# Log-Lattices for Atmospheric Flows

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## Abstract

**:**

## 1. Foreword by B. Dubrulle

## 2. Introduction

## 3. Log-Lattice Framework

#### 3.1. Energy Spectra

#### 3.2. Generalizations

#### 3.3. Limitations of Log-Lattices

## 4. Homogeneous Rotating Convections on Log-Lattices

#### 4.1. Definitions

#### 4.2. Non-Dimensional Numbers

- The Rayleigh number $\mathrm{Ra}=\alpha g{H}^{3}\Delta T/\left(\nu \kappa \right)$, which characterizes the forcing by the temperature gradient.
- The Prandtl number $\mathrm{Pr}=\nu /\kappa $, which is the ratio of the fluid viscosity to its thermal diffusivity.
- The Nusselt number $\mathrm{Nu}=JH/\kappa \Delta T$ that characterizes the mean total heat flux is the z direction is $J={\partial}_{z}<{u}_{z}\theta >-\kappa \Delta T$.
- The Ekman number $\mathrm{E}=\nu /\left(2\mathrm{\Omega}{H}^{2}\right)$, measuring the importance of the rotation with respect to the diffusive process.
- The Rossby number $\mathrm{Ro}=\sqrt{\alpha g\Delta T}/\left(2\mathrm{\Omega}\sqrt{H}\right)$, measuring the importance of the rotation with respect to buoyancy. In terms of other variables, we have $\mathrm{Ro}=\mathrm{E}\sqrt{\mathrm{Ra}}/\left(\sqrt{Pr}\right)$.
- The friction coefficient $F=f\sqrt{H}/\sqrt{\alpha g\Delta T}$, which provides the intensity of the Rayleigh damping.

#### 4.3. Equations on Log-Lattice

#### 4.4. Convection Onset

#### Onset at Zero Rotation

#### 4.5. Onset at Large Rotation

#### 4.6. Phenomenology When $F=0$

#### 4.6.1. Non-Rotating Case

- (I):
- When $\mathrm{Ra}\le {\mathrm{Ra}}_{c}$, we are in the laminar case. The fluid is at rest, $<{u}_{z}\theta >=0$, and the heat flux is only piloted by the Fourier law, so that $J=\kappa \Delta T/H$ and $Nu=1$.
- (II):
- Above the critical threshold for instability, when $\mathrm{Ra}>\sim {\mathrm{Ra}}_{c}$, convection sets in, $<{u}_{z}\theta >$ starts becoming positive, and we have $\mathrm{Nu}\sim {(\mathrm{Ra}-{\mathrm{Ra}}_{c})}^{\chi}$, where $\chi $ is an exponent characterizing the (super)-critical transition to convection.
- (III):
- When $\mathrm{Ra}\gg {\mathrm{Ra}}_{c}$, the turbulence becomes fully developed, and we are entering an “ultimate” regime (also called the Spiegel regime), in which the heat flux does not depend on the viscosity or the diffusivity any more. In that case, we have $\mathrm{Nu}\sim {\left(\mathrm{Ra}\mathrm{Pr}\right)}^{1/2}$ [20,21,22] and $Re\sim {(\mathrm{Ra}/\mathrm{Pr})}^{1/2}$ [23].

#### 4.6.2. Rotating Case

#### 4.6.3. Log-Lattice Simulation Details

## 5. Results

#### 5.1. Non-Rotating Case

#### 5.2. Rotating Case

#### Parameter Space and Critical Rayleigh Number

#### 5.3. Influence of Friction

#### 5.3.1. Laminar vs. Turbulent Regime

**Figure 6.**Ratio of the kinetic energy dissipated by friction, $F{U}^{2}$, to the kinetic energy dissipated by viscous processes, ${\u03f5}_{u}$, as a function of the Rayleigh number, Ra, and the Ekman number E. The points are color-coded by $log10\left(\mathrm{E}\right)$. The friction dominates for low-rotation and for $\mathrm{Ra}<{10}^{11}$.

**Figure 7.**Non-dimensional heat transfer Nu vs. Rayleigh number Ra in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The stars traces the conductive regime. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are highlighted by a black (respectively, white) square for the laminar (respectively, turbulent) regime. The black dashed line is $\mathrm{Nu}=20\sqrt{\mathrm{Ra}}$, corresponding to asymptotic non-rotating ultimate regime scaling. The red dotted line is $\mathrm{Nu}\sim {\mathrm{Ra}}^{3/2}$, corresponding to the geostrophic turbulent regime, see Figure 9 for an exact representation of the corresponding scaling law.

**Figure 8.**Vertical Reynolds number $Re=\sqrt{<{u}_{z}^{2}>}H/\nu $ as a function of Rayleigh number Ra in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number, E. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are highlighted by a black (respectively, white) square for the laminar (respectively, turbulent) regime. The black (respectively, red) dashed line follows the equation $y\sim {x}^{1/2}$ (respectively, $y\sim {x}^{1}$).

**Figure 9.**Universal law governing the heat transfer in the turbulent regime $\mathrm{Nu}\mathrm{E}$ as a function of $\mathrm{Ra}{\mathrm{E}}^{2}$ in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The symbols tagged by a white square trace the rotation-dominated regime, while the filled symbols trace the rotation-independent regime. The black dotted (respectively, dashed) line follows the equation $y\sim {x}^{3/2}$ (respectively, $y\sim {x}^{1/2}$).

#### 5.3.2. Influence of Rotation and Onset of Rotation-Dominated Regimes

#### 5.3.3. Temperature Fluctuation and Anisotropy

#### 5.3.4. Laminar and Turbulent Scaling Laws and GT Regimes

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Renormalized energy spectrum $E\left(k\right)/{\u03f5}^{2/3}{\eta}^{5/3}$ as a function of the renormalized wavenumber $k\eta $ for simulation of the Navier–Stokes equation on a ${20}^{3}$ log-lattice (black dotted line) and for a turbulent flow in Seymour Narrows (Ocean) (blue squares). The black dashed line corresponds to $E\left(k\right)\sim {k}^{-5/3}$. This figure was drawn by the authors using data extracted from the published graph shown in [1].

**Figure 2.**Non-dimensional heat transfer Nu vs. Rayleigh number Ra in 3D for $\mathrm{Pr}=0.7$ (circle) and $\mathrm{Pr}=1$ (squares). The dotted line corresponds to the empirical law: $\mathrm{Nu}=7{(\mathrm{Ra}-{\mathrm{Ra}}_{c})}^{3/2}/(\mathrm{Ra}/{\mathrm{Ra}}_{t}+1)$, with ${\mathrm{Ra}}_{c}=6\times {10}^{5}$ and ${\mathrm{Ra}}_{t}=5\times {10}^{6}$ that connects the near-convection onset regime to the asymptotic law $\mathrm{Nu}\sim 20\sqrt{\mathrm{Ra}}$ for large Ra, corresponding to asymptotic non-rotating ultimate regime scaling. This regime is itself split into a laminar regime (open symbols) and a turbulent regime (filled symbols).

**Figure 3.**Non-dimensional heat transfer Nu versus time t for $\mathrm{Ra}=3.46\times {10}^{11}$. The flow starts in a laminar regime, then abruptly transitions to a turbulent regime.

**Figure 4.**Parameter space covered by our log-lattice simulations at $\mathrm{Pr}=0.7$. The color of the symbols indicates the three possible regimes: conductive (white), transitional (yellow), and convective (black). The red line is the theoretical asymptotic prediction given by Equation (18). The magenta line is the theoretical convection threshold in the absence of rotation $\mathrm{Ra}=6\times {10}^{5}$. The green line has equation $\mathrm{Ra}=0.06{E}^{-2}$ and delineates regions of the parameter space where the turbulence is influenced by rotation (below the line) or not influenced by rotation (above the line), as diagnosed by the behavior of the kinetic energy dissipation, see Figure 5. The geostrophic turbulent regime is observed in between the red and the green line.

**Figure 5.**Non-dimensional energy dissipation ${\u03f5}_{u}=\nu <{(\nabla u)}^{2}>H/{U}^{3}=\mathrm{Ra}\mathrm{Nu}{\mathrm{Pr}}^{-2}/{\left(\mathrm{Ra}\mathrm{Pr}\right)}^{3/2}$ vs. vertical Rossby number ${\mathrm{Ro}}_{z}$ in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number, E. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are highlighted by a black (respectively, white) square for the laminar (respectively, turbulent) regime. The black (respectively, red) dotted line corresponds to ${\u03f5}_{u}\sim {\mathrm{Ro}}_{z}$ (respectively, ${\u03f5}_{u}\sim {\mathrm{Ro}}_{z}^{1/2}$), while the black (respectively, red) dashed line corresponds to ${\u03f5}_{u}=3.7$ (respectively, ${\u03f5}_{u}=1$).

**Figure 10.**Vertical Rossby number ${\mathrm{Ro}}_{z}$ vs. “turbulent” Rayleigh number $\mathrm{Ra}{E}^{2}$ in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are tagged by a black (respectively, white) square for the laminar (respectively, turbulent) regime. The black (respectively, red) dashed line follows $0.4{\mathrm{Ra}}^{1/2}E$ (respectively, $0.04{\mathrm{Ra}}^{1/2}E$).

**Figure 11.**Temperature fluctuations ${\theta}_{rms}=\sqrt{<{\theta}^{2}>}$ vs. Rayleigh number Ra in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are highlighted by a black (respectively, white) square for the laminar (respectively, turbulent) regime.

**Figure 12.**Velocity anisotropy $\sqrt{<{u}_{z}^{2}>}/\sqrt{<{u}^{2}>}$ vs. Rayleigh number Ra in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The open symbols trace the laminar regime, while the filled symbols trace the turbulent regime. The rotation-dominated regimes are highlighted by a black (respectively, white) square for the laminar (respectively, turbulent) regime.

**Figure 13.**Universal law governing the heat transfer in the laminar regime $\mathrm{Nu}{\mathrm{E}}^{2/3}$ as a function of $\mathrm{Ra}{\mathrm{E}}^{4/3}$ in 3D for $\mathrm{Pr}=0.7$ for rotating HRB simulations on log-lattices. The symbols are colored according to their Ekman number E. The open symbols containing by a black square trace the rotation-dominated regime, while the open symbols trace the rotation-independent regime. The red dotted line follows Equation (22), with $A=7$, ${\mathrm{Ra}}_{c}=22$ and ${\mathrm{Ra}}_{t}=5\times {10}^{2}$.

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**MDPI and ACS Style**

Pikeroen, Q.; Barral, A.; Costa, G.; Dubrulle, B.
Log-Lattices for Atmospheric Flows. *Atmosphere* **2023**, *14*, 1690.
https://doi.org/10.3390/atmos14111690

**AMA Style**

Pikeroen Q, Barral A, Costa G, Dubrulle B.
Log-Lattices for Atmospheric Flows. *Atmosphere*. 2023; 14(11):1690.
https://doi.org/10.3390/atmos14111690

**Chicago/Turabian Style**

Pikeroen, Quentin, Amaury Barral, Guillaume Costa, and Bérengère Dubrulle.
2023. "Log-Lattices for Atmospheric Flows" *Atmosphere* 14, no. 11: 1690.
https://doi.org/10.3390/atmos14111690