Nonlinear dynamics of forced baroclinic critical layers

Wang, Chen; Balmforth, N. J.

doi:10.1017/jfm.2019.829

Cited by 3 publications

(38 citation statements)

References 31 publications

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“…where the amplitudes, identified by the subscript 'I', satisfy a second-order differential equation (Wang & Balmforth 2020). However, the solution remains time dependent near the critical levels.…”

Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

“…, a practical concern is that the outer solution for the forced wave decays exponentially away from the wavemaker (see Wang & Balmforth 2020), which may ensure (5.6). However, the sustained secular growth of Φ, in combination with relatively large values for the other O(1) factors in (5.5) (namely α) implies that one is able to avoid the condition in (5.6) for typical parameter settings.…”

Section: Replicationmentioning

confidence: 99%

“…where ε, related to ε 0 , gauges the strength of the forcing at the baroclinic critical layer, so that |A| = 1 and A encodes a complex phase dictated byp I ( y) (Wang & Balmforth 2020).…”

Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

“…The second critical layer at y = −N is treated similarly, with the spatial symmetry of the problem ensuring that no separate considerations are required (Wang & Balmforth 2020). Note that δ is not prescribed for this derivation (and, indeed, (2.11) is independent of δ when expressed in terms of t and y); this scale becomes selected, however, in the developments of § 3.…”

Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

“…(2.17) Wang & Balmforth (2020) show that when t = O(ε −2/3 ) and δ = ε 2/3 , the mean flow feeds back on to the linear solution (2.13), halting the secular growth. However, in the next section, we will see that a secondary instability can arise for earlier times, and, in particular, when the mean vorticity gradient becomes order one.…”

Section: The Mean-flow Response In a Non-dissipative Critical Layermentioning

confidence: 99%

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Nonlinear dynamics of forced baroclinic critical layers II

Wang

Balmforth

2021

J. Fluid Mech.

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Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

Section: Replicationmentioning

confidence: 99%

“…where ε, related to ε 0 , gauges the strength of the forcing at the baroclinic critical layer, so that |A| = 1 and A encodes a complex phase dictated byp I ( y) (Wang & Balmforth 2020).…”

Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

Section: The Linear Non-dissipative Baroclinic Critical Layermentioning

confidence: 99%

Section: The Mean-flow Response In a Non-dissipative Critical Layermentioning

confidence: 99%

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Nonlinear dynamics of forced baroclinic critical layers II

Wang

Balmforth

2021

J. Fluid Mech.

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Dynamics of a stratified vortex under the complete Coriolis force: two-dimensional three-components evolution

Toghraei

Billant

2022

J. Fluid Mech.

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We study the dynamics of an initially axisymmetric and vertical Lamb–Oseen vortex in a stratified-rotating fluid under the complete Coriolis force on the $f$ plane, i.e. in the presence of a background rotation both along the vertical and horizontal directions. By a combination of direct numerical simulations and asymptotic analyses for small horizontal background rotation, we show that a critical layer appears at the radius where the angular velocity of the vortex is equal to the buoyancy frequency when the Froude number is larger than unity. This critical layer generates a vertical velocity that is invariant along the vertical and which first increases linearly with time and then saturates at an amplitude scaling like $Re^{1/3}/\widetilde{Ro}$ , where $Re$ is the Reynolds number and $\widetilde {Ro}$ the non-traditional Rossby number based on the horizontal component of the background rotation. In turn, a quasi-axisymmetric anomaly of vertical vorticity is produced at the critical radius through the non-traditional Coriolis force. Below a critical $\widetilde {Ro}$ depending on $Re$ , the Rayleigh's inflectional criterion is satisfied and a shear instability is subsequently triggered rendering the vertical vorticity fully non-axisymmetric. The decay of the angular velocity is then enhanced until it is everywhere lower than the buoyancy frequency. A theoretical criterion derived from the Rayleigh condition predicts well the instability. It shows that this phenomenon can occur even for a large non-traditional Rossby number $\widetilde {Ro}$ for large $Re$ . Hence, the non-traditional Coriolis force might have much more impact on geophysical vortices than anticipated by considering the order of magnitude of $\widetilde {Ro}$ .

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Baroclinic critical layer in a viscous stratified boundary layer flow on an undulated tilted surface

2023

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The present paper investigates theoretically and experimentally the boundary layer generated by a stably stratified fluid flowing horizontally along a surface tilted in the transverse direction and deformed by sinusoidal undulations with crests perpendicular to the flow direction. In the absence of undulations, a weak transverse velocity proportional to the normal velocity is created such that the flow remains purely horizontal. In the presence of undulations of amplitude $h$ , a stronger transverse flow is generated that exhibits a singular behaviour at the critical altitude where the frequency of the perturbation matches the buoyancy frequency of the fluid. This baroclinic critical layer was previously analysed by Passaggia et al. (J. Fluid Mech., vol. 751, 2014, pp. 663–684) for a boundary layer flow with a small sliding velocity on the surface. Here, the no-slip boundary condition of the experimental flow is applied. For this purpose, we solve the viscous sub-layer to obtain a complete theoretical model for the solution in the critical layer without any adjusting parameter. The theoretical predictions for the transverse velocity are compared with experimental measurements, and a good quantitative agreement is demonstrated. Compared with the sliding case, the no-slip boundary condition on the surface reduces the amplitude of the critical layer solution by a factor $Re^{-1/3}$ , where the Reynolds number Re is defined using the velocity at infinity and the thickness of the boundary layer. As a consequence, the transverse velocity has a maximum in the critical layer of order $h$ , but it still induces a shear rate of order $h\,Re^{1/3}$ .

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Nonlinear dynamics of forced baroclinic critical layers

Cited by 3 publications

References 31 publications

Nonlinear dynamics of forced baroclinic critical layers II

Nonlinear dynamics of forced baroclinic critical layers II

Dynamics of a stratified vortex under the complete Coriolis force: two-dimensional three-components evolution

Baroclinic critical layer in a viscous stratified boundary layer flow on an undulated tilted surface

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