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Onno Bokhove

Abstract

The problem of finding higher-order generalizations of the quasigeostrophic and semigeostrophic models, that is, higher-order balanced models, is investigated systematically by using a slaving principle. It is shown that most existing balanced models are based on an underlying slave equation with varying master variables and that this slaving approach arises as an alternative to the secular conventional Rossby number expansions. While in the conventional expansions all variables are expanded in a power series in the Rossby number, the slaving approach suggests modified expansion or iteration procedures in which only the fast variables (associated with gravity modes) are expanded or iterated. The slow variables (associated with vortical modes) are now the master variables in the system; a well-known choice for large-scale extratropical flows in geophysical fluid dynamics is potential vorticity.

To illustrate both the iteration and the modified expansion procedures, the slaving approach is applied to the rapidly rotating hydrostatic Boussinesq equations in a horizontally semi-infinite domain by using a Rossby number scaling. Higher-order generalizations of the quasigeostrophic equations result from modified expansions, and higher-order generalizations of the semigeostrophic equations result from nonlinear iterations. The validity of the latter for small Rossby numbers as well as small frontal parameters exemplifies the advantage nonlinear iterations have.

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Onno Bokhove and Vijaya Ambati

Abstract

Idealized laboratory experiments reveal the existence of forced–dissipative hybrid Rossby-shelf modes. The laboratory ocean consists of a deeper ocean (accommodating basin-scale Rossby modes) and a coastal step shelf (accommodating trapped shelf modes). Planetary Rossby modes are mimicked in the laboratory via a uniform topographic slope in the north–south direction. Hybrid modes are found as linear modes in numerical calculations, and similar streamfunction patterns exist in streak photography of the rotating tank experiments. These numerical calculations are based on depth-averaged potential vorticity dynamics with Ekman forcing and damping. Preliminary nonlinear calculations explore the deficiencies observed between reality and the linear solutions. The aim of the work is twofold: to show that idealized hybrid Rossby-shelf modes exist in laboratory experiments and to contribute in a general sense to the discussion on the coupling and energy exchange associated with hybrid modes between shallow coastal seas and deep-ocean basins.

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Onno Bokhove and E. R. Johnson

Abstract

Flows on coastal shelves and in the deep interior ocean are often considered separately, but transport of fluid between these two regions can have important biologial or environmental consequences. This paper considers a linear coupled coastal and deep interior-ocean model in the idealized context of a homogeneous two-dimensional cylindrical ocean with a rigid lid and axisymmetric step shelf topography. Both a semianalytical mode-matching approach and brute-force finite-element numerics have been used to analyze the linear dynamics. It is shown that hybrid planetary β-plane Rossby and topographic shelf modes emerge. The structure of these inviscid modes is clarified by considering their frequency dependence on shelf break radius, by contrasting the evolution of hybrid modes to the evolution of pure shelf and pure β-plane Rossby modes (considering streamfunction fields and particle paths), and by showing solutions of the initial value problem. Both “ocean” and “laboratory” parameter values are considered. Hybrid modes exchange information between the deep ocean and coastal shelves, especially at the intermediate frequencies where the separate planetary Rossby mode and topographic shelf mode dispersion curves overlap. The role of these modes is particularly clear in an initial value problem wherein a localized initial condition on the southern shelf later leads to large-scale interior ocean circulation. Forced–dissipative calculations reveal the sensitivity of resonantly generated hybrid Rossby–shelf modes to the strength of Ekman damping. For typical oceanic and laboratory parameter values hybrid modes are altered by increasing Ekman damping but do not disappear.

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Onno Bokhove and Theodore G. Shepherd

Abstract

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.

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Luca Cantarello, Onno Bokhove, and Steven Tobias

Abstract

An isentropic 1½-layer model based on modified shallow-water equations is presented, including terms mimicking convection and precipitation. This model is an updated version of the isopycnal single-layer modified rotating shallow water (modRSW) model. The clearer link between fluid temperature and model variables together with a double-layer structure make this revised, isentropic model a more suitable tool to achieve our future goal: to conduct idealized experiments for investigating satellite data assimilation. The numerical model implementation is verified against an analytical solution for stationary waves in a rotating fluid, based on Shrira’s methodology for the isopycnal case. Recovery of the equivalent isopycnal model is also verified, both analytically and numerically. With convection and precipitation added, we show how complex model dynamics can be achieved exploiting rotation and relaxation to a meridional jet in a periodic domain. This solution represents a useful reference simulation or “truth” in conducting future (satellite) data assimilation experiments, with additional atmospheric conditions and data. A formal analytical derivation of the isentropic 1½-layer model from an isentropic two-layer model without convection and precipitation is shown in a companion paper (Part II).

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Onno Bokhove, Luca Cantarello, and Steven Tobias

Abstract

In this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

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