A method to decompose geophysical flows into a balanced flow (defined by its potential vorticity, PV) and an imbalanced component (inertia–gravity waves, IGWs) is introduced. The balanced flow, called the optimal potential vorticity (OPV) balance, is a solution of an IGW-permitting dynamics in which the amount of IGWs is minimal. The residual IGWs are those spontaneously generated by the vortical flow during the numerical integration in which the PV anomaly grows slowly over a sufficiently long ramp period toward a prescribed PV field. The OPV balanced flow is obtained, iteratively, in a cycle of backward and forward integrations where IGWs are removed and PV is restored in every loop. The method is applied to the flow of unsteady vortices in the three-dimensional baroclinic non-hydrostatic dynamics on the $f$-plane and to the single-layer shallow-water dynamics on the sphere. Both applications show that the iterative method converges strongly, after only a few iterations, to the balanced flow.

The generation and propagation of a packet of small-amplitude inertia–gravity waves (IGWs) in a rotating stratified balanced flow is described. The initially balanced geophysical flow is an unstable baroclinic jet which breaks up into a street of cyclonic and anticyclonic vortices. The small-amplitude unbalanced component of the flow is extracted from the large-amplitude mesoscale balanced flow using the optimal potential vorticity balance approach. This analysis reveals that during the instability the balanced flow spontaneously emits bursts of IGWs. The emission occurs along two directions, into and out of the anticyclonic vortices. The inward-waves remain trapped inside the vortices while the outward-waves propagate away from them as a packet of small-amplitude IGWs with a three-dimensional helical structure. The wave packet emission is confirmed for different spatial resolutions ($128^3$, $160^3$, $192^3$ and $256^3$ grid points). The ratio between the balanced vertical and horizontal velocity components is of the order of $10^{-3}$, as is typical of mesoscale geophysical flows. The ratio between the unbalanced vertical and horizontal components is about 0.1. Since the unbalanced horizontal and the balanced vertical velocity components are of similar magnitude, the vertical velocity of the IGWs is about $10^{-4}$ times the balanced horizontal velocity. The IGWs are dominated by frequencies close to the inertial frequency and have a clockwise-rotating horizontal velocity, similar to plane wave solutions.

Results of a variety of numerical simulations are presented and the accuracy of quasigeostrophic (QG) and semigeostrophic (SG) vertical velocity estimates of the total vertical velocity is analyzed. The authors examine the dependence of the results on the potential vorticity (PV) anomaly of the flow, its time evolution, and the amount of numerical diffusion. The SG ω equation is solved in a novel way in the original physical coordinates rather than in geostrophic coordinates. A three-dimensional numerical model is used that explicitly conserves the PV on isopycnal surfaces through a contour-advective semi-Lagrangian (CASL) algorithm. The numerical simulations consist initially of one or two horizontal cylinders of anomalous PV: a shear zone that induces two or three counterflowing jets. These jets destabilize and break into cyclones or anticyclones. This is accompanied by enhanced vertical motion, which exhibits a dominantly balanced quadrupole pattern in horizontal cross sections, depending on the ellipticity of the gyres, together with weak second-order inertia–gravity waves. For flows containing only negative PV anomalies the magnitude of both the QG and SG vertical velocities are smaller than the magnitude of the total vertical velocity, while the opposite occurs for flows containing only positive PV anomalies. The reason for this behavior is that the QG ω equation misses a term proportional to the Laplacian of the horizontal velocity. A new, more accurate, ω equation is proposed to recover the vertical velocity when both experimental density and horizontal velocity data are available. The SG solution is nearly always more accurate than the QG solution, particularly for the largest vertical velocity values and when the flow has single-signed PV anomalies. For flows containing both positive and negative PV anomalies, for example, mushroomlike eddies, the QG vertical velocity is a better approximation to the total vertical velocity than the SG solution. The reason for this anomalous behavior lies in one additional assumption concerning the conservation of volume that is usually adopted to derive the SG ω equation.

High-resolution three-dimensional numerical experiments show that initially balanced (void of waves) geophysical flows, static and inertially stable, generate spiral patterns of small-amplitude inertia–gravity waves (IGWs). The spiral wave patterns are due to the spontaneous generation of IGW packets emitted from fluid volumes (the IGW sources) experiencing large local changes of potential vorticity. The IGW packets spread away from the vortical flow and cause spiral wave patterns of the same sense of spiralling, cyclonic or anticyclonic, as the moving IGW sources. The spiral patterns are noticeable in the vertical velocity in deep layers, away from the large-amplitude balanced vertical velocity. The generation of the spiral wave patterns is illustrated through several examples, namely, the single ellipsoidal vortex (cyclone and anticyclone), the merging of two spherical vortices, the dipole, and the anticyclonic shear instability.

Rotating stably stratified geophysical flows can exhibit a near ‘balanced’ evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both high-frequency acoustic waves and lower-frequency inertia–gravity waves (IGWs) contribute little to the flow evolution compared with the even-lower-frequency advection of PV. Moreover, this ‘slow’ PV-controlled balanced evolution appears unable to excite these higher-frequency waves in any significant way – i.e. balance persists.The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolution – over many inertial periods (days) – remains strongly controlled by PV advection.

Two fundamental modes of vertical velocity (w) in mesoscale subsurface eddies are described using the quasigeostrophic (QG) approximation and nonhydrostatic numerical modeling. The first mode of w (the spheroidal mode) arises when a spheroidal upright subsurface eddy acquires horizontal eccentricity and becomes an ellipsoid, still upright, vertically symmetric vortex. In this case, the vertical displacement of isopycnals vanishes at the middepth z = 0. Conservation of potential vorticity anomaly (PVA) on elliptical concave/convex isopycnals entails a three‐dimensional octupolar pattern of w which also vanishes at z = 0. The second mode of w (the tilted mode) arises when the eddy remains spheroidal but its vertical axis tilts relative to the vertical direction. In this case, the displacement of isopycnals is largest at the middepth z = 0 and has a dipolar distribution. The associated w is largest at the middepth and develops also a dipolar pattern. In both spheroidal and tilted modes, the vertical velocity pattern may be inferred from the fast advection of PVA conserving fluid particles on slower translating concave/convex or tilted isopycnals. This implies that the vertical velocity of both modes is approximately QG and may be correctly inferred from the QG omega equation as long as the Rossby number remains small. Under more general circumstances, the vortex is both spheroidal and tilted. In this case, both spheroidal and tilted modes coexist but remain, to a large extent, uncoupled, rotating with different and, at least at a first order of approximation, constant phase speeds.