A note on the derivation of precipitation
reduction inland in the linear model
By Idar Barstad
13 August 2014
Uni Research Computing
A Note on the derivation of precipitation reduction inland in the
linear model
By Idar Barstad, 13 Aug 2014
The linear model (i.e. Smith and Barstad, 2004; SB04) has been over-enthusiastic in producing
precipitation inland, away from the coast. This comes from the fact that the linear model (hereafter
LM) has no global water budget, assumes saturated background state and produces condensate and
thus rain from the perturbation principle. For description of the model, we refer to SB04; Smith and
Evans (2007; SE07) and Barstad and Schuller (2011;BS11).
To mitigate the problem with inland precipitation, SE07 introduced a vertically-integrated water
vapor flux correction of precipitation based on reduction of water in the airstream by precipitation.
From a background wind vector
the vertically-integrated water vapor flux is defined as
(1)
where the
is water vapor density and the scale height for water vapor Hw is defined as:
⁄
where
is the temperature in Kelvin,
is the gas constant for water vapor, L
the latent heat of condensation and the environmental lapse rate. See Figure 1 for illustration.
Figure 1: Specific humidity
(or water vapor density,
where is the air density ) as a
( )
function of height. The scale height for water vapor Hw is indicated. We may write
where
is the saturated water vapor density at the surface.
The trajectory-integrated precipitation at point (x,y) is defined as
1
(
∫
)(
)
.
(2)
The fraction of water removed from the airstream is
(3)
where
is the upstream flux vector. The remaining fraction is then:
.
(4)
When written in 2D, we have
∫
(5)
,
(6)
and rearrange to:
.
(7)
Using the reference drying ratio
and solving the ordinary differential equation, we
obtain the correction factor used in LM precipitation:
( )
( )
(8)
So that
(
)
(
)
(9)
See SE07 for further explanations.
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The Flux correction - applied on the flux
Here we do a similar approach as above, but for the flux, F. The local flux,
(10)
where the Fref is the saturated influx. Now, using (1 and 10) and assuming homogeneous drying
throughout the air column so that only the humidity
is influenced (checked and found
reasonable), we have:
(
)
.
(11)
Using definition of relative humidity
,
.
(12)
Using a very crude approximation, the expression (11) can also be written as:
(
)
(13)
and with (12):
.
(14)
is the lifting condensation level. Tests show that this formula becomes very inaccurate for high
temperatures and low Rh-values. A factor including latent heating need to be incorporated in some
way and is left for future work.
A simple version of the lifting condensation level is
(
)(
)
.
(15)
Tc is the temperature in degrees Celsius.
The source term expression for condensation between level z1 and z2 can, according to (16) in BS11,
be written as:
̂ (
)
∫
(
)
(16)
The superscript u refers to “upper level” in BS11. Applying the reduced flux approximation outlined
above and integrating vertically between lifting condensation level zLCL and half the vertical wave
length taken to be the penetration depth lz/2, we get the expression:
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̂(
)
̂
[
(
)
(
)
(
]
)
(17)
Here, the Fourier transformed of the terrain, ̂ ( ), is the substitute for the constant C.
, is the adiabatic lapse rate, and the intrinsic frequency
. The superscripts
are now dropped. In (17), we may approximate the lifting using the hydrostatic 2D flow
approximation letting lz/2 be
. If we in (16) instead integrated from zLCL to infinity and using
(12,14), our expression would take the form:
̂(
)
̂
[
]
(
) .
(18)
As it routes to (14), this expression is based on uncertain assumptions and should not be used. In the
proceeding, we use (17).
We have justified the integration limits leading up to (17) by assuming an “active layer” where the
wave dynamics controls condensation. If we add this to the reduced flux in (10), we have:
⁄
(
For zLCL= 0,
,
(
), or about 0.63.
).
(19)
=3000 m, N=0.01 s-1 and U=10 ms-1, the flux is modified with a factor ~
)
If the lifting does not reach saturation level, (
, there will not be condensation
)
and S(x,y)=0. Conversely, if the lifting (
, then the S-term will be activated. A
)
few comparisons with the full numerical model favors a higher threshold so that (
, where G=1.5 in our case. Other adjustments may be better to use, but have not been tested
yet.
Comparisons – results
In order to make the following figures, we have run a high resolution model, WRF with a 1km
horizontal grid and an advanced microphysical scheme (Morrison). This full model has been
compared against different versions of LM. A free-slip lower boundary condition is applied. The
terrain constitutes two ridges with heights 600 m and 1000 m, with a corresponding half-width of 35
km and 65 km, 100 km apart. This formation carries some resemblance to the west coast of Norway.
The flow comes in from the left. No Coriolis force is included. The flow is ideally constructed based on
the indicated environmental variables (U = [10 m/s, 15 m/s], Nm = 0.006 s-1, Rh=[100 %, 90 %] and
Tsfc = [272 K, 278 K, 282 K]. The atmosphere is constructed with a constant-with-height moist
Brunt-Vaisala frequency (Nm). The moisture flux is checked against LM and found to agree within 25
%. LM is consequently about 20 % too high, and the discrepancy comes from a reduced Rh towards
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the top of the full model. For the lower 2-3 km, the flux agrees well. Since precipitation is formed in
the lower part of the atmosphere, we accept the relatively large discrepancy in the upper level. The
time delays (tau’s) are calculated as follows: are constructed according to BS11; that is, taking into
account the depth of the precipitating layer and what type of hydrometeors that are produced (solid
or liquid). is set to 1500 s except when surface temperature is less than 278 K (
750 s) which
typically speeds up conversion, or when the freezing level is higher than the penetration depth
(
)(
2000 s) which slows down the conversion. The -values are highly uncertain estimates,
and can be regarded as a free parameter. All runs are integrated for 5 hrs. The versions (some
combinations of the above-mentioned adjustments) within the same run is indicated in the label in
the plot. They are:
Symbols
Prec_LMorg
Name
Precipitation from
original model
PrecLM
Precipitation from
reference run
Precipitation from run
compensating for flow
blocking effects
Precnu
Precwrf
PrecLMtheta
PrecLM2
Comments
As in SB04, but with tau adjustment according to
BS11 and –adjustments mentioned in the
text.
Using (17)
Reduce source term as flow tend to go around
for higher Nh/U; h is max(terrain). In addition to
(17), add adjustment where S=Sfull at Nh/U<=0.5
to S=0 at Nh/U>2.
www.wrf-model.org; ARW version, 3.5.1
Precipitation from the
full numerical model
Precipitation with
theta adjustment (9)
Precipitation as above,
but with lifting
requirements
As in (17) + added correction as by SE07, see (9).
As in (17)+ added correction as by SE07 and
using LCL-truncation of source term with a
factor G=1.5 (see text below (19)).
In all linear model runs, the pseudo-adiabatic lapse rate, , is taken as the mean of the value at 0 and
1 km altitude. The vertical extent of the displays are set according where the lowest ice particles are
found (qi>1e-5 kg/kg) indicated with heavy crosses. Flux and taus are indicated on the panels.
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Fig.B1: U=15 m/s and Rh = 90 %, displays show results for surface temperatures, T=[272, 278, 283] K.
Precipitation in mm/hr on y-axis and distance in km on x-axis. Thick solid line is terrain in deca
meters, light crosses indicate cloud water, heavy crosses ice.
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Fig.B2: As B1, but U=15 m/s and Rh=100 %, across T.
7
Fig.B3: As B1, but U=10 m/s and Rh=90 %, across T.
8
Fig.B4: As B1, but U=10 m/s and Rh=100 %, across T.
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Judging from the full model, the overall picture shows that our corrections improve the precipitation
in the LM. For saturated cases (Rh = 100 %), all LM simulations follow the full model closely, except
for warm and weak winds. All warm cases for weak winds have surprisingly low precipitation
maximum (Pmax) in the full model. The reason can be that the precipitation fluctuates in time for
warm cases and can hardly be regarded as in a steady state. The missing generation of ice aloft
upstream can be an explanation for the collapse in precipitation. The locations of the Pmax in the
stronger wind cases (U=15 m/s) are located somewhat upstream of the full model, and would favor
from longer taus, which would shift the distribution downstream. However, longer taus would also
lead to smaller amplitudes of Pmax, and thus for colder cases, longer taus would result in too small
amplitudes. For weaker winds, the location of Pmax seems to be good except for warmer cases
where the taus could preferably be longer.
Looking at the different LM-runs, we see that the “LM2” case is closer to the full model if judging by
the amplitude of Pmax. The upstream distribution is cut off due to the truncation of the source term
when the generated lifting is lower than the LCL. The precipitation in the full model probably comes
from lifting aloft, farther upstream. Generally, the “LMtheta” run shows an amplitude improvement
of Pmax at the second ridge. Most of the Pmax locations are too far upstream on the 2nd ridge and
could benefit from longer taus.
Conclusions
Results from our preliminary investigation show some prospects in reducing the linear model
precipitation inland. There are indications that further improvement can be obtained by including
more tau-dependencies (to wind speed and to a larger degree temperature). The Smith and Evans
(2007) flux correction (named “LM theta” herein) should be used. It is based on physical reasoning
and our results show that the LM improves. Further investigations are warranted before a decisive
recommendation on adjustments can be made. Better control over cloud ice production aloft and
control over steady state approximation are needed. We have taken the full numerical model as
representing the truth, but this may be quite doubtful -at least in cases where a firm steady state has
not been reached.
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References:
Barstad and Schüller, 2011: An extension of Smith's linear theory of orographic precipitation Introduction of vertical layers. Journal of Atmospheric Sciences, Vol. 68, 2695-2709.
Smith and Barstad 2004: A Linear Theory of Orographic Precipitation.
Journal of Atmospheric Sciences, Vol.61, 1377-1391.
Smith and Evans 2007: Orographic Precipitation and Water Vapor Fractionation over the
southern Andes. Journal of Hydrometeorology, Vol.8, 3-19.
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