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Considered here is the evolution of three-dimensional perturbations to the hydrostatic equations linearized about a two-dimensional base state U. Motivated by an argument by T. Warn, this study begins with the nonrotating, unstratified case, and draws analogies between the perturbation equations and equations describing evolution of material line elements and scalar gradients embedded in the same 2D flow. When U is chaotic, both scalar gradients and line elements are characterized by rapid growth, and this leads one to suspect that the perturbations behave similarly. A generalized Okubo-Weiss parameter is proposed, and it is argued that this gives a reasonable litmus test for identifying regions where growth is most probable. Rotation modifies the generalized Okubo-Weiss parameter and tends to curb growth of the perturbation fields, as expected. It is also pointed out that, in realistic geophysical settings, the stability parameter can be suggestive of growth locally, even when a globally defined Rossby number is small. Also considered is the effect of a constant stratification. The perturbation equations can then be separated into vertical modes that have simple sinusoidal structures. The equations describing the evolution of a given mode take a form analogous to the shallow water equations, linearized about U. Numerical simulations of these, assuming a simple but chaotic prescription of U, are carried out. For sufficiently strong stratification, a balance dynamics similar to that suggested by Riley, Metcalfe, and Weissman is recovered. For a given value of the buoyancy frequency N, however, this balance breaks down at high vertical wavenumbers. For high vertical wavenumbers, the modified Okubo--Weiss parameter once again appears to give a potentially useful indication of when growth should be expected. When the Rossby number is small, this criterion predicts stability, and growth occurs only when stratification effects are comparable to or larger than rotational effects. More specifically, growth is seen when the relevant Rossby radius is comparable to or larger than the characteristic length scale of U. It is also found in this limit that approximate geostrophic adjustment occurs prior to growth.

A large fraction of tropical cyclones (TCs) are generated near the intertropical convergence zone (ITCZ), and barotropic instability of the related wind shear has been shown to be an important generation mechanism. The latitudinal position of the ITCZ shifts seasonally and may shift poleward in response to global warming. Aquaplanet GCM simulations have shown TC-generation frequency to vary with position of the ITCZ. These results, and that moisture plays an essential role in the dynamics, motivate the present study on the growth rates of barotropic instability in ITCZ-like zonal wind profiles. Base-state zonal wind profiles are generated by applying a prescribed forcing (representing zonally averaged latent heat release in the ITCZ) to a shallow-water model. Shifting the latitudinal position of the forcing alters these profiles, with a poleward shift leading to enhanced barotropic instability. Next, an examination of how latent release impacts the barotropic breakdown of these profiles is considered. To do this, moisture is explicitly represented using a tracer variable. Upon supersaturation, precipitation occurs and the related latent heat release is parameterized as a mass transfer out of the dynamically active layer. Whether moisture serves to enhance or reduce barotropic growth rates is found to depend on how saturation humidity is represented. In particular, taking it to be constant or a function of the layer thickness (related to temperature) leads to a reduction, whereas taking it to be a specified function of latitude leads to an enhancement. Simple arguments are given to support the idea that moisture effects should lead to a reduction in the moist shallow-water model and that a poleward shift of the ITCZ should lead to an enhancement of barotropic instability. [ABSTRACT FROM AUTHOR]

The quasi-geostrophic energy equation is utilized to address both theoretically and numerically the insensitivity to turbulent closure that is exhibited by the wind-driven nonlinear Stommel-Munk model. When maximum velocities take place at least at inertial length scales, a Sverdrup interior is consistent with the small Rossby number assumption only if the frictional parameters are more than the critical values. If frictional parameters are smaller, valid solutions must lessen the source of energy. The ocean circulation model's steady-state behavior was also examined by using a pseudo-arclength continuation algorithm.

The effects of moisture on the energetics of a statistically stationary, baroclinically unstable jet representing the midlatitude atmosphere are examined using a two-layer, β-plane shallow-water model. Flow is driven by a relaxation of the interface between the two layers to a baroclinically unstable profile. Moisture is input to the lower layer by evaporation. When supersaturation occurs, precipitation is triggered and the related latent heat release drives a mass transfer between the two layers. A comparison between dry and moist reference atmospheres shows that precipitation reduces eddy kinetic energy. This is related to the meridional distribution of precipitation, which occurs on the poleward side of the jet (where the interface field is raised). This latitudinal structure of precipitation is related to a correlation between poleward flow and ascent, which is analyzed using a shallow-water analog to the ω equation. The precipitation effect on the energy budget is predominately due to zonal- and time-averaged terms. Because of this, dry simulations in which the thermal forcing is modified to mimic the effect of zonally averaged precipitation are carried out and compared with their precipitating counterparts. These simulations show a similar reduction of baroclinic eddy kinetic energy; however, the barotropic eddy kinetic energy response shows a larger difference. [ABSTRACT FROM AUTHOR]

The linear theory of vertically sheared convection is extended to deep‐atmosphere flows with arbitrary wind, stability, and diffusion profiles. Consistent with previous findings, reference single‐layer vertical channel flows show fastest growth for shear‐parallel roll circulations and much weaker growth for shear‐transverse circulations, causing the former to dominate. In a more realistic three‐layer setting, where the cloud layer lies between a mixed layer below and a stable free troposphere aloft, shear‐parallel rolls also dominate. However, shear‐transverse rolls grow much faster than before, which degrades the convective organization. An analysis focused on the vertical perturbation phase tilt leads to a novel interpretation of these results. Vertical shear imparts a downshear tilt, which acts to weaken the convective growth driven by dynamic and non‐hydrostatic buoyant vertical perturbation pressure gradients (VPPGs). Whether these VPPGs can maintain growth in the face of the shear depends largely on the Richardson number (Ri), with Ri≲−2 becoming a necessary condition for (inviscid) growth of shear‐perpendicular rolls in the short‐wave limit. In deeper, three‐layer atmospheres, longer vertical wavelengths are admitted, which fosters less tilted and faster growing perturbations. This effect, however, is partially offset by differential tilting between kinematic and thermal anomalies. Numerical simulations are used to verify the linear results and to explore the evolution of the convection into the nonlinear regime. As the nonlinearities grow, an initial preference for smaller‐scale, shear‐parallel circulations is ultimately overwhelmed by larger‐scale perturbations with no preferred orientation. Thus, the linear findings are most applicable to the early evolution of cloud layers undergoing turbulent transition. [ABSTRACT FROM AUTHOR]

The near-tropopause energy spectrum closely follows a -5/3 power law at mesoscales. Most theories addressing the mesoscale spectrum assume unbalanced dynamics but ignore the tropopause (near which the bulk of the data were collected). Conversely, it has also been proposed that the mesoscale spectrum results from tropopause-induced alterations of geostrophic turbulence. This paper seeks to reconcile these a priori mutually exclusive theories by presenting simulations that permit both unbalanced motion and tropopauseinduced effects. The model integrates the nonhydrostatic Boussinesq equations in the presence of a rapidly varying background stratification profile (an idealized tropopause). Decaying turbulence simulations were performed over a wide range of Rossby numbers. In the limit of weak flow (U ≲ 1ms-1), the essential features of the Boussinesq simulations are well captured by a quasigeostrophic version of the model: secondary roll-ups of filaments and shallow spectral slopes are observed near the tropopause but not elsewhere. However, these tropopause-induced effects rapidly disappear with increasing flow strength. For flow strengths more typical of the tropopause (U~10ms-1), the spectrum develops a shallow, near-5/3 tail associated with fast-time-scale, unbalanced motion. In contrast to weak flows, this spectral shallowing is evident at any altitude and regardless of the presence of a tropopause. Diagnostics of the fast component of motion reveal significant inertia-gravity wave activity at large horizontal scales (where the balanced flow dominates). However, no evidence points to such activity in the shallow range. That is, the mesoscale of the model is dominated by unbalanced turbulence, not waves. Implications and limitations of these findings are discussed. [ABSTRACT FROM AUTHOR]

Motivated by the substantial sensitivity of eddies in two-layer quasigeostrophic (QG) turbulence models to the strength of bottom drag, this study explores the sensitivity of eddies in more realistic ocean general circulation model (OGCM) simulations to bottom drag strength. The OGCM results are interpreted using previous results from horizontally homogeneous, two-layer, flat-bottom, f-plane, doubly periodic QG turbulence simulations and new results from two-layer, β-plane QG turbulence simulations run in a basin geometry with both flat and rough bottoms. Baroclinicity in all of the simulations varies greatly with drag strength, with weak drag corresponding to more barotropic flow and strong drag corresponding to more baroclinic flow. The sensitivity of the baroclinicity in the QG basin simulations to bottom drag is considerably reduced, however, when rough topography is used in lieu of a flat bottom. Rough topography reduces the sensitivity of the eddy kinetic energy amplitude and horizontal length scales in the QG basin simulations to bottom drag to an even greater degree. The OGCM simulation behavior is qualitatively similar to that in the QG rough-bottom basin simulations, in that baroclinicity is more sensitive to bottom drag strength than are eddy amplitudes or horizontal length scales. Rough topography therefore appears to mediate the sensitivity of eddies in models to the strength of bottom drag. The sensitivity of eddies to parameterized topographic internal lee wave drag, which has recently been introduced into some OGCMs, is also briefly discussed. Wave drag acts like a strong bottom drag in that it increases the baroclinicity of the flow, without strongly affecting eddy horizontal length scales. [ABSTRACT FROM AUTHOR]

Under quasigeostrophic (QG) dynamics, the presence of a sharp transition in the stratification profile leads to the formation of comparably sharp vertical gradients of buoyancy. When the QG potential vorticity is assumed to be confined to the jump region, as in surface quasigeostrophy (SQG), the sharp gradients are present initially in the buoyancy field. If smoothly varying initial conditions are considered instead, jump-scale features nevertheless emerge after a few turnover times. Inspection of the omega equation reveals that vertical velocity is vertically smoother than buoyancy. As such, the vertical velocity cannot compensate for the sharpness of the stratification jump in the buoyancy equation. Consequently, buoyancy evolves differently above and below the model tropopause, quickly generating sharp vertical gradients. This is confirmed by numerical simulations. The introduction of this small scale, h, characterizing the tropopause implies a larger Froude number, thereby undermining the validity of the quasigeostrophic approximation. For fixed h, scale analysis gives a characteristic horizontal velocity, U, above which not only does QG break down, but statically unstable conditions also develop. Using typical atmospheric values for the Brunt-Väisälä frequency, N = 0.01 s-1, and the jump width, h = 100 m, we argue that U must be less than about 1 ms-1 for static stability to hold (and smaller still for quasigeostrophy to be formally valid). Therefore, quasigeostrophic dynamics are consistent only with very weak near-tropopause flows and thus can hardly account for the observed wind profiles (e.g., the Nastrom and Gage spectral break). We also find that initially smooth flows exhibit secondary roll-up of filaments and shallow slopes near the model tropopause, reminiscent of SQG dynamics. These flows, however, are not SQG-like in the sense they have non-vanishing vertical velocities at the tropopause. [ABSTRACT FROM AUTHOR]

The classic baroclinic, wind-driven, double gyre problem is considered over a range of deformation radii, wind stress amplitudes, and bottom friction coefficients with the aim of better understanding the transfer of energy across scales. In this β-plane basin setting, significant differences are found with respect to classic studies of geostrophic turbulence, which generally assume zonal periodicity and for which the β term does not play a direct role in the energy transfers. In a basin geometry, the β term can play a direct role in the transfers; for example, it can be the dominant term allowing for energy transfer between the basin scale and the baroclinic mesoscale. It is also found that barotropization of baroclinic energy forces the barotropic mode near scales at which bottom drag damps this mode. Associated with this, the barotropic, nonlinear, inverse energy cascade does not extend between mesoscale injection and large-scale dissipation wavenumbers, as is often assumed. Instead, it is part of a 'double cascade' of barotropic energy in which the nonlinear inverse cascade is nearly offset by a forward cascade associated with the β term. This is particularly evident in weak bottom drag simulations, for which a time eddy-mean decomposition of the flow reveals the double cascade to be associated with the eddy-only terms. [ABSTRACT FROM AUTHOR]

The circumpolar transport of a wind-driven quasigeostrophic Antarctic Circumpolar Current is considered. Simple theory suggests transport in a strongly forced regime-the focus of this study-is largely determined by a partitioning of the southward Sverdrup flux into Drake Passage latitudes: some streamlines feed a 'basin contribution' to the circumpolar transport and others feed a large-scale recirculation gyre. Simulations assuming an idealized Scotia Ridge topography are considered to test for sensitivity to resolution. Considerable sensitivity to both vertical and horizontal resolution is found, and associated with this is a tight stationary eddy trapped on the western flank of the ridge. That is, this eddy is sensitive to resolution and exerts an influence that acts to reduce the circumpolar transport. Simulations using the Scotia Ridge-like topography are also compared to others using more realistic topography. In the idealized (ridge) topography experiments, there is only a single ridge against which topographic form drag can act to remove eastward momentum from the system; in the complex topography experiments, there are many. It is found that the experiments assuming realistic topography do not develop an analog to the single topographically trapped eddy prevalent in the Scotia Ridge topography simulations. Additionally, circumpolar transport in these simulations agrees better with the theory. Whether this agreement is simply fortuitous, however, is unclear. To address this, a series of simulations assumes topography that varies smoothly between the idealized ridge and realistic configurations. [ABSTRACT FROM AUTHOR]

Eddy-permitting simulations of a wind-driven quasigeostrophic model in an idealized Southern Ocean setting are used to attempt to describe what sets the wind-driven circumpolar transport of the Antarctic Circumpolar Current (ACC). For weak forcing, the transport is well described as a linear sum of channel and basin components. The authors' main focus is on stronger forcing. In this regime, an eddy-driven recirculation appears in the abyssal layer, and all time-mean circumpolar streamlines are found to stem from a Sverdrup-like interior. The Sverdrup flux into Drake Passage latitudes can then be thought of as the sum of one part that feeds the circumpolar current and another that is associated with the recirculation. The relative fractions of this partitioning depend on the bottom drag, the midchannel wind stress, and the wind stress curl. Increasing the strength of the bottom drag reduces the recirculation and increases circumpolar transport. Increasing a zero-curl eastward wind stress reduces the upper-layer expression of the recirculation and increases the transport. Increasing the curl-containing portion of the forcing (while holding the midchannel stress constant) increases the recirculation and decreases the transport. The weakly forced regime is also considered, as are the relative roles of large and small-scale eddies in transporting momentum vertically through the water column in the Drake Passage latitude band. It is found that the vertical momentum flux associated with transient structures can be used to distinguish between different regimes: these structures transmit momentum upward when the dynamics is dominated by the large-scale recirculation gyre and downward when it is not. [ABSTRACT FROM AUTHOR]

The classic wind-driven double-gyre problem for a homogeneous (unstratified) thin aspect ratio fluid is considered, but allowing for the flow to be depth dependent. Linear free modes for which the vertical wavenumber kz ≠ 0 are inertial oscillations, and they are excited with a large-scale stochastic forcing. This produces a background sea of near-inertial oscillations and their interaction with the vertically averaged flow is the focus of this study. In the absence of 3D forcing, the near-inertial motion vanishes and the barotropic quasigeostrophic system is recovered. With 3D forcing, 2D-to-3D energy transfers—coupled with a forward cascade of 3D energy and scale-selective dissipation—provide an energy dissipation mechanism for the gyres. The relative strength of this mechanism and a Rayleigh drag applied to the 2D flow depends on both the 3D forcing strength and the Rayleigh drag coefficient. [ABSTRACT FROM AUTHOR]

The idea that basinlike dynamics may play a major role in determining the Antarctic Circumpolar Current (ACC) transport is revisited. A simple analytic model is developed to describe the relationship between the wind stress and transport. At very low-wind stress, a nonzero minimum is predicted. This is followed by two distinct dynamical regimes for stronger forcing: 1) a Stommel regime in which transport increases linearly with forcing strength; and 2) a saturation regime in which the transport levels off. The baroclinic structure of the Sverdrup flux into the Drake Passage latitude band is central to the analytic model, and the geometry of characteristics, or geostrophic contours, is key to predicting the transition between the two regimes. A robustness analysis is performed using an eddy-permitting quasigeostrophic model in idealized geometries. Many simulations were carried out in large domains across a range of forcing strengths. The simulations agree qualitatively with the analytic model, with two main discrepancies being related to zonal jet structures and to a western boundary inertial recirculation. Eddy fluxes associated with zonal jets modify the baroclinic structure and lower the saturation transport value. Inertial effects increase the transport, although this effect is mainly limited to smaller domains. [ABSTRACT FROM AUTHOR]

The impact of eddy energy growth and radiation from a western North Pacific cyclone on the intensity of an eastern North Pacific cyclone a few days later is examined. Associated with the western cyclone is an upstream ridge and trough couplet, initially over Siberia on 8 March 1977. The amplitude of this couplet is perturbed in 5-day numerical simulations of the two marine cyclones. Balanced initial conditions are created by potential vorticity inversion. The magnitude of the upper-level couplet governs much of the subsequent growth of eddy energy in the western cyclone as well as the propagation of eddy energy between the two cyclones. This culminates in measurable changes in the maximum intensity of the eastern surface cyclone. The broader question of the sensitivity of this cyclone to upstream perturbations is also briefly addressed. [ABSTRACT FROM AUTHOR]

The sequential development of a western, and then an eastern, North Pacific cyclone is examined in terms of eddy energy and a phase-independent wave activity. Based on the propagation of both a contiguous wave activity center and eddy energy, the development of the western cyclone appears to influence its downstream neighbor. A quantitative comparison of these two diagnoses is made in terms of group velocity, and only minor differences are found during much of the initial evolution. It is only once the tropopause undulations lose their wavelike appearance (at which point, application of the group-velocity concept itself becomes quite tenuous) that the downstream propagation of eddy energy seems faster than that of wave activity. Conventional methods of tracking this wave packet are also briefly discussed. [ABSTRACT FROM AUTHOR]

It is pointed out that accounting for an ocean surface velocity dependence in the wind stress τ can lead to a significant reduction in the rate at which winds input mechanical energy to the geostrophic circulation. Specifically, the wind stress is taken to be a quadratic function of Ua - uo, where Ua and uo are the 10-m wind and ocean surface velocity, respectively. Because |Ua| is typically large relative to |uo|, accounting for a uo dependence leads only to relatively small changes in τ. The change to the basin-averaged wind power source, however, is considerably larger. Scaling arguments and quasigeostrophic simulations in a basin setting are presented. They suggest that the power source (or rate of energy input) is reduced by roughly 20%–35%. [ABSTRACT FROM AUTHOR]

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A spectral element shallow water model with an adaptive triangular mesh is used to examine the effects of basin geometry on the single gyre Munk problem. Specifically, we consider a circular basin shape, with and without the addition of a sinusoidal undulation in the coastline. We are particularly interested in how the curviness of the coastline might effect the well-known tendency of the solution to ‘runaway’ as the lateral viscous coefficient is made small. Results were dependent on boundary conditions. We consider both no slip and three variants of the free slip condition: zero vorticity at the coast, zero normal derivative of tangential velocity and zero normal flux of tangential momentum. For cases where the coastal undulations were large in amplitude, the no slip and the third free slip condition showed the least tendency towards runaway (i.e. towards developing unrealistically large velocities as the viscous coefficient was made small). In the circular geometry, by contrast, the differences between no slip and free slip were much greater, and of the variants on free slip, the zero vorticity condition showed the least tendency towards runaway. We also show that high-frequency variability (such as Kelvin waves) can differ markedly depending on which of the three free slip variants is used. [ABSTRACT FROM AUTHOR]

ABSTRACT Global vorticity budgets in C-grid shallow water (SW) and quasi-geostrophic (QG) models of wind-driven ocean circulation with free-slip boundary conditions are considered. For both models, it is pointed out that the discretized vorticity equation is defined only over a subdomain that excludes boundary grid nodes. At finite resolution, this implies an advective flux of vorticity across the perimeter of the discretized vorticity domain. For rectangular basins where grid axes are aligned with the basin walls, this flux tends to zero as resolution is increased. We also consider the case in which the grid is rotated with respect to the basin, so that a step-like coastline results. Increased resolution then leads to more steps and, because the advective flux of vorticity out of the domain is particularly large at steps, it is no longer obvious that increased resolution should reduce the advective flux. Results are found to be sensitive to numerical details. In particular, we consider different formulations for the non-linear terms (for both the SW and QG models) and two formulations of the viscous stress tensor for the SW model [the conventional five-point Laplacian and the δ–ζ stress tensor suggested by Madec et al. (J. Phys. Oceanogr . 21 , 1349–1371)]. For the SW model, the overall circulation and the behavior of the flux term are dependent on both the formulation of the viscous stress tensor and the non-linear terms. The best combination is found to be the δ–ζ tensor with an enstrophy-preserving advection scheme. With this combination, the circulation of the non-rotated basin is recovered in rotated basins and the advective flux tends to converge towards zero with increasing resolution. The poorest combination is the δ–ζ tensor with the conventional advective scheme. In this case, the advective flux term diverges with increasing resolution for some rotation angles and the model crashes for... [ABSTRACT FROM AUTHOR]