Quasigeostrophic Dynamics of a Finite-Thickness Tropopause

A model of tropopause dynamics is derived that is of intermediate complexity between the three-dimensional quasigeostrophic model and the surface quasigeostrophic (SQG) model. The model assumes that a sharp transition in stratification occurs over a small but finite tropopause region separating regions of uniform potential vorticity (PV). The model is derived using a matched-asymptotics technique, with the ratio of the thickness of the tropopause region to the typical vertical scale of perturbations outside as a small parameter. It reduces to SQG to leading order in this parameter but takes into account the next-order correction. As a result it remains three-dimensional, although with a PV inversion relation that is greatly simplified compared to the Laplacian inversion of quasigeostrophic theory. The model is applied to examine the linear dynamics of perturbations at the tropopause. Edge waves, described in the SQG approximation, are recovered, and explicit expressions are obtained for the corrections to their frequency and structure that result from the finiteness of the tropopause region. The sensitivity of these corrections to the stratification and shear profiles across the tropopause is investigated. In addition, the evolution of perturbations with near-zero vertically integrated PV is discussed. These perturbations, which are filtered out by the SQG approximation, are represented by a continuous spectrum of singular modes and evolve as sheared disturbances. The decomposition of arbitrary perturbations into edge-wave and continuous-spectrum contributions is discussed.

INVERSION AND SYMMETRY RELATIONS FOR A THREE-DIMENSIONAL SOLVABLE MODEL

The solvable sl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. We discuss the spatial symmetry properties of the model and show that its partition function per site obeys the inversion relation. The connection with the free boson model is also discussed.

ALGEBRAIC VARIETIES FOR THE CHIRAL POTTS MODEL

We describe the symmetries of the chiral checkerboard Potts model (duality, inversion relation, …) and write down the algebraic variety corresponding to the integrable case advocated by Baxter, Perk, Au-Yang. We examine some of its subvarieties, in different limits and for various lattices, with a special emphasis on q=3. This yields for q=3, a new algebraic variety where the standard scalar checkerboard Potts model is solvable. By a comparative analysis of the parametrization of the integrable four-state chiral Potts model and the one of the symmetric Ashkin-Teller model, we bring to light algebraic subvarieties for the q-state chiral Potts model which are invariant under the symmetries of the model. We recover in this manner the Fateev-Zamolodchikov points.

A STEP-BY-STEP APPROACH TO INTEGRABILITY

We try to elucidate the role played by different symmetries in simple models of statistical mechanics. Starting with obvious symmetries for the partition function and combining them with duality relations we obtain a set of constraints on the possible algebraic varieties relevant for the integrable manifolds and the phase diagram of the lattice model. When imposing in addition the inversion relation special polynomials are obtained, which are close and sometimes identical to the set of equations defining the parameter subspace of the integrable models. Our procedure is detailed on the q-state chiral Potts models on a square lattice, in particular for q=3 and 4.