Physics of Aquatic Systems

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1 Physics of Aquatic Systems. Turbulent Transport in Surface Waters Contents of Session : Transport 4.1 Turbulent (eddy) fusion 4. Measurement of eddy fusivity in lakes Tracer methods (artificial tracer) Budget gradient method Environmental tracer methods Modelling Werner Aeschbach Hertig 4.1 Turbulent Diffusion Transport processes in turbulent fluids: 1. Diffusion Transport by stochastic motion of dissolved molecules luss: D c ad Kon.:. Advection Transport by (mean) flow field luss: c v Kon.: ad DΔc ad ( cv) Advection Equation from Mass Conservation Mass balance of a conservative tracer in the volume V no sources or sinks of the tracer analogous to continuity equation but with c instead of ρ M c dv da cv da div ( cv) dv Vol Oberfl. Oberfl. Vol c div cv v c + c v v c Thus: Direct from advective derivative: or a conservative tracer holds: Gauß incompressible dc 0 + v c dt 4 Advection Diffusion Equation Modes of Transport: Advection and Diffusion Advection Diffusion ad c v D c 1 D: x x c v c + D c Advection Diffusion D: c v c + D Δ c Advection Diffusion 6 1

low in surface waters is turbulent Exchange model for turbulent fusion Turbulent exchange of water parcels creates transport: Effect as for

exchange of two parcels by turbulence Net flux: Qex L A Definition of turbulent fusions coefficient K : Qex K L A turbulent fusive flux: K 7 8

) Application of Reynolds decomposition to advective transport term: div ( cv) ( cv Reynolds i ) div i ( CV) + Ki x i decomposition i x i x i

fusion Using the local gradient approximation (closure step): C c' v i' Ki x Transport equation with tensor K of turbulent fusion

(eddy viscosity) as well as dissolved substances (eddy fusion).

fusivities depend on the scale of the system, but usually it holds: K h >> K >> D Turbulent fusion is a property of the flow field (eddies),

2 low in surface waters is turbulent Exchange model for turbulent fusion Turbulent exchange of water parcels creates transport: Effect as for fusion New transport parameter: Turbulent (eddy) fusivity K Turbulent Transport Exchange Model for Turbulent Diffusion Effect of vertical exchange of two parcels by turbulence Net flux: Qex L A Definition of turbulent fusions coefficient K : Qex K L A turbulent fusive flux: K 7 8 Turbulent Diffusion from Reynolds Decomposition Summary (details see exercise sheet!) Application of Reynolds decomposition to advective transport term: div ( cv) ( cv Reynolds i ) div i ( CV) + Ki x i decomposition i x i x i advection term mean advection + turb. fusion Using the local gradient approximation (closure step): C c' v i' Ki x Transport equation with tensor K of turbulent fusion coefficients: c + i ( cv) ( K c) (neglecting molecular fusion) Turbulent Diffusion Turbulence (eddies) leads to a fusive transport of momentum (eddy viscosity) as well as dissolved substances (eddy fusion). The turbulent fusion coefficients K i (turbulent fusivity, eddy fusivity) are analogous to the molecular fusion coefficients D, but (usually) much larger. The K i are in general anisotropic description by tensor Often reduction to components: K h (horiontal) and K (vertical) The turbulent fusivities depend on the scale of the system, but usually it holds: K h >> K >> D Turbulent fusion is a property of the flow field (eddies), therefore it is the same for all dissolved substances (or properties, e.g. heat) Transport in luids: Turbulent Diffusion Transport by stochastic motion of the fluid lux per unit time and area (analogous to ick's first law): 1-D vertical: -D: K Concentration change (ick's second law): 1-D: K K c Tensor, comp. K h and K -D: ( K c) Kh + Kh + K x x y y Scale Dependence of Turbulent Diffusion Larger scale ferent separation into mean and turbulence more contributions to K larger K Okubo: K L 4/ from Okubo, 1971, Deep Sea Res. 18:

4. Measurement of Turbulent Diffusion in Lakes Horiontal and Vertical Mixing Times Consider density stratified fluid: Isopycnals Lines of equal density rom energy consideration (also see later):

3 4. Measurement of Turbulent Diffusion in Lakes Horiontal and Vertical Mixing Times Consider density stratified fluid: Isopycnals Lines of equal density rom energy consideration (also see later): isopycnal mixing ( to isopycnals) >> diapycnal mixing ( to isopycnals) Lakes: Isopycnals horiontal (no N S T gradients) isopycnal horiontal, diapycnal vertical Therefore: K iso K x K y >> K dia K Typical assumption in lakes: horiontally completely homogeneous (mixed) only gradients in direction 1 D problem! Measurement of 1 vertical profile is sufficient Diffusion of a Peak of a Conservative Tracer K Measurement with Artificial Tracer Peak Vertical transport in stratified lakes: usually v 0 (no water in/outflow at depth) for uniform layer we assume K constant Transport equation (1 D): Solution for Initial condition of a δ peak C(,0) mδ( 0 ): C C K m C,t e πk t ( ) 0 4K t K 10 8 m s 1 < D th S 6 in mining lake Merseburg Ost Strong stratification by inflow of saline ground water Gaussian bell curve with standard deviation σ or variance σ : Diffusive mixing length L mix, fusive mixing time τ mix : σ σ L K t or K t K τ mix mix high stability, low turbulence Von Rohden and Ilmberger, 001, Aquat. Sci. 6: K h Measurement by Artificial Tracer Cloud K h Scaling with L or t in Lakes Real clouds, various lakes idealised, Lake Lucerne Diffusion: σ Kt Okubo: 4 K σ Thus: σ t Peeters et al., 1996, JGR C101: Dye cloud in Lake Lucerne (Switerland) K i increase with cloud: K h m s 1 (>> K ) σ i d dt K i Does not hold for single experiments, but fits well to the aggregated data Peeters et al., 1996, JGR C101:

4 1 0 0 m m Budget Gradient Method: Idea Mass balance of a conservative tracers (often T) below 0 No advection: Change only by turbulent fusive flux K ( 0 ) folllows from chance of balance below 0 and gradient at 0 0 m läche (A) A( 0 ) m Temperatur (T) K T t 1 Δ s Δ 0 0 Δ c pρ A() T d m t Sediment Budget Gradient Method: Equations for T Heat content below 0 : Temporal change: W 0,t K 0 0 pρ m W( ) c A()T()d ( 0 ) 0 pρ T(,t) W,t Change of content is due to turbulent fusive flux: Thus: c A() d m A A c ρk ( ) 0 0 th 0 0 p 0 T,t 0 W( T,t 0,t ) A() d m T(,t) A( ) c ρ A T,t 0 p Budget Gradient Method: Practical Use Budget Gradient Method: Example In practice one calculates the budget and the gradients from a few (at least ) vertical temperature profiles: with: K ( ) 0 0 W( ΔT 0 ) A() d Δ m Δt T A( ) c ρ A Δt T 0 p Δ t t t ; Δ W W,t W,t ; T ( C) M o n th D (c m/s ec) Method needs exact temperature data at ferent times It can be applied to seasonal cycle averaged over several years. Classical example: Lake Zurich, Switerland T(,t ) T 1 T,t Li, 197, Schwei. Z. Hydrol. : Institut für Depth Umweltphysik (m) Tracer to Determine Vertical Mixing Artificial tracers are good for small scales To determine large scale, long term mixing: use tracers that are already in the system ("environmental tracers", in principle also properties such as T, S) Dating tracers yield deep water residence times and vertical exchange rates Useful environmental tracers for the dating of lakes (see part II): Tritium( H or T) and H He CCs, S 6 Basics of the H He Method The product of H decay is a stable, conservative isotope: He Closed system: Sum of H + He is conserved ("stable H") Initial H known: Dating possible independent of input function concentration mother - daughter pair He tri ( t) () 1 He t ln1+ λ Ht The tritium bom peak was a good marker of precipitation of around 196 dating tool time H 4 4

H He Method in Lakes H He Ages in Holomictic Lakes Tritium He air Age of deep water should rise during stratification

Hypolimnion dhe dt λ H Data from Lake Lucerne Aeschbach Hertig et al., 1996, Limnol. Oceanogr.

flux upwards He: conservative tracer with source term +λ H Budget Gradient Method for H He In principle completely

changes due to fusion K ( ) 0 He,t A() H,t d m 0 A( 0 ) λ He,t 0 Imboden et al., 1981, Verh. Internat. Verein. Limnol.

8 days ormation in rocks from 6 Ra Lakes: lux from the sediment Activity: A λn Activity concentration: Decays per

dissolved Ra) Turbulent transport in higher water layers Transport + decay leads to stationary Rn distribution 1 D

5 H He Method in Lakes H He Ages in Holomictic Lakes Tritium He air Age of deep water should rise during stratification collapse during turnover Time Gas exchange He eq Epilimnion Thermocline Scheme Temperature H He He > He eq Hypolimnion dhe dt λ H Data from Lake Lucerne Aeschbach Hertig et al., 1996, Limnol. Oceanogr. 41: H He Dynamics: Example Lake Baldegg (CH) Incomplete mixing: Age decreases, but not to 0 Diffusive He flux upwards He: conservative tracer with source term +λ H Budget Gradient Method for H He In principle completely analogous to temperature, except: Source term +λ H has to be subtracted in the budget, in order to obtain only the changes due to fusion K ( ) 0 He,t A() H,t d m 0 A( 0 ) λ He,t 0 Imboden et al., 1981, Verh. Internat. Verein. Limnol. 1: Environmental Tracers: Radon 8 U-Zerfallsreihe Rn: Radioactive noble gas isotope Half life.8 days ormation in rocks from 6 Ra Lakes: lux from the sediment Activity: A λn Activity concentration: Decays per time and water volume Rn as a Tracer for Bottom Mixing in Lakes Diffusion from sediment: Excess Rn (not from dissolved Ra) Turbulent transport in higher water layers Transport + decay leads to stationary Rn distribution 1 D vertical fusion model, no advection, stationary: Solution: or: C h K C ( h) C ( 0) e λ ex ex Cex K λc ex 0 ex ln Cex h ln C ex 0 λ K h K folllows from slope in plot of lnc versus h 9 0

Radon: Example Lake Greifensee (CH) Rn: Example Lake Baldegg Complication: Influence of

9: 81 844 1 K from 1 D Vertical Lake Models K Determination from Numerical Lake Models 1 D

6 Radon: Example Lake Greifensee (CH) Rn: Example Lake Baldegg Complication: Influence of topography due to fast horiontal mixing A A A K + K x λa x K x K x K Imboden and Emerson, 1978, Limnol. Oceanogr. : Imboden and Joller, 1984, Limnol. Oceanogr. 9: K from 1 D Vertical Lake Models K Determination from Numerical Lake Models 1 D vertical fusion model: K is central model parameter for transport K profile determines shape of the tracer profiles K profile often unknown! K determination by inverse modeling: Describe K profile by few parameters Vary parameters (K profile) until best agreement between modelled and measured tracer profiles is reached Voraussetung: No other unknowns or sufficient data Model definition for H He 4 Numerical Models: Example Lac Pavin () Tracer and K Profiles in Models of Lac Pavin Aeschbach Hertig et al., GCA 6:7 7 und 00, Hydrobiol. 487:

7 Summary Turbulence leads to undirected transport Description by turbulent fusion Turbulent fusion Coefficients K i >> D Lakes are often described by 1 D vertical fusion K is central quantity for transport Measurement methods for K : Directly with tracer experiments (cloud spreading etc.) Budget gradient method with T, H He etc. rom Rn flux decay equilibrium above the sediment By inverse modelling of tracer time series 7 7

Physics of Aquatic Systems

Physics of Aquatic Systems 4. Turbulent Transport in Surface Waters Contents of Session 4: Transport 4.1 Turbulent (eddy) fusion 4. Measurement of eddy fusivity in lakes Tracer methods (artificial tracer)