A regime-dependent balanced control variable based on potential vorticity

In this paper it is argued that rotational wind is not the best choice of leading control variable for variational data assimilation, and an alternative is suggested and tested. A rotational wind parameter is used in most global variational assimilation systems as a pragmatic way of approximately representing the balanced component of the assimilation increments. In effect, rotational wind is treated as a proxy for potential vorticity, but one that it is potentially not a good choice in flow regimes characterised by small Burger number. This paper reports on an alternative set of control variables which are based around potential vorticity. This gives rise to a new formulation of the background error covariances for the Met Office's variational assimilation system, which leads to flow dependency. It uses similar balance relationships to traditional schemes, but recognises the existence of unbalanced rotational wind which is used with a new anti-balance relationship. The new scheme is described and its performance is evaluated and compared to a traditional scheme using a sample of diagnostics.

Inertia-Gravity Waves Emitted from Balanced Flow: Observations, Properties, and Consequences

This paper describes laboratory observations of inertia–gravity waves emitted from balanced fluid flow. In a rotating two-layer annulus experiment, the wavelength of the inertia–gravity waves is very close to the deformation radius. Their amplitude varies linearly with Rossby number in the range 0.05–0.14, at constant Burger number (or rotational Froude number). This linear scaling challenges the notion, suggested by several dynamical theories, that inertia–gravity waves generated by balanced motion will be exponentially small. It is estimated that the balanced flow leaks roughly 1% of its energy each rotation period into the inertia–gravity waves at the peak of their generation. The findings of this study imply an inevitable emission of inertia–gravity waves at Rossby numbers similar to those of the large-scale atmospheric and oceanic flow. Extrapolation of the results suggests that inertia–gravity waves might make a significant contribution to the energy budgets of the atmosphere and ocean. In particular, emission of inertia–gravity waves from mesoscale eddies may be an important source of energy for deep interior mixing in the ocean.

Learning a procedural task: effectiveness of multimedia presentations

Three experiments investigated the effectiveness of presenting procedural information through different media and their combinations. Experiment 1 examined the effectiveness of text, line drawings, text and line drawings, video. and video stills for learning a first aid task. The results showed an advantage of text and line drawings and of the video presentation over the other three conditions for both bandaging performance and answering questions about the task. Experiment 2 showed that the beneficial effect of the combination of text and pictures could not be accounted for simply in terms of a dual coding explanation. Rather, the effectiveness of the media and their combinations was influenced by the extent to which they conveyed action information. Finally, Experiment 3 showed no evidence of a contiguity effect: text and pictures were as effective when presented together on the same screen as when they were presented separately. Copyright © 2000 John Wiley & Sons, Ltd.

Precision of quasi-optical null-balanced bridge techniques for transmission and reflection coefficient measurements

The precision of quasioptical null-balanced bridge instruments for transmission and reflection coefficient measurements at millimeter and submillimeter wavelengths is analyzed. A Jones matrix analysis is used to describe the amount of power reaching the detector as a function of grid angle orientation, sample transmittance/reflectance and phase delay. An analysis is performed of the errors involved in determining the complex transmission and reflection coefficient after taking into account the quantization error in the grid angle and micrometer readings, the transmission or reflection coefficient of the sample, the noise equivalent power of the detector, the source power and the post-detection bandwidth. For a system fitted with a rotating grid with resolution of 0.017 rad and a micrometer quantization error of 1 μm, a 1 mW source, and a detector with a noise equivalent power 5×10−9 W Hz−1/2, the maximum errors at an amplitude transmission or reflection coefficient of 0.5 are below ±0.025.

Four-dimensional data assimilation and balanced dynamics

The problem of spurious excitation of gravity waves in the context of four-dimensional data assimilation is investigated using a simple model of balanced dynamics. The model admits a chaotic vortical mode coupled to a comparatively fast gravity wave mode, and can be initialized such that the model evolves on a so-called slow manifold, where the fast motion is suppressed. Identical twin assimilation experiments are performed, comparing the extended and ensemble Kalman filters (EKF and EnKF, respectively). The EKF uses a tangent linear model (TLM) to estimate the evolution of forecast error statistics in time, whereas the EnKF uses the statistics of an ensemble of nonlinear model integrations. Specifically, the case is examined where the true state is balanced, but observation errors project onto all degrees of freedom, including the fast modes. It is shown that the EKF and EnKF will assimilate observations in a balanced way only if certain assumptions hold, and that, outside of ideal cases (i.e., with very frequent observations), dynamical balance can easily be lost in the assimilation. For the EKF, the repeated adjustment of the covariances by the assimilation of observations can easily unbalance the TLM, and destroy the assumptions on which balanced assimilation rests. It is shown that an important factor is the choice of initial forecast error covariance matrix. A balance-constrained EKF is described and compared to the standard EKF, and shown to offer significant improvement for observation frequencies where balance in the standard EKF is lost. The EnKF is advantageous in that balance in the error covariances relies only on a balanced forecast ensemble, and that the analysis step is an ensemble-mean operation. Numerical experiments show that the EnKF may be preferable to the EKF in terms of balance, though its validity is limited by ensemble size. It is also found that overobserving can lead to a more unbalanced forecast ensemble and thus to an unbalanced analysis.

Free gravity waves and balanced dynamics

It is shown how a renormalization technique, which is a variant of classical Krylov–Bogolyubov–Mitropol’skii averaging, can be used to obtain slow evolution equations for the vortical and inertia–gravity wave components of the dynamics in a rotating flow. The evolution equations for each component are obtained to second order in the Rossby number, and the nature of the coupling between the two is analyzed carefully. It is also shown how classical balance models such as quasigeostrophic dynamics and its second-order extension appear naturally as a special case of this renormalized system, thereby providing a rigorous basis for the slaving approach where only the fast variables are expanded. It is well known that these balance models correspond to a hypothetical slow manifold of the parent system; the method herein allows the determination of the dynamics in the neighborhood of such solutions. As a concrete illustration, a simple weak-wave model is used, although the method readily applies to more complex rotating fluid models such as the shallow-water, Boussinesq, primitive, and 3D Euler equations.

Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Balanced excitation and inhibition: Model based analysis of local field potentials

We have developed a model of the local field potential (LFP) based on the conservation of charge, the independence principle of ionic flows and the classical Hodgkin–Huxley (HH) type intracellular model of synaptic activity. Insights were gained through the simulation of the HH intracellular model on the nonlinear relationship between the balance of synaptic conductances and that of post-synaptic currents. The latter is dependent not only on the former, but also on the temporal lag between the excitatory and inhibitory conductances, as well as the strength of the afferent signal. The proposed LFP model provides a method for decomposing the LFP recordings near the soma of layer IV pyramidal neurons in the barrel cortex of anaesthetised rats into two highly correlated components with opposite polarity. The temporal dynamics and the proportional balance of the two components are comparable to the excitatory and inhibitory post-synaptic currents computed from the HH model. This suggests that the two components of the LFP reflect the underlying excitatory and inhibitory post-synaptic currents of the local neural population. We further used the model to decompose a sequence of evoked LFP responses under repetitive electrical stimulation (5 Hz) of the whisker pad. We found that as neural responses adapted, the excitatory and inhibitory components also adapted proportionately, while the temporal lag between the onsets of the two components increased during frequency adaptation. Our results demonstrated that the balance between neural excitation and inhibition can be investigated using extracellular recordings. Extension of the model to incorporate multiple compartments should allow more quantitative interpretations of surface Electroencephalography (EEG) recordings into components reflecting the excitatory, inhibitory and passive ionic current flows generated by local neural populations.

Balanced Ocean-Data Assimilation near the Equator

The question is addressed whether using unbalanced updates in ocean-data assimilation schemes for seasonal forecasting systems can result in a relatively poor simulation of zonal currents. An assimilation scheme, where temperature observations are used for updating only the density field, is compared to a scheme where updates of density field and zonal velocities are related by geostrophic balance. This is done for an equatorial linear shallow-water model. It is found that equatorial zonal velocities can be detoriated if velocity is not updated in the assimilation procedure. Adding balanced updates to the zonal velocity is shown to be a simple remedy for the shallow-water model. Next, optimal interpolation (OI) schemes with balanced updates of the zonal velocity are implemented in two ocean general circulation models. First tests indicate a beneficial impact on equatorial upper-ocean zonal currents.