02 May 2023
 | 02 May 2023
Status: this preprint is open for discussion.

Geostrophic adjustment on the mid-latitude β-plane

Itamar Yacoby, Nathan Paldor, and Hezi Gildor

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 = 4Rd (‘narrow’ channel, where Rd is the radius of deformation) and Lᵧ=60Rd (‘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) Rd /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.

Itamar Yacoby et al.

Status: open (until 27 Jun 2023)

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Itamar Yacoby et al.

Itamar Yacoby et al.


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Short summary
The transition from an arbitrary initial sea surface height to a geostrophic balance in which the velocity is steady was solved last century for constant Coriolis frequency, f(y) where y is the latitude. This study extends the theory to the realistic case in which f(y) is linear with y. We find that the variation of f(y) simply translates the steady geostrophic state westward as low frequency Rossby waves that are harmonic in narrow domains and trapped near the equatorward boundary in wide ones.