We present a set of equations describing the nonlinear dynamics of flows
constrained by environmental rotation and stratification (Rossby numbers Ro∈[0.1,0.5] and Burger numbers of order unity). The fluid is assumed
incompressible, adiabatic, inviscid and in hydrostatic balance. This set of
equations is derived from the Navier Stokes equations (with the above
properties), using a Rossby number expansion with second order truncation.
The resulting model has the following properties: 1) it can represent motions
with moderate Rossby numbers and a Burger number of order unity; 2) it
filters inertia-gravity waves by assuming that the divergence of horizontal
velocity remains small; 3) it is written in terms of a single function of space
and time (pressure, generalized streamfunction or Bernoulli function); 4) it
conserves total (Ertel) vorticity in a Lagrangian form, and its quadratic norm
(potential enstrophy) at the model order in Rossby number; 5) it also
conserves total energy at the same order if the work of pressure forces
vanishes when integrated over the fluid domain. The layerwise version of the
model is finally presented, written in terms of pressure. Integral properties
(energy, enstrophy) are conserved by these layerwise equations. The model
equations agree with the generalized geostrophy equations in the appropriate
parameter regime. Application to vortex dynamics are mentioned.
McIntyre, M.E. (2001) Balance, Potential-Vorticity Inversion, Lighthill Radiation and the Slow Quasimanifold. IUTAM Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics, Limerick, 2-7 July 2000, 45-68.
McWilliams, J.C. and Gent, P.R. (1980) Intermediate Models of Planetary Circulations in the Atmosphere and Oceans. Journal of the Atmospheric Sciences, 37, 1657-1678. https://doi.org/10.1175/1520-0469(1980)037<1657:IMOPCI>2.0.CO;2
Cushman-Roisin, B. and Tang, B. (1990) Geostrophic Turbulence and the Emergence of Eddies beyond the Radius of Deformation. Journal of Physical Oceanography, 20, 97-113.
Cushman-Roisin, B., Sutyrin, G.G. and Tang, B. (1992) Two-Layer Geostrophic Dynamics. Part I. Governing Equations. Journal of Physical Oceanography, 22, 117-127. https://doi.org/10.1175/1520-0485(1992)022<0117:TLGDPI>2.0.CO;2
Norton, N.J., McWilliams, J.C. and Gent, P.R. (1986) A Numerical Model of the Balance Equations in a Periodic Domain and an Example of Balanced Turbulence. Journal of Computational Physics, 67, 439-471.
BenJelloul, M. and Carton, X. (2001) Asymptotic Models and Application to Vortex Dynamics. IUTAM Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics, Limerick, 2-7 July 2000, 105-110.
Sutyrin, G.G. (2004) Agradient Velocity, Vortical Motion and Gravity Waves in a Rotating Shallow-Water Model. Quarterly Journal of the Royal Meteorological Society, 130, 1977-1989. https://doi.org/10.1256/qj.03.54
Remmel, M. and Smith, L. (2009) New Intermediate Models for Rotating Shallow Water and an Investigation of the Preference for Anticyclones. Journal of Fluid Mechanics, 635, 321-359. https://doi.org/10.1017/S0022112009007897
Bertrand, C. and Carton, X. (1994) The Influence of Environmental Parameters on Two-Dimensional Vortex Merger. In: Van Heijst, G.J.F., Ed., Modelling of Oceanic Vortices, North Holland, Amsterdam, 125-134.