Geostrophic Vortices Research Articles

The Rotational and Divergent Kinetic Energy Spectra of Geostrophic Vortices and Inertia‐Gravity Waves

AbstractThe rotational and divergent kinetic energy (RKE and DKE) spectra of geostrophic vortices (Rossby waves; RWs) and inertia‐gravity waves (IGWs) in the global atmosphere are investigated with ERA5 reanalysis. The separation of RWs and IGWs in physical space is based on the normal‐mode decomposition, and the Helmholtz decomposition produces their RKE and DKE spectra at different layers, with a focus on spherical wavenumbers 10 ≤ l ≤ 100. In the upper troposphere and the middle and lower stratosphere, the RKE spectra of the total mode closely resemble the horizontal kinetic energy (HKE) spectra of RWs over most wavenumbers; the DKE spectra of the total mode are more comparable to the DKE spectra rather than the HKE spectra of IGWs, although their slopes are similar. The HKE of RWs is dominated by its rotational component, accounting for more than 80% of the HKE in most ranges of concern. Although the HKE of IGWs is dominated by its divergent component, its rotation component is also significant, with an average percentage exceeding 25% in all three vertical layers analyzed. Care must be taken when employing divergence as a proxy of IGWs, as the DKE may underestimate the HKE of IGWs. With the increase of altitude and the decrease of scale, the contribution of the divergent component increases in the horizontal circulation of both RWs and IGWs.

The interactions of the elliptical instability and convection

Elliptical instability is an instability of elliptical streamlines, which can be excited by large-scale tidal flows in rotating fluid bodies and excites inertial waves if the dimensionless tidal amplitude (ε) is sufficiently large. It operates in convection zones, but its interactions with turbulent convection have not been studied in this context. We perform an extensive suite of Cartesian hydrodynamical simulations in wide boxes to explore the interactions of elliptical instability and Rayleigh–Bénard convection. We find that geostrophic vortices generated by the elliptical instability dominate the flow, with energies far exceeding those of the inertial waves. Furthermore, we find that the elliptical instability can operate with convection, but it is suppressed for sufficiently strong convection, primarily by convectively driven large-scale vortices. We examine the flow in Fourier space, allowing us to determine the energetically dominant frequencies and wavenumbers. We find that power primarily concentrates in geostrophic vortices, in convectively unstable wavenumbers, and along the inertial wave dispersion relation, even in non-elliptically deformed convective flows. Examining linear growth rates on a convective background, we find that convective large-scale vortices suppress the elliptical instability in the same way as the geostrophic vortices created by the elliptical instability itself. Finally, convective motions act as an effective viscosity on large-scale tidal flows, providing a sustained energy transfer (scaling as ε2). Furthermore, we find that the energy transfer resulting from bursts of elliptical instability, when it operates, is consistent with the ε3 scaling found in prior work.

Interplay between geostrophic vortices and inertial waves in precession-driven turbulence

The properties of rotating turbulence driven by precession are studied using direct numerical simulations and analysis of the underlying dynamical processes in Fourier space. The study is carried out in the local rotating coordinate frame, where precession gives rise to a background shear flow, which becomes linearly unstable and breaks down into turbulence. We observe that this precession-driven turbulence is in general characterized by coexisting two-dimensional (2D) columnar vortices and three-dimensional (3D) inertial waves, whose relative energies depend on the precession parameter Po. The vortices resemble the typical condensates of geostrophic turbulence, are aligned along the rotation axis (with zero wavenumber in this direction, kz = 0), and are fed by the 3D waves through nonlinear transfer of energy, while the waves (with kz≠0) in turn are directly fed by the precessional instability of the background flow. The vortices themselves undergo inverse cascade of energy and exhibit anisotropy in Fourier space. For small Po < 0.1 and sufficiently high Reynolds numbers, the typical regime for most geo- and astrophysical applications, the flow exhibits strongly oscillatory (bursty) evolution due to the alternation of vortices and small-scale waves. On the other hand, at larger Po > 0.1 turbulence is quasi-steady with only mild fluctuations, the coexisting columnar vortices and waves in this state give rise to a split (simultaneous inverse and forward) cascade. Increasing the precession magnitude causes a reinforcement of waves relative to vortices with the energy spectra approaching the Kolmogorov scaling, and therefore, the precession mechanism counteracts the effects of the rotation.

Structure function of velocity in a geostrophic vortex under strong rotation

We consider analytically velocity pair structure function of turbulent pulsations on the background of a coherent geostrophic vortex in a fast rotating fluid. The statistics of the turbulent pulsations is determined by their dynamics that is the dynamics of inertial waves affected by the differential rotation in the vortex and weak viscous damping. Our consideration is restricted by the smallest scales, where the velocity field remains smooth. We establish the anisotropy of the structure function. The velocity gradient of the turbulent pulsations achieves its largest value for the nearly radial direction and its smallest value in the nearly streamwise direction. The inclination angle between the directions of extremum values of the gradient and the vortex-associated directions is determined by the interplay between the shear rate and the viscosity influence. The scaling laws resemble their behavior for a passive scalar in a steady shear flow.

Motion of three geostrophic Bessel vortices

This study investigates the dynamics of geostrophic vortices (described by modified Bessel functions). We focus on the three-vortex case, where the possibility of self-similar motion and general dynamics for arbitrary strengths is studied. It is found that self-similar motions are limited to rigid rotations and self-similar triple collapse is essentially impossible. For a general description, trilinear coordinates are adopted and extended to the model of geostrophic vortices. The similarities and differences with the point-vortex model are emphasized. The physical regions in the phase plane cannot be directly identified, unlike the point-vortex case where the physical regions are identified as conic sections. The boundary of the physical region approaches the vertices of the triangle in trilinear coordinates in geostrophic vortices, while making a tangent with the sides of the triangle in point vortices. For positive values of the three vortex strengths, the boundary of the physical region is a closed concave curve, whereas it is a circle or ellipse in the case of the point vortex. Importantly, we find a new type of the phase trajectory: an open curve not intersecting with the physical region boundary. We also demonstrate the physical motions and trajectories of vortices for several representative cases of vortex strengths from numerical computations.

Canyon Vortices: Application of the Theory of Topographic Vortices to the Phenomenon of Ice Rings in Baikal

A phenomenon of ice rings in Lake Baikal is considered. A hydrodynamic theory is proposed according to which geostrophic vortices form under ice in the area near the ice rings. These vortices are due to bed relief, namely, underwater canyons; therefore, they are referred to as canyon vortices. It is shown that a divergent ring vortex forms under ice in the ring domain at the crossing of a vertical cylindrical Stewartson layer on the lateral surface of the canyon geostrophic vortex and under-ice horizontal Ekman layer in the form of a horizontal vortex torus, and the melting of ice below in the form of a ring is due to the formation of this vortex torus.

Energy and enstrophy cascades in the geostrophic vortex with slowly varying inertia–gravity wave spirals

Although geostrophically balanced mesoscale vortices and unbalanced small-scale turbulence have been well studied, the link between them is not entirely clear, especially in the vertical dimension. The inertia–gravity wave (IGW) spiral plays an important role in the energy and enstrophy cascades between the geostrophic vortex and the small-scale turbulence. Since the vertical velocity cannot be measured in practice, the slowly varying IGW spiral formula in a vortex is used to distinguish the direction of vertical velocity. The vertical deformation of the vortex with IGW spirals results in a vertical semi-circulating cycle of the energy cascades: the energy cascades forward from the geostrophic vortex scale to small scales in the IGW spiral at one depth, flows along the IGW spiral, and then inversely cascades to the geostrophic vortex scale at another depth. Some small-scale energy at one depth eventually reaches large scales at another depth, which can prevent some energy from falling into small-scale dissipation, allowing the geostrophic vortex to continue for months in the ocean. A vertical full-circulating cycle is formed by connecting every IGW spiral in different geostrophic vortices and jets, similar to the energy exchange in the capillaries of the human body. The vertical closed zero-flux line of enstrophy cascades caused by the IGW spiral in a geostrophic vortex isolates the enstrophy at different scales to reduce the dissipation. Due to the IGW spiral, the energy and enstrophy spectra in the geostrophic vortex are −1 and 1 at small scales, respectively.

Transient generation of spiral inertia-gravity waves from a geostrophic vortex

The emission of inertia-gravity waves (IGWs) from an exact geostrophic vortex in a rotating and stratified fluid is investigated by three-dimensional numerical modeling. An initially balanced geostrophic vortex inevitably generates IGWs with spiral patterns within a short transient time period through an instability mechanism. This result reinforces the nonexistence of exactly invariant slow manifolds. The direction of the rotation of spiral IGWs is clockwise for both anticyclonic and cyclonic geostrophic vortices, which is consistent with the theoretical prediction. Spiral patterning can be regarded as a universal feature of IGWs, which occurs in the transient generation process. In the vertical direction, the energy of IGWs is dominated by mode-1 in the generation and propagation processes, leading to weak dissipation and long-distance propagation. A comparison of barotropic and baroclinic vortices suggests that horizontal nonzero strain and vorticity are essential for the occurrence of this instability mechanism, while the presence of vortex baroclinicity increases the intensity of the IGWs. The amplitude of the IGWs increases linearly with the Rossby number in the range of 0.04–0.1. Additionally, the IGWs emitted from an anticyclonic vortex are stronger than those radiated from a cyclonic vortex. Anticyclonic and cyclonic geostrophic vortices transfer approximately 0.54% and 0.41% of their kinetic energy to IGWs in this transient generation process, respectively. This transient generation of IGWs can supply an energy pathway from mesoscale eddies to diapycnal mixing processes in the interior of the oceans.

A generalized wave-vortex decomposition for rotating Boussinesq flows with arbitrary stratification

Abstract

Evidence of the Zakharov-Kolmogorov spectrum in numerical simulations of inertial wave turbulence

Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo an inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived analytically by Galtier (Galtier S., Phys. Rev. E, 68 (2003) 015301), thus offering numerical proof of the relevance of wave turbulence theory for three-dimensional, anisotropic waves. Lastly, we show that the same forcing leads to either geostrophic or wave turbulence depending on the initial conditions. Our results thus bring further evidence for bi-stability in rotating turbulent flows at low Rossby numbers.

Normal Mode Spectra of Idealized Baroclinic Waves

Abstract Normal modes are used to investigate the contributions of geostrophic vortices and inertia–gravity waves to the energy spectrum of an idealized baroclinic wave simulation. The geostrophic and ageostrophic modal spectra (GE and AE, respectively) are compared to the rotational and divergent kinetic energy (RKE and DKE, respectively), which are often employed as proxies for vortex and wave energy. In our idealized f-plane framework, the horizontal modes are Fourier, and the vertical modes are found by solving an appropriate eigenvalue problem. For low vertical mode number n, both the GE and AE spectra are steep; however, for higher n, while both spectra are shallow, the AE is shallower than the GE and the spectra cross. The AE spectra are peaked at the Rossby deformation wavenumber knR, which increases with n. Analysis of the horizontal mode equations suggests that, for large wavenumbers k≫knR, the GE is approximated by the RKE, while the AE is approximated by the sum of the DKE and potential energy. These approximations are supported by the simulations. The vertically averaged RKE and DKE spectra are compared to the sum of the GE and AE spectra over all vertical modes; the spectral slopes of the GE and AE are close to those of the RKE and DKE, supporting the use of the Helmholtz decomposition to estimate vortices and waves in the midlatitudes. However, the AE is consistently larger than the DKE because of the contribution from the potential energy. Care must be taken when diagnosing the mesoscale transition from the intersection of the vortex and wave spectra; GE and AE will intersect at a different scale than RKE and DKE, despite their similar slopes.

Cyclostrophic Corrections of AVISO/DUACS Surface Velocities and Its Application to Mesoscale Eddies in the Mediterranean Sea

AbstractMesoscale eddies, having a characteristic radius equal or larger than the local deformation radius, are generally considered to be geostrophic. Even if this is true for most of them, there are few cases where the ageostrophic velocity components induced by the local curvature of the streamlines are not negligible. In order to account for this ageostrophic part, we investigate the performance of an optimized iterative method which computes the cyclostrophic corrections starting from the geostrophic surface velocity of the AVISO (Archiving, Validation and Interpretation of Satellite Oceanographic data)/DUACS (Data Unification and Altimeter Combination System). We optimized the convergence of the iterative method using an intermediate cubic interpolation. The performance and the accuracy of the optimized iterative method is first evaluated on idealized eddies for which we can obtain their exact cyclogeostrophic solution. Mesoscale eddies of various shapes, intensities and different ellipticity are investigated. The iterative method is then applied to 15 years (2000–2015) of AVISO/DUACS geostrophic velocity fields, gridded at 1/8o for the Mediterranean Sea. We found that these ageostrophic corrections are needed for most of the mesoscale anticyclones that have a geostrophic vortex Rossby number larger than Ro>0.1. Both the Alboran and the Ierapetra eddies are frequently affected by the cyclostrophic corrections that may exceed 50 cm s−1. Lastly, the corrected velocity fields are compared with available in situ observations of velocity measurements (vessel‐mounted acoustic Doppler current profilers) performed within the Ierapetra eddy confirming the benefit of the proposed method.

Geostrophic Spirals Generated by the Horizontal Diffusion of Vortex Stretching in the Yellow Sea

Horizontal velocity spirals with a clockwise rotation (downward looking) rate of 1.7° m−1, on average, were observed in the western and northern Yellow Sea from December 2006 to February 2007. With the observed thermal wind relation, the beta-spiral theory was used to explain the dynamics of spirals. It was found that the horizontal diffusion of geostrophic vortex stretching is likely to be a major mechanism for generating geostrophic spirals. Vertical advection associated with surface/bottom Ekman pumping and topography-induced upwelling is too weak to support these spirals. Strong wind stirring and large heat loss in wintertime lead to weak stratification and diminish the effects of vertical advection. The cooling effect and vertical diffusion are offset by an overwhelming contribution of horizontal diffusion in connection with vortex stretching. The Richardson number-dependent vertical eddy diffusivity reaches a magnitude of 10−4 m2 s−1 on average. An eddy diffusivity of 2870 m2 s−1 is required for dynamic balance by estimating the residual term. This obtained value of 10−4 m2 s−1 is in good agreement with the estimation in terms of observed eddy activities. The suppressed and unsuppressed diffusivities in the observation region are 2752 and 2881 m2 s−1, respectively, which supports a closed budget for velocity rotation.

Can Large Oceanic Vortices Be Stable?

AbstractObservations show that radii of oceanic eddies often exceed the Rossby radius of deformation, whereas theoretical studies suggest that such vortices should be unstable. The present paper resolves this paradox by presenting a wide class of large geostrophic vortices with a sign‐definite gradient of potential vorticity (which makes them stable), in an ocean where the density gradient is mostly confined to a thin near‐surface layer (which is indeed the case in the real ocean). The condition of a thin “active” layer is what makes the present work different from the previous theoretical studies and is of utmost importance. It turns out that without it, the joint requirement that a vortex be large and have a sign‐definite potential vorticity gradient trivializes the problem by eliminating all vortices except nearly barotropic ones.

On the Spatial and Temporal Variability of Atmospheric Heat Transport in a Hierarchy of Models

Abstract The present study analyzes the spatial and temporal variability of zonally integrated meridional atmospheric heat transport due to transient eddies in a hierarchy of datasets. These include a highly idealized two-layer model seeded with point geostrophic vortices, an intermediate complexity GCM, and the European Centre for Medium-Range Weather Forecasts interim reanalysis (ERA-Interim) data. The domain of interest is the extratropics. Both the two-layer model and the GCM display a pronounced temporal variability in the zonally integrated meridional transport, with the largest values (or pulses) of zonally integrated transport being associated with extended regions of anomalously strong local heat transport. In the two-layer model these large-scale coherent transport regions, termed “heat transport bands,” are linked to densely packed baroclinic vortex pairs and can be diagnosed as low-wavenumber streamfunction anomalies. In the GCM they are associated with both the warm and cold sectors of midlatitude weather systems. Both these features are also found in ERA-Interim: the heat transport bands match weather systems and occur primarily in the storm-track regions, which in turn correspond to planetary-scale climatological streamfunction anomalies. The authors hypothesize that the temporal variability of the zonally integrated heat transport is partly linked to oscillatory variations in the storm-track activity but also contains a background red noise component. The existence of a pronounced variability in the zonally integrated meridional heat transport can have important consequences for the interplay between midlatitude dynamics and the energy balance of the high latitudes.

Winter positions of Arctic front during periods of cooling and warming

Winter positions of the Arctic front (AF) during the known periods of the climate cooling (1949–1980) and warming (1981–2012) were analyzed within the sector 10° W – 60° E. The AF positios were determined by the following indicators: 1) a surface pressure; 2) horizontal wind divergence; 3) geostrophic vortex; 4) geostrophic heat advection. The main extrema of these four dynamic characteristics coincide and fall on the latitude 72.5° N. This corresponds to the average position of the AF for a given resolution and confirms correctness of our choice of these characteristics as the AF indicators. Relative differences between mean profiles of all values of the above warm and cold periods were calculated using method of normalization of each value for the corresponding latitude by the standard deviation for the entire period (1949–2012). To study variability of the AF position we used mean yearly winter profiles of the variables under investigation together with the statistical analysis of positions of the extrema within the latitude degrees. For pressure and geostrophic advection positions of the absolute minima were determined while for geostrophic vortex and divergence – positions of the absolute maxima. The data show that according to different criteria the AF average positions for the period 1949–2012 lie within the zone 72.4–73.4 N. The interannual variability of the AF positions lies within the 1–2 degrees of latitude and corresponds to the range of the air temperature variability above the zone of maximal changes in the sea ice area. According to the standard deviation values of the divergence and the geostrophic vortex are the most stable in region of the AF passage. Comparison of differences of the studied characteristics between the warm and cold periods shows that the changes in the AF positions are not statistically significant (P(t) < 91% t‑criterion) unlike the changes in positions of isolines which characterize the warming (P(t) = 100%). Thus, despite significant changes in properties of the surface and the temperature regime to the north of 72.5 N (the warming), according to all the criteria the AF climatic position remains quasi‑stationary for 32‑year periods of averaging.

Forced Convection in a Stratified Rotating Medium Under the Influence of Vortex Shearing Stress on the Surface

The response of a semi-infinite stratified rotating medium to the action of an inhomogeneous vortex shearing stress on a horizontal surface is investigated analytically in a linear approximation. The previously unexplored region of the values of governing parameters is considered. It has been established that in the first approximation the indicated response represents a geostrophic vortex penetrating into a stably stratified medium to depths of the order of the horizontal scale of the shearing stress inhomogeneity and that the amplitude of the originating thermal perturbation is proportional to this scale and to the angular rotational velocity of the medium and inversely proportional to the coefficient of its viscosity.

Frictional Convergence in a Decaying Weak Vortex

Abstract Cross-isobaric flow and Ekman pumping are investigated in the frictional spindown of an initially barotropic vortex in a stratified atmosphere. Consistent with early work by Holton and others, it is found that the stratification limits the vertical penetration of the secondary circulation driven by friction, resulting in more rapid spindown than in the unstratified case. As a result, the cross-isobaric flow and Ekman pumping are weaker and shallower than classical calculations ignoring the stratification would lead one to believe. The effect of stability becomes stronger as the vortex becomes smaller for fixed boundary layer depth. For weak geostrophic vortices with horizontal scales of several hundred kilometers or less, such as tropical easterly waves, the reduction is particularly pronounced, which raises questions about the efficacy of Ekman pumping in forcing convection in such vortices. These results suggest a revised conceptual model for the role of Ekman pumping in the atmosphere. The theory as it stands is limited to weak, linear vortices in which geostrophic balance holds approximately, corresponding to small Rossby number, though extensions of the analytical theory to stronger vortices may be possible.

Numerical Study on the Asymptotic Equation for the Deformation of Vortex Lattice

We study a coarse-grained asymptotic equation which describes deformation of vortex lattices [Smirnov & Chukbar 2001]. It reads φt=φxxφyy–φ2xy, where φ denotes displacement of vortex locations. This equation is valid for a lattice with short-ranged interaction, e.g. geostrophic vortices with a screened potential. New self-similar blow-up solutions with infinite total energy are found. We ask whether or not finite-time blow-up can take place developing from smooth initial data with finite energy. The numerical simulations show finite-time blow-up in such a way that φ∈H3 but φ∉H4.

The potential and vortex traces of a tsunamigenic earthquake in the ocean

An equation system that describes the potential and vortex residual hydrodynamical fields that emerge in the rotating ocean during tsunami generation by co-seismic deformations of the bottom was obtained in terms of linear long-wave theory. For the model case of a cylindrically symmetrical deformation of the bottom, a completely analytical solution was found. On the basis of this solution, the structure of residual fields was analyzed and estimates of the residual horizontal displacements of water particles, of the speed of the vortex current, and of the energy of a geostrophic vortex were obtained under typical tsunami source conditions.