Oscillations of the $^7$Be solar neutrinos inside the Earth

We explore in detail oscillations of the solar $^7$Be neutrinos in the matter of the Earth. The depth of oscillations is about $(0.1 - 0.2)\%$ and the length $\approx 30$ km. The period of the oscillatory modulations in the energy scale is comparable with the width of the line determined by the temperature in the center of the Sun. The latter means that depending on the length of trajectory (nadir angle) one obtains different degree of averaging of oscillations. Exploring these oscillations it is possible to measure the width of the $^7$Be line and therefore the temperature of the Sun, determine precisely $\Delta m^2_{21}$, perform tomography of the Earth, in particular, measure the deviation of its form from sphere, and detect small structures. Studies of the Be neutrinos open up a possibility to test quantum mechanics of neutrino oscillations and search for the sterile neutrinos. Accuracy of these measurements with future scintillator (or scintillator uploaded) detectors of the $\sim 100$ kton mass scale is estimated.