Boundary conditions

Constant flux layer

Following Monin-Obukhov similarity theory (MOST) a constant flux layer can be assumed as boundary condition between the surface and the first grid level where scalars and horizontal velocities are defined (k = 1, z_MO = 0.5 Δz). It is then required to provide the roughness lengths for momentum z₀ and heat z_0,h. Momentum and heat fluxes as well as the horizontal velocity components are calculated using the following framework. The formulation is theoretically only valid for horizontally-averaged quantities. In PALM we assume that MOST can be also applied locally and we therefore calculate local fluxes, velocities, and scaling parameters.

Following MOST, the vertical profile of the horizontal wind velocity

is given in the surface layer by

where κ = 0.4 is the Von Kármán constant and Φ_m is the similarity function for momentum in the formulation of Businger-Dyer (see e.g. Panofsky and Dutton 1984)

Here, L is the Obukhov length, calculated as

(eq:L) 

The scaling parameters θ_* and q_* are defined by MOST as:

(eq:scales) 

with the friction velocity u_* defined as

In PALM, u_* is calculated from u_h at z_MO by vertical integration over z from z₀ to z_MO.

From the equations above it is possible to derive a formulation for the horizontal wind components, viz.

Vertical integration of the above equation over z from z₀ to z_MO then yields the surface momentum fluxes

The formulations above all require knowledge of the scaling parameters θ_* and q_*. These are deduced from vertical integration of

over z from z_0,h to z_MO. The similarity function Φ_h is given by

Note that this implementation of MOST in PALM requires the use of data from the previous time step. The following steps are thus carried out in sequential order. First of all, θ_* and q_* are calculated by integration using the value of z_MO/L from the previous time step. Second, the new value of z_MO/L is derived using the new values of θ_* and q_* but using u_* from the previous time step. Then, the new values of u_*, and subsequently the momentum fluxes are calculated by integration, respectively. At last, the new surface fluxes are derived from θ_* and q_*, and u_*. In the special case, when surface fluxes are prescribed instead of surface temperature and humidity, the first and last steps are omitted and θ_* and q_* are directly calculated from u_* and the surface fluxes.