Advanced Research WRF (ARW) Technical Note - MMM - University ...

If ∆x is less than the user-specified critical length scale lcr, then the length scale used for
calculating both Kh and Kv in (4.6) is lh,v = (∆x∆y∆z) 1/3 (and Kh = Kv = K). If ∆x is
greater than an critical length scale lcr, then the length scale lh = √ ∆x∆y in the calculation
of the horizontal eddy viscosity Kh using (4.6), and lv = ∆z for the calculation of the vertical
eddy viscosity Kv using (4.6).
Additionally, the eddy viscosities for scalar mixing are divided by the turbulent Prandtl
number Pr = 1/3.
Prognostic TKE Closure
For the predicted turbulent kinetic energy option (TKE; see section 4.1.4), the eddy viscosities
are computed using
√
e,
Kh,v = Cklh,v
where e is the turbulent kinetic energy (a prognostic variable in this scheme), Ck is a constant
(typically 0.15 < Ck < 0.25), and l is a length scale.
If the grid spacing ∆x is less than the critical length scale lcr, then
lh,v = min (∆x∆y∆z) 1/3 , 0.76 √ e/N
for N 2 > 0,
lh,v = (∆x∆y∆z) 1/3
for N 2 ≤ 0
(see section 4.1.4 for the calculation of N 2 ). Both the horizontal and vertical eddy viscosities
are multiplied by an inverse turbulent Prandtl number P −1
r = 1 + 2l/(∆x∆y∆z) 1/3 for scalar
mixing. In this case (lcr > ∆x) the horizontal and vertical eddy viscosities are equivalent.
If the grid spacing ∆x is greater than the critical length scale lcr, then lh = √ ∆x∆y for the
calculation of Kh. For calculating Kv,
lv = min ∆z, 0.76 √ e/N
for N 2 > 0,
lv = ∆z for N 2 ≤ 0.
The eddy viscosity used for mixing scalars is divided by a turbulent Prandtl number Pr. The
Prandtl number is 1/3 for the horizontal eddy viscosity Kh, and P −1
r = 1+2l/∆z for the vertical
eddy viscosity Kv.
4.1.4 TKE equation for the 1.5 Order Turbulence Closure
The prognostic equation governing the evolution of the turbulent kinetic energy e is
∂t(µde) + (∇ · Ve)η = µd( shear production + buoyancy + dissipation ). (4.7)
The time integration and the transport terms in (4.7) are integrated in the same manner as for
other scalars (as described in Chapter 3). The right-hand side source and sink terms for e are
given as follows.
Shear Production
The shear production term in (4.7) can be written as
shear production = KhD 2 11 + KhD 2 22 + KvD 2 33 + KhD2 xy
12 + KvD2 xη
13 + KvD2 yη
23 .
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