1. Ross Bannister Dept. of Meteorology, Univ. of Reading, UK Thanks to Mike Cullen, Stefano Migliorini, Mark Dixon Title: TBA (Transforms for a B-matrix in Assimilation) A ‘B-matrix’ for convective-scale variational data assimilation

2. Ross Bannister Durham August 2011 Page 2 of 12 Convective-scale data assimilation Short predictability timescales Data intensive Multiscale balances Diminishing geostrophic balance at small scales Diminishing hydrostatic balance at small scales (especially within convecting regions) High degree of flow-dependency Different multivariate relationships expected for precipitating and non-precipitating regions Highly anisotropic and inhomogeneous Boundary condition errors Phase errors

3. Ross Bannister Durham August 2011 Page 3 of 12 The story so far … Variational data assimilation (VAR) relies on a reasonable ‘B-matrix’ A reasonable ‘B-matrix’ resembles P f = MP a M T + Q of K.F. or P f = ) (x f - ) T > of very large ensemble B is modelled δx = B 1/2 δv, = I B implied = B 1/2 B T/2 B 1/2 is a prescribed operator that includes strong/weak balance relations Most operational VAR systems assume non-divergent wind is always geostrophically balanced, small Ro or small f pressure is always hydrostatically balanced Ro W/U vanishingly small

4. Ross Bannister Durham August 2011 Page 4 of 12 Typical B 1/2 in current generation VAR X ‘Balanced’ velocity potential regression L Linear balance operator (horizontal-only operator) G Regression operator (vertical-only operator) P Hydrostatic operator (vertical-only operator) B ½ s,x Spatial operator ← streamfunction ←‘unbalanced’ velocity potential ← ageostrophic pressure

5. Ross Bannister Durham August 2011 Page 5 of 12 What can go wrong with this? f not small or Ro not small (e.g. midlatitudes, L x small, Ro = U/fL x ) W large (non-hydrostatic) Q << L Part of current model isδp = L σ ψ δv ψ + σ pA δv pA Suppose reality is modelled well byδp = Q σ ψ δv ψ + σ pA δv pA

6. Ross Bannister Durham August 2011 Page 6 of 12 Improvements A. Allow mid-latitude f and non-small Ro simultaneously (diminishing geostrophic balance at small scales) B. Allow Ro W/L non-negligible (non-hydrostatic motion) B.1. Assume all δp is hydrostatic, but allow for non-hydrostatic δθ v B.2. Allow for non-hydrostatic δp and δθ v

7. Ross Bannister Durham August 2011 Page 7 of 12 A. Diminishing geostrophic balance at small scales Current model Replacement Large-scale pass filter Wavenumber Resembles current scheme at larger scales Simple modification to scheme No new control variables required

8. Ross Bannister Durham August 2011 Page 8 of 12 B.1. Non-hydrostatic δθ v Current model Replacement In matrix form One new control variable required Resembles current scheme at scales where extra control variable small No new complicated operators

9. Ross Bannister Durham August 2011 Page 9 of 12 Consequences of allowing diminishing geostrophic balance and non-hydrostatic δθ v Due to diminishing geostrophic balance at small scales, the implied covariances have smaller: δψ/δp covariances δψ/δθ v covariances δp/δθ v covariances δθ v variances δp variances Due to non-hydrostatic δθ v, the implied covariances have larger: δθ v variances

10. Ross Bannister Durham August 2011 Page 10 of 12 B.2. Non-hydrostatic δp and δθ v What about non-hydrostatic δp? How should we partition δp into hydrostatic and non-hydrostatic parts? Pielke R. Sr, 2002 gives a diagnostic equation for non-hydrostatic pressure given that the flow is in anelastic balance. This can be developed in incremental form δp nh non-hydrostatic pressure δα LS large-scale specific volume δp LS large-scale pressure δθ vLS large-scale virtual pot. temp δθ vnh conv-scale virtual pot. temp δu 3-D wind

11. Ross Bannister Durham August 2011 Page 11 of 12 B.2. Non-hydrostatic δp and δθ v Still just one new control variable required Need to invert Π at each VAR iteration Interpret current pressure as hydrostatic, add δp nh to this

12. Ross Bannister Durham August 2011 Page 12 of 12 Summary Models are gaining higher resolution (e.g. ~1km) High resolution is needed for quantitative precipitation forecasting Data assimilation methods not “one size fits all” The assumptions made for the data assimilation problem are different By using geostrophic and hydrostatic balances for small scales will lead to a sub-optimal analysis The B-matrix requires a rethink Propose two new schemes that: Allow for diminishing geostrophic balance Allow for non-hydrostatic temperature (one extra control variable) Allow for non-hydrostatic pressure (one extra control variable + anelastic balance) What needs to be done now? Determine validity of anelastic balance from data Refine the scheme and look at implied covariance stats and assimilation