Non-Traditional Parametrisation of GIWs

• Non-traditional parametrisation of GIWs introduces innovative frameworks that relax classical rotation and geometry constraints to improve simulation fidelity.
• It enables independent treatment of geometry and solution spaces, enhancing numerical adaptivity in isogeometric analysis and model order reduction.
• The approach advances asteroseismic diagnostics, robust fluid dynamics simulations, and generalized gauge formulations across astrophysical and computational mechanics domains.

Non-traditional parametrisation of GIWs (gravito-inertial waves) encompasses a diverse set of advanced frameworks in which classical assumptions and couplings—in geometry, physics, and functional space—are systematically relaxed. This paradigm extends beyond the traditional approximation of rotation (TAR) in stellar and planetary physics, and also encompasses generalised parameter representations in computational mechanics and geometric numerics. The aim is to maximise both physical fidelity and computational flexibility in describing, simulating, and parametrising systems dominated by inertia-gravity wave dynamics or their analogues.

1. Foundational Concepts and Scope

The classical (traditional) parametrisations of GIWs rely on tight coupling between physical symmetries and simulation spaces, or on simplifying assumptions such as the neglect of certain Coriolis terms (TAR), iso-parametric geometry-function spaces in isogeometric analysis, or reliance on rigid group symmetries in gauge theory. Non-traditional parametrisation refers to systematic relaxation or generalisation of these constraints, enabling new physics (e.g., in the presence of deformation, non-rigidity, or more general geometric evolutions) and/or advanced numerical adaptivity.

The domains of application currently include:

• Rotating stratified fluids in geophysics, planetary, and stellar interiors (Mathis, 23 Oct 2025, Mathis et al., 2019, Dhouib et al., 2021, Dhouib et al., 2021)
• Isogeometric analysis and simulation, especially in decoupling geometry from solution spaces (Atroshchenko et al., 2017)
• Quantum gravity and gauge theories, allowing generalised field splits, parameter spaces, and symmetries (Kotov et al., 2014, Gies et al., 2015)
• Computer-aided design integrated parametric solutions (Sevilla et al., 2019)

2. Mathematical Formulation: Breaking Traditional Constraints

a. Rotating Stratified Fluids and the Full Coriolis Model

The TAR neglects the horizontal (latitudinal) component of the Coriolis acceleration, valid only under $N \gg 2\Omega$. In the non-traditional parametrisation, the vertical and horizontal components are preserved: $$\vec{\Omega} = \Omega(\sin\Theta\,\hat{e}_y + \cos\Theta\,\hat{e}_z)$$ yielding:

• $f = 2\Omega\cos\Theta$ (vertical)
• $\tilde{f} = 2\Omega\sin\Theta$ (horizontal)

The equations of motion are: $$\begin{cases} \partial_t w - \tilde{f} u = -\frac{1}{\overline{\rho}} \partial_z p' + b + \nu \nabla^2 w \ \partial_t v + f u = -\frac{1}{\overline{\rho}} \partial_y p' + \nu \nabla^2 v \ \partial_t u - f v + \tilde{f} w = -\frac{1}{\overline{\rho}} \partial_x p' + \nu \nabla^2 u \ \end{cases}$$ and the resulting wave equation ("non-traditional Poincaré equation") couples vertical and horizontal wave dynamics explicitly (Mathis, 23 Oct 2025).

b. Geometry Independent Parametrisation (GIFT) in Isogeometric Analysis

Traditional isogeometric analysis (IGA) enforces that the function space for geometric representation and solution approximation is the same. Non-traditional parametrisation, as in the GIFT framework, employs independent function spaces:

• Geometry: $$x(\boldsymbol{\xi}) = \sum_{k \in I} C_k\, N_k(\boldsymbol{\xi})$$
• Solution: $$u(\boldsymbol{\xi}) = \sum_{j \in J} u_j\, M_j(\boldsymbol{\xi})$$

This enables, for example, geometry in NURBS, solution in PHT-splines—permitting local refinement and adaptivity independently of the global geometry (Atroshchenko et al., 2017).

c. Generalised Symmetry and Parametrisation in Gauge Theories

Traditional gauge theory requires invariance under group actions; the non-traditional approach generalises symmetry to regular foliations defined by vector fields $v_a$ satisfying an extended Killing equation: $v_{a(i;j)} = 0$ with gauge fields transforming under non-standard laws: $$\delta A^a = d\varepsilon^a + C^a_{bc} A^b\varepsilon^c + \omega^a_{b i} \varepsilon^b (dX^i - \rho^i_c A^c)$$ Thereby Lie groupoids (not just groups) become the structural symmetry (Kotov et al., 2014).

d. Parametric Model Order Reduction

Non-traditional parametrisation in model order reduction, specifically using NURBS control points as geometric parameters and PGD (proper generalised decomposition), allows direct, highly separated solutions for $u(x, \boldsymbol\mu)$ over vast parameter spaces (Sevilla et al., 2019).

3. Physical and Computational Implications

a. Rotation-Induced Effects and Beyond TAR

In rapidly rotating, weakly stratified, or sub-inertial regimes, non-traditional parametrisation significantly alters dynamics:

• Vertical wavenumber increases: $$k_V^2 = k_\perp^2 \left[ \frac{N^2 - \omega^2}{\omega^2 - f^2} + \left( \frac{\omega \tilde{f}_s}{\omega^2 - f^2} \right)^2 \right]$$
• Stronger damping and equatorial trapping of waves
• Momentum deposition occurs closer to excitation zones; classical TAR overestimates altitude of wave breaking and angular momentum transport efficiency
• For mean flow evolution: $$\overline{\rho} \,\frac{d\overline{U}}{dt} = \partial_z F_{\rm AM;V}(z)$$
• Improved inhibition of convective and shear-induced wave breaking, leading to weaker mixing and angular momentum redistribution than TAR-based estimates (Mathis, 23 Oct 2025)

b. Asteroseismic Diagnostics in Deformed, Rapidly Rotating Stars

Generalised TAR formulations using full spheroidal geometry and potentially differential rotation lead to:

• Generalised Laplace tidal equations (GLTE) with parametric dependence on radius and latitude, breaking full separability.
• Modified eigenvalue spectra and period spacings, forming the basis for new asteroseismic probes of internal rotation and shape (Mathis et al., 2019, Dhouib et al., 2021, Dhouib et al., 2021)
• The generalised TAR is globally valid only for $\Omega < 0.2\,\Omega_K$, i.e., modest rotation rates, for full radiative envelopes; in more rapid rotators, only inner domains are reliably described by current non-traditional methods (Dhouib et al., 2021).

c. Computational Efficiency and Adaptivity in Numerical Frameworks

Decoupling parametrisations unlocks:

• Adaptive local refinement without breaking CAD-exactness (GIFT, (Atroshchenko et al., 2017))
• Efficient model reduction and online evaluation for large geometric parameter spaces (PGD, (Sevilla et al., 2019))
• Robust numerical convergence even when patch tests fail and for non-nested basis choices
• Explicit mapping of geometric and parametric sensitivities, enabling robust uncertainty quantification.

d. Robustness in Gauge and Quantum Gravity Frameworks

Parametrisation dependence of the renormalization group flows can be minimised using the principle of minimum sensitivity, yielding physically meaningful, robust non-Gaussian fixed points. Universal quantities (such as the product of Newton and cosmological constants at the fixed point) show minimal variation at stationary points in the parametrisation space, validating asymptotic safety scenarios (Gies et al., 2015).

4. Analytical, Numerical and Physical Regime Transitions

Non-traditional parametrisation makes various analytical and numerical strategies tractable:

• Perturbative expansions in small deformation ($\varepsilon$) for generalized tidal equations
• Spectral methods for solving radially dependent eigenvalue problems
• Analytical and numerical identification of validity regimes for approximations (tested with 2D/3D stellar models, (Dhouib et al., 2021, Dhouib et al., 2021))
• Phase diagrams in gravity theory analytically mapped at parametrisation-insensitive points (Gies et al., 2015)

Transition regimes—between spherical and deformed, uniform and differential rotation, or strong versus weak stratification—are now analytically accessible, with explicit breakdown criteria for the validity of classical approximations.

5. Summary Table: Major Features of Non-Traditional Parametrisation of GIWs

Domain / Aspect	Traditional Parametrisation	Non-Traditional Parametrisation
Fluid wave dynamics	Neglects horizontal Coriolis, spherical geometry	Full Coriolis, spheroidal and/or differential
Geometry-function spaces	Identical (iso-parametric)	Independently chosen (GIFT)
Gauge/field theory	Lie group symmetries, global invariance	Lie groupoid/algebroid, extended Killing eq.
Model reduction	Problem-specific, mesh-dependent	CAD-based, control-point parametric (PGD)
Asteroseismic observables	Mode spacings by 1D equations	Mode spacings as function of radius & latitude
Adaptive refinement	Coupled to geometry, global mesh changes	Field-only refinement, geometry fixed (GIFT)
RG flow stability	Parametrisation-dependent	Stationary points for optimal universality

6. Implications, Limitations, and Application Domains

Non-traditional parametrisation of GIWs:

• Enables physically accurate and computationally viable modelling across regimes previously inaccessible using classical approximations
• Must be adopted in sub-inertial, weakly stratified, rapidly rotating, or differentially rotating systems—in both fluid dynamics and astrophysical settings (Mathis, 23 Oct 2025, Dhouib et al., 2021)
• Empowers numerically robust, geometry-exact, and adaptively refined simulation tools ((Atroshchenko et al., 2017), with seamless integration into CAD-centric workflows (Sevilla et al., 2019)
• In gauge theory and quantum gravity, clarifies the approach to universality and robustness with respect to nonphysical parametrisation choices, reinforcing asymptotic safety (Kotov et al., 2014, Gies et al., 2015)

A plausible implication is that further unification of these techniques—leveraging non-traditional parametrisation in both physical modelling and numerical implementation—will be central to tackling multi-resonant, multi-frequency, and multi-scale problems in future theoretical and applied research.