Examensarbete vid Institution för geovetenskaper ISSN 1650-6553 Nr 149
Numerical exploration of radiative-dynamic interactions in cirrus
Stina Sjöström
Abstract
An important factor in forecast models today is cirrus clouds, but not much are known about their dynamics which makes them hard to parameterize. In this study a new theory was derived to enable a more correct way to describe the interplay between radiative heating and dynamical motions in these clouds. This hypothesis was tested by performing three dimensional simula- tions of cirrus clouds, using the University of Utah Large Eddy Simulator (UULES). Eleven clouds of varying initial radius and ice water mixing ratio were examined, with the aim of find- ing a pattern in their dynamical features. The model was set up without short wave radiation from the sun, and without any precipitation affecting the clouds, leaving only terrestrial heating and atmospheric cooling to create motions in the clouds. Two categories of initial dynamics could be seen:
• Isentropic adjustment: The isentropes within the cloud are adjusting to the environment due to rising of the cloud. Causes horizontal spreading through continuity.
• Density current: A dominating initial feature is spreading in small mixed layers at the cloud top and bottom. Caused by the density difference between the cloud and its envi- ronment.
An interesting phenomenon showing up in the simulations was mammatus clouds, which were visible in two of the cases. The only instability available to create these clouds was the radiative heating difference, which does not agree with present theories for how they form.
Two dimensionless numbers S and C were derived to describe the nature of the spreading mo- tions and convection in the cloud. Both these numbers agreed with results.
Sammanfattning
Cirrusmoln har en viktig roll i dagens prognosmodeller, men är svåra att parametrisera på ett bra sätt eftersom man inte har tillräcklig kunskap om deras dynamik och utveckling. I denna studie togs en ny teori fram för att göra det möjligt att på ett mer korrekt sätt beskriva samspelet mellan strålningsuppvärmning och dynamiska rörelser i dessa moln. Hypotesen testades sedan genom att utföra tredimensionella simuleringar av cirrus moln med hjälp av University of Utah Large Eddy Simulator (UULES). Elva moln med varierande initiella radier och isvatteninnehåll undersöktes, med målet att finna ett mönster i dynamik och utveckling. UULES ställdes in så att miljön där molnen simulerades varken innehöll kortvågsstrålning från solen eller nederbörd.
Således fanns det bara en resterande faktor för att skapa rörelser i molnen; skillnaden i den infraröda strålningsuppvärmningen mellan molntopp och molnbas. Två kategorier av initiella rörelser uppstod i molnen:
• Justering av isotroper: Molnen stiger i höjd vilket gör att isotroperna inuti dem justeras till omgivningen. Detta orsakar horisontell spridning genom kontinuitet.
• Densitets ström: Horisontell spridning av molnen koncentrerad till mixade skikt i de övre och undre delarna. Orsakas av skillnad i densitet mellan moln och omgivning.
Ett intressant fenomen som visade sig i två av simuleringarna var mammatusmoln. Den enda instabiliteten tillgänglig för att skapa dessa moln var skillnaden i strålningsuppvärmning mellan molntopp och -bas. Detta stämmer inte överrens med nuvarande teorier för hur dessa moln skapas.
Två dimensionslösa tal, S och C togs fram för att indikera vilken av de initiella rörelserna som dominerar i molnet, samt vilken typ av konvektion som dominerar. Båda dessa tal stämde väl överrens med resultat.
Contents
1 Introduction 7
2 Theory 8
2.1 Radiative, microphysical and dynamical characteristics . . . 8
2.2 Classic approach . . . 8
2.3 New approach . . . 9
3 Methods 13 3.1 Background . . . 13
3.2 Model description . . . 14
3.2.1 Large Eddy Simulators (LES) in general . . . 14
3.2.2 The University of Utah Large Eddy Simulator (UULES) . . . 14
3.3 Simulations and model setup . . . 15
4 Results 19 4.1 h = 1250m . . . 19
4.2 h = 290m . . . 25
4.3 h = 29m . . . 31
4.4 Summarized results . . . 36
5 Conclusions 38
6 Future improvements 40
7 Acknowledgments 40
APPENDICES 41
A Model setup 41
B Cross-section figures 42
References 53
1 Introduction
Cirrus clouds play an important role in the earth’s climate, because they reflect a major part of the incoming radiation from the sun and also trap terrestrial radiation from beneath. This is well known. What has received less attention over the years are the dynamics and microstructures of these clouds. For instance, is there a relationship between the mesoscale dynamics and the microstructure in the cirrus clouds? It is known (Garrett et al. 2005, Garrett et al. 2006), that solar radiation along with the terrestrial radiation causes heating contrasts within the clouds, which allows it to work as a heat engine, starting internal dynamic motions. What is not known is how much this affects the evolution of the cloud, when taking away other initializing sources.
The purpose of this study is to investigate and explore parameters that are important in the evolution of a cirrus cloud. This investigation is performed using the highly advanced Univer- sity of Utah Large Eddy Simulator (UULES), developed by Zulauf (2001), with which three- dimensional simulations in time can be produced of an idealized cirrus deck. The analysis are simplified by approximating the clouds as homogeneous cylinders without initial dynamics.
Furthermore, the model is set up as neutrally buoyant, without precipitation, and with no short wave radiation from the sun. Thus, cloud motions are driven by terrestrial radiative heating alone. During the simulations, the anvils evolve differently depending on the values of certain parameters in the initial conditions. By analyzing the results, conclusions can be drawn about the dynamics and micro structures in the cloud.
Other similar studies have previously been made (Dobbie and Jonas 2001, Starr and Cox 1985).
However, those studies were set up differently than those covered by this report. The above mentioned articles included microphysics in their simulations, and their models were set up with a lower spatial resolution. The analysis in this report are the first using high resolution simulations in three dimensions, that have no precipitation, and that cover a wide range of cloud types.
Within the scope of this report, it is also studied whether there are circumstances favorable for producing mammatus clouds. The dynamics and characteristics of mammatus clouds have been investigated in an article by Schultz et al. (2006). This article states that some kind of instability has to occur to create these clouds, and a common conception is that the instability is precipitation-driven. Mammatus clouds have been successfully simulated (Kanak et al. 2006) in three dimensions, but in that case the model included precipitation but no radiation. Could it be that the radiative instability caused by heating from below and cooling from above is sufficient to create mammatus clouds?
Today, there is uncertainty in how to correctly parameterize cirrus clouds in climate and forecast models due to the fact that there are so many different processes that affect their development, evolution and structure. To be able to better represent these clouds in future models, there needs to be an increased understanding of how they work.
2 Theory
2.1 Radiative, microphysical and dynamical characteristics
Cirriform clouds contain a range of variability in ice crystal habits, e.g. columns, bullets, bullet- rosettes and plates. Depending on the sort of cloud the sizes of the ice crystals varies. In a study by Garrett et al. (2005) the effective radii were found to be ~10-50µm, this for a cirrus anvil. A value fitting into this range is used in the study described in this paper. The effective radii of ice crystals were here set to 20µm, which can be compared to the values of 50µm that Ackerman et al. (1988), was using. Thus, the clouds investigated in their study contained a larger mass of ice than the ones simulated here, even though their clouds had a smaller ice water mixing ratio.
Interactions between the radiation and the dynamics is strong within cirriform clouds and are more complicated than in other stratiform clouds, due to their mixture of ice particles. The heating from the sun is evenly divided throughout the layer but the heating from the infrared wavelengths yields a heating at the bottom of the cloud, due to terrestrial heating, and an at- mospheric cooling at the top of the cloud. Latent heating also takes place within the cloud.
However, at the altitudes where cirriform clouds form, this can be neglected.
2.2 Classic approach
A detailed theory describing cirrus dynamics was presented by D. K. Lilly (1988). It is based on earlier studies of the dynamics of stratocumulus clouds (Lilly, 1968), and which used an analytical model to reach the conclusion that the entire boundary layer consisted of a mixed layer. Applying this theory to stratified cirrus anvils means that they also embody a mixed layer and have a linear heating rate within the cloud as in Eq. 2.1:
H = BO(z − zm) − AO (2.1)
in which H [K/s] is the heating rate, BO and AO are constants and zm[m] is the height to the middle of the cloud. Lilly based this on an earlier article by Ackerman et al. (1988), in which linear heating profiles in cirrus anvils were assumed. A profile of the heating rate through the anvil would according to this assumption look something like Fig. 2.1.
−10 −5 0 5 0
2 4 6 8 10 12 14 16
Heating rate (K/day)
Height (km)
Figure 2.1: The classical view of the heating rate profile within an anvil
In the approach described above, Lilly assumed cirriform clouds to be one mixed layer, and that they would rise in altitude due to uniform heating from below.
There can be certain problems with this approach since it makes assumptions and approxima- tions that are based on theories for the dynamical evolution of stratocumulus clouds. The theory might be correct when studying very thin and wide cirrus anvils where the radiation is pene- trating evenly through the entire cloud. But for most cirrus anvils, which are very thick and practically considered as black bodies, there will be zero thermal heating in the interior of the clouds. By flight observations in an earlier study (Garrett et al., 2005), it has been shown that there is an atmospheric cooling at cloud top and a terrestrial heating at the cloud base, along with horizontal spreading and vertical thinning. Furthermore, that study also showed that the stratification and temperature inside the anvil were close to being constant. These discoveries do not agree with Lilly’s assumption that the entire cloud turns into a turbulent and unstable mixed layer. What is then actually happening in the cloud that would clarify these observations?
2.3 New approach
Strong long wave radiative heating/cooling at cloud base and top creates an unstable mixed layer within a penetration depth, h [m], at the top and bottom of the cloud that is much less then the cloud depth. Thus, the cloud interior is radiatively shielded.
The penetration depth, h is described as:
h = 1
γk(re)qiρair (2.2)
where γ=diffusivity factor ~1.7 , k(re)=absorption coefficient as a function of the effective ra- dius of ice crystals ~0.045 [m2g−1] , qi = ice water mixing ratio [g/kg], ρair= air density [kg/m3].
The value of k is taken from Ackerman et al. (1990), using re = 20µm.
The heating rate H, is related to qi through h, which can be shown starting with the first law of thermodynamics in Eq.2.3:
∂T
∂t = H = − 1 ρcp
∂Fnet
∂z (2.3)
where Fnetis the net flux, cp is the specific heat capacity and ρ is the air density:
∆Fnet= (Ft ↑ −Ft↓) − (Fi ↑ −Fi ↓) = σTt4
Ft= up-/downward flux from the top of the cloud, Fi= up-/downward flux from a depth ∆z = h into the interia of the cloud and Tt4= the temperature at the top of the cloud.
⇓
H = − 1 ρcp
σTt4 h
Lilly’s idea of cirrus dynamics and the new hypothesis is illustrated in Fig. 2.2 below. Since Lilly performed one dimensional model simulations, he had no ability to see if clouds were spreading. Thus, one difference is that his approach is applied to clouds with unlimited widths, while the new approach is applied to clouds with finite horizontal extents.
Figure 2.2: The classical and the new approach for how the mixed layer rises. The broken line denotes the mixed layer with height h, in which a rising occurs with the same speed wstrat as in the classic approach.
Heating causes cloud to rise with speed:
wstrat = H
dθ dz
(2.4)
Lilly’s theory was thus assuming that the heat could only go upward.
As mentioned above, there are results (Garrett et al. 2005, Garrett et al. 2006) that indicate that there are only small layers with depth h, at the top and bottom of the cloud that are affected by the heating; these layers also adjusts vertically with the speed wstrat .The continuity equation, where the density has been neglected with the Boussinesq approximation (absorption depth h
< < scaling depth H) is:
∂u
∂x +∂v
∂y + ∂w
∂z = 0 (2.5)
Thus, scale analysis gives the relation for clouds of radius L:
u L ∼ w
h
Where L[m] is the radius of the anvil and u the horizontal speed of a layer of depth h. If horizontal heating occurs while isentropes stay flat then from Eq.2.4, stratified motions follow:
ustrat ∼ wstratL
h ∼ HL
hdθdz (2.6)
Substituting the buoyancy frequency, N2 = (gdθ/θdz) , eq. 2.6 can be rewritten:
ustrat ∼ HLg hN2θ and
wstrat∼ Hg θN2
If ustratand wstratare not large enough to transport the heat in the layer, a density current forms.
This is similar to the process when potential energy converts to kinetic energy by gravity. The density current will have characteristic speeds, henceforth called umixand wmix, which can be derived by scale analysis of the conservation of energy equation and by using the continuity equation.
N2h2 = u2mix 2
⇓
umix∼ N h
To derive an expression for wmix, we combine the umixwith the continuity equation (2.5) : wmix∼ N h2
L
By knowing these two velocities, a new variable can be introduced; the spreading number, S, which is a measure of whether motions are driven primarily by isentropic adjustment or density currents:
S = ustrat
umix ∼ Hg L
θN3h2 (2.7)
in which it is expected that:
S << 1 isentropic adjustment and spreading
S >> 1 density currents and spreading
A similar number has been suggested earlier (Dobbie and Jonas, 2001); the ”radiation stability number”:
Rsn = N2θh
gHδt (2.8)
Radiative instability, occurs when 0 < Rsn< 1. The similarity of these two numbers makes it interesting to look at the ratio:
Rsn
S ∼ δtN h L
Convection exists if Rsn/S >1. By rearranging, we end up with a timescale that describes the time to lose heat by horizontal spreading in density currents. In some cases the cloud is too wide to be able to leak all the heat, which then gets trapped in the cloud and turns into convection.
The time needed for this to happen is defined in Eq.2.9 below.
δtmix < L
N h (2.9)
which suggests a second dimensionless number:
C = L h
If C > > 1, it is expected that heat is trapped and the mixed layers deepens beyond the initial depth h.
3 Methods
3.1 Background
The purpose of this study is to investigate cirriform clouds with different characteristics e.g.
initial radius and ice water mixing ratio, and explore how they respond dynamically to radiative heating. Some of the simulated clouds do not exist in the real atmosphere, but are interesting for what they reveal about the dynamics of clouds.
3.2 Model description
3.2.1 Large Eddy Simulators (LES) in general
Atmospheric motions are present over a wide range of scales covering six orders of magnitude, from the large scale eddies which can have sizes of the depth of the boundary layer, all the way down to the micro scale eddies. Energy is transported from the large scale to small scales of turbulence. In order to correctly resolve this wide range, an unresonable amount of calculation point would be needed in each direction. This would be numerically impossible to handle for the computers of today. The approach to handle this problem is based on the idea that large eddies are depending of the flow geometry while smaller eddies have more universal characteristics.
Thus, the small eddies are easier to parameterize than large ones. Therefore, the main principle of LES-modeling is that by separating the large eddies from the small, the large scale flow can be calculated directly, while the small scale turbulence effect is resolved using a Sub Grid Scale (SGS) model.
The advantages of an LES model compared to other models (e.g. those solving the Reynolds- averaged Navier-Stokes equations), is that they are not only numerically less expensive than direct numerical simulations, but furthermore gives a detailed high resolution and can resolve structures of turbulent flow well. This is especially of use in atmospheric science, where it is crucial to be able to parameterize turbulence and cloud dynamics.
3.2.2 The University of Utah Large Eddy Simulator (UULES)
The UULES model(Zulauf, 2001), is specially designed to examine small scale atmospheric flows, e.g: cumulus convection, turbulence and entrainment. The dynamical features of the model are based on a fully prognostic, compressible and non hydrostatic set of primitive equa- tions. The advantages of using a compressible system of prognostic equations is that it is rela- tively easy to modify the grid structuring, boundary conditions and differencing schemes. Fur- thermore, it makes implementation on a multi-processor machine easier.
However, there are also drawbacks with this method. When using a compressible system of equations, meteorologically unimportant sound waves are introduced. Since the speed of sound is much higher than the speed of the meteorologically important waves of interest, a very small and uneconomical time step is needed. The sound waves are taken care of in UULES by setting their speed to a lower value than normal, but larger than the speed of the meteorologically important signals. This is done by using the quasi-compressible (Droegemeier and Wilhelmson, 1987) approximation.
The model uses the Deardorff (1980) Sub-Grid Scale Turbulent Kinetic Energy (SGS TKE) closure, which employs a prognostic equation for SGS TKE. Sub grid fluxes of momentum and scalar quantities are decided using eddy viscosities derived from the SGS TKE.
The time differencing utilizes a time split second order Runge-Kutta (RK) scheme (Wicker and Skamarock, 1998), which allows the fast mode speed of sound and pressure terms to be integrated using a small time step. Thus, longer and more economical time steps can be used for the remaining forcing terms. This Runge-Kutta scheme consists of two steps when moving forward to the next time step. This differencing method does not require time filtering to ensure stability. The dependent variables are staggered in all three dimensions using a standard stencil, allowing for most spatial differences to be evaluated with a higher effective grid resolution.
In addition, a vertically stretched grid is used to increase the resolution near the areas of high importance, for instance in the middle of the cloud.
The parameterization of the radiative transfer is described in detail in Fu et al. (1995), it is a plane parallel broadband approach which uses 6 solar and 12 infrared bands. The parameteri- zation uses the δ-four stream scheme (Liou et al., 1988) to handle the solar and infrared regions of the spectrum, and it is based on the correlated k-distribution method as described by Fu and Liou (1992).
The momentum advection terms are differenced with a third order upwind-biased scheme (Wicker and Skamarock, 1998) , which has some advantages compared to second-order centered schemes.
Unlike the centered, the upwind scheme is stable when combined with the RK time differenc- ing, and no additional diffusion needs to be included. The modified UTOPIA scheme of Steven and Bretherton (1996) is used for scalar advection, and monotonicity is maintained using Zale- sak (1979)’s flux-corrected transport. All other terms utilize second order centered differences.
Periodic lateral boundary conditions are required, while the vertical boundary conditions are specified when initializing the runs. A sponge layer is specified near the model top to reduce the reflection of gravity waves. The code uses a standard MPI parallelization and is designed to be modular so that components can be easily swapped.
3.3 Simulations and model setup
For the experiments described here, the UULES was set up with no precipitation, background- flow or movements, and with no solar radiation effecting the cloud. Also, the clouds were initialized to be neutrally buoyant so that the difference in density between the cloud and the environment would not generate any dynamics. Thus, no external dynamic forces were affecting the clouds. All dynamic motions within the cloud were forced by initial differences in spatial heating.
The speed of sound was reduced to 50 m/s, and a cubic grid in the vertical direction made it possible to have a smaller grid size in the middle of the cloud at the height of 10,050 m. A sponge was placed at the top of the model, starting at 13,000 m, to prevent reflections of gravity waves in the domain.
Initial profiles of relative humidity and potential temperature are shown in figures : 3.1 and 3.2.
These are based on Garrett et al. 2005.
20 40 60 80 100 120 0
2 4 6 8 10 12 14 16 18
Relative Humidity (%)
Height (km)
The initial relative humidity profile Cloud Environment
Figure 3.1:Initial relative humidity profile for the cloud and the environment
3000 320 340 360 380 400
5 10 15
Potential temperature (K)
Height (km)
The initial potential temperature profile
(a)
339 339.5 340 340.5
9.7 9.8 9.9 10 10.1 10.2 10.3
Cloud Environment
(b)
Figure 3.2: (a)Initial potential temperature profile for the cloud and the environment,(b)Enlarged pic- ture, Dashed line: profile in cloud, Solid line: profile for environment
The dimensionless number C=L/h, the initial radius of the anvil: L, and the ice water mixing ratio: qi[g/kg], were all of great importance when deciding for which cases to run the model.
By combining these variables, a wide range of model runs was set up as in Table 3.1 below.
Table 3.1: The simulated cases marked with an X.
100m 1km 10km 100km
0.01g/kg X X X X
0.1g/kg X X X X
1g/kg X X X -
Note that it is only the initial value of the ice water mixing ratio (qi), that is being changed.
During the simulations, qi is varying.
The clouds were initialized as cylinders of varying radii, with the constant depth of 2.5 km in all cases (see Fig.(3.3) below). The cylinders were uniform, meaning that for example the initial values of ice water mixing ratio were homogeneously divided through the cloud. re, which is the effective ice crystal radius in the clouds, was set to 20 µm for all the simulated clouds. The model was set up in a way that there were no solar radiation, precipitation or initial dynamics to affect the clouds.
Figure 3.3:All clouds where initialized as homogeneous cylinders with the depth 2.5km
As the heating rate H, solely depend on the ice water mixing ratio (see section 2.3), there will only be three different heating rate profiles as seen in Fig.3.4. Notice how the profiles within the cloud varies from an almost linear transition (b), between the warmer cloud bottom to the cooler cloud top, to a very distinct and almost vertical transition (c). The more ice water the cloud contains, the less infrared radiation penetrates into the cloud. As for cirrus anvils, which
have an ice water content closer to 1g/kg, the profile would have the appearance of (c). It is interesting to compare these initial profiles with Lilly’s linear approximation of the heating rate within an anvil.
−10 −5 0 5
0 5 10 15
(a) qi=0.01g/kg
Heating rate (K/day)
z (km)
−1000 −50 0 50
5 10 15
(b) qi=0.1g/kg
Heating rate (K/day)
z (km)
−600 −400 −2000 0 200 400 5
10 15
(c) qi=1g/kg
Heating rate (K/day)
z (km)
Figure 3.4:Heating rate profiles (a)qi= 0.01g/kg , (b)qi = 0.1g/kg , (c)qi= 1g/kg
For each case described in Table (3.1), the domain size, number of grid points in each direction, and model time step were changed to get suitable resolution for the clouds. These were set with the aim to always have a resolution high enough to resolve the penetration depth h for each case.
To have the least computationally expensive setup, the largest time step possible without the model crashing was used, which would normally be between 1-10s. For cases where a high resolution was needed, the time step was smaller. For more detailed information about the set up of the model see Appendix A, Table A.1.
4 Results
For one of the cases where a full length run with the desired resolution would mean an unrea- sonably long time to simulate (L = 100km qi = 0.1g/kg), the setup was changed by reducing the cloud radius to a value that made the simulation feasible.
Below the results will be presented in sections with different values of h, calculated for each case with Eq.2.2. However, for the clouds were qi = 0.01g/kg, h would be 2900m if calcu- lated with this equation, which means that it would be larger than the total depth of the cloud.
Consequently, another more correct value of h had to be found for these cases. When studying the heating rate profile, (see Fig.3.4), there is an almost linear transition between cooling and heating in the cloud. Thus, the penetration depth, h, can be approximated to half of the total depth of the cloud i.e. 1250m.
4.1 h = 1250m
The cloud which was initialized with a radius of 100m and an ice water mixing ratio of 0.01g/kg, is shown in Fig. 4.1 below. This figure provides a 3D view of the cloud after full simulation time of 3600 seconds. No significant horizontal spreading takes place, though the top of the cloud appears to have vanished to the extent that almost half of the cloud has disappeared.
Furthermore the figure shows that only the edges of the cloud are left after 1 hour, leaving a shell shaped like a bowl. These characteristics are also distinct in the 2D cross-section figures in Appendix B, Fig.B.1, which shows the ice water mixing ratio in the cloud at two different time steps.
Fig.4.2 shows the equivalent potential temperature profiles at the same times as the 2D figures, and no signs of instability can be seen as suspected from the ice water mixing ratio. The profiles stay approximately the same with time, aside from a cooling and subsidence at cloud top which indicates that isentropic adjustment is the initial effect. S and C are in this cloud both much less than 1.
Figure 4.1: 3D-figure of cloud with L = 100m and qi = 0.01g/kg at 3600 seconds. S = 2 · 104 C = 0.08
337 338 339 340 341 342 343 344
8.5 9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=100m qi=0.01g/kg
0 s1800 s 3600 s
Figure 4.2: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 1800(cir- cles) and 3600(stars) seconds, L = 100m and qi = 0.01g/kg
When increasing the radius with a factor of 10, a different scenario shows up (see Fig.4.3 and Fig.B.2 (Appendix B)). This cloud has an appearance similar to the previous case, in which only the edges of the cloud remained after 1 hour.
The θe-profiles (Fig.4.4), point to the fact that the cloud top is vanishing, but not with the magnitude that would be expected with respect to the ice water mixing ratio-figures in Appendix A, this due to how θeis calculated. This calculation is performed by taking the mean value of θe
at every z-level where there is a cloud element. Hence, the really thin layers still existing at the cloud edges, will generate values of θe. Furthermore, the stability is somewhat constant with time according to the θe-profiles, and no mixed layers can be seen. Thus, the dominating effect in this cloud is isentropic adjustment, which would agree with the low values of S and C.
Figure 4.3: 3D-figure of cloud with L = 1km and qi= 0.01g/kg at 3600 seconds. S = 0.002C = 0.8
337 338 339 340 341 342 343 344 8.5
9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=1km qi=0.01g/kg
0 s2160 s 3600 s
Figure 4.4: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 2160(cir- cles) and 3600(stars) seconds, L = 1km and qi = 0.01g/kg
The next cloud, was set up with the radius 10km and the ice water mixing ratio 0.01g/kg (see Fig.4.5). This figure shows the cloud in three dimensions at time 3600 s. The initial cylindrical shape can still be seen after 1 hour, which indicates an insignificant amount of spreading. When studying the cross-section figures for this cloud (Fig.B.3 (Appendix B)), the prominant charac- teristic is the well defined boundary between the cloud’s lower parts with a relatively constant value of ice water mixing ratio and higher parts which are dissolving with time. Almost no spreading takes place in this case, but there are some vertical lifting, which however is small.
As seen in the equivalent potential temperature profiles, in Fig.4.6, the stability remains prac- tically constant with time as for the previous described clouds. The top is cooling which is connected to the vanishing cloud top but there are no indications of mixed layers in the cloud, this as a sign of the isentropic adjustment that is affecting the cloud.
Figure 4.5:3D-figure of cloud with L = 10km and qi = 0.01g/kg at 3600 seconds. S = 0.02 C = 8
337 338 339 340 341 342 343 344
8.5 9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=10km qi=0.01g/kg
0 s2880 s 3600 s
Figure 4.6: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 2880(cir- cles) and 3600(stars) seconds, L = 10km and qi = 0.01g/kg
Below (Fig.4.7 ) is a 3D- view of a very wide and tenuous cloud. The amount of spreading and lifting is in this case almost zero, and the topmost part of the cloud stays intact. An interesting observation is the bottom of the cloud, which is not entirely smooth due to a small amount of convection.
The 2D figure (Appendix B, Fig.B.4 )reveals that there has been a decrease in the ice water mixing ratio, mostly concentrated at the bottom and sideways areas of the cloud. The stability of the cloud (see Fig.4.8) is fairly constant with time except for a small cooling at cloud top, i.e.
this cloud is dominated by isentropic adjustment as the previous described clouds.
Figure 4.7:3D-figure of cloud with L = 100km and qi = 0.01g/kg at 3600 seconds. S = 0.2C = 80
336 338 340 342 344 8.5
9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=100km qi=0.01g/kg
0 s3600 s
Figure 4.8: Profiles of equivalent potential temperature in the cloud, θe[K], for 0s(solid line) and 3600s(stars), L = 100km and qi = 0.01g/kg
4.2 h = 290m
In the next cloud, with L = 100m and qi = 0.1g/kg (Fig.4.9), horizontal spreading is taking place on the top and bottom, leaving the cloud with a final structure shaped like a cone. When looking at the 2D plots for qiat times 1800 and 3600 seconds (Appendix B, Fig.B.5), it appears as if the ice water mixing ratio in the topmost part of the cloud, has reduced its value to nearly 0 g/kg. A visible rise of the cloud base has also occurred, on the order of a couple of hundred meters between 0-3600s. An interesting observation at time 1800 s is that the cloud base has subsided rather than risen, thus, the rising occur in the last 1800 seconds.
This rising is also shown in the θe-profiles in Fig.4.10, along with an even larger sinking motion in the cloud top which can be seen clearly. Furthermore, a cooling, mostly at cloud top occur.
The stability is relatively constant, except for a small mixed layer at cloud bottom visible most clearly in the last time step. Despite the mixed layer, the cloud is still dominated by isentropic adjustment.
Figure 4.9: 3D-figure of cloud with L = 100m and qi = 0.1g/kg at 3600 seconds.S = 0.02C = 0.34
337 338 339 340 341 342 343 344
8.5 9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=100m qi=0.1g/kg
0 s1800 s 3600 s
Figure 4.10: Profiles of the equivalent potential temperature in the cloudθe[K], for 0(solid line),1800(circles) and 3600(stars) seconds, L = 100m and qi = 0.1g/kg
The next case with the same value of h, has the initial radius and ice water content of 1km and 0.1g/kg respectively. Here, the top- and bottom-most part of the cloud starts spreading sideways in tongue shaped layers, which can be seen at time 1800s in Fig.B.6 in Appendix B. Furthermore, the ice water mixing ratio at that time has decreased from its initial value of 0.1g/kg to ~0.05g/kg. After 1 full hour has passed, a relatively large portion of the cloud top has disappeared. A vertical rising has also taken place. When looking at a 3D picture of the last time step (Fig.4.11 ), a bizarre structure is seen. The drop shaped top area of the cloud that now is visible in three dimensions, did not show up in the cross-section plots.
Both the rising and vanishing of the cloud are well represented in Fig.4.12, which shows the θe-profiles in the cloud with time. The curve for time 3600s, involves a few jumps, which would correspond to the clearly shown separate cloud layers in both the 2D (AppendixB) and 3D (4.11)figures. The stratification is oscillating with height, from stable to unstable showing several mixed layers at each time step. The mixed layers at the bottom of the cloud undergoes a rising with the speed wmix, this is clear when comparing the two time steps represented in the θe-profiles. These layers are a sign of density currents, which is dominating the initial dynamics in this cloud.
Figure 4.11:3D-figure of cloud with L = 1km and qi = 0.1g/kg at 3600 seconds. S = 0.3C = 3.4
336 338 340 342 344 8.5
9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=1km qi=0.1g/kg
0 s2520 s 3600 s
Figure 4.12: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 1800(cir- cles) and 3600(stars) seconds, L = 1km and qi = 0.1g/kg
With a qi and an L of 0.1g/kg and 10km respectively, this next case shows some different characteristics compared to the other clouds (see Fig.4.13). The cloud is slightly spreading and there is a small indication of lifting. But to get a full picture of what is happening, one must compare with the cross-section figures in Appendix B, Fig.B.7. These show both the dissipating top as well as the lifting in form of the rounded edges at the bottom. At time 2520s, small blobs of less dense air are starting to form at cloud top. As seen at time 3600s, this air entrainment continues until a major part of the cloud has vanished. Thus, some sort of convection is taking place, but only at the topmost parts.
The θe-profiles in Fig.4.14 are confirming the above mentioned characteristics of the clouds evolution. The top part of the cloud has subsided and lost a few hundred meters in height. A clear instability is also indicated by the steep slopes at the top of the cloud, where the entrain- ment and convection is taking place. Also at cloud bottom mixed layers can be seen in the profiles, which could explain the convection seen in the 3D figure. These mixed layers are an indication of the density current that is driving the motions in the cloud. Since there evidentially are mixed layers in this cloud, it is a density current driving the dynamical motions. Both of the dimensionless numbers, S and C are larger than 1 which indicates that there should be a density current taking place and convection in mixed layers.
Figure 4.13:3D-figure of cloud with L = 10km and qi = 0.1g/kg at 3600 seconds. S = 3C = 34.5
337 338 339 340 341 342 343 344
8.5 9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=10km qi=0.1g/kg
0 s2520 s 3600 s
Figure 4.14: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 2520(cir- cles) and 3600(stars) seconds,L = 10kmand qi = 0.1g/kg
As described above, it would be numerically too expensive to run the case L = 100km and qi = 0.1g/kg. Hence, a modification of this case was done, setting the initial radius to 25 km. The result at 3600s is seen in Fig.4.15, which shows that this cloud is being affected by entrainment from the surrounding air at the topmost parts of the cloud. A 2D view at the same time (Appendix B, Fig.B.8), confirms that the rising of the cloud is insignificant and that spreading in horizontal direction occurs after a short period of time.
The profiles for this case, shown in Fig.4.16, points at the instability and cooling that is oc- curring at cloud top. Density currents started the motions in this cloud which can be stated by studying the well defined mixed layer occuring at cloud top. There is also an indication of a mixed layer close to cloud base. The mixed layers and density currents, agrees with what is expected when looking at the values for S and C that are > >1.
Figure 4.15: 3D-figure of cloud with L = 25km and qi = 0.1g/kg at 3600 seconds. S = 7 C = 86.2
337 338 339 340 341 342 343 344 8.5
9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=25km qi=0.1g/kg
0 s3600 s
Figure 4.16: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line) and 3600(circles) seconds, L = 25km and qi = 0.1g/kg
4.3 h = 29m
This cloud was initialized with 1g/kg in ice water mixing ratio, 100m in radius and with the height of 2.5 km. That made its proportions highly unrealistic since clouds of that horizontal size normally have a smaller vertical extension. An 3D view can be found in Fig. 4.17, which shows the cloud as 720 seconds. The cloud has already started to spread horizontally in tongues at cloud top and bottom, an indication of the driving density currents. The motions in the cloud are introduced within a very short period of time and as seen in the 2D pictures in Appendix B Fig.B.9. By 2160s, a large portion of the cloud has spread out sideways and continues to the extent that after 1 hour, almost nothing is left of the cloud. The fast evolution of this cloud could be due to this cloud’s unrealistic aspect ratio.
The dramatic dissipation at cloud top along with the more moderate rising of the cloud is ob- vious when studying the equivalent temperature profiles for this case (see Fig.4.18). A rather significant cooling has occurred already at time 2160s, in comparison to the initial temperature.
The stratification has also experienced a change. At time 720 s, the top part of the θe-profile holds a steep slope, i.e. instability has started to take place at cloud top as an effect of the driving density currents. A mixed layer with a depth of a few hundred meters can be seen in the profiles. This almost vertical slope is also seen at time 2160s, but in a less vertically extended layer. Both S and C are > > 1, which agrees with the spreading in tongues.
Figure 4.17:3D-figure of cloud with L = 100m and qi= 1g/kg at 720 seconds. S = 15C = 3.4
336 338 340 342 344 346
8.5 9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=100m qi=1g/kg
0 s720 s 2160 s
Figure 4.18: Profiles of equivalent potential temperature in the cloud,θe[K], for 0(solid line), 720(cir- cles) and 2160(stars) seconds, L = 100m and qi = 1g/kg
For the cloud which was initialized with a radius of 1km and an ice water mixing ratio of 1g/kg, (see Fig.B.10 (Appendix B)), the most striking feature in its development is the fast horizontal spreading. This starts at the top and bottom layers and then continues along with a thinning of the cloud, caused by a density current. Furthermore, the bottom is rising a fair amount.
However, these cross-section figures does not reveal any particular convective activity in the bottom of the cloud, but when looking at the 3D-figure of this case (Fig.4.19), the instability is obvious. Large blobs are visible on the clouds surface, as signs of the on-going convection.
The θe-profiles (Fig.4.20) indicates that the stratification is changing during the evolution of the cloud. At 1440 seconds, there are two almost vertical sections of the slope, which denote the instability in the mixed layers that is obvious in the 3D figure. At time 3600 seconds, the squeezing of the cloud can be seen as the vertical extent obviously has reduced. The mixed layers that can be seen in the θe-profiles, are rising and deepening with time to the extent that at 3600 s, there is nearly complete overturning in the cloud. This can be stated since the mixed layers at cloud top and bottom have almost turned into one mixed layer. I this cloud the di- mensionless numbers, S and C are > > 1, and are thus a good measure of what is happening indicating both the dominating density currents and the convective overturning.
Figure 4.19:3D -figure of cloud with L = 1km and qi = 1g/kg at 1440 seconds. S = 205 C = 34.5
336 338 340 342 344 346 8.5
9 9.5 10 10.5 11 11.5
θe (K)
z (km)
θe profiles, L=1km qi=1g/kg
0 s1440 s 3600 s
Figure 4.20: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 1440(cir- cles) and 3600(stars) seconds, L = 1km and qi = 1g/kg
Fig.4.21, shows a cloud initialized with a radius of 10km and an ice water mixing ratio of 1g/kg at time 3600 seconds. Convection has clearly been introduced, which has caused small blobs to appear beneath the cloud. These have the magnitude of about 0.5km in horizontal diameter, which fits well with the average dimensions of mammatus clouds described by Schultz et al.
(2006). This article sums up the present theories, at time being, for how mammatus clouds form. According to this article, cirriform mammatus clouds have the average horizontal blob size of 0.5-8 km.
These pouch-like blobs are also clearly visible in a cross-section of the cloud, as seen in Ap- pendix B, Fig.B.11. After a relatively short time, a convective regime occurs both at the top and bottom of the cloud. In the top part, the convection penetrates deeper into the cloud than at the bottom part, and at time 3600s, the ice water mixing ratio is concentrated at the inner and lower layers of the cloud. Furthermore, horizontal spreading takes place along with a rising motion of the outer top and bottom edges, which leave the cloud with visible mixed layers in the form of tongues.
When looking at the profiles in Fig.4.22, both convective areas are visible. The cloud has become unstable in these parts, as indicated by the steep slopes of θe. The mixed layers at cloud top and base have deepened and risen, specially at cloud top where also a massive cooling occurs. The dominating force in this cloud is a density current. Furthermore, the vertical
Figure 4.21: 3D- figure of cloud with L = 10km and qi = 1g/kg at 3600 seconds. S = 1022C = 345
336 338 340 342 344 346
8.5 9 9.5 10 10.5 11 11.5 12
θe (K)
z (km)
θe profiles, L=10km qi=1g/kg
0 s1800 s 3600 s
Figure 4.22: Profiles of equivalent potential temperature in the cloud, θe[K], for 0(solid line), 1800(cir- cles) and 3600(stars) seconds, L = 10km and qi = 1g/kg
4.4 Summarized results
From the cases studied above, seven tables can be set up for different variables of interest.
As for the mean area of the cloud, it increases in all cases (see Table 4.1 ). As expected, the expansion is larger in the cases that show a great deal of spreading, these are situated in the lower left corners of the tables. For some of the clouds, the spreading is so extensive that the area have become 5 times larger than its initial value. The same pattern can be found in the change of position in the center of mass in each cloud, which is shown in Table 4.2 below. The increases in rc are much larger for the spreading cases than for the others.
After 1800 and 3600 seconds, the difference in total mass of each cloud from the initial state [%], has the appearance of Table 4.3. It shows that all clouds have lost some amount of mass at the final time step, but some clouds have lost more than others. Note that the cases that have the largest decrease, also are the cases in which the cloud top dissipated and where the spreading happened so fast that after 1 hour, the cloud basically was gone.
The change of the total amount of ice water present in an air column, Ice Water Path (IWP) [kg/m], is shown in Table 4.4 below. As for the mass, there are no cases in which IWP has increased after 1 hour. Here, the largest decreases are found for the narrow and dense clouds.
Table 4.5 shows the difference in the mass weighted potential temperature, in Kelvin for all the cases. Like a heat engine the temperatures varies from high to low values. There is only one cloud, in which the temperature is actually increasing with time, and that is for the cloud with L = 100km and qi = 0.01g/kg. This is very interesting, since present theories, (Lilly, 1988) suggest that heat from below is absorbed in the cloud and that the cloud then rises with a warmer temperature than it had initially. A slight tendence towards an increase can also be found at the case right below for time 1800s, but the temperature then decreases again. After one hour θ has settled at a negative value.
Table 4.1: The change in the area of the clouds, measured in percent at 1800 and 3600 seconds. One hour is noted within parenthesis
∆Area [%] 100m 1km 10km 100km
0.01g/kg ±0 (+2.5) ±0 (+3.2) +0.7 (+5.1) +0.06 (+0.2) 0.1g/kg +122 (+222) +78 (+198) +16 (+51) +7 (+24)
1g/kg +1133 (+668) +258 (+ 473) +24 (+72) -
Table 4.2: The movement of the center of mass of the clouds ,rc = (P qi · L/ P qi) measured in percent at 1800 and 3600 seconds. One hour is noted within parenthesis
∆rc[%] 100m 1km 10km 100km
0.01g/kg -4 (+0.7) +6.3 (+6.2) -0.03 (+1.3) +0.4 (-1.2)
Table 4.3: The change in the mass of the clouds measured in percent at 1800 and 3600 seconds. One hour is noted within parenthesis
∆M [%] 100m 1km 10km 100km
0.01g/kg -36 (-64) -25 (-57) -19 (-48) -17.7 (-28.5) 0.1g/kg +11 (-74) +13 (-17) -5.9 (-6.0) -8 (-13)
1g/kg -18 (-94) +35 (-4.6 ) -13 (-9.6) -
Table 4.4: The change in the Ice Water Path (IWP) of the clouds measured in percent at 1800 and 3600 seconds. One hour is noted within parenthesis
∆IW P [%] 100m 1km 10km 100km
0.01g/kg -36 (-65) -25 (-58) -19.5 (-51) -16.6 (-27.7) 0.1g/kg -50 (-92) -37 (-72) -19.2 (-38) -14 (-30)
1g/kg -93 (-99.2) -62 (-83) -30 (-47) -
Table 4.5: The change in the mass weighted potential temperature, (θm=P qi· θ/P qi), of the clouds measured in Kelvin at 1800 and 3600 seconds. One hour is noted within parenthesis
∆θm[K] 100m 1km 10km 100km
0.01g/kg -0.1 (-0.2) -0.11 (-0.28) -0.17 (-0.33) +0.33 (+0.43) 0.1g/kg -0.24 (-0.3) -0.82 (-1.4) -0.2 (-0.33) +0.02 (-0.04)
1g/kg -0.9 (-1.68) -0.34(-0.12 ) -0.08 (-0.1) -
In section 2.3, two dimensionless numbers were derived to better describe the nature of the dynamics in the clouds. The spreading number, S, varied for all the cases investigated in this study according to Table 4.6. Since the stratification is slightly different from top to bottom, the buoyancy frequency is not vertically uniform within the cloud. Thus, S is approximately twice as big at cloud top as at cloud bottom. Two scenarios were expected: S > > 1 density current and spreading that occurs in ”tongues” in mixed layers with depth h, at cloud top and bottom. Else, if S < < 1 isentropic adjustment and spreading is taking place. When looking at the table for S, there are cases with values both over and under 1. In the cases where convection occurred, the spreading numbers were all larger than 1 as expected.
The other dimensionless number, C, varied as in Table 4.7. When deriving C , it was expected that for values > > 1, the mixed layers in the density currents would deepen and turn in to convective overturning. For the cases where convective overturning did occur, the values of C filled this criteria.
Table 4.6: The spreading number, S = (gLH/N3h2θ), at the bottom part of the cloud for all the model runs. Showing cloud initial radius (L) and ice water mixing ratio (qi). (∗ initialized with the radius 25 km)
S 100m 1km 10km 100km
0.01g/kg 2 · 10−4 0.002 0.02 0.2
0.1g/kg 0.02 0.3 3 7∗
1g/kg 15 205 1022 -
Table 4.7: The dimensionless number C = L/h for all the simulated cases.(∗initialized with the radius 25 km)
C 0.1km 1km 10km 100km
0.01g/kg 0.034 0.34 3.4 34.5 0.1g/kg 0.34 3.4 34.5 86.2∗
1g/kg 3.4 34.5 345 -
5 Conclusions
One of the goals in this study, is to get a better picture of the dynamics in cirrus clouds depending on the initial characteristics of the cloud. By modelling 11 different clouds with the UULES model, three dimensional simulations are performed. When investigating the results presented in section 4, the clouds can be divided into two groups, following different scenarios:
• Isentropic adjustment (IA)
– For at least 4 of the cases, the cloud top partly or fully disappears. A possible explanation for this is that the top simply evaporates due to significant heating from below. For these clouds there are little or no horizontal spreading involved, and the isentropes within the cloud are adjusting to the environment as an effect from the terrestrial heating. This favors rising of the cloud. When studying the θv-profiles, there are no signs of instability concentrated to visible mixed layers. The spreading numbers S, were in these cases < < 1 and C< 1.
• Density current (DC)
– Initial motions in the clouds are induced in mixed layers at cloud top and bottom in the shape of tongues. These layers can be seen as instability in the θv-profiles in section 4. If the cloud hold the necessary characteristics, the mixed layers can deepen until parts of the cloud is mixed. When they are large enough, there is a good chance that convective overturning will take place, as an consequence of the fact that the heat can not leak out fast enough sideways. For the clouds that showed spreading and convective overturning S and C had values > > 1.
– The clouds in this category all have density currents as a driving force, although after some time isentropic adjustment can be introduced as a secondary effect. For these clouds, the spreading numbers were relatively close to 1 in comparison to other yielded values, e.g. 1,000.
An overview of the simulations and their categories are described in Table 5.1 below. Compare this classification with the values of the dimensionless numbers S and C in Table 4.6 and 4.7 respectively. These tables shows that a similar classification can be made for values greater and smaller than 1.
Table 5.1: An overview of the different scenarios found in this study, IA=isentropic adjustment, DC=Density current
100m 1km 10km 100km
0.01g/kg IA IA IA IA
0.1g/kg IA DC DC DC
1g/kg DC DC DC -
One of the initial questions when starting the study was whether mammatus clouds would form under any circumstances. Mammatus clouds did show up in cases where the clouds were both dense (qi =1g/kg) and wide (L=1 and 10km). This is intriguing, since present theories for how mammatus clouds form assumes that precipitation has to be involved to create the needed instability. In the simulations in this study, the model was set up without precipitation, which would mean that the mammatus clouds are a product of the instability generated by the heating difference between cloud top and base. The mammatus clouds were more pronounced in the case where C had a high value. Over all, one can say that for larger values of C, the mixed layers deepened, which was favorable to convectivre overturning. There was a small amount of convection at the bottom of the clouds for both qi = 0.1g/kg and qi = 0.01g/kg. But, in most cases the dominant convective feature was concentrated at the cloud top with the entrainment.
The spreading number S, and the dimensionless number C, proved to be fairly good indicators of what kind of dynamical motions that dominate the evolution of a cloud and if convection is taking place. When S and C < < 1, isentropic lifting without convection is the main feature, and when S and C > > 1, a density current is of most importance and convective overturning is favorable. However, for clouds with a spreading number closer to 1, both isentropic lifting and density currents take place.
6 Future improvements
Unfortunately UULES was set up slightly incorrectly regarding the stratification, such that the stability was not uniform throughout the cloud. This would obviously effect the buoyancy frequency, thus there was a difference in the behavior of the cloud top and base. This error was discovered too late in the time frame for this study to be able to change the code. Therefore, it would be interesting for future studies to make the necessary update of the model code and compare it with the results yielded herein.
To complete this investigation of dynamical effects in cirrus clouds, it would be of importance to cover all possibilities to create motions. Consequently, studies with full radiation (longwave and shortwave) turned on, would be desirable to compare with the results from this study. Also, it would be interesting to see what happens with the cloud after a longer time than one hour.
7 Acknowledgments
First of all, I would like to direct a big thank you to my supervisor Timothy Garrett, for all his help both before and during my stay at the University of Utah in Salt Lake City. Without your patience and hard work, this would not have been possible. I am forever grateful for your efforts.
Furthermore, thank you Michael Zulauf for all your help with the model and for answering my many questions. Thanks to all other faculty and staff members of the department of meteorology who have been helping me, especially Ruiyu Sun, Peter Bogenschutz and Steven Krueger. All graduate students, thank you for making me feel like home. I have found friends for life.
Thank you to the Department of Meteorology at Uppsala University, for giving me the oppor- tunity of doing my thesis abroad.
To my dear family and friends in Sweden, thank you for supporting and believing in me when I was struggling with this thesis. Knowing you are there for me has been invaluable. Thanks to Roger Sjöström for your comments when reading the paper, and Tomas Fernström for helping me with MATLAB.
APPENDICES A Model setup
Table A.1: (a)Time steps in seconds (b) Horizontal and vertical grid sizes in meters. Vertical within parenthesis
(a) Time steps [s] 100m 1km 10km 100km
0.01g/kg 1 4 10 10
0.1g/kg 1 4 10 10
1g/kg 1 2 10 -
(b) Grid sizes [m] 100m 1km 10km 100km 0.01g/kg 10(10) 40(30) 100(80) 500(80)
0.1g/kg 10(10) 45(20) 100(80) 200(80)
1g/kg 10(10) 30(20) 100(80) -
B Cross-section figures
(a)
(b)
Figure B.1: Ice water mixing ratio, qi[g/kg] in a cross-section through the center of the cloud, at time (a)1800 and (b)3600 seconds, L = 100m and qi(0) = 0.01g/kg
(a)
(b)
Figure B.2: Ice water mixing ratio, qi[g/kg], in a cross-section through the center of the cloud at time (a)2160 and (b)3600 seconds, L = 1km and qi(0) = 0.01g/kg
(a)
(b)
Figure B.3: Ice water mixing ratio, qi[g/kg], in a cross-section through the center of the cloud at times (a)2880 and (b)3600 seconds, L = 10km and qi(0) = 0.01g/kg
Figure B.4: Ice water mixing ratio, qi[g/kg], in a cross-section through the cloud at time 3600s, L = 100km qi(0) = 0.01g/kg
(a)
(b)
Figure B.5: Ice water mixing ratio, qi[g/kg] in a cross-section through the center of the cloud, at time (a)1800 and (b)3600 seconds, L = 100m and qi(0) = 0.1g/kg
(a)
(b)
Figure B.6: Ice water mixing ratio, qi[g/kg], in a cross-section through the cloud at times (a)1800 and (b)3600seconds, L = 1km and qi(0) = 0.1g/kg
(a)
(b)
Figure B.7: Ice water mixing ratio, qi[g/kg], in a cross-section through the cloud at times (a)2520s and (b)3600s, L = 10km and qi(0) = 0.1g/kg
Figure B.8:Ice water mixing ratio, qi[g/kg], in a cross-section through the center of the cloud at 3600s, L = 25km and qi(0) = 0.1g/kg
(a)
(b)
Figure B.9: Ice water mixing ratio, qi[g/kg], in a cross-section through the center of the cloud at times (a)720 and (b)2160 seconds, L = 100m and qi(0) = 1g/kg
(a)
(b)
Figure B.10: Ice water mixing ratio, qi[g/kg], in a cross-section through the center of the cloud at times (a)1440s and (b)3600 s, L = 1kmandqi(0) = 1g/kg
(a)
(b)
Figure B.11: Ice water mixing ratio, qi [g/kg] in a cross-section through the center of the cloud at (a)1800 and (b)3600 seconds, L = 10km and qi(0) = 1g/kg
References
Ackerman, S. A., Smith, W. L., Spinhirne, J. D., and Revercomb, H. E.: The 27-28 October 1986 FIRE IFO Cirrus Case Study: Spectral Properties of Cirrus Clouds in the 8-12Î14m Window, Mon. Wea. Rev., 118, 2377–2388, 1990.
Ackerman, T. P., Liou, K.-N., Valero, F., and Pfister, L.: Heating Rates in Tropical Anvils, J.
Atmos. Sci., 45, 1606–1623, 1988.
Deardorff, J. W.: Stratocumulus-capped mixed layers derived from a three dimensional model, Bound. -Layer. Meteor., 18, 495–527, 1980.
Dobbie, S. and Jonas, P.: Radiative influences on the structure and lifetime of cirrus clouds, Q.J.R.Meteorol.Soc., 127, 2663–2682, 2001.
Droegemeier, K. K. and Wilhelmson, R. B.: Numerical simulation of thunderstorm outflow dynamics. Part I: Outflow sensitivity experiments and turbulence dynamics, J. Atmos. Sci., 44, 1180–1210, 1987.
Fu, Q. and Liou, K. N.: On the correlated k-distribution method for radiative transfer in nonho- mogeneous atmospheres, J. Atmos. Sci., 49, 2139–2156, 1992.
Fu, Q., Krueger, S. K., and Liou, K. N.: Interactions of radiation and convection in simulated tropical cloud clusters, J. Atmos. Sci., 52, 1310–1328, 1995.
Garrett, T. J., Navarro, B. C., Baumgardner, D. G., Bui, P. T., Gerber, H., Herman, R. L., Heymsfield, A. J., Jensen, E. J., Lawson, P., Minnis, P., Nguyen, L., Poellot, M., Pope, S. K., Twohy, C. H., Valero, F. P. J., and Weinstock, E. M.: Evolution of a Florida cirrus anvil, J.
Atmos. Sci., 62, 2352–2372, in press, 2005.
Garrett, T. J., Zulauf, M., and Krueger, S. K.: Effects of Cirrus Near the Tropopause on Anvil Cirrus Dynamics, Geophys. Res. Lett., 33, submitted, 2006.
Kanak, K. M., Straka, J. M., and Schultz, D. M.: Numerical simulation of Mammatus-Like clouds in cirrus outflow anvils, 12th Conference of Cloud Physics, 2006.
Lilly, D. K.: Models of cloud-topped mixed layers under a strong inversion, Quart. J. Roy.
Meteor. Soc., 94, 292–309, 1968.
Lilly, D. K.: Cirrus Outflow Dynamics, J. Atmos. Sci., 45, 1594–1605, 1988.
Liou, K., Fu, Q., and Ackerman, T. P.: A simple formulation of the delta-four-stream approxi- mation for radiative transfer parameterizations, J. Atmos. Sci., 45, 1940–1947, 1988.
Schultz, D. M., Kanak, K. M., Straka, J. M., Trapp, R. J., Gordon, B. A., Zrnic, D. S., Bryan, G. H., Durant, A. J., Garrett, T. J., Klein, P. M., and Lilly, D. K.: The Mysteries of Mammatus Clouds: Observations and Formation Mechanisms, J. Atmos. Sci., 63, 2409–2435, 2006.
Starr, D. O. and Cox, S. K.: Cirrus clouds. Part I: A cirrus cloud model, J. Atmos. Sci., 42, 2663–2681, 1985.
Steven, D. E. and Bretherton, C. S.: A forward-in-time advection scheme and adaptive multi- level flow solver Modeling the Effects of Boundary Layer Circulations Generated by Cumu- lus COnvection and Leads on Large-Scale Surface Fluxes for nearly incompressible atmo- spheric flow, J. Comput. Phys., 129, 284–295, 1996.
Wicker, L. J. and Skamarock, W. C.: A time-splitting schemefor the elastic equations incorpo- rating second-order Runge-Kutta time differencing, Mon. Wea. Rev., 126, 1992–1999, 1998.
Zalesak, S. T.: Fully multidimensional flux-corrected transport algorithm for fluids, J. Comput.
Phys., 31, 335–362, 1979.
Zulauf, M. A.: Modelling the effects of boundary layer circulations generated by cumulus convection and leads on large scale surface fluxes, Ph.D. thesis, The University of Utah, 2001.
