On the spectrum of soil moisture from hourly to interannual scales

The spectrum of soil moisture content at scales ranging from 1 hour to 8 years is analyzed for a site whose hydrologic balance is primarily governed by precipitation (p), and evapotranspiration (ET). The site is a uniformly planted loblolly pine stand situated in the southeastern United States and is characterized by a shallow rooting depth (R_L) and a near-impervious clay pan just below R_L. In this setup, when ET linearly increases with increasing root zone soil moisture content (θ), an analytical model can be derived for the soil moisture content energy spectrum (E_s(f), where f is frequency) that predicts the soil moisture "memory" (taken as the integral timescale) as β₁^-1 ≈ ηR _L/ET_max, where ET_max is the maximum measured hourly ET and η is the soil porosity. The spectral model suggests that E_s(f) decays at f^-2-α at high f but almost white (i.e., f⁰) at low f, where α is the power law exponent of the rainfall spectrum at high f (α ≈ 0.75 for this site). The rapid E _s(f) decay at high f makes the soil moisture variance highly imbalanced in the Fourier domain, thereby permitting much of the soil moisture variability to be described by a limited number of Fourier modes. For the 8-year data collected here, 99.6% of the soil moisture variance could be described by less than 0.4% of its Fourier modes. A practical outcome of this energy imbalance in the frequency domain is that the diurnal cycle in ET can be ignored if β₁^-1 (estimated at 7.6 days from the model) is much larger than 12 hours. The model, however, underestimates the measured E_s(f) at very low frequencies (f ≪ β₁) and its memory, estimated from the data at 42 days. This underestimation is due to seasonality in ET_max and to a partial decoupling between ET and soil moisture at low frequencies.