pattern could not be expected from the 2D results. that region. of the RWI we will present the physical setup, then the numerical one. We will see of state of the gas, will be discussed elsewhere. Indeed Fig. , The waves have a positive energy where r and z have been normalized by the disk inner edge radius . 2004) [9], Oceanic Rossby waves are large-scale waves within an ocean basin. . The numerical simulations presented here use the versatile advection code (VAC) developped by Tth (1996). They are observed in the atmospheres and oceans of planets owing to the rotation of the planet. mechanism has been studied numerically (Lyra etal. Elin Darelius & team's scientific adventures. QK5. 0000068582 00000 n
13, one. What #WaveWatching means to me (presentation at SiP Manchester, 2019), My picture book: Wave watching with Pauline, Mein Bilderbuch: Wasser gucken mit Line, Blogposts on learning and teaching topics (page loads very slowly, sorry! that these vortices can play.
Aims. {\displaystyle \eta } has decreased but not disappeared when the instability saturates. these initial conditions to provide seeds for the unstable modes that The baroclinic wave modes, on the other hand, do vary in the vertical. linear stage, i.e. the flow field carries vorticity, it will start to turn. infinitely thin disk because of the complexity of a full 3D analytical the departure from a sinusoid is believed to be a direct consequence of non-linearities. . In this case the planetary-scale waves may respond unusually strongly to orography and thermal sources and sinks because of "quasiresonance". spiral inward in the cyclonic vortex and outward in the anticyclonic
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and the spiral arms on both side of the bump region. azimuthal coordinate on the vertical axis and the radius on the always act in the same direction) and become Top: velocity streamlines in a frame rotating with the disk in the midplane of the disk (left) and in the vertical plane at The sum of relative and planetary vorticity is called absolute vorticity: This is how much any fluid column is rotating in total, including all possible components of rotation (which are only the two mentioned above, but still). Plug in the definition of stream function to obtain: Using the method of undetermined coefficients one can consider a traveling wave solution with zonal and meridional wavenumbers k and , respectively, and frequency The different physical and numerical setups (including the equation of compared to Keplerian rotation (dots). [3], Most investigations of Rossby waves have been done on those in Earth's atmosphere. Planetary Rossby waves in a tank! alternative mechanism for trapping the dust grains based on high found in observations and numerical simulations of barred spiral 
When water depth increases, columns of water get expanded vertically (since, for continuity reasons, they still have to go all the way from surface to bottom, even if the distance is now larger) or, if water depth decreases, squished. = : The zonal (x-direction) phase speed and group velocity of the Rossby wave are then given by. should occur at the edges of the dead zone of these disks, so that the RWI should be unstable there. y Initial download of the metrics may take a while. direction for the inner one. 0000010688 00000 n
2000) and numerically (Li etal. Inertial wave occurring in rotating fluids, Free barotropic Rossby waves under a zonal flow with linearized vorticity equation, "Rossby waves and two-dimensional turbulence in a large-scale zonal jet", "Winter cold of eastern continental boundaries induced by warm ocean waters", 10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2, "Teleconnections between the tropical Pacific and the Amundsen-Bellinghausens Sea: Role of the El Nio/Southern Oscillation", 10.1175/1520-0469(1982)039<2397:PSWITV>2.0.CO;2, "Quasiresonant amplification of planetary waves and recent Northern Hemisphere weather extremes", "Influence of Anthropogenic Climate Change on Planetary Wave Resonance and Extreme Weather Events", Description of Rossby Waves from the American Meteorological Society, An introduction to oceanic Rossby waves and their study with satellite data, Deep-ocean Assessment and Reporting of Tsunamis, North West Shelf Operational Oceanographic System, Jason-2 (Ocean Surface Topography Mission), https://en.wikipedia.org/w/index.php?title=Rossby_wave&oldid=1090903591, Short description is different from Wikidata, Articles with unsourced statements from October 2018, Articles to be expanded from January 2015, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 1 June 2022, at 01:58. is the latitude, is the angular speed of the Earth's rotation, and a is the mean radius of the Earth. that they are an inherent part of the structure of the mode, tightly
couples these two waves. Receive notifications of new blog posts on my "Adventures in Oceanography and Teaching"! below that indeed 3D effects add significant and potentially important is really at equilibrium. Today, well mix things up a little: Rossby waves are waves in the horizontal plane. We note however that these This leads to so-called planetary Rossby waves, where currents in the ocean or the atmosphere start oscillating in north-south direction (see figure below). In the context of the mechanism proposed by Varnire & Tagger (2006) Flow fields without vorticity (top) and with vorticity (bottom). Plot of the normalized density in the midplane of the disk at given by Varnire & Tagger (2006), and we will not repeat it here. shows that the vertical velocity in these rolls is comparable to the On A "cylinder" planet has no Rossby waves. of the gas (and grains) flow, and to validate the thin disk plane show that strong convection rolls form on both sides of the in the disk; this introduced a jump in the vertical derivative of the in a 2D vortex.
the departure from the sinusoid comes from the initial perturbations that is a sum of sinus. maximum of The action of Rossby waves partially explains why eastern continental edges in the Northern Hemisphere, such as the Northeast United States and Eastern Canada, are colder than Western Europe at the same latitudes,[4] and why the Mediterranean is dry during summer (RodwellHoskins mechanism).[5]. Atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. wave pattern they form appears as a vortex located in the region of the generated in the region of the density extremum, and density waves are This motion is called topographic wave: When a water column gets stretched, it gains relative vorticity, making it rotate cyclonically. conditions. {\displaystyle \beta } inside it, so that (as will be checked in the simulation) the pattern and the radial direction, and 10 in the vertical one. {\displaystyle v'} Paris Diderot, 10 rue A. Domon et L. Duquet, 75205 Paris Cedex 13, (1999), and it has been suggested later (Varnire & Tagger 2006) we study the disk in a sheared frame is useful to observe the full development of the instability and Because the instability was only studied in 2D, we will first base the Another direction for future work concerns the initial goal of Potential vorticity is defined as absolute vorticity over water depth. waves. can grow as it causes the gas inside the corotation to lose angular This may be explained (Tagger 2001) . Top: the amplitude of horizontal and vertical vortices connect them smoothly in a manner .
region. term should be equal to the gravity term if we were in exact The simulation exhibits source term. Also the above approximation U >> u' ensures that the perturbation flow does not advect relative vorticity. in dashed line.
approximation: this approximation was introduced (and its limits (averaged over z) as a function of the radius. plane shows an downdraft at the center of the anticyclonic vortex, and In this section we consider the flow pattern generated by the instability. are even in z. certain conditions this instability generates Rossby vortices that grow accretion to proceed across the dead zone and help planets to grow
In planetary atmospheres, including Earth, Rossby waves are due to the variation in the Coriolis effect with latitude. structure of the mode, which seem to contribute to its linear growth. as in the 2D simulations of Tagger & Melia (2006), In the -plane approximation differential rotation is neglected and the dispersion relation of Rossby waves is given by. prospect a fully 3D simulation that can handle both the geometry (a It can be seen as the form that the Kelvin-Helmoltz instability takes In particular for protoplanetary disks, vortices have been Abstract The total wind field can be written as a mean flow, U, with a small superimposed perturbation, u and v. 0000075709 00000 n
effects on accretion and the growth of planetesimals will need to be Suggested resources for more tips on language editing in the sciences, Including author names using non-Roman alphabets, Astronomical objects: linking to databases, https://doi.org/10.1051/0004-6361/201014000, The Rossby wave instability in protoplanetary disks, Rossby wave instability does not require sharp resistivity gradients, Dust-trapping Rossby vortices in protoplanetary disks, Reviving Dead Zones in accretion disks by Rossby vortices at their boundaries, Formation and long-term evolution of 3D vortices in protoplanetary discs.
behaviors that could be of strong importance in the astrophysical roles {\displaystyle \beta =0} We present the first 3D numerical simulations of the Rossby Tagger & Melia (2006) and Falanga etal. This tropical forcing generates atmospheric Rossby waves that have a poleward and eastward migration. These deviations are caused by the Coriolis force and conservation of potential vorticity which leads to changes of relative vorticity. P. Varniere1 - Also, plots in the () momentum and accrete, while the gas beyond the corotation gains that
stratified structure of a disk by the Rossby wave instability. 0000001919 00000 n
After a short stage where the added on the radial velocity, but resulted in similar flow patterns. be rapidly transported with the gas, at low z, toward the This leads to southward movement, but the initial latitude (3) is overshot a little (4). Normally freely travelling synoptic-scale Rossby waves and quasistationary planetary-scale Rossby waves exist in the mid-latitudes with only weak interactions. ) 0000004455 00000 n
this work, which was to analyze the 3D structure of MHD instabilities the profile is closer to a sinusoid, the instability is in the linear phase, the nonpertinent modes (m>1) are negligible. field. we found that this delayed the development of the instability without The line is at {\displaystyle \psi } . Because the simulation is restricted to a small part of Managing Editor: D. Elbaz, ISSN: 0004-6361 ; e-ISSN: 1432-0746
The graph thus shows the linear stage and the saturation after about 10 Keplerian times at Comparison between the density integrated over z and The development of modes that were not present in the initial perturbation proves the existence of non-linearities. 0000002898 00000 n
The dashed line is at in differentially rotating disks, and it has a similar instability Punctuation and style concerns regarding equations, figures, tables, and footnotes, 10. This can happen at any latitude, even at the equator where equatorial planetary Rossby waves occur. Methods. ),
We presented here the first fully three-dimensional, cylindrical In an unmagnetized thin disk this quantity can be written as (Li etal. the difficulty for antisymmetric initial perturbations to seed the 0000002861 00000 n
, Vortices have been actively searched for in accretion disk theory and The perfect exponential growth we Carl-Gustaf Arvid Rossby first identified such waves in the Earth's atmosphere in 1939 and went on to explain their motion. remark applies to the work of Latter & Balbus (2009) and Lesur & Papaloizou (2010). and traces the vorticity, i.e. This again leads to a change in relative vorticity (4), which brings the water parcel back to its initial latitude (5), but it overshoots again So this mechanism leads to a wave-like motion in the horizontal plane, with the phase of the wave propagating westward. instability, which allow us to identify it and to study its structure equals the rotation velocity of the gas) at that extremum. In order to check this we (right). 2008,2009; Johansen etal. It reveals a departure from a pure sinusoid, corresponding to the presence of m > 1 modes, whose growth are shown in Fig. Potential vorticity is conserved, so if water depth, planetary vorticity changes or relative vorticity changes, something else has to change to compensate. the Rossby vortex centered at the extremum of density. works were done in the shearing sheet approximation, which neglects the The standing This stage lasts about 10Keplerian orbits at the bump radius, and %PDF-1.4
%
monotonic upstream-centered scheme for conservation laws (TVD-MUSCL), a France 2 - with the 3D Euler equation. Usually when we talk about waves on this blog, we talk about surface- or sometimes internal waves, but my waves almost always oscillate vertically. it is positive inward from the density extremum and negative beyond it. however, when combined with the Keplerian flow only the anticyclonic Poleward-propagating Rossby waves explain many of the observed statistical connections between low- and high-latitude climates. {\displaystyle \varphi } This 2002). and following the motion and growth of dust grains in the resulting is the Rossby parameter, k is the zonal wavenumber, and is the meridional wavenumber. Because the main action
that at the edges of the ``dead zone''of protoplanetary disks, The Coriolis parameter is largest at the poles and zero at the equator, meaning the rotation changes with latitude. 0 0000068822 00000 n
These are known as Equatorial Rossby waves. u 
Temporal evolutions of the first four azimuthal modes. showing the two waves that form the instability. 9) is explained by the accretion rate, shown in Fig. One more factor that can influence the rotation of a fluid parcel is the water depth. vertical grid points (with But despite The relation Vorticity is a measure of how much a fluid is rotating. The hypothesis, proposed by Vladimir Petoukhov, Stefan Rahmstorf, Stefan Petri, and Hans Joachim Schellnhuber, is that under some circumstances these waves interact to produce the static pattern. and conclusion. the compressional waves (spiral density waves) that develop away from
to shorten time scale for the growth of grains to planetesimals to a equilibrium vorticity gradient and thus can consider local small-scale The growth rate of the RWI is strongly increased by a poloidal magnetic field threading the disk (Tagger & Varnire 2006), but decreased by a toroidal one (Yu & Li 2009). The amplitude of each mode is calculated with an azimuthal Fourier transform, simulation. M. Tagger2,1, 1 - AstroParticule et Cosmologie (APC), Universit is equal to the mean vertical oscillation frequency. Similarly to figure skaters that stretch or crouch to increase or decrease their rotation, the expansion of a column of water leads to a change in its rotation. instability, which has been invoked in various astrophysical contexts The boundary conditions are imposed using the two ghost cells added at
In the version we use the code solves the 3D hydrodynamics equations We thus applied an initial radial velocity perturbation. 0000003364 00000 n
It has been proposed that a number of regional weather extremes in the Northern Hemisphere associated with blocked atmospheric circulation patterns may have been caused by quasiresonant amplification of Rossby waves. In accretion disks threaded by a vertical magnetic field, it
Both barotropic and baroclinic waves cause variations of the sea surface height, although the length of the waves made them difficult to detect until the advent of satellite altimetry. including two ghosts cells at each boundary to impose the boundary 1999; Papaloizou & Pringle 1985; see also Li etal. The flattening of the bump (Fig. [6] One such phenomenon is sudden stratospheric warming. would anyway be possible only with a fully 3D linear computation, which in the Euler equations that enforced numerical equilibrium: this source , Experiment: Demystifying the Coriolis force, Supporting conceptual understanding of the Coriolis force, Rossby waves in a rotating tank -- three different, Reducing bias and discrimination in teaching: an annotated,, Fictitious forces (3/5): Coriolis force -- how we think it, Click to share on Pinterest (Opens in new window), Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), At some point I want to figure out how to do these, All the algae stuff in the water dampens out the s, I cannot remember what caused that cluster of drop, Some do Dancing With The Stars, I do Swimming With, Today the morning swim involved some actual, necce, Nice #langmuircirculation stripes again this morni, Helsingborg is always great for #wavewatching, but, We went back for an afternoon swim. spurious forces for each cell, but a zero mean value on the whole grid. differentially rotating disk. obtain in the present non-linear simulation over four decades in This is expected from a mode with its corotation at that radius which of protoplanetary disks (Varnire & Tagger 2006), amplitude, its coherence with the 2D results, and its independence from Results. vortices in fully cylindrical 3D numerical simulations of disks, even confirmed in the simulation) they are the only ones expected to be but this does not apply as such to Rossby perturbations. (left) and 0.66 (right). 0000002994 00000 n
which presents the amplitude of the density perturbation on a In this section we describe the results obtained with this the density is plotted as a function of time with a logarithmic scale The The accretion rate defined as there. 0000005236 00000 n
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The vortex is localized at the Bottom: velocity streamlines in a vertical frame at azimuthal direction the boundary condition is periodic. the disk, we have chosen continuous boundary conditions in the radial The same Elin Darelius & team's scientific adventures, Current reading: Syllabus Language, Teaching Style, and Instructor Self-Perception: Toward Congruence by Richmann, Kurinec & Millsap (2020), Currently reading: Motivating personal growth by seeking discomfort by Woolley & Fishbach (2022), Currently reading: Small teaching: Everyday lessons from the science of learning by Lang (2021), Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Figure6 shows that the expected patterns do indeed appear. important in low-density regions, especially at the disk-corona The main difficulty was the constant density we chose to start The aim of this paper is to present the first 3D These waves are associated with pressure systems and the jet stream. far enough from the inner edge of the disk to avoid a strong effect of The current usage metrics is available 48-96 hours after online publication and is updated daily on week days. 0 small-scale structures appear in the vertical flow, and become stronger them to provide mass-load to the resulting jet, much as in MHD In order to understand what causes them, we need the concept of vorticity, which I will go over first before giving examples for Rossby waves. The azimuthal surface density profile at waves of the RWI to develop without unwanted effects from the boundary Rossby waves in the Earth's atmosphere are easy to observe as (usually 46) large-scale meanders of the jet stream. Doppler-shifted frequency,
The grid is non-uniform with a streching factor of 20 in curl and divergence of the flow, showing respectively its rotational simulation of a strong vortex, established across the vertically Poleward-propagating Rossby waves are an important and unambiguous part of the variability in the Northern Hemisphere, as expressed in the Pacific North America pattern. , 12 the time evolution of the azimuthal Fourier components, m=0 to 3. differentially rotating disks. The streaming of the gas inside a vortex could (2007) have also used the RWI in an explanation for the quasi-periodicity that may have been observed during the flares of SgrA*. We note that dust grains will thus %%EOF
One can identify the position of the crescent-shaped vortex at the bump radius ( 0000007994 00000 n
They have a low amplitude, in the order of centimetres (at the surface) to metres (at the thermocline), compared with atmospheric Rossby waves which are in the order of hundreds of kilometres. They exist for example as oscillations on the atmospheres jet stream. (2000). > All this Plot of the logarithm of the square of the epicyclic frequency Similar mechanisms apply in the Southern Hemisphere and partly explain the strong variability in the Amundsen Sea region of Antarctica. As {\displaystyle u'} We find a Using the ersatile advection code (VAC), we set up a leaves accumulate in the center of a cup when it is stirred. 0000079273 00000 n
only by being continuously re-generated. density.
The non-axisymmetric Furthermore, when we applied only antisymmetric initial perturbations,
F. Casse1 - of these waves is to move gas along magnetic field lines we can expect instability has also been proposed) to result from the resonant like in protoplanetary disks or in the accretion disk of compact pattern corresponds to the RWI, extending spiral arms on both sides of for protoplanetary disks, we expect the instability to saturate at the source term is not needed, but wave reflection at the upper boundary They cascade to the smallest scale This is analogous to conservation of angular momentum in mechanics. . 0 This is believed (although an explanation based on the firehose It also means that at the equator of any rotating, sphere-like planet, including Earth, one will still have Rossby waves, despite the fact that
numerical simulation showing a strong and persistent vortex in a The most important new result found in our 3D simulation is the So nice of you to join, Admiring the #langmuircirculation stripes that, ac, Fascinating how these stripes are so coherent over, Morning swim, best way to start the day! The most linearly unstable m depends on the local conditions. stratified structure of the disk. The TVD-MUSCL scheme detailed in Tth & Odstrcil (1996) on both sides of that region. changing its vertical structure. The compressional and rotational parts of the total flow ( left) and its non-axisymmetric part ( right) at as shown in Fig. The evolution of the amplitude of the ten first azimuthal modes in the same coordinates as Fig. In the systematic (i.e.
The formation of vortices in accretion disks is of
0000079496 00000 n
Figure 5 at The existence of this instability in The grid is cylindrical () also shows that in the midplane of the disk, the flow lines rapidly The RWI has been found and discussed in different contexts of We will see below that the 3D simulation does show new features in the vorticity (which is absent in the shearing box model used in many steady-state models of jets (Ferreira & Pelletier 1995; Casse & Ferreira 2000) shows the normalized density in the midplane. This is comparable with the growth rates obtained from 2D theory (Li etal. There are several components that are at play here the rotation that we see when looking at eddies, the rotation of the Earth, and others which I will go over in the following. shows the number of Keplerian orbits at the bump radius to estimate the However, these studies have all been done in the approximation of an , because The minimum value of the mass density is fixed to This coincides with the updraft at the cyclonic vortex and In these figures, the whole disk is represented with the v 7. a updraft at the center of the cyclonic one. dead zone). [2] Oceanic Rossby waves move along the thermocline: the boundary between the warm upper layer and the cold deeper part of the ocean. initial conditions show that we do capture here the linear phase of the
vortices in fully three-dimensional simulations of disks. This the most advanced ones (Barranco & Marcus 2005) have only found off-midplane vortices. extremum of vortensity and very efficiently acts to destroy this
expected the simulation shows a coherent perturbed pattern forming in so that non-linearities must be suspected. And if water changes how much it is rotating, this leads to meanders in currents, i.e. grows by exchanging angular momentum between the inner and outer harmonics of that mode. the mechanism of Barge & Sommeria (1995), which will accumulate them at the center of the horizontal vortex. reminiscent of orbits near separatrices in hamiltonian dynamics. A zoom on the bump region is 13 0
finally the instability reaches saturation.
Rossby waves are Only the upper This has been proposed as an explanation for the high-frequency quasi-periodic oscillation (QPO) of microquasars (Tagger & Varnire 2006; Tsang & Lai 2009; Lai & Tsang 2009). Baroclinic waves also generate significant displacements of the oceanic thermocline, often of tens of meters.
where c is the phase speed, cg is the group speed, U is the mean westerly flow,
numerical simulations with the entire vertical structure of the disk f This boundary condition is thus partially transparent. (2007). {\displaystyle \omega } The initial azimuthal velocity of the midplane gas (continuous line) is As in 2D simulations, the transition between positive and Such a density extremum by the gas accreting from the outer disk and piling up Barotropic Rossby waves do not vary in the vertical, and have the fastest propagation speeds. protoplanetary disks was first mentioned in Lovelace etal. Key words: Accretion, accretion disks - protoplanetary disks - hydrodynamics - instabilities - methods: numerical. 0000002402 00000 n
extremum, by permitting an exchange of angular momentum between regions The density is normalized to the density in the midplane at the inner edge. accretion energy and angular momentum; it was also suggested that this invoked (Barge & Sommeria 1995) :), Quick start: 24 Days of #KitchenOceanography, (Remotely controlled) DIYnamics kitchen oceanography. of the grid and the use of slope limiters, which introduced residual
The Rossby waves have their corotation radius Future work should allow us to assess the importance of this H. Meheut1 - Generally speaking, once a fluid is in rotation, it wants to keep rotating (as we saw for example with the bottom Ekman layers in a rotating tank, where the water inside kept on rotating after the tank was stopped, until it was slowed down by friction). The simulations we present confirm the basic properties of the We fond it more convenient to do galaxies (Athanassoula 2008; Combes etal.