Fourth Study Conference on BALTEX Scala Cinema Gudhjem

- 46 -
Is the Critical Bulk Richardson Number Constant ?
Sven-Erik Gryning 1 and Ekaterina Batchvarova 1,2
1) Wind Energy Department, Risø National Laboratory, DK-4000 Roskilde, DENMARK, Sven-Erik.Gryning@risoe.dk
2) National Institute of Meteorology and Hydrology, Bulgarian Academy of Sciences, 1784 Sofia, BULGARIA
1. Introduction
Water that evaporates from the sea surface into the
atmosphere becomes mixed throughout the boundary layer
up to the top that acts as a lid. The depth of the boundary
layer therefore is one of the parameters that controls the
water content in the air over the sea, and therefore has a
feedback on the evaporation from the water surface. It is
also one of the fundamental parameters used to characterise
the structure of the boundary layer.
The height of the boundary layer can be determined from
routinely available weather forecasts. Such forecasts are
based on simulations with Numerical Weather Prediction
models (NWP). The height of the boundary layer does not
form a part of the output from NWP models, but has to be
estimated from the available output data, usually by use of a
bulk Richardson number. Starting at the lower model level
the bulk Richardson number is determined at successive
greater heights by use of linear interpolation between
adjacent model levels. The boundary-layer top is assigned to
the height where the Richardson number exceeds a given
value, named the critical Richardson number and
traditionally taken to be 0.25. Here we discuss the value of
the critical Richardson number for the marine atmospheric
boundary layer.
2. NWP model
The study is based on output from The HIgh Resolution
Limited Area Model HIRLAM. Operationally, local
versions of the HIRLAM model are used, and in this study
we use HIRLAM data provided by the Swedish
Meteorological and Hydrological Institute. The horizontal
grid resolution is 22.5 times 22.5 km and there are 31
vertical levels. Output from the simulations includes hourly
profiles of wind (easterly u and northerly component v ),
temperature and humidity as function of the geopotential
height (given at the approximate levels 30, 150, 350, 600,
950, 1300, 1750, 2200, 2650…metres).
We compare two methods to extract the boundary-layer
height from the HIRLAM output data; both are based on a
bulk Richardson-number approach, but they differ in the
way the wind speed is taken into account. For both methods
the boundary-layer height is defined as the height where the
bulk Richardson number reaches a critical value.
Sørensen (1998) suggests the bulk Richardson number for
the layer between the surface and the height z above the
surface:
Ri
B
g z(
θv
( z)
−θ
v ( s))
= 2 2
θ ( s)(
u(
z)
+ v(
z)
)
v
. (1)
The quantities θ v (s)
and θ v (z)
are the virtual potential
temperatures at the surface (taken by Sørensen (1998) as the
lowest model level) and height z , respectively, u (z)
and
v (z)
are the horizontal wind components at height z , and
g is acceleration due to gravity.
Vogelezang and Holtslag (1996) suggest a bulk Richardsonnumber
where the wind is defined with respect to the lowest
model level s (here 30-m), and a term that accounts for
surface friction:
Ri
B
=
θ ( s)
v
g z(
θ ( z)
−θ
( s))
v v
2
2 2
[ ( u(
z)
− u(
s))
+ ( v(
z)
− v(
s))
+ bu
]
where b is a parameterisation constant, recommended to
be taken as 100.
Let us in gross terms consider the development of the
atmospheric boundary layer over land (high surface
roughness) and water (low surface roughness) for one and
the same heat fluxes. The height of the boundary layer
primarily is controlled by the heat flux as discussed by
Batchvarova and Gryning (1991). In both of the bulk
Richardson numbers, Eqs. (1) and (2), the effect of the
wind is accounted for in the denominator. Over water
owing to the small roughness length the wind speed is
typically higher than over land and with smaller friction
velocity. At a given height the Richardson number
suggested by Sørensen (1998) would tend to be smaller
over water than over land, and hence the critical
Richardson over water should be smaller than the critical
Richardson number over land The effect of surface
roughness on the critical Richardson number for the
Vogelezang and Holtslag (1996) method is less evident, as
this method accounts for the wind both through the wind
2
2
profile, [ ( u( z)
u(
s))
+ ( v(
z)
− v(
s))
]
2
*
velocity b u .
*
(2)
− and the friction
3. Observations
The study is based on observation of the height of the
marine atmospheric boundary layer carried out from 24
October to 5 November 1998, where a total of 24
radiosoundings were performed at Christiansø. Figure 1
shows a map of the southern part of the Baltic Sea with the
position of Christiansø marked with a cross.
NORTHING (km)
6300
6200
6100
Baltic Sea
6000
-50 0 50 100 150 200 250
EASTING (km)
Figure 1. Map of the southern part of the Baltic Sea with
land surfaces dotted. Bornholm is the island in the centre.
The cross shows the location of Christiansø east of
Bornholm. Co-ordinates refer to UTM34.