Shallow cumulus clouds

Transcription

1 Shaow cumuus couds A parameter study of the deveopment and stabiity of cumuus couds in the Atmospheric Boundary Layer September 2005 Bas Reintjes Supervisor: Drs. T. Heus Dr. H.J.J. Jonker Department of Muti Scae Physics Facuty of Appied Sciences Deft University of Technoogy

2 Tabe of Contents 1.Introduction Theory Governing equations Formation of couds The Turbuent Kinetic Energy budget equation Large Eddy Simuation Methodoogy Initia set up Important parameters Hypotheses Sensitivity experiments The BOMEX case Shifting the initia profie Adjusting the apse rate Subsidence veocity Resuts & Discussion Resuts from the simuations Scaing aws...30.concusion...32 Bibiography

3 1.Introduction Couds exist in a variety of formats and types. One cassification focuses on the height at which couds form. There are couds that form at high atitudes (generay above km) these are known as cirriform couds. Then there are atoform couds that form at mid-atitudes (between 2- km) and couds that form at even ower atitudes, the stratus and cumuus type couds. The focus of this research is on the owest coud ayer. The couds that form in the owest coud ayer form are a part of what is generay referred to as the Atmospheric (or Panetary) Boundary Layer (ABL). This is that part of the atmosphere that is infuenced by the earth's surface. Within the owest coud ayer a cassification of couds is once more possibe. There are couds that cover amost the entire sky. These couds are said to have a coud cover cose to unity. As one can imagine (and most ikey have observed in their daiy ives) there are aso couds with a ow coud cover (between 5 20%). These couds are referred to as shaow cumuus, or fair-weather cumuus as good weather normay accompanies this kind of coud. The shaow cumui amost never produce precipitation. The focus of this research is on these couds. Shaow cumuus pays an important roe in arge parts of our atmosphere. Most notaby in the (sub)tropics, but aso in more moderate cimates. Shaow cumuus is, when present, to a great extent responsibe for the transport of energy through the boundary ayer. When couds form heat is reeased (due to the condensation of water dropets) and this in turn changes important characteristics of the ABL. This indicates that shaow cumuus is important in weather forecasting and cimate prediction. Genera Circuation Modes (GCM's), which have a domain that cover the whoe gobe, use parametrizations to incorporate the effects of cumuus couds. The typica grid scae of a GCM is on the order of km, whereas a typica cumuus coud has a maximum size of about 1 km. Due to this fact cumuus couds cannot be directy impemented in GCM's and therefore parametrizations are needed. There are two ways to study cumuus couds and test parametrizations. One is through observations in the form of aircraft and radar measurements and weather baoons. The other approach is modeing coud dynamics using Large Eddy Simuations (LES) or Direct Numerica Simuation (DNS). As DNS is computationay very intensive because a turbuent scaes need to be resoved one frequenty takes refuge in LES, where ony the argest turbuent scaes are resoved. To improve on the modeing of coud dynamics it is important to know under what conditions the shaow cumuus simuations stay stabe and ink this to observations. This research aims at ooking into the causes and effects of shaow cumuus couds. So far itte is known about how changing arge scae forcings and initia profies infuences the stabiity and mean profies of shaow cumuus simuations. In most recent artices we known cases are used 3

4 without changing any of the initia vaues. These cases are e.g. BOMEX and ATEX. BOMEX is the Barbados Oceanographic and Meteoroogica Experiment (Hoand & Rasmusson (1973)), ATEX the Atantic Trade wind experiment (Augustein et a. (1973)). These experiments were conducted in the eary 70's and were extensive studies of trade wind regimes. As these cases are frequenty used one can ask the question in how far the modes are of use in other situations. So our main goa is to vary a imited number of initia parameters and see how this infuences the outcome of the simuations, most notaby mean profies. Furthermore it is interesting to see under which conditions the shaow cumuus stays stabe as this might tes us something about underying processes. Summing up what is being studied are the effects of the initia set up on the LES outcome. The objective of this study is therefore to try to give guideines which ead to shaow cumuus. Recenty scaing aws (e.g. Stevens (2004), Grant & Lock (2004) ) have been proposed that shoud ead to stabe cumuus. The scaing by Grant & Lock wi be criticay reviewed and tested in as far possibe. The study is rather heuristic in approach as many issues sti remain vague. As Siebesma et a. (2003) indicate the use of LES to deveop new fied campaigns is sti in its infancy. In this respect the research tries to give a modest Ansatz to the design of such a campaign. It does so by testing different initia set ups for shaow cumuus. An utimate goa is of course to deduce boundaries in initia profies and arge scae forcings for shaow cumuus couds to form, but this is considered beyond the scope (and scae) of this study. The next chapter gives some necessary theory and deveops an understanding of the formation of (shaow cumuus) couds. Furthermore a short introduction to LES wi be given. Chapter three focuses on the methodoogy we have adopted in conducting the research. Chapter four describes the actua set up of the sensitivity experiments. A concusion and discussion of the resuts is given in the fina chapter. 4

5 2.Theory This chapter gives a short overview of the reevant concepts. The first paragraph describes the reevant equations determining the formation of couds. The next paragraph wi focus on the actua formation of cumuus couds in a more quaitative way. Hereafter we wi shorty touch upon the turbuent kinetic energy (TKE) reation as it provides a way of understanding the turbuent fows that take pace in the atmospheric boundary ayer. We wi concude the chapter with a short introduction to the use of LES. 2.1.Governing equations In this research extensive use is made of eementary thermodynamics. We wi not give a fu overview of this subject but instead touch some of the ess known concepts. The reader not too famiiar with thermodynamics can use any eementary text book on thermodynamics. A good description of the use of thermodynamics in meteoroogica research can be found in Garratt (1992), Stu (1988) or De Roode (2004). When considering atmospheric thermodynamics one derives appropriate conserved variabes. Consider a voume of air that contains a certain amount of water vapor. The ratio between the water vapour and the tota mass of the voume of air is caed the water vapor specific humidity and denoted q v. In shaow cumuus couds there is aso an amount of iquid water, denoted as q. The tota specific humidity is now given by q t =q v q (2.1) where we have omitted the content of water in the ice state as this is negigibe in shaow cumuus couds. Obviousy the tota specific humidity is a conserved variabe as the water can change phase between water vapor and iquid water but cannot disappear. As ong as the parce of air is unsaturated, triviay, q t =q v. Now one can define a temperature scae that corrects the idea gas aw for the presence of moisture. The idea gas aw is given by pv =NkT =nrt (2.2) where p is the pressure, V is the voume N is the number of moecues k = 1.38x10-23 JK-1 is the Botzmann constant, n is the number of moes of gas and R= J mo-1k-1 is the universa gas constant. The corrected temperature scae is the so caed virtua temperature and is defined as R (2.3) T v=t [ 1 1 d q v q ] Rv where R d =7.0 J kg 1 K 1 is the specific gas constant for dry air and R v=41.5 J kg 1 K 1 is the gas constant for moist air. Tv is the temperature dry air must have to equa the density of moist air at the same pressure. As such it is aso referred to as the density temperature. The iquid water specific 5

6 humidity tends to diminish the virtua temperature, so in the absence of iquid water the virtua temperature is aways greater than the norma temperature. Another usefu temperature scae is the potentia temperature. In the atmosphere the pressure decreases with an increase in atitude. As a parce of air rises this has a direct effect on the temperature of the parce, because it expands due to the ower environmenta pressure at a higher atitude. To correct for these pressure effects the potentia temperature is used. This is the temperature an unsaturated parce of air woud have if brought adiabaticay from its initia state to a standard pressure, p0, typicay 1000 hpa. It reads as Rd p =T 0 c p (2.4) p where c p = J kg 1 K 1 is the specific heat capacity of dry air at a constant pressure. Of course one can substitute Tv for T to arrive at the virtua potentia temperature, v. The virtua potentia temperature can be seen as a measure for the buoyancy of an air parce, with respect ot its environment. Buoyancy is the important mechanism in the formation of couds. The buoyancy force is consequenty given by F b =g v, core v v (2.5) where v denotes the sab averaged virtua potentia temperature and v, core is the parces' virtua potentia temperature. When a parce is warmer than its environment it is positivey buoyant. One ast temperature scae is worth mentioning as it is a conserved quantity in reversibe adiabatic motion (in the absence of precipitation). This is the iquid water potentia temperature which is defined as = v q c p (2.) where v is the atent heat of vaporization and = pp R 0 d / cp is the Exner function. The approximate form of omits the Exner function. Having described the conserved variabes we turn ourseves to the equations that describe the ABL. These are the continuity equation (conservation of mass), the momentum equation (NavierStokes equations) and the equations for the conservation of therma energy (thermodynamic equation) and conservation of water vapour (humidity equation). For a thorough description of these equations the book by Tennekes and Lumey (1972) is a cassic, Garratt (1992), among others, provides an overview for these equations in the ABL.

7 In the present study we are making use of a fitered form of these equations1, whereby use is made of the Boussinesq approximation. The Reynods decomposition for the humidity equation is given by qt q t u i ' q t ' u i = S q t xi xi (2.7) t and the thermodynamic equation is given by u i ' ' u i = S t xi xi (2.8) The S are source/sink terms. The denotes an ensembe averaged (mean) quantity, whereas a prime denotes a fuctuating quantity. The ui and xi are the veocity- respectivey positioncomponents in the x,y,z (i,j,k) direction. In the thermodynamic equation the most important source/sink terms are the radiative cooing, the surface fuxes and the moecuar diffusion. In the humidity equation the sink term is the moecuar diffusion. The momentum equation reads asf 2 ui u i ui 1 p u j = g i3 2 ijk j u k t xj xi x 2j (2.9) where g is the gravitationa acceeration, i3 is the Kronecker deta, ijk is the aternating unit tensor and finay is the anguar veocity of the earth's rotation. The ast term in the momentum equation accounts for viscosity effects. To sove the governing equations some sort of cosure is needed. The LES-code used in this study (Cuijpers and Duynkerke (1993)) uses a one-and-a-haf order cosure based on eddy-diffusivities. 2.2.Formation of couds As this research is imited to shaow (fair-weather) cumuus couds the formation of couds as discussed here is imited to these type of couds. The formation of cumuus in the ABL is a process driven by turbuence. The formation of shaow cumuus can be understood by ooking at the apse rate. This is the rate at which a given quantity changes with respect to height, in this case virtua v potentia temperature. For active shaow cumuus to deveop it is important that is in between v the dry-adiabatic apse rate (which is zero by definition) and the wet-adiabatic apse rate (which is positive due to the reease of atent heat), that is, a so-caed conditionay unstabe apse rate. Conditiona instabiity means that the air is stabe under the condition that the air is not saturated, and unstabe if the air is saturated, as the air is saturated there is atent heat reease due to condensation. The situation is summarized in figure 2.1. When a parce of air ascends under the infuence of the heating of the surface it depends on the 1 This is due to the fact that we are making use of LES. 7

8 Figure 2.1: Potentia temperature apse rates. Couds wi form when the apse rate is between the dry- and wet-adiabatic apse rate. Aso depicted is the tota specific humidity profie. density of the surrounding air how far this parce wi rise. The rising of the parce is in genera caed a therma and the coud top marks the top of this therma. The density of the surrounding air is determined by the temperature and the moisture. The iquid water potentia temperature, discussed in paragraph 2.1, is a natura candidate to consider when ooking at the formation of couds. When discussing the formation of couds in terms of temperature we hereafter refer to uness stated expicity otherwise. At first the parce starts rising and foows the dry adiabatic apse rate. At a certain point the parce reaches its eve of neutra buoyancy (LNB), see figure 2.2. This means that the parce has the same density as its environment, i.e. if it had no momentum it woud remain at that height. But as the parce has a veocity in the z-direction it overshoots the LNB. It eventuay reaches the ifting condensation eve (LCL), if it has enough momentum in the zdirection. This is the eve where the parce woud become saturated, thus the eve at which the coud forms. From here there are two possibe scenarios. One is that the apse rate of the environment is arger than the wet adiabatic apse rate of the parce. In this case the parce remains negativey buoyant and the coud wi be imited in its growth. This is caed a forced coud and in this case the atmosphere is absoutey stabe(figure 2.2 eft part). In the other case however is smaer than v the wet adiabatic apse rate. This means that the parce is again positivey buoyant if it reaches the eve of free convection (LFC). This is the eve where a a parce of air ifted dry-adiabaticay unti saturated and saturation-adiabaticay thereafter woud first become warmer than its surroundings in a conditionay unstabe atmosphere. To reach the LFC the parce has to have enough vertica 8

9 Figure 2.2: Forced and active coud deveopment incuding appropriate eves. Soid ine is the environmenta apse rate. Dashed ine is the apse rate of the rising parce. momentum when it reaches the LCL. As the parce, which is now a coud parce, is positivey buoyant again it continues to rise and condensate unti another eve is reached, the imit of convection (LOC). This is the eve of neutra buoyancy for a saturated air parce. At this eve the temperature of the environment and the coudy air parce are once again equa. It is again possibe that this eve is overshot, and that the parce penetrates into the inversion ayer. The inversion ayer serves as a id on the mixed ayer, and is a very stabe ayer. Eventuay the parce reaches a eve where its momentum becomes negative and this marks the top of the coud. In the process at coud top there is mixing with the environment. As the parce penetrates into the inversion ayer it is possibe that air is entrained from the inversion into the coud ayer. This generay tends to ead to a deepening of the coud ayer. The formation of the coud and the height it reaches depend on the forcing of the rising therma, or as one coud argue on the properties of the iquid water potentia temperature and the tota specific humidity. The arger the forcing the taer the coud can become. One remark that has to be made is that the given expanation is highy ideaized. The main reason for this is that the properties of the therma are in reaity not conserved. There is to a certain extent atera mixing which eads to a bending of the profie of the rising parce in figure 2.2 to the environmenta curve. With the aid of the buoyancy a energy quantity can be derived that is beieved to be a measure for the vertica veocity in the coud. This is the Convective Avaiabe Potentia Energy (CAPE). It is aso the maximum (kinetic) energy avaiabe to a rising parce. The definition of CAPE is given by LOC g LOC (2.10) A= LFC F b dz = LFC v,core v dz v 9

10 Between the imit of free convection and the imit of convection the buoyancy force is positive. That is why these are used as the imits of integration. When one assumes that a potentia energy is converted into kinetic energy the foowing veocity scae resuts w CAPE = 2A (2.11) There are two major imitations to the use of CAPE. One being that atera mixing and pressure effects are negected, the other that the exact eves of LFC and LOC can make a big difference in the estimation of CAPE. 2.3.The Turbuent Kinetic Energy budget equation An important parameter in the study of the ABL is the turbuent kinetic energy (TKE) budget. This is so because the TKE budget is a measure of turbuence intensity and as such directy reated to the transport of momentum, heat and moisture through the atmosphere. Grant & Lock (2004) state that understanding what determines the magnitudes of the terms in the TKE budget is a key eement in understanding turbuent fows and thus the ABL. The TKE budget for a stationary, horizontay homogeneous boundary ayer is e = u' w ' u g w ' ' w ' e 1 w ' p' =0 v t z v, 0 z z (2.10) The brackets denote as before an ensembe averaged mean. In this equation the first term is production of TKE by mean shear, the second term is a buoyancy production term. The third and fourth term (between brackets) are vertica transport (redistribution) respectivey a pressure-veocity correation term which is aso responsibe for transport. The ast term is responsibe for moecuar transport of TKE and is given by u ' i 2 u' i x 2j. As one can see the TKE budget equation is nothing more than a summation of production and oss terms that are, in this specia case, at equiibrium. In convective situations the buoyancy fux is the most important producer of TKE. This is aso why CAPE is considered to be of use in the ABL. 2.4.Large Eddy Simuation In the modeing of the ABL there are eddies on a variety of ength scaes. They range from the Komogorov microscae (order of magnitude 1 mm) to the scae of the boundary ayer (order of magnitude 1 km). The energy in a turbuent fow is transported in what is termed the cascadeprocess. In this process energy is transported to successivey smaer eddies unti eventuay on the smaest eddy scae the energy is dissipated through viscous forces. To simuate a turbuent fow one woud favor the use of direct numerica simuation (DNS), where a turbuent scaes are resoved. However, as this woud require 1018 gridpoints (x, y and z-direction) for a cubice of air 10

11 with a dimension of 1 km3, this is computationay not feasibe. Common practice therefore is to ook at the arge scae eddies and resove them expicity.2 The smaer eddies are then parametrized, as opposed to DNS where a eddies are resoved. This is what is termed LES. The rationae for expicity resoving the argest eddies is that they can be thought of as being dependent on the arge scae forcings of the environment. They are aso responsibe for the transport of the buk of heat, moisture and momentum. For the smaer scae eddies it is generay assumed that they behave independenty and thus can be parametrized without oss of appicabiity of the (LES)-mode. The cosure that is needed for the sub-grid scae eddies is simper than a cosure scheme for the macro structure because the micro (or sub-grid) scae has a uniform structure. Furthermore simuation resuts show that the macro structure is rather insensitive to the chosen sub-grid cosure (Nieuwstadt (1998)). 2 This is in fact nothing ese than a fiter operation on the characteristic scae of the attice. 11

12 3.Methodoogy In this chapter the research aim wi be discussed. The first paragraph gives a short overview of the reevant arge scae forcings and initia profies that are specified under BOMEX, or for that matter any LES run of the ABL. The second paragraph focuses on the parameters that we think have the most infuence on the mean profies of the simuations. As a concusion to this chapter we wi formuate hypotheses regarding the outcome if one changes the initia data. 3.1.Initia set up For any LES to be run inputs have to be specified. These incude initia profies of the mean temperature and humidity and the horizonta veocity components. Furthermore arge scae forcings, describing processes that act on a arger scae than the domain that is considered in the LES, need to be taken into account. These forcings incude the radiative cooing of the atmosphere, the subsidence veocity, geostrophic winds3 and arge scae ow-eve drying due to advection. Finay consideration has to be given to the surface fuxes. Therefore the sensibe and atent heat and the momentum surface fux are prescribed. By imposing an initia set up to the mode the first two hours are generay considered to be fairy useess as the mode is in a so caed spin up state. This presents itsef most ceary in the strong peak in coud cover that appears after one hour (see Siebesma et a. (2003)). The spin-up comes about because initiay there is no resoved-scae turbuence that can generate horizonta variabiity in temperature and humidity. Therefore the first couds a arise simutaneousy, but after two hours this behavior dampens out as not a couds arise simutaneousy any onger. In BOMEX the initia profies are derived from measurements whereas the arge-scae forcings are based on budget studies (Hoand and Rasmusson (1973)). 3.2.Important parameters Having described the parameters used in the set up of the LES it is now time to pay attention to the most important parameters. As we are studying the mean profies and the way they transport energy through the ABL this dictates which parameters we wi change. Obviousy the mean profies itsef are of great importance, as a change in the initia profie is ikey to impact the resuts from the LES. In figure 3.1(a) the mean (iquid water) potentia temperature profie is sketched. There are four distinct parts. In the first 500 m a we mixed ayer is supposed, where the potentia temperature stays the same, the mixing is due to turbuence. Between 500 and 1500 m the potentia temperature raises due to the formation of couds. Then between 1500 and 2000 m we find a strong 3 The geostrophic wind is the horizonta wind for which the Coriois acceeration baances the horizonta pressure force. 12

13 Figure 3.1(a): Initia potentia temperature profie under BOMEX Figure 3.1(b): Initia specific humidity profie under BOMEX conditions stabe inversion where the potentia temperature increases even faster. Above 2000 m the potentia temperature raises in the same fashion as in the couds. The initia profie for the specific humidity is depicted in figure 3.1(b) and foows a simiar, but reversed, pattern (four distinct pieces) as the temperature. There are severa adjustments possibe in the initia profies. First one can shift the tota profie. Aso one can change the apse rate between 500 and 1500 m. Finay the atitude of the coud base and the inversion base can be atered. Another important parameter is the (arge scae) subsidence. This downward motion of air acts as a imitation on the formation of couds. The stronger the downward motion the ower couds wi form and the smaer they become. We therefore beieve that this can act as a strong infuence on the mean profies. The arge scae subsidence can emerge as part of the Hadey circuation, which is depicted in figure 3.2. As we study the second coud type from the eft in the Hadey circuation it is obvious that the subsidence is not imited to the Hadey circuation. An area of high pressure in itsef aso produces subsidence. We beieve the parameters discussed thus far can have a direct infuence on the mean profies. A more extensive understanding wi be deveoped in chapter 4. Other parameters are naturay of infuence to coud formation and stabiity of the ABL. These incude the surface fuxes, radiative cooing and arge scae advection. They are very important but by changing them the starting point and further evoution of the simuation are changed. We are most interested if a more subte change in initia parameters can ater the outcome of the mean profies. The impact of radiative cooing is at best responsibe for sma differences. In a recent study (Axesen (2005)) where the radiative cooing was set to zero, simiar resuts to BOMEX were found. This study is however imited because the runtime of the simuations was rather short. The rest of our research is therefore imited to the initia profies of temperature and specific humidity and the arge scae subsidence. 13

14 Figure 3.2: The Hadey circuation, the order of magnitude of the arge-scae subsidence is about 1 cm/s. Ev is the surface moisture fux. 3.3.Hypotheses It is reevant and usefu to deduce hypotheses of what we expect to happen when we ater the initia profies and the subsidence. A common approach in the study of turbuence is to resort to scaing aws and simiarity theory. Simiarity theory uses dimensiona anaysis to obtain reationships in non-dimensiona form, with the goa to revea scaing aws. This gives, besides a quaitative description, a more quantitative approach as to why we expect certain effects and how arge they wi be. This is an important way to improve on our knowedge of shaow cumuus. There is itte theory that describes the impact of atering the profies and arge-scae subsidence. A recent study by Grant & Lock (2004) based on simiarity theory by Grant & Brown (1999) used the foowing reation as a method for obtaining conditions for steady shaow cumuus w z 0.2 c 1/2 z cd A (3.1) where A stands for the convective avaiabe potentia energy (CAPE), w stands for the sub-coud ayer convective veocity scae defined by w = 1 /3 g z c w ' v ' v, 0. zc and zcd are the height of the ifting condensation eve respectivey the depth of the ayer between coud base and the base of the inversion. They further derive that w z ~ c 1/2 z cd A / 7 (3.2) This ooks ike the most promising (theoretica) avenue to assess whether shaow cumuus stays 14

15 stabe. One probem that needs to be soved is how they deduce a inear-aw respectivey a /7-aw for the same quantity. Grant and Lock indicate that the fomer reation is derived from a sma number of simuations and that the atter has wider appicabiity. Research by Neggers et a. (2004) shows there is an equiibrium soution for shaow cumuus. They use a simpified mass fux cosure, which introduces a moisture-convection feedback. They achieve this by retaining the coud fraction in their cosure. Neggers et a. give numerica resuts of their mode. An interesting resut of their paper is that for a constant (sea) surface temperature the subsidence veocity has hardy any infuence on the mixed ayer humidity. It is interesting to see if the above mentioned scaings work under the initia profies we are considering. 15

16 4.Sensitivity experiments This chapter discusses the actua set up of the adjustments that have been made to the standard BOMEX case, which wi be discussed shorty in paragraph 4.1. Paragraph 4.2 discusses the shift in initia profies, whereas paragraph 4.3 gives an overview of the adjustment of the apse rate in the coud ayer. Paragraph 4.4 concudes with the adjustment of the arge scae subsidence veocity. 4.1.The BOMEX case The BOMEX case is one of the odest and most studied weather cases in the meteoroogica word. It was conducted in June 199 near the coast of Barbados. At that time it was the most sophisticated and costy effort to understand weather in genera and trade wind regimes in particuar. In the Barbados region typica trade wind cumuus is frequenty observed and this was aso the case during phase 3 of BOMEX. One of the desirabe features of this phase is that it was remarkaby steady. Furthermore there was no observed precipitation or mesoscae fuctuations in fuxes. BOMEX was conducted on a square of approximatey 500 x 500 km. A detaied description can be found in Hoand and Rasmusson (1973) In most recent (LES)-studies that use BOMEX the average profies of two days of the most northern ship of the BOMEX square are used. This is so because averaging over greater tempora or spatia domains eves out the inversion that is found on the individua profies, see e.g. Siebesma et a. (2003). The actua data were obtained from rawinsondes, which produced observations for temperature, wind veocity, humidity and coudiness. They were compemented with surface eve data taken by ships and buoys. In the present study the simuation has a horizonta domain of.4 x.4 km, with the grid points eveny spaced at 100 m intervas. In the vertica (z-) direction we have a tota of 80 grid points with a mutua distance of 40 m, eading to a height of 3200 m. 4.2.Shifting the initia profie The initia profies for temperature and humidity were discussed in section 3.2. When shifting the profies for and q t one shoud expect that cumuus couds wi sti form. When the air becomes warmer whie q t remains constant there is increased evaporation eading to couds that shoud form higher. The opposite is true for a ower. Here one shoud expect to see couds at ower atitudes as a rising air parce is saturated earier. When the q t profie is changed something simiar is expected to happen. A higher tota humidity shoud ead to couds forming ower because the air is saturated earier. Furthermore something can be said about the top of the couds. As a parce reaches its LFC it is 1

17 thereafter positivey buoyant again. Once the parce reaches the LOC it is negativey buoyant but due to the overshoot it wi entrain some air of the environment, thereby deepening the coud ayer. Because the warmer air then eads to faster evaporation (due to the higher overa temperature) the ogica consequence is that the coud top of the coder profies shoud be higher than that of the warmer profies. Case name Surface temp (K) at 500 m (K) at 140 m (K) Runtime (hours) BOMEX % SHIFTT % SHIFTT % SHIFTT % SHIFTT % SHIFTT % SHIFTT BOMEX Shift Tabe 4.1: Shifted temperature profie. Note that remains unchanged, not mentioned vaues are simiar to BOMEX. Finay we can make some remarks about in the coud ayer. We expect, foowing the same reasoning as above, that the warmer profies ead to smaer couds (i.e. smaer vertica extent). In these couds however the apse rate is expected to be equa. For the humidity profie there shoud be no apparent change in the apse rate either. The profie was shifted by 0.3%, 0.% and 1% in either direction (see tabe 4.1), whereas the q t profie was shifted by 1%, 2% and 4% in either direction (see tabe 4.2). The shifts were chosen to disturb the initia profies significanty but not that much that ceary irreevant scenario's woud occur. To evauate whether a scenario is reevant typica (average) surface specific humidity and temperature were considered. Typica specific humidity (at sea eve) varies consideraby over the surface of the earth. The argest vaues being found in the tropics (~20 g/kg) and the owest vaues near the poes (~2 g/kg). The temperature variations foow the same pattern and ie between K. The chosen vaues are we within these ranges and aso within (sub)tropica ranges. 17

18 q t at 20 m (g/kg) q t at 500 m (g/kg) q t at 140 m (g/kg) Runtime (hours) BOMEX % SHIFTQ % SHIFTQ % SHIFTQ % SHIFTQ % SHIFTQ % SHIFTQ Case name BOMEX Shift q t Tabe 4.2: Shifted humidity profie. Note that q t remains unchanged, not mentioned vaues are simiar to BOMEX. 4.3.Adjusting the apse rate Besides shifting the initia profies and can aso be adjusted. As these are adjusted we expect that this shoud be persistent. As BOMEX is stabe with respect to the apse rate there is no a priori knowedge that does change this expectation for a changed apse rate. When the apse rate is changed there is more entrainment so we expect a deepening of the coud ayer with time. By increasing the apse rate we expect that the coud ayer deepens faster, because of the faster rise in temperature. By increasing the humidity apse rate the atmosphere in the coud ayer becomes drier. We therefore expect that the couds wi evaporate more quicky, thereby becoming ess deep. As to the coud base we expect no changes. The adjustment of the apse rates was made for the humidity and temperature. The percentage changes were +10%, +20%, +30% and -10% (see tabe 4.3 respectivey 4.4). The main rationae for increasing (as opposed to decreasing it) is that we expect to see a greater effect from warmer profies, because this brings the apse rate coser to the moist adiabat and thus coser to absoute stabiity, where no couds wi form. Case name Surface temp (K) at 500 m (K) at 140 m (K) Runtime (hours) BOMEX % LRT % LRT % LRT % LRT BOMEX adjustment Tabe 4.3: Adjustment of. Note that the sub-coud ayer is simiar to BOMEX for a simuations. 18

19 q t at 20 m (g/kg) q t at 500 m (g/kg) q t at 140 m (g/kg) Runtime (hours) BOMEX % LRQ % LRQ % LRQ % LRQ Case name BOMEX adjustment Tabe 4.4: Adjustment of q t. Note that the sub-coud ayer is simiar to BOMEX for a simuations. 4.4.Subsidence veocity The subsidence veocity acts as a arge scae forcing on the tota grid. We expect therefore that for increasing subsidence veocity couds wi become ess pronounced. The coud base wi most ikey be at the same height and as the initia profies remain the same the apse rate wi most ikey remain unchanged. On the other hand it can be argued on the basis of equation (2.7)-(2.9) something has to change in the apse rate. Because the situation is considered stabe (after spin-up) the time derivative can be set to zero. This impies that, as the ony change that has been made to (2.9) considers the subsidence (w-)veocity, and the apse rate is constant in time, the apse rate shoud adapt to the changed subsidence. If this is not the case one shoud expect that the other terms in (2.9) show adaptation, or the simuation cannot be considered stabe. With respect to the inversion we expect that a arger subsidence eads to a ower coud. Because of the arger subsidence the inversion is pushed down. So as a consequence, the arger the subsidence the smaer the coud wi be in time. The changed subsidences are given in tabe 4.5 (maximum vaue at 1500 m, according to Siebesma et a. (2003)). The changes are substantia but in the typica range of subsidence veocities of cm/s. Subsidence change Case name w at 1500 m (cm/s) BOMEX % SUB % SUB % SUB+30 Subsidence change Case name w at 1500 m (cm/s) BOMEX % SUB % SUB % SUB Tabe 4.5: Changed subsidence veocities. Initia profies are simiar to BOMEX. 19

20 5.Resuts & Discussion This chapter wi focus on the resuts that are obtained from the simuations. The first paragraph wi focus on the actua resuts and tries to give expanations for the obtained data. Paragraph 5.2 wi ink our resuts with the existing iterature and wi focus on appropriate scaing aws that were discussed in chapter 2. The fina paragraph concudes and gives some recommendations for future research. 5.1.Resuts from the simuations We dispay the profies that resut from the simuations in the same order as in chapter 4. We wi therefore start with the shifted profies. As the simuations are run for 3 hours we coud show a pot of the mean profies every 30 minutes of the simuation. As this woud resut in a very engthy report we wi imit the figures to pots of the mean profies and hours. Divergences from these profies in other hours of the simuation wi naturay be discussed. In the figures q and q t are shown aong with. Furthermore a pot of the deveopment of in the coud ayer over time is shown. The cacuation of is made on the basis of an interpoation between two points in the coud ayer. These points are chosen visuay for every simuation to be certainy within the coud ayer. This is done on the basis of the pots, where the apse rate appears to be uniform. A further support for the choice for these points is then obtained by ooking at the q pots to see if there is indeed a (substantia) amount of iquid water present. As is cear from figure 5.1 there are definite effects from shifting the initia profie. Most notaby there is an effect on the coud top. For the whoe simuation the coud top of the coder profies is above the coud top of the warmer profies. This can be understood, since the warmer profies are fuy evaporated at a ower atitude. A coder air parce starts to condensate earier. This can be seen in the ower right, where the coud base of the coder profies is ower than that of the warmer profies. But as the condensation takes pace at a ower temperature the evaporation is sower. Therefore the coud top wi be higher than that of a reativey warm profie. This is exacty what is the case in this shift. It is interesting to note that this eads to a faster drying of the atmosphere, aso in the sub-coud ayer. Apparenty there is a feedback mechanism from the coud ayer to the sub-coud ayer. Initiay of the warmer profies is substantiay higher than that of the cod profies. With time (see figure 5.2) a apse rates (except for the +1% case) converge to an average vaue of 2.7 K/km. Note that the -1% case is terminated before the end of simuation. This is due to the fact that the coud ayer deveops over the height of the grid (i.e. above 3180 m). The +1% case does not deveop any active cumuus and is not studied further. 20

21 Figure 5.1: Left top to bottom: q, q t and for the shifted profies after hours of simuation. Right top to bottom: q, q t and for the shifted profies after hours of simuation The second profie shift that we considered was a shift in the humidity profie. The profies after and hours of simuation are shown in figure 5.3. What immediatey draws attention is the strange shape of the +1% profie. It is not known what has caused this shape, but as is apparent from the iquid water content, no couds are formed in this case. We wi therefore negect this case, athough it is remarkabe that the +2% and +4% cases do not exhibit this behavior. The effect on the temperature is evident though ess pronounced than the effect of temperature on humidity. The same reasoning appies here as in the previous case. Aso the moisture feedback between coud and 21

22 sub-coud ayer seems to be present. After hours the drier profies have caused a decine in the surface temperature, whereas the more humid profies have increased the surface temperature. The coud base is in this set of simuations very steady. Among the different simuations there is hardy any difference, athough with time the sub-coud ayer deepens. This is understandabe as the surface temperature increases, which in turn is due to the arge scae ow-eve drying. This is 1.2 g/kg/day which is approximatey the shift that can be deduced from the two midde figures in figure 5.3. The inversion height of the drier profies graduay owers as was expected because the coud parces have ess water content and are thus fuy evaporated ower in the atmosphere as opposed to the humid profies where the coud ayer graduay deepens. The apse rate again decreases with time to a vaue of 2.70 K/km as can be seen in figure 5.4. To form some sort of criterion of the convergence we can take the bandwidth of the apse rate at a certain time. For figure 5.3 as we as figure 5.4 the convergence is evident, because the bandwidth at hour 12 is much arger than it is after 30 hours of simuation. What is furthermore interesting is that in the simuations the coder initia profies are much more coherent over the simuation horizon than the warmer profies Figure 5.2: Lapse rate for the shifted profies. Note that the +1% case is not dispayed, because the coud depth was too sma. -1% case ony run unti 23 hours. The strong convergence of the apse rate is cear. 22

23 Figure 5.3: Left top to bottom: q, q t and for the shifted q t profies after hours of simuation. Right top to bottom: q, q t and for the shifted q t profies after hours of simuation Having seen the resuts from the shifted profies we now turn attention to the profies with the adjusted and q. Figure 5.5 dispays the resuts for. After hours the temperature t profies are sti somewhat in ine with the origina perturbation. However after hours the apse rate is at the same eve as in the previous simuations (~2.70 K/km) whereas the coud base and coud top are shifted. Evidence of the feedback mechanism is aso present as the surface 23

24 Figure 5.4: Lapse rate for the shifted humidity profies. Note that the +1% case is not dispayed, because of the fact no couds were formed. The strong convergence of the apse rate is readiy evident. temperature has changed. Aso there is a drying tendency in the atmosphere and the coud depth for the coder profies has become deeper than that of the warmer profies. This can again be understood in terms of the evaporation. The change of over the simuated time is shown in figure 5.. Athough there is initiay a arge difference in the apse rates after about 10 hours of simuations it is at the same eve for a simuations. However after this time there is no further convergence. This eads to the concusions that there are forcings within the mode that act to deveop a certain apse rate. To achieve this apse rate other parameters are adjusted, such as the coud and inversion base and the surface temperature and humidity. The reason why the mode has such a strong preference for this apse rate is unknown. 24

25 Figure 5.5: Left top to bottom: q, q t and for the adjusted q, q t and for the adjusted after hours of simuation after hours of simuation. Right top to bottom: Now that we have ooked at the adjustment of we can focus on the adjustment of q. As is t evident from figure 5. the +30% profie crashed after 21 hours of simuation. It is not readiy apparent why this happened because the coud top was not very high and there were sti cumuus couds present at hour 20. It might have something to do with the top of the domain, where there is a sponge ayer to dampen disturbances. The humidity profie probaby had to much infuence from this sponge ayer. An indication for this is that there is a cear swing in the humidity profie at 2000 m at hours which graduay deveops over the simuation horizon. 25

26 Figure 5.: Lapse rate over time for the adjusted ceary evident.. The strong convergence of the apse rate in the first 10 hours is Furthermore one can notice that the same effects as in the previous simuations are present, abeit ess pronounced than in the case of a change in the temperature profie. After hours the change in the humidity profie is sti discernibe and the effect on temperature is sti sma. After hours the apse rate for temperature is on its usua eve. When one compares the temperature and humidity adjustment, the temperature seems to have a greater infuence on the humidity than the other way around. However this is purey a quaitative notion and needs to be tested further. The apse rate decreases in a simiar fashion as in the other profies to the vaue of 2.70 K/km. Lasty we ook at the adjustment of the subsidence. What directy draws attention is the simiarity of the surface temperature. Here there is no feedback present from the couds to the subcoud ayer. Or better said, the adjustment of the subsidence veocity gives no direct infuence on the surface vaues. This strengthens the beief that there is some kind of moisture feedback, because when we changed one of the two vaues that are of direct infuence on the initia profie there were changes in the sub-coud ayer vaues of temperature and humidity. The ony effect that can be deduced from the change in the subsidence veocity is an effect on the base of the inversion. This effect is as expected, the stronger the subsidence veocity the ower the 2

27 Figure 5.7: Left top to bottom: q, q t and for the adjusted q after hours of simuation. Right top to bottom:, q t and for the adjusted q after hours of simuation. The +30% profie crashed after hour 21. t q t coud ayer wi become (or: the inversion wi be pushed downwards stronger). There is no effect on the apse rate, and there was no effect on the fuxes is we. This is surprising in ight of the discussion in chapter 4. What is aso strange is the fact the +20%, -10% and -20% profies crash after 24 hours. There is once again no apparent reason, especiay because the +/-30% adjustments do not crash. It might be attributed to the top eve vaue of temperature and humidity. Concuding one can say that the trend is persistent among a simuations where the initia profies were changed that there is a moisture feedback. This was maybe unexpected, but confirms the 27

28 parametrization proposed by Neggers (2004). Furthermore it is interesting to note that the change in apse rate is not persistent. In the concusion we wi eaborate on this matter. Figure 5.8: Lapse rate over time for the adjusted humidity apse rate. The strong convergence of the apse rate is readiy evident. The +30% adjustment crashed after 21 hours of simuating. Figure 5.9: Lapse rate over time for the adjusted subsidence veocity. The strong convergence of the apse rate is evident, athough the +10% adjustment seems to be a bit off. +20%, -10% and -20% crash after 21 respectivey 24 hours.

29 Figure 5.10: Left top to bottom: iquid water content, tota water content and iquid water potentia temperature for the adjusted subsidence after hours of simuation. Right top to bottom: iquid water content, tota water content and iquid water potentia temperature for the subsidence after hours of simuation. 29

30 5.2.Scaing aws Here we wi shorty evauate the discussed scaing aws in chapter 3. We wi start with the Grant & Lock (2004) scaing aw, which is based on the coud depth, CAPE and sub-coud ayer veocity scae. The different vaues of the coud base, zc, coud depth, zcd, CAPE and sub-coud ayer convective veocity are given in tabe 1, for the and hour averages. What is readiy apparent is that the CAPE deveops with time. Especiay for active couds the difference between the and hour CAPE is significant. In contrast the sub coud convective veocity scae is rather constant during the simuation time. The coud depth tends to deepen for active couds, whereas the subcoud ayer deepens as we. This is in accord with the findings in the previous paragraph. More active couds carry a greater CAPE and this becomes more pronounced over the duration of the simuation. The proposed scaing is graphed in figure 5.11 for four points in time (-7, and 353 hours). It is apparent that the proposed scaing seems to work reasonaby we. Concuding we can say that the scaing evoves with time but is ceary present. The evovement with time is surprising because Grant & Lock propose their scaing as being independent of time and characteristic for shaow cumuus. Furthermore it is interesting that after about 20 hours of simuation the scaing is stabe with respect to time, a inear fit confirms this as the standard error drops consideraby with time. There is thus a imit on this scaing. Figure 5.11: Scaing for different times in the simuation. With time the scaing becomes more constant. The scaing is persistent over the whoe simuation horizon however. Linear scaing is potted, the /7-scaing gave simiar resuts. 30

31 hour z_c z_cd CAPE w* z_cd CAPE w* Lapse rate (theta) +10% Shift (theta) -0.3% Lapse rate (theta) +20% Shfit (theta) -0.% Lapse rate (theta) +30% Shift (theta) -1% Lapse rate (theta) -10% Shift (theta) +0.3% Lapse rate (q) +10% Shift (theta) +0.% Lapse rate (q) +20% Shift (q) +2% Lapse rate (q) +30% Shift (q) +4% Lapse rate (q) -10% Shift (q) +1% Subsidence +10% Shift (q) -2% Subsidence +20% Shift (q) -4% Subsidence +30% Subsidence -30% Subsidence -10% Subsidence -20% BOMEX hour z_c Tabe.1: Overview of the different simuations and key parameters. Coud base (z_c), depth of ayer between coud base and base of the inversion (z_cd), CAPE and sub-coud ayer convective veocity scae, w *. For some simuations vaues are missing. This is due to a crash (e.g. subsidence +20%) or the fact that no couds were present at that time (shift (theta) +0.%). Two simuations are not represented at a as they produced no cumuus couds. Those are the +1% shift of the humidity profie and the +1% shift of the temperature profie. 31