Large-eddy simulation of cumulus convection

TBcliNiS

-UNvjS-ITEi1

--Scheepshydre''ia

Archief

Mekeiweg 2,. 2628 CD Deift

-Tel: O15-i86873/Fax:781836

A,

£

U

Oh

,

A

kA

Laboratorium voor Scheepshydromechardca

Archiof

Mekelweg

2,2628 CD Deift

TeL 015.788873.Fax 015.781838

TECHNISCHE UNIVERSITEIT

Laboum voor

Scheepshydromechanlca Archief

Mek&weg 2.2628 CD Deift

TeL 015-788873 - Fac 015.781838

Large-eddy simulation of cumulus convection

PROEFSCHRIFT

ter verkrijging van de graad van doctor ann

de Technische Universiteit Deift, op gezag van de Rector MagiiificUs, prof. ir. F.K. Wakker in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen

op rnaandag 31 januari 1994 te 14.00 uur

door

Joannes Willibrordus Maria Cuijpers

doctorandus sterrenkunde, geboren te Amsterdam.

Prof. dr. ir. F. T. M. Nieuwstadt

Prof. dr. I. Oerlemans

CIP-GEGEVENS KONINKLUXE BIBUOTHEEK, DEN HAAG

Cuijpers, Joannes Wilhibrordus Maria

Large-eddy simulation of cumulus convection / Joannes Wilhibrordus Maria Cuijpers. - [S.!. : s.n.]. -111. Proefschrift Technische Universiteit Deift. - Met lit

opg.

ISBN 90-9006737-X

Trefw.: meteorologie I stroniingsleer.

All Rights Reserved.

Copyiight © 1994 by J.W.M. Cuijpers

No part of the material protected by this copyright notice may be reproduced Or Utilized in

any form orby any means, electronic or mechanical, including photocopying, recording or by any information storage and medeval system, without wr Uen permission from the copyright owner.

Clouds are pictures in the slcy

They stir the soul, they please the eye They bless the thirsty earth with rain,

which nurtures life fmm cell to brain

But no! They're demons, dark and dire, hurling hail, wind, flood, and fire Killing, scarring, cruel.rnasters Of destruction and disasters

Clouds have such diversity

Now blessed, now cursed, the best, the worst

But where would life without them be?

Vóllie Cotton

In: W.R. Cotton and R.A. Anthes, 1989

Stormand cloud dynamics

3.

The Puerto Rico Simulation

A shear and buoyancy driven boundary layer

43

3.1 IntroductiOn 43

3.2 Observations 43

CONTENTS

1.

General Introduction

1

1.1 Introduction 1

1.2 Stability criteria and cloud types 2

1.3 Climatology of cumulus clouds 7

1.4 The structure of the trade-wind regions 8 1.5 Physical processes controlling the structure of the trade-wind

boundary layer and cumulus clouds 10

1.6 Observations 12

1.6.1 ATEXandBOMEX 12

1.6.2 The Puerto Rico Field Experiment 13

1.6.3 GATE 14

1.6.4 Boundary Layer Experiment - 1983 15

1.6.5 ASTEX 16

1.7 LES models in meteorology 16

1.7.1 LES models applied to clear-air boundary layers 17 1.7.2 LES models applied to cloud-topped boundary layers 19

1.7.2.1 LES of stratocumulus-topped boundary layers 19 1.7.2.2 LES of cumulus-topped boundary layers 21

1.8 Aim and approach 22

1.9 Outline of this thesis 24

2. Model description

25

2.1 Introduction 25

2.2 Basic equations 25

2.3 The subgrid-scale model 27

2.4 CondensatiOn scheme 30

2.5 Radiation model 32

2.5.1 Longwave radiation 32

2.5.2 Shortwave radiation 34

2.6 Numerics 35

Appendix 2A: w O, as a function of w0 i and w'q1' 37

Contents V

3.2.1 Weather situation and flight plans 43

3.2.2 Aircraft instrumentation system 45

3.3 Initial and boundary conditions 46

3.4 Model resUlts 50

3.4.1 Mean profiles 52

3.4.2 Variances and fluxes 54

3.4.3 The turbulent kinetic energy budget 56 3.5 Dependence of fluxes and variances on cloud amount 59

3.6 The influence of a radiation scheme 62

3.7 Summary and conclusions 65

The GATE Simulation

A buoyancy driven boundary layer

67

4.1 Introduction 67

4.2 Observations 68

4.2.1 GATE Phase III 69

4.2.2 Daiasamplihgandinstrumentation 71

4.2.2.1 Aircraftdaia 71

4.22.2

Tethered balloon and structure sonde data 71

4.2.2.3 Shipsurfacedata 72

4.3 Initial and boundary conditions 72

4.4 Model results 79

4.4.1 Mean profiles 80

4.4.2

Variances and fluxes 82

4.4.3 The turbulent kinetic energy budget 87 4.5 Dependence of fluxes and variances on cloud amount 88

4.6 Summaryandconclusions 91

Analyses of the variance and flux budget terms

92

5.1 Introduction 92

5.2 Closure assumptions 94

5.3 The model simulations 96

5.4 The turbulent kinetic energy budget 98

5.4.1 Introduction 98

5.4.2 The TKE equation 98

5.4.3 The results 101

5.5 Budgets of the second and third order moments of the

vertical velocity 106

5.5.2 The vertical-velocity skewness budget 108

5.5.3 The influence of cloud water amount on the budgets 109

5.6 Budgets of heat, moisture and buoyancy fluxes 110

5.6.1 Introduction 110

5.6.2 The budget equations

ill

5.6.3 The heat, moisture and buoyancy flux budgets 112

5.7 Summaryand conclusions 119

Appendix.5A: Leibniz' nile and divergence theorem 123

6

Summary of conclusions and recommendations

124

6.1 Introduction 124 6.2 Summary of conclusions 124 6.3 Recommendations 127

References

129

List of Symbols

140

List of Abbreviations

- 144

Samenvatting

145

Dankwoord

149

Curriculum Vitae

150

CHAPTER 1

GENERAL INTRODUCTION

1.1

Introduction

In this thesis we will show the resuks of a study of shallow cumulus clouds over sea; These clouds are closely tied to the turbulent, i.e. randomly fluctuating, motions within the

Atmospheri Boundary Layer (ABL). The ABL is defined as the lower part of the atmosphere that is directly influenced by the presence of the earth's surface. Friction and the exchange of heat

and moisture at the surface influence the structure of the ABL. The turbulent motions can be

visualized as consisting of irregular swirls of motion called eddies, having different sizes. The largest eddies are of the cirder of the ABL height (typically a few kilometers on a sunny day over land and less than a kilometer over the tropical oceans), the smallest eddies are about 1 mm.

When the air near the earth's surface is wanner or moister than the atmosphere above, the

air will rise in so-called thermals. These thermals, which are in fact large eddies, produce

convective turbulence. Friction of the flow at the surface produces mechanical turbulence. Due to convectiveand mechanical turbulence, there is effective mixing of heat, moisture and momentum in a large part of the ABL, the so-called mixed layer (section 1.4).

During the upward motion of the air, the temperature of the air parcels will decrease and when sufflôient moisture is present, clouds will form in the upper part of the ABL. It depends on the vertical profiles of temperature and humidity what type of clouds will form (section 1.2). if the upper portion of the ABL is humid and well mixed stratocumulus clouds may form. They typically cover between 90 and 100% of the sky. Because stratocumulus shades the surface, it

reduces the surface heating during daytime. It thus influences directly the temperature in the ABL. The turbulent structure of these stratocumulus-topped boundary layers is driven by

radiative processes. Longwave radiative cooling at cloud top typically drives the convection in these boundary layers. Shortwave radiative heating may cause a decoupling of the cloud layer

from the layer underneath, when a stable layer (section 1.2) is formed near cloud base. This

leads to a cut off of the moisture supply from the surface causing a thinning and possibly break up of the cloud layer.

Cumulus clouds on the other hand cover a sinaller part of the sky. Therefore, the surface warms by absorption of solar radiation, that passes through the clear areas between the clouds.

New thermals will rise, on top of which other cumulus clouds may form. The convection in cumulus clouds is driven by the release of latent heat during condensation, Furthermore,

cumulus clouds influence the vertical transport of heat, moisture and pollutants from the mixed layer to the overlying so-called free atmosphere (section 1.4).

If we want to make weather or climate forecasts, the influence of cumulus clouds on the

thermodynamic structure of the atmosphere has to be taken into account. However,'cumulus

clouds have scales considerably less than those which can be explicitly resolved in forecastand climate models. Therefore, the effects of a cumulus cloud field on the large-scale atmospheric flow has to be parameterized, i.e. it has to be described in terms of the variables known in the

forecast modeL

In order to develop these parameterizations, one hasto understand the processes that are

involved in cumulus convection 1e the convective turbulent motions induces by cumulus

clouds. They can be investigated by observations or by using high resolution models, such as the

large-eddy simulation (LES) model used in this study (see Chapter 2).

In the next sections, we will discuss in more detail the different aspects related to

cumulus clouds. First, we will discuss some concepts related to the thermodynamics of moist air and clouds. Then we describe the different cloud types that can form in the ABL (section 1.2). With the LES model two simulations were made based on observations gathered in the

trade-wind areas and the Intertropical Convergence Zone (ITCZ), where cumulus clouds are often

found (section 1.3). The structure of the trade-wind boundary layer and the 1TCZ is discussed in section 1.4. The physical processes that control the structure of the trade-wind boundary layer and cumulus clouds are dicussed in section 1.5. After that, we will review some observational

studies, that were held in the past to gain more insight in cumulus-topped boundary layers

(section 1.6). A historical overview of the use of LES models in meteorology will be given in section 1.7. Finally, the aim and approach of this study and an outline of the remainder of this thesis are given in sections 1.8 and 1.9, respectively.

1.2

Stability criteria and cloud types

In the introduction we stated that if air near the surface is warmer or moister than the atmosphere above (i.e. the air is less dense), it will rise. How far it will rise depends on the

vertical profiles of temperature and humidity. In this section we will look into more detail to the so-called stability criteria, i.e. the conditiOns that determine whether an air parcel can rise or not.

In the following discussion, we will use the virtual potential temperature to obtain the stability criteria. The virtual potential temperature is defihed as the virtual temperature of an air parcelif it were expanded or compressed adiabatically (i.e. without heat exchange between the air parcel and its environment) from its existing pressure to a reference pressure (generally taken as

1000 mb, the pressure near the earth's surface). So, temperature variations due to pressure changes during ascent or descent of an air parcel are removed. The virtual temperature is the temperature that dry air must have to equal the density of moist air at the same pressure. For

instance, unsaturated moist air is less dense than dry air at the same temperature and pressure, resultingina higher virtual temperature When air is saturated (cloud air) the hquid water makes the air more dense. The suspension of liquid water in an air parcel, the so-called liquid water loading, reduces the virtual temperature. So, instead of looking to density differences, we will

look to virtual potential temperature differences. An air parcel at a certain height, that is less

dense than its environment, has a higher virtual potential temperature.

Height

4

Stability criteria and cloudtypes 3

over a small distance. It is assumed that the vertical motioii is adiabatic, there will be no mixing of moisture with the surrounding air nor condensation will take place and, at every instant, the

pressure of the air parcel and its environment at a given height are equal. Thus, during the vertical displacement of the air parcel, its virtual potenthi temperature remains the same; it

follows the line (p1-p 1') in Figuxe 1.1, which is called the moist adiabatic lapse rate. Given a lapse rate e. the change of virtual potential temperature with height) of the atmosphere indicated

by the line (ó-u'), the virtual potential temperature of the air parcel becomes larger than the

environment when the parcel is raised from o to p1'. The air parcel is positively buoyant (less dense than the environment) and will continue to rise. This situation is called unstable (the line

uu'). If, on the other hand, the atmospheric lapse rate is given by (o-c'), the virtual potential

temperature of the air parcel at p1' is less than the environment (c') and the airparcel is forced back to its original position (o). The atmosphere is called stable (the line c-c'). There is neutral stability when the virtual potential temperature of the air parcel and its environment are the same

(pi-pi ').

if condensation takes place during the rise of the air parcel, the virtual potential

temperature of the air parcel will change due to the release of latent heat, the decrease of moisture and an increase of liquid water. An ascending saturated air parcel will follow what is called the

Virtual potential temperature

Figure 1.1 Schematic plot, showing different environmental lapse rates (full lines) and the lapse rates of an unsaturated air parcel (thin broken line) and saturated air parcel (thick broken line).

saturated adiabatic lapse rate. It is indicated by the thick broken line (,p2-p2') in Figure 1.1. From this figure it is clear that an atmosphere with a lapse rate (c-c') is stable for an air parcel that is ascending without condensatiOn, but unstable when condensation takes place, because in the latter case the virtual potential temperature of the air parcel at p2' has become larger than the

environment (c'). Therefore, an atmosphere with a lapse rate as indicated by the line (c-c') is

called conditionally unstable With a lapse rate (s s) the atmosphere is called absolutely stable because it is stable for any vertical displacement of an air parcel, whether there is condensation or

not.

The stability conditions so far have been based on infinitesimal vertical displacements. Next, we will describe what happens when there is a buoyant air parcel near the surface We will assume that the virtual potential temperature profile of the ABL is given by the full thick lines in Figure 1.2. If an unsaturated buoyant air parcel or thermal ascends from p1, it will follow the moist adiabatic lapse rate (p l-pl '), as long as there is no condensation (Figure 1 .2a). At the top of the mixed layer (zj; see section 1.4), the thermal becomes negatively buoyant. However, it will overshoot into the more stable layer because of the momeñuun, it gained in the mixedlayer

Then, the negatively-buoyant air will sink back to height z, where its virtual potential

temperature equals the environmental virtual potential temperature. The layer, overwhich the thermal overshoots into the stable layer, is called the entrainment zone.

The level, where a thermal attains saturation during its ascent, is called the lifting

conden-Vimial potential temperature - Virtual potential temperature

-Figure 1.2 Schematic plOts showing a clear-air ABL (a) and an ABL with stratocumulus clouds (b). See text for further explanation.

aj

Height

f

zi

Virtual potential temperature

-b)

Height

zi

Stability criteria and cloud types 5

Virtual potential temperature

Figure 1.3 Schematic plots showing cumulus cloud layers with different vertical extent. See text for further explanation.

sation level (LCL). Because of vartations in temperature and moisture from thermal to thermal, there is a variation in the LCL height for different thermals. So, in the ABL there is a LCL-zone instead of one LCL height (Wilde et al.,1985).Above the LCL the thermal will continue to rise

following the saturated adiabatic lapse rate (p2-p2' in Figure l.2b). Assuming an absolutely

stable atmosphere above the LCL, the thermal will eventually terminate its rise. If the air is humid and well-mixed a stratocumulus deck may form.

Another situation is shown in Figure 1.3. It is assumed that the atmosphere is

conditionally unstable above Zj and thatthe LCL is in this conditionally unstable layer. Up to the LCL the air parcel will follow the moist adiabatic lapse rate (pi-pi'), but as soon as condensation takes place (at the LCL) it ascends with the saturated adiabatic lapse rate (p2-p2'). Depending on the amount of latent heat that is released during the condensation, the thermal will stop before (Fig. 1.3a) or after (Fig. l.3b) it becomespositively buoyant again. In the latter case, the thermal

will continue to rise well into the absolute stable layer, which is assumed to be on top of the

conditionally unstable layer. The level where the thermal attains positive buoyancy is called the level of free convection (LFC). It depends on the amount of moisture in the rising thermal as well

as the magnitude of the lapse rate in the conditionallyunstable layer.

The descriptions given above were all based on thermals rising through the mixed layer

without exchanging heat or moisture with the environment 1 e without lateral entrainment

Becanse in a real atmosphere lateral entrainmeiit will occur, the thermals do not exactly follow the moist adiabatic lapse rate. In the mixed layer, the virtual potential temperature of the environment is lower than that of the thermal, thus lateral entrainment will reduce the buoyancy of the thermal. The overshoot into the layer above the mixed layer will not be as far as indicated in Figures 1.2 and 1.3.

Because of the different vertical extents of cumulus clouds, they show differencesin their ability to exchange air between the mixed layer and the layers above. Stull (1985) proposed three categories, according to the nature of their interaction with the mixed layer forced, active and passive clouds.

Thermals in the mixed layer may overshoot into the conditionally unstable layer When these thermals rise above the LCL, so-called forced clouds form (Figure 1.4). The latent heat that is released during the condensation is insufficient to warm the air within the thermals so far that they become positively buoyant. However, the clouds can rise higher than thennals in which

no condensation takes place1. So, the clouds are able to transport mixed layer air to greater heights than thermals in which there is no condensation. Morphologically, these clouds are

cumulus hiimilis or cumulus mediocris (International Cloud Atlas, WMO, 1975).

Active clouds have their base at about the same height as forced clouds, since they also

form on top of overshooting mixed layer thermals. However, due to the release of sufficient

latent heat, they reach their level of free convection (LFC; Figure 1.4). These clouds can grow much further and peneirate well into the stable layer above, where the cloud tops are negatively buoyant again. Active cumulus clouds do mix air from the mixed layerinto the free atmosphere

and are therefore important in venting of pollutants (Ritter, 1984; Cho et at, 1984). Two

mechanisms ale possible for moving mixed layer air up into an active cloud base. First, there is continued forcing by the thermal that triggered the cloud formation. Secondly, there can be a cloud-induced updraft associated with negative pressure perturbations often found at the bottom

of a positive buoyancy region (Stull, 1985). Active clouds are classified as larger cumulus

mediocris and cumulus congestus.

Lastly, passive cumulus clouds are the decaying remnants of formerly active clouds

(Figure 1.4). These clouds have no interaction with the mixed layer anymore and although near cloud top buoyant rising air might still be present, these clouds Will eventually evaporate and

disappear. The- clOuds do not have a flat base -as is normally seen for forced and active clouds. In

a very humid environment, these clouds will evaporate slowly. According to Albrecht (1981) passive clOuds are responsible fOr a large fractiOn of the total cumulOs cloud cover in- the trade-wind areas.

1

Climatology of cwnulus clouds 7

free atmosphere

specific virtual potential humidity temperature

Figure 1.4 The multi-layered structure of the trade-wind boundary layer and the different cloud types.

1.3

Climatology of cumulus clouds

Routine weather observations from land stations over the period 1971 - 1981 (Warren et

aL, 1986) and from ships over the period 1952 - 1981 (Warren et aL, 1988) were used to

explore the global distribution of cloud cover and cloud type amounts. Warren et al. defined a frequency-of-occurrence as the number of times that a particular cloud type was reported present,

divided by the number of reports in which the level of that type was observable. Because all

observations were earth based, clouds at a high level may be obscured by clouds at lower levels.

Therefore, the restriction was made that the particular level had to be observable. All reports

within 5 x 50 latitude-longitude boxes were averaged. For cumulus clouds over ocean areas a frequency-of-occurrence of more than 65% and over land areas of more than 40% is frequently found in the5 x 5 boxes in the tropical regions. When all ocean reports are globally averaged the

frequency-of-occurrence is about 34%, while it is about 14% for the globally averaged land

reports.

-The amount when present or cloud cover is defined as the average fraction of the sky

which is covered by a particular cloud type, when it is present. The average cloud amount is defined as the product of frequency-ofoccunence and cloud cover. So, the average cloud amount is the amount of clouds averaged over space and time. The average cumulus cloud

amount in the trade wind regions is about 20% Globally averaged the cUmulus cloud amount over ocean and land is 12% and 4.5%, respectively. So, it was found that large areas of cumulus clouds exist in the trade-wind regions in each hemisphere and over the inid-lalitude conthients in summer In the tropical region, cumulus is often the most frequently occurring cloud type and moreover the cloud type contributing most to the total cloud cover.

A diurnal cycle of cumulus clOUds is often seen over the continents in summer. Starting

from a situation without clouds during the night, the cumulus cloud cover increases during

daytime to reach a maximum in the early afternoon. Over the oceans, the cumulus cloud fields are more persistent. The cloud cover varies by about 2%, with a maximum also at the beginning of the afternoon.

Warren et al (1988) reported an average cloud base height of cumulus at about 600

meters over ocean regions with a tendency of slightly increasing cloud base heights with latitude. Due to the generally stronger convection over land the average cloud base height is higher, about

l000m.

1.4

The structure of the trade-wind regions

Because in the trade-wind regions cumulus clouds are an often seen feature and one of our simulations is based on observations gathered in these regions, we will show in more detail the structure of the trade-wind areas. A schematic cross-section of the atmosphere through the trade-wind area and the Intertropical Convergence Zone is given in Figure 1.5.

Solar heating is strongly dependent on latitude, with a maximum at the equator and a minimum at the poles. The differential heating creates a circulatiOn, known as the Hadley

circulation. In the Hadley circulation, there is warm rising air near the equator and cold sinking air over the subtropical regions. In the lower part of the atmosphere, the air moves equatorward,

while at high altitudes the flow is directed towards the poles. Because of the earth's rotation, these lower and upper branches of the Hadley circulatiOn are not exactly north-south. In the lower branch there is an easterly component, while in the upper branch there is a westerly

component. The northeasterly winds in the northern hemisphere and the southeasterly winds in the southern hemisphere are called the trade winds. The region, where northeast and southeast

trades flow together is called the Intertropical Convergence Zone (1TCZ). The ITCZ is

characterized by strong upward motions and heavy rainfall.

In the trade-wind regions, there is a large-scale downward motion of air, the so-called subsidence. It influences the vertical development of the convection. When subsidence is strong,

The structure of the trade-wind regions 9

-1cn

cloudtse

AE

_

A

____

____

U V -I-! V

Equator

North ______

0' 30'N

Figure 1.5 Schematic cross-section through trades and !TCZ illustrating the circulation and moistening of the mixed layer and cloud layer. E is the water vapor flux, Sc is

stratocumulus and Cu is cumulus (After Tiedtke, 1987).

convection will be limited to smaller vertical extents (i.e. convection is suppressed) than when the subsidence is weak, in which situation the convection is said to be enhanced.

The layered structure of the trade-wind cumulus-topped boundary layer (CTBL) over the oceans can be described as follows (Figure 1.4; Malkus, 1956; Tiedtke, 1987):

a thin surface layer (a few tens of meters thick) with a superadiabatic temperature lapse rate and a sharply decreasing specific humidity;

a mixed layer up to about 600 m with an adiabatic temperature gradient and a slightly decreasing

specific humidity with height;

a conditionally unstable layer extending to the base of the inversion. This layer is called

conditionally unstable since it has a temperature lapse rate between the moist adiabatic lapse rate and the saturated adiabatic lapse rate (section 1.2). The specific humidity decreases with height. In this layer the clouds form and therefore we will also refer to it as the cloud layer

the trade wind inversion on top of the cloud layer shows a strong increase in temperature and decrease in humidity with height;

1.5

Physical processes controlling the structure of the

trade-wind boundary layer and cumulus clouds

The layeid structure of the CTBL over the oceans described m the previous section is a result of the interaction of several physical processes Here we will describe these processes and

discuss the important role of trade-wind cumulus convection in the atmosphenc energeucs

through intensifying the hydrological cycle. After that we will discuss the interaction between

cwnulus clouds and the surrounding air and the results of observations in relatively small

cumulus clouds.

The thermodynamical structure of a CTBL over the tropical oceans is controlled by (Betts and Ridgway, 1989) the convective fluxes, the surface wind, sea surface temperature, the cloud field, subsidence, and the radiation.field.

Above the CT.BL in the absence of turbulent fluxes, there is a close balance between

subsidence and the radiation field There is a radiational cooling while the subsidence

continually brings down warm and thy air By entrainment and cloud-induced mixing of this air the CTBL gets drier and warmer The drying of the CTBL is compensated by the evaporation at the surface and an upward flux of moisture from the surface. The evaporation is controlled by the surface wind and the humidity difference between the air just above the ocean and in the mixed layer.

The warming of the CTBL by entrainment and the surface sensible heat flux is largely in balance with the radiative cooling The balance is such that the mixed layer is always drier and cooler than the air near the ocean. Because of this surface-layer instability and the surface winds,

the surface sensible and latent heat fluxes are maintained. The balance between the surface sensible and latent heat fluxes and the net radiation determines on long time scales the sea

surface temperature.

The equilibrium temperature and moisture structure of the CTBL and especially the

associated, but continuously changing, cloud fields influence the longwave radiative cooling and the incoming shortwave radiation The cumulus clouds change the net radiation by absorption and reflection of solar (shortwave) radiation and absorption of the upward infrared (longwave)

aiation from the mixed layer.

A complicating factor in modeling the radiation in a cumulus cloOd field is that this field is

horizontally inhomogeneous. The shape, size and spatial distribution of the clouds all have an

effect on the radiation (Randall, 1989; Fouquart et aL, 1989 and references therein). FOr instance, in a broken cloud field, one cloud may be shadowed by a neighboring cloud. This

effect will of course depend on solar elevation and the distances between and the heights of the clouds. Harshvardan (1982) showed that, on global average, the influence of broken clouds on

the longwave radiative flux at the top of the atmosphere and the global albedo2 is different

compared with plane-parallel clouds. The latter are typically used in global dlithate models. Their

2 _{The albedoisdefined as the ratio of the reflected and in all di±eed scattered solar radiation to the incident}

Physical processes controlling the trade-wind boundary layer and cumulus 11

influence on the radiation budget is calculated using cloUd cover. Since it is not to be expected that more sophisticated radiation models will soon be used in climate models, one has to find an effective cloud cover for plane-parallel clouds, that would provide the same radiation budget as the broken cloud field.

The significance of cumulus convection in the trades for the atmospheric energetics is illustrated in the following way. Because of a constant input of moisture from the ocean and the

subsequent vertical transport of it by the cumulus ôlouds, the cloud layer (which deepens equatorward) is continuously accumulating moisture. Furthermore, cumulus clouds may occasionally penetrate into the inversion layer. There they will evaporate, thus cooling and

moistening the inversion layer. The increased amount of latent heat is transported into the tropics, where the clouds can grow much higher because of a weaker inversion layer. Diabatic heating due to the latent heat release and intense precipitation in these deep convection situations affects the atmospheric thermodynamic structure significantly. Thus cumulus convection in the trades is important because itintensifies the hydrological cycle.

During thgir life-cycle, cumulus clouds interact with the surrounding air. Entrainthent of this environmental air mfluences the hquid water content of cumulus clouds Cooper and Lawson

(1984) found that due to entrainment in cumulus clouds, sampled in the High Plains of the United States, the liquid water content decayed with an average decay time of 14 minutes. Furthermore, entrainment is also important in cloud chemistry. It can supply new chemical

species, thus promoting the continuation of chemical reactions inside the clouds (Chandler et al.,

1988).

Entrainment apparently occurs at all heights of cumulus clouds. In growing clouds,

entrainment occurs at the ascending cloud top. It was found from Observations and model results that vertical rather than horizontal mixing was most important in the development and dilution of

cumulus clouds (Blyth 1993) However it is still uncertain how environmental air is actually

entrained into and mixed with the cloud air. For instance, whether the environmental air is mixed with the cloudy air at the cloud boundarins or there is first a break up of air parcels into smaller ones, that mix with cloud air deeper inside the cloud is not clear.

It seems now well established that in cumulus clouds large fluctuations in liquid water content exists up to the smallest measurable scales (..- 1 cm; Baker, 1992). The edges of clouds are usually sharp and there is no clear systematic trend in the liquid water content from cloud edges towards the middle. A liquid water content, when calculated over several meters, larger than 2 g rn-3 is often found, although average values taken over a whole cloud penetration appear to be less than 1 g rn-3 (Blyth and Latham, 1990). The vertical profile of the liquid water content in a cloud is still uncertain; an approximately constant profile with height as well as decreasing

liquid water profiles have been found. However, Mason (1975) suggested that these profiles

may be influenced by different vigor and stages of development of clouds and characteristics of the environment.

1.6

Observations

In order to study the mechanisms that are involved in cumulus convection and to obtain data to validate numerical models several observational studies have been conducted. Among

them are the Atlantic Trade wind Expenment (ATEX) (Augstein et al

1973 1974) the

Barbados Oceanographic and MeteorOlogical Experiment (BOMEX) (Holland and Rasmusson, 1973 Nrna and Esbensen 1974) the Puerto Rico Field Experiment (Pennell and LeMone 1974

LeMone and Pennell, 1976), the GARP Atlantic Tropical Experiment (GATE) (Nicholls and

LeMone, 1980), the Boundary LayerExperilnent- 1983 (BLX83) (Stull and Elorante, 1984), and

the Atlantic Stratocumulus Transition Experiment (ASTEX) In this section we will briefly

discuss these experiments to show the results that were achieved.

1.6.1 ATEXandBOMEX

The Atlantic trade-wind experiment (ATEX) was conducted during February 1969 in the

Atlantic Northeast trade-wind region (11°N, 37'W; Augstein et al., 1973, 1974). Radiosonde and radar wind observations routine deck level observations (temperature humidity wind

velocity cloudiness etc ) from three ships that formed the corners of an equilateral triangle with

sides of about 750 kin, as well as turbulent surface fluxes from buoys near the ships were

obtained. During BOMEX (the Barbados Oceanographic and Meteorological Experiment) that was conducted during May and June 1969 over an area east of Barbados (15'N, 56'30'W), two

aircraft were flying around the perimeter of a 500-km square too (Holland and Rasmusson,

1973; Nitta and Esbensen, 1974). In this experiment four ships were located at the corners of the square.

The multi-layered structure of the traçle-win4 boundary layer, as described in section 1.4, which had been observed by several investigators (Bunker et al 1950 Garstang 1972) was confirmed in these two experiments Between the mixed layer and the cloud layer a so called

transition layer was assumed. It was a layer with an average thickness of 100 rn, with a nearly isothermal to slightly stable temperature distribution and a strong upward decrease of moisture, separating the regimes of cloud convection above from those of thy convection and mechanical mixing below. However, this transition layer could often not been detected.

In the analysis of the budgets of mass, water vapor and heat, the boundary layer up to

700 mb (- 4000 m) was divided into four sublayers (Holland and Rasmusson, 1973; Nitta and

Esbensen, 1974); the mixed layer, the cloud layer, the inversion layer and a layer from the inversion top to the 700 mb level The analysis showed that under well developed trade wind

conditions with little cloud activity (undisturbed periods) there is horizontal divergence in mixed

layer and cloud layer The horizontal divergent layer is separated by the trade wind inversion from a convergent layer above When cloud activity is larger the so-called disturbed periods

convergence exists in the mixed layer. During these periods, divergence is found aloft but the

magnitude is smaller than during undisturbed periods. The divergence is associated with downward motions and was found to be present over the whole height range between the sea

surface and the 700-mb level, with a maximum of 6 mm l at the inversion base during

undisturbed periods.

Analisis of the moisture budgets revealed a vertical flux of moisture from the sea surface

towards the inversion layer. The flux convergence counteracts the d±ying effect of the mean downward motion and preserves the strong humidity gradient in the inversion layer The

combined effects of mean downward motion of warm thy air (which prevents condensation) and the turbulent upward water vapor flUx.resulted in the accumulation of moisture in the trade-wind flow below the inversion.

In the inversion a large heat sink was found due to evaporation of cloud droplets which detrain near the top of the trade mversion Very few clouds were seen to penetrate into the upper layer above the inversion. During the disturbed period no sting heat sink or moisture source is

found at the same level as in the undisturbed periods. The enhanced cumulus convection weakens the inversion layer. Cumulus convection extends to greater heights, resulting in a

warming and diying in the whole layer between the top of the mixed layer and a height of about

3 1cm, while there is a cooling and moistening in the upper layer above 3 1cm, as a result of

detrainment effects of the clouds.

1.6.2 The Puerto Rico Field Experiment3

Iii order to obtain more insight in the structure of the boundary layer and the relation

between subcloud fluxes and the clouds the Puerto Rico Field Experiment was conducted. On 14

and 15 December 1972 data were collected with the NCAR DeHa'villand Buffalo gustprobe

aircraft in an area north of Puerto Rico In the mesoscale regions (-10-1001cm) the reasonably

uniform convection ranged from suppressed (section 1.4) with very little shallow clouds to

slightly enhanced with active (but:non-precipitating) trade cumulus (Pennell and LeMone, 1974; LeMone and Pennell, 1976). Surface winds were considerably (10 to 15 m s-1) from the east.

With the aircraft, measurements of the wind, temperature, humidity, variances, fluxes of

momentum, heat and moisture and overhead cloud occurrences were Obtained.

From the measurements in the mixed layer it was concluded that in both turbulent kinetic energy (TKE) and momentum flux budgets, the turbulent transport terms are quite important. They become the dominant source terms near the top of the mixed layer. At that time, an attempt was made to mprove the modeling of the turbulent boundaiy layer, by introducing the Reynolds

stress equation into the model and compute momentum transport explicitly. Shear,

pressure-gradient correlation and turbulent transport term had to be parameterized. The turbulent transport

term was assumed not to be crucial (Tennekes, 1973; Panofsky, 1973), particularly for

near-neutral conditions, but these observations showed that this was not the case.

Furthermore, it was shown (LeMone and Pennell, 1976) that a strong relationship

between the cloud distribution and subcloud layer structure and fluxes exists for

non-precipitating clouds. n the suppressed area, clouds formed from moist buoyant air parcels in the

Results of this experiment were used in the simulation described in Chapter 3.

upward moving regions of the quasi-two-dimensional roll vortices (i.e. these are forced clouds;

Stull 1985) In these upwefling regions turbulence and fluxes were greater The compensating

downward motions between these regions suppressed the upward motions of turbulent elements.

Cloud and turbulence distributions were considered by-products of the (two-dimensional)

subclóüd structure.

In the more enhanced regions the coupling between active clouds and subcloud layer of which the structure becomes more random is stronger The location of the clouds is still related

to the subcloud layer circulation, but fluxes are influenced by the cloud circulation. The

momentum flux was found to be significantly greater beneath patches of cloud than beneath clear air. Modest-sized clouds were fed from just below cloud base by a moisture flux nearly equal to the surface flux These enlarged fluxes could be detected 100- 300 meters below cloud base In the clear air, moisture fluxes were small at cloud base height. For patches of trade wind cumulus with heights of 2 to 3 km, the export of moisture fmmthe subcloud layer can be several times the evaporation rate. There must be a horizontal moisture convergence in the subcloud layer from an

area several times that of the cloud patch in order to have an equihbnum If not the subcloud

layer must thy out.

1.6.3 GATE4

During the GARP (Global Atmospheric R search Program) Atlantic Tropical Experiment

(GATE), which was held from Jutie to September 1974, four days were selected to study the

structure of the convective atmospheric boundary layer and the characteristics of the associated

thrbulent mixing processes (Nicholls and LeMonè, 1980). On three of these days there was significant cloud activity Since winds were light during these days, hence momentum fluxes

were small they concentrated on temperature and humidity proffles and fluxes of these variables in the mixed layer.

It was noted that on the days with cumulus activity, the absolute valOes of latent and

sensible heat fluxes near cloud base are similar to or greater than those at the surface. However,

in contrast to the findings of LeMone and Pennell (1976) in this case a direct relationship

between the subcloud fluxes and the amount of cloudiness overhead was not apparent.

Furthermore they could not detect any cloud roots (defined by a pronounced updraft, clearly related to a cloud above) below the cloud base It was assumed that the relationship found by LeMone and Pennell was strongly influenced by roll vortices present in the subcloud they

investigated, but that were absent on these four days of GATE. Although the vertical profiles of sensible and latent heat fluxes changed the vertical profile of virtual potential temperature was not affecied by changes in overhead cloudiness: -.

By conditionally sampling of temperature and humidity, it was found that the convective elements are relatively cooler and wetter than the mean temperature and humidity throughout

most of the mixed layer stressing the fact that over the tropical oceans it is mainly the water

vapor that maintains the positive buoyancy This Is in contrast with normal situations over land

where the rising elements are genetally wanner than their environment throughout the mixed

layer (Lenschow, 1970).

A spectral analysis of the subcloud layer structure revealed the presence of very long

wavelength 10 1cm) fluctuations. It was speculated that these large-scale fluctuations might be

the remnants of pre-existing spatial inhomogeneities produced by more active deeper

convection. Deep convection can strongly modify the boundary layer, and since on the days that were studied, the convective mixing processes were relatively weak, it will take some time before typical fair weather conditions are achieved. The existence of mesoscale structures, which have a

dynamical origin, possibly originating from within the mixed layer, was posed as another

explanation.

Finally, intermittent entrainment and may be the mixing induced by clotid formation

might have brought air with sufficient different properties into the mixed layer. Because of the weak mixing these intrusions wifi be able to maintain their identity for some time. But although

the effects of entrainment were clear in the upper part of the mixed layer, they were not sure

whether it could directly produce mesoscale (-10 km) fluctuations.

However, these longwave fluctuations may cause problems when simulating these boundary layers with LES models, because than the history of the boundary layer nor any

mesoscale variability is taken into account.

1.6.4 Boundary Layer Experiment - 1983

The observations discussed so far were all obtained above the oceans. Boundary Layer

Experiment - 1983 (BLX83; Stull and Elorante, 1984) was a field experiment over land, in

which the interactions between fair-weather cumulus clouds and the mixed-layer thermals was studied. In this experiment, remote sensors, surface observations, balloon platforms and aircraft measurements were used to -measUre the turbulent structure at the top of the boundary layer. Emphasis was placed on the-study of the entrainment zone and cloud-base fluxes.

In contrast to cumulus-topped boundary layers over the oceans, the cloud base height

over land exhibits a diurnal cycle Cloud bases are lower in the morning and nse to a peak in the late afternoon, apparently related to the diurnal cycle of the temperature. This-is in contrast with

the results of Warren et al. (1986), who found a maximum early in the afternoon. There was evidence of decoupling of the cumulus cloud bases from the boundary layer in the evening.

While the LCLs were decreasing in the evening, the cloud bases remained constant or even rose with time. The clouds dissipated shortly after this was observed.

Within a 25 km long region; the local LCL height was found to vary by 15 - 30% of-its average height. This leads to the definition of the "LCL zone", defined as the zone of variation, centered on the mean LCL height (Wilde et al., 1985). Cumulus clouds first form when the top of the entrainment zone reaches the bottom of the LCL zone.

The local LCL height varies because of variations in temperature and moisture from

thermal to thermal and lateral entrainment into thermals. However, during BLX83 some of the air Observations 1-5

at cloud base was observed to have the same LCL as that of the mean surface layer air, hence it had risen up to cloud base without or with relatively little dilution The thickness of the LCL zone is modulated by surface characteristics such as land use patterns albedo variations soil moisture variations and shading by clouds.

1.6.5 ASTEX

In June 1992 the Atlantic Stratocumulus Transition Experiment (ASTEX) was held,

Which focussed on the transition of stratocumulus to cumulus clouds (FIRE Phase II Research

Plan, 1989). During this campaign, data were collected using satellite, aircraft, ships,

radiosondes, buoys and instruments on islands off the coast of Africa (32'N, 21'W). Length

scales that were covered during the observations ranged frOm l0 to 106 in.

During a recently held workshop at ECMWF (ECMWF/GCSS Workshop on

Parameterization Of the Cloud Topped Boundary Layer, ECMWF, Reading, UK, 8-11 June

1993) the first results Of this campaign were shown. Among them were the diurnal cycle and possible break up of stratocumulus clouds instabilities at the top of the clouds the influence of drizzle on the formation of horizontal mhomogeneitiesinstratocumulus and decoupling of the cloud layer from the well mixed layer below.

Cumulus clouds were often seen to rise into the stratocumulus layer and their effects on maintaining (by recoupling itto the surface moisture source) or breaking up of the cloud layer (through enhanced entrainment of dry airfrom above the inversion) was monitored. The break up happened on time scales much longer than the circulation time scales of the stratocumulus eddies.

Dramatic changes in air mass were encountered, when northeasterly winds brought high ly polluted air from Europe. In the sO-called "dirty"-clouds thedroplets are much smaller, leading to a clear increase in reflectivity. It was speculated thata dOcrease of cloud dropletradius from 10 to 8 im is sufficient to undo the warniii g due to adOubliiig of the CO2-concentration.

1.7 LES models in meteorology

Besides observations such asmentioned above to study the physical processes in ABLs in genetal and CFBLs in particular, numerical models have helped to gain more insight..Among

these models there are the LES models as used in this study5. In this section, we will first

discuss the use of LES models in meteorology in general and then focuson the use of LES in the study of cumulus convection..

LES models in meteorology 17

1.7.1 LES models applied to clear-air boundary layers

In the late sia ties (Lilly, 1967) and early seventies (Deardorff, 1970, 1972a) LES models were first applied to study the ABL, The model of Deardorif had only a small number of grid

points (24 x 14 x 20 grid points in x, y and z directions, respectively) and was used to

investigate the idealized planetary boundary layer. This boundary layer is defined as a boundary layer having horizontal homogeneity of the mean wind and density, statistically steady states of

horizontally averaged quantities and which is free of buoyancy effects, i.e. there are no

temperature fluctuations.

The variables in the equations were made dimensionless6 with a length scale h, defined as the height slightly above the level where the wind first attains the geosirophic flow direction, the surface friction velocityu*, the time scale h/u* and the surface-level density p0.

In his first study, Deardorff discussed several aspects of the idealized planetary boundary

layer. For instance, he investigated whether the anelastic approximation with mean density a

function of height was a significant improvement over the assumption of constant density within the boundary layer. For that he compared the results from a simulation with constant density with one in which the density decreased linearly by 10% from the surface to the top of the model. The horizontally mean winds and stress components did not show any significant changes. The only

quantity that changed was the geostrophic wind. When the density was allowed to decrease

realistically With height the geostrophic wind increased correspondingly with height.

Furthermore, he found that the vertical transport terms in the TKE budget were relatively small in

the bulk of the model as was commonly assumed. Investigation of the turbulence patterns

revealed the elongation of the u-eddies in the x-direction near the surface and helical circulations (carrying momentum downwards) in planes normal to the mean flow near the surface. They were

assumed to be similar to the helical circulations found by Angell et al. (1968) during July

afternoons in Idaho.

The next step towards more realistic situations was achieved by introducing an equation for the potential temperature. Now a stratified boundary layer could be simulated too. Deatdorff applied his model to the unstable boundary layer (Deardorif, 1972a). The equations were still made dimensionless with the variables given before, where the length scale h now equals ;, the height of the inversion base below which the cOnvective mixing is cojifined. Because the domain extended to this height zj also, entrainment of warm air from the stable layer above could not be

simulated.

6 _{Since Deardorff swdied the neutral ABL, the choice of variables to make the equations dimensionless is}

unambiguous. However, when simulating convective ABLs, buoyancy as well as shear production may be

important and the choice of for instance a velocity scale is not obvious anymore. A summary of relevant scaling variables in ABLs without clouds is given by Holislag and Nieuwstadt (1986). Nowadays, the equations solving the atmospheric flow are no long dimensionless in LES models.

From the simulatiOns Deardorff found that in unstable situations the vertical velocity

variance has to be scaled with the convective velocity scale w =(()

Zi)

with ()o

the surface heat flux, rather than the friction velocity u*. The appropriate stability parameter

turned out to be -zj/L, with L the Monin-Obukhov length. With -Zj/L of order 45 and more, the elongated eddins fOund in the neutral boundaiy layer disappeared. Instead the turbulent structure was dominated entirely by convective updrafts without any preferred horinoñtal orientation. In a time series of the vertical-Velocity field the appearing, growth and merging of thermal elements into pre-existing large plumes were seen.

In contrast to his findings in the neutral case, in the unstable boundary layer the turbulent transport term was found to be more important, transporting turbulent energy from lower levels (below zIz1 = 0.3 - 0.4) to the upper levels. Above a height of 0.5 to 0.6 zj it even exceeded the rate of production due to buoyancy.

The unstable or convective atmosphenc boundary layer (CABL) dominated by large

scale structures is especially suitable for simulating with a LBS model (Moeng, 1984; Mason, 1989; Schmidt and Schumann, 1989). With this in mind, the choice of Deardorff to start with

the neutral boundary layer does not seem to be the most logical.. However, when computer

power and memory increased, new simulations of the neutral-static-stability planetary boundary layer (NSSPBL) could be made (Mason and Thomson, 1987). Mason and Thomson considered the influence of various domain sizes and model resolutiOns on the turbulent structure of the NSSPBL. As Deardorff did, they also found that the eddies had a generally elongated structure, which was particularly marked near the surface. In most cases the eddy elongatiOn orientation, changed such that the direction remained close to the direction of the wind shear vector at each height.

The flow statistics were found to depend significantly upon the range of scales

considered by the simulation (by enlarging the domain or increasing the resolution). This

explained the higher values of surface stress found by Mason and Thomson, with respect to the simulation of Deardorff. An analysis of the TKE budget confirmed most of Deardorff's results. There was a near balance between production and dissipationinthe bulk of the boundary layer Only in the upper part; transport of turbulence energy from lower levels becameimportant.

An ABL that is much more difficult to simulate than those discussed above is the stable boundary layer. A stably stratified ABL, that may arise by nocturnal radiative cooling, is usually

shallow (typically a few 100 m). Turbulence is generally less vigorous and sometimes highly

intermittent. Turbulence length scales are relatively small. Besides turbulence, wavelike

processes may be important too.

Since the basic idea of large-eddy simulations is that the dominant (large-scale motions)

are calculated explicitly and only the more isotropic smaller (subgrid) scales have to be

paranieterized, the small turbulence length scales restrict the resolution that can be used in the

LES models in meteorology 19 model. Mason and Derbyshire (1990) showed the results of successful simulations of stable boundary layers over uniform, flat terrain. A stable boundar' layer was obtained by applying

surface cooling to a fairly steady neutral-static-stability fully turbulent simulation.

The flow statistics were found to be consistent with local scaling as developed by Nieuwstadt (1984). Evidence to this approach was found, for instance, in the energy and

temperature variance budgets. In both budgets the productiOn terms were clearly in local balance with dissipation, since transport terms were found to be small

In their simulations no evidence for wavelike motions was found. The potentially non-local character of wavelike motions apparently was not strong enough to invalidate non-local scaling. If the waves are generated purely by turbulence, thisis to be expected as has been explained by Derbyshire (1990).

As noted above, a LES model is very well suited to simulate the CABL. In order to

investigate the influence of subgrid models, numerics and boundary conditions on the

performance of LES models four existing large-eddy codes were run for the same case of the CABL (Nieuwstadt et aL, 1991). From this comparison the conclusion was drawn that the large eddies are quite insensitive to the details of the subgrid model. The model results were within the scatter of available observations. Most differences in the model results could be attributed to the different values used for a parameter in the subgrid model. This parameter can be interpreted as the ratio of mixing or filter length to grid spacing. Differences in numerical methods and details of the staggered grid or lower boundary conditions were found hardly to effect the results.

1.7.2 LES models applied to cloud-topped boundary layers

By introducing an equation for the specific humidity and a condensation scheme, a LES model is capable of simulating a boundary layer with clouds as well. Since the clouds in these boundary layers alter the radiation budget (section 1.1) a.radiation scheme is often included as

well. In the following, we will discuss some results obtained from large-eddy simulations of cloud-topped boundary layers; first, stratocumulus-topped boundary layers and secondly

cumulus-topped boundary layers.

1.7.2.1 LES of stratocumulus-topped boundary layers

The first one who studied the stratocumulus-topped boundary layer (STBL) was Deardorff (1980). In his simulations, he varied the amount of heating from the surface. The

radiation cooling rate was prescribed and assumed to be non-zero near cloud top only. A number of situations were simulated, with clear sky, dry cloud8 and stratocumulus deck, including or omitting radiative effects and with varying cloud thickness.

The influence of these situations on the entrainment cOnstant, defined as the ratio of the

8 _{A dry cloud does not show liquid-vapor phase changes but there will be longwave radiative cooling at the top,}

entrainment buoyancy flux to the surface buoyancy flux, was investigated. Deardorif concluded

that the entrainment constant is influenced considerably by factors, such as magnitude and

vertical distribution of radiative cooling cloud water content, and capping-inversion thickness relative to the inversion base height

Moeng (1986) showed the results of LES which included the interaction of longwave

radiation and turbulence processes. Solar radiation was excluded in order to avOid complicated situations as detached or double cloud layers. A small surface buoyancy flux was assumed, thus turbulence is mainly driven by cloud-topped cooling from above. Because of this the relevant velocity scale was found to be given by the cloud-top height and the layer-averaged buoyancy flux inside the cloud layer.

Buoyancy was found to be the production term in the vertical component of the TKE iii

the cloud layer. In the subcloud layer, this component was produced mainly by import from

above. The surface buoyancy contributed ouly a minor fraction to the vertical energy component, although the proffles of the vertical velocity variance and kinetic energy flux in the STBL depend on the relative contributions of the surface heating and cloud-top cooling to the turbulence.

Another problem that was studied by Moeng (1987) was the determination of the entrainment rate. The results indicated that the entrainment ratio in a STBL is a factor of 1.5

larger than in a clear-air CABL.

A further analysis of the structure of CABLs and STBLs was shown by Schumann and Moeng (1991a,b). The convective circulations can be considered to consist of plumes that can be

decomposed into updfafts and downdrafts. Schumann and Moeng investigated the impact of

using vertical velocity, moisture fluctuations or a combination of both to discriminate between updrafts and downdrafts. By introducing a threshold value fOr the vertical velocity, the up- and downdrafts are separated by "environmental" air. Using the LES data of Moeng (1984, 1986)

they calculated mean vertical fluxes assuming top-hat profiles and compared them with the

actual fluxes. It was shown that if the up- and downdrafts were selected using the local vertical velocity, about 60% of the total fluxes of all relevant quantities was accounted for. The remaining 40% is transported by small-scale (subplume) motions, thatis not accountedfor when averaged over updrafts and downdrafts separately. This residue became smaller (10 to 30%) when a

non-zero threshold value (of about plus or minus 0.6 w1,) was used for the vertical velocity, thus

introducing environmental regions as well.

The w-plume description, using zero threshold values, was used to evaluate the budgets

of mass, momentum, heat, moisture and TKE. The budget equations contain a mixing term

representing the transport across the surface of updrafts at a given height towards downdrafts The budgets are generally controlled by divergence of vertical advection, but mixing between updrafts and downdrafts turned out to be essential to balance these advections. The nei mixing, due to small-scale turbulence, was most important at the inversion, especially in the cloudy case. It was concluded that a large portion of the TKE in the uppennost part of dOwndrafts is generated

iii the updrafts and does not come from buoyancy forcing. This should be taken into account

LES models in meteorology 21

1.7.2.2 LES of cumu&s-ropped boundary layers

Sommeria (1976) was the first one to use a LES model to study a cumulus-topped

boundary layer. In a new version of the model used by Deardorif (1972a), Sommeria included a condensation cycle with cloud formation and evaporation, a radiation computation of longwave radiative cooling and the influence of large-scale effects such as mean subsidence and advection. The simulation was based on meteorological conditions commonly observed in the undisturbed trades. No direct comparison with observations was made.

The clouds that formed in the simulation had horizontal dimensions of 100 to 400 m and extended vertically to about 800 m. The size of the clouds appeared to be independent of the grid size. It was noted that the momentum flux did not significantly alter by the existence of a cloud layer but the moisture flux seemed mainly driven by the cloud circulation Since the moisture flux increased with height from sea surface to cloud base, a slow diying of the mixed layer was

seen. This will increase the evaporation and it was assumed that the process might eventually lead to an equilibrium.

The combined effect of condensation and turbulent advection leads to a weak warming in the cloud formation region and a more important cooling in the evaporation region. This results in a destabilization of the cloud layer. It is counteracted by the subsidence. So, various physical processes are important in the heat and moisture budgets and it was noted that it is rather difficult

to reach a long-term steady state.

Sommeria and LeMone (1978) made direct comparison between the results of a LES model and the observations in the fair weather mixed layer during the Puerto Rico Field Experiment. In their simulation the radiative cooling was prescribed and the mesoscale

divergence had been neglected. The latter resulted in a slowly increase with time of the total

moisture content in the modeL

Since observations came from aircraft measurements of about 20 km long flight legs,

these data were filtered 'in order to match the spatial dimensions of the model (2 x 2 km2). The

agreement between the simulated and observed values of fluxes and variances was generally satisfactory. They found an underestimatiOn of the modeled moisture 'variance and an overestimation of the moisture flux at cloud base. However, the latter was found to be very

dependent on the cloud activity. The aircraft measurements came from areas with fewer clouds overhead, resulting in a lower cloud base moisture flux.

An analysis of the subcloud layer eddy structure revealed most of the features of the roll

structure, observed by LeMone and Pennell (1976), although the modeled rolls were slightly

larger in scale. It was assumed that the periodic boundary conditions, forcing the rolls to fit in the 2 km domain, was the reason for this. Another consequence of this rather small domain was a

large fluctuation in time of the cloud field.

Based on the observations made during GATE, Nicholls et al. (1982) showed the results of another LES, again excluding large-scale and diurnal effects and with a prescribed radiative flux divergence at each height in the model. This mean cooling rate was taken constant during the

agreed well with the measurements. The main effects of the turbulent mixing were the warming of the subcloud layer and a moistening and cooling of thecloud layer. The modeled cloud cover, cloud base and cloud top height were similar to those seen from the aiivraft. Although the vertical

velocity variance was well reproduced by the model other variances differed from the

observations Ths was a result of mesoscale contributions that were not accounted for by the

modeL

Furthermore, it was shown that potential and virtual potential temperature and the specific humidity were strongly affected by mixing occurring at cloud base, down to quite lOw levels in the subcloud layer Because of this cloud subcloud layer exchange the thermodynamic variables

vary smoothly throughout the subcloud layer, making the definition of a "transition" layer

(section 1.6.1) between a mixed layer and the cloud layer rather ambiguous.

While the fOrmer studies concentrated on the mixed layer structure and the influence of clouds on mixed layer fluxes and variances, Beniston and Sommeria (1981) investigated

trade-wind cumulus characteristics Such as cloud-radius-to-cloud-thickness ratio (they found an

average value of 0.4 ± 0.1), the linear relation between maximum area covered by a clOud and cloud life time and the statistical distribution of cloud radii They tested an exponential relation

between the number of clouds n(r) and their radius (r), given by Fraedrich (1976):

n(r) = K exp(-r), with K and coefficients. This relation fitted the model results reasonably

well for cloud radii between 100 and 600 m These cloud distributions are used in

parameterization schemes such as that of Arakawa and Schubert (1974).

The representation of thermodynamic fluxes by the product of a convective mass flux and the cloud-environment difference of the particular thermodynamic varlable (Betts, 1975) was fairly well fulfilled in the model, although there seems to be an inconsistency between the vertical profiles of the li4uid water static energy and the li4uid water potential temperature (which are closely related but the profiles differed considerably) as well as for water vapor fluxes inside clouds, shown in different figures.

Finally, they obtained empirical relations between the minimum of heat flux and the

maxima of moisture flux and buoyancy flux in the cloud layer as a function of cloud volume Since their simulation was made on a 2 x 2 km2 model domain, the cloud field showed a lot of variation in time. In the second half of the 3 hours of simulation, the cloud cover varied between

about 1 and 15%. This variability around a statistically steady state was used to obtain these

relations, but it was realized that several numerical simulations under various mean conditions being statistically steady would have been better and necessary to generalize their findings.

1.8 Aim and approach

In the previous sections, we discussed the structure of the trade-wind boundary layer and

the physical aspects involved in cumulus convection. The trade-wind boundary layer can be

divided into several sublayers with different characteristics: the free atmosphere, the cloud layer, and the mixed layer The most important physical processesineach layer can be summarized as

Aim and approach 23

-in the freeatmosphere, warming by subsidence is counteracted by radiative cooling,

-in the cloud layer, drying by subsidence is balanced by moistening due to cumulus convection, while radiative cooling balances the warming by subsidence;

-in the mixed layer, the warming by entrainment and the surface heat flux is largely in balance with the radiative cooling. The water vapor evaporated from the ocean surface is transported to the cloud layer. This moisture flUx is influenced by the clouds overhead.

Both observations and LES models have been used to gain more insight in the structure and physics of the CTBL. However, due to wetting of the insiruments in-cloud measurements were inaccurate. Most observational studies up to now have been concentrated on the turbulent

characteristics of the subcloud layer in a domain of a few tens of kilometers squared or

investigated the budgets of mass, water vapor and heat in the sublayers, mentioned above, within a 500-km square, as in ATEX and BOMEX. It is only recently that more detailed observations of the cloud layer, with measurements of quantities insideclouds as well as in the environment have become available (Smith and Jonas, 1993).

Most large-eddy simulations of CTBLs up to now have been concentrated on the

subcloud layer too. Many observed characteristics are confirmed in the LES models, although the small model domain leads to a clear shortcoming, since they can not represent the mesoscale

variability, seen in many observations. Mesoscale influences, such as subsidence and

temperature and moisture advection, have sometimes been included but only in the evolution of the horizontally averaged variables. In the models, the radiative flux divergence was generally prescribed and uncoupled from other processes. The .LES model has been used to obtain more msight m the cloud statistics and flux parameterizations used in mesoscale models

Two-dimensional numerical simulations, with models having horizontal dimensions of 30 kilometers have been report by Clark et al. (1986), who investigated the interaction between cumulus clouds and internal gravity waves. Three-dimensional simulations on larger horizontal

domains (24 - 32 kin)9, are reported by Kiemp and Wilhelmson (1978), who studied the

dynamic character of convective clouds and storms (i.e. convective systems with scales much

larger than we will investigate in this study). Tao et al. (1987) used a three-dimensional

simulation model to investigate statistical properties of ensembles of cumulus clouds, such as the vertical transport of mass, sensible heat, moisture and hdrizoñtal momentum.

The aim of this study is to develop a LBS model, that is able to simulate cumulus clouds. With this model, we can then investigate the physical processes that are important in cumulus convection and evaluate existing parameterizations, that has been used for CABLs and STBLs.

We will use the LES model to study the CTBL above the ocean. Such a boundary layer is easier to simulate than a boundary layer over land because over short time periods the surface

temperature can be considered constant and the surface moisture flux only depends on wind

speed and the specific humidity jump near the surface. Whereas over land the surface energy balance has to be taken into account as well.

These models have grid spacings on the order of 1 km and therefore do not resolve the turbulnce as in the