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Grooms I.,Courant Institute of Mathematical Sciences | Lee Y.,Courant Institute of Mathematical Sciences | Majda A.J.,Courant Institute of Mathematical Sciences | Majda A.J.,Center for Prototype Climate Modelling
Journal of Computational Physics | Year: 2014

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 -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. © 2014. Source


Majda A.J.,Courant Institute of Mathematical Sciences | Majda A.J.,Center for Prototype Climate Modelling | Grooms I.,Courant Institute of Mathematical Sciences
Journal of Computational Physics | Year: 2014

This is a research expository paper regarding superparameterization, a class of multi-scale numerical methods designed to cope with the intermittent multi-scale effects of inhomogeneous geophysical turbulence where energy often inverse-cascades from the unresolved scales to the large scales through the effects of waves, jets, vortices, and latent heat release from moist processes. Original as well as sparse space-time superparameterization algorithms are discussed for the important case of moist atmospheric convection including the role of multi-scale asymptotic methods in providing self-consistent constraints on superparameterization algorithms and related deterministic and stochastic multi-cloud parameterizations. Test models for the statistical numerical analysis of superparameterization algorithms are discussed both to elucidate the performance of the basic algorithms and to test their potential role in efficient multi-scale data assimilation. The very recent development of grid-free seamless stochastic superparameterization methods for geophysical turbulence appropriate for "eddy-permitting" mesoscale ocean turbulence is presented here including a general formulation and illustrative applications to two-layer quasigeostrophic turbulence, and another difficult test case involving one-dimensional models of dispersive wave turbulence. This last test case has randomly generated solitons as coherent structures which collapse and radiate wave energy back to the larger scales, resulting in strong direct and inverse turbulent energy cascades. © 2013 Elsevier Inc. Source


Grooms I.,Courant Institute of Mathematical Sciences | Majda A.J.,Courant Institute of Mathematical Sciences | Majda A.J.,Center for Prototype Climate Modelling
Journal of Computational Physics | Year: 2014

In this article we expand and develop the authors' recent proposed methodology for efficient stochastic superparameterization algorithms for geophysical turbulence. Geophysical turbulence is characterized by significant intermittent cascades of energy from the unresolved to the resolved scales resulting in complex patterns of waves, jets, and vortices. Conventional superparameterization simulates large scale dynamics on a coarse grid in a physical domain, and couples these dynamics to high-resolution simulations on periodic domains embedded in the coarse grid. Stochastic superparameterization replaces the nonlinear, deterministic eddy equations on periodic embedded domains by quasilinear stochastic approximations on formally infinite embedded domains. The result is a seamless algorithm which never uses a small scale grid and is far cheaper than conventional SP, but with significant success in difficult test problems.Various design choices in the algorithm are investigated in detail here, including decoupling the timescale of evolution on the embedded domains from the length of the time step used on the coarse grid, and sensitivity to certain assumed properties of the eddies (e.g. the shape of the assumed eddy energy spectrum). We present four closures based on stochastic superparameterization which elucidate the properties of the underlying framework: a 'null hypothesis' stochastic closure that uncouples the eddies from the mean, a stochastic closure with nonlinearly coupled eddies and mean, a nonlinear deterministic closure, and a stochastic closure based on energy conservation. The different algorithms are compared and contrasted on a stringent test suite for quasigeostrophic turbulence involving two-layer dynamics on a β-plane forced by an imposed background shear.The success of the algorithms developed here suggests that they may be fruitfully applied to more realistic situations. They are expected to be particularly useful in providing accurate and efficient stochastic parameterizations for use in ensemble-based state estimation and prediction. © 2013 Elsevier Inc. Source


Grooms I.,Courant Institute of Mathematical Sciences | Majda A.J.,Courant Institute of Mathematical Sciences | Majda A.J.,Center for Prototype Climate Modelling | Smith K.S.,Courant Institute of Mathematical Sciences | Smith K.S.,Center for Prototype Climate Modelling
Ocean Modelling | Year: 2014

Stochastic superparameterization, a stochastic parameterization framework based on a multiscale formalism, is developed for mesoscale eddy parameterization in coarse-resolution ocean modeling. The framework of stochastic superparameterization is reviewed and several configurations are implemented and tested in a quasigeostrophic channel model - an idealized representation of the Antarctic Circumpolar Current. Five versions of the Gent-McWilliams (GM) parameterization are also implemented and tested for comparison. Skill is measured using the time-mean and temporal variability separately, and in combination using the relative entropy in the single-point statistics. Among all the models, those with the more accurate mean state have the less accurate variability, and vice versa. Stochastic superparameterization results in improved climate fidelity in comparison with GM parameterizations as measured by the relative entropy. In particular, configurations of stochastic superparameterization that include stochastic Reynolds stress terms in the coarse model equations, corresponding to kinetic energy backscatter, perform better than models that only include isopycnal height smoothing. © 2014 Elsevier Ltd. Source

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