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Stratosphere Model Verification with Manufactured Geometry

Johannes Lawen, George Salman, Akshita Bhardwaj

TL;DR

This work introduces a manufactured-geometry approach to a stratosphere dynamical core by treating the tropopause as a moving lower boundary within a hydrostatic primitive-equation framework in $σ$–$p$ coordinates. The core idea is to couple tropospheric variability kinematically through a time-dependent boundary, employing an $η$-based hybrid mapping $p(η,\mathbf{x},t)=A(η) p_0+B(η) p_{\mathrm{trop}}(\mathbf{x},t)$ and an ALE+conservative-remap cycle to preserve layer positivity and tracer monotonicity. The authors develop a linear-response theory for boundary motion, derive a clear energy budget with boundary work $\mathcal{W}_{\mathrm{trop}}$, and link wave activity to mean-flow changes via EP-flux convergence within transformed Eulerian-mean theory. A structured verification plan combines fixed-boundary consistency checks, idealized forced-boundary tests, and reanalysis-driven boundary forcing to demonstrate numerical stability, dynamical fidelity, and budget closure. Overall, the MMG (Manufactured Geometry) framework provides a transparent, attribution-friendly platform for studying stratosphere–troposphere coupling without nudging, with potential extensions to two-way coupling and chemistry transport on moving boundaries.

Abstract

We propose an exact solution for a stratosphere dynamical core formulated in geopotential/pressure coordinates with a time-evolving lower boundary supplied by the troposphere. Rather than constraining the stratospheric circulation via specified dynamics (``nudging'') to a reanalysis, we treat the tropopause as a moving geometric boundary. The stratospheric domain thus expands, contracts, and undulates in response to tropospheric variability while preserving familiar hybrid $σ$--$p$ structure and pressure-gradient calculations. The approach integrates naturally with arbitrary Lagrangian--Eulerian (ALE) updates and conservative remap to maintain positive layer thickness and tracer monotonicity. We outline the formulation, highlight analytical properties (well-posedness, energetics, wave propagation), and sketch a verification/validation path based on modified standard test cases and reanalysis-driven experiments.

Stratosphere Model Verification with Manufactured Geometry

TL;DR

This work introduces a manufactured-geometry approach to a stratosphere dynamical core by treating the tropopause as a moving lower boundary within a hydrostatic primitive-equation framework in – coordinates. The core idea is to couple tropospheric variability kinematically through a time-dependent boundary, employing an -based hybrid mapping and an ALE+conservative-remap cycle to preserve layer positivity and tracer monotonicity. The authors develop a linear-response theory for boundary motion, derive a clear energy budget with boundary work , and link wave activity to mean-flow changes via EP-flux convergence within transformed Eulerian-mean theory. A structured verification plan combines fixed-boundary consistency checks, idealized forced-boundary tests, and reanalysis-driven boundary forcing to demonstrate numerical stability, dynamical fidelity, and budget closure. Overall, the MMG (Manufactured Geometry) framework provides a transparent, attribution-friendly platform for studying stratosphere–troposphere coupling without nudging, with potential extensions to two-way coupling and chemistry transport on moving boundaries.

Abstract

We propose an exact solution for a stratosphere dynamical core formulated in geopotential/pressure coordinates with a time-evolving lower boundary supplied by the troposphere. Rather than constraining the stratospheric circulation via specified dynamics (``nudging'') to a reanalysis, we treat the tropopause as a moving geometric boundary. The stratospheric domain thus expands, contracts, and undulates in response to tropospheric variability while preserving familiar hybrid -- structure and pressure-gradient calculations. The approach integrates naturally with arbitrary Lagrangian--Eulerian (ALE) updates and conservative remap to maintain positive layer thickness and tracer monotonicity. We outline the formulation, highlight analytical properties (well-posedness, energetics, wave propagation), and sketch a verification/validation path based on modified standard test cases and reanalysis-driven experiments.
Paper Structure (22 sections, 4 theorems, 66 equations)

This paper contains 22 sections, 4 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Solution
  3. Incompressible case: 2D specialization (manufactured geometry)
  4. Manufactured fields.
  5. Partition into $h_E$ and $h_B$.
  6. Closed form for $\mathcal{R}(\Theta)$.
  7. Remarks.
  8. Compressible Case
  9. Formulation overview (sketch)
  10. Coordinates, mapping, and prognostic state
  11. Mass continuity in the moving vertical coordinate
  12. ALE + conservative remap time step
  13. Analytical framework
  14. Linear response to a moving lower boundary
  15. Energetics, wave activity, and mean-flow modification
  16. ...and 7 more sections

Key Result

Proposition 1

Let the hybrid pressure mapping be with $A(\eta)$ and $B(\eta)$ continuously differentiable, $p_0>0$ a constant reference pressure, and $p_{\mathrm{trop}}(x,y,t)>0$ the tropopause pressure. Define If $\mathcal{J}(\eta,x,y,t) > 0$ for all $\eta\in[0,1]$ and all $(x,y,t)$, then $p(\eta,x,y,t)$ is strictly increasing in $\eta$ for every $(x,y,t)$. In particular, distinct $\eta$-levels cannot cross

Theorems & Definitions (8)

  • Proposition 1: Monotonicity of the hybrid $\eta$--$p$ mapping
  • proof
  • Proposition 2: Hydrostatic balance is unchanged by the moving boundary
  • proof
  • Proposition 3: Horizontal pressure-gradient force in hybrid form
  • proof
  • Proposition 4: Column mass budget and fixed-boundary limit
  • proof