Dynamics of Upward Jets with Newtonian Cooling
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STICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Dynamics of Upward Jets with Newtonian Cooling1 V. P. Goncharova,* and V. I. Pavlovb,** a
b
Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 109017 Russia UFR des Mathématiques Pures et Appliquées, Université de Lille, CNRS FRE 3723-LML, F-59000 Lille, France * e-mail: [email protected] ** e-mail: [email protected] Received September 4, 2017
Abstract—The Rayleigh–Taylor instability which is responsible for the occurrence of narrow upward jets is studied in the scope of the nonhydrostatic model with horizontally nonuniform density and the Newtonian cooling. As analysis shows, the total hierarchy of instabilities in this model consists of three regimes—collapse, algebraic instability, and inertial motion. Realization of these stages, mutual transitions, and interference depend on a ratio between two characteristic time scales—collapse time and cooling time. DOI: 10.1134/S106377611801003X
1. INTRODUCTION It is known that the Rayleigh–Taylor instability (RTI) appears when a higher density fluid is positioned above a fluid with lower density in a gravitational field or in a noninertial system when the fluid with lower density accelerates the fluid of higher density. In the last case, the instability arises due to “effective gravity” connected with acceleration or slowdown of the medium which contains domains with different density [1]. The physical cause of the initial layered density stratification can be various. Layers with temperature inhomogeneity, layers of salinity, inhomogeneous distribution of bubbles can be observed in geophysical conditions. Turbidity currents, whose the average density stratification derives from suspended mud or silt, can also exist. Collision and intersection of geophysical flows represent one more excellent example of noninertial systems with “effective gravity” in which the development of RTI arises at the vertical interfaces under the influence of the horizontal component of acceleration. The main aim of this work is to study the influence of cooling on the development of the RTI. In astrophysics, geophysical fluid dynamics and technical applications, numerous examples can be found when RTI is initiated by thermal irregularities localized in thin horizontal (perpendicular to gravity) layers. In boundary layers, such irregularities arise as a result of nonuniform heating and look like islets of more hot (light) fluid. 1 The article is published in the original.
The dynamics of these thermal islets is essentially nonlinear because their occurrence severely disturbs the balance in the fluid. (Let us remind that when a system is conservative and its initial state coincides or close to the unstable equilibrium then, at an early stage, its perturbations should evolve in the regime of exponential growth. Such behavior is predicted by the linear theory. That is how the instability develops in the classical problem [2] about the evolution of a plane interface separating two fluids when a heavier fluid lie
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