1. Ecohydrology
Fall 2017

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)
Zr 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 (sw) below which ET = 0
Field capacity or (sl) where ET = Etmax
ETmax is a fixed quantity, ET is a linear function of s
Spatial averaging makes this tenable
Max available water available wo = (sl – sw)*n*Zr
Define two dimensionless quantities:
γ = wo / α [soil storage per mean rainfall depth]
Di = ETmax / 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 - sw)/(sl – sw)
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
The value of γ that captures Budyko’s curve is At α = 1.5 cm (and sw = 0.2, sl = 0.85 and n = 0.4) this corresponds to a value of Zr of 35 cm which is approx. the global mean rooting depth
Fraction of rainfall lost to ET
Dryness Index

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 = wo / α
λ/η = λ* wo / ETmax

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