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 wind speed difference $\Delta U$ using the dimensionless momentum profile $\phi_m$,

\[ u_* = \frac{\kappa \Delta U}{F_m(\zeta, ...)},\]

where $F_m$ is the integrated stability correction function for momentum.

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:

\[ \theta_* = \frac{\kappa \Delta \theta}{F_h(\zeta, ...)}\]

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.

Velocity Variance ($\sigma_u^2$)

The variance of the horizontal wind speed components, computed by u_variance:

u_variance(param_set, Δz_eff, ustar, ζ)

The parameterization depends on stability:

Neutral/Stable ($\zeta \ge 0$): Proportional to $u_*^2$.
Unstable ($\zeta < 0$): Includes a contribution from the convective velocity scale $w_*$, which depends on the boundary layer height $z_i$ (taken to be a fixed parameter) and the heat flux at the effective height $\Delta z_{\text{eff}}$.

Scalar Variance ($\sigma_\phi^2$)

The variance of scalars (temperature, humidity), computed by scalar_variance:

scalar_variance(param_set, scale, ζ)