Physics of Aquatic Systems II

Transcription

1 Contents of Session 2 Physics of Aquatic Systems II 2. Introduction Hydrology/Aquatic Physics Werner Aeschbach-Hertig Institute of Environmental Physics University of Heidelberg Introduction to Hydrology Definition of Hydrology Global water cycle Hydrological systems, water balance Introduction to Physics of Aquatic Systems (AP I) Properties of water and density stratification Transport in a turbulent fluid (lakes, ocean) Basics of hydrogeology: hydraulic head, types of aquifers Darcy's law Groundwater flow and transport equations What is Hydrology? Science of water Water on the continents, above, on, and below the surface Physical science: Water balance, water fluxes Movement of water in space and time Chemical, biological, and technical/sociological aspects Engineering: water supply, flood protection, etc. Classical hydrological systems: Catchments of rivers/aquifers Related Disciplines, Subdisciplines: Hydrogeology and soil physics Physical Oceanography and Limnology Meteorology and (paleo)climatology Compartments of the Hydrosphere from Mook, 2001 Global Water Budget and Cycle The Global Hydrological Cycle 40 Fluxes in 10 3 km mm 1066 mm Continents Oceans km km 2 Whole Earth evaporation/precipitation flux: km 3 = 973 mm from Bear, 1979

2 Global Distribution of Water Resource Stress DIA/Q = Ratio of total (Domestic, Industrial, Agricultural) water demand to total sustainable water supply (river discharge Q) Human interference with the water cycle Drying up of the Aral Sea Global irrigation demand: ~ 25 % of available fresh water flux Vörösmarty et al., Science, 2000 Satellite image: MODIS Land Rapid Response Team, NASA/GSFC Basic Hydrological Unit: The River Basin Hydrologic Water Balance ET(t) Hydrological system: reservoir with - input P (precipitation) - output R (runoff) - output ET (evapotranspiration) Continuity/Mass conservation: Input Output = Change in storage ds P R ET = dt Simple equation, but individual terms are difficult to quantify Isotopes may help! Input/Output Approach: Hydrograph ("Ganglinie") Input Hydrological system (black box) Output Properties of Water Property Comparison Importance, consequences Specific heat Highest of all solids and 4180 J kg -1 K -1 liquids except liquid NH 3 Heat transport by water movement, heat buffering Heat of fusion Highest except NH 3 Thermostatic effect at freezing point J kg -1 Heat of evaporation J kg -1 Highest of all substances Heat and water transfer in the atmosphere ρ max at T > T freezing (~4 C at 0%, 1 atm) anomalous Density stratification of lakes, facilitates freezing ρ solid < ρ liquid anomalous Ice floats on water, freezing only at surface, weathering Surface tension Highest of all liquids Drop formation, capillary forces, soil water retention, cell physiology Dissolving power Very high Transport of dissolved substances Dielectric constant Highest of all liquids except H 2 O 2 and HCN High dissociation of dissolved salts

3 Density Anomaly and Stratification Vanishing of Density Anomaly in Saltwater ( C) 4 2 T ρmax ~ S Density ρ = gcm T= C T S ~ S S = Salinity S Fluid Dynamics: Navier-Stokes-Equation Balance of forces (accelerations) on fluid parcels in a fluid on the rotating Earth yields the Navier-Stokes-equation: dv v 1 = + ( v ) v = Φ p 2( Ω v) +ν v dt t ρ Turbulent Flow and Transport in Lakes Flow in lakes is turbulent (Re = UL/ν >> Re c ~ 1000) laminar turbulent gravity pressure gradient force Coriolis force friction Treatment (for description of transport): Split flow in mean current and statistical fluctuations Treat effect of small scale turbulence as diffusion New transport parameter: Turbulent (eddy) diffusivity K different in horizontal and vertical direction: K h >> K z Origin of Turbulent Diffusion: Small-Scale Eddies Transport in a Turbulent Fluid I Large scale: Advection Small scale: Turbulence Advection (movement of solute conc. c with mean flow v) Fad = MT L vc Molecular Diffusion (stochastic movement of solutes) Fdif = MT L Dm gradc Turbulent Diffusion (spreading due to turbulent flow) Ftur MT L = K gradc Molecular diffusion can be neglected or incorporated into K Total Mass Flux: F = vc K gradc tot

4 Transport in a Turbulent Fluid II Effect of Advection and Diffusion Mass balance: Changes in mass density in control volume V are due to fluxes across boundaries and sources/sinks. Application to water mass (ρv) and approximation of incompressibility yields the continuity equation: div v = 0 Application to tracer mass (ρvc) with source/sink J yields the transport equation: c = divftot = v c + ( K c) + J t Pulse Advection: Translation Diffusion: Broadening Front 1-D Vertical Diffusive Transport in a Stratified Lake Tracer experiment to determine K z Width: σ= 2K zt 2-D Horizontal Advective-Diffusive Transport in a Lake Tracer experiment to determine v, K x, K y Variance: 2 σ = 2K t z From: Von Rohden und Ilmberger, 2001, Aquat. Sci. 63: From: Peeters et al., 1996, J. Geophys. Res., C101, Basics of Hydrogeology Groundwater = Water in the saturated zone below the water table Vpore Laminar flow through porous medium with porosity θ V total Hydraulic Head Total energy at height z E = pv + mgz (Vadose Zone) (Phreatic Zone) Energy per weight E p h = z mg = ρg + Hydraulic head is measured by piezometers (open pipes) also called piezometric head, "Piezometerhöhe" from Fitts, Groundwater Science, Academic Press

5 Hydraulic head and types of aquifers Patterns and Time Scales of Groundwater Flow piezometer soil aquifer permeable unsaturated permeable saturated z = 0 p ρg z h Unconfined Aquifer h = water level in aquifer h = water table or phreatic surface piezometer soil Confined Aquifer aquitard aquifer impermeable screen z = 0 p ρg z h h > water level in aquifer h = potentiometric surface Darcy's Law Discussion of Terms in Darcy's Law from Bear, 1979 Empirical finding: Volume flux Q [L 3 T -1 ] through sand column is proportional to cross-sectional area A and head difference h = h 1 -h 2, and inversely proportional to length L: h Q = KA L Q with q = (spec.discharge) A dh h2 h1 and = (head gradient) dl L dh q = K dl q [LT -1 ]: Specific discharge, Darcy velocity (Filtergeschwindigkeit) The mean velocity v of water between 2 points is larger than q, because only a part of the cross-section is available for flow: q v = : Average linear velocity (Abstandsgeschwindigkeit) θ dh [ LT 1 ]: Head gradient in flow direction, driving force of flow dl K [LT -1 ]: hydraulic conductivity (Durchlässigkeitsbeiwert) K is the property of the porous medium that governs flow Porous media are usually anisotropic, i.e. K depends on flow direction. Replace K by conductivity tensor K: Darcy in 3-D: q= K grad h Properties of different porous media Sediment/Rock Grain size Porosity Hydraulic cond. d [mm] θ [%] K [m s -1 ] Gravel (Kies) > Distribution of hydraulic conductivity values Log-normal distribution: Large variability! Hard to estimate from hydraulic data: Tracers help! Sand Silt (Schluff) Clay (Ton) < Sandstone Crystalline rock From: Fitts, Groundwater Science, 2002

6 Equations of Groundwater Transport I Advection (movement of solutes with groundwater flow) Fad = q = θ v MT L c c Molecular Diffusion (stochastic movement of solutes) Fdif MT L = θ Dm gradc Dispersion (spreading due to different flow paths and velocities) Fdis = θ MT L D gradc Molecular diffusion can be neglected 1 Total Mass Flux: Ftot = vc D gradc θ Mechanical dispersion Origin of Dispersion Macrodispersion Equations of Groundwater Transport II Mass balance: Changes in mass density in control volume V are due to fluxes across boundaries and sources/sinks. Application to water mass (ρθv) with source/sink J w, assuming zero storativity/compressibility, and Darcy yields the flow equation: J w divq = div( K grad h) = Jw or div v = θ Application to tracer mass (ρθvc) with source/sink J c yields the transport equation: ( cθ ) = divf tot + Jc t c Jc J = w v c + ( D c) + + ( c in c) t θ θ Summary Stage: Global Hydrological Cycle and Water Resources Basics of Hydrology: Water balance in catchments Basics of Aquatic Physics: Mass Balances over control volume Water: Continuity equation (div v = 0 or J w ) Tracer: Transport equations ( c/ t = div F) Physics of Surface Water: Turbulent Flow Parameters: Advective velocity and turbulent diffusivity Physics of Groundwater: Dispersive Flow in porous medium Parameters 1: Advective velocity and dispersivity Parameters 2: Hydraulic conductivity and recharge rate