THE PARAMETERIZATION OF STABLE BOUNDARY LAYERS BASED ON CASES-99

1. THE PARAMETERIZATION OF STABLE BOUNDARY LAYERSBASED ON CASES-99 Zbigniew Sorbjan Marquette University, Milwaukee

2. Introduction: z Convective BL Gradient-based scaling: i Sw(z ) = w = w Sh (z ) = w / (i) 1/2 S(z) = Sh i Hi  IL and SBL are both - turbulent - stably stratified - with negative heat fluxes - with wind shear - with waves  Ho Ho Stable BL

3. Can the gradient-based scaling of the Interfacial Layer be applied in the Stable BL?

4. in the Stable BL:

5. CASES-99 Laminar Intermittent Turbulent After Steeneveld et al

6. CASES-99 Turbulent Intermittent Laminar

7. In the stable boundary layer, the Monin-Obukhov similarity predictions are: dT/dz ~ T*/L* ~ Ho2/o2 dq/dz ~ q*/L* ~ HoQo/o2 dU/dz ~ u*/L* ~ Ho/o • Note 1:”the z-less regime" • Note 2: the fluxes H, Q,  can be z-dependent (local scaling) • Note 3: the gradient Richardson number is constant and sub-critical: Ri = ( dT/dz) / (dU/dz) 2 ~  T* L* /u*2 ~ 1. Note 4: for Ho~ 0, o ~ 0 --> d/dz ~ b  Ho/o )2 ~ 0/0.

8. Note 5:

9. Consequently, an alternate set of governing parameterscan be considered: • thescalargradients: dT/dz, dq/dz, dU/dz, • the vertical velocity variance: w2 • the buoyancy parameter: g The 5 parameters above involve 4 independent units [m, s, K, kg]. Based on Buckingham's P-theorem, the following 4 local scales can be derived: un(z ) = w Ln (z ) = w / (dT/dz) 1/2 Tn(z) = Ln dT/dz qn (z) = Ln dq/dz and one non-dimensional parameter: the gradient Richardson number • Ri = (dT/dz)/(dU/dz)2

10. Buckingham's P-theorem:

11. A comparison: Flux-based scaling: U*(z ) = w1/2 L* (z ) = w3/2/ [H(z)] T*(z) = H(z)/U*(z) q*(z) = Q(z)/U*(z) Gradient-based scaling: un(z ) = w Ln (z ) = w / (dT/dz) 1/2 Tn(z) = Ln dT/dz qn (z) = Ln dq/dz in a “z-less” regime

12. Drawbacks of the M-O scaling in stable conditions: • (1) The M-O similarity is invalid, when the Richardson number varies • with height, and also when Ri is outside of the critical limit. • (2) The M-O similarity fails in theintermittent case, whenthe fluxes are small. • (3) When turbulence is weak, the fluxes are strongly contaminated by errors. • (4) The M-O similarity introduces self-correlation errors, i.e., the scaled variables andthe stability parameter z/L* depend on surface fluxes. • (5) Fluxes and variances can be influenced by non-turbulent motions, which do not follow the Monin-Obukhov scaling laws.

13. Advantages of the gradient-based scaling: (1) The velocity scale Un, defined by the vertical velocity variance, is less sensitive to sampling problems,compared to the flux-based scale. (2) The velocity scale Un is more robust, because the vertical velocity variance is relatively less sensitive to the choice of an averaging time-scale, and its probability distribution is nearly independent of Ri (e.g., Mahrt and Vickers, 2005). (3) The length scale Ln does not inherit the difficulty of measuring fluxes. (4) The measurements of temperature and humidity are more accurate than the evaluation of their fluxes (even though an appropriate calculation of gradients requires a sufficient vertical resolution of observations).

14. (5) Effectsofnon-stationarity and multiple layers within the SBL can be included and parametrically expressed in terms of the Richardson number Ri, which can vary with height, and can be larger than Ric. (6) The evaluation of the SBL height h is irrelevant in this approach. (7) The gradient-based scaling is valid in the stably-stratified interfacial layer above the CBL (Sorbjan, 2005 a, b, c).

15. EXPERIMENTAL DATA - CASES-99:source: Mahrt, L. and D. Vickers, 2005: "Extremely weak mixing in stable conditions” (to appear in Bound.- Layer. Meteor.)

16. Two composite cases: W- very WEAK turbulence (strongly stable case) S- STRONG turbulence (weakly stable case) _____________________________________ Compo- No. u* T* L site case records [m s-1] [K] [m] _____________________________________ W 22 0.04 0.07 0.51 _____________________________________ S 12 0.30 0.12 19.63 ______________________________________________ • 7x 2x 40x

17. Profiles of: the potential temperature, in the composite case W (open circles) and S (filled circles). The potential temperature is the deviation from the surface value. Weaker turbulence, stronger surface cooling

18. Profiles of the wind velocity, in case W (open circles) and case S (filled circles). Stronger turbulence, stronger wind

19. Profiles of the temperature flux, in case W (open circles) and case S (filled circles). 10x

20. Profiles of the Reynolds stress, in case W (open circles) and case S (filled circles). 100x

21. Profiles of the temperature variance, in case W (open circles) and case S (filled circles).

22. Profiles of the Richardson number Ri, in case W (open circles) and case S (filled circles). Overcritical region

23. Gradient-based Scales:

24. Profiles of length scales Ln, in case W (open circles) and case S (filled circles).

25. Profiles of temperature scales Tn, in case W (open circles) and case S (filled circles).

26. Profiles of velocity scales Un, in case W (open circles) and case S (filled circles).

27. Empirical similarity functions based on CASES-99

28. The dependence of the Reynolds stress on the Richardson number Ri, in case W (open circles) and case S (filled circles). W S Ric Neutral regime

29. Note: The dimensionless temperature flux in the SBL is bounded: H(z) ≥ - 0.3 UnTn = - 0.3 w2 N/, for any z and any Ri (Derrbyshire, 1990: Ho ≥ -0.14 G2 f/ -- not valid in the neutral limit)

31. Conclusions: 1) The "gradient-based" scaling produces consistent results in both, the weakly stable and very stable cases, with dimensionless parameters dependent on the Richardson number. 2) The "gradient-based" scaling is better suited for practical applications. It requires vertical profiles of wind and temperature, and also the vertical velocity variance. These characteristics of turbulence can be more accurately evaluated in the SBL, than the fluxes of heat, humidity, and momentum. 3) The "gradient-based" scaling seems to provide a useful framework for examining stably-stratified shear turbulence. Its general validity is being currently tested, based on a larger amount of empirical data.

32. References: • Sorbjan, Z., 2001: An evaluation of local similarity at the top of the mixed layer based on large-eddy simulations. Bound. Layer Meteor., 101, 183-207. • Sorbjan, Z., 2004: Large-eddy simulations of the baroclinic mixed layer. Bound. Layer Meteor.,112, 57-80. • Sorbjan, Z.:2005a, "Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part I: Free convection". Bound.-Layer Meteorol. (in print) • Sorbjan, Z.:2005b, "Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part I: Forced convection". Bound.-Layer Meteorol. (in print) • Sorbjan Z., 2005c: Similarity regimes in the stably-stratified surface layer. Submitted to Boundary-Layer Meteorology, • Sorbjan Z., 2005d: Comments of "Flux-gradient relationship, self-correlation and intermittency in the stable boundary layer". Submitted to Quarterly Journal of the Royal Meteorological Society, • Sorbjan Z., 2005e: Local structure of turbulence in stably-stratified boundary layers. Submitted to Journal of the Atmospheric Sciences.