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Linear Stability Analysis

This package includes tools for analyzing the linear stability of coupled barotropic-baroclinic systems with different time-stepping schemes. The stability analysis extends the modal decomposition approach to multi-stage Runge-Kutta schemes.

Overview

The stability analysis decomposes the system into barotropic and baroclinic modes and computes eigenvalues of the evolution matrix. For a two-dimensional (x, z) hydrostatic system with no rotation, the analysis considers:

  • Barotropic mode: Depth-independent mode
  • First baroclinic mode: Projection onto vertical mode φ = √2 cos(πz/H)

Key Findings

The analysis reveals critical constraints for split-explicit RK implementations:

  1. Transport Velocity Choice: Split-explicit RK schemes require the averaged barotropic transport velocity (w★) for tracer advection, not the previous substep's velocity (w^m), to maintain baroclinic stability.

  2. Stability Comparison: RK3-SE with correct coupling shows baroclinic mode amplification similar to the single-mode RK analysis, while incorrect coupling (w^m) leads to instabilities.

  3. Implicit vs. Split-Explicit: RK3-IM shows increased dissipation compared to RK3-SE for both barotropic and baroclinic modes, as expected from the implicit free surface treatment.

Using the Stability Analysis

The stability analysis is implemented in the linear_stability.ipynb notebook. Key functions compute:

  • Evolution matrices for different timesteppers
  • Eigenvalues for barotropic and baroclinic modes
  • Amplification factors as functions of CFL number and stratification

Running the Analysis

# See linear_stability.ipynb for full implementation
# The notebook computes eigenvalues for:
# - AB2-SE
# - RK3-SE with w^m (incorrect)
# - RK3-SE with w★ (correct)
# - RK3-IM

Mathematical Framework

The analysis linearizes the primitive equations around a rest state and decomposes variables as:

\[Y(x, z, t) = \sum_{q=0}^\infty Y_q(x, t) M_q(z)\]

where M_0 = 1 (barotropic) and M_1 = √2 cos(πz/H) (first baroclinic).

The discrete evolution matrix is constructed from the time-stepping scheme, and eigenvalues determine stability:

  • |λ| < 1: Stable (dissipative)
  • |λ| = 1: Neutral (non-dissipative)
  • |λ| > 1: Unstable

Results

The stability analysis shows:

  1. Baroclinic Mode: RK3-SE with w★ maintains stability similar to single-mode RK, while w^m leads to growing instabilities with increasing stratification.

  2. Barotropic Mode: Both RK3-SE and RK3-IM maintain stable barotropic modes, with RK3-IM showing more dissipation.

  3. CFL Dependence: RK3 schemes remain stable over larger CFL ranges than AB2, consistent with the amplification factor analysis.

References

The analysis methodology is described in detail in the paper, extending the approach of Demange et al. (2019) to multi-stage schemes. See the paper's Section 4 for the complete derivation.