Vertical Velocities and Available Potential Energy Generated

Vertical Velocities and Available Potential Energy Generated by LandscapeVariability—Theory

M. BALDI AND G. A. DALU

Institute of Biometeorology, IBIMET–CNR, Rome, Italy, and Department of Atmospheric Science, Colorado State University,Fort Collins, Colorado

R. A. PIELKE SR.

Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado

(Manuscript received 14 July 2006, in final form 14 May 2007)

ABSTRACT

It is shown that landscape variability decreases the temperature in the surface layer when, throughmesoscale flow, cool air intrudes over warm patches, lifting warm air and weakening the static stability ofthe upper part of the planetary boundary layer. This mechanism generates regions of upward verticalmotion and a sizable amount of available potential energy and can make the environment of the lowertroposphere more favorable to cloud formation. This process is enhanced by light ambient wind through thegeneration of trapped propagating waves, which penetrate into the midtropospheric levels, transportingupward the thermal perturbations and weakening the static stability around the top of the boundary layer.At moderate ambient wind speeds, the presence of surface roughness changes strengthens the wave activity,further favoring the vertical transport of the thermal perturbations. When the intensity of the ambient windis larger than 5 m s�1, the vertical velocities induced by the surface roughness changes prevail over thoseinduced by the diabatic flux changes. The analysis is performed using a linear theory in which the mesoscaledynamics are forced by the diurnal diabatic sensible heat flux and by the surface stress. Results are shownas a function of ambient flow intensity and of the wavelength of a sinusoidal landscape variability.

1. Introduction

It is well known that mesoscale processes can act toprovide adequate moisture and instability for convec-tion to initiate, and surface processes such as thosedriven by surface heterogeneity or soil moisture gradi-ents can play a fundamental role in the development ofconvection. Once initiated, the interaction of convec-tion with shear can enhance storm evolution and lead tosevere weather (Chang and Wetzel 1991). In this per-spective, the surface conditions (dry versus wet soil;presence of heterogeneities in surface conditions suchas dry land adjacent to wet land or alternating dry andwet patches) are important since they affect the parti-tioning between latent and sensible heat fluxes. Royand Avissar (2002) found that coherent mesoscale cir-culations were triggered by surface heterogeneities in

Amazonia. Otterman (1977, 31–41) speculated that an-thropogenic changes from bare soil to a complex veg-etated surface in regions of marginal rainfall can favorconvective precipitation. His hypothesis was supportedby an analysis of the rainfall patterns in Israel (Otter-man and Sharon 1979; Otterman et al. 1990). Moreover,specific humidity in the convective boundary layer isincreased for wet surfaces, leading to larger convectiveavailable potential energy (CAPE). In addition, modelsimulations have shown that a realistic vegetated soilhas an impact on the formation of drylines, since thevegetation is a source of moisture, and since the het-erogeneous distribution of the vegetation enhances thegradients of the surface fluxes providing the solenoidalforcing to local frontogenetic flows (Shaw et al. 1997).Moreover, observations show that in dryline regionsthe probability of deep convection initiation is high inthe late afternoon when the updraft forced by a meso-scale secondary circulation can locally break the bound-ary layer capping, and that the wind shear can modulatethe convective initiation in the mesoscale updrafts at

Corresponding author address: Marina Baldi, IBIMET–CNR,Via Taurini, 19-00185, Rome, Italy.E-mail: [email protected]

FEBRUARY 2008 B A L D I E T A L . 397

DOI: 10.1175/2007JAMC1539.1

© 2008 American Meteorological Society

JAM2629

the dryline (Ziegler and Rasmussen 1998). The devel-opment of thermally forced secondary circulations isfavored by the absence of ambient flow, since they aresuppressed by ambient flows with a wind speed exceed-ing 6 m s�1 for surface inhomogeneities larger than 50km, or by weaker winds for smaller inhomogeneities(Segal and Arritt 1992).

From a climatic point of view, land-use changes havean impact on the regional and global scale, since spa-tially heterogeneous land-use effects may be at least asimportant in altering the weather as changes in climatepatterns associated with greenhouse gases (Pielke et al.2002; Pielke 2005), and, while Pielke et al. (2007) dis-cussed the diverse role of land-use/land-cover changeon precipitation, Feddema et al. (2005) showed modelresults indicating that land use and land cover will con-tinue to have an important influence on climate for thenext century. Analyzing the relation between thechanges in the vegetation and the rainfall in the Sahe-lian region, Taylor et al. (2002) have demonstrated,through general circulation model (GCM) simulations,how these changes could cause substantial reductions ofthe precipitation and examined the hypothesis that an-thropogenic changes to the land surface caused thedrought conditions that have persisted since the late1960s; however, the recent historical land-use changesare not large enough to conclude that they are principalcause of the Sahel drought.

On a regional scale, the climate of the United Stateshas changed with vegetation in modern times, with sig-nificant changes of temperature and humidity distribu-tion up to 500 hPa (Bonan 1997). Model simulationsconfirm that significant atmospheric effects have beencaused by human modification of the landscape overthe U.S. central plains (Weaver and Avissar 2001). Astrong influence of the land use and land cover on thediurnal temperature range, with the greatest range overbare soil and the smallest range over forest-coveredsoil, has been observed by Gallo et al. (1996). In theU.S. central plains there is observational and modelevidence of seasonal temperature differences of 5°–10°C between patches of dry land and patches of culti-vated land with horizontal dimensions of the order of20–50 km, which can drive mesoscale flows similar tosea breezes (Segal and Arritt 1992), while Shaw andDoran (2001) observed that, in the U.S. central plains,the nonseasonal divergence patterns are related togentle topographic features rather than land-use fea-tures.

On a more local scale, Anthes (1984) found thatbandwidths of about 20 km or wider are needed inorder to produce vertical circulations extending toheights of 1 km or more in the atmosphere. Banta and

White (2003) observed that the mixing height differ-ence caused by small-scale landscape variability de-creases from 600 m to negligible values as the ambientwind increases from 1 to 6 m s�1. De Ridder and Gallée(1998) found that, when the soil moisture availabilityincreases in a semiarid region like southern Israel, thereis a reduction of the thermal diurnal amplitude, and anenhancement of the moist convection and precipitation.Perlin and Alpert (2001) conclude, through the analysisof two convective rain studies in this semiarid regiondone using model simulations, that there is “a positiveinfluence of the anthropogenic land use changes on theenhancement of thermal convection and associatedrainfall.” There are several studies on the regional im-portance of spatial and temporal variations in soil mois-ture and vegetation coverage, and several authors stud-ied the influence of landscape heterogeneity on cumu-lus convection including Segal et al. (1989a,b), Changand Wetzel (1991), Fast and McCorcle (1991), Chenand Avissar (1994a,b), Li and Avissar (1994), Clark andArritt (1995), Cutrim et al. (1995), Lynn et al. (1995a,b,1998), Rabin et al. (1990), Rabin and Martin (1996),and Wang et al. (2000). For the sake of completeness,Cotton and Pielke (2007) list papers regarding how re-gional weather patterns are affected by land-use andland-cover change.

The results found by Anthes (1984) using a linearmodel indicate that planting bands of vegetation insemiarid regions can, under favorable large-scale atmo-spheric conditions, enhance convective precipitation.Motivated by the above studies and further elaboratingon the work done by Anthes, we examine the impact ofmesoscale landscape variability on the tropospheric pa-rameters that may favor convective development. Sincethe work done by Anthes (1984) is focused on the ver-tical velocities forced by diabatic fluxes in the planetaryboundary layer (PBL) in the absence of ambient flow,we extend our analysis to the mechanism through whichweak ambient flow can favor development of convec-tion. Since most of the above studies are observational,numerical, or mixed observational and numerical, withthe exception of Anthes’ work, we approach the prob-lem using an analytical linear theory developed by Daluand Pielke (1993) and Dalu et al. (2003) for the ther-mally forced secondary flows, where, in the present pa-per, we have added the effects induced by spatial gra-dients of surface roughness, which can be important inthe presence of an ambient flow, since surface rough-ness changes contribute to the vertical fluxes up tomidtropospheric levels (Kustas et al. 2005).

The main motivation for using a theoretical approachis the apparent contradictions found in the results ofprevious authors. To illustrate, the thermally forced

398 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47

secondary circulations are weakened or suppressed bythe ambient flow (Segal and Arritt 1992), the ambientflow modulates the convective initiation (Ziegler andRasmussen 1998), the surface roughness plays no rolein the development of the convection (De Ridder andGallée 1998), the surface roughness changes contributeto the vertical fluxes up to midtropospheric levels (Kus-tas et al. 2005), the mixing height difference decreasesas the ambient wind increases (Banta and White 2003).To establish the range of validity of the above results asa function of wavelength and ambient flow intensity, avery large number of numerical simulations would benecessary. We therefore found it useful to approach theproblem with a theory where the results were expressedin terms of readable equations, showing, in fact, thatthere is no contradiction in the above results, but rangesof validity in terms of patch size and wind intensity.

In terms of simple solutions we revisit the main re-sults found in previous experimental and numericalstudies; namely, we evaluate the decrease of the tem-perature induced by the ambient flow and by the sec-ondary mesoscale flow in the lower layers of the PBL,and, in order to explore the environment modificationsthat may increase the chances of deep convection de-velopment, we study the intensity and the phase of theupdraft at the top of the PBL, the weakening of thestatic stability around the top of the boundary layer,and the amount of the potential energy available in thePBL after sunset.

We show the results as a function of the wavelengthof the landscape variability and of the intensity of theambient flow, focusing on a landscape variability withwavelengths between 10 and 100 km in the presence ofa wind intensity up to 10 m s�1, since the atmosphericresponse is generally weak outside this range. At largerwavelengths the gradients are small, and at smallerwavelengths (100 m–10 km), diffusion homogenizes thetemperature gradients above the surface layer, killingthe vertical velocities in the upper part of the boundarylayer (Anthes 1984; Dalu and Pielke 1993).

2. Governing equations

We use a linear theory to study the impact on thesecondary mesoscale flow induced by changes of thediabatic flux in the PBL and by changes of the surfaceroughness. Over the numerical model approach, theanalytic theoretical approach has the advantage thatthe results are expressed in terms of readable equationsthat summarize the equivalent results of many numeri-cal simulations, and that the perturbations induced bythe diabatic forcing and by the surface stress can beexamined separately, since the perturbation resultingfrom their combination is simply the sum of the pertur-

bations induced by each forcing separately, with thedisadvantage, however, that the validity of the resultsare limited to small perturbations. The governing 2DBoussinesq primitive nonhydrostatic linearized equa-tions are

L u � f � �x� � ��z�, L � � fu � 0, and

L w ��

�z� b � 0 or

��

�z� b, �1�

L b � N2w � Q and �xu � �zw � 0, �2�

with L � ��t � U�x � � � K�xx�. �3�

The mesoscale dynamics of the secondary flow is forcedby the diabatic sensible heat source Q and by the sur-face stress �. We assume that the vertical velocity isproportional to the gradient of the surface stress, w �x�; then, since u � �zw dx [second equation in Eq.(2)], u � � z� [first equation in Eq. (1)]. In addition weassume that over the nonvegetated patches Q is posi-tive and � is small, and that over the vegetated patchesQ is negative and � is large. In the time operator L, weadopt a horizontal diffusion coefficient K � 100 m2 s�1

and we neglect the vertical diffusion term Kzz, simpli-fying the mathematics (Dalu and Pielke 1993). We haveverified that the weakening of the horizontal gradientsat wavelengths smaller than 20 km are as those shownby Anthes (1984), who keeps both diffusion terms. Forsimplicity, the Rayleigh friction coefficient � is assumedthe same in the momentum and in the thermodynamicsequations; U is the ambient flow intensity; u, �, and ware the momentum components; is the geopotential;and b is the buoyancy perturbation. When the hydro-static approximation can be made (Dalu et al. 2003),the fourth equation can replace the third equation in 1.Here, N � 10�2 s�1 is the Brunt–Väisälä frequency andf � 10�4 s�1 is the inertia frequency.

We prescribe that the vertical divergence of the heatflux is constant through the depth of the convectiveboundary layer (hQ) and vanishes above it:

Q � Q0He�hQ � z� expi��0t � k0x�, �4�

with k0 �2�

L; �0 �

2�

1 day; Q0 � �0N2hQ;

N2 � g�z

�; hQ � Q

�1 � 1000 m. �5�

In Eq. (4), He(h � z) is the Heaviside function, whereHe(z � h) � 1 and He(z � h) � 0; L is the horizontalwavelength of the landscape variability, k0 is the hori-zontal wavenumber; �z is the vertical gradient of theambient potential temperature; � is the average poten-tial temperature of the PBL; and g is the gravitationalacceleration.


We do not adopt an exponential decaying diabaticsource as in Anthes (1984), because, with this choice,about one-third of the total sensible heat would beabove the PBL, which is about half of the heat withinthe PBL, and, since the secondary flow is driven by thehorizontal gradients of the heat source [Eq. (21)], theintensity of this flow above the PBL would be arbi-trarily enhanced:

Q exp�� z

hQ� ⇒

�hQ

�

Q dz

�0

�

Q dz

�1e�

13

, or

�hQ

�

Q dz

�0

hQ

Q dz

�1

e � 1�

12

. �6�

In the absence of ambient flow and of secondary cir-culation (U, u, �, w � 0), the potential temperatureperturbation is proportional to the time integral of theforcing (��0

� Q dt � L̂ �1� Q), where L̂� is the Fourier

transform of the operator in Eq. (3):

L̂� � �i�0 � �� K k02��. �7�

The temperature perturbation is vertically distributedassuming that the mixing due to the convective adjust-ment makes the PBL isentropic (Green and Dalu1980):

� � � �hQ � z

hQ�� He�hQ � z� expi�0�t � t0�,

�8�

� �

� �0

��0

|L̂�|�

�0

�� K k02�2 � �0

2, � �0

� hQ�z,

�9�

and t0 �1

�0tan�1� �0

� � K k02�. �10�

The time lag t0 decreases as the Rayleigh friction in-creases. In addition, small-scale features, smaller than20 km, are quicker in responding to the forcing, sincethe time lag is further reduced by diffusion processes athigh wavenumbers [Eq. (10) and Fig. 1a]. In the ab-sence of diffusion and Rayleigh friction t0 � �/(2�0),the time lag is 6 h; that is, if the maximum of the dia-batic heat occurs at noon, the maximum of the diabatictemperature occurs at 1800 LT. Therefore, in order toreduce the time lag to a more reasonable value, weadopt a value of the Rayleigh friction coefficient equal

to the inertia frequency, � � f. With this choice themaximum of the diabatic temperature occurs at about1430 LT (t0 � 2.4) over patches with a wavelengthlarger than 20 km [Eq. (11) and Fig. 1a]:

t0 �1

�0tan�1��0

� � � 2.4 h when � � f � 10�4 s�1.

�11�

In Eqs. (8)–(11) we used and hereinafter we will usethe definition of a complex number Z:

Z � a � ib � � exp�i�� with

� � |Z | and � � tan�1�b

a�, �12�

where � is the modulus and � is the angle (or argument)of the complex number Z.

a. Adopted wind stress and diabatic forcing

Since the deep convection is most likely to occur inthe late afternoon (Ziegler and Rasmussen 1998), inorder to evaluate the integrated daytime impact of thedynamics of the secondary flow on the atmospheric pa-rameters, we average in time the diabatic forcing in Eq.(4). The average is done through the half day of thediurnal time; this operation flattens the diabatic sourceto 2/� � 2/3 of its maximum. The mathematics is sim-plified without changing the inherent nature of this im-pact:

Q � Q0 He�hQ � z� expi�k0x�2

day

�0

0.5 day

Im�expi��0t�� dt. �13�

We study the dynamics of the secondary flow forced byQ and by �, with the assumption that they are a rea-sonable representation of the afternoon conditions:

Q�x, z� �2�

Q0 He�hQ � z� expi�k0x� and

�z��x, z� � ��0 He�h� � z� expi�k0x�, �14�

with �0 � CDU2; CD � 1.5 � 10�3;

h� � ��1 � 100 m; �15�

and ��x, z� � ��0��h� � z�� He�h� � z� expi�k0x�.

�16�

Here CD is the drag coefficient, and �0 is the amplitudeof the surface stress. In Eq. (16) we prescribed that thesurface stress linearly decreases through the depth (h�)of the surface layer (SL), where the vertical divergenceof the stress is constant [Eq. (14)], and that both vanish


above the SL. Since we assumed that the diabaticsource is positive over the nonvegetated patches andnegative over the vegetated patches, and that the sur-face stress is large over the vegetated patches and smallover the nonvegetated patches, Q(x, z) in Eq. (14) and�(x, z) in Eq. (16) are taken with opposite sign. Thepotential temperature perturbation in the absence ofsecondary flow is the balance between the stationarydiabatic forcing [Eq. (14)] and the dissipation due to theRayleigh friction and diffusion (�� L̂�1

� Q), where L̂� isthe Fourier transform of the operator in Eq. (3) forstationary forcing:

L̂� � �� K k02 � ik0U�, �17�

� � �2� �hQ � z

hQ�� He�hQ � z� expik0�x � x0�, �18�

with� �

� �0�

�

�� K k02�2 � �k0U�2

and

x0 �1k0

tan�1� k0U

� � K k02�. �19�

Here ��0 is the amplitude of the thermal contrast be-tween the vegetated and the nonvegetated patches

FIG. 1. (a) Time lag t0 between the diabatic source Q and the temperature perturbation ��. (b) Downwind displacement x0 of thediabatic temperature perturbation �� (solid line); xQ displacement of the vertical velocity wQ in the SL (dashed line); x� displacementof the vertical velocity w� in the SL (dotted line). (c) Decrease of the thermal contrast between the vegetated and the nonvegetatedpatches, ��. (d) Maximum of the updraft at the top of the boundary layer. Combined updraft induced by the diabatic flux and thesurface roughness, �(Q,�) (solid line); updraft induced by the diabatic flux only, wQ (dashed line); updraft induced by the surfaceroughness only, w� (dotted line).


when U � 0 and K � 0. The ambient flow reduces ��as shown in Fig. 1b. The reduction of the thermal con-trast is strongly enhanced by the diffusion processeswhen the patches are smaller than 20 km as in Anthes(1984), Fig. 1b, and Eq. (19). In addition, the ambientflow displaces the temperature perturbations down-stream a distance, x0. The displacement of the tempera-ture perturbation is a function of the ambient flow in-tensity and of the wavelength as shown in Fig. 1b [Eq.(19)]. At high wind speed the advection time is muchsmaller than the thermal equilibrium time, (k0 U)�1 K

��1, so we have the maximum displacement of the tem-perature perturbation,

k0U

�k 1 ⇒ x0 �

�

2k0�

L

4. �20�

The maximum displacement, x0, does not exceed one-fourth of a wavelength L (Fig. 1b).

b. Streamfunction equation and verticalwavenumber

Using the mass continuity [second equation in Eq.(2)] to define the streamfunction � (z� � u and x� ��w), we derive from the Navier–Stokes equations thestreamfunction equation (Dalu et al. 2003):

L̂��z�first equation in Eq. �1�� f�z�second equation in Eq. �1�� L̂��x�third equation in Eq. �1��

� �x�first equation in Eq. �2�� ⇒�L̂�

2 � f 2��zz� � �L̂�2 � N2��xx� � ��L̂��zz� � �xQ� �21�

�zz��z� � 12��z� � F �z� with ��x, z� � ��z� expi�k0x�, �22�

with 12 � k0

2� L̂ �2 � N2

L̂ �2 � f 2�, �23�

and F �z� �1

L̂�2 � f 2 ��

2�0L̂��h� � z� � ik0

2�

Q0He�hQ�z��. �24�

Here L̂� is defined in Eq. (17), �21 is the vertical wave-

number squared, and �(h� � z) is the Dirac function.Since �0 � U2 [Eq. (15)] the contribution of the surfacestress rapidly increases with the wind speed. In addi-tion, since L̂� � (� � K k2

0 � i k0 U), when the ambientflow is sufficiently large (U k � k�1

0 ), the main contri-bution to the secondary flow comes from the horizontalgradients of the surface stress (x�):

When U k 1.5 m s�1 and L � 100 km ⇒ L̂��

� ik0U�. �25�

The streamfunction �(z) is computed through theconvolution between the Green function, g(z, ), andthe forcing F( ):

��z� � �Q�z� � ��z� � {g�z, ��F �� G��}, g�z, �� exp�1|z � �|�

21

with G�z � 0� �1

L̂ �2 � f 2 ��

2�0L̂��h� � z� � ik0

2�

Q0 He�hQ � z��. �26�

In Eq. (26), the boundary condition of vanishing verti-cal velocity at the surface has been achieved by intro-

ducing in the lower semiplane a mirror image G(z) of thenonhomogeneous term F(z) as in Dalu and Pielke (1989):

��z� ��

2�0L̂�{exp�1|z � h�|� � exp�1�z � h��}

2k02�L̂�

2 � N2�; �27a�

�Q�z� � �iQo

�k0�L̂ �2 � N2�

{2�exp�1z� � He�hQ � z�� sign�z � hQ� exp�1|z � hQ|� � exp�1�z � hQ��}. �27b�


To have the correct confinement and propagation ofthe energy, in the exponential of the Green function,the real part of the vertical wavenumber is chosen witha negative sign and the imaginary part with a positivesign (Smith 1979):

1 � �|r | � i |i |. �28�

The modulus of the vertical wavenumber, |�1|, of thereal part, |�r|, and of the imaginary part, |�i|, are shownin Fig. 2. When the ambient flow is weak, the real partof the vertical wavenumber is much larger than the

imaginary part (|�1| � |�r|), and the secondary flow istrapped and confined in the PBL. When the ambientflow is strong, the imaginary part of the vertical wave-number is much larger than the real part (|�1| � |�i|),and the propagating inertia–gravity waves fill the entiretroposphere. In the presence of a moderate ambientflow, the secondary flow is made of propagatingtrapped waves, which fills the lower half of the tropo-sphere. In the absence of ambient flow, Rayleigh fric-tion, and diffusion, the ratio between the vertical wave-number and the horizontal wavenumber equals the ra-tio between the Rossby radius (R0) and the depth of theconvective layer (Dalu and Pielke 1989):

when U � 0 ⇒ i � 0 andr

k0�

R0

hQ�

N

f. �29�

In the presence of ambient flow with no dissipation, thevertical wavenumber is equal to the Scorer parameter(l) (Scorer 1949):

when U � 0 ⇒ r � 0 and i �N

U� l. �30�

3. Atmospheric response to a landscape variability

The vertical velocity forced by a sinusoidal diabaticsource in the absence of surface stress,

Q�x, z� �2�

Q0 He�hQ � z� cos�k0x�, �31�

is obtained by multiplying by i k0 the streamfunction[� � �Q expi(k0 x) in Eq. (22), where �Q is defined inEq. (27b)], and taking the real part

wQ�x, z� � �Re�ik0�Q�x, z�� Re�ik0�Q�z� expi�k0x�� Q0

��Q2 !cos�k0�x � xQ� � i�z � hQ�� exp��r�z � hQ��

� sign�z � hQ� cos�k0�x � xQ� � i|z � hQ|� exp��r |z � hQ|�"

�2Q0

��Q2 !He�hQ � z� cos�k0�x � xQ�� cos�k0�x � xQ� � iz� exp��rz�". �32�

The ambient flow U displaces the secondary flowthrough the imaginary part ik0U of the operator L̂� [Eq.(17)]. The displacement xQ is computed using the

argument of the complex part of the amplitudes of�ik0�Q,

ZQ � L̂�2 � N2 � �Q

2 expi�k0xQ�; �Q2 � |ZQ| and xQ �

1k0

tan�1�Im�ZQ�

Re�ZQ��. �33�

The vertical velocity forced by the surface stress is ob-tained by multiplying by i k0 the streamfunction [� � ��

expi(k0 x) in Eq. (22), where �� is defined in Eq. (27a)],and taking the real part

FIG. 2. Logarithm of vertical wavenumber and of its real and ofits imaginary part, as a function of the wavelength and of theambient flow intensity: log10(|�1|) (solid line); log10(|�r|) (dashedline); log10(|�i|) (dotted line).


w��x, z� � �Re�ik0��x, z�� Re�ik0��z� expi�k0x��

��

2�0

2k0��2 !cos�k0�x � x�� i |z � h�|� exp��r |z � h� |� � cos�k0�x � x�� i�z � h�� exp��r�z � h��",

�34�

with �z� � ��0 He�h� � z� cos�k0x� �35�

and Z� � �ZQ

ik0L̂�

� ��2 expi�k0x��; ��

2 � |��| and x� �1k0

tan�1� lm�Z��

Re�Z��. �36�

When the wavelength is sufficiently large, the hydro-static approximation can be made, and L̂2

� � N2 can bereplaced by N2 in the vertical momentum equation:

when hQ k0 K 1, then xQ � 0 and

x� �1k0

tan�1�� K k02

K0U �, �37�

where xQ and x� are the displacements of the verticalvelocities wQ and w� near the ground (z K hQ). There-fore, in the surface layer, the thermally forced hydro-static vertical velocity, wQ, is always in phase with thediabatic source [xQ � 0 in Eq. (37)], and, in addition,also w�(x, z) is in phase with the diabatic source whenk0 U k � � K k2

0. When the advection time equals thedissipation time due to the Rayleigh friction and diffu-sion processes (k0 U � � � K k2

0), w� is in phase with thediabatic thermal perturbation �� in Eq. (18).

When U # 0 and �i # 0, then above the surface layerthe vertical velocities are a composition of backwardtilted waves [Eqs. (32) and (34)]. The zero phase equa-tions for the diabatic forced waves and for the surfacestress forced waves are, respectively,

x � xQ �i

k0z and x � x� �

i

k0z. �38�

The horizontal momentum components are the ver-tical derivative of the streamfunction. These compo-nents are obtained multiplying by �1 [Eq. (28)] thestreamfunction and keeping the real part:

uQ�x, z� � Re�1�Q�x, z�� and

u��x, z� � Re�1��x, z��. �39�

When the diabatic source and the surface stress areboth present, the momentum components are, respec-tively,

w�Q,��x, z� � wQ�x, z� � w��x, z� and

u�Q,��x, z� � uQ�x, z� � u��x, z�. �40�

a. Vertical velocity at the top of the convectiveboundary layer

Since substantial updrafts can favor the formation ofclouds, we study the behavior of the vertical velocity atthe top of the convective boundary layer (CBL) as afunction of the wavelength and of the ambient flowintensity; the results are shown in Fig. 1d. When theambient wind is weak, the vertical velocity is mainlydiabatically forced. When the ambient wind is strong,the vertical velocity induced by the surface roughnessprevails; the latter occurs when the ambient flow inten-sity exceeds 5 m s�1. Since wQ and w� do not have thesame phase (xQ # x�), the maximum of w(Q,�) is notsimply the sum of the maximum of wQ with the maxi-mum of w�.

At very high wavenumber, the vertical velocities aresmall because of the diffusion as in Anthes (1984). Atlow wavenumber, the horizontal gradients are small,and the thermally forced vertical velocities are smalland less sensitive to the ambient flow. At intermediatewavenumber (10 � L � 30 km), the thermally inducedvertical velocities at the top of boundary layer arelarger in the presence of a moderate ambient flow thanin the absence of an ambient flow.

In the absence of ambient flow (U � 0), wQ is posi-tive over the warm nonvegetated patches and negativeover the cool vegetated patches. In the presence of am-bient flow (U # 0), wQ is in phase with the diabaticforcing only in the surface layer [Eq. (37)], while abovethe surface layer the updraft shifts toward the vegetatedpatches as the altitude increases [Eq. (38)]. At the topof the PBL wQ becomes positive over the vegetatedpatches when

ihQ � n�, n � 1, 3, 5, . . . or

N

UhQ � � ⇒ U �

N hQ

�� 3 �m s�1�. �41�

Therefore, the presence of moderate winds are benefi-cial to cloud formation since they enhance the verticalvelocity at the top of the PBL (Fig. 1d), shifting, in


addition, the updraft from the nonvegetated patchestoward the vegetated patches. In the approximation inEq. (41) we assumed n � 1 and �i � l, where l is theScorer parameter [Eq. (30)].

b. Decrease of the temperature in the surface layerand increase of the temperature in the upper halfof the boundary layer

The real part of the diabatic temperature perturba-tion in Eq. (18) is the thermal response to the sinusoidaldiabatic heat in Eq. (31):

� � �2� �hQ � z

hQ�� He�hQ � z� cosk0�x � x0�.

�42�

The thermal contrast between the vegetated and thenonvegetated patches, ��, is reduced and advected adistance x0 downstream by the ambient flow [Eq. (19);Figs. 1b,c).

The diabatic temperature perturbation is further per-turbed by the advection of the secondary flow forced bythe diabatic source,

� Q � � � �1

L̂�

�wQ��z � �z� �� uQ�x� ��, �43�

which horizontally averaged over one wavelength be-comes

� Q �1L �

�0.5L

0.5L

� Q dx. �44�

In Fig. 3a we show this temperature vertically averagedthrough the lower part of the boundary layer (��SL),and averaged through the upper half of the boundarylayer (��CBL),

� SL �5

hQ�

0

0.2hQ �� Q

h�� dz and

� CBL �2

hQ�

0.5hQ

hQ � � Q

hQ � h�� dz. �45�

The cool air, advected from the vegetated patches to-ward the nonvegetated patches, lifts and replaces thewarm air above nonvegetated patches. It results in asubstantial cooling of the horizontally averaged tem-perature of the surface layer and a warming of theboundary layer above it, as shown by ��SL and ��CBL inFig. 3a. When the wavelength is smaller than oneRossby radius (L � R0), at low wind speed, the warmair in the SL above the nonvegetated patches is entirelyreplaced by the cool air advected from the vegetatedpatches, efficiently decreasing the temperatures in the

SL. At high wind speed ��Q becomes negligible, since�� is negligible (Fig. 1c).

c. Weakening of the static stability around the topof the convective boundary layer

The diabatic temperature perturbation caused by theadvection of the secondary flow forced by the diabaticsource and by the surface stress is

� �Q,�� 1

L̂�

�w�Q,��z � �z� �� u�Q,��x� ��,

�46�

FIG. 3. (a) Cooling of the surface layer and warming of theboundary layer above the surface layer averaged over one wave-length where Q # 0 and � � 0: ��SL (solid line) and ��CBL (dottedline). (b) Brunt–Väisälä frequency around the top of the CBLaveraged over one wavelength where Q # 0 and � # 0, NCBL.


and its horizontal average over one wavelength is

� �Q,�� 1L �

� 0.5L

0.5L

� �Q,�� dx. �47�

The vertical profile of the horizontal Brunt–Väisälä fre-quency squared and averaged over one wavelength is

N2 �g

��z � �z� �Q,��. �48�

The perturbed Brunt–Väisälä frequency around the topof the convective boundary layer (NCBL) is shown inFig. 3b:

NCBL �1

0.4 hQ�

0.8hQ

1.2hQ

N dz. �49�

At low wind speed when the cool air over the vegetatedpatches intrudes under the warm air above the nonveg-etated patches, the static stability around the top of theconvective boundary layer weakens. At high windspeed, ��Q is small (Fig. 3a), the perturbation of thestatic stability around the top of the convective bound-ary layer is mainly due to the wave action induced bythe changes of surface stress.

In Figs. 4a,b we show the perturbed static stabilityaround the top of the CBL, averaged over a nonveg-etated patch and averaged over a vegetated patch, re-

FIG. 4. (a) NCBL averaged over a nonvegetated patch where Q # 0 and � # 0; (b) NCBL averaged over a vegetated patch whereQ # 0 and � # 0; (c) NCBL averaged over a nonvegetated patch where Q � 0 and � # 0; (d) NCBL averaged over a vegetated patchwhere Q � 0 and � # 0.


spectively, when the secondary flow is forced by thediabatic source and by the surface stress. At low windspeed the static stability over the nonvegetated patchesand over the vegetated patches is about the same. Athigh wind speed the static stability over the nonveg-etated patches increases, while the static stability overthe vegetated patches decreases. This behavior is due tothe change of the phase of the waves with altitude in thepresence of ambient flow [Eq. (38)].

To emphasize the role of the waves generated by thesurface stress and the change of the phase of thesewaves with the wind speed, we adopt � # 0 and Q � 0,in order to avoid the interference between the wavesgenerated by the surface roughness and the waves gen-erated by the diabatic source. We show the resultingperturbed static stability around the top of the CBLaveraged over a nonvegetated patch and averaged overa vegetated patch in Figs. 4c,d. At low wind speed theperturbation is negligible except at very high wavenum-ber [w� � L̂�� (� � K k2

0 � i k0U)�] [Eqs. (17), (34),and (36)], and � � U2 [Eq. (15)], in agreement withresults by De Ridder and Gallée (1998), who found thatthe convective processes are insensitive to the surfaceroughness. While at moderate wind speed, we find thatthere is a 10% perturbation of the static stability, whichis in agreement with the results by Kustas et al. (2005),who found that surface roughness changes contribute tothe vertical fluxes up to midtropospheric levels. In ad-dition, concerning the phase of these waves, we findthat, when U � 3 m s�1, the perturbation of the staticstability becomes positive over the smooth nonveg-etated patches and negative over the rough vegetatedpatches because of the tilt of the waves [Eq. (41)], withan increase of the stability over the nonvegetatedpatches equal to the decrease over vegetated patches.

d. Energetics

Since in summer after sunset the wind intensity oftenrapidly decreases [(Q, U) → 0], in this paragraph weevaluate the intensity of the updraft after sunset fromthe residual potential energy at sunset. Assuming thatat this time of the day the temperature perturbation is��(Q,�), the available potential energy (APE) is

APE �1

hQL ��0.5L

0.5L

� �Q,��0

hQ �gz��

� � dz� dx

�12�u2 � �2 � w2�, �50�

where z$ is the vertical distance of the air particle witha temperature perturbation, ��(Q,�), between its actuallevel and the level of neutral buoyancy, and where u, �,

and w are the modulus of the momentum componentsaveraged in the PBL over a wavelength L. From thecontinuity equation and from the second momentumequation we have

w � �ik0

1u and � � �

f

L̂u, or

w �k0

|u1| u and � �f

|L̂ |u; �51�

APE �12

u2�1 � � f

|L̂ |�2

� � k0

|1|�2��

12

w2�1 � � |1|k0�2�1 � � f

|L̂ |�2��. �52�

Results shows that after sunset there is still enoughpotential energy to drive substantial updrafts: uQ andwQ computed using ��Q in Eq. (43) are shown in Figs.5a,b, respectively, and u(Q,�) and w(Q,�) computed using��(Q,�) in Eq. (47) are shown in Figs. 5c,d, respectively.The intensity of the updraft, w(Q,�), decreases as thewind speed increases, since the temperature contrastbetween the patches weakens [Eq. (19) and Fig. 1c].While the contribution of the surface stress to the up-draft, w(Q,�), is negligible at low wind speed, it becomesimportant at moderate wind speed.

4. Conclusions

To summarize the results by different authors, wedeveloped a theory in which the results are formulatedin terms of mathematical solutions in which the pro-cesses, through which landscape variability modifies theenvironment, are easily identified, and in which therange of validity of the results are established as a func-tion of the size of the patches and of the intensity of thewind.

Our results show that, when the patches are a smallfraction of the Rossby radius, the reduced thermal di-urnal amplitude is due to the intrusion of cool air in thesurface layer, which lifts and replaces the warm airabove the warm nonvegetated patches. Concerning in-tentional changes of the surface characteristics in orderto modify the environment of the lower troposphere toenhance the chances of convective precipitation, thefollowing considerations can be useful.

The ambient wind greatly reduces the thermal con-trast and the thermally driven updraft; however, at thetop of the boundary layer, this updraft is more intensein the presence of a light ambient wind.

When the wind speed is moderate, the wave activityweakens the static stability at the top of the boundary


layer, and, since these waves are tilted, a significantupdraft is displaced over the vegetated patches at thetop of the boundary layer.

The contribution of the wind stress becomes relevantin the presence of a moderate wind speed, and, whenthe ambient flow is stronger than 5 m s�1, the verticalvelocity is mainly due to the mechanical effects inducedby the changes of surface roughness.

The residual available potential energy in the PBLcan drive significant secondary mesoscale flow aftersunset.

To summarize the main findings, a moderate ambientflow (3–4 m s�1) in the presence of a landscape vari-ability modifies the environment making it favorable tocloud formation through the enhancing of the updraft

at the top of the PBL (Fig. 1d) and the shifting of thisupdraft from the nonvegetated patches toward the veg-etated patches (Fig. 4d), where the static stability isweakened as a result (Fig. 4b). In addition, after sunset,when the ambient flow and the diabatic forcingweaken, the residual potential energy can drive eveningdeep convection (Fig. 5). The results are shown as afunction of the ambient flow intensity and of wave-length of a sinusoidal landscape variability.

Acknowledgments. This research was in part sup-ported by the African Monsoon MultidisciplinaryAnalyses (AMMA). Based on the French initiative,AMMA was built by an international scientific groupand is currently funded by a large number of agencies,

FIG. 5. Average flow intensity in the PBL after sunset, resulting from the release of the APE: (a) u where Q # 0 and � � 0; (b) wwhere Q # 0 and � � 0; (c) u where Q # 0 and � # 0; (d) w where Q # 0 and � # 0.


especially from France, the United Kingdom, theUnited States, and Africa. It has been the beneficiary ofa major financial contribution from the European Com-munity’s Sixth Framework Research Programme. Weacknowledge support from NSF Grant ATM-9910857.In addition, we are grateful to the two reviewers fortheir contributions, which greatly improved the manu-script.

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Vertical Velocities and Available Potential Energy Generated

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