Abstract
The empirical dependence of turbulence Prandtl number (Pr) on gradient Richardson number (Ri) is presented, derived so as to avoid the effects of self-correlation from common variables. Linear power relationships between the underlying variables that constitute both Pr and Ri are derived empirically from flux and profile observations. Pr and Ri are then reconstructed from these power laws, to indicate their interdependence whilst avoiding self-correlation. Data are selected according to the stability range prior to regression, and the process is iterated from neutral to higher stability until error analysis indicates the method is no longer valid. A Butterworth function is fitted to the resulting Pr −1(Ri) regression to give an empirical summary of the analysis. The form suggests that asymptotically Pr −1 decreases as Ri 3/2. Scatter in the data increases above Ri ~ 1, however, indicating additional constraints to Pr are not captured by Ri alone in this high stability regime. The Butterworth function is analytic for all Ri > 0, and may be included in suitable boundary-layer parameterisation schemes where the turbulent diffusivity for heat is derived from the turbulent diffusivity for momentum.
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Anderson PS (2005) Ice-shelf microtopography observed using satellite thermal imagery. J Glaciol 51(175): 528–538
Andre JC, Mahrt L (1982) The nocturnal surface inversion and influence of clear-air radiative cooling. J Atmos Sci 39(4): 864–878
Baas P, Steeneveld GJ, van de Wiel BJH, Holtslag AAM (2006) Exploring self-correlation in flux-gradient relationships for stably stratified conditions. J Atmos Sci 63(11): 3045–3054
Beare RJ, MacVean MK, Holtslag AAM, Cuxart J, Esau I, Golaz JC, Jimenez MA, Khairoutdinov M, Kosovic B, Lewellen D, Lund TS, Lundquist JK, McCabe A, Moene AF, Noh Y, Raasch S, Sullivan P (2006) An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol 118(2): 247–272
Derbyshire SH (1999) Boundary-layer decoupling over cold surfaces as a physical boundary-instability. Boundary-Layer Meteorol 90(2): 297–325
Garratt JR, Brost RA (1981) Radiative cooling effects within and above the nocturnal boundary-layer. J Atmos Sci 38(12): 2730–2746
Grachev AA, Andreas EL, Fairall CW, Guest PS, Persson POG (2007) On the turbulent Prandtl number in the stable atmospheric boundary layer. Boundary-Layer Meteorol 125(2): 329–341
Hoch SW, Calanca P, Philipona R, Ohmura A (2007) Year-round observation of longwave radiative flux divergence in Greenland. J Appl Meteorol Climatol 46(9): 1469–1479
Holtslag B (2006) Preface—GEWEX atmospheric boundary-layer study (GABLS) on stable boundary layers. Boundary-Layer Meteorol 118(2): 243–246
Jones AE, Wolff EW, Salmon RA, Bauguitte SJB, Roscoe HK, Anderson PS, Ames D, Clemitshaw KC, Fleming ZL, Bloss WJ, Heard DE, Lee JD, Read KA, Hamer P, Shallcross DE, Jackson AV, Walker SL, Lewis AC, Mills GP, Plane JMC, Saiz-Lopez A, Sturges WT, Worton DR (2008) Chemistry of the Antarctic boundary layer and the interface with snow: an overview of the CHABLIS campaign. Atmos Chem Phys 8(14): 3789–3803
King JC, Anderson PS (1994) Heat and water-vapor fluxes and scalar roughness lengths over an Antarctic ice shelf. Boundary-Layer Meteorol 69(1–2): 101–121
King JC, Mobbs SD, Rees JM, Anderson PS, Culf AD (1989) The stable Antarctic boundary layer experiment at Halley Base. Weather 44: 398–405
Klipp CI, Mahrt L (2004) Flux-gradient relationship, self-correlation and intermittency in the stable boundary layer. Q J Roy Meteorol Soc 130: 2087–2103
Lee X, Massman W, Law B (2004) Handbook of micrometeorology. Kluwer Academic Publishers, Dordrecht, pp 33–66
Louis JF, Tiedtke M, Geleyn J-F (1982) A short history of the operational PBL parameterization at ECMWF. Proceedings of the ECMWF Workshop on Boundary Layer Parameterisations. November 1981, ECMWF, Reading, UK1982, City, pp 59–80
Soomere T, Zilitinkevich S (2002) Supplement to “Third-order transport due to internal waves and non-local turbulence in the stably stratified surface layer”. Q J Roy Meteorol Soc 128: 1029–1031
Steeneveld GJ, Van de Wiel BJH, Holtslag AAM (2006) Modelling the Arctic stable boundary layer and its coupling to the surface. Boundary-Layer Meteorol 118(2): 357–378
Stull RB (1988) An introduction to boundary layer meteorology. Kluwer Academic Publishers, Dordrecht, pp 499–543
Zilitinkevich S (2002) Third-order transport due to internal waves and non-local turbulence in the stably stratified surface layer. Q J Roy Meteorol Soc 128: 913–925
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Anderson, P.S. Measurement of Prandtl Number as a Function of Richardson Number Avoiding Self-Correlation. Boundary-Layer Meteorol 131, 345–362 (2009). https://doi.org/10.1007/s10546-009-9376-4
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DOI: https://doi.org/10.1007/s10546-009-9376-4