Penetrative turbulence associated with mesoscale surface heat flux variations
aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Penetrative turbulence associated with mesoscale surface heat fluxvariations
Jahrul M Alam ∗ and M Alamgir HossainDepartment of Mathematics and StatisticsMemorial University, Canada Abstract
This article investigates penetrative turbulence in the at-mospheric boundary layer. Using a large eddy simu-lation approach, we study characteristics of the mixedlayer with respect to surface heat flux variations in therange from .
48 Wm − to .
92 Wm − , and ob-serve that the surface heterogeneity on a spatial scaleof km leads to downscale turbulent kinetic energycascade. Coherent fluctuations of mesoscale horizon-tal wind is observed at m above the ground. Sucha surface induced temporal oscillations in the horizon-tal wind suggest a rapid jump in mesocale wind fore-casts, which is difficult to parameterize using traditionalone-dimensional ensemble-mean models. Although thepresent work is idealized at a typical scale ( km) ofsurface heterogeneity, the results help develop effec-tive subgrid scale parameterization schemes for classicalweather forecasting mesoscale models. Earth’s surface is assumed to play a distinctive influencein weather and climate systems (Stull, 1976). The sur-face influence generates horizontal convection as a re-sult of differential heating along one horizontal bound-ary of the atmospheric boundary layer (ABL) (Lane,2008; Scotti and White, 2011). Coherent mesoscale mo-tions are reported by many studies, where the effect ofsurface heterogeneity are investigated numerically onscales of few hundred kilometers, using a grid spac- ∗ Corresponding author’s email: [email protected] ing of ∆ x ∼ O (10 km) (Skamarock and Klemp, 2008;Lin, 2007). However, in the geophysical fluid dynam-ics community, controversial opinions exist on whetherhorizontal convection due to differential heating at thesame level would lead to turbulence, or drive large scaleoverturning circulations (Paparella and Young, 2002;Scotti and White, 2011). A century old experimen-tal result by Sandström (1908) leads to the opinionthat a sustained circulation cannot occur if the level ofthe heating source is the same as that of the coolingsource. This hypothesis is supported theoretically by theanti-turbulence theorem of Paparella and Young (2002).However, Scotti and White (2011) employed a direct nu-merical simulation of horizontal convection at Rayleighnumber to show that a flow driven by the horizontalconvection exhibits the characteristics of a true turbulentflow. Clearly, some fundamental aspects of solar heatingat the earth’s surface is not fully understood.Much of our current understanding of surface inducedphenomena depends on numerical simulations and mea-surements/observations. For example, lightning dataover Houston, USA between years of 1989 and 2000indicates that highest flash densities occur over urbanareas. Measurements of turbulence for the years pe-riod from to in the North Atlantic, USA, andEuropean sectors conclude that clear-air turbulence in-creases in these regions by − % as a result of theurban induced impact on vertical transport and mixingof penetrative ABL turbulence. Note that the averagehorizontal flux of kinetic energy at a height ∼ mfrom the ground is about W m − . A destabiliza-tion of this huge energy flux due to differential heatingon the surface may cause catastrophic impact on the tur-1ulent atmosphere. Since penetrative turbulence in theatmosphere (see Stull, 1976) is primarily characterizedby solar heating, perturbations to boundary layer struc-tures by human activities ( e.g. urbanization) are sensi-tive to vertical mixing and transport (Bryan and Fritsch,2002; Lane, 2008).This article reports on a large eddy simulation (LES)based numerical model in order to characterize penetra-tive turbulence over heterogeneous land surface. Pri-mary objectives of this short article includes the in-vestigation of evidences whether horizontal convectiontransports large quantities of heat, as well as sustainslarge amounts of diapycnal mixing with a relativelysmall amount of dissipation. We demonstrate that dif-ferential surface heating on scales of a typical moderncity cascades kinetic energy downscale, which – in theabsence of shear – is sufficient to initiate a turbulentflow. The computational fluid dynamics approach usedin this work is fully detailed by Alam et al. (2014). Thismethod filters the governing equations using a multires-olution approach, and employs a Smagorinsky type eddyviscosity model for the subfilter scale processes. We usea sixth order weighted residual collocation method thathas no inherent numerical dissipation, and thus, there isno need to apply artificial dissipation to ensure numeri-cal stability. Penetrative turbulence occurs when an unstably strati-fied large body of fluid underlies a stably stratified fluidlayer. In this situation, turbulent eddies are driven ver-tically by the buoyancy force, and attempt to penetrateinto the overlying stable fluid (Stull, 1976). In the stableregion, when an eddy reaches its level of buoyancy, itreturns back downward into the unstable region. How-ever, due to the gained momentum, eddies often over-shoot their level of buoyancy, and thus, internal wavesare excited from the interface between the overlying sta-bly stratified fluid and the underling unstable mixinglayer (Stull, 1976; Lane, 2008). These waves transportskinetic energy to the upper atmosphere, and may initiateupper level clear air turbulence. Deardorff et al. (1969) investigated such penetrative turbulence experimentally.Here, we develop a large eddy simulation model to sim-ulate the above mentioned phenomena.
For a compressible atmospheric model, the temperature( T ) and the pressure ( p ) are represented by the poten-tial temperature ( θ ) and the Exner function ( π ′ ) (e.g.,Smolarkiewicz et al., 2014), θ = T (cid:18) p p (cid:19) R d /c p , π ′ = (cid:18) R d p ρθ (cid:19) R d / ( c p − R d ) , respectively, where p is a reference pressure, R d is thegas constant, c p is the specific heat at constant pres-sure, and ρ is the density. The continuity equation isreplaced with Eq (1). The notation ( x , x ) = ( x, z ) and ( u , u ) = ( u, w ) are adopted for simplicity. A spatialfilter is applied to the momentum and energy equations,where u i ( ≡ h u i i ) and θ ( ≡ h θ i ) represent filtered ve-locity and temperature, respectively. More specifically, ˜ u i = u i + u ′ i and ˜ θ = θ + ¯ θ ( z )+ θ + θ ′ (Deardorff, 1970,1980). Note the separation of the reference temperature θ from the background temperature ¯ θ ( z ) , which is con-venient for satisfying the surface condition. The follow-ing equations are solved in the present work; Dπ ′ Dt = − π ′ ∂u i ∂x i , (1) Du i Dt = − c p θ ∂π ′ ∂x i + gθθ δ i − ∂τ ij ∂x j , (2) DθDt = − w ∂ ¯ θ∂z − ∂τ θj ∂x j . (3)The simulation region is a vertical plane ( x, z ) that ex-tends km horizontally ( − ≤ x ≤ ) and kmvertically ( ≤ z ≤ ). A city of scale km exists for − ≤ x ≤ , which is surrounded by rural areas. Re-cent literature indicates that the most appropriate modelfor the near surface penetrative turbulence is not fullyunderstood. As a compromise, as discussed by Pope(2000), we have adopted a resolution that is finer thanthat is used by LES models of the ABL. The finest res-olution uses ∆ x = 97 . m and ∆ z = 3 . m. Thus, asignificant fraction of the energy containing large eddies2s resolved. Here, some advantages of the costly three-dimensional simulation are sacrificed for a high resolu-tion two-dimensional idealization (Lane, 2008). The subgrid scale turbulent stress is estimated bythe popular Smagorinsky (1963) model, τ ij = − C s ∆) | S | S ij , where | S | = p S ij S ij and S ij =(1 / ∂u i ∂x j + ∂u i ∂x j ) . We define the filter width by ∆ = √ ∆ x · ∆ z , and take C s = 0 . for the Smagorinskyconstant.According to Deardorff (1980), the subgrid scale eddycoefficients may be computed by K h = (1 + 2 l/ ∆) K m and K m = 2( C s ∆) | S | , where l ≤ ∆ is a subgrid scalemixing length. The length scale, l , is related to the sub-grid scale turbulence energy e ′ and the buoyancy fre-quency, N = gθ ∂ ¯ θ∂z , i.e. l = 0 . √ e ′ /N . If the scaleof resolved eddies is ∆ = 20 m, then a turbulent Prand-tle number P r = 0 . gives l ≈ m. The SGS flux forbuoyancy is τ θi = − K h ∂θ∂x i (Deardorff, 1980). We consider a time independent profile for the surfaceheat flux variation H sfc ( x ) = h H sfc i + H that is givenby H sfc ( x ) = h H sfc i + A [tanh ξ ( x + λ/ − tanh ξ ( x − λ/ . Here, h H sfc i = 0 . K m s − (about W m − ) repre-sents a domain average heat flux for all x in the rangefrom − to km, where there is no city inducedheat flux H . λ is the characteristic wavelength for thesurface heat flux variation H , where the heating re-gion (city) of the surface is from − to km, i.e. λ = 20 km. This choice for λ represents the resolvedscale for mesoscale numerical weather prediction mod-els. ξ = 100 is a dimensionless number that leads toa continuous sharp interface between the central heat-ing region and other part of the surface. Thus, the sur-face heat flux H sfc ( x ) takes approximately the form ofa square wave without sharp corners, and models the ef-fect of urban-rural heat flux variation. This article sum-marizes the sensitivity of surface heat flux on penetrative turbulence for values of H . A purpose of these simu-lations is to understand the sensitivity of high-amplitudesurface heat-flux heterogeneity for temporal oscillationin mesoscale atmospheric circulations. The total heat flux is composed of turbulent ( w ′ θ ′ ) andviscous ( α ∂θ∂z ) components, where the turbulent compo-nent vanishes on a flat surface. The dimensionless sur-face heat flux takes the form − √ RaP r ∂θ∂z | z =0 , where thepotential temperature is the same the surface tempera-ture.A scale analysis is necessary to compare the presentmodel with that presented by Dubois and Touzani(2009). With a fully developed turbulence in the convec-tive boundary layer, the vertical temperature profile sat-isfies ∂θ∂z = 0 in mixing region ( z > ) (Deardorff et al.,1969). In this case, natural convection heat transfer ischaracterized by the Rayleigh number R a = gH θ νκ H Hα ,where H is a vertical length scale, /θ is the coefficientof thermal expansion (1/K), ν is the kinematic viscos-ity (m /s), κ is thermal conductivity (W/m · K), g is theacceleration due to gravity (m/s ), and α is the thermaldiffusivity (m /s). In an LES, ν and α can be replacedwith K m and K h , respectively. So, we set H Hα = 10 o Kto match the adiabatic lapse rate ( o K/km) of the atmo-sphere so that ∂θ∂z = 0 . As a result, an increase of H bya factor of increases R a by a factor of . Thus, thevalues of H = 57 .
87 Wm − and .
74 Wm − repre-sents R a = 10 and , respectively. The comparisonis summarized in Table 1. In addition, we have com-pared the vertical profile of mean temperature with thatobtained from the Wangara day 33 experiment, whichshows an excellent agreement on ∂θ∂z = 0 between twodata sets. Fig 1 ( a ) demonstrates an example of surface inducedimpact on air pollution. The turbulent plume from thelower chimney ( m tall) moves toward the region of3
50 −40 −30 −20 −10 0 10 20 30 40 5000.511.52 x (km) z ( k m ) ( a ) ( b ) Figure 1: ( a ) Movement of plumes from two chimneys of unequal height as a result of differential surface heating.The figure is adapted from Google image database, and represents a pattern of expected circulation. ( b ) Low levelconverging flow and high level diverging flow is seen from this streamline at t = 6 . h for H = 115 . W m − .warmer surface (city area), and that from the higherchimney ( m tall) moves toward the region of coolersurface (urban, ocean). The atmosphere over the warmersurface has a decreased thermal stability, which causes alow level converging flow. Thus, the air parcels movehorizontally toward the center of the heated region,where they rises upward. The characteristic flow patternin Fig 1 is observed from our numerical simulation. Thecomparison in Fig 1 indicates that our numerical modelsimulates phenomena that is also observed in the nature.The streamline plot in Fig 1 ( b ) shows that air moves tothe left in the lower portion of the boundary layer above x > , and the air in the upper portion moves to theright. This flow pattern is computed at t = 6 . h fromthe simulation with H = 231 . W m − . To understand the mixing and turbulent transport inthe absence of shear or mechanically driven turbulence,we have initialized the flow with a stable stratification H .
87 Wm − .
74 Wm − Present D & T Present D & T θ min -0.0644 -0.0712 -0.1672 -0.1663 u max w max N u Ri = ( gθ ∂θ∂z | z =0 ) H /U controlsthe initial strength of unstable stratification. The flow isinitialized with Ri = − , and allowed to evolve natu-rally with time. Thus, the resulting circulations charac-terize penetrative turbulent convection (Deardorff et al.,1969). The time evolution of the streamlines in the en-tire domain is shown in Fig 2.The relative surface heat flux for − ≤ x ≤ is H = 925 . W m − ( ∼ . K m s − ), where thebackground potential temperature ¯ θ ( z ) in the stable re-gion has a gradient o / km, and the buoyancy fre-quency is N ≈ − s − . As seen from Fig 2, whenthe sun heats the surface, eddies begin to form and riseupward. However, they return back downward due tostable stratification in the upper atmosphere. This gen-erates an unstable mixing layer that is adjacent to thesurface. This turbulent mixed layer underlies a stableregion aloft. Based on the mean vertical profile of po-tential temperature (not shown), the depth of the mixedlayer for this simulation is approximately m.When an eddy loses buoyancy, it rises upward, anda negative horizontal buoyancy gradient occurs near itsleft edge, which generates a cyclonic circulation; sim-ilarly, anticyclonic circulation is formed on the otheredge of the eddy (Lane, 2008; Alam, 2011). The pat-tern of such a circulation and the time evolution of theassociated span-wise vorticity, during a penetrative tur-bulent convection, is realized from the streamlines inFig 2. This vortical pattern play its role as a heat transferagent, thereby causing a imbalance between the buoy-4 a ) t = 1 h ( b ) t = 2 h −40 −30 −20 −10 0 10 20 30 400.511.5 −40 −30 −20 −10 0 10 20 30 4000.511.5 ( c ) t = 4 h ( d ) t = 6 h −40 −30 −20 −10 0 10 20 30 4000.511.5 x (km) −40 −30 −20 −10 0 10 20 30 4000.511.5 x (km) Figure 2: The time evolution of span-wise vorticity is demonstrated using the streamline at t = 1 h, t = 2 h, t = 4 h,and t = 6 h for the simulation with H = 925 . W m − .ancy force and gravitational force. An important ques-tion of meteorological interest is whether the processleads to horizontal turbulent fluctuation, and if such fluc-tuation affects local weather prediction. At t = 1 h,cyclonic/anticyclonic eddies have been formed near theouter edges of the heating region, and have reached aheight of about m. Later, horizontal convectionis observed; i.e. turbulent eddies move horizontally,where turbulent eddies reach a maximum vertical heightof about m. Entrainment/detrainment occurs abovethis height. According to the Taylor’s hypothesis, if turbulent statis-tics is approximately stationary and homogeneous, thenthe turbulent field is advected over the time scales ofinterest. Under this hypothesis, time series of poten-tial temperature and horizontal velocity, as shown inFig 3, indicate that surface heterogeneity on a scale O (20 km) contributes toward downscale energy cas-cade. The growth of the horizontal potential temperaturegradient develops a horizontal pressure gradient, whichin turn generates horizontal wind. The time series of θ and u at several vertical locations have been analyzed to characterize penetrative turbulence.The sensitivity of surface heat flux on the temporalfluctuation of θ is clear from Fig 3. At z = 62 . m, fre-quency of oscillation increased at H = 925 .
92 Wm − compared to H = 462 .
96 Wm − . However, at z = 500 m, turbulence is seen fully developed in bothcases. This indicates that turbulent kinetic energy cas-cades downscale as the energy is transported by internalwaves. To verify that air flow over the simulated cityis characterized by a horizontal flow of numerous rotat-ing eddies, we present the horizontal velocity u , whichis the wind component that is parallel to the direction ofthe surface heat flux variation. The onset of turbulentfluctuations is compared between two surfaces fluxes inFig 4. In this article, we report some aspects of penetrativeturbulence in an idealized daytime boundary layer overa city that is surrounded by rural areas. As internalwaves excite from the interface between the mixed layerand stable layer, kinetic energy of boundary layer tur-bulence is transferred to upper label free atmosphere.A complete understanding of this mechanism remains5 = 462 .
96 Wm − , z = 62 . H = 925 .
92 Wm − , z = 62 . θ (t) (K) at z = 0.0625 km time, t (h) θ ( t ) ( K ) a t z = . k m H = 462 .
96 Wm − , z = 500 m H = 925 .
92 Wm − , z = 500 m time, t (h) θ ( t ) ( K ) a t z = . k m time, t (h) θ ( t ) ( K ) a t z = . k m Figure 3: Time series of potential temperature θ (0 , z, t ) near the surface ( z = 62 . m) and near top of the mixing layer( z = 500 m). 6 = 462 .
96 Wm − H = 925 .
92 Wm − u [ m / s ] u [ m / s ] Figure 4: The onset of turbulent fluctuation is shown through time series of horizontal velocity at z = 100 m from thesurface.a challenging future research topic. However, in thepresent article, we have shown that energy cascadesdownscale as turbulence penetrates upward. Our sim-ulations demonstrate that characteristics of the inducedhorizontal flow may be significantly different dependingon the surface heterogeneity from the perspective of theturbulence that is generated.The simulation is idealized. However, we have stud-ied characteristics of horizontal flow with surface heat-flux variation on the scale of
20 km , which is the typ-ical grid spacing in numerical weather prediction mod-els. One of our objectives is to understand if thermallyinduced mesoscale perturbation has a connection to pen-etrative turbulence. In other words, we show that dif-ferential heating on the same label leads to a sustainedturbulent flow. Note that we have compromised thethree-dimensional simulation with a two-dimensionalone in order to capture the energy containing large ed-dies. These results suggest that surface heterogeneityinduced differential heating causes subgrid scale turbu-lent fluctuations, and the impact of surface heterogene-ity could be more substantial on mesoscale atmosphericflows. In future studies, we plan to continue fully three-dimensional LES of penetrative turbulence. We plan to investigate how temporal oscillations characterize turbu-lent energy cascade, particularly with background windconditions and rough surface.
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