A Scale Aware Eddy Viscosity for the Roughness Sublayer of Tall Canopies: Implications to Biosphere-Atmosphere Mass, Energy and Momentum Exchanges (Invited & Awardee)

Background: In the roughness sublayer (RSL) above tall forests, the representation of biosphere-atmosphere mass, energy, and momentum exchanges in large-scale models relies upon the concept of eddy viscosity to link turbulent fluxes (including those of water vapor and carbon dioxide) from the biosphere to their mean gradients near the canopy-atmosphere interface (Bonan et al., 2018). These eddy viscosity formulations employ a single (or master) length scale to represent the effects of eddies on turbulent transport in the RSL. This single length scale is linked to the so-called shear length scale associated with Kelvin-Helmholtz instabilities dominating coherent structures near the canopy top (Harman and Finnigan, 2007). Yet, turbulent fluctuations span a wide range of eddy sizes that go from several kilometers (i.e. boundary layer depth) to a fraction of millimeter scale (Kolmogorov micro-scales) thereby making it difficult to prescribe all their effects in a single length. Recent approaches that utilize the co-spectral budget model (CSB) have been able to link the effective eddy viscosity to the shape of the vertical velocity spectrum. Using this link, the scale-wise integrated spectrum of the vertical velocity weighted by an appropriate relaxation time (Bos et al., 2004) was shown to recover numerous features of wall-bounded flows (Katul et al., 2013, 2014) including bulk friction features (Coscarella et al., 2021) and links between established phenomenological constants. This approach was extended to the RSL and shown to recover the so-called roughness sublayer correction functions to the law-of-the wall (Mortarini et al., 2023).

Problem statement: Recent experiments in the RSL of a tall canopy situated on complex terrain in Amazonia revealed surprising oscillatory behavior in the lagged auto-covariance function (or the Fourier pair of the energy spectrum) of the vertical velocity component (Mortarini et al., 2022). This oscillatory behavior was reported for near-neutral conditions thereby precluding the role of thermal stratification as a plausible explanation. The onset for such oscillatory behavior and its consequences to scale-aware eddy viscosity models remains open and frames the scope of the presentation.

What will be accomplished: A two-equation Lagrangian stochastic dispersion model (LSM) for the coupled turbulent vertical (w) and longitudinal (u) velocity increments is used to derive the vertical velocity auto-correlation Rww(τ) and the cross-correlation Ruw(τ) functions with time lag τ. Here, the τ's may be viewed as representing the characteristic time scale of turbulent eddies and this representation is preferred in the analysis because Taylor's frozen turbulence hypothesis is not required. The necessary conditions for oscillatory behavior are linked to a coefficient related to the modeled pressure-strain interaction term. For cases where the mean wind direction originates from flat terrain, no oscillations in Rww(τ) and Ruw(τ) are predicted or measured. On the other hand, for cases where the upwind direction originates from complex topography, damped oscillations are measured as predicted from the LSM. The agreement between LSM predictions and Eulerian velocity measurements of Rww(τ) and Ruw(τ) in the RSL and their implications to eddy viscosity models are discussed. The proposed mean strain distortion by topography identified here also plays a role in interpreting measured momentum, energy, and scalar fluxes in the RSL above tall forests, which are likely to be situated on non-flat terrain. In particular, the oscillations identified here can lead to 'long-range' memory that may be mis-interpreted as non-stationarity in the flow.

Future effort: Future efforts seek to (i) extend the present work to include temperature fluctuations so as to accommodate non-neutral atmospheric conditions within the RSL, (ii) develop a Langevin model for the turbulent kinetic energy fluctuations and explore its relation to recent Langevin models tested for the atmospheric surface layer across a variety of surfaces and atmospheric stability conditions, and (iii) explore the validity of this Langevin model and its expanded version to other forested sites and scalars (including water vapor and carbon dioxide) using publicly available high frequency data.

References:

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-W.J.T. Bos, H. Touil, L. Shao, and J.P. Bertogli. On the behavior of the velocity-scalar cross correlation spectrum in the inertial range. Physics of Fluids, 16:3818–3823, 2004.

-F. Coscarella, R. Gaudio, G.G. Katul, and C. Manes. Relation between the spectral properties of wall turbulence and the scaling of the Darcy-Weisbach friction factor. Physical Review Fluids, 6(5):054601, 2021.

- I.N. Harman and J.J. Finnigan. A simple unified theory for flow in the canopy and roughness sublayer. Boundary-Layer Meteorology, 123(2):339–363, 2007

-Katul, G.G., A. Porporato, C. Manes, and C. Meneveau. Co-spectrum and mean velocity in turbulent boundary layers. Physics of Fluids, 25(9):091702, 2013

-Katul, G. G., A. Porporato, S. Shah, and E. Bou-Zeid. Two phenomenological constants explain similarity laws in stably stratified turbulence. Physical Review E, 89(2), 023007, 2014.

-Mortarini, L., C.Q. Dias-J´unior, O. Acevedo, P.E.S. Oliveira, A. Tsokankunku, M. Sorgel, A. Ocimar Manzi, A.C. de Araujo, D.V. Brondani, I.M. Cely Toro, U.Giostra, and D.Cava. Vertical propagation of submeso and coherent structure in a tall and dense Amazon forest in different stability conditions. part II: Coherent structures analysis. Agricultural and Forest Meteorology, 322:108993, 2022.

-Mortarini, L., G.G. Katul, D. Cava, C.Q. Dias-Junior, N.L. Dias, A. Manzi, M. Sorgel, A.C. Araujo, and M. Chamecki. Adjustments to the law of the wall above an Amazon forest explained by a spectral link. Physics of Fluids, 35(2), 2023.