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Suppose that this temperature is T0 at the point x0, y0 , z0 and time t0 . If the balloon moves to the point x0 + δx, y0 + δy, z0 + δz in a time increment δt, then the temperature change recorded on the balloon, δT , can be expressed in a Taylor series expansion as δT = ∂T ∂t δt + ∂T ∂x δx + ∂T ∂y δy + ∂T ∂z δz + ( higher order terms) Dividing through by δt and noting that δT is the change in temperature following the motion so that DT δT ≡ lim δt→0 δt Dt we find that in the limit δt → 0 ∂T DT = + Dt ∂t ∂T ∂x Dx + Dt ∂T ∂y is the rate of change of T following the motion.

2, the longitudinal derivative of i can be expressed as ∂i = j sin φ − k cos φ ∂λ However, = (0, sin φ, Da i = Dt cos φ) so that (j sin φ − k cos φ) = ×i In a similar fashion, it can be shown that Da j/Dt = ×j and Da k/Dt = ×k. 2) Fig. 2 Resolution of δi in Fig. 1 into northward and vertical components. 3) Dt The left-hand side represents the rate of change of the absolute velocity Ua , following the motion as viewed in an inertial system. The right-hand side represents the sum of the real forces acting per unit mass.

1 Total Differentiation of a Vector in a Rotating System The conservation law for momentum (Newton’s second law of motion) relates the rate of change of the absolute momentum following the motion in an inertial reference frame to the sum of the forces acting on the fluid. For most applications in meteorology it is desirable to refer the motion to a reference frame rotating with the earth. Transformation of the momentum equation to a rotating coordinate system requires a relationship between the total derivative of a vector in an inertial reference frame and the corresponding total derivative in a rotating system.