A review is given of relevant work on the internal boundary layer (IBL) associated with:
 The major theme in all cases is on the downstream, modified profile form (wind and temperature), and on the growth relations for IBL depth.

The internal boundary layer — A review

In this paper, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky-Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.

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Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity

We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke–Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to ϵ = ±1, m = 0, 1. (I) ω = ϵ σ r , ϵ = ± 1 , m = 1 , 0 The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal (t → ∞) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters σ, r, b, ω and m.

Analytical and numerical investigation of two families of Lorenz-like dynamical systems

The WKB method has been used to develop an approximate solutionof the semi-geostrophic Ekman boundary layer with height-dependenteddy viscosity and a baroclinic pressure field The approximate solutionretains the same simple form as the classical Ekman solution Behavioursof the approximate solution are discussed for different eddy viscosityand the pressure systems These features show that wind structure inthe semi-geostrophic Ekman boundary layer depends on the interactionbetween the inertial acceleration, variable eddy viscosity and baroclinicpressure gradient Anticyclonic shear has an acceleration effect on theair motion in the boundary layer, while cyclonic shear has a decelerationeffect Decreasing pressure gradient with height results in a super-geostrophicpeak in the wind speed profile, however the increasing pressure gradient withheight may remove the peak Anticyclonic shear and decreasing the variableeddy viscosity with height has an enhanced effect on the peak

An Approximate Analytical Solution For The Baroclinic And Variable Eddy Diffusivity Semi-Geostrophic Ekman Boundary Layer

A review is presented of some Russian-language papers published as early as 1952–1954 that are unknown to most English-speaking readers. In these papers the influence of an abrupt change in the surface friction velocityv*, surface roughnessz0 and surface temperature or heat flux on the wind velocity profile and vertical motions has been investigated analytically and numerically. Most of the theories are based on the exchange-coefficient approximation for momentum and heat. In terms of this approximation, further generalization and development of the problem is discussed.

Wind profile and vertical motions above an abrupt change in surface roughness and temperature

A model of a stationary planetary boundary layer is proposed based on the equation of motion with the advective term retained. The latter is modeled by means of the so-called geostrophic momentum approximation in two versions — original and modified. New expressions for the vertical velocity W∞ at the top of the boundary layer are derived and analyzed. They underestimate W∞ compared to the classical expression.

A barotropic model of the Ekman planetary boundary layer based on the geostrophic momentum approximation

On the Lorenz System with Strengthened Nonlinearity

EN GENERAL, LOS SISTEMAS DINAMICOS NON LINEALES (NDS'S) DESCRITOS POR SISTEMAS DE ECUACIONES DIFERENCIALES ORDINARIAS NO LINEALES SON SENSIBLES A LOS CAMBIOS ESTRUCTURALES DE ESTAS. AUN CONSTANTES SIMPLES ADITIVAS PUEDEN CAMBIAR EL COMPORTAMIENTO GLOBAL DE UN SISTEMA DADO, POR EJEMPLO (1.4) MOSTRADO A CONTINUACION. PARA ESTUDIAR DIRECTAMENTE ESTE EFECTO ESCOGIMOS DOS (NDS'S) MUY CONOCIDOS: EL SISTEMA CLASICO DE LORENZ (1.2), (1.3) Y EL SISTEMA DE WIIN-NIELSEN (1.6) QUE TIENEN AMBOS VARIAS APLICACIONES METEREOLOGICOS. MAS AUN POSTULAMOS Y DAMOS UNA RESPUESTA AL PROBLEMA INUSITADO: ?CUAL DEBERA SER EL FORZAMIENTO ADITIVO CONSTANTE DE ESTOS SISTEMAS, A FIN DE TENER UN NUMERO DESEADO DE PUNTOS FIJOS REALES CON UNAS PROPIEDADES DE ESTABILIDAD DESEADAS? SE INCLUYEN TAMBIEN ALGUNOS RESULTADOS NUMERICOS PRELIMINARES.

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Reaction of two simple nonlinear dynamical systems to constant forcing.

Nonlinear dynamical systems (systems of 1 st order ordinary differential equations) capable of generating chaos are analytically nonintegrable. Despite of this fact, analytical tools can be used to extract useful information. In this paper the original Lorenz system and its modifications are reduced to single oscillatory type integral-differential equations with delayed argument. This yields to appearance of an ‘‘endogenous’’ term interpreted as memory for the past. Moreover, the equations are valid far from the initial instant (theoretically at t→∞), when the system eventually evolves on its attractor set. This corresponds to the numerical solutions when an appropriate initial part of the iterates is usually discarded to eliminate the transients. Besides, the form of the equations allows statistical treatment.

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S. Panchev

Papers

Wind profile and vertical motions above an abrupt change in surface roughness and temperature

A barotropic model of the Ekman planetary boundary layer based on the geostrophic momentum approximation

On the Lorenz System with Strengthened Nonlinearity

Reaction of two simple nonlinear dynamical systems to constant forcing.

The Lorenz chaotic systems as nonlinear oscillators with memory