# Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres

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## Abstract

**:**

**27**), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.

## 1. Introduction

## 2. Results

#### 2.1. Oceanic Dataset

#### 2.2. DAHD, DAH Power Spectrum and DAHMs

#### 2.3. DAH-MSLM Oceanic Emulators

## 3. Discussion

## 4. Models and Methods

#### 4.1. Mid-Latitude Ocean Model

#### 4.2. Data-Adaptive Harmonic Decomposition

#### 4.2.1. DAH Eigenelements and Power Spectrum

#### 4.2.2. DAH Coefficients

#### 4.3. Multilayer Stuart–Landau Models

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DAH | Data-adaptive harmonic |

DAHD | Data-adaptive harmonic decomposition |

DAHC | Data-adaptive harmonic coefficient |

DAHM | Data-adaptive harmonic mode |

EOF | Empirical orthogonal function |

HRC | Harmonic reconstructed component |

LFV | Low frequency variability |

MSLM | Multilayer Stuart–Landau model |

PCA | Principal component analysis |

PSD | Power spectral density |

RC | Reconstructed component |

RP | Ruelle–Pollicott |

SDE | Stochastic differential equation |

SL | Stuart–Landau |

## Appendix A. Time Variability of Stochastic Systems and Ruelle–Pollicott Resonances

**Figure A1.**(

**a**) The Ruelle–Pollicott (RP) resonances are the isolated eigenvalues of the Fokker–Planck operator, $\mathcal{A}$, defined in (A3); they are represented by red dots in (

**b**) and by black dots here. The rightmost vertical line represents the imaginary axis above which the power spectrum lies; see (

**a**) for another perspective. The rate of decay of correlations is controlled by the spectral gap $\tau $ (not to be confused with $\tau $ in (A6)); see [56,85]. (

**b**) The imaginary part of the RP resonances corresponds to the location of a peak in the PSD (black curve lying above the imaginary axis) and the real part to its width. In blue is represented a reconstruction of the PSD based on RPs; a discrepancy is shown here to emphasize that in practice, the RPs are very often only estimated/approximated; see [56,71] (courtesy of Maciej Zworski). (

**a**) Schematic of the spectrum of $\mathcal{A}$ given in (A3). (

**b**) Correspondence between the PSD and RP resonances according to (A5).

## Appendix B. Estimating Resonances from Time Series: The Reduced RP Resonances

**Figure A2.**Reduced RP resonances and their analytic approximation. Left panel: The pair of DAHCs analyzed. Right panel: Corresponding eigenvalues of P as estimated by using (A7), after application of the logarithm (in red). The blue vertical line corresponds to the imaginary axis. These eigenvalues correspond to approximations of the point spectrum of the averaged Fokker–Planck operator given in (A8). The analytic approximations provided by (A17) are shown as green dots.

## Appendix C. RP Resonances of Stuart–Landau Models: Analytic Approximations

## Appendix D. Multilayer Stuart–Landau Modeling of DAHCs

**Figure A3.**Reduced RP resonances and their analytic approximation. Same as in Figure A2, but for the frequency $f=0.061$ cycle/${y}^{-1}$.

**Figure A4.**Reduced RP resonances and their analytic approximation. Same as in Figure A2, but for the frequency $f=5.27$ cycle/${y}^{-1}$.

## Appendix E. Modeling Skills in the EOF Space

**Figure A5.**Same format as in Figure A7, but the comparison of harmonic reconstruction (red) and DAH-MSLM stochastic simulation (blue) in a full frequency band $f<9.1$ ${y}^{-1}$.

**Figure A6.**Same format as in Figure A8, but the comparison of harmonic reconstruction (red) and DAH-MSLM stochastic simulation (blue) in a full frequency band $f<9.1$ ${y}^{-1}$.

**Figure A7.**Probability density function (PDF) of the twelve leading PCs of upper-layer stream function anomalies, DAH-filtered in a frequency band $f<5.27$ ${y}^{-1}$: Red, harmonic reconstruction of QG data (HRC; cf. (23)); blue, DAH-MSLM stochastic simulation.

**Figure A8.**Autocorrelation function (ACF) of the twelve leading PCs of upper-layer stream function anomalies, DAH-filtered in a frequency band $f<5.27$ ${y}^{-1}$: Red, harmonic reconstruction of QG data (HRC; cf. (23)); blue, DAH-MSLM stochastic simulation.

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**Figure 1.**Upper-level stream function anomalies. (Left) Top: standard deviation; bottom: snapshot of instantaneous flow. (Right) Same, but in the truncated subspace of 30 leading spatial empirical orthogonal functions (EOFs). The nondimensional units are arbitrary, but are the same for all panels.

**Figure 2.**DAH power spectrum ${\mathcal{P}}_{\ell}$ of 30 leading PCs of the upper layer stream function anomalies. Left panel: Each discrete set ${\mathcal{P}}_{\ell}$ consists of 30 eigenelements (equal to number of input dataset’s channels), corresponding to the pairs of DAH eigenvalues $|{\lambda}_{j}|$ and eigenvectors ${\mathbf{W}}_{j}$ at a given temporal frequency ${f}_{\ell}$ (see Section 4.2.1 and (15)–(17)). Figure 3 and Figure 4 show DAHMs associated with four largest $|{\lambda}_{j}|$ at two selected frequencies: cyan, decadal LFV peak ${f}_{D}=0.061$ y${}^{-1}$ ($\approx 17$ y); green, interannual ${f}_{I}=0.674$ y${}^{-1}$ ($\approx 1.5$ y). The right panel shows a magnification of the low-frequency part of the spectrum.

**Figure 3.**Left and center panels: Space-time patterns of data-adaptive harmonic mode (DAHM) pairs corresponding to the four largest $|{\lambda}_{j}|$ (in descending order) at decadal frequency $f=0.061$ y${}^{-1}$ (cyan dots in the data-adaptive harmonic (DAH) power spectrum of Figure 2). Each of the modes in a pair is time-shifted by a quarter of a period, i.e., in exact phase quadrature; x-axis, time embedding dimension (in years); y-axis, spacial dimension (rank of principal component (PC)). Right panels: DAHCs obtained by projection of the input dataset of 30 PCs onto the DAHMs; see (20). A DAHC pair consists of narrowband time series at the same temporal frequency of the associated DAHMs, but modulated in amplitude.

**Figure 5.**Manifestation in the physical domain of the leading pair of DAHMs (see Figure 3) at the decadal variability $f=0.061$ y${}^{-1}$; i.e., corresponding to the largest $|{\lambda}_{j}|$ (top cyan dot) in Figure 2. The resulting pattern is periodic (with period $\approx 16.39$ y). Here, eight oscillation phases labeled by time are shown.

**Figure 7.**Decadal harmonic reconstruction component (HRC) and its reduced RP resonances for PC1: quasi-geostrophic (QG) model and multilayer Stuart-Landau model (MSLM). (

**a**,

**b**) show the sum of the first four HRCs on PC1 for the upper-layer stream function anomalies as simulated from the QG model (red curve) and its DAH-MSLM emulator (blue curve). In both panels, the PC1 is shown in black: in (

**b**), PC1 is obtained after simulation of the DAH-MSLM emulator, whereas in (

**a**), PC1 is obtained from simulation of the QG model. The HRCs are computed according to (23) in Section 4.2.2 below, from the corresponding simulated data. (

**c**) shows the corresponding reduced RP resonances (see Appendix B) as estimated from HRCs shown in (

**a**,

**b**); colors match across panels (

**a–c**).

**Figure 8.**Upper-level stream function anomalies and standard deviation: QG vs. MSLM. The QG’s standard deviation (SD) and its MSLM emulation are shown here in Panels (

**a**,

**b**), respectively, for the low-frequency range $0<f<{f}_{1}=0.18$ y${}^{-1}$. Panels (

**c**,

**d**) depict typical flow patterns underlying the SD patterns for the QG model and MSLM, respectively. Panels (

**e**,

**f**) (resp. (

**g**,

**h**)): same as for Panels (

**a**,

**b**) (resp. (

**c**,

**d**)), but for the intermediate frequency range ${f}_{1}=0.18$ y${}^{-1}<f<{f}_{2}=5.27$ y${}^{-1}$.

**Figure 9.**Upper-level stream function anomalies and standard deviation: QG vs. MSLM. The QG’s standard deviation (SD) and its MSLM emulation are shown here in Panels (

**a**,

**b**), respectively, for the high-frequency range ${f}_{2}=5.27$ y${}^{-1}<f<{f}_{3}=9.1$ y${}^{-1}$. Panels (

**c**,

**d**) depict typical flow patterns underlying the SD patterns for the QG model and MSLM, respectively. Panels (

**e**,

**f**) (resp. (

**g**,

**h**)): same as for Panels (

**a**,

**b**) (resp. (

**c**,

**d**)), but for the full range of frequencies $0<f<{f}_{3}=9.1$ y${}^{-1}$.

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Kondrashov, D.; Chekroun, M.D.; Berloff, P. Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres. *Fluids* **2018**, *3*, 21.
https://doi.org/10.3390/fluids3010021

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Kondrashov D, Chekroun MD, Berloff P. Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres. *Fluids*. 2018; 3(1):21.
https://doi.org/10.3390/fluids3010021

**Chicago/Turabian Style**

Kondrashov, Dmitri, Mickaël D. Chekroun, and Pavel Berloff. 2018. "Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres" *Fluids* 3, no. 1: 21.
https://doi.org/10.3390/fluids3010021