Geostrophic adjustment on the mid-latitude β-plane

Abstract. Analytical and numerical solutions of the Linearized Rotating Shallow Water Equations are combined to study the geostrophic adjustment on the mid-latitude β-plane. The adjustment is examined in zonal periodic channels of width Lᵧ = 4R_d (‘narrow’ channel, where R_d is the radius of deformation) and Lᵧ=60R_d (‘wide’ channel) for the particular initial conditions of a resting fluid with a step-like height distribution, η₀. In the one-dimensional case, where η₀ = η₀(y) we find that: (i) β affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when b = cot (Φ₀) R_d /R ≥ 0.5 (where Φ₀ is the channel’s central latitude and R is Earth’s radius); (ii) The energy conversion ratio varies by less than 10 % when b increases from 0 to 1; (iii) In ‘wide’ channels, β affects the waves significantly even for small b (e.g. b = 0.005). (iv) For b = 0.005, harmonic waves approximate the waves in ‘narrow’ channels, and trapped waves approximate the waves in ‘wide’ channels. In the two-dimensional case, where η₀ = η₀(x) we find that: (i) At short times the spatial structure of the steady solution is similar to that on the f-plane, while at long times the steady state drifts westward at the speed of Rossby waves – harmonic Rossby waves in ‘narrow’ channels and trapped Rossby waves in ‘wide’ channels; (ii) In ‘wide’ channels, trapped wave dispersion causes the equatorward segment of the wavefront to move faster than the northern segment; (iii) The energy of Rossby waves on the β-plane approaches that of the steady-state on the f-plane; (iv) The results outlined in (iii) and (iv) of the one-dimensional case also hold in the two-dimensional case.