## 1 Introduction
Note, that the buoyancy loss due to seawater and hull compressibility has been studied in [13, 15]. It is worth mentioning that an important feature of the ocean is stratification. Whereas here we restrict ourselves to the case of continuous variation of density there exist practical situations when the density has a step change. This can be exemplified by layers of fresh water contained between salted (sea) water and ice fields which can extend to several tens of meters depth-wise. The latter layer is formed due to ice melting into the sea. It can be easily demonstrated that, in spite of relatively small difference of the order of 2.5% between densities of sea and fresh water, the ascending UG cannot overcome this interface, enters into a hanging mode and may perform damped oscillations with frequency depending on relative difference of fresh and sea water densities. Some recent papers also discuss neutral buoyancy hulls which ensure possibility of depth-wise station keeping modes of operation which are known to have been widely employed for drifting buoys. One of the options is related to use of optimal neutrally-buoyant pressure hulls [17]. Discussed in [18] is flexible-bodied UG with neutral buoyancy so named _Flexible-Liquid-Rigid_ (FLR) composite hull. Note that with purpose of buoyancy control of the _diving buoys_ simplified engineering approaches are presented in [19] and [20] where in the authors state that the accuracy of measuring of physical parameters of water essentially depends on the speed of vertical motion of the _diving buoys_. In these papers variation of water density and hull deformation were not accounted for. The estimates have been made by the authors of the time for ascension and submersion in what they call _ordinary_ and _energy-saving_ modes. The motivation for preparing this paper was to construct a simple mathematical model that would allow to assess main effects related to water density variation and hull compression and explore influence of corresponding buoyancy control. To simplify the model we are using one-dimensional formulation reducing mathematics to an ordinary differential equation including terms accounting for depth-wise density gradient in combination with elementary _boiler formulae_ of structural mechanics for hull compression. The latter enable evaluation of the influence of the pressure hull geometry and material on the dynamics of submersion. Further glider applications would require consideration of lifting appendages (wings) although hopefully the effects in question would mostly depend on the depth effects studied herein for purely vertical motion. Advantages in applying a simplified model similar to what is discussed hereinafter include a possibility to explore energy expense required for compensation of buoyancy variation exemplified for the case of securing constant speed of submersion for a vehicle subject to influence of vertical density gradient and hull deformation. The present approach also allows to explore plausibility of an evident suggestion of counterbalancing nature of the accelerating effect of hull compression and decelerating impact of density growth (pycnocline effect). The model in question naturally produces a requirement of equal rate of compression of water and pressure hull as a condition of constant speed and easily yields a provision leading to _hanging modes_ whereby the vehicle stops and can be moved further through buoyancy control on board. _In what follows_: we formulate one dimensional mathematical description of the problem, then discuss analytical and numerical solutions for the speed and depth of submersion, exemplified by cylindrical and spherical pressure hulls with and without account of density gradient and hull compression. When the latter effects are present we explore conditions of occurrence of _hanging modes_ of the vehicle as well as buoyancy control for provision of constant speed of vertical motion with estimates of required energy expense and on-board buoyancy reserve.