Skip to main content
Log in

A new nonlinear analytical model for canopy flow over a forested hill

  • Original Paper
  • Published:
Theoretical and Applied Climatology Aims and scope Submit manuscript


A new nonlinear analytical model for canopy flow over gentle hills is presented. This model is established based on the assumption that three major forces (pressure gradient, Reynolds stress gradient, and nonlinear canopy drag) within canopy are in balance for gentle hills under neutral conditions. The momentum governing equation is closed by the velocity-squared law. This new model has many advantages over the model developed by Finnigan and Belcher (Quart J Roy Meteorol Soc 130: 1–29 2004, hereafter referred to as FB04) in predicting canopy wind velocity profiles in forested hills in that: (1) predictions from the new model are more realistic because surface drag effects can be taken into account by boundary conditions, while surface drag effects cannot be accounted for in the algebraic equation used in the lower canopy layer in the FB04 model; (2) the mixing length theory is not necessarily used because it leads to a theoretical inconsistency that a constant mixing length assumption leads to a nonconstant mixing length prediction as in the FB04 model; and (3) the effects of height-dependent leaf area density (a(z)) and drag coefficient (C d ) on wind velocity can be predicted, while both a(z) and C d must be treated as constants in FB04 model. The nonlinear algebraic equation for momentum transfer in the lower part of canopy used in FB04 model is height independent, actually serving as a bottom boundary condition for the linear differential momentum equation in the upper canopy layer. The predicting ability of the FB04 model is largely restricted by using the height-independent algebraic equation in the bottom canopy layer. This study has demonstrated the success of using the velocity-squared law as a closure scheme for momentum transfer in forested hills in comparison with the mixing length theory used in FB04 model thus enhancing the predicting ability of canopy flows, keeping the theory consistent and simple, and shining a new light into land-surface parameterization schemes in numerical weather and climate models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others


  1. Treatment of Cd near the ground in E6 remains the same as that in E4.


  • Allen T, Brown AR (2002) Large-eddy simulation of turbulent separated flow over rough hills. Boundary Layer Meteorol 102:177–198

    Article  Google Scholar 

  • Aubinet M, Heinesch B, Yernaux M (2003) Horizontal and vertical CO2 advection in a sloping forest. Boundary Layer Meteorol 108:397–417

    Article  Google Scholar 

  • Baldocchi DD, Meyers TP (1988) Turbulence structure in a deciduous forest. Boundary Layer Meteorol 43:345–364

    Article  Google Scholar 

  • Belcher SE, Hunt JCR (1998) Turbulent flow over hills and waves. Annu Rev Fluid Mech 30:507–538

    Article  Google Scholar 

  • Belcher SE, Newley TMJ, Hunt JCR (1993) The drag on an undulating surface induced by the flow of a turbulent boundary layer. J Fluid Mech 249:557–596

    Article  Google Scholar 

  • Bergen JD (1971) Vertical profiles of windspeed in a pine stand. For Sci 17:314–322

    Google Scholar 

  • Bradley EF (1980) An experimental study of the profiles of wind speed, shearing stress and turbulent intensities at the crest of a large hill. Quart J Roy Meteorol Soc 106:101–124

    Article  Google Scholar 

  • Brown AR, Hobson JM, Wood N (2001) Large-eddy simulation of neutral turbulent flow over rough sinusoidal ridges. Boundary Layer Meteor 98:411–441

    Article  Google Scholar 

  • Burns SP, Sun J, Lenschow DH, Oncley SP, Stephens BB, Yi C, Anderson DE, Hu J, Monson RK (2011) Atmospheric stability effects on wind fields and scalar mixing within and just above a subalpine forest in sloping terrain. Boundary Layer Meteorology 138:231–262. doi:10.1007/s10546-010-9560-6

    Article  Google Scholar 

  • Coceal O, Thomas TG, Castro IP, Belcher SE (2006) Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Boundary Layer Meteorol 121:491–519

    Article  Google Scholar 

  • Dupont S, Brunet Y, Finnigan JJ (2008) Large-eddy simulation of turbulent flow over a forested hill: Validation and coherent structure identification. Quart J Roy Meteorol Soc 134:1911–1929

    Article  Google Scholar 

  • Edburg SL, Allwine G, Lamb B, Stock D, Thistle H, Peterson H, Strom B (2010) A simple model to predict scalar dispersion within a successively thinned loblolly pine canopy. J Appl Meteorol Climate. doi:10.1175/2010JAMC2339.1,49,1913-1926

  • Feigenwinter C, Bernhofer C, Vogt R (2004) The influence of advection on the short term CO2-budget in and above a forest canopy. Boundary Layer Meteorol 113:201–224

    Article  Google Scholar 

  • Feigenwinter C, Bernhofer C, Eichelmann U, Heinesch B, Hertel M, Janous D, Kolle O, Lagergren F, Lindroth A, Minerbi S, Moderow U, Molder M, Montagnani L, Queck R, Rebmann C, Vestin P, Yernaux M, Zeri M, Ziegler W, Aubinet M (2008) Comparison of horizontal and vertical advective CO2 fluxes at three forest sites. Agr For Meteorol 148:12–24

    Article  Google Scholar 

  • Finnigan JJ, Belcher SE (2004) Flow over a hill covered with a plant canopy. Quart J Roy Meteorol Soc 130:1–29

    Article  Google Scholar 

  • Finnigan JJ (2008) An introduction to flux measurements in difficult conditions. Ecol Appl 18:1340–1350

    Article  Google Scholar 

  • Fischenich JC (1996) Velocity and resistance in densely vegetated floodways. Ph.D. thesis, Colorado State University, 203 pp.

  • Fons RG (1940) Influence of forest cover on wind velocity. J For 38:481–486

    Google Scholar 

  • Goulden ML, Munger JW, Fan SM, Daube BC, Wofsy SC (1996) Measurements of carbon sequestration by long-term eddy covariance: methods and a critical evaluation of accuracy. Glob Change Biol 2:169–182

    Article  Google Scholar 

  • Hunt JCR, Leibovich S, Richards KJ (1988a) Turbulent shear flows over low hills. Quart J Roy Meteorol Soc 114:1435–1470

    Article  Google Scholar 

  • Hunt JCR, Richards KJ (1984) Stratified airflow over one or two hills. Boundary Layer Meteorol 30:223–259

    Article  Google Scholar 

  • Hunt JCR, Richards KJ, Brighton PWM (1988b) Stably stratified shear flow over low hills. Quart J Roy Meteorol Soc 114:859–886

    Article  Google Scholar 

  • Hunt JCR, Carruthers DJ (1990) Rapid distortion theory and the 'problems' of turbulence. J Fluid Mech 212:497–532

    Article  Google Scholar 

  • Inoue E (1963) On the turbulent structure of air flow within crop canopies. J Meteor Soc Japan 41:317–326

    Google Scholar 

  • Jackson PS, Hunt JCR (1975) Turbulent wind flow over a low hill. Quart J Roy Meteorol Soc 101:929–955

    Article  Google Scholar 

  • Kaimal JC, Finnigan JJ (1994) Atmospheric boundary layer flows: their structure and measurements. Oxford University Press, Oxford

    Google Scholar 

  • Kutsch WL, Kolle O, Rebmann C, Knohl A, Ziegler W, Schulze E-D (2008) Advection and resulting CO2 exchange uncertainty in a tall forest in central Germany. Ecol Appl 18:1391–1405

    Article  Google Scholar 

  • Lalic, B. and D. T. Mihailovic (2002), A new approach in parameterization of momentum transport inside and above forest canopy under neutral conditions. Integrated Assessment and Decision Support, Proceedings of the1st biennial meeting of the International Environmental Modelling and Software Society, Switzerland, iEMSs: Manno, 139–154.

  • Landsberg JJ, James GB (1971) Wind profiles in plant canopies: studies on an analytical model. J Appl Ecol 8:729–741

    Article  Google Scholar 

  • Lee X (1998) On micrometeorological observations of surface-air exchange over tall vegetation. Agr For Meteor 91:39–49

    Article  Google Scholar 

  • Lemon E, Allen LH, Muller L (1970) Carbon dioxide exchange of a tropical rain forest. Bioscience 20:1054–1059

    Article  Google Scholar 

  • Mason PJ, King JC (1984) Atmospheric flow over a succession of nearly two-dimensional ridges and valleys. Quart J Roy Meteor Soc 110:821–845

    Article  Google Scholar 

  • Mason PJ, Sykes RI (1979) Flow over an isolated hill of moderate slope. Quart J Roy Meteor Soc 105:383–395

    Article  Google Scholar 

  • Massman WJ (1982) Foliage distribution in old-growth coniferous tree canopies. Can J For Res 12:10–17

    Article  Google Scholar 

  • Massman WJ (1997) An analytical one-dimensional model of momentum transfer by vegetation of arbitrary structure. Bound Layer Meteor 83:407–421

    Article  Google Scholar 

  • Massman WJ, Lee X (2002) Eddy covariance flux corrections and uncertainties in long-term studies of carbon and energy exchanges. Agr For Meteor 113:121–144

    Article  Google Scholar 

  • Meyers T, Paw U FT (1986) Testing of a higher-order closure-model for modeling air-flow within and above plant canopies. Bound Layer Meteor 37:297–311

    Article  Google Scholar 

  • Oliver HR (1971) Wind profiles in and above a forest canopy. Quart J Roy Meteor Soc 97:548–553

    Article  Google Scholar 

  • Pielke RA (2002) Mesoscale meteorological modeling, 2nd edn. Academic Press, San Diego, CA, p 676

    Google Scholar 

  • Poggi D, Katul GG (2007a) An experimental investigation of the mean momentum budget inside dense canopies on narrow gentle hilly terrain. Agr For Meteor 144:1–13

    Article  Google Scholar 

  • Poggi D, Katul GG (2007b) Turbulent flows on forested hilly terrain: the recirculation region. Quart J Roy Meteor Soc 133:1027–1039

    Article  Google Scholar 

  • Poggi D, Katul GG (2007c) The ejection-sweep cycle over gentle hills covered with bare and forested surfaces. Bound Layer Meteor 122:493–515

    Article  Google Scholar 

  • Poggi D, Katul GG, Finnigan JJ, Belcher SE (2008) Analytical models for the mean flow inside dense canopies on gentle hilly terrain. Quart J Roy Meteor Soc 134:1095–1112

    Article  Google Scholar 

  • Poggi D, Katul GG (2008) Turbulent intensities and velocity spectra for bare and forested gentle hills: flume experiments. Bound Layer Meteor 129:25–46

    Article  Google Scholar 

  • Queck R, Bernhofer C (2010) Constructing wind profiles in forests from limited measurements of wind and vegetation structure. Agric Forest Meteorol 150:724–735

    Article  Google Scholar 

  • Ross AN (2008) Large-eddy simulations of flow over forested ridges. Bound Layer Meteor 128:59–76

    Article  Google Scholar 

  • Ross AN, Vosper SB (2005) Neutral turbulent flow over forested hills. Quart J Roy Meteor Soc 131:1841–1862

    Article  Google Scholar 

  • Salmon JR, Teunissen HW, Mickle RE, Taylor PA (1988) The kettles hill project: field observations, wind-tunnel simulations and numerical model predictions for flow over a low hill. Bound Layer Meteor 43:309–343

    Article  Google Scholar 

  • Schlichting H (1960) Boundary layer theory, 4th edn. McGraw-Hill, New York, p 647

    Google Scholar 

  • Shaw RH (1977) Secondary wind speed maxima inside plant canopies. J Appl Meteor 16:514–521

    Article  Google Scholar 

  • Staebler RM, Fitzjarrald DR (2004) Observing subcanopy CO2 advection. Agr For Meteor 122:139–156

    Article  Google Scholar 

  • Sun J, Burns SP, Delany AC, Oncley SP, Turnipseed AA, Stephens BB, Lenschow DH, LeMone MA, Monson RK, Anderson DE (2007) CO2 transport over complex terrain. Agr For Meteor 145:1–21

    Article  Google Scholar 

  • Sun J, Oncley SP, Burns SP, Stephens BB, Lenschow DH, Campos T, Watt AS (2010) A multiscale and multidisciplinary investigation of ecosystem-atmosphere CO2 exchange over the Rocky Mountains of Colorado. Bull Amer Meteor Soc 91:209–230

    Article  Google Scholar 

  • Sutton OG (1953) Micrometeorology: a study of physical processes in the lowest layers of the earth's atmosphere. McGraw-Hill, New York, p 333

    Google Scholar 

  • Sykes RI (1978) Stratification effects in boundary layer flow over hills. Proe R Soc Lond A 361:225–243

    Article  Google Scholar 

  • Taylor PA, Teunissen HW (1987) The Askervein hill project: overview and background data. Bound Layer Meteor 39:15–39

    Article  Google Scholar 

  • Tennekes T, Lumley JL (1972) A first course in turbulence. MIT, Cambridge, MA, p 300

    Google Scholar 

  • Turnipseed AA, Anderson DE, Blanken PD, Baugh WM, Monson RK (2003) Airflows and turbulent flux measurements in mountainous terrain Part 1. Canopy and local effects. Agr For Meteor 119:1–21

    Article  Google Scholar 

  • von Kármán T (1930) Mechanische ähnlichkeit and yurbulenz. Nachr Ges Wiss Göettingen Math Phys Kl 68:58–76

    Google Scholar 

  • Wang W (2010) The influence of topography on single-tower-based carbon flux measurements under unstable conditions: a modeling perspective. Theor Appl Climatol 99:125–138

    Article  Google Scholar 

  • Wang W (2012) An analytical model for wind profiles in sparse canopies. Boundary Layer Meteorol 142(3):383–399. doi:10.1007/s10546-011-9687-0

  • Wang W, Rotach M (2010) Flux footprints over an undulating surface. Boundary Layer Meteorol 136(2):325–340. doi:10.1007/s10546-010-9498-8

    Article  Google Scholar 

  • Wang W, Davis K (2008) A numerical study of the influence of a clearcut on eddy-covariance flux measurements above a forest. Agr For Meteor 148:1488–1500. doi:10.1016/j.agrformet.2008.05.009

    Article  Google Scholar 

  • Wang W, Davis K, Cook BD, Yi C, Bakwin PS, Butler MP, Ricciuto DM (2005) Surface layer CO2 budget and advective contributions to measurements of net ecosystem-atmosphere exchange of CO2. Agr For Meteor 135:202–214

    Article  Google Scholar 

  • Wang W, Davis K, Yi C, Patton E, Butler M, Ricciuto D, Bakwin PS (2007) A note on the top-down and bottom-up gradient functions over a forested site. Bound Layer Meteor 124:305–314

    Article  Google Scholar 

  • Wang W, Davis K, Ricciuto DM, Butler MP, Cook BD (2006) Decomposing CO2 fluxes measured over a mixed ecosystem at a tall tower and extending to a region: a case study. J Geophys Res 111:G02005. doi:10.1029/2005JG000093

    Article  Google Scholar 

  • Wood N (2000) Wind flow over complex terrain: a historical perspective and the prospect for large-eddy modelling. Bound Layer Meteor 96:11–32

    Article  Google Scholar 

  • Wolfe GM, Thornton JA (2011) The Chemistry of Atmosphere-Forest Exchange (CAFE) model—part 1: model description and characterization. Atmos Chem Phys 11:77–101. doi:10.5194/acp-11-77-2011

    Article  Google Scholar 

  • Wolfe GM, Thornton JA, McKay M, Goldstein AH (2011) Forest-atmosphere exchange of ozone: sensitivity to very reactive biogenic VOC emissions and implications for in-canopy photochemistry. Atmos Chem Phys Discuss 11:13381–13424

    Article  Google Scholar 

  • Wyngaard JL (1973) On surface-layer turbulence. Workshop on micrometeorology. American Meteorological Society, Boston, pp 101–149

    Google Scholar 

  • Yi C (2008) Momentum transfer within canopies. J Appl Meteor Climatol 47:262–275

    Article  Google Scholar 

  • Yi C (2009) Instability analysis of terrain-induced canopy flows. J Atmos Sci 66:2134–2142. doi:10.1175/2009JAS3005.1

    Article  Google Scholar 

  • Yi C, Anderson DE, Turnipseed AA, Burns SP, Sparks J, Stannard D, Monson KR (2008) The contribution of advective fluxes to net ecosystem CO2 exchange in a high-elevation, subalpine forest ecosystem. Ecol Appl 18(6):1379–1390

    Article  Google Scholar 

  • Yi C, Monson RK, Zhai Z, Anderson DE, Lamb B, Allwine G, Turnipseed AA, Burns SP (2005) Modeling and measuring the nighttime drainage flow in a high-elevation, subalpine forest with complex terrain. J Geophys Res 110:D22303. doi:10.1029/2005JD006282

    Article  Google Scholar 

Download references


This work was financially supported by the National Science Foundation under grant no. ATM-0930015. The first author is supported by the National Natural Science Foundation of China under grant no. 41075039. The authors thank Drs. Ian Harman and John Finnigan from CSIRO for help with the calculation of wind field using the FB04 model. The authors also thank Christine Ramadhin for valuable comments on the manuscript.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Chuixiang Yi.



1.1 Wind over a flat forested surface

Over a flat forested surface, the turbulent stress in the layer above the canopy (Fig. 1a) is governed (Sutton 1953; Wyngaard 1973; Yi 2008) by

$$ \frac{{\partial {\tau_B}}}{{\partial z}} \approx 0, $$

where \( {\tau_B} = - \overline {u\prime w\prime } \) and is the kinematic turbulent stress, u′ and w′ are the fluctuations of velocity components in the horizontal and vertical, respectively. The mixing length model, in which

$$ {\tau_B}(z) = {\left( {\kappa \left( {z + d} \right)\frac{{\partial {U_B}}}{{\partial z}}} \right)^2}, $$

is valid in this layer, where d is the displacement height associated with the canopy and the origin of the vertical coordinate is taken at the canopy top. Based on (23), the mean velocity profile, U B , can be derived from (22), i.e,

$$ {U_B}(z) = \frac{{{u_{*}}}}{\kappa }\ln \left( {\frac{{z + d}}{{{z_0}}}} \right), $$

where, z 0 is the roughness length of the canopy, u * is the friction velocity.

Within the canopy layer (Fig 1a, z = −h to 0, where h is the canopy depth), according to the hypothesis in Yi (2008), the governing equation of the kinematic turbulent stress for a dense canopy can be given by

$$ \frac{{\partial {\tau_B}(z)}}{{\partial z}} = {F_d} \approx a(z){\tau_B}(z), $$

where a(z) is LAD and F d is the drag forcing exerted by the canopy. In Eq. 25, we have assumed that the drag on flow with a dense canopy is attributed largely to canopy elements except near the ground, i.e.,

$$ {\tau_B}(z) = {C_d}(z)U_B^2(z), $$

where, C d (z) is a bulk drag coefficient and is a function of height and canopy morphology. The analytical solution of the kinematic turbulent stress derived from Eq. 25 is

$$ {\tau_B}(z) = {\tau_B}(0){e^{{ - \left( {LAI - L(z)} \right)}}}, $$

where, \( {\tau_B}(0) = - \overline {u\prime w\prime } (0) = u_{*}^2 \) is the kinematic turbulent stress at the canopy top, LAI is the leaf area index, and

$$ L(z) = \int_{{ - h}}^z {a\left( {z\prime } \right)} dz\prime, $$

is the cumulative leaf area per unit ground area below height z. Equation 27 indicates that the turbulent stress can be predicted by LAD profile alone, which is in excellent agreement with observations (Yi 2008).

The mean wind profile within the canopy can be derived from Eqs. 26 and 27 as

$$ {U_B}(z) = {U_h}{\left( {\frac{{{C_d}(0)}}{{{C_d}(z)}}} \right)^{{\frac{1}{2}}}}{e^{{ - \frac{1}{2}\left( {LAI - L(z)} \right)}}}, $$

where, U h is the wind speed at the top of canopy.

Assuming that mean wind velocity and shear stress are continuous at the canopy top (z = 0), i.e., wind speed and its derivative with respect to z from (29) at the canopy top are equal to those from (24), and the shear stress from (27) at z = 0 is equal to that from (23), we have

$$ u_{*}^2 = {C_d}(0)U_h^2, $$
$$ {U_h} = \frac{{{u_{ * }}}}{\kappa }\ln \left( {\frac{d}{{{z_0}}}} \right), $$
$$ d = \frac{{2\sqrt {{{C_d}(0)}} }}{{\kappa \left[ { - \frac{1}{{{C_d}(0)}}\frac{{\partial {C_d}(0)}}{{\partial z}} + a(0)} \right]}}, $$
$$ {z_0} = d\exp \left( { - \frac{\kappa }{{\sqrt {{{C_d}(0)}} }}} \right) $$

If both a(z) and C d (z) are constant, Eqs. 32 and 33 are reduced to those in Eq. 6 of FB04.

Wind above the canopy over a gentle hill

As in FB04, the shape of a sinusoidal hill (Fig. 1b) is described in the rectangular coordinate system (X, Z) as

$$ {Z_s} = \frac{1}{2}H\cos \left( {kX} \right) - h $$

where, Z s is the surface height, H is the hill height, k is equal to π/(2L h ), L h is the hill half-length.

To obtain an analytical solution, two assumptions about the prescribed hill are made. First, the hill slope is sufficiently low, and perturbations to the background wind (U B) above the canopy can be solved with linearized equations. Second, the hill is long enough. This means that L h should be greater than 2L c (Poggi et al. 2008), where L c is a canopy adjustment length scale which is equal to 1/(C d 0 a 0), C d 0 and a 0 are the characteristic values for the canopy drag coefficient and LAD, respectively. In this case, the advection terms in the momentum equation may be negligible for a dense canopy over gentle terrain. This assumption has been supported by numerical experiments (Ross and Vosper 2005).

The same displaced coordinate system as in FB04 is used. The displaced (x, z) and the rectangular (X, Z) coordinate systems are related by,

$$ x = X + \frac{H}{2}\sin \left( {kX} \right){e^{{ - kZ}}}, $$
$$ z = Z - \frac{H}{2}\cos \left( {kX} \right){e^{{ - kZ}}}. $$

with this displaced (streamline) coordinate system, extra terms appear in the momentum equations (compared with those in a rectangular coordinate system), which are O(H 2 /L h 2) or smaller and, hence, may be negligible for the low slope hill (see FB04 for details). As a result, the streamwise (x direction) momentum equation can be written as,

$$ u\frac{{\partial u}}{{\partial x}} + w\frac{{\partial u}}{{\partial z}} = - \frac{{\partial p}}{{\partial x}} + \frac{{\partial \tau }}{{\partial z}} $$

where, u and w are the wind components in the x and z directions; receptively, p is the kinematic pressure, τ is the kinematic turbulent shear stress above the canopy, which is parameterized using the mixing length theory.

Under neutral conditions, the pressure perturbation in the inner region induced by the gentle sinusoidal hill is represented by

$$ \Delta p(x) = - \frac{1}{2}U_0^2Hk\exp \left( {ikx} \right), $$

(Jackson and Hunt 1975; Finnigan and Belcher 2004) and the horizontal PG forcing, driving the flow throughout the depth of the inner region (and canopy), is

$$ {\text{PG}} = {\rm Re} \left( {\frac{{\partial \Delta p}}{{\partial x}}} \right) = \frac{1}{2}U_0^2H{k^2}\sin \left( {kx} \right), $$

where, U 0 is the characteristic wind velocity in the outer region and is estimated as the background wind at the middle layer height h m . According to Hunt et al. (1988a, b), h m is given by,

$$ \frac{{{h_m}}}{{{L_h}}}{\left( {\ln \left( {{h_m}/{z_0}} \right)} \right)^{{1/2}}} = 1, $$

provided that L h is less than the boundary layer depth. The height of the inner region, h i , is defined by,

$$ \frac{{{h_i}}}{{{L_h}}}\ln \left( {{h_i}/{z_0}} \right) = 2{\kappa^2}. $$

Assuming that the wind perturbation induced by terrain is small compared to the background wind (i.e., wind over the corresponding flat surface), Eq. 37 can be linearized. The resulting approximate solution for the streamwise velocity in the inner region above the canopy is,

$$ u\left( {x,z} \right) = {U_B}(z) + \Delta u\left( {x,z} \right), $$


$$ \Delta u(x,z) = {\rm Re} \left\{ { - \frac{{\Delta p(x)}}{{{U_B}({h_i})}}\left[ {1 + \delta \left( {1 - \ln (\frac{{z + d}}{{{h_i}}}) - c{K_0}(2\sqrt {{ik{L_h}\frac{{z + d}}{{{h_i}}}}} )} \right)} \right]} \right\}, $$

δ = 1/ln(h i /z 0), and K 0 is the modified Bessel function of order zero.The turbulent stress is,

$$ \tau \left( {x,z} \right) = {\tau_B} \left( {z} \right) + \Delta \tau \left( {x, z} \right), $$


$$ \Delta \tau \left( {x,z} \right) = 2\kappa {u_{ * }}\left( {z + d} \right)\frac{{\partial \Delta u\left( {x,z} \right)}}{{\partial z}}. $$

The integration constant c is determined by coupling (42) and (43) to the solutions for flow within the canopy at z = 0 (canopy top). Assuming that turbulent stress and velocity are continuous at z = 0, respectively, we have,

$$ {C_d}(0){\left[ {{U_B}(0) + \Delta u\left( {x,0} \right)} \right]^2} = {\tau_B}(0) + \Delta \tau \left( {x,0} \right). $$

It is noticed that the exact value of constant c (that should be independent of position x and z) may not be achieved because the wind speed perturbation above the canopy is linear in PG (a function of x) while it is nonlinear within the canopy. This is due to different simplifications of the governing equation above and within the canopy. An approximate solution is provided here. Since the velocity perturbation is small compared with U B, the left side of the above equation can be approximated as \( {C_d}(0)\left[ {U_B^{{\,2}}(0) + 2{U_B}(0)\Delta u\left( {x,0} \right)} \right] \). Substituting (23), (24), (43), and (45) into (46), we have,

$$ c = \frac{{ - {C_d}(0){U_B}(0){U_B}\left( {{h_i}} \right)\left[ {1 + 1{\text{n}}\left( {{h_i}/{z_0}} \right) - 1{\text{n}}\left( {d/{h_i}} \right)} \right] - u_{ * }^21{\text{n}}\left( {{h_i}/{z_0}} \right)}}{{u_{ * }^2d\left( {{{{\partial {K_0}(g)}} \left/ {{\partial z}} \right.}} \right)1{\text{n}}\left( {{{{{h_i}}} \left/ {{{z_0}}} \right.}} \right) - {C_d}(0){U_B}(0){U_B}\left( {{h_i}} \right){K_0}(g)}}, $$

where \( g = 2\sqrt {{{\text{ik}}{L_h}\frac{{z + d}}{{{h_i}}}}} \). With the above approximate value of c, the resulting vertical profiles of wind and turbulent stress are approximately continuous but may not be smooth at the canopy top in some locations (i.e., their first derivatives with respect to z are not continuous at z = 0).

C d near the ground

For the no-canopy case under a neutrally stratified atmosphere, the drag coefficient is given by

$$ {C_d}(z) = {\left[ {\frac{\kappa }{{1{\text{n}}\left( {{{z} \left/ {{{z_{{g0}}}}} \right.}} \right)}}} \right]^2}, $$

where, z g0 is the roughness length of the ground. Equation 48 indicates that C d is infinite on the ground and decreases dramatically with height near the ground. Variations in C d are smaller at higher levels. For example, variation in C d is smaller than 0.03 for z between 10z g0 and 103 z g0. To account for significant variations in C d near the ground, where the drag effect exerted by the ground is superior to that by canopy, we assume that C d follows (48) below a level z L . Thus, we can rewrite (48) as

$$ {C_d}(z) = {C_d}\left( {{z_L}} \right){\left( {\frac{{1{\text{n}}\left( {{{{{z_L}}} \left/ {{{z_{{g0}}}}} \right.}} \right)}}{{1{\text{n}}\left( {{{z} \left/ {{{z_{{g0}}}}} \right.}} \right)}}} \right)^2}, $$

where, C d (z L )is the drag coefficient at z L and z g0 is taken to be 0.1 m in the study.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, W., Yi, C. A new nonlinear analytical model for canopy flow over a forested hill. Theor Appl Climatol 109, 549–563 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: