Abstract

In Wyngaard et al., 1971, a simple model was proposed to estimate Cn2 in the atmospheric surface layer, which only requires routine meteorological information (wind speed and temperature) as input from two heights. This Cn2 model is known to have satisfactory performance in unstable conditions; however, in stable conditions, the model only covers a relatively short range of atmospheric stabilities which significantly limits its applicability during nighttime. To mitigate this limitation, in this study we construct a new Cn2 model utilizing an extensive turbulence dataset generated by a high-fidelity numerical modeling approach (known as direct numerical simulation). The most distinguishing feature of this new Cn2 model is that it covers a wide range of atmospheric stabilities including the strongly stratified (very stable) conditions. To validate this model, approximately four weeks of Cn2 data collected at the Mauna Loa Observatory, Hawaii are used for comparison, and reasonably good agreement is found between the observed and estimated values.

© 2016 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Vol. 152 (SPIE, 2005).
    [Crossref]
  2. A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151), 163–187 (1954).
  3. R. R. Beland, Infrared and Electro-Optical Systems Handbook, Vol. 2 (SPIE, 1993).
  4. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic Publishers, 1988).
    [Crossref]
  5. P. S. Arya, Introduction to Micrometeorology (Academic, 2001).
  6. J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
    [Crossref]
  7. A. Dyer, “A review of flux-profile relationships,” Bound.-Layer Meteor. 7, 363–372 (1974).
    [Crossref]
  8. J. C. Wyngaard, Y. Izumi, A. Stuart, and J. R. Collins, “Behavior of the refractive-index-structure parameter near the ground,” J. Opt. Soc. Am. 61, 1646–1650 (1971).
    [Crossref]
  9. M. Wesely and E. Alcaraz, “Diurnal cycles of the refractive index structure function coefficient,” J. Geophys. Res. 78, 6224–6232 (1973).
    [Crossref]
  10. K. Davidson, G. Schacher, C. Fairall, and A. Goroch, “Verification of the bulk method for calculating overwater optical turbulence,” Appl. Opt. 20, 2919–2924 (1981).
    [Crossref] [PubMed]
  11. K. E. Kunkel and D. L. Walters, “Modeling the diurnal dependence of the optical refractive index structure parameter,” J Geophys. Res. 88, 10,999–11,004 (1983).
    [Crossref]
  12. E. L. Andreas, “Estimating Cn2 over snow and sea ice from meteorological data,” J. Opt. Soc. Am. A 5, 481–495 (1988).
    [Crossref]
  13. P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
    [Crossref]
  14. V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
    [Crossref]
  15. A. Tunick, “Cn2 model to calculate the micrometeorological influences on the refractive index structure parameter,” Environ. Model. Softw. 18, 165–171 (2003).
    [Crossref]
  16. J. Wyngaard and O. Coté, “The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer,” J. Atmos. Sci. 28, 190–201 (1971).
    [Crossref]
  17. Z. Sorbjan, “Gradient-based scales and similarity laws in the stable boundary layer,” Q. J. R. Meteorol. Soc. 136, 1243–1254 (2010).
  18. A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
    [Crossref]
  19. P. Moin and K. Mahesh, “Direct numerical simulation: a tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539–578 (1998).
    [Crossref]
  20. P. He and S. Basu, “Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: A direct numerical simulation approach,” Environ. Fluid Mech. 16, 373–399 (2015).
    [Crossref]
  21. P. He, “A high order finite difference solver for massively parallel simulations of stably stratified turbulent channel flows,” Comput. Fluids 127, 161–173 (2016).
    [Crossref]
  22. S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
    [Crossref]
  23. “Mauna loa seeing study. http://www.eol.ucar.edu/isf/projects/MLO_CN2/ ,” (2006).
  24. S. Oncley and T. Horst, “Calculation of Cn2 for visible light and sound from CSAT3 sonic anemometer measurements,” avaiable at https://www.eol.ucar.edu/system/files/files/field_project/DASH04/background.pdf (2004).
  25. R. B. Stull, Meteorology for Scientists and Engineers (Brooks Cole, 1999).

2016 (1)

P. He, “A high order finite difference solver for massively parallel simulations of stably stratified turbulent channel flows,” Comput. Fluids 127, 161–173 (2016).
[Crossref]

2015 (1)

P. He and S. Basu, “Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: A direct numerical simulation approach,” Environ. Fluid Mech. 16, 373–399 (2015).
[Crossref]

2013 (1)

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

2010 (1)

Z. Sorbjan, “Gradient-based scales and similarity laws in the stable boundary layer,” Q. J. R. Meteorol. Soc. 136, 1243–1254 (2010).

2003 (1)

A. Tunick, “Cn2 model to calculate the micrometeorological influences on the refractive index structure parameter,” Environ. Model. Softw. 18, 165–171 (2003).
[Crossref]

2000 (1)

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

1998 (1)

P. Moin and K. Mahesh, “Direct numerical simulation: a tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539–578 (1998).
[Crossref]

1995 (1)

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

1988 (1)

1983 (1)

K. E. Kunkel and D. L. Walters, “Modeling the diurnal dependence of the optical refractive index structure parameter,” J Geophys. Res. 88, 10,999–11,004 (1983).
[Crossref]

1981 (1)

1974 (1)

A. Dyer, “A review of flux-profile relationships,” Bound.-Layer Meteor. 7, 363–372 (1974).
[Crossref]

1973 (1)

M. Wesely and E. Alcaraz, “Diurnal cycles of the refractive index structure function coefficient,” J. Geophys. Res. 78, 6224–6232 (1973).
[Crossref]

1971 (3)

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

J. Wyngaard and O. Coté, “The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer,” J. Atmos. Sci. 28, 190–201 (1971).
[Crossref]

J. C. Wyngaard, Y. Izumi, A. Stuart, and J. R. Collins, “Behavior of the refractive-index-structure parameter near the ground,” J. Opt. Soc. Am. 61, 1646–1650 (1971).
[Crossref]

1954 (1)

A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151), 163–187 (1954).

1951 (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[Crossref]

Alcaraz, E.

M. Wesely and E. Alcaraz, “Diurnal cycles of the refractive index structure function coefficient,” J. Geophys. Res. 78, 6224–6232 (1973).
[Crossref]

Andreas, E. L.

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

E. L. Andreas, “Estimating Cn2 over snow and sea ice from meteorological data,” J. Opt. Soc. Am. A 5, 481–495 (1988).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Vol. 152 (SPIE, 2005).
[Crossref]

Arya, P. S.

P. S. Arya, Introduction to Micrometeorology (Academic, 2001).

Basu, S.

P. He and S. Basu, “Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: A direct numerical simulation approach,” Environ. Fluid Mech. 16, 373–399 (2015).
[Crossref]

Beland, R. R.

R. R. Beland, Infrared and Electro-Optical Systems Handbook, Vol. 2 (SPIE, 1993).

Bendall, C. S.

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

Bradley, E. F.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

Businger, J. A.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

Collins, J. R.

Corrsin, S.

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[Crossref]

Coté, O.

J. Wyngaard and O. Coté, “The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer,” J. Atmos. Sci. 28, 190–201 (1971).
[Crossref]

Czekits, C.

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Davidson, K.

Davidson, K. L.

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

Dirmhirn, I.

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Dyer, A.

A. Dyer, “A review of flux-profile relationships,” Bound.-Layer Meteor. 7, 363–372 (1974).
[Crossref]

Fairall, C.

Fairall, C. W.

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

Frederickson, P. A.

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

Goroch, A.

Grachev, A. A.

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

Guest, P. S.

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

He, P.

P. He, “A high order finite difference solver for massively parallel simulations of stably stratified turbulent channel flows,” Comput. Fluids 127, 161–173 (2016).
[Crossref]

P. He and S. Basu, “Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: A direct numerical simulation approach,” Environ. Fluid Mech. 16, 373–399 (2015).
[Crossref]

Izumi, Y.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

J. C. Wyngaard, Y. Izumi, A. Stuart, and J. R. Collins, “Behavior of the refractive-index-structure parameter near the ground,” J. Opt. Soc. Am. 61, 1646–1650 (1971).
[Crossref]

Karipot, A.

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Kunkel, K. E.

K. E. Kunkel and D. L. Walters, “Modeling the diurnal dependence of the optical refractive index structure parameter,” J Geophys. Res. 88, 10,999–11,004 (1983).
[Crossref]

Mahesh, K.

P. Moin and K. Mahesh, “Direct numerical simulation: a tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539–578 (1998).
[Crossref]

Moin, P.

P. Moin and K. Mahesh, “Direct numerical simulation: a tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539–578 (1998).
[Crossref]

Monin, A. S.

A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151), 163–187 (1954).

Obukhov, A. M.

A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151), 163–187 (1954).

Persson, P. O. G.

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Vol. 152 (SPIE, 2005).
[Crossref]

Poschl, P.

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Schacher, G.

Sorbjan, Z.

Z. Sorbjan, “Gradient-based scales and similarity laws in the stable boundary layer,” Q. J. R. Meteorol. Soc. 136, 1243–1254 (2010).

Stuart, A.

Stull, R. B.

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic Publishers, 1988).
[Crossref]

R. B. Stull, Meteorology for Scientists and Engineers (Brooks Cole, 1999).

Thiermann, V.

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Tunick, A.

A. Tunick, “Cn2 model to calculate the micrometeorological influences on the refractive index structure parameter,” Environ. Model. Softw. 18, 165–171 (2003).
[Crossref]

Walters, D. L.

K. E. Kunkel and D. L. Walters, “Modeling the diurnal dependence of the optical refractive index structure parameter,” J Geophys. Res. 88, 10,999–11,004 (1983).
[Crossref]

Wesely, M.

M. Wesely and E. Alcaraz, “Diurnal cycles of the refractive index structure function coefficient,” J. Geophys. Res. 78, 6224–6232 (1973).
[Crossref]

Wyngaard, J.

J. Wyngaard and O. Coté, “The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer,” J. Atmos. Sci. 28, 190–201 (1971).
[Crossref]

Wyngaard, J. C.

J. C. Wyngaard, Y. Izumi, A. Stuart, and J. R. Collins, “Behavior of the refractive-index-structure parameter near the ground,” J. Opt. Soc. Am. 61, 1646–1650 (1971).
[Crossref]

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

Zeisse, C. R.

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

Annu. Rev. Fluid Mech. (1)

P. Moin and K. Mahesh, “Direct numerical simulation: a tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539–578 (1998).
[Crossref]

Appl. Opt. (1)

Bound.-Layer Meteor. (2)

A. A. Grachev, E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, “The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer,” Bound.-Layer Meteor. 147, 51–82 (2013).
[Crossref]

A. Dyer, “A review of flux-profile relationships,” Bound.-Layer Meteor. 7, 363–372 (1974).
[Crossref]

Comput. Fluids (1)

P. He, “A high order finite difference solver for massively parallel simulations of stably stratified turbulent channel flows,” Comput. Fluids 127, 161–173 (2016).
[Crossref]

Environ. Fluid Mech. (1)

P. He and S. Basu, “Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: A direct numerical simulation approach,” Environ. Fluid Mech. 16, 373–399 (2015).
[Crossref]

Environ. Model. Softw. (1)

A. Tunick, “Cn2 model to calculate the micrometeorological influences on the refractive index structure parameter,” Environ. Model. Softw. 18, 165–171 (2003).
[Crossref]

J Geophys. Res. (1)

K. E. Kunkel and D. L. Walters, “Modeling the diurnal dependence of the optical refractive index structure parameter,” J Geophys. Res. 88, 10,999–11,004 (1983).
[Crossref]

J. Appl. Meteor. (1)

P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter (Cn2) over the ocean using bulk methods,” J. Appl. Meteor. 39, 1770–1783 (2000).
[Crossref]

J. Appl. Phys. (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[Crossref]

J. Atmos. Sci. (2)

J. Wyngaard and O. Coté, “The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer,” J. Atmos. Sci. 28, 190–201 (1971).
[Crossref]

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, “Flux-profile relationships in the atmospheric surface layer,” J. Atmos. Sci. 28, 181–189 (1971).
[Crossref]

J. Geophys. Res. (1)

M. Wesely and E. Alcaraz, “Diurnal cycles of the refractive index structure function coefficient,” J. Geophys. Res. 78, 6224–6232 (1973).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. SPIE (1)

V. Thiermann, A. Karipot, I. Dirmhirn, P. Poschl, and C. Czekits, “Optical turbulence over paved surfaces,” Proc. SPIE 2471, 197–203 (1995).
[Crossref]

Q. J. R. Meteorol. Soc. (1)

Z. Sorbjan, “Gradient-based scales and similarity laws in the stable boundary layer,” Q. J. R. Meteorol. Soc. 136, 1243–1254 (2010).

Tr. Akad. Nauk SSSR Geophiz. Inst. (1)

A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151), 163–187 (1954).

Other (7)

R. R. Beland, Infrared and Electro-Optical Systems Handbook, Vol. 2 (SPIE, 1993).

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic Publishers, 1988).
[Crossref]

P. S. Arya, Introduction to Micrometeorology (Academic, 2001).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Vol. 152 (SPIE, 2005).
[Crossref]

“Mauna loa seeing study. http://www.eol.ucar.edu/isf/projects/MLO_CN2/ ,” (2006).

S. Oncley and T. Horst, “Calculation of Cn2 for visible light and sound from CSAT3 sonic anemometer measurements,” avaiable at https://www.eol.ucar.edu/system/files/files/field_project/DASH04/background.pdf (2004).

R. B. Stull, Meteorology for Scientists and Engineers (Brooks Cole, 1999).

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Figures (4)

Fig. 1
Fig. 1

Dependence of similarity function (gT) on stability parameter (Rig). The plot is based on the similarity function proposed in W71 [8], i.e., Eqs. (4) and (5).

Fig. 2
Fig. 2

Dependence of similarity function (gT) on stability parameter (Rig) for stable conditions. The symbols with error bars are the ensemble results constructed from the DNS dataset. The lower error bars, the circles, and the higher error bars represent 25th, 50th, and 75th percentile values of the ensemble results, respectively. The blue dash-dot line is the regression fit, i.e., Eq. (12), based on the 50th percentile of the DNS data (r2 = 0.981 in logarithmic coordinates). The red dashed line is the similarity function proposed in W71 [8], i.e., Eqs. (4b) and (5b).

Fig. 3
Fig. 3

Time-series of C n 2 (15 m) at the Mauna Loa Observatory during July 1 to 28, 2006. The circles are the observational data measured at Mauna Loa Observatory, Hawaii. The black solid and hollow circles correspond to the cases Um > 2 m s−1 and Um < 2 m s−1, respectively, with Um being the magnitude of wind speed. The blue solid lines are the C n 2 values predicted using the DNS-based formulation, i.e., Eq. (12) for stable conditions and Eqs. (4a) and Eqs. (5a) for unstable conditions. The green dashed lines are the predicted C n 2 using the W71 formulation, i.e., Eqs. (4) and (5). The shaded areas denote the time periods when Rig is larger than 0.2. The mean correlation coefficients (RD and RN are for daytime and nighttime, respectively, calculated based on log 10 C n 2) between observation and DNS-based estimation during the four weeks are also shown in the bottom-left corner of each sub-figure. The data with Um < 2 m s−1 are excluded from the calculation of correlation coefficients. In addition, the data during July 28−29 are excluded due to the occurrence of rainfall.

Fig. 4
Fig. 4

(a) Correlation and (b) quantile-quantile plots between the observed and estimated C n 2 during July 1 to 28, 2006. RD and RN are the correlation coefficients (calculated based on log 10 C n 2 ) for daytime and nighttime, respectively. The calm wind data (Um < 2 m s−1) are excluded.

Equations (12)

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C T 2 z 4 / 3 ( θ ¯ z ) 2 = g T ( R i g ) .
R i g = g θ ¯ θ ¯ / z S 2 .
C n 2 = ( 7.9 × 10 5 P T 2 ) 2 C T 2 ( 1 + 0.03 β ) 2 ,
C T 2 z 4 / 3 ( θ ¯ z ) 2 = g T ( R i g ) = g T ( f T ( ζ ) ) = { 1.07 ( 1 9 ζ 1 + 0.5 | ζ | 2 / 3 ) 1 / 2 , ζ 0 , ( 4 a ) 0.79 ( 0.74 + 4.7 ζ ) ( 1 + 2.5 ζ 3 / 5 ) 1 / 2 , ζ > 0. ( 4 b )
R i g = f T ( ζ ) = { 0.74 ζ ( 1 15 ζ 1 9 ζ ) 1 / 2 , ζ 0 , ( 5 a ) ζ ( 0.74 + 4.7 ζ ) ( 1 + 4.7 ζ ) 2 , ζ > 0. ( 5 b )
u j x j = 0 ,
u i t + u i u j x j = p x i + 1 Re b x j u i x j + Δ P δ i 1 + R i b θ δ i 3 ,
θ t + θ u j x j = 1 Re b Pr x j θ x j .
C T 2 = 1.6 ε 1 / 3 χ ,
ε = ν u i x j u i x j , χ = 2 k θ i x j θ i x j .
1.6 ν u i x j u i x j 1 / 3 2 k θ i x j θ i x j z 4 / 3 ( θ z ) 2 .
C T 2 z 4 / 3 ( θ ¯ z ) 2 = g T ( R i g ) = 0.05 + 1.02 e 14.49 R i g , R i g > 0.

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