Department of Environmental Sciences

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Seminar Abstracts
Environmental Sciences Seminar Abstract            

  THE LINEAR ALGEBRA OF ATMOSPHERIC DEEP CONVECTION
Brian Mapes
University of Miami

The near-linearity of convective heating and drying profiles in response to temperature and moisture anomalies have powerful implications. A convecting column of air can be characterized by its sensitivity matrix (or response function) M, which indicates how that column would participate in larger-scale convectively-coupled flows. M can be displayed or appreciated in various ways, from simple image inspection (it is just a 2D array, albeit an abstract one), to eigenvalue-eigenvector format (showing the 'free' oscillations that convection could be expected to undergo), to time-dependent scenario forecasts from arbitrary initial conditions via matrix exponentiation: X(t) = X(0) exp(Mt). Another marvel of linear systems is that the matrix can be estimated in any basis (like the most convenient for observations), and then transformed to any other desired basis (the most powerful form for some application). Subtleties arise of course: linearized 'anomalies' must be defined around some convecting basic state, and estimation errors for M may feed into problems with its well-posedness (stiffness, underdetermination, instability, etc.). But if this linear framework catches on, these issues can be worked through systematically.


Last updated: 09/04/2011