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1 . Introduction
1.1 The planetary boundary layer
The planetary boundary layer (PBL) is the
region of the atmosphere near the surface where the influence of the surface
is felt through turbulent exchange of momentum, heat and moisture. The equations
which describe the largescale evolution of the atmosphere do not take into
account the interaction with the surface. The turbulent motion responsible
for this interaction is smallscale and totally subgrid and therefore needs
to be parametrized.
The transition region between the surface
and the free atmosphere, where vertical diffusion due to turbulent motion
takes place, varies in depth. The PBL can be as shallow as 100 m during
night over land and go up to a few thousand metres when the atmosphere is
heated from the surface. The PBL parametrization determines together with
the surface parametrization the surface fluxes and redistributes the surface
fluxes over the boundary layer depth. Boundary layer processes are relatively
quick; the PBL responds to its forcing within a few hours which is fast
compared to the time scale of the largescale evolution of the atmosphere,
in other words: the PBL is always in quasiequilibrium with the largescale
forcing.
1.2 Importance of the PBL in largescale
models
A number of reasons exists to have a realistic
representation of the boundary layer in a large scale model:

• The largescale budgets
of momentum heat and moisture are considerably affected by the surface
fluxes on time scales of a few days. 

• Model variables in the
boundary layer are important model products. 

• The boundary layer interacts
with other processes e.g. clouds and convection. 
The importance of the surface fluxes can
be illustrated by estimating the recycle time of the different quantities
on the basis of typical values of the surface fluxes. The numbers in Table
1 have been derived from a typical run with the ECMWF model or are simply
order of magnitudes estimations.
Already from these very crude estimates
it can be seen that the surface fluxes play an important role in forecasting
for the medium range. It is obvious that the surface fluxes are crucial
for the "climate" of the model. With regard to the momentum budget it should
be noted that the Ekman spindown time has to be considered as the relevant
time scale because it is an efficient mechanism for spinning down vorticity
in the entire atmosphere by frictional stress in the PBL only.
The second important reason to have a PBL
scheme in a largescale model is that forecast products are needed near
the surface. The temperature and wind at standard observation height (2
m and 10 m for temperature and wind respectively) are obvious products.
It is important to realize that the PBL scheme together with the land surface
and radiation scheme introduces the diurnal pattern in the surface fields.
Also the analysed and forecasted fields of surface fluxes (momentum, heat
and moisture) are becoming more and more important as input and verification
for wave models, air pollution models and climate models.
Table 1. Global budgets (order
of magnitude estimates)
Budget 
Total 
Surface flux 
Recycle time 
Water 
^{} J m^{2} 
80 W m^{2} 
10 days 
Internal +potential energy 
^{} J m^{2}(0.5% available) 
30 W m^{2} 
8 days 
Kinetic energy 
J m^{2} 
2 W m^{2} 
10 days 
Momentum 
kg m s^{1} 
0.1 N ^{m2} 
25 days(Eckman spindown time: 4 days 
Finally it has to be realized that other
processes can not be parametrized properly without having a PBL scheme.
Boundary layer clouds are an obvious example, but also convection schemes
often use surface fluxes of moisture in their closure.
1.3 Recommended literature
General textbook and introductory review
on most aspects of the PBL:

Stull, R.B. (1988): An introduction
to boundary layer meteorology. Kluwer publishers. 
Introduction to turbulence:

Tennekes, H. and Lumley, J.L.
(1972): A first course in turbulence. MIT press. 
Atmospheric turbulence:

Nieuwstadt, F.T.M. and Van Dop,
H. (eds. 1982): Atmospheric turbulence and air pollution modelling.
Reidel publishers. 

Haugen, D.A. (ed. 1973):
Workshop on micro meteorology. Am. Meteor. Soc. 

Monin, A.S. and Yaglom, A.M.
(1971): Statistical fluid dynamics. Vol I, MIT press. 

Panofsky, H.A. and Dutton, J.A.
(1984): Atmospheric turbulence: Models and methods for engineering
applications. John Wiley and sons. 
Surface fluxes:

Oke, T.R. (1978): Boundary
layer climates. Halsted press. 

Brutsaert, W. (1982): Evaporation
into the atmosphere. Reidel publishers. 
1.4 General characteristics of the planetary
boundary layer
Turbulence
The diffusive processes in the atmospheric
boundary are dominated by turbulence. The molecular diffusion can in general
be neglected except in a shallow layer (a few mm only) near the surface.
The time scale of the turbulent motion ranges from a few seconds for the
small eddies to about half an hour for the biggest eddies (see Fig. 1 ). The length scales vary from millimetres
for the dissipative fluctuations to a few hundred metres for the eddies
in the bulk of the boundary layer. The latter ones dominate the diffusive
properties of the turbulent boundary layer. It is clear that these scales
cannot be resolved by largescale models and need to be parametrized.
Figure 1 . Spectrum of the horizontal
wind velocity after Van Der Hoven (1957). Some experimental points
are shown.
Stability
The structure of the atmospheric boundary
layer is influenced by the underlying surface and by the stability of the
PBL (see Fig. 2 ).
The surface roughness determines to a certain
extent the amount of turbulence production, the surface stress and the shape
of the wind profile. Stability influences the structure of turbulence. In
an unstably stratified PBL (e.g. during daytime over land with an upward
heat flux from the surface) the turbulence production is enhanced and the
exchange is intensified resulting in a more uniform distribution of momentum,
potential temperature and specific humidity. In a stably stratified boundary
layer (e.g. during nighttime over land) the turbulence produced by shear
is suppressed by the stratification resulting in a weak exchange and a weak
coupling with the surface. The qualitative impact of these aspect is illustrated
in Fig. 2
Diurnal pattern
The typical evolution of the PBL over land
is illustrated in Fig. 3 for
a 24 hour interval. During daytime, with an upward heat flux from the surface,
the turbulent mixing is very strong, resulting in approximately uniform
profiles of potential temperature and wind over the bulk of the boundary
layer. The unstable boundary layer is therefore often called "mixed layer".
Near the surface we see a super adiabatic layer and a strong wind gradient.
The top of the mixed layer is capped by an inversion which inhibits the
turbulent motion (e.g. the rising thermals) to penetrate aloft. The inversion
height rises quickly early in the morning an reaches a height of a few kilometres
during daytime. When the heat flux from the surface changes sign at night
the turbulence in the mixed layer dies out and a shallow stable layer near
the surface develops. The nocturnal boundary layer has a height of typically
50 to 200 m dependent on wind speed and stability.
Figure 2 . The wind speed profile
near the ground including (a) the effect of terrain roughness, and (b)
to (e) the effect of stability on profile shape end eddy structure. In
(e) the profiles of (b) to (d) are replotted with a logarithmic height
scale. (Fig. 2.10 from Oke, 1978).
Figure 3 . (a) Diurnal variation
of the boundary layer on an "ideal" day. (b) Idealized mean profiles of
potential temperature ( ), wind speed (U) and vapour density (_{} ) for the daytime convective boundary layer. (c) Same
as (b) for nocturnal stable layer. The arrows indicate sunrise and sunset.
Surface fluxes
Figure 4 . Diurnal energy balance
of (a) a Scots and Corsican pine forest at Thetford (England) on 7 July
1971, and (b) a Douglas fir forest at Haney (B.C. Canada) on 10 July 1970,
including (c) the atmospheric vapour pressure deficit. In these figures
Q_{*} represents net radiation, Q_{H} the sensible heat
flux, Q_{E} the latent heat flux and Q_{s} the soil heat flux. (Fig. 4.24 from
Oke 1978).
The surface fluxes of momentum, heat an
moisture are of crucial importance to the model as they affect the climate
of the model, the model performance in the medium range and play an integral
role in a number of parametrizations. The diurnal pattern of the PBL over
land is mainly forced by the energy budget at the surface through the diurnal
evolution of the net radiation at the surface as illustrated in Fig.
4 . During daytime the net radiation is partitioned between the heat
flux into the ground, the sensible heat flux into the atmosphere and the
latent heat of evaporation. The ground heat flux is generally smaller than
10% of the net radiation during daytime. The partitioning of available net
radiation between sensible and latent heat is part of the land surface parametrization
scheme. The fluxes are much smaller during nighttime and are much less
important for the atmospheric budgets but equally relevant for the prediction
of boundary layer parameters.
The boundary layer over sea does not have
a distinct diurnal pattern but can be stable and unstable dependent of the
air type that is advected relative to the sea surface temperature. Strong
cold air advection over a relatively warm sea can lead to extremely high
fluxes of sensible and latent heat into the atmosphere. Warm air advection
over a cold sea leads to a stable PBL but does occur less frequently.
It is important to realize that the thermodynamic
surface boundary conditions over sea are very different from those over
land. Over sea the temperature and specific humidity are specified and kept
constant during the forecast. Over land the surface boundary condition for
temperature and moisture are nearly flux boundary conditions, because of
the constraint imposed by the surface energy budget. In the latter case,
the fluxes are determined by the net radiation at the surface. This means
that the total energy input into the atmosphere (sensible + latent heat)
is not so much determined by the flow or by the PBL parametrization but
by the net radiation at the surface. The sea however, is, with its fixed
SST boundary condition, an infinite source of energy to the model. The specification
of the PBL exchange with the sea surface is therefore extremely critical.
Beljaars and Miller (1991) give
an example of model sensitivity to the parametrization of surface fluxes
over tropical oceans.
PBL clouds
Cumulus clouds, stratocumulus clouds and
fog are very much part of the boundary layer dynamics and interact strongly
with radiation.
1.5 Conserved quantities and static stability
To describe vertical diffusion by turbulent
motion we have to select variables that are conserved for adiabatic processes.
In the hydrostatic approximation both potential temperature and dry static energy are conserved for dry adiabatic
ascent or descent. They are defined as
When moist processes are considered as well
it is necessary to use liquid water potential temperature
_{l} or the liquid water static energy s_{l}
and total water content q_{t}:
The static stability is determined by the
density of a fluid parcel moved adiabatically to a reference height in comparison
with the density of the surrounding fluid. The virtual potential temperature
_{v} and the virtual dry static energy
are often used for this purpose:
1.6 Basic equations
To illustrate the closure problem we derive
the Reynolds equations for momentum, starting from the three momentum equations
for incompressible flow in a rectangular coordinate system with z perpendicular
to the earth surface:
Decomposition of each variable into a mean
part and a fluctuating (turbulent) part:
After averaging, application of the Boussinesq
approximation (retain density fluctuations in the buoyancy terms only) and
applying the hydrostatic approximation:
Further simplifications are obtained by
neglecting viscous effects for large Reynolds numbers (UL/ >> 1, where L is a length scale) and by
assuming that the x, yscales are much larger than the zscales
(boundary layer approximation):
These equations describe the horizontal
momentum budget of the resolved flow and have extra terms due to the Reynolds
decomposition and averaging procedure. The terms and are known as the Reynolds stresses and represent
vertical transport of horizontal momentum by unresolved turbulent motion.
These are the terms that need to be parametrized. Similar equations exist
for potential temperature and specific humidity.
An equation that is important in some parametrization
schemes and gives insight in the dynamics of turbulence is the equation
for turbulent kinetic energy. To derive it, the equations for the fluctuating
quantities are taken by subtracting the equations for mean momentum from
the equations for total momentum. The kinetic energy of the fluctuations
is obtained by multiplying the equations for the different components by
the velocity fluctuation itself and by adding the three equations for the
three components. The result is:
The left hand side of this equation represents
time dependence and advection. The right hand side has source, sink and
transport terms. Terms I represent mechanical production of turbulence by
wind shear and convert energy of the mean flow into turbulence. Term two
is the production of turbulence by buoyancy and converts potential energy
of the atmosphere to turbulence or vice versa. Terms III and IV are transport
or turbulent diffusion terms because they equal zero when vertically integrated
over the domain. They represent the vertical transport of turbulence energy
by the turbulent fluctuations and the pressure fluctuations respectively.
The last term (V) is the dissipation term which converts turbulence kinetic
energy into heat by molecular friction at very small scales.
1.7 Ekman equation
To illustrate the mechanism of Ekman pumping
we consider the following equations for the steady state boundary layer:
For a simple closure with , and constant K we rewrite the equations in complex notation:
Figure 5 . (a) Hodograph of Ekman
spiral; the vectors indicate the increase of wind speed and its veering
with height, (b) Illustration of the ageostrophic wind in the PBL of a
cyclone, and (c) vertical velocity in cyclone due to Ekman pumping (Holton, 1979).
The solution with U = V = 0 at the
surface and U = U,V = V far away from the surface, reads:
The vertical velocity Wh at the top
of the PBL is derived from the continuity equation by integration from the
surface to boundary layer depth h, where h is large.
We see that the vertical velocity at the
top of the PBL is proportional to the curl of the surface stress and to
the curl of the geostrophic wind. This qualitative result is in fact independent
of how K is exactly specified. The vertical velocity can bee seen
as a boundary condition for the inviscid flow above the boundary layer.
The inflow of air in a cyclone due to friction at the surface causes an
ascending motion at the top of the PBL and spins down the vortex by vortex
compression (conservation of angular momentum). This mechanism is very important
because it transfers the effect of boundary layer friction to the entire
troposphere. Ekman pumping is illustrated in Fig.
5 .
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