Characterising the shape, size and orientation of cloud-feeding coherent boundary layer structures
Leif Denby, Steven J. Böing, Douglas J. Parker, Andrew N. Ross, Steven M. Tobias
RR E S E A R C H A R T I C L E
Q u a r t e r l y J o u r n a l o f t h e R o y a l M e t e o r o l o g i c a l S o c i e t y
The effect of ambient shear on coherent boundarylayer structures
Leif Denby | Steven J. Böing | Douglas J. Parker |Andrew N. Ross | Steven M. Tobias School of Earth and Environment,University of Leeds School of Mathematics, University ofLeeds
Correspondence
Leif Denby, Institute for Climate andAtmospheric Science, School of Earth andEnvironment, University of Leeds, Leeds,LS2 9JT, United KingdomEmail: [email protected]
Funding information
Met Office/NERC Grant NE/N013840/1
This paper presents two techniques for characterisation ofcloud-feeding coherent boundary layer structures throughanalysis of large-eddy simulations of shallow cumulus clouds,contrasting conditions with and without ambient shear. Thefirst technique is a generalisation of the two-point correla-tion function where the correlation length-scale as well asorientation can be extracted. The second technique consistsof decomposing the boundary layer air into individual co-herent structures and thereafter characterising each objectby size, orientation and moisture flux carried. Using thesetechniques it is found that the majority of vertical moistureflux is carried by plume-like structures with volume scalingwith the height of the boundary layer. This plume-like struc-ture is in apparent contrast with the assumptions of somemodelling systems, that fluxes are carried by thermals, andtherefore the conclusions are significant to parametrisationdevelopment for weather and climate models. The elonga-tion and orientation of boundary layer structures caused bythe introduction of ambient shear is also quantified, demon-strating the general applicability of the techniques for futurestudy of other boundary layer patterns.
Keywords — moist convection, coherent structures, con-vective triggering a r X i v : . [ phy s i c s . a o - ph ] A ug L EIF D ENBY | INTRODUCTION
Coherent boundary layer structures carry perturbations of temperature, moisture and vertical velocity necessaryto trigger convective clouds by overcoming the boundary layer top inversion in a conditionally unstable atmosphere.However, the degree to which the spatial distribution, morphology and perturbations carried by the coherent structuresaffect how clouds form is currently uncertain, as is which external drivers affect these properties of the coherentstructures. To study comprehensively the formation of clouds from coherent boundary layer structures we must first beable to identify and measure the properties of these structures, which is the aim of this paper.Coherent structures in the boundary layer carry so-called non-local (cannot simply be calculated from local scalarvalues) counter-gradient transport in the boundary layer (Deardorff, 1966), providing transport against the verticalmean gradient of moisture and heat (in contrast to smaller turbulent eddies doing down-gradient, diffusive, transport).Owing to the limited resolution available in Global Circulation Models and Numerical Weather Prediction models, it isnecessary to parameterise the unresolved sub-grid processes that provide vertical transport and lead to convectivecloud formation. The development of parametrisations of non-local transport has been key to improving boundary-layerparametrisations (Holtslag and Moeng, 1991; Brown and Grant, 1997). Over the past two decades, the Eddy-DiffusivityMass-Flux (EDMF) approach to boundary-layer parametrisation (e.g. Siebesma et al., 2007) has become popular: in thisapproach, local turbulent transport and transport by coherent structures (leading to the formation of convective clouds)are modelled separately. However, our current limited knowledge of the nature of these coherent structures limits ourability to refine models of the non-local transport, and thus in representing the genesis (formation) of convective cloudsin weather and climate simulations.As well as influencing the formation of individual clouds, these structures capture the convective state of theatmosphere, through their spatial organisation and by persisting sub-grid length-scales of motion over time, in a poorlyunderstood coupling between the sub-cloud and cloud layer. Representation of these sub-grid forms of organisation(a form of convective "memory") are largely absent in contemporary convection parameterisations, however theimportance of convective organisation in affecting, for example, the radiative properties of the atmosphere, and theimpact of limited representation of these processes in models, is becoming increasingly clear (Bony et al., 2017).Prior work has focused primarily on measuring coherence in the boundary layer as a whole, not looking at theproperties of individual coherent structures, but instead producing bulk length-scale estimates using spectral peaksin the autocorrelation and covariance spectrum to measure spatial and angular coherence (in the horizontal plane).Jonker et al. (1999) found in cloud-free Large-Eddy Simulations (LES) that the shortest correlation length-scale exists invertical velocity on the order of the boundary-layer depth, whereas the buoyancy providing field (potential temperature θ for dry LES, virtual potential temperature θ v when water vapour is included) typically attains larger steady-statelength-scales (Jonker et al., 1999; Salesky et al., 2017; de Roode et al., 2004). De Roode et al. (2004) in addition foundthat for passive tracers in dry and stratocumulus topped boundary layers the vertical profile of horizontal length-scalesis determined by the surface to boundary-layer top buoyancy flux ratio r = w (cid:48) θ (cid:48) vT w (cid:48) θ (cid:48) v , with minimum length-scales attainedwhen this ratio was r ≈ − . for dry convection (which is the classical buoyancy flux-ratio scale found for dry convectiveboundary layers) and r ≈ − for the stratocumulus case.In simulations where clouds are present the picture becomes more complicated for the dynamic scalars (Schumannand Moeng, 1991; de Roode et al., 2004), apart from vertical velocity which stays largely unchanged (restricted dynami-cally by the boundary-layer depth). The buoyancy ( θ v ) no longer necessarily reaches a steady-state length-scale profile,and water-vapour and potential temperature produce meso-scale variability (which appear to cancel in their contribu-tions to θ v ). The length-scale characteristics found in LES have been supported through similar analysis of observations(Nicholls and Lemone, 1980; Lenschow and Sun, 2007), with the relative scales of q and θ changing with Bowen ratio, EIF D ENBY producing narrower scales for the scalar dominating buoyancy. Identifying and characterising coherent structuresin observations Schumann and Moeng (1991) found that coherence length-scale typically peaks mid boundary-layer,where the number of structures reaches a minimum. Miao et al. (2006) found the plume spacing and width to be . h BL and . h BL respectively mid boundary-layer.In place of studying correlation in inverse distance (or wave-length) space, the present work studies correlation inreal space using cumulants (see history and review by Lauritzen, 2007). Cumulants have been applied by Lohou et al.(2000) to study thermals in a daytime boundary layer with weak wind shear over land, and characterise the influenceof anisotropy on vertical transport. Schmidt and Schumann (1989) used cumulants to study coherence in the verticalrather than the horizontal plane, and identified both large-scale plumes and transient thermals in a convective boundarylayer. They have also been used to expand the prognostic equations (Ait-Chaalal et al., 2016). In modelling by Lohouet al. (2000) and Schmidt and Schumann (1989) correlation is studied in real space, rather than inverse distance, orwave-length space. In this work we develop this technique further by utilising cumulants and producing vertical profilesof integral length-scale and orientation of coherence in real space.With respect to identifying coherent structures in the boundary layer prior work focused on using limit valueson vertical velocity or water vapour concentration (either separately or in combination) to define object masks (Bergand Stull, 2004; Grant and Brown, 1999; Nicholls and Lemone, 1980; Schumann and Moeng, 1991). More recentlyEfstathiou et al. (2020) developed a masking technique to maximise the vertical transport carried by the selected regionof the boundary layer. Couvreux et al. (2010) noted that object masks based on the physical fields (vertical velocity,water vapour, temperature) had the drawback that they poorly capture transport through the boundary layer topinversion and into the cloud-layer, and proposed a technique based on a surface-released decaying passive tracer totrack the rising boundary layer structures. This tracer technique has been used to identify coherent structures in cloudyboundary layers (Dawe and Austin, 2012; Park et al., 2016; Brient et al., 2019). The current work uses the tracer incombination with the object-splitting technique described in Park et al. (2018) to identify individual structures that arecharacterised by a prominent local maximum of the vertical velocity in the boundary layer.The aim of this paper is to demonstrate the use of new techniques to identify and characterise coherent structuresin the boundary layer and to provide the means to separate out and quantify the non-local transport done by coherentboundary layer structures. The identification method presented here seeks to identify coherent structures with proper-ties necessary to trigger clouds without formulating constraints on the thermodynamic and dynamic properties, andinstead utilise a passive tracer advected with the field. The spatial form and orientation of these coherent structures arequantified so that the characteristic properties of structures which carry the majority of the vertical (cloud-triggering)transport may be identified. Knowing the form and scales of cloud-triggering boundary-layer structures will informdevelopment of new models of convective triggering and provides a framework for studying the transient developmentof scales of organisation.As a means of investigating the extent to which the methods discussed herein are able to unpick and quantifyboundary layer transport and its influence on clouds in different environmental conditions, we will use two simulationsof shallow convection, with and without shear, as a demonstration of large-scale influence on coherent boundary layerstructures and convective clouds. The modelling setup for the simulations are discussed in Sec. 2. The methods usedto identify coherent boundary layer structures and quantify their properties will be discussed in Sec. 3. Finally theapplication of these methods will be shown in Sec. 4 and a discussion of this analysis will be given in Sec. 5. L EIF D ENBY
10 5 0 u [m/s] he i gh t [ m ] v [m/s]
200 220 240 wind direction [deg] wind-speed [m/s] no shear, t=0hrno shear, t=6hr with shear, t=0hrwith shear, t=6hr
F I G U R E 1
Vertical profiles of horizontal mean wind at times t = 0 h and t = 6 h in meridional ( v ) and zonal ( u )directions together with wind direction φ (measured from x -axis) for cases with and without shear. Mean winddirection in sub-cloud layer after t = 6 h is at φ ≈ ◦ in simulation with shear. | MODELLING SETUP
Simulations were carried out with the non-hydrostatic UCLA-LES Large-Eddy Simulations model (Stevens et al., 2005)with two-moment warm-rain microphysics scheme (Stevens and Seifert, 2008) on a km × km × km double-periodicdomain with an isotropic grid-spacing of ( ∆ x , ∆ y , ∆ z ) = ( , , ) m.The simulation setup is based on the Rain in Cumulus over the Ocean (RICO) field study (Rauber et al., 2007) andassociated LES model inter-comparison study (VanZanten et al., 2011). The RICO setup is characterised by shallowcumuli developing from moisture-dominated fluxes from the ocean surface, with the clouds constrained in growthby a prescribed large-scale subsidence aloft and large-scale advection of moisture out of the domain. In the originalinter-comparison study, the simulation settles into a quasi-steady state after a short ( ≈ h) rapid response to the initialcondition, after which convection slowly (over ≈ h) aggregates into larger cloud clusters by precipitation-induced coldpools. As the process leading to formation of these cloud clusters is not the focus of this study, we will be consideringonly the stage of cloud development before these large clusters have developed (here using t = 6 h.In this work there are two key differences to the original RICO setup as published in Seifert et al. (2015). Firstly, inorder to study the effect of ambient wind-shear on the coherent boundary layer structures, two simulations were run,one with and one without shear (see profiles in Fig. 1). In the former the wind-profile from RICO was left unchangedand in the latter the meridional and zonal wind components were set to zero. Secondly, because the near-surfacehorizontal velocity differs between the two simulations, the bulk aerodynamic parameterisation of surface flux wasreplaced with a fixed sensible ( F s = 7 W / m ) and latent heat flux ( F v = 150 W / m ) so that the two conditions have thesame fluxes provided from the surface. The surface flux values were estimated from the original RICO simulation oncenear-equilibrium conditions have been reached (at t ≈ h).As seen in the horizontal cross-sections of vertical velocity in Fig. 2, the presence of ambient shear causes bothboundary-layer structures and clouds to become organised into elongated structures instead of convective cells. Thisis noticeable through the elongated line-like regions of high vertical velocity in simulations with shear, causing theclouds (cloud-base is at z ≈ m) to organise into structures resembling cloud streets instead of (as in the case withoutshear) at the nodes of boundary-layer convective cells. The development of cloud streets under conditions with ambientshear is consistent with prior studies (see reviews by Etling and Brown, 1993; Young et al., 2002), as is the presence ofsheet-like coherent structures attached to the surface extending into the bulk of the boundary layer (as in Khanna andBrasseur, 1998). EIF D ENBY h o r z . d i s t . [ k m ] condition = no shear z t = . condition = with shear10 5 0 5 10horz. dist. [km]10.07.55.02.50.02.55.07.510.0 h o r z . d i s t . [ k m ]
10 5 0 5 10horz. dist. [km] z t = . w [ m / s ] F I G U R E 2
Horizontal cross-sections of vertical velocity through the boundary-layer middle ( z ≈ . m, top) andat cloud-base height ( z ≈ . m, bottom) for simulations with shear (right) and without (left). The presence of ambientshear is seen to clearly break the geometry of convective cells and create elongated coherent boundary structures. | METHODS
Before the properties of coherent boundary layer structures that trigger clouds can be measured, it is necessary todefine exactly what we mean by a coherent structure. In contrast to the cloud layer, where we can define a coherentstructure purely on the concentration of water droplets, in the boundary layer there are at least three scalar fields thatcarry the perturbation that eventually triggers a cloud: moisture, temperature and vertical velocity.We first investigate the length-scales of variability in these fields as a bulk-property of the fluid. If these fields wereto vary over similar length-scales it would be relatively straightforward to define threshold criteria on either of thescalar fields that would apply to all fields. Unfortunately this is not the case (section 3.1) and so we instead develop amethod which identifies air with properties similar to that which triggers convective clouds (section 3.2.1), by trackingair entering newly-formed convective clouds. | Characteristic length-scales - cumulant analysis
As an alternative to moments, cumulants provide a means to summarise the statistical correlation between one or morevariables (Lauritzen, 2007). Similarly to Tobias and Marston (2017), where cumulants were used to identify and measurecoherent structures in 3D rotating Couette flow, we here utilise the second cumulant (two-point correlation function),which for fields ψ and ϕ at height z (here z = z = z in contrast to Tobias and Marston, 2017) is given by c ψϕ ( ξ , ν , z ) =1 L x L y ∫ L x ∫ L y ψ (cid:48) ( x , y , z ) ϕ (cid:48) ( x + ξ , y + ν , z ) dxdy , where ψ (cid:48) and ϕ (cid:48) are deviations from the horizontal mean of ψ and ϕ respectively, and L x and L y are the lengths of thedomain in the x - and y -direction. The positions are wrapped around in the x - and y -direction exploiting the periodic L EIF D ENBY boundary conditions of the simulation.An example of this method applied to the spatial correlation of vertical velocity ( ψ = w ) and water vapour ( ϕ = q v )in the middle of the boundary layer in a simulation with ambient shear is shown in Fig. 3. In cases such as this where anexternal forcing is causing boundary-layer and cloud structures to develop in a preferential direction, the characteristiclength-scale will be longer in this direction. To quantify this asymmetry we identify a principal and perpendicular directionof coherence (measured in terms of the angles θ p and θ ⊥ ) of the central part of the cumulant. This central part ˆ c ψϕ isdefined as the connected region at the origin with the same sign as at the origin of the cumulant c ψϕ . Treating ˆ c ψϕ as a2D mass-distribution, we then estimate the orientation angle as the principal axis (eigenvector with largest eigenvalue)of the moment of inertia tensor: I = (cid:34)∫ ˆ c ψϕ ( ξ , ν ) ν d ξdν ∫ ˆ c ψϕ ( ξ , ν ) ξν d ξdν ∫ ˆ c ψϕ ( ξ , ν ) ξν d ξdν ∫ ˆ c ψϕ ( ξ , ν ) ξ d ξdν (cid:35) . (1)The cumulant can then be sampled in this horizontal plane along the principal and perpendicular directions ofcoherence (as seen in Fig. 3 right) so that the coherence can be quantified in these directions. The presence of ambientshear is for example seen to cause elongation in the direction of the ambient wind (this will be discussed in detail insection 4.1).Once the direction of principal coherence ( θ p ) has been identified, a characteristic length-scale ( L p ) of coherencemay be estimated in this direction and similarly in the perpendicular direction ( L ⊥ ). These length-scales are computedthrough a cumulant-weighted integral of distance ( l = (cid:112) ξ + ν ) from the cumulant origin: L ψ , ϕδ = ∫ ∞∞ l ˆ c δψ , ϕ ( l ) d l ∫ ∞∞ ˆ c δψ , ϕ ( l ) d l , (2)where δ ∈ [ p , ⊥] (for either the principal or perpendicular length-scale), with the cumulant along a particular directiongiven by ˆ c δψ , ϕ ( l ) = ˆ c ψ , ϕ ( ξ = l cos ( θ δ ) , ν = l sin ( θ δ )) (3)evaluated at an arbitrary point using piece-wise linear interpolation.As will be discussed in section 4.1, the presence of ambient shear creates a marked elongation of this coherencelength-scale across all scalar fields, and this direction coincides with ambient wind direction.As well as providing a means to quantify the length-scale of correlation (and thus a single characteristic length-scalefor all coherent boundary layer structures), the shape of the cumulant can provide insight into the dynamical structure ofcoherent structures by quantifying the relative spatial distribution of different scalar fields. This is possible by studyingtwo different aspects of the cumulant produced from two different fields, specifically the offset of the cumulant peakvalue from the origin and the skewness of the distribution around the origin. An offset of the cumulant from the originindicates that the extreme values of two different scalar fields are located spatially offset from each other and wouldsuggest something is driving a separation between two fields. Similarly, skewness in the cumulant distribution indicatesthat the two different scalar fields have differently spatially skewed distributions, e.g one field may appear spatiallyGaussian, but another may be skewed relative to this. The corollary to this is that a 2 nd cumulant, between two differentscalar fields, which is centered on and symmetric around the origin, indicates that these two fields on average are EIF D ENBY y - d i s t a n c e [ k m ] p = 31.2 a) c o v a r i a n c e [ m / s g / k g ]
1e 2 b) p = 31.2= p + 905.02.50.02.55.0 C ( w , q v ) [ m / s g / k g ]
1e 3
F I G U R E 3 a) Cumulant of vertical velocity and water vapour (i.e. the horizontal moisture flux) in horizontal plane at z = 300 m with principal direction of coherence identified by the red dashed line, and b) the same cumulant sampledalong (in blue) and perpendicular (in orange) to the principal axis with characteristic (integral) width indicated withvertical lines. The elongated nature of the coherence as seen on the left is quantified by a significantly larger ( ≈ mvs ≈ m) characteristic length along the direction of shear.spatially distributed identically (in terms of skewness) around their respective peak values and their peak values are, onaverage, co-located. | Object-based analysis
To gain a more comprehensive understanding of transport by coherent structures in the boundary layer, we transitionfrom looking at the boundary layer air in a bulk sense, to studying transport by individual coherent structures that maytrigger clouds. This requires identifying the regions of the boundary layer that contribute to transport into convectiveclouds, splitting these regions into individual coherent structures and finally formulating methods to quantify theproperties of these structures. | Object identification
To quantify the characteristic properties of individual coherent structures carrying out vertical transport, these struc-tures must first be identified. This was achieved by first producing a 3D mask to pick out the part of the atmospherethought to contain coherent structures, and thereafter splitting this mask into individual 3D objects.The 3D mask was produced from the concentration of a passive tracer ( φ ) decaying with a time-scale τ , which wasreleased from the surface (as first used in Couvreux et al., 2010). Specifically the time evolution of the tracer is given by ∂φ∂t = − φτ . (4)The decay time-scale was set to τ = 15 min in this study as this represents the typical overturning time-scale ofboundary-layer eddies in the simulations used (see appendix A).From the scalar φ a mask is created using its standard deviation in a horizontal cross-section ( σ φ ( z ) ) and its localdeviation from the horizontal mean ( φ (cid:48) ( x , y , z ) = φ ( x , y , z ) − φ ( z ) ) by requiring that the local deviation is n standard L EIF D ENBY deviations from the mean, i.e. the mask m ( x , y , z ) is given by m ( x , y , z ) = (cid:40) if φ (cid:48) ( x , y , z ) > n σ φ ( x , y , z ) , otherwi se (5)here n = 2 was used as this was found to produce closest agreement between the properties of air entering clouds andthose identified to belong to coherent structures (see section 4.2). The choice of decay time-scale and limit value for n issimilar to those ( τ = 15 min and n = 2 . ) identified by Chinita et al. (2018) to be optimal when studying shallow moistconvection in the BOMEX (Barbados Oceanographic and Meteorological Experiment) case (Siebesma et al., 2003).The constructed 3D mask was observed to identify boundary layer air with thermodynamic properties similar to airentering into recently formed clouds (see section 4.3 for details), making it a suitable method to distinguish verticaltransport by local diffusive mixing (small eddies) from transport by larger non-local eddies carrying fluxes leading tocloud-formation.Using the method of Park et al. (2018), individual three-dimensional objects were identified from the mask by firstlabelling contiguous regions of the mask as individual proto-objects and further splitting these proto-objects where theywere deemed to be comprised of individual objects, for example as two individual rising thermals which were touching.The sub-division of the proto-objects was done by first creating fragments from each proto-object and merging thesefragments into individual objects. The identification and merging of fragments was done through the use of a secondscalar field ( γ ), here vertical velocity ( γ = w ) with the aim to identify individual rising structures. The fragments werecreated by assigning a unique label to each cell within a proto-object which had a local maximum in γ , and associatingthe remaining proto-object cells to these labels by iteratively copying the labels of neighbouring cells in the direction ofsteepest local gradient in γ . Neighbouring fragments were then merged, if for two fragments the following criterion wassatisfied γ l − γ c γ l − γ m ≤ f , (6)where γ l is the smallest value of the two fragment local maxima, γ c the lowest value along their interface (the "col"), γ m is the minimum value encountered in either of the two fragments and f is a tunable parameter controlling how shallowthe col must be for two fragments to be merged. Here f = 0 . was used as in Park et al. (2018).To measure the thermodynamic properties of air causing the formation of clouds, individual clouds were identifiedfrom 2D column-integrated liquid water path ( m lwp ) with a threshold value of m lwp > . kg / m and tracked by spatialoverlap in consecutive time-steps using the method detailed in Heus and Seifert (2013). This method identifies active clouds as ones with at least one buoyant core (identified from the virtual potential temperature θ v ) and passive clouds asthose without a buoyant core. In addition, this cloud-tracking method splits clouds with multiple buoyant cores intosmaller sub-clouds with the non-buoyant regions defined as outflow . In this work we only consider the properties of airentering single-core active clouds as these are likely to have the strongest and clearest connection to boundary layervariations. To facilitate selecting clouds which recently formed we also keep track of the age of each cloud ( t age ) bystoring the time of appearance for each tracked cloud. | Object characterisation
Once individual 3D coherent structures have been identified, a method is needed to calculate characteristic propertiesof these objects. Here we detail techniques to compute characteristic length-scales and orientation of each coherent
EIF D ENBY structure. Topological measures - Minkowski functionals
Instead of attempting to fit a parameterised shape (for example an ellipsoid) to each object with the intention ofestimating an object’s scale (length, width and thickness), we instead calculate a set of characteristic scales using theso-called
Minkowski functionals (Minkowski, 1903) that measure the topology of arbitrary structures in N -dimensionalspace (see review by Mecke (2000) for details). These have been used in other physical applications to characterisefor example dissipative structures in magnetohydrodynamic turbulence (Zhdankin et al., 2014), galaxy distribution(Schmalzing and Buchert, 1997) and cosmological structure formation (Sahni et al., 1998; Schmalzing et al., 1999). Inthree dimensions the Minkowski functionals are V = V = ∫ dV , (7) V = A ∫ dS , (8) V = H π = − π ∫ (cid:43) · ˆ n dS = 16 π ∫ ( κ + κ ) dS , (9) V = 14 π ∫ ( κ κ ) dS , (10)where ˆ n is the surface normal, and κ , κ are the maximum and minimum local curvature. To evaluate these integralsnumerically on the discrete output from the large-eddy simulations, we use Crofton’s formula (see Appendix B), whichprovides discrete approximations for terms (for example the surface normal) which are otherwise difficult to evaluateon objects constructed from individual cubic volumes of the simulations underlying grid. For reference, the Minkowskifunctionals for a parameterised spheroid and ellipsoid will be shown, using the analytical expressions (where available,and otherwise numerical integration) for surface area and mean curvature given in Schmalzing et al. (1999) and Poelaertet al. (2011).From these functionals a characteristic length ( L m ), width ( W m ) and thickness ( T m ) can be calculated as L m = 3 V V , W m = 2 V πV , T m = V V , (11)where the normalisation is such that all of these measures equal the radius when applied to a sphere. These may befurther summarised by computing the filamentarity ( F m ) and planarity ( P m ) F m = L m − W m L m + W m , P m = W m − T m W m + T m , (12)which in turn indicate whether an object is more stick -like (large filamentarity) or pancake -like (large planarity).The Minkowski functionals thus enable the quantification of an object’s shape, making it possible to, for example,investigate whether the objects that are the primary contributors to vertical transport have a characteristic shape,which is key in deciding how to model this transport.To provide a reference for the range of values which may be expected for coherent structures in the atmosphericboundary layer, Fig. 4 is a filamentarity vs planarity plot for a number of synthetically created, numerically integrated,sheared and stretched spheroids. The centerline length was kept constant across all shapes while varying the width andshearing distance. Each data point is marked by an outline of the shape’s structure by plotting the vertical cross-sectionthrough each shape’s symmetry plane. As a means of reference for the numerically integrated shapes the analytical EIF D ENBY = 20.00 0.05 0.10 0.15 0.20 0.25planarity0.00.10.20.30.40.5 f il a m e n t a r i t y r spheroid F I G U R E 4
Filamentarity vs planarity plot for synthetically created 3D thermal structures with centerline length m and varying shear and distance. For each structure the data point is marked with a vertical cross-section. Forreference the analytical solution for a spheroid with varying major to minor-axis ratio ( λ ) is included in solid bluetogether with reference lines for deforming a spheroid through a general ellipsoid (dotted blue) with fixed major axisand volume while changing aspect ratio between the remaining two axes ( α ).functions for filamentarity and planarity are plotted for a spheroid, by integrating the analytical forms of the Minkowskifunctionals for a spheroid while varying the aspect ratio between one axis and the two others. In addition to the spheroidreference lines the deformation of a spheroid through a general ellipsoid while keeping the volume and major axis lengthconstant is provided, including the aspect ratio of the two remaining axis as α (it can for example be seen that a prolatespheroid with λ = 2 becomes an oblate spheroid with λ = 1 / ). Object tilt and orientation
As the Minkowski functionals (detailed in the previous section) only provide measures of scale, but not orientation, ofindividual objects, we introduce a means of calculating the x y -plane orientation ( φ , measured from the x -axis) and tilt ( θ ,measured from the z -axis) of an individual object (see Figure 5). The characteristic tilt and orientation for an individualcoherent structure is calculated by first forming a centerline through the centroids of vertically adjacent slices of a givenstructure, and then computing the area-weighted angular average of z -tilt and x y -orientation along this centreline (seeappendix C for details). | RESULTS
In this section we analyse the sheared and non-sheared simulations with the methods detailed above. The analysis firstfocuses on extracting characteristic length-scales of different fields in the boundary layer as a whole without attemptingto identify individual coherent structures. Later the boundary layer air likely feeding convective clouds is identified.And lastly, properties of individual coherent structures are studied with the aim of revealing the properties of coherentstructures that dominate the vertical moisture flux.
EIF D ENBY x [ m ] y [ m ]
250 0 250 z [ m ] x [ m ] y [ m ] z [ m ] F I G U R E 5
Voxel-rendering of single coherent structure from sheared simulation from two viewing angles, withobject orientation angles θ (tilt from z -axis) and φ (xy-orientation from x -axis) and centerline (red) indicated. | Vertical profiles of characteristic horizontal scales
To study the vertical transport produced by coherent structures in the boundary layer we must first formulate how todefine these structures. Unfortunately the different scalar fields that are connected with transport relevant to moistconvection (vertical velocity, buoyancy, moisture and heat content) vary on very different length-scales and thesescales change with height in the boundary layer. To demonstrate this variability Figure 6 and 7 show the 2 nd cumulantof vertical velocity with itself ( c w , w , the auto-correlation) and the 2 nd cumulant of vertical velocity and liquid waterpotential temperature ( c w , θ l ) in horizontal plane mid boundary-layer ( z = 300 m).Considering first the auto-correlation of vertical velocity ( c w , w ), we note that vertical velocity features are elongatedwith ambient shear and axisymmetric without shear, as expected. The coherence length-scale (slashed vertical line) islargest in the middle of the boundary layer where vertical velocity peaks before thermals are decelerated becomingnegatively buoyant in the relatively warm and dry layer below cloud.Considering instead c w , θ l , there is narrow length-scale of positive correlation until z ≈ m embedded within alarger-scale negative correlation. The positive correlation is provided by buoyancy in turn induced by sensible surfaceheat fluxes. However given that RICO represents shallow convection over the ocean (making the Bowen ratio small),the buoyancy becomes dominated by water vapour, and above z ≈ m, the correlation with potential temperaturebecomes negative. This transition causes the correlation length-scale to increase with height until z ≈ m, abovewhich the larger-scale negative correlation takes over. In simulations with ambient shear the correlation betweenvertical velocity and temperature is not only elongated in the direction of shear, but the correlation is asymmetric in thedirection of shear. This means that potential temperature features are displaced in the downwind direction relative tothe vertical velocity, suggesting that the similarity solutions that assume radially symmetric distributions of differentscalar fields (i.e. all scalar fields are assumed to be a function of a single radius r ), as most plume-based models do (e.g.Devenish et al., 2010), may not be valid in conditions where shear is present.To examine the variation in correlation length-scale with height, the cumulant-based coherence calculation wascarried out at every model level in the boundary layer. Fig. 8 shows the autocorrelation length-scale as a functionof height for both the sheared and non-sheared simulations. In the discussion below, subscript S and N S will beused to denote properties extracted from the shear and non-sheared simulations respectively. Comparing first thecharacteristic length-scales across different fields the vertical velocity is consistently confined to narrower features,whereas moisture and sensible heat organise on larger scales. The characteristic length-scale of the radioactive tracer
EIF D ENBY y - d i s t a n c e [ k m ] p =39.7z=12.5m no shear 2000 0 2000distance [m]0.02.55.07.5 c o v a r i a n c e [ m / s m / s ] ×10 y - d i s t a n c e [ k m ] p =30.1 with shear 2000 0 2000distance [m]0246 c o v a r i a n c e [ m / s m / s ] ×10 y - d i s t a n c e [ k m ] p =38.9z=262.5m 2000 0 2000distance [m]012 c o v a r i a n c e [ m / s m / s ] ×10 y - d i s t a n c e [ k m ] p =30.3 2000 0 2000distance [m]01 c o v a r i a n c e [ m / s m / s ] ×10 y - d i s t a n c e [ k m ] p =41.4z=512.5m 2000 0 2000distance [m]0.00.51.0 c o v a r i a n c e [ m / s m / s ] ×10 y - d i s t a n c e [ k m ] p =31.0 2000 0 2000distance [m]0.00.51.0 c o v a r i a n c e [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 c w , w [ m / s m / s ] ×10 F I G U R E 6 nd cumulant of vertical velocity with vertical velocity at increasing heights in the boundary layer insimulations without shear (leftmost two columns) and with shear (rightmost two columns). At each height the cumulant(left, measuring coherence as a function of distance) is associated with the same cumulant sampled along the identifiedprinciple and perpendicular direction of coherence (right) with the calculated coherence length-scale indicated withslashed vertical lines. Note the magnitude of coherence changes with height. y - d i s t a n c e [ k m ] p =98.3z=12.5m no shear 2000 0 2000distance [m]05 c o v a r i a n c e [ m / s K ] ×10 y - d i s t a n c e [ k m ] p =30.4 with shear 2000 0 2000distance [m]0246 c o v a r i a n c e [ m / s K ] ×10 y - d i s t a n c e [ k m ] p =11.9z=262.5m 2000 0 2000distance [m]10 c o v a r i a n c e [ m / s K ] ×10 y - d i s t a n c e [ k m ] p =25.3 2000 0 2000distance [m]2101 c o v a r i a n c e [ m / s K ] ×10 y - d i s t a n c e [ k m ] p =7.3z=512.5m 2000 0 2000distance [m]7.55.02.50.0 c o v a r i a n c e [ m / s K ] ×10 y - d i s t a n c e [ k m ] p =31.7 2000 0 2000distance [m]5.02.50.0 c o v a r i a n c e [ m / s K ] ×10 c w , l [ m / s K ] ×10 c w , l [ m / s K ] ×10 c w , l [ m / s K ] ×10 c w , l [ m / s K ] ×10 c w , l [ m / s K ] ×10 c w , l [ m / s K ] ×10 F I G U R E 7 nd cumulant of vertical velocity with liquid water potential temperature (panels as in Figure 6) EIF D ENBY h e i g h t [ m ] c w , w c q v , q v c l , l c q c , q c c l , v , l , v c , no shear t=6hr perpendicularno shear t=6hr principle with shear t=6hr perpendicularwith shear t=6hr principle F I G U R E 8
Horizontal coherence length-scales of vertical velocity ( w ), water vapour ( q v ), liquid water potentialtemperature ( θ l ), cloud water ( q c ), virtual liquid water potential temperature ( θ l , v ) and the radiative passive tracer ( φ ),for simulations with shear (solid lines) and without shear (dashed lines).field is generally larger than that for vertical velocity, possibly owing to the tracer concentration retaining high values inover-turning vortices of thermals within which the vertical velocity becomes negative.In both simulations, with and without shear, the vertical velocity length-scale increases with height from thesurface as thermals are accelerated by the buoyancy provided through surface fluxes, reaching a maximum scale of L pw , w (cid:12)(cid:12) N S ≈ m and L pw , w (cid:12)(cid:12) S ≈ m at z ≈ m. Above this height, velocity scale stagnates (no shear) or decreases(with shear) with height, as the buoyancy of the rising coherent structures becomes negative and the structures begindecelerating through the CIN-layer. This effect of buoyancy on the correlation length-scale is more clearly seen wheninvestigating the cross-correlations with vertical velocity (see below). Comparing the moisture and virtual liquidpotential temperature (buoyancy) length-scales with the radioactive tracer, it is notable that although the buoyancyis water-vapour dominated, the length-scales of correlation of water vapour ( L pq v , q v (cid:12)(cid:12) N S ≈ m and L pq v , q v (cid:12)(cid:12) S ≈ mat z ≈ m) are significantly larger than those of both the radioactive tracer ( L pφ , φ (cid:12)(cid:12) N S ≈ m and L pφ , φ (cid:12)(cid:12) S ≈ m at z ≈ m) and than the buoyancy features ( L pθ l , v , θ l , v (cid:12)(cid:12) N S ≈ m and L pθ l , v , θ l , v (cid:12)(cid:12) S ≈ m at z ≈ m) in the lower-halfof the boundary layer (until z ≈ m). This means that larger-scale water vapour variability is producing buoyancy ona shorter length-scale, which in turn is accelerating boundary layer air on an even shorter length-scale. This is importantfor modelling, as the variability of the buoyancy scalar (here water vapour) cannot simply be used to infer the scale ofcoherent rising structures. Lastly, the maximum correlation length-scale of cloud water (a measure of cloud-size) ison the order of L pq c , q c (cid:12)(cid:12) N S ≈ m and L pq c , q c (cid:12)(cid:12) S ≈ m at z ≈ m (near cloud-base), showing most similarity to theradioactive tracer coherence length-scale at cloud-base.The direction of longest correlation distance (see Fig. 9) is for all fields (including cloud water) oriented withthe ambient wind-direction, which is also the principle direction of shear, throughout the boundary-layer and intocloud-base. In the absence of shear, some fields demonstrate some degree of turning with height, but examining multipletimesteps this was seen to be a transient feature, so that no preferential orientation can be discerned.Applying the same analysis to the vertical fluxes of heat, moisture, buoyancy and radioactive tracer (Fig. 10) wesee that these features are generally narrower than the scalar being transported, suggesting that the scale of verticalvelocity is dominating the length-scales of vertical transport. All flux fields show clear elongation by ambient shear. Theheights at which the correlation is negative is marked by minus-sign markers ("-") showing again the transitions for heat(at z ≈ m the heat flux changes sign) and buoyancy (at z ≈ m rising structures are no longer buoyant) wherethe correlation-length for both collapses. The moisture and radioactive tracer flux correlations ( c w , q v and c w , φ ) havenear-monotonic increases in size with height until cloud-base is approached. EIF D ENBY
50 25 0 25principle axis [deg]0100200300400500600700 h e i g h t [ m ] c w , w
50 25 0 25principle axis [deg] c q v , q v
50 25 0 25principle axis [deg] c l , l
50 25 0 25principle axis [deg] c q c , q c
50 25 0 25principle axis [deg] c l , v , l , v
50 25 0 25principle axis [deg] c , no shear t=6hr principle orientation with shear t=6hr principle orientation F I G U R E 9
Direction of longest coherence as angle from x-axis (for the variables shown in Fig. 8). In the presence ofambient shear all fields are elongated in the direction of shear. In particular, note that cloud structures are elongated inthe same direction as boundary layer structures. h e i g h t [ m ] c w , q v c w , l c w , l , v c w , q c c w , no shear t=6hr perpendicularno shear t=6hr principle with shear t=6hr perpendicularwith shear t=6hr principle F I G U R E 1 0
Horizontal coherence length-scales, in simulations with (orange lines) and without (blue lines) shear, ofvertical velocity with water vapour, liquid water potential temperature, buoyancy (virtual liquid water potentialtemperature), cloud condensate and radioactive tracer effectively producing a coherence length-scale for the verticalflux for each scalar field. Heights at which anti-correlation occurs are marked with a minus-sign ("-").
EIF D ENBY L i q u i d W a t e r P o t e n t i a l t e m p e r a t u r e [ K ] no shear L i q u i d W a t e r P o t e n t i a l t e m p e r a t u r e [ K ] no shear masked byradioactive tracer-based envelope ( > 2.0 ( )) z=12.5mz=112.5mz=212.5mz=312.5mz=412.5mz=512.5mz=612.5minto cloudbase(tracked clouds)14.5 15.0 15.5 16.0Total water mixing ratio[g/kg]297.7297.8297.9298.0 L i q u i d W a t e r P o t e n t i a l t e m p e r a t u r e [ K ] with shear L i q u i d W a t e r P o t e n t i a l t e m p e r a t u r e [ K ] with shear masked byradioactive tracer-based envelope ( > 2.0 ( )) F I G U R E 1 1
Joint distributions of total water vapour specific humidity and liquid potential temperature (same aspotential temperature in the sub-saturated boundary layer below cloud) in horizontal cross-sections at increasingheights in the boundary layer together with properties of air entering through cloud-base of newly formed( t age < min) clouds in simulation with (bottom) and without (top) shear, both without (left) and with (right) theradioactive tracer mask applied. For each height and cloud-base properties the inner and outermost contours identifythe regions which cumulatively contain the % and % highest point density, with the number of bins in θ l and q t scaling by the number of points, N bins = ( N x × N y ) / . From a relatively moist and warm near-surface state thedistribution first dries and cools into the bulk of the boundary layer, then warms and dries with height as air subsidingthrough the boundary layer top imparts a stronger influence. | Cross-correlation of scalar fields
As shown above, different scalar fields show different length-scales of coherence and so we now turn to investigatingthe extent to which these fields are correlated, not in space but in the distributions of their scalar values. In Fig. 11 forthe simulations with (right) and without shear (left) the joint distribution of water vapour and potential temperaturethrough horizontal cross-sections at increasing heights are plotted. At each height a bivariate histogram was created,the bins ranked by number of points and contours drawn around the bins with 10% (inner contour, solid lines) and 90%(outer contour, stippled lines) cumulative point count. Constructed in this way the set of contours at each height identifythe centre and width of the joint distribution for the scalar fields visualised (forming a 2D box-and-whisker plot). Forreference, the boundary layer mean values for potential temperature and moisture are included. In addition to thedistributions in horizontal cross-sections, the joint distribution for points immediately below (grid-spacing ∆ z = 25 m)cloud of newly-formed ( t age < min) clouds (identified by cloud-tracking) is included.The distributions have similar characteristics in the two simulations; near the surface the boundary layer is warmand moist relative to the boundary layer mean, through the super-adiabatic near-surface the distribution rapidly EIF D ENBY becomes cooler and drier with height (until z ≈ m). With further increase in height the distribution is stretchedto drier and warmer values as mixing with warm dry air subsiding through the boundary layer top has an increasinginfluence.In both simulations, the distribution of air entering newly formed clouds (in red) coincides with the coldest and mosthumid part of the boundary layer joint distribution. The cloud-base distribution contains a larger range of moisturevalues in the simulation with shear, including drier parts of the bulk of the boundary layer distribution. This may bebecause ambient shear increases mixing into the rising coherent structures, carrying drier air in the regions with highvertical momentum or because the lower overall virtual potential temperature (buoyancy) of the boundary layer causesmore air to be buoyant enough to rise to the level of condensation.In addition, the distributions in the two simulations differ through a translation in potential temperature by approxi-mately ≈ . K, which although small is on the order of the width of the distributions in both cases. This suggests that inthe presence of shear the surface fluxes and subsiding air are less effective at heating the bulk of the boundary layer.This offset indicates that even in conditions where the prescribed surface fluxes are exactly the same, the dynamics ofboundary layer transport can vary to a degree that alters the property of air that forms clouds, and so thresholds basedsolely on the thermodynamic fields (e.g. θ l and q t ) are inadequate in identifying air that will form clouds.It is clear from these joint distributions that if the aim is to characterise coherent structures which actually causethe formation of clouds, it is inadequate to simply construct a conditional sampling based on threshold values of thescalar fields causing transport. Instead a method that tracks air transported from the surface layer is required and tothis end we in the next section employ a radioactively decaying tracer. a l t i t u d e [ m ] nu m o b j e c t s [ ] . . . horz. mean flux [m/s g/kg]0.0000.002 h o r z . m e a n f l u x [ m / s g / k g ] no shear (4055 objects) a l t i t u d e [ m ] nu m o b j e c t s [ ] . . . horz. mean flux [m/s g/kg] samplingfull domainall objects r equiv >200 m and r equiv <400 m objects0.0000.002 h o r z . m e a n f l u x [ m / s g / k g ] with shear (9910 objects) F I G U R E 1 2
Vertical moisture flux decomposed by height and object equivalent radius, with size distribution ofobjects (top) and moisture flux profile (right). In the moisture flux profile the total domain mean (orange) is showntogether with the moisture flux contributed by the selected objects (blue). The flux is carried, in aggregate, primarily byobjects r equiv ≈ m (the mean moisture-flux profile for objects in this range, m > r equiv > m, is given ingreen) and not by the order of magnitude more numerous smaller objects or relatively few larger objects. EIF D ENBY | Identifying cloud-feeding coherent structures
We now turn to characterising the coherent structures that have the potential to trigger clouds. The joint distributionsof the previous section are conditionally sampled (Fig. 11 right) by requiring that the concentration of decaying passivetracer is at least 2 standard deviations from the horizontal mean (see Sec. 3.2.1 for details).Using this method for both the sheared and non-sheared simulations, distributions collapse down to align near-perfectly with that of air entering through cloud-base, indicating that the radioactive tracer is picking exactly the airthat may trigger clouds. In addition, for both cases the means of the distributions appear to be near-linearly translatedwith height suggesting that these coherent cloud-triggering structures mix with the bulk of the boundary layer at asimilar rate for both potential temperature and water vapour. Finally, the widths of the joint distributions appear nearlyunchanged with height. All of these facts are encouraging for the prospect of parameterisation of the mixing intocoherent updrafts in the boundary layer, by which the properties of cloud-triggering air may be predicted based on thesurface fluxes (and other external forcing factors).Although the radioactive tracer method identifies air with the same statistical properties as air that is triggeringclouds, it is not guaranteed that every volume of this boundary layer air will actually trigger a cloud. Some structuresmay simply be too small to survive the journey to the condensation level without being completely mixed into the bulkboundary layer air. This can be addressed in further work by tracking boundary layer structures and identifying whichones actually trigger clouds (beyond the scope of this study). Here, we instead identify the structures that dominate thevertical moisture transport. | Height and object-scale decomposition of vertical flux
Key to understanding the formation of clouds is to learn which coherent structures in the boundary layer collectivelytransport most of the vertical moisture flux. At either extreme one might believe that either the very few largest struc-tures dominate, but equally the many orders of magnitude more numerous smallest structures could also collectivelydominate the vertical flux. In Figure 12 the moisture flux is decomposed by object size (here equivalent spherical radius, r equiv = (cid:112) V / π ) and as a function of height, with the number of objects given along the top and total moisture fluxprofile along the right-hand side (total moisture flux in blue and flux carried by coherent structures denoted by orangemarkers). Note that an individual object of a given size will likely contribute to the flux at a number of different heights,and the figure simply shows at any given height which scale of objects contribute to the total moisture flux.Examining first the horizontal mean flux, it is clear that approximately half of the vertical moisture flux is carried bythe coherent structures which have been identified (the non-local transport), with a near-constant flux-profile withheight, lending more credibility to the use of the radioactive tracer as a means to identify coherent structures in theboundary layer. The moisture flux is near-constant with height above z ≈ m, below which it appears to drop to zeronear the surface. The moisture flux is in fact non-zero below this height, but is not carried by the resolved flow andis instead parameterised in the LES simulation as a sub-grid flux, providing the surface moisture source of the oceansurface.By looking at the flux decomposition it can be seen that, above the surface-layer, objects of equivalent radius in therange r equiv ∈ [ m , m ] dominate the moisture flux, carrying, in aggregate, ≈ % and ≈ % (in the simulationswithout and with shear respectively) of the vertical moisture flux between z = 100 m and z = 500 m. And so althoughthere are only on the order of hundreds of objects of this size, they transport the majority of the vertical moisture flux. EIF D ENBY o b j e c t d e n s i t y [ / m ] F I G U R E 1 3
Distribution of Minkowski length, width and thickness for individual coherent structures identified insimulations with and without shear. Shear is seen to cause objects to become longer and thinner on average. | Minkowski characteristics of coherent structures
Having identified the coherent structures with correct thermodynamic properties (for triggering clouds) and carryingthe majority of the vertical moisture, we next calculate characteristic properties of each structure and attempt toquantify the difference in these structures with and without the presence of ambient shear. To do this we calculate acharacteristic length, width and thickness for each object using the so-called Minkowski functionals (detailed in Sec.3.2.2).The distributions of these (Figure 13) show that, both with and without shear, the moisture flux transport is carriedprimarily by structures which are long and thin, and that these structures become longer and thinner in the presence ofshear. In both simulations some objects have a Minkowski length longer than the depth of the boundary layer depthbecause they extend in both the horizontal and vertical direction.This shift in the distribution can be succinctly captured by computing the filamentarity and planarity (measuring how pencil -like or disc -like object each is) as seen in Figure 14, showing that ambient shear causes the coherent structures tobecome more elongated and planar at the same time, which can be seen by a ≈ % increase in filamentarity ( F N S ≈ . and F S ≈ . ) and ≈ % increase in planarity ( P N S ≈ . and P S = 0 . ). As a reference (in black), the filamentarity andplanarity of an ellipsoid with varying elongation (parameterised as the aspect ratio between one axis and the remainingtwo) the coherent structures can be seen to move from being more cylindrical to sheet-like in shape. | Object orientation
In addition to knowing the characteristic length-scales and shape of individual coherent structures it is instructive todetermine the tilt and orientation of each object to be able to formulate an integral model to represent transport bycoherent structures. In Figure 15, the tilt and orientation angles for all objects present at t = 6 h in both the simulationwith and without shear are plotted. This shows the presence of ambient shear ( ≈ m s − change in wind-speed over thesub-cloud layer) caused the mean tilt of individual objects to change from θ no , shear ≈ ◦ (near-vertical, given the nearsymmetrical angular spread in orientation angle) to θ shear ≈ ◦ and changed the structures from having no preferentialhorizontal orientation, to being oriented with direction of wind shear ( φ shear ≈ ◦ ) . The principal orientation directionin the presence of shear was found to agree with the direction of coherence elongation found using cumulants. Thissuggests that individual coherent structures tilt in the same direction that the structures spatially align along (as thecumulant-based technique cannot separate elongation of individual structures from spatial organisation of individualstructures). EIF D ENBY =2 =3 =40.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45planarity [1]0.00.20.40.60.8 f il a m e n t a r i t y [ ] r no shear: 424 objects with shear: 285 objects F I G U R E 1 4
Filamentarity vs planarity for coherent structures dominating the vertical moisture flux( r equiv > m and r equiv < m) in simulations with (red) and without shear (blue), together with spheroid reference(solid black) with independent axis parameterised by λ and deformation of a spheroid through a general ellipsoid withfixed major axis and varying aspect between the remaining two axes ( α ). xy - p l a n e a n g l e [ d e g ] no shear: 424 objects with shear: 285 objects F I G U R E 1 5 xy-orientation angle ( φ ) against z-axis slope angle ( θ ) for objects dominating the vertical moisture flux( r equiv > m and r equiv < m) present at t = 6 h in simulations with (red) and without shear (blue), together withdistributions in each along plot margins EIF D ENBY w q v [W/m^2]0100200300400500600 z [ m ] w q v [W/m^2]no shear0 500 1000 w q v [W/m^2]0100200300400500600 z [ m ] w q v [W/m^2]with shear samplingfull domainmaskobjects: z min < 100 m objects: z min > 100 m objects: z min < 100 m and z max > 500 m objects: r equiv > 200 m and r equiv < 400 m F I G U R E 1 6
Vertical profiles of mean moisture flux, area fraction and mean total moisture flux conditionallysampled by passive tracer mask, objects attached and detached from surface and entire domain for simulation with(bottom) and without (top) shear at t = 6 h. | Plume vs thermal - vertical extent vs height
Having decomposed the coherent transport in the boundary layer into contributions by individual coherent structuresthe question of whether these structures are more thermal- or plume-like may be investigated by considering thevertical extent of each structure. As plumes must be continuously fed from below, these objects must extend the fulllength from the surface layer to the cloud-base to constitute plume-like objects.Figure 16 explores this (for the simulations with and without shear respectively) by plotting the in-object meanmoisture flux profile, area-fraction and contribution to the domain mean moisture flux. Together with the domain-wideand passive tracer mask mean, the mean moisture flux contributed by objects attached ( z min < m) and detached( z min > m) from the surface are plotted, as well as the mean moisture flux for objects which extend the full height ofthe boundary layer ( z min < m and z max > m). Comparing first the profiles for objects attached and detachedfrom the surface it is clear that very little moisture flux is actually carried by thermal-like coherent structures detachedfrom the surface. Further, the plume-like coherent structures that are attached to the surface and extend to cloud basecarry > % of the moisture flux carried by coherent structures, with only a small fraction carried by shorter structuresbelow the middle of the boundary layer. The contribution by different coherent structures to the total moisture fluxcan largely be explained by change in fractional area covered by different groups of coherent structures, whereas thechange in mean in-object moisture flux is nearly unchanged between the different groups of structures.In both simulations the horizontal area-fraction covered by all identified coherent structures decreases with heightthrough the boundary layer (from σ (cid:12)(cid:12) S ( z ≈ m ) ≈ % to σ (cid:12)(cid:12) S ( z ≈ m ) ≈ %) as does the mean in-structure moisture EIF D ENBY flux ( w (cid:48) q (cid:48) v (cid:12)(cid:12) S ( z ≈ m ) ≈ W m − to w (cid:48) q (cid:48) v (cid:12)(cid:12) S ( z ≈ m ) ≈ W m − ), although for the case without shear thesedecreases are less pronounced. These decreases together lead to a decrease in the contribution to domain-meanmoisture flux contribution from coherent structures with height. The small area-fraction of coherent structures here(as compared to work by Efstathiou et al., 2020) is likely due to using a standard deviation scaling ( n = 2 here) in linewith work by Chinita et al. (2018).When considering only the the plume-like coherent structures the area-fraction covered increases with height (from σ (cid:12)(cid:12) S ( z ≈ m ) ≈ % to σ (cid:12)(cid:12) S ( z ≈ m ) ≈ %) as these structures entrain boundary layer air, but this is compensatedby a decrease with height in mean in-structure moisture flux (as the coherent structures dry through mixing with theboundary layer air) to give a near-constant moisture flux profile with height.For comparison, the mean moisture flux for the flux-dominating coherent structures ( r equiv > m and r equiv < m) is given, showing the same near-constant profile and accounting for ≈ / of the flux carried by plume-likecoherent structures. | DISCUSSION
The pre-cloud coherent boundary layer structures (identified using a decaying passive tracer) which in aggregatecarry a large fraction of the vertical moisture flux ( ≈ % and ≈ % of the moisture flux between z = 100 m and z = 500 m in the simulations without and with shear respectively), have volumes equivalent to a spherical radius of m < r equiv < m. In addition, the structures that dominate the vertical moisture flux extend from the surface tothe cloud layer (in both simulations with and without shear), suggesting they are more plume-like than thermal-likein morphology. The presence of ambient shear is seen to tilt the coherent objects from being nearly vertical to beingoriented at ◦ from the z-axis in the direction of the ambient wind. The presence of shear also causes the structures tobecome elongated and more planar, by ≈ % and ≈ % respectively. These findings suggest that models representingnon-local transport by coherent boundary layer structures, should chose the plume model as the fundamental startingpoint and may find that boundary layer transport can be adequately represented by a plume with characteristic widthon the scale of the boundary layer depth. The effect of ambient shear appears to be to organise vertical transportinto coherent structures which are elongated (which would increase entrainment by increasing the surface area) andstretched planar, and at the same time organised into linear features (which may decrease entrainment by limitingthe dry air reaching invidiual plumes). Understanding the competing effects of shear in future work and allowing fornon-radially symmetric plumes is key to allowing for better representation in parameterisations.The moisture flux profiles, when considering all coherent structures identified, decrease with height in bothsimulations, but to a larger extent in the simulation with shear ( ≈ W m − vs ≈ W m − ), which arises by a markeddecrease in area fraction in both simulations and in-object mean flux in the case with ambient shear. However, whenonly considering the objects that dominate the moisture flux transport ( m < r equiv < m) their area-fractionincreases with height, compensating for the in-structure drop in mean moisture-flux, resulting in a almost identicalmoisture flux profile for the two simulations when only considering these flux-dominating structures. This suggests thatalthough the spatial organisation of coherent boundary layer structures is different in the simulations with and withoutshear, and the in-structure flux and area fraction is affected by shear, the total flux contribution by coherent structuresappears relatively unchanged under the introduction of shear.The two methods presented here, the first focusing on bulk-measures of coherence in the boundary layer (usingcumulants) and the second on identifying and quantifying properties of individual coherent structures (using Minkowskifunctionals, tilt/orientation calculation and flux decomposition), have complementary strengths. EIF D ENBY
The principal difference between the two approaches is that the cumulant-based method produces a length-scaleestimate through considering the spatial coherence throughout the fluid, whereas identifying and characterisingindividual objects gives an estimate of scale for individual coherent structures. Two individual objects in close proximitywill increase the cumulant length-scale estimate, meaning that the cumulant length-scale conflates the object size andobject organisation (specifically the inter-object distance). In addition, in cases where there are multiple populations ofcoherent structures, the cumulant method will conflate these into one integrated measure, and the relevant details ofeach separate population may be lost. In particular, in the upper part of the boundary layer which is characterised byascending buoyant thermals and descending entrainment flows, the cumulant analysis will not necessarily give a "clean"description of either of these.Another aspect by which the two approaches differ is how the cumulant method estimates length-scales in thehorizontal plane (the method requires translational symmetry to study coherence as a function of displacement and socannot be applied vertically without picking a reference height), whereas the length-scales of individual objects are notconstrained in the orientation in which these length-scales are calculated. This firstly means that the two measuresof length cannot be directly compared, as the Minkowski length-scales are not measured in the horizontal plane andnecessitate the calculation of object orientation to interpret the length-scales calculated.Finally, the cumulant-based method does produce a bulk estimate of the orientation of coherent structures (orthe direction of their relative positioning), but in the horizontal plane and so this estimate lacks information aboutthe tilt of individual coherent structures. The Minkowski functionals do not provide measures of direction, only scale,another reason why the use of Minkowski functionals was supplemented by a separate method to estimate the tilt andorientation of individual objects. | CONCLUSIONS AND FURTHER WORK
This paper has demonstrated two methods by which to measure cloud-feeding coherent structures in the atmosphericboundary layer. The first method quantifies the horizontal orientation and length-scale of coherence between any twoscalar fields, and through this makes it possible to measure the coherence in the boundary layer as a whole. The secondmethod identifies cloud-feeding coherent structures using a decaying passive tracer and is able to quantify length-scalesand orientation (both vertical and horizontal) for each of these objects, allowing for a more instructive decompositionwhere the non-local transport by individual coherent structures can be studied.We have demonstrated the use of cumulants to measure characteristic length-scales for different scalar fieldsand fluxes of these fields in the bulk of the boundary layer. This showed that vertical velocity features are significantlynarrower ( ≈ m) than the moisture and potential temperature fields ( ≈ m). This method is also able to quantifythe elongation of these coherent structures and calculate the elongation direction, showing how the presence ofambient wind shear causes elongation of the vertical velocity and a less-pronounced change to the heat and moisturefields.We additionally demonstrated how a surface-released radioactive tracer may be used to identify air with the samestatistical properties as air entering through cloud-base. This allowed for the development of a method to identifyindividual cloud-feeding coherent structures, which were then characterised using Minkowski functionals (producing acharacteristic length, width and thickness for each object) and a technique for calculating an object’s tilt and horizontalorientation. With these methods it was shown that the coherent structures which dominate the vertical moisture fluxare plume-like in nature (being attached to the surface and reaching the cloud-layer), and that the presence of ambientshear causes significant elongation of these structures while causing them to tilt and orient in the direction of ambient EIF D ENBY A | TIME-SCALE OF CONVECTIVE OVERTURNING IN BOUNDARY LAYER
Using the sub-cloud characteristic velocity scale w ∗ (as in Holtslag and Nieuwstadt, 1986, but corrected for the con-tribution to buoyancy from water vapour) and the boundary layer depth z BL we can calculate a sub-cloud convectiveoverturning time-scale τ BL as τ BL = z BL w ∗ (13)The sub-cloud convective velocity scale is given as w ∗ = (cid:16) w (cid:48) b (cid:48) (cid:12)(cid:12)(cid:12) z BL (cid:17) / , (14)with buoyancy flux w (cid:48) b (cid:48) (cid:12)(cid:12)(cid:12) = gT v , w (cid:48) θ (cid:48) v (cid:12)(cid:12)(cid:12) , (15)and virtual potential temperature flux (approximately) given by (de Roode et al., 2004) w (cid:48) θ (cid:48) v (cid:12)(cid:12)(cid:12) = w (cid:48) θ (cid:48) (cid:12)(cid:12)(cid:12) + R v R d θ w (cid:48) q (cid:48) (cid:12)(cid:12)(cid:12) , (16)where z BL is the boundary-layer depth, g the gravitational acceleration and | denotes surface values. With surfacefluxes for sensible w (cid:48) θ (cid:48) (cid:12)(cid:12)(cid:12) = F s ρ c p , d and latent heat θ w (cid:48) q (cid:48) (cid:12)(cid:12)(cid:12) = F v ρ L v , surface moisture q v , = 15 g kg − , surface tempera-ture T = 300 K and diagnosed boundary-layer depth z BL = 650 m (here taken as the cloud-base height) the convectiveoverturning time-scale becomes τ BL = 16 min. The constants used above are those for density of dry air ρ = 1 . kg m − ,latent heat of vaporisation L v = 2 . × J kg − , specific heat capacity of dry air c p , d = 1005 J kg − K − , gas constantsfor dry air R d = 287287 J kg − K − and water vapour R v = 461 J kg − K − . EIF D ENBY B | CROFTON’S FORMULA FOR DISCRETE INTEGRALS
Numerically evaluating the integrals of the Minkowski functionals in 3D (equations 7 to 10) is non-trivial on discrete 3Dmasks as these structures are necessarily blocky and so constructing, for example, the surface normal at a corner ispoorly defined.Instead of approximating the surface normals, the Minkowski integrals can be approximated discretely usingCrofton’s formula. This amounts to counting the number of vertices ( N ), edges ( N ), faces ( N ) and cells ( N ) on boththe interior and exterior of a given object mask. With these the Minkowski functionals in 3D are given as V = N , V = 2 N − N x , V = 2 N − N + 3 N x , V = N − N + N − N ∆ x . It can be shown for shapes where analytical forms for the Minkowski functionals exist that these approximatedefinitions converge to the true values when ∆ x → ∞ . Note that the above approximations assume the underlying gridto have isotropic grid spacing ( ∆ x = ∆ y = ∆ z ). C | CALCULATION OF SLOPE AND ORIENTATION OF INDIVIDUAL OBJECTS
The x y -orientation angle ( φ measured from the x -axis) and tilt angle θ (measured from the z -axis) are calculated fromcharacteristic slope scales ∆ x , ∆ y , ∆ z : φ = arctan ( ∆ y , ∆ x ) , θ = arctan ( ∆ l xy , ∆ z ) , ∆ l xy = (cid:113) ∆ x + ∆ y , which are evaluated as area-weighted changes in the centroid position ∆ x = ∆ zV (cid:213) k A ( z k ) x c ( z k +1 ) − x c ( z k − ) , ∆ y = ∆ zV (cid:213) k A ( z k ) y c ( z k +1 ) − y c ( z k − ) , ∆ z = ∆ zV (cid:213) k A ( z k ) z c ( z k +1 ) − z c ( z k − ) , where x c ( k ) , y c ( k ) and A ( k ) are the centroid x and y position, and area at height-index k respectively, and V the volume,given by A ( z k ) = (cid:213) i , j m ( x i , y j , z k ) ∆ x ∆ y , V = (cid:213) i , j , k m ( x i , y j , z k ) ∆ x ∆ y ∆ zx c ( z k ) = (cid:205) i , j x i m ( x i , y j , z k ) (cid:205) i , j m ( x i , y j , z k ) , y c ( z k ) = (cid:205) i , j y i m ( x i , y j , z k ) (cid:205) i , j m ( x i , y j , z k ) for an individual object defined by the mask m and grid-spacing ∆ x , ∆ y , ∆ z . EIF D ENBY R E F E R E N C E S
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