Vladimir Zeitlin

Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f -1 0 and (f 0 Ro) -1 respectively, where f 0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction. The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H 0 , where AH and H 0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H 0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t ≤ (f 0 Ro) -1 . We find modifications to this equation for longer times t ≤ (f 0 Ro 2 ) -1 . The fast component consists mainly of linear inertia-gravity waves rapidly propagating outward from the initial disturbance. For large relative elevations (ΔH/H 0 Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrodinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations

0 and (f0Ro) −1 respectively, where f0 is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times t � (f0Ro) −1 ). In the FG case the fast component is ap acket of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schr¨ odinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.

Internal gravity wave emission from a pancake vortex: An example of wave-vortex interaction in strongly stratified flows

At small Froude numbers the motion of a stably stratified fluid consists of a quasisteady vortical component and a propagating wave component. The vortical component is organized into layers of horizontal motions with well-pronounced vertical vorticity and often takes the form of so-called “pancake” vortices. An analytical model of such a vortex that is a solution of the Euler–Boussinesq equations at a vanishing Froude number is constructed as a superposition of horizontal two-dimensional Kirchhoff elliptic vortices. This vortex is nonstationary and internal gravity waves are, therefore, excited by its motion. The radiation properties are studied by matching the vortex field with the far internal gravity wave field according to the procedure applied in acoustics to determine vortex sound. The structure of the gravity wave field is completely quantified. By calculating energy and angular momentum fluxes carried by outgoing waves and attributing them to the adiabatic change of the vortex parameters, we calc...

Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory

The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with non-negative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity.

Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations

High-resolution shock-capturing finite-volume numerical methods are applied to investigate nonlinear geostrophic adjustment of rectilinear fronts and jets in the rotating shallow-water model. Numerical experiments for various jet/front configurations show that for localized initial conditions in the open domain an adjusted state is always attained. This is the case even when the initial potential vorticity (PV) is not positive-definite, the situation where no proof of existence of the adjusted state is available. Adjustment of the vortex, PV-bearing, part of the flow is rapid and is achieved within a couple of inertial periods. However, the PV-less low-energy quasi-inertial oscillations remain for a long time in the vicinity of the jet core. It is demonstrated that they represent a long-wave part of the initial perturbation and decay according to the standard dispersion law

Hamilton’s principle for quasigeostrophic motion

{\sim}t^{-1/2}

Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow-water model. Part 1. Passive lower layer

. For geostrophic adjustment in a periodic domain, an exact periodic nonlinear wave solution is found to emerge spontaneously during the evolution of wave perturbations allowing us to conjecture that this solution is an attractor. In both cases of adjustment in open and periodic domains, it is shown that shock-formation is ubiquitous. It takes place immediately in the jet core and, thus, plays an important role in fully nonlinear adjustment. Although shocks dissipate energy effectively, the PV distribution is not changed owing to the passage of shocks in the case of strictly rectilinear flows.

Nonlinear development of inertial instability in a barotropic shear

We show that the equation of quasigeostrophic (QG) potential vorticity conservation in geophysical fluid dynamics follows from Hamilton’s principle for stationary variations of an action for geodesic motion in the f-plane case or its prolongation in the β-plane case. This implies a new momentum equation and an associated Kelvin circulation theorem for QG motion. We treat the barotropic and two-layer baroclinic cases, as well as the continuously stratified case.

Moist versus Dry Barotropic Instability in a Shallow-Water Model of the Atmosphere with Moist Convection

Buoyancy-driven coastal currents, which are bounded by a coast and a surface density front, are ubiquitous and play essential role in the mesoscale variability of the ocean. Their highly unstable nature is well known from observations, laboratory and numerical experiments. In this paper, we revisit the linear stability problem for such currents in the simplest reduced-gravity model and study nonlinear evolution of the instability by direct numerical simulations. By using the collocation method, we benchmark the classical linear stability results on zero-potential-vorticity (PV) fronts, and generalize them to non-zero-PV fronts. In both cases, we find that the instabilities are due to the resonance of frontal and coastal waves trapped in the current, and identify the most unstable long-wave modes. We then study the nonlinear evolution of the unstable modes with the help of a new high-resolution well-balanced finite-volume numerical scheme for shallow-water equations. The simulations are initialized with the unstable modes obtained from the linear stability analysis. We found that the principal instability saturates in two stages. At the first stage, the Kelvin component of the unstable mode breaks, forming a Kelvin front and leading to the reorganization of the mean flow through dissipative and wave–mean flow interaction effects. At the second stage, a new, secondary unstable mode of the Rossby type develops on the background of the reorganized mean flow, and then breaks, forming coherent vortex structures. We investigate the sensitivity of this scenario to the along-current boundary and initial conditions. A study of the same problem in the framework of the fully baroclinic two-layer model will be presented in the companion paper.

Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection

Inertial instability is investigated numerically in a two-dimensional setting in order to understand its nonlinear stage and saturation. To focus on fundamental mechanisms, a simple barotropic shear U(y)=tanh y on the f-plane is considered. The linear stability problem is first solved analytically, and the analytical solutions are used to benchmark numerical simulations. A simple scenario of the nonlinear development of the most unstable mode was recurrently observed in the case of substantial diffusivity: while reaching finite amplitude the unstable mode spreads laterally, distorting the initially vertical instability zone. This process produces strong vertical gradients which are subsequently annihilated by diffusion, making the flow barotropic again but with the shear spread over a wider region. In the course of such evolution, unexpectedly, strong negative absolute vorticity anomalies are produced. In weakly diffusive simulations, the horizontal spreading of the unstable motions and the enhancement of...