Analytical Properties of Dispersion Relations of the Equation of Internal Gravity Waves

In this work, the analytical properties of dispersion relations of the equation of internal gravity waves with benchmark and arbitrary distributions of buoyancy frequency are investigated. To solve the problem analytically, the benchmark distribution of the buoyancy frequency is used, which is known from applied oceanological calculations in the presence of seasonal thermocline. Implicit forms of dispersion dependences are obtained; they are expressed through the Bessel function of real index. For nonzero wave numbers, an asymptotic method of studying the dispersion relation is proposed based on constructing uniform asymptotics of the Bessel functions for large values of real index and argument, expressed through the Airy functions. For an arbitrary distribution of buoyancy frequency, the asymptotic representations of dispersions relationships at small wavenumbers are obtained by means of the perturbation method and the WKB method. The solutions constructed in this work allow further computing the amplitude-phase characteristics for the fields of internal gravity waves with benchmark and arbitrary distributions of the buoyancy frequency.