Quasi-geostrophic Potential Vorticity Research Articles

Wind stress curl as a driving force of annual waves in the upper ocean for interpreting energetics at all latitudes

The dynamics of waves and eddies in the upper ocean plays an important role in the climate variation of tropical and subtropical regions. Previous diagnoses for annual Rossby waves in oceanic model outputs manifested zonally alternating signals (ZASs) in the time-averaged distributions of wind input as well as pressure-flux divergence terms in the budget equation of wave energy. This is the case when the annual mean of the wind input is estimated as the inner product of simulated velocity vector and wind stress vector in previous studies. The present study proposes a new mathematical expression for estimating the wind input that is analogous to one derived from the quasi-geostrophic potential vorticity equation. Namely, the wind input is estimated as the negative of the product of pseudo-streamfunction and wind stress curl, the latter of which is associated with the horizontal divergence of Ekman velocity. This can be interpreted as replacement of kinetic energy input with gravitational potential energy input. Pseudo-streamfunction in the present study is inverted from Ertel’s potential vorticity anomaly and is seamlessly available at all latitudes. This contrasts with the quasi-geostrophic streamfunction which is singular at the equator. The new expression enables reducing ZASs in the horizontal distributions of both wind input and pressure-flux divergence terms, without harming the qualitative advantage of energy flux vectors to indicate the group velocity of waves at all latitudes.

Tropospheric Thermal Forcing of the Stratosphere through Quasi-Balanced Dynamics

Abstract The steady response of the stratosphere to tropospheric thermal forcing via an SST perturbation is considered in two separate theoretical models. It is first shown that an SST anomaly imposes a geopotential anomaly at the tropopause. Solutions to the linearized quasigeostrophic potential vorticity equations are then used to show that the vertical length scale of a tropopause geopotential anomaly is initially shallow, but significantly increased by diabatic heating from radiative relaxation. This process is a quasi-balanced response of the stratosphere to tropospheric forcing. A previously developed, coupled troposphere–stratosphere model is then introduced and modified. Solutions under steady, zonally symmetric SST forcing in the linear β-plane model show that the upward stratospheric penetration of the corresponding tropopause geopotential anomaly is controlled by two nondimensional parameters: 1) a dynamical aspect ratio and 2) a ratio between tropospheric and stratospheric drag. The meridional scale of the SST anomaly, radiative relaxation rate, and wave drag all significantly modulate these nondimensional parameters. Under Earthlike estimates of the nondimensional parameters, the theoretical model predicts stratospheric temperature anomalies 2–3 larger in magnitude than that in the boundary layer, approximately in line with observational data. Using reanalysis data, the spatial variability of temperature anomalies in the troposphere is shown to have remarkable coherence with that of the lower stratosphere, which further supports the existence of a quasi-balanced response of the stratosphere to SST forcing. These findings suggest that besides mechanical and radiative forcing, there is a third way the stratosphere can be forced—through the tropopause via tropospheric thermal forcing. Significance Statement Upward motion in the tropical stratosphere, the layer of atmosphere above where most weather occurs, is thought to be controlled by weather disturbances that propagate upward and dissipate in the stratosphere. The strength of this upward motion is important since it sets the global distribution of ozone. We formulate and use simple mathematical models to show the vertical motion in the stratosphere can also depend on the warming in the troposphere, the layer of atmosphere where humans live. We use the theory as an explanation for our observations of inverse correlations between the ocean temperature and the stratosphere temperature. These findings suggest that local stratospheric cooling may be coupled to local tropospheric warming.

On the Quartic Korteweg–de Vries hierarchy of nonlinear Rossby waves and its dynamics

A long-standing challenge in exploring the dynamics of large-scale atmospheric or oceanic motions is to find more appropriate theoretical models. This paper aims at the mechanisms of nonlinear Rossby waves by a new obtained nonlinear evolutionary equation hierarchy approach. The multiple scales method and weak nonlinear perturbation method are adopted based on the quasi-geostrophic potential vorticity conservation law for large scale atmospheric or oceanic motions. It is the first time that a Quartic Korteweg–de Vries model equation is derived to describe the propagation of nonlinear Rossby waves amplitude, which provides a new acceptable theoretical model for the study of atmospheric blocking phenomena. Qualitative and quantitative analysis for the new Quartic KdV model are finished to disclose the evolutionary mechanisms of nonlinear Rossby waves. The corresponding results declares that the β effect, topographic features and detuning parameter are all important factors on the nonlinear Rossby waves.

Variable coefficient extended cKP equation for Rossby waves and its exact solution with dissipation

In this article, the variable coefficient (2 + 1)-dimensional extended cylindrical Kadomtsev–Petviashvili (cKP) equation describing Rossby waves was derived from the quasi-geostrophic potential vorticity equation. It is difficult for the variable coefficient cKP equation with dissipation to calculate the exact solution. For obtaining the exact solution, a new transformation was constructed for the first time to reduce the extended cKP equation to the extended KP equation. We emphasize that the exact solution, and not just approximate solution, in Rossby waves flow field can be obtained when dissipation is included. The exact lump and interaction solutions with dissipative effect are given according to the modified Hirota bilinear method, and physics for the evolution of Rossby waves is analyzed based on the obtained solutions. When the dissipative parameter μ0 increases, the structure of the amplitude A changes in the spatial scale y. And when the dissipative parameter increases to a certain value, the structure of Rossby waves tends to be stable. It is pointed out that the dissipative parameter μ0 determines not only the amplitude A of Rossby waves but also structures of Rossby waves flow field, with μ0 acting on the spatial scale y and the timescale t.

Filaments of uniform quasi-geostrophic potential vorticity in pure strain

Three-dimensional filaments of quasi-geostrophic potential vorticity are generic features of atmospheric and oceanic flows. They are often generated during the strong interactions between three-dimensional quasi-geostrophic vortices. They contribute to a direct cascade of enstrophy in spectral space. These filaments correspond to shear zones. Therefore they may be sensitive to shear instabilities akin to the Kelvin–Helmholtz instability of the classical two-dimensional vorticity strip. They are, however, often subjected to a straining flow induced by the surrounding vortices. This straining flow affects their robustness. This paper focuses on a simplified model of this situation. We consider the effect of a pure strain on a three-dimensional filament of uniform quasi-geostrophic potential vorticity. We first consider a quasi-static situation where the strain, assumed small, only affects the cross-sectional shape of the filament, but not the velocity field. We address the linear stability of the filament in that context and also show examples of the filament's nonlinear evolution. We then consider the linearised dynamics of the filament in pure strain. In particular we focus on the maximum perturbation amplification observed in the filament. We conclude that small to moderate strain rates are efficient at preventing a large perturbation growth. Nonlinear effects can nevertheless leads to the roll-up of weakly strained filaments.

A direct derivation of the Gent–McWilliams/Redi diffusion tensor from quasi-geostrophic dynamics

The transport induced by ocean mesoscale eddies remains unresolved in most state-of-the-art climate models and needs to be parametrized instead. The natural scale separation between the forcing and the emergent turbulent flow calls for a diffusive parametrization, where the eddy-induced fluxes are related to the large-scale gradients by a diffusion tensor. The standard parametrization scheme in climate modelling consists in adopting the Gent–McWilliams/Redi (GM/R) form for the diffusion tensor, initially put forward based on physical intuition and educated guesses before being put on firm analytical footing using a thickness-weighted average (TWA). In the present contribution, we provide a direct derivation of this diffusion tensor from the quasi-geostrophic (QG) dynamics of a horizontally homogeneous three-dimensional patch of ocean hosting a large-scale vertically sheared zonal flow on the $\beta$ plane. The derivation hinges on the identification of a useful cross-invariant defined as the product of the buoyancy and QG potential vorticity fluctuations. While less general than the TWA approach, the present QG framework leads to rigorous constraints on the diffusion tensor. First, there is no diapycnal diffusivity arising in the QG GM/R tensor for low viscosity and small-scale diffusivities. The diffusion tensor then involves only two vertically dependent coefficients, namely the GM transport coefficient $K_{GM}(z)$ and the Redi diffusivity $K_R(z)$ . Second, as identified already by previous authors, the vertical structures of the two coefficients are related by the so-called Taylor–Bretherton relation. Finally, while the two coefficients differ generically in the interior of the water column, we show that they are equal to one another near the surface and near the bottom of the domain for low-enough dissipative coefficients. We illustrate these findings by simulating numerically the QG dynamics of a horizontally homogeneous patch of ocean hosting a vertically sheared zonal current resembling the Antarctic Circumpolar Current.

Cautionary tales from the mesoscale eddy transport tensor

The anisotropic mesoscale eddy transport tensor is diagnosed using passive tracers advected online in both an idealized 101-member mesoscale-resolving quasi-geostrophic (QG) double-gyre ensemble, and a realistic 24-member eddying (1/12∘) ensemble of the North Atlantic. We assert that the Reynold’s decomposition along the ensemble dimension, rather than the spatial or temporal dimension, allows us to capture the intrinsic spatiotemporal variability of the mean flow and eddies. The tensor exhibits good performance in reconstructing the eddy fluxes of passive tracers, here defined as fluctuations about the ensemble thickness-weighted averaged (TWA) mean. However, the inability of the tensor to reconstruct eddy fluxes of QG potential vorticity, which encapsulates the eddy-mean flow interaction, and other active tracers raises the question: To what extent can the diagnosed tensor be applied to inform the parametrization of mesoscale dynamics?

Compensation between Resolved Wave Forcing and Parameterized Orographic Gravity Wave Drag in the Northern Hemisphere Winter Stratosphere Revealed in NCEP CFS Reanalysis Data

Abstract Compensation between the resolved wave (RW) forcing and the parameterized orographic gravity wave drag (OGWD) accompanying barotropic/baroclinic (BT/BC) instability in the realistic atmosphere is investigated using Climate Forecast System Reanalysis data in the Northern Hemisphere winter stratosphere. When sufficiently narrow and/or strong negative OGWD drives instability, RWs are generated in situ, providing positive Eliassen–Palm flux divergence that compensates for the parameterized OGWD enhancement; this is consistent with the findings of previous studies based on the idealized general circulation models. However, dependence of the compensation rate on RW forcing differs from the nearly complete compensation in the previous studies, implying that an additional mechanism operates for the compensation: the refractive-index modification by BT/BC instability. The negative meridional gradient of the quasigeostrophic potential vorticity leads to the negative refractive index squared for RWs with phase speeds less than the zonal-mean zonal wind. This prevents RWs from entering the destabilized areas, resulting in the divergence of Eliassen–Palm fluxes that cancels out the parameterized OGWD perturbation. Although both mechanisms act simultaneously, the refractive-index modification plays an important role in the compensation processes in the stratosphere where RWs are dominated by the planetary-scale waves.

Effect of the Quadric Shear Zonal Flows and Beta on the Downstream Development of Certain Unstable Baroclinic Waves

Many factors are affecting the downstream development of baroclinic waves, among which zonal shear flow is one of the factors that need to be considered. In this paper, the influence of zonal shear flow and <em>β</em> on the downstream development of unstable chaotic baroclinic waves is studied from the two-layer model in a wide channel controlled by quasi-geostrophic potential vorticity equation. Through the obtained Lorentz equation, We concentrated on the influence of zonal shear flow (the second derivative of baseline zonal flow is not zero) on the downstream development of baroclinic waves. In the absence of zonal shear flow, chaotic behavior along feature points would occur, and the amplitude would change rapidly from one feature to another, that is, it would change very quickly in space. When zonal shear flow is introduced, the influence of zonal shear flow on the downstream development of unstable baroclinic waves is examined categorically. And from Lorentz’s final equation, we’re investigating a change in his solution. It is found that the zonal shear flow smoothes the solution of the equation and reduces the instability, and with the increase of zonal shear flow, the stability in space will increase gradually. The second derivative of the zonal shear flow (the quadrical shear flow) therefore has a major influence on the stability of space.

A new (2+1)-D mZK-Burgers model for non-linear Rossby waves aswell as the analytical solution

In the paper, based on the quasi-geostrophic potential vorticity equation with topography effect, we derived a modified ZakharovKuznetsor (mZK)-Burgers equation by employing multiscale analysis and perturbation method. The model can be described the propagation of the nonlinear long wave and solitary eddy. The exact solutions are given by virtue of the (G'/G)-expansion method to analyze wave propagation characteristics.

Forced solitary wave and vorticity with topography effect in quasi-geostrophic modelling

A forced KdV equation including the special topography effect is derived to describe nonlinear long wave and solitary eddy based on the quasi-geostrophic potential vorticity model. We obtain the theoretical solution of the equation and the concrete form of stream function through perturbation theory and multi-scale analysis methods. It is found that the joint effect of weak shear basic flow and topography can change the cyclone and anticyclone structure of eddy, and in the meantime topographic structure affects the East-West propagation direction of solitary wave. Finally, according to the interaction between nonlinear long wave and topography by pseudo spectral numerical method, the topographic height is related to the amplitude, wavelength and wave velocity of the excited wave train, and the topography affects not only the spatial structure of wave, but also the amplitude of wave.

A Multi‐Layer Linear Rossby Wave Dispersion Relation for Vertical Tilt of Mesoscale Eddies

AbstractVertical tilt of mesoscale eddies has been frequently observed in world oceans. The observations of composite eddies, individual eddies from an oceanic reanalysis product, and idealized numerical experiments of mesoscale eddies all show that the eddy vertical tilt is gradually increased with time in eddy mature stage, implying that the eddy translation speed changes with depth. However, eddy vertical tilt cannot be explained by the classic linear Rossby wave dispersion relation, since the classic zonal translation speed () of linear Rossby waves is depth‐independent from the quasi‐geostrophic potential vorticity (QGPV) equation. To deduce a depth‐dependent linear Rossby wave dispersion relation, we solved the QGPV equation by assuming that the stratified water column in oceans can be separated into a number of fine layers with constant stratification in each layer, whereas the stratification changes among layers. A multi‐layer linear Rossby wave (MLRW) dispersion relation in stratified oceans is then deduced. The MLRW dispersion relation shows that changes with depth. In each layer, is proportional to stratification and layer thickness but inversely proportional to Earth's rotation. The estimated structure from the MLRW dispersion relation is consistent with the structure of eddies in numerical experiments and the observed composite eddies. Noticeably, there are some discrepancies between the theoretical estimates and those from observations and numerical simulations, primarily due to the sensitivity of the theoretical estimates to layer thickness. The study attempts to provide an approach to describe and predict mesoscale eddy vertical tilt.

A Three-Dimensional Inertial Model for Coastal Upwelling along Western Boundaries

Abstract A three-dimensional inertial model that conserves quasigeostrophic potential vorticity is proposed for wind-driven coastal upwelling along western boundaries. The dominant response to upwelling favorable winds is a surface-intensified baroclinic meridional boundary current with a subsurface countercurrent. The width of the current is not the baroclinic deformation radius but instead scales with the inertial boundary layer thickness while the depth scales as the ratio of the inertial boundary layer thickness to the baroclinic deformation radius. Thus, the boundary current scales depend on the stratification, wind stress, Coriolis parameter, and its meridional variation. In contrast to two-dimensional wind-driven coastal upwelling, the source waters that feed the Ekman upwelling are provided over the depth scale of this baroclinic current through a combination of onshore barotropic flow and from alongshore in the narrow boundary current. Topography forces an additional current whose characteristics depend on the topographic slope and width. For topography wider than the inertial boundary layer thickness the current is bottom intensified, while for narrow topography the current is wave-like in the vertical and trapped over the topography within the inertial boundary layer. An idealized primitive equation numerical model produces a similar baroclinic boundary current whose vertical length scale agrees with the theoretical scaling for both upwelling and downwelling favorable winds.

Necessary conditions for the instability of quasigeostrophic waves induced by trace shortwave radiative absorbers

Necessary conditions for radiative–dynamical instability of quasigeostrophic waves induced by trace shortwave radiative absorbers are derived. The analysis pivots on a pseudomomentum conservation equation that is obtained by combining conservation equations for quasigeostrophic potential vorticity, thermodynamic energy, and trace absorber mixing ratio. Under the assumptions that the absorber-induced diabatic heating rate is small and the zonal-mean basic state is hydrodynamically neutral, a perturbation analysis of the pseudomomentum equation yields the conditions for instability. The conditions, which only require knowledge of the zonally averaged background distributions of wind and absorber, expose the physical processes involved in destabilization—processes not exposed in previous analytical and modeling studies of trace absorber-induced instabilities. The simplicity of instability conditions underscores their utility as a tool that is both interpretive and predictive. The conditions for instability, which have broad application to synoptic-scale waves in Earth's and other planetary atmospheres, are discussed in light of previous instability studies involving stratospheric ozone and Saharan mineral dust aerosols.

A new nonlinear integral-differential equation describing Rossby waves and its related properties

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.

Asymmetrical meridional expansion of bright clouds from Saturn's 2010 great white storm

Saturn's recurring great white storms play an important role in modifying its atmosphere. In 2010, such a storm with clouds encircling the planet occurred in the northern hemisphere. An interesting phenomenon of this storm is that the associated bright clouds expanded asymmetrically with respect to latitude, such that the southern boundary of the bright clouds moved ~2.7 times as far as the northern boundary during an 8-month period. Based on the wind and temperature fields retrieved from the Cassini visible and infrared observations, we explore the mechanism behind this asymmetrical expansion. Our analysis shows that the northern edge, which quickly stopped moving, coincides with the largest meridional gradient of the quasi-geostrophic potential vorticity (PV) in the region of interest, which is coincident with the strongest jet in the region, suggesting this forms an effective barrier to meridional transport, much like a polar vortex. In contrast, the storm's southern edge, which kept moving, passed through weaker PV gradients and jets. For the threshold value of the meridional gradient of PV needed to form an effective barrier to meridional transport in Saturn's mid-latitude upper-troposphere, we estimate lower and upper bounds of ~2.1 × 10−11 m−1 s−1 and ~3.6 × 10−11 m−1 s−1.

A new three dimensional dissipative Boussinesq equation for Rossby waves and its multiple soliton solutions

A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a three dimensional dissipative Boussinesq equation by the reduced perturbation method, i.e.utt+e1uxx+e2(u2)xx+e3utxy+e4uxxxx+e5uxxyy=0. It is emphasized that the new equation is different from the existing Boussinesq equations, which describe the three dimensional nonlinear Rossby waves in the atmosphere. Moreover, we explore the dispersion relation of the linear wave through the new equation. Using the travelling wave method and simplest equation method, the general solution and soliton solutions of the equation are obtained successfully respectively. Finally, the formation mechanism of Rossby waves is discussed by multiple soliton solutions.

Role of Gravity Waves in a Vortex-Split Sudden Stratospheric Warming in January 2009

AbstractThe role of gravity waves (GWs) in a sudden stratospheric warming (SSW) event that occurred in January 2009 (SSW09) is investigated using the MERRA-2 dataset. Nearly 2 weeks prior to the central date (lag = 0), at which the zonal-mean zonal wind at 10 hPa and 60°N first becomes negative, westward GW drag (GWD) is significantly enhanced in the lower mesosphere and stratosphere. At 5 days before lag = 0, planetary waves (PWs) of zonal wavenumber 2 (ZWN-2) in the stratosphere are enhanced, while PWs of ZWN-1 are weakened, which are evident from the amplitudes of the PWs and their Eliassen–Palm flux divergence (EPD). To examine the relationship between PWs and GWs, a nonconservative GWD (NCGWD) source term of the linearized quasigeostrophic potential vorticity equation is considered. A ZWN-2 pattern of the NCGWD forcing is developed around z = 55–60 km with a secondary peak around z = 40 km just before the PWs of ZWN-2 in the stratosphere began to enhance. A significant positive correlation between the NCGWD forcing in the upper stratosphere and lower mesosphere (USLM; 0.3–0.1 hPa in the present data) and the PWs of ZWN-2 in the stratosphere (5–1 hPa) exists. This result demonstrates that the amplification of the PWs of ZWN-2 in the stratosphere before the onset of SSW09 is likely related to the generation of PWs by GWD in the USLM, which is revealed by the enhanced downward-propagating PWs of ZWN-2 into the stratosphere from above.

Solitary waves of nonlinear barotropic–baroclinic coherent structures

This study describes the evolutionary mechanisms of nonlinear barotropic–baroclinic interactions, especially, on the excitations, propagations, and decreases of nonlinear coherent structures. Starting from the classical two-layer quasi-geostrophic potential vorticity conservation model equations, the barotropic and baroclinic model equations are derived from the classical work of Pedlosky and Thomson [J. Fluid Mech. 490, 189–215 (2003)]. By considering the effects of bottom topography and beta-plane approximation, the coupled nonlinear Korteweg–de Vries model equations for the evolutions of barotropic and baroclinic coherent structures are obtained by using the methods of multiple scales and perturbation expansions, respectively. Solitary wave solutions are given according to the method of elliptic function expansions, and the physical mechanisms for the evolutions of the nonlinear barotropic–baroclinic interactive coherent structures are analyzed based on the obtained solitary wave solutions. It will be potentially useful for further theoretical investigations on atmospheric blocking phenomena or wave–flow interactions.

The barotropic Rossby waves with topography on the earth’sδ-surface

AbstractThe Rossby solitary waves in the barotropic vorticity model which contains the topography on the earth’sδ-surface is investigated. First, applying scale analysis method, obtained the generalized quasi-geostrophic potential vorticity equation (QGPVE). Using The Wentzel–Kramers–Brillouin (WKB) theory, the evolution equation of Rossby waves is the variable-coefficient Korteweg–de Vries (KdV) equation for the barotropic atmospheric model. In order to study the Rossby waves structural change to exist in some basic flow and topography on theδ-surface approximation, the variable coefficient of KdV equation must be explicitly, Chebyshev polynomials is used to solve a Sturm-Liouville-type eigenvalue problem and the eigenvalue Rossby waves, these solutions show that the parameterδusually plays the stable part in Rossby waves and slow down the growing or decaying of Rossby waves with the parameterβ.