Two TKE budget field studies conducted over sites of different roughness (Frenzen & Vogel, 1992; 1997) have observed dissipation rates in neutral and unstable conditions which were 15 to 20% too small to balance the total rates of TKE production. The difference is sufficiently well defined to write
(Phi-epsilon) = 0.8[(Phi-m) - z/L] for z/L < 0 , (1)
for the dissipation function (Phi-epsilon) in unstable conditions. Further, by using the larger values of the Kolmogorov constant shown for weaker turbulence intensities by Praskovsky and Oncley (1994) and observed by Oncley et al. (1995) to re-analyze spectra recorded in stable conditions, similar but somewhat smaller deficits are implied in the stable regime. An approximate relation for the near-neutral and stable cases can therefore be written by simply increasing the constant in (1):
(Phi-epsilon) = 0.85[(Phi-m) - z/L] for z/L >= 0 . (2)
These expressions provide an improved closure for what amounts to a complete set of approximate Phi functions by substituting the concept of the near-constant dissipation deficit for the local dissipation assumption. Once rates of TKE production by wind shear above a given site have been expressed as normalized functions of z/L, the remaining terms of the budget can be written as combinations of Phi-m(z/L) and/or the independent variable itself: Phi-epsilon can be expressed as weighted sums of Phi-m and z/L as described above, while the buoyancy production (Phi-B) and turbulence transport (Phi-T) terms can be written in terms of z/L alone. Thus we have +/- Phi-B = +/- z/L by definition, while field experiments have shown Phi-T = +z/L for z/L < 0 and Phi-T = 0 for z/L > 0 to be reasonably good approximations (Wymgaard and Cote, 1971).
Together, the approximate Phi functions represent the TKE budget, with the implied imbalance term giving some indication of relative accuracy. Individually, the relations for Phi-m and Phi-epsilon could be applied to particular problems. The Phi-m functions could be used (in the usual way) in conditions for which they had been scaled, that is, above smooth sites when scaled with U*o or above rougher sites when scaled with U*max (see Frenzen & Vogel, 1997). Similarly, the new Phi-epsilon relations could be used in short-range diffusion models.
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REFERENCES
Frenzen, P. and Vogel, C. A. : 1992a, 'The TKE Budget in the
Atmospheric Surface Layer: A Review and Reexamination in
the Field', Boundary-Layer Meteorol. 60, 49-76.
Frenzen, P., and Vogel, C. A. : 1997, 'Scaling the TKE Budget above a Roughness Sublayer', in 12th Symp. on Boundary Layers and
Turbc., Vancouver, BC, Canada, July, 1997, pp. 279-280
Oncley, S. P., Horst, T. W., Praskovsky, A., and Wilczak, J. M.:
1995, 'The TKE Budget from the FLAT Experiment', in 11th Symp. on
Boundary Layers and Turbc.', Charlotte, NC, Mar. 1995, pp. 5-8.
Praskovsky, A., and Oncley, S. P.: 1994, 'Measurements of the
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Reynolds Numbers', Phys. Fluids 6, 2886-2888.
Vogel, C. A., and Frenzen, P. : 1992, 'The TKE Budget in the
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Transport Terms', in 10th Symp. on Turbc. and Diffusion,
Portland, OR, Sept. 1992, pp. 161-164.
Wyngaard, J. C., and Coté, O. R. : 1971, 'The Budgets of TKE and
Temperature Variance in the Atmospheric Surface Layer', J. Atmos.
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