Rotating Shallow Water Research Articles

A Unified Formulation of Quasi‐Geostrophic and Shallow Water Equations via Projection

AbstractThis paper introduces a unified model for layered rotating shallow‐water (RSW) and quasi‐geostrophic (QG) equations, based on the intrinsic relationship between these two sets of equations. We propose a novel formulation of the QG equations as a projection of the RSW equations. This formulation uses the same prognostic variables as RSW, namely velocity and layer thickness, thereby restoring the proximity of these two sets of equations. It provides direct access to the ageostrophic velocities embedded within the geostrophic velocities resolved by the QG equations. This approach facilitates the study of differences between QG and RSW using a consistent numerical discretization. We demonstrate the effectiveness of this formulation through examples including vortex shear instability, double‐gyre circulation, and a simplified North Atlantic configuration.

MODEL STRUCTURAL INFERENCE USING LOCAL DYNAMIC OPERATORS

This paper focuses on the problem of quantifying the effects of model-structure uncertainty in the context of time-evolving dynamical systems. This is motivated by multi-model uncertainty in computer physics simulations: developers often make different modeling choices in numerical approximations and process simplifications, leading to different numerical codes that ostensibly represent the same underlying dynamics. We consider model-structure inference as a two-step methodology: the first step is to perform system identification on numerical codes for which it is possible to observe the full state; the second step is structural uncertainty quantification (UQ), in which the goal is to search candidate models "close" to the numerical code surrogates for those that best match a quantity-of-interest (QOI) from some empirical dataset. Specifically, we: (1) define a discrete, local representation of the structure of a partial differential equation, which we refer to as the "local dynamical operator" (LDO); (2) identify model structure non-intrusively from numerical code output; (3) non-intrusively construct a reduced order model (ROM) of the numerical model through POD-DEIM-Galerkin projection; (4) perturb the ROM dynamics to approximate the behavior of alternate model structures; and (5) apply Bayesian inference and energy conservation laws to calibrate a LDO to a given QOI. We demonstrate these techniques using the two-dimensional rotating shallow water (RSW) equations as an example system.

A quasi-Hamiltonian discretization of the thermal shallow water equations

The rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model.

The Influence of Stratification on the Instabilities in an Idealized Two-Layer Ocean Model

Abstract This work investigates the instability of a two-layer Bickley jet in the context of the rotating shallow water (RSW) model. This provides a general context in which the instability of oceanographic jets with simple stratification can be investigated. The three objectives of this work are as follows: First, the study investigates the morphology of unstable modes that can occur in this two-layer model. This is done by performing a linear stability analysis to investigate different types of flows both with and without vertical shear. Second, the authors study how the growth rates of the unstable modes are affected by changes in the stratification. Third, this study looks at the nonlinear evolution of some of these instabilities to determine how easy it is for nonprimary instabilities to develop. This is motivated by the fact that in the literature there have been many investigations that have found a multitude of unstable modes in this model, and it is not evident as to how easily they can be generated in oceanographic flows.

Reply to “Comments on ‘Analogies of Ocean/Atmosphere Rotating Fluid Dynamics with Gyroscopes: Teaching Opportunities’”

I t is fascinating to learn that the analogy between gyroscopes and rotating shallow-water (RSW) dynamics extends so far. We were unaware of the original work on gyroscope–fluid dynamics analogies of Arnol’d or Obukhov or of the interesting recent developments by Gluhovsky, Tong, and their collaborators. Our focus was on the analogies between gyroscopic motion (precession and nutation) and linear RSW modes dominated by Coriolis forces (geostrophic flow and inertial oscillation, respectively). We were aware that close similarities exist between the nonlinear Euler equations for the gyroscope [(S19) and (S20) in our supplement] and for the nonlinear equations describing a frictionless particle sliding on a rotating sphere [(S32) and (S33)]. However, we had no idea that the equations for a system of coupled Volterra gyrostats provide a generic basis to construct nonlinear low-order models (LOMs) of fluid dynamics. Such LOMs with quadratic nonlinearity are very common in geophysical fluid dynamics. They apply to a vast range of phenomena because the ubiquitous convective acceleration is a quadratic nonlinearity. Some important oceanic and atmospheric processes involve other nonlinearities, however, like radiative transfer, phase changes, or the seawater equation of state. It is unclear if coupled Volterra gyrostats extend to include LOMs of these cases. Our initial suspicion that a link exists between RSW dynamics and gyroscopes was based on intuition and play. Perhaps the coupled Volterra gyrostat–fluids analogy can be exploited further to construct real physical machines that demonstrate other principles of ocean/atmosphere fluid dynamics—that would be a thought-provoking application of the important discoveries that Gluhovsky and Tong describe. DOI:10.1175/BAMS-D-13-00239.1

Comments on “Analogies of Ocean/Atmosphere Rotating Fluid Dynamics with Gyroscopes: Teaching Opportunities”

W e welcome the interesting recent paper by Haine and Cherian (2013, hereafter HC13), which introduced a physical analogy between the motion of a mechanical gyroscope and two geophysical flow phenomena that occur in the rotating shallow-water (RSW) equations: namely, geostrophic f low and inertial oscillations. HC13 found, for instance, that geostrophy corresponds to gyroscope precession and that inertial oscillations correspond to gyroscope nutation. They also pointed to other examples of physical analogies, including those between the celebrated Lorenz (1963) low-order model (LOM) of thermal convection and phenomena occurring in lasers, the dynamics of a thermohaline loop oscillator, and El Nino–Southern Oscillation (HC13, p. 682). HC13 did not mention, however, that an earlier LOM of barotropic flow (Lorenz 1960) established an analogy between atmospheric dynamics and gyroscopes (Obukhov and Dolzhansky 1975). Besides, the formal correspondence between the theories of fluid mechanics and of rigid-body mechanics, based on Lie algebras, was revealed by Arnol’d (1989). Finally, systems of equations having the form of several coupled such gyroscopes, as models for fluid flow, had been discussed even earlier by A. N. Kolmogorov during a 1958 seminar in Moscow (see Arnol’d 1991) and by A. M. Obukhov and collaborators in a series of papers beginning with Obukhov (1969). In this Comment we wish to expand the discussion of HC13. Note first that the Lorenz (1963) model was shown (Gluhovsky 1982) to be equivalent to the simplest Volterra gyrostat in a forced, dissipative regime. A gyrostat is a system of bodies whose relative motion does not alter the mass distribution of the system: for example, a gyroscope attached to an axisymmetric rotor; in a Volterra (1899) gyrostat, the rotor spins at a constant angular velocity relative to its “carrier” body (Wittenburg 2008). Second, the Euler gyroscope in the presence of gravity is described by the Euler–Poisson equations (Leimanis 1965). Geostrophic models thereof were considered by Gluhovsky and Dolzhansky (1980) in a study of convection within a rotating ellipsoid under horizontally nonuniform heating. When the gravity acts along the axis of rotation, the ordinary gyroscope (Euler) equations result. When there is an angle between the axis of rotation and the direction of gravity, a three-model chaotic system with three nonlinear terms was obtained, which in the case of a symmetric ellipsoid proved equivalent to the Lorenz (1963) model. Third, the use of LOMs has an established tradition in the study of fluid dynamics and atmospheric flows. Extending the analogy between hydrodynamic LOMs and gyroscopes to include LOMs in the form of coupled gyrostats ensures that energy is conserved in the limit of no dissipation and forcing (Gluhovsky and Tong 1999). This fundamental property of the original partial differential equations is not necessarily retained in general LOMs, which may lead to unphysical behaviors (e.g., Thiffeault and Horton 1996; Gluhovsky and Tong 1999). LOMs for RSW systems (Lorenz 1986; Bokhove and Shepherd 1996) can also be presented as coupled gyrostats, as well as LOMs found in other applications, including shell models of turbulence and Hamiltonian LOMs in geophysical f luid dynamics (Gluhovsky and Tong 1999; Gluhovsky et al. 2002; Gluhovsky 2006; Tong and Gluhovsky 2002, 2008).

Analogies of Ocean/Atmosphere Rotating Fluid Dynamics with Gyroscopes: Teaching Opportunities

The dynamics of the rotating shallow-water (RSW) system include geostrophic f low and inertial oscillation. These classes of motion are ubiquitous in the ocean and atmosphere. They are often surprising to people at first because intuition about rotating f luids is uncommon, especially the counterintuitive effects of the Coriolis force. The gyroscope, or toy top, is a simple device whose dynamics are also surprising. It seems to defy gravity by not falling over, as long as it spins fast enough. The links and similarities between rotating rigid bodies, like gyroscopes, and rotating fluids are rarely considered or emphasized. In fact, the dynamics of the RSW system and the gyroscope are related in specific ways and they exhibit analogous motions. As such, gyroscopes provide important pedagogical opportunities for instruction, comparison, contrast, and demonstration. Gyroscopic precession is analogous to geostrophic flow and nutation is analogous to inertial oscillation. The geostrophic adjustment process in rotating fluids can be illustrated using a gyroscope that undergoes transient adjustment to steady precession from rest. The controlling role of the Rossby number on RSW dynamics is reflected in a corresponding nondimensional number for the gyroscope. The gyroscope can thus be used to illustrate RSW dynamics by providing a tangible system that behaves like rotating fluids do, such as the large-scale ocean and atmosphere. These relationships are explored for their potential use in educational settings to highlight the instruction, comparison, contrast, and demonstration of important fluid dynamics principles.

Inertial, barotropic, and baroclinic instabilities of the Bickley jet in two-layer rotating shallow water model

We undertake a detailed study of inertial instability of the barotropic Bickley jet and its nonlinear saturation in the 2 layer rotating shallow water (RSW) model on the f-plane and compare it with the classical barotropic and baroclinic instabilities. We start with analytical and numerical investigation of the linear stability problem under hypothesis of strict homogeneity in the along-flow direction (“symmetric instability”). The unstable modes are identified and their parameters are determined. The dependence of the instability on Rossby and Burger numbers of the jet is investigated. The nonlinear development of the instability is then studied with the help of high-resolution well-balanced finite-volume numerical code recently developed for multi-layer RSW, which is initialized with the most unstable mode found from the linear stability analysis. It is shown that symmetric inertial instability is saturated by reorganization of the mean flow, without full homogenization of the anticyclonic region, where the unstable modes reside. We then study along the same lines the fully two-dimensional (2D) problem and compare the results with the one-dimensional analysis. The barotropic instability competes with inertial instability in this case. We show that, for sufficiently strong anticyclonic shears, the inertial instability still has the dominant growth rate in the long-wave sector. The “symmetric” one-dimensional inertial instability turns to be a liming case of the ageostrophic baroclinic instability. Yet, the “asymmetric” inertial instability at small but non-zero wavenumbers has the higher growth rate. We study again the nonlinear development of the most unstable inertial mode and show that homogenization of the region of strong anticyclonic shear of the flow takes place on average. The reorganization of the flow reveals a high degree of complexity, with coherent structure formation and inertia-gravity wave emission, both giving rise to the substantial enhancement of ageostrophic motions. We compare these processes for inertial baroclinic and barotropic instabilities.

New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbersRoin the range 0.1 <Ro< 1 (away from the QG parameter regimeRo→ 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting thema priorito any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.

Shallow-water modons on thef-plane

Solutions for steadily translating localized vortical structures, or modons, are sought in the framework of a 1½-layer rotating shallow-water (RSW) model on thef-plane. In this model, the fluid is assumed to rotate at a constant rate and to be composed of an active finite-depth layer and a passive infinitely deep layer. The focus is on the smooth intense modons, in which the potential vorticity field is continuous, and the pressure (hence, the active-layer thickness) and velocity are smooth, while inertial effects and deviations of the active-layer thickness from the static level are considerable. The problem is solved numerically employing a Newton--Kantorovich iterative procedure, Fourier–Chebyshev spectral expansion and collocations. The numerics are preceded by a theoretical modon design discussion that includes: derivation of fundamental modon invariants; distinction between the flow in the trapped-fluid region and the flow outside it; and the boundary conditions at the separatrix, the streamline demarcating the two regions. Also, some basic distinctions from the quasi-geostrophic modons are discussed, and an asymptotic analysis of the RSW modon far-field characteristics is carried out. This analysis reveals that an RSW modon must propagate more slowly than inertia–gravity waves. In smooth modons, the requirement that the active-layer thickness should be positive imposes an even stronger restriction on the allowed translational speed. To enable the use of Fourier–Chebyshev series, only the modons with circular separatrices are considered. The numerical iterative procedure is initialized by an analytical quasi-geostrophic dipolar modon solution; accordingly, the obtained RSW modons appear as cyclone–anticyclone pairs. Computations show that the allowed maximal translational speed monotonically decreases as a function of the modon size and, for reasonable sizes, is appreciably smaller than the gravity-wave limit. As distinct from quasi-geostrophic modons, the RSW modons with circular separatrices display nonlinearity of the potential vorticity (PV)vs. streamfunction relation, and the cyclone–anticyclone asymmetry: while the integral mass anomaly in the modon is zero, the cyclone is more intense and compact than the anticyclone.

Long-Range Dynamics of a Shallow Water Triad: Renormalization, Modulation, and Cyclogenesis

Long-time-range atmospheric variability has many sources, and nonlinear mode interactions could play an important role. This paper sets out to explore such internal mechanisms of the long-range variability for the rotating shallow water (RSW) system, at its lowest spectral resolution (triad), but for long (decadal) timescales, about 105 Coriolis units. A small Rossby–Froude number ε sets multiple timescales for the system, ranging from the fast gravity–oscillation scale t to the slow eddy-turnover scale τ = εt and yet slower (climatological) scales. A two-step averaging/renormalization procedure is developed to study underlying mechanisms of long-range variability. It explains the basic phases (relaxation, intensification) and the role of wave–vortex interactions, and reveals a novel mechanism of long-range variability, due to modulation of vortex motions [potential vorticity (PV) modes] by gravity waves. The long cycles [on timescales O(104)] are characterized by relative value of the zonal PV component of the flow. They include longer phases of high zonal PV, intermittent with shorter transitional “blocking” phases, dominated by nonzonal modes. Theoretical results are compared with numeric simulations to assess validity and predictive skill of the renormalized systems.