On instabilities in data assimilation algorithms

Data assimilation algorithms are a crucial part of operational systems in numerical weather prediction, hydrology and climate science, but are also important for dynamical reconstruction in medical applications and quality control for manufacturing processes. Usually, a variety of diverse measurement data are employed to determine the state of the atmosphere or to a wider system including land and oceans. Modern data assimilation systems use more and more remote sensing data, in particular radiances measured by satellites, radar data and integrated water vapor measurements via GPS/GNSS signals. The inversion of some of these measurements are ill-posed in the classical sense, i.e. the inverse of the operator H which maps the state onto the data is unbounded. In this case, the use of such data can lead to significant instabilities of data assimilation algorithms. The goal of this work is to provide a rigorous mathematical analysis of the instability of well-known data assimilation methods. Here, we will restrict our attention to particular linear systems, in which the instability can be explicitly analyzed. We investigate the three-dimensional variational assimilation and four-dimensional variational assimilation. A theory for the instability is developed using the classical theory of ill-posed problems in a Banach space framework. Further, we demonstrate by numerical examples that instabilities can and will occur, including an example from dynamic magnetic tomography.

Coupled Kelvin-wave and mirage-wave instabilities in semigeostrophic dynamics

A weak instability mode, associated with phase-locked counterpropagating coastal Kelvin waves in horizontal anticyclonic shear, is found in the semigeostrophic (SG) equations for stratified flow in a channel. This SG instability mode approximates a similar mode found in the Euler equations in the limit in which particle-trajectory slopes are much smaller than f/N, where f is the Coriolis frequency and N > f the buoyancy frequency. Though weak under normal parameter conditions, this instability mode is of theoretical interest because its existence accounts for the failure of an Arnol’d-type stability theorem for the SG equations. In the opposite limit, in which the particle motion is purely vertical, the Euler equations allow only buoyancy oscillations with no horizontal coupling. The SG equations, on the other hand, allow a physically spurious coastal “mirage wave,” so called because its velocity field vanishes despite a nonvanishing disturbance pressure field. Counterpropagating pairs of these waves can phase-lock to form a spurious “mirage-wave instability.” Closer examination shows that the mirage wave arises from failure of the SG approximations to be self-consistent for trajectory slopes f/N.

Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

Anthropogenic warming of the oceans: Observations and model results

Observations show the oceans have warmed over the past 40 yr. with appreciable regional variation and more warming at the surface than at depth. Comparing the observations with results from two coupled ocean-atmosphere climate models [the Parallel Climate Model version 1 (PCM) and the Hadley Centre Coupled Climate Model version 3 (HadCM3)] that include anthropogenic forcing shows remarkable agreement between the observed and model-estimated warming. In this comparison the models were sampled at the same locations as gridded yearly observed data. In the top 100 m of the water column the warming is well separated from natural variability, including both variability arising from internal instabilities of the coupled ocean-atmosphere climate system and that arising from volcanism and solar fluctuations. Between 125 and 200 m the agreement is not significant, but then increases again below this level, and remains significant down to 600 m. Analysis of PCM's heat budget indicates that the warming is driven by an increase in net surface heat flux that reaches 0.7 W m(-2) by the 1990s; the downward longwave flux increases bv 3.7 W m(-2). which is not fully compensated by an increase in the upward longwave flux of 2.2 W m(-2). Latent and net solar heat fluxes each decrease by about 0.6 W m(-2). The changes in the individual longwave components are distinguishable from the preindustrial mean by the 1920s, but due to cancellation of components. changes in the net surface heat flux do not become well separated from zero until the 1960s. Changes in advection can also play an important role in local ocean warming due to anthropogenic forcing, depending, on the location. The observed sampling of ocean temperature is highly variable in space and time. but sufficient to detect the anthropogenic warming signal in all basins, at least in the surface layers, bv the 1980s.

HOx observations over West Africa during AMMA: impact of isoprene and NOx

Aircraft OH and HO2 measurements made over West Africa during the AMMA field campaign in summer 2006 have been investigated using a box model constrained to observations of long-lived species and physical parameters. "Good" agreement was found for HO2 (modelled to observed gradient of 1.23 ± 0.11). However, the model significantly overpredicts OH concentrations. The reasons for this are not clear, but may reflect instrumental instabilities affecting the OH measurements. Within the model, HOx concentrations in West Africa are controlled by relatively simple photochemistry, with production dominated by ozone photolysis and reaction of O(1D) with water vapour, and loss processes dominated by HO2 + HO2 and HO2 + RO2. Isoprene chemistry was found to influence forested regions. In contrast to several recent field studies in very low NOx and high isoprene environments, we do not observe any dependence of model success for HO2 on isoprene and attribute this to efficient recycling of HOx through RO2 + NO reactions under the moderate NOx concentrations (5–300 ppt NO in the boundary layer, median 76 ppt) encountered during AMMA. This suggests that some of the problems with understanding the impact of isoprene on atmospheric composition may be limited to the extreme low range of NOx concentrations.

Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability

Waves with periods shorter than the inertial period exist in the atmosphere (as inertia-gravity waves) and in the oceans (as Poincaré and internal gravity waves). Such waves owe their origin to various mechanisms, but of particular interest are those arising either from local secondary instabilities or spontaneous emission due to loss of balance. These phenomena have been studied in the laboratory, both in the mechanically-forced and the thermally-forced rotating annulus. Their generation mechanisms, especially in the latter system, have not yet been fully understood, however. Here we examine short period waves in a numerical model of the rotating thermal annulus, and show how the results are consistent with those from earlier laboratory experiments. We then show how these waves are consistent with being inertia-gravity waves generated by a localised instability within the thermal boundary layer, the location of which is determined by regions of strong shear and downwelling at certain points within a large-scale baroclinic wave flow. The resulting instability launches small-scale inertia-gravity waves into the geostrophic interior of the flow. Their behaviour is captured in fully nonlinear numerical simulations in a finite-difference, 3D Boussinesq Navier-Stokes model. Such a mechanism has many similarities with those responsible for launching small- and meso-scale inertia-gravity waves in the atmosphere from fronts and local convection.

Shear banding in molecular dynamics of polymer melts

In order to establish constitutive equations for a viscoelastic fluid uniform shear flow is usually required. However, in the last 10 years S. Q. Wang and co-workers have demonstrated that some entangled polymers do not flow with the uniform shear rate as usually assumed, but instead choose to separate into fast and slow flowing regions. This phenomenon, known as shear banding, causes flow instabilities and in principle invalidates all rheological measurements when it occurs. In this Letter we report the first observation of shear banding in molecular dynamics simulations of entangled polymer melts. We show that our observations are in a very good agreement with the phenomenology developed by Fielding and Olmsted. Our findings provide a simple way of validating the empirical macroscopic phenomenology of shear banding. © 2012 American Physical Society

Large-scale dynamics of the mesosphere and lower thermosphere: an analysis using the extended Canadian Middle Atmosphere Model

The extended Canadian Middle Atmosphere Model is used to investigate the large-scale dynamics of the mesosphere and lower thermosphere (MLT). It is shown that the 4-day wave is substantially amplified in southern polar winter in the presence of instabilities arising from strong vertical shears in the MLT zonal mean zonal winds brought about by parameterized nonorographic gravity wave drag. A weaker 4-day wave in northern polar winter is attributed to the weaker wind shears that result from weaker parameterized wave drag. The 2-day wave also exhibits a strong dependence on zonal wind shears, in agreement with previous modeling studies. In the equatorial upper mesosphere, the migrating diurnal tide provides most of the resolved westward wave forcing, which varies semiannually in conjunction with the tide itself; resolved forcing by eastward traveling disturbances is dominated by smaller scales. Nonmigrating tides and other planetary-scale waves play only a minor role in the zonal mean zonal momentum budget in the tropics at these heights. Resolved waves are shown to play a significant role in the zonal mean meridional momentum budget in the MLT, impacting significantly on gradient wind balance. Balance fails at low latitudes as a result of a strong Reynolds stress associated with the migrating diurnal tide, an effect which is most pronounced at equinox when the tide is strongest. Resolved and parameterized waves account for most of the imbalance at higher latitudes in summer. This results in the gradient wind underestimating the actual eastward wind reversal by up to 40%.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat “patchy” connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a “lattice-directed” traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Optical characterization of the growth and spatial structure of a substorm onset arc

We present a detailed case study of the characteristics of auroral forms that constitute the first ionospheric signatures of substorm expansion phase onset. Analysis of the optical frequency and along-arc (azimuthal) wave number spectra provides the strongest constraint to date on the potential mechanisms and instabilities in the near-Earth magnetosphere that accompany auroral onset and which precede poleward arc expansion and auroral breakup. We evaluate the frequency and growth rates of the auroral forms as a function of azimuthal wave number to determine whether these wave characteristics are consistent with current models of the substorm onset mechanism. We find that the frequency, spatial scales, and growth rates of the auroral forms are most consistent with the cross-field current instability or a ballooning instability, most likely triggered close to the inner edge of the ion plasma sheet. This result is supportive of a near-Earth plasma sheet initiation of the substorm expansion phase. We also present evidence that the frequency and phase characteristics of the auroral undulations may be generated via resonant processes operating along the geomagnetic field. Our observations provide the most powerful constraint to date on the ionospheric manifestation of the physical processes operating during the first few minutes around auroral substorm onset.

On nonlinear symmetric stability and the nonlinear saturation of symmetric instability

A nonlinear symmetric stability theorem is derived in the context of the f-plane Boussinesq equations, recovering an earlier result of Xu within a more general framework. The theorem applies to symmetric disturbances to a baroclinic basic flow, the disturbances having arbitrary structure and magnitude. The criteria for nonlinear stability are virtually identical to those for linear stability. As in Xu, the nonlinear stability theorem can be used to obtain rigorous upper bounds on the saturation amplitude of symmetric instabilities. In a simple example, the bounds are found to compare favorably with heuristic parcel-based estimates in both the hydrostatic and non-hydrostatic limits.

Rossby number expansions, slaving principles, and balance dynamics

We consider the problem of constructing balance dynamics for rapidly rotating fluid systems. It is argued that the conventional Rossby number expansion—namely expanding all variables in a series in Rossby number—is secular for all but the simplest flows. In particular, the higher-order terms in the expansion grow exponentially on average, and for moderate values of the Rossby number the expansion is, at best, useful only for times of the order of the doubling times of the instabilities of the underlying quasi-geostrophic dynamics. Similar arguments apply in a wide class of problems involving a small parameter and sufficiently complex zeroth-order dynamics. A modified procedure is proposed which involves expanding only the fast modes of the system; this is equivalent to an asymptotic approximation of the slaving relation that relates the fast modes to the slow modes. The procedure is systematic and thus capable, at least in principle, of being carried to any order—unlike procedures based on truncations. We apply the procedure to construct higher-order balance approximations of the shallow-water equations. At the lowest order quasi-geostrophy emerges. At the next order the system incorporates gradient-wind balance, although the balance relations themselves involve only linear inversions and hence are easily applied. There is a large class of reduced systems associated with various choices for the slow variables, but the simplest ones appear to be those based on potential vorticity.

Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere

The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjørtoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered.