IMPROVEMENT OF THE K-PROFILE MODEL FOR THE PLANETARY
BOUNDARY LAYER BASED ON LARGE EDDY SIMULATION DATA
Y. NOH
, W. G. CHEON and S. Y. HONG
Department of Atmospheric Sciences, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul
120-749, Korea
S. RAASCH
Institute of Meteorology and Climatology, University of Hannover, Germany
(Received in final form 9 August 2002)
Abstract. Modifications of the widely used K-profile model of the planetary boundary layer (PBL),
reported by Troen and Mahrt (TM) in 1986, are proposed and their effects examined by comparison
with large eddy simulation (LES) data. The modifications involve three parts. First, the heat flux
from the entrainment at the inversion layer is incorporated into the heat and momentum profiles, and
it is used to predict the growth of the PBL directly. Second, profiles of the velocity scale and the
Prandtl number in the PBL are proposed, in contrast to the constant values used in the TM model.
Finally, non-local mixing of momentum was included. The results from the new PBL model and the
original TM model are compared with LES data. The TM model was found to give too high PBL
heights in the PBL with strong shear, and too low heights for the convection-dominated PBL, which
causes unrealistic heat flux profiles. The new PBL model improves the predictability of the PBL
height and produces profiles that are more realistic. Moreover, the new PBL model produces more
realistic profiles of potential temperature and velocity. We also investigated how each of these three
modifications affects the results, and found that explicit representation of the entrainment rate is the
most critical.
Keywords: K-profile model, Large eddy simulation (LES), Non-local mixing, Planetary boundary
layer (PBL), PBL model.
1. Introduction
The transport by large eddies plays an important role in the vertical mixing of heat,
momentum and moisture in the convective boundary layer. The usual downgradient
parameterization of vertical mixing is not appropriate in this case, and the non-local
mixing associated with the bulk properties of the planetary boundary layer (PBL)
has been increasingly applied (Deardorff, 1966; Therry and Lacarrere, 1983; Stull,
1984; Troen and Mahrt, 1986; Holtslag and Moeng, 1991; Chrobok et al., 1992;
Abdella and McFarlane, 1997).
The inclusion of the effects of non-local mixing was pursued in various dif-
ferent ways. For example, Stull (1984) developed the transilient model in which
various non-local mixings are explicitly treated, and Therry and Lacarrere (1983)
E-mail: noh@atmos.yonsei.ac.kr
Boundary-Layer Meteorology 107: 401–427, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
402 Y. NOH ET AL.
and Abdella and McFarlane (1997) included non-local terms in the second-order
turbulence closure model. Meanwhile, Troen and Mahrt (1986) developed the K-
profile model in which the profiles of eddy diffusivity and viscosity are presumed
a priori, based on the overall mixing property of the PBL.
In the Troen and Mahrt model (TM model, hereafter) the non-local mixing in
the kinematic heat flux is represented by
−
w
θ
= K
h
∂θ
∂z
− γ
h
. (1)
Here K
h
is the eddy diffusivity for potential temperature θ and γ
h
is the counter-
gradient term representing the non-local mixing due to large convective eddies.
The TM model also assumes the profile of the eddy viscosity as
K
m
= kw
s
z
1 −
z
h
2
, (2)
where k is the von Karman constant (= 0.4), z is the distance from the surface, and
h is the boundary-layer height. Note that in the TM model h represents the height
where the heat flux disappears, i.e.,
w
θ
= 0atz = h, which is located above the
height of the minimum heat flux. The eddy diffusivity K
h
in (1) is computed from
K
m
by using the Prandtl number Pr,asK
h
= Pr
−1
K
m
.
The velocity scale w
s
in (2) is represented by the value scaled at the top of the
surface layer, w
s0
such as
w
s0
= (u
3
∗
+ 7εkw
3
∗
)
1/3
, (3)
where
w
∗
=[(g/T
0
)(w
θ
0
h)]
1/3
(4)
is the convective velocity scale given the surface buoyancy flux (g/T
0
)w
θ
0
,and
the ratio of the surface layer height to the PBL height, ε is arbitrarily specified to
be 0.1.
Here the velocity scale w
s0
is made to be coincident with the velocity scale at
the top of the surface boundary layer (z = εh) as
w
s0
= u
∗
φ
−1
m
, (5)
where u
∗
is the surface friction velocity and φ
m
is the wind profile function
evaluated at the top of the surface layer.
The value of φ
m
is obtained by satisfying the compatibility with the surface
boundary similarity. The profile functions for momentum and heat, φ
m
and φ
h
,are
given respectively by
φ
m
= φ
h
= 1 + 4.7z/L (6)
IMPROVEMENT OF THE K-PROFILE MODEL 403
for stable conditions, and
φ
m
= (1 − 16z/L)
−1/4
, (7)
φ
h
= (1 − 16z/L)
−1/2
, (8)
for unstable conditions, from Businger et al. (1971), where L is the Obukhov length
(≡−u
3
∗
T
0
/kgw
θ
0
). To determine (5) in unstable conditions, however, Troen and
Mahrt (1986) assumed the following function;
φ
m
= (1 − 7z/L)
−1/3
. (9)
The exponent of −1/3 in (9) was chosen to ensure the free-convection limit for
z L. With the coefficient chosen to be 7, the values obtained from (7) and (9)
differ less than 6% for z/L < 2. We can obtain the value of w
s0
, given by (3), by
substituting (9) into (5).
The countergradient term γ
h
in (1) is given by
γ
h
= b
0
w
θ
0
w
s0
h
, (10)
where b
0
is a coefficient of proportionality. Troen and Mahrt (1986) suggested the
valueofthiscoefficientasb
0
= 6.5.
The boundary-layer height is determined by specifying a critical value of the
bulk Richardson number Ri
b
defined by
h = Ri
b
T
0
|U(h)|
2
g(θ(h) − θ
s
)
, (11)
where U(h) and θ(h) are the horizontal wind speed and the potential temperature
at z = h. The temperature scale of the boundary layer θ
s
is defined as
θ
s
= θ(z
1
) + θ
T
(12)
with
θ
T
= b
0
w
θ
0
w
s0
. (13)
Here θ(z
1
) is the potential temperature at the lowest atmospheric level in the model
z
1
. For the critical value of the bulk Richardson number, Ri
b
= 0.5 was suggested.
For the actual prediction of the growth of h in the TM model, the criterion (11)
can be rewritten as
θ(h) = θ
s
+ Ri
b
T
0
|U(h)|
2
gh
, (14)
404 Y. NOH ET AL.
Using (14), h is determined at each time step, after the potential temperature profile
is modified by the vertical diffusion given by (1).
The Prandtl number is given by the value Pr
0
obtained at the top of the surface
layer (z = εh) as (Troen and Mahrt, 1986),
Pr
0
=
φ
h
φ
m
+ b
0
εk. (15)
In the TM model the Prandtl number is assumed constant over the whole boundary
layer.
The TM model has been applied to climate and weather forecast models (Hong
and Pan, 1996; Holtslag and Boville, 1993; Lüpkes and Schlünzen, 1996), and
it was found to improve the prediction of the PBL effectively by incorporating
the non-local mixing, notwithstanding its relative simplicity. The results showed
that the TM model can transport heat and moisture more effectively away from
the surface compared to a local mixing model, and thus reproduces more realistic
temperature and humidity profiles.
The performance of the TM model was examined recently using large eddy
simulation (LES) data. Ayotte et al. (1996) evaluated the performance of various
PBL models by comparing various integrated quantities from the models with those
from LES. Although the TM model was found to perform better than other PBL
models in general, its prediction of the entrainment rate is not satisfactory. In par-
ticular, it was found that the entrainment is overestimated for a PBL with strong
shear and underestimated for free convection in the TM model. Meanwhile, Brown
(1996) found that the TM model produces too strong shear within the convective
boundary layer from examinations of mean temperature and velocity profiles ob-
tained from LES, although the TM model still shows improved results compared
to the local mixing scheme.
The above-mentioned investigations of the TM model using LES were mainly
concerned with the mean variables such as potential temperature and velocity or
the integrated properties. However, for more rigorous assessment of the validity of
the model we also need to examine the various assumptions introduced into the
TM model under various circumstances, such as the heat flux profile of (1) and the
critical bulk Richardson number used to determine the boundary-layer height in
(11).
Meanwhile, a few questions are raised with regard to the TM model from a
theoretical viewpoint.
First, the entrainment is not represented explicitly in the heat flux profile, and
the vertical mixing and the growth of the PBL are treated separately. This is rather
absurd, considering that both the vertical mixing and the growth of the PBL repres-
ent the vertical heat transfer. They are usually treated simultaneously in turbulence
closure models, e.g., the Mellor–Yamada model (Mellor and Yamada, 1982).
Second, the model is not consistent with the case of free convection. The K-
profile of (2) does not obey the free convection limit, K
h
∝ z
4/3
for z h
IMPROVEMENT OF THE K-PROFILE MODEL 405
(Panofsky and Dutton, 1984), as pointed out by Holtslag and Moeng (1991). Fur-
thermore, both the velocity scale and the Prandtl number Pr, which are evaluated
at the top of the surface layer by (3) and (15), are assumed to remain invariant
throughout the PBL. However, the factors used to evaluate w
s
and Pr,suchas
turbulent kinetic energy (TKE) and stability, vary substantially with height within
the PBL.
Finally, it still needs to be determined whether we can really neglect the non-
local mixing of momentum in the convective boundary layer, while the non-local
mixing of heat is applied. Brown and Grant (1997) and Frech and Mahrt (1996)
suggested the inclusion of the non-local mixing of momentum for more realistic
prediction of the wind profile in the convective boundary layer. However, it is not
yet clear how the inclusion of the non-local mixing of momentum can improve the
general performance of a PBL model such as the growth of h.
In this paper, we suggested a new PBL model by modifying the TM model as
to the above-mentioned three factors with the help of LES data. Having developed
a new PBL model, we evaluated the performance of both TM model and the new
PBL model by comparing the resultant various profiles and boundary-layer heights
with the LES data. In addition, we analyzed how each of these three modifications
in the new PBL model affects the results.
2. LES Model and Simulations
The various PBL flows used in this study were generated from the LES code
described in Raasch and Etling (1991). Subgrid-scale turbulence is modelled ac-
cording to Deardorff (1980). A prognostic equation is solved for the subgrid-scale
TKE, which is used to parameterize the subgrid-scale fluxes. Between the sur-
face and the first computational grid level Monin–Obukhov similarity is assumed.
Periodic boundary conditions are used in both lateral directions. The present LES
model has been successfully applied to various PBL problems (Raasch and Etling,
1998; Raasch and Harbusch, 2001; Schröter et al., 2000).
The numerical scheme is a standard, second-order finite difference scheme
using the absolutely-conserving scheme of Piacsek and Williams (1970) for
the nonlinear advection term. The prognostic equations are time-advanced by a
leapfrog scheme. A weak time filter is applied to remove the time-splitting instabil-
ity of the leapfrog method (Asselin, 1972). During the integration the time step is
adjusted so that it never exceeds one tenth of the allowed value due to the CFL
(Courant–Friedrichs–Lewy) and diffusion criteria. Incompressiblity is applied by
means of the Poisson equation for pressure, which is solved by an FFT method. Re-
cently the code has been parallelized, and the performance of the new parallellized
code is found to be excellent on an SGI/Cray-T3E with an almost linear speed-up
up to a very large number of processors (Raasch and Schröter, 2001).
