1. Filtered eqs And Turbulence
Chp 3
Filtered eqs
And Turbulence

2. Why filter eqs? The governing eqs permit motions on all scales:
Planetary scales: 1000’s of km
Cb’s: 10km
Turbulent eddies:100’s of m
Viscous eddies: 1mm
A numerical model cannot cover the entire range of scales

3. What is turbulence?
Difficult to define but you know it when you see it or feel it.
Turbulent flows have large Reynold’s numbers

4. Fig. 3.1. Photograph of a cumulonimbus cloud showing turbulent eddies of many scales.

15. First-order Turbulence Closure

22. The TKE eq

28. The Reynold’s Stress eq

34. Scalar turbulent covariance eqs

36. LES

52. Subgrid partial cloudiness

59. Fig Estimated probability density functions for  (left) and rv (right), superimposed on a dashed curve of the standard normal density. Data are from 28 July for a flight leg over South Park, Colorado, at a height of 50 m off the surface, starting at 1152 MDT. Averaging interval was 1000 m. [From Banta (1979).]

60. Fig Estimated probability density function for rt – rs superimposed on standard normal curve (dashed line). Data are from the same flight leg as in Fig [From Banta (1979).]

62. Fig Histograms of t from 3-D data at three levels inside the cloud layer. The three theoretical models have also been plotted: (a) 550 m, (b) 1000 m, and (c) 1250 m. [From Bougeault (1981a).]

63. Fig One- and two-dimensional PDFs from a flight through a stratocumulus cloud. Shown in the lower-right corner are the flight-level data. The dotted line in the qt - l panel corresponds to saturation (i.e., points above are saturated, points below are sub-saturated). [From Larson et al. (2002).]

64. Fig As in Fig.3.7 but from a flight through a field of cumulus clouds. [From Larson et al. (2002).]

65. Fig. 3. 9. A double-Gaussian analytic fit to the data shown in Fig. 3
Fig A double-Gaussian analytic fit to the data shown in Fig The arrow in the top-right panel denotes a delta function. [From Larson et al. (2002).]