Potential vorticity aspects of the MJO

Considering linearized motion about a resting basic state, we derive analytical solutions of the equatorial ββ-plane primitive equations under the assumption that the flow is steady in a reference frame moving eastward with a diabatic forcing resembling a Madden–Julian Oscillation (MJO) convective envelope. The solutions are analyzed in terms of potential vorticity (PV) dynamics. Because the diabatic source term for PV contains a factor βyβy, the diabatic heat source is ineffective at generating a PV anomaly at the equator but maximizes the PV response near the poleward edges of the heat source. In this way a moving heat source can produce two ribbons of lower tropospheric PV anomaly, a positive one off the equator in the northern hemisphere and a negative one off the equator in the southern hemisphere, with oppositely signed PV anomalies in the upper troposphere. Associated with these PV anomalies are geopotential anomalies that are shifted several hundred kilometers poleward. In the lower troposphere these zonally elongated geopotential anomalies resemble ITCZ trough zones, which demonstrates the close connection between the MJO wake dynamics and the formation of double ITCZs.To demonstrate that the MJO wake response can be described by simple PV dynamics, we propose an invertibility principle relating the PV to the streamfunction, which in turn is locally related to the geopotential. This equatorial invertibility principle accurately recovers the balanced wind and mass fields found in the MJO wake in the primitive equation model. However, while the invertibility principle highlights the ability of simple PV dynamics to accurately describe the flow in the wake of an MJO convective envelope, it also clearly illustrates the inability of such dynamics to describe the Kelvin-like flow pattern ahead of the convection.