Quasi-geostrophic Theory Research Articles

Global Wavenumber Spectra of Sea Surface Salinity in the Mesoscale Range Using Satellite Observations

Sea surface salinity (SSS) variability at mesoscales has become an important research topic in recent decades, thanks to satellite missions enabling observations of SSS with global capacity and mesoscale resolution. Here, we analyze the near-global data of the Aquarius/SAC-D along-track SSS, focusing on the slopes of SSS variance spectra in the mesoscale range from 180 to 430 km. In the vast extratropics, the spectral slope is close to −2, indicating a dynamical regime for the inverse cascade of depth-integrated energy identified by the surface quasi-geostrophic theory. However, the spectral slopes in regions near the mouths of the largest rivers are steeper than −2, reaching −3, possibly due to the large river freshwater flux. In addition, data from high-resolution thermosalinograph are used to validate satellite measurements and show good consistency in terms of SSS variance spectral slopes.

Multi-scale dynamics of the interaction between waves and mean flows: From nonlinear WKB theory to gravity-wave parameterizations in weather and climate models

The interaction between small-scale waves and a larger-scale flow can be described by a multi-scale theory that forms the basis for a new class of parameterizations of subgrid-scale gravity waves (GW) in weather and climate models. The development of this theory is reviewed here. It applies to all interesting regimes of atmospheric stratification, i.e., also to moderately strong stratification as occurring in the middle atmosphere, and thereby extends classic assumption for the derivation of quasi-geostrophic theory. At strong wave amplitudes a fully nonlinear theory arises that is complemented by a quasilinear theory for weak GW amplitude. The latter allows the extension to a spectral description that forms the basis of numerical implementations that avoid instabilities due to caustics, e.g., from GW reflection. Conservation properties are discussed, for energy and potential vorticity, as well as conditions under which a GW impact on the larger-scale flow is possible. The numerical implementation of the theory for GW parameterizations in atmospheric models is described, and the consequences of the approach are discussed, as compared to classic GW parameterizations. Although more costly than the latter, it exhibits significantly enhanced realism, while being considerably more efficient than an approach where all relevant GWs are to be resolved. The reported theory and its implementation might be of interest also for the efficient and conceptually insightful description of other wave-mean interactions, including those where the formation of caustics presents a special challenge.

Scaling Approaches to Quasigeostrophic Theory for Moist, Precipitating Air

Abstract Quasigeostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG–Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely, Smith and Stechmann’s family of precipitating quasigeostrophic (PQG) models on the one hand, and the extension of QG–Ekman for dry air by a strongly diabatic layer (DL) of intermediate height (QG–DL–Ekman) on the other hand. To this end, Smith and Stechmann’s PQG equations for soundproof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors. Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG–DL–Ekman theory. The detailed derivation of a moist QG–DL–Ekman model is deferred to a future publication.

Scale interactions between the meso‐ and synoptic scales and the impact of diabatic heating

AbstractFor both the meso‐ and synoptic scales, reduced mathematical models give insight into their dynamical behaviour. For the mesoscale, the weak temperature gradient approximation is one of several approaches, while for the synoptic scale the quasigeostrophic theory is well established. However, the way these two scales interact with each other is usually not included in such reduced models, thereby limiting our current perception of flow‐dependent predictability and upscale error growth. Here, we address the scale interactions explicitly by developing a two‐scale asymptotic model for the meso‐ and synoptic scales with two coupled sets of equations for the meso‐ and synoptic scales respectively. The mesoscale equations follow a weak temperature gradient balance and the synoptic‐scale equations align with quasigeostrophic theory. Importantly, the equation sets are coupled via scale‐interaction terms: eddy correlations of mesoscale variables impact the synoptic potential vorticity tendency and synoptic variables force the mesoscale vorticity (for instance due to tilting of synoptic‐scale wind shear). Furthermore, different diabatic heating rates—representing the effect of precipitation—define different flow characteristics. With weak mesoscale heating relatable to precipitation rates of , the mesoscale dynamics resembles two‐dimensional incompressible vorticity dynamics and the upscale impact of the mesoscale on the synoptic scale is only of a dynamical nature. With a strong mesosocale heating relatable to precipitation rates of , divergent motions and three‐dimensional effects become relevant for the mesoscale dynamics and the upscale impact also includes thermodynamical effects.

“Eddy” Saturation of the Antarctic Circumpolar Current by Standing Waves

Abstract It is now well established that changes in the zonal wind stress over the Antarctic Circumpolar Current (ACC) do not lead to changes in its baroclinicity nor baroclinic transport, a phenomenon referred to as “eddy saturation.” Previous studies provide contrasting dynamical mechanisms for this phenomenon: on one extreme, changes in the winds lead to changes in the efficiency with which transient eddies transfer momentum to the sea floor; on the other extreme, structural adjustments of the ACC’s standing meanders increase the efficiency of momentum transfer. In this study the authors investigate the relative importance of these mechanisms using an idealized, isopycnal channel model of the ACC. Via separate diagnoses of the model’s time-mean flow and eddy diffusivity, the authors decompose the model’s response to changes in wind stress into contributions from transient eddies and the mean flow. A key result is that holding the transient eddy diffusivity constant while varying the mean flow very closely compensates for changes in the wind stress, whereas holding the mean flow constant and varying the eddy diffusivity does not. This implies that eddy saturation primarily occurs due to adjustments in the ACC’s standing waves/meanders, rather than due to adjustments of transient eddy behavior. The authors derive a quasigeostrophic theory for ACC transport saturation by standing waves, in which the transient eddy diffusivity is held fixed, and thus provides dynamical insights into standing wave adjustment to wind changes. These findings imply that representing eddy saturation in global models requires adequate resolution of the ACC’s standing meanders, with wind-responsive parameterizations of the transient eddies being of secondary importance.

A Theory of Standing Meanders of the Antarctic Circumpolar Current and Their Response to Wind

Abstract Standing meanders of the Antarctic Circumpolar Current (ACC) and associated eddy hotspots play an important role for the meridional heat flux and downward momentum transfer in the Southern Ocean. Previous modeling studies show that the vorticity balance characterizing standing meanders in the upper ocean is dominated by advection of relative vorticity and stretching. Through the adjustment of this vorticity balance, standing meanders have been suggested to provide a pathway for the transfer of the momentum input by the wind from the surface to the bottom, leading to stronger bottom flows and energy dissipation. However, the dynamics governing the meander formation and its adjustment to wind remain unclear. Here we develop a quasigeostrophic theory and combine it with a regional model of the Macquarie Ridge region and an idealized channel model to explore the dynamics and vertical structure of standing meanders of the ACC. The results show that the entire vertical structure of the meander, including its dynamics in the upper ocean, is controlled by the bottom flow interacting with topography. Based on our results, we suggest a novel mechanism for the response of the ACC to wind in which “flexing” of the meander, or change in its curvature, is a response to changes in the bottom (barotropic) flow. Stronger bottom flow in response to stronger wind interacts with topography and generates a larger-amplitude Rossby wave propagating into the upper ocean. The ACC mean shear aloft amplifies the Rossby wave and leads to a larger-amplitude meander in the upper ocean dominated by advection of relative vorticity and stretching.

Surface Quasigeostrophic Turbulence in Variable Stratification

Abstract Numerical and observational evidence indicates that, in regions where mixed layer instability is active, the surface geostrophic velocity is largely induced by surface buoyancy anomalies. Yet, in these regions, the observed surface kinetic energy spectrum is steeper than predicted by uniformly stratified surface quasigeostrophic theory. By generalizing surface quasigeostrophic theory to account for variable stratification, we show that surface buoyancy anomalies can generate a variety of dynamical regimes depending on the stratification’s vertical structure. Buoyancy anomalies generate longer-range velocity fields over decreasing stratification and shorter-range velocity fields over increasing stratification. As a result, the surface kinetic energy spectrum is steeper over decreasing stratification than over increasing stratification. An exception occurs if the near-surface stratification is much larger than the deep-ocean stratification. In this case, we find an extremely local turbulent regime with surface buoyancy homogenization and a steep surface kinetic energy spectrum, similar to equivalent barotropic turbulence. By applying the variable stratification theory to the wintertime North Atlantic, and assuming that mixed layer instability acts as a narrowband small-scale surface buoyancy forcing, we obtain a predicted surface kinetic energy spectrum between k−4/3 and k−7/3, which is consistent with the observed wintertime k−2 spectrum. We conclude by suggesting a method of measuring the buoyancy frequency’s vertical structure using satellite observations.

Non-linear intensification of monsoon low-pressure systems by the BSISO

Abstract. More than half of the rainfall brought to the Indian subcontinent by the summer monsoon is associated with low-pressure systems (LPSs). Yet their relationship with the boreal summer intraseasonal oscillation (BSISO) – the dominant intraseasonal forcing on the monsoon – is only superficially understood. Using reanalysis data, we explore the relationship between the BSISO and LPS intensity, propagation and precipitation, and associated underlying mechanisms. The BSISO has a large impact on mean monsoon vorticity and rainfall as it moves northward – maximising both in phases 2–3 over southern India and phases 5–6 over northern India – but a much weaker relationship with total column water vapour. We present evidence that LPS genesis also preferentially follows these phases of the BSISO. We identify significant relationships between BSISO phase and LPS precipitation and propagation: for example, during BSISO phase 5, LPSs over northern India produce 51 % heavier rainfall and propagate northwestward 20 % more quickly. Using a combination of moisture flux linearisation and quasi-geostrophic theory, we show that these relationships are driven by changes to the underlying dynamics rather than the moisture content or thermodynamic structure of the monsoon. Using the example of LPSs over northern India during BSISO phase 5, we show that the vertical structure of anomalous vorticity can be split into contributions from the BSISO background circulation and the non-linear response of the LPS to anomalous BSISO circulation. Complementary hypotheses emerge about the source of this non-linear vorticity response: non-linear frictional convergence and secondary barotropic growth. We show that both are important. The BSISO imparts greater meridional shear on the background state, supporting LPS intensification. The BSISO background and non-linear LPS response both contribute significantly to anomalous boundary layer convergence, and we show through vortex budget arguments that the former supports additional LPS intensification in boundary layer, while the latter supports faster westward propagation. This work therefore yields important insights into the scale interactions controlling one of the dominant synoptic systems contributing to rainfall during the monsoon.

Wind-Forced Variability of the Zonal Overturning Circulation

AbstractThe mechanisms of wind-forced variability of the zonal overturning circulation (ZOC) are explored using an idealized shallow water numerical model, quasigeostrophic theory, and simple analytic conceptual models. Two wind-forcing scenarios are considered: midlatitude variability in the subtropical/subpolar gyres and large-scale variability spanning the equator. It is shown that the midlatitude ZOC exchanges water with the western boundary current and attains maximum amplitude on the same order of magnitude as the Ekman transport at a forcing period close to the basin-crossing time scale for baroclinic Rossby waves. Near the equator, large-scale wind variations force a ZOC that increases in amplitude with decreasing forcing period such that wind stress variability on annual time scales forces a ZOC ofO(50) Sv (1 Sv ≡ 106m3s−1). For both midlatitude and low-latitude variability the ZOC and its related heat transport are comparable to those of the meridional overturning circulation. The underlying physics of the ZOC relies on the influences of the variation of the Coriolis parameter with latitude on both the geostrophic flow and the baroclinic Rossby wave phase speed as the fluid adjusts to time-varying winds.Significance StatementThe purpose of this study is to better understand how large-scale winds at mid- and low latitudes move water eastward or westward, even in the deep ocean that is not in direct contact with the atmosphere. This is important because these currents can shift where heat is stored in the ocean and if it might be released into the atmosphere. It is shown that large-scale winds can drive rapid cross-basin transports of water masses, especially so at low latitudes. The present results provide a guide on what controls this motion and highlight the importance of large-scale ocean waves on the water movement and heat storage.

The summer Asia–North America teleconnection and its modulation by ENSO in Community Atmosphere Model, version 5 (CAM5)

Seasonal forecasts of summer continental United States (CONUS) rainfall have relatively low skill, partly due to a lack of consensus about its sources of predictability. The East Asian monsoon (EAM) can excite a cross-Pacific Rossby wave train, also known as the Asia–North America (ANA) teleconnection. In this study, we analyze the ANA teleconnection in observations and model simulations from the Community Atmospheric Model, version 5 (CAM5), comparing experiments with prescribed climatological SSTs and prescribed observed SSTs. Observations indicate a statistically significant relationship between a strong EAM and increased probability of positive precipitation anomalies over the US west coast and the Plains-Midwest. The ANA teleconnection and CONUS rainfall patterns are improved in the CAM5 experiment with prescribed observed SSTs, suggesting that SST variability is necessary to simulate this teleconnection over CONUS. We find distinct ANA patterns between ENSO phases, with the La Niña-related patterns in CAM5 disagreeing with observations. Using linear steady-state quasi-geostrophic theory, we conclude that incorrect EAM forcing location greatly contributed to CAM5 biases, and jet stream disparities explained the ENSO-related biases. Finally, we compared EAM forcing experiments with different mean states using a simple dry nonlinear atmospheric general circulation model. Overall, the ANA pattern over CONUS and its modulation by ENSO forcing are well described by dry dynamics on seasonal-to-interannual timescales, including the constructive (destructive) interference between El Niño (La Niña) modulation and the ANA patterns over CONUS.

On the Discrete Normal Modes of Quasigeostrophic Theory

AbstractThe discrete baroclinic modes of quasigeostrophic theory are incomplete, and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal-mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and the fact that their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin-space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert-space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly nonlinear wave-interaction theory of geostrophic turbulence that takes topography into account.

Balanced ellipsoidal vortex equilibria in a background shear flow at finite Rossby number

We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, $h/r$ , as well as three parameters characterising the background flow, the strain rate, $\gamma$ , the ratio of the background rotation rate to the strain, $\beta$ , and the angle from which the flow is applied, $\theta$ . However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, $f/N$ ( $f$ and $N$ are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, $\gamma _c$ , beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain $\gamma _m$ at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of $\gamma _m$ occur as $\beta \to 1$ . For large values of $f/N$ , changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.

On the Structure and Kinematics of an Algerian Eddy in the Southwestern Mediterranean Sea

An Algerian Eddy, anticyclonic vortex generated by the instability of the Algerian Current in the southwestern Mediterranean Sea, is studied using data provided by drifters (surface currents), Argo floats (temperature and salinity profiles), environmental satellites (absolute dynamic topography maps and ocean color images) and operational oceanography products. The eddy was generated in May 2018 and lasted as an isolated vortex until November 2018. Its morphology and kinematics are described in June–July 2018 when drifters were trapped in its core. During that period, the eddy was slowly moving to the NE (~2 km/day), with an overall diameter of about 200 km (slowly growing with time) and maximal surface swirl velocity of ~50 cm/s at a radius of ~50 km. Geostrophic currents derived from satellite altimetry data compare well with low-pass filtered drifter velocities, with only a slight overestimation, which is expected as its maximum vorticity corresponds to a small Rossby number of ~0.6. Satellite ocean color images and some drifters show that the eddy has an elliptical spiral structure. The looping tracks of the drifters trapped in the eddy were analyzed using two statistical methods: least-squares ellipse fitting and wavelet ridge analysis, revealing a typical eccentricity of about 0.5, a wide range of inclination and a rotation period between 3 and 10 days. Clusters of drifters on the northeastern limb of the eddy were also considered to estimate divergence and vorticity. The results indicate convergence (divergence) and downwelling (upwelling) at scales of 20–50 km near the northeastern (northwestern) edge of the eddy, in agreement with the quasi-geostrophic theory. Vertically, the eddy extends mostly down to 250 m depth, with a warm, low-salinity and low-density signature and with geostrophic currents near 50 cm/s in the top layer (down to ~80 m) reducing to less than 10 cm/s near 250 m. Near the surface, colder water is advected into it.

Numerical study on interaction between eddies and the Kuroshio Current east of Taiwan, China

We investigated the interaction between mesoscale eddies and the Kuroshio Current east of Taiwan, China, using a fine-resolution regional general circulation model. Mesoscale eddies are injected into a region east of Taiwan, China, according to the quasi-geostrophic theory of stratified fluids. Modeled eddies propagated westward at the velocity of the first baroclinic mode Rossby wave. When eddies collide with the Kuroshio Current east of Taiwan, China, the spatial structure and volume transport of the Kuroshio Current shows a significant variation. The upper 600 m of the anticyclonic eddy cannot cross the Kuroshio Current to reach the region west of the Kuroshio Current; rather, these waters flow northward along the eastern side of the Kuroshio Current. The upper water carried by the anticyclonic eddies cannot reach the shelf of the East China Sea (ECS). In contrast, the waters in the upper layer of the cyclonic eddy reach the western side of the Kuroshio Current and then flow northward. The dynamic mechanism analysis shows that the interaction between the Kuroshio Current and the cyclonic (anticyclonic) eddy decrease (increase) the horizontal potential vorticity (PV) gradient, or PV barrier, whereby the cyclonic (anticyclonic) eddy can (cannot) cross the Kuroshio Current. This study implies that the continental shelf could potentially be influenced by cyclonic eddies in the open ocean, which can transport heat and material from the upper open ocean across the Kuroshio Current to the shelf waters.

Nongeostrophic Baroclinic Instability over Sloping Bathymetry: Buoyant Flow Regime

AbstractBaroclinic instabilities are important processes that enhance mixing and dispersion in the ocean. The presence of sloping bathymetry and the nongeostrophic effect influence the formation and evolution of baroclinic instabilities in oceanic bottom boundary layers and in coastal waters. This study explores two nongeostrophic baroclinic instability theories adapted to the scenario with sloping bathymetry and investigates the mechanism of the instability suppression (reduction in growth rate) in the buoyant flow regime. Both the two-layer and continuously stratified models reveal that the suppression is related to a new parameter, slope-relative Burger number Sr ≡ (M2/f2)(α + αp), where M2 is the horizontal buoyancy gradient, α is the bathymetry slope, and αp is the isopycnal slope. In the layer model, the instability growth rate linearly decreases with increasing Sr {the bulk form Sr = [U0/(H0f)](α + αp)}. In the continuously stratified model, the instability suppression intensifies with increasing Sr when the regime shifts from quasigeostrophic to nongeostrophic. The adapted theories are intrinsically applicable to deep ocean bottom boundary layers and could be conditionally applied to coastal buoyancy-driven flow. The slope-relative Burger number is related to the Richardson number by Sr = δrRi−1, where the slope-relative parameter is δr = (α + αp)/αp. Since energetic fronts in coastal zones are often characterized by low Ri, that implies potentially higher values of Sr, which is why baroclinic instabilities may be suppressed in the energetic regions where they would otherwise be expected to be ubiquitous according to the quasigeostrophic theory.

Balanced ellipsoidal vortex at finite Rossby number

ABSTRACT Here we examine the motion of an isolated ellipsoidal vortex in a rotating stratified fluid. We derive an analytical solution to a set of balanced equations at the next order to quasi-geostrophic theory, providing insights into geophysical vortices at finite Rossby number ε. This is achieved through the solution of a set of complicated Poisson equations. Though complicated, the analytical solution give rise to a velocity field that depends linearly on the spatial coordinates inside the vortex, and, thus preserves the ellipsoidal form. From this general solution, we determine a number of equilibria where the vortex rotates steadily about the vertical axis and examine their stability. At the next order to QG, one finds asymmetry in the behaviour of cyclonic and anti-cyclonic vortices, with anti-cyclonic vortices rotating faster and generally more unstable than cyclonic vortices.

Shelf–Open Ocean Exchange Forced by Wind Jets

AbstractThe general problem of exchange from a shallow shelf across sharp topography to the deep ocean forced by narrow, cross-shelf wind jets is studied using quasigeostrophic theory and an idealized primitive equation numerical model. Interest is motivated by katabatic winds that emanate from narrow fjords in southeast Greenland, although similar topographically constrained wind jets are found throughout the world’s oceans. Because there is no net vorticity input by the wind, the circulation is largely confined to the region near the forcing. Circulation over the shelf is limited by bottom friction for weakly stratified flows, but stratification allows for much stronger upper-layer flows that are regulated by weak coupling to the lower layer. Over the sloping topography, the topographic beta effect limits the deep flow, while, for sufficient stratification, the upper-layer flow can cross the topography to connect the shelf to the open ocean. This can be an effective transport mechanism even for short, strong wind events because damping of the upper-layer flow is weak. A variety of transients are generated for an abrupt onset of winds, including short topography Rossby waves, long topographic Rossby waves, and inertial waves. Using parameters representative of southeast Greenland, katabatic wind events will force an offshore transport of O(0.4) Sv (1 Sv ≡ 106 m3 s−1) that, when considered for 2 days, will result in an offshore flux of O(5 × 1010) m3.

Circulation Induced by Isolated Dense Water Formation over Closed Topographic Contours

AbstractThe problem of localized dense water formation over a sloping bottom is considered for the general case in which the topography forms a closed contour. This class of problems is motivated by topography around islands or shallow shoals in which convection resulting from brine rejection or surface heat loss reaches the bottom. The focus of this study is on the large-scale circulation that is forced far from the region of surface forcing. The authors find that a cyclonic current is generated around the topography, in the opposite sense to the propagation of the dense water plume. In physical terms, this current results from the propagation of low sea surface height from the region of dense water formation anticyclonically along the topographic contours back to the formation region. This pressure gradient is then balanced by a cyclonic geostrophic flow. This basic structure is well predicted by a linear quasigeostrophic theory, a primitive equation model, and in rotating tank experiments. For sufficiently strong forcing, the anticyclonic circulation of the dense plume meets this cyclonic circulation to produce a sharp front and offshore advection of dense water at the bottom and buoyant water at the surface. This nonlinear limit is demonstrated in both the primitive equation model and in the tank experiments.

Super- and sub-rotating equatorial jets in shallow water models of Jovian atmospheres: Newtonian cooling versus Rayleigh friction

Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

Balance dynamics in rotating stratified turbulence

If classical quasigeostrophic (QG) flow breaks down at smaller scales, it gives rise to questions of whether higher-order nonlinear balance can be maintained, to what scale and for how long. These are naturally followed by asking how this is affected by stratification and rotation. To address these questions, we perform non-hydrostatic Boussinesq simulations where the initial data is balanced using the Baer–Tribbia nonlinear normal mode initialization scheme (NNMI), which is accurate to second order in the Rossby number, as the next-order improvement to first-order QG theory. The NNMI procedure yields an ageostrophic contribution to the energy spectrum that has a very steep slope. However, as time passes, a shallow range emerges in the ageostrophic spectrum when the Rossby number is large enough for a given Reynolds number. It is argued that this shallow range is the unbalanced part of the motion that develops spontaneously in time and eventually dominates the energy at small scales. If the initial flow is not nonlinearly balanced, the shallow range emerges at even lower Rossby number and it appears at larger scales. Through numerous simulations at different rotation and stratification, this study gives a clear picture of how energy is cascaded in different initially balanced regimes of rotating stratified flow. We find that at low Rossby number the flow mainly consists of a geostrophic part and a balanced ageostrophic part with a steep spectrum. As the Rossby number increases, the unbalanced part of the ageostrophic energy increases at a rate faster than the balanced part. Hence, the total energy spectrum displays a shallow range above a transition wavenumber. This wavenumber evolves to smaller values as rotation weakens.