Recent literature points to an attempt to understand the theoretical dynamics behind the MJO as seen in these two publications - Dynamics moisture mode vs. moisture mode in MJO dynamics and A general theoretical framework for understanding essential dynamics of Madden–Julian oscillation
As concluded in the original paper by Madden and Julien oscillation the key is to observe that original MJ observations were either in a in phase relationship of the surface pressure and the zonal wind(850 hPa) or out of phase relationship. No such relationship was observed with the meridional wind.
So the "geostrophic balance" in the MJO(in it's non dimensional form) is simply this equation
$$ \beta * y*u = -(\frac{\partial \Phi}{\partial y}) $$
So $\Phi$ here represents the first baroclinic mode (the meaning of baroclinicity in the tropics is quite different to it's meaning in the mid latitudes). In the tropics what it means is that vertical structures of winds and currents that reverse directions (cf. MJO review). So the MJO has a baroclinic mode of vibration because the winds reverse direction from 850 hPa to 200 hPa.
So in the assumption (proven by observations) that the MJO can be characterized as a low frequency wave with an anisotropic horizontal length scale where the zonal length scale is much larger than the meridional scale by a magnitude of 10.So the meridional acceleration in the v-momentum equations can be neglected. With this approximation the zonal wind is in approximate geostrophic balance with the meridional pressure gradient and all high frequency inertio gravity waves, mixed Rossby-gravity waves as well as short Rossby waves are all filtered out. So that only leaves the moist Kelvin and long Rossby waves as well as the MJO.
Alternatively if one were to consider the Rossby radius of deformation in the case of the MJO the horizontal wavelength of the disturbance if equal to the Rossby radius of deformation then both the velocity and mass fields undergo mutual adjustment as illustrated in these slides - Waves in Atmosphere on page 52.
Similar conclusions were arrived at to derive balanced dynamics in the tropics by assuming linear theory and a shallow water model - Balanced dynamics in the tropics
UPDATE
Looking at other peer reviewed references that attempt to provide an explanation of the MJO it appears that an assumption of linear theory wrt to the MJO is not universally accepted and that non linear advection does have a role to play in the slowly propagating planetary scale disturbance(cf.Simple model of the 40-50 oscillation. A later paper by Madden(cf.Seasonal variations of the 40-50 oscillations in the tropics reports that there can be substantial meridional wind perturbations bringing into question whether the assumption of geoostrophic balance is universally valid or not in the case of the MJO.
The original paper by MJ in the light of the observations obtained then perhaps made a conclusion that the structure and dynamics of the signal could resemble a Kelvin wave in which case the geostrophic balance of the Equatorial Kelvin wave (no large scale meridional wind perturbations and the equator acts as a wave guide ) could be used to explain the geostrophic balance in the case of the MJO. Later observations have proved this assumption incomplete.
