On the Extent of Applicability of Various Non-linear Similarity Functions for Computation of Surface Fluxes under Stable Conditions in Numerical Models

In this study, a systematic mathematical analysis has been presented for the extent of applicability of various non-linear similarity functions for momentum \(({{\upvarphi }}_{{\text{m}}})\) $({\mathrm{\phi }}_{\text{m}})$ and heat \(({{\upvarphi }}_{{\text{h}}})\) $({\mathrm{\phi }}_{\text{h}})$ under stable conditions to compute surface turbulent fluxes in numerical models. The investigation is carried out for equal and unequal momentum \(({{\text{z}}}_{0})\) $({\text{z}}_0)$ and heat \(({{\text{z}}}_{{\text{h}}})\) $({\text{z}}_{\text{h}})$ roughness lengths. The study reveals that \({{\upvarphi }}_{{\text{m}}}\) ${\mathrm{\phi }}_{\text{m}}$ and \({{\upvarphi }}_{{\text{h}}}\) ${\mathrm{\phi }}_{\text{h}}$ utilized in the National Centre for Atmospheric Research Community Atmosphere Model version 5 (NCAR-CAM5) (Holtslag et al. in Mon Weather Rev 118:1561–1575, 1990) have several restrictions on their applicability in moderately to strongly stable cases. If the ratios of \({{\text{z}}}_{0}\) ${\text{z}}_0$ and \({{\text{z}}}_{{\text{h}}}\) ${\text{z}}_{\text{h}}$ to the height \(({\text{z}})\) $(\text{z})$ from the surface (i.e., \(\frac{{{\text{z}}}_{0}}{{\text{z}}}\) $\frac{{\text{z}}_0}{\text{z}}$ and \(\frac{{{\text{z}}}_{{\text{h}}}}{{\text{z}}}\) $\frac{{\text{z}}_{\text{h}}}{\text{z}}$ ) lie in the range \((0.2, 1)\) $(0.2,1)$ , the functions are valid for a limited range of \(\upzeta \) $\mathrm{\zeta }$ (stability parameter) in strong stable conditions \(\left(\upzeta >1\right)\) $\left(\mathrm{\zeta },>,1\right)$ ; however, when \(\frac{{{\text{z}}}_{0}}{{\text{z}}}\le 0.2\) $\frac{{\text{z}}_0}{\text{z}}\le 0.2$ and \(\frac{{{\text{z}}}_{{\text{h}}}}{{\text{z}}}\le 0.2\) $\frac{{\text{z}}_{\text{h}}}{\text{z}}\le 0.2$ , the validity of functions is unrestricted. In terms of bulk Richardson number \(\left({{\text{Ri}}}_{{\text{B}}}\right)\) $\left({\text{Ri}}_{\text{B}}\right)$ , the functions are valid for a limited range of moderately to strongly stable conditions. These theoretically derived upper limits have also been validated using observations from the UK Meteorological Office’s Cardington and Cooperative Atmosphere-Surface Exchange Study-99 datasets. On the other hand, similarity functions based on Cheng and Brutsaert (Boundary-Layer Meteorol 114:519–538, 2005), Grachev et al. (Boundary-Layer Meteorol 124:315–333, 2007), Srivastava et al. (Meteorol Appl 27, 2020), and Gryanik et al. (J Atmos Sci 77:2687–2716, 2020) are found to be theoretically valid for all values of \(\upzeta \) $\mathrm{\zeta }$ and \({{\text{Ri}}}_{{\text{B}}}\) ${\text{Ri}}_{\text{B}}$ . The efforts have also been made to implement these functions in the Weather Research and Forecasting as well as global scale models.