# Ensemble Kalman filters for dynamical systems with unresolved turbulence

## Abstract

Ensemble Kalman filters are developed for turbulent dynamical systems where the forecast model does not resolve all the active scales of motion. Coarse-resolution models are intended to predict the large-scale part of the true dynamics, but observations invariably include contributions from both the resolved large scales and the unresolved small scales. The error due to the contribution of unresolved scales to the observations, called ‘representation’ or ‘representativeness’ error, is often included as part of the observation error, in addition to the raw measurement error, when estimating the large-scale part of the system. It is here shown how stochastic superparameterization (a multiscale method for subgridscale parameterization) can be used to provide estimates of the statistics of the unresolved scales. In addition, a new framework is developed wherein small-scale statistics can be used to estimate both the resolved and unresolved components of the solution. The one-dimensional test problem from dispersive wave turbulence used here is computationally tractable yet is particularly difficult for filtering because of the non-Gaussian extreme event statistics and substantial small scale turbulence: a shallow energy spectrum proportional to k{sup −5/6} (where k is the wavenumber) results in two-thirds of the climatological variance being carried by the unresolved small scales.more »

- Authors:

- Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 (United States)
- (United Arab Emirates)

- Publication Date:

- OSTI Identifier:
- 22382110

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 273; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENERGY SPECTRA; ERRORS; FILTERS; INCLUSIONS; MATHEMATICAL MODELS; MATHEMATICAL SOLUTIONS; ONE-DIMENSIONAL CALCULATIONS; RESOLUTION; STATISTICS; STOCHASTIC PROCESSES; TURBULENCE

### Citation Formats

```
Grooms, Ian, E-mail: grooms@cims.nyu.edu, Lee, Yoonsang, Majda, Andrew J., and Center for Prototype Climate Modelling, NYU Abu Dhabi, Abu Dhabi.
```*Ensemble Kalman filters for dynamical systems with unresolved turbulence*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.05.037.

```
Grooms, Ian, E-mail: grooms@cims.nyu.edu, Lee, Yoonsang, Majda, Andrew J., & Center for Prototype Climate Modelling, NYU Abu Dhabi, Abu Dhabi.
```*Ensemble Kalman filters for dynamical systems with unresolved turbulence*. United States. doi:10.1016/J.JCP.2014.05.037.

```
Grooms, Ian, E-mail: grooms@cims.nyu.edu, Lee, Yoonsang, Majda, Andrew J., and Center for Prototype Climate Modelling, NYU Abu Dhabi, Abu Dhabi. Mon .
"Ensemble Kalman filters for dynamical systems with unresolved turbulence". United States. doi:10.1016/J.JCP.2014.05.037.
```

```
@article{osti_22382110,
```

title = {Ensemble Kalman filters for dynamical systems with unresolved turbulence},

author = {Grooms, Ian, E-mail: grooms@cims.nyu.edu and Lee, Yoonsang and Majda, Andrew J. and Center for Prototype Climate Modelling, NYU Abu Dhabi, Abu Dhabi},

abstractNote = {Ensemble Kalman filters are developed for turbulent dynamical systems where the forecast model does not resolve all the active scales of motion. Coarse-resolution models are intended to predict the large-scale part of the true dynamics, but observations invariably include contributions from both the resolved large scales and the unresolved small scales. The error due to the contribution of unresolved scales to the observations, called ‘representation’ or ‘representativeness’ error, is often included as part of the observation error, in addition to the raw measurement error, when estimating the large-scale part of the system. It is here shown how stochastic superparameterization (a multiscale method for subgridscale parameterization) can be used to provide estimates of the statistics of the unresolved scales. In addition, a new framework is developed wherein small-scale statistics can be used to estimate both the resolved and unresolved components of the solution. The one-dimensional test problem from dispersive wave turbulence used here is computationally tractable yet is particularly difficult for filtering because of the non-Gaussian extreme event statistics and substantial small scale turbulence: a shallow energy spectrum proportional to k{sup −5/6} (where k is the wavenumber) results in two-thirds of the climatological variance being carried by the unresolved small scales. Because the unresolved scales contain so much energy, filters that ignore the representation error fail utterly to provide meaningful estimates of the system state. Inclusion of a time-independent climatological estimate of the representation error in a standard framework leads to inaccurate estimates of the large-scale part of the signal; accurate estimates of the large scales are only achieved by using stochastic superparameterization to provide evolving, large-scale dependent predictions of the small-scale statistics. Again, because the unresolved scales contain so much energy, even an accurate estimate of the large-scale part of the system does not provide an accurate estimate of the true state. By providing simultaneous estimates of both the large- and small-scale parts of the solution, the new framework is able to provide accurate estimates of the true system state.},

doi = {10.1016/J.JCP.2014.05.037},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 273,

place = {United States},

year = {2014},

month = {9}

}