Thermal wind is a concept in meteorology that refers to the change in wind speed and direction that is associated with differences in temperature between air masses. It is not a physical wind in the sense of a local wind like a sea breeze or mountain breeze but rather a theoretical concept used to describe the influence of thermal gradients on the wind pattern in the atmosphere.
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Thermal-wind balance relates vertical shear of geostrophic flow to horizontal density or temperature gradients. With constant Coriolis parameter and reference density ,for the sign conventions used here.