# Mean Flow from Phase Averages in the 2D Boussinesq Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

- The unchanged data from the numerical solution of the 2D Boussinesq equations $\mathbf{u}$;
- The data after applying the transformation, yielding the mapped solution $\mathbf{w}$;
- The data after applying the transformation and the averaging procedure, which give the averaged solution $\overline{\mathbf{w}}$;
- The data after applying the inverse transformation $\overline{\mathbf{u}}$.

#### 2.1. Numerical Computation of Solutions to the Boussinesq Equations

#### 2.2. Fourier Analysis and the Computation of the Exponential of $\mathcal{L}$ with the Divergence Constraint

#### 2.3. Averaging with a Convolution Integral

## 3. Results and Discussion

#### 3.1. Impact of Mapping and Averaging

#### 3.2. Definition of a Mean Flow

#### 3.3. Potential for Use in Numerical Methods

## 4. Conclusions

- The nonlinear averaging procedure depends strongly on the value of $\eta $, but could potentially provide a reasonable representation of the time-mean of the solutions, whether or not N is large. Further, the average ‘modulates’ the waves, as in (2). This has the effect that the mean $\overline{\mathbf{u}}$ depends on how fast the oscillations are in the exponential operator (2), which could be interesting for further studies. Further work is needed to assess whether this would be a good definition of a mean flow in fluid dynamics. For example, it would be interesting to examine a case that admits a low-frequency solution such as the 2D Boussinesq equations with forcing, as in [12] and examples shown in [8];
- We also observed that the method could potentially be used to take larger time steps, as has been shown in simple examples [21], and that the ability to do so critically depends on the value of $\eta $.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PDE | partial differential equation |

ODE | ordinary differential equation |

## Appendix A. Computational Details for the Exponential

#### Appendix A.1. The Eigenvalues and Eigenvectors

**First case:**${k}_{1}=0,{k}_{3}=0$,

**Second case:**${k}_{1}=0,{k}_{3}\ne 0$,

**Third case:**${k}_{1}\ne 0,{k}_{3}=0$,

**Fourth case:**${k}_{1}\ne 0,{k}_{3}\ne 0$,

#### Appendix A.2. Diagonal Matrix or Jordan Normal Form

**First case:**${k}_{1}=0,{k}_{3}=0$,

**Second case:**${k}_{1}=0,{k}_{3}\ne 0$,

**Third case:**${k}_{1}\ne 0,{k}_{3}=0$,

**Fourth case:**${k}_{1}\ne 0,{k}_{3}\ne 0$,

#### Appendix A.3. Transformation Matrix and Inverse

**First case:**${k}_{1}=0,{k}_{3}=0$,

**Second case:**${k}_{1}=0,{k}_{3}\ne 0$,

**Third case:**${k}_{1}\ne 0,{k}_{3}=0$,

**Fourth case:**${k}_{1}\ne 0,{k}_{3}\ne 0$,

#### Appendix A.4. The Matrices with Divergence Constraint

**First case:**${k}_{1}=0,{k}_{3}=0$,

**Second case:**${k}_{1}=0,{k}_{3}\ne 0$,

**Third case:**${k}_{1}\ne 0,{k}_{3}=0$,

**Fourth case:**${k}_{1}\ne 0,{k}_{3}\ne 0$,

## References

- Verhulst, S.F.; Murdock, J. Averaging Methods in Nonlinear Dynamical Systems, 2nd ed.; Springer: New York, NY, USA, 2007. [Google Scholar] [CrossRef]
- Herring, J.R. Approach of axisymmetric turbulence to isotropy. Phys. Fluids
**1974**, 17, 859–872. [Google Scholar] [CrossRef] - Herring, J.R.; Métais, O. Numerical experiments in forced stably stratified turbulence. J. Fluid Mech.
**1989**, 202, 97–115. [Google Scholar] [CrossRef] - Métais, O.; Herring, J.R. Numerical simulations of freely evolving turbulence in stably stratified fluids. J. Fluid Mech.
**1989**, 202, 117–148. [Google Scholar] [CrossRef] - Babin, A.; Mahalov, A.; Nicolaenko, B. Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Eur. J. Mech. B Fluids
**1996**, 15, 291–300. [Google Scholar] - Babin, A.; Mahalov, A.; Nicolaenko, B.; Zho, Y. On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations. Theor. Comput. Fluid Dyn.
**1997**, 9, 223–251. [Google Scholar] [CrossRef] - Bogoliubov, N.; Mitropolsky, Y. Asymptotic Methods in the Theory of Nonlinear Oscillations; Gordon and Breach: New York, NY, USA, 1961. [Google Scholar]
- Embid, F.; Majda, J. Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial. Differ. Equ.
**1996**, 21, 619–658. [Google Scholar] [CrossRef] - Klainerman, S.; Majda, A.J. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.
**1981**, 34, 481–524. [Google Scholar] [CrossRef] - Schochet, S. Fast Singular Limits of Hyperbolic PDEs. J. Differ. Equ.
**1994**, 114, 476–512. [Google Scholar] [CrossRef] - Newell, A. Rossby wave packet interactions. J. Fluid Mech.
**1969**, 35, 255–271. [Google Scholar] [CrossRef] - Smith, L.M. Numerical study of two-dimensional stratified turbulence. Contemp. Math.
**2001**, 283, 91–106. [Google Scholar] - Smith, L.; Waleffe, F. Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech.
**2002**, 451, 145–168. [Google Scholar] [CrossRef] - Kafiabad, H.A.; Vanneste, J.; Young, W.R. Wave-averaged balance: A simple example. J. Fluid Mech.
**2021**, 911, R1. [Google Scholar] - Wagner, G.L.; Young, W.R. Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech.
**2015**, 785, 401–424. [Google Scholar] - Yamazaki, H.; Cotter, C.; Wingate, B. Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere. Q. J. R. Meteorol. Soc.
**2023**. accepted. [Google Scholar] [CrossRef] - Majda, A.; Embid, P. Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data. Theoret. Comput. Fluid Dyn.
**1998**, 11, 155–169. [Google Scholar] [CrossRef] - Whitehead, J.P.; Haut, T.; Wingate, B. The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: Variations from quasi-geostrophy. Theor. Comput. Fluid Dyn.
**2018**, 32, 713–732. [Google Scholar] [CrossRef] - Mu, P.; Ju, Q. Three-scale singular limits of the rotating stratified Boussinesq equations. Appl. Anal.
**2021**, 100, 2405–2417. [Google Scholar] [CrossRef] - Jones, D.A.; Mahalov, A.; Nicolaenko, B. A Numerical Study of an Operator Splitting Method for Rotating Flows with Large Ageostrophic Initial Data. Theor. Comput. Fluid Dyn.
**1999**, 13, 143. [Google Scholar] [CrossRef] - Haut, T.; Wingate, B. An asymptotic parallel-in-time method for highly oscillatory pdes. SIAM J. Sci. Comput.
**2014**, 36, A693–A713. [Google Scholar] [CrossRef] - Peddle, A.G.; Haut, T.; Wingate, B. Parareal convergence for oscillatory pdes with finite time-scale separation. SIAM J. On Scientific Comput.
**2019**, 41, A3476–A3497. [Google Scholar] [CrossRef] - Rosemeier, J.; Haut, T.; Wingate, B. Multi-level Parareal algorithm with Averaging for Oscillatory Problems. SIAM J. Sci. Comput.
**2023**. submitted. [Google Scholar] - Majda, A. Introduction to P.D.E.’s and Waves for the Atmosphere and Ocean; Courant Lecture Notes; New York University, Courant Institute of Mathematical Sciences and American Mathematical Society: New York, NY, USA, 2002; Volume 9. [Google Scholar]
- Burns, K.J.; Vasil, G.M.; Oishi, J.S.; Lecoanet, D.; Brown, B.P. Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods. Phys. Rev. Res.
**2020**, 2, 023068. [Google Scholar] [CrossRef] - Weinan, E.; Engquist, B. Multiscale modeling and computation. Not. Am. Math. Soc.
**2003**, 50, 1062–1070. [Google Scholar] - Engquist, B.; Tsai, R. Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput.
**2005**, 74, 1707–1742. [Google Scholar] [CrossRef]

**Figure 1.**The figure illustrates how the data sets for the mean flow representation are created. The gray boxes represent the data sets. The arrows show the operations which are applied to the data sets.

**Figure 2.**The total, kinetic, and potential energy (black lines) compared to their mean values (blue lines). The panel on the left shows the case for $N=10$ and $\eta $ = 0.1. The panel on the right shows the case for less frequent oscillations when $N=1$ and $\eta $ = 0.2. Solid lines are the total energy, dotted lines are the kinetic energy, and dashed lines are the potential energy. The solutions oscillate between kinetic and potential energy as they decay. The principal effect of the averaging window on the mapped and averaged solution is to track the frequency of the energetic exchanges between kinetic and potential energy and to reduce the magnitude of the oscillations. The case $N=10$ on the left suggests that the phase-averaged mean flow shows an oscillatory total energy.

**Figure 3.**This figure demonstrates the effect of the averaging window on oscillatory functions. In the left panel, the averaging window $\eta $ is smaller than the period of the oscillations. The data are barely altered. In the right panel, the averaging window $\eta $ is larger than the period of the oscillations. Therefore, the oscillations are damped and the mean value is between the peaks and valleys of the signal.

**Figure 4.**Time series at a single point for the 3rd component of each vector field. The top row is for $N=20$, middle for $N=10$, and bottom for $N=1$. Panels on the left show the time evolution of the data in the moving frame, while, on the right, they depict the evolution in the ordinary frame. Within each graph, colored lines showing the effect of changing the averaging window. As N decreases, there is less difference between the dynamics in the moving frame and the ordinary frame, an effect of the moving frame oscillating at a slower rate over the time scale of the simulation. There are more oscillations in the moving frame, where the averaging takes place, than in the ordinary frame.

**Figure 5.**Time evolution of the 3rd component of the 4 data sets described in Figure 1 for $N=10$. The top left panel shows $\mathbf{u}$ in the ordinary frame. The oscillatory pattern here is also seen in the potential energy shown in Figure 2. The large-scale pattern of oscillations between positive and negative values can be seen even as higher frequency waves appear. There are approximately 2 times more oscillations in $\mathbf{w}$ than in the ordinary domain. The appearance of the waves is more noticeable starting at approximately $t=1$. The lower right panel shows the time evolution of the average in the moving frame $\overline{\mathbf{w}}$, which smooths the higher frequency oscillations. Finally, mapping back to the ordinary frame, the upper right panel shows the result of the mapping and averaging on the variables in the ordinary frame, $\overline{\mathbf{u}}$.

**Figure 6.**Time evolution of the 3rd component of the 4 data sets described in Figure 1 for $N=1$. The difference between the ordinary frame (

**top left**panel) and mapped frame (

**bottom left**frame) is less pronounced than the corresponding frames for $N=10$ (Figure 5) because the oscillations in the exponential operator for $N=1$ are less frequent. The higher frequencies that appear later in the simulation are smoothed by the averaging, with the result in the top right panel, which shows the result of the mapping and averaging in the ordinary domain $\overline{\mathbf{u}}$.

**Figure 7.**This figure shows the departure from the mean for three different averaging windows, $\eta $, for the case when $N=10$, in the moving frame. As the averaging window, $\eta $, is increased, the excursions from the mean flow also increase. The excursions also show a periodic pattern.

**Figure 8.**This figure shows the time rate of change of the 3rd component of $\mathbf{w}$ and $\overline{\mathbf{w}}$. The wider the value of $\eta $, the smaller the amplitude of the time rate of change, and, in some cases, the less frequent the oscillations. While this could be useful for taking large time steps, the degree of regularity depends strongly on the value of $\eta $.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wingate, B.A.; Rosemeier, J.; Haut, T.
Mean Flow from Phase Averages in the 2D Boussinesq Equations. *Atmosphere* **2023**, *14*, 1523.
https://doi.org/10.3390/atmos14101523

**AMA Style**

Wingate BA, Rosemeier J, Haut T.
Mean Flow from Phase Averages in the 2D Boussinesq Equations. *Atmosphere*. 2023; 14(10):1523.
https://doi.org/10.3390/atmos14101523

**Chicago/Turabian Style**

Wingate, Beth A., Juliane Rosemeier, and Terry Haut.
2023. "Mean Flow from Phase Averages in the 2D Boussinesq Equations" *Atmosphere* 14, no. 10: 1523.
https://doi.org/10.3390/atmos14101523