Semi-geostrophic Dynamics Research Articles

Scientific Challenges of Convective-Scale Numerical Weather Prediction

AbstractAfter extensive efforts over the course of a decade, convective-scale weather forecasts with horizontal grid spacings of 1–5 km are now operational at national weather services around the world, accompanied by ensemble prediction systems (EPSs). However, though already operational, the capacity of forecasts for this scale is still to be fully exploited by overcoming the fundamental difficulty in prediction: the fully three-dimensional and turbulent nature of the atmosphere. The prediction of this scale is totally different from that of the synoptic scale (103km), with slowly evolving semigeostrophic dynamics and relatively long predictability on the order of a few days.Even theoretically, very little is understood about the convective scale compared to our extensive knowledge of the synoptic-scale weather regime as a partial differential equation system, as well as in terms of the fluid mechanics, predictability, uncertainties, and stochasticity. Furthermore, there is a requirement for a drastic modification of data assimilation methodologies, physics (e.g., microphysics), and parameterizations, as well as the numerics for use at the convective scale. We need to focus on more fundamental theoretical issues—the Liouville principle and Bayesian probability for probabilistic forecasts—and more fundamental turbulence research to provide robust numerics for the full variety of turbulent flows.The present essay reviews those basic theoretical challenges as comprehensibly as possible. The breadth of the problems that we face is a challenge in itself: an attempt to reduce these into a single critical agenda should be avoided.

A study of surface semi-geostrophic turbulence: freely decaying dynamics

In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical flows.

Implications of the Semigeostrophic Nature of Rossby Waves for Rossby Wave Packet Detection

Abstract Upper-tropospheric Rossby wave packets have received increased attention recently. In most previous studies wave packets have been detected by computing the envelope of the meridional wind field using either complex demodulation or a Hilbert transform. The latter requires fewer choices to be made and appears, therefore, preferable. However, the Hilbert transform is fraught with a significant problem, namely, a tendency that fragments a single wave packet into several parts. The problem arises because Rossby wave packets show substantial deviations from the almost-plane wave paradigm, a feature that is well represented by semigeostrophic dynamics. As a consequence, higher harmonics interfere with the reconstruction of the wave envelope leading to undesirable wiggles. A possible cure lies in additional smoothing (e.g., by means of a filter) or resorting to complex demodulation (which implies smoothing, too). Another possibility, which does not imply any smoothing, lies in applying the Hilbert transform in semigeostrophic coordinate space. It turns out beneficial to exclude planetary-scale wavenumbers from this transformation in order to avoid problems in cases when the wave packet travels on a low wavenumber quasi-stationary background flow.

Impact of the SST–wind stress coupling on the dynamics and stability of ocean current inside and outside SST frontal zones

The robust relationship between the sea surface temperature (SST) and wind stress is used to examine the impact of the SST–wind stress coupling on the dynamics and stability of ocean currents inside and outside SST frontal zones based on the wave-activity equations for the semi-geostrophic (SG) and quasi-geostrophic (QG) models.The results show that the coupling has fundamentally different effects on the ocean current inside and outside SST fronts. Outside the frontal zones the coupling only redistributes the surface temperature perturbation but does not change the total pseudo-momentum and pseudo-energy. Therefore the QG pseudo-momentum and pseudo-energy are still conserved under the impact of the coupling, although the pseudo-energy is slightly modified. The stability conditions, more general than the ones obtained by Spall (2007), are obtained from the conservation of the wave-activities. Inside the SST frontal zones, however, the coupling, along with the nonlinear Ekman effect, destroys the conservation of the SG pseudo-momentum and pseudo-energy. Therefore no stability condition is available, indicating that the current inside the frontal zone may be unstabilized by the coupling or nonlinear Ekman transport effect. It is also found that different dynamical features inside and outside frontal zones cause the different modulation of the coupling strength. The drastic differences are closely associated with the different assumptions underlying QG and SG dynamics about the density/temperature gradient and isotropy of motions.The different impacts of the coupling are further illustrated analytically by comparing the normal mode growth rates for the SG and QG Eady models. The comparisons show that ocean current outside the SST frontal zone is stable under any normal mode disturbances when the coupling factor σQG<−U¯s where U¯s is the basic zonal flow at the surface, but it is unstable under certain normal modes when σQG>−U¯s. However the ocean current inside the frontal zone is always unstable under the influence of the SST–wind coupling or nonlinear Ekman effect. It is found that the negative coupling (when surface wind blows from cold water to warm water) tends to generate strong instability. In addition, the growing/decaying normal modes are non-dispersive and propagate only in one direction (eastward) outside the frontal zone, but they are dispersive and can propagate in both directions inside frontal zone.

Surface semi-geostrophic dynamics in the ocean

The surface quasi-geostrophic approximation is re-written in an oceanic context using the two-dimensional semi-geostrophic approximation. The new formulation allows to take into account the presence of out-of-balance flow features at scales comparable to or smaller than the Rossby radius of deformation and for small bulk Richardson numbers. Analytical solutions show that, while the surface quasi-geostrophic approximation tends to underestimate the buoyancy anomaly, the inclusion of finite Rossby number allows for larger values of the buoyancy anomaly at depth. The projection of the surface semi-geostrophic solution on the first baroclinic modes is calculated. The result of the projection is a functional form that decreases with the values of the Rossby number and toward smaller scales. Solutions for constant and exponential profile for the background potential vorticity are compared. Results of the comparison show that, in agreement with the results found for balanced flows, even for large Rossby number the exponential profile for the background potential vorticity retains smaller values for the buoyancy anomaly at depth than the solution found using a constant potential vorticity profile. †Previously at: Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ, USA.

Current separation and upwelling over the southeast shelf of Vietnam in the South China Sea

The high‐resolution, unstructured grid Finite‐Volume Community Ocean Model (FVCOM) was used to examine the physical mechanisms that cause current separation and upwelling over the southeast shelf of Vietnam in the South China Sea (SCS). Process‐oriented experiments suggest that the southwesterly monsoon wind is a key physical mechanism for upwelling in that area but not a prerequisite to cause current separation. With no wind forcing, current separation in summer can occur as a result of the encounter of a southward along‐shelf coastal current from the north and northeastward buoyancy‐driven and stratified tidal‐rectified currents from the southwest. The southward current can be traced upstream to the Hong River in the Gulf of Tonkin. This current is dominated by semigeostrophic dynamics and is mostly confined to the narrow shelf along the northern Vietnamese coast. The northeastward currents are generated by tidal rectification and are intensified by the Mekong River discharge and southwesterly monsoon wind forcing. The dynamics controlling this current are fully nonlinear, with significant contributions from advection and vertical turbulent mixing. Upwelling in the current separation zone can be produced by a spatially uniform constant wind field and can be explained using simple wind‐induced Ekman transport theory. This finding differs from previous theory in which the regional dipole wind stress curl is claimed as a key mechanism for current separation and upwelling in this coastal region. Our SCS FVCOM, driven by the wind stress, river discharge, and tides, is capable of reproducing the location and tongue‐like offshore distribution of temperature as those seen in satellite‐derived sea surface temperature imagery.

Wave activity and baroclinic stability theorems for a two-dimensional stratified semi-geostrophic model

Arnold's method is used to derive two finite-amplitude wave-activity invariants – pseudo-momentum and pseudo-energy – based on the Hamiltonian structure for the two-dimensional baroclinic semi-geostrophic (SG) model in physical coordinates. The corresponding Liapunov stability theorems are established. For small-amplitude disturbances the pseudo-momentum and pseudo-energy can be greatly simplified by introducing the quasi-potential-vorticity perturbation. The corresponding linear stability conditions, expressed explicitly in terms of the basic state, are obtained. The lateral boundary conditions are found to have a big impact on the stability of SG dynamics in both linear and nonlinear cases. The results show that nonlinear stability criteria are more complicated than linear stability criteria, in that they contain more conditions in the interior resulting from the inclusion of potential temperature and the nonlinearity of SG potential vorticity in the wave activity. It is also found that the nonlinear stability criteria for the SG model are very different from those for the quasi-geostrophic (QG) model. The failure to extend the Liapunov stability theorems to the three-dimensional SG model is discussed. It is found that this failure results from the third-order nonlinear terms in the three-dimensional SG potential vorticity. Since the variation of these terms in the wave activity is not sign-definite, nonlinear stability conditions cannot be obtained. Copyright © 2011 Royal Meteorological Society

Surface Boundaries of the Southern Plains: Their Role in the Initiation of Convective Storms

Abstract The nature of the different types of surface boundaries that appear in the southern plains of the United States during the convectively active season is reviewed. The following boundaries are discussed: fronts, the dryline, troughs, and outflow boundaries, The boundaries are related to their environment and to local topography. The role these boundaries might play in the initiation of convective storms is emphasized. The various types of boundary-related vertical circulations and their dynamics are discussed. In particular, quasigeostrophic and semigeostrophic dynamics, and the dynamics of solenoidal circulations, density currents, boundary layers, and gravity waves are considered. Miscellaneous topics pertinent to convective storms and their relationship to surface boundaries such as along-the-boundary variability, boundary collisions, and the role of vertical shear are also discussed. Although some cases of storm initiation along surface boundaries have been well documented using research datasets collected during comprehensive field experiments, much of what we know is based only on empirical forecasting and nowcasting experience. It is suggested that many problems relating to convective-storm formation need to be explored in detail using real datasets with new observing systems and techniques, in conjunction with numerical simulation studies, and through climatological studies.

The Vorticity–Velocity Gradient Cofactor Tensor and the Material Invariant of the Semigeostrophic Theory

Abstract A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted Ξ̃) is derived. The specific potential Ξ̃, that is, Ξ̃ in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential Ξ̃. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and f-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential Ξ̃) differs conceptually from the Beltrami–Rossby–Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the f-plane approximation and on the horizontal nature of the geostrophic velocity.

Finite-Amplitude Wave-Activity Invariants and Nonlinear Stability Theorems for Shallow Water Semigeostrophic Dynamics

Abstract Motivated by the boundary contributions to the wave-activity invariants and stability theorems of a class of Salmon’s L1-like Hamiltonian balance models, Arnold’s method is applied in this work to derive finite-amplitude wave-activity invariants and corresponding stability theorems for shallow water semigeostrophic (SWSG) dynamics. It is shown that the Jacobian term in the potential vorticity of the SWSG model affects the stability properties in two ways: it generates stability constraints in the interior, and it makes the stability condition of cyclonic shear of basic flow at boundaries inescapable even when Ripa’s “subsonic” condition is satisfied in the interior. The latter effect makes the stability properties of the SWSG model different from those of the L1-like Hamiltonian balance models for which the condition of cyclonic shear of basic flow on the boundaries is not necessary when Ripa’s “subsonic” condition is satisfied. The physical reason for this difference is given and the implication...

Legendre-Transformable Semigeostrophic Theories

Abstract For semigeostrophic (SG) theories derived from the Hamiltonian principles suggested by Salmon it is known that a duality exists between the physical coordinates and geopotential, on the one hand, and isentropic geostrophic momentum coordinates and geostrophic Bernoulli function, on the other hand. The duality is characterized geometrically by a contact structure. This enables the idealized balanced dynamics to be represented by horizontal geostrophic motion in the dual coordinates while the mapping back to physical space is determined uniquely by requiring each instantaneous state to be the one of minimum energy with respect to volume-conserving rearrangements within the physical domain. It is found that the generic contact structure permits the emergence of topological anomalies during the evolution of discontinuous flows. For both theoretical and computational reasons it is desirable to seek special forms of SG dynamics in which the structure of the contact geometry prohibits such anomalies. It...

Barotropic Instability Due to Kelvin Wave–Rossby Wave Coupling

Abstract The stability of a vertically bounded, piecewise linear shear profile in a channel is analyzed using both a quasigeostrophic (QG) and primitive equation (PE) model. The choice of a finite depth domain thus allows us to consider more realistic flows in which the jet is vertically bounded. A potential vorticity discontinuity in the QG model can give rise to an isolated shear-generated Rossby mode that remains stable in the absence of other mean flow discontinuities. The finite depth assumption in the QG model is of little consequence as the vertical scale of the basic state enters the dynamical equations in a trivial manner. Solving this problem in the PE model, on the other hand, leads to unstable modes not present in the QG limit. Using semigeostrophic (SG) dynamics the authors are able to identify two modes of instability. One occurs as a result of a Kelvin wave–Kelvin wave coupling and the other is a product of Kelvin wave–Rossby wave coupling. It is also found that resonance between the Rossby...

Linear stability theorems for shallow water semi-geostrophic dynamics

The linear stability of zonal basic flow in the shallow water semi-geostrophic (SWSG) model is examined with an emphasis on the contributions of lateral boundaries to the stability properties of the model. Two conservation equations for wave activities, analogous to the pseudomomentum and pseudoenergy densities in the shallow water quasi-geostrophic (SWQG) theory, are obtained. Two linear stability theorems are established based on the pseudoenergy conservation equation. The first theorem, corresponding to positive definite pseudoenergy, requires the gradient of basic state potential vorticity to be negatively correlated with basic flow in the interior, and the shear of basic flow to be cyclonic at lateral boundaries. The second theorem, corresponding to negative definite pseudoenergy, requires the gradient of basic state potential vorticity to be positively correlated with basic flow in the interior and the shear of basic flow to be anti-cyclonic at lateral boundaries, plus some additional conditions. In both theorems, Ripa's subsonic condition U ≤ √gH is found unnecessary, and the unnecessity is found to be associated with the dynamic structure of the model. The general properties of normal mode disturbances are discussed. Bounds on the phase speed and growth rate of unstable normal modes are estimated. The contributions of lateral boundaries to the instability of constant shear and cosine type basic flows are investigated numerically. It is shown that the contributions crucially depends on the value of the Rossby number. When stability criterion in the interior is violated, unstable growth rate can be significantly enhanced by the boundary contributions. The underlying physical interpretation is given.

Coupled Kelvin-Wave and Mirage-Wave Instabilities in Semigeostrophic Dynamics

Abstract A weak instability mode, associated with phase-locked counterpropagating coastal Kelvin waves in horizontal anticyclonic shear, is found in the semigeostrophic (SG) equations for stratified flow in a channel. This SG instability mode approximates a similar mode found in the Euler equations in the limit in which particle-trajectory slopes are much smaller than f/N, where f is the Coriolis frequency and N > f the buoyancy frequency. Though weak under normal parameter conditions, this instability mode is of theoretical interest because its existence accounts for the failure of an Arnol’d-type stability theorem for the SG equations. In the opposite limit, in which the particle motion is purely vertical, the Euler equations allow only buoyancy oscillations with no horizontal coupling. The SG equations, on the other hand, allow a physically spurious coastal “mirage wave,” so called because its velocity field vanishes despite a nonvanishing disturbance pressure field. Counterpropagating pairs of these w...

Two discrete models for semi-geostrophic dynamics

We discuss two dynamical systems, each intended to represent a discrete analogue of the infinite-dimensional f-plane semi-geostrophic equations. The first model, the lattice model, gives rise to a Hamiltonian system for which the Hamiltonian vector field is discontinuous. This poses a problem with uniqueness of solutions. The second model, the doubly stochastic model, is a C 1 generalisation of the lattice model. We outline both models and completely solve both models in the simplest possible case. Finally, we discuss an interpretation of the dynamics of each model from the viewpoint of air parcels in an idealised atmosphere.

A generalized Charney-Stern theorem for semi-geostrophic dynamics

A generalized Charney-Stern theorem for semi-geostrophic dynamics

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir &amp; Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2. Pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy.A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir &amp; Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

A generalized Charney-Stern theorem for semi-geostrophic dynamics

A stability theorem for zonal flow through a channel bounded by zonally invariant topography is derived for semi-geostrophic (SG) dynamics. The stability theorem generalizes the quasi-geostrophic linear stability theorem of Charney and Stern. The SG stability theorem takes the form of a conservation law for a quadratic disturbance quantity whose sign-definiteness yields necessary and sufficient criteria for the basic flow to be stable. The disturbance quantity generalizes the linearized SG pseudomomentum wave activity for a rectangular channel derived by Kushner and Shepherd to a channel with more general topography. The conservation law is obtained by direct manipulation of the SG equations after they have been transformed to isentropic and geostrophic coordinates. The stability criteria depend on the basic state's meridional potential-vorticity gradients in the interior, and on the topographic slope and the shear of the basic-state velocity at bounding material surfaces, but do not depend on the form of the small-amplitude disturbance. DOI: 10.1034/j.1600-0870.1995.00103.x

Rossby number dependence on meander dynamics of a potential vorticity front

Semi-geostrophic dynamics of jets are studied using a potential vorticity front in an equivalent barotropic model. Meandering processes of the front are examined in the thin-jet limit on a β-plane by a curvilinear coordinate system. For calculated along-front velocity fields, asymmetrical profiles are caused by meandering. This asymmetry of the velocity profile is enhanced as the Rossby number becomes large. Using the along-front velocity fields, the normal velocity of front is expressed so that the Rossby number is explicitly included. This expression can be rewritten in the form of the mKdV equation.