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Geoffrey K. Vallis

Abstract

Geostrophic balance is shown to be the minimum energy state, for a given linear potential vorticity field, for small deviations of the height field around a resting state, in the shallow-water equations. This includes (but is not limited to) the linearized shallow-water equations. Quasigeostrophic motion is evolution on the slow manifold defined by advection of linear potential vorticity by the velocity field that minimizes that energy. Other linear and nonlinear arguments suggest that geostrophic adjustment is a process whereby the energy of a flow is minimized consistent with the maintenance of the potential vorticity field. A variational calculation that minimizes energy for a given potential vorticity field leads to a balance relationship that for the unapproximated shallow-water equations is similar but not identical to geostrophic balance. Preliminary numerical evidence, involving the inversion of potential vorticity for a simple model, indicates that this balance is a somewhat better approximation to the primitive equations than geostrophy.

It is also shown how the process of geostrophic adjustment may be significantly accelerated, or parameterized, in the primitive equations by the addition of certain terms to the equations of motion. Application of the parameterization to an unbalanced state in a primitive equation model is very effective in achieving a balanced state and in continuously filtering gravity waves. It is more accurate and less sensitive to tunable parameters than pure divergence damping, and may also be a useful and much simpler alternative to nonlinear normal-mode schemes whenever those may be inappropriate.

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Geoffrey K. Vallis

Abstract

The equilibrium statistics and predictability properties of one- and two-layer quasi-geostrophic flow are examined with the aid of a numerical model. The effect of beta in one-layer flow is to slow the transfer of energy into larger scales and to increase the predictability. In two-layer flow, when beta is zero, energy caters the system via baroclinic instability of the mean Row at very large scales and most energy transfer is confined to low wavenumbers. When beta is non-zero, energy enters at higher wavenumber (in baroclinic modes mainly) before cascading preferentially to lower wavenumber zonal barotropic modes. The predictability of two-layer flow is not significantly altered by beta, because beta increases the range of wavenumber over which significant nonlinear energy transfer occurs. The predictability times of the long waves are found to be always larger than those of the short waves, even when the initial error is spread evenly acres wavenumbers. Reducing the mean baroclinicity increases the predictability time. Two-layer flow is lest predictable than one-layer flow of the same barotropic energy, because of the effects of barolinic instability and the transfer of energy from baroclinic modes.

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Geoffrey K. Vallis

Abstract

The spectral integration of the quasi-geostrophic equations is reexamined for simple boundary conditions in Cartesian geometry. For doubly-periodic flow, it is shown that the mean shear must be constant in time or its evolution specified; grid transform methods are then appropriate. In a channel model, a suitable choice of spectral expansion allows the mean shear to be determined internally, i.e., from the model equations, and also allows the meridional gradient of potential vorticity at the boundaries to be specified. However, the use of conventional transform techniques will lead to aliasing and energy nonconservation, and use must be made either of interaction coefficients or a combination of appended zero grid transforms and analytic Fourier expansions.

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Geoffrey K. Vallis

Abstract

The combined effects of wind, geometry, and diffusion on the stratification and circulation of the ocean are explored by numerical and analytical methods. In particular, the production of deep stratification in a simply configured numerical model with small diffusivity is explored.

In the ventilated thermocline of the subtropical gyre, the meridional temperature gradient is mapped continously to a corresponding vertical profile, essentially independently of (sufficiently small) diffusivity. Below this, as the vertical diffusivity tends to zero, the mapping becomes discontinuous and is concentrated in thin diffusive layers or internal thermoclines. It is shown that the way in which the thickness of the main internal thermocline (i.e., the diffusive lower part of the main thermocline), and the meridional overturning circulation, scales with diffusivity differs according to the presence or absence of a wind stress. For realistic parameter values, the ocean is in a scaling regime in which wind effects are important factors in the scaling of the thermohaline circulation, even for the single hemisphere, flat-bottomed case.

It is shown that deep stratification may readily be produced by the combined effects of surface thermodynamic forcing and geometry. The form of the stratification, but not its existence, depends on the diffusivity. Such deep stratification is efficiently produced, even in single-basin, single-hemisphere simulations, in the presence of a partially topographically blocked channel at high latitudes, provided there is also a surface meridional temperature gradient across the channel. For sufficiently simple geometry and topography, the abyssal stratification is a maximum at the height of the topography. In the limit of small diffusivity, the stratification becomes concentrated in a thin diffusive layer, or front, whose thickness appears to scale as the one-third power of the diffusivity. Above and below this diffusive abyssal thermocline are thick, largely adiabatic and homogeneous water masses. In two hemisphere integrations, the water above the abyssal thermocline may be either “intermediate” water from the same hemisphere as the channel, or “deep” water from the opposing hemisphere, depending on whether the densest water from the opposing hemisphere is denser than the surface water at the equatorward edge of the channel. The zonal velocity in the channel is in thermal wind balance, thus determined more by the meridional temperature gradient across the channel than by the wind forcing. If the periodic channel extends equatorward past the latitude of zero wind-stress curl, the poleward extent of the ventilated thermocline, and the surface source of the mode water, both then lie at the equatorial boundary of the periodic circumpolar channel, rather than where the wind stress curl changes sign.

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Rong Zhang and Geoffrey K. Vallis

Abstract

In this paper, it is shown that coherent large-scale low-frequency variabilities in the North Atlantic Ocean—that is, the variations of thermohaline circulation, deep western boundary current, northern recirculation gyre, and Gulf Stream path—are associated with high-latitude oceanic Great Salinity Anomaly events. In particular, a dipolar sea surface temperature anomaly (warming off the U.S. east coast and cooling south of Greenland) can be triggered by the Great Salinity Anomaly events several years in advance, thus providing a degree of long-term predictability to the system. Diagnosed phase relationships among an observed proxy for Great Salinity Anomaly events, the Labrador Sea sea surface temperature anomaly, and the North Atlantic Oscillation are also discussed.

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Junyi Chai and Geoffrey K. Vallis

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This paper discusses the sensitivity of the horizontal and vertical scales of extratropical eddies when criticality is varied in a dry, primitive-equation, general circulation model. Criticality is a measure of extratropical isentropic slope and when defined appropriately its value is often close to 1 for Earth’s climate. The model is forced by a Newtonian relaxation of temperature to a prescribed temperature profile, and criticality is increased by increasing the thermal relaxation rate on the mean flow. When criticality varies near 1, it is shown that there exists a weakly nonlinear regime in which the eddy scale increases with criticality without involving an inverse cascade, while at the same time the Rossby radius may in fact decrease. The quasigeostrophic instability of the Charney problem is revisited. It is demonstrated that both the horizontal and vertical scales of the most unstable wave depend on criticality, and simple estimates for the two scales are obtained. The authors reconcile the opposite trends of the eddy scale and Rossby radius and obtain an estimate for the eddy scale in terms of the Rossby radius and criticality that is broadly consistent with simulations.

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Miles Sundermeyer and Geoffrey K. Vallis

Abstract

Low-order primitive equation and balanced models are compared by evaluating the correlation dimension of each over a range of Rossby numbers. The models are the nine-component primitive equation model of Lorenz and the corresponding three-component balance model. Both models display behavior ranging from stable fixed points and limit cycles to chaotic dynamics. At low Rossby number, the correlation dimensions of the models are (to the accuracy of the calculation) very similar, even in the presence of strange attractors. At higher Rossby number, the behavior differs: in some regions where the balance model goes into a limit cycle the primitive equation model displays chaotic behavior, with a correlation dimension greater than three. This appears to be caused by the (somewhat intermittent) appearance of gravity waves. Since here the calculated correlation dimension is higher than the number of slow modes, the gravity waves cannot be slaved to the slower geostrophic activity.

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Yu Zhang and Geoffrey K. Vallis

Abstract

Ocean heat uptake is explored with non-eddying (2°), eddy-permitting (0.25°), and eddy-resolving (0.125°) ocean circulation models in a domain representing the Atlantic basin connected to a southern circumpolar channel with a flat bottom. The model is forced with a wind stress and a restoring condition for surface buoyancy that is linearly dependent on temperature, both being constant in time in the control climate. When the restore temperature is instantly enhanced regionally, two distinct processes are found relevant for the ensuing heat uptake: heat uptake into the ventilated thermocline forced by Ekman pumping and heat absorption in the deep ocean through meridional overturning circulation (MOC). Temperature increases in the thermocline occur on the decadal time scale whereas, over most of the abyss, it is the millennial time scale that is relevant, and the strength of MOC in the channel matters for the intensity of heat uptake. Under global, uniform warming, the rate of increase of total heat content increases with both diapycnal diffusivity and strengthening Southern Ocean westerlies. In models with different resolutions, ocean responses to uniform warming share similar patterns with important differences. The transfer by mesoscale eddies is insufficiently resolved in the eddy-permitting model, resulting in steep isopycnals in the channel and weak lower MOC, and this in turn leads to weaker heat uptake in the abyssal ocean. Also, the reduction of the Northern Hemisphere meridional heat flux that occurs in a warmer world because of a weakening MOC increases with resolution. Consequently, the cooling tendency near the polar edge of the subtropical gyre is most significant in the eddy-resolving model.

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Riccardo Farneti and Geoffrey K. Vallis

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The variability and compensation of the meridional energy transport in the atmosphere and ocean are examined with the state-of-the-art GFDL Climate Model, version 2.1 (CM2.1), and the GFDL Intermediate Complexity Coupled Model (ICCM). On decadal time scales, a high degree of compensation between the energy transport in the atmosphere (AHT) and ocean (OHT) is found in the North Atlantic. The variability of the total or planetary heat transport (PHT) is much smaller than the variability in either AHT or OHT alone, a feature referred to as “Bjerknes compensation.” Natural decadal variability stems from the Atlantic meridional overturning circulation (AMOC), which leads OHT variability. The PHT is positively correlated with the OHT, implying that the atmosphere is compensating, but imperfectly, for variations in ocean transport. Because of the fundamental role of the AMOC in generating the decadal OHT anomalies, Bjerknes compensation is expected to be active only in coupled models with a low-frequency AMOC spectral peak. The AHT and the transport in the oceanic gyres are positively correlated because the gyre transport responds to the atmospheric winds, thereby militating against long-term variability involving the wind-driven flow. Moisture and sensible heat transports in the atmosphere are also positively correlated at decadal time scales. The authors further explore the mechanisms and degree of compensation with a simple, diffusive, two-layer energy balance model. Taken together, these results suggest that compensation can be interpreted as arising from the highly efficient nature of the meridional energy transport in the atmosphere responding to ocean variability rather than any a priori need for the top-of-atmosphere radiation budget to be fixed.

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Rong Zhang and Geoffrey K. Vallis

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The mechanisms affecting the path of the depth-integrated North Atlantic western boundary current and the formation of the northern recirculation gyre are investigated using a hierarchy of models, namely, a robust diagnostic model, a prognostic model using a global 1° ocean general circulation model coupled to a two-dimensional atmospheric energy balance model with a hydrological cycle, a simple numerical barotropic model, and an analytic model. The results herein suggest that the path of this boundary current and the formation of the northern recirculation gyre are sensitive to both the magnitude of lateral viscosity and the strength of the deep western boundary current (DWBC). In particular, it is shown that bottom vortex stretching induced by a downslope DWBC near the south of the Grand Banks leads to the formation of a cyclonic northern recirculation gyre and keeps the path of the depth-integrated western boundary current downstream of Cape Hatteras separated from the North American coast. Both south of the Grand Banks and at the crossover region of the DWBC and Gulf Stream, the downslope DWBC induces strong bottom downwelling over the steep continental slope, and the magnitude of the bottom downwelling is locally stronger than surface Ekman pumping velocity, providing strong positive vorticity through bottom vortex-stretching effects. The bottom vortex-stretching effect is also present in an extensive area in the North Atlantic, and the contribution to the North Atlantic subpolar and subtropical gyres is on the same order as the local surface wind stress curl. Analytic solutions show that the bottom vortex stretching is important near the western boundary only when the continental slope is wider than the Munk frictional layer scale.

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