Wind and remnant tree sway in forest cutblocks. III. a windflow model to diagnose spatial variation

Abstract

Root-mean-square tree sway angle (σ_θ) can be related to the r.m.s. value (σ_u|u|) of the wind `force' u|u|, specified nearby. Therefore the spatial pattern of wind statistics over the landscape, if known, arguably maps the relative risk of windthrow – suggesting a wind model can provide the basis to interpret spatial patterns of windthrow, and guide strategies with respect to that concern. To test this idea, we adopt a simple flow model able to describe both the mean wind (U) and kinetic energy of the turbulence (k), viz. Reynolds' equations closed using eddy-viscosity K∝λk^1/2, where λ is the turbulence lengthscale. We first compare this model with others' measurements of wind near forest edges, then simulate our own observations, which spanned arrays of cutblocks and intervening forest blocks in the Boreal forest (periodic spacing 1.7h or 6.1h, where h is mean tree height). We show that the model predicts well the spatial variation of the mean windspeed and turbulent kinetic energy, these being the wind statistics having greatest impact upon tree sway. Model-implied spatial patterns of r.m.s. wind force (σ_u|u|) agree closely with those observed, and in conjunction with a tree-motion model, imply tree sway (σ_θ).

Introduction

Management trials in the boreal mixedwood forest of northern Canada are evaluating felling practises that, at the time of aspen harvest, preserve the spruce understory (`released spruce') for later cutting. It is considered necessary to leave uncut or partially-cut forest strips to shelter the selectively-cut zones, because the previously-sheltered remnant spruce are very vulnerable to windthrow. An overall description of this long-term, practically-oriented project, which is being carried out near Manning (Alberta) by Forestry Canada and partners, is given by Navratil et al. (1994). The work we report here (and in companion papers, Flesch and Wilson, 1999aFlesch and Wilson, 1999b) was initiated in the hope of interpreting the observed spatial variation of tree windthrow across such arrays of cutblocks, so as to permit generalisation. It involves our own measurements of turbulence and tree sway in two cutblocks, each the leeward member of a periodic series; an analysis of the response of instrumented trees to the wind forcing; and, in this paper, an attempt to establish a framework for generalisation of our findings by numerically modelling the winds.

The link between tree sway statistics and wind statistics is discussed at length by Flesch and Wilson (1999b). Briefly, we treated the tree as a rigid rod, free to swing about a ground-level pivot in response to the wind force, but under the moderation of an angular spring and damper. We derived a transfer function relating the short-term power spectrum S_θ of tree angular displacement (θ) to the concurrent power spectrum S_u|u| of the wind force u|u|, where u is the instantaneous alongwind velocity component, measured nearby. By `short-term' statistics, we mean statistical properties (standard deviations of tree sway angle σ<sub>θ</sub> and of wind force σ<sub>u|u|</sub>, variance spectra S<sub>θ</sub>, S<sub>u|u|</sub>, etc.) defined by a sample taken over about 15 to 60 min., such intervals being sufficiently short that `external' or large-scale conditions are roughly constant, but sufficiently long that many `cycles' of the rapid turbulent variations are captured. Thus the `view' of our windthrow analysis consists of (say) 30 min. snapshots, from which a longer-term view may be constructed by integration; and the fluctuating (turbulent) variables, e.g. the alongwind velocity u, are decomposed

$$\text{u(x,y,z,t)=U(x,y,z)+u′(x,y,z,t)}$$

into sums of the average (in this case, the mean alongwind velocity U, a function of position only) and the instantaneous deviation (or fluctuation, here u′) from it. This terminology (upper-case for mean values; prime for fluctuation from average) will apply throughout our paper.

Returning to our tree sway model, the wind-force spectrum S_u|u| was observed not at the `subject' remnant tree, but merely, at the same alongwind location relative to the upwind edge of the cutblock, and at a convenient, arbitrary height (9 m). We found that normalized wind force spectra S_u|u|/σ²_u|u| were similar at all points across the cutblocks (i.e. practically invariant), so that the sway (σ_θ) of a tree, no matter where located, could be definitively related to σ²_u|u| at that point. But in turn, the force-variance σ²_u|u| can be determined from the lowest order statistics of the wind

$$\text{σ}{}^2{}_{\mathrm{u|u|}}\text{≈σ}{}^2{}_{\mathrm{uu}}\text{≡σ}{}^4{}_{\mathrm{u}}\text{Kt}{}_{\mathrm{u}}\text{−1+4}\text{U}\text{σ}{}_{\mathrm{u}}{}^2\text{+4}\text{U}\text{σ}{}_{\mathrm{u}}\text{Sk}{}_{\mathrm{u}}$$

where the right-hand equality is exact. Within our framework then, the wind statistics governing root-mean-square (r.m.s.) tree sway are: mean U, variance σ²_u, skewness Sk_u, and kurtosis Kt_u. A sensitivity analysis (see Appendix A) indicates that under conditions typical of the flow in our Manning cutblocks (Sk_u ≈ 1, Kt_u ≈ 4), spatial modulation of the wind-force variance (and thus of σ²_θ) is controlled, in order of importance, by spatial variation in the velocity variance σ²_u (one component of the turbulent kinetic energy¹), and in the mean velocity U. Thus for the remainder of this paper, our focus is on modelling the spatial variation, around and about forest edges, of these principal wind and turbulence statistics, U and k. As regards the connexion of our method to the practical issue of tree windthrow, by hypothesizing that the spatial pattern of wind statistics implies the corresponding spatial pattern of windthrow, we obviously are assuming that the key factor in the cross-landscape variation of treefall susceptibility is the wind forcing, rather than any systematic variation in soil conditions, rooting depth, tree health, etc. It is also implicit that we presume extreme tree displacements (or wind forces) scale with the standard deviation σ_θ (or with σ_u|u|).

Having motivated our focus on mean wind (U) and turbulence (k) in forest clearings, we now review previous efforts to model forest edge flows; describe a numerical windflow model, developed by Wilson et al. (1998) specifically for the description of disturbed canopy winds; compare simulations using that model against others' observations of forest-edge flows; and finally simulate the spatial pattern of the wind and turbulence in our cutblocks at Manning for comparison with our observations.