Application of a Simple Diffusivity Formulation to Examine Regime Transition and Jet–Eddy Energy Partitioning in Quasi-Geostrophic Turbulence

Shih-Nan Chen

doi:10.1175/JPO-D-23-0110.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. A summary of theories to be applied and evaluated
3. Methods
- a. Numerical experiments
- b. Metrics and model validation
4. Regime transition and a diffusivity formulation
5. Utility of the diffusivity formulation
6. Discussion
- a. A quantification of the zonostrophic regime
- b. Implications for the eddy parameterization problem
7. Summary and outlook
Acknowledgments.
Data availability statement.
APPENDIX A
- An Approximate Solution of Chen’s (2023) f-Plane Theory for Eddy Diffusivity
APPENDIX B
- Applicability to Linear Drag and Two-Dimensional Turbulence
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Fig. 1.

(a) Responses of dimensionless thermal diffusivity ${\tilde{D}}_{τ}$ to varied strength of β suppression (as measured by $β_{*}$ ) for selected values of drag strength $μ_{*}$ . An asymptotic theory of f-plane diffusivity ${\tilde{D}}_{f}$ , given by Eq. (1), is denoted by the horizontal line for different $μ_{*}$ , whereas a drag-independent scaling of β-controlled diffusivity ${\tilde{D}}_{β}$ proposed by Lapeyre and Held (2003) is indicated by the gray dashed curve. For each $μ_{*}$ , the intersect of two asymptotic theories defines a critical transition (e.g., star symbols for $μ_{*} = 0.003 and 0.3$ ). The transition point is later defined in section 5a and used to differentiate the drag- and β-controlled flow regimes in Fig. 4. (b) Here, ${\tilde{D}}_{τ}$ normalized by ${\tilde{D}}_{β}$ is plotted against $β_{*}$ . This is to illustrate that cases tend to approach an approximately drag-insensitive state (i.e., ${\tilde{D}}_{τ} / {\tilde{D}}_{β} \approx 1$ ) as $β_{*}$ increases.

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Fig. 2.

Structure of baroclinic streamfunction (i.e., proportional to temperature) at equilibrium for (a),(b) $μ_{*} = 0.003$ and (c),(d) $μ_{*} = 0.3$ . The left and right column correspond to $β_{*} = 0.01 and 0.3$ , respectively. Meridional profiles of zonal mean flow $\bar{u}$ are shown in the side panels. Note that, at low drag, flow structure shows marked changes, from isotropic fluctuations in (a) to anisotropic eddy–jet combination in (b), when $β_{*}$ increases from 0.01 to 0.3. Such changes are not seen in large drag cases (bottom panels), suggesting that regime transition is drag dependent.

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Fig. 3.

Evaluations of the (a),(b) diffusivity formulation of Eq. (10) and (c),(d) GF21’s theory of Eq. (6). (left) The ${\tilde{D}}_{τ}$ values diagnosed from the whole experiments are plotted against $β_{*}$ , with different $μ_{*}$ indicated by colors. The same color scheme is used throughout this work. The dashed curves in (a) and (c) are from Eqs. (10) and (6), respectively. Best-fit scaling coefficients are used for GF21’s theory (see a summary in Table 1). (right) One-to-one plots of experiments vs predictions.

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Fig. 4.

A diagram to classify the turbulent flow regimes in $μ_{*} – β_{*}$ parameter space. The background color contours indicate ${log}_{10} {\tilde{D}}_{τ}$ predicted by (10), with the experiments denoted by the color-coded circles. The thick white curve is the transition point $β_{*}_{, crit}$ defined by ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 1$ (see text). This curve separates the flows into drag- and β-controlled regimes. In these regimes, the contours of constant ${log}_{10} {\tilde{D}}_{τ}$ are largely horizontal and vertical, indicating strong sensitivity to drag strength and $β_{*}$ , respectively. The white dashed curve indicates the isoline of ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 10$ . The space to the right of this curve is away from the transition and is where β effects are expected to dominate over drag [i.e., ${\tilde{D}}_{τ} \approx {\tilde{D}}_{β}$ via Eq. (10); see text].

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Fig. 5.

Test of the β scaling of Eq. (2) in the β-controlled regime as identified using the regime diagram of Fig. 4. In both panels, the experiments in the β-controlled regime (i.e., ${\tilde{D}}_{f} / {\tilde{D}}_{β} > 1$ ) are denoted by filled circles, with the same color scheme for different $μ_{*}$ as in Fig. 3: (a) eddy mixing length and (b) barotropic velocity [defined in Eq. (8)]. The open circles correspond to cases in the drag-controlled regime. The scaling coefficients (1.9 and 2.6 for $l_{e}$ and V_e, respectively) are determined from the best fits. The color-coded tick marks on the right axis are predictions using the f-plane theory of Eq. (1).

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Fig. 6.

Evaluations of an energy-balance-based theory for zonal jet speed. (a) Predictions of rms barotropic jet speed from Eq. (14) are compared with the simulation results. The cases are all in the β-controlled regime ( ${\tilde{D}}_{f} / {\tilde{D}}_{β} > 1$ ) where zonal jets are known to emerge spontaneously. (b),(c) The responses of jet and eddy velocities ( ${\bar{u}}_{jet} / U$ and V_e/U) to varied drag strength $μ_{*}$ when $β_{*} (= 0.6)$ is held fixed. (d),(e) As in (b) and (c), but with varying $β_{*}$ and fixed $μ_{*} = 0.003$ . In (b) and (d), jet speed predictions with and without considering eddy dissipation [Eqs. (14a) and (16)] are indicated by the solid and dashed curves, respectively. In (c) and (e), the dashed curves denote the drag-free scaling for eddy velocity in Eq. (15). In (c), values of ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ are given to show that eddy statistics become approximately drag-insensitive when ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ .

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Fig. 7.

Evaluations of (a) jet fraction of energy dissipation and drag-independent properties in (b) eddy diffusivity and (c) eddy velocity. In these panels, ε_jet/ε, ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ , and V_e normalized by a drag-free scaling in Eq. (15) are plotted against ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ . The cases with $β_{*} = 0.8$ are excluded to allow the jet dissipation hypothesis to be tested when drag-free eddy statistics are more strictly satisfied (see text). In (b) and (c), when ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ (i.e., right side of the vertical dashed line), both ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ and $V_{e} / [4.3 β_{*}^{- 1} (1 - β_{*})]$ approach one, suggesting that the eddies are approximately drag insensitive. Under this condition, estimates of ε_jet/ε in (a) show that zonal jets do not dominate the energy dissipation, with ε_jet/ε = 0.40 ± 0.18. In (a), the goodness of dissipation estimates using jet–eddy decomposition in Eq. (12) is confirmed by showing that ε_approx/ε is near one for all cases (diamonds).

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Fig. 8.

(a) Quantification of jet–eddy dissipation ratio, ε_jet/ε_e, in the $μ_{*} – β_{*}$ parameter space. The use of dissipation is consistent with the diffusivity-based regime diagram in Fig. 4, as a steady energy balance is ${\tilde{D}}_{τ} = \tilde{ε}$ . Only ε_jet/ε_e in the β-controlled regime is plotted because jet emergence requires β effects to be important. A subspace referred to as the zonostrophic regime where zonal jets dominate over eddies energetically is identified with ε_jet/ε_e ≥ 1 (i.e., at the lower-right corner). The boundary of zonostrophic conditions are estimated by Eqs. (14) and (17), denoted by the gray and white dashed curves, respectively. (b) The scaling of Eq. (17), with $ε_{jet} / ε_{e} \sim β_{*} {(1 - β_{*})}^{- 1 / 2} / μ_{*}$ , is shown to provide a crude estimate for ε_jet/ε_e in the simulations. A linear fit is indicated by the gray line.

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Fig. A1.

Evaluations of an approximate solution for Chen’s (2023) f-plane theory. The experimentally derived (a) barotropic eddy velocity, (b) mixing length, and (c) diffusivity are plotted against the drag strength $μ_{*}$ for cases with nearly vanishing $β_{*} (= 0.01)$ . These quantities are made dimensionless using U and λ (see text). In each panel, predictions from the numerical solution of Eq. (A3) and approximation solution Eq. (A4) are shown as the solid and dashed curves, respectively.

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Fig. B1.

As in Fig. 3 in the main text, but evaluating the diffusivity formulation of Eq. (10) when applied to (a),(b) 2LQG turbulence with linear drag and to (c),(d) 2D turbulence with quadratic drag. In (a) and (b), ${\tilde{D}}_{f}$ used in Eq. (10) is from Eq. (A6) for linear drag. In (c) and (d), the numerical data are obtained by digitizing KJ17’s Fig. 3. In (c), predictions from Eq. (10) and KJ17’s theory [i.e., their Eq. (42)] are indicated by the thick black and gray curves, respectively.

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Editorial Type:: Article

Article Type:: Research Article

Application of a Simple Diffusivity Formulation to Examine Regime Transition and Jet–Eddy Energy Partitioning in Quasi-Geostrophic Turbulence

Shih-Nan Chen

Shih-Nan Chen ^aInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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Online Publication:: 29 Jan 2024

Print Publication:: 01 Feb 2024

DOI:: https://doi.org/10.1175/JPO-D-23-0110.1

Page(s):: 557–576

Received:: 13 Jun 2023

Final Form:: 04 Dec 2023

Accepted:: 09 Dec 2023

Published Online:: 29 Jan 2024

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

This study uses a simple diffusivity formulation to examine flow regime transition and jet–eddy energy partitioning in two-layer quasigeostrophic turbulence. Guided by simulations, the formulation is empirically constructed so that the diffusivity is bounded by a f-plane asymptote (D_f) in the limit of vanishing β (termed drag-controlled) while reduced to a drag-independent scaling (D_β) of Lapeyre and Held toward large β (termed β-controlled). Good agreement is found for diffusivities diagnosed from simulations with both quadratic and linear drag and in 2D turbulence. From the formulation, a regime diagram is readily constructed, with D_f/D_β = 1 separating the drag-controlled and β-controlled regimes. The diagram also sets the parameter range where an eddy velocity scaling is applicable. The quantitative representations of eddy variables then enable a reasonably skillful theory for zonal jet speed to be developed from energy balance. It is shown that, using D_f/D_β ≥ 10, a state where eddy statistics are approximately drag insensitive could be identified and interpreted using wave-damping competitions in slowing an inverse cascade. However, contrary to an existing hypothesis, the energy dissipation in such a state is not dominated by zonal jets. A modest revision for a way to maintain balance while keeping eddies drag insensitive is proposed. In the regime diagram, a subspace of zonostrophic condition, defined as jet dissipation surpassing eddy, is further quantified. It is demonstrated that a rough scaling could help interpret how the relative importance of jet and eddy dissipation varies across the parameter space.

© 2024 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: Shih-Nan Chen, schen77@ntu.edu.tw

Abstract

Corresponding author: Shih-Nan Chen, schen77@ntu.edu.tw

Keywords: Eddies; Turbulence; Quasigeostrophic models

1. Introduction

Meridional heat transport due to geostrophically balanced, swirling motions relative to mean flows contributes significantly to the oceanic and atmospheric energy balance (Wunsch and Ferrari 2018; Held 2019). In oceanographic context, these swirling flows are referred to as mesoscale eddies. When viewing these eddies as macroturbulence (e.g., Rhines 1977; Salmon 1980; Held 1999), one may represent their transport by the turbulent diffusion formulation, with an eddy diffusivity encapsulating the bulk effects of eddy stirring that ultimately lead to net transport [see, e.g., Jansen et al. (2015) for a recent synthesis]. To constrain the turbulent transport, an important task is then to understand how the diffusivity (or equivalently the eddy velocity and length scales via the mixing length theory) is related to mean flows and environmental conditions (Vallis 2021).

To develop understanding for the eddy diffusivity, a reduced-physics system, known as homogeneous two-layer quasigeostrophic (2LQG) turbulence, has been used as a testbed. The 2LQG turbulence aims to represent a patch of eddy field in which spatial and temporal variations of the mean flow are sufficiently small such that eddy statistics may be considered to be locally homogeneous. It is thus justified to break the full inhomogeneous problem into piecewise uniform sections where one focuses on relating the eddy properties with an externally imposed, locally uniform mean state. There is supporting evidence for the applications of this local approach to global inhomogeneous flows (e.g., Pavan and Held 1996; Chang and Held 2019; Gallet and Ferrari 2021).

For 2LQG turbulence, there is an extensive literature on the scaling theories of eddy diffusivity in two asymptotic regimes. From a perspective of turbulent cascade, these two regimes correspond to different stopping mechanisms of a barotropic inverse cascade: by damping due to bottom drag (e.g., Larichev and Held 1995; Held 1999; Arbic and Flierl 2004; Chang and Held 2019; Chen 2023) and by the β effects that channel energy into Rossby waves/zonal modes (e.g., Vallis and Maltrud 1993; Held and Larichev 1996; Lapeyre and Held 2003). In this study, these two regimes are referred to as drag-controlled and β-controlled, respectively. The general idea is that the diffusivity is set by the halting scale because it represents the energy-containing eddies. Theories were then built upon balance constraints between energy production, dissipation, and cascade [e.g., motivated by Salmon’s (1980) energy pathways], with these quantities expressed in terms of the halting scale and associated barotropic energy level. Much of the theoretical development has focused on the limits where one of the cascade-halting mechanisms dominates. Note that there were also recent advances in a different perspective where the diffusivity was predicted based on interactions between pairs of dilute vortices (referred to as the vortex-gas model; Gallet and Ferrari 2020, 2021; Thompson and Young 2006). In this study, we take on the cascade viewpoint as a followup of Chen (2023), but comparisons with the vortex-gas model will be made.

The theories targeting asymptotic scaling behaviors clearly will encounter problems when applying to transitions where bottom drag and β influences are both at work. An apparent need for a more complete theory has motivated further developments. Recently, Chang and Held (2021, hereafter CH21) and Gallet and Ferrari (2021), hereafter GF21) both proposed theories for the diffusivity that have explicit drag and β dependencies. The central idea is motivated by the work of Ferrari and Nikurashin (2010): The diffusivity is bounded by the f-plane value and is suppressed by β, with the suppression strength governed by some dimensionless measures of Rossby wave phase speed or wave-turbulence crossover scale (see section 2c). The advances made by CH21 and GF21 have improved the overall representation of the eddy diffusivity. In both studies, the predicted diffusivities showed quantitative agreement with numerical simulations.

In comparison with the extensive works on diffusivity formulations, less attention has been given to understanding the quantitative properties of zonal jets that are known to emerge from β-plane turbulence. In particular, the understanding for the relative strength of jets and eddies as well as their relative contributions to dissipation across the parameter space is still quite limited.

For instance, a hypothesis of jet dominance in energy dissipation was proposed to help defend a scaling prediction that eddy properties are drag independent. But the validity of this hypothesis is largely unverified. The drag-free prediction is derived from a theory, referred to as the β-scaling, put forth by Held and Larichev (1996, hereafter HL96). The β-scaling was formulated assuming the existence of a Kolmogorovian spectrum with a dissipation ε setting the rate at which energy flows through the system and that the eddy length scale is limited by Rossby wave-turbulence crossover phenomenology (see section 2b). It gives analytical expressions for barotropic eddy velocity, mixing length, and diffusivity in terms of ε and β and has shown predictive skills when applied to a variety of turbulent flows (e.g., Barry et al. 2002; Lapeyre and Held 2003; Zurita-Gotor 2007; Chang and Held 2022). However, HL96 demonstrated that, when the scaling is combined with energy balance, the explicit dependency on ε can be eliminated. The resulting dimensionless eddy scales become only a function of β [see Eqs. (3) and (4) in section 2b]. The complete lack of drag sensitivity in the scaling is surprising and has been a subject of further examinations (e.g., Thompson and Young 2007). It is surprising because true termination of inverse cascade ultimately requires energy dissipation. But the dissipative effects are not explicit in the scaling formulations. A defense for the drag insensitivity is to hypothesize that a majority of energy dissipation occurs in zonal jets, thereby shielding eddies from a direct damping effect of bottom drag (HL96; see section 2b for mathematical expressions). This argument is referred to as the jet dominance hypothesis in this work. Note that, while the hypothesis is conceptually appealing, its validity has not been carefully examined.

For zonal jet energetics, it is well established that the standard QG theory of eddy–mean-flow interactions provides a useful framework to diagnose the generation of zonal flows from a background eddy field. From this framework, a central result is that the bottom drag exerted on a zonal mean flow is balanced by the vertical integral of eddy potential vorticity (PV) fluxes (e.g., see Vallis 2017, ch. 15.2–15.4; Galperin and Read 2019). Here the eddy PV fluxes represent the divergence of eddy momentum fluxes in the horizontal and form stresses in the vertical. The vertical integral therefore subsumes the momentum transfer due to eddies over a column that could force zonal jets [e.g., a classic example of this is the atmospheric eddy-driven jet; see again Vallis (2017)]. However, an estimate of jet velocity needs knowledge of eddy PV fluxes, which are generally not known a prior. This complicates the use of the momentum balance constraint to interpret the jet regime’s parameter sensitivity. On the other hand, a global (domain integrated) energy balance could provide an integral constraint to be exploited. If quantitative formulations for diffusivity and eddy velocity are available, for example from the β-scaling, and jet/eddy dissipation can be separated, it seems possible to infer jet speed from the global energy balance. In this study we will explore such a possibility, aiming to better understand jet’s parameter sensitivity.

Here we set out to develop a simple representation of thermal diffusivity in 2LQG turbulence on a β plane. The goals are to use the diffusivity formulation to 1) quantify the turbulent flow regimes, 2) develop a theory for zonal jet speed from energy balance, and 3) use the velocity scale estimates to better understand jet/eddy contributions to energy dissipation. The numerical experiments described in GF21 are expanded. Guided by the experiments, an empirical combination of two asymptotic theories leads to a diffusivity formulation that is quite simple and robust. From the formulation, a diagram separating the drag- and β-controlled regimes is readily constructed. The regime diagram also helps identify the parameter range over which the β scaling for eddy scales is applicable. We aim to show that the quantification of flow regimes, diffusivity, and eddy velocity allows a reasonably skillful estimate of jet speed to be developed and interpreted. It permits evaluations of the jet dominance hypothesis, from which a modest revision is proposed. The relative contributions of jets and eddies to energy dissipation across the parameter space are also explored. The rest of this paper is organized as follows. In section 2, we briefly review the asymptotic as well as modern diffusivity theories to be used and compared against. In section 3, numerical experiments, diagnostic metrics, and model validations are described. Section 4 presents a simple diffusivity formulation. In section 5, the formulation is then used to tackle the tasks of quantifying turbulent flow regimes and obtaining theoretical estimates for eddy and jet velocities in β-controlled turbulence. The roles of jets and eddies in energy dissipation are also examined. Last, in section 6, mapping of a subspace referred to as the zonostrophic regime is presented, along with a brief discussion of the implications for eddy parameterization.

2. A summary of theories to be applied and evaluated

a. Drag-controlled (f-plane) regime

Following the work of Ferrari and Nikurashin (2010), we expect an f-plane theory to provide a useful upper bound for the diffusivity. In this limit, damping due to bottom drag is the main cascade-halting mechanism, and the eddy scales are functions of drag strength (e.g., Thompson and Young 2006). For a f-plane theory, our starting point is the one described in Chen (2023). This theory is an extension of Held’s (1999) model in that the barotropization assumption was relaxed to allow eddies to be baroclinic, and the inertial range assumption was modified to empirically incorporate a drag-dependent cascade fraction. It is shown in appendix A that the following approximate solutions for eddy scales can be found in the case of quadratic drag:

{\tilde{V}}_{f} \approx (\frac{c_{0} c_{2}}{1.2 \sqrt{2}}) μ_{*}^{- 1 + m} + 2,

(1a)

{\tilde{l}}_{f} \approx c_{2} μ_{*}^{m} {\tilde{V}}_{f},

(1b)

{\tilde{D}}_{f} \approx c_{0} {\tilde{V}}_{f} {\tilde{l}}_{f} .

(1c)

In Eq. (1) and throughout this study, a tilde denotes a dimensionless variable; the barotropic eddy velocity V and mixing length

l

are nondimensionalized by the mean flow U and deformation radius λ, respectively; and the diffusivity D is expressed as

V l

via the mixing length theory. The dimensionless drag strength is

μ_{*} (\equiv μ λ)

, where μ is a quadratic drag coefficient (unit of m⁻¹; see GF20 and Chen 2023). We may also interpret

μ_{*}

as an inverse measure for the extent of inverse cascade by noting that μ⁻¹ and λ represent the friction-halting and energy injection scales (Held 1999). Equation (1) has been tested over a wide drag range of

μ_{*} = O (10^{- 3} – 1)

. Coefficients c₀, c₂, and m are determined from experiments. Values as in Chen (2023) (c₀ = 0.31, c₂ = 2, m = 1/7) are used throughout this study. Note that evaluations of Eq. (1) in the limit of vanishing β and interpretations of solution properties are given in the appendix. A corresponding theory for linear drag can be found there as well.

b. β-controlled regime (β scaling)

Another limiting theory for the diffusivity is the β scaling proposed by HL96 and Lapeyre and Held (2003). In this limit, the presence of β and the excitation of Rossby waves/zonal modes was suggested to provide the primary halting mechanism for the inverse cascade. The central idea is that the upscale energy transfer would become inefficient when the eddy turnover rate becomes lower than the Rossby wave frequency. Equating the wave frequency (ω ∼ β/k) with the turnover rate (

τ_{k}^{- 1} \sim ε_{c}^{1 / 3} k^{2 / 3}

, where k is a wavenumber and ε_c is a cascade rate) yields a crossover scale

l_{β}

(see Vallis 2017, ch. 12). Assuming that the barotropic energy is concentrated near the crossover scale in a Kolmogorovian spectrum and invoking the mixing length theory, one obtains the β scaling of

V_{β} \sim ε^{2 / 5} β^{- 1 / 5}, l_{β} \sim ε^{1 / 5} β^{- 3 / 5}, D_{β} \sim ε^{3 / 5} β^{- 4 / 5} .

(2)

In Eq. (2), an equilibrium inertial range is assumed so that a three-way balance between the production, dissipation, and cascade rates (ε_p = ε ∼ ε_c) is satisfied. Consistent with the inertial range picture, the above scaling can also be derived from dimensional analysis using ε and β as the repeating variables.

In 2LQG turbulence, energy is produced by eddy stirring that fluxes heat downgradient (e.g., Salmon 1998). Expressed in terms of a thermal diffusivity D_τ, the production rate is ε_p = D_τU²/λ² (see section 3b for the energy equation). Following HL96 to assign D_β in Eq. (2) to the role of the thermal diffusivity and applying ε_p = ε, one can eliminate ε in Eq. (2). The scaling becomes

{\tilde{V}}_{β} \sim β_{*}^{- 1}, {\tilde{l}}_{β} \sim β_{*}^{- 1}, {\tilde{D}}_{β} \sim β_{*}^{- 2},

(3)

where

β_{*} (\equiv β λ^{2} / U)

is a dimensionless measure for the strength of β suppression (or the inverse of supercriticality as defined in HL96). However, it is not entirely clear why D_β should represent the thermal diffusivity: There are two other possibilities in 2LQG flows, namely the diffusivities of upper- and lower-layer potential vorticity (PV). In fact, Lapeyre and Held (2003) argued that a more sensible choice is to use D_β as the lower-layer diffusivity D₂ because the background PV gradients are weaker there so that the lower-layer PV behaves more like a passive tracer. Relating D₂ (∼D_β) with D_τ through the Taylor–Bretherton relationship [i.e., Eq. (6) in Thompson and Young (2007)] and repeating the above algebra, Lapeyre and Held (2003) obtained an alternative expression for the diffusivity:

{\tilde{D}}_{β} \approx c_{3}^{5 / 2} β_{*}^{- 2} {(1 - β_{*})}^{5 / 2},

(4)

where c₃ is an O(1) coefficient that is determined experimentally. Thompson and Young (2007) reported agreement between this alternative scaling and numerical data for a parameter range characterized by low drag and intermediate

β_{*}

(with c₃ = 1.65). Note that, toward small

β_{*}

values, the asymmetry in the background PV gradients between the layers becomes small. The three diffusivities approach one another, and the scaling in Eq. (4) reduces to Eq. (3).

An important feature shared by Eqs. (3) and (4) is the lack of an explicit drag dependency. If the dissipation in energy balance depends only on drag coefficient and eddy velocity (i.e., $ε_{p} = D_{β} U^{2} / λ^{2} = ε \sim μ V_{β}^{3}$ ), one can see that the drag-free property of D_β and V_β cannot be satisfied simultaneously. As discussed in the introduction, the apparent inconsistency has led to a hypothesis that ε is controlled by zonal jets, not eddy velocity (e.g., $ε \sim μ V_{β}^{3}$ ). A main goal of this study is to quantify the contributions of jets and eddies to the dissipation (see section 5d), aiming to better understand how the drag-free eddy statistics may be maintained.

c. Combined effects

As introduced in section 1, theories recently proposed by CH21 and GF21 considered the thermal diffusivity with a general drag and β dependencies. These theories represent the state of knowledge. Predictions from them will be used as benchmarks for comparisons (see section 4). Below we summarize the relevant formulations and parameters. Detailed derivations shall refer to the original papers.

For CH21, the diffusivity was written as the f-plane diffusivity D_f_,CH21 multiplied by a suppression function F in the following form:

\frac{{\tilde{D}}_{τ}}{{\tilde{D}}_{f,CH21}} = F^{3 / 2}, F = \frac{5 λ_{0}^{5 / 3}}{3} \int_{λ_{0}}^{\infty} \frac{κ^{- 8 / 3}}{1 + λ_{1} σ^{10 / 3} κ^{- 8 / 3}} d κ,

(5)

where

{\tilde{D}}_{f,CH21} \sim μ_{*}^{- 5 / 4}

and

σ \equiv ε_{p}^{- 1 / 5} β^{3 / 5} μ^{- 2 / 5} λ^{3 / 5}

is a dimensionless parameter controlling the relative importance of drag and β. The construction of F can be traced back to Ferrari and Nikurashin (2010, hereafter FN10), where these authors considered suppression of meridional eddy mixing due to the presence of a zonal mean flow. FN10 suggested that the diffusivity is suppressed because meridional displacement could lose coherence when eddies begin to propagate relative to the mean. They derived a function F for this effect that depends on a ratio of Doppler-shifted wave frequency and eddy turnover rate. Noting that the controlling ratio is scale-dependent, the single-wavenumber picture of FN10 was later extended to a diffusivity spectrum for two-dimensional turbulence by Kong and Jansen (2017, hereafter KJ17) and for 2LQG turbulence by CH21. In Eq. (5), the parameter σ thus plays the role of FN10’s time scale ratio (in fact, CH21 interpreted it as a ratio of modified friction to β-halting length; see CH21 for discussion).

To obtain a prediction, one first determines the scaling coefficient for ${\tilde{D}}_{f,CH21}$ experimentally. As noted by CH21, Eq. (5) needs to be solved by iterations because σ depends on the diffusivity through ε_p = D_τU²/λ². CH21 found λ₀ = 1 (or 0.7 for their Fig. 4b) and λ₁ = 1.25 to best-fit their numerical experiments.

For GF21, the predictive formulations also involve a suppression function F, with

\frac{{\tilde{D}}_{τ}}{{\tilde{D}}_{*,vg}} = F (B) = \frac{1}{{(1 + a_{2} B^{γ / 2})}^{2}}, B = \frac{β_{*} l_{*,vg}}{\ln^{3 / 2} (l_{*,vg})} = \frac{β_{*} (a_{0} / \sqrt{μ_{*}})}{\ln^{3 / 2} (a_{0} / \sqrt{μ_{*}})},

(6)

where

{\tilde{D}}_{*, vg} = a_{1} μ_{*}^{- 1}

and

l_{*, vg} = a_{0} / \sqrt{μ_{*}}

are the f-plane vortex-gas scalings for diffusivity and mixing length in Gallet and Ferrari (2020) (a₀ and a₁ are the scaling coefficients). Here B is a squared ratio of the energy-containing scale in the f-plane vortex gas and the Rhines scale. GF21 proposed B as a control parameter to gauge the relative importance of arrest mechanisms due to drag and β. The function F(B) was then empirically constructed based on two asymptotic constraints: when B is small, the diffusivity is expected to be set by the vortex-gas scaling and hence F(B) → 1. Conversely, when B is large, the Rhines scale sets the eddy mixing length. In this limit, GF21 assumed the diffusivity to obey a drag-independent power law of

{\tilde{D}}_{τ} \sim β_{*}^{- γ}

. They showed by matching the diffusivity at the transition that the power law is approximately equal to setting F(B) → B^−γ, with γ ≈ 3.64.

To obtain a prediction, one first determines the scaling coefficients a₀ and a₁ for the f-plane mixing length and diffusivity and then apply Eq. (6). GF21 used a₀ = 2.5 and a₁ = 2. They reported that the empirical suppression function F(B) in Eq. (6) with a₂ = 2.5 captures the behavior of ${\tilde{D}}_{τ}$ for both quadratic and linear drag. To compare with GF21’s predictions, we will use these original coefficients as well as coefficients obtained from best fits (see section 4b and Table 1). For linear drag, the form of B is given by GF21’s Eq. (11).

3. Methods

a. Numerical experiments

Design of the numerical experiments follows Chen (2023) but is extended to consider the effects of β suppression. The evolution of 2LQG turbulence is governed by the standard two-layer quasigeostrophic potential vorticity (QGPV) equations on a β plane. We consider a basic state of two equal-thickness layers imposed with constant zonal mean shear of ±U. Dissipative terms identical to those in GF21 and Chen (2023) are included to the evolution equations: A bottom drag term is confined to the lower layer, which parameterizes the Ekman spindown to allow eddy fields to equilibrate. For numerical stability, hyperviscous damping is added for both layers to remove enstrophy accumulation at grid scales [see GF21’s Eqs. (1)–(4)]. Simulations are carried out in a 2πL × 2πL doubly periodic box. Note that, as in Chen (2023), both quadratic and linear drag forms are considered, but the focus is on the former.

The governing equations are made dimensionless using U and L and are integrated forward in time using a well-validated spectral solver Dedalus (Burns et al. 2020). The dimensionless equations are identical to those in Chen (2023) (see his appendix A), except that the inclusion of β modifies the background PV gradients. In the dimensionless form, the upper- and lower-layer PV gradients are $(L^{2} / λ^{2}) (1 + β_{*})$ and $(L^{2} / λ^{2}) (- 1 + β_{*})$ , respectively. As demonstrated by Thompson and Young (2007), the problem of β-plane 2LQG turbulence is governed by two dimensionless parameters, $μ_{*}$ and $β_{*}$ , under the following conditions: The simulation domain is sufficiently large to prevent confinement effects, and the hyperviscosity is sufficiently small to not significantly alter the energy balance. Chen (2023) found L/λ = 50 for a 512 × 512 grid and dimensionless hyperviscosity $\tilde{ν} = 10^{-}^{17}$ to be appropriate choices. These values are used throughout. The experiments are then designed to sweep the $μ_{*} – β_{*}$ parameter space. Specifically, we consider $μ_{*} = 0.00 3–1 .0$ and $β_{*} = 0.0 1– 0.8$ , a range similar to GF21. Typical discrete parameter values are $μ_{*} = 0.003, 0.01, 0.03, 0.1, 0.3, 1.0$ , combining with $β_{*} = 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 0.6, 0.8$ . For linear drag, the combinations are $κ_{*} \equiv κ λ / U = 0.2, 0.3, 0.5, 0.7, 1.0$ with $β_{*} = 0.01, 0.05, 0.1, 0.2, 0.5$ , where κ is a linear drag coefficient. Each experiment is run to a steady state over a period of 2000T_E, with the Eady time scale T_E (≡λ/U) being the time unit. Equilibrium eddy statistics are obtained by averaging over the last 400T_E.

b. Metrics and model validation

Following CH21 and GF21, we diagnose the thermal diffusivity as

D_{τ} = ⟨ τ \partial_{x} ψ ⟩ / U

(7)

from the experiment. In Eq. (7), ψ [≡(ψ₁ + ψ₂)/2] and τ [≡(ψ₁ − ψ₂)/2] are the standard barotropic and baroclinic streamfunction, respectively, and the angle brackets denote a horizontal average. Recognizing that τ represents temperature via the hydrostatic relation, barotropic perturbation velocity (u, υ) = (−∂_yψ, ∂_xψ), and −U is proportional to the meridional gradient of background temperature, Eq. (7) is equivalent to a formulation of diffusive heat transport. Since one of our aims is to quantify dissipation due to zonal jets that are known to emerge from a background eddy field (e.g., Panetta 1993), we further decompose perturbation variables into zonal mean (denoted by an overbar) and eddy components (denoted by a subscript e). For example, the barotropic streamfunction becomes

ψ = \bar{ψ} + ψ_{e}

, which gives

(u, υ) = (\bar{u} + u_{e}, υ_{e})

, where

\bar{u}

corresponds to zonal jets, and

\bar{υ} = \partial_{x} \bar{ψ} = 0

. With the above decomposition, we define the root-mean-square (rms) barotropic velocity and mixing length for eddies as

V_{e} = \sqrt{⟨ υ_{e}^{2} ⟩}, l_{e} = \sqrt{⟨ τ_{e}^{2} ⟩} / U .

(8)

Note that these definitions of eddy scales are consistent with the mixing length theory. By Eq. (7) and using the jet–eddy decomposition, we have ⟨τυ⟩ = ⟨τ_eυ_e⟩ = D_τU. Using V_e and

l_{e} U

in Eq. (8) for the scales of meridional stirring velocity and τ_e, we obtain

D_{τ} \sim V_{e} l_{e}

as expected from the mixing length theory.

With the metrics defined above, we carry out two tests to ensure that the simulations of 2LQG turbulence are conducted correctly. The first test is to evaluate the domain-averaged total energy budgets. Since the terms involving β drop out of energy integrals, the total energy equation is identical to that described in Chen (2023) [see his Eqs. (A5)–(A10)]. At a steady state, the balance is between production ε_p and two dissipative sinks due to bottom drag ε and hyperviscosity

0 = \underset{ε_{p}}{\underset{︸}{⟨ τ \partial_{x} ψ ⟩ U / λ^{2}}} \underset{ε}{\underset{︸}{- 0.5 μ ⟨ {| u_{2} |}^{3} ⟩}} + hyper,

(9)

where ε is written for the quadratic drag. Direct calculations give an accurately closed energy budget, with −ε/(ε_p + hyper) = 1.0001 ± 0.008. The dissipation due to hyperviscosity is negligible so that the production and dissipation due to bottom drag are nearly in balance as shown by −ε/ε_p = 0.97 ± 0.01. The second test is to compare the diffusivity diagnosed from Eq. (7) with GF21’s results. We digitized GF21’s Fig. 4b and calculated the ratio of D_τ between this study and GF21’s. The resulting ratio is 1.12 ± 0.18, reasonably close to one (not shown). Note that one should not expect exactly identical diffusivities from the two studies because the choices of hyperviscosity are likely different [i.e., GF21 did not report the specific values, but their values are likely larger; e.g., Gallet and Ferrari (2020) used

\tilde{ν} = 1 0^{-}^{13} versus 1 0^{-}^{17}

in Chen (2023)]. Nevertheless, there is an overall agreement with GF21’s numerical data, with a mean difference of less than 13%. We take the agreement as a verification of the numerical implementations in this work.

4. Regime transition and a diffusivity formulation

a. Drag-dependent regime transition

We first use the experiments to illustrate that the diffusivity is approximately constrained by the drag- and β-controlled asymptotes. In Fig. 1a, the responses of diffusivity to varied $β_{*}$ are plotted for selected values of drag strength $μ_{*}$ . Two limiting theories predicted by Eqs. (1c) and (4) are superposed. As $β_{*}$ decreases, the diffusivity for a given $μ_{*}$ tends to approach the drag-controlled limit denoted by the horizontal lines. For example, ${\tilde{D}}_{τ}$ at $μ_{*} = 0.01$ (blue circles) levels off toward smaller $β_{*}$ and is roughly capped by the blue reference line. Conversely, as $β_{*}$ increases, the diffusivity becomes approximately limited by Lapeyre and Held’s (2003) drag-free scaling: Most of the data points are bounded by the gray dashed curve, except for the cases with large $β_{*} (= 0.8)$ .

Fig. 1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

A different way to evaluate the bound provided by Lapeyre and Held’s (2003) drag-free scaling is to plot the ratio of ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ (Fig. 1b). It can be seen that, as $β_{*}$ increases, there is a tendency for different cases to converge toward ${\tilde{D}}_{τ} / {\tilde{D}}_{β} \approx 1$ , suggesting a transition toward the β-controlled limit. There is also an indication for a shift toward an approximately drag insensitive behavior. For a constant $β_{*}$ value, ${\tilde{D}}_{β}$ is fixed [Eq. (4)]. The vertical spread of ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ in Fig. 1b is thus due entirely to ${\tilde{D}}_{τ}$ differences among cases. At $β_{*} = 0.01$ , the diffusivity has strong drag dependency: ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ for $μ_{*} = 0.003 and 0.3$ (i.e., red and yellow circles) are separate vertically by around a factor of 100. By a sharp contrast, at $β_{*} = 0.5 and 0.6$ , data points become largely overlapped, indicating much reduced drag sensitivity. The ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ variation over a 100-fold change in $μ_{*}$ is within a factor of 2.

Note however that the drag sensitivity does not stay minimal as $β_{*}$ increases. At $β_{*} = 0.8$ , the cases show a moderate increase in drag sensitivity, with ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ vertical spread in Fig. 1b becoming a factor of 4.5, and the case with larger $μ_{*}$ in fact having a larger diffusivity. This behavior occurring as $β_{*} \to 1$ was reported by Thompson and Young (2007) and CH21, but the causes for this moderate change remain uncertain (CH21). To focus on the leading-order effects, we will proceed by considering $β_{*} = 0.8$ cases approximately constrained by the ${\tilde{D}}_{β}$ asymptote. A caveat to be kept in mind is that these cases may ultimately require different treatments.

Given that the diffusivity for a given $μ_{*}$ is approximately constrained by two asymptotes (Fig. 1a), the points where the two limiting theories intersect (e.g., black stars for two $μ_{*}$ values in Fig. 1a) then conveniently mark the transition that separates the drag- and β-controlled regimes. Consistent with this picture, the horizontal structure of eddy fields at different sides of the transition point differ significantly. For example, we may compare two cases with $μ_{*} = 0.003$ (red circles in Fig. 1). These cases have $β_{*} = 0.01 and 0.3$ , located at the opposite sides of the transition point. At $β_{*} = 0.01$ , the horizontal structure of τ (proportional to temperature fluctuations) is largely isotropic (Fig. 2a). This is indicative of horizontally homogeneous turbulence where β has a negligible effect, and the eddy field is in the drag-controlled regime. By a sharp contrast, at $β_{*} = 0.3$ , the baroclinic streamfunction clearly shows zonally elongated patterns (Fig. 2b). These patterns can be qualitatively understood as a result of increased β effects that preferentially suppress meridional motion (e.g., Vallis and Maltrud 1993). In particular, five zonal jets as indicated by the peaks of $\bar{u}$ profiles (side panel) have emerged from the background turbulence. Enhanced τ gradients are aligned with the eastward jets by virtue of the thermal wind balance.

Fig. 2.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

However, the value of critical $β_{*}$ that characterizes the transition depends on the drag strength. If we again compare two cases with $β_{*} = 0.01 and 0.3$ but for larger drag with $μ_{*} = 0.3$ (yellow circles in Fig. 1), now, unlike Fig. 2b, the eddy field at $β_{*} = 0.3$ does not show clear signatures of β suppression. It can be seen from Figs. 2c and 2d that, in spite of a 30-fold increase of $β_{*}$ , the baroclinic streamfunction remains isotropic. There is no sign of zonal jet formation in the zonal-mean velocity profiles. The fact that these two cases exhibit isotropic turbulence is consistent with Fig. 1. At $μ_{*} = 0.3$ , the transition point occurs roughly at $β_{*} > 0.4$ . These two cases therefore are both in the drag-controlled regime where the β effects are relatively unimportant.

The main points to be made here are the following. First, for a given $μ_{*}$ , the thermal diffusivity D_τ is approximately bounded by two asymptotes. The intersect of the two asymptotic theories [i.e., by equating Eqs. (1c) and (4); see section 5a] provides a straightforward quantification of the transition point that separates the isotropic, drag-controlled and anisotropic, β-controlled flow regimes. Second, the transition is clearly drag-dependent. The critical $β_{*}$ of the transition point increases with drag strength (e.g., black stars in Fig. 1). Qualitatively, we may understand this positive drag-dependency by arguing that the transition occurs when the Rossby wave frequency matches the damping rate. As $μ_{*}$ increases, a greater damping rate needs to be overcame by a higher wave frequency, thereby requiring a larger $β_{*}$ for the transition to occur (see section 4c).

b. A simple formulation for eddy diffusivity applicable to 2LQG and 2D turbulence

The observation that the diffusivity is bounded by two asymptotic theories in Fig. 1 suggest a simple D_τ representation. We follow the approach of GF21 to develop an empirical function for β-suppression relative to a f-plane diffusivity. A simple functional form consistent with Fig. 1 is

\frac{{\tilde{D}}_{τ}}{{\tilde{D}}_{f}} = \frac{1}{1 + {\tilde{D}}_{f} / {\tilde{D}}_{β}},

(10)

where

{\tilde{D}}_{f}

and

{\tilde{D}}_{β}

are again obtained by Eqs. (1) and (4). Note that, when

{\tilde{D}}_{f} ≪ {\tilde{D}}_{β}

(e.g., left side of a black star in Fig. 1), damping due to bottom drag limits the eddy scales so that

{\tilde{D}}_{τ} \approx {\tilde{D}}_{f}

. Conversely, when

{\tilde{D}}_{f} ≫ {\tilde{D}}_{β}

(e.g., right side of a black star), β effects become dominant, and

{\tilde{D}}_{τ} \approx {\tilde{D}}_{β}

. Moreover, the regime transition occurs at

{\tilde{D}}_{f} = {\tilde{D}}_{β}

, in line with the analyses shown in Figs. 1 and 2.

The diffusivity representation of Eq. (10) is tested against numerical experiments. Despite its simplicity, the agreement over the entire range of D_τ is very good. In Fig. 3a, for a given $μ_{*}$ value, the prediction shown by the dashed curve generally captures the smooth decrease of D_τ as $β_{*}$ increases. In the one-to-one plot of Fig. 3b, the data points are tightly located on the diagonal line, suggesting that there is no systematic bias in the predicted D_τ. The exceptions are the few cases with lowest diffusivities. These cases are with $β_{*} = 0.8$ . As discussed in section 4a (associated with Fig. 1b), these cases exhibit moderate drag sensitivity that leads to noticeable deviations from the drag-free ${\tilde{D}}_{β}$ scaling. The cause for the deviations is not well understood (CH21), but this problem is beyond the scope of this article.

Fig. 3.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

For comparisons, the same numerical dataset is used to evaluate the predictions of GF21’s and CH21’s theories [Eqs. (5) and (6)], with the statistics summarized in Table 1. The comparisons with GF21 are shown in Figs. 3c and 3d. As described in section 2c, the coefficients in Eq. (6) are obtained from best fits. It can be seen in Fig. 3 that GF21’s predictions are in agreement with the data, but the comparison is not as tight as the predictions using Eq. (10). The points are clearly more spread out from the diagonal line in Fig. 3d than in Fig. 3b.

The comparisons are made more quantitative by evaluating the root-mean-square error (RMSE). The RMSEs for ${log}_{10} {\tilde{D}}_{τ}$ between the predictions and numerical data are calculated as in Chen (2023) and are summarized in Table 1. An RMSE value of 0.1 may be interpreted as an average drift by a factor of 10^0.1 = 1.26. For the entire dataset, the RMSE for the predictions using Eq. (10) is smaller than GF21’s using Eq. (6), with a value of 0.194 versus 0.302 for best-fit and 0.356 for original coefficients. This result is consistent with the tighter one-to-one comparison shown by Fig. 3b. Since the f-plane vortex-gas scaling used by GF21 was developed for low-drag conditions (see Chen 2023), we repeat the above calculations for low-drag cases with $μ_{*} \leq 0.1$ . For $μ_{*} \leq 0.1$ , the RMSEs of the two predictions are reduced and become closer to each other, with an RMSE of 0.159 versus 0.195. For the predictions using CH21’s theory [Eq. (5)], if the parameters used for CH21’s Fig. 4b are adopted (i.e., λ₀ = 0.7), the RMSE is much larger than the other two formulations, having a value of 0.578 and hence a mean drift by almost a factor of 4. If we use λ₀ as a tuning parameter, the smallest RMSE is 0.252. This drift is close to but still larger than that using Eq. (10).

Table 1.

A summary of root-mean-square error (RMSE) among various predictions of ${log}_{10} ({\tilde{D}}_{τ})$ . The first, second, and third rows are for 2LQG turbulence with quadratic drag, linear drag, and 2D turbulence, respectively. For quadratic drag, predictions using formulations from this study [Eq. (10)], GF21 [Eq. (6)], and CH21 [Eq. (5)] are compared. For CH21’s predictions, the parameter λ₀ = 0.7 is from the original paper, but a value of 0.5 is found to give the lowest RMSE when applied to our experiments. For GF21’s, two sets of scaling coefficients are used. “Original” refers to the values as in their paper, with a₀ = 2.5, a₁ = 2.0, and a₂ = 2.5 in Eq. (6). For best fits, the coefficients of a₀ = 2.7 and a₁ = 1.8 are first determined by applying the f-plane vortex-gas scaling for $l_{*, vg}$ and ${\tilde{D}}_{*, vg}$ to our $β_{*} \to 0$ cases. Then, the suppression function F(B) with a₂ = 2.7 is found by minimizing the RMSE. For linear drag and 2D turbulence, the comparisons and figures are described in appendix B. To briefly summarize the formulations, predictions for linear drag are from Eq. (10) and GF21’s Eqs. (11) and (17). A procedure identical to that for quadratic drag is used to determine the best-fit coefficients. Specifically, GF21 formulations are $l_{*,vg} = 2.9 e^{0.36 / κ_{*}}, l_{*,vg} = 1.8 e^{0.72 / κ_{*}}$ , and a₂ = 1.5 in F(B). For 2D turbulence, KJ17’s Eq. (42) is used.

The same representation of D_τ in Eq. (10) is applicable to cases with linear drag and to forced-dissipative two-dimensional (2D) turbulence. For linear drag and 2D turbulence, we compare the predictions using Eq. (10) with GF21’s and KJ17’s theories, respectively. Details are described in appendix B. Here we only report the RMSEs to summarize the comparisons: For linear drag, the RMSE is generally comparable to GF21’s theory, with a value of 0.120 versus 0.103 for the best-fit coefficients. For 2D turbulence, the RMSE is also comparable to KJ17’s, having small RMSEs of 0.065 and 0.068, respectively.

c. Physical interpretations

The above comparisons lead us to conclude that the simple diffusivity representation in Eq. (10), albeit empirical, is quite robust and reasonably accurate. It is robust because it is constrained by two well-established asymptotes. The robustness is further supported by the fact that the same functional form is applicable to 2LQG for both drag forms and to 2D turbulence. The empiricism involved in Eq. (10) is far from satisfactory, but some degree of empiricism is not uncommon in diffusivity formulations [e.g., GF21; CH21; see comments by Vallis (2021)]. A heuristic argument for how the empirical form may arise will be discussed in section 7.

With this caveat in mind, we proceed to discuss the physical meaning of the proposed formulation. From Eq. (10), the ratio

{\tilde{D}}_{f} / {\tilde{D}}_{β}

emerges as a controlling parameter because the diffusivity is bounded by the smaller of the two (as shown in Fig. 1). The physical meaning of

{\tilde{D}}_{f} / {\tilde{D}}_{β}

may be further elucidated with the aid of prior studies. Specifically, following the work of FN10,

{\tilde{D}}_{f} / {\tilde{D}}_{β}

may be interpreted as a ratio of Rossby wave frequency to a bottom-drag damping rate. To see this, one uses Eqs. (1) and (4) to write

\frac{{\tilde{D}}_{f}}{{\tilde{D}}_{β}} \sim \frac{μ_{*}^{m} {(μ_{*}^{- 1 + m} + 2)}^{2}}{β_{*}^{- 2} {(1 - β_{*})}^{5 / 2}} \sim \frac{μ_{*}^{3 m} μ_{*}^{- 2}}{β_{*}^{- 2}},

(11)

where, for simplicity, we have assumed low-drag conditions to neglect the correction for eddy baroclinicity. Since regime transition tends to occur with

β_{*} \sim O (0.1)

(see Fig. 1a),

{\tilde{D}}_{β}

is approximated by

β_{*}^{- 2}

. Noting that

μ_{*}^{3 m}

is a small correction (e.g., m = 1/7), the ratio

{\tilde{D}}_{f} / {\tilde{D}}_{β}

is mainly controlled by

β_{*} / μ_{*}

, which may be further expressed as

(β_{*} / μ_{*}) \sim μ_{*}^{- m} [β l / (μ V)]

by use of Eq. (1b). Here the numerator represents the Rossby wave frequency at the energy-containing scale, while the denominator is a scale-selective damping rate.

Given that ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ is proportional to the ratio of Rossby wave frequency and damping rate, the suppression of diffusivity in Eq. (10) may be interpreted via the classic phenomenology of wave interference of isotropic turbulence (e.g., Rhines 1975; Vallis and Maltrud 1993; Sukoriansky et al. 2007; Ferrari and Nikurashin 2010, etc.). That is, when the wave frequency is low compared with the damping rate (i.e., ${\tilde{D}}_{f} / {\tilde{D}}_{β} ≪ 1$ ), the eddy turnover rate that governs the cascade is limited by damping. Turbulence is largely unaffected by β to remain horizontally isotropic, thereby characterized with a diffusivity approaching the f-plane asymptote. On the other hand, when the wave frequency is high (i.e., ${\tilde{D}}_{f} / {\tilde{D}}_{β} ≫ 1$ ), cascade processes become affected by waves, which favors transferring energy to zonal modes. The resulting turbulence is therefore anisotropic, characterized by weakened meridional flows and a suppressed diffusivity.

5. Utility of the diffusivity formulation

As stated in the introduction, the main goals of this study are to use the diffusivity formulation to help 1) quantify the turbulent flow regimes, 2) develop a global estimate of zonal jet speed, and 3) better understand jet/eddy contributions to energy dissipation. With the ${\tilde{D}}_{τ}$ formulation in Eq. (10), we are in a position to tackle these main tasks. In the following subsections, we will first show that the formulation enables a straightforward quantification of drag- and β-controlled flow regimes in $μ_{*} – β_{*}$ parameter space, with ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ being a controlling parameter. The regime diagram also sets the parameter range over which the β-scaling for barotropic eddy velocity and mixing length in Eq. (2) is applicable. With the quantitative representations for eddy variables, a theory for zonal jet speed will then be developed from the energy balance, with an aim to understand jet’s parameter sensitivity.

a. A regime diagram and eddy scales

From Eq. (10) and Fig. 1, we have seen that the transition between drag- and β-controlled asymptotes occurs at ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 1$ . By equating Eqs. (1c) and (4), we can determine the transition point, denoted as $β_{*, crit}$ , for a given $μ_{*}$ (i.e., the black stars in Fig. 1). In Fig. 4, $β_{*, crit}$ (thick white curve) is plotted onto a map of predicted diffusivity in the $μ_{*} – β_{*}$ parameter space. It separates the diffusivity into drag-controlled and β-controlled regimes. It can be seen that, to the left of the $β_{*, crit}$ curve, the contours of ${log}_{10} {\tilde{D}}_{τ}$ is largely horizontal, meaning that the diffusivity depends only on the drag strength as expected for the drag-controlled limit. By contrast, to the right of the $β_{*, crit}$ curve, the contours become nearly vertical, reflecting a strong sensitivity to $β_{*}$ indicative of the β-controlled limit. The $β_{*, crit}$ curve lies along the bends of each contour, delineating the regime transition. The transition curve has a positive slope. This means that $β_{*, crit}$ increases with drag strength, because a larger damping rate needs to be matched by a higher Rossby wave frequency and hence a greater $β_{*}$ [see Eq. (11)]. It should be noted that the separation curve in Fig. 4 clearly does not represent an abrupt change between drag and β influences. The space near ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 1$ is a transition region where both effects are at work [i.e., smooth bends; see also Eq. (10)]. To further differentiate the condition of β dominance, in Fig. 4 we indicate ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 10$ as the white-dashed curve. To the right of this curve, we expect ${\tilde{D}}_{τ} \approx {\tilde{D}}_{β}$ via Eq. (10). This condition away from the transition region will be useful for identifying the parameter space where eddy statistics are approximately drag insensitive in section 5d.

Fig. 4.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

The regime diagram in Fig. 4 has identified the parameter range over which β effects exert a stronger control over frictional processes in setting the eddy diffusivity (i.e., to the right of $β_{*, crit}$ curve). By virtue of the mixing length theory, next we examine if the same parameter range also sets where the β-scaling for eddy velocity and length scales is applicable. The answer is positive. The scaling works well for the cases located in the preidentified β-controlled regime (filled circles in Fig. 5) but not elsewhere. The barotropic velocity and mixing length diagnosed via Eq. (8) largely follows the scaling relationships of 2.6V_β and $1.9 l_{β}$ as indicated by the diagonal lines. The agreement for the mixing length is better than that for the barotropic velocity. There are notable deviations from the diagonal line when the eddy velocity is relatively weak (i.e., large $β_{*}$ ) in Fig. 5b. Nevertheless, the RMSEs of the scaling predictions are small. The mean drifts for $l_{e}$ and V_e are a factor of 1.15 and 1.25, respectively. By a sharp contrast, out of the β-controlled regime, the β-scaling clearly does not apply. In both panels, the cases in the drag-controlled regime (open circles) tend to increasingly deviate from the diagonal lines as the influences of β weaken relative to drag (i.e., ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ decreases). Toward the limit of ${\tilde{D}}_{f} / {\tilde{D}}_{β} ≪ 1$ , cases with the same drag strength (i.e., same color) tend to align horizontally. These horizontal lines correspond to the drag-controlled theory of Eqs. (1a) and (1b) as indicated by the colored tick marks on the right axis.

Fig. 5.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

b. A jet speed theory from energy balance

Having the quantitative representations for diffusivity and barotropic eddy velocity in the β-controlled regime, we can now develop a simple prediction for the zonal jet speed from energy balance. By Eq. (9), Eq. (7), and neglecting the hyperviscous damping, the production-to-dissipation balance is ε_p = D_τU²/λ² = ε. Following Chen (2023) and using the decomposition described in section 3, the dissipation may be divided into zonal jet and eddy parts as

ε = 0.5 μ ⟨ {| u_{2} |}^{3} ⟩ \approx ε_{approx} = 0.5 (1.2) μ {(⟨ {\bar{u}}_{2}^{2} + 2 υ_{2, e}^{2} ⟩)}^{3 / 2} .

(12)

In Eq. (12), the factor of 1.2 comes from relating the average of velocity cubed with cubed of rms velocity, and the factor of 2 is to assume an isotropic eddy field. The goodness of the approximation is confirmed by that the values of ε_approx/ε are close to one for all experiments, with a mean of 0.93 ± 0.017.

With D_τU²/λ² = ε and Eq. (12), the task is to use the diffusivity formulation in Eq. (10) for D_τ and to relate the β scaling in Eq. (2) with υ_2,_e so as to obtain an expression for the jet velocity. However, the scaling relationship of V_e ≈ 2.6V_β is for barotropic eddy velocity. As shown by Chen (2023), using barotropic velocity to represent the lower-layer value υ_2,_e could greatly overestimate the dissipation because eddies have significant baroclinic structure particularly under moderate to large drag. Chen (2023) showed further that, by representing the barotropic–baroclinic velocity covariance as V_eU, the variance of lower-layer eddy velocity can be expressed in terms of V_e via

⟨ υ_{2, e}^{2} ⟩ \approx ⟨ υ_{e}^{2} ⟩ - 2 \sqrt{⟨ υ_{e}^{2} ⟩} U = V_{e}^{2} \underset{f_{corr} (V_{e} / U)}{\underset{︸}{[1 - 2 / (V_{e} / U)]}},

(13)

where f_corr is a correction function for incomplete barotropization [i.e., baroclinic velocity would vanish in complete barotropization so that f_corr → 1; see Eq. (A1)]. It is important to note here that, while it was applied to an f-plane theory in Chen (2023), the correction for eddy baroclinicity using f_corr is expected to be valid for β-plane cases. This is because f_corr was developed from an identity of modal velocity decomposition. The validity of Eq. (13) mainly hinges on a significant inverse cascade so that the total KE is dominated by the barotropic component and on general scaling/spectral relationships that are applicable to both f- and β-plane scenarios. Details shall refer to the texts leading up to Chen’s (2023) Eq. (10).

Combining D_τU²/λ² = ε with Eqs. (12) and (13), we obtain an expression for the jet velocity of

\frac{{\bar{u}}_{jet}}{U} \approx {[{(\frac{2 {\tilde{D}}_{τ}}{1.2 μ_{*}})}^{2 / 3} - 2 {(\frac{V_{e}}{U})}^{2} f_{corr}]}^{1 / 2},

(14a)

where

{\bar{u}}_{jet}

(

\equiv \sqrt{⟨ {\bar{u}}^{2} ⟩}

) is the rms barotropic jet velocity, and the rms eddy velocity is

V_{e} / U \approx 2.6 {\tilde{D}}_{τ}^{2 / 5} β_{*}^{- 1 / 5},

(14b)

in which the scaling of V_e ≈ 2.6V_β (i.e., Fig. 5b) and the dimensionless form of energy balance,

{\tilde{D}}_{τ} \approx \tilde{ε}

, have been applied. One can then use Eq. (10) for

{\tilde{D}}_{τ}

to obtain velocity predictions.

In Eq. (14a) we have assumed zonal jets to be approximately barotropic. Analyses suggest that this assumption is a reasonable starting point. We calculate the ratio of rms lower-layer and barotropic zonal velocities ( $\sqrt{⟨ {\bar{u}}_{2}^{2} ⟩} / {\bar{u}}_{jet}$ ) as a measure for jet baroclinicity. In the β-controlled regime where jets could emerge, the ratio has a mean value of 0.87 ± 0.11, meaning that the barotropic approximation has a fairly small error of 13% on average. However, further examinations show that the jet baroclinicity does increase with drag strength (but not too sensitive to β). For example, at low drag with $μ_{*} \leq 0.01$ , zonal jets are almost completely barotropic, with $\sqrt{⟨ {\bar{u}}_{2}^{2} ⟩} / {\bar{u}}_{jet} = 0.94 \pm 0.04$ . But, at a larger drag of $μ_{*} \geq 0.1$ , the ratio decreases to 0.71 ± 0.08. Similar sensitivity of jet baroclinicity to drag was found by Thompson and Young (2007) (e.g., their Fig. 1). The possible cause of this drag dependency does not appear to be well understood and is left for further studies. Given that the ratio is generally above 0.7, we consider the barotropic jets as an adequate leading-order approximation, especially for low drag conditions.

The above velocity predictions can be further simplified, but the simplification requires different conditions and shall be discussed separately. For barotropic eddy velocity, Eq. (14b) may reduce to a drag-free form under

{\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10

(i.e., to the right of the white dashed curve in Fig. 4, away from transition zone). Under this condition,

{\tilde{D}}_{τ}

[

\approx {\tilde{D}}_{β}

via Eq. (10)] is approximately drag free. Inserting into Eq. (14b) then yields

V_{e} / U \approx 2.6 {\tilde{D}}_{β}^{2 / 5} β_{*}^{- 1 / 5} \approx 4.3 β_{*}^{- 1} (1 - β_{*}) .

(15)

For jet speed, if zonal jets dominate the energy dissipation as HL96 hypothesized, we may neglect eddy dissipation in Eq. (14a). Anticipating that jet dominance also requires

{\tilde{D}}_{f} / {\tilde{D}}_{β} ≫ 10

(see section 5c), Eq. (14a) can be further simplified to

\frac{{\bar{u}}_{jet}}{U} \approx {(\frac{2 {\tilde{D}}_{β}}{1.2 μ_{*}})}^{1 / 3} \approx 1.8 μ_{*}^{- 1 / 3} β_{*}^{- 2 / 3} {(1 - β_{*})}^{5 / 6} .

(16)

To summarize briefly, in the β-controlled regime (

{\tilde{D}}_{f} / {\tilde{D}}_{β} > 1

) where zonal jets could emerge, jet and eddy velocities may be estimated by Eqs. (14a) and (14b), respectively. If

{\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10

, Eq. (14b) for eddy velocity could be further approximated by a drag-free form in Eq. (15). Under an even more restrictive condition of jet dominating the dissipation (to be seen in section 6a), Eq. (14a) for jet speed simplifies to Eq. (16).

c. Testing the jet speed estimates

In Fig. 6, the jet speed predictions in Eqs. (14) and (16) are tested against the simulations. In Fig. 6a, there is reasonable agreement between the simulated and predicted ${\bar{u}}_{jet}$ . Numerical data mostly locate near the diagonal line, except for the case with the largest drag (i.e., the brown circle corresponds to $μ_{*} = 1.0$ ). A linear fit gives a slope of 0.86 and R² of 0.9. The RMSE for ${log}_{10} ({\bar{u}}_{jet})$ is 0.136, equivalent to a reasonably small drift by a factor of 1.37. The large overestimate of ${\bar{u}}_{jet}$ for $μ_{*} = 1.0$ is due to an underestimate of υ_2,_e. The same problem was reported in Chen (2023) (i.e., his Fig. 2a) where it was suggested that f_corr needs further refinement for O(1) drag when baroclinic and barotropic velocities become comparable. Nevertheless, the overall agreement supports the use of Eq. (14) as a leading-order estimate for the rms jet speed. By contrast, when using Eq. (16), which neglects eddy dissipation, the agreement deteriorates significantly (not shown). The linear-fit slope becomes 0.56, indicating significant overprediction of ${\bar{u}}_{jet}$ . In particular, the RMSE almost triples (=0.360), giving a mean drift by a factor of 2.29.

Fig. 6.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

To illustrate the parameter dependency, next we show examples of how ${\bar{u}}_{jet} / U$ responds when a specific parameter is varied. The response of V_e/U is contrasted. When $β_{*}$ is held fixed, the dimensionless jet speed responds negatively to increasing drag strength (Fig. 6b). This negative response is captured well by Eq. (14a) (black curve). The approximate theory of Eq. (16), as indicated by the dashed line, collapses to Eq. (14a) toward low drag but increasingly overestimates ${\bar{u}}_{jet} / U$ as $μ_{*}$ increases. Toward low drag, ${\bar{u}}_{jet}$ increasingly exceeds V_e (i.e., comparing Figs. 6b,c). Neglecting eddy dissipation in Eq. (16) is thus a good approximation so that Eqs. (14a) and (16) collapse to one curve. Conversely, toward large drag, ${\bar{u}}_{jet}$ tends to drop below V_e. The jet speed estimates without eddy dissipation therefore deteriorate.

In contrast to the jet’s clear negative response to drag variations, the eddy velocity stays largely unchanged (Fig. 6c). The drag-free scaling of Eq. (15) represents the eddy velocity well (dashed line). Inserting $β_{*} = 0.6$ into Eq. (15) yields V_e/U ≈ 2.9, capturing the level of eddy velocity. The approximately drag-insensitive behavior of eddy velocity occurs because these cases are deep into the β-controlled regime. Specifically, they have ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ and hence ${\tilde{D}}_{τ} \approx {\tilde{D}}_{β}$ by Eq. (10), therefore giving drag-insensitive diffusivity and eddy velocity.

Physically, the different drag sensitivity of jets and eddies may be understood as follows. For cases deep into the β-controlled regime ( ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ ), the arrest of an inverse cascade that sets the barotropic eddy energy level is more effectively controlled by Rossby wave/zonal mode excitation than by bottom drag [Eq. (11) and section 4c]. The eddy properties, including V_e and D_τ, are thus approximately independent of drag. This means that, as $μ_{*}$ is varied, the energy production ε_p stays largely unchanged, and so does ε. The jet speed must then decrease with increasing drag to maintain the balance [e.g., Eq. (14)].

When $μ_{*}$ is held fixed, both jet and eddy velocities tend to decrease with increasing $β_{*}$ , with the latter responding more sensitively (Figs. 6d,e). As $β_{*}$ increases, an inverse cascade terminates at smaller length scales and at a lower barotropic energy level. The eddy velocity thus decreases, following largely Eq. (15), as denoted by the dashed curve in Fig. 6e. Via Eqs. (12) and (15), we may scale the eddy part of dissipation as $ε_{e} = 0.5 (1.2) μ {⟨ 2 υ_{2, e}^{2} ⟩}^{3 / 2} \sim μ_{*} β_{*}^{- 3} {(1 - β_{*})}^{3} f_{corr}^{3 / 2} (U^{3} / λ)$ . If we compare with $ε_{p} = {\tilde{D}}_{τ} (U^{3} / λ) \approx {\tilde{D}}_{β} (U^{3} / λ)$ , one sees that ε_e drops faster than ε_p (i.e., $β_{*}^{- 3}$ vs $β_{*}^{- 2}$ ). Toward large $β_{*}$ , eddy dissipation would become relatively unimportant (i.e., dashed and solid curves collapse in Fig. 6d). As a result, the jet speed largely follows the dependency of energy production to decrease with increasing $β_{*}$ [Eq. (16)].

There is a subtle point to note here. The largely negative dependency of ${\bar{u}}_{jet} / U$ on $β_{*}$ may appear to contradict the observation that zonal jets are more apparent as $β_{*}$ increases (e.g., Fig. 2). The contradiction is resolved by noting in Figs. 6d and 6e that, while the jets tend to weaken, the ratio of ${\bar{u}}_{jet} / V_{e}$ increases with $β_{*}$ because V_e drops faster than ${\bar{u}}_{jet}$ . That is, zonal jets tend to be more dominant toward larger $β_{*}$ . This point will be elaborated in section 6a where we use the ratio to quantify the zonostrophic limit, a jet-dominating subspace within the β-controlled regime.

d. Jet versus eddy contributions to energy dissipation: A revision of jet dominance hypothesis

In Figs. 6c and 6b, we have seen hints for the departure of HL96’s jet dominance hypothesis. In Fig. 6c where all cases satisfy ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ , the eddy velocity is approximately drag insensitive, with V_e/U varying by less than a factor of 1.4 over a 33-fold change in $μ_{*}$ . Yet, among these cases, three of them (i.e., $μ_{*} \geq 0.03$ ) have ${\bar{u}}_{jet} / V_{e} \leq 1$ , meaning that energetically zonal jets do not dominate. Consistent with this result, the theory neglecting eddy dissipation in Eq. (16) overestimates the jet speed (dashed line in Fig. 6b). Below, a need of revising HL96’s hypothesis is made more explicit by directly estimating the contributions of zonal jets to energy dissipation. From Eq. (12), the dissipation due to zonal jets (ε_jet) and eddies (ε_e) are calculated as $0.5 (1.2) μ {⟨ {\bar{u}}_{2}^{2} ⟩}^{3 / 2}$ and $0.5 (1.2) μ {⟨ 2 υ_{2, e}^{2} ⟩}^{3 / 2}$ , respectively. The fraction of jet dissipation is ε_jet/ε.

To examine the jet dissipation fraction when eddy properties are drag insensitive, we may restrict our attention to ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ with an additional constraint of $β_{*} < 0.8$ . This excludes $β_{*} = 0.8$ cases that are known to exhibit moderate drag sensitivity even though they are deep into the β-controlled regime (see Fig. 1 and section 4a). This allows the drag-insensitive behavior to be more strictly satisfied. Indeed, as confirmed in Figs. 7b and 7c, all cases under the condition of ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ and $β_{*} < 0.8$ are nearly drag independent. At the right side of the vertical dashed line, values of ${\tilde{D}}_{τ} / {\tilde{D}}_{β}$ and ${\tilde{V}}_{e} / {\tilde{V}}_{β}$ { $= (V_{e} / U) / [4.3 β_{*}^{- 1} (1 - β_{*})]$ via Eq. (15)} distribute tightly along 1, indicating a drag-independent response in diffusivity and eddy velocity.

Fig. 7.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

However, among these cases, energy dissipation is often not dominated by zonal jets. In Fig. 7a, the jet fraction ε_jet/ε can be well below 0.5. It varies between 0.07 and 0.68, with a mean of 0.40 ± 0.18. The calculations suggest that eddies play a nonnegligible role in energy dissipation, inconsistent with HL96’s jet dominance hypothesis. Note that the above conclusion does not change if we include $β_{*} = 0.8$ cases. Adding $β_{*} = 0.8$ cases gives ε_jet/ε of 0.38 ± 0.21, qualitatively similar to the values without them.

To sum up, the analyses in Fig. 7 and in section 5c suggest a modest modification of HL96’s jet dominance hypothesis. For cases with ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ and $β_{*} < 0.8$ , we do find the barotropic eddy velocity and diffusivity to be nearly drag independent (Figs. 7b,c). However, under such a condition, zonal jets often do not dominate the dissipation, as suggested by ε_jet/ε = 0.40 ± 0.18 (Fig. 7a). How does the turbulent flow maintain its energy balance while keeping eddies approximately drag insensitive? When the drag strength $μ_{*}$ is varied and $β_{*}$ is held fixed, ε and V_e stay unchanged because ε = ε_p ≈ D_βU²/λ² and V_e ≈ V_β and both D_β and V_β are drag independent. By Eq. (12), to keep ε and υ_2,_e [i.e., only a function of V_e via Eq. (13)] fixed, one does not need jets to dominate. Balance can be achieved by having ${\bar{u}}_{jet}$ that varies inversely with $μ_{*}$ , as shown in Fig. 6b and by Eq. (14a).

6. Discussion

a. A quantification of the zonostrophic regime

The analyses in section 5d have suggested that jet dominance in dissipation is not a requirement to maintain drag-insensitive eddy statistics. Yet, a regime of jet dominance has attracted considerable attention in the literature, for its highly anisotropic flow fields and energy spectrum. Such a regime, referred to as zonostrophic, was typically identified using a length scale ratio for 2D turbulence (e.g., Galperin et al. 2006; Sukoriansky et al. 2007; Scott and Dritschel 2012). Here we will use jet–eddy dissipation ratio as a direct quantification for 2LQG flows. A connection between dissipation and length scale ratios will be made.

Using the jet–eddy partitioning described in section 5d, we map out the dissipation ratio ε_jet/ε_e in the $μ_{*} – β_{*}$ parameter space (Fig. 8a). The zonostrophic regime is defined as ε_jet/ε_e ≥ 1 (i.e., to the right of the thick black curve) to reflect jet dominance. The use of dissipation as opposed to energy is consistent with the mapping of drag- and β-controlled regime using diffusivity (Fig. 4), as the energy balance in a dimensionless form is ${\tilde{D}}_{τ} = \tilde{ε}$ . It can be seen that ε_jet/ε_e tends to increase with increasing $β_{*}$ and with decreasing drag strength $μ_{*}$ . The zonostrophic regime occupies a relatively small portion of the parameter space at the lower-right corner, with $β_{*} \geq 0.4$ and $μ_{*} \leq 0.04$ .

Fig. 8.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

The jet theory developed in section 5b could be used to estimate the regime transition and help interpret the parameter dependency. By Eqs. (12) and and (13), ε_jet and ε_e are estimated as $0.5 (1.2) μ {\bar{u}}_{jet}^{3}$ and $0.5 (1.2) 2^{3 / 2} μ V_{e}^{3} f_{corr}^{3 / 2}$ , respectively. Using Eqs. (14) and (10), their ratio can then be readily calculated in the parameter space, with the estimated ratio of 1 denoted by the gray dashed curve in Fig. 8a. For relatively low drag conditions ( $μ_{*} \leq 0.03$ ), the jet theory gives a reasonable estimate for the boundary of the zonostrophic regime, as the black and gray dashed curves largely coincide. However, beyond $μ_{*} = 0.03$ , the estimates worsen, as shown by the gray curve bending away from the actual transition points. The problem lies mainly in that using barotropic jet velocity, albeit being a reasonable approximation, still overestimates the lower-layer value. By comparison, the baroclinicity in eddy velocity is largely accounted for with f_corr in Eq. (13). The net result is an increasing overestimation of ε_jet as drag increases. Further consideration of jet baroclinicity is needed here. Nevertheless, the above comparison suggests that the jet theory is useful for estimating the transition into the zonostrophic regime over a significant drag range.

By invoking additional assumptions, we may obtain a rough but physically more illuminating scaling for ε_jet/ε_e. If we assume energy production is primarily balanced by jet dissipation and

⟨ υ_{2, e}^{2} ⟩ \approx V_{e}^{2}

in the low-drag zonostrophic regime, the dissipation ratio may be simplified to

\frac{ε_{jet}}{ε_{e}} \approx 2^{- 3 / 2} {(\frac{{\bar{u}}_{jet}}{V_{e}})}^{3} \approx 2^{- 3 / 2} {[\frac{1.8 μ_{*}^{- 1 / 3} β_{*}^{- 2 / 3} {(1 - β_{*})}^{5 / 6}}{4.3 β_{*}^{- 1} (1 - β_{*})}]}^{3} \approx 0.026 \frac{β_{*} {(1 - β_{*})}^{- 1 / 2}}{μ_{*}},

(17)

where we have used Eqs. (16) and (15) for

{\bar{u}}_{jet}

and V_e, respectively. For a given

β_{*}

value, we can then easily calculate the corresponding

μ_{*}

that satisfies ε_jet/ε_e = 1. In Fig. 8a, this rough estimate is indicated by the white dashed line. As expected, over the range where

μ_{*} \leq 0.01

and

β_{*} = 0.1 \sim 0.3

, the neglect of eddy dissipation causes significant overestimates of

{\bar{u}}_{jet}

and hence of ε_jet/ε_e such that the white dashed line has a shallower slope [note that neglecting eddy baroclinicity in the denominator of Eq. (17) is less erroneous due to low-drag conditions]. But this simple scaling does provide a rough delineation of the zonostrophic limit. Moreover, when we compare Eq. (17) with the diagnosed dissipation ratio in Fig. 8b, there is a reasonable linear relationship between ε_jet/ε_e and

β_{*} {(1 - β_{*})}^{- 1 / 2} / μ_{*}

for ε_jet/ε_e > 1 where the scaling is valid. Quite surprisingly, there is also a suggestion that the scaling relationship may extend below ε_jet/ε_e ∼ 1, although the numerical data below the horizontal reference line are visibly more scattered.

From the rough scaling, the parameter dependency of the zonostrophic regime could be interpreted as follows. For cases in the β-controlled regime satisfying ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ , zonal jet strength varies inversely with drag strength [i.e., the numerator of Eq. (17)], whereas eddy velocity is approximately drag insensitive (i.e., the denominator). A lower drag thus gives a greater ε_jet/ε_e, favoring a transition toward a zonostrophic state. On the other hand, as $β_{*}$ increases, both ${\bar{u}}_{jet}$ and V_e are weakened because of reduced energy production, but the latter drops more steeply (i.e., $β_{*}^{- 2 / 3}$ vs $β_{*}^{- 1}$ ; see Figs. 6d,e). A greater $β_{*}$ thus favors zonal jets to be more dominant, again promoting the zonostrophic condition.

It is interesting to note that the scaling criterion of Eq. (17) could be converted to a length scale ratio established to identify 2D zonostrophic turbulence. In 2D flows, zonostrophic has been characterized by a highly anisotropic kinetic energy (KE) spectrum, with a spectral peak of zonal flow KE surpassing that of residual eddies (Galperin et al. 2006; Sukoriansky et al. 2007). If the zonal flow and eddy KE spectrum take a k⁻⁵ and k^−5/3 slope, respectively [see Eq. (13.21) in a review by Galperin and Read (2019)], it is easy to verify that the two spectra intersect at a wavenumber k_β [ $\sim l_{β}^{- 1}$ in Eq. (2)], and the spectral peak of zonal flow KE is concentrated near the Rhines wavenumber k_R[ $\sim {(β / E_{jet}^{1 / 2})}^{1 / 2}$ , where E_jet is the zonal flow KE]. Because the zonal spectrum is steeper than the eddy, a condition of R_β = k_β/k_R > 1 is equivalent to having a spectral peak of zonal KE surpassing the eddy’s. The length scale ratio R_β thus serves as an indicator for a zonostrophic state. If we assume E_jet dominating the total energy to write $ε \approx μ E_{jet}^{3 / 2}$ , the length scale ratio becomes R_β ∼ [β³/(εμ⁵)]^1/30. This expression is a quadratic-drag version of Eq. (12.19) in Vallis (2017). Replacing ε by D_βU²/λ² and after some straightforward algebra, one arrives at a scaling relationship of $R_{β} \sim {[β_{*} {(1 - β_{*})}^{- 1 / 2} / μ_{*}]}^{1 / 6} \sim {(ε_{jet} / ε_{e})}^{1 / 6}$ . That is, in the barotropic limit, the scaling of ε_jet/ε_e for 2LQG zonostrophic turbulence in Eq. (17) is linked with an existing length scale ratio for 2D scenarios. The consistency therefore lends additional support for the use of Eq. (17) as a crude estimate for the zonostrophic regime.

b. Implications for the eddy parameterization problem

The scaling estimate for ε_jet/ε_e in Eq. (17) and shown in Fig. 8b may have implications for oceanic eddy parameterizations. From the perspectives of climate simulations, an adequate representation of oceanic eddy fluxes, typically cast in a form of eddy diffusivity times mean gradients, is critically needed. This is because state-of-the-art climate models that target long-term simulations cannot resolve those eddies (e.g., see Jansen et al. 2015). Like the approaches described in section 2, modern eddy parameterizations often used energy balance to constrain the diffusivity (e.g., Cessi 2008; Marshall and Adcroft 2010; Jansen et al. 2015; Mak et al. 2018). A common ingredient is to express the energy dissipation as a function of eddy kinetic energy itself (e.g., ∼μE^3/2 or κE² where E is barotropic eddy energy) or to use a linear formulation with an unknown damping rate that subsumes all dissipative processes (e.g., Mak et al. 2022). However, if the results of 2LQG turbulence carry over, the unresolved fluctuations will include spontaneously generated zonal jets, especially under conditions of weak drag and large β (i.e., toward a zonostrophic regime). This implies that expressing dissipation in terms of eddy energy alone may overlook energy sinks through zonal jets.

The scaling in Eq. (17) may offer a crude way to account for the overlooked energy sinks. For example, with Eq. (17), the total dissipation including jet and eddy contributions may be expressed in terms of ε_e as $ε_{e} [1 + (ε_{jet} / ε_{e})] \approx ε_{e} [1 + b_{1} β_{*} {(1 - β_{*})}^{- 1 / 2} μ_{*}^{- 1}]$ . Since $β_{*}$ and $μ_{*}$ are external parameters (or depending only on resolved flows), this expression can be readily applied to the energy balance. Given the rough agreement between the diagnosed ε_jet/ε_e and Eq. (17) in Fig. 8b, one may hope the above expression will capture the leading-order contribution of ε_jet, pending the determination of a scaling coefficient. Details aside, the general idea of considering jet dissipation is in line with a recent finding of Jansen et al. (2015). These authors reported large overestimates of diffusivity when strong, lower-layer zonal jets are present in their simulations (see their Fig. 9). They suggested that accounting for ε_jet can potentially lower the eddy energy levels and hence reduce the errors. Obviously, the above considerations only provide a hypothesis. Actual utility of the possible ε_jet parameterization needs implementations and tests that warrant further investigations.

7. Summary and outlook

This study uses a simple diffusivity formulation to examine flow regime transition and eddy–jet energy partitioning in homogeneous 2LQG turbulence on a β plane. An estimate of the root-mean-square zonal jet speed is developed from a global energy balance.

The formulation for thermal diffusivity is a combination of two existing asymptotic theories, empirically constructed to satisfy the constraints as suggested by numerical experiments. In the limit of vanishing β (referred to as the drag-controlled regime), the diffusivity approaches an upper bound of a f-plane theory ${\tilde{D}}_{f}$ for a given drag strength $μ_{*}$ . Toward the large β limit (referred to as the β-controlled regime), the diffusivity is increasingly suppressed and reduces to a drag-independent scaling ${\tilde{D}}_{β}$ put forth by Lapeyre and Held (2003). Good agreement is found when the formulation [Eq. (10)] is applied to 2LQG turbulence with both drag forms and to 2D turbulence (see Fig. 3, Table 1, and Fig. B1).

The simple formulation enables a straightforward quantification of flow regimes, whereby the parameter range appropriate for an existing eddy scaling is readily identified. Via Eq. (10), the dimensionless diffusivity is mapped out in $μ_{*} – β_{*}$ parameter space, with the ratio of asymptotic diffusivities, ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ , being a controlling parameter. The drag- and β-controlled regimes are characterized by ${\tilde{D}}_{f} / {\tilde{D}}_{β} < 1 and > 1$ , respectively (Fig. 4). Interpreting ${\tilde{D}}_{f} / {\tilde{D}}_{β}$ as a ratio of Rossby wave frequency to a damping rate [Eq. (11)], the two regimes correspond to scenarios in which cascades become limited by bottom-drag damping and wave/zonal mode excitation, respectively. The regime transition, delineated by ${\tilde{D}}_{f} / {\tilde{D}}_{β} = 1$ , is thus drag-dependent (i.e., $β_{*, crit}$ , the thick white curve in Fig. 4): As $μ_{*}$ increases, a transition from isotropic turbulence to anisotropic eddy/zonal jet structure is delayed to a greater $β_{*}$ value, consistent with the observed flow patterns (Fig. 2). It is shown further that the identified β-controlled regime also sets the parameter range over which the existing β scaling for eddy velocity and mixing length is applicable (Fig. 5).

With the quantitative representations for diffusivity and eddy velocity in hand, the rms speed of zonal jets emerging from β-controlled turbulence can be inferred from a global energy balance [Eqs. (14) and (16)]. The theory is able to provide a reasonable leading-order estimate for the jet speed in the simulations (Fig. 6) and clarify the jet’s parameter dependency: the jet speed scales inversely with drag strength and β. As $μ_{*}$ increases, energy production changes little (i.e., ε_p ≈ D_βU²/λ², and D_β is drag insensitive). The jet speed must decrease to keep dissipation roughly unchanged [Eq. (16) and Figs. 6b,c]. As $β_{*}$ increases, the production drops and so must ${\bar{u}}_{jet}$ [Eq. (16) and Fig. 6d]. Note however that, because eddy velocity responds more sensitively to $β_{*}$ than jets [Eqs. (15) and (16)], zonal flows appear more dominant toward larger $β_{*}$ (e.g., Fig. 8a).

Building from the foregoing theoretical considerations, we proceed to investigate two questions concerning the jet–eddy energy partitioning in β-controlled turbulence. One is about whether jet dominance in dissipation is a requirement to keep eddy statistics drag-free (i.e., the jet dominance hypothesis; in section 5d). The other is to quantify and estimate the parameter space in which zonal flows are really energetically dominant relative to eddies—a subspace referred to as the zonostrophic regime (in section 6a).

The jet dominance hypothesis is first evaluated. Held and Larichev (1996) argued that, if dissipation depends only on eddy velocity as typically formulated, then the drag-free property of diffusivity and eddy velocity derived from the β scaling [Eqs. (3) and (4)] cannot be simultaneously satisfied in energy balance. This has led to a hypothesis that dissipation must occur in zonal jets, not eddies. It is shown here that, using a criterion of ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ such that ${\tilde{D}}_{τ} \approx {\tilde{D}}_{β}$ , a state where eddy properties are approximately drag insensitive can indeed be identified (Figs. 6c and 7b,c). Physically, this criterion implies cascades to be primarily limited by wave/zonal mode excitation [Eq. (11)], and eddies are thus relatively drag-insensitive. However, in this state, dissipation is often not dominated by zonal jets (Fig. 7a). The jet dissipation fraction ε_jet/ε has a mean value of 0.38 ± 0.21, suggesting nonnegligible contributions of eddy dissipation on average. A modest revision for HL96’s hypothesis is then proposed: Drag-insensitive eddy statistics do not require jet dominance in dissipation. Balance can be maintained simply by having jet speed that varies inversely with drag strength [Eq. (14)].

We are also able to characterize the zonostrophic regime. This regime has been described for its highly anisotropic KE spectrum and identified by a length scale ratio for 2D flows, but these basic properties have not been fully examined for 2LQG turbulence. Again in the $μ_{*} – β_{*}$ parameter space, we map out the jet–eddy dissipation ratio, ε_jet/ε_e, so as to use ε_jet/ε_e ≥ 1 to identify the zonostrophic condition (Fig. 8a). It is found that the zonostrophic condition only occupies a small portion of the parameter space, tending to occur toward large $β_{*}$ and low drag (i.e., $β_{*} \geq 0.4$ and $μ_{*} \leq 0.04$ ). A rough scaling for ε_jet/ε_e is developed [Eq. (17) and Fig. 8b], which facilitates a crude estimate for the boundary of the zonostrophic regime (i.e., ε_jet/ε_e = 1, denoted by the white dashed line in Fig. 8a). It is shown further that the ε_jet/ε_e scaling is consistent with a length scale ratio established as a measure of zonostrophic turbulence in 2D flows.

There are a number of unresolved issues that are worth pointing out here. An obvious one is the empiricism involved in the diffusivity formulation. Equation (10) was not derived from first principles but empirically constructed to satisfy the asymptotic behavior seen in the simulations. That is, as $β_{*} \to 0$ , $D_{τ} \approx D_{f}$ , whereas toward large $β_{*}$ and small $μ_{*}$ , $D_{τ} \approx D_{β}$ (Fig. 1a). Noting in Eq. (11) that $D_{f} / D_{β} \sim {(β_{*} / μ_{*})}^{2}$ , any formulation with a form of D_τ/D_f = 1/[1 + (D_f/D_β)ⁿ]^1/ⁿ, where n is arbitrary, would satisfy the asymptotic behavior. It seems possible to obtain a suppression function similar to the above form from energy balance. For illustrative purposes, we start from Held’s (1999) scaling but incorporate zonal jet dissipation. We may express $ε \sim μ {V^{3} [1 + f (μ_{*}, β_{*})}$ , where we have used an unknown function f( $μ_{*}$ , $β_{*}$ ) to grossly represent a jet-to-eddy velocity ratio (e.g., see section 6b). If we proceed as in Held (1999) to write $ε_{p} \sim (V l) U^{2} / λ^{2} \sim ε \sim μ {V^{3} [1 + f (μ_{*}, β_{*})}$ combining with $l / V \sim λ / U$ [e.g., see Eq. (3) in Chen 2023], we would obtain a scaling for the diffusivity of $D_{τ} / D_{f, H 99} = 1 / {[1 + f (μ_{*}, β_{*})]}^{2}$ . Note that there is a suppression function that has a generic form not unlike those in Ferrari and Nikurashin (2010), GF21 [Eq. (6)], and Eq. (10). Of course, it may not be analytically tractable to express f( $μ_{*}$ , $β_{*}$ ) in terms of D_f/D_β (note that ε_jet/ε does scale neatly with D_f/D_β in Fig. 7a). Nevertheless, the above exercise may serve as a partial justification for the use of an empirical form like Eq. (10) for the diffusivity.

Another open question is the link between the energy-based estimate of jet speed and QG eddy-mean interaction theory for zonal mean flows. In section 5b, we have developed a theory for the rms jet speed from the global energy balance [see Eqs. (12)–(14)]. There is a well-established local constraint that is not used in this study. Here, global and local refer to domain average and zonal mean properties, respectively. The local constraint comes from momentum balance of zonal mean flows in the standard QG eddy–mean interaction framework. Specialized to two-layer QG flows, the requirement of no net meridional transport gives a constraint of $H_{1} \bar{υ_{1, e} q_{1, e}} + H_{2} \bar{υ_{2, e} q_{2, e}} = γ H_{2} {\bar{u}}_{2}$ , where γ is a linear bottom drag coefficient, H is the undisturbed layer thickness, and $\bar{υ_{e} q_{e}}$ is the eddy PV flux for each layer [e.g., see section 15.2 and TL.1–4 in Vallis (2017)]. One could see immediately that this relationship gives another handle for the lower-layer jet speed ${\bar{u}}_{2}$ . Some aspects of this local constraint have been explored by Thompson and Young (2007) and Thompson (2010). However, we would argue that the PV fluxes are generally not known, which complicates the use of the momentum constraint to interpret jet sensitivity. For example, we may again consider a condition when eddies are approximately drag insensitive (i.e., ${\tilde{D}}_{f} / {\tilde{D}}_{β} \geq 10$ ). In this case, the PV fluxes are unlikely to be drag-free. This is because, if expressing the fluxes in terms of diffusive transport, the zonal mean PV gradient depends on the jet structure itself such that the fluxes would have drag dependency [e.g., $\partial {\bar{q}}_{1} / \partial y = β + U / λ^{2} - \partial^{2} {\bar{u}}_{1} / \partial y^{2} + ({\bar{u}}_{1} - {\bar{u}}_{2}) / (2 λ^{2})$ , where $\bar{u}$ is drag-dependent]. From the local constraint, it is not clear to us what the drag dependency of ${\bar{u}}_{2}$ would look like if the left-hand terms are also drag dependent. More fundamentally, to incorporate the local momentum constraint into descriptions of global properties, it seems that linking the local eddy fluxes with global energy balance is an important but challenging task remaining to be solved.

Last, we should also point out that the drag-free diffusivity scaling proposed by Lapeyre and Held (2003) [Eq. (4)] provides a better asymptotic bound than that by HL96 [Eq. (3)]. In Fig. 1a, there is a steeper drop in diffusivity toward larger $β_{*}$ in a manner consistent with Eq. (4). By comparison, Eq. (3) would predict a straight line with a slope of −2. As described in section 2b, the two scalings differ in their choices of relating the β scaling [Eq. (2)] with a specific diffusivity. Lapeyre and Held (2003) justified their choice of lower-layer PV diffusivity by arguing that lower-layer flow is more turbulent (diffusion-like) and less wavelike because of a weaker background PV gradient. Qualitatively, this picture is consistent with our experiments. For a case with $μ_{*} = 0.003$ and $β_{*} = 0.5$ in which $β_{*}$ is large enough for the scalings in Eqs. (3) and (4) to differ, we find the meridional displacements of PV to be larger in the lower layer than in the upper layer (not shown). However, processes leading to the different appearance of PV structure are not examined further. This means that a rigorous justification for choosing lower-layer PV diffusivity as in Lapeyre and Held (2003) still eludes us.

Acknowledgments.

Two anonymous reviewers provided constructive comments that improved this work substantially. Discussions with CY Chang (GFDL) and Kaushik Srinivasan (UCLA) about an early version of this work were helpful. This work is supported by the National Science and Technology Council of Taiwan through Grants MOST 108-2611-M-002-022-MY4 and NSTC 112-2611-M-002-015.

Data availability statement.

The source code for solving the QGPV equations by Dedalus and simulation outputs are available upon request from the author.

APPENDIX A

An Approximate Solution of Chen’s (2023) f-Plane Theory for Eddy Diffusivity

Here we briefly describe an approximate solution of Chen’s (2023) f-plane theory for eddy diffusivity in 2LQG turbulence. In what follows, we will develop approximations, test the solution against numerical experiments under a condition of vanishing β, and discuss its linkages with prior studies.

Following Chen (2023), the balance constraints for barotropic eddy velocity V and mixing length

l

may be written as

ε_{p} = ε : (c_{0} V_{f} l_{f}) \frac{U^{2}}{λ^{2}} = \frac{1}{2} μ ⟨ {| u_{2} |}^{3} ⟩ \approx \sqrt{2} (1.2) μ {⟨ υ_{2}^{2} ⟩}^{3 / 2} \approx \sqrt{2} (1.2) μ {(V_{f}^{2} - 2 V_{f} U)}^{3 / 2},

(A1a)

ε_{c} / ε_{p} \sim μ_{*}^{- 2 m} : l_{f} / V_{f} = c_{2} μ_{*}^{m} λ / U,

(A1b)

where a subscript f denoting the f plane is used to differentiate from the β scaling (see main text) and c₀, c₂, and m are coefficients determined from experiments.

In Eq. (A1), the first equation is a production-dissipation balance, with a correction of partial barotropization incorporated into the dissipation estimates (i.e., −2V_fU term, allowing eddies to be baroclinic; see Chen 2023) and μ being a quadratic drag coefficient. The second describes a cascade-production balance, modified to empirically allow the cascade fraction ε_c/ε_p to increase with cascade extent. In Eq. (A1b), the dimensionless drag strength $μ_{*} (\equiv μ λ)$ can be interpreted as an inverse measure for the extent of an inverse cascade. Note that μ⁻¹ and λ represent the spatial scale near which a cascade is stopped by drag (Held 1999) and baroclinic energy is converted to barotropic mode (Salmon 1998). Therefore, in Eq. (A1b), the cascade fraction increases as the cascade extent (∼1/μ) increases (i.e., m is a positive empirical constant).

Using the mean shear U and deformation radius λ as velocity and length scales, Eq. (A1) are made dimensionless as

(c_{0} {\tilde{V}}_{f} {\tilde{l}}_{f}) \approx \sqrt{2} (1.2) μ_{*} {({\tilde{V}}_{f}^{2} - 2 {\tilde{V}}_{f})}^{3 / 2},

(A2a)

{\tilde{l}}_{f} = c_{2} μ_{*}^{m} {\tilde{V}}_{f} .

(A2b)

Plugging Eq. (A2b) into Eq. (A2a), one obtains the following cubic equation:

\frac{c_{0} c_{2}}{\sqrt{2} (1.2)} μ_{*}^{- 1 + m} = {\tilde{V}}_{f} {(1 - \frac{2}{{\tilde{V}}_{f}})}^{3 / 2},

(A3)

which was solved numerically by Chen (2023) for the dimensionless barotropic velocity

{\tilde{V}}_{f}

It is shown here that Eq. (A3) has an approximate solution. With

2 / {\tilde{V}}_{f}

being a small number for an extensive inverse cascade, we may expand

{(1 - 2 / V_{f})}^{3 / 2} \approx 1 - 3 / {\tilde{V}}_{f} + 3 / 2 (1 / {\tilde{V}}_{f}^{2}) + O (1 / {\tilde{V}}_{f}^{3})

. If

{\tilde{V}}_{f} ≫ 1

, we can just retain the first two terms. However, toward O(1) drag where

{\tilde{V}}_{f}

also becomes O(1), the second-order correction [

O ({\tilde{V}}_{f}^{- 2})

] becomes nonnegligible. We find empirically that a compromise is to take

{(1 - 2 / {\tilde{V}}_{f})}^{3 / 2} \approx 1 - 2 / {\tilde{V}}_{f}

(see Fig. A1). Plugging this into Eq. (A3) then gives a closed form expression for the barotropic velocity as

{\tilde{V}}_{f} \approx \frac{c_{0} c_{2}}{\sqrt{2} (1.2)} μ_{*}^{- 1 + m} + 2.

(A4)

The full solutions for eddy scales (

{\tilde{V}}_{f}

and

{\tilde{l}}_{f}

) and diffusivity are obtained by combining Eq. (A4) with Eq. (A2b) and

{\tilde{D}}_{f} = c_{0} {\tilde{V}}_{f} {\tilde{l}}_{f}

. These are given in Eq. (1) of the main text.

Fig. A1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

The goodness of the approximate solution is confirmed in Fig. A1. To obtain predictions from Eqs. (A3) and (A4), we use coefficients identical to Chen (2023), with c₀ = 0.31, c₂ = 2, and m = 1/7. The numerical experiments with $β_{*} = 0.01$ , which approach the f-plane limit, are chosen for the evaluations. It can be seen that the predictions capture the responses of eddy velocity, mixing length, and diffusivity to varied drag strength $μ_{*}$ , consistent with the results of Chen (2023). Importantly, the prediction curves from Eqs. (A3) and (A4) lie close to each other, supporting the goodness of the approximate solution in Eq. (A4). Toward weak drag (e.g., $μ_{*} \leq 0.01$ ), the two curves collapse. This occurs when an extensive cascade renders the partial barotropizaation correction negligible [i.e., neglecting $- 2 / {\tilde{V}}_{f}$ in Eq. (A3) and +2 in Eq. (A4)] such that ${\tilde{V}}_{f}$ in Eqs. (A3) and (A4) collapse to a straight power-law line. Conversely, toward large drag, the presence of a nonnegligible +2 term in Eq. (A4) weakens the drag sensitivity. Physically, this represents an increase in eddy baroclinicity due to a reduced extent of cascade (e.g., Charney 1971). Eddies are able to buffer the changes in dissipation by adjusting the lower-layer velocity, thereby keeping the barotropic flow relatively insensitive to drag changes (Chen 2023). Overall, the comparisons in Fig. A1 suggest that we can use Eq. (A4) as an approximate f-plane theory for the diffusivity, at least over $μ_{*} = 0.003 – 1$ [see Eq. (1) in the main text]. This frees us from the need to solve Eq. (A3) numerically.

It is worth pointing out that the above theory can recover the scaling results of prior studies. When assuming complete eddy barotropization and existence of an inertial range, we neglect the dissipation correction [i.e., +2 term in Eq. (A4)] and set m = 0. The above theory is reduced to Held’s (1999) scaling, with ${\tilde{V}}_{f} \sim μ_{*}^{- 1}, {\tilde{l}}_{f} \sim μ_{*}^{- 1}$ , and ${\tilde{D}}_{f} \sim μ_{*}^{- 2}$ [using Eqs. (A4) and (A2b), and ${\tilde{D}}_{f} = c_{0} {\tilde{V}}_{f} {\tilde{l}}_{f}$ ]. If we stick with the barotropization assumption but now consider a variable cascade fraction with m > 0 for a lack of an inertial range, Eq. (A4) can recover Chang and Held’s (2021) scaling. Specifically, taking m = 0.25 and neglecting the +2 term, we obtain ${\tilde{V}}_{f} \sim μ_{*}^{- 3 / 4}, {\tilde{l}}_{f} \sim μ_{*}^{- 1 / 2}$ and ${\tilde{D}}_{f} \sim μ_{*}^{- 5 / 4}$ , identical to CH21’s Eq. (18).

Finally, for completeness, we list Chen’s (2023) f-plane theory in the case of linear bottom drag. It can be shown that an equation for

{\tilde{V}}_{f}

similar to Eq. (A3) is

\frac{c_{0} c_{2}}{2} κ_{*}^{- 1 + m} = (1 - \frac{2}{{\tilde{V}}_{f}}),

(A5)

from which

{\tilde{V}}_{f}

and subsequently

{\tilde{l}}_{f}

and

{\tilde{D}}_{f}

can be obtained as

\begin{array}{l} {\tilde{V}}_{f} = 2 / (1 - \frac{c_{0} c_{2}}{2} κ_{*}^{- 1 + m}), \\ {\tilde{l}}_{f} = c_{2} κ_{*}^{m} {\tilde{V}}_{f}, \\ {\tilde{D}}_{f} = c_{0} {\tilde{V}}_{f} {\tilde{l}}_{f}, \end{array}

(A6)

where

κ_{*} (\equiv κ λ / U)

is the dimensionless drag strength defined for a linear drag coefficient κ. Equation (A6) is used for Fig. B1 in appendix B and described in section 4b of the main text.

APPENDIX B

Applicability to Linear Drag and Two-Dimensional Turbulence

The analyses below aim to show that the same diffusivity (D_τ) formulation in Eq. (10) is applicable to cases with linear bottom drag and to forced-dissipative two-dimensional (2D) turbulence. For 2LQG turbulence with linear drag, we have carried out new experiments that are similar to those in GF21 but expand the drag range to O(1), as described in section 3a. For 2D turbulence, numerical data are obtained by digitizing KJ17’s Fig. 3. Predictions of Eq. (10) are then compared with GF21’s and KJ17’s theories. The predictions are obtained as follows. In the case of linear drag, ${\tilde{D}}_{f}$ is calculated from Eq. (A6), and ${\tilde{D}}_{β}$ is from Eq. (4). Parameters used here are nearly identical to Chen (2023), with c₀ = 0.27, c₂ = 2, and m = 0.26. A lower c₃ (=1.2) is adopted to better represent the β-controlled asymptote. In the case of 2D turbulence, dimensional D_τ is used in Eq. (10) because turbulence is forced by imposing the injection rate ε, not by the mean shear ±U. Scalings of the asymptotic diffusivity are $D_{f} = 0.016 ε^{1 / 3} C_{D}^{- 4 / 3}$ and $D_{β} = 3.96 ε^{3 / 5} β^{- 4 / 5}$ , where the coefficients are estimated from KJ17’s Fig. 3 (i.e., their red and green lines) and C_D is a quadratic drag coefficient equivalent to μ used in this study.

Like for the quadratic drag shown in Fig. 3, the variations of ${\tilde{D}}_{τ}$ for linear drag are well represented by the simple form of Eq. (10). For a fixed drag strength $κ_{*} (\equiv κ λ / U)$ , the predictive curves (dashed) are able to capture the flattening of ${\tilde{D}}_{τ}$ as $β_{*} \to 0$ (i.e., the f-plane asymptote ${\tilde{D}}_{f}$ ) and a smooth decrease due to β suppression as $β_{*}$ increases (Fig. B1a). Overall, the numerical data are located along the diagonal line in Fig. B1b, suggesting a reasonably good agreement between the predictions and experiments. It is noticeable in Fig. B1a that, for $κ_{*} = 1$ , the use of Eq. (10) overestimates the diffusivity, as the yellow dashed curve is above the data points. This overestimation is attributable to the overestimation of ${\tilde{D}}_{f}$ (i.e., a vertical offset), which can be improved if a more accurate f-plane theory is used.

Fig. B1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0110.1

Again we can calculate the RMSE and compare it with GF21’s theory. Like for quadratic drag, we try two sets of scaling coefficients to obtain diffusivity predictions from GF21’s Eqs. (11) and (17): original values and values from best fits following the procedure of GF21 (see section 2c and Table 1’s caption). The RMSE using Eq. (10) is 0.120, equivalent to a mean drift by a factor of 1.32 (Table 1). This error is comparable to GF21’s RMSE of 0.103 and 0.175, with best-fit and original scaling coefficients, respectively. Overall, the comparisons suggest that, for linear bottom drag, the diffusivity formulation in Eq. (10) has predictive skills similar to GF21’s.

To represent tracer diffusivity in 2D turbulence, the formulation in Eq. (10) also shows skills comparable to the more elaborated theory of Kong and Jansen (2017). Figure B1c reproduces KJ17’s Fig. 3, where the tracer diffusivity normalized by the f-plane scaling is plotted against a controlling dimensionless number σ ( $= ε^{- 1 / 5} β^{3 / 5} C_{D}^{- 1}$ ). This parameter represents a ratio of friction-arrested to wave-turbulence crossover length scale (see KJ17). We note that the criticality parameter D_f/D_β in Eq. (10) is convertible to σ through D_f/D_β ∼ σ^4/3. In Fig. B1c, the predicted curve using Eq. (10) (black curve) captures the flattening of diffusivity toward small σ and its monotonic decrease as σ increases. When σ (and equivalently D_f/D_β) ≪ 1, eddy scales are controlled by bottom drag, and hence D/D_f levels off. Conversely, when σ ≫ 1, the crossover length imposes a scale limit, and D/D_f ∼ D_β/D_f ∼ σ^−4/3, indicating that the black curve in Fig. B1c approaches a slope of −4/3. From Fig. B1d, the predictions using Eq. (10) and data clearly follow a tight one-to-one relationship, although Eq. (10) tends to slightly underestimate the diffusivity. Quantitatively, Eq. (10) yields a small RMSE of 0.065 and a mean drift by a factor of 1.16. These values are comparable to those obtained by KJ17’s theory, with a RMSE of 0.068 [i.e., using their Eq. (42); see also the gray curve in Fig. B1c].

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Journal of Physical Oceanography

Abstract
1. Introduction
2. A summary of theories to be applied and evaluated
3. Methods
- a. Numerical experiments
- b. Metrics and model validation
4. Regime transition and a diffusivity formulation
5. Utility of the diffusivity formulation
6. Discussion
- a. A quantification of the zonostrophic regime
- b. Implications for the eddy parameterization problem
7. Summary and outlook
Acknowledgments.
Data availability statement.
APPENDIX A
- An Approximate Solution of Chen’s (2023) f-Plane Theory for Eddy Diffusivity
APPENDIX B
- Applicability to Linear Drag and Two-Dimensional Turbulence
REFERENCES

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Application of a Simple Diffusivity Formulation to Examine Regime Transition and Jet–Eddy Energy Partitioning in Quasi-Geostrophic Turbulence

Abstract

Abstract

1. Introduction

2. A summary of theories to be applied and evaluated

a. Drag-controlled (f-plane) regime

b. β-controlled regime (β scaling)

c. Combined effects

3. Methods

a. Numerical experiments

b. Metrics and model validation

4. Regime transition and a diffusivity formulation

a. Drag-dependent regime transition

b. A simple formulation for eddy diffusivity applicable to 2LQG and 2D turbulence

c. Physical interpretations

5. Utility of the diffusivity formulation

a. A regime diagram and eddy scales

b. A jet speed theory from energy balance

c. Testing the jet speed estimates

d. Jet versus eddy contributions to energy dissipation: A revision of jet dominance hypothesis

6. Discussion

a. A quantification of the zonostrophic regime

b. Implications for the eddy parameterization problem

7. Summary and outlook

Acknowledgments.

Data availability statement.

APPENDIX A

An Approximate Solution of Chen’s (2023) f-Plane Theory for Eddy Diffusivity

APPENDIX B

Applicability to Linear Drag and Two-Dimensional Turbulence

REFERENCES

Share Link

AMS Publications

Get Involved with AMS

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Related Content

Application of a Simple Diffusivity Formulation to Examine Regime Transition and Jet–Eddy Energy Partitioning in Quasi-Geostrophic Turbulence

Abstract

Abstract

1. Introduction

2. A summary of theories to be applied and evaluated

a. Drag-controlled (f-plane) regime

b. β-controlled regime (β scaling)

c. Combined effects

3. Methods

a. Numerical experiments

b. Metrics and model validation

4. Regime transition and a diffusivity formulation

a. Drag-dependent regime transition

b. A simple formulation for eddy diffusivity applicable to 2LQG and 2D turbulence

c. Physical interpretations

5. Utility of the diffusivity formulation

a. A regime diagram and eddy scales

b. A jet speed theory from energy balance

c. Testing the jet speed estimates

d. Jet versus eddy contributions to energy dissipation: A revision of jet dominance hypothesis

6. Discussion

a. A quantification of the zonostrophic regime

b. Implications for the eddy parameterization problem

7. Summary and outlook

Acknowledgments.

Data availability statement.

APPENDIX A

An Approximate Solution of Chen’s (2023) f-Plane Theory for Eddy Diffusivity

APPENDIX B

Applicability to Linear Drag and Two-Dimensional Turbulence

REFERENCES

Share Link

Headings

References

Figures

Metrics

Related Content

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us