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1
Pesnell, W. D. "Brunt-Vaisala frequency and semiconvection." Astrophysical Journal 301 (February 1986): 204. http://dx.doi.org/10.1086/163887.
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Wüst, Sabine, Michael Bittner, Jeng-Hwa Yee, Martin G. Mlynczak, and James M. Russell III. "Variability of the Brunt–Väisälä frequency at the OH* layer height." Atmospheric Measurement Techniques 10, no. 12 (2017): 4895–903. http://dx.doi.org/10.5194/amt-10-4895-2017.
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Abstract. In and near the Alpine region, the most dense subnetwork of identical NDMC (Network for the Detection of Mesospheric Change, https://www.wdc.dlr.de/ndmc/) instruments can be found: five stations are equipped with OH* spectrometers which deliver a time series of mesopause temperature for each cloudless or only partially cloudy night. These measurements are suitable for the derivation of the density of gravity wave potential energy, provided that the Brunt–Väisälä frequency is known. However, OH* spectrometers do not deliver vertically resolved temperature information, which is necessary for the calculation of the Brunt–Väisälä frequency. Co-located measurements or climatological values are needed. We use 14 years of satellite-based temperature data (TIMED-SABER, 2002–2015) to investigate the inter- and intra-annual variability of the Brunt–Väisälä frequency at the OH* layer height between 43.93–48.09° N and 5.71–12.95° E and provide a climatology.
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Dolas, Prakash M., and Karanam Kishore Kumar. "Retrieval of static stability parameter from the radiosonde/rawinsonde ascent rate profiles: a wavelet approach." Annales Geophysicae 27, no. 2 (2009): 547–53. http://dx.doi.org/10.5194/angeo-27-547-2009.
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Abstract. In the present communication a novel method is presented to derive the altitude profile of Brunt-Väisälä period from the ascent rate profile of sounding balloons. The basic premise of the present method is that the oscillations in the ascent rate of the balloon will have the signature of Brunt-Väisälä frequency, which can be retrieved by using sophisticated spectral tools. We employ wavelet transforms to arrive at the Brunt-Väisälä period profile. Comparison of retrieved Brunt-Väisälä periods with the values derived from the temperature data available from the same radiosonde ascent shows good agreement. Retrieving the atmospheric temperature from the height profile of Brunt-Väisälä period is also discussed in the present communication. We have shown that it is possible to estimate the Brunt-Väisälä period and temperature profiles from the rawinsonde ascent rate data alone where temperature sounding is not available.
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Haverkort, J. W., H. J. de Blank, and B. Koren. "The Brunt–Väisälä frequency of rotating tokamak plasmas." Journal of Computational Physics 231, no. 3 (2012): 981–1001. http://dx.doi.org/10.1016/j.jcp.2011.03.016.
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Millard, R. C., W. B. Owens, and N. P. Fofonoff. "On the calculation of the Brunt-Väisäla frequency." Deep Sea Research Part A. Oceanographic Research Papers 37, no. 1 (1990): 167–81. http://dx.doi.org/10.1016/0198-0149(90)90035-t.
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Weber, Jan Erik H., Kai H. Christensen, and Göran Broström. "Stokes Drift in Internal Equatorial Kelvin Waves: Continuous Stratification versus Two-Layer Models." Journal of Physical Oceanography 44, no. 2 (2014): 591–99. http://dx.doi.org/10.1175/jpo-d-13-0135.1.
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Abstract The Stokes drift in long internal equatorial Kelvin waves is investigated theoretically for an inviscid fluid of constant depth. While the Stokes drift in irrotational waves is positive everywhere in the fluid, that is, directed along the phase velocity, this is not always the case for internal Kelvin waves, which possess vorticity. For constant Brunt–Väisälä frequency, the Stokes drift in such waves is sinusoidal in the vertical with a negative value in the middle of the layer for the first baroclinic mode. For a pycnocline that is typical of the equatorial Pacific, this study finds for the first mode that the largest negative Stokes drift velocity occurs near the depth where the Brunt–Väisälä frequency has its maximum. Here, estimated drift values are found to be on the same order of magnitude as those observed in the Pacific Equatorial Undercurrent at the same level. In contrast, a two-layer model with constant density in each layer yields a positive Stokes drift in both layers. This contradicts the fact that, as shown in this paper, the vertically integrated Stokes drift (the Stokes flux) must be zero for arbitrary Brunt–Väisälä frequency.
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Doake, C. S. M., H. F. J. Corr, and A. Jenkins. "Polarization of radio waves transmitted through Antarctic ice shelves." Annals of Glaciology 34 (2002): 165–70. http://dx.doi.org/10.3189/172756402781817572.
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AbstractThe polarization behaviour of radar waves transmitted through two Antarctic ice shelves has been investigated using a step frequency radar with a centre frequency of 300 MHz and a bandwidth of 150 MHz. One site was on Brunt Ice Shelf at a site near Halley station, and 17 sites were on George VI Ice Shelf near the southern ice front. Birefringence in the ice dominated the behaviour on Brunt Ice Shelf, where the anisotropy in the effective permittivity was found to be about 0.14%. On George VI Ice Shelf, a highly anisotropic reflecting surface was the controlling feature, suggesting a fluted ice-shelf base formed by oceanographic currents.
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Grandoni, Giovanni, Maria Cristina Mammarella, and Maurizio Favaron. "CLIMATOLOGY OF THE BRUNT-VÄISÄLÄ FREQUENCY OVER MILAN, ITALY." GEOGRAPHY, ENVIRONMENT, SUSTAINABILITY 3, no. 1 (2010): 16–24. http://dx.doi.org/10.24057/2071-9388-2010-3-1-16-24.
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Grandoni, Giovanni, Maria Cristina Mammarella, and Maurizio Favaron. "CLIMATOLOGY OF THE BRUNT-VÄISÄLÄ FREQUENCY OVER MILAN, ITALY." GEOGRAPHY, ENVIRONMENT, SUSTAINABILITY 3, no. 1 (2010): 16–24. http://dx.doi.org/10.15356/2071-9388_01v03_2010_02.
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Barnes, G., and P. S. Cally. "Frequency Dependent Ray Paths in Local Helioseismology." Publications of the Astronomical Society of Australia 18, no. 3 (2001): 243–51. http://dx.doi.org/10.1071/as01040.
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AbstractThe surface of the Sun is continually oscillating due to sound waves encroaching on it from the interior. Measurements of the surface velocity are used to infer some of the properties of the regions through which the sound waves have propagated. Traditionally, this has been done by using a modal decomposition of the surface disturbances. However, the use of ray descriptions, in the form of acoustic holography or time–distance helioseismology, provides an alternative approach which may reveal more detailed information about the properties of local phenomena such as sunspots and active regions. Fundamental to any such treatment is determining the correct ray paths in a given atmosphere. In the simplest approach, the ray paths are constructed to minimise the travel time between two points (Fermat's principle). However, such an approach is only valid in the high frequency limit, ω » ωc, N, where ωc is the acoustic cut-off and N the Brunt-VÄisÄlÄ frequency. Although ωc is often included in time– distance calculations, and N occasionally, the same is not true of acoustic holography. We argue that this raises concerns about image sharpness. As illustrations, representative ray paths are integrated in a realistic solar model to show that the Fermat approximation performs poorly for frequencies of helioseismic interest. We also briefly discuss the importance of the Brunt-VÄisÄlÄ frequency to the time–distance diagram.
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Ryndina, V. V. "The recovery of the Brunt–Väisälä frequency using dispersion curves." Journal of Applied Mathematics and Mechanics 70, no. 2 (2006): 245–50. http://dx.doi.org/10.1016/j.jappmathmech.2006.06.006.
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Jones, R. Michael. "On using ambient internal waves to monitor Brunt-Väisälä frequency." Journal of Geophysical Research 100, no. C6 (1995): 11005. http://dx.doi.org/10.1029/95jc00139.
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Houry, Sabine, Eric Dombrowsky, Piepre De Mey, and Jean-Francois Minster. "Brunt-Väisälä Frequency and Rossby Radii in the South Atlantic." Journal of Physical Oceanography 17, no. 10 (1987): 1619–26. http://dx.doi.org/10.1175/1520-0485(1987)017<1619:bvfarr>2.0.co;2.
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Passaggia, Pierre-Yves, Patrice Meunier, and Stéphane Le Dizès. "Response of a stratified boundary layer on a tilted wall to surface undulations." Journal of Fluid Mechanics 751 (June 25, 2014): 663–84. http://dx.doi.org/10.1017/jfm.2014.296.
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AbstractThe structure of a stratified boundary layer over a tilted bottom with a small streamwise undulation is studied theoretically and numerically. We show that the tilt of the boundary can induce strong density variations and wall-transverse velocities in the critical layer when the frequency of the forcing by the topography $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}kU(z_c)$ is equal to the transverse Brunt–Väisälä frequency $N \sin \alpha $ ($N$ being the vertical Brunt–Väisälä frequency). The viscous solution in the critical layer, obtained and compared with direct numerical simulation results, is in good agreement for both the scaling and the spatial structure. The amplitude of the transverse velocity response is also shown to exhibit quasi-resonance peaks when the stratification strength is varied.
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Bukatov, А. A., N. M. Solovei, and E. A. Pavlenko. "Climate Trend Estimation of the Powell Basin Hydrophysical Characteristics." Океанология 63, no. 4 (2023): 539–47. http://dx.doi.org/10.31857/s0030157423040020.
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The trend assessment of the dynamic state of the Powell Basin waters in the Weddell Sea was made on the basis of the hydrological data of the 79th cruise of the R/V AkademikMstislavKeldysh (January 16–February 6, 2020) and the World Ocean Database-2018 data for January-February from 1975 to 2020. At each node of the quarter-degree grid, a linear trend was constructed for the calculated values of the maximum buoyancy frequency and the maximum amplitude of the vertical component of the internal wave velocity. It is shown that the southwestern and northwestern parts of the Powell Basin differ significantly in their hydrophysical characteristics. In the northwest of the basin, the linear trend of the maximum buoyancy frequency is negative, the trends of the depths of the maximum values of the Väisälä-Brent frequency and the amplitude of the vertical velocity component are positive. In the southwestern part of the basin, the opposite is true: the trend of the maximum buoyancy frequency is positive, the trends in the depths of the maximum values of the Väisälä-Brunt frequency and the amplitude of the vertical velocity component are negative.
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Zakinyan, Robert G., Alaa H. Kamil, Vladislav A. Svetlichny, and Arthur R. Zakinyan. "On the Frequency of Internal Gravity Waves in the Atmosphere: Comparing Theory with Observations." Atmosphere 16, no. 1 (2025): 73. https://doi.org/10.3390/atmos16010073.
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This paper is devoted to the dynamics of the propagation of non-planetary scale internal gravity waves (IGWs) in the stratified atmosphere. We consider the system of equations describing internal gravity waves in three approximations: (1) the incompressible fluid approximation, (2) the anelastic gas (compressible fluid) approximation, and (3) a new approximation called the non-Boussinesq gas approximation. For each approximation, a different dispersion relation is given, from which it follows that the oscillation frequency of internal gravity waves depends on the direction of propagation, the horizontal and vertical components of the wave vector, the vertical gradient of the background temperature, and the background wind shear. In each of the three cases, the maximum frequency of internal gravity waves is different. Moreover, in the anelastic gas approximation, the maximum frequency is equal to the Brunt–Väisälä buoyancy frequency, and in the incompressible fluid approximation, it is larger than the Brunt–Väisälä frequency by a factor of . In the model proposed in this paper, the value of the maximum frequency of internal gravity waves occupies an intermediate position between the above limits. The question arises: which of the above fluid representations adequately describe the dynamics of internal gravity waves? This paper compares the above theories with observational data and experiments.
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Bradley, P. A., та J. A. Guzik. "Suppressing g Modes in Shell Hydrogen-Burning δ Scuti Stars". International Astronomical Union Colloquium 176 (2000): 444. http://dx.doi.org/10.1017/s0252921100058346.
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For shell hydrogen-burning δ Scuti stars, many more unstable modes are predicted than are observed. For example, for 4 CVn and δ Scuti itself, only 18 and 6 modes are observed, respectively, whereas several hundred ℓ = 0, 1 and 2 rotationally-split modes are predicted. The predicted modes have a mixed p- and g-mode character, with many g-type nodes present in the H-exhausted core and shell H-burning region, where the Brunt-Väisälä frequency is large.Here we explore whether the predicted frequency spectrum can be made to agree better with the observed spectrum if the g-mode character of the pulsation modes is partially suppressed. Additional motivation for this approach is provided by the observation by Breger et al. (1999) that 4 modes in 4 CVn identified as ℓ = 1 have nearly equal frequency spacings of ∼14 μHz, reminiscent of consecutive radial orders from a pure p-mode spectrum. We consider 2.1- and 2.3-M⊙ evolutionary models that match the observed Teff, L, and log g, plus an identified radial mode frequency of 4 CVn and δ Scuti. For these models, we experimented with setting the Brunt-Väisälä frequency to zero from the model center to the edge of or beyond H-depleted core, so that the g-type portion of the waves becomes evanescent in this region.
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Vijayalakshmi, A.R.*. "STUDY OF EVOLUTION OF LINEARIZED DISTURBANCES IN A STRATIFIED BOUNDED COUETTE FLOW." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 6 (2017): 464–71. https://doi.org/10.5281/zenodo.814790.
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Using initial value problem approach the evolution of linearized disturbances in a stratified shear flow is studied. The resulting equation in time posed by using Fourier transform is solved for the Fourier amplitudes for the case of bounded couette flow with point source of the field of transverse velocity and density as the initial distributions. For small values of Brunt frequency the perturbation solutions are obtained.
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VORONOVICH, ALEXANDER G. "Strong solitary internal waves in a 2.5-layer model." Journal of Fluid Mechanics 474 (January 10, 2003): 85–94. http://dx.doi.org/10.1017/s0022112002002744.
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A theoretical model for internal solitary waves for stratification consisting of two layers of incompressible fluid with a constant Brunt–Väisälä frequency and a density jump at the boundary between layers (‘2.5-layer model’) is presented. The equation of motion for solitary waves in the case of a constant Brunt–Väisälä frequency N is linear, and nonlinearity appears due only to boundary conditions between layers. This allows one to obtain in the case of long waves a single ordinary differential equation for an internal solitary wave profile. In the case of nearly homogeneous layers the solitons obtained here coincide with the solitons calculated by Choi & Camassa (1999), and in the weakly nonlinear case they reduce to KdV solitons. In the general situation strong 2.5-layer solitons can correspond to higher modes. Sufficiently strong solitons could also possess a recirculating core (at least, as a formal solution).The model was applied to the data collected during the COPE experiment. The results are in reasonable agreement with experimental data.
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Tsuda, Toshitaka, Thomas E. VanZandt, Masahiro Mizumoto, Susumu Kato, and Shoichiro Fukao. "Spectral analysis of temperature and Brunt-Väisälä frequency fluctuations observed by radiosondes." Journal of Geophysical Research 96, no. D9 (1991): 17265. http://dx.doi.org/10.1029/91jd01944.
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Ratnasingam, R. P., P. V. F. Edelmann, D. M. Bowman, and T. M. Rogers. "On the Geometry of the Near-core Magnetic Field in Massive Stars." Astrophysical Journal Letters 977, no. 1 (2024): L30. https://doi.org/10.3847/2041-8213/ad95f8.
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Abstract It is well-known that the cores of massive stars sustain a stellar dynamo with a complex magnetic field configuration. However, the same cannot be said for the field's strength and geometry at the convective–radiative boundary, which are crucial when performing asteroseismic inference. In this Letter, we present 3D magnetohydrodynamic (MHD) simulations of a 7 M ⊙ mid-main-sequence star, with particular attention given to the convective–radiative boundary in the near-core region. Our simulations reveal that the toroidal magnetic field is significantly stronger than the poloidal field in this region, contrary to recent assumptions. Moreover, the rotational shear layer, also important for asteroseismic inference, is specifically confined within the extent of the Brunt–Väisälä frequency peak. These results, which are based on the inferred properties of HD 43317, have widespread implications for asteroseismic studies of rotation, mixing, and magnetism in stars. While we expect our results to be broadly applicable across stars with similar Brunt–Väisälä frequency profiles and stellar masses, we also expect the MHD parameters (e.g., Rem) and the initial stellar rotation rate to impact the geometry of the field and differential rotation at the convective–radiative interface.
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ERSHKOVICH, ALEXANDER I., and PETER L. ISRAELEVICH. "On the MHD analogue of the Brunt/Väisälä frequency in a magnetized plasma." Journal of Plasma Physics 64, no. 2 (2000): 195–200. http://dx.doi.org/10.1017/s0022377800008588.
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The MHD analogue of the Brunt/Väisälä frequency, NB, in a magnetized, ideally conducting plasma is obtained, with the vertical component of the magnetic field, Br, taken into account. The magnetic field vector (Br, Bθ, Bϕ) is assumed to satisfy the condition B·∇B ≈ BrdB/dr, which holds in many cases of interest. The frequency NB happens to depend, generally speaking, on the magnetic field orientation relative to the direction of gravity. However, for an isentropic gas, the convective instability criterion is governed by the magnetic field strength (rather than by the orientation of B). In general, the magnetic field has a stabilizing (destabilizing) effect if B/ρ grows (decreases) along the vertical axis r. This conclusion seems not to depend on the specific magnetic field configuration.
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Wüst, Sabine, Michael Bittner, Jeng-Hwa Yee, Martin G. Mlynczak, and James M. Russell III. "Variability of the Brunt–Väisälä frequency at the OH<sup>∗</sup>-airglow layer height at low and midlatitudes." Atmospheric Measurement Techniques 13, no. 11 (2020): 6067–93. http://dx.doi.org/10.5194/amt-13-6067-2020.
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Abstract. Airglow spectrometers, as they are operated within the Network for the Detection of Mesospheric Change (NDMC; https://ndmc.dlr.de, last access: 1 November 2020), for example, allow the derivation of rotational temperatures which are equivalent to the kinetic temperature, local thermodynamic equilibrium provided. Temperature variations at the height of the airglow layer are, amongst others, caused by gravity waves. However, airglow spectrometers do not deliver vertically resolved temperature information. This is an obstacle for the calculation of the density of gravity wave potential energy from these measurements. As Wüst et al. (2016) showed, the density of wave potential energy can be estimated from data of OH∗-airglow spectrometers if co-located TIMED-SABER (Thermosphere Ionosphere Mesosphere Energetics Dynamics, Sounding of the Atmosphere using Broadband Emission Radiometry) measurements are available, since they allow the calculation of the Brunt–Väisälä frequency. If co-located measurements are not available, a climatology of the Brunt–Väisälä frequency is an alternative. Based on 17 years of TIMED-SABER temperature data (2002–2018), such a climatology is provided here for the OH∗-airglow layer height and for a latitudinal longitudinal grid of 10∘×20∘ at midlatitudes and low latitudes. Additionally, climatologies of height and thickness of the OH∗-airglow layer are calculated.
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Hatta, Yoshiki. "Semi-analytical Expression of G-mode Period Spacing: The Case of Brunt–Väisälä Frequency with Not a Jump but a Ramp." Astrophysical Journal 950, no. 2 (2023): 165. http://dx.doi.org/10.3847/1538-4357/acd4b9.
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Abstract To decipher complex patterns of gravity-mode period spacings observed for intermediate-mass main-sequence stars is an important step toward a better understanding of the structure and dynamics in the deep radiative region of the stars. In this study, we apply the JWKB approximation to derive a semi-analytical expression of the g-mode period spacing pattern, for which the gradient in the Brunt–Väisälä frequency is taken into account. The formulation includes the term P −1 B ⋆, where P and B ⋆ represent the g-mode period and degree of the structural variation, the latter of which especially is related to the steepness of the gradient of the Brunt–Väisälä frequency. Tests with one-dimensional stellar models show that the semi-analytical expression derived in this study is useful for inferring the degree of the structural variation B ⋆ with an accuracy of ∼10 % in the case of relatively massive intermediate-mass models with the mass M larger than 3 M ⊙. The newly formulated expression will possibly allow us to put further constraints on, e.g., mixing processes inside intermediate-mass main-sequence g-mode pulsators such as β Cep, SPB, and γ Dor stars that have been principal targets in asteroseismology.
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Brown, S. N., and H. K. Cheng. "The response of a stratified rapidly rotating flow to a pulsating topography." Journal of Fluid Mechanics 177 (April 1987): 359–79. http://dx.doi.org/10.1017/s0022112087000995.
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A theoretical study is made of the disturbance produced by an oscillating, shallow topographical feature in horizontal relative motion in a rapidly rotating, linearly stratified, unbounded fluid. For a sinusoidal surface oscillation, an explicit solution is obtained in terms of wavenumber spectra of the topography. The oscillating far-field behaviour is shown to consist of a large-scale, cyclonic component above the topography and a system of inertial waves behind the caustics, which spreads predominantly in the downstream direction. A significant property of the flow field is its dependence on a frequency threshold familiar from classical works on internal gravity waves in the absence of rotation, determined by the Brunt-Väisälä value. When the frequency is supercritical, a prominent circle of maximum disturbance appears in the far field, which provides the transition boundary between two distinct cyclonic structures and an upstream barrier to the propagating waves ahead of the obstacle. The circle has a radius depending on the relative magnitude of the pulsating frequency and the Brunt-Väisälä value, and is distinctly marked also by a phase jump in pressure and velocities. These features are substantiated by numerical examples of the full solution at a large but finite distance above the obstacle at supercritical frequencies. The circle of maximum disturbance signifies a preferential direction for energy propagation unaccounted for by group velocity. Its relation to the classical result of Görtler in the homogeneous case and that in the classical internal-gravity-wave theory are examined.
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Candelier, Fabien, Rabah Mehaddi, and Olivier Vauquelin. "The history force on a small particle in a linearly stratified fluid." Journal of Fluid Mechanics 749 (May 15, 2014): 184–200. http://dx.doi.org/10.1017/jfm.2014.219.
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AbstractThe hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a weakly density-stratified fluid is investigated theoretically. The study is carried out under the Oberbeck–Boussinesq approximation and in the limit of small Reynolds and small Péclet numbers. The force acting on the particle is obtained by using matched-asymptotic expansions. In this approach, the small parameter is given by $a/\ell $, where $a$ is the particle radius and $\ell $ is the stratification length, as defined by Ardekani & Stocker (Phys. Rev. Lett., vol. 105, 2010, article 084502), which depends on the Brunt–Väisälä frequency, on the fluid kinematic viscosity and on the thermal or the concentration diffusivity (depending on the case considered). The matching procedure used here, which is based on series expansions of generalized functions, slightly differs from that generally used in similar problems. In addition to the classical Stokes drag, it is found that the particle experiences a memory force given by two convolution products, one of which involves, as usual, the particle acceleration and the other one, the particle velocity. Owing to the stratification, the transient behaviour of this memory force, in response to an abrupt motion, consists of an initial fast decrease followed by a damped oscillation with an angular frequency corresponding to the Brunt–Väisälä frequency. The perturbation force eventually tends to a constant which provides us with correction terms that should be added to the Stokes drag to accurately predict the settling time of a particle in a diffusive stratified fluid.
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Pinçon, C., M. Takata, and B. Mosser. "Evolution of the gravity offset of mixed modes in RGB stars." Astronomy & Astrophysics 626 (June 2019): A125. http://dx.doi.org/10.1051/0004-6361/201935327.
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Context.Observations of mixed modes in evolved low-mass stars enable us to probe the properties of not only the outer envelope of these stars, but also their deep layers. Among the seismic parameters associated with mixed modes, the gravity offset, denoted withεg, is expected to reveal information on the boundaries of the inner buoyancy resonant cavity. This parameter was recently measured for hundreds of stars observed by theKeplersatellite and its value was shown to change during evolution.Aims.In this article, we theoretically investigate the reasons for such a variation in terms of structure properties, focusing only on the red giant branch.Methods.Using available asymptotic analyses and a simple model of the Brunt–Väisälä and Lamb frequencies in the upper part of the radiative zone, we derived an analytical expression ofεgfor dipolar modes and compared its predictions to observations.Results.First, we show that the asymptotic value ofεgwell agrees with the mean value observed at the beginning of the ascent of the red giant branch, which results from the high density contrast between the helium core and the base of the convective envelope. Second, we demonstrate that the predicted value also explains the sharp decrease inεgobserved for the more luminous red giant stars of the sample. This rapid drop turns out to occur just before the luminosity bump and results from the kink of the Brunt–Väisälä frequency near the upper turning point associated with the buoyancy cavity as stars evolve and this latter nears the base of the convective envelope. The potential ofεgto probe the value and slope of the Brunt–Väisälä frequency below the base of the convective region is clearly highlighted.Conclusions.The observed variation inεgand its link with the internal properties on the red giant branch are now globally understood. This work motivates further analyses of the potential of this parameter as a seismic diagnosis of the region located between the hydrogen-burning shell and the base of the convective envelope, and of the local dynamical processes associated for instance with core contraction, the migration of the convective boundary, or overshooting.
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Van Buren, Tyler, Owen Williams, and Alexander J. Smits. "Turbulent boundary layer response to the introduction of stable stratification." Journal of Fluid Mechanics 811 (December 13, 2016): 569–81. http://dx.doi.org/10.1017/jfm.2016.775.
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The response of an initially neutral rough-wall turbulent boundary layer to a change in wall temperature is investigated experimentally. The change causes a localized peak in stable stratification that diffuses and moves away from the wall with downstream distance. The streamwise and wall-normal components of turbulent velocity fluctuations are damped at similar rates, even though the stratification only directly impacts the wall-normal component. The Reynolds shear profiles reveal the growth of an internal layer that scales approximately with the bulk Brunt–Väisälä frequency.
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Shapiro, Alan, Evgeni Fedorovich, and Stefan Rahimi. "A Unified Theory for the Great Plains Nocturnal Low-Level Jet." Journal of the Atmospheric Sciences 73, no. 8 (2016): 3037–57. http://dx.doi.org/10.1175/jas-d-15-0307.1.
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Abstract A theory is presented for the Great Plains low-level jet in which the jet emerges in the sloping atmospheric boundary layer as the nocturnal phase of an oscillation arising from diurnal variations in turbulent diffusivity (Blackadar mechanism) and surface buoyancy (Holton mechanism). The governing equations are the equations of motion, mass conservation, and thermal energy for a stably stratified fluid in the Boussinesq approximation. Attention is restricted to remote (far above slope) geostrophic winds that blow along the terrain isoheights (southerly for the Great Plains). Diurnally periodic solutions are obtained analytically with diffusivities that vary as piecewise constant functions of time and slope buoyancies that vary as piecewise linear functions of time. The solution is controlled by 11 parameters: slope angle, Coriolis parameter, free-atmosphere Brunt–Väisälä frequency, free-atmosphere geostrophic wind, radiative damping parameter, day and night diffusivities, maximum and minimum surface buoyancies, and times of maximum surface buoyancy and sunset. The Holton mechanism, by itself, results in relatively weak wind maxima but produces strong jets when paired with the Blackadar mechanism. Jets with both Blackadar and Holton mechanisms operating are shown to be broadly consistent with observations and climatological analyses. Jets strengthen with increasing geostrophic wind, maximum surface buoyancy, and day-to-night ratio of the diffusivities and weaken with increasing Brunt–Väisälä frequency and magnitude of minimum slope buoyancy (greater nighttime cooling). Peak winds are maximized for slope angles characteristic of the Great Plains.
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Ostrovsky, Lev, Irina Soustova, Yuliya Troitskaya, and Daria Gladskikh. "Evolution of small-scale turbulence at large Richardson numbers." Nonlinear Processes in Geophysics 31, no. 2 (2024): 219–27. http://dx.doi.org/10.5194/npg-31-219-2024.
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Abstract. The theory of stratified turbulent flow developed earlier by the authors is applied to data from different areas of the ocean. It is shown that turbulence can be amplified and supported even at large gradient Richardson numbers. The cause of that is the exchange between kinetic and potential energies of turbulence. Using the profiles of Brunt–Väisälä frequency and vertical current shear given in Forryan et al. (2013), the profiles of the kinetic energy dissipation rate are calculated. The results are in reasonable agreement with the experimental data.
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Fedorenko, A. K., A. V. Bespalova, O. K. Cheremnykh, and E. I. Kryuchkov. "A dominant acoustic-gravity mode in the polar thermosphere." Annales Geophysicae 33, no. 1 (2015): 101–8. http://dx.doi.org/10.5194/angeo-33-101-2015.
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Abstract. The article presents a summary of the main findings of the systematic study of acoustic-gravity waves (AGWs) in the polar thermosphere. This study was based on the in situ measurements made by the Dynamics Explorer 2 (DE2) spacecraft late in its mission when it descended low enough (250–400 km). It was found out that AGWs in the polar thermosphere are observed within a narrow frequency band close to the Brunt–Väisälä frequency and with horizontal wavelengths about 500–600 km. The broadband spectrum of travelling ionospheric disturbance (TID) frequencies observed by radars is caused by the Doppler effect. The AGW amplitudes do not depend on the altitude, but grow almost linearly with the wind velocity. They propagate towards the wind.
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Smeyers, P., T. Van Hoolst, I. De Boeck, and L. Decock. "A New Asymptotic Treatment of g-modes of a Star." International Astronomical Union Colloquium 155 (1995): 285–86. http://dx.doi.org/10.1017/s025292110003712x.
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An asymptotic representation of low-frequency, linear, isentropic g-modes of a star is developed without the usual neglect of the Eulerian perturbation of the gravitational potential. Our asymptotic representation is based on the use of asymptotic expansions adequate for solutions of singular perturbation problems (see, e.g., Kevorkian & Cole 1981).Linear, isentropic oscillation modes with frequency different from zero are governed by a fourth-order system of linear, homogeneous differential equations in the radial parts of the radial displacement ξ(r) and the divergence α(r). These equations take the formThe symbols have their usual meaning. N2 is the square of the frequency of Brunt-Väisälä. The functions K1 (r), K2 (r), K3 (r), K4 (r), depend on the equilibrium model, e.g.,We introduce the small expansion parameterand assume, for the sake of simplification, N2 to be positive everywhere in the star so that the star is everywhere convectively stable.
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Pérez, I. A., M. L. Sánchez, M. Á. García, and B. de Torre. "Boundary layer structure and stability classification validated with CO<sub>2</sub> concentrations over the Northern Spanish Plateau." Annales Geophysicae 27, no. 1 (2009): 339–49. http://dx.doi.org/10.5194/angeo-27-339-2009.
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Abstract. A description of the lower boundary layer is vital to enhance our understanding of dispersion processes. In this paper, Radio Acoustic Sounding System sodar measurements obtained over three years were used to calculate the Brunt-Väisälä frequency and the Monin-Obukhov length. The Brunt-Väisälä frequency enabled investigation of the structure of this layer. At night, several layers were noticeable and the maximum was observed at the first level, 40 m, whereas during the day, it was present at about 320 m. The Monin-Obukhov length was calculated with the four first levels measured, 40–100 m, by an original iterative method and used to establish four stability classes: drainage, extremely stable, stable and unstable. Wind speed and temperature median profiles linked to these classes were also presented. Wind speeds were the lowest, but temperatures were the highest and inversions were intense at night in drainage situations. However, unstable situations were linked to high wind speeds and superadiabatic temperature profiles. Detrended CO2 concentrations were used to determine the goodness of the classification proposed evidencing values which under drainage at night in spring were nearly 28 ppm higher than those corresponding to unstable situations. Finally, atmosphere structure was presented for the proposed stability classes and related with wind speed profiles. Under extremely stable situations, low level jets were coupled to the surface, with median wind speeds below 8 m s−1 and cores occasionally at 120 m. However, jets were uncoupled in stable situations, wind speed medians were higher than 11 m s−1 and their core heights were around 200 m.
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Audard, N., and J. Provost. "About Seismological Properties of Intermediate Mass Stars." International Astronomical Union Colloquium 137 (1993): 544–46. http://dx.doi.org/10.1017/s025292110001839x.
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AbstractStars more massive than about 1.2M⊙ are characterised by a convective core, which induces at its frontier a rapid variation of the sound speed and of the Brunt-Väissälä frequency, close to a discontinuity. We present preliminary results about the search of the signature this core could have on the p-mode spectra. For a set of frequencies of three stars of 1, 1.5 and 2 M⊙, we study in particular the small frequency separation Δν0,2, ~ νn,l, νn-1,l+2 for high order and low degree oscillation modes, which is particularly sensitive to the interior of stars. We underline characteristic behaviours of the 1.5 and 2 M⊙ stars, through the comparison between computed frequencies and their approximation obtained by asymptotic and polynomial fittings, and also through second order quantities relatively to the frequencies.
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Kandaswamy, Palani G., B. Tamil Selvi, and Lokenath Debnath. "Propagation of Rossby waves in stratified shear flows." International Journal of Mathematics and Mathematical Sciences 12, no. 3 (1989): 547–57. http://dx.doi.org/10.1155/s0161171289000682.
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A study is made of the propagation of Rossby waves in a stably stratified shear flows. The wave equation for the Rossby waves is derived in an isothermal atmosphere on a beta plane in the presence of a latitudinally sheared zonal flow. It is shown that the wave equation is singular at five critical levels, but the wave absorption takes place only at the two levels where the local relative frequency equals in magnitude to the Brunt Vaisala frequency. This analysis also reveals that these two levels exhibit valve effect by allowing the waves to penetrate them from one side only. The absorption coefficient exp(2πμ)is determined at these levels. Both the group velocity approach and single wave treatment are employed for the investigation of the problem.
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Sarma, SBSS. "Energy Dissipation Rates of Turbulence with an Airborne Microwave Refractometer." Australian Journal of Physics 44, no. 4 (1991): 435. http://dx.doi.org/10.1071/ph910435.
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Estimates of turbulence energy dissipation rates have been obtained in the free atmosphere of the planetary boundary layer. These are derived in terms of the variance of radio refractive index fluctuations {Lln2 }, the Brunt-Vaisala frequency N, the mean vertical gradient of generalised potential refractive index M, and the turbulence structure parameter C~, using an airborne microwave refractometer. The energy dissipation rates thus derived are comparable with the results from other experiments. The inferred energy dissipation rates (on a near real-time basis) from the airborne microwave refractometer. are useful for detection and warning of wind shears in the atmosphere.
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Ter-Krikorov, A. M. "Fundamental solution of the internal-wave equation for a medium with a discontinuous brunt-väisälä frequency." Journal of Applied Mathematics and Mechanics 61, no. 4 (1997): 601–7. http://dx.doi.org/10.1016/s0021-8928(97)00077-4.
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Doostmohammadi, A., S. Dabiri, and A. M. Ardekani. "A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratified fluid." Journal of Fluid Mechanics 750 (May 30, 2014): 5–32. http://dx.doi.org/10.1017/jfm.2014.243.
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AbstractIn this paper, the transient settling dynamics of a spherical particle sedimenting in a linearly stratified fluid is investigated by performing fully resolved direct numerical simulations. The settling behaviour is quantified for different values of Reynolds, Froude and Prandtl numbers. It is demonstrated that the transient settling dynamics is correlated to the induced Lagrangian drift of flow around the settling particle. A simplified model is provided to predict the maximum velocity of the settling particle in linearly stratified fluids. The peak velocity can be followed by the oscillation of the settling velocity and the particle can even reverse its direction of motion before reaching to its neutrally buoyant level. The frequency of oscillation of settling velocity scales with the Brunt–Väisälä frequency and the motion of the particle can lead to the formation of secondary and tertiary vortices following the primary vortex.
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Raevskii, M. A., and V. G. Burdukovskaya. "The Influence of Random Internal Waves on the Characteristics of a Horizontal Antenna in a Shallow Sea." Акустический журнал 69, no. 5 (2023): 584–94. http://dx.doi.org/10.31857/s0320791922600469.
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The effect of random internal waves on the gain and pattern of a horizontal antenna array in a shallow sea has been investigated. An algorithm for calculating the correlation matrix of the point source field at the grating aperture is proposed. The antenna gain is analyzed for different spatial processing methods: the beam forming method, the optimal linear processing method, and the optimal quadratic processing method. The effect of internal waves on the directivity pattern is described by the dispersion of the angular response of the grating. Numerical simulation results are presented for a model summer-type waveguide and an exponential Brunt–Väisälä frequency profile. The empirical internal wave spectrum proposed earlier from the SWARM95 experiment is used. The dependences of the antenna array characteristics on the number of its elements, orientation with respect to the source, radiation frequency, and the acoustic characteristics of the bottom are analyzed.
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Reissmann, J. H. "On the representation of regional characteristics by hydrographic measurements at central stations in four deep basins of the Baltic Sea." Ocean Science 2, no. 1 (2006): 71–86. http://dx.doi.org/10.5194/os-2-71-2006.
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Abstract. In this work the eddy resolving data sets of salinity, temperature, and oxygen content aquired in the framework of the German-Russian project MESODYN (MESOscale DYNamics) in the Arkona Basin, the Bornholm Basin, the Stolpe Furrow, and the Eastern Gotland Basin during summer and winter stratification situations are utilized to examine to which extent the observations at the central monitoring stations within these basins are representative for the spatial mean state of the corresponding region with respect to comparative monitoring purposes of the whole Baltic Sea. The investigation covers profiles of salinity, potential temperature, oxygen content, potential density, and squared buoyancy or Brunt-Väisälä frequency. Moreover, some parameters of the halocline, namely its depth, thickness, and upper and lower boundaries, and the first baroclinic Rossby radii are subject to the investigation. The profiles match best for the squared buoyancy or Brunt-Väisälä frequency. The profiles of salinity match best in the Eastern Gotland Basin and worst in the Arkona Basin both for summer and winter stratification situations. The overall agreement for the halocline parameters is good. The baroclinic Rossby radii match their spatial mean values well, if the depth range considered for their calculation is restricted to the mean depth in each region at the bottom side. In doing so they also match the spatial mean values of the first baroclinic Rossby radii calculated considering the whole depth range at each station. Overall, the regional characteristics of the investigated quantities and parameters are represented well by the hydrographic measurements at the central stations in the four regions in spite of some significant differences between the spatial mean states and the observations at the central stations. In particular, the observations at the central stations seem to be usefull for comparisons between these regions. However, the observed differences may affect regional investigations covering just a single region.
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Reissmann, J. H. "On the representation of regional characteristics by hydrographic measurements at central stations in four deep basins of the Baltic Sea." Ocean Science Discussions 2, no. 4 (2005): 363–98. http://dx.doi.org/10.5194/osd-2-363-2005.
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Abstract. In this work the eddy resolving data sets of salinity, temperature, and oxygen content aquired in the framework of the German-Russian project MESODYN (MESOscale DYNamics) in the Arkona Basin, the Bornholm Basin, the Stolpe Furrow, and the Eastern Gotland Basin during summer and winter stratification situations are utilized to examine to which extent the observations at the central monitoring stations within these basins are representative for the spatial mean state of the corresponding region. The investiation covers profiles of salinity, potential temperature, oxygen content, potential density, and squared buoyancy or Brunt-Väisälä frequency. Moreover, some parameters of the halocline, namely its depth, thickness, and upper and lower boundaries, and the first baroclinic Rossby radii are subject to the investigation. The profiles match best for the squared buoyancy or Brunt-Väisälä frequency. The profiles of salinity match best in the Eastern Gotland Basin and worst in the Arkona Basin both for summer and winter stratification situations. The overall agreement for the halocline parameters is good. The baroclinic Rossby radii match their spatial mean values well, if the depth range considered for their calculation is restricted to the mean depth in each region at the bottom side. In doing so they also match the spatial mean values of the first baroclinic Rossby radii calculated considering the whole depth range at each station. Overall, the regional characteristics of the investigated quantities and parameters are represented well by the hydrographic measurements at the central stations in the four regions in spite of some significant differences between the spatial mean states and the ovservations at the central stations. In particular, the observations at the central stations seem to be usefull for comparisons between these regions.
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Garcia, O. E., E. Leer, H. L. Pécseli, and J. K. Trulsen. "Magnetic field-aligned plasma currents in gravitational fields." Annales Geophysicae 33, no. 3 (2015): 257–66. http://dx.doi.org/10.5194/angeo-33-257-2015.
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Abstract. Analytical models are presented for currents along vertical magnetic field lines due to slow bulk electron motion in plasmas subject to a gravitational force. It is demonstrated that a general feature of this problem is a singularity in the plasma pressure force that develops at some finite altitude when a plasma that is initially in static equilibrium is set into slow motion. Classical fluid models thus do not allow general steady-state solutions for field-aligned currents. General solutions have to be non-stationary, varying on time scales of many periods of a plasma equivalent to the Brunt–Väisälä frequency. Except for very special choices of parameters, a steady-state solution exists only in an average sense. The conditions at large altitudes turn out to be extremely sensitive to even small changes in parameters at low altitudes. Low frequency fluctuations detected at large altitudes in the polar regions need not be caused by local low frequency instabilities, but merely reflect small fluctuations in conditions at low altitudes.
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Ryndina, V. V. "Some problems of the uniqueness of the re-establishment of the Brunt-Väisälä frequency using dispersion curves." Journal of Applied Mathematics and Mechanics 64, no. 1 (2000): 117–22. http://dx.doi.org/10.1016/s0021-8928(00)00032-0.
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Montalbán, Josefina. "The Generation of Internal Waves." International Astronomical Union Colloquium 137 (1993): 278–80. http://dx.doi.org/10.1017/s0252921100017917.
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The generation of internal waves in the radiatively stable stellar region by the turbulent motion at the boundary of the overlaying convective zone is similar to the same case in the deep ocean or in the earth atmosphere (Townsend, 1965), and can be described in a simple way as following: When an turbulent fluid element arrives at the boundary of the convective region with a non-zero momentum, it beats and it deforms the interface between both regions. This disturbance of the equilibrium state excites a train of internal waves propagating below the convective zone in the horizontal and vertical directions for the frequencies lower than the characteristic one for the stable stratification (Brunt-Väisälä frequency).
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Shi, Yunlong, Baoshu Yin, Hongwei Yang, Dezhou Yang, and Zhenhua Xu. "Dissipative Nonlinear Schrödinger Equation for Envelope Solitary Rossby Waves with Dissipation Effect in Stratified Fluids and Its Solution." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/643652.
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We solve the so-called dissipative nonlinear Schrödinger equation by means of multiple scales analysis and perturbation method to describe envelope solitary Rossby waves with dissipation effect in stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency, andβeffect are important factors to form the envelope solitary Rossby waves. By employing trial function method, the asymptotic solution of dissipative nonlinear Schrödinger equation is derived. Based on the solution, the effect of dissipation on the evolution of envelope solitary Rossby wave is also discussed. The results show that the dissipation causes a slow decrease of amplitude of envelope solitary Rossby waves and a slow increase of width, while it has no effect on the propagation velocity. That is quite different from the KdV-type solitary waves. It is notable that dissipation has certain influence on the carrier frequency.
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Augier, Pierre, Sébastien Galtier, and Paul Billant. "Kolmogorov laws for stratified turbulence." Journal of Fluid Mechanics 709 (August 31, 2012): 659–70. http://dx.doi.org/10.1017/jfm.2012.379.
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AbstractFollowing the Kolmogorov technique, an exact relation for a vector third-order moment $\mathbi{J}$ is derived for three-dimensional incompressible stably stratified turbulence under the Boussinesq approximation. In the limit of a small Brunt–Väisälä frequency, isotropy may be assumed which allows us to find a generalized $4/ 3$-law. For strong stratification, we make the ansatz that $\mathbi{J}$ is directed along axisymmetric surfaces parameterized by a scaling law relating horizontal and vertical coordinates. An integration of the exact relation under this hypothesis leads to a generalized Kolmogorov law which depends on the intensity of anisotropy parameterized by a single coefficient. By using a scaling relation between large horizontal and vertical length scales we fix this coefficient and propose a unique law.
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Harsono, Gentio, Budi Purwanto, Anindya Wirasatriya, Sri Murtiana, and Rifqi N. Agassi. "Percampuran Vertikal Massa Air Lapisan Pertengahan Perairan Lifamatola pada Bulan Maret 2009." Buletin Oseanografi Marina 12, no. 3 (2023): 365–78. http://dx.doi.org/10.14710/buloma.v12i3.56350.
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Perairan Lifamatola menarik dikaji selain sebagai lintasan Arlindo juga proses percampuran massa airnya terjadi secara intensif. Tujuan penelitian ini adalah mengkaji percampuran turbulen di Perairan Lifamatola pada Maret 2009. Data salinitas, temperatur dan kedalaman diperoleh dari program Tropical Ocean Climate Study (TOCS) tahun 2009, kerjasama Badan Pengkajian Penerapan Teknologi (BPPT) dan Jamstec (Japan of Marine Science Earth and Technology) Jepang menggunakan RV. Kaiyo. Analisis meliputi analisis skala Thorpe (LT), frekuensi Brunt Vaisala (N), disipasi energi kinetik turbulen (ε) dan estimasi nilai difusivitas eddy vertikal (KZ). Hasil identifikasi karakteristik massa air Samudera Pasifik, North Pacific Intermediate Water (NPIW) pada perairan ini S-max tidak terlihat karena diduga sudah bertranformasi menjadi massa air dengan karakteristik yang baru akibat proses percampuran. Hasil perhitungan terhadap nilai frekuensi Brunt Vaisala, pada lapisan termoklin diperoleh nilai N2 tertinggi yaitu 0,05-0,15 s-1 pada kedalaman 100 m dan kedalaman 200 m, sementara lapisan tercampur berkisar 0,0- 0,01 s-1 dan lapisan dalam 0-0,005 s-1. Lapisan termoklin memiliki nilai Td -5 m sampai 3 m, lapisan dalam berkisar -4 m sampai 5 m dan lapisan dalam berkisar -13m sampai 19 m. Nilai rata-rata disipasi energi kinetik turbulen (ε) dari semua lapisan yaitu sebesar 3.29 x 10-9 Wkg-1. Hasil dari rata-rata menunjukkan bahwa semakin bertambahnya kedalaman nilai disipasi energi kinetik turbulen (ε) semakin menurun. Nilai difusivitas paling tinggi terjadi pada lapisan termoklin dan menurun dengan semakin bertambah nya kedalaman. Nilai Kz pada lapisan permukaan antar sampling berbeda-beda, terjadi akibat Gelombang Internal dan aktivitas pasang surut. Lifamatola waters are interesting to study apart from being the Arlindo trajectory as well as the process of intensive mixing. The aim of this research is to examine turbulent mixing in Lifamatola Waters in March 2009. Salinity, Temperature and Depth data were obtained from the Tropical Ocean Climate Study (TOCS) program in 2009, a collaboration between the Agency for the Assessment of the Application of Technology (BPPT) and Japan of Marine Science Earth and Technology uses RV. Kaiyo. The analysis includes Thorpe scale analysis (LT), Brunt Vaisala frequency (N), turbulent kinetic energy dissipation (ε) and estimation of vertical eddy diffusivity (KZ) values. The results of identifying the characteristics of the Pacific Ocean wate mass, North Pacific Intermediate Water (NPIW) in these waters, S-max is not visible because it is thought to have been transformed into a water mass with new characteristics due to the mixing process. The results of the calculation of the Brunt Vaisala frequency value, in the thermocline layer, the highest N2 value was obtained, namely 0.05-0.15 s-1 at a depth of 100 m and a depth of 200 m, while the mixed layer ranged from 0.0-0.01 s-1 and inner layer 0-0.005 s-1. The thermocline layer has a Td value of -5 m to 3 m, the inner layer ranges from -4 m to 5 m and the inner layer ranges from -13m to 19 m. The average value of turbulent kinetic energy dissipation (ε) from all layers is 3.29 x 10-9 Wkg-1. The results of the average show that as the depth increases the dissipation value of turbulent kinetic energy (ε) decreases. The highest diffusivity value occurs in the thermocline layer and decreases with increasing depth. The Kz value in the surface layer varies between sampling, occurring due to internal waves and tidal activity.
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Marquet, Pascal, and Jean-François Geleyn. "On a general definition of the squared Brunt-Väisälä frequency associated with the specific moist entropy potential temperature." Quarterly Journal of the Royal Meteorological Society 139, no. 670 (2012): 85–100. http://dx.doi.org/10.1002/qj.1957.
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BILLANT, PAUL. "Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations." Journal of Fluid Mechanics 660 (July 21, 2010): 354–95. http://dx.doi.org/10.1017/s0022112010002818.
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In stratified and rotating fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and rotating fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b. These equations have the same form as those obtained for vortex filaments in homogeneous fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous fluids when the fluid is strongly stratified: |max| < N (where N is the Brunt–Väisälä frequency and max the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-rotating in contrast to homogeneous fluids. When the stratification is weaker, i.e. |max| > N, the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency.In contrast to previous theories, which apply only to strongly stratified non-rotating fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt–Väisälä frequency: Γ/(2πb2) ≪ N, where Γ is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and rotating fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids will be demonstrated to originate from the sign reversal of the self-induction function.
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Yim, Eunok, and Paul Billant. "On the mechanism of the Gent–McWilliams instability of a columnar vortex in stratified rotating fluids." Journal of Fluid Mechanics 780 (September 2, 2015): 5–44. http://dx.doi.org/10.1017/jfm.2015.426.
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In stably stratified and rotating fluids, an axisymmetric columnar vortex can be unstable to a special instability with an azimuthal wavenumber $m=1$ which bends and slices the vortex into pancake vortices (Gent & McWilliams Geophys. Astrophys. Fluid Dyn., vol. 35 (1–4), 1986, pp. 209–233). This bending instability, called the ‘Gent–McWilliams instability’ herein, is distinct from the shear, centrifugal or radiative instabilities. The goals of the paper are to better understand the origin and properties of this instability and to explain why it operates only in stratified rotating fluids. Both numerical and asymptotic stability analyses of several velocity profiles have been performed for wide ranges of Froude number $\mathit{Fr}_{h}={\it\Omega}_{0}/N$ and Rossby number $\mathit{Ro}=2{\it\Omega}_{0}/f$, where ${\it\Omega}_{0}$ is the angular velocity on the vortex axis, $N$ the Brunt–Väisälä frequency and $f$ the Coriolis parameter. Numerical analyses restricted to the centrifugally stable range show that the maximum growth rate of the Gent–McWilliams instability increases with $\mathit{Ro}$ and is independent of $\mathit{Fr}_{h}$ for $\mathit{Fr}_{h}\leqslant 1$. In contrast, when $\mathit{Fr}_{h}>1$, the maximum growth rate decreases dramatically with $\mathit{Fr}_{h}$. Long axial wavelength asymptotic analyses for isolated vortices prove that the Gent–McWilliams instability is due to the destabilization of the long-wavelength bending mode by a critical layer at the radius $r_{c}$ where the angular velocity ${\it\Omega}$ is equal to the frequency ${\it\omega}$: ${\it\Omega}(r_{c})={\it\omega}$. A necessary and sufficient instability condition valid for long wavelengths, finite Rossby number and $\mathit{Fr}_{h}\leqslant 1$ is that the derivative of the vertical vorticity of the basic vortex is positive at $r_{c}$: ${\it\zeta}^{\prime }(r_{c})>0$. Such a critical layer $r_{c}$ exists for finite Rossby and Froude numbers because the real part of the frequency of the long-wavelength bending mode is positive instead of being negative as in a homogeneous non-rotating fluid ($\mathit{Ro}=\mathit{Fr}_{h}=\infty$). When $\mathit{Fr}_{h}>1$, the instability condition ${\it\zeta}^{\prime }(r_{c})>0$ is necessary but not sufficient because the destabilizing effect of the critical layer $r_{c}$ is strongly reduced by a second stabilizing critical layer $r_{c2}$ existing at the radius where the angular velocity is equal to the Brunt–Väisälä frequency. For non-isolated vortices, numerical results show that only finite axial wavenumbers are unstable to the Gent–McWilliams instability.