Bibliography - Geoffrey K Vallis

1. Gerber, E P., and Geoffrey K Vallis, February 2009: On the zonal structure of the North Atlantic Oscillation and annular modes. Journal of the Atmospheric Sciences, 66(2), doi:10.1175/2008JAS2682.1.
   [ Abstract ]

   The zonal structure and dynamics of the dipolar patterns of intraseasonal variability in the extratropical atmosphere—namely, the North Atlantic Oscillation (NAO) and the so-called annular modes of variability—are investigated in an idealized general circulation model. Particular attention is focused on the relationships linking the zonal structure of the stationary waves, synoptic variability (i.e., the storm tracks), and the zonal structure of the patterns of intraseasonal variability. Large-scale topography and diabatic anomalies are introduced to modify and concentrate the synoptic variability, establishing a recipe for a localized storm track. Comparison of the large-scale forcing, synoptic variability, and patterns of intra-seasonal variability suggests a nonlinear relationship between the large-scale forcing and the variability. It is found that localized NAO-like patterns arise from the confluence of topographic and diabatic forcing and that the patterns are more localized than one would expect based on superposition of the responses to topography and thermal forcing alone. The connection between the eddy life cycle of growth and decay and the localization of the intraseasonal variability is investigated. Both the termination of the storm track and the localization of the intraseasonal variability in the GCM depend on a difluent region of weak upper-level flow, where eddies break and dissipate rather than propagate energy forward through downstream development. The authors' interpretation suggests that the North Atlantic storm track and the NAO are two manifestations of the same phenomenon. Conclusions from the GCM study are critiqued by comparison with observations.

2. Zurita-Gotor, Pablo, and Geoffrey K Vallis, April 2009: Equilibration of baroclinic turbulence in primitive equations and quasigeostrophic models. Journal of the Atmospheric Sciences, 66(4), doi:10.1175/2008JAS2848.1.
   [ Abstract ]

   This paper investigates the equilibration of baroclinic turbulence in an idealized, primitive equation, two-level model, focusing on the relation with the phenomenology of quasigeostrophic turbulence theory. Simulations with a comparable two-layer quasigeostrophic model are presented for comparison, with the deformation radius in the quasigeostrophic model being set using the stratification from the primitive equation model. Over a fairly broad parameter range, the primitive equation and quasigeostrophic results are in qualitative and, to some degree, quantitative agreement and are consistent with the phenomenology of geostrophic turbulence. The scale, amplitude, and baroclinicity of the eddies and the degree of baroclinic instability of the mean flow all vary fairly smoothly with the imposed parameters; both models are able, in some parameter ranges, to produce supercritical flows. The criticality in the primitive equation model, which does not have any convective parameterization scheme, is fairly sensitive to the external parameters, most notably the planet size (i.e., the f/β ratio), the forcing time scale, and the factors influencing the stratification. In some parameter settings of the models, although not those that aremost realistic for the earth's atmosphere, it is possible to produce eddies that are considerably larger than the deformation scales and an inverse cascade in the barotropic flow with a −5/3 spectrum. The vertical flux of heat is found to be related to the isentropic slope.

3. Vallis, Geoffrey K., and E P Gerber, 2008: Local and hemispheric dynamics of the North Atlantic Oscillation, annular patterns and the zonal index. Dynamics of Atmospheres and Oceans, 44(3-4), doi:10.1016/j.dynatmoce.2007.04.00.
   [ Abstract ]

   In this paper we discuss the atmospheric dynamics of the North Atlantic Oscillation (NAO), the zonal index, and annular patterns of variability (also known as annular modes). Our goal is to give a unified treatment of these related phenomena, to make explicit how they are connected and how they differ, and to illustrate their dynamics with the aid of an idealized primitive equation model. Our focus is on tropospheric dynamics. We first show that the structure of the empirical orthogonal functions (EOFs) of the NAO and annular modes follows, at least in part, from the structure of the baroclinic zone. Given a single baroclinic zone, and concomitantly a single eddy-driven jet, the meridional structure of the EOFs follows from the nature of the jet variability, and if the jet variability is constrained to conserve zonal momentum then the observed structure of the EOF can be explained with a simple model. In the zonal direction, if the baroclinic zone is statistically uniform then so is the first EOF, even though there may be little correlation of any dynamical fields in that direction. If the baroclinic activity is zonally concentrated, then so is the first EOF. Thus, at the simplest order of description, the NAO is a consequence of the presence of an Atlantic storm track; the strong statement of this would be that the NAO is the variability of the Atlantic storm track. The positive phase of the NAO corresponds to eddy momentum fluxes (themselves a consequence of wave breaking) that push the eddy-driven jet polewards, separating it distinctly from the subtropical jet. The negative phase of the NAO is characterized by an equatorial shift and, sometimes, a weakening of the eddy fluxes and no separation between sub-tropical and eddy-driven jets. Variations in the zonal index (a measure of the zonally averaged zonal flow) also occur as a consequence of such activity, although the changes occurring are not necessarily synchronous at different longitudes, and the presence of annular modes (i.e., the associated patterns of variability) does not necessarily indicate zonally symmetric dynamics. The NAO, is not, however, a consequence of purely local dynamics, for the storm tracks depend for their existence on patterns of topographic and thermal forcing of near hemispheric extent. The Atlantic storm track in particular is a consequence of the presence of the Rocky mountains, the temperature contrast between the cold continent and warm ocean, and the lingering presence of the Pacific storm track. The precise relationship between the NAO and the storm tracks remains to be determined, as do a number of aspects of storm track dynamics, including their precise relation to the stationary eddies and to the regions of largest baroclinicity. Similarly, the influences of the stratosphere and of sea-surface temperature anomalies, and the causes and predictability of the inter-annual variability of the NAO remain open problems.

4. Zhao, R, and Geoffrey K Vallis, 2008: Parameterizing mesoscale eddies with residual and Eulerian schemes, and a comparison with eddy-permitting models. Ocean Modelling, 23(1-2), doi:doi:10.1016/j.ocemod.2008.02.0.
   [ Abstract ]

   In this paper we explore and test certain parameterization schemes that aim to represent the effects of unresolved mesoscale eddies on the larger-scale flow. In particular, we examine a scheme based on the residual or transformed Eulerian mean formulation of the equations, in which the eddies are parameterized by a large vertical viscosity in the momentum equations, with no skew flux parameterization appearing in the tracer (e.g., temperature or salinity) evolution equations, although terms that parameterize diffusion along isopycnal surfaces remain. The residual scheme is compared both to a conventional parameterization that uses a skew diffusion (or equivalently advection by a skew velocity), and to eddy-permitting calculations. Although in principle almost equivalent to certain forms of skew flux schemes, the residual formulation is found to have certain practical advantages over the conventional scheme in some circumstances, and in particular near the upper boundary where conventional schemes are sensitive to the choice of tapering but the residual scheme is less so. The residual scheme also enables the horizontal viscosity – which is mainly applied to maintain model stability – to be reduced. Finally, the residual scheme is somewhat easier to implement, and the tracer transport is easier to interpret. On the other hand, the residual scheme gives, at least formally, a transformed velocity, not the Eulerian velocity and will not be appropriate in all circumstances

5. Cash, B A., P J Kushner, and Geoffrey K Vallis, 2007: Comment on. Geophysical Research Letters, 34, L03707, doi:10.1029/2006GL027274.
6. Fučkar, N S., and Geoffrey K Vallis, 2007: Interhemispheric influence of surface buoyancy conditions on a circumpolar current. Geophysical Research Letters, 34, L14605, doi:10.1029/2007GL030379.
   [ Abstract ]

   This study shows that the surface buoyancy conditions in the Northern Hemisphere may influence the stratification and transport of the Antarctic Circumpolar Current (ACC). We use a course-resolution ocean general circulation model (OGCM) in an idealized single-basin configuration with a circumpolar channel. A decrease in the magnitude of the surface temperature meridional gradient in the Northern Hemisphere reduces production of the deep water, affecting the interhemispheric Meridional Overturning Circulation (MOC) and deepening the thermocline in both hemispheres. The induced change of stratification in the Southern Hemisphere circumpolar region increases the zonal volume transport of circumpolar current because of an increase in the local meridional density gradient and the associated thermal wind shear, which is the dominant baroclinic component of the total volume transport. The result is robust to variations in the background vertical mixing and the parameterization scheme for mesoscale eddies.

7. Garner, Stephen T., D M W Frierson, Isaac Held, O M Pauluis, and Geoffrey K Vallis, June 2007: Resolving convection in a global hypohydrostatic model. Journal of the Atmospheric Sciences, 64(6), doi:10.1175/JAS3929.1.
   [ Abstract ]

   Convection cannot be explicitly resolved in general circulation models given their typical grid size of 50 km or larger. However, by multiplying the vertical acceleration in the equation of motion by a constant larger than unity, the horizontal scale of convection can be increased at will, without necessarily affecting the larger-scale flow. The resulting hypohydrostatic system has been recognized for some time as a way to improve numerical stability on grids that cannot well resolve nonhydrostatic gravity waves. More recent studies have explored its potential for better representing convection in relatively coarse models. The recent studies have tested the rescaling idea in the context of regional models. Here the authors present global aquaplanet simulations with a low-resolution, nonhydrostatic model free of convective parameterization, and describe the effect on the global climate of very large rescaling of the vertical acceleration. As the convection expands to resolved scales, a deepening of the troposphere, a weakening of the Hadley cell, and a moistening of the lower troposphere is found, compared to solutions in which the moist convection is essentially hydrostatic. The growth rate of convective instability is reduced and the convective life cycle is lengthened relative to synoptic phenomena. This problematic side effect is noted in earlier studies and examined further here.

8. Gerber, E P., and Geoffrey K Vallis, 2007: Eddy–Zonal Flow Interactions and the Persistence of the Zonal Index. Journal of the Atmospheric Sciences, 64(9), doi:10.1175/JAS4006.1.
   [ Abstract ]

   An idealized atmospheric general circulation model is used to investigate the factors controlling the time scale of intraseasonal (10–100 day) variability of the extratropical atmosphere. Persistence on these time scales is found in patterns of variability that characterize meridional vacillations of the extratropical jet. Depending on the degree of asymmetry in the model forcing, patterns take on similar properties to the zonal index, annular modes, and North Atlantic Oscillation. It is found that the time scale of jet meandering is distinct from the obvious internal model time scales, suggesting that interaction between synoptic eddies and the large-scale flow establish a separate, intraseasonal time scale. A mechanism is presented by which eddy heat and momentum transport couple to retard motion of the jet, slowing its meridional variation and thereby extending the persistence of zonal index and annular mode anomalies. The feedback is strong and quite sensitive to model parameters when the model forcing is zonally uniform. However, the time scale of jet variation drops and nearly all sensitivity to parameters is lost when zonal asymmetries, in the form of topography and thermal perturbations that approximate land–sea contrast, are introduced. A diagnostic on the zonal structure of the zonal index provides intuition on the physical nature of the index and annular modes and hints at why zonal asymmetries limit the eddy–mean flow interactions.

9. Zhang, Rong, and Geoffrey K Vallis, 2007: The role of bottom vortex stretching on the path of the North Atlantic Western Boundary Current and on the Northern Recirculation Gyre. Journal of Physical Oceanography, 37(8), doi:10.1175/JPO3102.1.
   [ Abstract ]

   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 recirulation 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.

10. Vallis, Geoffrey K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge, UK: Cambridge University Press, 745 pp.
11. Zhang, Rong, and Geoffrey K Vallis, 2006: Impact of Great Salinity Anomalies on the Low-Frequency Variability of the North Atlantic Climate. Journal of Climate, 19(3), doi:10.1175/JCLI3623.1.
    [ 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.

12. Cash, B A., P J Kushner, and Geoffrey K Vallis, 2005: Zonal asymmetries, teleconnections, and annular patterns in a GCM. Journal of the Atmospheric Sciences, 62(1), doi:10.1175/JAS-3361.1.
    [ Abstract ]

    The influence of zonally asymmetric boundary conditions on the leading modes of variability in a suite of atmospheric general circulation models is investigated. The set of experiments consists of nine model configurations, with varying degrees and types of zonal asymmetry in their boundary conditions. The structure of the leading EOF varies with the zonal asymmetry of the base state for each model configuration . In particular, a close relationship is found between the structure of the EOF and the model storm tracks. An approximately linear relationship is found to hold between the magnitude of the zonal asymmetry of the leading EOF and of the storm tracks in the models. It is shown that this linear relationship extends to the observations. One-point correlation maps centered on the regions where the EOFs reach their maximum amplitude show similar structures for all configurations. These structures consist of a north-south dipole, resembling the observed structure of the North Atlantic Oscillation (NAO). They are significantly more zonally localized than the leading EOF, but do resemble one-point correlation maps and sector EOFs calculated for a simulation with zonally symmetric boundary conditions. Thus, the leading EOF for each simulation appears to represent the longitudinal distribution of zonally localized NAO -like patterns. This longitudinal distribution appears to be tied to the distribution of high-frequency eddies, as represented by the storm tracks. A detailed momentum budget for each case confirms that high-frequency eddies play a crucial role in producing the NAO-like patterns. Other dynamical processes also play an important role, but vary with the details of the simulation.

13. Dewar, W K., R M Samelson, and Geoffrey K Vallis, 2005: The ventilated pool: A model of subtropical mode water. Journal of Physical Oceanography, 35(2), doi:10.1175/JPO-2681.1.
    [ Abstract ]

    An analytical model of subtropical mode water is presented, based on ventilated thermocline theory and on numerical solutions of a planetary geostrophic basin model. In ventilated thermocline theory, the western pool is a region bounded on the east by subsurface streamlines that outcrop at the western edge of the interior, and in which additional dynamical assumptions are necessary to complete the solution. Solutions for the western pool were originally obtained under the assumption that the potential vorticity of the subsurface layer was homogenized. In the present theory, it is instead assumed that all of the water in the pool region is ventilated and, therefore, that all the Sverdrup transport is carried in the uppermost, outcropping layer. The result is the formation of a deep, vertically homogeneous, fluid layer in the northwest corner of the subtropical gyre that extends from the surface to the base of the ventilated thermocline. This ventilated pool is an analog of the observed subtropical mode waters. The pool also has the interesting properties that it determines its own boundaries and affects the global potential vorticity–pressure relationship. When there are multiple outcropping layers, ventilated pool fluid is subducted to form a set of nested annuli in ventilated, subsurface layers, which are the deepest subducted layers in the ventilated thermocline.

14. Gerber, E P., and Geoffrey K Vallis, 2005: A stochastic model for the spatial structure of annular patterns of variability and the North Atlantic oscillation. Journal of Climate, 18(12), doi:10.1175/JCLI3337.1.
    [ Abstract ]

    Meridional dipoles of zonal wind and geopotential height are found extensively in empirical orthogonal function (EOF) analysis and single-point correlation maps of observations and models. Notable examples are the North Atlantic Oscillation and the so-called annular modes (or the Arctic Oscillation). Minimal stochastic models are developed to explain the origin of such structure. In particular, highly idealized, analytic, purely stochastic models of the barotropic, zonally averaged zonal wind and of the zonally averaged surface pressure are constructed, and it is found that the meridional dipole pattern is a natural consequence of the conservation of zonal momentum and mass by fluid motions. Extension of the one-dimensional zonal wind model to two-dimensional flow illustrates the manner in which a local meridional dipole structure may become zonally elongated in EOF analysis, producing a zonally uniform EOF even when the dynamics is not particularly zonally coherent on hemispheric length scales. The analytic system then provides a context for understanding the existence of zonally uniform patterns in models where there are no zonally coherent motions. It is also shown how zonally asymmetric dynamics can give rise to structures resembling the North Atlantic Oscillation. Both the one- and two-dimensional results are manifestations of the same principle: given a stochastic system with a simple red spectrum in which correlations between points in space (or time) decay as the separation between them increases, EOF analysis will typically produce the gravest mode allowed by the system’s constraints. Thus, grave dipole patterns can be robustly expected to arise in the statistical analysis of a model or observations, regardless of the presence or otherwise of a dynamical mode.

15. Gnanadesikan, Anand, Richard D Slater, P S Swathi, and Geoffrey K Vallis, 2005: The energetics of ocean heat transport. Journal of Climate, 18(14), doi:10.1175/JCLI3436.1.
    [ Abstract ]

    A number of recent papers have argued that the mechanical energy budget of the ocean places constraints on how the thermohaline circulation is driven. These papers have been used to argue that climate models, which do not specifically account for the energy of mixing, potentially miss a very important feedback on climate change. This paper reexamines the question of what energetic arguments can teach us about the climate system and concludes that the relationship between energetics and climate is not straightforward. By analyzing the buoyancy transport equation, it is demonstrated that the large-scale transport of heat within the ocean requires an energy source of around 0.2 TW to accomplish vertical transport and around 0.4 TW (resulting from cabbeling) to accomplish horizontal transport. Within two general circulation models, this energy is almost entirely supplied by surface winds. It is also shown that there is no necessary relationship between heat transport and mechanical energy supply.

16. Henning, C C., and Geoffrey K Vallis, 2005: The effects of mesoscale eddies on the stratification and transport of an ocean with a circumpolar channel. Journal of Physical Oceanography, 35(5), doi:10.1175/JPO2727.1.
    [ Abstract ]

    The effects of eddies in a primitive equation ocean model configured in a single hemisphere domain with circumpolar channels at their poleward ends are investigated; in particular, two regimes for the mass balance in the channel are investigated. With small overlying winds, the channel stratification is largely set by diffusion operating in the gyre portion of the domain: the depth scale varies with a fractional power of the diffusivity but has little dependence on the wind stress. As the winds are increased, the depth becomes increasingly controlled by a tendency toward small residual circulation. In this limit, a scaling theory is derived for the stratification in the channel that predicts the overall depth of the thermocline as a power of the wind stress and that allows the eddy length scale to differ from the channel length scale. The predicted depth depends on the details of the closure chosen for the eddy buoyancy flux, but in general it varies as some fractional power of the wind stress, and a channel-only numerical simulation agrees well with this prediction. When a gyre region is added to the channel, vertical diffusion in the gyre exerts some control on the channel stratification even at higher winds, forcing the mass balance into a mixed regime in which both eddy and diffusive effects are important. The depth scale varies less with the wind stress than in a channel-only configuration, and the residual mean circulation in the channel is maintained by the convergence of cross-isopycnal eddy buoyancy fluxes.

17. Loving, J L., and Geoffrey K Vallis, 2005: Mechanisms for climate variability during glacial and interglacial periods. Paleoceanography, 20, PA4024, doi:10.1029/2004PA001113.
    [ Abstract ]

    This paper suggests and explores mechanisms relevant to millennial-scale climate variability during glacial periods. In particular, we present the results of model studies that are able to reproduce many aspects of observed glacial climate variability (e.g., Dansgaard-Oeschger oscillations) without external forcing and that provide a natural explanation for the prevalence of high-amplitude variability in glacial climates and the relative stability of the Holocene. We show that the role of sea ice is critical to cold climate variability because of the effective reduction in the high-latitude meridional sea surface temperature gradient resulting from sea ice expansion and the associated role of sea ice in inhibiting heat flux from the ocean to atmosphere. Thus as sea ice expands in a cooler climate, the high-latitude oceanic heat loss to the atmosphere is inhibited, the thermohaline circulation weakens, and the sinking regions move equatorward, leading to a shallower and weaker deep circulation. This weak circulation is unstable, and intermittent high-amplitude oscillations occur on a timescale and with a spatial structure very similar to Dansgaard-Oeschger cycles. Consistent results are found using both a three-dimensional ocean circulation model coupled to an energy balance atmospheric model and with a much simpler ocean box model. In general, freshening plays a secondary role in the weakening of the North Atlantic thermohaline circulation. Significant freshening is required to alter the stable northern deepwater formation that occurs in a warm climate such as today's Holocene, but once this freshening threshold is achieved, the thermohaline circulation shifts to reverse overturning with sinking in the tropics.

18. Scaife, A A., J R Knight, Geoffrey K Vallis, and C K Folland, 2005: A stratospheric influence on the winter NAO and North Atlantic surface climate. Geophysical Research Letters, 32, L18715, doi:10.1029/2005GL023226.
    [ Abstract ]

    The North Atlantic Oscillation (NAO) has a profound effect on winter climate variability around the Atlantic basin. Strengthening of the NAO in recent decades has altered surface climate in these regions at a rate far in excess of global mean warming. However, only weak NAO trends are reproduced in climate simulations of the 20th Century, even with prescribed climate forcings and historical sea-surface conditions. Here we show that the unexplained strengthening of the NAO can be fully simulated in a climate model by imposing observed trends in the lower stratosphere. This implies that stratospheric variability needs to be reproduced in models to fully simulate surface climate variations in the North Atlantic sector. Despite having little effect on global mean warming, we show that downward coupling of observed stratospheric circulation changes to the surface can account for the majority of change in regional surface climate over Europe and North America between 1965 and 1995.

19. Cash, B A., P J Kushner, and Geoffrey K Vallis, 2004: Reply. Journal of the Atmospheric Sciences, 61(8), 954-956.
20. Grianik, N, Isaac Held, K S Smith, and Geoffrey K Vallis, 2004: The effects of quadratic drag on the inverse cascade of two-dimensional turbulence. Physics of Fluids, 16(1), 73-78.
    [ Abstract ]

    We explore the effects of a quadratic drag, similar to that used in bulk aerodynamic formulas, on the inverse cascade of homogeneous two-dimensional turbulence. If a two-dimensional fluid is forced at a relatively small scale, then an inverse cascade of energy will be generated that may then be arrested by such a drag at large scales. Both scaling arguments and numerical experiments support the idea that in a statistically steady state the length scale of energy-containing eddies will not then depend on the energy input to the system; rather, the only external parameter that defines this scale is the quadratic drag coefficient itself. A universal form of the spectrum is suggested, and numerical experiments are in good agreement. Further, the turbulent transfer of a passive tracer in the presence of a uniform gradient is well predicted by scaling arguments based solely on the energy cascade rate and the nonlinear drag coefficient.

21. Henning, C C., and Geoffrey K Vallis, 2004: The effects of mesoscale eddies on the main subtropical thermocline. Journal of Physical Oceanography, 34(11), 2428-2443.
    [ Abstract PDF ]

    The effects of mesoscale eddies on the main subtropical thermocline are explored using a simply configured wind- and buoyancy-driven primitive equation numerical model in conjunction with transformed Eulerian mean diagnostics and simple scaling ideas and closure schemes. If eddies are suppressed by a modest but nonnegligible horizontal diffusion and vertical diffusion is kept realistically small, the model thermocline exhibits a familiar two-regime structure with an upper, advectively dominated ventilated thermocline and a lower, advective– diffusive internal thermocline, and together these compose the main thermocline. If the horizontal resolution is sufficiently high and the horizontal diffusivity is sufficiently low, then a vigorous mesoscale eddy field emerges. In the mixed layer and upper-mode-water regions, the divergent eddy fluxes are manifestly across isopycnals and so have a diabatic effect. Beneath the mixed layer, the mean structure of the upper (i.e., ventilated) thermocline is still found to be dominated by mean advective terms, except in the "mode water" region and close to the western boundary current. The eddies are particularly strong in the mode-water region, and the low-potential-vorticity pool of the noneddying case is partially eroded away as the eddies try to flatten the isopycnals and reduce available potential energy. The intensity of the eddies decays with depth more slowly than does the mean flow, leading to a three-way balance among eddy flux convergence, mean flow advection, and diffusion in the internal thermocline. Eddies subduct water along isopycnals from the surface into the internal thermocline, replenishing its water masses and maintaining its thickness. Just as in the noneddying case, the dynamics of the internal thermocline can be usefully expressed as an advective–diffusive balance, but where advection is now by the residual (eddy-induced plus Eulerian mean) circulation. The eddy-induced advection partially balances the mean upwelling through the base of the thermocline, and this leads to a slightly thicker thermocline than in the noneddying case. The results suggest that as the diffusivity goes to zero, the residual circulation will go to zero but the thickness of the internal thermocline may remain finite, provided eddy activity persists.

22. Vallis, Geoffrey K., E P Gerber, P J Kushner, and B A Cash, 2004: A mechanism and simple dynamical model of the North Atlantic Oscillation and annular modes. Journal of the Atmospheric Sciences, 61(3), 264-280.
    [ Abstract PDF ]

    A simple dynamical model is presented for the basic spatial and temporal structure of the large-scale modes of intraseasonal variability and associated variations in the zonal index. Such variability in the extratropical atmosphere is known to be represented by fairly well-defined patterns, and among the most prominent are the North Atlantic Oscillation (NAO) and a more zonally symmetric pattern known as an annular mode, which is most pronounced in the Southern Hemisphere. These patterns may be produced by the momentum fluxes associated with large-scale midlatitude stirring, such as that provided by baroclinic eddies. It is shown how such stirring, as represented by a simple stochastic forcing in a barotropic model, leads to a variability in the zonal flow via a variability in the eddy momentum flux convergence and to patterns similar to those observed. Typically, the leading modes of variability may be characterized as a mixture of “wobbles” in the zonal jet position and “pulses” in the zonal jet strength. If the stochastic forcing is statistically zonally uniform, then the resulting patterns of variability as represented by empirical orthogonal functions are almost zonally uniform and the pressure pattern is dipolar in the meridional direction, resembling an annular mode. If the forcing is enhanced in a zonally localized region, thus mimicking the effects of a storm track over the ocean, then the resulting variability pattern is zonally localized, resembling the North Atlantic Oscillation. This suggests that the North Atlantic Oscillation and annular modes are produced by the same mechanism and are manifestations of the same phenomenon. The time scale of variability of the patterns is longer than the decorrelation time scale of the stochastic forcing, because of the temporal integration of the forcing by the equations of motion limited by the effects of nonlinear dynamics and friction. For reasonable parameters these produce a decorrelation time of the order of 5–10 days. The model also produces some long-term (100 days or longer) variability, without imposing such variability via the external parameters except insofar as it is contained in the nearly white stochastic forcing.

23. Cash, B A., P J Kushner, and Geoffrey K Vallis, 2002: The structure and composition of the annular modes in an aquaplanet general circulation model. Journal of the Atmospheric Sciences, 59(23), 3399-3414.
    [ Abstract PDF ]

    The annular mode simulated by an atmospheric general circulation model with a zonally symmetric lower boundary is investigated. The annular mode, defined as the leading empirical orthogonal function (EOF) of the zonal-mean surface pressure, has a meridional structure consisting of a north–south dipole, similar to observations. The leading EOF of the zonally varying surface pressure has the same meridional structure and is also zonally symmetric. Because the lower boundary is zonally symmetric, composites of days with high projection onto the mode have, to within sampling error, no zonal structure. However, individual periods during which the zonal-mean surface pressure projects strongly onto the annular mode are dominated by zonally localized structures. Thus, the model annular mode represents a zonally homogeneous distribution of zonally localized events with a similar meridional structure, rather than a zonally symmetric mode of variability per se. Individual annular-mode events typically show a north–south teleconnection pattern whose meridional structure closely resembles the annular mode and whose zonal structure extends 60° to 90° in longitude, with a slight northwest–southeast offset between its centers of action. Similar structures are found for EOFs calculated over a subset of the domain corresponding to the width of the Atlantic basin. The spatial structure of both the teleconnection pattern and the regional EOFs resemble the observed North Atlantic Oscillation (NAO) pattern.

24. Smith, K S., G Boccaletti, C C Henning, I Marinov, C-Y Tam, Isaac Held, and Geoffrey K Vallis, 2002: Turbulent diffusion in the geostrophic inverse cascade. Journal of Fluid Mechanics, 469, 13-48.
    [ Abstract PDF ]

    Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes. When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When [beta] is significant, we find a clear distinction between the tracer mixing scale, which depends on [beta] but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and [beta]. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

25. Smith, K S., and Geoffrey K Vallis, 2002: The scales and equilibration of midocean eddies: Forced-dissipative flow. Journal of Physical Oceanography, 32(6), 1699-1720.
    [ Abstract PDF ]

    The statistical dynamics of midocean eddies, generated by baroclinic instability of a zonal mean flow, are studied in the context of homogeneous stratified quasigeostrophic tubulence. Existing theory for eddy scales and energies in fully developed turbulence is generalized and applied to a system with surface-intensified stratification and arbitrary zonal shear. The theory gives a scaling for the magnitude of the eddy potential vorticity flux, and its (momentum conserving) vertical structure. The theory is tested numerically by varying the magnitude and mode of the mean shear, the Coriolis gradient, and scale thickness of the stratification and found to be partially successful. It is found that the dynamics of energy in high ( m > 1) baroclinic modes typically resembles the turbulent diffusion of a passive scalar, regardless of the stratification profile, although energy in the first mode does not. It is also found that surface-intensified stratification affects the baroclinicity of flow: as thermocline thickness is decreased, the (statistically equilibrated) baroclinic energy levels remain nearly constant but the statistically equilibrated level of barotropic eddy energy falls. Eddy statistics are found to be relatively insensitive to the magnitude of linear bottom drag in the small drag limit. The theory for the magnitude and structure of the eddy potential vorticity flux is tested against a 15-layer simulation using profiles of density and shear representative of those found in the mid North Atlantic; the theory shows good skill in representing the vertical structure of the flux, and so might serve as the basis for a parameterization of eddy fluxes in the midocean. Finally, baroclinic kinetic energy is found to concentrate near the deformation scale. To the degree that surface motions represent baroclinic eddy kinetic energy, the present results are consistent with the observed correlation between surface eddy scales and the first radius of deformation.

26. Huck, T, Geoffrey K Vallis, and A C de Verdière, 2001: On the robustness of the interdecadal modes of the Thermohaline Circulation. Journal of Climate, 14(5), 940-963.
    [ Abstract PDF ]

    Ocean models in box geometry forced by constant surface fluxes of density have been found to spontaneously generate interdecadal oscillations of the thermohaline circulation. This paper analyzes the sensitivity of these oscillations to various physical effects, including the presence of mesoscale turbulence, various thermal surface boundary conditions, and the presence of wind forcing or bottom topography. The role of unstable long baroclinic waves is also reexamined in an attempt to understand the oscillation period. In idealized geometry, it is found that the low-frequency variability of the thermohaline circulation under quasi-constant surface fluxes is a robust feature of the large-scale circulation. It is not strongly affected by energetic mesoscale turbulence; the oscillation period is relatively invariant with respect to varying resolution and momentum and tracer horizontal mixing coefficients, although it loses some regularity as shorter and longer periods of variability emerge when the mesoscale activity increases in strength with smaller mixing coefficients. The oscillations are also retained as the ocean model is coupled to an interactive atmospheric energy balance model: the thermohaline modes are robust to a range of exchange coefficients that widens with the amplitude of the mean circulation. The presence of an additional wind-forced component generally weakens the oscillation, and depending on the relative strength of thermodynamic and dynamic forcings, the oscillation may be completely killed. A simple interpretation is given, highlighting the role of upward Ekman pumping in damping density anomalies. Finally, the interaction of these baroclinic modes with bottom topography depends strongly on the relative directions of the mean topographic features and the mean currents and baroclinic waves, but usually results in a damping influence.

27. Smith, K S., and Geoffrey K Vallis, 2001: The scales and equilibration of midocean eddies: Freely evolving flow. Journal of Physical Oceanography, 31(2), 554-571.
    [ Abstract PDF ]

    Quasigeostrophic turbulence theory and numerical simulation are used to study the mechanisms determining the scale, structure, and equilibration of mesoscale ocean eddies. The present work concentrates on using freely decaying geostrophic turbulence to understand and explain the vertical and horizontal flow of energy through a stratified, horizontally homogenous geostrophic fluid. It is found that the stratification profile, in particular the presence of pycnocline, has significant, qualitative effects on the efficiency and spectral pathways of energy flow. Specifically, with uniform stratification, energy in high baroclinic modes transfers directly, quickly (within a few eddy turnaround times), and almost completely to the barotropic mode. By contrast, in the presence of oceanlike stratification, kinetic energy in high baroclinic modes transfers intermediately to the first baroclinic mode, whence it transfers inefficiently (and incompletely) to the barotropic mode. The efficiency of transfer to the barotropic mode is reduced as the pycnocline is made increasingly thin, The β effect, on the other hand, improves the efficiency of barotropization, but for oceanically realistic parameters this effect is relatively unimportant compared to the effects of nonuniform stratification. Finally, the nature of turbulent cascade dynamics is such as to lead to a concentration of first baroclinic mode kinetic energy near the first radius of deformation, which, in the case of a nonuniform and oceanically realistic stratification, has a significant projection at the surface. This may in part explain recent observations of surface eddy scales by TOPEX/Poseidon satellite altimetry, which indicate a correlation of surface-height variance with the scale of the first deformation radius.

28. Vallis, Geoffrey K., 2000: Large-scale circulation and production of stratification: Effects of wind, geometry, and diffusion. Journal of Physical Oceanography, 30(5), 933-954.
    [ Abstract PDF ]

    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 continuously 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.

29. Smith, K S., and Geoffrey K Vallis, 1999: Linear baroclinic instability in extended regime geostrophic models. Journal of the Atmospheric Sciences, 56(11), 1579-1593.
    [ Abstract PDF ]

    The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number is small are investigated. The models are the "L₁" dynamics, the "geostrophic potential vorticity" equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes, in a highly idealized and limited set of cases. The L₁ and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravity waves, except possibly at solid boundaries), valid on the B plane, and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the deformation and planetary scales of motion. The L₁ model also includes the "frontal geostrophic" equations as a third limit. In fact, the choice to investigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion. The models are cast in multilayer form, and the dispersion properties and eigenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the L₁ and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic and planetary geostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.

30. Wells, M L., Geoffrey K Vallis, and E A Silver, 1999: Tectonic processes in Papua New Guinea and past productivity in the eastern equatorial Pacific Ocean. Nature, 398, 601-604.
    [ Abstract PDF ]

    Phytoplankton growth in the eastern equatorial Pacific Ocean today accounts for about half of the 'new' production—the fraction of primary production fuelled by externally supplied nutrients—in the global ocean. The recent demonstration that an inadequate supply of iron limits primary production in this region supports earlier speculation that, in the past, fluctuations in the atmospheric deposition of iron-bearing dust may have driven large changes in productivity. But we argue here that only small (2 nM) increases in the iron concentration in source waters of the upwelling Equatorial Undercurrent are needed to fuel intense diatom production across the entire eastern equatorial Pacific Ocean. Episodic increases in iron concentrations of this magnitude or larger were probably frequent in the past because a large component of the undercurrent originates in the convergent island-arc region of Papua New Guinea, which has experienced intensive volcanic, erosional and seismic activity over the past 16 million years. Cycles of plankton productivity recorded in eastern equatorial Pacific sediments may therefore reflect the influence of tectonic processes in the Papua New Guinea region superimposed on the effects of global climate forcing.

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