organizing committee - research institute for … · organizing committee: michio yamada (kyoto...
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Organizing Committee:
Michio Yamada (Kyoto University, Chair) Shigeo Yoden (Kyoto University)
Kazunari Shibata (Kyoto University) Yukio Kaneda (Aichi Insititute of Technology)
Yoshi-Yuki Hayashi (Kobe University) Keiichi Ishioka (Kyoto University)
Shin-ichi Takehiro (Kyoto University) Email:[email protected]
Program
November 6 (Wed)
9:15 -- 9:30 Opening
9:30 -- 10:30 R. K. Scott (Univ. St Andrews)
Zonal jet formation by potential vorticity mixing at large
and small scales
10:50 -- 11:50 E. Kartaschova (J. Kepler University)
Exact resonant triads of Rossby waves as a basic model for
intra-seasonal oscillations in the Earth's atmosphere
(Lunch)
13:20 -- 14:20 L. M. Smith (Univ. Wisconsin)
On the role of near resonances for the formation of jets
and vortices in geophysical flows
14:40 -- 15:40 W. R. Young (Scripps Inst. Oceanography)
Zonostrophic Instability
16:00 -- 17:00 Short oral presentations for Posters
November 7 (Thu)
9:30 -- 10:30 M. Heimpel (Univ. Alberta)
The depth and structure of zonal flow from numerical models
of giant planets
10:50 -- 11:50 E. Dormy (Ecole Normale Superieure)
Zonal flows, dipole collapse and dynamo waves in numerical
dynamos
(Lunch)
13:20 -- 14:20 G. I. Ogilvie (Univ. Cambridge)
The evolution of warped astrophysical discs due to
angular-momentum transport and unstable internal waves
14:40 -- 15:40 A. S. Brun (CEA-Saclay)
On differential rotation and magnetism of solar-like stars
15:40 -- 17:30 Core time for poster presentations
(Banquet)
November 8 (Fri)
9:30 -- 10:30 J. Aurnou (UCLA)
Experimental studies of turbulent rotating convection
systems
10:50 -- 11:50 P. Williams (Univ. Reading)
Numerical simulation of banded jets in the laboratory
(Lunch)
13:20 -- 14:20 D. R. Durran (Univ. Washington)
Estimating the response of mid-latitude Orographic
precipitation to global warming
14:40 -- 15:40 T. G. Shepherd (Univ. Reading)
Complexities of the atmospheric jet stream
16:00 -- 17:00 G. K. Vallis (GFDL/Princeton Univ)
Zonal jets and equatorial superrotation in terrestrial
atmospheres
17:00 -- 17:15 Closing
List of poster presentations P1: K. Hori, S. Takehiro, H. Shimizu
Waves and linear stability of magnetoconvection in a Rotating cylindrical
annulus.
P2: Takahiro Iwayama, Takeshi Watanabe
Universal spectrum in the infrared range of two-dimensional turbulent flows
P3: Umin LEE
Formation mechanism of viscous decretion discs around rapidly rotating
stars
P4: Noboru Nakamura, Clare Huang
Finite amplitude dispersion relation and breaking of Rossby waves on a
sphere
P5: Satoshi Noda, Masaki Ishiwatari, Kensuke Nakajima, Yoshiyuki O.
Takahashi, Shin-ichi Takehiro, Masanori Onishi,,George L. Hashimoto,
Kiyoshi Kuramoto, Yoshi-Yuki Hayashi
Atmospheric general circulations of synchronously rotating water-covered
exoplanets: Dependence on planetary rotation rate
P6: Kiori Obuse, Shin-ichi Takehiro, Michio Yamada
Merging and disappearing processes of zonal jets in forced two-dimensional
turbulence on a rotating sphere
P7: Izumi Saito, Keiichi Ishioka
Angular distribution of energy spectrum in two-dimensional beta-plane
turbulence in the long-wave limit
P8: Eiichi Sasaki, Shin-ichi Takehiro, Michio Yamada
Approximation of chaotic zonal flow on a sphere
P9: Youhei SASAKI, Shin-ichi Takehiro, Ken-suke Nakajima, Yoshi-Yuki Hayashi
Surface zonal flows induced by thermal convection in a rapidly rotating
thin spherical shell
P10: N. Sugimoto, K. Ishioka, H. Kobayashi, Y. Shimomura
Spontaneous gravity wave radiation from a co-rotating vortex pair in an
unbounded f-plane shallow water system
P11: Yuki Yasuda, Kaoru Sato, Norihiko Sugimoto
A Theoretical Examination for the Spontaneous Radiation of
Inertia-gravity Waves Using the Renormalization Group Method
P12: Keiichi Ishioka
A Proof for the Equivalence of Two Upper Bounds for the Growth of
Disturbances from Barotropic Instability
1 SCOTT, Richard Kirkness Univ. St Andrews School of Mathematics and Statistics [email protected]
2 SMITH, Leslie Morgan Univ. Wisconsin Department of Mathematics & Engineering Physics [email protected]
3 YOUNG, William R. University of California, San Diego Scripps Institution of Oceanography [email protected]
4 SHEPHERD Ted. G. Univ. Reading Department of Meteorology [email protected]
5 VALLIS, Geoff GFDL/Princeton Univ. Program in Atmospheric and Oceanic Sciences [email protected]
6 DORMY, Emmanuel Ecole Normale Superieure Département de Physique [email protected]
7 HEIMPEL, Moritz Univ. Alberta Department of Physics [email protected]
8 OGILVIE, Gordon Univ. Cambridge Department of Applied Mathematics and Theoretical Physics [email protected]
9 AURNOU, Jonathan UCLA Department of Earth and Space Sciences [email protected]
10 WILLIAMS, Paul Univ. Reading Department of Meteorology and the National Centre for Atmospheric Science [email protected]
11 KARTASCHOVA, Elena J. Kepler Univ. Research Institute for Symbolic Computation [email protected]
12 Brun, Allan Sacha CEA Saclay Laboratory on Dynamics of Stars and theirs Environments [email protected]
13 Durran, Dale R. Univ. Washington Department of Atmospheric Sciences [email protected]
14 Aoki, Kunihiro Hokkaido University Faculty of Environmental Earth Science [email protected]
15 Amemiya, Arata University of Tokyo Faculty of Earth and Planetary Science [email protected]
16 Hayashi, Michiya University of Tokyo AORI [email protected]
17 Hohokabe, Hirotaka Kyushu University Department of Earth and Planetary Sciences [email protected]
18 Hori, Kumiko University of Tokyo Earthquake Research Institute [email protected]
19 Iwayama, Takahiro Kobe University Department of Earth and Planetary Sciences [email protected]
20 Lee, Umin Tohoku University [email protected]
21 Nakamura, Noboru University of Chicago Department of the Geophysical Sciences [email protected]
22 Noda, Satoshi Japan Meteorological Agency Meteorological Research Institute [email protected]
23 Noguchi, Shunsuke Kyoto University Graduate school of Science [email protected]
24 Obuse, Kiori Tohoku University WPI-AIMR [email protected]
25 Ohfuchi, Wataru Japan Agency for Marine Science and Technology [email protected]
26 Saito, Izumi Kyoto University Graduate school of Science [email protected]
27 Sasaki, Eiichi Kyoto University Graduate school of infomatics [email protected]
28 Sasaki, Youhei Kyoto University Department of Mathematics [email protected]
29 Shimomura, Yutaka Keio University [email protected]
30 Sugimoto, Norihiko Keio University [email protected]
31 Yasuda, Yuki University of Tokyo Graduate school of Science [email protected]
32 Yoshida, Shigeo Kyushu University Department of Earth and Planetary Sciences [email protected]
33 Yamada, Michio Kyoto University Research Institute for Mathematical Sciences [email protected]
34 Hayashi, Yoshi-Yuki Kobe University Department of Earth and Planetary Sciences [email protected]
35 Takehiro, Shin-ichi Kyoto University Research Institute for Mathematical Sciences [email protected]
36 Ishioka, Keiichi Kyoto University Graduate school of Science [email protected]
37 Nakajima, Kensuke Kyushu University Department of Earth and Planetary Sciences [email protected]
38 Yoden, Shigeo Kyoto University Graduate school of Science [email protected]
39 Dutrifoy, Alexandre Université Libre de Bruxelles FNRS [email protected]
40 Takahashi, Yoshiyuki Kobe University Center for Planetary Science [email protected]
List of Participants
Name University etc. Institution e-mail address
Zonal jet formation by potential vorticity mixing at large and small scales
Richard SCOTTEmail: [email protected]
School of Mathematics, University of St AndrewsSt Andrews, Scotland
A striking feature of the large-scale turbulent motions of planetary atmospheres and oceans is the presence of well-defined zonal jets, coexisting with a background turbulent flow. Here, we examine the formation of such zonal jets in geostrophic turbulence with an emphasis on the inhomogeneous potential vorticity mixing by turbulent eddy and wave motions. We examine different jet/turbulence regimes in the simplest possible system of a mid-latitude beta-plane, using high-resolution, long-time numerical integrations. Three key results will be discussed: (i) the emergence of strong jet motions when dynamical forcings are weak; (ii) the independence of the jet formation process from the two-dimensional inverse energy cascade; (iii) the identification of two distinct types of potential vorticity mixing, one dominated by turbulent eddies, the other by the action of localized Rossby wave critical layers. Jet regimes are found to depend in a simple way upon two non-dimensional parameters, which may be related to three natural length scales of the system: the Rhines scale, the forcing scale, and a length scale relating the strength of the forcing to the background potential vorticity gradient.
Two cases are considered, in which forcing scales are either (i) much smaller than, or (ii) comparable to the scale of the emerging jets. In the first case, the late-time distribution of potential vorticity is found to depend in a simple way on a single non-dimensional parameter that may be considered as a measure of the strength of the forcing. Jet strength increases as the forcing strength decreases and the limiting case of the potential vorticity staircase, comprising a monotonic, piecewise-constant profile in the north-south direction, is approached when the forcing is so weak that eddy intensities are substantially weaker than potential vorticity jumps across jet cores. For stronger forcing, in contrast, eddy intensities become sufficiently intense that they continually disrupt the steepening of potential vorticity gradients in jet cores, preventing strong jets from developing.
In the second case, in which the forcing scale is comparable to the scale of the emerging jets, the potential vorticity is again found to organize into a staircase-like profile, provided only that the Rhines scale is at least of the same order as the forcing scale. That strong jets are observed even when these two scales are similar demonstrates, in particular, that jet formation may be considered completely independently from dynamical processes associated with the two-dimensional turbulent inverse energy cascade. More generally, the character of potential vorticity mixing is found to depend on a parameter involving forcing strength, forcing scale, and planetary vorticity gradient, occurring either predominantly in localized critical layers, or through the more uniform small-scale turbulent eddy mixing. In the former case, care must be taken with the form of the dynamical forcing to ensure that the material advection of potential vorticity is not obscured. A combined condition for the formation of strong zonal jets is given summarizing the overall dependence on all parameters.
Nonlinear Resonance Analysis for Hasegawa-Mima Equation in Bounded Domains
Elena KARTASHOVA
Email: [email protected] IFA, Johannes Kepler University
Linz, 4040, Austria
In weakly nonlinear dispersive wave systems, during the last years Nonlinear Resonance Analysis has become the methodology of choice for characterizing energy transport due to exact resonances. For studying resonant dynamics one has to compute the solutions of the resonance conditions, construct the corresponding resonance clusters (in Fourier space) and solve – analytically or numerically – the set of dynamical systems corresponding to these clusters. We illustrate the power of this approach regarding the Hasegawa-Mima equation (HME) with two different boundary conditions. Example 1: HME with zero boundary conditions on a sphere (“rigid lid” approximation). The resonance conditions in this case read 𝑚𝑚1/𝑛𝑛1(𝑛𝑛1 + 1) + 𝑚𝑚2/𝑛𝑛2(𝑛𝑛2 + 1)= 𝑚𝑚3/𝑛𝑛3(𝑛𝑛3 + 1), 𝑚𝑚1 + 𝑚𝑚2 = 𝑚𝑚3, | 𝑛𝑛1 − 𝑛𝑛2| ≤ 𝑛𝑛3 ≤ 𝑛𝑛1 + 𝑛𝑛2, where integers 𝑚𝑚𝑗𝑗 and (𝑛𝑛𝑗𝑗 − 𝑚𝑚𝑗𝑗)= are longitudinal and latitudinal wave numbers. The resonance clusters in the meteorologically relevant domain 𝑚𝑚𝑗𝑗,𝑛𝑛𝑗𝑗 ≤ 21 are 4 isolated triads, 3 clusters of two triads connected by one joint mode and one cluster consisting of six interconnected triads. We show that intra-seasonal oscillations (IOs) of Earth’s atmospheric flow can be modeled by the four triads of resonantly interacting planetary waves, isolated from each other and from all other planetary waves. The model does not depend on the topography (mountains, etc.), gives a natural explanation of IOs in both the Northern and the Southern Hemispheres and allows interpreting interaction between tropical and mid-latitude IOs, etc. Example 2: HME with non-zero periodic boundary conditions on a β-plane, in the limit of infinite Rossby deformation radius. Resonance conditions in this case read 𝑚𝑚1/(𝑛𝑛12 + 𝑚𝑚1
2) + 𝑚𝑚2/(𝑛𝑛22 +𝑚𝑚22)= 𝑚𝑚3/(𝑛𝑛32 + 𝑚𝑚3
2), 𝑚𝑚1 + 𝑚𝑚2 = 𝑚𝑚3, 𝑛𝑛1 + 𝑛𝑛2 = 𝑛𝑛3, where integers 𝑚𝑚𝑗𝑗, 𝑛𝑛𝑗𝑗 are wave numbers. We demonstrate that in this case resonance clustering differs substantially from characteristic clustering in other 3-wave systems: instead of a usual set of isolated triads and a few bigger clusters, there exist no isolated triads. Resonant triads are interconnected in a complicated way and the smallest cluster consists of 6 connected triads and is formed by 6 distinct modes only. This specific clustering is due to the form of the dispersion function which allows in the general case 12 symmetries. We discuss consequences of this finding for programming the HME to study it numerically. We also argue that this result reveals a special mechanism of energy confinement by exact resonant modes which is lacking in other presently known three wave systems and even in the HME with other boundary conditions.
On the role of near resonances for the formation of jets and vortices in geophysical flows
Leslie SMITH
Email: [email protected] University of Wisconsin, Madison
Madison, WI 53706 USA The generation and propagation of coherent structures is investigated through the lens of wave-
wave and wave-vortical interactions. The framework is flow representation by superposition of the of linear eigenmodes, which form a complete basis: Rossby waves on the beta-plane; inertial waves within the f-cube; gravity waves and vortical modes for rotating shallow-water and 3D Boussinesq flows. Different types of reduced models are studied with goal to understand the most essential interactions for the formation and evolution of jets and vortices.
For purely rotating flows (beta-plane and f-cube), reduced models consisting of near resonant three-wave interactions are the natural step beyond weak turbulence theory of three-wave exact resonances. We demonstrate that near resonant interactions are primarily responsible for the generation of zonal flows on the beta-plane, and large-scale vortical columns within the f-cube (Figure 1). Furthermore, the near resonances induce east/west asymmetry on the beta-plane, and cyclonic/anticyclonic asymmetry within the f-cube.
Figure 1: A reduced model of near resonances generates zonal flows on the beta-plane (left:
zonally averaged velocity vs meridional direction) and vortical columns on the 3D f-plane (right: vertically averaged vorticity contours in the (x,y)-plane).
With stratification, the presence of the vortical eigenmode leads to richer set of exact and near
resonant interactions underlying flow evolution. Reduced models constructed by retaining an entire class (or several classes) of wave-vortical interactions keep selected classes of near resonant interactions, and they are PDEs in physical space. To improve upon QG, we consider a reduced model consisting of QG vortical-vortical-vortical interactions together with vortical-vortical-wave interactions. The latter model is closely related to higher-order PV-inversion schemes. One test case consists of an initially balanced dipole with localized region of high-speed flow in between the poles - a simplified model of a jet streak. In 3D, this and other test cases suggest that vortical-vortical-wave interactions critically influence the evolution of balanced flows, while vortical-wave-wave interactions are essential for structure formation.
−10 −5 0 5 100
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near resonances, t = 350
Zonostrophic Instability
William R. Young Email: [email protected]
Scripps Institution of Oceanography, University of California, San Diego 9500 Gilman Dr. #0213, La Jolla, CA 92093-0213, USA
A beta-plane flow driven by spatially homogeneous stochastic forcing spontaneously develops
strong zonal jets that break translational symmetry in the meridional direction. I show that the emergence of these jets can be understood as the linear instability of an underlying spatially homogeneous jetless flow i.e., homogeneous beta-plane turbulence is linearly unstable to the formation of jets. Analytic results, such as a calculation of the stability boundary, are obtained by neglecting the nonlinear eddy-eddy interactions and constructing deterministic equations for the evolution of the two-point single-time correlation function of the vorticity. The relation of this formulation to the stochastic structural stability theory of Farrel and Ioannou and the cumulant expansions of Marston and Tobias is clarified.
The Depth and Structure of Zonal Flow From Numerical Models of Giant Planets
Moritz Heimpel Email: [email protected]
University of Alberta, Department of Physics Edmonton, Alberta, Canada
Observations of cloud motions reveal the surface winds of all four giant planets in the solar system to be primarily zonal. In particular, Jupiter and Saturn exhibit strong prograde equatorial flow and weaker but well-defined bands of East-West flow in alternating directions at higher latitudes. The depth to which these flows penetrate has long been debated and is still an unsolved problem. The coexistence of zonal flows and vortices is also of interest. While the zonal flows of the gas giants alternate in latitude between cyclonic and anticyclonic shear zones and have a well-defined range of wavelengths that seem to be stable over long timescales, vortices on Jupiter show a clear asymmetry between cyclones and anticyclones, and both Jupiter and Saturn have vortices with a large range of observed sizes and lifetimes. We model rotating deep convection for relatively thin spherical shells using the benchmarked 3D spherical anelastic convection code MAGIC. Under both Boussinesq and compressible conditions, the models can reproduce the general surface zonal flow structure. In both types of models the dominant Coriolis force leads to cylindrically symmetric deep zonal flows. Whereas the equatorial flows span the northern and southern hemispheres outside the tangent cylinder, high latitude zonal flows are truncated at the inner boundary of the rotating convection model. To investigate the coexistence of zonal flow and vortices vary the radial background density variation and the thermal boundary conditions. We find that, whereas the formation of jets is relatively insensitive to these conditions, vortex formation and dynamics depends strongly on them. For constant entropy difference between top and bottom boundaries, models with strong radial density gradients form alternating, zonal flows, and very small-scale flow and vorticity structures at the outer boundary. Models driven by constant convective entropy flux at the bottom, and small convective, or reversed entropy flux at the top, also can produce multiple zonal jets. These models, with a nearly neutral or stably stratified background vertical thermal structure near the outer boundary, can produce vortices of larger scale and lifetime.
Figure 1: Development of cool anticyclonic pancake vortices in a stably stratified thin layer with strong zonal flow. Temperature at the outer boundary is shown in a, where warm is red and relatively cool is blue. Radial vorticity is shown in b (and magnified in c), where red (blue) is cyclonic (anticyclonic) radial vorticity in the northern hemisphere, while the sign is reversed in the southern hemisphere (red is anticyclonic). The axial vorticity in a meridional slice that coincides with the large vortex in c is shown in d, where red (blue) axial vorticity is cyclonic (anticyclonic).
Zonal flows, dipole collapse and dynamo waves in numerical dynamos
Emmanuel DORMY Email: [email protected]
CNRS/ENS/IPGP, Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
I will review some of the recent progress on modeling planetary and stellar dynamos. Particular attention will be given to the dynamo mechanisms and the resulting properties of the field. I will present direct numerical simulations using a simple Boussinesq model. These simulations are interpreted using the classical mean-field formalism. I then investigate the transition from steady dipolar to multipolar dynamo waves solutions varying different control parameters, and discuss the relevance to stellar magnetic fields. I will show that owing to the role of the strong zonal flow, this transition is hysteretic. In the presence of stress-free boundary conditions, the bistability extends over a wide range of parameters. I will then compare the Boussinesq and anelastic models. Scaling relations for Boussinesq and anelastic models are found to be very similar and the effects of compressibility do not appear to be essentials.
Figure 1 Time evolution of the radial magnetic field averaged in longitude (for an aspect ratio of 0.65) for a Boussinesq Model. The initial dipole field survives for a few diffusion times, and then vanishes to yield a butterfly-like diagram (after Goudard & Dormy, 2008).
The evolution of warped astrophysical discs due toangular momentum transport and unstable internal waves
Gordon OGILVIEEmail: [email protected]
DAMTP, University of CambridgeCMS, Wilberforce Road, Cambridge CB3 0WA, UK
Astrophysical discs, which include circumstellar discs in which planets are formed and high-energy accretion discs around black holes, consist of a continuous distribution of material in orbital motion around a central massive body. In a standard, planar disc, the “zonal flow” is fixed by the orbital dynamics, except inasmuch as radial pressure gradients can support small departures from pure orbital motion. The radial transport or vertical removal of angular momentum leads to a radial accretion flow.
A warped disc, however, is one in which the plane of the orbital motion varies freely with radius and time. Radial transport or external exchange of the vector angular momentum leads to evolution of both the shape and the mass distribution of the disc. There is strong theoretical and observation motivation for considering warped discs in several situations. Current interest, for example, focuses on black holes in galactic nuclei, which grow by accreting gas that is supplied in different orientations, and protoplanetary systems, where spin-orbit misalignments have been discovered, suggesting that a warped disc may have been involved.
I introduce a new local model for the study of warped astrophysical discs. This generalizes the well known shearing sheet of Goldreich & Lynden-Bell by imposing the local curvature of the orbital plane in addition to shear and rotation. The simplest hydrodynamic solutions in the local model are horizontally homogeneous laminar flows that oscillate at the orbital frequency. These determine the large-scale evolution of the shape and mass distribution of the disc through their hydrodynamic stresses. I present a simpler and independent derivation of the basic equations for warped discs obtained previously by Ogilvie and describe the resonance that leads to the anomalous behaviour of Keplerian warped discs.
I also analyse the hydrodynamic stability of the laminar flows and find widespread instability deriving (at least initially) from parametric resonances of inertial waves. Other internal waves could play a similar role in stably stratified discs. I present preliminary results of numerical simulations (by S.-J. Paardekooper) of the nonlinear outcome of the instability. Very small warps in nearly Keplerian discs of low viscosity can be expected to generate hydrodynamic turbulence, or at least sustained wave activity, by this mechanism. As well as modifying the dynamics of the warp by moderating the internal flows and generating additional stresses, this dynamics could have important consequences for the processes of planet formation.
On Differential Rotation and Magnetism of solar-like stars
Allan Sacha BRUN Email: [email protected]
AIM, CEA-Saclay, France and RIMS, Kyoto University, Japan
We have performed a systematic study of the differential rotation realized in 3-D anelastic numerical simulation of solar-like G & K stars (i.e stars with mass ranging from 0.5 to 1.1 Msol). By varying the rotation rate from 1 times solar to 5 times solar we show how the differential rotation ca be either, anti-solar (slow equator, fast poles), solar-like (fast equator-slow pole with a conical profile at mid latitude) or cyclindrical (with alternance of prograde and retrograde jets; Jupiter-like). We discuss the orgin of this change of behavior in terms of angular momentum redistribution within the convective shells and thermal wind effects. We deduce scaling law for the differential rotation and associated meridional circulation (resulting from so-called gyroscoping pumping) and uses these laws to model in 2-D activity cycles in solar-like stars comparing the cycle dependence to rotation rate to observations. We then discuss dynamo action in 3-D simulations and the existence of intense magnetic wreaths that may become unstable and form for the first time a « spot »-dynamo as seen in the Sun.
Figure 1 3 types of differential rotation realized in simulations of G & K depending on their Rossby number
Laboratory Models of Rapidly Rotating Convective Turbulence
Jonathan AURNOU Email: [email protected]
UCLA Earth & Space Sciences Los Angeles, CA, 90095-1567 USA
Rapidly-rotating, convective turbulence underlies the generation of large-scale zonal flows in almost all geophysical and astrophysical fluid systems. In this talk, I will review laboratory experimental simulations of thermal convection in rotating, axially-aligned, cylindrical containers. These cylindrical experiments are essentially local-scale models of geophysical turbulence: without boundary curvature zonal flows do not form in these experiments, thus allowing us to focus on the small-scale dynamics. I will focus on the differing rotating convective behaviors that develop in water (a moderate Prandtl fluid) as well as in molten gallium (a low Prandtl fluid). In addition, throughout the talk, laboratory experimental results will be compared with the results of direct numerical simulations and asymptotically-reduced models of rapidly rotating convection (Figure 1).
Figure 1: Images from direct numerical simulations (DNS), laboratory experiments and asymptotically-reduced models of rapidly rotating convection in water. This figure shows three representative cases from the convective Taylor column regime. The DNS and reduced modeling images show renderings of the temperature anomaly field. Kalliroscope flakes are used to image shear structures in the laboratory experiment. The DNS and laboratory experiments are both carried out at Ekman number E = 1e-7, and the reduced model is scaled to that E value. All three cases have Rayleigh number values that are less than ten times supercritical.
Numerical simulations of banded jets in the laboratory
Paul WILLIAMS Email: [email protected]
Department of Meteorology, University of Reading Earley Gate, Reading, RG6 6BB, UK
It is natural to want to study banded jets in the laboratory, so that controlled and repeatable
experiments can be performed in a wide range of dynamical regimes. The potential vorticity gradient required for zonation, which is supplied on a rotating sphere by the planetary vorticity gradient, is typically supplied in laboratory experiments by using either sloping topography or the parabolic shape of the free surface associated with solid-body rotation. Several such laboratory experiments have been performed to study zonation. Due to their small geometric dimensions, however, laboratory experiments have typically been unable to access dynamical regimes at the high Reynolds numbers relevant to zonation in planetary atmospheres and oceans.
In an attempt to access dynamical regimes at higher Reynolds numbers, Read et al. (2004, 2007) performed laboratory experiments on the French Coriolis platform, which has a diameter of 14 m. The experiments were performed in a rotating annulus, with an inner radius of 2 m and an outer radius of 6.5 m. The bottom of the tank sloped downwards with increasing radius, providing a topographic beta-effect. The local Reynolds numbers (based on the jet scales) were estimated to exceed 2000, which is at least an order of magnitude greater than in previous experiments. There was clear evidence of zonal jet formation on the Rhines scale, although there was substantial meandering of the jets and occasional splitting and mergers.
This talk will present results from the first study to attempt to simulate the laboratory experiments of Read et al. (2004, 2007). Fifteen high-resolution, long-duration numerical simulations have been performed using a two-layer quasi-geostrophic model, at various combinations of the baroclinic deformation radius and the Rhines scale. The numerical results will be compared with the laboratory experiments, in an attempt to gain an improved dynamical understanding of the jet features observed, and to investigate the impacts of finite baroclinic deformation radii on the jet spacing.
Figure 1 Baroclinic interface height perturbation in a quasi-geostrophic numerical simulation of a rotating annulus experiment performed in the laboratory.
The Role of the Jet Stream in the Response ofMid-Laititude Orographic Precipitation to Increases in CO2
byDale DURRAN and Xiaoming SHI
Email: [email protected] of Atmospheric Sciences, University of Washington, Seattle
Seattle, WA, 98195, USA
Instabilities in the midlatitude jet streams of the Earth’s atmosphere are responsible fora significant fraction of the mid-latitude precipitation. General circulation models suggestthere will be changes in the mean position of the jets and storm tracks in a warmer world,which are expected to produce commensurate changes in the zonally-averaged distributionof midlatitude precipitation. In this talk, we will examine the possibility that shifts in theposition of the jet stream might lead to much more dramatic changes in local precipitationon the windward sides of major north-south mountain barriers.
The sensitivity of stratiform mid-latitude orographic precipitation to global mean tem-perature is investigated through numerical simulations with a general circulation model.The individual terrain elements are idealizations of the Earth’s major north-south mountainchains and occupy four islands equally spaced around the northern hemisphere of a planetotherwise covered by ocean. Although mountains have only a modest influence on the sen-sitivity of the zonally averaged precipitation to changes in global mean surface temperature,the precipitation on the windward slopes of the ridges themselves is highly sensitive to suchchanges. For ridges extending from 40N to 60N, the windward-slope hydrological sen-sitivity exceeds the Clausius-Clapeyron scaling of ∼ 7 %K−1 over the northern half of thebarrier. When CO2 is doubled, our simulations generate a global mean temperature increaseof 2.56 K, leading to substantial precipitation changes, particularly at the northern ends ofthe mountains, where the annual precipitation increases by 36%.
The changes in orographic precipitation are linked to northward shifts in the storm tracks,which alter both zonal mean properties and the the frequency with which storms strike themountains. The annual accumulated precipitation is modified by changes in the mean pre-cipitation intensity and the annual mean hours during which precipitation occurs. A simplediagnostic model reveals the primary factors modifying the mean orographic precipitationintensity are changes in (1) the moist adiabatic lapse rate of saturation specific humidity, (2)the wind speed perpendicular to the mountain, and (3) the vertical displacement of saturatedair parcels above the windward slope. The strong dependence of (2) and (3) on latitude fur-ther confirms the important influence of storm track shifts on the response of mid-latitudeorographic precipitation to global warming.
Complexities of the atmospheric jet stream
Theodore G. SHEPHERD Email: [email protected]
Department of Meteorology, University of Reading Reading, Berks. RG6 6BB U.K.
The zonal flow in Earth’s atmosphere is known as the jet stream. It represents a fundamental aspect of the atmospheric circulation and is also linked with weather phenomena such as storm tracks. Two basic ingredients of the jet stream are the tropical Hadley circulation, which leads to a strong subtropical jet, and upper tropospheric momentum fluxes from midlatitude baroclinic eddies, which can themselves induce an “eddy driven” jet. In practice the Hadley-driven and eddy-driven jet co-exist, sometimes together and sometimes distinctly. Moreover there are strong longitudinal asymmetries in the jet, especially in the Northern Hemisphere, due to asymmetries in surface conditions, and resulting inter-hemispheric differences. Understanding the structure of the atmospheric jet stream, its seasonal dependence, its variability, and its response to external forcing (including climate change), at a quantitative level severely challenges our theoretical understanding of atmospheric dynamics. It appears that the jet stream behaves very nonlinearly, which means that the implications of biases in climate models need to be understood before their results can be used with confidence.
Jets and Superrotation in Idealized Atmospheres
Geoffrey VALLIS With Sam Potter and Jonathan Mitchell
Email: [email protected] University of Exeter, UK
We explore the mechanisms of zonal jets, and in particular equatorial superrotation, using very idealized numerical models using the primitive equations on the sphere. It is well-known that zonal jets robustly arise in rotating atmospheres there is a wavemaker at a particular latitude. Rossby waves are then generated that propagate away, and eastward momentum converging on the source region producing a zonal jet. Such a mechanism produces the jet stream on Earth and, most likely, the jets on giant planets including the equatorial superrotation. However, on slowly rotating terrestrial atmospheres it seems unlikely that superrotation is produced by that mechanism. Rather, simulations indicate that, at small thermal Rossby number, a mechanism involving equatorial Kelvin waves is involved. We will describe a number of simulations with dry dynamical cores exploring this phenomenon.
.
Waves and linear instability of magnetoconvection in a rotating cylindrical annulus
Kumiko Hori *1, Shin-ichi Takehiro2, Hisayoshi Shimizu1 * Email: khori @ eri.u-tokyo.ac.jp
1) Earthquake Research Institute, University of Tokyo, Tokyo, 113-0032, Japan2) RIMS, Kyoto University, Kyoto, 606-8502, Japan
Magnetohydrodynamic waves in a rapidly rotating planetary core can cause the magnetic secular variation. To strengthen the physical basis of such waves, we revisit the linear stability analyses on thermal convection in a quasi-geostrophic rotating cylindrical annulus applied by a magnetic field and extend investigation of the oscillatory modes to a broader range of the parameters. Particular attention is paid to influence of thermal boundary conditions, either fixed temperature or heat-flux conditions. While the non-dissipative approximation yields a slow wave propagating retrograde (westward), termed as an MC(Magnetic-Coriolis)-Rossby wave, dissipative effects make them various. When magnetic diffusion is much stronger than thermal diffusion, a very slow wave propagating prograde (eastward) can be excited. Retrograde-travelling slow waves set in when magnetic diffusion is weaker. Emergence of the slow modes helps the convection to occur at lower critical Rayleigh numbers than in the nonmagnetic case. The convection onsets with the prograde-propagating slow wave for strong magnetic diffusion, whereas a slow MC mode conducts the critical convection for weak magnetic diffusion. Fixed heat-flux boundary conditions have profound effects on marginal curves, which monotonically increase with the horizontal wavenumber and lead to larger length scales at the onset, provided proper field strength for the Lorentz force to balance with the Coriolis force. The effect however becomes less clear as magnetic diffusion is weakened, where various magnetohydrodynamic waves set in.
P1
Universal spectrum in the infrared range of two-dimensional turbulent flows
Takahiro IWAYAMA1 and Takeshi WATANABE2 Email1: [email protected]
Department of Earth and Planetary Sciences, Kobe University Kobe, 657-8501, Japan
Email2: [email protected] Department of Scientific Engineering and Simulation, Nagoya Institute of Technology
Nagoya, 466-8555, Japan
The low-wavenumber behavior of decaying turbulence governed by the generalized two-dimensional (2-D) fluid system, the so-called α-turbulence system, is investigated theoretically and through direct numerical simulation. This system is governed by the nonlinear advection equation for an advected scalar q and is characterized by the relationship between q and the stream function φ: 𝑞 = −(−∇2)𝛼/2 𝜑. Here, the parameter α is a real number that does not exceed 3. The enstrophy
transfer function in the infrared range (k → 0) is theoretically derived to be 𝑇𝛼Ｑ(𝑘 → 0) ~ 𝑘5 using
a quasi-normal Markovianized model of the generalized 2D fluid system. This leads to three canonical cases of the infrared enstrophy spectrum, which depend on the initial conditions: 𝑄𝛼(𝑘 →0)~𝐽𝑘, 𝑄𝛼(𝑘 → 0)~𝐿𝑘3, and 𝑄𝛼(𝑘 → 0)~𝐼𝑘5, where J, L, and I are various integral moments of two-point correlation for q. The prefactors J and L are shown to be invariants of the system, while I is an increasing function of time. The evolution from a narrow initial enstrophy spectrum exhibits a universal infrared enstrophy spectrum of the form 𝑄𝛼(𝑘 → 0)~𝐼𝑘5, which is independent of α. These results are verified by direct numerical simulations of the generalized 2D fluid system.
P2
Formation Mechanism of Viscous Decretion Discs around Rapidly Rotating Stars
Umin LEE Email: [email protected]
Astronomical Institute, Tohoku University, Sendai, Miyagi 980-8578, Japan
Be stars are spectral B-type stars close to the main sequence that exhibit line emission over the photospheric spectrum. Emission lines observed in Be stars are produced in circumstellar gaseous discs around the stars, most of which are known to be rapidly rotating at speeds close to the break-up one. Discs around Be stars are believed to be viscous Keplerian decretion discs, but their formation mechanism has not yet been identified. In this poster, I discuss a formation mechanism for steady viscous Keplerian decretion discs around rapidly rotating stars. B-type main-sequence stars are also known as a pulsator; low frequency g-modes and r-modes in slowly pulsating B stars and high frequency p-modes in beta-Cephei stars are excited by the opacity bump mechanism. Here, I assume that low frequency modes, which may be excited by the opacity bump mechanism, convective motion in the core, or tidal force if the star is in a binary system, can transport a sufficient amount of angular momentum to the region close to the stellar surface. For stellar rotation (zonal flows), I employ a theory of wave-meanflow interaction to derive a meanflow equation, which describes the angular momentum transfer by waves. Under these assumptions, I construct a star-disc system, in which there forms a viscous decretion disc around a rapidly rotating star because of the angular momentum supply. I find a series of solutions of steady viscous decretion discs around a rapidly rotating star that extend to more than 10 times the radius of the star, depending on the amount of angular momentum supply.
P3
P4 Finite amplitude dispersion relation and breaking
of Rossby waves on a sphere
Noboru Nakamura and Clare Huang [email protected]
Department of the Geophysical Sciences, University of Chicago
Breaking of Rossby waves is an important mechanism by which the angular momentum density of the zonal-mean flow is modified in the atmosphere. Whilst wave breaking is routinely associated with a critical line in shear flows, where the zonal phase speed of the wave matches the speed of the zonal mean flow, Rossby waves often break before reaching a critical line or even in the absence of one, exposing a gap between the theory and observation. The authors attempt to fill this gap by extending the concepts of zonal phase speed and critical lines for finite-amplitude barotropic Rossby waves on a sphere using the conserved quantities such as pseudomomentum and pseudoenergy. The approximate dispersion relation reduces to the linear dispersion relation at small amplitude. At the critical line the angular speed of the mean flow matches the modified frequency of the wave. Numerical simulations of freely decaying barotropic Rossby waves on a sphere are used to demonstrate the agreement of the critical lines and the location of wave breaking (maximum pseudomomentum dissipation).
Atmospheric general circulations of synchronously rotatingwater-covered exoplanets: Dependence on planetary rotation rate
S. Noda (1), M. Ishiwatari (2,3), K. Nakajima (4), Y. O. Takahashi (3,5),S. Takehiro (6), Masanori Onishi (5), George L. Hashimoto (7), Kiyoshi Kuramoto (2,3), and Y.-Y. Hayashi (3,5)
(1) Meteorological Research Institute, Tsukuba, Japan, (2) Hokkaido University, Sapporo, Japan,(3) Center for Planetary Science, Kobe, Japan, (4) Kyushu University, Fukuoka, Japan, (5) Kobe University, Kobe,
Japan, (6) Kyoto University, Kyoto, Japan, (7) Okayama University, Okayama, Japan
Email: [email protected]
In order to investigate circulation patterns and en-
ergy transport of synchronously rotating terrestrial
planets (One side of these plenets have perpetual day
and the other side of them have perpetual night.), a
parameter experiment for the dependence on plan-
etary rotation rate is performed by using a general
circulation model (GCM) with simplified hydrologic
and radiative processes. The value of planetary rota-
tion rate is varied from zero to the Earth’s value,
while other parameters such as planetary radius,
global mean surface pressure, and solar constant are
set to the Earth’s values. In all runs, statistically
equilibrium states or oscillating states are obtained,
and the runaway greenhouse state does not emerge.
While circulation patterns and amounts of sensible
and latent heat transport from the dayside to the
nightside change significantly according to values of
planetary rotation rate, summation of sensible heat
transport and latent heat transport almost remains
unchanged. The reason is that the outgoing longwave
radiation in broad area of the dayside is bounded by
the radiation limit of moist atmospheres. Three at-
mospheric states emerge in ascending order of value
of planetary rotation rate as follows: (1) states with
dayside-nightside direct circulation in the cases with
small rotation rate (Fig.1a) and weak super rota-
tion in the cases with large rotation rate (Fig.1b),
(2) states with strong super rotation and oscillating
meridionally asymmetric patterns (Fig.1c), (3) states
with mid-latitudinal jets (Fig.1d). The atmospheric
state changes gradually from state (1) to state (2)
with increasing planetary rotation rate from zero.
With further increase of planetary rotation rate, the
multiple equilibrium solutions of state (2) and state
(3) emerge. For the cases with planetary rotation
rate near Earth’s value, only state (3) emerges.
(a)
(b)
(c)
(d)
Figure 1: Wind vector [m/s] and geopotential [J/kg] at
σ = 0.17. The size of the unit vector is 50 m/s. Contour
interval of geopotential is 500 J/kg. Subsolar point is set
to be 90 degrees of longitude at the equator. (a) Ω = 0,
(b) Ω = 0.15 ΩE , (c) Ω = 0.5 ΩE , (d) Ω = ΩE , where Ω
is planetary rotation rate and ΩE is the rotation rate of
the Earth (2π/86400 [s−1] = 7.272× 10−5 [s−1]).
P5
Fig1. Example of long-time development of zonal-mean zonal angular momentum. A multiple zonal-band structure emerges, and then experiences intermittent mergers/disappearances of zonal jets.
Fig2. Real parts of leading eigenvalues (dots). All the zonal jets are linearly unstable.
Fig.3 Time derivatives of the distance between two jets.
Merging and disappearing processes of zonal jets in forced two-dimensional turbulence on a rotating sphere
Kiori OBUSE*, Shin-ichi TAKEHIRO, and Michio YAMADA Email: [email protected]
WPI-AIMR, Tohoku University Sendai, 980-8577, Japan
In forced two-dimensional turbulence on a rotating sphere, it is well known that a multiple zonal-band structure, i.e. a structure with alternating eastward and westward jets, emerges in the course of time development. The multiple zonal-band structure then experiences intermittent
mergers/disappearances of zonal jets, and a structure with only a few large-scale zonal jets is realised as an asymptotic state (Fig.1, Obuse et al., 2010). With the view of understanding the long-time behaviour of the zonal jets in two-dimensional turbulence in rotating systems, we consider large-scale zonal flows superposed upon a homogeneous zonal flow and a small-scale sinusoidal transversal flow on a β plane, which is the model originally introduced by Manfroi and Young (1999), and investigate the merging/disappearing processes of zonal jets.
First, we analytically obtain solutions of steady isolated zonal jet of the evolution equation of such zonal flows. Then it is shown that these steady zonal jet solutions are all linearly unstable (Fig.2). The numerical time integration of the
evolution equation also confirms that the final state of a perturbed unstable steady solution is a uniform flow. These results suggest that mergers/disappearances of zonal jets in two-dimensional turbulence on β plane and on a rotating sphere might be due to the instability of the zonal jets caused by the effect of turbulence (Obuse et al., 2011).
Utilising the analytical solution of steady isolated zonal jet stated above, the weak interaction between two neighbouring zonal jets is also studied. The time derivative of the distance between two identical zonal jets is estimated by a perturbation method, confirming that two zonal jets placed apart attract each other, and the attraction becomes stronger as the distance between them gets shorter. The estimated time derivative of the distance between two zonal jets is in agreement with that obtained from the numerical time integration of the evolution equation (Fig.3).
It is also found by numerical simulation that the two zonal jets then merge to a new steady isolated zonal jet of different parameters. Because of the linear instability of the new steady zonal jet, the final state is expected to be a uniform flow. These results are consistent with gradual mergers/disappearances of zonal jets seen in forced two-dimensional turbulence on β plane and on a rotating sphere, and imply the importance of the weak interaction between neighbouring zonal jets for the long-time behaviour of zonal jets in forced two-dimensional turbulence in rotating systems (Obuse et al., 2013).
P6
Angular distribution of energy spectrum in two-dimensional β-plane turbulence in the long-wave limit
Izumi SAITO and Keiichi ISHIOKA Email: [email protected]
Graduate School of Science, Kyoto University Kyoto, 606-8502, Japan
The time evolution of two-dimensional decaying turbulence governed by the long-wave limit, in which LD/L →0, of the quasi-geostrophic equation is investigated numerically. Here, LD is the Rossby radius of deformation, and L is the characteristic length scale of the flow. In this system, the ratio of the linear term that originates from the β-term to the nonlinear term is estimated by a dimensionless number, γ = βLD
2/U, where β is the latitudinal gradient of the Coriolis parameter, and U is the characteristic velocity scale. As the value of γ increases, the inverse energy cascade becomes more anisotropic. When γ≧1, the anisotropy becomes significant and energy accumulates in a wedge-shaped region where |l|> √3 |k| in the two-dimensional wavenumber space. Here, k and l are the longitudinal and latitudinal wavenumbers, respectively. When γ is increased further, the energy concentration on the lines of l=±√3 k is clearly observed. These results are interpreted based on the conservation of zonostrophy, which is an extra invariant other than energy and enstrophy and was determined in a previous study. Considerations concerning the appropriate form of zonostrophy for the long-wave limit and a discussion of the possible relevance to Rossby waves in the ocean are also presented.
P7
Approximation of chaotic zonal flows on a non-rotating/rotating sphere
Eiichi SASAKI†, Shin-ichi Takehiro‡ and Michio YAMADA‡
†Email: [email protected]
†Graduated school of informatics, Kyoto Univ., Kyoto, 606-8502, Japan
‡RIMS, Kyoto Univ., Kyoto, 606-8502, Japan
We study properties of 2D Navier -Stokes flow on a non-rotating/rotating sphere forced by a
steady zonal forcing. Our problem setting is similar to Kolmogorov problem in the point that both
the forcing and a trivial solution are expressed by a single eigenfunction of Laplacian of the
manifold: a sphere and a torus.
At first we study the bifurcation structure arising from the trivial solution. In the non-rotating
case, as the Reynolds number increases, the trivial solution becomes linearly unstable and a steady
traveling wave solution bifurcates through the Hopf bifurcation. As the Reynolds number further
increases, other steady traveling wave solutions bifurcate from the trivial solution and the steady
traveling wave solutions. These steady solutions become Hopf unstable at high Reynolds numbers.
On the other hand, in some rotating cases, we find saddle-node bifurcation, closed-loop branches
and a linearly stable solution at high Reynolds numbers. Therefore these results show that
bifurcation structure changes drastically as the rotation rate increases.
Next we carry out time integration at high Reynolds numbers. The unsteady solution is turbulent
and seems to wander around the unstable steady solutions. The turbulent flow appears related to
these unstable steady solutions. We then approximate the turbulent flow field by a linear
combination of these steady solutions, and find that the linear combination agrees well with the
turbulent flow within about 1% error (see Figure 1). This result suggests that the turbulent flow at
high Reynolds numbers almost lies in a linear space spanned by the steady solutions bifurcating at
low Reynolds numbers.
When the sphere is rotating, the linear combination does not give a good approximation of the
turbulent flow, which suggests that the turbulent solution is significantly outside the linear space
spanned by the steady solutions.
Figure 1 Snapshot of streamfunctions . (a), (b) and (c) are the turbulent flow, while (d), (e) and (f) are
the linear combination of the steady solutions.
P8
Surface zonal flows induced by thermal convection in a rapidly rotating thin spherical shell
Youhei Sasaki1, Shin-ichi Takehiro2, Kensuke Nakajima3, and Yoshi-Yuki Hayashi4
1Department of Mathematics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan2Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
3Faculty of Sciences, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan4Center for Planetary Science/ Faculty of Science, Kobe University, Nada-ku, Kobe 657-8501, Japan,
Introduction
Surface flows of Jupiter and Saturn are characterized by the broad prograde jets around the equator and the narrow alternatingjets in mid- and high-latitudes. It is not yet clear whether those surface jets are produced by convective motions in the “deep”region, or are the result of fluid motions in the “shallow” weather layer. Recently, Heimpel and Aurnou (2007) consider a thinspherical shell model and show that the equatorial prograde jets and alternating jets in mid- and high-latitudes can producesimultaneously when the Rayleigh number is sufficiently large and convection becomes active inside the tangent cylinder.However, in this study, eight-fold symmetry in the longitudinal direction is assumed and fluid motion is calculated only inthe one-eighth sector of the whole spherical shell. In the present study, we perform numerical simulations in the whole thinspherical shell domain.
Model and experimental setup
We consider Boussinesq fluid in a spherical shell rotating with angular velocity Ω. The non-dimensionalized governingequations consist of equations of continuity, motion, and temperature. The non-dimensional parameters appearing in thegoverning equations are the Prandtl number, Pr = ν/κ, the Ekman number, E = ν/(ΩD2), and the modified Rayleigh number,Ra∗ = αgo∆T/(ΩD2), where ν,D, κ,α,ro ,go , and ∆T are kinematic viscosity, the shell thickness, thermal diffusivity, the outerradius of the shell, the thermal expansion coefficient, the acceleration of gravity at the outer boundary, and the temperaturecontrast between the boundaries, respectively. The spherical shell geometry is defined by the radius ratio, χ = ri/ro , whereri is the inner radius of the shell. Both inner and outer boundaries are isothermal, impermeable and stress free. The initialconditions are zero velocity field with 4th-fold symmetry temperature perturbation. The control parameters used in this studyare E = 10−4,Pr = 0.1, χ = 0.75, and the modified Rayleigh number is 0.05 or 0.1. Time integration was performed over4000 planetary rotations (100000 time step).
Results
Left two panels of fig.1 show the azimuthally-averaged zonal velocity profiles at the outer surface for two different values ofthe Rayleigh number. Alternating zonal jets emerge in mid- and high-latitudes when Ra∗ = 0.1 while they cannot be observedin the case of Ra∗ = 0.05. Right two panels of fig.1 show azimuthally-averaged zonal velocity distributions in a meridionalcross section. It can be seen that zonal flows are almost uniform along the rotation axis. Convective motions can be observedin the tangent cylinder when Ra∗ = 0.1 while they are not active when Ra∗ = 0.05. Even though the spatial resolution andthe rotation rate are reduced and radius ratio is slightly increased, we have confirmed that the equatorial prograde flows andalternating jets in mid- and high-latitudes are produced simultaneously.
-500 -400 -300 -200 -100 0 100 200 300 400 500(1)
Reynolds number
-80
-60
-40
-20
0
20
40
60
80
(1)
Lat
itude
Zonal velocity (Ra*=0.05)
-15 -10 -5 0 5 10 15
(×100 1)
Reynolds number
-80
-60
-40
-20
0
20
40
60
80
(1)
Lat
itude
Zonal velocity (Ra*=0.1)
Zonal velocity (Ra* = 0.05)
-400 -200 0 200 400
Zonal velocity (Ra* = 0.1)
-1500 -750 0 750 1500
Fig.1 Snapshot of azimuthally-averaged zonal velocity profiles at the outer surface and in a meridional crosssection. The dashed lines in the left two panels show the latitude at which the tangent cylinder intersects theouter surface.
References
Heimpel,M.H. and Aurnou,J.M.,2007 Icarus, 187, 540–557.
P9
Spontaneous gravity wave radiation from a co-rotating vortex pair
in an unbounded f-plane shallow water system
Norihiko Sugimoto1)
, Keiichi Ishioka2)
, Hiromichi Kobayashi1)
, and Yutaka Shimomura1)
Email: [email protected]
(1)Dept. Phys., Hiyoshi, Keio University, Yokohama, 223-8521, Japan
(2)Dept. Geophys., Kyoto University, Kyoto, 606-8502, Japan
Spontaneous gravity wave radiation from vortical flows is investigated analytically and
numerically in an unbounded f-plane shallow water system. It is well known that gravity waves play
very important roles on the atmosphere and ocean by driving global circulation, especially in the
middle atmosphere, since they propagate far away from the source region and put significant
amount of momentum and energy. Recently, observational studies suggest that gravity waves are
radiated from vortical flows, such as polar night jet, sub-tropical jet, and tropical cyclone. Since
gravity waves are spontaneously radiated from nearly balanced vortical flows, this radiation process
is called as a spontaneous gravity wave radiation. Although there are several numerical studies, this
process has not been fully understood.
In the present study, we use the most simplified system of f-plane shallow water that includes
both gravity waves and vortical flows. In order to discuss the conditions of gravity wave radiation,
we use the analogy with the theory of the aero-acoustic sound wave radiation (Lighthill theory). We
derive analytical estimation of far field of gravity waves spontaneously radiated from a co-rotating
vortex pair (figure 1). We also perform numerical simulations of the same experimental setting as
the analytical study. This numerical method enables to treat an unbounded domain. The results
excellently agree with each other (figure 2). In the poster, we will discuss cyclone and anti-cyclone
asymmetry due to the effect of the earth’s rotation by the parameter sweep experiments.
(1)Sugimoto, N., Ishioka, K., and Ishii, K., “Parameter sweep experiments on spontaneous gravity
wave radiation from unsteady rotational flow in an f-plane shallow water system,” J. Atmos. Sci.,
Vol. 65, (2008), pp.234-249.
(2)Sugimoto, N., and Ishii, K., “Spontaneous gravity wave radiation in a shallow water system on a
rotating sphere,” J. Meteor. Soc. Japan, Vol. 90, (2012), pp.101-125
(3)Sugimoto, N., Ishioka, K., Kobayashi., H., and Shimomura, Y., “Spontaneous gravity wave
radiation from a co-rotating vortex pair in an unbounded f-plane shallow water system,” J. Comput.
Phys., to be submitted.
Figure 1 Snapshot of
analytical estimation of far
field of gravity waves.
Figure 2 Far field of gravity waves for a cyclone pair (left) and anti-cyclone pair
(right) for analytical estimation (solid line) and numerical simulation (broken line).
P10
A Theoretical Study on the Spontaneous Radiation of Inertia-gravity Waves Using the Renormalization Group Method
Yuki Yasuda*, Kaoru Sato, and Norihiko Sugimoto Email: [email protected]
Department of Earth and Planetary Science, The University of Tokyo, Tokyo, 113-0033, Japan
Using the renormalization group (RG) method (Chen et al. 1996), which is a singular perturbation method, the interaction between the vortical flow and the Doppler-shifted inertia-gravity waves (GWs) which both have slow time-scales is formulated for the hydrostatic Boussinesq equations on the 𝑓 plane. In general, the RG method enables us to extract slowly-varying components systematically and naturally from the system containing multiple time-scale motions. The derived time evolution equations (RG equations, referred to as RGEs) describe the spontaneous radiation of GWs from the components slaved to the vortical flow through a quasi-resonance (see Fig. 1(a)) together with the GW radiation reaction on the large-scale vortical flow. The quasi-resonance occurs when the space and time scales of slaved components are comparable to those of GWs.
The RGEs are validated using numerical simulations of the vortex dipole by Japan Meteorological Agency Nonhydrostatic Model. The flow near the center of the dipole is quite strong due to the confluence, which is similar to a localized jet stream in the atmosphere. GW distribution obtained by the RGE integration accords well with the numerical simulation (Figs. 1(b) and 1(c)). This result supports the validity of our theory. It is shown that the GW sources can be interpreted physically by either the mountain-wave-like mechanism (McIntyre 2009) or the velocity-variation mechanism (Viúdez 2007). The shear of the vortical flow determines which mechanism is dominant. The distribution of GW momentum flux (𝑢′𝑤′) is examined based on the numerical simulation data. The GWs propagating energy upward (downward) from the jet have the negative (positive) 𝑢′𝑤′ . In addition, it is shown that the magnitude of 𝑢′𝑤′ is approximately proportional to the sixth power of Rossby number. This result is consistent with the theoretical estimate by the linearized second-order RGE, indicating that higher-order than the second-order RGEs are not necessary for the description of GW spontaneous radiation.
Figure 1. (a) Schematic illustration for the GW spontaneous radiation from a localized jet stream. Vertical flow (𝑤) is slaved to the vortical flow consisting of the jet. The vertical flow couplets are a compensation flow for the jet. Doppler-shifted GWs are radiated due to a quasi-resonance with the slaved components including these couplets when the space and time scales of the GWs and the slaved components are comparable. (b) GWs in the 𝑥-𝑧 section of the vortex dipole obtained by the numerical model simulation. Colors show the horizontal divergence, while contours show the zonal flow. (c) Same as (b) but for the horizontal divergence obtained theoretically by the RGE integration.
P11
A Proof for the Equivalence of Two Upper Bounds for
the Growth of Disturbances from Barotropic Instability
Keiichi ISHIOKA
Email: [email protected]
Graduate School of Science, Kyoto University
Kyoto, 606-8502, Japan
If a flow is unstable, disturbances grow, but their growth does not continue indefinitely. Rather,
there exists some upper bound to the growth. Shepherd (1988) proposed a method by which to
calculate a fully nonlinear rigorous upper bound for the growth of disturbances from barotropic
instability using the conservation of the pseudo-momentum density based on the nonlinear stability
theorem given by Arnol'd (1966).
The upper bound, however, was not the tightest bound under the constraints of the conservation
of all of the considered invariants. A tighter bound was obtained by Ishioka and Yoden (1996) by
revising the method of Shepherd (1988). Ishioka and Yoden (1996) also proposed a new method by
which to calculate the tightest bound under the constraints of the conservation of all of the
considered invariants. They applied these two methods, namely, the revised version of the method
of Shepherd (1988) and the method of calculating the tightest bound, to several basic flow profiles
and showed that the values of the two upper bounds calculated using these two methods were
approximately equivalent, within a relative error of 1%. This implies that the revised version of the
method of Shepherd (1988) can yield the tightest bound under the considered constraints. No proof
for the equivalence, however, has yet been reported.
In the present study, a proof for the equivalence is presented by developing an annealing-like
procedure to reach the absolute vorticity profile that corresponds to the upper bound. The procedure
also provides a more efficient method by which to calculate the upper bound.
References
Arnol'd, V. I., 1966: On an a priori estimate in the theory of hydrodynamical stability. Izv.
Vyssh. Uchebn. Zaved. Mathematika, 54, no. 5, 3--5. (English transl.: Amer. Math. Soc.
Transl., Series 2, 79, 267-269(1969).)
Ishioka, K. and S. Yoden, 1996: Numerical methods of estimating bounds on the non-linear
saturation of barotropic instability. J. Meteor. Soc. Japan, 74, 167-174.
Shepherd, T. G., 1988: Rigorous bounds on the nonlinear saturation of instabilities to parallel
shear flows. J. Fluid Mech., 196, 291-322.
P12