# A multi-wave model for the Quasi-Biennial Oscillation: Plumb’s model extended

^{1}CNRS, Aix-Marseille Université, IRPHE, France (leard@irphe.univ-mrs.fr)^{2}Princeton Center for Theoretical Science, Princeton, USA^{3}CNRS, Aix-Marseille Université, IRPHE, France

In the Earth’s stratosphere, equatorial zonal winds reverse from easterlies to westerlies with a period of roughly 28 months. This phenomenon, known as Quasi-Biennial Oscillation (QBO), is driven by internal gravity waves (IGWs) propagating in the stratosphere and interacting with the ambient large-scale flow. Those waves are generated by the turbulent motions of the troposphere. In 1977, an idealised model describing the generation of a reversing large-scale flow by two counter-propagating monochromatic internal gravity waves was developed by Plumb [1]. In 1978, the famous Plumb & McEwan’s experiment [2] validated this model using oscillating membranes to force a standing wave pattern at the boundary of a linearly stratified salty-water layer in a cylindrical shell container.

Recently, the effects of the wave dissipation and wave energy were studied by Renaud et al. [3] using the Plumb model in order to explain the QBO disruption observed in 2016. It was found that as the Reynolds number increases, bifurcations from periodic to non-periodic regimes are seen for the large-scale flow oscillations.

Here, we present the results obtained from an extended version of the Plumb’s model, taking into account the stochastic generation of IGWs in Nature. Our new model includes a wide spectrum of waves as forcing for the large-scale flow. A gaussian distribution of energy is used in order to compare monochromatic forcing results (characterised by a gaussian energy spectrum with a small standard deviation) with multi-wave forcing results (large standard deviation). Unexpectedly, we find that in a large parameter domain, gathering the energy of the forcing into one frequency results in non-periodic oscillations for the QBO while spreading the same amount of energy among many frequencies results in periodic oscillations. We also investigate more realistic distribution of energy for the forcing including classical convective spectra, with or without rotation. We find that different forcings result in very similar reversals. This result is quite relevant for Global Circulation Models (GCMs) where internal gravity waves are parameterised in order to drive a realistic QBO. However, our study suggests that driving a QBO with realistic characteristics (amplitude, period) does not involve that the input forcing (i.e. the wave spectrum characteristics) is realistic as well.

**References:**

[1] R. A. Plumb, « The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation », Journal of the Atmospheric Sciences, 1977.

[2] R. A. Plumb and A. D. McEwan, « The instability of a forced standing wave in a viscous stratified fluid: a laboratory analogue of the quasi biennial oscillation », Journal of the Atmospheric Sciences, 1978.

[3] A. Renaud, L.-P. Nadeau, and A. Venaille, « Periodicity Disruption of a Model Quasibiennial Oscillation of Equatorial Winds », Phys. Rev. Lett., vol. 122, n^{o} 21, p. 214504, 2019.

**How to cite:**
Léard, P., Lecoanet, D., and Le Bars, M.: A multi-wave model for the Quasi-Biennial Oscillation: Plumb’s model extended, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-7375, https://doi.org/10.5194/egusphere-egu2020-7375, 2020

## Comments on the display material

AC: Author Comment |CC: Community Comment | Report abuseDisplay material version 1– uploaded on 07 May 2020CC1: Question, Haruka Okui, 08 May 2020I am sorry that I send my incomplete question by mistake in the end of session. May I ask you after all?

My question is about the relation between your results and aspects suggested by Renaud et al. (2019). Can the effect of Reynolds number and that of forcing spectrum on the generation of periodic oscillation be correlated to each other or explained by a comprehensive way?

I am sorry if I'm asking a misguided question because of my poor understanding. I'd be grateful if you could answer the question for my own information.

AC2: Reply to CC1, Pierre Léard, 13 May 2020Well I think that the results from our study and the study of Renaud et al. are indeed connected. Renaud found that a low Reynolds number for the forcing wave results in a periodic reversing flow. With increasing wave Reynolds number, the mean-flow regime bifurcates towards non-periodic regime.

In our study, we want to have the same total energy on the forcing for every spectrum we consider. Therefore, for a large spectrum, the energy per wave is weaker than the monochomatic case where the energy is focused in one frequency. For a large spectrum, this results in a low Reynolds number for each wave in the spectrum and giving periodic oscillations for the mean-flow.

Even if we consider several waves for the forcing, for the cases shown in the slides, the Reynolds number associated to each wave is small and the oscillations are periodic, even if the forcing is a superposition of many different frequency waves.

CC6: Reply to AC2, Haruka Okui, 14 May 2020The answer was clear, so I understood it well. Thank you very much.

CC7: Reply to CC6, Paul PUKITE, 14 May 2020"Can the effect of Reynolds number and that of forcing spectrum on the generation of periodic oscillation be correlated to each other or explained by a comprehensive way?"For a forced response, the output will contain a remnant of the input forcing, and unless the system has a high-Q resonance, the natural response will damp. For the QBO, the hint that the system has a forced response is that the layer above the QBO shows a strict Semi-Annual Oscillation -- and thus is called the SAO (without a quasi in front of it). The only question then is what aligns the 28-month QBO cycle. It's a simple exercise to show that the nodal lunar tide amplified by the semi-annual cycle is the only candidate that can generate the necessary periodic pulses to trigger predictable reversals.

And what would ordinarily be inexplicable (the 2015-2016 QBO anomaly) actually reinforces this idea, since the QBO cycle is once again synchronized to the predicted wave-train. Whatever caused this perturbation is analogous to a tsunami amongst ocean tidal cycles -- i.e. it will not influence the long-term synchronization.

CC2: Comment on EGU2020-7375, Andrew Bushell, 08 May 2020OS4.3@Pierre Leard: I am sorry that your slot was rather washed-out by the session

chat overrunning its schedule but I thought the slides you presented were very

interesting.

I published some work a few years ago where I replaced the fixed NOGW source in

an AGCM with a variable (PPN dependent) source. In this case, for comparison I

set the launch flux scaling so that the mean over several years and a broad

latitude range (intermittent and spatially varying) matched the almost invariant

mean of the original 'fixed source' scheme. The result, as with your model, was

a very similar period despite the much altered nature of the source variability:

https://journals.ametsoc.org/doi/full/10.1175/JAS-D-15-0022.1

In addition, recent comparisons made of AGCMs in the QBOi project (see

presentations D3238, D3265 and D3266 in EGU202 session AS1.18 for more details

and references) in particular D3266 about the simple future climate tests show a

possibility of the AGCMs to leave the quasi-periodic regime - although your

Plumb-type model is naturally quite simple it would be interesting if its

behaviour were sufficiently analagous to indicate why some of the predictions

might diverge this way.

By the way, I do not know if you had already heard about a workshop proposed on

the subject of 'QBO at 60 years' - it should have taken place this summer but

has been postponed due to the coronavirus ... however there is a contact email

if you would be interested in receiving email when a new schedule is proposed:

qboworkshop@ncas.ac.uk

Regards,

Andrew Bushell

CC5: Reply to CC2, Paul PUKITE, 14 May 2020The gravity waves generating the QBO arise from interaction of the semiannual cycle with the lunar nodal cycle (13.606 days). This creates an impulse that reverses direction every 365.242/(365.242/27.212-13)=865 days=28.4 months. At higher altitudes (above the QBO) the semiannual thermal tide forces the SAO due to the lower atmsopheric density.

AC3: Reply to CC2, Pierre Léard, 14 May 2020I thank you for all these references and I will take a close look at them.

I've learnt quite recently about the QBOi project and I really need to dive in these studies. I really lack knowledge about the atmospheric community studies on the Plumb model since I'm from a physicist community. I think I still have a lot to learn from the investigations made with GCMs and this project seems to gather a mass of useful information

CC3: Comment on EGU2020-7375, Paul PUKITE, 10 May 2020I never understood the rationale for the Plumb experiment to act as a validation for the QBO mechanism. First of all, the Plumb apparatus is a rotating cylinder and not a rotating sphere, so it does not allow the apparently important Coriolis effect. Secondly, the lab-sized nature of the apparatus prevents the emulation of any gravitational effect. There are centrifugal acceleration aspects but these aren't balanced by gravity, and what's more, any gravitational effects are along the length of the cylinder and not radial. Overall and at best, it may help explain certain stratification properties, but validates nothing about the fundamental QBO oscillation.

AC1: Reply to CC3, Pierre Léard, 13 May 2020The cylinders are not rotating in the the Plumb and McEwan experiment. The QBO appears at equatorial latitude and the Coriolis effects are negligible at the equator. Therefore, there is no need to consider rotation in the model

CC4: Reply to AC1, Paul PUKITE, 14 May 2020Thanks. The cancellation of the Coriolis force at the equator allows an analytical solution to Laplace's Tidal Equation along that topological boundary region [1]. Forcing the solution with lunisolar cycles of nodal symmetry will generate precisely the 28-month QBO period.

Mathematical Geoenergy(Wiley, 2018). doi:10.1002/9781119434351