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MONTHLY WEATHER REVIEW
VOLUME 129
Simple Inclusion of z-less Turbulence within and above the Modeled Nocturnal
Boundary Layer
KYUNG-JA HA
Department of Atmospheric Sciences, Pusan National University, Pusan, South Korea
L. MAHRT
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
(Manuscript received 27 June 2000, in final form 19 October 2000)
ABSTRACT
In simple models, the nocturnal boundary layer is generally formulated in terms of eddy diffusivities, which
are functions of height above ground and boundary layer depth. This approach is modified in this study to
include ‘‘z-less’’ turbulence, which is formulated in terms of a stability-dependent mixing length and a Prandtl
number, which are not tied to the existence of a boundary layer depth. These modifications have been guided
by analysis of nocturnal boundary layer data from the Cooperative Atmospheric-Surface Exchange Study–1999.
The modeled turbulence sometimes exhibits an upside-down structure with turbulence generated by shear
associated with nocturnal flow acceleration at higher levels. The modified model produces a more realistic
smoother wind speed maximum (nocturnal jet) and generates layers of turbulence above the boundary layer.
Remaining shortcomings of the new model are discussed.
1. Introduction
With stable conditions, layers of turbulence may occur within the surface inversion layer, or in the overlying
residual layer, that are at least temporarily decoupled
from the surface. Such vertical structure of the turbulence does not satisfy the formal definition of a boundary layer. The decoupled turbulence, however, may be
indirectly related to boundary layer processes through
the collapse of the daytime convective boundary layer
and subsequent flow acceleration associated with inertial
oscillations. Such inertial oscillations may induce a lowlevel jet. Layers of turbulence associated with sheargenerated turbulence and wave instability, initially decoupled from the surface, may later couple to the surface
through downward bursting (Nappo 1991; Neu 1995;
Mahrt 1999; Cuxart et al. 2000). The detached turbulence redistributes scalars and momentum above the
boundary layer in the residual layer, which later can
become coupled to the boundary layer through downward bursting. Detached turbulence may additionally be
induced by shear associated with surface cold air drainage and meandering motions. Significant turbulence
may also occur above stable boundary layers generated
by advective processes (Smedman 1988, 1991; Ohya et
al. 1997).
Modeling turbulence in the stable boundary layer can
be classified in terms of (a) a traditional approach based
on height with respect to the ground surface and the
boundary layer depth; (b) local scaling (Nieuwstadt
1984), which still requires the definition of the boundary
layer depth as a governing parameter; and (c) z-less
stratification where the fluxes are parameterized as purely local with no recognition of the ground surface or
boundary layer top (Wyngaard 1973). The model developed in this study employs a combination of the traditional approach and the z-less formulation.
While recognizing the importance and complexity of
detached generation of turbulence, this study emphasizes simplification for those model applications that
cannot accommodate the complexity or computer time
requirements of boundary layer models based on the
turbulence energy equation or other higher-moment
equations. As an aside, our attempts to model the very
stable cases with a turbulence kinetic energy equation
have revealed major deficiencies in the model, and future modifications are needed.
2. Model
Corresponding author address: Dr. Larry Mahrt, College of Oceanic and Atmospheric Sciences, Ocean Admin Bldg 104, Oregon State
University, Corvallis, OR 97331-5503.
E-mail: mahrt@oce.orst.edu
q 2001 American Meteorological Society
We begin with the one-dimensional model of the atmospheric boundary layer, plant canopy, and soil based
on Troen and Mahrt (1986), Pan and Mahrt (1987),
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HA AND MAHRT
For the very stable case (z/L . 1.0), we set z/L 5 1.0
as suggested by Kondo et al. (1978).
The eddy diffusivity for heat K h (m 2 s 21 ) is related
to the eddy diffusivity for momentum in terms of the
turbulent Prandtl number (P r ):
asymptotic mixing length (l 0 ) at neutral stability (Ri 5
0) to vanishingly small values at large Ri. The original
formulation of Kim and Mahrt produces too much mixing within our one-dimensional model in that it eliminated the observed early evening very stable period
(sections 4–5) and unrealistically led to well-mixed potential temperature and wind in some nocturnal conditions. To improve the qualitative comparison with the
data (section 5), l 0 is reduced from 50 m in their study
to 15 m in the present study. The ’’optimal‘‘ value of
l 0 probably depends on grid size. This value for the
asymptotic mixing length is of comparable magnitude
to the mixing length (14.2 m) for heat for neutral stratification using the formulation of Louis et al. (1981).
The coefficients c1, c2, and c3 remain fixed at 28.5,
0.15, and 3.0. The corresponding mixing length decreases more slowly with the Richardson number than
in the formulation of Louis et al. (1981).
The original formulation of Kim and Mahrt was applied only to the upper 70% of the boundary layer. Here,
the z-less formulation is not considered if the asymptotic
mixing length is larger than the mixing length predicted
by Monin–Obukhov similarity theory for neutral stratification, expressed as
K h 5 K m /P r .
l 0 . kz.
Holtslag et al. (1990), Holtslag and Moeng (1991), and
Ek and Mahrt (1991a). We refer to these papers and
Chang et al. (1999) for a description of the model and
here describe only parts of the model where changes
are made.
a. Eddy diffusivity
The eddy diffusivity for momentum K m (m 2 s 21 ) is
formulated as
K m 5 u*fm21
12 1
2
p
z
z
z
hk 1 2
,
L
h
h
(1)
where z is height above ground, L is the Obukhov length,
h is the boundary-layer depth, k is the von Karman
constant, p 5 2.0, and
f m 5 1 1 4.7z/L.
(2)
(3)
Here, P r is determined as the value at the top of the
surface layer using surface layer similarity theory.
b. Local z-less turbulent mixing
Modifications to the above model are presented in the
next three sections. Values of needed coefficients introduced below are determined by comparisons (section 5)
with the data described in section 3. Rigorous comparisons could not be made because the time and height
dependence of the pressure gradient field could not be
adequately determined, particularly with respect to mesoscale variations.
Local z-less generation of turbulence not directly related to surface fluxes will be formulated in terms of a
mixing length, l h , based on the gradient Richardson
number, Ri, such that
) )
dVh
Kh 5 l
,
dz
(4)
K m 5 K h Pr ,
(5)
2
h
where | dV h /dz | is the magnitude of the vector wind
shear. The z-less mixing length is formulated using the
same general relationship in Kim and Mahrt (1992):
[
l h 5 l 0 exp(2c1Ri) 1
Pr 5 1.5 1 3.08Ri.
]
c2
,
Ri 1 c3
(6)
(7)
With Eq. (6) and the coefficients reported below, the
mixing length decreases from slightly greater than the
(8)
With this condition, the z-less formulation is not considered below 37.5 m. We tested a stability-dependent
form of Eq. (8) that is physically more appealing. However, this approach led to additional nonlinear interactions, which complicate interpretation of the model.
If the inequality Eq. (8) is met, then the z-less formulation predicts interaction with the surface and is
therefore inconsistent and not applied. The z-less formulation is applied at all levels, both within and above
the boundary layer, subject to the near-surface constraint
[Eq. (8)]. If this inequality is not satisfied, the model
chooses the larger of the eddy diffusivities predicted by
the original boundary layer formulation (z/h dependence) and the local z-less formulation.
c. Determination of the boundary layer depth
The boundary layer depth is diagnosed in terms of a
specified critical bulk Richardson number (Rc). The
boundary layer depth (h) is the lowest height where the
bulk Richardson number Ri across the entire stable
boundary layer equals the critical Richardson number
such that
h5
R cuy 0 [u(h) 2 1 y (h) 2 ]
,
g[uy (h) 2 uy 0 ]
(9)
where u y 0 is the virtual potential temperature at the lowest model level and u and y are the horizontal velocity
components.
To diagnose the height of boundary layer, the computation is initiated by calculating the bulk Richardson
number Ri between the first model level and level l
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TABLE 1. The model with z-less turbulence and initialization from
HAPEX-MOBILHY.
TABLE 2. The modified model with initialization based on
HAPEX-MOBILHY.
Parameter
Value
Parameter
Critical Ri number
Geostrophic wind
Surface roughness lengths
Model
0.3
Vg 5 5 m s21
z0,m 5 1.0 m, z0,h 5 0.1 m
z-less
Critical Ri number
Geostrophic wind
Surface roughness lengths
Model
where l is sequentially increased until Ri exceeds Rc.
Then the boundary layer depth is computed by interpolating the bulk Richardson number between this final
level where Ri . Rc and the model level immediately
below, so that h is the level where the bulk Richardson
number equals the critical value. Unless otherwise noted, Rc is assigned to be 0.3. Even though the boundary
layer may temporarily collapse in very stable conditions,
such a collapse is not likely to occur over an entire grid
area and we will retain the lowest model level as the
minimum boundary layer depth.
d. Turbulence decay
The traditional boundary layer formulation and the
local z-less formulation are both diagnostic relationships. As a result, the eddy diffusivity may decrease
discontinuously with time. In reality, the turbulence cannot decrease faster than the turbulence decay rate. For
neutrally stratified flow, this decay rate is inversely related to the size of the turbulence eddies as simulated
in Nieuwstadt and Brost (1986) and others. Here we
impose a simple condition on the maximum allowed
rate of decay,
1 t 2,
K m (z, t 1 d t) . K m (z, t) exp 2
dt
(10)
where dt is the model time step and t is the turbulence
decay timescale. This test value [right-hand side of Eq.
(10)] is the minimum allowed value for the eddy diffusivity. If the usual diagnostic formulation predicts a
smaller value of the eddy diffusivity, then Eq. (10) is
activated. Although in reality, t depends on the initial
properties of the turbulence and the stratification, we
specify a constant decay rate corresponding to t 5 300
s. This value is based on casual comparisons with the
observations in section 5.
3. Data
We initialize the model with data from 0600 Central
Standard Time 16 June 1986 from the Hydrological Atmospheric Pilot Experiment–Modélisation du Bilan
Hydrique (HAPEX–MOBILHY; Ek and Mahrt 1991b)
for the sensitivity tests in section 4. On this day, skies
were clear and synoptic horizontal gradients were weak.
Roughness lengths of 1.0 m for momentum and 0.1 m
for heat are applied (Holtslag and Ek 1996). The geo-
VOLUME 129
Value
0.3
Time–height-independent Vg 5 5 m s21
z0,m 5 1.0 m, z0,h 5 0.1 m
z-less and decay condition
strophic wind was assumed to be constant from the south
at 5 m s 21 with a prescribed subsidence of 1.0 cm s 21
at 2 km, which decreases linearly to zero at the surface.
We also analyze observations of the nocturnal boundary layer (NBL) over a grassland from the Cooperative
Atmosphere-Surface Exchange Study–1999 (CASES99) data (Blumen 1999). This field study was conducted in south-central Kansas (37.638N, 96.778W).
Mean profiles are constructed from six levels of aspirated thermistors, four levels of cup anemometers, and
eight levels of sonic anemometers on a 55-m tower.
Momentum and buoyancy flux profiles are constructed
from eight levels of sonic anemometers. The roughness
length for momentum is estimated to be 0.03 m based
on data from the nearby Microfronts experiment over a
similar surface (Sun 1999), and the thermal roughness
length is chosen to be an order of magnitude smaller
with a value of 0.003 m.
The asymptotic mixing length and decay timescale
(section 2) were subjectively selected by comparing the
general behavior of the model with data from CASES99.
Tuning coefficients or quantitative evaluation of the
model is not possible because the observations correspond to complex time–height-dependent pressure gradient fields, which cannot be adequately estimated from
the observations. Transient mesoscale motions are always present but are not captured by the model. As a
result, finetuning of the coefficients would actually be
compensating for inadequate knowledge of the timedependent pressure gradient field. Furthermore, individual physical mechanisms cannot be isolated from the
data. For example, the turbulence is never in a pure state
of decay, even in the residual layer immediately after
the collapse of the daytime boundary layer.
4. Numerical simulations
This section performs sensitivity tests for the HAPEX-MOBILHY case. We first examine the influence of
vertical resolution and critical Richardson number on
TABLE 3. Comparison of the modified model with CASES99.
Parameter
Critical Ri number
Geostrophic wind
Surface roughness lengths
Model
Value
0.3
Height-dependent Vg
z0,m 5 0.03 m, z0,h 5 0.003 m
z-less, decay condition
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FIG. 2. The height of boundary layer top for near-surface
resolution models of 2.5, 5, 10, and 20 m during nighttime.
FIG. 1. Vertical grid structure below 200 m for grids with nearsurface resolution of 2.5, 5, 10, and 20 m.
the original model and sequentially add the modifications in section 4b.
a. Influence of vertical resolution and critical Ri in
NBL
In very stable conditions, the boundary layer may
become quite thin and the vertical resolution of the model becomes a limiting factor. We therefore compare results using different vertical resolution near the surface
(Fig. 1). With significant stability, finer resolution predicts thinner nocturnal boundary layers. For fine vertical
resolutions, the boundary layer depth is thinnest in the
early evening, corresponding to the early evening very
stable period observed by Businger (1973), Mahrt et al.
(1998), and others. The 2.5-, 5-, and 10-m resolutions
capture the early evening very stable period while the
coarser vertical resolutions tend to miss this feature (Fig.
2). Since the 2.5- and 5-m resolutions produce similar
results, we will use the 5-m vertical grid spacing. The
maximum eddy diffusivity for the 2.5- and 5-m resolution models is much larger and occurs at a lower level
compared to the coarser resolutions (Fig. 3). Fine vertical resolution is essential for modeling the very stable
boundary layer since the very thin stable boundary layer
corresponds to rapid change of the mean variables with
height, especially near the surface.
The specified value for the critical Richardson num-
FIG. 3. The eddy diffusivity profile for the different model resolutions for 0000 and 0600 LST.
The solid, dotted, and dashed lines denote 5-, 10-, and 20-m resolution models, respectively.
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FIG. 4. Time dependence of the boundary layer depth for critical
Richardson numbers of 0.3, 0.5, and 1.0.
ber (Rc) controls the boundary layer depth through relationship Eq. (15). We now compare the results for Rc
5 0.5, 1.0 with the results for the control case where
Rc 5 0.3. For Rc 5 1.0, the boundary layer depth is
significantly larger and the early evening very stable
period is not captured (Fig. 4).
b. Modifications to the model
The local z-less formulation is now added above and
within the boundary layer, subject to the constraint on
the mixing length [Eq. (8)]. The z-less formulation
above the boundary layer generates significant turbulence (eddy diffusivity) just above the nocturnal boundary layer due to developing shear and very weak stratification (Fig. 5), as also observed in Cuxart et al.
(2000). Consider the region of lightest shading (strongest turbulence) above the boundary layer in Fig. 5. The
bottom of this layer increases with time due to the decay
and buoyancy destruction of the turbulence. However,
the top of this layer also increases with time due to
increased shear generation associated with the accelerating flow above the boundary layer. The layer of
significant turbulence becomes thickest around midnight
when the low-level jet is close to maximum strength.
The original Kim and Mahrt formulation for z-less
mixing eliminates the early evening very stable period
(Fig. 6). Our modified formulation captures this period
of weak mixing and strong surface stratification, even
though the turbulence of the z-less formulation produces
greater mixing than the original unmodified model. The
locally generated z-less turbulence leads to a more realistic smooth wind maximum in contrast to the traditional model, which develops a sharp peak in the wind
profile (Fig. 7).
The eddy diffusivity/mixing length formulations can
lead to unrealistic instantaneous collapse of the turbulence. Application of the decay formulation [Eq. (10)]
allows for a more realistic collapse of the turbulence
over a finite time period, as occurred in Fig. 5. With
the finite decay condition, the turbulence collapse in the
early evening is less dramatic and significant turbulence
FIG. 5. The time–height cross section of the eddy diffusivity K
(m 2 s 21 ) with local z-less turbulence. The thick dotted line denotes
the boundary layer top, which plays a minor role due to the dominance
of z-less turbulence. The ‘‘S’’ symbols on the x axis show sunset and
sunrise.
is more likely to occur detached from the surface (Fig.
8). However, the overall features with and without the
finite decay condition are similar.
The number of levels where each condition is activated is shown in Fig. 9 as a function of time. In this
case, the z-less condition (triangles) is activated at almost every level under stable conditions, except near
the surface. With stronger geostrophic wind speed or
less net radiative cooling, the z-less condition is activated less frequently. The decay condition (plusses) is
activated in the early evening and intermittently
throughout the night.
5. Application to CASES99 data
To summarize, the modified model includes the possibility of local z-less mixing subject to the near-surface
FIG. 6. The time evolution of eddy diffusivities for the Kim and
Mahrt model (KM; dotted), the original model with no modifications
(solid), and with z-less turbulence added (dashed).
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FIG. 9. The number of model levels where the z-less formulation
(D) and decay condition (1) are activated. The model contains 83
levels.
FIG. 7. The wind speed V (m s 21 ) profile for the unmodified model
(solid line) and with z-less turbulence (dotted line) at 0300 LST.
constraint [Eq. (8)] and the finite decay condition [Eq.
(10)]. To evaluate the utility of the modified model and
to identify additional problems, we now compare the
model to the CASES99 data. On some of the nights,
the observed stress increased with height across the 55m tower layer. Traditional boundary layer models cannot
simulate this situation. As an example, we compare re-
FIG. 8. The time–height section for eddy diffusivity K (m 2 s 21 )
simulated by the modified model with z-less turbulence and the finite
decay condition. The dotted line indicates the height of boundarylayer top. The ‘‘S’’ symbols on the x axis show sunset and sunrise,
respectively.
sults from the modified model with the data from the
evening beginning on 20 October (Fig. 10). Initial profiles are taken from slow ascent radiosonde data (http:
//www.atd.ucar.edu/sssf/projects/cases99) at 0600 local
time. To produce vertical structure of the wind field,
which on average is in rough agreement with the observed wind field, the time-independent geostrophic
wind is specified to increase from 3 m s 21 at the surface
to 6 m s 21 at 150 m.
Between 2100 and 0100 local time, the observed
FIG. 10. The observed and modeled nocturnal u* (m s 21 ) for 20
Oct 1999 of CASES99.
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ardson number increases. The turbulence then decays
and the shear rebuilds, and so on. The timescale for this
intermittency is approximately 1 h. The observed stress
also exhibits variations on this timescale, although they
are largely masked by mesoscale variations on a timescale greater than 1 h.
The early evening very stable period for the modified
model is less well defined compared to the data. While
the model does show an early evening minimum in the
eddy diffusivity, it is not as evident in terms of the
friction velocity.
6. Conclusions
FIG. 11. Same as in Fig. 10 but for wind speed V (m s 21).
stress increases with height throughout the tower layer
(Fig. 10), apparently due to enhanced shear (Fig. 11)
associated with the nocturnal low-level jet, which occurs
several hundred meters above the ground surface. The
corresponding vertical convergence of the downward
momentum flux accelerates the low-level flow. The original unmodified model (not shown) is incapable of producing this feature of the flow. In the modified model,
the momentum flux intermittently increases with height
(Fig. 10) but not as systematically as in the observations.
The observed data show significant mesoscale variations on a timescale of several hours (Figs. 10–11).
The shear associated with such motions apparently generates significant turbulence independently of surface
processes. The model wind field does not include this
transient behavior and consequently predicts weaker
shear above 25 m and generally weaker momentum flux
(Fig. 11). Adjusting model coefficients to compare more
closely to the data is not attempted since the differences
are in part due to absence of these mesoscale variations
in the model. A subsequent study will focus on the
influence of transient motions, which were present during all of the nights. Since such transient motions may
not be captured by even regional models, modeling the
details of such nocturnal boundary layers is a difficult
endeavor.
The model does produce intermittent z-less turbulence
due to the internal relationship between the shear and
mixing. As the shear increases due to flow acceleration,
turbulence is generated. The resulting mixing reduces
the shear and stratification. Since the Richardson number depends quadratically on the mean shear, the Rich-
We have modified a simple model of the nocturnal
boundary layer where the diffusivity was originally
specified only as a function of z/h, where h is the boundary layer depth. The modifications allow development
of z-less turbulence, which depends neither on height
above ground nor on the boundary layer depth. The
modified model generates turbulence above the boundary layer. The modifications provide for a finite decay
rate, which replaces the unrealistic instantaneous collapse occurring in the usual diagnostic relationships for
the eddy diffusivity. The finite decay condition allows
some persisting turbulence in the residual layer in the
early evening, not present in the traditional model. The
modeled z-less turbulence produces a smoother and
more realistic nocturnal wind maximum, without an unrealistic sharp peak, characteristic of the traditional
model. A vertical resolution of 5 m or finer is required
to produce the observed early evening very stable period.
The above model improvements were possible without a significant increase in complexity or computer
time. However, the model remains oversimplified and
major differences between the model and the observations remain. The boundary layer sometimes seems nonexistent or undefinable from the observations in that all
of the significant turbulence is detached from the surface. The locally generated z-less turbulence in the model can produce an ‘‘upside-down’’ structure where the
momentum transport increases with height. However,
this feature is weak and too transient in the model compared to the observations, apparently due to elevated
shear induced by mesoscale motions not captured by
the model. The observed turbulence is strongly modulated by mesoscale variations of the wind speed on a
timescale of a few hours, which is not captured in the
model. This inadequacy is currently under investigation.
Acknowledgments. We gratefully acknowledge Jinwon Kim and Michael Ek for helpful discussions and
the useful comments of the Natalie Perlin and the reviewers on the manuscript. This material is based upon
work supported by Grant DAAD 19-9910249 from the
Army Research Office and Grant 9807768-ATM from
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HA AND MAHRT
the Physical Meteorology Program of the National Sciences Foundation.
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