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1975
Druyan, LM, Somerville RCJ, Quirk WJ.  1975.  Extended-Range Forecasts with GISS Model of Global Atmosphere. Monthly Weather Review. 103:779-795.   10.1175/1520-0493(1975)103<0779:erfwtg>2.0.co;2   AbstractWebsite
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Galchen, T, Somerville RCJ.  1975.  Numerical-Solution of Navier-Stokes Equations with Topography. Journal of Computational Physics. 17:276-310.   10.1016/0021-9991(75)90054-6   AbstractWebsite

A finite difference scheme for solving the equations of fluid motion in a generalized coordinate system has been constructed. The scheme conserves mass and all the first integral moments of the motion. The scheme also advectively “almost conserves” second moments, in that the magnitude of implicit numerical smoothing is typically about an order smaller than explicit viscosity and diffusion. Calculations with the model support the theoretical conjecture that the difference scheme is stable whenever the analogous Cartesian scheme is stable. The scheme has been used to calculate dry atmospheric convection due to differential heating between top and bottom of mountainous terrain. The general small-scale characteristics of mountain up-slope winds have been simulated. In addition, the results have demonstrated the crucial role played by the eddy diffusivities and the environmental stability, in determining both the quantitative and the qualitative features of the circulation.

Galchen, T, Somerville RCJ.  1975.  On the Use of a Coordinate Transformation for Solution of Navier-Stokes Equations. Journal of Computational Physics. 17:209-228.   10.1016/0021-9991(75)90037-6   AbstractWebsite

The equations of fluid motion have been formulated in a generalized noncartesian, non-orthogonal coordinate system. A particular coordinate transformation, which transforms a domain with an irregular lower boundary into a cube, has been constructed. The transformed system, unlike the original one, has flat boundaries and homogeneous boundary conditions. Where the topography is flat, the original and transformed systems are identical, and extra terms do not appear. A finite difference scheme for solving the transformed equations has been constructed and will be described in a subsequent issue of this journal.

1974
Stone, PH, Quirk WJ, Somervil.Rc.  1974.  Effect of Small-Scale Vertical Mixing of Horizontal Momentum in a General Circulation Model. Monthly Weather Review. 102:765-771.   10.1175/1520-0493(1974)102<0765:teossv>2.0.co;2   AbstractWebsite

Several experiments are described in which the sub-grid-scale vertical eddy viscosity in the GISS global general circulation model was varied. The results show that large viscosities suppress large-scale eddies in middle and high latitudes, but enhance the circulation in the tropical Hadley cell and increase the extent of the tropical easterlies. Comparison with observations shows that the GISS model requires eddy viscosities 1 m2/s or less to give realistic results for middle and high latitudes, and eddy viscosities 100 m2/s to give realistic results for low latitudes. A plausible mechanism for the implied increase in small-scale mixing in low latitudes is cumulus convection.

Somervil.Rc, Stone PH, Halem M, Hansen JE, Hogan JS, Druyan LM, Russell G, Lacis AA, Quirk WJ, Tenenbau.J.  1974.  GISS Model of Global Atmosphere. Journal of the Atmospheric Sciences. 31:84-117.   10.1175/1520-0469(1974)031<0084:tgmotg>2.0.co;2   AbstractWebsite

A model description and numerical results are presented for a global atmospheric circulation model developed at the Goddard Institute for Space Studies (GISS). The model version described is a 9-level primitive-equation model in sigma coordinates. It includes a realistic distribution of continents, oceans and topography. Detailed calculations of energy transfer by solar and terrestrial radiation make use of cloud and water vapor fields calculated by the model. The model hydrologic cycle includes two precipitation mechanisms: large-scale supersaturation and a parameterization of subgrid-scale cumulus convection.Results are presented both from a comparison of the 13th to the 43rd days (January) of one integration with climatological statistics, and from five short-range forecasting experiments. In the extended integration, the near-equilibrium January-mean model atmosphere exhibits an energy cycle in good agreement with observational estimates, together with generally realistic zonal mean fields of winds, temperature, humidity, transports, diabatic heating, evaporation, precipitation, and cloud cover. In the five forecasting experiments, after 48 hr, the average rms error in temperature is 3.9K, and the average rms error in 500-mb height is 62 m. The model is successful in simulating the 2-day evolution of the major features of the observed sea level pressure and 500-mb height fields in a region surrounding North America.

1973
Somerville, RCJ, Lipps FB.  1973.  A Numerical Study in Three Space Dimensions of Bénard Convection in a Rotating Fluid. Journal of the Atmospheric Sciences. 30:590-596.: American Meteorological Society   10.1175/1520-0469(1973)030<0590:ansits>2.0.co;2   AbstractWebsite

The primitive, nonlinear, Boussinesq equations of motion, continuity and thermodynamic energy are integrated numerically in three space dimensions and time to study convection driven by unstable vertical density gradients and subject to Coriolis forces. Parameter values are chosen to permit quantitative comparison with data from laboratory experiments for rotating Bénard convection in water. The model realistically simulates the structure of the convection cells, their horizontal scale, and the mean vertical heat transport. The experimentally observed phenomenon of a non-monotone dependence of heat transport on rotation rate is reproduced and shown to be a consequence of the rotational constraint on the wavelength of the cells.

1972
Willis, GE, Deardorff JW, Somerville RCJ.  1972.  Roll-diameter dependence in Rayleigh convection and its effect upon the heat flux. Journal of Fluid Mechanics. 54:351-367.   10.1017/S0022112072000722   Abstract

The average roll diameter in Rayleigh convection for 2000 < R < 31000, where R is the Rayleigh number, has been measured from photographs of three convecting fluids: air, water and a silicone oil with a Prandtl number σ of 450. For air the average dimensionless roll diameter was found to depend uniquely upon R and to increase especially rapidly in the range 2000 < R < 8000. The fluids of larger σ exhibited strong hysteresis but also had average roll diameters tending to increase with R. The increase in average roll diameter with R tended to decrease with σ. Through use of two-dimensional numerical integrations for the case of air it was found that the increase in average roll diameter with R provides an explanation for the usual discrepancy in heat flux observed between experiment and two-dimensional numerical calculations which prescribe a fixed wavelength.

1971
Somerville, RCJ.  1971.  Bénard convection in a rotating fluid. Geophysical Fluid Dynamics. 2:247-262.: Taylor & Francis   10.1080/03091927108236061   AbstractWebsite

Abstract The steady nonlinear regime of Bénard convection in a uniformly rotating fluid is treated using a two-dimensional primitive-equation numerical model with rigid boundaries. Quantitative comparisons with laboratory heat transport data for water are made in the parameter ranges for which the experimental flows are approximately two-dimensional and steady. When an experimentally realistic spatial periodicity is imposed upon the numerical solution, the model simulates the experimental determinations of Nusselt number fairly accurately. In particular, it predicts the observed non-monotonic dependence on Taylor number. When spatial periodicities corresponding to those of the linear stability problem are specified, however, the accuracy of the simulation is less and the Taylor number dependence is monotonic.The steady nonlinear regime of Bénard convection in a uniformly rotating fluid is treated using a two-dimensional primitive-equation numerical model with rigid boundaries. Quantitative comparisons with laboratory heat transport data for water are made in the parameter ranges for which the experimental flows are approximately two-dimensional and steady. When an experimentally realistic spatial periodicity is imposed upon the numerical solution, the model simulates the experimental determinations of Nusselt number fairly accurately. In particular, it predicts the observed non-monotonic dependence on Taylor number. When spatial periodicities corresponding to those of the linear stability problem are specified, however, the accuracy of the simulation is less and the Taylor number dependence is monotonic.

Lipps, FB, Somervil.Rc.  1971.  Dynamics of Variable Wavelength in Finite-Amplitude Benard Convection. Physics of Fluids. 14:759-&.   10.1063/1.1693502   AbstractWebsite

The finite‐amplitude Bénard convection problem is investigated by numerical integration of the rigid‐boundary Boussinesq equations in two and three space dimensions. Solutions are obtained for a wide range of Prandtl numbers and at moderate Rayleigh numbers for which the flow is observed to approach a two‐dimensional steady state. Detailed quantitative comparisons are made with experimental data in an effort to explain the observed increase of cell wavelength with Rayleigh number and to determine the effect of changing cell size on the heat transport. The three‐dimensional model shows good evidence of being able to yield realistic values of the cell wavelength, while the two‐dimensional models yield wavelengths that are much too short. These results strongly suggest that the increase in wavelength is determined by a three‐dimensional transient process, while the convection tends to a two‐dimensional steady state. The increase in cell size is shown to be responsible for a substantial part of the discrepancy between previous theoretical‐numerical and experimental determinations of Nusselt number. It also provides a plausible explanation for the experimentally observed dependence of heat transport on Prandtl number.

1967
Somerville, RCJ.  1967.  A Nonlinear Spectral Model of Convection in a Fluid Unevenly Heated from Below. Journal of the Atmospheric Sciences. 24:665-676.: American Meteorological Society   10.1175/1520-0469(1967)024<0665:ansmoc>2.0.co;2   AbstractWebsite

A two-dimensional form of the Boussinesq equations is integrated numerically for the case of a rectangular channel with a temperature gradient maintained along the bottom. The side walls are insulating, the top wall has a constant temperature, and the velocity obeys free boundary conditions on all four walls. The fields of stream function and temperature departure are represented by truncated double Fourier series, and integration of the initial-value problem for the spectral amplitudes results in steady states which agree qualitatively with those of previous experimental and theoretical investigations.Calculations are presented at two levels of truncation (wave numbers 2 and 3) for a wide range of Prandtl numbers and a moderate range of horizontal Rayleigh numbers and top temperatures. For sufficiently large gravitational stability, a single asymmetric convection cell develops. Its intensity and asymmetry increase markedly with increasing horizontal Rayleigh number, decrease with increasing top temperature, and respond very slightly to changes in Prandtl number. As the top temperature is decreased below the temperature of the warm side of the bottom, however, the possibility is indicated that the single cell may be modified by a Bénard-like multi-cellular structure.