Thermal wind

Thermal-wind balance relates vertical shear of geostrophic flow to horizontal density or temperature gradients. With constant Coriolis parameter $f_0$ and reference density ${\rho }_0$,for the sign conventions used here.

For a steady density field sampled by two ships at angles $\theta$ and zero, write $\mathbf{y}=C\left(\begin{array}{cc}\sin \theta & -\cos \theta \\ 0& -1\end{array}\right)(u,v{)}^{\mathsf{T}}$. Independent measurement errors of variance ${\sigma }_y^2$ giveNearly parallel tracks poorly constrain $u$ and induce positive component covariance for $0<\theta <\pi /2$. Orthogonal tracks remove this covariance and give equal component precision under the stated independent-noise model.

Combining geostrophic balance, $fu=-{\Phi }_y$, $fv={\Phi }_x$, with hydrostatic balance, ${\Phi }_p=-RT/p$, and commuting the horizontal and vertical derivatives proves the displayed equations. Horizontal derivatives hold pressure fixed. This relates vertical shear to horizontal temperature gradients away from the equator; the derivation requires nonzero Coriolis parameter and the validity of geostrophic balance.

For $b=A(y)+B(z)$, constant nonzero $f_0$, and zero basic velocity at $z=0$, thermal-wind balance givesWith full buoyancy represented by $f_0{\bar{\psi }}_z$, the constant-coefficient three-dimensional quasi-geostrophic potential vorticity is $\bar{q}=f_0+z{A}^{\prime \prime }/f_0+f_0{B}^{\prime }/N^2$. The reference-buoyancy convention in quasi-geostrophic potential vorticity changes only the uniform reference contribution.