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The parametrization of the planetary
boundary layer
May 1992
By Anton Beljaars
European Centre for Medium-Range Weather
Forecasts
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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 large-scale evolution of the atmosphere do not take into account the interaction with the surface. The turbulent motion responsible for this interaction is small-scale and totally sub-grid 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 large-scale evolution of the atmosphere, in other words: the PBL is always in quasi-equilibrium with the large-scale forcing.
1.2 Importance of the PBL in large-scale models
A number of reasons exists to have a realistic representation of the boundary layer in a large scale model:
| • The large-scale 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 spin-down 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 large-scale 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.
| 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 m-2 | 25 days(Eckman spin-down 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 large-scale models and need to be parametrized.
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 day-time 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 night-time 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.
), 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
Qs 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 night-time 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
|
(1) |
|
(2) |
When moist processes are considered as well it is necessary to use liquid water potential temperature
l or the liquid water static energy sl
and total water content qt:
|
(3) |
|
(4) |
|
(5) |
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:
|
(6) |
|
(7) |
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:
|
(8) |
|
(9) |
Decomposition of each variable into a mean part and a fluctuating (turbulent) part:
|
(10) |
After averaging, application of the Boussinesq approximation (retain density fluctuations in the buoyancy terms only) and applying the hydrostatic approximation:
|
(11) |
|
(12) |
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, y-scales are much larger than the z-scales
(boundary layer approximation):
|
(13) |
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:
|
(14) |
|
(15) |
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:
|
(16) |
|
(17) |
For a simple closure with
, 
and constant K we rewrite the equations in complex notation:
|
(18) |
The solution with U = V = 0 at the surface and U = U,V = V far away from the surface, reads:
|
(19) |
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.
|
(20) |
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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