^{1}

^{2}

^{2}

^{3}

^{2}

^{4}

^{4}

^{1}

^{2}

^{3}

^{4}

In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.

With the rapid development of computer technology and continuous improvement of global observing system’s performance in recent decades, many new reanalysis datasets (also known as meteorological big data) from advanced operational and research centers are built, upgraded, and opened to the meteorology community. Nowadays, it is still a great challenge to gain maximum valuable information and find new motion mechanisms from reanalysis datasets using a new method or powerful algorithm, although many scientists have made much progress. It is very important to accurately describe and understand the structural changes and instability mechanisms of the complex atmospheric motion in both operational numerical weather predictions and scientific researches [

The POD was introduced to the fluid dynamics community first by Lumley [

In DMD, the flow field is decomposed into a set of modes whose dynamics are governed by their corresponding eigenvalues, and the isolated eigenvalues define the temporal frequency and growth/decay rate of each mode [

The remainder of this article is organized as follows. In Section

Herein, a short review of the POD method is provided, whose details can be found in [

In applications of POD to fluid flow, we start with the vector field

The POD method aims to seek the optimal set of basis functions that can best represent the given flow field data, which can be obtained by solving the following eigenvalue problem:

The solution of equation (

Generally, only a subset of the POD modes retained to express the flow; the energy of these modes can be presented by

With the determination of the relative important POD modes, the flow field can be represented in a reduced dimensionality:

For practical applications, the decomposition can be carried out using the classical method or the snapshot method. In matrix form, the POD can be realized by the standard singular value decomposition (SVD) which is a robust approach to determine the POD modes [

In the field of atmospheric/marine science, whether the forecast field data generated by the high-resolution numerical forecasting model or the analysis field produced by the global variational data assimilation system, they are distributed in the regular space grid at regular intervals. And they only describe the state of atmospheric motion at a certain moment, that is, strictly speaking, an instantaneous state field (actually, the state changes little before and after the moment) which is equivalent to store using the camera after taking pictures of the atmospheric motion state. Therefore, it can be named as the atmospheric motion snapshot analogy to the concept of flow snapshot in fluid mechanics. Thus, in this paper, we use the modal decomposition methods to analyze the snapshots of atmospheric motion over some time.

Here, we collect data from a two-dimensional stratospheric jet flow velocity field

The key idea of the POD is to reduce the data dimensions by projecting high-dimensional data to a low-dimensional space while reserving the variation in the original variables as much as possible. It can be seen as an application of singular value decomposition (SVD) in a finite-dimensional space, and a brief review of the algorithm is shown as follows.

Consider the given snapshot matrix

The snapshots matrix

where

In practical applications, one usually retains

With the determination of the most important POD modes, the jet flow field can be represented only using the truncated modes, as follows:

To obtain the matrix

In this section, we first give a brief overview of the Koopman operator theory, which is the basis for DMD analysis, and then describe the basic procedure for the standard DMD and sparsity-promoting DMD method; the latter one leads to an optimal selection of the original DMD modes.

First, we also consider the discrete-time dynamical system mentioned in Section

The Koopman operator

The Koopman operator steps forward in time an observable. Note that

According to Rowley et al. [

Thus, the function

Under the action of Koopman operator, the evolution of vector function

Thus, the eigenvalue

In DMD architecture, consider finite measurements at times

In practice application, the state can be directly measured; thus,

It is easy to prove that the eigenvalues of the operator

Arrange the

The locally linear approximation in equation (

The matrix

For this reason, DMD circumvents the eigendecomposition of

Thus, the matrix

By projecting the high-dimensional matrix

The matrix

The matrix

Therefore, the snapshot at a time

Rewrite equation (

As there are no explicit ways for DMD to select modes, unlike the POD modes which are selected based on the rank of energy from the singular values. The sparsity-promoting DMD method is proposed by Jovanovic et al. which induces a sparse structure of the DMD modes by augmenting the

After the sparse structure of

A powerful and well-suited algorithm named Alternating Direction Method of Multipliers (ADMM) for solving the optimization problems of equation (

The sparsity-promoting DMD is a variant of the standard DMD method; here, the main steps of the SPDMD algorithm are summarized as follows:

First, a matrix

The SPDMD aims to solve the optimization problem in equation (

where

here

in which the

in which

in which

and the solution is determined by

where

The data used in this paper is the ERA5 daily reanalysis dataset, the horizontal resolution of the data is 0.25° × 0.25°, and the time resolution is 1 hour [

Before applying the POD and SPDMD to the reanalysis dataset, it seems instructive to treat the raw data to make ourselves familiar with the features of its output modes. For this reason, four snapshots of the original stratospheric 200 hPa jet are depicted to give a first impression of the flow field. In Figure

Four snapshots of the stratospheric 200 hPa jet at time. (a)

The POD and SPDMD methods are applied to decompose the snapshots of the stratospheric 200 hPa jet. Figure

(a) Percentage of the energy of each mode (b) and cumulative energy distribution of the 200 hPa jet.

The first eight POD spatial modes of the 200 hPa jet are shown in Figure

The first eight POD modes extracted from the 200 hPa jet (a–h correspond to the 1–8th modes; wind velocity visualized by vector plots and the background color contours visualize the vertical vorticity component).

Figure

The absolute value of the DMD amplitudes

In SPDMD, as our emphasis on sparsity increases, a smaller number of modes with nonzero amplitudes are obtained, and the fidelity of the reduced-order model decreases, as quantified in Figure

Performance loss with the number of retained DMD modes resulting from SPDMD for the 200 hPa jet.

As illustrated in Figure

Figure

Distribution of eigenvalues of 200 hPa jet: open circles represent full DMD eigenvalues; filled circles that are numbered with 1–5 are the modes isolated by the SPDMD: (a) full spectrum, (b) zoom on the isolated modes.

We make a change of units by taking the logarithm of

Growth rate and frequency of the five SPDMD modes of the 200 hPa jet flow field.

DMD mode | Growth rate | Frequency |
---|---|---|

1 | 8.4299 | 0 |

2 | 2.3828 | 4.1590 |

3 | −2.0152 | 1.7497 |

4 | −1.3161 | 3.2941 |

5 | −1.6680 | 5.3176 |

The real part of the DMD modes identified in Figure

The five DMD modes identified by the SPDMD from the 200 hPa jet flow (a–e correspond to the 1–5th modes, velocity is visualized by vector plots, and the background color contours visualize the vertical vorticity component).

To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, we consider constructing the POD/SPDMD reduced-order models, respectively. In POD, to reconstruct 99% of the total fluctuation energy, the POD modes are truncated at _{1,} and DM_{2,3,4,5} with their complex conjugates) attempted. The initial transient flow field at the sampling period

Initial velocity field of 200 hPa jet flow at (a)

The POD/SPDMD reconstruction flow fields at

POD reconstruction of the 200 hPa jet flow field (26 POD modes) at (a)

SPDMD reconstruction of the 200 hPa jet flow field (11 SPDMD modes) at (a)

The average relative error of the POD/SPDMD reduced-order model.

Number of modes | Error of | Error of |
---|---|---|

11 modes | 0.0971 | 0.1100 |

17 modes | 0.0671 | 0.0843 |

25 modes | 0.0593 | 0.0640 |

47 modes | 0.0443 | 0.0383 |

As shown in Table

In this paper, an advanced modal decomposition method named sparsity-promoting DMD (SPDMD) has been introduced into atmospheric motion flow-field analysis. And its performance and advantages are demonstrated on sequence snapshots of the stratospheric 200 hPa jet flow firstly and compared with the POD method. The SPDMD method can successfully identify the most important dynamic modes associated with their respective frequencies and growth rates, and the characteristics of each model, including the energy distribution and stability. The different dynamic modes with different spatial scales, which represent the motion components of different spatial scales of the jet flow, and the superposition of these modes can represent the main characteristics of the jet flow. Compared with the traditional modal decomposition method POD, fewer sparsity-promoting DMD modes can capture nearly the same coherent structures and provide a high-quality approximation of the original jet flow field.

The ERA5 daily reanalysis dataset used to support the findings of this study is provided from the Center for Medium-Range Weather Forecasts (ECMWF) (website:

The authors declare that they have no conflicts of interest regarding the publication of this paper.

This research was funded by the National Key R&D Program of China (Grant no. 2018YFC1506704) and the National Natural Science Foundation of China (Grant no. 41475094).