Intermittent turbulence and oscillations in the stable boundary layer:
a system dynamics approach
Bas J.H. van de Wiel* , Arnold F. Moene, Oscar K. Hartogensis,
Reinder J. Ronda, Henk A.R. De Bruin, Albert A.M. Holtslag
Wageningen University, The Netherlands
* Corresponding author address: Bas van de Wiel,
Dept. of Meteor. and Air Qual. Duivendaal 2, 6701
AP, Wageningen, The Netherlands;
The stable boundary layer (SBL) is often
characterised by turbulence which is not continuous in
space and time. This so-called intermittent turbulence
may affect the whole depth of the SBL. It is often
reflected by oscilatory behaviour of the near surface
temperature and wind speed. In the past this
intermittency phenomenon has been studied mostly
numerically (e.g. Revelle 1993, McNider 1995). In this
study we use both a numerical as an analytical
We focus on clear nights over land formed by
cooling due to the strong surface radiation. In such
cases often stability develops faster than shear, which
causes the Richardson number to increase. As a
consequence air becomes decoupled from the
surface. Soon, however, air will be accelerated by the
omnipresent pressure force until shear is strong
enough to break down the stability causing a
turbulence burst. Because of this strong mixing, shear
is rapidly reduced and stability takes over. The
situation has returned to its begin and the mechanism
starts over again, causing intermittent bursts of
turbulence and oscillations in the mean variables
Our goal is to describe and predict the above
mentioned mechanism. We will use a simple physical
SBL model, which is integrated numerically. Also, the
model is studied analytically from a system dynamics
point of view. This approach results in a
dimensionless parameter, which is a function of
external parameters controlling the system. With this
parameter the equilibrium behaviour of the model (i.e.
oscillatory or non-oscillatory) can be predicted.
2. MODEL DESCRIPTION
Figure 1: The model system: state variables, fluxes
and model domain
Point of departure for the SBL model are the
conservation equations for momentum and heat.
Complexity of the model system is kept to a minimum,
while preserving most important physical processes.
By doing so the structure and dynamic interaction
between the governing equations can be studied.
The conservation equations for momentum and heat
for our system are given by:
Some model characteristics/assumptions are:
1) Assumption of a fixed boundary layer height: no
turbulent exchange with air above boundary
2) No coriolis effects and no moisture effects.
3) First order turbulence closure, exchange
coefficient depending on bulk Ri.
4) Emissivity approach for radiation.
5) Soil heat flux parameterized by vegetation
resistence law (alternatively by a force-restore
method for bare soil).
Due to this assumptions the generality of the present
results may be limited. In future research the present
framework will be extended to more general cases.
The equations are integrated over the
domain of interest, which results in a system of three
coupled non-linear differential equations describing
the time evolution of the integrated values: U,Ta and
Ts. This system is used in both the numerical and
3. MODEL RESULTS
The model presented in section 2 was integrated
numerically. The time development of the surface
temperature during a 10 hour transient run is shown in
A general decrease in surface temperature is seen, as
is commonly observed in nocturnal conditions. After
some time the temperature increases suddenly and
drops back to the general trend after a short time.
This result follows the results reported by Revelle
(1993), who used a multi-layer model with comparable
flux parametrisations for turbulence, soil heat flux and
radiation, however incorporating Coriolis effects.
The time between the temperature peaks of about 1-2
hours is comparable with the time between
temperature peaks reported by Revelle (i.e. 30-240
min.). The peak height of 4-5 K agrees with the peak
height of the near surface temperature of about 5 K as
in Revelle. Thus, the truncated model presented here
essentially shows the same type of behaviour as the
more complex model.
Figure 2: calculated time developement of surface
temperature in a 10 hour transient run.
As an illustration in Fig. 3 also the intermittent
behaviour of the turbulent fluxes is shown for the case
given above (note that stationary solutions after 30
hrs. are shown).
30 32 34 3638 40
Sens. heat flux
Sens. heat flux [W/m2]
Figure 3: modelled fluxes in an equilibrium situation.
Furthermore, uppon a varying pressure gradient
(keeping the other parametes at a constant value), in
an equilibrium situation three regimes seems to exist
in the present model:
a) The pressure gradient is weak. The
equilibrium solution is non-oscillating, with
weak turbulence resulting in a low surface
The pressure gradient is strong. The
equilibrium solution is non-oscillating, with
strong turbulence resulting in a relatively high
The pressure gradient is moderate. The
equilibrium solution is oscillating, with
intermittent turbulence and intermediate, but
oscillating surface temperatures.
Of course, for each combination of external
parameters such as cloud cover and surface
roughness, the transition between the three regimes
occurs at different values of pressure gradient. In the
following a method to predict the equilibrium
behaviour of the model as a function of external
parameters is presented.
4. ANALYTICAL ANALYSIS
Our symplified system is governed by three coupled
non-linear differential equations containing three
unknown (internal) variables U, Ta and Ts. Analytical
stability analysis of the system equilibria shows that
the transitions between the three flow regimes are
manifestations of two Hopf-bifurcations connecting a
non-oscillatory solution (weakly turbulent case) to an
oscillatory solution (intermittent case) and connecting
this oscillatory solution in turn to a non-oscillatory
solution (strongly turbulent case). Application of
analytical bifurcation theory results in a dimensionless
parameter (denoted with Pi), which is a function of the
external model parameters such as air emissivity,
cloud cover and surface roughness. By evaluation of
this dimensionless parameter, the occurence of a
Hopf-bifurcation in the model can be predicted. Thus
the equilibrium behaviour of internal model variables
(i.e. oscillating or non-oscillating), can be predicted
from the evaluation of the external parameters. Due
to the very complex structure of this dimensionless
parameter (i.e.over one page length) , its detailed
form is not presented here. The exact form will be
reported in Van de Wiel et al. (2002b). It can be
shown that the critical value of Pi is equal to the
numerical value one. Thus, the equilibrium solutions
of the model can be devided in:
Pi <1; oscillatory equilibrium behaviour
Pi ≥1; non- oscillatory equilibrium behaviour
5. COMPARISON OF ANALYTICAL AND
In Fig. 4, for a typical situation, the value of Pi is
shown as a function of the pressure gradient (other
external parameters constant). Our analytical analysis
reveals that a transition in flow behaviour is expected
at Pi=1. In case of Fig. 4, Pi equals 1 for two different
values of pressure gradient. This means that, if the
pressure gradient is gradually increased from low to
high values, two transitions in flow behaviour are
Pressure Gradient (*10e-4) [m/s^2]
Figure 4: Dimensionless value of Pi as a function of
the applied pressure gradient.
In Figs. 5a and 5b, the results of five numerical runs
(A to E) are shown, corresponding to five different
values of pressure gradient as depicted in Fig. 4.
4042 444648 50
Ts [K] run A
Ts [K] run B
Fig. 5a: Surface temperature in an equilibrium
situation for different values of pressure gradient
4042 44 4648 50
Ts [K] run C
Ts [K] run D
Ts [K] run E
Fig. 5b: Surface temperature in an equilibrium
situation for different values of pressure gradient
A comparison of Figs.5a and 5b with Fig. 4, learns
that indeed there seems to occur a transition in flow
behaviour at two positions as was predicted
independently by the analytical model. Several
hundreds of additional runs were made, varying all
other external parameters. For all cases this again
resulted in oscillating behaviour for Pi <1and non-
oscillating behaviour for Pi ≥1.
6. SBL CLASSIFICATION
We propose to use the dimensionless Pi parameter
as a classification parameter dividing equilibrium
behaviour in: oscillatory behaviour ( Pi <1) and non-
oscillatory behaviour ( Pi ≥1).
Two important parameters determining the equilibrium
model behaviour are the pressure gradient and the
IsoThermal net Radiation (ITR). The isothermal net
radiation is defined as the net radiation which would
occur at the surface when the surface layer is
isothermal, i.e. Ts=Ta (Holtslag and De Bruin 1988). It
is the maximum value (in absolute sense) for the net
radiation that may occur in the SBL and is determined
by the emissivities of the surface and the atmospher.
As an illustration in Fig. 6 the dependende of Pi on the
ITR (for convenience the fraction of cloud cover
corresponding to the ITR values are given) and the
pressure gradient is given in a contour plot.
Figure 6: Contourplot of Pi =1 as a function of
pressure gradient and isothermal net radiation
All points within the contour line Pi=1 correspond to
the oscillatory cases.
In Fig. 6 it is seen that, up to a certain level of
moderate ITR, three regimes exist cf. section 3. In fact
the oscillating regime appears to split a single regime
of non-periodic flow. Because Pi depends on all
external parameters, the shape of Fig. 6 also depends
on the other external parameters. Sensitivity analysis
shown that the, surface roughness, the boundary
layer heat and the soil heat capacity are important
parameters determining the equilibrium behaviour.
In Fig. 7 an example of intermittent turbulence is
given. The intermittent turbulence (sonic 5 min. avg.)
was observed on a clear night (4/5 Oct. 1999) by the
Wageningen University Group during the extensive
field campaign CASES99, Kansas (Poulos et al.
2002). From Fig. 7 it occurs that both the amplitude
and the period of the turbulent periods show an
irregular behaviour. Some turbulent events have an
amplitude of 5 [W/m2] and a duration of 30 minutes,
others an amplitude of 25 [W/m2] and a duration of 4
It was found that the amplitudes and duration of
turbulent and the quiet periods simulated by the
model are comparable to the typically observed
amplitudes and periods (compare Figs. 7,8 and 3).
Of course, in each modelled case the exact periods
and amplitudes depend on the particular values of the
imposed external variables. Due to the simplicity of
the model (e.g. homogeity, stationarity of external
variables) the modelled flux pattern is purely regular,
Pressure gradient (*10e-4) [m/s^2]
Isotherm al net radiation [W /m^2]
Cloud cover [-]
Non-oscillating lim it behaviour
lim it behaviour
contrary to the observed fluxes, which show variation Download full-text
in periods and amplitudes.
Local time [hr]
Figure 7: near-surface turbulent heat flux and net
radiation on 4/5 Oct.,CASES99.
An interesting result is given by the net-radiation
graph in Fig. 7. As expected the (abs. value) of the
net radiation decreases during the night due to the
strong surface cooling, which dimishes the outgoing
longwave radiation. However, superimposed on this
general trend the net-radiation shows small
From Fig. 7 it appears that the oscilations are highly
correlated with the fluctuations of the turbulent heat
flux. The oscillations in the net-radiation are a direct
consequence of oscillation in the surface temperature.
These oscillations of the surface temperature in turn
affect the stability of the near surface atmosphere
leading to an important feedback mechanism between
radiation and turbulence (section 1).
For the whole period of CASES99 (1month, Oct. ’99)
we classified the behaviour of near surface turbulence
using time series of the sensible heat flux and net-
radiation as in Fig. 7 (in combination with time-series
of the surface stress). It occurred that the
observations could be classified in three sub-
classes/regimes (Fig. 8):
1) A continuous turbulent regime
2) An intermittent regime
3) A radiative regime
To give an impression of the frequency of occurrence:
from the set of 27 nights 9 were classified as
continuous turbulent, 6 as intermittent, 4 as radiative
and 8 nights did not show a clear single regime (often
transitions between the different regimes), or had
incompleet data. Remark: due to the fact that a large
number of nights had clear-sky conditions the number
of intermittent and radiative nights is rather high. The
few cloudy nights showed continuous turbulent
behaviour. The existence of three subclasses/regimes
in the observations agrees (at least qualitatively) with
the predictions of the simplified model.
In order to interpret the observations of Fig. 8 in terms
of the classification of Fig. 6, we plotted the three
observed cases in Fig. 6 by (R)adiative, (I)ntermittent
and (T)urbulent. Because only clear-sky cases were
shown, they appear in the lower part of the graph.
Apart from this preliminary qualitative comparison the
authors are currently carrying out a quantitative
comparison between the model predictions and the
Local time [hr]
Heat flux [W/m2]
Figure 8: near-surface turbulent heat flux on three
clear-sky nights during CASES99.
8. CONCLUSIONS AND RECOMMENDATIONS
1) We conclude that the intermittency mechanism as
described in section 1, can be explained from a
system dynamics point of view as a Hopf bifurcation
connecting an oscillatory and non-oscillatory state of
the system. This property can be used in predicting
the equilibrium behaviour of the system.
2) The theoretical framework predicts the existence of
three regimes, with decreasing pressure gradient: a
continuous turbulent regime, an intermittent regime
and a radiative regime.
3) Observations seem to confirm the existence of
Both observational and theoretical work are in
progress, in order to study the intermittency
mechanism described in this text. The model
assumptions and restrictions need further attention
with respect to their theoretical and practical
consequences. Finally, to get a better picture of
intermittent turbulence occuring in the SBL, the
possible interaction of near-surface intermittency with
other processes such as wave formation and with
turbulence produced by elevated shear layers, needs
to be investigated.
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H. A. R. De Bruin, and A. A. M. Holtslag, 2002:
Intermittent turbulence and oscillations in the stable
boundary layer over land :
-Part I: A bulk model. J. Atmos. Sci. 59, 942-958.
-Part II:A symstem dynamics approach. J. Atmos.
Sci., in press.