Overtransmission of Rossby Waves at a Lower-Layer Critical Latitude in the Two-Layer Model

2020 ◽  
Vol 77 (3) ◽  
pp. 859-870 ◽  
Author(s):  
Matthew T. Gliatto ◽  
Isaac M. Held
Keyword(s):  
Rossby Waves ◽  
Lower Layer ◽  
Scattering Problem ◽  
Lower Troposphere ◽  
Vertical Shear ◽  
Mean Flow ◽  
External Mode ◽  
The Mean ◽  
Critical Latitude ◽  
Pv Gradient

Abstract Rossby waves, propagating from the midlatitudes toward the tropics, are typically absorbed by critical latitudes (CLs) in the upper troposphere. However, these waves typically encounter CLs in the lower troposphere first. We study a two-layer linear scattering problem to examine the effects of lower CLs on these waves. We begin with a review of the simpler barotropic case to orient the reader. We then progress to the baroclinic case using a two-layer quasigeostrophic model in which there is vertical shear in the mean flow on which the waves propagate, and in which the incident wave is assumed to be an external-mode Rossby wave. We use linearized equations and add small damping to remove the critical-latitude singularities. We consider cases in which either there is only one CL, in the lower layer, or there are CLs in both layers, with the lower-layer CL encountered first. If there is only a CL in the lower layer, the wave’s response depends on the sign of the mean potential vorticity gradient at this lower-layer CL: if the PV gradient is positive, then the CL partially absorbs the wave, as in the barotropic case, while for a negative PV gradient, the CL is a wave emitter, and can potentially produce overreflection and/or overtransmission. Our numerical results indicate that overtransmission is by far the dominant response in these cases. When an upper-layer absorbing CL is encountered, following the lower-layer encounter, one can still see the signature of overtransmission at the lower-layer CL.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

The propagation of atmospheric Rossby gravity waves in latitudinally sheared zonal flows

1977 ◽  
Vol 287 (1340) ◽  
pp. 115-143 ◽  
Keyword(s):  
Gravity Waves ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Flow Speed ◽  
Wave Number ◽  
Mean Flow ◽  
Zonal Flows ◽  
Planetary Scale ◽  
The Mean ◽  
Critical Latitude

Using the B-plane approximation we formulate the equations which govern small perturbations in a rotating atmosphere and describe a wide class of possible wave motions, in the presence of a background zonal flow, ranging from ‘moderately high’ frequency acoustic-gravity-inertial waves to ‘low’ frequency planetary-scale (Rossby) waves. The discussion concentrates mainly on the propagation properties of Rossby waves in various types of latitudinally sheared zonal flows which occur at different heights and seasons in the earth’s atmosphere. However, it is first shown that gravity waves in a latitudinally sheared zonal flow exhibit critical latitude behaviour where the ‘intrinsic ’ wave frequency matches the Brunt-Vaisala frequency (in contrast to the case of gravity waves in a vertically sheared flow where a critical layer exists where the horizontal wave phase speed equals the flow speed) and that the wave behaviour near such a latitude is similar to that of Rossby waves in the vicinity of their critical latitudes which occur where the ‘intrinsic’ wave frequency approaches zero. In the absence of zonal flow in the atmosphere the geometry of the planetary wave dispersion equation (which is described by a highly elongated ellipsoid in wave-number vector space) implies that energy propagates almost parallel to the /--planes. This feature may provide a reason why there seems to be so little coupling between planetary scale motions in the lower and upper atmosphere. Planetary waves can be made to propagate eastward, as well as westward, if they are evanescent in the vertical direction. The W.K.B. approximation, which provides an approximate description of wave propagation in slowly varying zonal wind shears, shows that the distortion of the wave-number surface caused by the zonal flow controls the dependence of the wave amplitude on the zonal flow speed. In particular it follows that Rossby waves propagating into regions of strengthening westerlies are intensified in amplitude whereas those waves propagating into strengthening easterlies are diminished in amplitude. A classification of the various types of ray trajectories that arise in zonal flow profiles occurring in the Earth’s atmosphere, such as jet-like variations of westerly or easterly zonal flow or a belt of westerlies bounded by a belt of easterlies, is given, and provides the conditions giving rise to such phenomena as critical latitude behaviour and wave trapping. In a westerly flow there is a tendency for the combined effects on wave propagation of jet-like variations of B and zonal flow speed to counteract each other, whereas in an easterly flow such variations tend to reinforce each other. An examination of the reflexion and refraction of Rossby waves at a sharp jump in the zonal flow speed shows that under certain conditions wave amplification, or over-reflexion, can arise with the implication that the reflected wave can extract energy from the background streaming motion. On the other hand the wave behaviour near critical latitudes, which can be described in terms of a discontinuous jump in the ‘wave invariant’, shows that such latitudes can act as either wave absorbers (in which case the mean flow is accelerated there) or wave emitters (in which case the mean flow is decelerated there).

scholarly journals Ageostrophic instability in rotating, stratified interior vertical shear flows

2014 ◽  
Vol 755 ◽  
pp. 397-428 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Claire Ménesguen
Keyword(s):  
Energy Source ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Gravity Waves ◽  
Shear Flows ◽  
Vertical Shear ◽  
Mean Flow ◽  
Efficient Energy Transfer ◽  
The Mean ◽  
Inertia Gravity Waves

AbstractThe linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.

scholarly journals The interaction of free Rossby waves with semi-transparent equatorial waveguide – wave-mean flow interaction

2009 ◽  
Vol 16 (3) ◽  
pp. 381-392 ◽  
Author(s):  
G. M. Reznik ◽  
V. Zeitlin
Keyword(s):  
Rossby Waves ◽  
Landau Equation ◽  
Spatial Modulation ◽  
Equilibrium Solutions ◽  
Mean Flow ◽  
Ginzburg Landau ◽  
Baroclinic Waves ◽  
Ginzburg Landau Equation ◽  
The Mean ◽  
Mean Current

Abstract. Nonlinear interactions of the barotropic Rossby waves propagating across the equator with trapped baroclinic Rossby or Yanai modes and mean zonal flow are studied within the two-layer model of the atmosphere, or the ocean. It is shown that the equatorial waveguide with a mean current acts as a resonator and responds to barotropic waves with certain wavenumbers by making the trapped baroclinic modes grow. At the same time the equatorial waveguide produces the barotropic response which, via nonlinear interaction with the mean equatorial current and with the trapped waves, leads to the saturation of the growing modes. The excited baroclinic waves can reach significant amplitudes depending on the magnitude of the mean current. In the absence of spatial modulation the nonlinear saturation of thus excited waves is described by forced Landau-type equation with one or two attracting equilibrium solutions. In the latter case the spatial modulation of the baroclinic waves is expected to lead to the formation of characteristic domain-wall defects. The evolution of the envelopes of the trapped Rossby waves is governed by driven Ginzburg-Landau equation, while the envelopes of the Yanai waves obey the "first-order" forced Ginzburg-Landau equation. The envelopes of short baroclinic Rossby waves obey the damped-driven nonlinear Schrodinger equation well studied in the literature.

Precursor Signals and Processes Associated with MJO Initiation over the Tropical Indian Ocean*

2013 ◽  
Vol 26 (1) ◽  
pp. 291-307 ◽  
Author(s):  
Chongbo Zhao ◽  
Tim Li ◽  
Tianjun Zhou
Keyword(s):  
Indian Ocean ◽  
Rossby Wave ◽  
Lower Troposphere ◽  
Equatorial Indian Ocean ◽  
Specific Humidity ◽  
Boreal Winter ◽  
Mean Flow ◽  
Wave Response ◽  
The Mean ◽  
Convection Initiation

Abstract The precursor signals of convection initiation associated with the Madden–Julian oscillation (MJO) in boreal winter were investigated through the diagnosis of the 40-yr ECMWF Re-Analysis (ERA-40) data for the period 1982–2001. The western equatorial Indian Ocean (WIO) is a key region of the MJO initiation. A marked increase of specific humidity and temperature in the lower troposphere appears 5–10 days prior to the convection initiation. The increased moisture and temperature cause a convectively more unstable stratification, leading to the onset of convection. A diagnosis of lower-tropospheric moisture (heat) budgets shows that the moisture (temperature) increase is caused primarily by the horizontal advection of the mean specific humidity (temperature) by the MJO flow. The anomalous flow is primarily determined by the downstream Rossby wave response to a preceding suppressed-phase MJO over the eastern Indian Ocean, whereas the upstream Kelvin wave response to the previous eastward-propagating convective-phase MJO is not critical. An idealized numerical experiment further supports this claim. The Southern Hemisphere (SH) midlatitude Rossby wave train and associated wave activity flux prior to the MJO initiation were diagnosed. It is found that SH midlatitude Rossby waves may contribute to MJO initiation over the western Indian Ocean through wave energy accumulation. Idealized numerical experiments confirm that SH midlatitude perturbations play an important role in affecting the MJO variance in the tropics. A barotropic energy conversion diagnosis indicates that there is continuous energy transfer from the mean flow to intraseasonal disturbances over the initiation region, which may help trigger MJO development.

Tidal Currents and Estuarine-Type Circulation in Johnstone Strait, British Columbia

1976 ◽  
Vol 33 (10) ◽  
pp. 2242-2264 ◽  
Author(s):  
Richard E. Thomson
Keyword(s):  
Lower Layer ◽  
Small Variation ◽  
Tidal Mixing ◽  
Mean Flow ◽  
Western Basin ◽  
Nonlinear Advection ◽  
Tide Height ◽  
The Mean ◽  
Stratified Estuary ◽  
Tidal Signal

Four months of current meter observations across the western basin of Johnstone Strait have been examined, with particular attention given to the mean flow and to variations at tidal frequencies. We show that the time-averaged motions are typical of a moderately stratified estuary driven by tidal mixing and nonlinear advection. Steady currents are nearly unidirectional at all depths with the net outflow in the upper layer essentially balanced by a net inflow in the lower layer to order 103 m3∙s−1. In addition, the relatively small variation in residual current speed is found to decrease with depth and to be associated mostly with the quasi-fortnightly tidal cycle. Near the surface the variance in the residual flow appears to be related to along-channel winds whose speeds and duration exceed 6 m∙s−1 and 24 h, respectively. Time-dependent motions are dominated by the tidal signal which is mixed, predominately semidiurnal. Maximum speeds of order 1 m∙s−1 are found at depth and are generally 1.5–1.7 times larger than in the upper layer. There is also a strong correlation between the tidal current speeds below 150-m depth and the local tide height lagged by 6 h. It is suggested that these large lower layer currents are associated with baroclinic motions being generated by the barotropic tide propagating over the rapidly shoaling bathymetry to the east of the observation region.

Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer

2010 ◽  
Vol 665 ◽  
pp. 209-237 ◽  
Author(s):  
J. GULA ◽  
V. ZEITLIN ◽  
F. BOUCHUT
Keyword(s):  
Shallow Water ◽  
Linear Stability ◽  
Nonlinear Evolution ◽  
Lower Layer ◽  
Short Wave ◽  
Finite Volume Scheme ◽  
Coastal Currents ◽  
Mean Flow ◽  
The Mean ◽  
Rotating Shallow Water

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linear stability problem for coastal currents and study the nonlinear evolution of the instabilities with the help of high-resolution direct numerical simulations. We show how nonlinear saturation of the ageostrophic instabilities leads to reorganization of the mean flow and emergence of coherent vortices. We follow the same lines as in Part 1 and, first, perform a complete linear stability analysis of the baroclinic coastal currents for various depths and density ratios. We then study the nonlinear evolution of the unstable modes with the help of the recent efficient two-layer generalization of the one-layer well-balanced finite-volume scheme for rotating shallow water equations, which allows the treatment of outcropping and loss of hyperbolicity associated with shear, Kelvin–Helmholtz type, instabilities. The previous single-layer results are recovered in the limit of large depth ratios. For depth ratios of order one, new baroclinic long-wave instabilities come into play due to the resonances among Rossby and frontal- or coastal-trapped waves. These instabilities saturate by forming coherent baroclinic vortices, and lead to a complete reorganization of the initial current. As in Part 1, Kelvin fronts play an important role in this process. For even smaller depth ratios, short-wave shear instabilities with large growth rates rapidly develop. We show that at the nonlinear stage they produce short-wave meanders with enhanced dissipation. However, they do not change, globally, the structure of the mean flow which undergoes secondary large-scale instabilities leading to coherent vortex formation and cutoff.

Eddy fluxes and topography in stratified quasi-geostrophic models

1999 ◽  
Vol 380 ◽  
pp. 59-80 ◽  
Author(s):  
WILLIAM J. MERRYFIELD ◽  
GREG HOLLOWAY
Keyword(s):  
Layer Thickness ◽  
Stream Function ◽  
Potential Vorticity ◽  
Lower Layer ◽  
Stratified Flow ◽  
Mean Flow ◽  
Random Forcing ◽  
Flow Over Topography ◽  
Eddy Fluxes ◽  
The Mean

Turbulent stratified flow over topography is studied using layered quasi-geostrophic models. Mean flows develop under random forcing, with lower-layer mean stream-function positively correlated with topography. When friction is sufficiently small, upper-layer mean flow is weaker than, but otherwise resembles, lower-layer mean flow. When lower-layer friction is larger, upper-layer mean flow reverses and can exceed lower-layer mean flow in strength. The mean interface between layers is domed over topographic elevations. Eddy fluxes of potential vorticity and layer thickness act in the sense of driving the flow toward higher entropy. Such behaviour contradicts usual eddy parameterizations, to which modifications are suggested.

scholarly journals Strong Intraseasonal Variability of Meridional Currents near 5°N in the Eastern Indian Ocean: Characteristics and Causes

2017 ◽  
Vol 47 (5) ◽  
pp. 979-998 ◽  
Author(s):  
Gengxin Chen ◽  
Weiqing Han ◽  
Yuanlong Li ◽  
Michael J. McPhaden ◽  
Ju Chen ◽  
...  
Keyword(s):  
Indian Ocean ◽  
Rossby Waves ◽  
Intraseasonal Variability ◽  
Upper Ocean ◽  
Mean Flow ◽  
Eastern Boundary ◽  
Typical Period ◽  
The Indian Ocean ◽  
The Mean

AbstractThis paper reports on strong, intraseasonal, upper-ocean meridional currents observed in the Indian Ocean between the Bay of Bengal (BOB) and the equator and elucidates the underlying physical processes responsible for them. In situ measurements from a subsurface mooring at 5°N, 90.5°E reveal strong intraseasonal variability of the meridional current with an amplitude of ~0.4 m s−1 and a typical period of 30–50 days in the upper 150 m, which by far exceeds the magnitudes of the mean flow and seasonal cycle. Such prominent intraseasonal variability is, however, not seen in zonal current at the same location. Further analysis suggests that the observed intraseasonal flows are closely associated with westward-propagating eddylike sea surface height anomalies (SSHAs) along 5°N. The eddylike SSHAs are largely manifestations of symmetric Rossby waves, which result primarily from intraseasonal wind stress forcing in the equatorial waveguide and reflection of the equatorial Kelvin waves at the eastern boundary. Since the wave signals are generally symmetric about the equator, similar variability is also seen at 5°S but with weaker intensity because of the inclined coastline at the eastern boundary. The Rossby waves propagate westward, causing pronounced intraseasonal SSHA and meridional current in the upper ocean across the entire southern BOB between 84° and 94°E. They greatly weaken in the western Indian Basin, but zonal currents near the equator remain relatively strong.

scholarly journals Alignment of Hurricane-like Vortices on f and β Planes

2009 ◽  
Vol 66 (6) ◽  
pp. 1779-1792 ◽  
Author(s):  
Robert W. Jones ◽  
Hugh E. Willoughby ◽  
Michael T. Montgomery
Keyword(s):  
Rossby Waves ◽  
Critical Radius ◽  
Swirling Flow ◽  
Baroclinic Instability ◽  
Necessary Condition ◽  
Initial Condition ◽  
The Mean ◽  
Nonlinear Solutions ◽  
The Waves ◽  
Pv Gradient

Abstract A nonlinear, two-layer, vortex-tracking semispectral model (i.e., Fourier transformed in azimuth only) is used to study the evolution of dry, but otherwise hurricane-like, initially tilted vortices in quiescent surroundings on f and β planes. The tilt projects onto vorticity asymmetries that are dynamically vortex Rossby waves. Since the swirling wind in the principal mean vortex used here decays exponentially outside the eyewall, it has an initial potential vorticity (PV) minimum. The resulting reversal of PV gradient meets the necessary condition for inflectional (i.e., barotropic or baroclinic) instability. Thus, the vortex may be inflectionally stable or unstable. On an f plane, the tilt precesses relatively slowly because the critical radius, where the phase speeds of the waves match the mean swirling flow, is far from the center. An alternative Gaussian-like PV monopole that has a monotonic outward decrease of PV is stable to inflectional instability. It has a smaller critical radius and rapid tilt precession. Generally, vortices with fast tilt precession are more stable, as are stronger vortices in higher latitudes. On a β plane, the interaction between the symmetric vortex and the planetary PV gradient induces β gyres that push the vortex poleward and westward. The interaction between the β gyres and the planetary PV gradient may either create a PV minimum or intensify a minimum inherited from the initial condition. Thus, the nonlinear β effect reduces the ability of the vortex to recover from initial tilt, relative to the same vortex on an f plane. This result contrasts with previous studies of barotropic vortices on f planes, where the linear and nonlinear solutions were nearly identical.

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