(1) Introduction to Hydrodynamic Instability in Nature
a. Gravity currents in the atmosphere, rivers, and oceans, avalanches, pyroclastic flows, and finger instability of gravity currents on a slope
b. The surface-heated and double-diffusive convections (e.g. thermal and salinity stratifications in oceans)
c. Wind-ocean interaction (Kelvin-Helmholtz instability)
d. Surface and internal waves
d. Spatial instability (e.g., smoke, plume, boundary layer, wake, jet)

(2) Basic Concepts and Preliminaries
a. The matrix theory (eAt, Jordan normal form, matricant, 2-norm, normal matrix, Floquet theory)
b. The Fluid Mechanics—e.g., deriving the linearized perturbation equation and base-state solution of plane-Couette flow

(3) The Thermal Instability of a Fluid Layer Heated from Below
a. The equations of continuity, motion, and energy
b. The Boussinesq approximation
c. Non-dimensionalization (Rayleigh, Grashof and Prandtl number)
d. The base-state solution and linearized perturbation equation
e. The free-free and rigid-rigid boundary conditions
f. The Fourier transform and spectral method
g. Long-wave approximation (e.g., layering mode)
h. Solving the dynamical system
i. Stable, unstable, and neutrally stable
j. The marginal stability curve and stationary/oscillatory onset
k. Drawing the flow pattern at the instability onset—the critical-state stream function
l. The kinetic energy equation of perturbations
m. The transient instability for distinct and repeated eigenvalues
n. Three-dimensional analysis
o. Time-periodic heating at boundaries: the time-periodic basic state and perturbation equations, Floquet theory and numerical methods for obtaining the Floquet exponent, Mathieu’s equation, what if there are repeated eigenvalues ?
p. Special lecture: oscillating box—resonance in a gravity-modulated instability (a brief introduction).

(4) The Double-Diffusive Instability
a. Introduction to ‘double-diffusive’
b. The governing equations and Boussinesq approximation
c. The Rayleigh numbers for temperature and solute stratifications
d. The base-state solution and linearized perturbation equation
e. The boundary conditions for the ‘diffusive’ and ‘finger’ regimes
f. The stability diagram
g. Oscillatory/stationary onset
h. The formation of diffusive- and finger- regime layers
i. Related observations in oceans

(5) The Instability of Newtonian Fluids on an Inclined Plane
a. The governing equations for perturbations
b. The linearization of dynamic and kinematic boundary conditions
c. The Orr-Sommerfeld equation
d. First and second long-wave approximations
e. Approximation for small Reynolds number
f. The shallow-water equation
g. The instability induced by one-D wavy bottom—Floquet theory
h. Whether there can be a theory of ‘Band-gap’ in a Newtonian, mud, granular, or debris flow on a slope with two-D wavy bottom

(6) Perturbations in Plane-Sheared Dense Granular Flows (DGF)
a. The visco-plastic constitutive law
b. The inertial number
c. The base-state solution and perturbation equations

(7) Advanced Topic—Secondary Instability Theory