Interior Potential Vorticity Gradient Research Articles

The role of surface potential vorticity in the vertical structure of mesoscale eddies in wind-driven ocean circulations

Abstract The vertical structure of ocean eddies is generally surface-intensified, commonly attributed to the dominant baroclinic modes arising from the boundary conditions (BCs). Conventional BC considerations mostly focus on either flat- or rough-bottom conditions. The impact of surface buoyancy anomalies—often represented by surface potential vorticity (PV) anomalies—has not been fully explored. Here, we study the role of the surface PV in setting the vertical distribution of eddy kinetic energy (EKE) in an idealized adiabatic ocean model driven by wind stress. The simulated EKE profile in the extra-tropical ocean tends to peak at the surface and have an e-folding depth typically smaller than half of the ocean depth. This vertical structure can be reasonably represented by a single surface quasi-geostrophic (SQG) mode at the energy-containing scale resulting from the large-scale PV structure. Due to isopycnal outcropping and interior PV homogenization, the surface meridional PV gradient is substantially stronger than the interior PV gradient, yielding surface-trapped baroclinically unstable modes with horizontal scales comparable to or smaller than the deformation radius. These surface-trapped eddies then grow in size both horizontally and vertically through an inverse energy cascade up to the energy-containing scale, which dominates the vertical distribution of EKE. As for smaller horizontal scales, the EKE distribution decays faster with depth. Guided by this interpretation, an SQG-based scale-aware parameterization of the EKE profile is proposed. Preliminary offline diagnosis of a high-resolution simulation shows the proposed scheme successfully reproducing the dependence of the vertical structure of EKE on the horizontal grid resolution.

Stirring of interior potential vorticity gradients as a formation mechanism for large subsurface-intensified eddies in the Beaufort Gyre

Abstract The Beaufort Gyre (BG) is hypothesized to be partially equilibrated by those mesoscale eddies that form via baroclinic instabilities of its currents. However, our understanding of the eddy field’s dependence on the mean BG currents and the role of sea ice remains incomplete. This theoretical study explores the scales and vertical structures of eddies forming specifically due to baroclinic instabilities of interior BG flows. An idealized quasi-geostrophic model is used to show that flows driven only by the Ekman pumping contain no interior potential vorticity (PV) gradients and generate weak and large eddies, ℴ(200km) in size, with predominantly barotropic and first baroclinic mode energy. However, flows containing realistic interior PV gradients in the Pacific halocline layer generate significantly smaller eddies of about 50 km in size, with a distinct second baroclinic mode structure and a subsurface kinetic energy maximum. The dramatic change in eddy characteristics is shown to be caused by the stirring of interior PV gradients by large-scale barotropic eddies. The sea ice-ocean drag is identified as the dominant eddy dissipation mechanism, leading to realistic sub-surface maxima of eddy kinetic energy for drag coefficients higher than about 2×10−3. A scaling law is developed for the eddy potential enstrophy, demonstrating that it is directly proportional to the interior PV gradient and the square root of the barotropic eddy kinetic energy. This study proposes a possible formation mechanism of large BG eddies and points to the importance of accurate representation of the interior PV gradients and eddy dissipation by ice-ocean drag in BG simulations and theory.

Genesis and Decay of Mesoscale Baroclinic Eddies in the Seasonally Ice-Covered Interior Arctic Ocean

AbstractObservations of ocean currents in the Arctic interior show a curious, and hitherto unexplained, vertical and temporal distribution of mesoscale activity. A marked seasonal cycle is found close to the surface: strong eddy activity during summer, observed from both satellites and moorings, is followed by very quiet winters. In contrast, subsurface eddies persist all year long within the deeper halocline and below. Informed by baroclinic instability analysis, we explore the origin and evolution of mesoscale eddies in the seasonally ice-covered interior Arctic Ocean. We find that the surface seasonal cycle is controlled by friction with sea ice, dissipating existing eddies and preventing the growth of new ones. In contrast, subsurface eddies, enabled by interior potential vorticity gradients and shielded by a strong stratification at a depth of approximately 50 m, can grow independently of the presence of sea ice. A high-resolution pan-Arctic ocean model confirms that the interior Arctic basin is baroclinically unstable all year long at depth. We address possible implications for the transport of water masses between the margins and the interior of the Arctic basin, and for climate models’ ability to capture the fundamental difference in mesoscale activity between ice-covered and ice-free regions.

Baroclinic instability and the mesoscale eddy field around the Lofoten Basin

AbstractThe vigorous mesoscale eddy field around the Lofoten Basin west of northern Norway is thought to be related to eddy shedding from the Norwegian Atlantic Current flowing along the Norwegian coast. Here we study baroclinic instability in the current with a particular focus on the influence of topography. The flow over the steepest part of the continental slope is found to be the most unstable. The growth characteristics cannot be understood from Eady theory alone but require the consideration of interior potential vorticity gradients. A study of the fully developed macroturbulent field shows that eddy kinetic energy is advected away from the generation regions and that nonlinear effects likely draw the eddy statistics away from the linear growth regime.

Baroclinic instability and eddy tracer transport across sloping bottom topography: How well does a modified Eady model do in primitive equation simulations?

Theoretical tracer diffusivities given by linear Eady theory that accounts for non-zero bottom slopes are compared with diffusivities diagnosed from primitive equation simulations of thermally forced flows over an idealized continental slope. The behavior is discussed in terms of a bottom slope parameter δ ¯ , the ratio of the bottom slope to an expression roughly representing the depth-averaged isopycnal slope. The theoretical diffusivities, scaled by the total thermal wind shear and the first internal deformation radius, agree qualitatively with diagnosed diffusivities for δ ¯ ≲ 0 , the parameter regime appropriate for buoyant boundary currents flowing over continental slopes. But whereas Eady theory would suggest maximum diffusivities for moderate positive slopes, δ ¯ ≃ 0.5 , the diagnosed diffusivities are highest for δ ¯ = 0 , i.e. for flat bottoms. Finally, whereas Eady diffusivities should drop to zero for δ ¯ ≳ 1 , i.e. when the bottom becomes steeper than the mean isopycnal slope, the diagnosed diffusivities do not. Similarities and differences are discussed in terms of more general linear stability theory applied to the background density profile over the central slope region. It is found that interior potential vorticity gradients, neglected by Eady theory, both cause significant modification to the Eady mode and also enable non-Eady instabilities that are responsible for the non-zero diffusivities for δ ¯ ≳ 1 . Furthermore, estimates of kinetic energy spectral fluxes suggest that an inverse kinetic energy cascade is present, and it is speculated that this is responsible for the diagnosed maximum diffusivity for flat bottoms.

The Continuous Spectrum in Baroclinic Models with Uniform Potential Vorticity Gradient and Ekman Damping

Abstract Analytic solutions of the continuous spectrum are obtained for quasigeostrophic models in which the basic-state meridional potential vorticity (PV) gradient is uniform but nonzero. The modes that form the continuous spectrum—continuum modes—are baroclinically neutral, singular solutions of the linear dynamical equations, which have an interior critical level. Although most inviscid shear flows support continuous spectra, previous studies have largely been restricted to models in which the interior PV gradient is zero. In that situation, continuum modes are formed by a superposition of boundary potential temperature anomalies and an interior PV δ function at the critical level. Here it is shown how a nonzero interior PV gradient and Ekman damping nontrivially affect the vertical structure of continuum modes. A consideration of the perturbation PV further rationalizes their vertical structure.

An Interpretation of Baroclinic Initial Value Problems: Results for Simple Basic States with Nonzero Interior PV Gradients

Abstract In the Eady model, where the meridional potential vorticity (PV) gradient is zero, perturbation energy growth can be partitioned cleanly into three mechanisms: (i) shear instability, (ii) resonance, and (iii) the Orr mechanism. Shear instability involves two-way interaction between Rossby edge waves on the ground and lid, resonance occurs as interior PV anomalies excite the edge waves, and the Orr mechanism involves only interior PV anomalies. These mechanisms have distinct implications for the structural and temporal linear evolution of perturbations. Here, a new framework is developed in which the same mechanisms can be distinguished for growth on basic states with nonzero interior PV gradients. It is further shown that the evolution from quite general initial conditions can be accurately described (peak error in perturbation total energy typically less than 10%) by a reduced system that involves only three Rossby wave components. Two of these are counterpropagating Rossby waves—that is, generalizations of the Rossby edge waves when the interior PV gradient is nonzero—whereas the other component depends on the structure of the initial condition and its PV is advected passively with the shear flow. In the cases considered, the three-component model outperforms approximate solutions based on truncating a modal or singular vector basis.

Quasigeostrophic Turbulence with Explicit Surface Dynamics: Application to the Atmospheric Energy Spectrum

Abstract The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior. The system exhibits three fundamental types of baroclinic instability: interactions between the upper and lower surfaces (Eady type), interactions between one surface and the interior (Charney type), and interactions between the barotropic and baroclinic interior modes (Phillips type). The turbulent steady states that result from each of these instabilities are distinct, and those of the former two types yield shallow kinetic energy spectra at small scales along those boundaries where mean temperature gradients are present. When both mean interior and surface gradients are present, the surface spectrum reflects a superposition of the interior-dominated −3 slope cascade at large scales, and the surface-dominated −5/3 slope cascade at small scales. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient, and scales with the inverse of the deformation scale when β = 0.

Potential Momentum and the equilibration of short Charney waves

This paper proposes a new formalism for studying the extratropical circulation, based on the concept of potential momentum. Mathematically, the potential momentum is defined as the zonal momentum profile that produces the same potential vorticity distribution as the baroclinic term. Physically, easterly (westerly) potential momentum is associated with isentropic layers that open up (close down) with latitude. The former is the case in the extratropical troposphere, and gives a negative contribution to the interior potential vorticity gradient. Likewise, the surface temperature variance is interpreted in this framework as a reservoir of easterly potential momentum at the surface. With this redefinition of the mean flow, the wave-mean flow interaction problem can be recast in a form that is reminiscent of the barotropic framework. In particular, the eddy PV flux is the only wave-mean flow interaction term, and exchanges eddy pseudomomentum and mean flow momentum (which includes both the standard momentum and potential momentum) locally. On the other hand, the partition of the mean flow momentum between its physical and potential components is determined remotely, subject to the thermal wind constraint. This constraint is enforced by a residual circulation that exchanges both terms without affecting the total momentum or the full PV gradient. The potential momentum framework relates thermal homogenization at the surface to the net interior eddy absorption, regardless of whether this results from a meridional or a vertical EP convergence. This helps explain previous results by the same authors that a limited interior PV gradient does not constrain thermal homogenization at the surface, since in the 3D problem the waves can also refract meridionally to be absorbed at other latitudes.

The Equilibration of Short Charney Waves: Implications for Potential Vorticity Homogenization in the Extratropical Troposphere

In this paper the authors discuss the vertical distribution of the interior potential vorticity gradients in the extratropical troposphere in an idealized setting. This structure is characterized by large positive gradients at the top of the boundary layer and a narrow homogenized region in the interior. To understand this partial homogenization, the authors have looked at how Charney waves of different scales equilibrate. It is shown that when the interior gradients of potential vorticity are small compared to the delta function at the ground, the modes can equilibrate without eliminating the lower-level shear. As is the case in the Eady model, these neutral states can exist even though they violate the Charney–Stern condition for stability. The equilibration process also produces a concentration of the potential vorticity gradients at heights of the order of the Rossby depth, which can be regarded as a boundary to the interior mixing of potential vorticity. The extent of interior potential vorticity mixing required for equilibration depends on the ratio between the boundary and interior potential vorticity gradients. It is implicit in the analysis of previous authors that this ratio is close to 1 in midlatitudes, which implies that an Eady model with full homogenization of the interior potential vorticity gradients is not an appropriate paradigm for the midlatitude troposphere. Nevertheless, it is argued that localized interior potential vorticity mixing could be sufficient for neutralizing the shortest waves.

Generalized Charney–Stern and Fjortoft Theorems for Rapidly Rotating Vortices

Abstract A generalized Charney–Stern theorem for rapidly rotating (large Rossby number) baroclinic vortices, such as hurricanes, is derived based on the asymmetric balance (AB) approximation. In the absence of dissipative processes, a symmetrically stable baroclinic vortex is shown to be exponentially stable to nonaxisymmetric perturbations if a generalized potential vorticity gradient on theta surfaces remains single signed throughout the vortex. The generalized potential vorticity gradient involves the sum of an interior potential vorticity gradient associated with the symmetric vortex and surface contributions associated with the vertical shear of the tangential wind. The AB stability formulation is then shown to yield Fjortoft's theorem as a corollary. In the modem view of shear instabilities the theorems admit simple interpretation. The Charney–Stern theorem represents a necessary condition for the existence of counterpropagating Rossby waves associated with the radial potential vorticity gradient, w...

Potential Vorticity Constraints on the Dynamics and Hydrography of the Southern Ocean

Abstract Constraints on the hydrography and geostrophic velocity shear of the Southern Ocean implicit in its potential vorticity field are discussed and illustrated by diagnostic study of observed and modeled potential vorticity fields. A stress-driven, thermodynamically inactive, eddy-resolving quasigeostrophic model of the Southern Ocean suggests that, through the systematic erosion of potential vorticity gradients by geostrophic eddies, the large-scale flow equilibrates toward a state in which interior potential vorticity gradients are small. Observations of the large-scale isopycnal distribution of potential vorticity (IPV), deduced from climatological hydrographic data, reveal a much richer structure. The most striking feature of the IPV field is the presence of large, near-surface gradients of IPV (a vortocline) coinciding with the axis of the Antarctic Circumpolar Current (ACC). To the south of this front potential vorticity (PV) is large; to its north, PV has low values, and relative to those foun...

Optimal Growth Rates in the Quasigeostrophic Initial Value Problem

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

Stability properties of an arbitrarily oriented mean flow

The linear properties of eddies growing in a mean flow that is oriented in any horizontal direction are analyzed. The mean flow only varies with height. The growth rate, phase speed and structure of the waves change dramatically with different orientations of the mean flow and the waves. These properties are related to the vertical profile of the mean total interior potential vorticity gradient, Q?. Q? is largest for those orientations where the Coriolis terms are most positive; it is smallest when those terms are most negative. When Q? is maximized; the growth rates are largest, the propagation speed is least, the upper level eddy amplitude is enhanced and the sharpest vertical tilts of the trough and ridge axes occur at lower levels. The opposite properties result when Q? is minimized. Limiting the horizontal extent of the wave (by using side-wall boundaries) causes cross-flow asymmetry in the wave structure and induces horizontal tilts to the wave axes. DOI: 10.1111/j.2153-3490.1981.tb01743.x

External Rossby Modes on a β–Plane with Realistic Vertical Wind Shear

Abstract Free oscillations of small amplitude and low frequency in an isothermal atmosphere at rest have the vertical structure of an edge wave, energy density decaying exponentially away from the ground. In this paper we treat these external Rossby modes on a middle latitude β-plane to see how their vertical structure and phase speed are changed by the presence of a winter zonal wind with realistic vertical structure. We find that for planetary-scale modes the basic state interior potential vorticity gradient and the temperature gradient at the lower boundary produce roughly equal but opposite changes in phase speed. Energy density decays less rapidly with altitude when the basic state temperature gradient is included in the lower boundary condition. As wavenumber is increased, the phase speed of the external Rossby mode is found to approach the speed of the zonal wind at the surface, and there is a limiting wavenumber beyond which the mode does not exist. At this limiting value of wavenumber, the mode b...

Generalized Eady waves

Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution.The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency.