Compressible vs. anelastic (Elliptic equation example) ATM 562 Fovell Fall, 2015

Problem statement MT3 involves construction of a thermal perturbation and also a pressure perturbation obtained by solving the perturbation hydrostatic equation – This is an attempt to adjust the environment against the provocation represented by the perturbation. In a neutral atmosphere, that’s accomplished solely by sound waves. – However, the hydrostatic equation is a 1D concept, and can only adjust one column at a time: a grid point is adjusted based on what lies above or below it. – Sound wave time scale so short that even the far distant environment “feels” the thermal perturbation rather quickly. – If the adjustment really is so fast, and the details don’t matter, why not make it infinitely fast?

Solution Since we don’t care about the details of the acoustic adjustment – we just want it done – an alternative is to invoke the anelastic approximation – This approximation makes sound waves infinitely fast – The pressure tendency equation disappears, and we are left with an elliptic equation for  ’. Next slides compare dimensional p’ predictions from compressible (MT3/MT5) and anelastic models – The compressible simulation uses c s = 50 m/s

Initial condition (same bubble, different p’) MT3/MT5 Initial condition Anelastic approach

Initial condition p’ amplitude large; area narrow Every grid point already impacted to some degree Initial condition (same bubble, different p’) Instantaneous adjustment above and far beyond thermal

Note very little change in 1 st minute Something is emerging… p’ > 0 appearing

Animation