1. Ecohydrology Fall 2015

2. Core Questions Primary production is controlled by time-varying soil moisture Stochastic rainfall inputs Soil physical properties control storage and “overflow” Moisture-dependent ET loss What patterns emerge from a simple model of this rainfall-soil moisture-vegetation interactions? Rainfall delivery patterns Emergent patterns of soil moisture distributions

3. Model Formulation n is soil porosity (constant, dimensionless) Z r is the rooting depth (cm) s is soil moisture (dimensionless) R(t) is stochastic rainfall (marked Poisson process with mean depth α in cm and recurrence interval λ in day -1 ) ET[s(t)] is evaporation as a function of s (cm d -1 ) LQ[s(t), t] is excess rainfall loss via runoff and infiltration below the root zone (cm d -1 )

4. Stochastic Rainfall Two parameters describe time variation in rainfall Mean depth α Mean of an exponential distribution Frequency λ Number of rainstorms per day (days between rain events)

5. Soil Moisture Dynamics Moisture thresholds at: Wilting point (s w ) below which ET = 0 Field capacity or (s l ) where ET = Et max ET max is a fixed quantity, ET is a linear function of s Spatial averaging makes this tenable Max available water available w o = (s l – s w )*n*Z r Define two dimensionless quantities: γ = w o / α [soil storage per mean rainfall depth] D i = ET max / α*λ [dryness index]

6. Solving for pdfs of s Propagate stochastic rainfall through the “filter” of soil water storage and use to get a pdf of soil moisture (effective moisture “x” = (s - s w )/(s l – s w ) Involves gamma and truncated gamma distributions parameterized with the governing parameters (λ, α, η, γ)

7. Reproducing Macroscale Behavior Captures characteristic behavior of the semi- empirical Budyko curve (dots) Increasing γ Fraction of rainfall lost to ET Dryness Index The value of γ that captures Budyko’s curve is 5.5. At α = 1.5 cm (and s w = 0.2, s l = 0.85 and n = 0.4) this corresponds to a value of Z r of 35 cm which is approx. the global mean rooting depth

8. Patterns of Vegetation Stress From pdf of soil moisture classify water stress based on the mode of x (x * is water stress threshold) γ = w o / α λ/η = λ* w o / ET max

9. Predicts Effects of Experimental Change in Rainfall Pattern (not amount)

10. Take Home Message Simple analytical model can capture the key elements of the rainfall-vegetation-soil moisture system Stress responses can arise from changes in rainfall pattern, not just amount Interactions between soil storage and rainfall create geographic variation in water stress