A surface-aware projection basis for quasigeostrophic flow

Abstract: Recent studies indicate that altimetric observations of the ocean's mesoscale eddy field reflect the combined influence of surface buoyancy and interior potential vorticity anomalies. The former have a surface-trapped structure, while the latter have a more grave form. To assess the relative importance of each contribution to the signal, it is useful to project the observed field onto a set of modes that separates their influence in a natural way. However, the surface-trapped dynamics are not well-represented by standard baroclinic modes; moreover, they are dependent on horizontal scale. Here we derive a modal decomposition that results from the simultaneous diagonalization of the energy and a generalisation of potential enstrophy that includes contributions from the surface buoyancy fields. This approach yields a family of orthonomal bases that depend on two parameters: the standard baroclinic modes are recovered in a limiting case, while other choices provide modes that represent surface and interior dynamics in an efficient way. For constant stratification, these modes consist of symmetric and antisymmetric exponential modes that capture the surface dynamics, and a series of oscillating modes that represent the interior dynamics. Motivated by the ocean, where shears are concentrated near the upper surface, we also consider the special case of a quiescent lower surface. In this case, the interior modes are independent of wavenumber, and there is a single exponential surface mode that replaces the barotropic mode. We demonstrate the use and effectiveness of these modes by projecting the energy in a set of simulations of baroclinic turbulence.

• The paper introduces a new surface-aware modal decomposition that captures both surface-buoyancy and interior potential vorticity dynamics through simultaneous diagonalization of energy and extended enstrophy.
• It employs a tunable parameter framework to recover classical baroclinic and exponential SQG-like modes, enabling efficient hybrid representations of diverse flow regimes.
• Numerical simulations demonstrate that optimal selection of weights improves energy reconstruction in regimes with strong surface gradients, enhancing reduced modeling and data assimilation.

Surface-Aware Modal Decomposition for Quasigeostrophic Flow

Introduction and Motivation

The paper "A surface-aware projection basis for quasigeostrophic flow" (1206.4468) addresses a foundational gap in the modal analysis of oceanic and atmospheric flows governed by quasigeostrophic (QG) dynamics. Classic modal decompositions, notably baroclinic modes, are widely used for projecting vertical structures of geostrophic flows. However, compelling observational and theoretical evidence shows that surface-intensified motions—such as those represented by Surface Quasigeostrophic (SQG) dynamics—are poorly captured by standard baroclinic projections, especially in the presence of strong surface buoyancy anomalies and at finescale (submesoscales). The canonical baroclinic modes assume zero surface buoyancy, resulting in inefficient and slow convergence when expanding realistic flows with significant near-surface gradients.

This motivates the need for a new spectral basis capable of representing both surface-trapped and interior dynamics, providing an orthonormal decomposition that naturalizes the separation between surface (buoyancy-driven) and interior (potential vorticity-driven) signals. The new basis should diagonalize both energy and an extended potential enstrophy, accommodating non-zero surface and bottom buoyancies.

Theory: Generalized Modal Basis via Simultaneous Diagonalization

The proposed method introduces a "surface-aware" modal decomposition constructed through the simultaneous diagonalization of two quadratic invariants: the QG energy and a generalized potential enstrophy that explicitly incorporates the variances of surface and bottom buoyancy fields. This leads to the definition of a generalized potential vorticity vector $\mathbf{Q}$, which aggregates surface and bottom buoyancies and interior PV as independent variables.

Given a horizontally periodic domain and after horizontal Fourier decomposition, the continuous (and discretized) system considers the following invariants at each horizontal wavenumber $\kappa$:

• Total energy $E_\kappa$
• Potential enstrophy $Z_\kappa$
• Surface buoyancy variances $B_\kappa^\pm$

By parameterizing a generalized "potential enstrophy" functional as $P_\kappa = Z_\kappa + \alpha_+ B^+ + \alpha_- B^-$, with tunable non-dimensional weights $(\alpha_+, \alpha_-)$, the authors formulate an eigenproblem that seeks modes simultaneously orthogonal with respect to energy and diagonal in this extended enstrophy. The resulting eigenvalue problem for the modal streamfunctions $\phi_n(z)$ is non-standard due to the eigenvalue appearing in both the interior operator and Robin-type boundary conditions, and the structure of the modes becomes dependent on $\kappa$ and the chosen weights $\alpha_\pm$.

The classical baroclinic modes are recovered for $\alpha_\pm \rightarrow \infty$, corresponding to zero surface buoyancy. In the opposite limit ($\alpha_\pm \ll 1$), purely surface and bottom-trapped, exponential SQG-like modes are recovered. Intermediate and asymmetric choices yield "hybrid" modes that efficiently represent flows with both significant surface and interior activity.

Analytical Properties, Special Cases, and Limiting Regimes

In the case of constant stratification ($N$ constant), the eigenproblem admits analytical solutions. For large $\alpha$, the basis converges to the standard baroclinic set, with oscillatory vertical structure and boundary conditions $\phi'_n = 0$. For small $\alpha$, two evanescent (surface/bottom-intensified) modes emerge, with explicit exponential (or hyperbolic) vertical structure and strong $\kappa$-dependence.

A critical case arises when the lower boundary is assumed quiescent (as is typical in oceanic contexts), i.e., $\alpha_- \rightarrow \infty$, $\alpha_+$ finite or small. Here, the basis separates into a set of wavenumber-independent interior modes (Dirichlet modes vanishing at the upper surface, Neumann at the bottom), and a single wavenumber-dependent surface mode that encapsulates the exponential surface dynamics. These limit cases are of particular relevance for practical oceanographic applications, as they closely correspond to observed flow decompositions.

Numerical Validation: Projection of Simulated Turbulent QG Flows

The utility and parsimony of the surface-aware basis are empirically demonstrated by projection of high-resolution QG simulations with three prototype mean states:

• Interior-driven (baroclinic mode-1) instability: Energy is efficiently captured by classical baroclinic modes (large $\alpha$ optimal).
• Surface-driven (Eady-type) instability: Energy is concentrated at the surfaces and is poorly resolved in the baroclinic basis but is efficiently captured by the surface-aware modes (small $\alpha$ optimal).
• Semi-realistic oceanic state: Characterized by surface-intensified dynamics and quiescent bottom, the corresponding projection demonstrates that a single surface mode (with $\alpha_+ \ll 1, \alpha_- \gg 1$) plus standard interior modes captures the vast majority of energy, agreeing with theoretical expectations.

Strong numerical evidence underscores that the optimal choice of weights $\alpha_\pm$ is regime-dependent: extreme values isolate pure surface or interior regimes, while moderate values are beneficial for coexisting signals. The relative energy content captured by the first two modes provides an operational criterion for selecting optimal weights.

Implications and Prospects

This modal framework addresses longstanding obstacles in connecting surface altimetric observations to the full-depth, three-dimensional structure of oceanic turbulence—crucial for interpreting present and forthcoming high-resolution satellite data. The theoretical clean separation between PV-driven and surface-buoyancy-driven signals has practical utility for reduced-order modeling, data assimilation, and physical interpretation of remote sensing products.

A notable theoretical implication is that the resulting basis is inherently horizontal-scale-dependent ($\kappa$-dependent) for surface modes, a necessary feature to retain the correct SQG dynamics. This precludes a strict separation-of-variables representation for all modes but is unavoidable for representing realistic boundary-driven flows.

With the flexibility provided by the parameters $(\alpha_+, \alpha_-)$, the method allows adaptive selection of the basis for specific dynamical regimes or even a wavenumber-dependent tuning. Quantitative diagnostics such as the ratio of projected $Z_\kappa$ to $B_\kappa^+$, or connections to large-scale mean gradients, can be used for systematic parameter selection.

Conclusion

This work provides a rigorous, generalizable framework for modal decomposition of QG flows that incorporates both surface and interior dynamics in an energy-optimal, orthogonal manner. For oceanographic applications, particularly in regimes dominated by strong surface gradients, this yields a basis capable of much more efficient (lower-rank) reconstructions compared to standard baroclinic expansions. The practical implications are substantial for the interpretation of satellite altimetry, development of surface-to-interior inference schemes, and construction of dynamically efficient reduced models.

Future extensions may consider dynamic tuning of weights based on flow diagnostics or explore generalizations to more complex stratifications, non-rigid boundaries, or domains with varying bathymetry. The projection methodology has potential for widespread adoption in analysis and parameterization frameworks targeting the energetic submesoscale regime observed in contemporary ocean data.