Geostrophic adjustment on the midlatitude <i>β</i> plane
<p>Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude <span class="inline-formula"><i>β</i></span> plane. The adjustment is examined in zonal periodic channe...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Copernicus Publications
2023-08-01
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| Series: | Ocean Science |
| Online Access: | https://os.copernicus.org/articles/19/1163/2023/os-19-1163-2023.pdf |
| Summary: | <p>Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude <span class="inline-formula"><i>β</i></span> plane. The adjustment is examined in zonal periodic channels of width <span class="inline-formula"><i>L</i><sub><i>y</i></sub>=4<i>R</i><sub>d</sub></span> (narrow channel, where <span class="inline-formula"><i>R</i><sub>d</sub></span> is the radius of deformation) and <span class="inline-formula"><i>L</i><sub><i>y</i></sub>=60<i>R</i><sub>d</sub></span> (wide channel) for the particular initial conditions of a resting fluid with a step-like height distribution, <span class="inline-formula"><i>η</i><sub>0</sub></span>. In the one-dimensional case, where <span class="inline-formula"><i>η</i><sub>0</sub>=<i>η</i><sub>0</sub>(<i>y</i>)</span>, we find that (i) <span class="inline-formula"><i>β</i></span> affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M10" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>b</mi><mo>=</mo><mi mathvariant="normal">cot</mi><mo>(</mo><msub><mi mathvariant="italic">ϕ</mi><mn mathvariant="normal">0</mn></msub><mo>)</mo><mstyle displaystyle="false"><mfrac style="text"><mrow><msub><mi>R</mi><mi mathvariant="normal">d</mi></msub></mrow><mi>R</mi></mfrac></mstyle><mo>≥</mo><mn mathvariant="normal">0.5</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="96pt" height="18pt" class="svg-formula" dspmath="mathimg" md5hash="5213142684fbb2a8704964ae01e88b82"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="os-19-1163-2023-ie00001.svg" width="96pt" height="18pt" src="os-19-1163-2023-ie00001.png"/></svg:svg></span></span> (where <span class="inline-formula"><i>ϕ</i><sub>0</sub></span> is the channel's central latitude, and <span class="inline-formula"><i>R</i></span> is Earth's radius); (ii) the energy conversion ratio varies by less than <span class="inline-formula">10</span> % when <span class="inline-formula"><i>b</i></span> increases from 0 to 1; (iii) in wide channels, <span class="inline-formula"><i>β</i></span> affects the waves significantly, even for small <span class="inline-formula"><i>b</i></span> (e.g., <span class="inline-formula"><i>b</i>=0.005</span>); and (iv) for <span class="inline-formula"><i>b</i>=0.005</span>, harmonic waves approximate the waves in narrow channels, and trapped waves approximate the waves in wide channels. In the two-dimensional case, where <span class="inline-formula"><i>η</i><sub>0</sub>=<i>η</i><sub>0</sub>(<i>x</i>)</span>, we find that (i) at short times the spatial structure of the steady solution is similar to that on the <span class="inline-formula"><i>f</i></span> 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 <span class="inline-formula"><i>β</i></span> plane approaches that of the steady state on the <span class="inline-formula"><i>f</i></span> plane; and (iv) the results outlined in (iii) and (iv) of the one-dimensional case also hold in the two-dimensional case.</p> |
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| ISSN: | 1812-0784 1812-0792 |