Physics of Aquatic Systems

Size: px

Start display at page:

Download "Physics of Aquatic Systems"

Eunice Stewart
6 years ago
Views:

1 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) Budget gradient method Environmental ltracer methods Modelling (inverse modeling of tracer distributions) Observation of small scale turbulence (microstructure) Werner Aeschbach Hertig 4.1 Turbulent Diffusion Advection Diffusion Equation Transport processes in turbulent fluids: Advection Diffusion 1. Diffusion Transport by stochastic motion of dissolved molecules luss: Dc Kon.: c t Dc ad c v Dc. Advection Transport by (mean) flow field luss: ad cv Kon.: c ad cv t ad 1 D: t x x c v c D c Advection Diffusion D: c v c D c t Advection Diffusion 4 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 c L A Definition of turbulent fusions coefficient K : Qex K L A turbulent fusive flux: c K 6 1

Turbulent Diffusion from Reynolds Decomposition Summary Application of Reynolds decomposition to advective transport term: div cv cv

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

the fluid lux per unit time and area (analogous to ick's first law): c 1-D vertical: -D: K Concentration change (ick's second law):

K h and K c -D: K c t c c c Kh Kh K x x y y 7 8 Scale Dependence of Turbulent Diffusion Larger scale ferent separation into mean and

Measurement of Turbulent Diffusion in Lakes Consider density stratified fluid: Isopycnals = Lines of equal density rom energy

(no N S T gradients) isopycnal = horiontal, diapycnal = vertical Therefore: K iso = K x = K y >> K dia = K Typical assumption in

2 Turbulent Diffusion from Reynolds Decomposition Summary Application of Reynolds decomposition to advective transport term: div cv cv Reynolds c 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 t i cv K c (neglecting molecular fusion) Transport in luids: Turbulent Diffusion Transport by stochastic motion of the fluid lux per unit time and area (analogous to ick's first law): c 1-D vertical: -D: K Concentration change (ick's second law): 1-D: c t c K Kc Tensor, comp. K h and K c -D: K c t c c c Kh Kh K x x y y 7 8 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: Measurement of Turbulent Diffusion in Lakes 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 Horiontal and Vertical Mixing Times Diffusion of a Peak of a Conservative Tracer 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 peak C(,0) = m( 0 ): C C K t m C,t e K t 0 4K t Gaussian bell curve with standard deviation or variance : K t or K t Diffusive mixing length L mix, fusive mixing time mix : L K mix mix 11 1

K Measurement with Artificial Tracer Peak K h Measurement by Artificial Tracer Cloud S 6 in mining lake Merseburg Ost Strong stratification by inflow of saline ground water Real clouds, various lakes

3 K Measurement with Artificial Tracer Peak K h Measurement by Artificial Tracer Cloud S 6 in mining lake Merseburg Ost Strong stratification by inflow of saline ground water Real clouds, various lakes idealised, Lake Lucerne K 10 8 m s 1 < D th high stability, low turbulence Von Rohden and Ilmberger, 001, Aquat. Sci. 6: Peeters et al., 1996, JGR C101: Dye cloud in Lake Lucerne (Switerland) K i increase with cloud: d dt K h m s 1 (>> K ) i K i 1 14 K h Scaling with L or t in Lakes Diffusion: Okubo: Thus: Kt 4 K t Does not hold for single experiments, but fits well to the aggregated data Peeters et al., 1996, JGR C101: Tracers to Determine Vertical Mixing Artificial tracers are good for small scales (in space and time) To determine large scale, long term mixing: Use tracers that are already in the system: so called "environmental tracers", including properties such as T, S Such tracers should behave conservatively (no sinks/sources) or have well known sinks and sources ( balance possible), at least in some part of the profile (e.g. deep water) These tracers should have vertical gradients and welldefined boundary conditions at the surface The tracers should be easy to measure with good resolution A frequently used tracer for K determination is temperature 1 16 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 change of balance below 0 and gradient at 0 0 m läche (A) A( 0 ) m Temperatur (T) = K T t 1 t s 0 0 = c p A() T t d m Sediment Budget Gradient Method: Equations for T Heat content below 0 : Temporal change: 0 p 0 W( ) c A()T()d m 0 0 p T,t W,t t Change of content is due to turbulent fusive flux: Thus: W 0,t t K 0 c A() d t m A A c K 0 th 0 0 p 0 T,t 0 W T,t 0,t A() d m t T,t A c A 0 p t T,t

profiles: with: K 0 0 W T 0 A() d t m T A c A t T 0 p 0 0 0 t t t ; W W,t W,t ; 1 0 0 0 1 1 0 0 0 T ( C) 0 1 1 0 1 4 6 7 8 9 10 111 1 M o n th D (c m/s ec) 0.0 1 0.

Classical example: Lake Zurich, Switerland T,t T 1T,t 1 0 0 0 4 0 0 6 0 7 0 8 0 9 0 1 0 0 10 9 8 7 6 11+ 4 Li, 197, Schwei. Z. Hydrol.

4 1 0 0 m m 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 T 0 A() d t m T A c A t T 0 p t t t ; 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 1T,t Li, 197, Schwei. Z. Hydrol. : Institut für Depth Umweltphysik (m) 0 Age Tracers to Determine Vertical Mixing Dating tracers (see part II) yield deep water residence times and therefore also vertical exchange rates Useful environmental tracers for the dating of lakes (see part II): Tritium( H or T) and H He CCs, S 6 General concept of "ideal water age" : Measures isolation time of (deep) water from atmosphere Surface boundary condition: = 0 at surface Sinks/sources: Increase in the deep water in the absence of exchange/mixing (is simply d/dt = 1 (yr/yr) Scheme Data from Lake Lucerne H He Ages in Holomictic Lakes Age of deep water should rise during stratification collapse during turnover Aeschbach Hertig et al., 1996, Limnol. Oceanogr. 41: H He Dynamics: Example Lake Baldegg (CH) Incomplete mixing: Age decreases, but not to 0 Diffusive flux of "age" upwards Age: conservative tracer with source term +1 Budget Gradient Method for "Ideal Age" In principle completely analogous to temperature, except: Source term +1 has to be subtracted in the budget, in order to obtain only the changes due to fusion K 0 0 m,t A() () 1d t,t A0 0 Imboden et al., 1981, Verh. Internat. Verein. Limnol. 1:

5 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 = N 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: Cex C t h K C h C 0e ex ex K C ex 0 ex ln Cex h lnc ex 0 K h K folllows from slope in plot of lnc versus h 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 A t 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 Requirement: No other unknowns or sufficient data Model definition for H He 9 0

Numerical Models: Example Lac Pavin () Tracer and K Profiles in Models of Lac Pavin

1 Eddy Correlation (Eddy Covariance) The vertical turbulent flux j of a species with

Upward movement: Concentration (on average) somewhat lower vt Downward movement:

of fluctuations c c() t Eddy Correlation: Instrumentation Sonic Anemometer: Measurement

Simultaneous, highly resolved measurement of velocity and concentration is required.

Microstructure Microstructure Method for K Determination Eddies in a vertical T

6 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: Eddy Correlation (Eddy Covariance) The vertical turbulent flux j of a species with concentration c is given by: c j cv K In case of downward turbulent flux (c/ > 0): Upward movement: Concentration (on average) somewhat lower vt Downward movement: Concentration (on average) somewhat higher Eddy covariance method: Direct measurement of fluctuations c c() t Eddy Correlation: Instrumentation Sonic Anemometer: Measurement of v (Ultrasound speed) Infrared Gas Analyser Measurement of c (IR spectroscopy) Simultaneous, highly resolved measurement of velocity and concentration is required. More common in aquatic systems: Microstructure techniques 4 4 Turbulence and Microstructure Microstructure Method for K Determination Eddies in a vertical T gradient lead to T fluctuations: T T L Imboden and Wüest, Mixing mechanisms in lakes, in Lerman et al. (eds.), 199. Wüest et al., 001, Eawag News 6 6

Microstructure: Cox Number Method Measure T fluctuations T' with high resolution (~ 1mm) Calculate gradients T'/ and their variance (T'/) Variance is produced by turbulence (eddies) and decreased by

7 Microstructure: Cox Number Method Measure T fluctuations T' with high resolution (~ 1mm) Calculate gradients T'/ and their variance (T'/) Variance is produced by turbulence (eddies) and decreased by smoothing due to molecular fusion. In equilibrium, turbulent and molecular fusivity are related by T K D T Cox Number T The Cox number is the factor by which turbulent fusion is stronger than molecular fusion See: Osborn and Cox, 197: Oceanic fine structure. Geophys. luid Dyn. : 1-4. Microstructure: Dissipation Method Measure T fluctuations T' with high resolution (~ 1mm) Calculate spectrum of T fluctuations Use Kolmogorov theory of turbulence and its power spectrum to calculate TKE dissipation rate, e.g. 1 4 Kolmogorov length scale: LK Batchelor length scale: DT LB 1 4 smallest velocity fluctuations ti smallest Temp. fluctuations Use mixing efficiency mix 0.1 to calculate K from (see later): N mix K 7 8 Kolmogorov Turbulence Spectrum Ek c k Microstructure Results for K Microstructure measurements show that turbulence is strongest near the boundaries (lake surface and bottom shear stress). Energy input ) E(k) Inertial subrange Dissipation L K Wüest et al., 001, Eawag News 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 Directly from microstructure measurements 41 7

Physics of Aquatic Systems

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)