## 1. Introduction

[2] The fact that Numerical Weather Prediction Models (NWPMs) and General Circulation Models (GCMs) need stochastic parameterizations of the physical processes that occur at the subgrid scales is now well established [*Palmer et al.*, 2005]. A practical motivation is that without stochastic parameterizations, the multiple forecasts done with NWPMs to make ensemble predictions do not spread enough. There is also more physical reasons: the stochastic parameterizations represent better the unpredictable aspects of the subgrid-scale dynamics. As an illustration, the fact that the GWs observed in situ in the low stratosphere are very intermittent [*Hertzog et al.*, 2008], justifies the introduction of stochastic effects in the non-orographic GW parameterizations done by*Piani et al.* [2004] and *Eckermann* [2011].

[3] In the work by *Piani et al.* [2004] the inclusion of stochastic effects is done on the Doppler spread parameterization of *Hines* [1997], which belongs to the GW schemes that treat globally and at each time an entire spectrum of GWs, a technique that aims to better take into account the nonlinear nature of the GWs breaking. They found that when the *Hines* [1997] parameterization is made stochastic, the model simulates better the QBO: the stochastic approaches can also help to improve individual climate simulations. *Eckermann* [2011], introduced stochastic effects in the parameterization summarized by *Garcia et al.* [2007], and which is a “multiwave” parameterization, in the sense that it represents the GW field as the superposition of independent monochromatic GWs. The method used by *Eckermann* [2011] consists in treating only one GW at each “physical” time step, and by choosing its amplitude and spectral characteristics randomly. With this technique, the multiwaves schemes become more computationally efficient, a clear progress since these schemes need to take into account a large number of GWs.

[4] Nevertheless, the *Eckermann* [2011]approach has a conceptual defect. It considers that each GW acts during one model “physical” time step only, that is often less than one period of the GW considered, and certainly much less than the lifetime of GW packets. This undermines the time-Fourier analysis which is at the basis of the parameterization of non-orographic GWs.*Eckermann* [2011]also noticed that his method can have the more practical defect of producing grid-scale noise. Also, and may be because they were too expansive before being made stochastic the multiwave parameterizations have not been tested in the context of the simulation of the QBO. This is an important issue since the QBO dynamics involves critical levels interactions between GWs and the large-scale flow, and we know that this critical layer dynamics can necessitate a very good spectral resolution to be well solved [*Martin and Lott*, 2007].

[5] Since the late 1990's, there have been many dedicated GCM simulations that produce QBO-type oscillations [see,*Takahashi*, 1999], and some climate models now routinely produces a realistic QBO [*Scaife et al.*, 2000; *Giorgetta et al.*, 2002]. According to these papers, two key factors are at least needed, namely a sufficiently good vertical resolution, and a parameterization of the non-orographic gravity waves supposedly triggered by convection. Accordingly, if we want others to adopt our proposed stochastic GW parameterization, a good motivation is the demonstration that it can help a GCM to simulate a QBO. On top of the QBO, it is also important to measure the impact of the GW parameterization on the resolved waves. One obvious reason is that these waves also contribute to the QBO dynamics [*Holton and Lindzen*, 1972], a second important reason is that the gravest among those waves, like the near 10 day Kelvin waves and the 4–5 day RGWs dominate the tropical variability of the stratosphere at the synoptic scales.

[6] The purpose of the present paper is to give a formalism that generalizes the stochastic method of *Eckermann* [2011], and in order that at each time step and at each place a multiwaves parameterization can represent a very large number of waves at a very low cost. We will then show that the parameterization we propose can help to produce a QBO, and to improve the explicit simulation of the gravest equatorial waves.