A Baroclinic Instability that Couples Balanced Motions and Gravity Waves

2005 ◽  
Vol 62 (5) ◽  
pp. 1545-1559 ◽  
Author(s):  
Riwal Plougonven ◽  
David J. Muraki ◽  
Chris Snyder
Keyword(s):  
Gravity Waves ◽  
Baroclinic Instability ◽  
Normal Modes ◽  
Lower Boundary ◽  
Vertical Shear ◽  
Rossby Number ◽  
Edge Waves ◽  
Spatial Coupling ◽  
The Waves ◽  
Upper Boundary

Abstract Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves. The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

scholarly journals On the appearance of internal wave attractors due to an initial or parametrically excited disturbance

2013 ◽  
Vol 714 ◽  
pp. 283-311 ◽  
Author(s):  
Janis Bajars ◽  
Jason Frank ◽  
Leo R. M. Maas
Keyword(s):  
Gravity Waves ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Harmonic Oscillators ◽  
Two Dimensional ◽  
Boussinesq Model ◽  
Resonant Modes ◽  
The Waves ◽  
Shaped Domain ◽  
Parametric Forcing

AbstractIn this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler–Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors.

Inertia–Gravity Waves Generated within a Dipole Vortex

2007 ◽  
Vol 64 (12) ◽  
pp. 4417-4431 ◽  
Author(s):  
Chris Snyder ◽  
David J. Muraki ◽  
Riwal Plougonven ◽  
Fuqing Zhang
Keyword(s):  
Gravity Waves ◽  
Initial Conditions ◽  
Initial Period ◽  
Potential Temperature ◽  
Leading Edge ◽  
Vertical Shear ◽  
Coriolis Parameter ◽  
Dipole Vortex ◽  
The Waves ◽  
Inertia Gravity Waves

Abstract Vortex dipoles provide a simple representation of localized atmospheric jets. Numerical simulations of a synoptic-scale dipole in surface potential temperature are considered in a rotating, stratified fluid with approximately uniform potential vorticity. Following an initial period of adjustment, the dipole propagates along a slightly curved trajectory at a nearly steady rate and with a nearly fixed structure for more than 50 days. Downstream from the jet maximum, the flow also contains smaller-scale, upward-propagating inertia–gravity waves that are embedded within and stationary relative to the dipole. The waves form elongated bows along the leading edge of the dipole. Consistent with propagation in horizontal deformation and vertical shear, the waves’ horizontal scale shrinks and the vertical slope varies as they approach the leading stagnation point in the dipole’s flow. Because the waves persist for tens of days despite explicit dissipation in the numerical model that would otherwise damp the waves on a time scale of a few hours, they must be inherent features of the dipole itself, rather than remnants of imbalances in the initial conditions. The wave amplitude varies with the strength of the dipole, with waves becoming obvious once the maximum vertical vorticity in the dipole is roughly half the Coriolis parameter. Possible mechanisms for the wave generation are spontaneous wave emission and the instability of the underlying balanced dipole.

scholarly journals Baroclinic Waves with Parameterized Effects of Moisture Interpreted Using Rossby Wave Components

2010 ◽  
Vol 67 (9) ◽  
pp. 2766-2784 ◽  
Author(s):  
Hylke de Vries ◽  
John Methven ◽  
Thomas H. A. Frame ◽  
Brian J. Hoskins
Keyword(s):  
Rossby Wave ◽  
Large Scale ◽  
Initial Conditions ◽  
Normal Modes ◽  
Lower Boundary ◽  
Vertical Shear ◽  
Zonal Flows ◽  
Baroclinic Waves ◽  
Strength Of Interaction ◽  
Type C

Abstract A theoretical framework is developed for the evolution of baroclinic waves with latent heat release parameterized in terms of vertical velocity. Both wave–conditional instability of the second kind (CISK) and large-scale rain approaches are included. The new quasigeostrophic framework covers evolution from general initial conditions on zonal flows with vertical shear, planetary vorticity gradient, a lower boundary, and a tropopause. The formulation is given completely in terms of potential vorticity, enabling the partition of perturbations into Rossby wave components, just as for the dry problem. Both modal and nonmodal development can be understood to a good approximation in terms of propagation and interaction between these components alone. The key change with moisture is that growing normal modes are described in terms of four counterpropagating Rossby wave (CRW) components rather than two. Moist CRWs exist above and below the maximum in latent heating, in addition to the upper- and lower-level CRWs of dry theory. Four classifications of baroclinic development are defined by quantifying the strength of interaction between the four components and identifying the dominant pairs, which range from essentially dry instability to instability in the limit of strong heating far from boundaries, with type-C cyclogenesis and diabatic Rossby waves being intermediate types. General initial conditions must also include passively advected residual PV, as in the dry problem.

scholarly journals Characteristics of gravity waves generated in a baroclinic instability simulation

2015 ◽  
Vol 15 (22) ◽  
pp. 32639-32678
Author(s):  
Y.-H. Kim ◽  
H.-Y. Chun ◽  
S.-H. Park ◽  
I.-S. Song ◽  
H.-J. Choi
Keyword(s):  
Life Cycle ◽  
Gravity Waves ◽  
Baroclinic Instability ◽  
Mature Stage ◽  
Velocity Spectrum ◽  
Two Dimensional ◽  
Baroclinic Wave ◽  
Upper Troposphere ◽  
Westerly Jet ◽  
The Waves

Abstract. An idealized baroclinic instability case is simulated using a ~ 10 km resolution global model to investigate the characteristics of gravity waves (GWs) generated in the baroclinic life cycle. Three groups of GWs (W1–W3) appear around the high-latitude surface trough at the mature stage of the baroclinic wave. They have horizontal and vertical wavelengths of 40–400 and 2.9–9.8 km, respectively, in the upper troposphere. The two-dimensional phase-velocity spectrum of the waves is arc-shaped with a peak at 17 m s−1 eastward, which is difficult for the waves to propagate upward through the tropospheric westerly jet. At the breaking stage of the baroclinic wave, a midlatitude surface low is isolated from the higher-latitude trough, and two groups of quasi-stationary GWs (W4 and W5) appear near the surface low. These waves have horizontal and vertical wavelengths of 60–400 and 4.9–14 km, respectively, and are able to propagate vertically for long distances. The generation mechanism of the simulated GWs is discussed.

Baroclinic instability in a reduced gravity, three-dimensional, quasi-geostrophic model

2000 ◽  
Vol 403 ◽  
pp. 1-22 ◽  
Author(s):  
P. RIPA
Keyword(s):  
Kinetic Energy ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Normal Modes ◽  
Lower Boundary ◽  
External Potential ◽  
Present System ◽  
Integrals Of Motion ◽  
The Difference

The classical quasi-geostrophic model in an active layer with an arbitrary vertical structure is modified by adding a boundary condition at the interface with a passive (motionless) lower layer: the difference between isopycnal and interface elevations is a Lagrangian constant, so that a particle in this boundary remains there and conserves its density. The new model has the appropriate integrals of motion: in particular, a free energy quadratic and positive definite in the deviation from a state with a uniform flow, made up of the internal and ‘external’ potential energies (due to the displacement of the isopycnals and the interface) and the kinetic energy.Eady's model of baroclinic instability is extended with the present system, i.e. including the effect of the free lower boundary. The integrals of motion give instability conditions that are both necessary and sufficient. If the geostrophic slope of the interface is such that density increases in opposite directions at the top and bottom boundaries, then the basic flow is nonlinearly stable. For very weak internal stratification (as compared with the density jump at the interface) normal modes instability is similar to that of a simpler model, with a rigid but sloping bottom. For stronger stratification, though, the deformation of the lower boundary by the perturbation field also plays an important role, as shown in the dispersion relation, the structure of growing perturbations, and the energetics of the instability. The energy of long growing perturbations is mostly internal potential, whereas short ones have an important fraction of kinetic energy and, for strong enough stratification, external potential.

Transverse Heterogeneity Effects in the Dissipation-Induced Instability of a Horizontal Porous Layer

2011 ◽  
Vol 133 (12) ◽  
Author(s):  
A. Barletta ◽  
M. Celli ◽  
A. V. Kuznetsov
Keyword(s):  
Linear Stability ◽  
Porous Layer ◽  
Normal Modes ◽  
Lower Boundary ◽  
Boussinesq Approximation ◽  
Darcy Law ◽  
Main Task ◽  
Heterogeneity Effects ◽  
Upper Boundary

The linear stability of a parallel flow in a heterogeneous porous channel is analyzed by means of the Darcy law and the Oberbeck–Boussinesq approximation. The basic velocity and temperature distributions are influenced by the effect of the viscous dissipation, as well as, by the boundary conditions. A horizontal porous layer bounded by impermeable and infinitely wide walls is considered. The lower boundary is assumed to be thermally insulated, while the upper boundary is assumed to be isothermal. A transverse heterogeneity for the permeability and for the thermal conductivity is taken into account. The main task of this work is to investigate the role of this heterogeneity in changing the threshold for the onset of instability. A linear stability analysis by means of the normal modes method is performed. The onset of instability against oblique rolls is studied. The eigenvalue problem is solved numerically.

Linear baroclinic instability in the presence of heat inflow from the lower boundary

1995 ◽  
Vol 13 (4) ◽  
pp. 419-426
Author(s):  
M. Fantini
Keyword(s):  
Energy Exchange ◽  
Baroclinic Instability ◽  
Normal Modes ◽  
Lower Boundary ◽  
Maximum Growth ◽  
High Wave ◽  
Explosive Cyclones ◽  
Wave Numbers ◽  
Heat Influx ◽  
Heat Inflow

Abstract. A linear Eady model with a parameterization of heat influx from the lower boundary is studied analytically in order to obtain the characteristics of baroclinic normal modes modified by this non-adiabatic source. The results display a secondary maximum of growth rate at high wave numbers and a range of absolutely unstable waves, thus suggesting that the property observed among mid-latitude explosive cyclones of being near-stationary in the phase of maximum growth may be captured by this representation of the air-sea energy exchange.

On the effects of frontogenetic strain on symmetric instability and inertia–gravity waves

2012 ◽  
Vol 711 ◽  
pp. 620-640 ◽  
Author(s):  
Leif N. Thomas
Keyword(s):  
Gravity Waves ◽  
Richardson Number ◽  
Low Frequency ◽  
Deformation Field ◽  
Vertical Shear ◽  
Geostrophic Flow ◽  
Ageostrophic Circulation ◽  
The Waves ◽  
Symmetric Instability ◽  
Inertia Gravity Waves

AbstractThe dynamics of symmetric instability and two-dimensional inertia–gravity waves in a baroclinic geostrophic flow undergoing frontogenesis is analysed. A frontogenetic strain associated with a balanced deformation field drives an ageostrophic circulation and temporal variations in the basic state that significantly affect the properties of perturbations to the background flow. For stable stratification, perturbations to the basic state result in symmetric instability or inertia–gravity waves, depending on the sign of the Ertel potential vorticity and the magnitude of the Richardson number of the geostrophic flow. The kinetic energy (KE) of both types of motion is suppressed by frontogenetic strain due to the vertical shear in the ageostrophic circulation. This is because the perturbation streamlines tilt with the ageostrophic shear causing the disturbances to lose KE via shear production. The effect can completely dampen symmetric instability for sufficiently strong strain even though the source of KE for the instability (the vertical shear in the geostrophic flow) increases with time. Inertia–gravity waves in a baroclinic flow undergoing frontogenesis simultaneously lose KE and extract KE from the deformation field as they decay. This is because the horizontal velocity of the waves becomes rectilinear, resulting in a Reynolds stress that draws energy from the balanced flow. The process is most effective for waves of low frequency and for a geostrophic flow with low Richardson number. However, even in a background flow that is initially strongly stratified, frontogenesis leads to an exponentially fast reduction in the Richardson number, facilitating a rapid energy extraction by the waves. The KE transferred from the deformation field is ultimately lost to the unbalanced ageostrophic circulation through shear production, hence the inertia–gravity waves play a catalytic role in loss of balance. Given the large amount of KE in low-frequency inertia–gravity waves and the ubiquitous combination of strain and baroclinic geostrophic currents in the ocean, it is estimated that this mechanism could play a significant role in the removal of KE from both the internal wave and mesoscale eddy fields.

Evolution of Tsunami-Induced Internal Acoustic–Gravity Waves

2015 ◽  
Vol 72 (6) ◽  
pp. 2303-2317 ◽  
Author(s):  
Chen Wei ◽  
Oliver Bühler ◽  
Esteban G. Tabak
Keyword(s):  
Remote Sensing ◽  
Gravity Waves ◽  
Numerical Study ◽  
Lower Boundary ◽  
Radiation Condition ◽  
Initial Value ◽  
Acoustic Gravity Waves ◽  
Linear Waves ◽  
Remote Sensing Methods ◽  
The Waves

Abstract The authors present an idealized theoretical and numerical study of tsunami-induced internal waves in the atmosphere. These are gravity waves modified by acoustic effects that can propagate rapidly from the ocean surface up to the ionosphere, where they are well known to leave a detectable fingerprint in airglow patterns and other remote sensing observables. Accurate modeling of the wave propagation is a prerequisite for being able to detect and decode this transient observational fingerprint by remote sensing methods. The authors study this problem by formulating the initial-value problem for linear waves forced by an idealized tsunami at the lower boundary and then employing a semianalytic Fourier–Laplace method to solve it. This approach allows them to compute the detailed time evolution of the waves while ensuring that the correct radiation condition in the vertical is satisfied at all times, a nontrivial matter for these transient waves. The authors also compare the predictions of an anelastic model with that of a fully compressible model in order to discern the importance of acoustic effects. The findings demonstrate that back-reflection at the tropopause is a significant factor for the structure of these waves and that the earliest observable signal in the ionosphere is, in fact, a fast acoustic precursor wave generated by the nearly impulsive formation of the tsunami itself.

Sign in / Sign up

Export Citation Format

Share Document