Skip to main content
SpringerLink
Log in

Direct Solution of Various Micrometeorological Problems via the Lambert-W Function

  • Notes and Comments
  • Published:
Boundary-Layer Meteorology Aims and scope Submit manuscript

Abstract

For a family of relatively common problems in micrometeorology, whose forms previously required iterative or indirect treatment, analytical solution is possible via the Lambert-W function. Use of this function reduces computational time and the need for approximations, while offering a clearer view of the character and solution of such problems. This note provides a generalized form for such problems, and demonstrates its use for three applications: internal boundary-layer growth due to a change in surface roughness; flow displacement over a canopy; and determination of friction velocity and roughness length over the sea.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. In computational software (e.g., the ProductLog function in Mathematica or scipy.special.lambertw in python) the branch index is given as an optional argument. Note \(\mathcal {W}(x)\) is available in many programming languages and analysis software, to high precision and via lookup table  (Johansson 2019).

  2. Depending on the particular IBL model, the characteristic \(z_0\) in Eq. 6 may be the downwind or upwind roughness, or some functional combination of the two. Some models take \(c_r\) or \(c_\ell \) to be other than 0 or 1; see Table 1 of Savelyev and Taylor Savelyev and Taylor (2005) for model formulae and values of \(\{C,c_\ell ,c_r\}\).

  3. These values of \(c_\ell \) and \(c_r\) have been used for decades; in wind engineering, for example, \(c_\ell =1\) and \(C=0.9\) following (Troen and Petersen 1989). The value of \(c_r\) taken by various parametrizations is irrelevant for \(x\gg z_0\), unless \(c_r\gg C\) (as in Raabe (1983),where \(c_r\) is the ratio of upwind and downwind roughness lengths).

  4. Previously in Sogachev and Kelly (2016) I suggested a \(\mathcal {W}\) solution for displaced flow over a forest canopy, but that included an error; Eq. 3 in that work is missing a negative sign and exponent of \(-1\), and should be replaced by Eq. 9.

  5. The value of \(z_0=0.2\) mm coincidentally corresponds to the maximum value of the real part of (13) for \(U<0\), though use of the Charnock form for wave-driven winds is generally not recommended (and such adaptation is beyond the scope of this note).

References

  • Bou-Zeid E, Meneveau C, Parlange M (2004) Large-eddy simulation of neutral atmospheric boundary layer flow over heterogeneous surfaces: blending height and effective surface roughness. Water Resour Res 40(2):W025, 051-18

    Article  Google Scholar 

  • Calaf M, Meneveau C, Meyers J (2010) Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys Fluids 22(015):110

    Google Scholar 

  • Charnock H (1955) Wind stress on a water surface. Q J R Meteorol Soc 81:639–644

    Article  Google Scholar 

  • Corless R, Gonnet G, Hare D, Jeffrey D, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5(4):329–359. https://doi.org/10.1007/BF02124750

    Article  Google Scholar 

  • Hayes B (2005) Why W? Am Sci 93(2):104–108

    Article  Google Scholar 

  • Jackson PS (1976) The propagation of modified flow downstream of a change in roughness. Q J R Meteorol Soc 102:924–933

    Google Scholar 

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

    Article  Google Scholar 

  • Johansson F (2019) Computing the Lambert W function in arbitrary-precision complex interval arithmetic. Numer Algorithm. https://doi.org/10.1007/s11075-019-00678-x

    Article  Google Scholar 

  • Miyake M (1965) Transformation of the atmospheric boundary layer over inhomogeneous surfaces. Dept. Atmos. Sci., Univ. Washington, Seattle, WA, USA, PhD thesis AD 626630

  • Panofsky H, Dutton J (1984) Atmospheric turbulence. Wiley, Hoboken

    Google Scholar 

  • Raabe A (1983) On the relation between the drag coefficient and fetch above the sea in the case of off-shore wind in the near-shore zone. Z Fur Meteorol 33(6):363–367

    Google Scholar 

  • Rotach MW (1995) Profiles of turbulence statistics in and above an urban street canyon. Atmos Environ 29(13):1473–1486. https://doi.org/10.1016/1352-2310(95)00084-C

    Article  Google Scholar 

  • Savelyev SA, Taylor PA (2001) Notes on internal boundary-layer height formula. Boundary-Layer Meteorol 101:293–301

    Article  Google Scholar 

  • Savelyev SA, Taylor PA (2005) Internal boundary layers: I. Height formulae for neutral and diabatic flows. Bound-Layer Meteorol 115(1):1–25. https://doi.org/10.1007/s10546-004-2122-z

    Article  Google Scholar 

  • Sogachev A, Kelly M (2016) On displacement height, from classical to practical formulation: stress, turbulent transport and vorticity considerations. Boundary-Layer Meteorol 158(3):361–381. https://doi.org/10.1007/s10546-015-0093-x

    Article  Google Scholar 

  • Thom A (1971) Momentum absorption by vegetation. Q J R Meteorol Soc 97:414–428

    Article  Google Scholar 

  • Troen I, Petersen EL (1989) European wind atlas. Risø National Laboratory, Roskilde

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Wind Energy, Risø Lab / Campus, Danish Technical University [DTU], Frederiksborgvej 399, Building 125, Roskilde, 4000, Denmark

    Mark Kelly

Authors
  1. Mark Kelly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Kelly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kelly, M. Direct Solution of Various Micrometeorological Problems via the Lambert-W Function. Boundary-Layer Meteorol 179, 163–168 (2021). https://doi.org/10.1007/s10546-020-00592-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10546-020-00592-z

Keywords

Search

Navigation