A ‘B-matrix’ for convective-scale variational data assimilation

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

2. 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. The story so far … • Variational data assimilation (VAR) relies on a reasonable ‘B-matrix’ • A reasonable ‘B-matrix’ resembles • Pf = MPaMT + Q of K.F. or • Pf = < ( xf - < xf> ) (xf - < xf> )T > of very large ensemble • B is modelled • δx = B1/2δv, < δvfδvfT > = I • Bimplied = B1/2BT/2 • B1/2 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. Typical B1/2 in current generation VAR ← streamfunction ←‘unbalanced’ velocity potential ← ageostrophic pressure X ‘Balanced’ velocity potential regression L Linear balance operator (horizontal-only operator) G Regression operator (vertical-only operator) P Hydrostatic operator (vertical-only operator) B½s,x Spatial operator

5. What can go wrong with this? • f not small or Ro not small (e.g. midlatitudes, Lx small, Ro = U/fLx) • W large (non-hydrostatic) Q << L

6. 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 δθv B.2. Allow for non-hydrostatic δp and δθv

7. A. Diminishing geostrophic balance at small scales Current model Replacement Large-scale pass filter • Resembles current scheme at larger scales • Simple modification to scheme • No new control variables required Wavenumber

8. B.1. Non-hydrostatic δθv 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. Consequences of allowing diminishing geostrophic balance and non-hydrostatic δθv • Due to diminishing geostrophic balance at small scales, the implied covariances have smaller: • δψ/δp covariances • δψ/δθvcovariances • δp/δθvcovariances • δθv variances • δp variances • Due to non-hydrostatic δθv, the implied covariances have larger: • δθv variances

10. B.2. Non-hydrostatic δp and δθv • 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 δpnh non-hydrostatic pressure δαLS large-scale specific volume δpLS large-scale pressure δθvLS large-scale virtual pot. temp δθvnhconv-scale virtual pot. temp δu 3-D wind

11. B.2. Non-hydrostatic δp and δθv Interpret current pressure as hydrostatic, add δpnh to this • Still just one new control variable required • Need to invert Π at each VAR iteration

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