Atmospheric Circulation and Dynamics

Ocean Circulation and Dynamics

Climate Dynamics and Ocean-Atmosphere Interaction

Fluid Dynamics, including GFD and Turbulence

Convection and Moist Dynamics

To see the papers in chronological order click here.

Articles may appear under more than one heading, and not all my publications are present. If the PDF of an article is not available, please send me an email.

Thomson, S. I. and Vallis, G. K., 2019. Hierarchical Modelling of Solar System Planets with Isca

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We describe the use of Isca for the hierarchical modelling of solar system planets, with particular attention to Earth, Mars and Jupiter. Isca is a modelling framework for the construction and use of models of planetary atmospheres at varying degrees of complexity, from featureless planets with an atmosphere forced by a thermal relaxation back to a specified temperature, through aquaplanets with no continents (or no ocean) with a simple radiation scheme, to near-comprehensive models with a multi-band radiation scheme, a convection scheme and configurable continents and topography. By a judicious choice of parameters and parameterization schemes the model may be configured for fairly arbitrary planets, with stellar radiation input determined by astronomical parameters, taking into account the planet's obliquity and eccentricity. In this paper we describe models at varying levels of complexity for Earth, Mars and Jupiter using the primitive equations and/or the shallow water equations.

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.

Local PDF file.

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 useability.

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., 2019. The Effects of Gravity on the Climate and Circulation of a Terrestrial Planet.

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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.

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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.

Read, P., Lewis, S. and Vallis, G. K. 2018. Atmospheric Dynamics of Terrestrial Planets. In H. J. Deeg, J. A. Belmonte (eds.), Handbook of Exoplanets, https://doi.org/10.1007/978- 3-319-30648-3_50-2.

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Here we review the atmospheric dynamics of terrestrial planets, with attention to both solar system planets and exoplanets. We discuss the importance of a number of key dimensionless parameters, for example, the thermal Rossby and the Burger numbers as well as nondimensional measures of the frictional or radiative timescales, in defining the type of circulation regime to be expected in a prototypical planetary atmosphere subject to axisymmetric driving. These considerations help to place each of the solar system terrestrial planets into an appropriate dynamical context and also lay the foundations for predicting and understanding the climate and circulation regimes of (as yet undiscovered) Earth-like extrasolar planets. However, as recent discoveries of “super-Earth” planets around some nearby stars are beginning to reveal, this parameter space is likely to be incomplete, and other factors, such as the possibility of tidally locked rotation and tidal forcing, may also need to be taken into account for some classes of extrasolar planet.

Thomson, S. I. and Vallis, G. K., 2017. Atmospheric Response to SST anomalies. Part 1: Background-state dependence, teleconnections and local effects in winter (Part 1) and in Summer (Part 2) Submitted to

PDF file, Part 1.

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.

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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.

Vallis, G. K., Zurita-Gotor, P., Cairns, C., and Kidston, J. 2015. Response of the large-scale structure of the atmosphere to global warming.

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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 discussing 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.

Chai, J, Jansen, M. and Vallis, G. K., 2016. Equilibration of a baroclinic planetary atmosphere in the limit of vanishing bottom friction

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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.

Vallis, G. K., 2016. Geophysical Fluid Dynamics: Whence, Whither and Why?

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.

Mitchell, J. L., Vallis, G. K. and Potter, S., 2014. Effects of seaonal cycle on superrotation in planetary atmospheres

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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.

Vallis, G. K., 2016. Geophysical Fluid Dynamics: Whence, Whither and Why?

Click here for published version

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.

Maher P, Gerber EP, Medeiros B, Merlis TM, Sherwood 5, Sheshadri A, Sobel AH, Vallis GK, Voigt A and Zurita-Gotor, P. 2019. Model Hierarchies for Understanding Atmospheric Circulation.

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In this review, we highlight the complementary relationship between simple and comprehensive models in addressing key scientific questions to describe Earth’s atmospheric circulation. The systematic representation of models in steps, or hierarchies, connects our understanding from idealized systems to comprehensive models, and ultimately the observed atmosphere. We define three interconnected principles that can be used to characterize the model hierarchies of the atmosphere. We explore the rich diversity within the governing equations in the dynamical hierarchy, the ability to isolate and understand atmospheric processes in the process hierarchy, and the importance of the physical domain and resolution in the hierarchy of scale. We center our discussion on the large scale circulation of the atmosphere and its interaction with clouds and convection, focusing on areas where simple models have had a significant impact. Our confidence in climate model projections of the future is based on our efforts to ground the climate predictions in fundamental physical understanding. This understanding is, in part, possible due to the hierarchies of idealized models that afford the simplicity required for understanding complex systems.

Potter, S., Vallis, G. K. and Mitchell, J. L. 2013. Spontaneous superrotation and the role of Kelvin waves in an idealized dry GCM

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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.

Jucker, M, Fueglistaler, S, and Vallis, G. K. 2014. Stratospheric sudden warmings in an idealized GCM

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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

Jucker, M, Fueglistaler, S, and Vallis, G. K. 2013. Maintenance of stratospheric structure in an idealized general circulation model

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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.

O'Rourke, A. and Vallis, G. K., 2013. Jet Interaction and the Influence of a Minimum Phase Speed Bound on the Propagation of Eddies.

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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.

Mitchell, J. and Vallis, G. K. 2010. The transition to superrotation in terrestrial atmospheres

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We explore the transition to superrotation in terrestrial atmospheres (i.e, atmospheres with a solid surface below, such as those of 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.

Zurita-Gotor, P. and Vallis, G. K., 2010. Circulation sensitivity to heating in a simple model of baroclinic turbulence.

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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.

Zurita-Gotor, P. and Vallis, G. K. 2011. Dynamics of mid-latitude tropopause height in an idealized model.

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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.

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,* 44, 184-212.

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A discussion and synthesis of the fundamental dynamics of the NAO its relation to storm tracks and annular modes/patterns. It also illustrates the phenomena with numerical simulations and with stochastic models. It also provides context for the more detailed studies below.

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.

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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.
Gerber, E. P. and Vallis, G. K. 2007. Eddy-Zonal flow interactions and the persistence of the zonal index. *J. Atmos. Sci.,* , 69, 3296-3311.

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Discusses the dynamics of the zonal index, with particular reference to its timescale and interaction with baroclinic eddies, using an idealized GCM (specifically a primitive equation dry dynamical core).

Cash, B., P. Kushner and G. K. Vallis. 2002. The structure and composition of the annular
modes in an aquaplanet GCM. *J. Atmos. Sci., * 59, 3399-3314.

Cash_KV02.pdf

By way of numerical simulations with a simplified GCM, the papers suggests 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.

Cash, B., P. Kushner and G. K. Vallis, 2005. Zonal asymmetries, teleconnections and annular modes in a GCM. *J. Atmos. Sci., * 62, 207-219.

Cash-KV05.pdf

Following the above 2002 paper, we introduce asymmetries to the problem. We explore 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., * 61, 264-280.

Vallis_NAO05.pdf

This paper is an attempt to get at the heart of the dynamics of the NAO and annular patterns. It offers a simple dynamical model - perhaps the simplest possible - of the NAO and its relationship to the storm tracks and so-called annular modes.

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, * 18, 2102-2118.

Gerber-Vallis05.pdf

This paper further abstracts the mechanisms underlying the NAO and annular patterns. In particular, we demonstrate that the spatial structures -- for example the dipolar structure of the EOFs -- of the NAO can be captured by a simple stochastic model.

Thomson, S. I. and Vallis, G. K., 2017. Atmospheric Response to SST anomalies. Part 1: Background-state dependence, teleconnections and local effects in winter (Part 1) and in Summer (Part 2) Submitted to *J. Atmos. Sci.*

PDF file, Part 1.

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.

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.

Scaife-KVF.pdf

Over the late 20th century, the observed NAO index was observed to increase. This has generally not been simulated by GCMs. Here we show that these changes in the (tropospheric) NAO can be simulated properly if the stratosphere is accurately simulated (by means of an artificial forcing, confined to the stratosphere). It doesn't demonstrate that the stratosphere forces the troposphere, but it does demonstrate a link.

Kidston, J. and Vallis, G. K. 2010. Relationship between eddy-driven jet latitude and width * Geophys. Res. Lett*, doi:10.1029/2010GL044849

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Shows that more poleward jets tend to be wider, in both idealized and comprehensive models, and that barotropic instability might account for this.

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

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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.

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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.

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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.

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This paper presents a minimal model of the Madden--Julian Oscillation (MJO), isolating a robust mechanism that leads to the observed characteristic pattern and eastward propagation. A localized heat source due to condensation at the equator leads to a Gill-like pattern in the geopotential, which in turn induces moisture convergence and further condensation. Over a wide range of parameters the moisture convergence is found to be slightly to the east of the heat source. This convergence leads to condensation and hence a heat source that also is east of the original one, thus causing the pattern itself to propagate east. The speed of the ensuing eastward propagation is limited by the ability of the moisture convergence to remain east of the moving condensation heat source. If the pattern moves too quickly, the moisture convergence cannot keep up; the propagation then slows and/or the pattern itself may dissolve. The speed of propagation thus scales with the fluid speed that is induced by the condensation itself, and thus in turn with the strength of the condensational heating, and not with a gravity wave speed. The speed also increases with the distance between the initial heating source and the subsequent condensation. In the real world this distance is determined not only by the location of moisture convergence but also by the complex physics of convection in a conditionally unstable environment, and thus cannot be accurately determined in any simple model. Thus, even though the underlying MJO mechanism is not complicated its reproduction will necessarily depend rather sensitively on model parameters in numerical simulations.

Vallis, G. K., 2020. The Trouble with Water: Condensation, Circulation and Climate

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This perspectives article discusses a few of the problems that arise in geophysical fluid dynamics and climate that are associated with the presence of moisture in the air, its condensation and release of latent heat. Our main focus is Earth's atmosphere but we also discuss how these problems might manifest themselves on other planetary bodies, with particular attention to Titan where methane takes on the role of water. It is published in a special issue of the European Journal on Physics on 'Fundamental Issues in Climate Dynamics'

Vallis, G. K. and Penn, J. 2020. Convective Organization and Eastward Propagating Equatorial Disturbances in a Simple Excitable System.

Click to download PDF file (2 MB)

We describe and illustrate a mechanism whereby convective aggregation and eastward propagating equatorial disturbances, similar in some respects to the Madden--Julian oscillation, arise. We construct a simple, explicit system consisting only of the shallow water equations plus a humidity variable. For a broad range of parameters the system is excitable and self-sustaining, even if linearly stable, with condensation producing convergence and gravity waves that, acting together, trigger more condensation. On the equatorial beta-plane the convection first organizes near the equator, generating patterns related to those in the Matsuno--Gill problem. However, the pattern is unsteady and more convection is triggered on its eastern edge, leading to a precipitating disturbance that progresses eastward. The effect is enhanced by westward prevailing winds that increase the evaporation east of the disturbance. The pattern is confined to a region within a few deformation radii of equator because here the convection can best create the convergence needed to organize into a self-sustaining pattern. Formation of the disturbance preferentially occurs where the surface is warmer and sufficient time (a few tens of days) must pass before conditions arise that enable the disturbance to reform. The speed of the disturbance depends on the efficiency of evaporation and the heat released by condensation, and is typically a few meters per second, much less than the Kelvin wave speed.

Vallis, G. K., Parker, D, J. and Tobias, S. M. 2019. A Simple System for Moist Convection: The Rainy Benard Model.

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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. Subnmitted to

Local PDF file.

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. Maher P, Vallis GK, Sherwood SC, Webb MJ and Sansom PG. 2018. Impact of Parameterized Convection on Climatological Precipitation in Atmospheric Global Climate Models.

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Convective parameterizations are widely believed to be essential for realistic simulations of the atmosphere. However, their deficiencies also result in model biases. The role of convection schemes in modern atmospheric models is examined using Selected Process On/Off Klima Intercomparison Experiment simulations without parameterized convection and forced with observed sea surface temperatures. Convection schemes are not required for reasonable climatological precipitation. However, they are essential for reasonable daily precipitation and constraining extreme daily precipitation that otherwise develops. Systematic effects on lapse rate and humidity are likewise modest compared with the intermodel spread. Without parameterized convection Kelvin waves are more realistic. An unexpectedly large moist Southern Hemisphere storm track bias is identified. This storm track bias persists without convection schemes, as does the double Intertropical Convergence Zone and excessive ocean precipitation biases. This suggests that model biases originate from processes other than convection or that convection schemes are missing key processes.

Vallis, G. K., G. J. Shutts, G. and M. E. B. Gray. 1997. Balanced mesoscale motion and stratified turbulence forced by convection.

Vallis-etal97.pdf

Looks at the possible generation of larger scales of motion from convective forcing. In particular, the paper explores the generation of an inverse cascade in low Froude number flow, with the forcing coming from resolved convection at small scales. The paper is relevent to the presence of the observed -5/3 range at the hundred-kilometer scale in the atmosphere. (Other theories exist for that regime too.)

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.

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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.

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The above two papers explore the so-called hypohydrostatic rescaling, which results in a the material derivative of vertical velocity being multiplied by a large numerical factor. This has the effect of making convection occur at larger scales, and thus leads to a possible 'parameterization' of convection: that is, the convective scale is made larger, so the model can resolve it. The results were mixed; in the Garner et al paper the technique was demonstrated to work in a global model, but in the Pauluis et al paper the technique was compared to convective-resolving simulations with less satisfactory results. The jury is still out.

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Performs a linear stability analysis of flow over topography and of stationary waves, using a long-wave approximation and triad interactions. The paper also shows that Charney-Stern-like criterion for instability in a 2-layer model also applies to finite-amplitude disturbances: that is, it is a nonlinear stability result.

Vallis, G.K. and J.O. Roads, Large-scale stationary and turbulent flow over topography.

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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.

Vallis, G.K. and J.O. Roads, 1986. 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.

Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, Propagation of barotropic modons over topography.

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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.

Zurita-Gotor, P. and Vallis, G. K. 2009. Equilibration of baroclinic turbulence in primitive equation and quasi-geostrophic models.

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Explores the nonlinear equilibration of baroclinic eddies in PE and QG models. We find that quasi-geostrophic theory (and in particular geostrophic turbulence theory) can do a reasonable job, with qualitative and sometimes quantitative agreement between the PE model and QG theory. We also find that supercritical states and an inverse energy cascade can be found in some parameter regimes (although not those most corresponding the the Earth's atmosphere).

Vallis, G.K., A numerical study of transport properties in eddy resolving and parameterized models.

Vallis_QJ88.pdf

Studies how the heat transport varies with the imposed temperature gradient in a numerical model of QG turbulence, and compares to possible parameterization schemes. Shows that the heat transport increases faster than linearly with temperature gradient, but that supercritical flows can exist. Thus, in this model, baroclinic-adjustment like arguments do not apply.

Vallis, G.K. and J.O. Roads, Large-scale stationary and turbulent flow over topography.

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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.

PDF file.

Observations of warm past climates and projections of future climate change show that the Arctic warms more than the global mean, particularly during winter months. Past warm climates such as the early Eocene had above-freezing Arctic continental temperatures year-round. In this paper, we show that a reduced Arctic land seasonality with increased greenhouse gases is a robust consequence of the smaller surface heat capacity of land (compared to ocean), without recourse to other processes or feedbacks. We use a General Circulation Model (GCM) with no clouds or sea ice and a simple representation of land. The equator-to-pole surface temperature gradient falls with increasing CO2, but this is only a near-surface phenomenon and occurs with little change in total meridional heat transport. The high-latitude land has about twice as much warming in winter than in summer, whereas high-latitude ocean has very little seasonality in warming. A surface energy balance model shows how the combination of the smaller surface heat capacity of land and the nonlinearity of the temperature dependence of surface longwave emission gives rise to the seasonality of land surface temperature change. The atmospheric temperature change is surface-enhanced in winter as the atmosphere is near radiative-advective equilibrium, but more vertically homogeneous in summer as the Arctic land gets warm enough to trigger convection. While changes in clouds, sea ice and ocean heat transport undoubtedly play a role in high latitude warming, these results show that enhanced land surface temperature change and surface-enhanced atmospheric temperature change in winter can happen in their absence for very basic and robust reasons.

Vallis, G. K., Zurita-Gotor, P., Cairns, C., and Kidston, J. 2014. Response of the large-scale structure of the atmosphere to global warming.

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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 discussing 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.

Kidston, J., Dean, S.M., Renwick, J. A. and Vallis, G. K. 2009. A robust increase in the eddy length scale in the simulation of future climates.

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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.

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Meridional Overturning Circulation in a Multibasin Model. Part II: Sensitivity to diffusivity and wind in warm and cool climates,

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These two papers examine the various roles of buoyancy forcing, winds and diffusion on the MOC in a multi-basin model, with relevance to the climate of today and past glacial climates. Increased Southern Ocean winds, and increased diapycnal diffusion, both act to strengthen the MOC, but the extent to which they do so depends on the buoyancy forcing. In a today's climate, the AMOC is driven not only by Southern Ocean winds but by diffusion in the Pacific, and the MOC takes a winding path between the basins. In glacial climates the Atlantic and Pacific are decoupled and the NADW is shallower, and Southern Ocean winds play a more dominant role in the Atlantic overturning circulation.

Samelson, R. and G.K. Vallis, 1997. Large-scale circulation with small diapycnal diffusivity: the two-thermocline limit.

Samelson-Vallis.pdf

Theories for the thermocline had hitherto generally fallen into two camps, adiabatic theories (like the ventilated thermocline model) and diffusive theories (the thermocline as an internal boundary layer). In this paper we suggest that in fact the main thermocline has two dynamical regimes: an adiabatic regime lying above an intrinsically diffusive regime. We presented various scaling arguments, some supporting numerical calculations, and a few comparisons with observations.

Vallis, G. K., 2000. Large-scale circulation and production of stratification: effects of wind, geometry and diffusion.

Vallis2000.pdf

A follow-on and extension of the above paper, exploring the effects on interhemispheric circulation and an ACC, using an idealized, primitive equation, OGCM. The paper shows that the ACC has a profound effect on the circulation, and explores the influence of surface boundary conditions and interhemispheric asymmetries. The main subtropical thermocline still has adiabatic and diffusive components, although details differ from the single-hemisphere case.

Dewar, W. D., R. S. Samelson and G. K. Vallis. 2005. The ventilated pool: A model of subtropical mode water.

Dewar-SV05.pdf

Observations show the presence of a large pool of homogeneous water in the north-west areas of subtropical gyres, known as

Samelson, R. and G.K. Vallis, 1997. A simple fictional and diffusive scheme for the planetary geostrophic equations in a closed basin.

Samelson-Vallis.pdf

Suggests how to make the planetary-geostrophic equations well-posed and numerically efficient in a closed domain. Such a model was used in some of the above modelling studies, notably the

Henning, C and Vallis, G. K., 2004. The Effect of mesoscale eddies on the main subtropical thermocline.

Henning_Vallis04.pdf

Discusses and simulates the effects of mesoscale eddies on the main thermocline. Shows that the fundamental structure of the Samelson-Vallis model seems okay, but that eddies do have a notable impact on thermocline structure, especially the internal thermocline.

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.

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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.

Nikurashin, M. and Vallis, G. K. 2012. A Theory of the Interhemispheric Deep Overturning Circulation and Associated Stratification.

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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.

Nikurashin, M. and Vallis, G. K. 2011. A Theory of Deep Stratification and Overturning Circulation in the Ocean.

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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).

Venaille, A, Vallis, G. K. and Smith, K. S. 2011. Baroclinic turbulence in the ocean: analysis with primitive equation and quasi-geostrophic simulations.

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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.

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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).

Nikurashin, M. and Vallis, G. K. 2012. A Theory of the Interhemispheric Deep Overturning Circulation and Associated Stratification.

PDF file

The paper extends the one listed above 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. Watson, A., Vallis, G. K., and Nikurashin, M. 2015. Southern Ocean Buoyancy Forcing of Ocean Ventilation and Glacial Atmospheric CO

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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 CO

Zhang, Y and Vallis, G. K. 2013. Ocean Heat Uptake in Eddying and Non-eddying Ocean Circulation Models in a Warming Climate

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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.

Xie, P. and Vallis, G. K. 2011. The Passive and Active Nature of Ocean Heat Uptake in Idealized Climate Change Experiments

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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.

Ilicak, M. and Vallis, G. K. 2012. Simulations and scaling of horizontal convection.

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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.

Huck, T., and G. K. Vallis. 2001. Linear stability analysis of the three-dimensional thermally-driven ocean circulation: application to interdecadal oscillations.

Huck-Vallis01.pdf

Carries out a three-dimensional linear instability calculation for the THC. Shows that there are unstable modes that resemble the variability in a corresponding 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.

Huck-Vallis-CdV01.pdf

Explores the robustness of interdecadal variability of the THC.

Vallis, G. K., 2000. Large-scale circulation and production of stratification: effects of wind, geometry and diffusion.

Vallis2000.pdf

Explores the meridional overturning circulation in a single-basin, two-hemisphere model with an ACC, using an idealized, primitive equation, OGCM. The paper shows that the ACC has a profound effect on the circulation, and explores the influence of surface boundary conditions and interhemispheric asymmetries. The main subtropical thermocline still has adiabatic and diffusive components, although details differ from the single-hemisphere case.

Loving, Jolene L., and Geoffrey K. Vallis, 2005. Mechanisms for climate variability during glacial and interglacial periods.

loving-vallis-paleo05.pdf

In glacial climates the temperature of the North Atlantic fluctuated strongly on millennial timescales, and these fluctuations have become known as Dansgaard-Oeschger oscillations. In this paper we offers a mechanism of these oscillations involving an instability of the thermohaline circulation; sea-ice plays an important role.

Fuckar, N. S. and Vallis, G. K. 2007. Interhemispheric influence of surface buoyancy conditions on a circumpolar current.

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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. An increase in temperature of a 5 K can cause the transport to change from 50 Sv to 100 Sv. This result may have some paleo relevance, as on millennial timescales there is observed to be some connection between the high latitudes of the two hemispheres.

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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.

Ferrari, R., Griffies, S. M., Nurser, G. and Vallis, G. K. 2009. A Boundary-Value Problem for the Parameterized Mesoscale Eddy Transport. In press in

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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.

Henning, C and Vallis, G. K. 2005. The Effects of Mesoscale Eddies on the Stratification and Transport of an Ocean with a Circumpolar Channel.

Henning-Vallis05.pdf

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,

Henning, C and Vallis, G. K., 2005. The Effect of mesoscale eddies on the main subtropical thermocline.

Henning_Vallis04.pdf

Discusses and simulates the effects of mesoscale eddies on the main thermocline.

Smith, K. S. and G. K. Vallis. 2002. Scales and equilibration of mid-ocean eddies: Forced-dissipative flow.

Smith-Vallis02.pdf

Discusses and simulates the scales of mid-ocean eddies, from the point of view of force-dissipative geostrophic turbulence.

Smith, K. S. and G. K. Vallis. 2001. Scales and equilibration of mid-ocean eddies. Freely decaying flow.

Smith-Vallis01.pdf

Discusses and simulates the scales of mid-ocean eddies, from the point of view of decaying geostrophic turbulence.

Zhao, R and Vallis, G. K. 2008. Parameterizing mesoscale eddies with residual and Eulerian schemes, and a comparison with eddy-permitting models.

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The equations of motion are cast in 'residual' or TEM form, in which the effects of eddies appear as a PV flux in the momentum equations, which may then be parameterized straightforwardly (if not necessarily accurately) as a PV diffusion. Results from parameterized models are then compared with results from eddy permitting models, with encouraging results.

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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.

Wang, J. and G.K. Vallis, 1994. Emergence of Fofonoff states in inviscid and viscous ocean circulation models.

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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., 1993. Statistical mechanics, turbulence, and ocean circulation. in Statistical Methods in Physical Oceanography,

Discussion, in the Aha Hulikoa way, of the application of some statistical mechanical ideas to ocean circulation.

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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.

Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean.

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.

Anadadesikan, A, Swathi, P, Slater, R. S. and Vallis, G. K. 2005. Energetics of large-scale ocean circulation.

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Discusses the energetics of the large-scale circulation and its relation to Sandstrom's effect. Sandstrom's effect is much misunderstood: it is not a useful theorem, because the conditions for its satisfaction are not obeyed in the ocean, but nevertheless there is a real effect. I have a discussion of its foundations in 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,

The following papers are oceanographic, but motivated by possible influences on climate.

Watson, A., Vallis, G. K., and Nikurashin, M. 2015. Southern Ocean Buoyancy Forcing of Ocean Ventilation and Glacial Atmospheric COPDF file.

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 CO

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.

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In this paper we show that topographic effects and bottom vortex stretching, and so the deep western boundary current, can influence the Gulf Stream path. I would never have thought that such a mild paper could be so controversial, but it evidently hit the preconceived notions of a reviewer who vehemently disliked it. It was over two years in review.

Zhang, R. and Vallis, G. K. 2006. Impact of Great Salinity Anomalies on the low frequency variability of the North Atlantic Climate.

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Shows that Great Salinity anomalies can potentially alter the climate of the North Atlantic region. They do this by affecting deep convection, which in turn affects the deep western boundary current, and thence the Gulf Stream path and so the climate of the North Atlantic region.

Loving, Jolene L., and Geoffrey K. Vallis, 2005. Mechanisms for climate variability during glacial and interglacial periods.

loving-vallis-paleo05.pdf

In glacial climates the temperature of the North Atlantic fluctuated strongly on millennial timescales, and these fluctuations have become known as Dansgaard-Oeschger oscillations. In this paper we offers a mechanism of these oscillations involving an instability of the thermohaline circulation; sea-ice plays an important role.

Huck, T., and G. K. Vallis. 2001. Linear stability analysis of the three-dimensional thermally-driven ocean circulation: application to interdecadal oscillations.

Huck-Vallis01.pdf

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.

Huck-Vallis-CdV01.pdf

Explores the robustness of interdecadal variability of the THC.

PDF file.

Observations of warm past climates and projections of future climate change show that the Arctic warms more than the global mean, particularly during winter months. Past warm climates such as the early Eocene had above-freezing Arctic continental temperatures year-round. In this paper, we show that a reduced Arctic land seasonality with increased greenhouse gases is a robust consequence of the smaller surface heat capacity of land (compared to ocean), without recourse to other processes or feedbacks. We use a General Circulation Model (GCM) with no clouds or sea ice and a simple representation of land. The equator-to-pole surface temperature gradient falls with increasing CO2, but this is only a near-surface phenomenon and occurs with little change in total meridional heat transport. The high-latitude land has about twice as much warming in winter than in summer, whereas high-latitude ocean has very little seasonality in warming. A surface energy balance model shows how the combination of the smaller surface heat capacity of land and the nonlinearity of the temperature dependence of surface longwave emission gives rise to the seasonality of land surface temperature change. The atmospheric temperature change is surface-enhanced in winter as the atmosphere is near radiative-advective equilibrium, but more vertically homogeneous in summer as the Arctic land gets warm enough to trigger convection. While changes in clouds, sea ice and ocean heat transport undoubtedly play a role in high latitude warming, these results show that enhanced land surface temperature change and surface-enhanced atmospheric temperature change in winter can happen in their absence for very basic and robust reasons.

Geen, R., Lambert, F. H., and Vallis, G. K. 2017. Regime Change Behavior During Asian Monsoon Onset.

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In which we explore the dynamics of the monsoon, and the relatioship 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.

Vallis, G. K. and Farneti, R. 2009. Meridional Energy Transport in the Atmosphere-Ocean System. Scaling and Numerical Experiments.

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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.)

Farneti, R. and Vallis, G. K. 2012. Meridional Energy Transport in the Coupled Atmosphere-Ocean system: Compensation and Partitioning

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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.

Padilla, L., Vallis, G. K. and Rowley, C. 2011. Probabilistic estimates of transient climate sensitivity subject to uncertainty in forcing and natural variability.

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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).

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.

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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., Dean, S.M., Renwick, J. A. and Vallis, G. K. 2009. A robust increase in the eddy length scale in the simulation of future climates.

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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.

Watson, A., Vallis, G. K., and Nikurashin, M. 2015. Southern Ocean Buoyancy Forcing of Ocean Ventilation and Glacial Atmospheric CO

PDF file.

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 CO

Vallis, G. K. 2009. Mechanisms of climate variability from years to decades. In press in

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A review article and essay on climate variability. Not so much a literature survey but a discussion of mechanisms, illustrated by examples.

Farneti, R. and Vallis, G. K. 2009. An intermediate complexity climate model based on the GFDL Flexible Modelling System.

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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.

Farneti, R. and Vallis, G. K. 2008. Mechanisms of interdecadal climate variability and the role of ocean-atmosphere coupling

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Explores the mechanisms of decadal climate variability using a simplified, but still fully dynamical, 3D coupled ocean-atmosphere-climate model.

Loving, Jolene L., and Geoffrey K. Vallis, 2005. Mechanisms for climate variability during glacial and interglacial periods.

loving-vallis-paleo05.pdf

Offers a mechanism of Dansgaard-Oeschger oscillations. Specifically, the paper demonstrates an instability/oscillation of the meridional overturning circulation that is present only during glacial climates, and that would cause large variations in high-latitude atmospheric temperatures on millenial timescales. Shows how sea-ice plays an important role.

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, 398, 601-604. (1999)

Wells-Silver-Vallis99.pdf

Vallis, G.K., Conceptual models of El Nino and the Southern Oscillation.

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Discusses whether El Nino and ENSO is a chaotic or stochastic system, and proposes example models of both. Twenty years on the jury remains hung as to whether El Nino is chaotic is the usual sense or stochastic. (Note that stochasticity is really just unresolved, often high-dimensional, chaos.)

Vallis, G.K., El Nino: A chaotic dynamical system? Science, 232, 243-245 (1986).

Vallis86.pdf

An early paper suggesting that El Nino might be a chaotic dynamical system, and presents a simple model to illustrate this. The model is meant to be illustrative rather than particularly realistic. There have been many subsequent attempts to make more realistic models demonstrating chaos in the ENSO system.

Local PDF file.

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.

Roads, J.O., G.K. Vallis, and L. Remer, 1984. 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).

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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 that had otherwise idealized dynamics, in particular specified winds. Thus, the model is much simpler than a GCM, but less parameterized than some other simple models. One goal of the study was to look at the factors determining cloud cover and its relatioon to relative humidity.

Roads, J.O. and G.K. Vallis, 1984. An energy balance model with cloud feedbacks.

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Most EBMs have specified clouds. This is an attempt to construct and learn something from an EBM in which clouds are predicted.

Vallis, G.K., 1982. A statistical dynamical climate model with a simple hydrology cycle.

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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.

Tsang, Y-K, and Vallis, G. K. 2018. A Stochastic Lagrangian Basis for a Probabilistic Parameterization of Moisture Condensation in Eulerian Models. Subnmitted to

Local PDF file.

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., 2016. Geophysical Fluid Dynamics: Whence, Whither and Why?

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.

Venaille, A, Nadeau,, L-P, and Vallis, G. K., 2014. Ribbon Turbulence.

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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.

Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean.

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.

Ferrari, R., Griffies, S. M., Nurser, G. and Vallis, G. K. 2009. A Boundary-Value Problem for the Parameterized Mesoscale Eddy Transport. Submitted to

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A variation on the Gent-McWilliams scheme for parameterizing mesoscale eddies. The new scheme enforces a low mode vertical structure for the parameterized streamfunction, as motivated by geostrophic turbulence theory, for example in the papers by Smith and Vallis below

Maltrud, M. and G.K. Vallis, 1991. Energy spectra and coherent structures in forced two-dimensional and geostrophic turbulence.

Maltrud-Vallis91.pdf

Shows that the -5/3 inverse energy spectrum of 2D turbulence can be robustly simulated, and obtains an approximate value for the Kolomogorov-Kraichnan constant for this range. The paper also explores the 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.

Vallis, G.K. and M. E. Maltrud., 1993. Generation of mean flows and jets on a beta-plane and over topography.

Vallis-Maltrud.pdf

Explores the effects of beta in two-dimensional turbulence, and in particular shows how the beta effect combines with two-dimensional turbulence to give rise to zonal jets. The mechanism differs somewhat from that of Rhines, who invoked weakly-nonlinear theory. Here, we show that the presence of Rossby waves will prevent certain waveumbers from being efficiently excited, producing a dumbbell shaped region in spectral space that is nearly void of energy. The natural consequence is the production of zonal flows and jets. The paper proposes a scaling for the jet scale, which differs somewhat from the Rhines scale. The paper also notes that friction will be necessary for the flow to equilibrate, and this complicates matters. Similar phenomena occur for flow over topography.

Maltrud. M. and G.K. Vallis, 1993. Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence.

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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.

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.

Smith_etal02.pdf

A rather broad-ranging paper, discussing and simulating the transport properties of the inverse cascade in various flavors of 2D turbulence, including surface-geostrophic turbulence.

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.

Grianik_HSV04.pdf

The paper shows that the halting scale of the inverse cascade in 2D turbulence is independent of the strength of the turbulence, if a nonlinear drag (such as is common in boundary layer schemes) is used.

Smith, K. S. and G. K. Vallis. 2002. Scales and equilibration of mid-ocean eddies: Forced-dissipative flow.

Smith-Vallis02.pdf

Discusses and simulates the scales of mid-ocean eddies, from the point of view of force-dissipative geostrophic turbulence.

Smith, K. S. and G. K. Vallis. 2001. Scales and equilibration of mid-ocean eddies. Freely decaying flow.

Smith-Vallis01.pdf

Discusses and simulates the scales of mid-ocean eddies, from the point of view of decaying geostrophic turbulence.

Oetzel, K. and G. K. Vallis. 1997. Strain, vortices, and the enstrophy inertial range in two-dimensional turbulence. Phys. Fluids 9, 2991-3004.

Oetzel-Vallis97.pdf

Shows that a -3 enstrophy inertial range can in fact 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. The paper thus provided some rather unexpected support for the Kraichnan -3 enstrophy range. Subsequent much higher resolution studies have found similar results.

Vallis, G.K., A numerical study of transport properties in eddy resolving and parameterized models.

Vallis_QJ88.pdf

Studies how the heat transport varies with the imposed temperature gradient in a numerical model of QG turbulence, and compares to possible parameterization schemes. Shows that the heat transport increases faster than linearly with temperature gradient, but that supercritical flows can exist. Thus, in this model, 'baroclinic adjustment' or 'marginal supercriticality' arguments do not hold.

Zurita-Gotor, P. and Vallis, G. K. 2009. Equilibration of baroclinic turbulence in primitive equation and quasi-geostrophic models.

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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, with qualitative and sometimes quantitative agreement between the PE model and QG theory. 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). Note, though, that although the stratification can adjust in the PE model, the lack of vertical resolution means that the tropopause does not have complete freedom to adjust its height.

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Review of 2D turbulence. Much of this material (that is, most of the theoretical development, if not the numerical simulations) is now incorporated into textbooks, in particular this one, to which the interested reader may refer.

Vallis, G.K. From laminar flow to turbulence. 1996. In:

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A longish review of various pathways to turbulence (period-doubling, etc.) and of fully-developed turbulence itself.

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Discusses and simulates how baroclinic instability and the beta effect will affect predictability. Shows the beta effect can enhance predictability.

Vallis, G.K., 1985. Remarks on the predictability properties of two- and three-dimensional flow.

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Gives a theoretical discussion of the predictability properties of 2D and 3D turbulence. Some of this material is now incorporated into my AOFD book.

Carnevale, G.F. and G.K. Vallis, 1983. Applications of entropy to predictability theory.

An early discussion of the use of entropy, applied to 2D inviscid flow.

Vallis, G. K., 1983. Barotropic and baroclinic predictability in geostrophic turbulence.

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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.

Vallis, G.K., 1985. Instability and flow over topography.

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Performs a linear stability analysis of flow over topography and of stationary waves, using a long-wave approximation and triad interactions. The paper also shows that Charney-Stern-like criterion for instability in a 2-layer model also applies to finite-amplitude disturbances; that is, it is a nonlinear stability result.

Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, 1988. The role of initial conditions in flow stability, with applications to modons.

Warn_BSV.pdf

Shows how to construct balanced models of arbitrarily high order, by 'slaving' fields to a single slolwy evolving variable, such as potential vorticity.

Mundt, M. G.K. Vallis and J. Wang, 1997. Balanced models for the large- and meso-scale circulation.

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In practice, balanced models may achieve higher accuracy not by going to higher order in a formal asymptotic expansion, but by spanning different regimes of flow, such as quasi-geostrophy and planetary-geostrophy. In this paper use PV inversion to construct such models and show how accurate they can be, in an oceanic setting. Care must be takenwhen using such models, as they may not preserve all the invariants of the original set.

Vallis, G. K. 1996. Approximate geostrophic models for large-scale flow in the ocean and atmosphere.

Vallis, G.K., 1996. Potential vorticity inversion and balanced equations of motion for rotating and stratified flows.

Vallis96.pdf

Shows how PV inversion can be used to construct higher-order balanced models for stratified flow.

Sundermeyer, M. and G.K. Vallis, 1993. Correlation dimension of primitive equation and balanced models.

Sundermeyer-Vallis.pdf

Vallis, G.K., 1992. Mechanisms and parameterizations of geostrophic adjustment and a new model for balanced flow.

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Shows that geostrophic balance is the minimum energy state for a given field of potential vorticity, in the linear approximation. Provides a nonlinear extension.

Smith, K. S. and G. K. Vallis. 1998. Linear wave and instability properties of extended range geostrophic models.

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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.

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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. The method can be used to construct Arnold stable states.

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.

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Carnevale, G., M. Briscolini, R. Kloosterziel, and G.K. Vallis. 1997. Three dimensionally perturbed vortex tubes in a rotating flow.

Carnevale, G.F., R. Purini, M. Briscolini, and G.K. Vallis, 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 (review article) (1989).

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).

Carnevale, G. F., R. Purini, M. Briscolini, and G.K. Vallis, 1998. Numerical experiments on modon stability to topographic perturbations.

Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, 1998. Propagation of barotropic modons over topography.

Schonbek, M. and Vallis, G. K. 1999. Energy Decay of Solutions to the Boussinesq, Primitive and Planetary Geostrophic Equations.

Schonbek-Vallis99.pdf

A brief foray into pure mathematics, proving rigorously certain properties of the primitive and similar equations.

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.

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Describes a fast method to solve non-separable elliptic equations in irregular domains. Combines the capacitance matrix method with a fast iteration.

Vallis, G.K.,1985. On the spectral integration of the quasi-geostrophic equations for doubly-periodic and channel flow.

Vallis, G.K. and B.-L. Hua, 1988. Eddy viscosity of the anticipated potential vorticity method.