A note on the derivation of precipitation reduction inland in the linear model

Transcription

1 A note on the derivation of precipitation reduction inland in the linear model By Idar Barstad 13 August 2014 Uni Research Computing

2 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

3 ( )( ). (2) The fraction of water removed from the airstream is (3) where is the upstream flux vector. The remaining fraction is then: When written in 2D, we have. (4) (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. 2

4 The Flux correction - applied on the flux Here we do a similar approach as above, but for the flux, F. The local flux, where the F ref 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: (10) ( ). (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 z 1 and z 2 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 z LCL and half the vertical wave length taken to be the penetration depth l z /2, we get the expression: 3

5 ( ) ( ) ( ) ( ) (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 l z /2 be. If we in (16) instead integrated from z LCL 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: ( ). (19) For z LCL = 0,, =3000 m, N=0.01 s -1 and U=10 ms -1, the flux is modified with a factor ~ ( ), or about 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 = 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 4

6 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 Name Comments Prec_ LMorg Precipitation from As in SB04, but with tau adjustment according to original model BS11 and adjustments mentioned in the text. Prec LM Precipitation from Using (17) reference run Prec nu Precipitation from run compensating for flow blocking effects 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=S full at Nh/U<=0.5 to S=0 at Nh/U>2. Prec wrf Precipitation from the full numerical model ARW version, Precipitation with As in (17) + added correction as by SE07, see (9). Prec LMtheta Prec LM2 theta adjustment (9) Precipitation as above, but with lifting requirements 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. 5

7 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. 6

8 Fig.B2: As B1, but U=15 m/s and Rh=100 %, across T. 7

9 Fig.B3: As B1, but U=10 m/s and Rh=90 %, across T. 8

10 Fig.B4: As B1, but U=10 m/s and Rh=100 %, across T. 9

11 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 2 nd 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. 10

12 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, Smith and Barstad 2004: A Linear Theory of Orographic Precipitation. Journal of Atmospheric Sciences, Vol.61, Smith and Evans 2007: Orographic Precipitation and Water Vapor Fractionation over the southern Andes. Journal of Hydrometeorology, Vol.8,