1. The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard University

2. Physical importance of convection vertical flux of heat and moisture – i.e., fluxes of sensible and latent heat » (momentum ignored here) Really we care about flux convergences – or heating and moistening rates Q1 and Q2 are common symbols for these » (with different sign and units conventions)

3. Kinds of atm. convection – Dry Subcloud turbulent flux – may include or imply surface molecular flux too – Moist (i.e. saturated, cloudy) shallow (cumulus) – nonprecipitating: a single conserved scalar suffices middle (congestus) & deep (cumulonimbus) – precipitating: water separates from associated latent heat – Organized ‘coherent structures’ even in dry turbulence ‘multicellular’ cloud systems, and mesoscale motions

4. “Deep” convection Inclusive term for all of the above – Fsurf, dry turbulence, cu, cg, cb – multicellular/mesoscale motions in general case Participates in larger-scale dynamics – Statistical envelopes of convection = weather one line of evidence for near-linearity – Global models need parameterizations need to reproduce function, without explicitly keeping track of all that form or structure

5. deep convection: cu, cg, cb, organized http://www.rsmas.miami.edu/users/pzuidema/CAROb/ http://metofis.rsmas.miami.edu/~bmapes/CAROb_movies_archive/?C=N;O=D

6. Q1 and Q2 in deep convection

7. interacts statistically with large scale waves 1998 CLAUS Brightness Temperature 5ºS-5º N straight line = nondispersive wave

8. A convecting “column” (in ~100km sense) is like a doubly-periodic LES or CRM Contains many clouds – an ensemble Horizontal eddy flux can be neglected – vertical is

9. Atm. column heat and moisture budgets, as a T & q column vector equation: LS dynamics rad. water phase changes eddy flux convergence Surface fluxes and all convection (dry and moist). What a cloud resolving model does.

10. Outline Linearity (!) of deep convection (anomalies) The system in matrix form – M mapped in spatial basis, using CRM & inversion (ZK) – math checks, eigenvector/eigenvalue basis, etc. Can we evaluate/improve M estimate from obs?

11. Linearity (!) of convection My first grant! (NSF) Work done in mid- 1990s Hypothesis: low-level (Inhibition) control, not deep (CAPE) control (Salvage writeup before moving to Miami. Not one of my better papers...)

12. A much better job of it

13. Low-level Temperature and Moisture Perturbations Imposed Separately Courtesy Stefan Tulich (2006 AGU) Inject stimuli suddenly Q1 anomalies Q1

14. Linearity test of responses: Q 1 (T,q) = Q 1 (T,0) + Q 1 (0,q) ? T q T q sum, assuming linearity both Courtesy Stefan Tulich (2006 AGU) both sum Yes: solid black curve resembles dotted curve in both timing & profile

15. Linear expectation value, even when not purely deterministic Ensemble Spread Courtesy Stefan Tulich (2006 AGU)

16. Outline Linearity (!) of deep convection (anomalies) The system in matrix form – in a spatial basis, using CRM & inversion (ZK) could be done with SCM the same way... – math checks, eigenvector/eigenvalue basis, etc. Can we estimate M from obs? – (and model-M from GCM output in similar ways)?

17. Multidimensional linear systems can include nonlocal relationships Can have nonintuitive aspects (surprises)

18. Zhiming Kuang, Harvard: 2010 JAS

19. Splice together T and q into a column vector (with mixed units: K, g/kg) Linearized about a steadily convecting base state 2 tried in Kuang 2010 – 1. RCE and 2. convection due to deep ascent-like forcings

20. Zhiming’s leap: Estimating M -1 w/ long, steady CRM runs

21. Zhiming’s inverse casting of problem ^^ With these, build M -1 column by column Invert to get M! (computer knows how) Test: reconstruct transient stimulus problem

22. Specify LSD + Radiation as steady forcing. Run CRM to ‘statistically steady state’ (i.e., consider the time average) All convection (dry and moist), including surface flux. Total of all tendencies the model produces.  0 for a long time average Treat all this as a time- independent large-scale FORCING applied to doubly-periodic CRM (or SCM) 0

23. Forced steady state in CRM Surface fluxes and all convection (dry and moist). Total of all tendencies the CRM/ SCM produces. T Forcing (cooling) q Forcing (moistening) CRM response (heating) CRM response (drying)

24. Now put a bump (perturbation) on the forcing profile, and study the time-mean response of the CRM Surface fluxes and all convection (dry and moist). Total of all tendencies the CRM/ SCM produces. T Forcing (cooling) CRM time-mean response: heating’ balances forcing’ (somehow...)

25. How does convection make a heating bump? anomalous convective heating 1. cond 1. Extra net Ccondensation, in anomalous cloudy upward mass flux How does convection ‘know’ it needs to do this? 2. EDDY: Updrafts/ downdrafts are extra warm/cool, relative to environment, and/or extra strong, above and/or below the bump =

26. How does convxn ‘know’? Env. tells it! by shaping buoyancy profile; and by redefining ‘eddy’ (T p – T e ) Kuang 2010 JAS cond 0... and nonlocal (net surface flux required, so surface T &/or q must decrease)... and nonlocal (net surface flux required, so surface T &/or q must decrease) 0 0 0 (another linearity test: 2 lines are from heating bump & cooling bump w/ sign flipped) via vertically local sounding differences..

27. ... this is a column of M -1 M -1

28.  a column of M -1

29. 0 -4 Image of M -1 Units: K and g/kg for T and q, K/d and g/kg/d for heating and moistening rates Tdot  T’qdot  T’ Tdot  q’qdot  q’ z z 0 Unpublished matrices in model stretched z coordinate, courtesy of Zhiming Kuang

30. 0 -4 Interpreting biggest feature: Tdot  T’qdot  T’ Tdot  q’qdot  q’ z z 0 To make an upper-level moistening spike, the CRM state adjusts until it has 0. A surface flux anomaly >0 (T’ <0, q’ <0 at surface) 1. An extra-steep lapse rate (just a moist adiabat tied to surface conditions?) 2. Dry layer at target level (so eddy flux w’q’ converges there?)

31. M -1 and M from 3D 4km CRM from 2D 4km CRM ( ^^^ these numbers are the sign coded square root of |M ij | for clarity of small off- diag elements)

32. M -1 and M from 3D 4km CRM from 3D 2km CRM ( ^^^ these numbers are the sign coded square root of |M ij | for clarity of small off- diag elements)

33. Finite-time heating and moistening has solution: So the 6h time rate of change is:

34. M [exp(6h×M)-I] /6h Spooky action at a distance: How can 10km Q1 be a “response” to 2km T? (A: system at equilibrium, calculus limit) Fast-decaying eigenmodes are noisy (hard to observe accurately in equilibrium runs), but they produce many of the large values in M. Comforting to have timescale explicit? Fast-decaying eigenmodes affect this less.

35. Low-level Temperature and Moisture Perturbations Imposed Separately Courtesy Stefan Tulich (2006 AGU) Inject stimuli suddenly Q1 anomalies Q1

36. Columns: convective heating/moistening profile responses to T or q perturbations at a given altitude 0 square root color scale 0 1 2 5 8 12 z (km) 0 1 2 5 8 12 z (km)

37. Image of M Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot 0 T’  Tdot q’  Tdot T’  qdot q’  qdot square root color scale FT  PBL PBL  FT 0 1 2 5 8 12 z (km) 0 1 2 5 8 12 z (km)

38. Image of M (stretched z coords, 0-12km) Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot 0 z z T’  Tdot q’  Tdot T’  qdot q’  qdot square root color scale positive T’ within PBL yields... positive T’ within PBL yields... +/-/+ : cooling at that level and warming of adjacent levels (diffusion) +-++-+

39. Off-diagonal structure is nonlocal (penetrative convection) 0 square root color scale

40. Off-diagonal structure is nonlocal (penetrative convection) 0 T’  Tdot q’  Tdot T’  qdot q’  qdot square root color scale + effect of q’ on heating above its level T’+ (cap)  qdot+ below T’ cap inhibits deepcon 0 1 2 5 8 12 z (km) 0 1 2 5 8 12 z (km)

41. Integrate columns in top half x dp/g 0 square root color scale

42. sensitivity: d{Q1}/dT, d{Q1}/dq

43. Integrate columns in top half x dp/g of [exp(6h×M)-I] /6h 0 compare plain old M

44. A check on the method prediction for the Tulich & Mapes ‘transient sensitivity’ problem initial ‘stimuli’ in the domain- mean T,q profiles

45. Example: evolution of initial warm blip placed at 500mb in convecting CRM Appendix of Kuang 2010 JAS Doing the actual experiment in the CRM: Computed as [exp( (M -1 ) -1 t)] [Tinit(p),0]:

46. Example: evolution of initial moist blip placed at 700mb in convecting CRM Appendix of Kuang 2010 JAS Doing the actual experiment in the CRM: Computed via exp( (M -1 ) -1 t):

47. Example: evolution of initial moist blip placed at 1000mb in convecting CRM Appendix of Kuang 2010 JAS Doing the actual experiment in the CRM: Computed via exp( (M -1 ) -1 t):

48. Outline Linearity (!) of deep convection (anomalies) The system in matrix form – mapped in a spatial basis, using CRM & inversion (ZK) – math checks, eigenvector/eigenvalue basis Can we estimate M from obs? – (and model-M from GCM output in similar ways)?

49. 10 Days 1 Day 2.4 h 3D CRM results 2D CRM results There is one mode with slow (2 weeks!) decay, a few complex conj. pairs for o(1d) decaying oscillations, and the rest (e.g. diffusion damping of PBL T’, q’ wiggles) Eigenvalues Sort according to inverse of real part (decay timescale) 15 min

50. Column MSE anomalies damped only by surface flux anomalies, with fixed wind speed in flux formula. A CRM setup artifact.* *(But T/q relative values & shapes have info about moist convection?) 3D CRM results 2D CRM results Slowest eigenvector: 14d decay time 0

51. 3D CRM results 2D CRM results eigenvector pair #2 & #3 ~1d decay time, ~2d osc. period congestus-deep convection oscillations

52. Singular Value Decomposition Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form M = UΣV* where U is an m×m real or complex unitary matrix, Σ is an m×n diagonal matrix with nonnegative real numbers on the diagonal, and V* (the conjugate transpose of V) is an n×n real or complex unitary matrix. The diagonal entries Σ i,i of Σ are known as the singular values of M. The m columns of U and the n columns of V are called the left singular vectors and right singular vectors of M, respectively.unitary matrixdiagonal matrixconjugate transposesingular values

53. Outline Linearity (!) of deep convection (anomalies) The system in matrix form – mapped in a spatial basis, using CRM & inversion anything a CRM can do, an SCM can do – math checks, eigenvector/eigenvalue basis, etc. Evaluation with observations

54. Put in anomalous sounding [T’,q’]. Get a prediction of anomalous rain.

55. COARE IFA 120d radiosonde array data M prediction can be decomposed since it is linear. Most of the response is to q anomalies.

57. Convection is a linearizable process – can be summarized in a timeless object M – tangent linear around convecting base states How many? as many as give dynamically distinct M’s. This unleashes a lot of math capabilities ZK builds M using steady state cloud model runs – can we find ways to use observations? Many cross checks on the method are possible – could be a nice framework for all the field’s tools » GCM, SCM, CRM, obs. Wrapup