Biglobal analysis of baroclinic instability in a current-undercurrent oceanic system

In this work, baroclinic instability in a current-undercurrent system is analyzed using the biglobal instability analysis (BIA). Idealized model flows are considered and the flow parameters are estimated from the Western North Pacific circulation system. Compared with the prevailing one-dimensional linear stability analysis (1D-LSA), BIA can deal with the basic flow of continuously nonuniform vertical shear and strong horizontal variation within the framework of the Boussinesq equation to account for nongeostrophic effects. The basic velocity gradually changes from the vertically linear Eady type to a more realistic distribution, which destabilizes the Phillips-type mode due to the strengthening of the vertical shear. With an increasing zonal velocity variation, the mode is more of a barotropic type in the long-wave range and is more of a baroclinic type in the short-wave range. Moreover, the high vertical shear near the vertical boundaries supports a series of Charney-type modes in the confined boundary region. The Charney modes are severely affected by the boundary constraint, leading to a wide unstable wave-number range deep into the small-scale region and enhanced ageostrophic motions. The top and bottom boundary layers due to viscosity and diffusivity can be destabilizing and sustain the modal growth for short waves. In comparison, 1D-LSA can overestimate the growth rate of baroclinic modes and cannot quantify the coupling effects with the horizontal shear for the present case.

Figure 1

Bathymetry ($\mathrm{m}$) contours, and the schematic of the main upper-layer currents (black arrows) and middle-layer undercurrent system (dashed arrows) in the Western North Pacific [32]. MUC, Mindanao Undercurrent; NEUC, North Equatorial Undercurrent; KC, Kuroshio Current; LUC, Luzon Undercurrent.

Figure 2

One-dimensional velocity case ($r_x=0$): Vertical distribution of the basic-flow (a) meridional velocity and (b) buoyancy frequency (at $x=0$) with different $r_z$ and the (c) buoyancy frequency contour in the $r_z=0.5$ case.

Figure 3

Two-dimensional velocity case: Contours of the basic-flow (a) meridional velocity, (b) buoyancy, and (c) buoyancy frequency with $r_z=0.5, r_x=0.4$.

Figure 4

Growth rates of the (a) Phillips and (b) Charney modes using different zonal BCs ($r_z=1, r_x=0$). The two variables for each case in the legend denote those of equal values at $x=0$ and 1 as zonal BCs.

Figure 5

Normal-mode spectrums from (a) inviscid and (b) viscous ($\text{Re}={10}^6, \text{Pr}=1$) calculations using 1D-LSA with different grid numbers. The basic flow is ${\overline{V}}_{\text{lin}}\left(z\right)$ with $N^2=1$, and the case parameters are from Molemaker et al. [13] as $\text{Ro}=\text{Fr}=1/\sqrt{2}, A_R=1, k_x=0$, and $k_y=2$ with $k_x$ the zonal wave number as in Eq. (1).

Figure 6

(a) Vertical distribution of the basic-flow potential vorticity ($x=0$) and its gradient with different $r_z$, and (b) the global mode spectrum from BIA with $\text{Ro}=0.01$ and no viscosity ($k_y=4.0, r_z=0.5$).

Figure 7

Growth rates of the Phillips mode (a) from BIA and (b) from both the approximate 1D-LSA ($k_x=0$) and BIA for varying $r_z$ cases. For 1D-LSA, the upper and lower growth-rate envelopes are shown with $x_{\text{base}}\in [0,1]$.

Figure 8

Growth rate contributions of different terms to the (a) KE, (b) PE, and (c) TE of the Phillips mode ($r_z=0.5$). The symbols in the legend are defined in Eq. (15), namely $\mathcal{P}$ (velocity shear production), $\mathrm{\Pi }$ (pressure work), $\mathcal{F}$ (buoyancy flux), $\mathcal{V}$ and $\mathcal{D}$ (viscous and diffusion terms), and ${\mathcal{B}}_x$ and ${\mathcal{B}}_z$ (buoyancy production).

Figure 9

Basic-flow (a) velocity and (e) its vertical gradient and [(b)–(d) and (f)–(h)] the contours of different energy-budget terms for the most unstable mode ($k_y=2.41$) in the $r_z=0.5$ case: (b) KE, (c) $\mathcal{F}$, (d) $\mathrm{\Pi }$, (f) PE, (g) ${\mathcal{B}}_x$, and (h) ${\mathcal{B}}_z$. The terms are normalized by $2{\widehat{E}}_{\text{int}}$ [see Eq. (17)], and their symbols are the same as those in Fig. 8.

Figure 10

Growth rate contributions of different terms to the TE of the Phillips mode in the cases ($\mathit{a}$) $r_z=0$, ($\mathit{b}$) $r_z=0.5$, and ($\mathit{c}$) $r_z=1.0$. See Fig. 8 for the term symbols.

Figure 11

Growth rate of the Phillips mode with different meridional wave numbers in different $r_x$ cases.

Figure 12

Growth rate contributions of different terms to the [(a)–(c)] KE and [(d)–(f)] TE of the unstable mode in the cases [(a) and (d)] $r_x=0.2$, [(b) and (e)] $r_x=0.6,$ and [(c) and (f)] $r_x=1.0$. See Fig. 8 and Eq. (15) for the term symbols.

Figure 13

Contours of the velocity components and KE of the modes at [(a)–(d)] $k_y=1.3$, [(e)–(h)] $k_y=2.7$, and [(i)–(l)] $k_y=3.9$ in the $r_x=0.6$ case: [(a), (e), and (i)] zonal velocity, [(b), (f), and (j)] meridional velocity, [(c), (g), (k)] vertical velocity, and [(d), (h), and (l)] KE. The black dashed lines in [(d), (h), and (l)] are the contours of $\overline{V}$ (uniform levels hereinafter).

Figure 14

(a) Global mode spectrum at $k_y=4.5$, and the (b) growth rates, (c) phase velocities, and [(d)–(i)] contours of the normalized vertical velocity ($|{\widehat{w}}^{\prime }|/max(|{\widehat{w}}^{\prime }\left|\right)$) for different Charney modes (inviscid, $\text{Ro}=0.01, r_z=1, r_x=0$). [(d) and (g)] For mode Cb-1, [(e)–(h)] for Cb-2, and [(f)–(i)] for Cb-3. $\overline{V}\left(z\right)$ is shown in (j) for reference.

Figure 15

Growth rates of the Cb-1 mode in the cases (a) with different $\text{Ro}$ ($\text{Fr}=\text{Ro}$ holds) and $r_z$ (inviscid) and (b) with and without viscosity ($\text{Ro}=0.1, r_z=1$).

Figure 16

Contours of [(a)–(d)] KE and [(f)–(i)] PE for the most unstable Cb-1 mode in the cases [(a) and (f)] $\text{Ro}=0.01, r_z=0.5$; [(b) and (g)] $\text{Ro}=0.01, r_z=1.0$; [(c) and (h)] $\text{Ro}=0.1, r_z=0.5$; and [(d) and (i)] $\text{Ro}=0.1, r_z=1.0$. The vertical direction is stretched for better illustration. Panels (e) and (j) give the basic-flow velocity and its gradient for reference.

Figure 17

Growth-rate contributions of different terms to (a) KE, (b) PE, and (c) TE for the Cb-1 mode in the benchmark case $r_z=1$. The term symbols in the legend are the same as those in Fig. 8.

Figure 18

(a) Growth rates and (b) phase velocities at different $k_y$, and [(c)–(h)] contours of the normalized vertical velocity $\left[\right|{\widehat{w}}^{\prime }|/max(|{\widehat{w}}^{\prime }\left|\right)]$ for different Charney modes (${\omega }_r<0$) in the inviscid case ($\text{Ro}=0.01, r_z=r_x=1$). [(c) and (f)] For mode Cb-1, [(d) and (g)] for mode Cb-2, and [(e) and (h)] for mode Cb-3.

Figure 19

Contours of [(a) and (e)] vertical velocity, [(b) and (f)] pressure, [(c) and (g)] KE, and [(d) and (h)] PE for the most unstable Cb-1 mode in the case [(a)–(d)] $\text{Ro}=0.01$ and [(e)–(h)] $\text{Ro}=0.1$ ($r_z=r_x=1$). The vertical direction is stretched for better illustration. The black dashed lines in (c) and (g) are the contours of $\overline{V}$.

Figure 20

Isosurfaces of the disturbance [(a) and (c)] meridional velocity and [(b) and (d)] buoyancy for the most unstable [(a) and (b)] Phillips ($k_y=2.74$) and [(c) and (d)] Cb-1 mode ($k_y=8.32$) in the $r_z=r_x=1$ case. The contours on $y$ cross sections are the basic flow. The white dashed lines are the intersection of the isosurfaces and plane contours.

Figure 21

Upper row: Growth rates at different (a) aspect ratios ($\text{Ri}=0, k_y=0$) and (b) meridional wave numbers ($\text{Ri}=0.92, A_R=0$) in the case by Zemskova et al. [41]. Lower row: (c) Growth rates with different $k_x$ (0, 0.5, 1.0, 1.5, 2.0) and (d) the shape function of the $k_x=2$ mode ($k_y=0.96$) from BIA and 1D-LSA for the $r_z=0$ case. Here $\Im$ denotes the imaginary part of complex.