A Scaling Theory of the MJO Horizontal Wavelength

Yang and Ingersoll (2013, Triggered Convection, Gravity Waves, and the MJO: A Shallow-Water Model. J. Atmos. Sci., 70, 2476-2486.) use a shallow water model to simulate the MJO. In this model, we parameterize radiation as a large-scale mass sink, and parameterize convection as a small-scale mass source that is only triggered when the layer thickness is lower than a certain threshold. Over a wide parameter range, the model gives a strong MJO-like signal - a large-scale pattern that drifts slowly to the east, together with wind and pressure fields similar to those observed. Based on the simulation results, we propose that the MJO is an interference pattern of the westward and eastward inertia-gravity (WIG and EIG) waves. The MJO propagation speed is determined by the difference in the phase speeds of the EIG and WIG waves. In this study, we try to quantitatively understand what controls the MJO horizontal scale using the same shallow water model. There are four parameters in the model - the Kelvin wave speed (c), the convective timescale (t_c), the size of the convection (r_c), and the number density (s_c) of convection. We perform numerical simulations to examine the sensitivity of the MJO zonal wavenumber (k) to these parameters: k increases when s_c and r_c increase and when c decreases; k does not depend on the convective timescale. The scaling between k and the model parameters obeys a power law within an inertial range. A scaling argument will be presented based on the equatorial beta-plane scaling. This scaling argument helps to understand the climate dynamics of the MJO, e.g., how the MJO responds to global warming. The fidelity of this scaling argument can be examined by performing systematic global warming simulations using super-parameterized 3D climate models that reproduce the MJO signal in the current climate.