scholarly journals Theoretical Studies of Convectively Forced Mesoscale Flows in Three Dimensions. Part II: Shear Flow with a Critical Level

2010 ◽  
Vol 67 (3) ◽  
pp. 694-712 ◽  
Author(s):  
Ji-Young Han ◽  
Jong-Jin Baik
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Wind Direction ◽  
Critical Level ◽  
Deep Convection ◽  
Vertical Wind Shear ◽  
Finite Depth ◽  
Vertical Displacement ◽  
Two Dimensions ◽  
Three Dimensions

Abstract Convectively forced mesoscale flows in a shear flow with a critical level are theoretically investigated by obtaining analytic solutions for a hydrostatic, nonrotating, inviscid, Boussinesq airflow system. The response to surface pulse heating shows that near the center of the moving mode, the magnitude of the vertical velocity becomes constant after some time, whereas the magnitudes of the vertical displacement and perturbation horizontal velocity increase linearly with time. It is confirmed from the solutions obtained in present and previous studies that this result is valid regardless of the basic-state wind profile and dimension. The response to 3D finite-depth steady heating representing latent heating due to cumulus convection shows that, unlike in two dimensions, a low-level updraft that is necessary to sustain deep convection always occurs at the heating center regardless of the intensity of vertical wind shear and the heating depth. For deep heating across a critical level, little change occurs in the perturbation field below the critical level, although the heating top height increases. This is because downward-propagating gravity waves induced by the heating above, but not near, the critical level can hardly affect the flow response field below the critical level. When the basic-state wind backs with height, the vertex of V-shaped perturbations above the heating top points to a direction rotated a little clockwise from the basic-state wind direction. This is because the V-shaped perturbations above the heating top is induced by upward-propagating gravity waves that have passed through the layer below where the basic-state wind direction is clockwise relative to that above.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

Velocity distribution function for a dilute granular material in shear flow

1997 ◽  
Vol 340 ◽  
pp. 319-341 ◽  
Author(s):  
V. KUMARAN
Keyword(s):  
Distribution Function ◽  
Shear Flow ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Particle Diameter ◽  
Velocity Distribution Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Binary Collisions

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

scholarly journals Discrete Propagation in Numerically Simulated Nocturnal Squall Lines

2006 ◽  
Vol 134 (12) ◽  
pp. 3735-3752 ◽  
Author(s):  
Robert G. Fovell ◽  
Gretchen L. Mullendore ◽  
Seung-Hee Kim
Keyword(s):  
Gravity Waves ◽  
High Frequency ◽  
Deep Convection ◽  
Vertical Wind Shear ◽  
Lower Troposphere ◽  
Propagation Speed ◽  
Surface Fluxes ◽  
Squall Line ◽  
Radiative Processes ◽  
Squall Lines

Abstract Simulations of a typical midlatitude squall line were used to investigate a mechanism for discrete propagation, defined as convective initiation ahead of an existing squall line leading to a faster propagation speed for the storm complex. Radar imagery often shows new cells appearing in advance of squall lines, suggesting a causal relationship and prompting the search for an “action-at-a-distance” mechanism to explain the phenomenon. In the simulations presented, the identified mechanism involves gravity waves of both low and high frequency generated in response to the latent heating, which subsequently propagate out ahead of the storm. The net result of the low-frequency response, combined with surface fluxes and radiative processes, was a cooler and more moist lower troposphere, establishing a shallow cloud deck extending ahead of the storm. High-frequency gravity waves, excited in response to fluctuations in convective activity in the main storm, were subsequently ducted by the storm’s own upper-tropospheric forward anvil outflow. These waves helped positively buoyant cumulus clouds to occasionally form in the deck. A fraction of these clouds persisted long enough to merge with the main line, invigorating the parent storm. Discrete propagation occurred when clouds developed into deep convection prior to merger, weakening the parent storm. The ducting conditions, as diagnosed with the Scorer parameter, are shown to be sensitive to vertical wind shear and radiation, but not to the microphysical parameterization or simulation geometry.

scholarly journals No steady water waves of small amplitude are supported by a shear flow with a still free surface

2013 ◽  
Vol 717 ◽  
pp. 523-534 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Finite Depth ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Positive Coefficients

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

scholarly journals Theoretical Studies of Convectively Forced Mesoscale Flows in Three Dimensions. Part I: Uniform Basic-State Flow

2009 ◽  
Vol 66 (4) ◽  
pp. 947-965 ◽  
Author(s):  
Ji-Young Han ◽  
Jong-Jin Baik
Keyword(s):  
Steady State ◽  
Early Stage ◽  
Deep Convection ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Convective Heating ◽  
Stationary Mode ◽  
Transient Heating ◽  
Downward Motion ◽  
Updraft Region

Abstract Convectively forced mesoscale flows in three dimensions are theoretically investigated by examining the transient response of a stably stratified atmosphere to convective heating. Solutions for the equations governing small-amplitude perturbations in a uniform basic-state wind with specified convective heating are analytically obtained using the Green function method. In the surface pulse heating case, it is explicitly shown that the vertical displacement at the center of the 3D steady heating decreases as fast as t−1 for large t. Hence, unlike in two dimensions, the steady state is approached in three dimensions. In the finite-depth steady heating case, the perturbation vertical velocity field in the stationary mode shows a main updraft region extending over the heating layer and V-shaped upward and downward motions above and below the heating layer. Including the third dimension results in a stronger updraft at an early stage, a weaker compensating downward motion, and a weaker stationary gravity wave field in a quasi-steady state than in the case of two dimensions. An examination of flow response fields for various vertical structures of convective heating indicates that stationary gravity waves above the main updraft region become strong in intensity as the height of the maximum convective heating increases. In response to the transient heating, a main updraft region extending over the heating layer no longer appears at a dissipation stage of deep convection. Instead, alternating regions of upward and downward motion with an upstream phase tilt appear.

Frequency downshift in three-dimensional wave trains in a deep basin

1997 ◽  
Vol 352 ◽  
pp. 359-373 ◽  
Author(s):  
KARSTEN TRULSEN ◽  
KRISTIAN B. DYSTHE
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Wave Breaking ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Deep Basin ◽  
Weakly Nonlinear ◽  
Wave Trains ◽  
Dimensional Wave

The conservative evolution of weakly nonlinear narrow-banded gravity waves in deep water is investigated numerically with a modified nonlinear Schrödinger equation, for application to wide wave tanks. When the evolution is constrained to two dimensions, no permanent shift of the peak of the spectrum is observed. In three dimensions, allowing for oblique sideband perturbations, the peak of the spectrum is permanently downshifted. Dissipation or wave breaking may therefore not be necessary to produce a permanent downshift. The emergence of a standing wave across the tank is also predicted.

THREE-DIMENSIONAL WAVE-FREE POTENTIALS IN THE THEORY OF WATER WAVES

2013 ◽  
Vol 55 (2) ◽  
pp. 175-195 ◽  
Author(s):  
HARPREET DHILLON ◽  
B. N. MANDAL
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Condition ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Finite Depth ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Thin Elastic Plate ◽  
Linearized Theory

AbstractProblems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.

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