In this course, some general concepts regarding the properties of nonlinear waves, vortices, and turbulence are introduced by focusing on the study of a hydrodynamical problem of wide astrophysical interest. In addition to building the general framework, the lectures are the occasion to present several new important results.

The correct 2D equations describing the dynamics of astrophysical fluid disks (both for the case where the self-gravity of the disk is important and for the case where it is negligible) are derived from the full set of 3D hydrodynamical equations. Based on a WKB approximation, some stationary solutions of the 2D equations are shown to originate solitons that propagate in the form of large-scale spiral arms in the disk. In the geostrophic approximation (applicable when the pattern rotates slowly in comparison with the angular velocity of the disk) stationary solutions give rise to solitary vortices, in the case of a "scalar" nonlinearity, and to double vortices ("modons"), in the case of a "vector" nonlinearity. Nonstationary solutions describe the Rossby wave turbulence. The relevance of the results presented in the course to observational data will be briefly discussed.

1. Introduction.
   • Linear phenomena.
     • Oscillations and waves of small amplitude.
     • Vortices.
     • Instabilities.
   • Nonlinear phenomena.
     • Saturating and explosive instabilities.
     • Wave-breaking. Wave packet spreading. Wave solutions.
       • Wave breaking.
       • Wave packet spreading.
       • Wave solitons.
     • Nonlinear interaction of waves and wave turbulence. Interaction of solitons.
       • The necessary condition for the turbulence.
       • Some definitions and Kolmogorov-Obukhov hypothesis.
       • Three-wave interaction.
       • Four-wave interaction.
       • Interaction of solitons.
     • Nonlinear vortices. Vortex turbulence and its difference from wave turbulence.
       • Different kinds of nonlinear vortices in astrophysical disks.
       • Vortex turbulence and its difference from wave turbulence.
2. Derivation and applicability of 2D models for astrophysical fluid disks.
   • Astrophysical disks are one of the most interesting dynamical system in the Universe. Some unsolved problems.
   • Traditional conditions for 2D description of the astrophysical disks.
   • Basic equations for "volume" 3D functions.
     • Basic dynamical equations.
     • The equation of state.
   • Derivation of the appropriate equations for "flat" 2D functions.
     • Ordering of the relevant terms in the basic equations.
     • Closed system of integro-differential equations for a barotropic disk.
     • Two limiting cases of astrophysical disks.
   • The closed set of differential equations for a politropic disk in the external gravitational field.
     • Derivation of the 2D equations.
     • On the application C = const.
   • The closed set of differential equations for a politropic self-gravitating disk.
     • The derivation of 2D equations.
     • Why the gradient of the flat pressure has not the physical sense of a force.
   • 3D waves of the small amplitude simmetrical with respect to the disk plane.
     • Perturbation theory.
     • Zero approximation.
     • First approximation. Closed set of the six linearized equations.
     • The derivation of the closed system of integro-differential equations for 3D perturbations.
     • The derivation of a dispersion relation, describing 3D perturbations.
       • The derivation of a dispersion relation for 3D isothermal disk in the strong external gravitational field.
       • Self-gravitating disk.
3. Envelope solitons and explosive instability in self-gravitating politropic disks.
   • Basic equations for "flat" functions.
   • Zero epproximation.
   • Derivation of nonlinear equation describing the dynamics of arbitrary amplitude perturbations in the WKB-approximation.
   • The derivation of an equation with a cubic nonlinearity describing the rotatioing disk dynamics on the boundary of gravitational instability.
   • Linear approximation: on the boundary of gravitational instability.
   • The generation of multiple harmonics.
   • Nonlinear dispersion relation. Nonlinear stabilization.
   • Envelope solitons.
   • Explosive instability.
   • Nonlinear waves in a viscous medium.
   • Shock waves, their stability.
   • Astrophysical applications: galactic and accretion disks.
4. Rossby vortices in rotating astrophysical objects.
   • Introduction.
   • Basic initial equations.
   • The geostrophic (epicyclic) approximation. Rossby number. Iteration method.
   • The derivation of a nonlinear dynamical equation.
   • Investigation of nonlinear dynamical equation.
     • The proof of the infinitesimal of terms in the right-hand side in comparison with terms in the left-hand side.
     • Some specific cases.
   • The canonical form of dynamical equation with a vector nonlinearity.
   • Linear approximation: the Rossby waves in the atmosphere and ocean.
   • The canonical form of dynamical equation with a vector and scalar nonlinearities.
   • Single vortices: cyclons and anticyclons.
   • Double vortices (modons).
   • Vortex turbulence.
   • Astrophysical applications: active and interacting galaxies
5. Turbulence in rotating astrophysical objects.
   • Analysis of observational data.
   • Model. Basic equations.
   • Derivation of kinetic equation for interecting waves
   • Agreement of weak Rossby wave turbulence spectrum with that for interstellar medium.
6. Conclusion.

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