InternalModesExponentialStratification

Solve the vertical mode problem for exponential stratification.

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Declaration

classdef InternalModesExponentialStratification < InternalModesBase

Overview

InternalModesExponentialStratification provides the closed-form benchmark for the exponential profile used throughout the manuscript:

\[N^2(z) = N_0^2 e^{2 z / b}\]

together with

\[\bar{\rho}(z) = \rho_0 \left(1 + \frac{b N_0^2}{2 g} \left(1 - e^{2 z / b}\right)\right).\]

The resulting vertical modes are expressed in terms of Bessel functions, matching the analytical benchmark used in the validation sections of Early, Lelong, and Smith (2020).

im = InternalModesExponentialStratification(N0=5.2e-3, b=1300, zIn=[-5000 0], zOut=zOut, latitude=33);
[F, G, h, omega] = im.modesAtWavenumber(2*pi/1000);

Topics

• Create and initialize modes
  • InternalModesExponentialStratification Initialize the exponential-stratification analytical solver.
• Inspect grids and stratification
  • N0 Surface buoyancy frequency \(N_0\) in radians per second.
  • b Exponential e-folding depth \(b\) in meters.
• Compute modes
  • bottomModesAtWavenumber Return the analytical bottom SQG mode for exponential stratification.
  • surfaceModesAtWavenumber Return the analytical surface SQG mode for exponential stratification.
• Inspect analytical solutions
  • BarotropicEquivalentDepthAtFrequency Estimate the barotropic equivalent depth for fixed \(\omega\).
  • BarotropicEquivalentDepthAtWavenumber Estimate the barotropic equivalent depth for fixed \(K\).
  • FSolution Exact analytical \(F(z,\omega,c)\) solution handle.
  • GSolution Exact analytical \(G(z,\omega,c)\) solution handle.
  • IsStratificationExponential Test whether a supplied profile is close to the exponential benchmark.
  • N2Function Background buoyancy-frequency function handle.
  • NormalizedModesForOmegaAndC Evaluate and normalize analytical mode functions at zOut.
  • rhoFunction Background density function handle.

Developer Topics

These items document internal implementation details and are not part of the primary public API.

• Developer topics
  • FSolutionApprox Approximate analytical \(F(z,\omega,c)\) solution handle.
  • FindRootsInRange Find analytical eigenvalue roots over a bounded search interval.
  • GSolutionApprox Approximate analytical \(G(z,\omega,c)\) solution handle.
  • ModeFunctionsForOmegaAndC Select the exact or approximate analytical mode functions.
  • kConstantNormalizationForOmegaAndC Return the kConstant normalization for the analytical exponential solution.
  • nInitialSearchModes Number of trial roots used when expanding the analytical search interval.
  • shouldApproximate Predicate that chooses between exact and approximate Bessel forms.

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