Rotating convection and flows with horizontal kinetic energy backscatter

Abstract

Numerical simulations of large scale geophysical flows typically require unphysically strong dissipation for numerical stability. Towards energetic balance various schemes have been devised to re-inject this energy, in particular by horizontal kinetic energy backscatter. In a set of papers, some of the authors have studied this scheme through its continuum formulation with momentum equations augmented by a backscatter operator, e.g. in rotating Boussinesq and shallow water equations. Here we review the main results about the impact of backscatter on certain flows and waves, including some barotropic, parallel and Kolmogorow flows, as well as internal gravity waves and geostrophic equilibria. We particularly focus on the possible accumulation of injected energy in explicit medium scale plane waves, which then grow exponentially and unboundedly, or yield bifurcations in the presence of bottom drag. Beyond the review, we introduce the rotating 2D Euler equations with backscatter as a guiding example. For this we prove the new result that unbounded growth is a stable phenomenon occurring in open sets of phase space. We also briefly consider the primitive equations with backscatter and outline global well-posedness results.

Figures (8)

• Figure 1: Samples of spectra of the backscatter operator $\boldsymbol{B}$ with wave vector $\boldsymbol{k}=(k_x,k_y)\in\mathbb{R}^2$ in (b), wave length $k=|\boldsymbol{k}|$ ( Euclidean norm) and growth rate $\lambda$ in (a). Parameters are $d=1$ and $b=0.6$ (black), $b=1.6$ (blue) and $b=3$ (red). Spectra of $\boldsymbol{B}$ for domain $\mathbb{R}^2$ are curves in (a) and disks in (b), and for domain $\mathbb{T}^2$ points with wave vectors on the axes (circles) and in $(1,1)$-direction (squares). The unstable spectrum ($\lambda>0$) in (b) is located inside the disks with the respective radii $\sqrt{b/d}$, except for the neutrally stable spectrum at the origin for any choice of parameters.
• Figure 2: (Fig. 1 in PRY2023.) Samples of spectra of $\mathcal{L}$ from \ref{['e:linop']} in the isotropic case $d_1=d_2 =1$, $b_1=b_2 =2$ and $f=0.3$, $g=9.8$, $H_0=0.1$ so that ${C}_{\rm c}=0.1$ and $k_{\rm c}=1$. (a) The real (blue) and complex (red) spectrum (magnification in inset) are simultaneously critical for ${C}={C}_{\rm c}$ at $|\boldsymbol{k}|= k_{\rm c} =1$; the real spectrum is also zero at $\boldsymbol{k}=0$ and approaches zero as $|\boldsymbol{k}|\to \infty$. (b) Stable case at ${C} = 0.12$ (dotted) and an unstable case at ${C} = 0.08$ (dashed).
• Figure 3: (Taken from Figs. 1, 8 in PRY22.) We plot part of the wave vector plane where an associated growth rate is $\lambda>0$ (red), $\lambda=0$ (black), $\lambda<0$ (blue). Panels (a,b) concern the loci of explicit solutions \ref{['sol: RSWB2']} with fixed parameters $d_1=1.0,\, d_2=1.04,\, b_1=1.5,\, b_2=2.2,\, f=0.3,\, g = 9.8,\, H_0 = 0.1,\, \alpha_1=1.0$ and $\alpha_2=-0.5$. White curves: loci of solutions with $\alpha_2=0$; white bullets: loci of steady solutions; black dashed: loci of solutions satisfying \ref{['cond: RSWB2b']} only. Panel (c) concerns \ref{['cond: B-intsupa']} with $d_1, d_2,b_2$ as in (a,b) and $b_1=1.1$ so white bullets mark the loci of steady solutions. In (b,c) the solutions corresponding to intersections with an arbitrary gray line in (b) or arbitrary circle in (c), marked by white circles and bullets, can be linearly superposed and still yield explicit solutions.
• Figure 4: Simulation of 2D Euler equations with kinetic energy backscatter $b=0.0015$, $d=0.001$ on $[0,2\pi]^2$ with periodic boundaries by truncated Fourier series with $128\times128$ modes based on mitcode with Crank-Nicholson time-step $0.1$. Top rows: initial data and final data at time $t=8500$ showing $u,v$ and vorticity $w$. Bottom row shows absolute values of Fourier coefficients for relatively large scale modes. The final state is essentially a linear combination of $\boldsymbol{e}_j$, $j=1,\ldots,4$.
• Figure 5: (Fig. 4 in PRY2023.) Samples of the loci of explicit flows \ref{['sol: RSWB2']} (red: $\lambda>0$; white: $\lambda<0$; black: $\lambda=0$ (except $\boldsymbol{k}=0$)) with $\alpha_2=s=0$, $d_1=1$, $d_2=1.04$, $b_1=1.5$, $b_2=2.2$, $f=0.3$, $g=9.8$, $H_0=0.1$, $Q=0$ so that ${C}_{\rm c}\approx0.116$. Blue: the wave vectors satisfying \ref{['cond: RSWB2b']} with $\alpha_2=0$. The intersections of blue and black curves are loci of steady flows, those within red regions are loci of unboundedly growing flows.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 4.1
• Remark 4.2
• proof : Proof of Theorem \ref{['thm:stabgrow']}
• Remark 5.1: Anisotropy
• Theorem 5.2: Bifurcation of GE PRY2023
• Remark 5.3
• Theorem 5.4: Bifurcation of GWs for $Q\neq 0$ PRY2023