three-dimensional numerical experiments on convectively forced internal gravity waves

Q. J . R. Meteorol. SOC. (1989), 115, pp. 309-333 551.51 1.331.558

Three-dimensional numerical experiments on convectively forced internal gravity waves

By THOMAS HAUFL and TERRY L. CLARK National Center for Atmospheric Research2, Boulder, Colorado, V.S.A.

(Received 26 February 1988; revised 29 September 1988)

SUMMARY Results are presented of thermally forced dry and moist convection and the associated gravity wave fields

from three-dimensional numerical simulations using a non-hydrostatic anelastic model. This paper extends earlier two-dimensional simulations to include effects of the third spatial dimension employing a very similar environmental speed-shear case for the study. The present simulations produce scattered fair weather cumuli in agreement with observations. In many important respects, the physical response is quite similar to that obtained in the earlier two-dimensional calculations. The near-uniform surface sensible heat flux results in Rayleigh modes filling the convective boundary layer (CBL) to begin with, whereas later, after convective motions start interacting with the overlying stable layer, larger horizontal scale deep modes become evident and in some cases dominant. The eigenfunction structure of these dominant forced normal modes consists of boundary layer eddies in the lower levels and gravity waves above. They are important organizers of the cumulus convection. As in the earlier two-dimensional simulations, the efficiency of gravity wave excitation was found to be very sensitive to the mean wind shear in the region spanning the CBL and the overlying stable layer. The dominant horizontal wavelength in the shear direction ranges between 10 and 15 km in the free atmosphere whereas it peaks at about 6 km in the CBL.

The strong difference between the preferred directions of alignment for the eddies in the CBL (rolls aligned with the mean shear) and the overlying waves (aligned with lines of constant phase normal to the shear) results in overall broken conditions. The boundary layer motions are organized in broken 'varicose-like' rolls aligned approximately with the mean shear. The overlying waves show a somewhat more scattered pattern. This scattered-type dominant forced modal response combined with the nonlinear effect of the clouds themselves results in a cloud pattern revealing a high degree of randomness. This cloud field randomness occurs in spite of a near-zero horizontal wavenumber structure to the surface sensible heat flux.

Exchanges of momentum between convective and mean motions in the CBL result in strongly curved stress profiles and a mixing out of the initial boundary layer shear. Sensitivity tests were performed where the mixing of momentum was partially compensated by the addition of low-level pressure gradient terms.

1. INTRODUCTION

Consider the problem of a stable atmosphere overlying a neutrally stratified convective boundary layer over flat terrain as illustrated in Fig. 1. The wind is assumed to increase with height with its vertical shear exceeding 5 m s-lkm-'. Convective motions are forced by a surface sensible heat flux. The type of motion generally scales with the boundary layer depth. It is determined by the transfer of heat from the ground to greater heights thus maintaining a well-mixed convective boundary layer. Convective eddies which penetrate the stable layer experience a differential wind speed in a sheared environment. Mason and Sykes (1982) suggested that, in the presence of shear, these convective eddies act as small hills, in that air is forced to flow over or around them exciting gravity waves. Both interfacial waves located at the inversion base and deep tropospheric gravity waves can develop. Interfacial waves are trapped in the inversion layer near the top of the convective boundary layer. These have been studied by Carruthers and Moeng (1987). This shallow and small-scale feature, however, is beyond the scope of this paper. The deeper propagating waves extract energy from the boundary layer and transport it to possibly stratospheric heights providing there are no intervening critical or trapping levels. Gravity waves can travel away from their region of excitation

Permanent affiliation: Institut fur Physik der Atmosphare, DFVLR, D-8031 Oberpfaffenhofen, F.R.G. The National Center for Atmospheric Research is sponsored by the National Science Foundation.

309

310 T. HAUF and T. L. CLARK

- 5 E s N Y

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TIME (hrs)

Figure 1. Sketch of the basic physical processes based on CHKs two-dimensional simulations in a convective boundary layer (CBL) with strong environmental shear. Region @ shows thermals scaling with the depth of the CBL develop. Region @ shows appearance of deep modes with wide-spaced eddies and overlying waves. Region @ shows formation of clouds with tropospheric waves travelling upwind at a higher phase speed than

the thermals in the CBL. Region @ shows waves which may propagate away from their source region.

to the far-field (region @ in Fig. 1). Many gravity wave observations refer to the latter case where the origin of the observed waves is not known nor is their source considered the subject of investigation. Furthermore, gravity waves excited by deep convective systems penetrating the tropopause and showing a quite distinct behaviour with pulse- like ground signals of several 100pb (Bull and Neisser 1976) are not the subject of this study. In contrast, in this paper we focus on near-field solutions of fair weather convection forced by a surface heat flux.

Early observations of convectively forced internal gravity waves, also referred to as convection waves, reaching to several kilometres height, were made from gliders. They used smooth wave updraughts to gain altitudes unachievable using boundary layer motions alone (see references in Kuettner et al. (1987), and Clark et al. (1986), hereafter referred to as KHC and CHK, respectively). Using an instrumented aircraft, KHC studied convectively forced internal gravity waves over the Great Plains. Waves were found whenever convective activity prevailed in the boundary layer and moderate to strong vertical wind shear was present (3-10m s-lkm-'). Typically the horizontal wavelength varied between 5 and 15 km with vertical wave motions of + 1 to 3 m s-l. A quite unexpected observation was the vertical extent of the wave system which generally seemed to be more than 9 km.

The case on the 12 June 1984 was studied in CHKs two-dimensional nonlinear numerical simulations and in the linear model simulations of Clark and Hauf (1986) and is also the subject of the present paper. These numerical simulations investigated the nature of the waves and the controlling parameters. The excitation efficiency was found to be governed by the strength and vertical distribution of the shear. The model was initiated with a wind profile shown in Fig. 1 where the shear is uniform within the convective boundary layer and some part of the overlying stable layer but zero otherwise. Decreasing only the magnitude of the shear resulted in smaller wave amplitudes as well as a broader power spectrum of the vertical wave motion. Changing the profile so that the shear was confined to the boundary layer (dashed profile in Fig. 1) considerably reduced the wave amplitude. Strong shear layers in the region spanning the upper portions of the boundary layer and overlying stable layer appear to define the most favourable situation for the excitation of gravity waves. Gravity waves, showing good agreement with the observations, were simulated with and without clouds. The main

CONVECTIVELY FORCED GRAVITY WAVES 311

effect of clouds in the CHK study appears to be that of introducing a sufficient level of nonlinearity and affecting the relative horizontal motions between the eddies and waves. Both nonlinear dry simulations by CHK and the linear simulations of Clark and Hauf (1986) show strong vertical coherence between the eddies and waves. Boundary layer fields in the presence of waves were quite different from those in an isolated boundary layer. This effect can be described in terms of a weak interaction between the boundary layer and the free atmosphere. As the waves move over the boundary layer eddies certain scales are enhanced whereas others are suppressed resulting in a preference of certain spectral modes of the gravity wave system which is understood as a selection of the dominant eigenmodes of the troposphere (0 in Fig. 1). The wave-generation process generally is described in terms of normal or eigenmodes, where a normal mode is understood (see e.g. Gill 1982) to mean a field’s spatial and temporal structure corresponding to a particular point in horizontal wavenumber space. Deep modes in this context are understood to consist of eddies in the boundary layer and gravity waves above. The impact of waves on the boundary layer dynamics is of special interest as the latter can no longer be considered as being governed only by local boundary layer parameters.

In the 12 June 1984 case clouds were observed in a shallow 800m-deep layer below the capping inversion at 2.3 km, above which the atmosphere was unconditionally stable, thus inhibiting any deep convection. Clouds grew in the stable air just above the CBL and, therefore, were exposed to the impacts of thermals from below and gravity waves from above (region @ in Fig. 1). In CHK, the upshear propagation of gravity waves with respect to the thermals resulted in a preferred upshear growth of the clouds. The growth was observed to occur where smaller clouds developed in the wave updraught and were consequently advected into the main cloud. It was concluded that dry and shallow moist convection in the presence of convection waves is a highly non-local problem due to the influence of the dominant deep normal modes. If, in contrast to the 12 June 1984 case, a conditionally unstable environment exists, convection waves can provide enough ascent to both organize and trigger deep convection. This aspect is studied by Balaji and Clark (1988) in a companion paper. The generality of the conclusions drawn from CHK’s numerical simulations is limited by the two-dimensionality of this study. The inclusion of the third dimension introduces directional effects which can alter the competition between the shallow Rayleigh modes confined to the CBL and the deep modes containing gravity waves. Also, there are significant differences between two- and three-dimensional dynamics caused by the inclusion of the vortex tilting and stretching terms. This difference directly affects the vertical mixing of momentum within the CBL, resulting in the erosion of the mean wind structure for the threeidimensional case. In two dimensions the shear profile was maintained more or less unaffected over the two-hour simulation time.

The basic objectives of the present simulations are to test the representitivity of the two-dimensional findings, and to identify the basic three-dimensional features of convection waves. In an attempt to directly compare the two- and three-dimensional results we tried to compensate for the effect of changing shear profile especially near the boundary layer top and applied additional horizontal pressure gradients. Such mesoscale pressure gradients can be understood as the result of differential heating as observed over sloping terrain (Lettau 1967) or of horizontal surface inhomogeneities. In this manner we could partially balance the effect of mixing on the shear profile over sufficiently long times to study its influence on wave generation. Other mechanisms with potential to compensate for the effect of mixing, such as horizontal momentum advection or the direct simulation of sloping terrain, may be considered in future research. As a conse- quence of the coarse resolution, clouds were only poorly resolved and questions address-


ing the upshear growth of cumulus clouds in the presence of gravity waves were not investigated.

The paper is organized as follows. After a description of the model and the numerical experiments in section 2 we study in section 3 the temporal development of the various experiments in detail. Waves and wave growth are investigated in section 4. Perspective views of isothermal surfaces illustrate the structure of the gravity wave field. Series of two-dimensional w-power spectra at successive times show how the various spectral modes develop with time. From the spectral results, conclusions about the directional effects of the forcing can be drawn. The effects of additional horizontal pressure gradients are also studied. The role of clouds is the subject of section 5 and an example is presented illustrating the wave effect on cloud ensembles. Momentum transport by convection waves is treated in section 6 and conclusions are presented in section 7.

2. THE NUMERICAL EXPERIMENTS

(a) The numerical model The three-dimensional non-hydrostatic anelastic model of Clark (1977, 1979) with

grid nesting as described in Clark and Farley (1984) and advection scheme improvements as described in Smolarkiewicz and Clark (1986) is used in this study. An earlier version of this model was used in CHK in two dimensions. As this finite difference model is already adequately dscribed in the literature we refer the interested reader to the cited papers for further details.

Various sensitivity studies were performed in CHK, where in experiments DRY5 of that paper the horizontal domain was increased from 30 to 45 km and in DRY4 the vertical domain was increased from 15 to 25 km. The first experiments did reveal a weak quantization effect of the horizontal domain on the developing waves whereas the second experiments indicated that the 15 and 25 km vertical domain extents gave very similar results for that particular problem. These later experiments highlighted the refraction of the vertically propagating waves at the tropopause. Table I describes the various parameters of the three-dimensional experiments. Based upon the results of CHK we chose a domain of 60x60km in the horizontal for the present experiments. The nesting capabilities of the model were used in the vertical in order to consider a reasonably deep atmosphere and allow for finer tropospheric resolution. The outermost domain was chosen as 60X60x30km to represent a deep atmosphere and the second domain was chosen as 6 0 x 6 0 ~ 15 km to represent the troposphere, as illustrated in Fig. 2. To minimize spurious wave reflection, the top of domain 2 was chosen so as to lie above the tropopause, which was at 10 km above the ground, and an absorber was placed in the uppermost 15 km of the outer domain to absorb vertically propagating gravity waves reaching those stratospheric heights. A third domain was added in order to better resolve the surface layer and to improve the simulation of the strong shear confined to the lowest few hundred metres.

(b) Boundary and initial conditions Our intention is to study the long-time solutions of a near-field problem, i.e. a

problem where the solutions remain in the region of forcing. Therefore, as in CHK, cyclic boundary conditions have been applied in the horizontal. Open boundary conditions imply a finite fetch for perturbation motion development and consequently a finite Lagrangian time scale for wave development. It is impractical to consider a sufficiently large domain to attain the desired long-time forced solutions. Our near-field forcing was taken as a surface sensible heat flux of 300Wm-* with a 1% white noise variability uniformly applied at the lower boundary for the duration of the experiment.

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The initial profiles of wind, potential temperature and humidity are shown in Fig. 3 and correspond to the observed case of 12 June 1984 in the late afternoon. The experiments were initiated using these profiles and assuming horizontally homogeneous flow. Thermodynamic variables are seen to be well mixed up to 1.3 km above the ground with weak stability appearing between 1.3 and the cloud base level of 1.6 km. Owing to the coarse vertical resolution of the model, a strong inversion observed in the initial sounding at 2.3 km is represented by a layer of only slightly increased stability between 1.75 and 2.25 km. Above 2.3 km the atmosphere is unconditionally stable. The winds are westerlies with a weak mid-tropospheric jet and a small northward component in the lowest levels. Below about 4 km the winds were taken from the aircraft measurements of KHC, whereas rawinsonde data were used above this level. Measured winds were rotated clockwise by 40 degrees with the mean wind parallel to the x axis to minimize truncation errors.

It should be noted that the initial wind profiles differ substantially from the idealized profiles used in CHK. The atmosphere is highly baroclinic with a speed shear of the geostrophic wind. The observed boundary layer wind structures may be due to a combination of baroclinicity and other large-scale forcing such as thermal gradients from fronts or sloping terrain effects. The actual cause of the low-level environmental structure is considered to be outside the scope of the present paper.

This initial wind field was taken to be in geostrophic balance throughout both the free atmosphere and the boundary layer. To assess the effect of this assumption within the boundary layer we performed a simulation with a more realistic extension of the geostrophic wind into the boundary layer. The results from this experiment showed no significant differences in the final wind profiles. Observations, for example by Arya and Wyngaard (1975) and by Kaimal et af. (1976), and analytical studies (Griesseier 1972), indicate that convective mixing is rapid and limits the wind-shear magnitude in the boundary layer. This suggests that the Coriolis force is of only minor importance in such cases, which probably explains the lack of sensitivity to our questionable assumption about the initial boundary layer geostrophic wind structure.


8(K) V(m/s)

Figure 3. The initial profiles of wind components u and u, potential temperature, 8, and water vapour mixing ratio, qv.

(c ) Description of the experiments Two sets of experiments are considered in this paper. The first set comprises the

standard moist (ST-M), standard dry (ST-D), high-resolution moist (HI-M) and high- resolution dry (HI-D). Most of the analysis refers to ST-M. These experiments were designed to test the sensitivity of the forced wave response to horizontal and vertical spatial resolution within the troposphere as well as to the presence of clouds and the accompanying latent heat effects. As part of these experiments, a third domain was put in the lowest 500m to better resolve the dynamics and thermodynamics of the surface layer. A surface drag formulation for the momentum flux was applied at the surface using a drag coefficient CD = 0.01. This third domain also resulted in an enhanced efficiency of energy transfer between the surface sensible heat flux and the perturbation kinetic energy in the lowest model levels.

The shear profile in the upper regions of the boundary layer and lower regions of the free atmosphere was found by CHK to represent an important physical variable governing the character of the thermally forced deep normal mode response. A problem with these experiments is that, without some sort of parametrized large-scale forcing, the convection in three-dimensional space rapidly produces a well-mixed boundary layer. This can be seen in Fig. 4 from the initial and late stage wind, wind shear, and potential temperature profiles. Convective motions also entrain air into the stable layer immediately above and transfer momentum as well as heat down into the CBL. The height increase of the CBL can be seen in Fig. 4. The mixing decreases the initial shear in the upper boundary layer, and increases the shear in the stable layer above. As shown in Fig. 5, shear in the boundary layer is now confined to the surface layer which is a well-known and common feature of convective boundary layers (Wyngaard et al. 1974; Arya and Wyngaard 1975). The strong modification of the shear profile within the boundary layer by the convection dramatically changes the problem from that represented by the initial profiles. This long-term variability of the physical parameter space severely limits our ability to intercompare directly the present three-dimensional results with the two-


-15 -10 -5 0 0 5

- T = 0 Min 4 ---T=200 Min

-2 0 2 300 305 310

V (mh) dV/dZ ( I h * 10'3) 8 (K )

Figure 4. Initial and late stage (t = 200min) wind, wind shear, and temperature profiles from experiment ST-M.

dimensional results of CHK. The present results should be taken as representative of those one might expect in nature for the final mean environmental profiles of the various experiments.

In order to offset the mixing effects of the convection on the boundary layer shear a parametrized boundary layer forcing was introduced into the second set of experiments FC-0, FC-1 and FC-2 (see Table 1) using mesoscale horizontal thermal gradients. These experiments allow us to perform sensitivity studies on the efficiency of gravity wave excitation with respect to shear near boundary layer top. The motivation for using thermal forcing stems from Lettau (1967) who showed that thermal forcing is a common feature over the Great Plains, caused by the slight inclination of the topography. He demonstrated that thermal forcing can provide a mesoscale source term for shear in the boundary layer and these gradients can influence the nocturnal low-level jet. Westerly gradients of 0.00, 0.02 and 044Kkm-' were chosen for FC-0, FC-1 and FC-2, respectively. Assuming a well-mixed hydrostatic boundary layer for this thermal gradient portion of the boundary

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nested domain. Figure 5. The velocity profile in the lowest 450m at t = 200rnin in experiment ST-M taken from the third

CONVECTNELY FORCED GRAVlTY WAVES 317

layer temperature, we have parametrized an additional pressure gradient term as

d +,)I& = - Rp( 1 - z / h ) d T / h (1) which applies only in the boundary layer between z = 0 and h where h = 2 km. The term in (1) was appropriately added to the horizontal equations of motion. This type of formulation allows for the addition of horizontal thermal gradient forcing using cyclic boundary conditions. As shown in Fig. 6, the long-term effect of these additional mesoscale pressure gradients is an acceleration of the mean flow below 2.5 km by up to 13 m s-l in FC-2, and 6 m s-' in FC-1 at t = 200 min. Owing to the action of the Coriolis force, wind speed in the y direction increases as well. Above 3km the profiles are identical. Minor increases of wind shear occur between z = 1 and 1.5 km whereas significant increases occur between 1.5 and 3 km. Convective mixing, however, still maintains a nearly well-mixed boundary layer against the additional pressure forcing in the weak-forcing experiment FC-1. The directional effect of the forcing was minor. The magnitudes and directions of the shear between z = 1.5 and 2km are 4.8, 11.6 and 14 .8~10-~s - ' directed 15, 30 and 28 degrees south of east for experiments FC-0, FC-1 and FC-2, respectively. Although the shear could be preserved to some degree in FC-2, the forcing value of O-MKkm-' seems rather large. The same holds for the resulting accelerations.

( d ) Shear in the upper convective boundary layer Convection mixes momentum within the CBL very quickly so that, even with an

additional pressure gradient, only weak shear can be maintained. Knowledge of the conditions under which mean environmental shear is maintained in the upper part of a

-25 -20 -15 - 10 -5 0 I I 5 I I I I

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Figure 6. Initial and late stage (t = UW)min) profiles of the I( and u wind components for experiments FC-0, FC-1 and FC-2.


convective boundary layer could further our understanding about the occurrence of strong events of convectively forced internal gravity waves. Wyngaard (1985) discusses the problem of an unmixed CBL having considerable shear in its upper half. He suggested that this is likely to result from strong temporal changes, differential advection, entrainment and baroclinicity. He also presents a case of an unmixed CBL over sloping terrain.

Observations are rare and usually restricted to the lower boundary layer. Kusano et al. (1969) showed from tower measurements below 250m that strong shear during daytime and in summer is a short-lived phenomenon lasting in general not more than two hours. Long-lived shear events were related to advective situations. We hypothesize that the same holds also for the shear in the upper boundary layer. Lenschow et al. (1980) argued that advection was the most likely cause for shear within a convective boundary layer. Entrainment and baroclinicity in our experiments did not provide sufficient momentum transport to counteract the convective mixing. Numerical simulation of larger-scale momentum advection excludes the direct use of cyclic boundary conditions. Typical advective situations where one might expect strong convectively forced gravity waves are found, for instance, in the cold air after a front (Brodhun et al. 1976), or over sloping terrain.

3. TEMPORAL DEVELOPMENT OF KINETIC ENERGY

Although we are primarily interested in the long-time solutions, a study of the initial evolution is necessary to interpret some of the results.

The numerical experiments are classified into three periods using volume integrals of the perturbation kinetic energy. Integrals over two different domains are considered, where the first domain extends from z = 0 to 3 km and the second from z = 3 to 15 km. The first integration domain completely encompasses the boundary layer and extends about 1 km into the free atmosphere. It integrates the perturbation kinetic energy of shallow modes confined to the boundary layer as well as the lowest 3 km portion of any deep modes. The second integration domain lies well above the top of the boundary layer and as a result measures the gravity wave contribution to the kinetic energy in this domain but underestimates its total value.

Figure 7 shows a time series of the integrated perturbation kinetic energy for experiments ST-M, HI-M and HI-D. The three different time periods are called the initialization period, the transition period and the developed period. The first two periods are primarily model states which are to a large extent influenced by the experimental design. It is the developed period, or the long-time forced response, which is of main physical interest.

The initialization stage is characterized by perturbation motions being confined to the boundary layer. Initially the boundary layer has neutral stability with a shear of about 5 m s-lkm-l, which results in longitudinal rolls in agreement with linear stability analysis (Asai 1970, 1972; Asai and Nakasuji 1973) for a thermally unstable shear flow. During the first 50 minutes the dynamics responds mainly fo the white noise portion of the surface heat flux. The temperature structure, however, becomes increasingly superadiabatic, as the surface heat flux is distributed over the lowest 500m.

The potential energy build-up during the initial period of development is finally released at about 50 minutes. This results in a near-constant exponential growth rate of kinetic energy in the boundary layer until at about 60 minutes a new growth rate is established where kinetic energy is developing simultaneously in both the boundary layer and the free atmosphere. This 10-min delay suggests that the waves are forced by the


n

f x > (3

W z W

Y

a

Figure 7. Time series of volume-integrated perturbation kinetic energies for experiments ST-M. HI-M and HI-D shown by squares, triangles and circles, respectively. The open symbols (dashed traces) are for the integration limits of z = 0 to 3 km. The filled symbols (solid traces) are for the integration limits of I = 3 to

15 km.

convection. The overshoot in the maximum of the kinetic energy in the boundary layer suggests a nonlinear saturation of the Rayleigh modes. The maximum of w, for example, typically peaks at about 8ms-’ and then levels off to about half this value during the developed period. A wide range of scales is excited during this transition period. The end of the transition is probably at about 90min, when a positive heat flux is well established and the boundary layer maintains a near-neutral stability profile.

An intercomparison of experiments ST-M and HI-M in Fig. 7 reveals some effects of spatial resolution upon the evolution of the kinetic energy. Increasing the resolution results in an earlier release of the potential energy build-up and consequently less of an overshoot in the maximum kinetic energy time series during the transition period. The time series of perturbation kinetic energy in the free atmosphere changes structure between ST-M and HI-M. In the latter case there is ten times less energy in the wave field during the transition time compared with ST-M. At later times the higher-resolution experiment has approximately half the wave field energy of ST-M.


This is also reflected in all wave-power spectra (not shown) which should be taken into account when comparing high- and low-resolution spectra. Despite different amplitudes, a nearly-equal energy increase with time is found at later times. A late time intercomparison of the spectra shows little difference between ST-M and HI-M in the spectral structure for either the l-D spectra or the height contour plots, except in amplitudes.

The energy conversion from surface heat flux to boundary layer kinetic energy is also unaffected by resolution. In HI-M and ST-M the spacing of the longitudinal boundary layer rolls is approximately five times the horizontal grid resolution and decreases with increasing resolution, in line with the arguments of Mason and Thomson (1987) where they hypothesized that the predominance of roll vortices is related to the coarse resolution. Consequently, the number of rolls within the domain rises. In conclusion, the dominant effect of the low resolution is to increase the energy transfer into the waves during the transition period with little influence on the long-time spectral structure of the deep modes.

At about t = 120min the model atmosphere has set up dynamical structures in the boundary layer as well as in the free atmosphere which are representative of the long- time response to the surface sensible heat flux, despite the changes of total kinetic energy as discussed above.

To ascertain the effects of moisture on the initiation of gravity waves, experiments HI-D and HI-M are compared (Fig. 7). The kinetic energy traces for these experiments are nearly identical for both boundary layer energy as well as wave energy, with values for HI-D smaller in amplitude and shifted by about 10 minutes towards later times. The earlier transition is due to the release of latent heat which increases the vertical exchange of energy. The resulting slight energy level differences do not affect the vertical structure of the modes between HI-M and HI-D (not shown). Clouds have little influence on the structure of the gravity wave field, in line with the findings of CHK.

4. WAVES AND WAVE GROWTH

(a) The gravity waue character For all the experiments there is the expected phase shift of 90 degrees between 8

and w resulting in negligible free atmosphere heat flux at levels above any moist convection. Figure 8 shows time series of 8 and w at various heights in the middle of the domain, showing their strong oscillations. Noting that the oscillations start at 66 min at 4.5 km and at 78min at 7-5 km, the onset of wave motion is estimated to propagate vertically with a speed of 4.2ms-'. The amplitudes for both 8 and w decrease with height. The variability of w and 8 is about +1 m s-' and +O-5 K at 4.5 km, and 20.3 ms-' and k0.3 K at 10.5 km. At the lowest height of 1.25 km, which is well within the CBL, we observe a steady increase in the average temperature, caused by the surface heat flux. The oscillations in the boundary layer are due to the convection and are much stronger and more irregular. They are shown here to demonstrate the difference between the more disorganized convective motions and the more regular wave structure. The polychromatic wave pattern shows a meangeriod (relative to the ground) of approximately 16 minutes. Noting that at z = 4.5 km U = 2-2ms-' and L = 12km we get a corresponding intrinsic frequency of about 8 x 10-4s-1 which is considerably smaller than the Brunt-Vaisiila frequency of approximately 9 x iO-3s-i.

Three-dimensional perspective views of w (not shown) reveal a scattered field of single updraughts extending well above the boundary layer. The waves appearing at 60min are vertically oriented and are nearly cylindrical with diameter increasing with


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I I I

W z = 10.5 km - + I s

Figure 8. Time series of w (solid) and P (dashed) at four heights in the middle of the horizontal domain from experiment ST-M.

height. The latter effect is probably due to density decreasing with height (Hines 1972). At about 120min the waves can be seen to extend into the stratosphere and are leaning slightly into the mean wind. From plots at each 100s (not shown here) one can observe updraught fields detached from the top of the CBL. This indicates vertical propagation of the waves consistent with the lack of vertical coherence in the time series at various heights as shown in Fig. 8.

Figure 10 shows three-dimensional perspective views of 8 surfaces for the 8 values of 304 K, 310 K and 320 K. These surfaces correspond to average heights of 3.0,5-2 and 8.5 km. Four different times are displayed 200s apart. The time increment between two subsequent figures enables us to follow single waves as indicated by the arrows, which show that they are propagating against the mean wind. Owing to their close proximity to the heat source the lower-level 8 surfaces are rougher, with a broader spectrum of scales as shown in Fig. 9 for the w field. The wave amplitudes shown in Fig. 10 are somewhat smaller than those observed by KHC, where variations between +1 and 2 3 m s-l were observed for w.


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(d ) .

0.1 1. 10. 1ro. k,

Figure 9(i). Power spectra of w field at t = 60min for experiment ST-M. (a) and (b) show two-dimensional spectra in terms of height and one wavenumber component where an integration was performed over the other component. Logarithmic contour intervals (lo-’, . . .) are used. (c) and (d) show one-dimensional spectra

as either k,E(k,) versus kx, (c), or kyE(k,,) versus k,, (d).

(6 ) Structural aspects of the wave field The scattered structures of w and 8 in Fig. 10 are explicable in terms of wave

fronts modulated in the cross-shear direction by boundary layer motions. According to Wegener’s hypothesis (e.g. Gossard and Hooke 1975) wave fronts tend to form perpendicular to the wind shear in a pure speed-shear situation. Applied to our case, the shear in the stable layer overlying the boundary layer would tend to support wave fronts aligned in the north-south direction. Continuity of the dynamical fields requires a similar structure to be maintained in the boundary layer if such wave fronts are to exist without cross-stream modulation. In our case this low-level condition never exists as this is primarily a speed-shear case. At early times we have east-west longitudinal rolls which at later times appear to be broken. This boundary layer structure results in a strong modulation of the wave fronts but this does not appear to affect the horizontal phase


15.

10. A

?I 5 N

5.

Figure 9(ii). As (i) but at t = 200 min. Lower curves in (c) and (d) are for L = 3 to 15 km domain whereas upper curves are for the entire domain energy. Wavenumber k, or ky = 1 refers to 60km.

velocity vector of the waves, which maintains an approximate westward speed of 10 m s-'. Had there been a preference for longitudinal rolls aligned in the north-south direction then banded convection and wave fronts would have been supported. This requires a nearly 90" angle between the mean boundary layer shear and the shear in the overlying stable layer. This is the type of case treated by Balaji and Clark (1988) and which was observed by Jaeckisch (1968, 1972).

The interplay between the CBL and free atmospheric dynamics can be seen in the power spectra of the vertical wind velocity which are shown at 60 and 200min for experiment ST-M in Fig. 9. At t = 60min the energy is confined to the boundary layer with structures elongated in the x direction. The y-spectrum energy is centred around wavenumbers 13-15, corresponding to wavelengths of about 4-5 km (the roll spacing). The x spectrum is centred around wavenumber 7, corresponding to 8-9 km. This initial stage spectral distribution of energy seems to be in line with boundary layer stability analysis.


I - -

Figure 10. Perspective view of tJ surfaces for 304,310 and 320 K for experiment ST-M. The horizontal domain is 60x60km and the peaks scale +600m. The mean heights of the isosurfaces are 3.0, 5.2 and M k m ,

respectively. Arrows point to one particular wave peak.


The later time spectra in Fig. 9, at t = 200min, show a much broader distribution of boundary layer energy in both horizontal wavenumbers k, and k,,. The extension of the x spectra to high wavenumbers is caused by the break-up of the rolls. The flow structure is similar to what has been called ‘varicose-like’ in direct simulations of Benard convection (Grotzbach 1982). There does remain even at t = 200 min an elongation of the boundary layer structures in the x direction. There is a much more dramatic broadening of the y- direction boundary layer spectra than for the x direction. Most noteworthy is the development of low wavenumber scales.

In contrast to the broad distribution of horizontal scales in the boundary layer eddies, the range of scales for the gravity waves is much narrower. The gravity wave energy ends up centred at k, = 4 (12 km) and ky = 8 (7.5 km). The logarithmic increments in the height-spectrum contour plots emphasize that all modes which are present in the boundary layer also appear in the free atmosphere but are evanescent for higher wavenumbers, with higher wavenumbers decaying the fastest. Wavenumbers lower than 7, however, seem to be strongly decaying only above 10 km. It should be emphasized that the deep modes (k, = 4) are present from the beginning of the wave motion, indicating a linear forcing mechanism. The decay of shorter modes (k, < 4) and the continuous increase of the deep mode amplitude which is accompanied by a shift of the spectral maxima, however, reveal a nonlinear saturation of the linear modes. The selection mechanism of the deep modes with time is the main point of interest rather than the instant wave excitation at the boundary layer/free atmosphere interface. It is interesting to note that major differences between the one-dimensional spectra of the eddies and waves are confined to the k, spectrum. These differences increase somewhat with time as a second maximum (not shown) in the k, spectrum near k, = 7 disappears. The ky spectra for waves and boundary layer structures are similar for t > 120 min. Waves are simply modulated by the latter in the ‘non-active’ shear direction. Figure 9 also shows that it is the longer waves which have the highest amplitudes in the upper regions of the troposphere.

(c) The effect of horizontal thermal gradients As previously discussed, the purpose of the horizontal thermal gradient forcing in

the boundary layer is to test the sensitivity of the gravity wave response to variations in the mean wind shear across the top of the CBL. Figure 11 shows that there is a systematic increase in the boundary layer kinetic energy due to the thermal gradient forcing. By the end of the experiments there is over 15 times more energy in the boundary layer eddies for FC-2 than in the reference experiment FC-0. The kinetic energy of the wave field shows a much less pronounced response to the thermal gradient forcing. Experiments FC-0 and FC-1 have similar kinetic energies in the w field above z = 3 km for the entire experimental period. It is only FC-2 with a gradient of 0.04Kkm-* where we see a substantial increase in the wave field energy for t > 160 rnin and by t = 240 min there is about three times as much wave energy in FC-2 as in either FC-0 or FC-1. As seen from w-power spectra in Fig. 12, high wavenumber contributions in the x direction contribute to the energy increase in the boundary layer, whereas in they direction all scale amplitudes are increased. Wave response, however, is quite different as low wavenumbers are preferred. Most energy in the x direction goes into wavenumber k, = 4 (15 km) and in the y direction into ky = 7 (8-5 km). Differences between wave spectra and boundary layer spectra are most pronounced for experiment FC-2. Height-contour plots of the spectra reveal the same structure for all experiments FC-0, FC-1 and FC-2 (shown for FC-2 at t = 240min only). The main effect of additional thermal gradients is to broaden the range of scales into the high-k, region and simultaneously raise the amplitude of all


0 20 40 60 80 100 120 140 160 180 200220240 1 0 4 L 1 I I 1 I 1 1 I I I I I I

TIME ( m i d Figure 11. Time series of total perturbation kinetic energy as in Fig. 7. The triangles, circles and squares

refer to experiments FC-0, FC-1 and FC-2, respectively.

scales. Also, the maximum of the y spectrum is now found at low wavenumbers for all heights.

5 . THE ROLE OF CLOUDS

An important dynamical interaction was noted by CHK who found that the horizontal phase velocities of the boundary layer eddies and overlying gravity waves were decoupled in the presence of clouds. The resulting character of the dominant normal mode solution was found to strongly influence the cloud dynamics. As the wave field moves upwind over the thermal field the coincidence of a thermal and a wave updraught leads to a region of enhanced cloud growth. This point is further analysed in Balaji and Clark (1988). An analysis of the horizontal phase velocities for k, = 4 and ky = 8 eigenmodes in HI-D and HI-M resulted in similar vertical structures. In both cases the mode moved westward at about 10 m s-l relative to the ground. Below z = 4 km, the phase velocity calculation was rather transient in character, due possibly to shallow waves near the top of the boundary layer, making it difficult to determine an unambiguous mean structure. With our present analysis it was not possible to ascertain whether or not there was any systematic horizontal phase velocity decoupling in the presence of clouds as noted in the previous two studies. The dynamical impact of the fair weather cumuli is not sufficient to influence the overall kinetic energy levels as evidenced in the time series of Fig. 7. The current three-dimensional numerical simulations have much coarser horizontal resolutions and the dynamical interplay between the waves and the clouds is not well resolved. Owing to effects of poor resolution, the standard experiment ST-M showed only a short-lived cumulus population. In experiment HI-M, with a horizontal resolution of 500m, a cloud population lasted over the whole integration time, similar to that in Fig. 13. The average diameter of the fully developed clouds is about 2-4 km with a depth varying between 500 and lOOOm, in agreement with the environmental profiles of


0

0.20

0.00

Figure 12. The power spectra of w as in Fig. 9 at r = 240min from FC-2.

temperature and humidity which showed unconditionally stable stratification and low humidity above the cloud layer. The drift speed of the clouds at 2 km height determined from the positions at two consecutive time steps is nearly equal to the average wind speed at 2 km. In the vicinity of clouds, locally confined wind velocity deviations from the horizontal mean of the order of +3 m s-l are found, in agreement with the observations by KHC. It is emphasized that the cloud population is a result of the interaction between waves and thermals, rather than determined from the boundary layer thermals alone. The latter idea is favoured by many authors such as Hill (1977), Stull (1985), Yau and Michaud (1982) and Reuter and Yau (1987). The presence of waves can lead to considerable lifting on the order of a hundred metres (Fig. 10) resulting in a strong influence on cloud initiation. The important dynamical interplay among the waves, eddies and clouds makes the present type of fair weather cumulus population prediction a highly non-local problem. Some parameters which seem to govern the cumulus cloud population are the horizontal wavelength of the dominant deep modes, the relative phase speed between the waves and eddies, the size and lifetime of the thermals and the time scale of cloud dissipation. These parameters, as well as nonlinear processes, result in a rather random-looking field of clouds as shown in Fig. 13. This randomness is not the result of the variability of the surface forcing. These results suggest that the simulation of such


Figure 13. Horizontal cross-section through the cloud liquid water field at z = 2 k m for f = 150 and 155 min, for experiment HI-M. Cloud outlines (dashed) refer to q. = 0.01 g kg-'. Contour interval is 0.03125 g kg-'. The

arrows point to the same cloud.


fair weather cases requires the treatment of the full depth of the troposphere in order to capture the required spectrum of deep normal modes.

Convectively forced gravity waves in three dimensions were simulated by Yau and Michaud (1982). Due to a limited vertical domain size of 6km, wave development was suppressed and consequently had no impact on cumulus cloud growth, which was then found to be determined by local parameters only. The development of deep convection in the presence of waves is treated by Balaji and Clark (1988) and is omitted here.

6. MOMENTUM TRANSPORT

It was argued by KHC that thermally forced waves might contribute to the large- scale vertical momentum budget. Although locally weaker in amplitude than mountain waves they are probably more frequent and could cover broader areas. Figure 14(a) shows the profiles of the resolved Reynolds stresses z, and zy for experiments ST-M, HI- D and HI-M. The maximum momentum flux is found in the convective boundary layer with values of about 3 dyncm-2 decreasing to a local maximum of about 0.3 dyn cm-2 at z = 3.5 km in the overlying stable atmosphere. Locally these values appear to be small compared with those caused by mountain waves. Bretherton (1969), for example, found typical values of 3 dyn cm-2 in Wales where the topographic variation was only 500m. It is still an open question, though, as to the relative importance of thermal waves on the large-scale momentum budget as frequency of occurrence and areal coverage, among other factors, must be assessed. Momentum flux components for the high-resolution experiments HI-D and HI-M are nearly identical, demonstrating the negligible influence of clouds on the momentum flux. Compared with ST-M they are smaller in amplitude but reveal almost the same vertical structure. This is probably due to the earlier times at which values were taken for HI-D and HI-M. Wyngaard (1985) showed that a well-mixed horizontal momentum profile in the CBL is mainly achieved through stress curvature. It is interesting to note in Fig. 14(b) the strong turning of the horizontal stress vector between 0.5 and 4.0 km by about 125". The wind shear vector shows a similar behaviour and turns by about the same amount. Directional dependence of the stress vector with height is inherently three-dimensional and is typical for baroclinic boundary layers (Wyngaard et al. 1974; Kaimal et al. 1976; Lenschow et al. 1980). The simulated subgrid-scale momentum fluxes are negligible in magnitude compared with the resolved fluxes. Note also in Fig. 14(b) that at lower heights shear and stress vectors are antiparallel whereas at greater heights they are nearly orthogonal, suggesting a com- plicated relationship between stress and strain.

7. CONCLUSIONS

Three-dimensional numerical simulations of convectively forced internal gravity waves for both dry and fair-weather-cumulus conditions were performed using environmental initial conditions of 12 June 1984. This was the case CHK treated using a two- dimensional model. As in CHK, significant internal gravity wave amplitudes resulted as a response to the long-term forcing. The present results support many of the earlier interpretations of CHK and extend their findings. Some of the major differences introduced by including the third spatial dimension are: a fully three-dimensional deep mode was selected, resulting in scattered fields of shallow clouds; the boundary layer dynamics produced much more well-mixed profiles of dynamic and thermodynamic variables; and the nonlinear saturation of the Rayleigh modes occurred in three dimensions prior to the deep modes achieving dominance.

330

2 - n r(,

T. HAUF and T. L. CLARK

I I I 1 I 1 1

- 0.2 ( b)

5.0

4.0

n

-* E

I- r (3 r

3.0

w

2 .o

I .o

(8)

0.2 0. I 0 -0.1 -0.2 -0.3 T x TY kg/(s2m 1

h c - Y

N rp \ > rp

0.5 km

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 r, (kg/s2m)

0.1 < x In

0 2 N 3

-0.1 - -0.2

Figure 14. (a) Profiles of Reynolds stresses z13 = z, and zu = ty for experiments ST-M, HI-D and HI-M. (b) Horizontal stress (dashed) and shear vectors (solid) of experiment ST-M at I = 200 min for heights L =

0.5, 1.0, . . ., 4-0 km.


The early-time modal structures obtained in the boundary layer are in line with the Rayleigh modes one might expect from typical stability analysis. Subsequent to a transition period, however, deep modes appear which consist of boundary layer eddies and overlying waves. The horizontal scales associated with these deep modes are much larger than the Rayleigh modes and their amplitudes are comparable at the later times for the near-field forced problem. As in CHK, the vertical structure of these deep modes was found to be sensitive to the mean shear in the region spanning the CBL and overlying stable layer. Experimental results support the earlier findings that increased magnitudes of CBL/stable layer shear result in comparable increases in wave amplitude. Variability of shear structure can also strongly influence the horizontal wavenumber selection of the dominant deep modes, as discussed by CHK and Clark and Hauf (1986). However, the range in mean shear conditions was not adequate in the present experiments to address such questions. The selection of the dominant deep modal structure has been described earlier, and also in this paper, as an interaction between boundary layer eddies and gravity waves. One might view the selection of these free modes as a weak resonance. As the deep modes can be identified very early in the simulations their selection is most likely represented in terms of a linear model. The final amplitudes, especially of the shorter modes (k, > 5), however, appear to be saturated which is certainly associated with nonlinear fluid dynamics. This topic has been dealt with in some detail in the companion paper of Balaji and Clark (1988).

The general character of the wave and eddy field response resulted in scattered fields. This is interpreted to be a result of the competition between the preferred longitudinal boundary layer rolls and waves oriented with their phases aligned normal to the shear direction. For the present speed shear case, these two directions of alignment are nearly orthogonal resulting in the eddies and waves destroying any banded structures. The final structure was that of a ‘varicose-like’ structure to the eddies and an overlying scattered wave field. The along-shear wave spacing peaked at k, = 4 (15 km) and the cross-shear spacing peaked at the larger wavenumber of k,, = 8 (7.5 km). A directional shear case was considered by Balaji and Clark (1988) where the two directions of alignment were well matched. As a result they obtained an overall banded structure for the waves.

The results of the present paper emphasize the importance of considering cumulus convection as a near-field forced problem for undisturbed environmental conditions. The deep normal modes require a considerable atmospheric depth for their definition and considerable time to develop. Wave amplitudes of ? 1 m sK1 were obtained with shear direction spacing of ==12km in the lower free atmosphere levels. The moist experiments indicate that these waves are important in determining the relative cloud-scale spacings. The clouds had a negligible effect on the overall wavenumber selection and vertical structure of the waves for the present fair-weather case. Owing to the noisy character of the lower-level horizontal phase velocities, it was not possible to determine any effect of clouds on this parameter. CHK emphasized the important role of the deep forced modes on the dynamics of the cumulus clouds themselves. The spatial resolution of the present experiments was not adequate to address this issue directly. The experiments do, however, support the concept, as suggested by CHK, that cumulus convection and individual cumulus dynamics should be considered as a field problem as opposed to the concept of treating cumulus in isolation by using bubbles as initial conditions.

The momentum flux vector changed direction significantly with height in the lower atmosphere indicating nonlinear interaction of convective motions with the background flow. Above 4.5 km, momentum is transported by waves downward at -0-3 dyn cm-’. The directional changes of momentum flux vector are inherently three-dimensional and


are typical for baroclinic boundary layers. The wave momentum flux for these experiments is low compared with estimates of mountain wave fluxes such as that of Bretherton (1969) for waves over hills in Wales. Any contribution of these waves to the larger-scale momentum budget would be small and confined to the first 4 km above the surface. The degree to which thermally forced shallow convection contributes to the mean momentum budget for other situations is a subject of current studies. The heat flux in the wave region is negligibly small.

ACKNOWLEDGMENTS

This work was done when one of us (T.H.) was visiting scientist at NCAR’s Mesoscale and Microscale Meteorology Division. Support from NCAR and the DFVLR are grate- fully acknowledged. The authors are especially indebted to Bill Hall for his continuous and indefatigable help in running the model.

Arya, S. P. S. and Wyngaard, J. C. 1975

Asai, T. 1970

Asai, T. and Nakasuji, I.

Balaji, V. and Clark, T. L.

Bretherton, F. P.

Brodhun, D., Bull, G. and

Bull, G. and Neisser, J.

Carruthers, D. J. and

Clark, T. L.

Neisser, J.

Chin-Hoh Moeng

1972

1973

1988

1969

1976

1976

1987

1977

1979

Clark, T. L. and Farley, W. R. 1984

Clark, T. L. and Hauf, T. 1986

Clark, T. L., Hauf, T. and CHK 1986 Kuettner J. P.

Gill, A. E. 1982 Gossard, E. E. and Hooke, W. H. 1975 Griesseier. H. 1972

Grotzbach, G. 1982

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