Physical Scales
In Monin-Obukhov Similarity Theory (MOST), turbulent fluxes are parameterized using characteristic physical scales. These scales represent the turbulent fluctuations of velocity and scalars in the surface layer.
Friction Velocity ($u_*$)
The friction velocity $u_*$ is the characteristic velocity scale of the turbulence, related to the surface kinematic momentum flux (stress) $\tau/\rho$:
\[ u_*^2 = \frac{|\tau|}{\rho} = \left( (\overline{u'w'})^2 + (\overline{v'w'})^2 \right)^{1/2}.\]
In SurfaceFluxes.jl, this is computed by compute_ustar. $u_*$ is derived from the effective wind speed difference $\Delta U = U_{\text{eff}}$ (gustiness included) using the dimensionless momentum profile $F_m$,
\[ u_* = \frac{\kappa \Delta U}{F_m(\zeta, ...)},\]
where $F_m = \ln(\Delta z_{\text{eff}}/z_{0m}) - \psi_m(\zeta) + \psi_m(\zeta z_{0m}/\Delta z_{\text{eff}})$ is the dimensionless profile integral for momentum (plus the roughness-sublayer correction, if configured). With gustiness ($U_{\text{eff}} > |\Delta\mathbf{u}|$), the stress the code returns is smaller than $\rho u_*^2$: $|\tau|/\rho = u_*^2 |\Delta\mathbf{u}| / U_{\text{eff}}$.
The Obukhov length and the stability parameter follow from $u_*$ and the surface buoyancy flux $B$: $L = -u_*^3/(\kappa B)$ (obukhov_length) and $\zeta = (\Delta z - d)/L$.
Scalar Scales ($\theta_*, q_*$)
Similar scales are defined for potential temperature ($\theta$) and specific humidity ($q$).
Temperature Scale ($\theta_*$): Related to the kinematic potential temperature flux $\overline{w'\theta'}$:
\[ u_* \theta_* = -\overline{w'\theta'}.\]
Computed as:
\[ \theta_* = \frac{\kappa \Delta \theta}{F_h(\zeta, ...)},\]
with the interior-minus-surface difference $\Delta\theta = [c_{pd}(T_{\text{int}} - T_{\text{sfc}}) + g(\Delta z - d)]/c_{pm}$ of the dry static energy, divided by the moist heat capacity.
In SurfaceFluxes.jl, this scale is computed by compute_theta_star.
Humidity Scale ($q_*$): Related to the kinematic specific humidity flux $\overline{w'q'}$ (evaporation):
\[ u_* q_* = -\overline{w'q'}.\]
Computed similarly to $\theta_*$ using the same heat stability function $F_h$. See compute_q_star.
Variances
SurfaceFluxes.jl also provides functions to estimate the variances of turbulent fluctuations, which are useful for higher-order closure models or statistical analysis.
The variance and TKE functions are convective surface-layer closures and are independent of the chosen flux-profile parameterization: Businger, Gryanik, and Grachev all return the same values, because Grachev et al. (2007) and Gryanik et al. (2020) do not define variance similarity functions. On the stable side they reduce to constant (neutral) values. This is a crude approximation — observed $\sigma_u/u_*$ increases with stability rather than staying constant, and Monin–Obukhov scaling of the horizontal-velocity variances breaks down in stable stratification. Use the stable-side variances as rough estimates only; they are not validated stable-boundary-layer similarity, even when a stable-boundary-layer parameterization (Gryanik, Grachev) is selected for the fluxes.
Turbulent Kinetic Energy
The function surface_tke returns the turbulent kinetic energy (TKE) following Tan et al. (2018):
surface_tke(param_set, Δz_eff, ustar, ζ)Returns the TKE, $(u_* \phi)^2$. The parameterization depends on stability:
- Neutral/Stable ($\zeta \ge 0$): $\text{TKE} = 3.75 u_*^2$.
- Unstable ($\zeta < 0$): $\text{TKE} = 3.75 u_*^2 + 0.2 w_*^2 + u_*^2 (-\zeta)^{2/3}$, where the convective velocity scale $w_*$ depends on the boundary layer height $z_i$ (taken to be a fixed parameter) and the buoyancy flux implied by $\zeta$.
The streamwise variance $\sigma_u^2 = (u_* \phi_{\sigma u})^2$ (Panofsky et al. 1977) is available separately through the universal function phi(uf, ζ, MomentumVariance()). Panofsky et al. express $\phi_{\sigma u}$ in terms of the mixed-layer stability $z_i/L$; this function is evaluated at the local $\zeta$.
Scalar Variance ($\sigma_\phi^2$)
The variance of scalars (temperature, humidity), computed by scalar_variance:
scalar_variance(param_set, scale, ζ)With the scalar scale $s_*$ (for example, $\theta_*$ or $q_*$):
- Stable: $\sigma_s^2 = 4 s_*^2$.
- Unstable: $\sigma_s^2 = 4 s_*^2 (1 - 8.3\zeta)^{-2/3}$ (Tan et al. 2018), whose free-convection limit $\sigma_s \approx 0.99 |s_*| (-\zeta)^{-1/3}$ matches Wyngaard et al. (1971).
References
- Panofsky, H. A., Tennekes, H., Lenschow, D. H., & Wyngaard, J. C. (1977). The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary-Layer Meteorology, 11, 355–361. DOI: 10.1007/BF02186086
- Tan, Z., Kaul, C. M., Pressel, K. G., Cohen, Y., Schneider, T., & Teixeira, J. (2018). An extended eddy-diffusivity mass-flux scheme for unified representation of subgrid-scale turbulence and convection. Journal of Advances in Modeling Earth Systems, 10, 770–800. DOI: 10.1002/2017MS001162
- Wyngaard, J. C., Coté, O. R., & Izumi, Y. (1971). Local free convection, similarity, and the budgets of shear stress and heat flux. Journal of the Atmospheric Sciences, 28, 1171–1182. DOI: 10.1175/1520-0469(1971)028<1171:LFCSAT>2.0.CO;2