Properties and baroclinic instability of stratified thermal upper-ocean flow

TL;DR

This work analyzes a stratified, rotating shallow-water quasigeostrophic model $IL^{(0,1)}QG$ with a Lie–Poisson Hamiltonian structure to examine baroclinic instability in the upper ocean. It derives the three prognostic fields $(\bar{ξ}, ψ_σ, ψ_{σ^2})$, formulates their dynamics with an elliptic inversion for the stream function, and establishes a Kelvin circulation theorem, invariant subspaces, and both Casimir and weak Casimir invariants. Linear theory reveals Rossby waves and a neutral force-compensating mode, while stability is explored via spectral, formal, and Lyapunov criteria; a priori nonlinear bounds are obtained using Shepherd’s method, constraining perturbation growth even when states are spectrally unstable. Direct nonlinear simulations show that stratification does not universally suppress submesoscale activity, challenging some prior numerical results and highlighting the role of Lyapunov-stable states in bounding nonlinear growth. Overall, the paper advances the geometric-mechanical understanding of stratified upper-ocean flow and provides rigorous bounds and insights that inform submesoscale dynamics and their modeling.

Abstract

We study the properties of, and investigate the stability of a baroclinic zonal current in, a thermal rotating shallow-water model, sometimes called \emph{Ripa's model}, featuring stratification for quasigeostrophic upper-ocean dynamics. The model has Lie--Poisson Hamiltonian structure. In addition to Casimirs, the model supports weak Casimirs forming the kernel of the Lie--Poisson bracket for the potential vorticity evolution independent of the details of the buoyancy as this is advected under the flow. The model sustains Rossby waves and a neutral model, whose spurious growth is prevented by a positive-definite integral, quadratic on the deviation from the motionless state. A baroclinic zonal jet with vertical curvature is found to be spectrally stable for specific configurations of the gradients of layer thickness, vertically averaged buoyancy, and buoyancy frequency. Only a subset of such states was found Lyapunov stable using the available integrals, except the weak Casimirs, whose role in constraining stratified thermal flow remains to be understood. The existence of Lyapunov-stable states enabled us to \emph{a priori} bound the nonlinear growth of perturbations to spectrally unstable states. Our results do not support the generality of earlier numerical evidence on the suppression of submesoscale wave activity as a result of the inclusion of stratification in thermal shallow-water theory, which we supported with direct numerical simulations.

Figures (7)

• Figure 1: Cartoons depicting the density structure on a vertical plane in the thermal shallow-water model, known as Ripa's model or IL$^0$ (left), and its stratified version analyzed in this paper, denoted as IL${^{(0,1)}}$ (right). A reduced-gravity setting is assumed, featuring one active layer bounded from above by a rigid lid and from below by a soft interface, which interfaces with an inert layer.
• Figure 2: (left) Emergent cascade of submesoscale vorticity filament rollups in a reduced-gravity direct numerical simulation of the IL$^0$QG with "topographic" forcing in a doubly periodic domain $\mathbb R/R\mathbb Z \times \mathbb R/R\mathbb Z$ of the $\beta$-plane, where $R \approx 25$ km is the baroclinic Rossby radius of deformation. (right) Ocean color image acquired by VIIRS (Visible Infrared Imaging Radiometer Suite) on 1 January 2015 west of the Drake Passage in the Southern Ocean, revealing vortices with diameters ranging from a couple of km to a couple of hundred km. Image credit: NASA Ocean Color Web (https://oceancolor.gsfc.nasa.gov/gallery/447/).
• Figure 3: Stability of the basic state family \ref{['eq:BS']}, representing baroclinic zonal jets, in the $\alpha := \bar{U}/U_\sigma$ vs $\mu := U_{\sigma^2}/U_\sigma$ space, respectively measuring the vertical linear shear and curvature of a jet in the family. In the shaded regions, corresponding to the set $\mathbb S$ defined by $\alpha (1 - \frac{2}{3}\mu) \le 0$, the phase speed of an infinitesimally-small normal-mode perturbation is real for every wavenumber. Within the hatched region, the subset $\mathbb L:= \{\mu < 0\} \cap \{\alpha < 0\}$ of $\mathbb S$, there is stability for finite-size perturbations of arbitrary structure. Moreover, in $\mathbb L$ the distance, in an $L^2$ sense, of a perturbation to the basic state is at all times bounded by a multiple of its initial distance. This means that in $\mathbb L$ there is Lyapunov stability. Spectral stability is possible inside the white regions, the complement of $\mathbb S$, $\mathbb S^c$. The curves labeled by the normalized wavenumber magnitude $\kappa := R{\sqrt{k^2 + l^2}}$ bound the $(\alpha,\mu)$-subregions of $\mathbb S^c$ where there is spectral instability for the normalized $\beta$ and stratification parameters, $b := \beta/U_\sigma R^2$ and $S := N^2_\mathrm{r} H_\mathrm{r}/2g_\mathrm{r}$, respectively, as indicated.
• Figure 4: As a function of normalized wavenumber, normalized phase speed for a normal-mode perturbation on the baroclinic zonal jet defined by \ref{['eq:BS']} with parameters as indicated. Asymptotic dispersion relation curves as $\kappa\uparrow\infty$ and the corresponding limiting value are included.
• Figure 5: (right) Growth rate as a function of zonal wavenumber in the limit of weak stratification ($S\downarrow 0)$ with basic state parameters $\alpha$ and $b$ as indicated for three values of $\mu$. The IL$^0$QG result corresponds to the $\mu = 0$ curve as the velocity in that model can only include linear shear (implicitly, by the thermal-wind balance). (middle) Zonal velocity (implicit) vertical profiles leading to the growth rates in the left panel. (left) Upper bound on the growth rate in $(\alpha,\mu)$-space for weak stratification. The thick line is the corresponding result for the IL$^0$Q.

Theorems & Definitions (13)

• Remark 1: On notation
• Remark 2: Implicit vertical shear
• Remark 3
• Definition 1: Functional, functional derivatives, and variations
• Remark 4
• Theorem 1: Noether for noncanonical Hamiltonians
• Remark 5
• Remark 6
• Lemma 1
• proof