Geoscientific Model Development Discussions www.geosci-model-dev-discuss.net 1991-9611 1991-962X 1 1 2008 10.5194/gmdd-1-187-2008 http://www.geosci-model-dev-discuss.net/1/187/2008/ http://www.geosci-model-dev-discuss.net/1/187/2008/gmdd-1-187-2008.html http://www.geosci-model-dev-discuss.net/1/187/2008/gmdd-1-187-2008.pdf 187 241 2008-09-05 QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments P. D. Williams T. W. N. Haine P. L. Read S. R. Lewis Y. H. Yamazaki National Centre for Atmospheric Science, Department of Meteorology, University of Reading, Reading, UK Department of Earth and Planetary Sciences, Johns Hopkins University, MD, USA Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford, UK Department of Physics and Astronomy, Open University, Milton Keynes, UK QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments. Appleby, J. C.: Comparative theoretical and experimental studies of baroclinic waves in a two-layer system, Ph.D. thesis, University of Leeds, 1982. Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow, J Comp Phys., 1, 119–143, 1966. Boas, M. L.: Mathematical methods in the physical sciences, Wiley, second edn., 1983. Brugge, R., Nurser, A. J. G., and Marshall, J. C.: A quasi-geostrophic ocean model: some introductory notes, Tech. rep., Blackett Laboratory, Imperial College, London, 1987. Carrigan, C. R.: Instability of a two-layer baroclinic flow in a channel, Tellus, 30, 468–471, 1978. Cattaneo, F. and Hart, J. E.: Multiple states for quasi-geostrophic channel flows, Geophys Astrophys Fluid, 54, 1–33, 1990. Charney, J. G.: The use of the primitive equations of motion in numerical prediction, Tellus, 7, 22–26, 1955. Charney, J. G., Fj\ortoft, R., and von Neumann, J.: Numerical integration of the barotropic vorticity equation, Tellus, 2, 237–254, 1950. Daley, R.: A non-iterative procedure for the time integration of the balance equations, Mon. Weather Rev., 110, 1821–1830, 1982. Davey, M. K.: Recycling flow over bottom topography in a rotating annulus, J Fluid~Mech., 87, 497–520, 1978. Douglas, J. F. and Gasiorek, J. M. K.: Fluid Mechanics, Pearson Higher Education, 911 pp., 2000. Gill, A E.: Atmosphere-Ocean Dynamics, Academic Press, 662 pp., 1982. Haltiner, G. J. and Williams, R. T.: Numerical prediction and dynamic meteorology, Wiley, 2nd edn., 496 pp., 1980. Hide, R.: Experiments with rotating fluids, Q. J. Roy Meteor. Soc. 103, 1–28, 1977. Hide, R., Mason, P. J., and Plumb, R. A.: Thermal convection in a rotating fluid subject to a horizontal temperature gradient: spatial and temporal characteristics of fully developed baroclinic waves, J Atmos Sci., 34, 930–950, 1977. Hignett, P., White, A. A., Carter, R. D., Jackson, W. D. N., and Small, R. M.: A comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus, Q J Roy Meteorol Soc., 111, 131–154, 1985. King, J. C.: Instabilities and nonlinear wave interactions in a two-layer rotating fluid, Ph.D. thesis, University of Leeds, 1979. Kwon, H. J. and Mak, M.: On the equilibration in nonlinear barotropic instability, J Atmos Sci., 45, 294–308, 1988. Lewis, S. R.: A quasi-geostrophic numerical model of a rotating internally heated fluid, Geophys Astrophys Fluid, 65, 31–55, 1992. Lovegrove, A. F.: Bifurcations and instabilities in rotating two-layer fluids, Ph.D. thesis, Oxford University, 1997. Lynch, P.: The slow equations, Q. J. Roy Meteor. Soc. 115, 201–219, 1989. McIntyre, M. E.: Convection and baroclinic instability in rotating fluids, Ph.D. thesis, Cambridge University, 1967. McIntyre, M. E. and Norton, W A.: Potential vorticity inversion on a hemisphere, J Atmos Sci., 57, 1214–1235, 2000. McWilliams, J. C. and Gent, P. R.: Intermediate models of planetary circulations in the atmosphere and ocean, J Atmos Sci., 37, 1657–1678, 1980. Mesinger, F. and Arakawa, A.: Numerical methods used in atmospheric models, Global Atmospheric Research Programme Publications Series No 17, 1976. Moura, A. D.: The eigensolutions of the linearized balance equations over a sphere, J Atmos Sci., 33, 877–907, 1976. Mundt, M. D., Brummell, N. H., and Hart, J. E.: Linear and nonlinear baroclinic instability with rigid sidewalls, J Fluid~Mech., 291, 109–138, 1995. Pedlosky, J.: The stability of currents in the atmosphere and the ocean: Part I, J Atmos Sci., 21, 201–219, 1964. Pedlosky, J.: Finite-amplitude baroclinic waves, J Atmos Sci., 27, 15–30, 1970. Pedlosky, J.: Finite-amplitude baroclinic waves with small dissipation, J Atmos Sci., 28, 587–597, 1971. Pedlosky, J.: Limit cycles and unstable baroclinic waves, J Atmos Sci., 29, 53–63, 1972. Pedlosky, J.: Geophysical Fluid Dynamics, Springer-Verlag, 710 pp., 1987. Phillips, N. A.: Energy Transformations and Meridional Circulations associated with simple Baroclinic Waves in a two-level, Quasi-geostrophic Model, Tellus, 3, 273–286, 1954. Phillips, N. A.: The general circulation of the atmosphere: a numerical experiment, Q. J. Roy Meteor. Soc., 82, 123–164, 1956. Phillips, N. A.: Geostrophic motion, Rev Geophys., 1, 123–176, 1963. Read, P. L., Collins, M., Früh, W.-G., Lewis, S L., and Lovegrove, A F.: Wave interactions and baroclinic chaos: a paradigm for long timescale variability in planetary atmospheres, Chaos, Solitons Fractals, 9, 231–249, 1998. Reynolds, O.: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Phil Trans R Soc London, 174, 935–982, 1883. Robert, A. J.: The integration of a low order spectral form of the primitive meteorological equations, J Meteor Soc Japan, 44, 237–245, 1966. Salmon, R. and Talley, L. D.: Generalizations of Arakawa's Jacobian, J Comp Phys., 83, 247–259, 1989. Smith, R. K.: On Limit Cycles and Vacillating Baroclinic Waves, J Atmos Sci., 31, 2008–2011, 1974. Smith, R. K.: On a theory of amplitude vacillation in baroclinic waves, J Fluid~Mech., 79, 289–306, 1977. Smith, R. K. and Pedlosky, J.: A note on a theory of vacillating baroclinic waves, J Atmos Sci., 32, 2027, 1975. Stephen, A. V.: POD methods in baroclinic flows, Ph.D. thesis, University of Oxford, 1998. Vettin, F.: Experimentelle Darstellung von Luftbewegungen unter dem Einfluss von Temperatur-Unterschieden und Rotations-Impulsen, Meteor. Z., 1, 227–230, 1884. White, A. A.: Documentation of the finite difference schemes used by the Met~O 21 2D Navier-Stokes model, Tech. rep., Geophysical Fluid Dynamics Laboratory, UK Meteorological Office, internal Report Met O 21 IR86/3, 1986. Williams, G. P.: Planetary Circulations: 2. The Jovian Quasi-Geostrophic Regime, J Atmos Sci., 36, 932–968, 1979. Williams, P. D.: Nonlinear interactions of fast and slow modes in rotating, stratified fluid flows, Ph.D. thesis, Oxford University, 2003. Williams, P. D., Read, P. L., and Haine, T. W. N.: Spontaneous generation and impact of inertia-gravity waves in a stratified, two-layer shear flow, Geophys Res Lett., 30, 2255, 2003. Williams, P. D., Haine, T. W. N., and Read, P. L.: Stochastic resonance in a nonlinear model of a rotating, stratified shear flow, with a simple stochastic inertia-gravity wave parameterization, Nonlin. Processes Geophys., 11, 127–135, 2004. Williams, P. D., Haine, T. W. N., and Read, P. L.: On the generation mechanisms of short-scale unbalanced modes in rotating two-layer flows with vertical shear, J Fluid~Mech., 528, 1–22, 2005. Yoshida, A. and Hart, J. E.: A numerical study of baroclinic chaos, Geophys Astrophys Fluid, 37, 1–56, 1986.