₀)

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Backward procedure

   −∂_∂CT∗ t − v · ∇C ∗ T = ∇ · h K · ∇C_T∗ i C_T∗(T,x) = δ_ω(x), C_T∗ (t, x) = 0 on ∂ω

• _{Must be integrated backward in time.}

• _{A single run of the adjoint model provides}

m(T ;t₀,x₀) for a range of (t₀,x₀).

But m(t₀ + τ;t₀, x₀) is required for all τ > 0,

⇒ solve the adjoint problem for a range of ‘initial conditions’ C_t∗

0+τ(t0 +τ,x) = δω(x) (unless the hydrodynamics is constant)

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Backward procedure (2)

Definition : D(t,τ,x) = C_t∗_+τ(t,x) = m(t + τ;t,x)    ∂D ∂t − ∂D ∂τ + v · ∇D + ∇ · h K · ∇D i = 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).

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Mean residence time

¯ θ(t,x) = Z ∞ 0 m(t + τ;t, x)dτ = Z ∞ 0 D(t, τ, x)dτ    ∂D ∂t − ∂D ∂τ + v · ∇D + ∇ · h K · ∇Di = 0 D(t, 0,x) = δ_ω(x), D(t, τ,x) = δ(τ) on ∂ω Z ∞ 0 . . . dτ, assuming lim τ_→∞D(t,τ,x) = 0 ⇒      ∂θ¯ ∂t +δω +v · ∇ ¯ θ + ∇ _· hK · ∇θ¯ i = 0 ¯ θ(t,x) = 0 on ∂ω

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Boundary conditions

∂D ∂t − ∂D ∂τ + v · ∇D + ∇ · h K · ∇D i = 0 or ∂θ¯ ∂t + δω + v · ∇ ¯ θ + ∇ _· hK · ∇θ¯ i = 0

At open boundaries of the control region ω prescribe

• _D(_t,τ,_x) = δ(τ − ₀)

or

• θ¯(_t,_x) = ₀

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1D Exemple - Residence time

u = Const.(> 0) κ = Const. ω ¯ θ ∂(_·) ∂t = 0      δ_]−L,L[ + ud ¯θ dx +κ d2θ¯ dx2 = 0 ¯ θ(+L) = θ¯(−L) = 0

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Initial conditions

∂θ¯ ∂t + δω + v · ∇ ¯ θ + ∇ _· hK · ∇θ¯ i = 0

Must be solved by backward integration from ‘initial conditions’ given at some time T .

¯

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Initial conditions (2)

t τ T T − ∆T ∆T ? assume D = 0 ⇒ θ¯(T,x) = 0 D(t,τ,x) known ¯ θ(t, x) = Z ∞ 0 D(t,τ,x)dτ = Z T 0 D(t,τ,x)dτ only material with RT < ∆T is taken into account

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Initial conditions (3)

Concentration of material with RT < ∆T is given by ˜ C_T = 1 −C_T∗ where    −∂_∂CT∗ t − v · ∇C ∗ T = ∇ · h K · ∇C_T∗ i C_T∗(T,x) = δ_ω(x)

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Summary

     ∂θ¯ ∂t + δω +v · ∇ ¯ θ + ∇ _· hK · ∇θ¯ i = 0 ¯ θ(T, x) = 0, θ¯(t,x) = 0 on ∂ω

⇒ mean residence time

   ∂C_T∗ ∂t + v · ∇C ∗ T + ∇ · h K · ∇C_T∗ i = 0 C_T∗ (T, x) = δ_ω(x), C_T∗(t,x) = 0 on ∂ω ⇒ convergence if ˜C_T = 1 −C_T∗ ≈ 1

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NWECS model

• ∆_x = ∆_z = ₁₀′_{, 10} σ_-levels

• _{free-surface, baroclinic, k turbulence model}

• _{10 tidal constituents, NCEP Reanalysis met. data}

356˚ 356˚ 358˚ 358˚ 0˚ 0˚ 2˚ 2˚ 4˚ 4˚ 49˚ 49˚ 50˚ 50˚ 51˚ 51˚ 52˚ 52˚ France U.K. Belgium North Sea English Channel Dover Strait A B Control region 356˚ 356˚ 358˚ 358˚ 0˚ 0˚ 2˚ 2˚ 4˚ 4˚ 49˚ 49˚ 50˚ 50˚ 51˚ 51˚ 52˚ 52˚

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Residence time (days)

358˚ 358˚ 359˚ 359˚ 0˚ 0˚ 1˚ 1˚ 2˚ 2˚ 49˚ 49˚ 50˚ 50˚ 51˚ 51˚ France U.K. Residence time 10₀ 20 30 40 50 60 70 80 90 100 110 120 130 140 150 days

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Exposure time (days)

358˚ 358˚ 359˚ 359˚ 0˚ 0˚ 1˚ 1˚ 2˚ 2˚ 49˚ 49˚ 50˚ 50˚ 51˚ 51˚ France U.K. Exposure time 10₀ 20 30 40 50 60 70 80 90 100 110 120 130 140 150 days

CART - 2007

V

a

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a

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ty

M o d el st ar t In iti al iz at io n 0 20 40 60 80 100 120

Basin average RT & ET (days)

0 20 40 60 80

100 120

Basin average RT & ET (days)

0 20 40 60 80

100 120

Basin average RT & ET (days)

1/1/1983 1/5/1983 1/9/1983 1/1/1984 1/5/1984 1/9/1984 0 ₁ 0 ₁ R es id en ce tim e E x p o su re tim e C o n v er g en ce ch ec k C A R T -the C ons tit ue nt O rie nt ed A ge and R es ide nc e tim e T he or y – p. 71

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RT in the mixed layer

Does turbulence increase or decrease the residence time in the surface mixed layer of settling particles ?

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RT in the mixed layer : set-up

m ix ed la y er d ep th = h air-sea interface pycnocline w = Cte. κ(z)    ∂C ∂t = ∂ ∂z wC + κ(z)∂θ ∂z C(0,z) = δ(z −z₀) wC + κ∂C ∂z = 0 κ∂C ∂z = 0 d dz wθ − κ(z)dθ dz = 1 wC − κ∂θ ∂z = 0 κ∂θ ∂z = 0

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RT in the mixed layer : results

θ(z) = z w + 1 w Z h z exp −w Z ζ z dζ κ(ζ) dζ z w No diffusion < θ(z) < h w Infinite mixing 1 2 h w < ¯ θ = 1 h Z h 0 θ( z)dz < h w Factor 2 only !

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Conclusion of part II

• _{Useful diagnostic for numerical models} • _{Flexibility : residence / exposure time} • _{Can be generalized to tracers with linear}

dynamics.