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Du3
Dt
Du1
Dt
Du2
Dt
The three moment equations:
∂ρ
= 0 = −1 +fu2 −
ρ∂x1
∂
(u
∂x3
′ 1u′ 3 )
∂ρ
= 0 = −1 +fu1 −
ρ∂x2
∂
∂x3
(u ′ 2 u′ 3
∂ρ
= 0 = −1 −2Ω(η1u2 −η2u1)
ρ∂x3
∂
(u
∂x3
′ 3u′ 3 )
The scalar equations: (1)
DT ∂
= 0 = (−u
Dt ∂x3
′ 3θ′ )(= ∂T
∂t +u1
∂T
∂t +u2
∂T
)
∂x2
Dq ∂
= 0 = (−u
Dt ∂x3
′ 3q′ )
Equation1summarizes the equationform the mean flowforourpseudohomogeneousABL.
Asmentionedtheconstantpressuregradientisnecessary,butlimitsthehorizontalscaleforthe
model. The temperature equation further illustrates the meaning of the substantial derivative,
for all the variables. The equation for u3 is not important in this approximation and will be
neglected in the following.
Additionally, in eq. 1 our simplified ABL is situated on the rotating planet Earth, reflected
by appearance of the Coriolis parameter, and the Earth’s rotation rate Ω, with f = 2Ωsinϕ
and with ϕ being the latitude on the globe. Further it is seen that we have introduced the
symbol θ, the so called potential temperature. This is a modified temperature including the
fact that the average pressure and density in the atmosphere decreases with height, due to
gravity. Thismeans thatan adiabaticallymovingairpacketcoolsmovingup and heatsmoving
down,but willremaininequilibriumwithsurroundingsandat thesamepotentialtemperature.
If θ increases with height the air is denser than equilibrium at the bottom and therefore stable
against vertical perturbations. Conversely for θ decreasing with height, the air is lighter than
equilibrium at the bottom and hence unstable, if perturbed vertically. Within the boundary
layer, θ is often approximated by:
θ = T +Γ·z,with Γ = g
With Γ being about 0.01 K/m. It is noted that the only difference between T and θ is the
linear height variation.
Equation 1 shows that the vertical fluxes of scalars are constant with height, while the
momentum fluxes are slightly more complicated. Focusing on the two first equations for
the horizontal velocity components, we define the geostrophic wind, G, from the pressure
gradient, and perpendicular to the direction of this gradient:
G = (U1G,U2G) = (− 1 ∂p
,
fρ ∂x2
1 ∂p
) = (−
fρ ∂x1
1 ∂p 1 ∂p
, ) (3)
fρ ∂y fρ ∂x
The two first equations in eq. 1 now take the form:
Cp
0 = f(u2 −U2G)− ∂
∂x3 (u′ 1 u′ 3 )
0 = −f(u1 −U1G)− ∂
∂x3 (u′ 2u′ 3 ) (4)
This equation shows that the wind velocity approaches the geostrophic wind at the top of
the boundary layer, where the turbulence disappears. Down through the boundary layer the
DTU Wind Energy-E-Report-0029(EN) 29
(2)