The FEST Model for Testing the Significance of Hysteresis in Hydrology.

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The FEST Model for Testing the Significance of Hysteresis in Hydrology J. Philip O'Kane Division of Common and Ecological Designing, Natural Exploration Organization UCC Int. Workshop on HYSTERESIS and MULTI-SCALE ASYMPTOTICS, College School Plug, Ireland, Walk 17-21, 2004 Substance
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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

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Content 1. Presentation soil material science 2. The BASE model bare soil with dissipation and waste 3. The FEST model fully vegetated soil chunk with transpiration 4. The structure of FEST feedback structure bifurcation

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1. Presentation 1. Hysteresis in hydrology, climatology, ecohydrology Is it noteworthy? For what questions? 2. Hysteresis in open channel stream Rate subordinate 3. Hysteresis in soil material science Rate free 4. Strategy Build “test rigs” to answer the inquiries BASE model - pde - soil material science FEST model - tribute - conceivable soil bio-physical science

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Some dirt physical science

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Soil: a multi-stage material Each stage has mass M and volume V The REV – agent rudimentary volume 1 cm Air M a , V a Water M w , V w Soil-solids M s , V s 1 cm 1 cm

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Ratios depict the multi-stage material Total porosity f = (V a + V w )/(V a + V w + V s ) Void ratio e = (V a + V w )/V s [m 3 m - 3 ] Particle thickness r s = M s/V s Dry mass thickness r b = M s/(V a + V w + V s ) Water thickness r w = M w/V w [Mg m - 3 ] Air M a , V a Water M w , V w Soil-solids M s , V s

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Moisture content Volumetric wetness q = V w/(V a +V w +V s ) In mud soils the dirt lattice swells , V s = f(V w ), q has no very much characterized most extreme worth In rock, sand and residue, the dirt network is “rigid” q has a greatest at immersion 0 < q < q s < 1, at immersion V a = 0 Mass wetness w = M w/M s q = w r b/r w in inflexible soils

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Potential vitality of soil water A mass m of soil water of volume V and thickness  w = m/V is proceeded onward a self-assertive way through a vertical separation z by a power mg =  w Vg The dissipationless work done against the power of gravity is mgz = (  w Vg)z There are three option methods for speaking to the potential vitality of this water as dissipationless work (a) per unit mass, (b) per unit volume, and (c) per unit weight

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Potential vitality per unit mass, volume and weight

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Total potential is a total of fractional possibilities y = y g + y m + y o + y p + y a + y W y g gravitational potential y m matric potential y o osmotic potential y p hydrostatic potential y an environmental y W overburden potential

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Partial possibilities with regular reference state - free water at z=0

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Soil-dampness trademark - matric potential, soil suction or drying  m =  m (  ),  m   e < 0, 0 <    s ,  e air-passage potential,  =  s  =  (  m ) inverse capacity Specific water limit C(  ) = d /d  m Drying and wetting are diverse - hysteresis - normally disregarded !

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Y m (z) allotments q( z ) into fluid and vapor parts h(z) relative mugginess of soil-air M w is the molar mass of water (0.018 kg/mol) , R the molar gas consistent (8.314 J/mol K) T the steady temperature in degrees Kelvin (293 K at 20 0 C) .

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The BASE model - exposed soil with vanishing and waste

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T E P Water stream in a section of soil Vertical direction starting from the earliest stage positive downwards to the watertable (no air) 0 Soil 1 I Soil 2 10 m z Soil 3 1 m 1 m C D

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Conservation of water mass in one measurement f l is the flux thickness of fluid water ( kg m - 2 s - 1 ) f v is the flux thickness of water vapor ( kg m - 2 s - 1 ), in the bearing of positive z i.e. downwards,

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Generalized Darcy’s Law Philip, 1955 Buckingham, 1907

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Philip-Richards mathematical statement – ψ structure Solutions looked for in the space of persistent capacities y m (z,t) Discontinuities permitted in q (z,t) to coordinate spasmodic soil skylines Philip 1955, Richards, 1931

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Boundary conditions & driving Flux Boundary conditions Precipitation Evaporation Overland stream - overlook at first Interflow - disregard in one measurement Potential Boundary condition Ponded invasion Fixed water table Mixed Boundary condition Evaporation Drainage to a moving water table Forcing capacity Transpiration

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Standard hydrological inquiries Infiltration & surface spillover Evaporation Transpiration Redistribution Capillary ascent Drainage

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Two sets of exchanged limit conditions -air or soil control of fluxes? External pair - fluxes at potential rates Raining or drying atmosphere control Inner pair - fluxes at littler real rates Surface ponding or stage 2 drying soil control

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The sprinkling and drying cycle t d Potential vanishing Actual dissipation Soil drying starts Ea<Ep Ea=Ep t E t Q Soil wetting starts q 0 <q R q 0 =q R Actual invasion Potential penetration t p

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Alternating control t d Potential vanishing Actual dissipation Atmosphere control! Ea=Ep Soil control ? Ea<Ep Soil drying starts t E t Q Soil wetting starts Soil control ? q 0 <q R Atmosphere control! q 0 =q R Actual invasion Potential penetration t p

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Richards mathematical statement – θ structure

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Infiltration - environment control D \ K constant K linear K non-straight K delta capacity D Mein & Larson (1973) consistent D Breaster Breaster Clothier et al (1973) (1973) (1981) ‘Fujita D’ Knight & Rogers et al. Sander et al. Philip (1983) (1988) (1974)

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Infiltration - soil control D \ K constant K linear K non-direct K delta capacity D Green & Ampt (1911) steady D Carslaw & Philip Philip Jaeger (1969) (1974) (1946) ‘Fujita D’ Fujita not solved not comprehended (1952)

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Evaporation - climate control D \ K constant K linear K non-straight K delta capacity D not relevant consistent D Breaster* Breaster* Kühnel (1973) (1973) (1989 [C]) ‘Fujita D’ Knight & Sander & Sander & Philip* Kühnel Kühnel (1974) (19**) (19**) *complementary to penetration arrangement

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Evaporation - soil control D \ K constant K linear K non-straight K delta capacity D not pertinent steady D Carslaw & Kühnel Jaeger* Sander (1989 [C]) (1946) (19**) ‘Fujita D’ Fujita* not solved not illuminated (1952) *complementary to invasion arrangement

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The FEST model - completely vegetated soil piece with transpiration Goal: from conceivable biophysics a tribute - for testing hysteresis administrators

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FEST common differential comparison Uniform dampness in the root zone Gradients in potential get to be contrasts Brooks-Corey-Campbell parametric expressions for the matric potential and pressure driven conductivities of soils Square wave barometrical constraining

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Transpiration Roots totally enter the uniform root zone A 3-D wick sucks water from the uniform roots to a uniform shelter Leaf potential is matric capability of soil water in addition to change in gravitational potential between the roots and covering Potential transpiration ( given ) drives genuine transpiration

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Potential transpiration - given The Philip limit condition Leaf vanishing is corresponding to the distinction in stickiness between (a) the air, and (b) the stomatal air in “thermodynamic” balance with its plant water in the overhang

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Actual transpiration drops beneath the potential rate when stomates close at leaf possibilities between some higher worth (e.g. - 5,000cm) and the shriveling potential (e.g. - 10,000cm)

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Feedback circles for transpiration

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Transpiration circles with soil material science parameters

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Infiltration Actual invasion is the precipitation\'s base rate and the potential penetration rate Infiltration is accepted to happen all through the dirt section through particular ways because of worm openings, creature tunnels and dead roots showing the invading water consistently to the dirt framework .

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Potential penetration rate is equivalent to the pressure driven conductivity at the dirt water potential times the contrast between that potential and the air passage capability of the downpour separated by a self-assertive pore dispersing

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Infiltration input circles

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Infiltration with parameters

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Feedback circles for seepage/fine ascent with soil material science parameters

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Feedback circles for waste, slim ascent

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Cut the criticism circles Multiple equilibria Bifurcation

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Bifurcation to leave

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Titles Bifurcation on e over p = 1.2, 1, 0.8; period 10 Bifurcation on e over p = 1.2, 1, 0.8; period 20 Bifurcation on e over p = 1.2, 1, 0.8; period 40 Bifurcation on theta(0)

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Insertion of Preisach administrator One insertion makes everything hysteretic Extension in space horizontally with

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