Balanced shallow-water energy as a signed potential-vorticity pairing ($E_b=-\frac{gH}{2f}\int \eta qdA$)

With constant density divided out, a decaying or periodic geostrophic balance on an f-plane haswhere $q=\zeta -f\eta /H$ is the signed linearized shallow-water potential-vorticity anomaly. The stationary height equation and integration by parts prove the identity. Replacing $q$ by $|q|$ destroys it: an odd height paired with symmetric opposite-sign sheets would then misleadingly give zero.