## 1. Introduction

Two-layer quasigeostrophic (QG) turbulence has been studied as an idealization for geostrophic eddy field in the atmosphere and ocean (Salmon 1980; Haidvogel and Held 1980; Hua and Haidvogel 1986; Larichev and Held 1995; Held and Larichev 1996; Smith and Vallis 2002; Arbic and Flierl 2004; Thompson and Young 2006, hereafter TY06; Thompson and Young 2007; Chang and Held 2019, hereafter CH19; Gallet and Ferrari 2020, hereafter GF20; Gallet and Ferrari 2021). Compared with that in comprehensive general circulation models, turbulence in such QG flows can be more readily explored because fast waves are filtered out, thereby allowing for efficient numerical integrations and parameter space exploration. Yet, key factors influencing the atmospheric and oceanic eddies are retained: the constraint imposed by rapid rotation and stratification (represented by two vertical modes) for balanced motion are built into QG formulations. In particular, when forced by zonal mean flows with imposed shear—a crude representation for the mean state of a patch of interior ocean and midlatitude atmosphere—turbulent eddies fed by the baroclinically unstable mean flow undergo cascade processes and are ultimately dissipated. Studies of this equilibration have shed insights into the roles of eddies in the climate system. For example, an energy diagram deduced from two-layer QG turbulence by Salmon (1980, 1998) has guided our evolving view on how oceanic mesoscale eddies are energized and the routes to dissipation (e.g., see reviews in Ferrari and Wunsch 2009; Klein et al. 2019). Scaling theories for eddy length and velocity scales in the same turbulent flow have informed parameterizations of oceanic mesoscale eddy fluxes that are unresolved in coarse-resolution climate models (e.g., Griffies et al. 2000; Cessi 2008; Jansen et al. 2015).

It is well established that two-layer QG turbulence at equilibrium exhibits a dual cascade phenomenology (Salmon 1980; Larichev and Held 1995; Scott and Arbic 2007; see Fig. 2 in TY06). On scales much greater than the Rossby deformation radius *λ*, the baroclinic eddy energy is produced by barotropic eddies fluxing heat downgradient (see section 2). The baroclinic energy, which is predominately in a potential energy (PE) form, is transferred to smaller scales in a direct cascade. This direct cascade, however, cannot proceed further beyond the deformation scale because the flows in two layers become decoupled. A majority of the baroclinic energy is thus argued to be converted to barotropic mode and, by an analogy to 2D flow, transfers upscale until this inverse cascade is halted due to the action of bottom drag (e.g., Larichev and Held 1995; Held 1999), *β* effect (e.g., Rhines 1975; Held and Larichev 1996), or their combination.

Under this dual cascade scenario, the eddy heat flux is of fundamental importance because it sets the rate of energy flowing through the system, but it is an internal property, with its dependence on environmental variables unknown a priori. On large scales, Salmon (1980) has shown that the baroclinic potential vorticity behaves as a passive tracer. A main deduction from this is that the baroclinic streamfunction, which is proportional to temperature, is advected passively by barotropic flow. The resulting tracer (i.e., heat) flux may then be regarded as diffusive, provided that there is sufficient scale separation between the mean gradient and the barotropic stirring agent (Held 1999). Therefore, from the perspective of a mixing length theory, a key step toward a quantitative theory for the eddy heat flux requires understanding for the mixing length near which temperature variability is generated by barotropic stirring and for the velocity scale typical of this stirring. Note that given a Kolmogorov spectrum for barotropic inverse cascade (or a spectrum with a steep slope), a bulk of barotropic kinetic energy (KE) would be contained in large eddies (Held and Larichev 1996). A reasonable estimate of the eddy velocity scale responsible for temperature variability is then the root-mean-square velocity of the barotropic flow (see section 2).

However, even under the simplest setting, the two-layer QG turbulence is still complex enough to prevent us from establishing a thorough understanding. The simplest form of two-layer QG turbulence is that on the *f* plane with equal layer thickness. This *f*-plane case is of considerable interest because it represents a limit where bottom drag dominates over the *β* effect in stopping the inverse cascade, and this friction-controlled regime has been suggested to be relevant to midocean and atmospheric eddies (Arbic and Flierl 2004; Arbic and Scott 2008; Held 1999). Yet, for this *f*-plane limit, there is still a lack of consensus on the physical model and scaling theories appropriate for constraining the eddy scale.

Early work by Larichev and Held (1995) and Held (1999) proposed a scaling theory for the largest eddy size (i.e., cascade-halting scale) and barotropic velocity scale, based on the turbulent cascade phenomenology. As will be described further in section 2, the theory (termed the Held model) invoked a three-way balance between the baroclinic energy input, barotropic inverse cascade, and dissipation. The main underlying assumptions are that, first, the eddies are barotropized such that dissipation due to bottom drag can be approximated using barotropic velocity (referred to as the barotropization approximation hereafter); second, existence of an inertial range requires the energy cascade rate to match the input and to be expressed as spectrally local interactions; and third, the halting scale is also the mixing length of heat. Putting the above together led to scaling predictions for the mixing length and barotropic velocity that have a negative power-law relation with a dimensionless quadratic bottom drag parameter [e.g., Eq. (16) in Held 1999]. Subsequent studies, however, showed that the proposed power-law scaling overpredicted the sensitivity to bottom drag (Arbic and Scott 2008; CH19). Quantitatively, CH19 found the power-law exponents for mixing length and velocity to be −0.58 and −0.78, significantly weaker than the Held prediction of −1 power. The apparent discrepancy has motivated CH19 to seek corrections. These authors argued that the energy conversion from baroclinic to barotropic mode is not concentrated at the deformation scale but spreads over a range of wavenumbers. The spread led to a modification of the inertial range assumption by allowing the cascade rate to have a wavenumber dependence, as opposed to being a constant per input. This cascade correction can be incorporated into the Held scaling, but the proposed wavenumber dependence is nevertheless empirical.

Recently, GF20 proposed an alternative scaling for the eddy scales, based on interactions between coherent vortices. This model is referred to as the vortex gas model. Contrasting the spectral space view of turbulence cascade taken by Held et al. the vortex gas model emphasized a physical space view of vortex interactions in heat transport and dissipation. This viewpoint is motivated by the work of TY06, who found emergence of coherent vortices with distinct temperature signatures in two-layer QG turbulence and speculated about their roles in heat transport. GF20 took a step further to hypothesize that a barotropic vortex dipole is a sufficient model to describe the transport and dissipation in the turbulent flow. From dipole simulations, they were able to relate dipole spacing (as the mixing length) and mutual advective velocity with heat fluxes. Applying these relationships then allowed them to deduce scalings for the mixing length and heat diffusivity (see section 2b for details). These alternative scaling predictions were shown to capture the correct drag sensitivity for both quadratic and linear drag, therefore representing a significant improvement over the Held model.

Despite the recent advances, there are aspects of these eddy scale theories that merit further investigations. Both the Held (and its extension like in CH19) and vortex gas models applied the barotropization approximation to express the dissipation in terms of pure barotropic velocity. Although this approximation is likely valid in the asymptotic limit of weak drag, one expects the approximation errors to increase with drag strength because bottom drag preferentially dampens the lower-layer velocity and thus tend to increase the top–bottom velocity differences to oppose barotropization. For example, Jansen et al. (2015) reported that, in their unstable zonal jet simulations, the bottom to barotropic eddy KE ratio drops quite sharply from 1 to 0.2 when a dimensionless drag strength increases from 0.01 to 0.5 (i.e., inverse of the length scale ratio in their Fig. 8). This suggests that, if using the barotropic velocity to approximate the lower-layer value, the dissipation could be overestimated by over a factor of 10 as the drag strength approaches *O*(1) (i.e., dissipation ∝ KE^{3/2} for quadratic drag). It is not entirely clear the extent to which the barotropization approximation is applicable. Also, if we were to apply the eddy scalings to a geophysically relevant setting where the drag is of significant importance (e.g., Held 1999; Arbic and Flierl 2004), it seems necessary to incorporate partial barotropization of the eddies into the existing theories. While empirical parameterizations like Jansen et al. (2015) exist, there is still a lack of a theory that explicitly accounts for eddy partial barotropization. Furthermore, it has been shown by Smith and Vallis (2002) that the Held model cannot be applied to linear drag because the balance constraints become degenerate (see section 2 for details). Yet, numerical experiments in Arbic and Flierl (2004), TY06, and GF20 showed that baroclinic eddies did equilibrate with linear drag. The degeneracy problem in Held-type, turbulent cascade models remains largely unresolved.

The objectives of this study are to propose modifications to the Held model and to examine their effects on the response of eddy scales to varied bottom drag strength. We choose to build upon the Held model for its relatively straightforward extension, but quantitative comparisons with the vortex gas scaling are also carried out. The target eddy scales are the mixing length and rms barotropic velocity, which combines to characterize the diffusive eddy heat transport. The modifications account for eddy partial barotropization, to relax the commonly invoked barotropization approximation, and consider a drag-dependent cascade rate per energy input, to correct for the lack of an inertial range as in CH19. The proposed theory is tested against simulations of *f*-plane, two-layer QG turbulence where the halting of inverse cascade by quadratic and linear drag are considered. We aim to show that incorporation of partial barotropization allows the Held scaling to capture the progressively weakened sensitivity in eddy scales to increased drag strength. It also partly resolves the degeneracy problem. Overall, the proposed theory enables the eddy scale predictions to be extended to *O*(1) quadratic drag and has skill comparable to the vortex gas scaling for linear drag. However, toward the weak linear drag limit, the proposed theory breaks down but the vortex gas scaling performs well. Properties of vortex-pair transport and structure in this limit are explored.

This work is organized as follows. Section 2 reviews the Held and vortex gas models to be examined in this study. Section 3 describes the numerical experiments, validations, and eddy scale diagnostics. In section 4, errors associated with the barotropization approximation and inertial range assumption are identified. Corrections for them are then incorporated into the Held model and tested against simulations with both quadratic and linear drag. Spectral energy budgets are also analyzed to support the proposed cascade correction. In section 5, quantitative comparisons with the vortex gas scaling are presented. A case in the weak linear drag limit where the proposed theory is unable to apply is examined. Finally, section 6 provides a summary of the main findings. Implications and a number of unresolved issues are discussed.

## 2. Scaling theories

*V*that set the diffusivity

*D*of eddy heat transport (i.e.,

*ε*comes from a release of mean flow potential energy by downgradient eddy heat fluxes:

_{p}*ψ*is the barotropic meridional velocity, the baroclinic streamfunction

_{x}*τ*is proportional to temperature, and the zonal mean flow magnitude

*U*represents the background temperature gradient (i.e.,

*ε*) and the other due to hyperviscous term (hyper). Here the bottom drag is expressed using the quadratic law, with a quadratic drag coefficient

*μ*.

^{8}operator is designed to operate at small scales [see Eq. (A9)]. The energy dissipation due to the latter should thus be relatively small, provided that the value of hyperviscosity

*υ*is small enough. The second assumption is barotropization, which approximates the lower-layer velocity by the barotropic velocity. That is,

*O*(0.1). Using these two assumptions, a simplified energy balance becomes

### a. Held’s turbulent cascade model

The Held model is reviewed thoroughly by CH19. Here we only focus on the parts relevant to this study.

*k*

_{0}may be interpreted simply as the peak wavenumber of a barotropic energy spectrum, and a barotropic eddy velocity. Assuming that

*k*

_{0}also sets the mixing length of heat (i.e.,

*V*(see section 3 for precise definitions for

*V*), one can scale the energy production as

*ε*

_{bt}∼

*μV*

^{3}. The above then allows the simplified energy balance (2) to be expressed in terms of two unknowns of

*V*. For a second balance condition, the existence of an inertial range requires the rate of barotropic inverse cascade

*ε*to match the dissipation/production. i.e.,

_{c}*ε*∼

_{c}*ε*≈

*ε*. In the inertial range, dimensional consideration yields a cascade rate scaling of

_{p}*ε*∼

_{c}*V*

^{3}

*k*

_{0}(e.g., Smith and Vallis 2002; CH19). Combining the above leads to a closed set of equations

*T*(

_{e}*T*

_{Eady}(≡

*λ*/

*U*), and

*T*is approximated by the ratio of mixing length to the rms barotropic velocity. Because

_{e}*T*

_{Eady}is an imposed mean-flow quantity, the mixing length and eddy velocity must then grow in proportion to maintain a constant ratio. From (3), one can easily obtain the following eddy scale predictions,

*μλ*) is a dimensionless parameter gauging the strength of quadratic bottom drag.

It has been known that the Held scaling cannot be applied to linear drag because the two balances in (3) reduce to the same functional form (e.g., Smith and Vallis 2002). This is referred to as the degeneracy problem in this study. For linear drag, the dissipation under the barotropization approximation scales as *ε*_{bt} ∼ *κV*^{2}. Substituting it into the energy balance yields *κλ*/*U*) is the dimensionless drag strength defined using a linear drag coefficient *κ*. Note that this form of energy balance is identical to the cascade rate constraint [second equation in (3)], thereby causing the degeneracy (see section 4d).

### b. GF20 vortex gas model

GF20 proposed a conceptual model that treats the barotropic turbulent flow as a collection of dilute vortices, thus referred to as vortex gas. These authors hypothesize that a barotropic vortex dipole is a sufficient model to capture the turbulent transport properties. The heat transport and energy dissipation in (2) are then modeled as a result of mutual advection of a dipole over a background temperature gradient.

*V*

_{iv}and

*V*

_{iv}at radius

*r*equal to the intervortex distance

*D*scales as

*r*

^{−1}profile to preserve the circulation like a point vortex [i.e.,

**u**

_{bt}|

^{3}〉 in (2) does not simply scale as

*D*

_{GF}/(

*Uλ*) will then be the same as those in the Held model. Therefore, considering the

*r*

^{−1}vortex velocity profile has a key effect in weakening the drag sensitivity of the eddy scaling (i.e.,

*V*

_{GF}is the rms barotropic velocity, not the vortex mutual advective velocity

*V*

_{iv}at

## 3. Methods

### a. Numerical experiments

The design of numerical experiments largely follows GF20, but the parameter range is extended beyond *O*(1) quadratic and linear drag. To validate the numerical implementations, we first repeat GF20’s experiments using the same parameters. New sets of experiments with lower hyperviscosity are then carried out, as it is found that the hyperviscosity used by GF20 yields too large of hyperviscous dissipation. The new sets with an extended drag range are then used to evaluate the theory proposed in this study. Details are as follows.

The standard two-layer quasigeostrophic potential vorticity (QGPV) equations, with equal layer thickness and imposed mean shear on an *f* plane, are solved using a spectral PDE solver Dedalus (Burns et al. 2020). The equation sets, including the nondimensionalization and parameterizations of bottom drag and hyperviscosity, are identical to those in GF20. A summary is given in appendix A [Eqs. (A1)–(A3), (A11), and (A12)]. The simulations are carried out in a 2*πL* × 2*πL*, doubly periodic square domain. The basic state is a purely zonal mean flow of +*U* and −*U* for the upper and lower layer, respectively. The imposed shear is thermal wind balanced by an interface tilt which provides the source of potential energy for the eddy field.

The experiments are designed to study the responses of eddy scales to varied bottom drag strength. In the case of quadratic bottom drag, the response is governed by three dimensionless parameters, *λ*/*L*, *μℓ*), and *UL*^{7}), where *ν* is the hyperviscosity [see Eqs. (A11) and (A12)]. These parameters characterize the domain size relative to the deformation radius *λ*, bottom drag strength, and the rate at which enstrophy is removed at small scales. With a sufficiently small value of *λ*/*L* so that the equilibrated eddy size is not limited by domain confinement, we are left with only one dynamically relevant parameter in

The experiments are summarized in Table 1. The setting all uses *U* = 1 and *L* = 1 in a 512^{2} horizontal grid unless otherwise noted. The set “GF” is the repeat experiments that use *λ*/*L* = 0.02 (i.e., L002), ^{−17}, much lower than GF20’s, is found (see section 3c below). Setting *λ*/*L* but has ^{−3}–10 for quadratic drag and ^{−1}–2.5. The eddy scales derived from the combined experiments then provide the benchmark for the proposed theory to test against (see sections 4 and 5). Finally, the BT drag 1 and 2 experiments have pure barotropic velocity for bottom drag in the QGPV equations. They are designed to illustrate the effects of complete barotropization. Details are described in the analyses.

A summary of numerical experiments designing to investigate the responses of eddy scales to varied drag strength (

### b. Eddy scale definitions and model validation

*V*, and the diffusivity

*D*are diagnosed directly from simulations using the definitions of

*ψ*and

*τ*are the barotropic and baroclinic streamfunctions, respectively, the subscript denotes partial derivative, and the angle bracket denotes domain average (see appendix A for details). These quantities will be used for the theories to compare against. Given that the eddy velocity in

*f*-plane cases is horizontally isotropic (e.g., Larichev and Held 1995), the square of rms meridional velocity

*V*

^{2}in (7) is equivalent to the barotropic energy level that is expected to be mostly contained in large scales near the mixing length. Noting again that

*U*represents the background temperature gradient, the definitions of

*D*are consistent with mixing length and downgradient heat diffusion:

*τ*

^{2}) is generated via stirring of background gradient, whereas

*D*is from the gradient transport formulation (TY06).

As a validation of the QG simulations, we compare the calculations of mixing length and diffusivity in (7) against the values reported by GF20. In Figs. 1a and 1b, the dimensionless *D* are shown for different bottom drag strength

Since our focus is to examine the barotropization approximation across different drag strength, that is, approximating *ε* by *ε*_{bt} in (2) for different *ε*_{bt}; orange) overestimates the true dissipation by more than a factor of 1.5 (see more in section 4). The accuracy of dissipation estimates is also confirmed for other cases. We evaluate the steady-state energy balance by averaging the budget terms over the last 200 dimensionless time units. This is equivalent to 200 *T*_{Eady}, where *T*_{Eady} ≡ *λ*/*U*. In Fig. 1b, the steady-state values of −*ε*/(*ε _{p}* + hyper) are close to one for all cases (cross symbols), again suggesting a closed three-way balance in (1) and supporting the accuracy of the dissipation estimates.

### c. Reducing hyperviscosity

A problem revealed by the energy budget in Fig. 1c is that the hyperviscous dissipation is not negligible in the GF cases. As described in section 2, both vortex gas and Held models assume hyperviscosity to be negligible as compared with bottom drag in (1). This limit is not yet reached with the value of ^{−17} a suitable choice. In Table 2, the ratio of hyperviscous to total dissipation [i.e., second and third terms in Eq. (1)] decreases from 0.25 to 0.026 as ^{−17}. The mixing length and barotropic eddy velocity, both normalized by the values with

Sensitivity tests aiming to find suitable value of *ε* + hyper)]. These tests have

Quite surprisingly, though, the error of neglecting hyperviscosity has a minor effect on the eddy scales. In Figs. 1a and 1b, when we use *ε _{p}* and bottom drag dissipation

*ε*at the steady state for all cases.

## 4. Partial barotropization and modifications of held scaling

With the diagnostics laid out in sections 2 and 3 and the experiments that satisfy *ε _{p}* ≈

*ε*at a steady state, we are in a position to explore the scaling theories further. Below we will illustrate the problem of barotropization approximation and modify the Held model by allowing for partial barotropization. An additional correction of drag-dependent cascade rate per energy input and application to linear drag will also be considered.

### a. Problem of the barotropization approximation

We first illustrate the problem of the barotropization approximation by comparing the ratio of true and approximate dissipation among the L002 cases. The true dissipation defined in (1) is *ε*_{bt}/*ε* to be roughly a constant among different cases.

In Fig. 2a, the dissipation ratio *ε*_{bt}/*ε* is plotted against *O*(0.1) drag with *ε* by a factor of 2.2 and 3.7, respectively. Note that this strong *ε* ∼ *ε*_{bt} in (3) and (5).

Qualitatively, the increase of *ε*_{bt}/*ε* with drag strength can be understood as a result of partial barotropization. The ratio *ε*_{bt}/*ε* is controlled by the differences between barotropic velocity |**u**_{bt}| and lower-layer velocity |**u**_{2}| (see more in section 4c below). As bottom drag increases, the lower-layer velocity is preferentially damped so that the difference between |**u**_{bt}| and |**u**_{2}| increases. In other words, eddies become increasingly partial barotropized (i.e., more baroclinic), which reflects in an increase in *ε*_{bt}/*ε*.

We may estimate the influence of barotropization error on the scaling theories. The barotropization error refers to the degree of *ε*_{bt}/*ε* deviated from one. For simplicity, we represent the *n* is obtained by power-law fits in Fig. 2a. For the Held scaling, when combining *ε* ≈ *ε _{p}* in (1) and

*ε*

_{bt}∼

*μV*

^{3}and

*n*= 0, and the scalings revert back to the Held and GF20 result in (4) and (6), respectively. In Fig. 2a, we try two separate power-law fits (dashed curves). For two drag ranges of

*n*= 0.14 and 0.36. This simple exercise then suggests that the error of making the barotropization approximation can lead to a 14% to 36% reduction in the scaling slope, with an average of 25% if the fit is for the whole range

### b. Checking other assumptions in the Held model

In additional to the barotropization approximation, the Held model also invoked a diffusive representation for the baroclinic energy production *ε _{p}* and an assumption of inertial range for barotropic inverse cascade. These assumptions are examined as well.

In (3), the heat flux 〈*ψ _{x}τ*〉 in the baroclinic energy production

*ε*is expressed as downgradient diffusive transport, with a diffusivity

_{p}*λ*,

*τ*is passively stirred by barotropic flows. The tracer fluxes could thus be regarded as diffusive (Held 1999).

*ε*∼

_{c}*ε*in (3). Together with the assumption that the cascade-halting scale also sets the mixing length (

_{p}*ε*/

_{c}*ε*was assumed to be a constant, yielding

_{p}*ε*per input. When the drag strength increases, the mixing length

_{c}*ε*/

_{c}*ε*(see more below).

_{p}### c. Proposed corrections and eddy scale predictions

#### 1) Partial barotropization

*ε*in terms of rms barotropic velocity

*V*is needed. We can express the dissipation using the rms, lower-layer meridional velocity

*υ*

_{2}as

^{3/2}comes from the assumption of horizontal isotropy, whereas the factor of 1.2 is to relate

*υ*

_{2}magnitude cubed with the cube of rms

*υ*

_{2}. It can be shown that, for a simple sine wave, the factor is equal to (4/3

*π*)2

^{3/2}= 1.2. Calculating the factor directly in the experiments yields a value of around 1.3. For the purpose of developing an approximate theory, we simply use a constant value of 1.2.

*υ*

_{2}in (9) may be written in terms of

*V*using the standard modal decomposition and its spectral properties. For equal-thickness layers without approximation, we have

*λ*to be barotropized (e.g., Charney 1971) and the barotropic energy spectrum to peak strongly at the halting wavenumber

*k*

_{0}(i.e.,

*υ*

_{bt}

*υ*

_{bc}〉 by the covariance near

*k*

_{0}, with

*k*

_{0}(e.g., Smith and Vallis 2002). We may further express

*k*

_{0}take on the value of

*V*and

*U*, respectively. Note that scaling

*k*

_{0}. On the other hand, the scaling relation of

*VU*), the dissipation is equivalent to

*ε*

_{bt}in (2) which has been shown to increasingly overestimate the true dissipation

*ε*as the drag strength increases (i.e.,

*ε*

_{bt}/

*ε*in Fig. 2a). With the correction, the dissipation estimates improve significantly. The corrected ratio

*ε*

_{corr}/

*ε*as denoted by diamond symbols in Fig. 2a is now mostly near 1, suggesting that the biases are largely removed.

While the above correction has significantly improved the dissipation estimates, its performance degrades as *υ*_{bt}*υ*_{bc}〉 by only the covariance near *k*_{0}. Note that the scaling of *λ*. Such a condition is most likely violated toward *λ* (see below). Nevertheless, we consider the correction in (10) a starting point to account for partial barotropization. Further improvements are left for future works.

*ε*with the corrected dissipation

_{p}*ε*

_{corr}and invoking the cascade rate constraint [i.e.,

*ε*/

_{c}*ε*= constant in (8)] result in a closed set of equations for

_{p}*V*and

*c*

_{0}and

*c*

_{1}for setting the magnitude of energy production and eddy turnover time, respectively. These constants are determined from the experiments. Once they are set, the solution of (12) gives

*V*and

*c*

_{0}= 0.31 from Fig. 2b and

*c*

_{1}= 1.2 as an averaged time scale ratio in Fig. 2c. Note that these constants are relatively well constrained. TY06 reported

*c*

_{0}= 0.25 in their experiments (see their Fig. 7), generally consistent with our value. It will be shown below that these constants also carry over to linear drag cases.

*V*made dimensionless by the mean flow

*U*

*V*/

*U*is found, we can obtain

In Fig. 3, the eddy scales predicted from (12) are tested against the simulations. It can be seen that the predicted rms barotropic velocity (gray curve in Fig. 5a) is in good agreement with the simulations (filled symbols). Compared with the Held scaling (gray line), adding the barotropization correction reduces the oversensitivity to bottom drag, allowing the eddy velocity predictions to correspond better with the simulations. This effect is particularly pronounced at large drag (e.g., *ε*_{bt}/*ε* > 2 in Fig. 2a), and the correction reduces the steep drop in *V*/*U*.

Similar results are found for the mixing length

Since the diffusivity is well described by the mixing length scaling (i.e.,

A tentative conclusion we may draw from the comparisons in Fig. 3 is that the weakening of drag sensitivity toward large drag is largely attributable to a decrease in the degree of barotropization (i.e., *ε*_{bt}/*ε* increases with *ε*_{bt}/*ε* has a value of 1.06 and 1.24, not too far from one. The correction for partial barotropization in (12a) then becomes less important [i.e.,

However, correcting for partial barotropization alone is clearly not enough to adequately represent the eddy scales. We have seen in Fig. 3b that the prediction from (12) still overestimates the drag sensitivity of the mixing length. It also quite severely overpredicts the diffusivity toward weak drag. Note that, under the inertial range assumption *ε _{c}* ∼

*ε*,

_{p}*V*/

*U*and

*c*

_{1}in (12b). Yet, simulations do suggest the two to scale differently, with

*ε*/

_{c}*ε*is allowed to vary with

_{p}#### 2) Adding a cascade correction

*c*

_{1}= 1.2) and the Held model [Eq. (3)]. To make a simple empirical correction, we fit the data in Fig. 2c with a power law to obtain

*c*

_{2}= 2 and

*m*= 1/7. Physically, we may interpret this positive

*λ*where the baroclinic energy is converted to barotropic mode in Salmon’s (1980) energy diagram and Held’s frictional arrest scale

*μ*

^{−1}near which the barotropic inverse cascade is stopped by bottom drag [i.e.,

*ε*/

_{c}*ε*). Analyses of spectral energy fluxes to be shown in the following subsection will provide further support for this drag-dependent cascade fraction.

_{p}*V*/

*U*

*m*= 0, this equation is reduced to the same functional form as (13). We can again solve (15) numerically to find

*V*/

*U*and obtain

Adding a cascade rate correction improves the mixing length prediction significantly. The updated prediction, shown as the black curve in Fig. 3b, is clearly in better agreement with the simulations than that without the cascade correction (gray curve). The improvement in the mixing length prediction also yields a better agreement for the diffusivity, especially toward the weak drag end.

If one were to apply only the cascade correction without considering partial barotropization (in the same spirit as CH19), one can obtain a scaling by combining (14) with Held’s energy balance, *ε _{p}* ≈

*ε*

_{bt}, in (3). This leads to relations of

*O*(0.1) or smaller.

In section 4d, we will apply both the partial barotropization and cascade rate correction to linear drag cases. Quantitative comparisons with the vortex gas scaling will be delayed to section 5 where the quadratic and linear drag cases can be discussed together.

#### 3) Cascade from spectral energy budgets

The empirical relation of *ε _{c}*/

*ε*) when the drag strength increases, which manifests as an increase in eddy turnover time (i.e., less efficient eddy–eddy interactions to drive cascade). To check this interpretation, we compare the spectral barotropic energy budgets among the L002 cases. The budget calculations and notations follow Larichev and Held [1995, their Eq. (5a)], and the formulation is written out in (A15). An example with

_{p}Following Scott and Wang (2005), we integrate the flux convergence (term I) from a wavenumber *K* to infinity to obtain the barotropic flux at that wavenumber Π(*K*) [see (A16)]. This barotropic flux is normalized by the domain-averaged production [i.e., *K*)/*ε _{p}*, is shown in Fig. 4b for different cases. Taking the

*K*= 1), confirming that the barotropic flux only redistributes the energy across scales. For

*K*greater than 20, the net energy input (i.e., blue curve in Fig. 4a) requires a diverging flux (i.e., black curve; I = −∂Π/∂

*K*< 0) to move the excessive input to lower wavenumbers. The converging flux at the low-wavenumber range (I > 0 for

*K*< 20) then supports the net energy sink (red curve).

*ε*shows no extended wavenumber range where the normalized flux is near −1, contrary to the expectation for an inertial range. However, the magnitude of Π/

_{p}*ε*does increase with decreasing drag strength (filled circles in Fig. 4c), approximately following a power-law relation of

_{p}_{max}/

*ε*.

_{p}Importantly, the power law (16) obtained from direct quantification of barotropic flux is consistent with the empirical relation *ε _{c}* represents the magnitude of barotropic flux, we can substitute (16) into (8) to find

*m*(= 1/7) obtained from the fit in (14). The consistency therefore supports the use of (14) to represent the variable barotropic flux per input when the inertial range is not present. Limitations of this empirical approach will be discussed further in section 6c.

### d. Applying to linear drag

*ε*=

*κ*〈|

**u**

_{2}|

^{2}〉. Note here that the bottom stresses are parameterized as 2

*κ|*

**u**

_{2}

*|*

**u**

_{2}, following the formulation and notations in GF20. Assuming horizontal isotropy and applying the correction in (10) give

*V*/

*U*,

*V*/

*U*as

*m*= 0, and Eqs. (20) and (21) revert back to the same form of (18) and (19), respectively. To obtain the eddy scale predictions below, we take

*c*

_{0}= 0.29,

*c*

_{1}= 1.2, and

*c*

_{2}= 1.9, following the same procedures as in Fig. 2. These coefficients are nearly identical to the quadratic drag cases. The only notable difference is the power-law exponent

*m*. The linear drag cases have

*T*/

_{e}*T*

_{Eady}that is more sensitive to drag strength (

*m*= 0.26 versus 1/7 for quadratic drag). This is due mainly to the fact that, to obtain a comparable dynamical range of inverse cascade, one needs to vary the quadratic drag more. For example, at

*V*/

*U*takes a value of around 22. This energy level is roughly comparable to

*V*/

*U*of 17 at

*T*/

_{e}*T*

_{Eady}(or equivalently

*ε*/

_{c}*ε*) ought to be more sensitive to changes in linear drag strength.

_{p}Like Fig. 3, the eddy scales derived from simulations and predictions are compared in Fig. 5 for linear drag. The predictions perform reasonably well, except for the lowest drag case (*V*/*U* and the weakened drag sensitivity as *O*(1) and beyond (gray curve in Fig. 5a). But, the predicted

Why does the above theory fail for *ε*_{bt}/*ε* are labeled in Fig. 5a for a number of cases. The *ε*_{bt}/*ε*= 1.01, meaning that using the barotropic velocity to estimate dissipation has a negligible error of only 1%. Neglecting the correction for partial barotropization (−2*VU*) in (17), the energy balance *ε _{p}* =

*ε*becomes

The weak-drag asymptote aside, the above analyses show that the proposed corrections can add significantly to the original Held model. The incorporation of partial barotropization via (17) helps resolve the degeneracy problem for linear bottom drag. When the partial barotropization and cascade corrections are considered together, there is reasonably good agreement between the predicted and simulated eddy scales. Note that the predictions are obtained using model coefficients (i.e., *c*_{0}, *c*_{1}, *c*_{2}) that are nearly identical to the quadratic drag cases. These are encouraging signs for the robustness of the proposed theory.

## 5. Comparisons with vortex gas scaling

We have seen in section 4 that, for both quadratic and linear drag, the Held scaling can be augmented by considering partial barotropization and drag-dependent cascade rate per input. By comparison, the vortex gas scaling proposed by GF20 also represents a significant advance over the Held model (e.g., Figs. 1a,b). In particular, the vortex gas does not suffer from the degeneracy problem we face for weak linear drag (e.g.,

### a. Root-mean-square error

We first compare the root-mean-square error (RMSE) between the proposed theory and vortex gas scaling. The errors are calculated as misfits against simulations for log10(variable) and are summarized in Table 3. For example, a value of 0.1 may be interpreted as an average drift by a factor of 10^{0.1} ≈ 1.26 from simulations. The vortex gas predictions are obtained using (6) for quadratic drag and GF20’s Eqs. (15) and (16) for linear drag, with scaling coefficients determined by best fits. The exact formulations are given in Table 3. In general, the scaling coefficients agree with GF20’s values to within 10%.

Summary of root-mean-square errors (RMSEs) when the proposed theory and vortex gas scaling are compared against simulations. The RMSE is calculated for log_{10}(variable). Full range refers to *c*_{0}, *c*_{2}, and *m*, are determined from the experiments, with *c*_{0} = 0.31 (0.29), *c*_{2} = 2.0 (1.9), and *m* = 1/7 (0.26) for quadratic (linear) drag.

For quadratic drag, we separate the comparisons into two ranges: *V*/*U* and diffusivity, having RMSEs that are smaller by around a factor of 3. This result is consistent with Fig. 3. The proposed theory (thick black curves) and vortex gas (black dashed lines) are close to each other. But the proposed theory does capture the increased *V*/*U* and *O*(1) drag and beyond (i.e., black and gray curves). Note that the better performance of the proposed theory is not surprising, as the vortex gas scaling is strictly valid only for low-drag conditions. Nevertheless, the agreement in Table 3 and Fig. 3 demonstrates that the proposed theory can add to the existing models by extending the eddy scale predictions to *O*(1) drag.

For linear drag, the comparison results are not entirely consistent with quadratic drag. We first note that Eq. (21) cannot be applied over the full drag range because the predicted eddy velocity becomes negative when *O*(1) drag. From Table 3 for *O*(1) drag is intriguing due to the presence of the barotropization errors. Like in the quadratic drag cases, invoking the barotropization approximation leads to large overestimations of dissipation for *O*(1) drag, e.g., *ε*_{bt}/*ε* = 3.7 and 9.1 for *O*(1) *O*(1)

Focusing back to the low-drag condition, an obvious advantage of the vortex gas scaling over the proposed theory is that it remains applicable toward the weak-drag asymptote: in Fig. 5, the vortex gas describes the eddy scales well at

### b. Vortex gas properties at weak linear drag

One possible interpretation for the superior performance of vortex gas at

As reviewed in section 2, the vortex gas model was constructed based on two main assumptions: 1) a vortex dipole is sufficient to represent the bulk property of turbulent transport (see GF20’s Fig. 2) and 2) the vortex velocity scales with *r*^{−1} like a point vortex. The latter is crucial because it leads to enhanced dissipation,

Overall, the results show no clear indication that eddy transport is controlled by vortex interactions at *r*^{−1} structure. We average the velocity of all identified vortices to obtain composite-mean profiles (see Fig. B2c). Within around *r*^{−1} slope, tending closer to the *r*^{−0.5} slope instead (see Fig. B2d). It is then argued that a flatter profile could significantly change the scaling behavior of the vortex gas (see appendix B). Therefore, it remains unclear to us what the appropriate physical model would be for the weak linear-drag asymptote. This issue will be discussed in section 6c.

## 6. Summary and discussion

In this study, modifications of the Held scaling (Held 1999), which allows energy-containing eddies to be partially barotropized and the cascade rate per input to vary with drag, are proposed. The proposed theory is applied to forced-dissipative simulations of *f*-plane, two-layer QG turbulence where bottom drag provides the main mechanism to stop the inverse cascade. The target eddy scales are the mixing length *V*, which combines to characterize the meridional heat transport via a diffusivity *O*(1) drag conditions.

The main motivations are that 1) the Held scaling is known to be oversensitive to the quadratic drag strength *O*(0.1) drag. The use of pure barotropic velocity can overestimate the dissipation by more than a factor of 2 for

When compared with the Held scaling, incorporation of partial bartropization allows capturing the progressively weakened sensitivity to increased quadratic drag (Fig. 3). It also partly resolves the degeneracy problem: the same correction is applicable to linear drag, except for a case in the asymptotic weak-drag limit where nearly complete barotropization of the eddies renders the correction not useful (Fig. 5). This limiting case aside, considering partial barotropization alone leads to good predictions of eddy velocity *V*/*U*, although not of mixing length *D*/(*Uλ*) (Figs. 3 and 5). Representations for the mixing length and diffusivity are improved when an additional correction for a drag-dependent cascade rate per energy input is considered. Spectral balance of the barotropic energy shows that this drag dependence can be understood as a change in spectral room for the barotropic inverse cascade (Fig. 4). When drag increases, there is increased overlap between the energy input and dissipation spectra, which reduces the required cascade fraction and manifests as increased eddy turnover time relative to the Eady time scale (Fig. 2c).

When compared with the vortex gas scaling, the theory with proposed corrections can extend the eddy scale predictions to *O*(1) quadratic drag and generally has comparable skills for linear drag (Figs. 3 and 5 and Table 3). However, the vortex gas scaling has an advantage in that it remains applicable in the limit of weak linear drag (i.e., dashed and solid curves diverge for

The analyses described in this study have a number of implications and unresolved issues that warrant further discussion:

### a. The degree of eddy barotropization

*ε*

_{bt}/

*ε*is equivalent to a decrease in KE

_{bot}/KE

_{bt}, where the ratio of lower-layer kinetic energy (KE

_{bot}) to barotropic KE serves as a measure for the degree of barotropization (i.e., a ratio of 1 means complete barotropization). The theory described in section 4c allows for a theoretical estimate for the KE ratio. Using (10), the KE ratio is

Quite surprisingly, the KE ratios in Fig. 6a are in overall agreement with prior primitive equation simulations. Jansen et al. (2015) studied the properties of baroclinic eddies spawned from unstable surface-intensified zonal jets on a *β* plane. The KE ratios in response to varied quadratic drag were reported. Noting that our dimensionless drag strength *L _{d}*/

*L*(see their Table 1), we overlay their results onto Fig. 6a as open symbols. Despite the apparent differences in the mean-flow settings (e.g., presence of

_{f}*β*, mean-flow shear), the KE ratios from the two separate studies show general similarity. In particular, their

*f*-plane cases (red circles), which are more comparable to our setting, have KE ratios agreeing closely with ours. An empirical fit for the whole experiments used by Jansen et al. (2015) [red dashed curve; their Eqs. (12) and (13)] also lay close to the theoretical curves. On the one hand, the general agreement suggests the potential use of the theory for eddy parameterization: The need of an empirical fit to constrain the near-bottom eddy velocity like in Jansen et al. (2015) may by relaxed by using (22) or (10). Yet, on the other hand, the agreement needs explanations. According to Charney (1971), the degree of barotropization is tied with the extent of inverse cascade. It is thus anticipated that, in the case of strong

*β*and weak drag, the cascade and hence barotropization may be strongly limited by the

*β*effect (e.g., Rhines 1975). However, from Fig. 6a, the presence of

*β*appears to exert a secondary control on the vertical structure when compared with surface friction. It is possible that the observed behavior is specific to the parameters considered by Jansen et al. (2015). Nevertheless, the generic behavior of friction versus

*β*control on the eddy vertical structure is a subject that merits further investigations.

The analyses in section 4 suggest that the progressive weakening of drag sensitivity in the eddy scales is largely attributable to a decrease in barotropization. It is shown in Figs. 3a and 3b that, by accounting for partial barotropization via (10), the Held scaling as indicated by the straight lines in the log–log plots becomes concave upward toward large drag (i.e., gray curve for **u**_{2}| in Eq. (1) by |**u**_{bt}|; see appendix A]. When the effect of partial barotropization is removed, the concave upward shape in the *V*/*U* versus

Why is the weakening of drag sensitivity more pronounced toward large drag? Toward large drag, both limited spectral room and strong dissipation weaken the barotropic velocity. As *V* approaches *U* (e.g., *V*/*U* < 5.5 for *ε* as compared with *ε*_{bt} in Eq. (12a)]. This effect “shields” the barotropic flow from feeling the drag changes (e.g., Jansen et al. 2015), manifested in the weakened drag sensitivity.

### b. Relevance of O(1) drag regime

The comparisons in Figs. 3 and 5 have shown that, by simply incorporating partial barotropization through (11) and (17), the predictability for barotropic eddy velocity is extended to *O*(1) drag and beyond. With an additional correction for the cascade rate, the prediction for mixing length over the same drag range is also quite good. Being able to cover the *O*(1) drag regime is important because it is likely a relevant regime for atmospheric and oceanic eddies (Held 1999; Arbic and Scott 2008). Following Held (1999), the quadratic drag parameter may be casted in terms of background static stability *N* via *C _{D}* = 10

^{−3}–10

^{−2}and

*N*of 1–2 × 10

^{−2}(s

^{−1}) for both oceanic and atmospheric basic flows (i.e., oceanic value is likely toward larger end because of stronger stratification), a typical

*ε*

_{bt}greatly overestimates the true dissipation

*ε*. This is also where the eddy scales show progressively weakened drag sensitivity—a feature both vortex gas and Held scalings unable to represent due to their employment of the barotropization assumption. The modifications proposed here therefore enable studies of baroclinic eddies in a dynamically relevant parameter range (albeit the apparent need to include the

*β*effect).

We may convert the above estimate of drag range to linear drag. Following CH19, we define an effective linear drag in terms of the quadratic drag coefficient via *κ* = *μ*|**u**_{2}|. The conversion between *υ*_{2}|, and via (10), the conversion becomes *V*/*U* ≈ 5.4 and 2.25 for *ε*_{bt}/*ε* for

### c. Unresolved issues

There are a number of unresolved issues raised in this study. The most apparent one is the large discrepancy toward the limit of weak linear drag: the proposed theory (black and gray curves in Fig. 5) cannot represent the eddy scales for

It was pointed out by Smith and Vallis (2002) and demonstrated in section 4d that the Held scaling cannot be applied to linear drag because the balance constraints are degenerate. This has led Smith and Vallis (2002) to argue that linear drag alone cannot stop the inverse cascade. Such an argument may be understood from the perspective of scale dependency in the drag form. Without *β*, inverse cascade may be arrested at scales near which the damping rate exceeds eddy turnover rate that sets the cross-scale energy transfer. In the Held scaling, the turnover rate is an imposed quantity via (3) and is thus held fixed as the barotropic energy is transferred and accumulated toward large scales (i.e., *R* scales as *R*_{quadratic} ∼ *μV* and *R*_{linear} ∼ *κ* for quadratic and linear drag. Evidently, *R*_{quadratic} can increase as cascade proceeds to increase both *V* and *R*_{linear} is scale independent, thereby unable to slow the cross-scale energy transfer.

The inclusion of partial barotropization changes the above picture. With the correction in (17), the damping rate for linear drag becomes *R*_{linear,corr} ∼ *κ*(1 − 2*U*/*V*). It has scale dependency analogous to *R*_{quadratic}. As *V*/*U* increases, *R*_{linear,corr} also increases because eddies become more barotropized (i.e., smaller −2*U*/*V* correction) and thus are more efficiently dissipated by surface friction. This effect could then provide the needed scale-dependent damping to stop the inverse cascade, thereby avoiding the degeneracy problem in the Held scaling.

However, the scale dependency induced by partial barotropization would eventually vanish toward the weak-drag limit. As *V*/*U*)^{2} would continue to rise, causing the correction to ultimately become negligible. We can see in Fig. 5a that, *V*/*U* ≈ 150 at *R*_{linear,corr} ≈ *R*_{linear} ∼ *κ*. Then, as argued by Smith and Vallis (2002), the inverse cascade at

Physically speaking, the above interpretation has attributed the discrepancy to the possibility that the theory lacks representations of certain cascade-stopping mechanisms. At

Following the above reasoning, if we are to proceed with a turbulent cascade model like the one proposed here, we will need to face a lingering question: What are the missing cascade-stopping mechanisms that may help explain the data-theory discrepancy at

As emphasized above, the evidence we provide for domain limitation is only suggestive. To properly evaluate this effect, one will need to conduct additional experiments by increasing the domain size (i.e., expanding the range of *λ*/*L*) and further reducing the drag strength. It can be anticipated that, by removing the domain size limitation, both barotropic energy and mixing length in the simulation would be allowed to grow, thereby reducing the difference between numerical data and predictions in Fig. 5. However, given the steepness of the prediction curves for

Finally, the consideration of drag dependency in the eddy turnover time in (14) is largely empirical. The main idea there is to incorporate a variable cascade fraction (i.e., |Π|_{max}/*ε _{p}* as represented by

*ε*/

_{c}*ε*) for the absence of an inertial range (see Fig. 4). By recognizing

_{p}_{max}/

*ε*) and the separation between the input and dissipation spectra (in Figs. 4b,c). Note that the incorporation of drag-dependent cascade rate via (14) is in the same spirit as the approach adopted by CH19. In their work, it was argued that the barotropic flux should vary with wavenumbers. This is because, unlike the classic Salmon’s (1980) diagram, the energy input to barotropic mode spreads out in wavenumber space such that the upscale energy transfer can pick up local conversion. This spreading of energy conversion can be clearly seen from term II in Fig. 4a. Based on this argument, CH19 obtained

_{p}*x*= 0.72 from empirical fits and the scaling for mixing length [their Eq. (16)], one arrives at

_{max}/

*ε*, would exceed one and hence become unphysical. From Fig. 4c, the cascade fraction has not reached this limit at

_{p}## Acknowledgments.

Two anonymous reviewers provided insightful comments that improves this work substantially. Interpretations of the degeneracy problem in terms of scale-independent damping was suggested by one of the reviewers. Discussion with Chiung-Yin Chang (Princeton University) was helpful. This work is supported by the Ministry of Science and Technology of Taiwan through Grant MOST 108-2611-M-002-022-MY4.

## Data availability statement.

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

## APPENDIX A

### QGPV and Total Energy Equations

*U*in equal-thickness layers on an

*f*plane, evolves following

*ψ*

_{1}and

*ψ*

_{2}denote the eddy part of the streamfunction [e.g., (

*u*

_{1},

*υ*

_{1}) = (−∂

_{y}ψ_{1}, ∂

_{x}ψ_{1})],

*λ*is the Rossby radius of deformation,

*J*denotes a Jacobian operator representing the advective terms, and

*ν*is the hyperviscosity. In (A2), the bottom drag is parameterized using the quadratic and linear drag formulations as

*μ*(unit of m

^{−1}) and

*κ*(unit of s

^{−1}) being the respective drag coefficients.

*ψ*

_{1}, (A2) by −

*ψ*

_{2}, integrating horizontally over the doubly periodic domain, and taking the thickness-weighted (i.e., equal weight of 1/2 for our case) sum of the two. This leads to

*E*consists of KE (first and second terms) and PE (third term) components,

*ε*is the baroclinic energy input (see main text), and

_{p}*ε*and hyper are the energy loss due to bottom drag and hyperviscous dissipation, respectively. Note that (A7) is written in modal form, using the barotropic and baroclinic streamfunctions

*ψ*≡ (

*ψ*

_{1}+

*ψ*

_{2})/2 and

*τ*≡ (

*ψ*

_{1}−

*ψ*

_{2})/2.

*L*that characterize a square domain length of 2

*πL*and the mean flow velocity scale

*U*. Denoting the dimensionless variable with a tilde, the dimensionless governing equations in the case of quadratic drag become

*L*/

*λ*measuring the domain size relative to the deformation radius and

*μℓ*) and

*ν*/(

*UL*

^{7})]) characterizing the strength of bottom drag and hyperviscous damping, respectively. The experimental designs refer to the main text.

*ε*by

*ε*

_{bt}in Eq. (2)], we also carry out experiments in which

*ε*is formulated using pure barotropic velocity (i.e., imposing

*ε*=

*ε*

_{bt}). These experiments are labeled as BT drag in Table 1. By comparing, for instance, L002 and BT drag 1, we will be able to examine the response of eddy scales when partial barotropization is permitted (i.e.,

*ε*<

*ε*

_{bt}; L002) or removed (i.e.,

*ε*=

*ε*

_{bt}; BT drag). For implementation, it is easier to impose

*ε*=

*ε*

_{bt}in the modal form of QGPV equations:

*q*

_{bt}= ∇

^{2}

*ψ*and

*q*

_{bc}= ∇

^{2}

*τ*−

*τ*/

*λ*

^{2}. Then, for BT drag experiments, we simply set +drag/2 → −

*μ*[∂

*(|∇*

_{x}*ψ*|∂

*) + ∂*

_{x}ψ*(|∇*

_{y}*ψ*|∂

*)] in the barotropic equation and −drag/2 → 0 in the baroclinic equation. Note that, because barotropic drag is felt equally by both layers, it only shows up in the barotropic mode. We have checked that, when using the lower-layer drag, numerical integrations using layer and modal formations are identical as it should be.*

_{y}ψ*K*= (I) following Scott and Wang (2005), we can obtain the barotropic flux by integrating this expression from a wavenumber

*K*to infinity:

## APPENDIX B

### Examinations of Vortex Gas Properties at Weak Linear Drag

Here we examine two guiding assumptions of the vortex gas model using a vortex finding algorithm. The assumptions are that 1) a vortex dipole is sufficient to capture the qualitative transport properties in turbulent flow. The heat flux is thus expected to be preferentially enhanced within vortex pairs in a manner similar to GF20’s Fig. 2; 2) the vortex velocity scales with *r*^{−1} like a point vortex. The increase of velocity toward a vortex core enhances the vortex-averaged dissipation [i.e.,

For the above item 1, we compare the heat fluxes inside and outside vortex pairs, with an expectation that the former would be preferentially enhanced. To check this picture, we apply the vortex finding algorithm of TY06 to the barotropic vorticity field. A vortex center is identified to have magnitude of Okubo–Weiss (OW) parameter greater than a threshold value and magnitude of vorticity greater than its surrounding points (see appendix B in TY06 for details). Vortex pairs are found as vortices whose intervortex distance is shorter than the mixing length (i.e., * _{c}* = 0.05, 0.02, 0.01, and 0.005. For example, a value of OW

*= 0.05 as in TY06 indicates a threshold equal to 5% of the maximum OW magnitude in the field.*

_{c}In Fig. B1, an example of instantaneous barotropic vorticity ∇^{2}*ψ*, baroclinic streamfunction *τ*, and heat fluxes *ψ _{x}τ* is shown. From Fig. B1a, the vortex finding algorithm with OW

*= 0.02 successfully picks out vortices that have large vorticity extrema and are generally round shaped. When compared with the map of baroclinic streamfunction (∝ temperature), we see that cyclones (cyan circles) and anticyclones (red circles) tend to have a cold and warm core, respectively, consistent with the findings of TY06. In the heat flux map, vortex pairs and their range of influences are identified by the yellow contours, with the dashed contours denoting*

_{c}We can quantify the heat fluxes inside and outside regions of vortex interactions. Since the heat flux is proportional to energy production *ε _{p}*, we calculate the area averaged production as

*A*is chosen to be inside or outside vortex pairs (i.e., yellow contours in Fig. B1c). This conditionally averaged production is then normalized by the domain-averaged

*ε*and plotted against different choices of OW

_{p}*in Fig. B2a. The main feature to note is that, despite the use of different OW*

_{c}*values for vortex-finding and different ranges for vortex influences, the averaged heat fluxes inside (circles) and outside (triangles) vortex pairs remain similar, consistent with the visual inspections of Fig. B1. In Fig. B2b, we see that the area fraction occupied by vortex pairs varies greatly with OW*

_{c}*and ranges. A larger range gives a larger area fraction (filled and open symbols for*

_{c}*allows more vortices to be found, leading to a larger area occupation. For example, at OW*

_{c}*= 0.005, the entire surface area is within vortex range of influences. We thus see the values of averaged flux/*

_{c}*ε*inside vortex pairs being one (i.e., red circles in Fig. B2a), and the flux outside loses its meaning (i.e., no triangles; we exclude flux calculations when the corresponding area fraction is less than 0.1). In sharp contrast to the large variations in vortex-pair area fraction, the averaged heat fluxes inside and outside vortex pairs stay close to each other (Fig. B2a). The values of averaged flux/

_{p}*ε*are all near one, varying only between 0.87 and 1.1. The averaged fluxes inside vortex pairs tend to be slightly larger than outside, but the differences are within 15%. The above results therefore suggest no clear tendency of heat flux enhancement within ranges of vortex interactions.

_{p}For the above item 2, that is, the approximate *r*^{−1} profile, we examine the composite vortex velocity structure. To obtain the composite, we interpolate the meridional velocity *υ*/*U* for all vortices identified during the steady period onto a common normalized grid (* _{c}* are shown in Fig. B2c. One can see that the use of larger OW

*picks out stronger vortices, which gives a stronger composite velocity (e.g., red versus orange curves). The shading denoting one standard deviation for OW*

_{c}*= 0.02 shows that individual vortex profiles do spread significantly around the mean. But the averaged profiles are robust: when multiplied by a factor of −1, the profiles of anticyclones are nearly identical to those of cyclones (not shown).*

_{c}Over the region where the vortex velocity is relatively strong, the averaged velocity appears to vary more gently than *r*^{−1}. To test the *r* dependence, in Fig. B2d we examine the averaged profiles normalized by the velocity at *r*^{−1} profile, we expect * _{c}* values considered, the profile slopes are flatter than the solid reference line, tending closer to the

*r*

^{−0.5}slope instead.

The deviation from *r*^{−1} profile, due presumably to vortex interactions, could modify the scaling behavior of the vortex gas. If we assume vortex velocity to have a general profile of *n* = 1, we recover the vortex gas results, with *n* < 1, the vortex gas scaling is significantly modified. With *n* ≠ 1, the above integral becomes *n* < 1 and *r*^{−1} shape.

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