This paper addresses the problem of behavior modes of baroclinic geostrophic vortices with ellipsoid-shaped cores in horizontal flows with vertical shear. In such flows, a vortex core is confined between two stationary horizontal planes that the vortex contacts with its upper and lower points. Under the background flow effect, the length of all ellipsoid axes can change in space, as do vortex orientation angles. We identify three vortex behavior modes. The first mode is vortex survival in a shear flow where the vortex undergoes finite oscillations of its semi-axes for an indefinite period of time and can exhibit complicated behavior in
terms of its orientation angles. This mode corresponds to strong vortices. In the second mode, the vortex elongates along the flow from the beginning, remaining with finite horizontal dimensions perpendicular to the flow,
and is compressed vertically. This mode involves the destruction of the vortex by the flow, resulting finally in a
thin vertical ocean structure formed from the vortex. Weak vortices undergo this type of evolution. This type is referred to as the “unlimited elongation mode.” Finally, the third mode is called the “finite lifetime mode”: for a finite period of time, the vortex behaves similarly to the survival mode (its shape is deformed in a finite manner and it rotates or oscillates in space), but eventually the vortex elongates indefinitely in a manner similar to the
destruction mode. We outline the region of each mode on a dimensionless parameter plane of the problem and
define the boundaries between the above-mentioned vortex behavior modes.