Publications, Papers and Preprints by G. K. Vallis

Organized thematically

To see the articles arranged chronologically click here.

The papers below are, somewhat arbitrarily, divided into the following four main categories. Click on the link to go there directly.

Atmospheric Circulation and Dynamics
Ocean Circulation and Dynamics
Climate Dynamics and Ocean-Atmosphere Interaction
Fluid Dynamics, including GFD and Turbulence

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.

Books

Atmospheric and Oceanic Fluid Dynamics
Cambridge University Press, by G. K. Vallis.
Graduate level textbook with a few monograph-like aspects. Contains material on fundamentals of GFD; waves, instabilities and turbulence; general circulation of atmosphere; general circulation of ocean.
Click here for new material
Click here for book web site


Climate and the Oceans, 2011
Princeton University Press, by G. K. Vallis.
Short undergraduate book due to appear in late 2011. It may also be useful as an overview for graduate students and scientists in other fields.

Atmospheric Circulation and Dynamics

General Circulation and Planetary Atmospheres

Vallis, G. K., Zurita-Gotor, P., Cairns, C., and Kidston, J. 2014. Response of the large-scale structure of the atmosphere to global warming. Q. J. Roy. Meteor. Soc. submitted.
PDF file (Revised, June 2014).
This is a discussion paper concerned with the response of the large-scale structure of the atmosphere to increased concentrations of greenhouse gases. We present results from CMIP5 integrations as well as reviewing various arguments for how the atmosphere might change, and we discuss what is robust and what is not. Topics include the height of the tropopause, the expansion of the Hadley Cell and the shift of the westerlies. Comments by email to me at my Exeter email address (g.vallis at exeter etc) are welcome.

Potter, S., Vallis, G. K. and Mitchell, J. L. 2013. Spontaneous superrotation and the role of Kelvin waves in an idealized dry GCM J. Atmos. Sci, submitted.
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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. 2013. Maintenance of stratospheric structure in an idealized general circulation model J. Atmos. Sci, in press.
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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. J. Atmos. Sci., doi: 10.1175/JAS-D-12-0303.1.
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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 J. Geophys. Res. (Planets), v115, E12008, doi:10.1029/2010JE003587.
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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. J. Atmos. Sci.,, 67, 1543--1588.
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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. J. Atmos. Sci., (in press)
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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.

North Atlantic Oscillation, Annular Patterns etc

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.

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.

Convection

Vallis, G. K., G. J. Shutts, G. and M. E. B. Gray. 1997. Balanced mesoscale motion and stratified turbulence forced by convection. Quart. J. Roy. Meteor. Soc., 123, 1621-1652.
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. Theor. Comput. Fluid Dynam., 20,} 485-499.
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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. J. Atmos. Sci. 64, 2061-2075.
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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.

Stationary Waves and Flow over Topography

Vallis, G.K., 1985. Instability and flow over topography. Geophys. Astrophys. Fluid Dyn., 34, 1-38.
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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. J. Atmos. Sci., 41, 3255-3271 (1984).
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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. Geophys. Astrophys. Fluid Dynam., 41, 45-101 (1988).

Eddy Transport and Eddy Effects on the General Circulation

Zurita-Gotor, P. and Vallis, G. K., 2010. Circulation sensitivity to heating in a simple model of baroclinic turbulence. J. Atmos. Sci.,, (in press).
PDF file
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. J. Atmos. Sci., (in press)
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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. Quart. J. Roy. Meteor. Soc., 114, 183-204 (1988).
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. J. Atmos. Sci., 41, 3255-3271 (1984).
PDF file
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.

Climate change and the atmosphere

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. Geophys. Res. Lett, 37, L03806, doi:10.1029/2009GL041615.
PDF file
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.

Ocean Circulation and Dynamics

Thermocline theory and circulation theory

Samelson, R. and G.K. Vallis, 1997. Large-scale circulation with small diapycnal diffusivity: the two-thermocline limit. J. Mar. Res. 55, 1-54.
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. J. Phys. Oceanogr., 30, 933-954.
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. J. Phys. Oceanogr., 35, 137-150.
Dewar-SV05.pdf
Observations show the presence of a large pool of homogeneous water in the north-west areas of subtropical gyres, known as mode water; he North Atlantic has a particularly well-studied example. In this paper we offer an analytic theory of such mode waters, set in the context of thermocline theory and the large-scale circulation

Samelson, R. and G.K. Vallis, 1997. A simple fictional and diffusive scheme for the planetary geostrophic equations in a closed basin. J. Phys. Oceanogr. 27, 186-194.
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 J. Marine Research paper of S & V.

Henning, C and Vallis, G. K., 2004. The Effect of mesoscale eddies on the main subtropical thermocline. J. Phys. Oceanogr., 34, 2428-2443.
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. J. Phys. Oceanog., 42,1652-1667.
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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. J. Phys. Oceanog., v41. doi: 10.1175/2010JPO4529.1
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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. J. Phys. Oceanogr., (in press)
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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.

Meridional Overturning Circulation

Nikurashin, M. and Vallis, G. K. 2011. A Theory of Deep Stratification and Overturning Circulation in the Ocean. J. Phys. Oceanog., v41. doi: 10.1175/2010JPO4529.1
PDF file
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. J. Phys. Oceanog., 42,1652-1667.
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.

Zhang, Y and Vallis, G. K. 2013. Ocean Heat Uptake in Eddying and Non-eddying Ocean Circulation Models in a Warming Climate J. Phys. Ocean, in press.
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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 Climate Dynamics, DOI 10.1007/s00382-011-1063-8
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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. Tellus, A , 64, http://dx.doi.org/10.3402/tellusa.v64i0.18377.
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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. Tellus, 53A, 526-545.
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. J. Climate, 14, 940-963.
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. J. Phys. Oceanogr., 30, 933-954.
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. Paleoceanography, 20, PA4024, doi:10.1029/2004PA00111
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. Geophysical Research Letters 34, L14605, doi:10.1029/2007GL030379.
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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.

Effects of Mesoscale Eddies in the Ocean

Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nature Geosciences doi: 10.1038/NGEO1657.
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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 Ocean Modelling, doi:10.1016/j.ocemod.2010.01.004.
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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. J. Phys. Oceanogr., 35, 880-896.
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, and the deep circulation away from the channel underneath the main subtropical thermocline.

Henning, C and Vallis, G. K., 2005. The Effect of mesoscale eddies on the main subtropical thermocline. J. Phys. Oceanogr., 34, 2428-2443.
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. J. Phys. Oceanogr., 32, 1699-1721.
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. J. Phys. Oceanogr., 31, 554≠571.
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. Ocean Modelling, doi:10.1016/j.ocemod.2008.02.005
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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.

Circulation theory (misc)

Ilicak, M. and Vallis, G. K. 2011. Simulations and scaling of horizontal convection. Tellus, submitted.
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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. J. Mar. Res., 82, 85-127.
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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, `Aha Hulikoa' proceedings, ed. P. Muller and G. Holloway. 473-491.
Discussion, in the Aha Hulikoa way, of the application of some statistical mechanical ideas to ocean circulation.

Ocean Energetics

Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nature Geosciences doi: 10.1038/NGEO1657. PDF file
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. J. Climate, 18, 2604--2616.
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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, J. Phys. Oceanog., 21, 1534-1552.

Oceans and Climate

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

Zhang, R. and Vallis, G. K. 2007. The role of bottom vortex stretching on the path of the North Atlantic western boundary current and on the northern recirculation gyre. J. Phys. Ocean, (in press).
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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. J. Climate, 19, 470-482.
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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. Paleoceanography, 20, PA4024, doi:10.1029/2004PA00111
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. Tellus, 53A, 526-545.
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. J. Climate, 14, 940-963.
Huck-Vallis-CdV01.pdf
Explores the robustness of interdecadal variability of the THC.

Climate Dynamics and Ocean-Atmosphere Interaction

Farneti, R. and Vallis, G. K. 2012. Meridional Energy Transport in the Coupled Atmosphere-Ocean system: Compensation and Partitioning J. Climate, submitted.
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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. J. Climate , doi: 10.1175/2011JCLI3989.1
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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. J. Climate, 23, 2418-2427, doi10.1175/2009JCLI3466.1.
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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. 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.

Vallis, G. K. and Farneti, R. 2009. Meridional Energy Transport in the Atmosphere-Ocean System. Scaling and Numerical Experiments. Quart. J. Roy. Meteor. Soc., 135, 1643-1660, doi:10.1002/qj.498
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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.)

Vallis, G. K. 2009. Mechanisms of climate variability from years to decades. In press in Stochastic Physics and Climate Modelling, T. Palmer and P. Williams, Eds.
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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. Geosci. Model Dev. Disc. 2, 34-383.
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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 Clim. Dyn., DOI 10.1007/s00382-009-0674-9.
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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. Paleoceanography, 20, PA4024, doi:10.1029/2004PA00111
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. J. Geophys. Res. (Oceans), 93, 13979-13991 (1988).
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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.

Clouds and hydrology

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. Tellus, 36A, 236-250 (1984)
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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. Tellus, 34, 211-227 (1982).
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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.

Fluid Dynamics, including GFD and Turbulence

Turbulence Theory and Simulation

Nikurashin, M, Vallis, G.K. and Adcroft, A. 2013. Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nature Geosciences doi: 10.1038/NGEO1657.
PDF file
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 Ocean Modelling.
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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. J. Fluid Mech., 228, 321-342 (1991).
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. J. Phys. Oceanog., 23, 1346-1362.
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. Physics of Fluids A, 5, 1760-1775.
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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. J. Fluid Mech., 469, 13-48.
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. J. Phys. Oceanogr., 32, 1699-1721.
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. J. Phys. Oceanogr., 31, 554≠571.
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. Quart. J. Roy. Meteor. Soc., 114, 183-204 (1988).
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. J. Atmos. Sci., 66, 837-863.
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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.

Reviews and Synthesis
Vallis, G.K., 1992. Problems and phenomenology in two-dimensional turbulence. In: Non-linear Phenomena in Atmospheric and Oceanic Sciences, eds. G. Carnevale and R. Pierrehumbert. 1-25. Springer-Verlag (refereed chapter in book).
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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: An Introduction to Nonlinear Physics, ed. L. Lam. Springer-Verlag, pp. 308-357 (chapter in book) (1996).
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A longish review of various pathways to turbulence (period-doubling, etc.) and of fully-developed turbulence itself.

Predictability Theory

Vallis, G.K., 1983. On the predictability of quasigeostrophic flow: the effects of beta and baroclinicity. J. Atmos. Sci., 40, 10-27.
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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. Quart. J. Roy. Meteor. Soc., 111, 1039-1047.
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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. Am. Inst. Physics Proceedings, "The Predictability of Fluid Motions." 577-593, eds. G. Holloway and B. West.
An early discussion of the use of entropy, applied to 2D inviscid flow.

Vallis, G. K., 1983. Barotropic and baroclinic predictability in geostrophic turbulence. Am. Inst. Physics Proceedings, "The Predictability of Fluid Motions." 117-133, eds. G. Holloway and B. West.

Waves and instabilities

Smith, K. S. and G. K. Vallis, 1998. Linear wave and instability properties of extended range geostrophic models. J. Atmos. Sci., 56, 1579-1593.
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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. Geophys. Astrophys. Fluid Dyn., 34, 1-38.
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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. Phys. Fluids, 31(9), 2567-2572.

Balance and Intermediate Models

Warn, T, O. Bokhove, T.G. Shepherd, and G.K. Vallis, 1995. Rossby number expansions, slaving, and balance dynamics. Quart. J. Roy. Meteor. Soc., 121, 723-739.
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. J. Phys. Oceanogr. 27, 1133-1152.
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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. Physica D. 98, 647-651.

Vallis, G.K., 1996. Potential vorticity inversion and balanced equations of motion for rotating and stratified flows. Q. J. Roy. Met. Soc. 122, 291-322.
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. J. Atmos. Sci., 50, 2556-2564.
Sundermeyer-Vallis.pdf

Vallis, G.K., 1992. Mechanisms and parameterizations of geostrophic adjustment and a new model for balanced flow. J. Atmos. Sci., 49, 1144-1160.
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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. J. Atmos. Sci., 56, 1579-1593.
PDF file
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.

Fundamental Fluid and Vortex Dynamics

Vallis, G.K., G.F. Carnevale, and W.R. Young, 1989. Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech., 203, 133-152 (1989).
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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. J. Fluid Mech., 213, 549-571 (1990).
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Carnevale, G., M. Briscolini, R. Kloosterziel, and G.K. Vallis. 1997. Three dimensionally perturbed vortex tubes in a rotating flow. J. Fluid Mech., 341:127-163.

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. Phys. Fluids, 31(9), 2562-2567.

Carnevale, G.F., G.K. Vallis, R. Purini, and M. Briscolini, 1998. Propagation of barotropic modons over topography. Geophys. Astrophys. Fluid Dynam., 41, 45-101.

Schonbek, M. and Vallis, G. K. 1999. Energy Decay of Solutions to the Boussinesq, Primitive and Planetary Geostrophic Equations. J. Math. Analysis & Applics. 234, 457-481.
Schonbek-Vallis99.pdf
A brief foray into pure mathematics, proving rigorously certain properties of the primitive and similar equations.

Numerical Methods

Cummins, P. and G.K. Vallis, 1994. Solvers for separable and non-separable elliptic problems in irregular domains. Assoc. Comput. Machinery, Trans. Math. Software. 20, 247-253.

Pares-Sierra, A. and G.K. Vallis, 1989. A fast semi-direct method for the numerical solution of non-separable elliptic equations in irregular domains. J. Comp. Physics, 82, 398-412.
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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. J. Atmos. Sci., 42, 95-99.

Vallis, G.K. and B.-L. Hua, 1988. Eddy viscosity of the anticipated potential vorticity method. J. Atmos. Sci., 45, 617-627.