Revising Z: part II - logical development

This is the second of two related papers. In "Revising Z: Part I - logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility

Revising Z: part I - logic and semantics

This is the first of two related papers. We introduce a simple specification logic ZC comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtain a sound logic for Z, including a basic schema calculus

Results on formal stepwise design in Z

Stepwise design involves the process of deriving a concrete model of a software system from a given abstract one. This process is sometimes known as refinement. There are numerous refinement theories proposed in the literature, each of which stipulates the nature of the relationship between an abstract specification and its concrete counterpart. This paper considers six refinement theories in Z that have been proposed by various people over the years. However, no systematic investigation of these theories, or results on the relationships between them, have been presented or published before. This paper shows that these theories fall into two important categories and proves that the theories in each category are equivalent

An analysis of total correctness refinement models for partial relation semantics I

This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Wookcock which underlines data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others

Z Logic and its Consequences

This paper provides an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have co-written since 1997. We provide an informal as well as formal introduction to Z logic and show how it may be used, and extended, to investigate issues such as equational logic, the logic of preconditions, the issue of monotonicity and both operation and data refinement

Particle flux in the oceans: Challenging the steady state assumption

Atmospheric carbon dioxide levels are strongly controlled by the depth at which the organic matter that sinks out of the surface ocean is remineralized. This depth is generally estimated from particle flux profiles measured using sediment traps. Inherent in this analysis is a steady state assumption; that export from the surface does not significantly change in the time it takes material to reach the deepest trap. However, recent observations suggest that a significant fraction of material in the mesopelagic zone sinks slowly enough to bring this into doubt. We use data from a study in the North Atlantic during July/August 2009 to challenge the steady state assumption. An increase in biogenic silica flux with depth was observed which we interpret, based on vertical profiles of diatom taxonomy, as representing the remnants of the spring diatom bloom sinking slowly (<40 m d-1). We were able to reproduce this behaviour using a simple model using satellite-derived export rates and literature-derived remineralization rates. We further provide a simple equation to estimate ‘additional’ (or ‘excess’) POC supply to the dark ocean during non-steady state conditions, which is not captured by traditional sediment trap deployments. In seasonal systems, mesopelagic net organic carbon supply could be wrong by as much as 25% when assuming steady state. We conclude that the steady state assumption leads to misinterpretation of particle flux profiles when input fluxes from the upper ocean vary on the order of weeks, such as in temperate and polar regions with strong seasonal cycles in export

Geographical, seasonal, and depth variation in sinking particle speeds in the North Atlantic

Particle sinking velocity is considered to be a controlling factor for carbon transport to the deep sea and thus carbon sequestration in the oceans. The velocities of the material exported to depth are considered to be high in high-latitude productive systems and low in oligotrophic distributions. We use a recently developed method based on the measurement of the radioactive pair 210Po-210Pb to calculate particle sinking velocities in the temperate and oligotrophic North Atlantic during different bloom stages. Our estimates of average sinking velocities (ASVs) show that slowly sinking particles (&lt;100?m?d?1) contribute significantly to carbon flux at all the locations except in the temperate regions during the bloom. ASVs appear to vary strongly with season, which we propose is caused by changes in the epipelagic community structure. Our results are the first field data to confirm the long-standing theory that particle sinking velocities increase with depth, with increases of up to 90% between 50 and 150?m depth

Ocean nutrient pathways associated with passage of a storm

Storms that affect ocean surface layer dynamics and primary production are a frequent occurrence in the open North Atlantic Ocean. In this study we use an interdisciplinary dataset collected in the region to quantify nutrient supply by two pathways associated with a storm event: entrainment of nutrients during a period of high wind forcing and subsequent shear-spiking at the pycnocline due to interactions of storm generated inertial currents with wind. The post-storm increase in surface layer nitrate (by ~20 mmol m?2) was predominantly driven by the first pathway: nutrient intrusion during the storm. Alignment of post-storm inertial currents and surface wind stress caused shear instabilities at the ocean pycnocline, forming the second pathway for nutrient transport into the euphotic zone. During the alignment period, pulses of high turbulent nitrate flux through the pycnocline (up to 1 mmol m?2 day?1; approximately 25 times higher than the background flux) were detected. However, the impact of the post-storm supply was an order of magnitude lower than during the storm due to the short duration of the pulses. Cumulatively, the storm passage was equivalent to 2.5-5 % of the nitrate supplied by winter convection and had a significant effect compared to previously reported (sub)-mesoscale dynamics in the region. As storms occur frequently, they can form an important component in local nutrient budgets