Initial and Transient Growth of Symmetric Instability

Satoshi Kimura

doi:10.1175/JPO-D-23-0048.1

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Fig. 1.

Comparing the fastest-growing initial and normal mode growth rates on a Ro–Ri space. (a) Fastest-growing initial mode growth rate σ_FGM. (b) Growth rate ratio, $σ_{FGM} / λ_{FGM}$ .

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Fig. 2.

Energy contribution of the initial modes on a Ro–Ri plane. (a) Signs of GSP_σ. (b) LSP contribution with respect to GSP, ${LSP}_{σ} / {GSP}_{σ}$ . (c) MB contribution with respect to GSP, ${MB}_{σ} / {GSP}_{σ}$ . We do not consider Ri below 0.25, as Ri < 0.25 is a necessary but not sufficient condition for the Kelvin–Helmholtz instability to grow.

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Fig. 3.

Summary of the dominant energy source in the initial mode and normal mode on a Ro–Ri plane. The growth of the normal mode occurs for $Ri < 1 / (1 + Ro)$ , where the dominant source is either GSP_λ or LSP_λ. In addition to these shear production terms, the initial mode withdraws its energy from the meridional buoyancy flux MB_σ. The growth of the initial mode occurs with a combination of MB_σ, GSP_σ, and LSP_σ.

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Fig. 4.

SI growth rates vs baroclinic growth rate in a flow Ro = 0. The variable λ_b denotes the approximation of the fastest-growing baroclinic growth rate derived by (Stone 1966), which is defined as $λ_{b} = \sqrt{5 / [54 (1 + Ri)]}$ having the fastest-growing wavenumber, $k_{max} = \sqrt{5 / [2 (1 + Ri)]}$ .

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Fig. 5.

Transient growth rates in a SI unstable flow (Ri = 0.7, Ro = −0.5, and δ = 0.1) with respect to the nondimensional time. The abscissa represents the nondimensional time, scaled by f, and the ordinate is the nondimensional growth rates. The right side of the ordinate indicates the initial and normal mode growth rates, also shown in the horizontal dashed and dotted lines. The black and blue dots indicate the half-life of ζ and ζ_MB, which are 1.3 and 1 nondimensional time units, respectively. The transient growth rates are calculated with a nondimensional time increment of 0.01.

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Fig. 6.

Half-life of ζ_MB (t_1/2) on a Ro–Ri plane. The half-life is computed for a flow unstable to the normal mode, $1 / Ri - Ro - 1 > 0$ with varying Ri and Ro. The parameter ranges are 0 < Ri < 5 and −3 < Ro < 3. The magenta dot represents the combination of Ro and Ri used in Fig. 5. The transient growth rate ζ_MB is computed with a time increment of 0.01, which is the time resolution of t_1/2.

View raw image

Fig. 7.

Plot of t_1/2 as a function of σ_FGM. Black dots represent t_1/2 computed in Fig. 6. The fastest-growing initial SI growth rate σ_FGM is defined in (33).

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Fig. 8.

Comparing σ_GSP and λ_GSP on a Ro–Ri plane. The gray region indicates ζ_GSP(t) increases in time, which satisfies (38), whereas the black regions indicate ζ_GSP(t) decreases in time. The white region represents that Ri is above the marginal stability curve, so the normal mode SI does not grow. The magenta dot represents the combination of Ro and Ri used in Fig. 5.

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Editorial Type:: Article

Article Type:: Research Article

Initial and Transient Growth of Symmetric Instability

Satoshi Kimura

Satoshi Kimura ^aJapan Agency for Marine-Earth Science and Technology, Yokosuka, Japan

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https://orcid.org/0000-0003-1689-1605

Online Publication:: 21 Dec 2023

Print Publication:: 01 Jan 2024

DOI:: https://doi.org/10.1175/JPO-D-23-0048.1

Page(s):: 115–129

Received:: 16 Mar 2023

Final Form:: 24 Oct 2023

Accepted:: 26 Oct 2023

Published Online:: 21 Dec 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The mechanism of initial and transient perturbations of symmetric instability (SI) in a hydrostatic flow with lateral shear is analyzed by applying the generalized stability analysis. It is well known that the SI’s most rapidly growing motion is along isopycnals, and the growth rates consist of growing, neutral, and decaying modes. The eigenvectors of these three modes are not orthogonal to each other, hence the initial and transient perturbations bear little resemblance to the normal mode. Our findings indicate that the emergence of normal modes occurs within a time span of 1–3 inertial periods, which we refer to as the transient state. The overall growth of perturbation energy is divided into three components: geostrophic shear production (GSP), lateral shear production (LSP), and meridional buoyancy flux (MB). During the transient state, the perturbation energy is partly driven by MB, contrary to the normal mode which has zero MB. The relative energy contribution is evaluated through the ratio to GSP. While the MB-to-GSP ratio of the initial mode is higher than that of the normal mode, the LSP-to-GSP ratio remains constant. In the absence of the fastest-growing normal mode, MB can serve as the predominant initial energy source. The precise transition in the energy regime is contingent upon the geostrophic Richardson number and Rossby number.

Significance Statement

Fronts can be unstable to instabilities, which generate disturbance growth and lead to the mixing of water masses. We wanted to understand the initial and transient development of disturbance growth leading to the well-known exponentially growing state. While the exponentially growing disturbance is dominant in the long run, the disturbance growth may not have enough time to achieve the exponentially growing state. We find that the initial disturbance growth bears little resemblance to the exponentially growing state. Capturing the complete spectrum of front evolution remains challenging, and observations have thus far been limited to short-term records. The insights learned from this study can aid in better characterizing the disturbance growth captured in these short-term records.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Satoshi Kimura, skimura@jamstec.go.jp

Abstract

Significance Statement

Corresponding author: Satoshi Kimura, skimura@jamstec.go.jp

Keywords: Baroclinic flows; Potential vorticity; Stability; Singular vectors; Fronts

1. Introduction

The ocean surface is filled with fronts that separate different types of water, often identified by a sharp change in the surface color of the ocean. One way to generate perturbation energy in fronts is through the exponential growth of normal mode instabilities (Eady 1949; Stone 1966; Haine and Marshall 1998; Boccaletti et al. 2007). One such instability that operates in submesoscale (roughly 1–100 km) is symmetric instability (SI), which is the mode structure across the front direction and invariant along the front (Thomas 2008; Arobone and Sarkar 2015; Skyllingstad et al. 2017; Bachman et al. 2017). The linear theory of SI has mainly focused on the stability of exponentially growing normal modes to determine the necessary condition for instability (Ooyama 1966; Hoskins 1974). These analyses have shown that the fastest-growing mode is along isopycnals, and the mode grows by extracting kinetic energy from horizontal currents.

The fastest-growing normal mode describes the long-term behavior of the perturbation state. The fastest-growing normal mode emerges after the transient state. The normal modes cannot describe the initial and transient state, unless the linearized dynamical operator is classified as a normal operator (the operator commutes with its Hermitian transpose). It is well known that the linearized dynamical operator of shear flow, in general, is nonnormal and its eigenvectors are not orthogonal to each other (Farrell 1984; Trefethen et al. 1993). The structure of the perturbation spawned from a nonnormal operator during the initial state bears little resemblance to the long-term behavior predicted by the normal modes (Farrell and Ioannou 1996; Whitaker and Sardeshmukh 1998; Schmid 2007). The linear operator of SI is nonnormal, raising the question of whether such initial and transient state is also relevant in the context of SI.

There are two potential contexts in which the nonnormal behavior of SI could be important to motivate this work. First, initial SI can manifest beyond the known necessary condition for instability based on the normal mode analysis. A flow is unstable to the normal mode when the Ertel potential vorticity (PV) of a geostrophic current has the opposite sign to the Coriolis parameter f (Hoskins 1974). SI unstable flows are typically observed in the surface and bottom boundary layers of the ocean (Allen and Newberger 1998; Thomas and Lee 2005; D’Asaro et al. 2011; Savelyev et al. 2018). Surface winds and cooling at the ocean surface can reduce the PV, thereby sustaining a flow susceptible to SI (Thomas and Lee 2005; D’Asaro et al. 2011; Yu et al. 2019). Changes in the wind stress impose perturbations on the ocean surface, which can trigger the Ekman inertial instability to push the momentum downward with the viscous stress (Grisouard and Zemskova 2020). In the bottom boundary layer, the downslope Ekman flow along the steep topographic slope creates an environment susceptible to the normal mode SI (Naveira Garabato et al. 2019). While these environments are unstable to the normal mode SI, the stratification throughout most of the ocean interior is relatively strong for the normal mode to grow. There is a possibility that the parameter regime in the ocean interior can support the initial SI mode. In fact, Zemskova et al. (2020) have performed a few simulations of fronts stable to SI and found that the initial/transient SI mode dominates over the normal mode baroclinic instability. Their simulations suggest that the transient growth is the dominant mechanism for strongly stratified front; however, there has been no systematic exploration of the initial SI mode.

The second question pertains to the duration required for the normal mode SI to fully develop. The observation of SI requires to sample oceanic flows down to the submesoscale, which entails complex sampling methods involving ships, satellites, and autonomous instruments (e.g., D’Asaro et al. 2011; Poje et al. 2014; Shcherbina et al. 2015; Sarkar et al. 2016; Adams et al. 2017; Haney et al. 2021). However, even with these sophisticated approaches, submesoscale motions may not be fully resolved due to their rapidly evolving temporal scales. The observations have provided short-term records of energetics at the oceanic submesoscale from specific oceanic events. Certain exceptions are measurements of seasonal cycles from mooring arrays and repeated ship tracks, which indicate robust submesoscale flows in winter, associated with the deepening of mixed layers (Callies et al. 2015; Buckingham et al. 2019; Zhang et al. 2021). In almost all instances of reported turbulence, observations have been conducted at mid-to-high latitudes (higher than 30°N/S), where the sites are characterized by a deep wintertime mixed layer due to strong cooling or/and strong current systems (e.g., in the proximity of the western boundary currents or the Antarctic Circumpolar Current). It is probable that normal modes can rapidly grow in these sites; nevertheless, the time required to develop the normal mode (saturation time of SI) has not been adequately addressed. Normal mode analysis determines the stability and its spatial structure in the limit of t → ∞, where t represents time (Trefethen 1997). Given that frontal intensification and changes in the wind direction can occur within several inertial periods (Thomas and Lee 2005; Thomas et al. 2016; Adams et al. 2017; Yu et al. 2019), it is appropriate to consider the stability in finite time as well as in the limit of t → 0. Of particular interest is determining the duration required for the perturbation to reach the normal mode state.

The previous studies on the generalized stability analysis of SI have assumed the absence of lateral shear (Xu 2007; Xu et al. 2007; Heifetz and Farrell 2008; Zemskova et al. 2020). These studies have provided valuable insights into the transient energy growth associated with SI. In our study, we expand upon the previous studies by incorporating the influence of lateral shear into the generalized stability analysis of SI. By considering the lateral shear, we aim to enhance our understanding of interactions between SI and eddies, which are prevalent features in the ocean. We present the initial growth rate of SI in a hydrostatic flow in the presence of lateral shear and discuss the energy sources. Section 2 outlines the governing equations and generalized stability analysis, which describes the stability of the initial and transient growth. The linearized SI problem is summarized in section 3. Section 4 applies the generalized stability analysis to SI and compares the stability in the limits of t → 0 (initial mode) and t → ∞ (normal mode). The transient state is discussed in section 5. Discussion and summary are provided in section 6.

2. Methods

a. Governing equation

We begin with the inviscid hydrostatic Boussinesq equations of motion in a rotating Cartesian coordinate system {x, y, z}. A constant Coriolis parameter, f, and no horizontal component of Earth’s rotation are assumed. The resulting equations are

\frac{D u}{D t} - f υ = - \frac{\partial π}{\partial x}, \frac{D υ}{D t} + f u = - \frac{\partial π}{\partial y}, 0 = - \frac{\partial π}{\partial z} + b, \frac{D b}{D t} = 0, \nabla \cdot u = 0,

(1)

where D/Dt is the material derivative, and u = {u, υ, w} are the {x, y, z} components of the velocity. The variable π represents the pressure scaled by the uniform characteristic density ρ₀. Buoyancy is defined as b = −g(ρ–ρ₀)/ρ₀, where g is the acceleration due to gravity. The PV is defined as q = (fe_z + ∇ × u) ⋅ ∇b, where e_z is the vertical unit vector. The velocity, buoyancy and pressure terms are separated into two parts, a background profile and a perturbation:

u = U (y, z) e_{x} + ϵ u^{'} (y, z, t), b = B (y, z) + ϵ b^{'} (y, z, t), and π = Π + ϵ π^{'} (y, z, t),

(2)

where e_x represents the horizontal unit vector in the alongfront direction (x). The variables denoted with primes indicate the perturbation state, and ϵ is an infinitesimally small number representing the magnitude away from the background state. We assume that all the motions are uniform in the x direction. The perturbation terms are assumed to take the sinusoidal form in the across-front (y) and vertical (z) direction:

ϕ^{'} (y, z, t) = \hat{ϕ} (t) e^{ily + imz},

(3)

where ϕ represents the state variables, either u, υ, w, π, or b, and

\hat{ϕ} (t)

is the corresponding structure function. The variables l and m denote the wavenumbers in the across-front and vertical directions, respectively.

We derive the background and perturbation equations by substituting (2) with the perturbation form (3) into (1), then collect the O(1) and O(ϵ) terms. The background state is in the thermal wind balance:

f \frac{\partial U}{\partial z} = - \frac{\partial B}{\partial y},

(4)

which links the functional forms of U and B. Here, we assume that the background profiles are linear in the across-front direction and vertical direction:

U (y, z) = U_{y} y + U_{z} z and B (y, z) = B_{y} y + B_{z} z .

(5)

The thermal wind balance links B_y = −fU_z, and the PV of the background flow is

Q = (f - U_{y}) B_{z} - \frac{{[B_{y}]}^{2}}{f} .

(6)

The resulting perturbation equations for the alongfront momentum, across-front momentum, vertical momentum, buoyancy, and continuity are

\frac{\partial \hat{u}}{\partial t} + U_{y} \hat{υ} + U_{z} \hat{w} - f \hat{υ} = 0, \frac{\partial \hat{υ}}{\partial t} + f \hat{u} = - i l \hat{π}, 0 = - i m \hat{π} + \hat{b},

(7)

\frac{\partial \hat{b}}{\partial t} + B_{y} \hat{υ} + B_{z} \hat{w} = 0, and i l \hat{υ} + i m \hat{w} = 0.

(8)

The pressure (

\hat{π}

) and vertical velocity (

\hat{w}

) are eliminated by the vertical momentum balance and the continuity equation. The perturbation equations are written in the matrix form, where the state vector ψ consists of the horizontal velocities and buoyancy:

\frac{\partial}{\partial t} ψ = A ψ, where ψ = [\begin{matrix} \frac{\hat{u}}{\sqrt{2}} \\ \frac{\hat{υ}}{\sqrt{2}} \\ \frac{\hat{b}}{\sqrt{2 B_{z}}} \end{matrix}] and A = [\begin{matrix} 0 & f - U_{y} - s U_{z} & 0 \\ - f & 0 & s \sqrt{B_{z}} \\ 0 & \frac{- B_{y} - s B_{z}}{\sqrt{B_{z}}} & 0 \end{matrix}] .

(9)

The variable s represents the ratio of the across-front to vertical wavenumber,

s = - l / m

. The buoyancy perturbation in ψ is divided by

\sqrt{2 B_{z}}

, which makes the dimensional unit meters per second. The norm of the state vector (ψ ⋅ ψ^†) corresponds to the total perturbation energy:

E = \frac{1}{2} (\hat{u} {\hat{u}}^{†} + \hat{υ} {\hat{υ}}^{†} + \frac{\hat{b} {\hat{b}}^{†}}{B_{z}}),

(10)

where the superscript dagger indicates the Hermitian transpose.

b. Nondimensional linear equations

The linearized problem (9) is nondimensionalized by a time scale f⁻¹ and length scale H, where H represents the depth of the front. Governing nondimensional parameters are the geostrophic Richardson number, Rossby number (with sign), and the ratio of the Coriolis parameter to the thermal wind shear:

Ri = \frac{B_{z}}{U_{z}^{2}}, Ro = \frac{- U_{y}}{f}, and δ = \frac{f}{U_{z}} .

(11)

The background potential vorticity consists of the lateral shear (U_y) and the Coriolis parameter (f): ξ_g = f–U_y. The ambient cyclonic eddy retains the potential vorticity larger than the Coriolis parameter,

ξ_{g} / f > 1

, which results in Ro > 0. The anticyclonic eddy in the background has Ro < 0. A measure of the aspect ratio of the front is

\frac{H}{L} = ‖ δ Ro ‖,

(12)

where L is the characteristic horizontal length scale of the front. The front in consideration is in hydrostatic balance meaning ∥δRo∥ is small (∥δRo∥ ≪ 1). A typical value of δ can reach up to 0.1 in an ocean mixed layer at midlatitudes, and Ro and Ri are O(1) (Molemaker et al. 2005; Boccaletti et al. 2007; Callies et al. 2015; Nagai et al. 2015; Sarkar et al. 2016; Kimura et al. 2023). The scaling makes the nondimensional Coriolis parameter to

f^{*} = 1

, where the superscript asterisk indicates nondimensional variable. The nondimensional background flow parameters are represented as

U_{y}^{*} = - Ro, U_{z}^{*} = \frac{1}{δ}, B_{y}^{*} = - \frac{1}{δ}, and B_{z}^{*} = \frac{Ri}{δ^{2}} .

(13)

The nondimensional background PV is

Q^{*} = (1 + Ro) Ri δ^{- 2} - δ^{- 2}

. A set of nondimensionalized linear equations for SI is derived by applying (13) to (9):

\frac{\partial}{\partial t^{*}} ψ^{*} = A^{*} ψ^{*}, where ψ^{*} = [\begin{matrix} \frac{1}{\sqrt{2}} {\hat{u}}^{*} \\ \frac{1}{\sqrt{2}} {\hat{υ}}^{*} \\ \frac{δ}{\sqrt{2 Ri}} {\hat{b}}^{*} \end{matrix}] and A^{*} = [\begin{matrix} 0 & 1 + Ro - \frac{s}{δ} & 0 \\ - 1 & 0 & \frac{s \sqrt{Ri}}{δ} \\ 0 & \frac{1}{\sqrt{Ri}} - \frac{s \sqrt{Ri}}{δ} & 0 \end{matrix}],

(14)

where

A^{*}

and

ψ^{*} = 1

are the linearized nondimensional SI operator and the state vector for its perturbation, respectively. We describe the stability of the system in the nondimensional representation. Henceforth, we exclude the superscript asterisk from the variables to indicate the nondimensional variable. The stability matrix and variables (

A

\hat{u}

\hat{υ}

\hat{b}

, t, and ψ) are nondimensional, unless stated otherwise. The inner product of ψ is the nondimensional perturbation energy of the system,

ψ \cdot ψ^{†} = \frac{1}{2} [\hat{u} {\hat{u}}^{†} + \hat{υ} {\hat{υ}}^{†} + (δ^{2} / Ri) \hat{b} {\hat{b}}^{†}]

c. Generalized stability analysis

Stability analysis is a method to quantify the behavior of a linear differential equation of the following form:

\frac{\partial ψ}{\partial t} = A ψ,

(15)

where ψ represents the state of the system at time t and

A

is the linearized differential operator. The operator

A

is generally derived by expanding a set of nonlinear partial differential equations about a mean state. The vector ψ contains a set of perturbation variables arising from the mean state. The operator

A

is classified as normal, if

A

commutes with its Hermitian transpose (

A A^{†} = A^{†} A

A classical approach in geophysical fluid dynamics is to use normal mode stability analysis. The normal mode stability analysis identifies a set of minimum critical parameters that give positive real eigenvalues of the operator $A$ (conditions for exponentially growing perturbations). While a set of exponentially growing perturbations dominate in the limit of t → ∞, it does not represent the perturbation growth by the nonnormal operators in finite time (e.g., Trefethen et al. 1993; Farrell and Ioannou 1996; Heifetz and Farrell 2003; Eisenman 2005).

An alternative to the normal mode analysis is the generalized stability analysis. The solution of (15) is ψ(t) = e^{$A$ t}ψ(0), where the matrix exponential e^{$A$ t} advances the system forward in time. The generalized stability analysis aims to identify the initial condition ψ(0) that maximizes the energy of the perturbation after t. Such initial and evolved conditions can be identified by the singular value decomposition (SVD) of e^{$A$ t}:

e^{A t} = U Σ V^{†},

(16)

where Σ is a diagonal matrix of growth factors that specifies the energy amplification (Farrell and Ioannou 1996; Schmid 2007). The column of the matrix

U

is the evolved perturbations from the initial condition in the column of

V

, multiplied by the associated growth factor Σ after time t.

The diagonal elements of Σ are in descending order, so the evolved condition with the maximum energy amplification at a given time t is specified in the first column of $U$ . The first column of $V$ corresponds to the initial condition that results in the maximum energy amplification. The derivative of Σ with respect to t represents the growth rate, which converges to the normal mode growth rate in the limit of t → ∞. The SVD of e^{$A$ t} is equivalent to the eigenvalue problem of $e^{A^{†} t} e^{A t}$ . Taylor expansion of $e^{A^{†} t} e^{A t}$ reveals that the maximum eigenvalue of $(A^{†} + A) / 2$ and its eigenvector determines the growth rate and structure in the limit of t → 0 (Farrell and Ioannou 1996).

We track the time evolution of the growth rate (transient growth rate) based on the energy of the evolved perturbation [

U_{i}^{†} (t) \cdot U_{i} (t) / 2

ζ_{i} (t) = \frac{1}{2 U_{i}^{†} (t) \cdot U_{i} (t)} \frac{d}{d t} [U_{i}^{†} (t) \cdot U_{i} (t)],

(17)

where the subscript indicates the ith column of the matrix

U

. The fastest-growing mode corresponds to i = 1 that is ζ₁(t). In this study, we are interested in the fastest-growing mode, i = 1. The subscript is omitted and ζ(t) implies ζ₁(t) in the subsequent sections. The transient growth rate can be partitioned into sources and sink terms, which can be useful in assessing the relevant mechanism of the instability. For example, we can track the relative importance of the shear production to buoyancy production in the context of the linearized SI dynamics. In summary, the transient growth rate in the limit of t → ∞ and t → 0 corresponds to the eigenvalue problem of the normal mode and initial mode:

\lim_{t \to \infty} ζ_{i} (t) = {eig}_{i} (A) and \lim_{t \to 0} ζ_{i} (t) = {eig}_{i} (\frac{A^{†} + A}{2}),

(18)

where eig_i(

X

) represents the operation that computes the ith leading eigenvalue of a matrix

X

. The transient growth identifies a set of perturbations that maximizes the norm |ψ|. In our case, the norm represents the total perturbation energy. The norm can take various forms as long as it can be expressed using a vector norm with a matrix norm kernel (Schmid and Henningson 2001).

3. Linearized SI problem

Many literature have discussed the linear normal modes arising from (14) (e.g., Sawyer 1956; Eliassen 1962; Ooyama 1966; Mooers 1975; Hua et al. 1997; Colin de Verdière 2012). Normal modes correspond to the eigenvectors of the linearized operator

A

, and their growth rates are the eigenvalues. Previous studies have identified a set of Ri and Ro, which lead to monotonically growing solutions (positive real eigenvalues), and the energy source of the instability (Stone 1966; Hoskins 1974; Haine and Marshall 1998). A normal mode describes the growth rate of the linear system at t → ∞, whereas an initial mode represents the growth rate at the initial state t → 0. If the linearized operator is normal, the behavior of the solution at these stages is identical. However, the operator is a nonnormal matrix (

A A^{†} \neq A^{†} A

), the nonnormal effects can change the behavior of the solution at the initial stage. We formulate the normal mode and initial mode eigenvalue problems by following (18):

λ ψ_{λ} = [\begin{matrix} 0 & 1 + Ro - \frac{s}{δ} & 0 \\ - 1 & 0 & \frac{s \sqrt{Ri}}{δ} \\ 0 & \frac{1}{\sqrt{Ri}} - \frac{s \sqrt{Ri}}{δ} & 0 \end{matrix}] ψ_{λ} and σ ψ_{σ} = [\begin{matrix} 0 & \frac{1}{2} (Ro - \frac{s}{δ}) & 0 \\ \frac{1}{2} (Ro - \frac{s}{δ}) & 0 & \frac{1}{2 \sqrt{Ri}} \\ 0 & \frac{1}{2 \sqrt{Ri}} & 0 \end{matrix}] ψ_{σ} .

(19)

The eigenvalues λ and σ represent the growth rate of the normal and initial modes, respectively. The structures (

\hat{u}

\hat{υ}

\hat{b}

) of the normal and initial modes are specified by vectors ψ_λ and ψ_σ. The rate of change in energy is partitioned into three terms:

\begin{array}{l} \frac{\partial E}{\partial t} = GSP + LSP + MB, where \\ E = \frac{1}{2} (\hat{u} {\hat{u}}^{†} + \hat{υ} {\hat{υ}}^{†} + \frac{\hat{b} {\hat{b}}^{†} δ^{2}}{Ri}) (total perturbation energy), \\ GSP = - \frac{s Ri}{δ^{2}} {\hat{u}}^{†} \hat{w} (geostrophic shear production), \\ LSP = {\hat{u}}^{†} \hat{υ} Ro (lateral shear production), and \\ MB = \frac{δ}{Ri} \hat{υ} {\hat{b}}^{†} (meridional buoyancy flux) \cdot \end{array}

(20)

Our energy budget analysis has both similarities and differences in comparison to the previous analyses conducted by Stamper and Taylor (2017), Zemskova et al. (2020), and Wienkers et al. (2021a). The analyses of Stamper and Taylor (2017) and Wienkers et al. (2021a) solely characterize the kinetic energy budget. In contrast, our analysis and that of Zemskova et al. (2020) consider the total energy budget. The vertical buoyancy flux is linked to MB, as we assume no variations in the alongfront component (

\hat{w} = s \hat{υ}

from the continuity). The vertical buoyancy flux is equal to MB along the isopycnal surface (

s = δ / Ri

). An additional component in our analysis is LSP, the advection of the horizontal Reynolds stress by the lateral shear. The influence of lateral shear on the flow was not considered in all the previous studies listed above. The net temporal growth rate ζ consists of three temporal partial growth rates:

\begin{array}{l} ζ (t) = ζ_{GSP} (t) + ζ_{LSP} (t) + ζ_{MB} (t), where \\ ζ_{GSP} (t) = \frac{GSP}{2 E}, ζ_{LSP} (t) = \frac{LSP}{2 E}, and ζ_{MB} (t) = \frac{MB}{2 E} . \end{array}

(21)

The growth rates are determined by computing the SVD of e^{$A$ t} for each t, following (16) and (17). In the limit of t → 0 and t → ∞, the net temporal growth rate converges to both the initial mode growth rate and the normal mode growth rate, respectively:

\lim_{t \to 0} ζ (t) = σ and \lim_{t \to \infty} ζ (t) = λ .

(22)

These limits are also valid for each individual partial growth rate.

4. Results: Stability in t → 0 and t → ∞

a. Normal mode solution for t → ∞ with varying s

While the normal mode SI has been extensively studied (e.g., Stone 1966; Ooyama 1966; Hoskins 1974; Mooers 1975; Xui and Clark 1985; Colin de Verdière 2012), we briefly review the solutions in order to compare them with the initial mode. The characteristic equation of the normal mode growth rate from (19) is

λ^{2} = - \frac{Ri}{δ^{2}} s^{2} + 2 \frac{s}{δ} - Ro - 1.

(23)

The largest real λ specifies the growth rate of the well-known fastest-growing SI mode. The fastest-growing mode is aligned with isopycnals

s = δ / Ri = - B_{y} / B_{z}

, which holds under the hydrostatic and inviscid approximation (e.g., Hoskins 1974):

λ^{SI} = [\begin{matrix} \sqrt{\frac{1}{Ri} - Ro - 1} \\ 0 \\ - \sqrt{\frac{1}{Ri} - Ro - 1} \end{matrix}] and ψ_{λ}^{SI} = [\begin{matrix} - \sqrt{\frac{1}{Ri} - Ro - 1} & \frac{1}{\sqrt{Ri}} & \sqrt{\frac{1}{Ri} - Ro - 1} \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}] .

(24)

It should be noted that SI motions significantly diverge from the isopycnals in a nonhydrostatic flow (Wienkers et al. 2021a). The growing, neutral, and decaying modes correspond to the first, second, and third columns of

ψ_{λ}^{SI}

, and the rates are in λ^SI. All the modes have zero growth rates in the marginal state,

1 / Ri - Ro - 1 = 0

. The fastest-growing mode grows with the rate λ_FGM when the PV has the sign opposite to the Coriolis parameter:

λ_{FGM} = {\begin{array}{l} \sqrt{\frac{1}{Ri} - Ro - 1}, & if \frac{1}{Ri} - Ro - 1 > 0, \\ 0, & otherwise . \end{array}

(25)

The structure and energy of the fastest-growing mode are

[\begin{matrix} {\hat{u}}_{λ} \\ {\hat{υ}}_{λ} \\ {\hat{b}}_{λ} \end{matrix}] = [\begin{matrix} - \sqrt{2 (\frac{1}{Ri} - Ro - 1)} \\ \sqrt{2} \\ 0 \end{matrix}] and E_{λ} = (\frac{1}{Ri} - Ro) .

(26)

The subscripts λ indicate that the variable corresponds to the fastest-growing normal mode with growth rate λ_FMG. The instability is driven by the energy contributions from GSP and LSP:

{GSP}_{λ} = {\begin{array}{l} \frac{2}{Ri} \sqrt{\frac{1}{Ri} - Ro - 1} \\ 0 \end{array} {and LSP}_{λ} = {\begin{array}{l} - 2 Ro \sqrt{\frac{1}{Ri} - Ro - 1}, & if \frac{1}{Ri} - Ro - 1 > 0 \\ 0 & otherwise \cdot \end{array}

(27)

The rate of the fastest-growing mode λ_FGM is expressed in terms of partial growth rates:

λ_{FGM} = λ_{GSP} + λ_{LSP} where λ_{GSP} = \frac{1}{2} \frac{{GSP}_{λ}}{E_{λ}} and λ_{LSP} = \frac{1}{2} \frac{{LSP}_{λ}}{E_{λ}} .

(28)

The fastest-growing mode does not have buoyancy perturbation, so the partial growth rate of the meridional buoyancy flux is zero in the normal mode, λ_MB = 0. The energy contribution from LSP normalized by GSP is

\frac{{LSP}_{λ}}{{GSP}_{λ}} = - RoRi if \frac{1}{Ri} - Ro - 1 > 0.

(29)

This energy analysis holds in the absence of the alongfront variations. Increasing the alongfront variations results in increasing the contribution from MB, thereby the energy budget resembles the baroclinic instability (Stamper and Taylor 2017).

b. Initial mode solution for t → 0 with varying s

The initial mode is solved by the deriving the characteristic equation of σ from (19):

σ^{2} = \frac{1}{4} [\frac{1}{Ri} + {(Ro - \frac{s}{δ})}^{2}] .

(30)

The characteristic equation forms a quadratic polynomial and maintains concave upward across all values of s. The characteristic equation (30) attains a minimum value given by

σ_{min} = \frac{1}{2 \sqrt{Ri}} at s_{min} = δ Ro = - \frac{U_{y}}{U_{z}} .

(31)

The initial growth rate reaches its minimum along the isovelocity slope (s = δRo), whereas the normal mode growth rate achieves its maximum along the isopycnal slope (

s = δ / Ri

). The isovelocity and isopycnal slopes are aligned when RiRo = 1; however, these slopes do not align with each other in the presence of monotonically growing normal modes (

λ_{FGM}^{2} > 0

). There are no combinations of Ri and Ro that support the slowest-growing initial mode and the fastest-growing normal mode.

c. Initial mode solution for t → 0 along isopycnals (initial mode SI)

We analyze the initial growth along isopycnals to better characterize the nonnormal effects on the fastest-growing normal mode. The initial growth rates along isopycnals are obtained by substituting

s = δ / Ri

into (19):

\begin{array}{l} σ^{SI} = [\begin{matrix} \frac{1}{2} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} \\ 0 \\ - \frac{1}{2} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} \end{matrix}] and \\ ψ_{σ} = [\begin{matrix} Ro (Ro - \frac{1}{Ri}) \sqrt{Ri} & - {Ri}^{1 / 4} {(Ro - \frac{1}{Ri})}^{- (1 / 2)} & (Ro - \frac{1}{Ri}) \sqrt{Ri} \\ {\sqrt{Ri}}^{} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} & 0 & - {\sqrt{Ri}}^{} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} \\ 1 & 1 & 1 \end{matrix}] . \end{array}

(32)

There are three types of initial modes, growing, neutral and decaying modes, which are specified in σ^SI. The first, second, and third columns of

ψ_{σ}^{SI}

correspond to their perturbation structures. The fastest-growing initial mode along isopycnals has

σ_{FGM} = \frac{1}{2} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} .

(33)

The initial mode retains a positive growth rate, even if the PV condition does not support the normal mode growth (Fig. 1a). The minimum σ_FGM lies along

Ri = 1 / Ro

when the monotonically growing normal mode is absent (

λ_{FGM}^{2} < 0

). The initial growth rate exceeds the normal mode growth rate, i.e., the growth rate ratio

σ_{FGM} / λ_{FGM}

is larger than unity (Fig. 1b). The ratio increases toward the marginal stability of the normal mode, near λ_FGM = 0, highlighting the importance of the initial mode in marginally stable fronts.

Fig. 1.

Comparing the fastest-growing initial and normal mode growth rates on a Ro–Ri space. (a) Fastest-growing initial mode growth rate σ_FGM. (b) Growth rate ratio, $σ_{FGM} / λ_{FGM}$ .

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

The fastest-growing initial mode has the following structures and the energy E_σ:

[\begin{matrix} {\hat{u}}_{σ} \\ {\hat{υ}}_{σ} \\ {\hat{b}}_{σ} \end{matrix}] = [\begin{matrix} \sqrt{2 Ri} (Ro - \frac{1}{Ri}) \\ \sqrt{2 Ri} \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} \\ \sqrt{\frac{2 Ri}{δ^{2}}} \end{matrix}] and E_{σ} = 2 [Ri {(\frac{1}{Ri} - Ro)}^{2} + 1]

(34)

The GSP, LSP, and MB terms are written as

\begin{array}{l} {GSP}_{σ} = 2 (\frac{1}{Ri} - Ro) \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}}, \\ {LSP}_{σ} = - 2 RiRo (\frac{1}{Ri} - Ro) \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}}, and \\ {MB}_{σ} = 2 \sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} . \end{array}

(35)

These terms are used to express the rate of energy growth in terms of the partial growth rates:

σ_{FGM} = σ_{GSP} + σ_{LSP} + σ_{MB}, where σ_{GSP} = \frac{1}{2} \frac{{GSP}_{σ}}{E_{σ}}, σ_{LSP} = \frac{1}{2} \frac{{LSP}_{σ}}{E_{σ}}, and σ_{MB} = \frac{1}{2} \frac{{MB}_{σ}}{E_{σ}} .

(36)

The GSP of the initial mode (GSP_σ) can be negative for a flow

1 / Ri < Ro

, and the term is a sink of energy in the absence of the monotonically growing normal mode,

λ_{FGM}^{2} < 0

(Fig. 2a). One of the significant differences between the normal and initial modes is in the buoyancy perturbation. There is no buoyancy perturbation in the normal mode, whereas the initial mode has nonzero buoyancy perturbation. The MB is one of the energy sources in the initial mode.

Fig. 2.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

The relative contributions of MB and LSP with respect to GSP are

\frac{{MB}_{σ}}{{GSP}_{σ}} = {(\frac{1}{Ri} - Ro)}^{- 1} and \frac{{LSP}_{σ}}{{GSP}_{σ}} = - RoRi .

(37)

The above ratios describe the energy budget of the initial state normalized by the geostrophic shear production. The LSP is the dominant sources of energy in anticyclonic flows with

Ri > - 1 / Ro

(Fig. 2b). In the cyclonic flow, the normal mode is sustained by GSP_λ, whereas LSP_λ acts as the energy sink term. The relative contribution of LSP to GSP in both the initial and normal modes are identical,

{LSP}_{σ} / {GSP}_{σ} = {LSP}_{λ} / {GSP}_{λ}

, so the ratio presented in Fig. 2b holds for the normal mode as well.

There is an upper limit on the ratio ${MB}_{σ} / {GSP}_{σ}$ . The ratio ${MB}_{σ} / {GSP}_{σ}$ can be expressed in terms of the fastest-growing normal mode growth rate, ${MB}_{σ} / {GSP}_{σ} = 1 / (λ_{FGM}^{2} + 1)$ . Thus, in the presence of the normal mode (λ_FGM > 0), the MB contribution to the initial mode never exceeds the GSP contribution, ${MB}_{σ} / {GSP}_{σ} < 1$ . The initial MB contribution can be more significant when the normal mode has slower growth. However, in the absence of the normal mode, the MB_σ contribution can be dominant (Fig. 2c). In such cases, the dominant energy source is either MB_σ or LSP_σ. The relative contribution of MB_σ to LSP_σ is assessed by the ratio ${LSP}_{σ} / {MB}_{σ} = {RiRo}^{2} - Ro$ from (37), where LSP_σ is dominant over MB_σ in $Ri > (1 + Ro) / {Ro}^{2}$ .

We summarize the dominant energy source of the initial and normal modes within a Ro–Ri plane (Fig. 3). The eigenvector representation of the energy terms can be useful in categorizing the instability type. In a stably stratified environment, the normal mode SI manifests in three known types: pure SI, inertial SI, and centrifugal/inertial instability (Thomas et al. 2013). Within the realm of pure SI, the GSP_λ is the dominant energy source. The pure SI can grow in both anticyclonic and cyclonic flows (Fig. 3). In contrast, the inertial SI develops in the anticyclonic flow with contributions from GSP_λ and LSP_λ when $1 < Ri < 1 / (1 + Ro)$ (Thomas et al. 2013). Our analysis pinpoints the parameter regimes where LSP_λ exceeds GSP_λ, and the LSP_λ can be the dominant energy source when Ri < 1 and Ro < −1 (Fig. 3). In this context, we refer to the evolving instability driven by LSP_λ as the inertial SI. The LSP_λ is an energy sink in cyclonic flows, thereby leading to the absence of the inertial SI. The centrifugal/inertial instability is a type of instability that grows in the absence of geostrophic shear, which is not considered in this study. The other types of instability that are not considered in this study are symmetric/gravitational and gravitational instabilities in an unstably stratified environment (Ri < 0). The energy analysis presented in this study is not directly applicable to the symmetric/gravitational and gravitational instabilities because the fastest-growing modes in Ri < 0 do not necessarily grow along the isopycnal slopes.

Fig. 3.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

d. Comparison to the baroclinic instability when Ro = 0

We found that the initial SI mode can grow when Ri > 1 in the absence of the lateral shear Ro = 0. It is well known that the normal mode baroclinic instability dominates if Ri > 0.95, while the normal mode SI is the dominant mode of instability in 0.25 < Ri < 0.95 (Stone 1966). The baroclinic instability grows in the alongfront direction, whereas SI grows along isopycnals in the across front direction. Notably, the initial mode SI demonstrates a higher growth rate compared to the normal mode baroclinic growth rate predicted by Stone (1966) (Fig. 4). In the parameter regime where baroclinic instability is considered dominant, the initial mode SI may play a role. It is important to acknowledge that an aspect missing from this comparison is the initial growth rate of the nongeostrophic baroclinic instability, which requires a further investigation in order to identify the fastest-growing initial instability.

Fig. 4.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

5. Results: Transient growth

The transient growth refers to the state in between the initial and normal modes. The transient state is quantified by the transient growth rate of the evolved perturbation ζ and its partial growth rates in (21). The transient growth occurs when the normal mode is monotonically growing, λ_FGM > 0. Otherwise, the transient behavior results in an oscillation with a frequency associated with the imaginary part of λ_FGM. The transient growth rate converges to the initial mode growth rate at t → 0 and to the normal mode growth rate toward t → ∞. One way to assess the time scale of convergence is the half-life of ζ(t), which specifies the time it takes for ζ(t) to adjust to σ/2. The half-life of ζ and ζ_MB have nondimensional time scale of 1.3 and 1, respectively (Fig. 5). Increasing ζ_GSP and ζ_LSP make the half-life of ζ longer than ζ_MB.

Fig. 5.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

We assess the time it takes to reach the normal-mode state by the half-life of ζ_MB denoted as t_1/2 in a Ro–Ri plane. The adjustment to the normal mode can take a few nondimensional time units if not longer for weaker normal modes (Fig. 6). During the adjustment period, the meridional buoyancy flux contributes to the perturbation energy, unlike the normal mode. A weaker initial mode takes longer to reach the normal mode state. The half-life of ζ_MB is parameterized by $t_{1 / 2} = 1.2 σ_{FGM}^{- 1.3}$ , which tends to underestimate the decaying time (Fig. 7). A motivation of this form is to capture the behavior in which t_1/2 decreases with increasing σ_FGM.

Fig. 6.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

Fig. 7.

Plot of t_1/2 as a function of σ_FGM. Black dots represent t_1/2 computed in Fig. 6. The fastest-growing initial SI growth rate σ_FGM is defined in (33).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

The initial mode growth rate generally exceeds the normal mode growth rate (σ_FGM > λ_FGM) (e.g., Fig. 1b). This implies that ζ(t) generally decreases over time. Another term that decreases over time is ζ_MB(t), which converges to zero in the normal mode state. While both ζ(t) and ζ_MB(t) decrease over time, the growth rates ζ_GSP(t) and ζ_LSP(t) can either increase or decrease. We evaluate the evolution of ζ_GSP(t) with the condition for σ_GSP < λ_GSP:

\frac{1}{Ri} > \frac{1}{4} \frac{{(λ_{FGM}^{2} + 1)}^{4}}{λ_{FGM}^{2}} - {(λ_{FGM}^{2} + 1)}^{2} .

(38)

This condition relies on the parameter Ri and the fastest-growing normal mode growth rate λ_FGM. The derivation is described in the appendix. When Ri satisfies (38), ζ_GSP(t) increases over time on average. There are two parameter ranges where ζ_GSP(t) decreases: 1) near the marginal stability of the normal mode [along

Ri = 1 / (1 + Ro)

] and 2) low-Ri anticyclones (Fig. 8).

Fig. 8.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0048.1

6. Summary and discussion

The nonnormal characteristics of hydrostatic SI in a geostrophically maintained front are analyzed using the generalized stability analysis. We study the initial and transient states leading to the well-known normal mode state. In particular, we derive for the first time the expression for the initial growth rate. It is well known that the normal mode has the fastest-growing mode along isopycnals (s = −B_y/B_z). However, the initial mode does not have the fastest-growing mode, the initial mode reaches the minimum growth rate [ $σ_{min} = 1 / (2 \sqrt{Ri})$ ] at the isovelocity slopes ( $s = - U_{y} / U_{z}$ ). This difference in the growth rate extremum is due to the concavity of the eigenvalue equation in s. The eigenvalue equation for the initial mode is concave upward in s, while the eigenvalue equation for the normal mode is concave downward. The growth rate of the initial mode increases as s increases.

We analyze the initial growth along isopycnals to better understand the nonnormal effects on the fastest-growing normal mode SI. The evolution of the total perturbation energy depends on three terms: geostrophic shear production (GSP), lateral shear production (LSP), and meridional buoyancy flux (MB) terms. We determine the primary energy source for both the initial and normal modes within a Ro–Ri plane (Fig. 3). The relative contribution of LSP compared to GSP remains to be equal between the initial and normal modes ( ${LSP}_{λ} / {GSP}_{λ} = {LSP}_{σ} / {GSP}_{σ} = - RoRi$ ). One noteworthy distinction between the initial mode and the normal mode lies in the existence of nonzero buoyancy perturbations. This implies that SI at the initial stage can partly acquire energy from the MB, unlike the normal mode. In the absence of the normal mode, the MB contribution can become the primary energy source, which we specify the flow regime in terms of Ri and Ro as $1 / (1 + Ro) < Ri < (1+Ro) / {Ro}^{2}$ .

We assess the time it takes to reach the normal mode state by assessing the evolution of the MB contribution to the total perturbation energy. The evolution is quantified by ζ_MB(t), which starts from σ_MB and converges toward λ_MB = 0 at t → ∞ (the normal mode state). We describe the time scale through the half-life of ζ_MB, which specifies the time required to transition from σ_MB to σ_MB/2. The half-life is parameterized as $t_{1 / 2} = 1.2 σ_{FGM}^{- 1.3}$ , where σ_FGM is the initial net growth rate. A front with Ri = 1 in the absence of lateral shear (Ro = 0) has t_1/2 = 1.9, based on this parameterization. In general, t_1/2 takes about 1–3 nondimensional time units. The half-life offers a valuable representation in describing the decay of ζ_MB. After two half-lives (2 t_1/2), one-fourth (25%) of the σ_MB persists, and so on. The half-life presents an expression of halving during successive half-life periods: $ζ_{MB} (t = n t_{1 / 2}) = σ_{MB} / 2^{n}$ , where n denotes an integer. We anticipate that 1.5% of σ_MB (n = 6) persists after 6–18 nondimensional time units, corresponding to approximately 1–3 inertial periods. This means that a flow takes about 1–3 inertial periods to reach the normal mode state.

The normal mode SI increases the shearing of the isopycnals, leading to Kelvin–Helmholtz (KH) instability to facilitate the forward cascade of energy (Thomas et al. 2008; Taylor and Ferrari 2009, 2010; Grisouard 2018; Wienkers et al. 2021b). Simulations of SI typically begin with a front without lateral shear, corresponding to Ro = 0 in our analysis. These simulations demonstrate that the saturation of SI and subsequent development of the KH instability require a few inertial periods, consistent with our finding on the time required for the normal mode SI to develop. Some simulations have revealed a transition from SI to baroclinic instability as the bulk Richardson number of the flow increases over time, the emergence of a horizontal shear instability in the alongfront direction (Stamper and Taylor 2017). The initial mode may play a role in setting the vertical buoyancy flux once the SI unstable front is stabilized by the secondary circulations. Certain simulations have taken into account a SI stable front (Ri = 2 and Ro = 0) (e.g., Zemskova et al. 2020). In this simulation, the KH instability is suppressed; however, the energy growth rate remains nonzero at the beginning of the simulation for approximately 1–2 nondimensional time units (time scaled by f), highlighting the significance of the initial mode in sustaining the energy growth. Our analysis suggests that such energy growth is fueled by the MB growth rate for a flow regime $1 / (1 + Ro) < Ri < (1 + Ro) / {Ro}^{2}$ and the LSP growth rate for $Ri > (1 + Ro) / {Ro}^{2}$ .

One of the remaining questions is on the implications of the initial mode to oceanographic observations. Many observations have identified SI unstable flows, using the stability criteria derived from the normal mode analysis (e.g., Garabato et al. 2017; Naveira Garabato et al. 2019; Yu et al. 2019). These observations show that the SI unstable conditions are found near the surface mixed layer and the bottom boundary layer, while the interior of the ocean is generally stable to the normal mode SI. We envisage that the initial mode contributes to the perturbation energy in the ocean interior and the onset of the SI near the boundaries. A set of continuous horizontal measurements of temperature, salinity, and velocity is useful in evaluating the effect of the initial modes. Such measurements remain challenging with ocean profiles. Moorings offer an alternative approach; however, the relative motion of moorings introduces uncertainty in lateral gradients. These gradients are averaged over a 24-h period (e.g., Buckingham et al. 2019), which may not provide adequate temporal resolution to assess the initial mode.

The spatial and temporal scales of submesoscale flow align with internal waves, thereby resulting in substantial uncertainties when attempting to extract submesoscale features from short-term records (McWilliams 2016; Rocha et al. 2016; Torres et al. 2018; Cao et al. 2019). Moreover, the development of submesoscale phenomena in the ocean is influenced by external forcing factors such as wind and sea ice cover (Hamlington et al. 2014; Haney et al. 2015; von Appen et al. 2018). Our analysis and the simulations of Zemskova et al. (2020) demonstrate that the lifespan of the initial mode is short, and the normal mode takes about 1–3 inertial periods to develop. It remains uncertain whether the normal mode SI can develop without being disrupted by external forcing factors within these temporal scales. A flow governed by a nonnormal operator has been recognized for its ability to maintain or amplify perturbation energy through stochastic forcing, even in the absence of monotonically growing modes (Farrell and Ioannou 1996; Schmid 2007). A SI stable front may resonate with a forcing frequency, which can maintain the perturbation energy. Future investigations are needed to address the maintenance of perturbation energy by stochastic forcing.

Acknowledgments.

Ryuichiro Inoue is thanked for his careful reading that identified several inconsistencies in an earlier draft. This work also benefited from discussions with Eyal Heifetz. Constructive comments from the two reviewers have improved the presentation of the manuscript. SK received support from the Arctic Challenge for Sustainability II (ArCS II), Program Grant JPMXD1420318865.

Data availability statement.

This work does not utilize any observational data. The transient growth rate, described in section 2d, is computed by the following Python code: https://github.com/skimura04/SI_JPO23/blob/main/main_energy_evolution_FIG.py.

APPENDIX

A Flow Condition Satisfying σ_FGM > λ_FGM

We derive a range of Ri in which the transient GSP growth rate increases in time (σ_GSP < λ_GSP). The partial growth rates λ_GSP and σ_GSP are

λ_{GSP} = \frac{1}{Ri} \frac{\sqrt{\frac{1}{Ri} - Ro - 1}}{\frac{1}{Ri} - Ro} and σ_{GSP} = \frac{1}{2} \frac{\frac{1}{Ri} - Ro}{Ri} \frac{\sqrt{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}}}{{(\frac{1}{Ri} - Ro)}^{2} + \frac{1}{Ri}} .

(A1)

These partial growth rates are functions of the fastest-growing normal mode growth rate λ_FGM, where

λ_{FGM} = \sqrt{1 / Ri - Ro - 1}

. The above expressions can be written with λ_FGM:

λ_{GSP} = \frac{1}{Ri} \frac{λ_{FGM}}{λ_{FGM}^{2} + 1} and σ_{GSP} = \frac{1}{2} \frac{λ_{FGM}^{2} + 1}{Ri} \frac{\sqrt{{(λ_{FGM}^{2} + 1)}^{2} + \frac{1}{Ri}}}{{(λ_{FGM}^{2} + 1)}^{2} + \frac{1}{Ri}} .

(A2)

We want to know Ri that satisfies the inequality, σ_GSP < λ_GSP:

\frac{1}{2} \frac{λ_{FGM}^{2} + 1}{Ri} \frac{\sqrt{{(λ_{FGM}^{2} + 1)}^{2} + \frac{1}{Ri}}}{{(λ_{FGM}^{2} + 1)}^{2} + \frac{1}{Ri}} < \frac{1}{Ri} \frac{λ_{FGM}}{λ_{FGM}^{2} + 1} .

(A3)

Taking the square of both sides results in the inequality presented in (38):

\frac{1}{Ri} > \frac{1}{4} \frac{{(λ_{FGM}^{2} + 1)}^{4}}{λ_{FGM}^{2}} - {(λ_{FGM}^{2} + 1)}^{2} .

(A4)

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Fig. 1.

Comparing the fastest-growing initial and normal mode growth rates on a Ro–Ri space. (a) Fastest-growing initial mode growth rate σ_FGM. (b) Growth rate ratio, $σ_{FGM} / λ_{FGM}$ .

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Fig. 2.

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Fig. 3.

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Fig. 4.

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Fig. 5.

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Fig. 6.

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Fig. 7.

Plot of t_1/2 as a function of σ_FGM. Black dots represent t_1/2 computed in Fig. 6. The fastest-growing initial SI growth rate σ_FGM is defined in (33).