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## The FEST Model for Testing the Importance of Hysteresis in Hydrology

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**The FEST Model**for Testing the Importance of Hysteresis in Hydrology J. Philip O’Kane Department of Civil & Environmental Engineering, Environmental Research Institute UCC Int. Workshop on HYSTERESIS & MULTI-SCALE ASYMPTOTICS, University College Cork, Ireland, March 17-21, 2004**Content**1. Introduction soil physics 2. The BASE model bare soil with evaporation and drainage 3. The FEST model fully vegetated soil slab with transpiration 4. The structure of FEST feedback structure bifurcation**1. Introduction**1. Hysteresis in hydrology, climatology, ecohydrology Is it significant? For what questions? 2. Hysteresis in open channel flow Rate dependent 3. Hysteresis in soil physics Rate independent 4. Method Build “test rigs” to answer the questions BASE model - pde - soil physics FEST model - ode - plausible soil bio-physics**Soil: a multi-phase material**Each phase has mass M and volume V The REV – representative elementary volume 1 cm Air Ma, Va Water Mw, Vw Soil-solids Ms, Vs 1 cm 1 cm**Ratios describe the multi-phase material**Total porosity ff = (Va + Vw)/(Va + Vw + Vs ) Void ratio e = (Va + Vw)/Vs[m3m-3] Particle densityrs = Ms/Vs Dry bulk densityrb = Ms/(Va + Vw + Vs ) Water densityrw = Mw/Vw [Mgm-3] Air Ma, Va Water Mw, Vw Soil-solids Ms, Vs**Moisture content**Volumetric wetnessq = Vw /(Va +Vw +Vs ) In clay soilsthe soil matrix swells, Vs = f(Vw), q has no well-defined maximum value In gravel, sand and silt, the soil matrix is “rigid” q has a maximum at saturation 0 < q < qs < 1, at saturation Va= 0 Mass wetnessw = Mw /Ms q = wrb/rw in rigid soils**Potential energy of soil water**• A mass m of soil water of volume Vand density w = m/V • is moved on an arbitrary path through a vertical distance z • by a force • mg = wVg • The dissipationless work done against the force of gravity is • mgz = (wVg)z • There are three alternative ways of representing the • potential energy of this water as dissipationless work • (a) per unit mass, (b) per unit volume, and (c) per unit weight**Total potential**is a sum of partial potentials y = yg + ym + yo + yp + ya + yW yggravitational potential ymmatric potential yo osmotic potential yp hydrostatic potential ya atmospheric yW overburden potential**Partial potentials with common reference state - free water**at z=0**Soil-moisture characteristic - matric potential, soil**suction or drying m = m(), m e < 0, 0 < s, e air-entry potential, = s = (m) inverse function Specific water capacity C() = d/dm Drying and wetting are different - hysteresis - usually ignored !**Ym(z) partitions q(z)into liquid and vapour fractions**• h(z) relative humidity of soil-air • Mwis the molar mass of water (0.018 kg/mol), • R the molar gas constant (8.314 J/mol K) • T the constant temperature in degrees Kelvin (293 K at 200C).**T**E P Water flow in a column of soil Vertical coordinate from the ground surface positive downwards to the watertable (no air) 0 Soil 1 I Soil 2 10 m z Soil 3 1 m 1 m C D**Conservation of water mass in one dimension**fl is the flux density of liquid water (kg m-2s-1) fv is the flux density of water vapour (kg m-2s-1), in the direction of positive zi.e. downwards,**Generalised Darcy’s Law**Philip, 1955 Buckingham, 1907**Philip-Richards equation – ψ form**Solutions sought in the space of continuous functions ym(z,t) Discontinuities allowed inq(z,t) to match discontinuous soil horizons Philip 1955, Richards, 1931**Boundary conditions & forcing**Flux Boundary conditions Precipitation Evaporation Overland flow - ignore initially Interflow - ignore in one dimension Potential Boundary condition Ponded infiltration Fixed water table Mixed Boundary condition Evaporation Drainage to a moving water table Forcing function Transpiration**Standard hydrological questions**Infiltration & surface runoff Evaporation Transpiration Redistribution Capillary rise Drainage**Two pairs of switched boundary conditions - atmosphere or**soil control of fluxes? Outer pair - fluxes at potential rates Raining or drying atmosphere control Inner pair - fluxes at smaller actual rates Surface ponding or phase 2 drying soil control**The raining and drying cycle**td Potential evaporation Actual evaporation Soil drying begins Ea<Ep Ea=Ep tE tQ Soil wetting begins q0<qR q0=qR Actual infiltration Potential infiltration tp**Alternating control**td Potential evaporation Actual evaporation Atmosphere control! Ea=Ep Soil control ? Ea<Ep Soil drying begins tE tQ Soil wetting begins Soil control ? q0<qR Atmosphere control! q0=qR Actual infiltration Potential infiltration tp**Infiltration - atmosphere control**D \ K constant K linear K non-linear K delta function D Mein & Larson (1973) constant D Breaster Breaster Clothier et al (1973) (1973) (1981) ‘Fujita D’ Knight & Rogers et al. Sander et al. Philip (1983) (1988) (1974)**Infiltration - soil control**D \ K constant K linear K non-linear K delta function D Green & Ampt (1911) constant D Carslaw & Philip Philip Jaeger (1969) (1974) (1946) ‘Fujita D’ Fujita not solved not solved (1952)**Evaporation - atmosphere control**D \ K constant K linear K non-linear K delta function D not applicable constant D Breaster* Breaster* Kühnel (1973) (1973) (1989 [C]) ‘Fujita D’ Knight & Sander & Sander & Philip* Kühnel Kühnel (1974) (19**) (19**) *complementary to infiltration solution**Evaporation - soil control**D \ K constant K linear K non-linear K delta function D not applicable constant D Carslaw & Kühnel & Kühnel Jaeger* Sander (1989 [C]) (1946) (19**) ‘Fujita D’ Fujita* not solved not solved (1952) *complementary to infiltration solution**The FEST model -fully vegetated soil slab with transpiration**Goal: from plausible biophysics an ode - for testing hysteresis operators**FEST ordinary differential equation**• Uniform moisture in the root zone • Gradients in potential become differences • Brooks-Corey-Campbell parametric expressions for the matric potential and hydraulic conductivities of soils • Square wave atmospheric forcing**Transpiration**• Roots completely penetrate the uniform root zone • A 3-D wick sucks water from the uniform roots to a uniform canopy • Leaf potential is matric potential of soil water plus change in gravitational potential between the roots and canopy • Potential transpiration (given) drives actual transpiration**Potential transpiration - given**The Philip boundary condition Leaf evaporation is proportional to the difference in humidity between (a) the atmosphere, and (b) the stomatal air in “thermodynamic” equilibrium with its plant water in the canopy**Actual transpiration**• drops below • the potential rate • when stomates close • at leaf potentials between • some higher value (e.g. -5,000cm) • and the wilting potential (e.g. -10,000cm)**Infiltration**Actual infiltration is the minimum of the rainfall rate and the potential infiltration rate Infiltration is assumed to occur throughout the soil slab through preferential paths due to worm holes, animal burrows and dead roots presenting the infiltrating water uniformly to the soil matrix.**Potential infiltration rate**is equal to the hydraulic conductivity at the soil water potential times the difference between that potential and the air entry potential of the rain divided by an arbitrary pore spacing**Feedback loops for drainage/capillary rise with soil physics**parameters**Cut the feedback loops**Multiple equilibria Bifurcation