The papers are arranged chronologically here. To see them thematically
For my books
, see the column to the lower left.
Papers and Preprints
Thomson, S. I. and Vallis, G. K., 2019. The Effects of Gravity on the Climate and Circulation of a Terrestrial Planet. Quart. J. Roy. Met. Soc.
The climate and circulation of a terrestrial planet are governed by, among other things, the distance to its host star, its size, rotation rate, obliquity, atmospheric composition and gravity. Here we explore the effects of the last of these, the Newtonian gravitational acceleration, on its atmosphere and climate. We first demonstrate that if the atmosphere obeys the hydrostatic primitive equations, which are a very good approximation for most terrestrial atmospheres, and if the radiative forcing is unaltered, changes in gravity have no effect at all on the circulation except for a vertical rescaling. That is to say, the effects of gravity may be completely scaled away and the circulation is unaltered. However, if the atmosphere contains a dilute condensible that is radiatively active, such as water or methane, then an increase in gravity will generally lead to a cooling of the planet because the total path length of the condensible will be reduced as gravity increases, leading to a reduction in the greenhouse effect. Furthermore, the specific humidity will decrease, leading to changes in the moist adiabatic lapse rate, in the equator-to-pole heat transport, and in the surface energy balance because of changes in the sensible and latent fluxes. These effects are all demonstrated both by theoretical arguments and by numerical simulations with moist and dry general circulation models.
Colyer, G. and Vallis, G. K., 2019. Zonal-mean atmospheric dynamics of slowly-rotating terrestrial planets. J. Atmos. Sci
The zonal-mean atmospheric flow of an idealized terrestrial planet is analyzed using both numerical simulations and zonally symmetric theories, focusing largely on the limit of low planetary rotation rate. Two versions of a zonally symmetric theory are considered, the standard Held–Hou model, which features a discontinuous zonal wind at the edge of the Hadley cell, and a variant with continuous zonal wind but discontinuous temperature. The two models have different scalings for the boundary latitude and zonal wind. Numerical simulations are found to have smoother temperature profiles than either model, with no temperature or velocity discontinuities even in zonally symmetric simulations. Continuity is achieved because of the presence of an overturning circulation poleward of the point of maximum zonal wind, which allows the zonal velocity profile to be smoother than the original theory without the temperature discontinuities of the variant theory. Zonally symmetric simulations generally fall between the two sets of theoretical scalings, and have a faster polar zonal flow than either. Three-dimensional simulations that allow for eddy motion fall closer to the scalings of the variant model. At very low rotation rates the maximum zonal wind falls with falling planetary rotation rate, even in the three-dimensional simulations, and collapses completely at zero rotation. Nevertheless, the low-rotation limit of the overturning circulation is strong enough to drive the temperature profile close to a state of nearly constant potential temperature.
Vallis, G. K., Parker, D, J. and Tobias, S. M. 2019. A Simple System for Moist Convection: The Rainy Benard Model. J. Fluid Mech
. 862, pp. 162–199.
Rayleigh–Benard convection is one of the most well-studied models in fluid mechanics. Atmospheric convection, one of the most important components of the climate system, is by comparison complicated and poorly understood. In this paper we present an idealized model of moist Rayleigh–Bénard convection by taking the Boussinesq limit of the ideal gas equations and adding a condensate that obeys a simplified Clausius–Clapeyron relation, so allowing moist convection to be studied at a fundamental level. The model has an exact, Rayleigh-number independent ‘drizzle’ solution in which the diffusion of water vapour from a saturated lower surface is balanced by condensation, with the temperature field (and so the saturation value of the moisture) determined self-consistently by the heat released in the condensation. This state is the moist analogue of the conductive solution in the classical problem. We numerically determine the linear stability properties of this solution as a function of Rayleigh number and a nondimensional latent-heat parameter. We also present some time dependent, nonlinear solutions at various values of Rayleigh number and the nondimensional condensational parameters.
Tsang, Y-K, and Vallis, G. K. 2018. A Stochastic Lagrangian Basis for a Probabilistic Parameterization of Moisture Condensation in Eulerian Models. J. Atmos. Sci,
75, 3925-3941, 2018.
We describe the construction of an efficient probabilistic parameterization that could be used in a coarse-resolution numerical model in which the variation of moisture is not properly resolved. An Eulerian model using a coarse-grained field on a grid cannot properly resolve regions of saturation---in which condensation occurs---that are smaller than the grid boxes. Thus, in the absence of a parameterization scheme, either the grid box must become saturated or condensation will be underestimated. On the other hand, in a stochastic Lagrangian model of moisture transport, trajectories of parcels tagged with humidity variables are tracked and small-scale moisture variability can be retained. One way to introduce subgrid-scale saturation into an Eulerian model is to assume the humidity within a grid box has a probability distribution. To close the problem, this distribution is conventionally determined by relating the required subgrid-scale properties of the flow to the grid-scale properties using a turbulence closure. Here, instead, we determine an assumed probability distribution by using the statistical moments from a stochastic Lagrangian version of the system. The stochastic system is governed by a Fokker--Planck equation and we use that, rather than explicitly following the moisture parcels, to determine the parameters of the assumed distribution.
Vallis, G. K. and co-authors, 2018. Isca: A Framework for the Global Modelling of the Atmospheres of Earth and Other Planets at Varying Levels of Complexity. Geosci. Model Devel.
Isca is a framework for the idealized modelling of the global circulation of planetary atmospheres at varying levels of complexity and realism, from dry dynamical cores to moist models with continents and complex radiation and convection schemes. Thus, for example, options are available in the dry thermal relaxation scheme to account for the effects of obliquity and eccentricity as well as different atmospheric optical depths and a surface mixed layer. Moving to more complexity, an idealized gray radiation scheme, a two-band scheme and a multi-band scheme are all available, all with simple moist effects (or not, at your choice) and astronomically-based solar forcing. At the complex end of the spectrum the framework provides a direct connection to comprehensive atmospheric general circulation models. A Python front end aids configurability and usability.
For Earth modeling, options include an aqua-planet and a simple land model with configurable continental outlines and topography. Planetary atmospheres may be configured by changing planetary size and mass, solar forcing, atmospheric mass, radiative, and other parameters. The following four papers (Thomson & Vallis, 1,2; Penn & Vallis; Geen et al) all use Isca for Earth and exoplanets. All software is open source and is on github under Isca.
Thomson, S. I. and Vallis, G. K., 2018a. Atmospheric Response to SST anomalies:
Background-state dependence, teleconnections and local effects in winter. J. Atmos. Sci., 75, 4107-4124
PDF file, Part 1.
Thomson, S. I. and Vallis, G. K., 2018b. Atmospheric Response to SST anomalies: Part II: Background-state dependence, teleconnections and local effects in summer J. Atmos. Sci., 75, 4125-4138
PDF file, Part 2.
These are a pair of papers that investigate the atmospheric response to SST anomalies. Our goals are to understand what, if any, anomalies are robust, and what are the underlying causes of differences between robust and non-robust responses, in winter and summer and in the tropics and extra-tropics. We do this by analyzing the response of an idealized atmospheric model to SST anomalies using two slightly different configurations of continents and topography. These configurations give rise to different background wind fields and variability within the same season, and therefore give a measure of how robust a response is to small changes in the background-state. By and large winter responses are more robust than summer responses and tropical responses are more robust than mid-latitude responses, and these manuscripts explore the mechanisms that give rise to that.
Penn, J. and Vallis, G. K. 2017. The Thermal Phase Curve Offset on Tidally- and Non-Tidally-Locked Exoplanets: A Shallow Water Model. Astrophysical J.
We show that the peak of an exoplanet thermal phase curve is, in general, offset from secondary eclipse when the planet is rotating. That is, the planetary hot-spot is offset from the point of maximal heating (the substellar point) and may lead or lag the forcing; the extent and sign of the offset is a function of both the rotation rate and orbital period of the planet. We also find that the system reaches a steady-state in the reference frame of the moving forcing. The model is an extension of the well studied Matsuno-Gill model into a full spherical geometry and with a planetary-scale translating forcing representing the insolation received on an exoplanet from a host star.
Geen, R., Lambert, F. H., and Vallis, G. K. 2017. Regime Change Behavior During Asian Monsoon Onset. J. Climate.
In which we explore the dynamics of the monsoon, and the relationship between the monsoons produced by idealized models and those in the real world, or at least those produced by less-idealized models. As the ITCZ moves off the equator the Hadley circulation appears to transition from an equinoctial regime with two near symmetric, significantly eddy-driven cells, to a monsoon-like regime with a strong, thermally direct cross-equatorial cell, intense low-latitude precipitation, and a weak summer hemisphere cell. We investigate the relevance of this behavior to monsoon onset by using a primitive-equation model in two aquaplanet configurations and in a configuration with a realistic configuration of Earth’s continents and topography.
Aluie, H, Hecht, M. and and Vallis, G. K. 2018. Mapping the Energy Cascade in the North Atlantic Ocean: The Coarse-graining Approach. J. Phys. Oceanog,
A coarse-graining framework is implemented to analyze nonlinear processes, measure energy transfer rates and map out the energy pathways from simulated global ocean data. Traditional tools to measure the energy cascade from turbulence theory, such as spectral flux or spectral transfer rely on the assumption of statistical homogeneity, or at least a large separation between the scales of motion and the scales of statistical inhomogeneity. The coarse-graining framework allows for probing the fully nonlinear dynamics simultaneously in scale and in space, and is not restricted by those assumptions. This paper describes how the framework can be applied to ocean flows.
Energy transfer between scales is not unique due to a gauge freedom. Here, it is argued that a
Galilean invariant subfilter scale (SFS) flux is the suitable quantity to measure energy scale-transfer in the Ocean. It is shown that the SFS definition can yield answers that are qualitatively different from traditional measures that conflate spatial transport with the scale-transfer of energy. The paper presents geographic maps of the energy cascade that are both local in space and allow quasi-spectral, or scale-by-scale, dynamics to be diagnosed. Utilizing a strongly eddying simulation of flow in the North Atlantic Ocean, it is found that an upscale inverse cascade does not hold everywhere. Indeed certain regions, near the Gulf Stream and in the Equatorial Counter Current have a marked downscale transfer. Nevertheless, on average an upscale transfer is a reasonable mean description of the extra-tropical energy scale-transfer over regions of $O(10^3)$ kilometers in size.
Vallis, G. K., 2016.
Geophysical Fluid Dynamics: Whence, Whither and Why? Proc. Roy. Soc
Click here for published version
Local PDF file.
This article discusses the role of GFD in understanding the natural environment, and in particular the dynamics of atmospheres and oceans on Earth and elsewhere. GFD is a branch of science that deals with complex interacting systems and thus in some ways resembles condensed matter physics or aspects of biology - we may seek explanations and understanding by constructing theories or making simple models of the system as a whole. However, in many fluid systems of interest, these days we can also obtain predictions for how the system behaves by nearly direct numerical simulation from the governing equations, something that is often impossible in biology or condensed matter physics. Such simulations, as manifested in complicated General Circulation Models, have been in some ways extremely successful and one may reasonably now ask whether understanding a complex geophysical system is necessary for predicting it. Here we discuss such issues and the roles that GFD has played in the past and will play in the future.
Chai, J, Jansen, M. and Vallis, G. K., 2016.
Equilibration of a baroclinic planetary atmosphere in the limit of vanishing bottom friction. J. Atmos. Sci.
This paper discusses whether and how a baroclinic atmosphere can equilibrate in the limit of vanishing bottom friction in a dry, primitive equation, general circulation model. By decreasing friction by four orders of magnitudes, the model equilibrates in a new regime which exhibits very different energy and momentum budgets compared to Earth’s atmosphere. In this regime, baroclinic eddies convert potential energy into kinetic energy similar to Earth’s atmosphere. However, kinetic energy is converted back into potential energy by circulations at the largest scales, thus closing the energy cycle.
Watson, A., Vallis, G. K., and Nikurashin, M. 2015. Southern Ocean Buoyancy Forcing of Ocean Ventilation and Glacial Atmospheric. CO2
. Nature Geosciences.
During past glacial periods carbon dioxide levels in the atmosphere were significantly lower than pre-industrial levels (and much lower than today), and indeed there is a strong correlation between temperature and atmospheric CO2
levels over the past 400,000 years at least. In this paper we propose a mechanism for the drawn-down of carbon-dioxide into the ocean during glacial periods. We do this by coupling a carbon-cycle model to a semi-analytic model of the ocean overturning circulation. When forced with glacial boundary conditions the deep abyssal cell expands at the expense of the upper cell, and upwelling in the Southern Ocean occurs further north. This enables carbon dioxide to be drawn down into the southern ocean and sequestered in the abyss.
Vallis, G. K., Zurita-Gotor P., Cairns, C and Kidston, J. 2015. Response of the large-scale structure of the atmosphere to global warming. Q. J. Roy. Meteor. Soc.
This paper examines the response of the large-scale structure of the atmosphere to increased concentrations of greenhouse gases. We present results from CMIP5 integrations as well as reviewing various arguments for how the atmosphere might change, and we discuss what is robust and what is not. Topics include the height of the tropopause, the expansion of the Hadley Cell and the shift of the westerlies.
Venaille, A, Nadeau,, L-P, and Vallis, G. K., 2014. Ribbon Turbulence. Phys. FLuids
We investigate the non-linear equilibration of a two-layer quasi-geostrophic flow in a channel with an initial eastward baroclinically unstable jet in the upper layer, paying particular attention to the role of bottom friction. In the limit of low bottom friction, classical theory of geostrophic turbulence predicts an inverse cascade of kinetic energy in the horizontal with condensation at the domain scale and barotropization in the vertical. By contrast, in the limit of large bottom friction, the flow is dominated by ribbons of high kinetic energy in the upper layer. These ribbons correspond to meandering jets separating regions of homogenized potential vorticity.
Jucker, M, Fueglistaler, S, and Vallis, G. K. 2014. Stratospheric sudden warmings in an idealized GCM J. Geophys. Res.
An idealized general circulation model (GCM) with an analytically described Newtonian cooling term is employed to study the occurrence rate of sudden stratospheric warmings (SSWs) over a wide range of parameters. In particular, the sensitivity of the SSW occurrence rates to orographic forcing and both relaxation temperature and damping rate is evaluated. The stronger the orographic forcing and the weaker the radiative forcing (in both temperature and damping rate), the higher the SSW frequency. The separate effects of the damping rates at low and high latitudes are somewhat more complex. Aside from the study of SSWs, the paper also provides an idealized radiative forcing for the stratosphere
Mitchell, J. L., Vallis, G. K. and Potter, S., 2014. Effects of seaonal cycle on superrotation in planetary atmospheres Astrophys. J.
The dynamics of dry atmospheric general circulation model simulations forced by seasonally varying Newtonian relaxation are explored over a wide range of two control parameters and are compared with the large-scale circulation of Earth, Mars, and Titan in their relevant parameter regimes. We find that a large seasonal cycle (small thermal inertia) prevents model atmospheres with large thermal Rossby numbers from developing superrotation by the influences of (1) cross-equatorial momentum advection by the Hadley circulation and (2) hemispherically asymmetric zonal-mean zonal winds that suppress instabilities leading to equatorial momentum convergence. We also demonstrate that baroclinic instabilities must be sufficiently weak to allow superrotation to develop.
Potter, S., Vallis, G. K. and Mitchell, J. L. 2013. Spontaneous superrotation and the role of Kelvin waves in an idealized dry GCM J. Atmos. Sci,
The nondimensional parameter space of an idealized dry primitive equations model is explored to find superrotating climate states. The model has no convective parameterization and is forced using a simple thermal relaxation to a prescribed radiative equilibrium temperature. It is demonstrated that of four nondimensional parameters that determine the model’s state only the thermal Rossby number has a significant effect on superrotation. The mode that drives the transition to superrotation in an intermediate thermal Rossby number atmosphere is shown to behave like a Kelvin wave in the tropics.
Zhang, Y and Vallis, G. K. 2013.
Ocean Heat Uptake in Eddying and Non-eddying Ocean Circulation Models in a Warming Climate J. Phys. Ocean,
Ocean heat uptake is explored with non-eddying, eddy-permitting and eddy-resolving (0.125$\dg$) ocean circulation models in an idealized domain crudely representing the Atlantic basin connected to a southern circumpolar channel. 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 timescale 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.
Farneti, R. and Vallis, G. K. 2013. Meridional Energy Transport in the Coupled Atmosphere-Ocean system: Compensation and Partitioning J. Climate,
(doi: http://dx.doi.org/10.1175/JCLI-D-12-00133.1, in press)
We explore the partitioning of heat transport between the ocean and atmosphere in a hierarchy of models ranging from a coupled GCM to an energy balance model. We find that Bjerknes compensation works in some but not all situations, and works imperfectly. We argue that compensation can best be interpreted as arising from the highly efficient nature of the energy transport in the atmosphere rather than any a priori need for the top-of-atmosphere radiation budget to be fixed.
Jucker, M, Fueglistaler, S, and Vallis, G. K. 2013. Maintenance of stratospheric structure in an idealized general circulation model J. Atmos. Sci,
We explore the factors maintaining the structure of the stratosphere, including the tropical cold point, using an idealized general circulation model. We look at the effects of mountain-forced stationary waves, baroclinic waves, and the actual stratospheric thermal forcing. All of these are important in their own way; none may be neglected if some degree of realism is desired.
Zurita-Gotor, P. and Vallis, G. K. 2013.
The determination of extratropical tropopause height in an idealized gray-radiation model J. Atmos. Sci.,
(doi: http://dx.doi.org/10.1175/JAS-D-12-0209.1, in press)
Explores the joint effects of baroclinic turbulence and diabatic forcing in determining the height of the tropopause, using an idealized multi-level primitive equation model with a gray radiation scheme.
By and large we find that radiative constraints constrain the tropopause height and prevents an adjustment to marginal criticality. We also find that a mixing barrier forms at the jet maximum when baroclinicity has a finite vertical scale.
O'Rourke, A. and Vallis, G. K., 2013. Jet Interaction and the Influence of a Minimum Phase Speed Bound on the Propagation of Eddies. J. Atmos. Sci.,
We investigate the interaction between analogs of the subtropical jet and the eddy-driven jet using a barotropic model. Cospectral analysis of eddy momentum flux convergence indicating that eddy activity is generally restricted by both a minimum and maximum phase-speed bound: the wavenumber-dependent minimum phase speed represents a turning line for meridionally propagating waves, and the upper bound representing a critical line. The authors vary the separation distance between the relaxation and stirring regions and find that a sustained, double-jet state is achieved when either a critical or turning latitude forms in the interjet region. The eddy-driven and subtropical jets have a tendency to merge if waves can propagate through the interjet region.
Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nature Geosciences
In an attempt to understand how energy is dissipated in the ocean we performed simulations that simultaneously resolve the mesoscale, submesoscale, and the internal waves generated by topography. Most of the energy is converted from geostrophic eddies to smaller-scale motions in the abyssal ocean, catalyzed by rough, small-scale topography. Although most of the energy is dissipated in the bottom boundary layer, about 20% is radiated into the ocean interior where it becomes the main source of turbulent mixing.
Nikurashin, M. and Vallis, G. K. 2012. A Theory of the Interhemispheric Deep Overturning Circulation and Associated Stratification. J. Phys. Oceanog.,
The paper extends the one listed below to include the mid-depth circulation (i.e., 'NADW'). Thus, we offer a simple theoretic model of the meridional overturning circulation and associated deep stratification in an interhemispheric, single-basin ocean with a circumpolar channel, including the mid-depth circulation (NADW) and the abyssal circulation (AABW). The theory includes the effects of wind, eddies, and diapycnal mixing, and predicts the deep stratification and overturning streamfunction in terms of the surface forcing and other problem parameters. It is a very different model from the traditional Stommel-Arons-Munk type models that rely on mixing. Rather, it tries to quantify and encapsulate recent ideas about the role of the Southern Ocean winds and eddies, and quantify their importance relative to diapycnal mixing.
Ilicak, M. and Vallis, G. K. 2012. Simulations and scaling of horizontal convection. Tellus, A
, 64, http://dx.doi.org/10.3402/tellusa.v64i0.18377.
We describe some simulations of horizontal convection at Rayleigh numbers of up to 10^11. We explore whether Sandstom's so-called theorem appears to be valid, whether certain scaling relations are satisfied, and what the effects of a stress at the surface are.
Padilla, L., Vallis, G. K. and Rowley, C. 2011. Probabilistic estimates of transient climate sensitivity subject to uncertainty in forcing and natural variability. J. Climate ,
We explore the impact of uncertainty in past forcing and of unforced variability in the climate record on estimates of climate sensitivity. Using a Kalman filter to estimate parameters in an EBM, we provide a range of probabilistic estimates of the transient climate sensitivity (TCS).
Nikurashin, M. and Vallis, G. K. 2011. A Theory of Deep Stratification and Overturning Circulation in the Ocean. J. Phys. Oceanog.,
v41. doi: 10.1175/2010JPO4529.1
We offer a simple theoretical model of the deep stratification and meridional overturning circulation in an idealized single-basin ocean with a circumpolar channel. The theory includes the effects of wind, eddies, and diapycnal mixing, predicts the deep stratification in terms of the surface forcing and other problem parameters, makes no assumption of zero residual circulation, and consistently accounts for the interaction between the circumpolar channel and the rest of the ocean (or so it seems to us).
Zurita-Gotor, P. and Vallis, G. K. 2011. Dynamics of mid-latitude tropopause height in an idealized model. J. Atmos. Sci.,
Explores the joint effects of baroclinic turbulence and diabatic forcing in determining the height of the tropopause, using an idealized multi-level primitive equation model.
It is found that when the vertical redistribution of heat is important the radiative constraint tightly constrains the tropopause height and prevents an adjustment to marginal criticality.
Venaille, A, Vallis, G. K. and Smith, K. S. 2011. Baroclinic turbulence in the ocean: analysis with primitive equation and quasi-geostrophic simulations. J. Phys. Oceanogr.,
This paper examines the factors determining the distribution, length scale, magnitude
and structure of mesoscale oceanic eddies. We find that typically there is a modest
transfer of energy (an `inverse cascade') to larger scales in the horizontal, with the
length scale of the resulting eddies typically comparable to or somewhat larger than
the wavelength of the most unstable mode. The eddies are, however, manifestly nonlinear
and in many locations the turbulence is fairly well-developed.
Xie, P. and Vallis, G. K. 2011. The Passive and Active Nature of Ocean Heat Uptake in Idealized Climate Change Experiments Climate Dynamics,
We explore the extent to which the uptake of heat by the ocean is similar to that of a passive tracer, and to what extent changes in the circulation pattern are important. We find that over a wide range of values of parameters heat uptake is nearly always determined to a greater degree by the existing heat reservoir redistribution than by the nearly passive uptake of temperature due to changes in the surface boundary conditions.
Mitchell, J. and Vallis, G. K. 2010. The transition to superrotation in terrestrial atmospheres J. Geophys. Res. (Planets),
v115, E12008, doi:10.1029/2010JE003587.
We explore the transition to superrotation in terrestrial atmospheres (i.e, atmospheres with a solid surface below, such as those on Earth, Mars or Titan). We show that as the thermal Rossby increases, because the planet rotates more slowly or its radius decreases, baroclinic instability in mid-latitudes decreases and barotropic instability increases, and this leads to superrotation.
Kidston, J. and Vallis, G. K. 2010. Relationship between eddy-driven jet latitude and width Geophys. Res. Lett
Shows that more poleward jets tend to be wider, in both idealized and comprehensive models, and that barotropic instability might account for this.
Held, I. M, Winton, M., Takahashi, K., Delworth, T., Zeng, F., and Vallis, G. K. 2010. Probing the Fast and Slow Components of Global
Warming by Returning Abruptly to Preindustrial Forcing. J. Climate,
23, 2418-2427, doi10.1175/2009JCLI3466.1.
The fast and slow components of global warming in a comprehensive climate model are isolated by examining the response to an instantaneous return to preindustrial forcing. The response is characterized by an initial fast exponential decay with an e-folding time smaller than 5 years, leaving behind a remnant that evolves more slowly.
Kidston, J., Vallis, G. K., Dean, S.M. and Renwick, J. A. 2011. Can the increase in the eddy length scale under global warming cause the poleward shift of the jet streams? J. Climate
, doi: 10.1175/2010JCLI3738.1
This manuscript offers a mechanism whereby an increase in eddy length scales can cause a poleward shift of the eddy-driven jets and surface westerlies, such as is simulated in a large number of comprehensive climate models under global warming.
Kidston, J., Dean, S.M., Renwick, J. A. and Vallis, G. K. 2010. A robust increase in the eddy length scale in the simulation of future climates. Geophys. Res. Lett
, 37, L03806, doi:10.1029/2009GL041615.
This paper shows that a large number of comprehensive climate models (in fact all models in CMIP 3) exhibit an increase in the eddy length scale in the future compared with the simulation of 20th Century climate. The increase in length scale is on the order of 5% by the end of the 21st century, and the Southern Hemisphere exhibits a larger increase than the Northern Hemisphere.
Kidston, J., Frierson, D. M. W., Renwick, J. A. and Vallis, G. K. 2010. Observations, Simulations, and Dynamics of Jet Stream Variability and Annular Modes J. Climate
, 23, 6186-6199.
The characteristics of the dominant pattern of extra-tropical variability (the so-called annular modes) are examined, and it is shown that there is genuine hemispheric symmetry in the variation of the zonal wind in the Southern Hemisphere but not the Northern Hemisphere, and that the mechanism of the annular mode is baroclinic in origin.
Ferrari, R., Griffies, S. M., Nurser, G. and Vallis, G. K. 2010. A Boundary-Value Problem for the Parameterized Mesoscale Eddy Transport. Ocean Modelling
A new method for parameterizing mesoscale eddies in ocean models. The new method, which may be regarded as a variation on the GM scheme, enforces a low mode vertical structure for the parameterized streamfunction, as motivated by geostrophic turbulence theory as for example in the papers by Smith and Vallis below. The scheme also satisfies appropriate boundary conditions at the top and bottom of the ocean without ad hoc tapering.
Zurita-Gotor, P. and Vallis, G. K., 2010. Circulation sensitivity to heating in a simple model of baroclinic turbulence. J. Atmos. Sci.,
, 67, 1543--1588.
A study of sensitivity of the circulation of an idealized primitive-equation two-level model on the form and strength of the heating, aiming to understand the qualitatively different sensitivity of the isentropic slope (marginally supercritical, highly supercritical, etc) on differential heating reported by previous idealized studies when different model formulations are used. Although motivated in part by the differences that moisture evidently makes, we used a dry model.
Vallis, G. K. and Farneti, R. 2009. Meridional Energy Transport in the Atmosphere-Ocean System. Scaling and Numerical Experiments.
Quart. J. Roy. Meteor. Soc.
, 135, 1643-1660, doi:10.1002/qj.498
A discussion of the mechanisms of energy transport in the coupled atmosphere-ocean system, with some scaling estimates and numerical experiments using an idealized coupled ocean-atmosphere model. (The model has the full three-dimensional primitive equations but has simplified physical parameterizations and geometry.)
Gerber, E. P. and Vallis, G. K., 2009. On the zonal structure of the North Atlantic Oscillation and Annular Modes.
J. Atmos. Sci.,
, 66, 332-352.
A numerical and theoretical study of the zonal structure and dipolar patterns
in the extratropical atmosphere, and in particular the NAO and annular
patterns. The dynamics of such patterns are discussed and 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.
Zurita-Gotor, P. and Vallis, G. K. 2009. Equilibration of baroclinic turbulence in primitive equation and quasi-geostrophic models.
J. Atmos. Sci.,
Explores the nonlinear equilibration of baroclinic eddies in two-layer PE and QG models. We find that geostrophic turbulence theory has some predictive power for the energy levels etc of the equilibrated turbulent states. We also find that supercritical states and an inverse energy cascade can be found in some parameter regimes (although not those most corresponding to the Earth's atmosphere).
Farneti, R. and Vallis, G. K. 2009. An intermediate complexity climate model based on the GFDL Flexible Modelling System. Geosci. Model Dev.
This paper discusses the formulation of a coupled ocean-atmosphere-land-ice model that is much simpler than a full GCM, but still has three-dimensional dynamics and thermodynamics in the atmosphere and ocean.
Vallis, G. K. 2009. Mechanisms of climate variability from years to decades. In press in
Stochastic Physics and Climate Modelling
, T. Palmer and P. Williams, Eds.
A review article and essay on climate variability. Not so much a literature survey as a discussion of mechanisms, illustrated by examples.
Farneti, R. and Vallis, G. K. 2009. Mechanisms of interdecadal climate variability and the role of ocean-atmosphere coupling Clim. Dyn.,
Explores the mechanisms of decadal climate variability using a simplified, but still fully dynamical, 3D coupled ocean-atmosphere-climate model.
Vallis, G. K. and Gerber, E. P. 2008. Local and Hemispheric Dynamics of the North Atlantic Oscillation, Annular Patterns and the Zonal Index. Dyn. Atmos. Oceans,
A synthesis of the fundamental dynamics of the NAO and its relation to storm tracks and annular modes/patterns. It also illustrates the phenomena with numerical simulations and with stochastic models.
Zhao, R., and Vallis, G. K. 2008. Parameterizing mesoscale eddies with residual and Eulerian schemes, and a comparison with eddy-permitting models. Ocean Modelling,
Discusses the use of 'residual' or TEM form of the equations of motion, in an ocean context, especially with regard to parameterizing mesoscale eddies.
Gerber, E. P. and Vallis, G. K. 2007. Eddy-Zonal flow interactions and the persistence of the zonal index. J. Atmos. Sci.,
Discusses the dynamics of the zonal index, with particular reference to its timescale and interaction with baroclinic eddies.
Zhang, R. and Vallis, G. K. 2007. The role of bottom vortex stretching on the path of the North Atlantic western boundary current and on the northern recirculation gyre.
J. Phys. Ocean,
Shows that bottom vortex stretching, and the deep western boundary current, influence the Gulf Stream path.
Fuckar, N. S. and Vallis, G. K. 2007. Interhemispheric influence of surface buoyancy conditions on a circumpolar current.
Geophysical Research Letters
34, L14605, doi:10.1029/2007GL030379.
Demonstrates, using an idealized ocean GCM, that the boundary conditions on surface temperature in the North Atlantic have a strong influence on the zonal transport of the ACC.
Zhang, R. and Vallis, G. K. 2006. Impact of Great Salinity Anomalies
on the low frequency variability of the North Atlantic Climate. J.
Shows that Great Salt anomalies can potentially alter the Gulf Stream path and influence the climate of the North Atlantic
Pauluis, O. M., Frierson, D. M. W., Garner, S. T., Held, I. M. and G. K.
Vallis, 2006. The Hypo-hydrostatic Rescaling and Its Impacts on Modeling of Atmospheric Convection. Theor. Comput. Fluid Dynam.,
Garner, S. T., Frierson, D. M. W., Held, I. M., Pauluis, O. M. and G. K.
Vallis. 2007. Resolving Convection in a Global Hypohydrostatic Model. J. Atmos. Sci.,
The above two papers explore the so-called hypohydrostatic rescaling, which has the effect of making convection occur at larger scales, and thus leads to a possible 'parameterization' of convection.
Anadadesikan, A, Swathi, P, Slater, R. S. and Vallis, G. K. 2005.
Energetics of large-scale ocean circulation. J. Climate.
Discusses the energetics of the large-scale circulation and its relation to Sandstrom's effect.
Loving, Jolene L., and Geoffrey K. Vallis, 2005. Mechanisms for climate
variability during glacial and interglacial periods. Paleoceanography,
20, PA4024, doi:10.1029/2004PA00111
Offers a mechanism of Dansgaard-Oeschger oscillations. Shows how sea-ice plays an important role.
Dewar, W. D., R. S. Samelson and G. K. Vallis. 2005. The ventilated pool: A model of subtropical mode water. J. Phys. Oceanogr.,
Offers an analytic theory of mode water, in the context of thermocline theory and the large-scale circulation
Scaife, A. A., Knight, J. R., Vallis, G. K. and Folland, C. K. 2005.
Simulation of observed changes in the North Atlantic Oscillation and
surface climate in the latter half of the 20th Century.
Geophysical Research Letters,
32, L18715, doi:10.1029/2005GL023226.
Changes in the NAO over the late 20th century can be simulated properly if the stratosphere is accurately simulated.
Henning, C. and Vallis, G. K. 2005. The Effects of Mesoscale Eddies on the Stratification and Transport of an Ocean with a Circumpolar Channel. J. Phys. Oceanogr.,
Shows, using an idealized eddying ocean model, that mesoscale eddies have a fundamental effect on the stratification of a circumpolar channel, of which the ACC is an example, and
the deep circulation away from the channel underneath the main subtropical thermocline.
Gerber, E. P. and Vallis, G. K., 2005. A stochastic model of the
spatial structure of the
annular patterns of variability and the NAO. J. Climate,
Demonstrates that the spatial structure -- for example the dipolar structure of the EOFs -- of the NAO can be captured by a simple stochastic model.
Henning, C and Vallis, G. K., 2004. The Effect of mesoscale eddies on the main subtropical thermocline. J. Phys. Oceanogr.,
Discusses and simulates the effects of mesoscale eddies on the main thermocline.
Cash, B., P. Kushner and G. K. Vallis, 2005. Zonal asymmetries, teleconnections and annular modes in a GCM. J. Atmos. Sci.,
Discusses and simulates the relationship between 'annular modes', zonal asymmetries, the storm track, etc., using a simplified GCM.
Vallis, G. K., E. Gerber, P. Kushner and B. Cash. 2004. A mechanism and simple model of the North Atlantic Oscillation and Annular Modes. J. Atmos. Sci.,
Offers a simple model of the NAO, and its relationship to the storm tracks and so so-called annular modes.
Griannik, N., I. Held, K.S. Smith and Vallis, G. K. 2004. Effect of nonlinear drag on the inverse cascade. Phys. Fluids 16, 73-78.
The halting scale of the inverse cascade in 2D turbulence is independent of the strenght of the turbulence, if a nonlinear drag (such as is common in boundary layer schemes) is used.
Vallis, G. K. Mean and Eddy Dynamics of the Main Thermocline. 2003. In Nonlinear Processes in Geophysical Fluid Dynamics. O. U. Velasco Fuentes, J. Sheinbaum and J. Ochoa (editors). Kluwer Academic Publishers, pp141-173.
A review and discussion paper on thermocline dynamics, summarizing the work in my papers with Roger Samelson and with Cara Henning, as well as the subject more generally.
Cash, B., P. Kushner and G. K. Vallis. 2002. The structure and composition of the annular
modes in an aquaplanet GCM. J. Atmos. Sci.,
By way of numerical simulations with a simplified GCM, shows that 'annular modes' do not necessarily contain annular dynamics. It is just the statistics
of the dynamics that need be annular in order to give zonally symmetric EOFs and annular mode-like properties.
Smith, K. S., G. Boccaletti, C. C. Henning, I. Marinov, C. Y. Tam., I. M. Held and G. K. Vallis. 2002.
Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech.,
Discusses and simulates the transport properties of the inverse cascade.
Smith, K. S. and G. K. Vallis. 2002. Scales and equilibration of mid-ocean eddies: Forced-
dissipative flow. J. Phys. Oceanogr.,
Discusses and simulates the scales of mid-ocean eddies, from the point of view of force-dissipative geostrophic turbulence.
Huck, T., and G. K. Vallis. 2001. Linear stability analysis of the three-dimensional thermally-driven ocean circulation: application to interdecadal oscillations. Tellus,
Carries out a three-dimensional linear instabiliot calculation for the THC. Shows that there are unstable modes that resemble the variability in the nonlinear, time-dependent model
Huck, T., G. K. Vallis, and A. Colin de Verdiere. 2001. On the robustness of the inter-decadal modes of the thermohaline circulation. J. Climate,
Explores the robustness of interdecadal variability of the THC.
Smith, K. S. and G. K. Vallis. 2001. Scales and equilibration of mid-ocean eddies. Freely decaying flow. J. Phys. Oceanogr.,
Discusses and simulates the scales of mid-ocean eddies, from the point of view of decaying geostrophic turbulence.
Vallis, G. K. 2000. Thermocline Theories and WOCE: A Mutual Challenge. WOCE Newsletter,
39, 30-34, (2000).
Brief discussion and overview of thermocline theory and how it might be tested.
Vallis, G. K., 2000. Large-scale circulation and production of stratification: effects of wind, geometry and diffusion. J. Phys. Oceanogr.,
Discusses and simulates the interhemispheric circulation in an idealized, primitive equation, GCM. Shows that the ACC has a profound effect on the circulation. Explores the influence of surface boundary conditions and interhemispheric asymmetries.
Wells, M., G. K. Vallis and E. Silver. Influence of Tectonic Processes in Papua New Guinea on Past Productivity in the Eastern Equatorial Pacific Ocean. Nature,
Schonbek, M. and Vallis, G. K. Energy Decay of Solutions to the Boussinesq, Primitive and Planetary Geostrophic Equations. J. Math. Analysis & Applics.
Smith, K. S. and G. K. Vallis. Linear wave and instability properties of extended range geostrophic models. J. Atmos. Sci.,
56, 1579-1593. (1998)
Calculates the linear wave and baroclinic instability properties of various types of geostrophic model, including quasi-geostrophy, planetary geostrophy, and the so-called geostrophic potential vorticity model that spans QG and PG.
Oetzel, K. and G. K. Vallis. Strain, vortices, and the enstrophy inertial range in two-dimensional turbulence. Phys. Fluids 9, 2991-3004. (1997)
Shows that a -3 enstrophy inertial range can emerge if the resolution is sufficiently high. Offers a theory for the co-existence of coherent structures with turbulence, and shows that at small scales the coherent vortices will be strained away.
Vallis, G. K., G. J. Shutts, G. and M. E. B. Gray. 1997. Balanced mesoscale motion and stratified turbulence forced by convection. Quart. J. Roy. Meteor. Soc.,
Looks at the possible generation of an inverse cascade in low Froude number flow, with he forcing coming from resolved convection at small scales.
Camevale, G., M. Briscolini, R. Kloosterziel, and G.K. Vallis. Three dimensionally perturbed vortex tubes in a rotating flow. J. Fluid Mech.,
Samelson, R. and G.K. Vallis, 1997. Large-scale circulation with small diapycnal diffusivity: the two-thermocline limit. J. Mar. Res.
Proposes that the main thermocline has two dynamical regimes: an adiabatic regime (aka the ventilated themocline), lying above an intrinsically diffusive regime (the internal thermocline). Presents supporting numerical calculations.
Samelson, R. and G.K. Vallis, 1997. A simple fictional and diffusive scheme for the planetary geostrophic equations in a closed basin. J. Phys. Oceanogr.
Suggests how to make the planetary-geostrophic equations well-posed and numerically efficient in a closed domain. Such a model that was used in the previous reference.
Mundt, M. G.K. Vallis and J. Wang, Balanced models for the large- and meso-scale cir-
culation. J. Phys. Oceanogr.
27, 1133-1152 (1997).
Shows how accurate certain PV-based balanced models can be, especially one that bridges the parameter regime between quasi-geostrophy and planetary-geostrophy.
Vallis, G. K. Approximate geostrophic models for large-scale flow in the ocean and atmosphere. Physica D.
98, 647-651 (1996).
Vallis, G.K., Potential vorticity inversion and balanced equations of motion for rotating and stratified flows. Q. J. Roy. Met. Soc.
122, 291-322 (1996). Vallis96.pdf
Shows how PV inversion can be used to construct higher-order balanced models for strtified flow.
Vallis, G.K. From laminar flow to turbulence. 1996. In: An Introduction to Nonlinear Physics,
ed. L. Lam. Springer-Verlag, pp. 308-357 (chapter in book) (1996).
Review of various pathways to turbulence (period-doubling, etc.) and of fully-developed turbulence itself.
Warn, T, O. Bokhove, T.G. Shepherd, and G.K. Vallis, Rossby number expansions, slaving, and balance dynamics. Quart. J. Roy. Meteor. Soc.,
121, 723-739 (1995).
Shows how to construct balanced models of arbitrarily high order, by 'slaving' fields to a single evolving variable, such as potential vorticity.
Cummins, P. and G.K. Vallis, Solvers for separable and non-separable elliptic problems in irregular domains. Assoc. Comput. Machinery, Trans. Math.
Software. 20, 247-253 (1994).
Wang, J. and G.K. Vallis, Emergence of Fofonoff states in inviscid and viscous ocean circulation models. J. Mar. Res., 82, 85-127 (1994).
The so-called Fofonoff solution is the maximum entropy state for two-dimensional flow. This paper explores the statistical mechanical equilibrium of unforced and inviscid ocean models in closed domains, and shows that Fofonoff states do indeed emerge as the time-averaged flow in long integrations.
Vallis, G.K., Statistical mechanics, turbulence, and ocean circulation. in Statistical Methods in Physical Oceanography, `Aha Hulikoa' proceedings, ed. P. Muller and G. Holloway. 473≠491 (1993).
Vallis, G.K. and M. E. Maltrud., 1993. Generation of mean flows and jets on a beta-plane and over topography. J. Phys. Oceanog.
, 23, 1346-1362.
Shows how the beta effect combines with two-dimensional turbulence to give rise to zonal jets. Presents some supporting numerical simulations.
Maltrud. M. and G.K. Vallis, Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence. Physics of Fluids A,
5, 1760-1775 (1993).
Explores in some detail the energy and enstrophy inertial ranges in 2D turbulence. In particular, the enstrophy transfer is found to be quite nonlocal, in spectral space.
Sundermeyer, M. and G.K. Vallis, Correlation dimension of primitive
equation and balanced models. J. Atmos. Sci.,
50, 2556-2564 (1993).
Vallis, G.K., Mechanisms and parameterizations of geostrophic adjustment and a new model for balanced flow. J. Atmos. Sci.,
49, 1144-1160 (1992).
Shows that geostrophic balance is the minimum energy state for a given field of potential vorticity, in the linear approximation. Provides a nonlinear extension.
Vallis, G.K., 1992. Problems and phenomenology in two-dimensional turbulence. In: Non-linear Phenomena in Atmospheric and Oceanic Sciences, eds. G. Carnevale and R. Pierrehumbert. 1-25. Springer-Verlag (refereed chapter in book).
Review of 2D turbulence. Much of this material is now incorporated into textbooks, for example my AOFD book.
Auad, G., A. Pares-Sierra, and G.K. Vallis, 1991. Energetics and diagnostics of a model of the circulation in the California Current System, J. Phys. Oceanog.
, 21, 1534-1552.
Maltrud, M. and G.K. Vallis, 1991. Energy spectra and coherent structures in forced two-dimensional and geostrophic turbulence. J. Fluid Mech.,
228, 321-342 (1991).
Shows that the -5/3 inverse energy spectrum of 2D turbulence can be robustly simulated, and determines a value for the Kolmogorov-Kraichnan constant. Also explores effects of simultaneously forcing the fluid at two distinct scales. Shows that an upscale energy spectrum can co-exist, over the same wavenumber range, with a downscale enstrophy cascade.
Carnevale, G.F. and G.K. Vallis, 1990. Iso-vortical energy variation in two-dimensional flows. In: Topological Fluid Mechanics, Proceedings of the IUTAM Symposium, eds. H.K. Moffatt and A. Tsinober. 294-303 (refereed conference proceedings).
Vallis, G.K., G.F. Carnevale, and T.G. Shepherd, 1990. A natural method for the stable states of Hamiltonian systems. In: Topological Fluid Mechanics, Proceedings of the IUTAM Symposium, eds. H.K. Moffatt and A. Tsinober. 429-439 (refereed conference proceedings).
Carnevale, G.F. and G.K. Vallis, 1990.
Pseudo-advective relaxation to stable two-dimensional
states. J. Fluid Mech.,
213, 549-571 (1990).
Vallis, G.K., G.F. Carnevale, and W.R. Young, 1989.
Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech.,
203, 133-152 (1989).
Shows how a modification of the Euler equations of motion can be made that causes the modified system to monotonically increase in energy, while keeping the Casimirs (e.g. enstrophy) constant. Thus can be used to construct Arnold stable states.
Pares-Sierra, A. and G.K. Vallis, 1989. A fast semi-direct method for the numerical solution of non-separable elliptic equations in irregular domains. J. Comp. Physics,
Describes a fast method to solve non-separable elliptic equations in irregular domains. Combines the capacitance matrix method with a fast iteration.
Carnevale, G.F., R. Purini, M. Briscolini, and G.K. Vallis, 1989. Influence of topography on modon propagation and survival. In: Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, eds. J.C.J. Nihoul and B.M. Jamart. Elsevier Science Publishers (invited review article).
Vallis, G.K., 1988. Conceptual models of El Nino and the Southern Oscillation. J. Geophys. Res. (Oceans),
Discusses whether El Nino and ENSO is a chaotic or stochastic system, and proposes example models of both.
Carnevale, G. F., R. Purini, M. Briscolini, and G.K. Vallis, Numerical experiments on modon stability to topographic perturbations. Phys. Fluids,
31(9), 2562-2567 (1988).
Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, 1988. The role of initial conditions in flow stability, with applications to modons. Phys. Fluids,
Vallis, G.K., 1988. A numerical study of transport properties in eddy resolving
and parameterized models. Quart. J. Roy. Meteor. Soc.,
Studies how the heat transport varies with the imposed temperature gradient in a numerical model of QG turbulence, and compares to possible parameterization schemes.
Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, Propagation of barotropic modons over topography. Geophys. Astrophys. Fluid Dynam.,
41, 45-101 (1988).
Vallis, G.K. and B.-L. Hua, Eddy viscosity of the anticipated potential vorticity method.
J. Atmos. Sci.,
45, 617-627 (1988).
Vallis, G.K., El Nino: A chaotic dynamical system? Science, 232, 243-245 (1986).
Suggests that El Nino might be a chaotic dynamical system, and presents a simple model to illustrate this.
Vallis, G.K. and J.O. Roads, Turbulent effects in large scale flow over topography. In:
Proceedings of Second International Symposium on Tibet Plateau and Mountain Meteorology, Beijing, eds. Z. Baozhen and E. Reiter. 390-407, Academia Sinica, China
(conference proceedings) (1986).
Vallis, G.K., Instability and flow over topography.
Geophys. Astrophys. Fluid Dyn.,
34, 1-38 (1985).
Performs a linear stability analysis of flow over topography and of stationary waves, using a long-wave approximation and triad interactions. Also shows that the Charney-Stern-like criterion for instability in a 2-layer model also applies to finite-amplitude disturbances.
Vallis, G.K., Remarks on the predictability properties of two- and three-dimensional flow.
Quart. J. Roy. Meteor. Soc.,
111, 1039-1047 (1985).
Provides a theoretical discussion of the predictability properties of 2D and 3d turbulence. Some of this material is now incorporated into my AOFD book
Vallis, G.K., On the spectral integration of the quasi-geostrophic equations for doubly-periodic and channel flow. J. Atmos. Sci.,
42, 95-99 (1985).
Vallis, G.K. and J.O. Roads, 1984. Large-scale stationary and turbulent flow over topography. J. Atmos. Sci.,
In this paper we compare linear theory of flow over topography with the results from a time-dependent, nonlinear, unsteady, simulation of the same flow. Generally, the instabilities extract energy from the stationary waves, reducing the amplitude of the response from that given by linear theory.
Roads, J.O., G.K. Vallis, and L. Remer, Cloud/climate sensitivity experiments.
In: Climate Processes and Climate Sensitivity, 92-107.
Geophysical Monograph 29, eds. J. Hansen and T. Takahashi.
American Geophysical Union (refereed article in book)
This paper looked at the relationship between the large-scale circulation and large-scale clouds, using a model that had explicit evolution equations for water vapour and cloud water but otherwise idealized dynamics.
Roads, J.O. and G.K. Vallis, 1984.
An energy balance model with cloud feedbacks. Tellus,
Most EBMs have specified clouds. This is an attempt to construct and learn something from an EBM in which clouds are predicted.
Carnevale, G.F. and G.K. Vallis, Applications of entropy to predictability theory. Am. Inst. Physics Proceedings,
"The Predictability of Fluid Motions." 577-593, eds. G. Holloway and B. West (1983).
Vallis, G. K., Barotropic and baroclinic predictability in geostrophic turbulence. Am. Inst. Physics Proceedings,
"The Predictability of Fluid Motions." 117-133, eds. G. Holloway and B. West (1983).
Vallis, G.K., On the predictability of quasigeostrophic flow: the effects of beta and baroclinicity. J. Atmos. Sci.,
40, 10-27 (1983).
Discusses and simulates how baroclinic instability and the beta effect will affect predictability.
Cayan, D., R. Harnack, and G.K. Vallis, Report on workshop on weather and climate variability during winter 1981-1982. Bull. Am. Met. Soc.,
63, 1079-1183 (1982).
Vallis, G.K., A statistical dynamical climate model with a simple hydrology cycle. Tellus,
34, 211-227 (1982).
Describes a zonally-averaged near-primitive-equation (actually semi-geostrophic) model, using Green-like parameterizations for eddy transport. Adds an equation for moisture (i.e., a hydrology cycle) and explores the effects.