Efficient non-linear data assimilation in geophysical fluid dynamics

New ways of combining observations with numerical models are discussed in which the size of the state space can be very large, and the model can be highly nonlinear. Also the observations of the system can be related to the model variables in highly nonlinear ways, making this data-assimilation (or inverse) problem highly nonlinear. First we discuss the connection between data assimilation and inverse problems, including regularization. We explore the choice of proposal density in a Particle Filter and show how the ’curse of dimensionality’ might be beaten. In the standard Particle Filter ensembles of model runs are propagated forward in time until observations are encountered, rendering it a pure Monte-Carlo method. In large-dimensional systems this is very inefficient and very large numbers of model runs are needed to solve the data-assimilation problem realistically. In our approach we steer all model runs towards the observations resulting in a much more efficient method. By further ’ensuring almost equal weight’ we avoid performing model runs that are useless in the end. Results are shown for the 40 and 1000 dimensional Lorenz 1995 model.

Resonant gravity-wave drag enhancement in linear stratified flow over mountains

High-drag states produced in stratified flow over a 2D ridge and an axisymmetric mountain are investigated using a linear, hydrostatic, analytical model. A wind profile is assumed where the background velocity is constant up to a height z1 and then decreases linearly, and the internal gravity-wave solutions are calculated exactly. In flow over a 2D ridge, the normalized surface drag is given by a closed-form analytical expression, while in flow over an axisymmetric mountain it is given by an expression involving a simple 1D integral. The drag is found to depend on two dimensionless parameters: a dimensionless height formed with z_1, and the Richardson number, Ri, in the shear layer. The drag oscillates as z_1 increases, with a period of half the hydrostatic vertical wavelength of the gravity waves. The amplitude of this modulation increases as Ri decreases. This behaviour is due to wave reflection at z_1. Drag maxima correspond to constructive interference of the upward- and downward-propagating waves in the region z < z_1, while drag minima correspond to destructive interference. The reflection coefficient at the interface z = z_1 increases as Ri decreases. The critical level, z_c, plays no role in the drag amplification. A preliminary numerical treatment of nonlinear effects is presented, where z_c appears to become more relevant, and flow over a 2D ridge qualitatively changes its character. But these effects, and their connection with linear theory, still need to be better understood.

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Linear viscoelasticity from molecular dynamics simulation of entangled polymers

The linear viscoelastic (LVE) spectrum is one of the primary fingerprints of polymer solutions and melts, carrying information about most relaxation processes in the system. Many single chain theories and models start with predicting the LVE spectrum to validate their assumptions. However, until now, no reliable linear stress relaxation data were available from simulations of multichain systems. In this work, we propose a new efficient way to calculate a wide variety of correlation functions and mean-square displacements during simulations without significant additional CPU cost. Using this method, we calculate stress−stress autocorrelation functions for a simple bead−spring model of polymer melt for a wide range of chain lengths, densities, temperatures, and chain stiffnesses. The obtained stress−stress autocorrelation functions were compared with the single chain slip−spring model in order to obtain entanglement related parameters, such as the plateau modulus or the molecular weight between entanglements. Then, the dependence of the plateau modulus on the packing length is discussed. We have also identified three different contributions to the stress relaxation:  bond length relaxation, colloidal and polymeric. Their dependence on the density and the temperature is demonstrated for short unentangled systems without inertia.

Linear amplification of marginally neutral baroclinic waves

Baroclinic wave development is investigated for unstable parallel shear flows in the limit of vanishing normal-mode growth rate. This development is described in terms of the propagation and interaction mechanisms of two coherent structures, called counter-propagating Rossby waves (CRWs). It is shown that, in this limit of vanishing normal-mode growth rate, arbitrary initial conditions produce sustained linear amplification of the marginally neutral normal mode (mNM). This linear excitation of the mNM is subsequently interpreted in terms of a resonance phenomenon. Moreover, while the mathematical character of the normal-mode problem changes abruptly as the bifurcation point in the dispersion diagram is encountered and crossed, it is shown that from an initial-value viewpoint, this transition is smooth. Consequently, the resonance interpretation remains relevant (albeit for a finite time) for wavenumbers slightly different from the ones defining cut-off points. The results are further applied to a two-layer version of the classic Eady model in which the upper rigid lid has been replaced by a simple stratosphere.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

Decadal predictability of the Atlantic Ocean in a coupled GCM: forecast skill and optimal perturbations using Linear Inverse Modelling

The decadal predictability of three-dimensional Atlantic Ocean anomalies is examined in a coupled global climate model (HadCM3) using a Linear Inverse Modelling (LIM) approach. It is found that the evolution of temperature and salinity in the Atlantic, and the strength of the meridional overturning circulation (MOC), can be effectively described by a linear dynamical system forced by white noise. The forecasts produced using this linear model are more skillful than other reference forecasts for several decades. Furthermore, significant non-normal amplification is found under several different norms. The regions from which this growth occurs are found to be fairly shallow and located in the far North Atlantic. Initially, anomalies in the Nordic Seas impact the MOC, and the anomalies then grow to fill the entire Atlantic basin, especially at depth, over one to three decades. It is found that the structure of the optimal initial condition for amplification is sensitive to the norm employed, but the initial growth seems to be dominated by MOC-related basin scale changes, irrespective of the choice of norm. The consistent identification of the far North Atlantic as the most sensitive region for small perturbations suggests that additional observations in this region would be optimal for constraining decadal climate predictions.

Stratospheric influence on baroclinic lifecycles and its connection to the Arctic Oscillation

Using an idealized primitive equation model, we investigate how stratospheric conditions alter the development of baroclinic instability in the troposphere. Starting from the lifecycle paradigm of Thorncroft et al., we consider the evolution of baroclinic lifecycles resulting from the addition of a stratospheric jet to the LC1 initial condition. We find that the addition of the stratospheric jet yields a net surface geopotential height anomaly that strongly resembles the Arctic Oscillation. With the additional modification of the tropospheric winds to resemble the high-AO climatology, the surface response is amplified by a factor 10 and, though dominated by the tropospheric changes, shows similar sensitivity to the stratospheric conditions.