On the hydrostatic approximation in rotating stratified flow
<p>Hydrostatic models were and still are the workhorses for realistic simulations of ocean dynamics, especially for climate applications. Introducing a Fourier space projection method and using the Heisenberg–Gabor limit, a formalism is developed to systematically evaluate the role of flatnes...
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| Format: | Article |
| Language: | English |
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Copernicus Publications
2025-07-01
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| Series: | Nonlinear Processes in Geophysics |
| Online Access: | https://npg.copernicus.org/articles/32/261/2025/npg-32-261-2025.pdf |
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| Summary: | <p>Hydrostatic models were and still are the workhorses for realistic simulations of ocean dynamics, especially for climate applications. Introducing a Fourier space projection method and using the Heisenberg–Gabor limit, a formalism is developed to systematically evaluate the role of flatness, stratification, rotation and friction for the fidelity of the hydrostatic approximation. The hydrostatic approximation is formally first order in <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi mathvariant="italic">γ</mi><mo>=</mo><mi>H</mi><mo>/</mo><mi>L</mi></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="42pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="5695540f6a11504171dfc14e10380a1d"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="npg-32-261-2025-ie00001.svg" width="42pt" height="14pt" src="npg-32-261-2025-ie00001.png"/></svg:svg></span></span>, where <span class="inline-formula"><i>H</i></span> is the vertical and <span class="inline-formula"><i>L</i></span> the horizontal scale of the phenomenon considered. For linear (low-amplitude) and unforced stratified rotating flow, the dynamics can be separated into balanced flow and wave motion. It is shown that for the linear balanced motion the hydrostatic approximation is exact and for wave motion it is second order, obtaining the leading prefactors. The fidelity of the hydrostatic approximation therefore also relies on the ratio of the amplitude of wave motion to balanced motion. This ratio adds considerably to the quality of the hydrostatic approximation for larger-scale flows in the atmosphere and the ocean.</p>
<p>Imposing the divergenceless condition is a linear projection of the dynamical variables into the subspace of divergenceless vector fields, for both the Navier–Stokes and the hydrostatic formalism. Both projections are local in Fourier space. The former is well known, while the latter, developed here, asks for an extension of the dynamical space to four dimensions. The projection is followed by a time-evolution operator, which differs in the wave frequencies only. Combining the projection and the linear evolution operators in both formalisms leads to the linear projection-evolution operator.</p>
<p>Calculating the difference of the two projection-evolution operators, the expression of the error, scaling and prefactors done by the hydrostatic approximation is obtained. Analyzing the eigenspace of the projector-evolution operators, it is shown that for rotating buoyant vortical flow, the hydrostatic approximation is of third order for buoyant forcing, second order for horizontal and first order for vertical dynamical forcing. Balanced dynamics is in the kernel of the linear projection-evolution operator, and conservation of potential vorticity is expressed by the kernel of its adjoint.</p>
<p>Using the Heisenberg–Gabor limit, it is shown that for large-scale ocean dynamics, the difference of the dynamics of the projection-evolution operator between the two formalisms is insignificant. It is shown that the hydrostatic approximation is appropriate for realistic ocean simulations with vertical viscosities larger than <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M4" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>≈</mo><msup><mn mathvariant="normal">10</mn><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="35pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="81bd1de4a2b1867f0c066a1ce421c440"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="npg-32-261-2025-ie00002.svg" width="35pt" height="13pt" src="npg-32-261-2025-ie00002.png"/></svg:svg></span></span> m<span class="inline-formula"><sup>2</sup></span> s<span class="inline-formula"><sup>−1</sup></span>. A special emphasis is on unveiling the physical interpretation of the calculations.</p> |
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| ISSN: | 1023-5809 1607-7946 |