One of the major discoveries during TOGA-COARE was that both the convective activity and the surface heat flux anomaly lag behind the (peak of) surface convergence along the propagation of Madden-Julian oscillations. This phase relationship is consistent with neither wave-CISK nor linear WISHE theories. For its explanation, an extension of the analytical solution to the Gill problem is sought, which is thermodynamically self-consistently defined. For the convective closure, the balance of the boundary--layer thermodynamic budget is considered, in which the surface fluxes are approximately balanced with cooling and drying by convective downdrafts on the large scales. This balance is computed by assuming a simple mass-flux representation of cumulus convection. States that maximize this balance in a global variance measure are sought by searching in the phase velocity and the second harmonics of the longitudinal heating profile. These identified states are to be compared with Madden-Julian oscillations, and other observed convective propagating modes.
When an equatorially confined, stationary heating is applied, westerly wind bursts arise along the equator to the west of the heating, which results in an enhanced surface flux and creates a tendency for westward propagation to balance the boundary--layer thermodynamic budget. However, this hypothetical westward movement leads to an enhanced surface flux off the equator due to the enhanced Rossby-wave response to the west, which overthrows the improved balance along the equator. An eastward propagation with a phase velocity of 5 m/s is found to give the best balance. Additional modes are identified to the case with an equatorially symmetric dipolar heating, and a narrower equatorial heating than the standard case. A minimum of the boundary-layer thermodynamic imbalance is found at the fast eastward (12m/s) and westward (-5m/s) propagating modes. These three modes appear to correspond well to Madden-Julian oscillations, the super-clusters (Kelvin mode), and the cloud clusters (Rossby modes), respectively.
A relatively large remaining imbalance for these identified modes suggests the intrinsic importance of higher frequency transients in maintaining thermodynamic balance for this phenomenon.