Towards a evaluation of a tensor eddy-diffusivity model for the terra incognita grey gray zone Omduth Coceal 1, Mary-Jane Bopape 2, Robert S. Plant 2 1 National Centre for Atmospheric Science (NCAS), University of Reading 2 University of Reading a work-in-progress

Motivation: sub-filter models in the grey zone Coarse-grid LES (~100m) and high-resolution NWP (<~1km) are both characterised by poorly-resolved turbulence Usual sub-grid schemes are no longer appropriate Sub-grid model needs to do more than in well-resolved LES So need to re-examine appropriateness of assumptions Wyngaard (2004) and Hatlee & Wyngaard (2007) suggested an approach based on conservation equations for subfilter scale (SFS) fluxes and stresses Proposed to include extra SFS production terms – their inclusion leads to a generalised tensor eddy-diffusivity formulation We illustrate the approach and resulting model for SFS scalar flux Here we focus on the “coarse LES” end, with aim to meet NWP

SFS scalar flux for a conserved scalar Apply spatial filter to flow variables and equations: Simplified evolution equation/model for SFS flux (Wyngaard, 2004): SFS flux SFS stressSFS turbulence timescale Tilting production Gradient production Pressure destruction

A tensor eddy-diffusivity model Neglect the lhs terms and one or more of the rhs terms: Scalar eddy-diffusivity (Lilly-Smagorinsky) Tensor eddy-diffusivity (stress-gradient) Tensor eddy-diffusivity (tilting-stress-gradient)

Questions How important are the extra SFS production terms? What is the effect of including these extra terms on the SFS fluxes? What is behaviour in grey zone regime?

Method Follow procedure of a priori testing of SFS models Compute SFS scalar flux directly Compare with model-computed values Use instantaneous 3D data within whole BL Need high resolution data close to truth

LES of a convective boundary layer Met Office Large Eddy Model (LEM) Convective BL with heat flux of 30 Wm -2, z 0 = m, z H = m Smagorinsky SGS scheme Domain size of 5km x 5km x 2km 512 x 512 x 512 grid points Δx = Δy = 10m, Δz = 4m

Vertical velocity in x-z plane

z = 20 m

z = 80 m

z = 200 m

z = 800 m

SFS fluxes Filter size = 80 m Tensor model with tilting predicts SFS fluxes better than scalar model through most of BL depth

SFS production terms Filter size = 80 m Little anisotropic gradient contribution compared to gradient diffusion, except close to surface for f 1 Tilting term significant through BL, dominates over gradient term close to surface Advection term is important

Effect of filter length Decreasing filter size reduces SFS fluxes Little effect on tilting and gradient production terms but increases advection term

Results from DNS – SFS production DNS of neutral flow over large roughness (Coceal et al., 2006, 2007, Branford et al. 2011) Performed analysis similar to Wyngaard (2004) with HATS data Ratio of full production term (tilting + gradient) to gradient diffusion term is significantly larger than 1

Results from DNS – SFS fluxes Again tensor model with tilting term performs someehat better than scalar model

Conclusions Direct calculation of SFS production terms from high res 3D data A priori evaluation of their effect on the resulting SFS fluxes Extra production terms are significant through the whole BL Inclusion of these extra terms in a resulting diagnostic tensor model shows some improvement compared to scalar model Other terms are also usually important (advection, buoyancy production, time rate of change) Analyse other DNS/LES datasets from different flow regimes