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Parametric subharmonic instability of inertial shear at ocean fronts
- James P. Hilditch, Leif N. Thomas
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- Journal:
- Journal of Fluid Mechanics / Volume 966 / 10 July 2023
- Published online by Cambridge University Press:
- 05 July 2023, A34
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The two-dimensional stability of vertically sheared inertial oscillations at ocean fronts is explored through a linear stability analysis and nonlinear simulations. Baroclinic effects reduce the minimum frequency of inertia-gravity waves to an extent determined by the balanced Richardson number ${{Ri}}$ of the front. Below a critical value of ${{Ri}}$, which depends on the strength of the inertial shear, the inertial oscillations become unstable to parametric subharmonic instability (PSI) resulting in growing perturbations that oscillate at half the inertial frequency $f$. Since the critical value is always greater than 1, PSI can occur at fronts stable to symmetric instability. Although modest in weak inertial shear, growth rates exceeding $f/2$ can be achieved for inertial shear greater than or equal to the thermal wind shear. Our formulation allows for non-hydrostatic perturbations and can be applied to initially unstratified geostrophic adjustment problems. We find that PSI will almost totally damp the transient oscillations that arise during geostrophic adjustment. The perturbations gain energy at the expense of the inertial oscillations through ageostrophic shear production. The perturbations then themselves become unstable to secondary Kelvin–Helmholtz instabilities creating a pathway by which the inertial oscillations can be dissipated rapidly. In contrast to symmetric and baroclinic instabilities that draw on a front's kinetic or potential energy, PSI acts to increase the energy stored in the balanced front as the convergence and divergence of the eddy-momentum fluxes set up a secondary circulation in the sense to stand up the front.
Critical and near-critical reflections of near-inertial waves off the sea surface at ocean fronts
- Nicolas Grisouard, Leif N. Thomas
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- Journal:
- Journal of Fluid Mechanics / Volume 765 / 25 February 2015
- Published online by Cambridge University Press:
- 20 January 2015, pp. 273-302
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In a balanced oceanic front, the possible directions of the group velocity vector for internal waves depart from the classic Saint Andrew’s cross as a consequence of sloping isopycnals and the associated thermal wind shear. However, for waves oscillating at the Coriolis frequency $f$, one of these directions remains horizontal, while the other direction allows for vertical propagation of energy. This implies the existence of critical reflections from the ocean surface, after which wave energy, having propagated from below, cannot propagate back down. This is similar to the reflection of internal waves, propagating in a quiescent medium, from a bottom that runs parallel to the group velocity vector. We first illustrate this phenomenon with a series of linear Boussinesq numerical experiments on waves with various frequencies, ${\it\omega}$, exploring critical (${\it\omega}=f$), forward (${\it\omega}>f$), and backward (${\it\omega}<f$) reflections. We then conduct the nonlinear equivalents of these simulations. In agreement with the classical case, backward reflection inhibits triadic resonances and does not exhibit prominent nonlinear effects, while forward reflection shows strong generation of harmonics that radiate energy away from the surface. Surprisingly though, critical reflections are associated with oscillatory motions that extend down from the surface. These motions are not freely propagating waves but instead take the form of a cluster of non-resonant triads which decays with depth through friction.
Damping of inertial motions by parametric subharmonic instability in baroclinic currents
- Leif N. Thomas, John R. Taylor
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- Journal:
- Journal of Fluid Mechanics / Volume 743 / 25 March 2014
- Published online by Cambridge University Press:
- 04 March 2014, pp. 280-294
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A new damping mechanism for vertically-sheared inertial motions is described involving an inertia–gravity wave that oscillates at half the inertial frequency, $f$, and that grows at the expense of inertial shear. This parametric subharmonic instability forms in baroclinic, geostrophic currents where thermal wind shear, by reducing the potential vorticity of the fluid, allows inertia–gravity waves with frequencies less than $f$. A stability analysis and numerical simulations are used to study the instability criterion, energetics, and finite-amplitude behaviour of the instability. For a flow with uniform shear and stratification, parametric subharmonic instability develops when the Richardson number of the geostrophic current nears $Ri_{PSI}=4/3+\gamma \cos \phi $, where $\gamma $ is the ratio of the inertial to thermal wind shear magnitude and $\phi $ is the angle between the inertial and thermal wind shears at the initial time. Inertial shear enters the instability criterion because it can also modify the potential vorticity and hence the minimum frequency of inertia–gravity waves. When this criterion is met, inertia–gravity waves with a frequency $f/2$ and with flow parallel to isopycnals amplify, extracting kinetic energy from the inertial shear through shear production. The solutions of the numerical simulations are consistent with these predictions and additionally show that finite-amplitude parametric subharmonic instability both damps inertial shear and is itself damped by secondary shear instabilities. In this way, parametric subharmonic instability opens a pathway to turbulence where kinetic energy in inertial shear is transferred to small scales and dissipated.
Nonlinear stratified spindown over a slope
- Jessica A. Benthuysen, Leif N. Thomas
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- Journal:
- Journal of Fluid Mechanics / Volume 726 / 10 July 2013
- Published online by Cambridge University Press:
- 05 June 2013, pp. 371-403
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Nonlinear stratified spindown of an along-isobath current over an insulated slope is shown to develop asymmetries in the vertical circulation and vertical relative vorticity field. During spindown, cyclonic vorticity is weakened to a greater extent than anticyclonic vorticity near the boundary because of buoyancy advection. As a consequence, Ekman pumping is weakened over Ekman suction. Momentum advection can weaken Ekman pumping and strengthen Ekman suction. Time-dependent feedback between the geostrophic flow and the frictional secondary circulation induces asymmetry in cyclonic and anticyclonic vorticity away from the boundary. Buoyancy advection over a slope can modify the secondary circulation such that anticyclonic vorticity decays faster than cyclonic vorticity outside the boundary layer. In contrast, momentum advection can cause cyclonic vorticity to spin down faster than anticyclonic vorticity. A scaling and analytical solutions are derived for when buoyancy advection over a slope can have a more significant impact than momentum advection on these asymmetries. In order to test this scaling and analytical solutions, numerical experiments are run in which both buoyancy and momentum advection are active. These solutions are contrasted with homogeneous or stratified spindown over a flat bottom, in which momentum advection controls the asymmetries. These results are applied to ocean currents over continental shelves and slopes.
On the effects of frontogenetic strain on symmetric instability and inertia–gravity waves
- Leif N. Thomas
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- Journal:
- Journal of Fluid Mechanics / Volume 711 / 25 November 2012
- Published online by Cambridge University Press:
- 20 September 2012, pp. 620-640
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The dynamics of symmetric instability and two-dimensional inertia–gravity waves in a baroclinic geostrophic flow undergoing frontogenesis is analysed. A frontogenetic strain associated with a balanced deformation field drives an ageostrophic circulation and temporal variations in the basic state that significantly affect the properties of perturbations to the background flow. For stable stratification, perturbations to the basic state result in symmetric instability or inertia–gravity waves, depending on the sign of the Ertel potential vorticity and the magnitude of the Richardson number of the geostrophic flow. The kinetic energy (KE) of both types of motion is suppressed by frontogenetic strain due to the vertical shear in the ageostrophic circulation. This is because the perturbation streamlines tilt with the ageostrophic shear causing the disturbances to lose KE via shear production. The effect can completely dampen symmetric instability for sufficiently strong strain even though the source of KE for the instability (the vertical shear in the geostrophic flow) increases with time. Inertia–gravity waves in a baroclinic flow undergoing frontogenesis simultaneously lose KE and extract KE from the deformation field as they decay. This is because the horizontal velocity of the waves becomes rectilinear, resulting in a Reynolds stress that draws energy from the balanced flow. The process is most effective for waves of low frequency and for a geostrophic flow with low Richardson number. However, even in a background flow that is initially strongly stratified, frontogenesis leads to an exponentially fast reduction in the Richardson number, facilitating a rapid energy extraction by the waves. The KE transferred from the deformation field is ultimately lost to the unbalanced ageostrophic circulation through shear production, hence the inertia–gravity waves play a catalytic role in loss of balance. Given the large amount of KE in low-frequency inertia–gravity waves and the ubiquitous combination of strain and baroclinic geostrophic currents in the ocean, it is estimated that this mechanism could play a significant role in the removal of KE from both the internal wave and mesoscale eddy fields.
Contributors
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- By Basem Abdelmalak, Joseph Abdelmalak, Alaa A. Abd-Elsayed, David L. Adams, Eric E. Adelman, Maged Argalious, Endrit Bala, Gene H. Barnett, Sheron Beltran, Andrew Bielaczyc, William Bingaman, James M. Blum, Alina Bodas, Vera Borzova, Richard Bowers, Adam Brown, Chad M. Brummett, Alexandra S. Bullough, James F. Burke, Juan P. Cata, Neeraj Chaudhary, Michael J. Claybon, Miguel Cruz, Milind Deogaonkar, Vikram Dhawan, Thomas Didier, D. John Doyle, Zeyd Ebrahim, Hesham Elsharkawy, Wael Ali Sakr Esa, Ehab Farag, Ryen D. Fons, Joseph J. Gemmete, Matt Giles, Phil Gillen, Goodarz Golmirzaie, Marcos Gomes, Lisa Grilly, Maged Guirguis, David W. Healy, Heather Hervey-Jumper, Shawn L. Hervey-Jumper, Paul E. Hilliard, Samuel A. Irefin, George K. Istaphanous, Teresa L. Jacobs, Ellen Janke, Greta Jo, James W. Jones, Rami Karroum, Allen Keebler, Stephen J. Kimatian, Colleen G. Koch, Robert Scott Kriss, Andrea Kurz, Jia Lin, Michael D. Maile, Negmeldeen F. Mamoun, Mariel Manlapaz, Edward Manno, Donn Marciniak, Piyush Mathur, Nicholas F. Marko, Matthew Martin, George A. Mashour, Marco Maurtua, Scott T. McCardle, Julie McClelland, Uma Menon, Paul S. Moor, Laurel E. Moore, Ruairi Moulding, Dileep R. Nair, Todd Nelson, Julie Niezgoda, Edward Noguera, Jerome O’Hara, Aditya S. Pandey, Mauricio Perilla, Paul Picton, Marc J. Popovich, J. Javier Provencio, Venkatakrishna Rajajee, Mohit Rastogi, Stacy Ritzman, Lauryn R. Rochlen, Leif Saager, Vivek Sabharwal, Oren Sagher, Kenneth Saliba, Milad Sharifpour, Lesli E. Skolarus, Paul Smythe, Wolf H. Stapelfeldt, William R. Stetler, Peter Stiles, Vijay Tarnal, Khoi D. Than, B. Gregory Thompson, Alparslan Turan, Christopher R. Turner, Justin Upp, Sumeet Vadera, Jennifer Vance, Anthony C. Wang, Robert J. Weil, Marnie B. Welch, Karen K. Wilkins, Erin S. Williams, George N. Youssef, Asma Zakaria, Sherif S. Zaky, Andrew Zura
- Edited by George A. Mashour, Ehab Farag
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- Book:
- Case Studies in Neuroanesthesia and Neurocritical Care
- Published online:
- 03 May 2011
- Print publication:
- 03 February 2011, pp x-xvi
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Nonlinear stratified spin-up
- LEIF N. THOMAS, PETER B. RHINES
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- Journal:
- Journal of Fluid Mechanics / Volume 473 / 10 December 2002
- Published online by Cambridge University Press:
- 13 December 2002, pp. 211-244
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Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.