Two-layer Quasi-geostrophic Model Research Articles

Nonlinear Cascades of Surface Oceanic Geostrophic Kinetic Energy in the Frequency Domain*

Abstract Motivated by the ubiquity of time series in oceanic data, the relative lack of studies of geostrophic turbulence in the frequency domain, and the interest in quantifying the contributions of intrinsic nonlinearities to oceanic frequency spectra, this paper examines the spectra and spectral fluxes of surface oceanic geostrophic flows in the frequency domain. Spectra and spectral fluxes are computed from idealized two-layer quasigeostrophic (QG) turbulence models and realistic ocean general circulation models, as well as from gridded satellite altimeter data. The frequency spectra of the variance of streamfunction (akin to sea surface height) and of geostrophic velocity are qualitatively similar in all of these, with substantial variance extending out to low frequencies. The spectral flux Π(ω) of kinetic energy in the frequency ω domain for the QG model documents a tendency for nonlinearity to drive energy toward longer periods, in like manner to the inverse cascade toward larger length scales documented in calculations of the spectral flux Π(k) in the wavenumber k domain. Computations of Π(ω) in the realistic model also display an “inverse temporal cascade.” In satellite altimeter data, some regions are dominated by an inverse temporal cascade, whereas others exhibit a forward temporal cascade. However, calculations performed with temporally and/or spatially filtered output from the models demonstrate that Π(ω) values are highly susceptible to the smoothing inherent in the construction of gridded altimeter products. Therefore, at present it is difficult to say whether the forward temporal cascades seen in some regions in altimeter data represent physics that is missing in the models studied here or merely sampling artifacts.

The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

AbstractIn the Nastrom–Gage spectrum of atmospheric turbulence, we observe a${k}^{\ensuremath{-} 3} $energy spectrum that transitions into a${k}^{\ensuremath{-} 5/ 3} $spectrum, with increasing wavenumber$k$. The transition occurs near a transition wavenumber${k}_{t} $, located near the Rossby deformation wavenumber${k}_{R} $. The Tung–Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length scale range. We show that, in a temperature-forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber${k}_{t} $near${k}_{R} $. We also show that, if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behaviour contrary to what occurs in two-dimensional turbulence. As a result, the ratio$\eta / \varepsilon $of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber${k}_{t} $further. Using a random Gaussian forcing model, we reach the same conclusion, under the modelling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung–Orlando theory can account for the approximate coincidence between${k}_{t} $and${k}_{R} $. We also identify certain open questions that require further investigation via numerical simulations.

Interactions of surface and deep anticyclonic eddies in the Bay of Biscay

In 1990, satellite observations revealed that an anticyclonic surface eddy (a SWODDY, for Slope Water Oceanic eDDY) followed a cycloidal trajectory north of the Iberian coast in the Bay of Biscay. To understand the mechanisms underlying such a trajectory, we study the evolution of an idealized surface eddy in a two-layer flat-bottom quasi-geostrophic model. The effect of several processes is studied, notably the presence of deep anticyclonic vorticity. This deep vorticity may result either from the tilting of the swoddy itself, or from the presence of an anticyclonic eddy of different origins, such as a meddy (Mediterranean Water EDDY). We also study the influence of a zonal coast south of the swoddy, via the “mirror effect”.Firstly, a point-vortex model is used on the f-plane. When the surface and deep vortices lie much farther away from the coast than from each other, their motion is close to the addition of a mutually induced rotation and of a quasi-uniform zonal drift induced by the mirror vortices. The sensitivity of the rotation and translation characteristics to vortex position, strength, thickness and to a surrounding flow, are investigated. Such a surface-deep vortex interaction can reasonably well represent the observed motions of the swoddy, if the deep vortex is far enough from the surface one, and if they have comparable strengths.Then, a numerical code of the two-layer quasi-geostrophic equations is used to model finite-area vortices, again on the f-plane. Vertical alignment of the surface and deep vortices or vortex pairing with the mirror image, are not observed. Vortex splitting due to mutual shearing effects occurs only for vortices with very different strengths. Vortex trajectories similar to the ones observed are reproduced by the model for equal strength vortices.Complementary data, from the CONGAS experiments in 2004–2007, and from the ARGO profiling float database, are used to show that meddies (or at least coherent salinity anomalies at 1000m depth) can originate from the continental slope near 45°N, 8°W and move regularly northeastward toward the region of swoddy generation and drift. These data also show that meddy–swoddy (“deep vortex–surface vortex”) encounters may occur in the southeastern Bay of Biscay.

Two-layer quasi-geostrophic model of contour dynamics for a round basin

A quasi-geostrophic contour dynamics model permitting one to study flows induced by a system of vortex patches in a two-layer ocean with round shore boundaries in the presence of specified background flows caused by the bottom relief, the β effect, and sources and sinks at the boundaries is proposed. The principal relations of the model are presented and the algorithm of its numerical realization is described. Some experimental results of the study of the evolution of unstable two-layer vortices are demonstrated.

Simplified two-layer models of precipitating atmosphere and their properties

We derive a two-layer rotating shallow-water model for a moist atmosphere with water vapor condensation and related diabatic heating. Moist convection is represented by additional mass exchanges between the layers, which are determined from the moist enthalpy conservation principle, and related to the precipitation. Various boundary conditions at the lower and upper boundaries may be used. We show that the model reproduces in appropriate limits the main simplified models used previously in the literature for description of the large-scale moist-convective dynamics and precipitation fronts, namely the linear two-layer baroclinic models, the nonlinear two-layer quasigeostrophic model, and the nonlinear one-layer moist-convective rotating shallow-water model. We study the properties of the equations of the model, with special attention to the hyperbolicity loss, which is inherent to multilayer shallow-water models, and to the front propagation. Numerical illustrations of these properties are given with the help of a recently proposed high-resolution finite-volume numerical scheme with precipitation sources/sinks.

Lie symmetry analysis and exact solutions of the quasigeostrophic two-layer problem

The quasigeostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras, we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Wherever possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.

Baroclinic multipole evolution in shear and strain

In a two-layer quasi-geostrophic model, the evolution of a symmetric baroclinic multipole, composed of a central vortex with strength μκ in the upper layer, and A satellites with strength κ in the lower layer, is studied. This multipole is imbedded in a center-symmetric shear/strain field, either steady or time-periodic. Special attention is given to the case of the tripole (A = 2). The stability of this tripole is assessed and its oscillations in the external field are analyzed. Conditions for resonance of these oscillations are derived and transition to chaos is described.

Estimating the size of the regular region of a topographically trapped vortex

We formulate a dynamically consistent two-layer quasigeostrophic model of geophysical flow using the concept of background currents which are characterized by a constant potential vorticity minimizing the energy. An incident current over a delta-like isolated topographic feature generates a topographically trapped vortex in the bottom layer with a singular elliptic point, and one with a regular elliptic point in the upper layer. Such vortices are finite regions of recirculation which occur in the vicinity of isolated topographic features. The corresponding Hamiltonian equations of motion for a fluid particle are known to produce chaotic advection under the presence of the periodic incident current. When a periodic incident current is superimposed, fluid is entrained and detrained from the neighbourhood of the vortex and chaotic particle motion occurs. At a small amplitude periodic incident current we have a near-integrable case of weak chaotization in narrow stochastic layer near separatrix. At a finite amplitude periodic incident current we have the case of strong chaos, which will be investigated here. For the bottom layer of the flow, there is always a regular region (regular vortex core) around the singular elliptical point even in the case of a finite amplitude periodic incident current. Particles in this regular region have regular trajectories and will remain there permanently, whereas particles with chaotic trajectories will be emanated from the vortex by the incident current. Using the Chirikov criterion of resonance overlap, we estimate the radius of this regular vortex core and a range of the optimal frequencies, i.e. those frequencies of the incident current which provide a maximal possible stochastization of fluid particle trajectories.

Linear baroclinic and parametric instabilities of boundary currents

The linear baroclinic and parametric instabilities of boundary currents with piecewise-constant potential vorticity are studied in a two-layer quasi-geostrophic model. The growth rates of both the exponential modes and of the optimal perturbations are calculated for the baroclinic instability of steady coastal currents. We show that the growth rates of the exponential modes are maximal for a vertically symmetric flow. Furthermore, the vertical asymmetries induced by different layer thicknesses, the presence of a barotropic potential vorticity or bottom topography, all act to dampen the growth rates and favor growth at shorter wavelengths. It is shown that this behavior can be predicted from the conditions for vertical resonance of Rossby waves on the two potential vorticity fronts. Also, the baroclinic instability of the optimal perturbations has larger growth rates at shorter wavelengths and shorter time scales. As well, the presence of a sloping bottom of moderate amplitude favors the growth of these optimal perturbations. Finally, we compute the growth rates of parametric instability of oscillatory coastal flows. We show that subharmonic resonance is the most unstable mode of growth. In addition, a second region of parametric instability is found (for the first time) away from marginality of exponential-mode baroclinic instability. It is shown that the functional dependency of the growth rates of parametric instability, for optimal excitation, are similar to that of the optimal perturbations of baroclinic instability. To explain this a mechanism for parametric instability, involving the rapid growth of short-wave optimal perturbations, is proposed.

Dynamics of separating western boundary currents in ocean models

Western boundary current (WBC) separation dynamics are investigated in a two-layer quasigeostrophic model of a wind-driven subtropical ocean gyre. The results suggest that the vorticity in the thin cyclonic layer next to the coast plays a key role in the WBC separation mechanism. It is found that the outflow from the western boundary current is a region where this cyclonic vorticity diminishes rapidly downstream, through the combined action of Laplacian viscosity and eddy vorticity flux divergence. These results are consistent with earlier work on laminar, unstratified WBC dynamics, and extend those findings to the more realistic case of turbulent, stratified flow.

Dissipative Energization of Baroclinic Waves by Surface Ekman Pumping

Abstract A two-layer quasigeostrophic model is used to study the effect of lower boundary Ekman pumping on the energetics of baroclinic waves. Although the direct impact of the Ekman pumping is to damp the total eddy energy, either the eddy available potential energy (EAPE) or the eddy kinetic energy (EKE), individually, can grow because of the Ekman pumping. Growth of EAPE is favored if the phase difference between the upper and lower wave fields is less than a quarter wavelength, and EKE is favored if the phase difference is greater than a quarter wavelength. A numerical model calculation shows that the EAPE growth occurs directly through the Ekman pumping and that the increased EAPE can in turn lead to further growth by strengthening the baroclinic energy conversion from zonal available potential energy to the EAPE. Through this indirect effect, the Ekman pumping can increase the net production of total eddy energy.

Baroclinic multipole formation from heton interaction

In a two-layer quasi-geostrophic model, the interaction between two opposite-signed hetons (baroclinic vortex pairs) is studied analytically and numerically, for singular and finite-area vortices.For point vortices, using trilinear coordinates, it is shown that the possible evolutions depend on the deformation radius Rd: for large Rd, the layers decouple, vortices pair in each layer and their trajectories are open; for medium Rd, the exchange of opposite-sign partners between layers becomes possible; for small Rd, two other regimes appear: one where hetons remain unaltered during their evolution but follow open trajectories, and one where hetons occupy only a bounded subdomain of space at all times. Conditions for invariant co-rotation of the heton pair are derived and analyzed.Then, the nonlinear evolutions of finite-area heton pairs, with piecewise-constant vorticity, are computed with contour dynamics. When the central cyclonic vortex is initially aligned vertically, a transition occurs between three nonlinear regimes as layer coupling increases: for weak coupling, the vortices pair horizontally and drift away in opposite directions; for moderate layer coupling, the core vortex splits into two parts, one of which remains as a tilted columnar vortex at the center; for stronger layer coupling, each anticyclone pairs with part of the cyclone in each layer, thus forming an L-shaped dipole, a new coherent structure of two-layer flows. When the initial distance between the central and satellite vortices is increased, the velocity shear at the center decreases and the central vortex remains vertically aligned, thus forming a Z-shaped tripole, also a newly observed vortex compound. Such tripoles also compete with oscillating states, in which the core vortex periodically aligns and tilts, a regime observed when layer coupling is moderate and as vortices become closer in each layer. This Z-shaped tripole forms for various values of stratification and of initial distances between vortices, and is therefore a robust vortex compound in two-layer quasi-geostrophic flows.

Filtering Turbulent Sparsely Observed Geophysical Flows

AbstractFiltering sparsely turbulent signals from nature is a central problem of contemporary data assimilation. Here, sparsely observed turbulent signals from nature are generated by solutions of two-layer quasigeostrophic models with turbulent cascades from baroclinic instability in two separate regimes with varying Rossby radius mimicking the atmosphere and the ocean. In the “atmospheric” case, large-scale turbulent fluctuations are dominated by barotropic zonal jets with non-Gaussian statistics while the “oceanic” case has large-scale blocking regime transitions with barotropic zonal jets and large-scale Rossby waves. Recently introduced, cheap radical linear stochastic filtering algorithms utilizing mean stochastic models (MSM1, MSM2) that have judicious model errors are developed here as well as a very recent cheap stochastic parameterization extended Kalman filter (SPEKF), which includes stochastic parameterization of additive and multiplicative bias corrections “on the fly.” These cheap stochastic reduced filters as well as a local least squares ensemble adjustment Kalman filter (LLS-EAKF) are compared on the test bed with 36 sparse regularly spaced observations for their skill in recovering turbulent spectra, spatial pattern correlations, and RMS errors. Of these four algorithms, the cheap SPEKF algorithm has the superior overall skill on the stringent test bed, comparable to LLS-EAKF in the atmospheric regime with and without model error and far superior to LLS-EAKF in the ocean regime. LLS-EAKF has special difficulty and high computational cost in the ocean regime with small Rossby radius, which creates stiffness in the perfect dynamics. The even cheaper mean stochastic model, MSM1, has high skill, which is comparable to SPEKF for the oceanic case while MSM2 has significantly worse filtering performance than MSM1 with the same inexpensive computational cost. This is interesting because MSM1 is based on a simple new regression strategy while MSM2 relies on the conventional regression strategy used in stochastic models for shear turbulence.

The Linear Stability of Time-Dependent Baroclinic Shear

Abstract This article aims to advance the understanding of inherent randomness in geophysical fluids by considering the particular example of baroclinic shear flows that are spatially uniform in the horizontal directions and aperiodic in time. The time variability of the shear is chosen to be the Kubo oscillator, which is a family of time-dependent bounded noise that is oscillatory in nature with various degrees of stochasticity. The author analyzed the linear stability of a wide range of temporally periodic and aperiodic shears with a zero and nonzero mean to get a more complete understanding of the effect of oscillations in shear flows in the context of the two-layer quasigeostrophic Phillips model. It is determined that the parametric mode, which exists in the periodic limit, also exists in the range of small and moderate stochasticities but vanishes in highly erratic flows. Moreover, random variations weaken the effects of periodicity and yield growth rates more similar to that of the time-averaged steady-state analog. This signifies that the periodic shear flows possess the most extreme case of stabilization and destabilization and are thus anomalous. In the limit of an f plane, the linear stability problem is solved exactly to reveal that individual solutions to the linear dynamics with time-dependent baroclinic shear have growth rates that are equal to that of the time-averaged steady state. This implies that baroclinic shear flows with zero means are linearly stable in that they do not grow exponentially in time. This means that the stochastic mode that was found to exist in the Mathieu equation does not arise in this model. However, because the perturbations grow algebraically, the aperiodic baroclinic shear on an f plane can give rise to nonlinear instabilities.

Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

Abstract Satellite altimetry and high-resolution ocean models indicate that the Southern Ocean comprises an intricate web of narrow, meandering jets that undergo spontaneous formation, merger, and splitting events, as well as rapid latitude shifts over periods of weeks to months. The role of topography in controlling jet variability is explored using over 100 simulations from a doubly periodic, forced-dissipative, two-layer quasigeostrophic model. The system is forced by a baroclinically unstable, vertically sheared mean flow in a domain that is large enough to accommodate multiple jets. The dependence of (i) meridional jet spacing, (ii) jet variability, and (iii) domain-averaged meridional transport on changes in the length scale and steepness of simple sinusoidal topographical features is analyzed. The Rhines scale, ℓβ = 2πVe/β, where Ve is an eddy velocity scale and β is the barotropic potential vorticity gradient, measures the meridional extent of eddy mixing by a single jet. The ratio ℓβ /ℓT, where ℓT is the topographic length scale, governs jet behavior. Multiple, steady jets with fixed meridional spacing are observed when ℓβ ≫ ℓT or when ℓβ ≈ ℓT. When ℓβ < ℓT, a pattern of perpetual jet formation and jet merger dominates the time evolution of the system. Zonal ridges systematically reduce the domain-averaged meridional transport, while two-dimensional, sinusoidal bumps can increase transport by an order of magnitude or more. For certain parameters, bumpy topography gives rise to periodic oscillations in the jet structure between purely zonal and topographically steered states. In these cases, transport is dominated by bursts of mixing associated with the transition between the two regimes. Topography modifies local potential vorticity (PV) gradients and mean flows; this can generate asymmetric Reynolds stresses about the jet core and can feed back on the conversion of potential energy to kinetic energy through baroclinic instability. Both processes contribute to unsteady jet behavior. It is likely that these processes play a role in the dynamic nature of Southern Ocean jets.

Finite-Amplitude Equilibration of Baroclinic Waves on a Jet

Abstract A two-layer quasigeostrophic model is used to study the equilibration of baroclinic waves. In this model, if the background flow is relaxed toward a jetlike profile, a finite-amplitude baroclinic wave solution can be realized in both supercritical and subcritical regions of the model’s parameter space. Analyses of the model equations and numerical model calculations indicate that the finite-amplitude wave equilibration hinges on the breaking of Rossby waves before they reach their critical latitude. This “jetward” wave breaking results in an increase in the upper-layer wave generation and a reduction in the vertical phase tilt. This change in the phase tilt has a substantial impact on the Ekman pumping, as it weakens the damping on the lower-layer wave for some parameter settings and enables the Ekman pumping to serve as a source of wave growth at other settings. Together, these processes can account for the O(1)-amplitude wave equilibration. From a potential vorticity (PV) perspective, the wave breaking reduces the meridional scale of the upper-layer eddy PV flux, which destabilizes the mean flow. This is followed by a strengthening of the lower-layer eddy PV flux, which weakens the lower-layer PV gradient and constrains the growth of the lower-layer eddy PV. The same jetward wave breaking focuses the upper-layer PV flux toward the jet center where the upper-layer PV gradient is greatest. This results in an intensification of the upper-layer eddy PV relative to lower-layer eddy PV. Because of this large ratio, the upper-layer eddy PV plays the primary role in inducing the upper- and lower-layer eddy streamfunction fields, decreasing the vertical phase tilt. As a result, the Ekman pumping on the eddies is weakened, and for some parameter settings the Ekman pumping can even act as a wave source, contributing toward O(1)-amplitude wave equilibration. By reducing the horizontal shear of the zonal wind, the same wave breaking process weakens the barotropic decay, which also contributes to the wave amplification.

A Wave Amplitude Transition in a Quasi-Linear Model with Radiative Forcing and Surface Drag

Abstract A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave amplitude bifurcation—in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution can be changed from one type of equilibration to the other.

Explosive instability of geostrophic vortices. Part 2: parametric instability

In a two-layer quasi-geostrophic model, a baroclinic vortex is submitted to a periodic forcing of its mean baroclinic azimuthal velocity. It is shown that parametric effects could stabilize a vortex which is baroclinically unstable in the absence of forcing. Conversely, parametric resonance can destabilize a baroclinically stable vortex, under conditions on the vortex parameters, on the ratio of layer thicknesses or on the forcing frequency.

Synchronization in a coupled two-layer quasigeostrophic model of baroclinic instability – Part 1: Master-slave configuration

Abstract. Synchronization is studied using a pair of diffusively-coupled, two-layer quasi-geostrophic systems each comprising a single baroclinic wave and a zonal flow. In particular, the coupling between the systems is in the well-known master-slave or one-way configuration. Nonlinear time series analysis, phase dynamics, and bifurcation diagrams are used to study the dynamics of the coupled system. Phase synchronization, imperfect synchronization (phase slips), or complete synchronization are found, depending upon the strength of coupling, when the systems are either in a periodic or a chaotic regime. The results of investigations when the dynamics of each system are in different regimes are also presented. These results also show evidence of phase synchronization and signs of chaos control.

Equilibration of Baroclinic Turbulence in Primitive Equations and Quasigeostrophic Models

Abstract This paper investigates the equilibration of baroclinic turbulence in an idealized, primitive equation, two-level model, focusing on the relation with the phenomenology of quasigeostrophic turbulence theory. Simulations with a comparable two-layer quasigeostrophic model are presented for comparison, with the deformation radius in the quasigeostrophic model being set using the stratification from the primitive equation model. Over a fairly broad parameter range, the primitive equation and quasigeostrophic results are in qualitative and, to some degree, quantitative agreement and are consistent with the phenomenology of geostrophic turbulence. The scale, amplitude, and baroclinicity of the eddies and the degree of baroclinic instability of the mean flow all vary fairly smoothly with the imposed parameters; both models are able, in some parameter ranges, to produce supercritical flows. The criticality in the primitive equation model, which does not have any convective parameterization scheme, is fairly sensitive to the external parameters, most notably the planet size (i.e., the f /β ratio), the forcing time scale, and the factors influencing the stratification. In some parameter settings of the models, although not those that are most realistic for the earth’s atmosphere, it is possible to produce eddies that are considerably larger than the deformation scales and an inverse cascade in the barotropic flow with a −5/3 spectrum. The vertical flux of heat is found to be related to the isentropic slope.