Revisiting the Baroclinic Eddy Scalings in Two-Layer, Quasigeostrophic Turbulence: Effects of Partial Barotropization

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

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Abstract

In this study, modifications of the Held scaling (Held) are proposed and tested against simulations of f-plane, two-layer quasigeostrophic turbulence. The aim is to better constrain the eddy mixing length and rms barotropic velocity in response to varied quadratic and linear bottom drag. The proposed modifications allow eddies to be partially barotropized, to relax the commonly invoked barotropization approximation, and consider a drag-dependent cascade rate per energy input, to account for the lack of an inertial range. Quantitative comparisons with the vortex gas scaling are also carried out. It is shown that the progressively weakened sensitivity in eddy scales to increased drag strength is mainly a result of eddy partial barotropization. For both drag forms except toward the limit of weak linear drag, accounting for partial barotropization alone leads to good predictions of eddy velocity, although not of mixing length. It also partly resolves the degeneracy of balance constraints for linear drag because partial barotropization acts like scale-dependent damping. Adding a cascade correction, which is interpreted as allowing for changes in spectral room for cascade, further improves the mixing length representation. Overall, the proposed theory can augment the existing scalings by extending the eddy scale predictions to O(1) quadratic drag and has skills generally comparable to the vortex gas scaling for linear drag. However, toward the weak linear drag limit where eddies approach complete barotropization, the proposed theory breaks down but the vortex gas performs well. Potential issues concerning the applicability of vortex gas to this limit are discussed.

© 2023 American Meteorological Society. 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

In this study, modifications of the Held scaling (Held) are proposed and tested against simulations of f-plane, two-layer quasigeostrophic turbulence. The aim is to better constrain the eddy mixing length and rms barotropic velocity in response to varied quadratic and linear bottom drag. The proposed modifications allow eddies to be partially barotropized, to relax the commonly invoked barotropization approximation, and consider a drag-dependent cascade rate per energy input, to account for the lack of an inertial range. Quantitative comparisons with the vortex gas scaling are also carried out. It is shown that the progressively weakened sensitivity in eddy scales to increased drag strength is mainly a result of eddy partial barotropization. For both drag forms except toward the limit of weak linear drag, accounting for partial barotropization alone leads to good predictions of eddy velocity, although not of mixing length. It also partly resolves the degeneracy of balance constraints for linear drag because partial barotropization acts like scale-dependent damping. Adding a cascade correction, which is interpreted as allowing for changes in spectral room for cascade, further improves the mixing length representation. Overall, the proposed theory can augment the existing scalings by extending the eddy scale predictions to O(1) quadratic drag and has skills generally comparable to the vortex gas scaling for linear drag. However, toward the weak linear drag limit where eddies approach complete barotropization, the proposed theory breaks down but the vortex gas performs well. Potential issues concerning the applicability of vortex gas to this limit are discussed.

© 2023 American Meteorological Society. 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

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 ∝ KE3/2 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

Two scaling theories to be examined in this study, namely, the Held and vortex gas models, are briefly reviewed here. The theories seek after the mixing length lm and barotropic velocity scale V that set the diffusivity D of eddy heat transport (i.e., DVlm). The two theories were built from different physical models: turbulence cascade versus coherent vortices, but they applied the same energy balance constraint. The domain-averaged energy balance is derived in appendix A [see Eqs. (A5)(A10)], and the notations follow GF20. At a steady state, the balance is between production (i.e., baroclinic energy input) and dissipation,
0=ψxτUλ2εpμ2|u2|3ε+hyper.
The production εp comes from a release of mean flow potential energy by downgradient eddy heat fluxes: ψxτU=ψxτ(τ¯/y)>0, noting that ψx is the barotropic meridional velocity, the baroclinic streamfunction τ is proportional to temperature, and the zonal mean flow magnitude U represents the background temperature gradient (i.e., U=τ¯/y via thermal wind; overbar denotes zonal mean). The total dissipation, on the other hand, has two parts: one due to the retarding action of bottom drag on eddy motion (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.
Both Held and vortex gas scalings invoked two assumptions to simplify Eq. (1). The first is to neglect the hyperviscous term. The rationale is that inverse cascade tends to concentrate energy toward large scales, whereas the hyperviscous term with a highly scale-selective ∇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, ε=(μ/2)|u2|3(μ/2)|ubt|3 in (1). This approximation is often assumed to be valid in low-drag conditions when a friction-arrest scale is well separated from the injection scale so that an extensive inverse cascade allows energy to accumulate in the barotropic mode (e.g., Charney 1971). We will show below that it could lead to significant error when the bottom drag strength is of O(0.1). Using these two assumptions, a simplified energy balance becomes
0ψxτUλ2εpμ2|ubt|3εbt,
where a subscript bt is used to emphasize the barotropization approximation applied to the bottom drag dissipation.

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.

As described in the introduction, the Held model was developed from Salmon’s (1980) dual cascade scenario under the assumption of an inertial range for the barotropic inverse cascade. The premise is a three-way balance between the energy production, cascade rate, and dissipation. The balance is formulated in terms of a cascade-halting length scale k01, where k0 may be interpreted simply as the peak wavenumber of a barotropic energy spectrum, and a barotropic eddy velocity. Assuming that k0 also sets the mixing length of heat (i.e., k01lm; lm is the mixing length) and using a rms barotropic velocity V (see section 3 for precise definitions for lm and V), one can scale the energy production as εp=ψxτU/λ2=(DU)U/λ2(Vlm)U2/λ2. With the barotropization approximation, the dissipation is simply εbtμV3. The above then allows the simplified energy balance (2) to be expressed in terms of two unknowns of lm and V. For a second balance condition, the existence of an inertial range requires the rate of barotropic inverse cascade εc to match the dissipation/production. i.e., εcεεp. In the inertial range, dimensional consideration yields a cascade rate scaling of εcV3k0 (e.g., Smith and Vallis 2002; CH19). Combining the above leads to a closed set of equations
εpεbt:(Vlm)U2/λ2μV3εcεp:Tek01/Vlm/Vλ/U.
In (3), we see that the cascade constraint gives a well-known result that the eddy turnover time Te (k01/V) is set by the Eady time scale TEady (≡ λ/U), and Te is approximated by the ratio of mixing length to the rms barotropic velocity. Because TEady 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,
lm,Held/λμ*1VHeld/Uμ*1DHeld/(Uλ)μ*2,
where μ* (≡ 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 εbtκV2. Substituting it into the energy balance yields (lm/V)κ*(λ/U), where κ* (≡ 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.

The vortex gas model was formulated in terms of two variables, Viv and liv, that describe the dipole interactions. A dipole consists of two vortices of opposite circulation, characterized by mutual advection velocity Viv at radius r equal to the intervortex distance liv. Two balance conditions are invoked. One is the simplified energy balance in (2), while the other is a constraint for energy conversion that relates potential energy release by slantwise parcel movement over mixing length lm with kinetic energy gain, yielding Vivlm(U/λ) (see GF20 for details). The following scaling relations are then used to close the system: (i) the mixing length is set by the intervortex distance: lmliv and the diffusivity D scales as Vivliv and (ii) the vortex velocity increases toward the core, following an r−1 profile to preserve the circulation like a point vortex [i.e., υ/Viv(r/lm)1; see section 5 for direct estimates]. The relation (ii) is crucial because it amplifies the domain-averaged velocity moments and thus modifies the drag sensitivity. Specifically, the third-order moment 〈|ubt|3〉 in (2) does not simply scale as Viv3 but is instead amplified by a correction factor [Viv3(liv/λ)]. Using the above allows rewriting the two balance conditions as
εpεbt:(Vivliv)U2/λ2μViv3(liv/λ),Vivliv(U/λ),
which leads to eddy scalings as a function of drag strength μ*
lm,GF/λμ*0.5,VGF/U(Viv/U)ln(liv/λ)(lm,GF/λ)ln(lm,GF/λ),DGF/(Uλ)μ*1.
By comparing (5) with (3), we can see that, without the amplification of velocity moments, the equation sets (5) will have the identical form as (3). The scalings for lm,GF/λ and DGF/() 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., μ*1 versus μ*0.5). Note also that VGF is the rms barotropic velocity, not the vortex mutual advective velocity Viv at riv. In the case of linear bottom drag, one can obtain eddy scaling following similar procedures as above, but the scaling depends exponentially on the inverse of linear drag strength κ* [see GF20’s Eqs. (15) and (16)].

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 2πL × 2πL, 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, μ* (≡ μℓ), and ν˜ (≡ UL7), 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 ν˜ and a suitable choice 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 μ* (e.g., TY06; GF20).

The experiments are summarized in Table 1. The setting all uses U = 1 and L = 1 in a 5122 horizontal grid unless otherwise noted. The set “GF” is the repeat experiments that use λ/L = 0.02 (i.e., L002), ν˜=1013, and μ*=0.0031 as in GF20. The set “hyper” is to find a suitable value of ν˜ that yields negligible hyperviscous dissipation as compared with bottom drag. A ν˜ value of 10−17, much lower than GF20’s, is found (see section 3c below). Setting ν˜=1017, the base experiments are denoted as L002 for both quadratic and linear drag. These cases are used for analyses throughout this work. There are also set L004 and L001 that uses different λ/L but has μ* values overlapped with L002. These sets are to ensure the robustness of numerical solutions and to expand μ* range. The combination of L001, L002, and L004 gives a wide μ* range of 10−3–10 for quadratic drag and κ* of 10−1–2.5. 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.

Table 1

A summary of numerical experiments designing to investigate the responses of eddy scales to varied drag strength (μ* and κ* are for quadratic and linear drag, respectively). See main text and appendix A for the definitions of the three dimensionless parameters.

Table 1

b. Eddy scale definitions and model validation

Following TY06 and CH19, the mixing length lm, barotropic velocity V, and the diffusivity D are diagnosed directly from simulations using the definitions of
V=ψx2,lm=U1τ2,D=U1ψxτ,
where 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 V2 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 lm, and D are consistent with mixing length and downgradient heat diffusion: lm represents the scale near which temperature variance (i.e., τ2) 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 lm and D are shown for different bottom drag strength μ*. It can be seen that our results (set GF in Table 1) are essentially identical to the GF20 values (black circles versus filled red squares), thereby validating the numerical implementations. It is also confirmed here that the vortex gas scaling (thin black line) performs better than the Held scaling (dashed line). In Figs. 1a and 1b, both the model-derived mixing length and diffusivity are in better agreement with the vortex gas scaling. The slopes of the Held predictions are too steep. This means that the Held scaling overestimates the sensitivity to bottom drag, consistent with prior findings in Arbic and Scott (2008) and CH19.

Fig. 1.
Fig. 1.

Responses of (a) mixing length lm and (b) diffusivity D to varied dimensionless quadratic drag strength μ*. The experiments reported in GF20 are denoted by the filled red squares. The black and gray open circles correspond to GF and L002 in Table 1, with the GF set designed to repeat GF20’s results. The solid and dashed lines denote the vortex gas [Eq. (6)] and Held scaling [Eq. (4)], respectively. (c) An example of eddy energy budget [Eq. (A10)] with μ*=10 in the GF set. The dissipation approximated using pure barotropic velocity [εbt in Eq. (2)] is shown as the orange curve. The steady-state quantities are obtained by averaging over the last 200 time units throughout this work. Steady balances between the baroclinic production εp and dissipation due to bottom drag ε and hyperviscosity for different cases are shown as the cross symbols in (b).

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

Since our focus is to examine the barotropization approximation across different drag strength, that is, approximating ε by εbt in (2) for different μ*, we need to assess the accuracy of our dissipation estimates. The good accuracy is assured by the fact that our eddy energy budget is closed. An example is shown in Fig. 1c for the case with μ*=0.1 from the GF set. It can be seen that, at the steady state, there is a balance between energy production (black) and two dissipative sinks due to bottom drag (red) and hyperviscosity (blue), as expected in (1). The residual denoted by the gray curve is essentially zero. For this particular case, the bottom drag dissipation calculated using barotropic velocity (εbt; 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 TEady, where TEadyλ/U. In Fig. 1b, the steady-state values of −ε/(εp + 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 ν˜=1013 used by GF20. In Fig. 1c, the hyperviscous term (blue) accounts for an uncomfortably large fraction, 25%, of the total energy dissipation at the steady state. Sensitivity tests suggest reducing ν˜ by four orders of magnitude to 10−17 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 ν˜ is reduced to 10−17. The mixing length and barotropic eddy velocity, both normalized by the values with ν˜=1013, also show sign of convergence.

Table 2

Sensitivity tests aiming to find suitable value of ν˜ that yields negligible hyperviscous energy dissipation as compared with that due to bottom drag [i.e., small hyper/(ε + hyper)]. These tests have μ*=0.03.

Table 2

Quite surprisingly, though, the error of neglecting hyperviscosity has a minor effect on the eddy scales. In Figs. 1a and 1b, when we use ν˜ that is four orders of magnitude smaller (ν˜=1017; gray circles), the dependence of mixing length and diffusivity on μ* do not show significant changes. This is because the relative contribution of hyperviscous dissipation is insensitive to drag changes. For μ* varying from 0.003 to 0.3, the fraction of hyperviscous dissipation stays nearly constant among the GF cases, ranging between 23.2% and 25.2% with a very small standard deviation of 0.7% (not shown). Understanding the cause behind the nearly constant fraction of hyperviscous dissipation is beyond the scope of this study. From this point forward, all experiments have ν˜=1017 or lower. With a sufficiently low ν˜, we have confirmed that there is a tight balance between production εp 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 εpε 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=(μ/2)|u2|3, whereas the approximation in (2) has εbt=(μ/2)|ubt|3. If the barotropization assumption is valid, we expect the ratio of εbt/ε to be roughly a constant among different cases.

In Fig. 2a, the dissipation ratio εbt/ε is plotted against μ* (gray circles). Inconsistent with the barotropization assumption, the ratio is not approximately constant across different drag strength. It increases steeply from 1.06 at μ*=0.001 to 3.7 at μ*=0.03. This means that, at μ*=0.001, the error of making the barotropization approximation is negligible at only 6%. However, toward O(0.1) drag with μ*=0.1and0.3, the error becomes a strong overestimation of ε by a factor of 2.2 and 3.7, respectively. Note that this strong μ* dependence was not considered in the Held and vortex gas scalings as both simply took εεbt in (3) and (5).

Fig. 2.
Fig. 2.

Evaluations of three assumptions in the Held model for different drag strength μ*: (a) the barotropization approximation (εεbt), (b) diffusive representation of heat fluxes in terms of mixing length and barotropic velocity (DVlm), and (c) existence of an inertial range [εcεp; Eq. (8)] In (a), a corrected dissipation using Eq. (11) is also plotted as open diamonds for comparison. The cases correspond to the L002 set with ν˜=1017.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

Qualitatively, the increase of εbt/ε with drag strength can be understood as a result of partial barotropization. The ratio εbt/ε is controlled by the differences between barotropic velocity |ubt| and lower-layer velocity |u2| (see more in section 4c below). As bottom drag increases, the lower-layer velocity is preferentially damped so that the difference between |ubt| and |u2| increases. In other words, eddies become increasingly partial barotropized (i.e., more baroclinic), which reflects in an increase in εbt/ε.

We may estimate the influence of barotropization error on the scaling theories. The barotropization error refers to the degree of εbt/ε deviated from one. For simplicity, we represent the μ* dependence as a power law εbt/εμ*n, where the exponent n is obtained by power-law fits in Fig. 2a. For the Held scaling, when combining εbt/εμ*n with εεp in (1) and εbtμV3 and lm/Vλ/U in (3), we obtain a scaling that incorporates the barotropization error for the mixing length as lm/λμ*(1n). Similarly, for vortex gas, the modified scaling becomes lm/λliv/λμ*0.5(1n). Note that, when the barotropization error is ignored, we set 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 μ*=0.0010.03 and μ*=0.030.3, the corresponding exponents are 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 μ*=0.0010.03. Such a slope change is significant especially for larger drag. A correction for this will be incorporated into the Held model below.

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 εp and an assumption of inertial range for barotropic inverse cascade. These assumptions are examined as well.

In (3), the heat flux 〈ψxτ〉 in the baroclinic energy production εp is expressed as downgradient diffusive transport, with a diffusivity DVlm. This diffusive representation is well supported by the simulations. In Fig. 2b, we compare the proportionality coefficient for the diffusive transport, ψxτ/(VlmU), among different cases. The coefficient is indeed nearly a constant, having a mean of 0.31 and showing only a small range of variations between 0.29 and 0.33 for μ* varied from 0.001 to 1. The robustness of diffusive representation is consistent with TY06 (i.e., their Fig. 7) and is perhaps not too surprising: on scales greater than λ, τ is passively stirred by barotropic flows. The tracer fluxes could thus be regarded as diffusive (Held 1999).

As discussed in section 2a, if an inertial range for barotropic inverse cascade exists, the balance requirement yields a scaling of cascade rate matching the dissipation and production, εcεp in (3). Together with the assumption that the cascade-halting scale also sets the mixing length (k01lm), the balance reduces to setting the eddy turnover time to the Eady time scale. By (3), we may write out this condition explicitly as
TeTEadylm/Vλ/U(1εc/εp)1/2.
In the Held scaling, the ratio of εc/εp was assumed to be a constant, yielding lm/Vλ/U in (3). In Fig. 2c, we diagnose this time scale ratio among the cases. The ratio is clearly not a constant. It increases with the drag strength, from 0.7 at μ*=0.001 to 1.7 at μ*=0.3. From (8), the increase may be interpreted as a result of reduced cascade rate εc per input. When the drag strength increases, the mixing length lm and the halting scale decrease (Fig. 1a), allowing less spectral room for inverse cascade and hence reducing εc/εp (see more below).

c. Proposed corrections and eddy scale predictions

1) Partial barotropization

To correct for the barotropization error in the Held model, an improved representation for the true dissipation ε in terms of rms barotropic velocity V is needed. We can express the dissipation using the rms, lower-layer meridional velocity υ2 as
ε=μ2|u2|3μ223/2|υ2|3μ21/21.2(υ22)3.
In (9), the factor of 23/2 comes from the assumption of horizontal isotropy, whereas the factor of 1.2 is to relate υ2 magnitude cubed with the cube of rms υ2. It can be shown that, for a simple sine wave, the factor is equal to (4/3π)23/2 = 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.
The rms, lower-layer meridional velocity υ2 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 υ22=υbt22υbtυbc+υbc2. By inverse cascade, we anticipate eddies on scales much greater than λ to be barotropized (e.g., Charney 1971) and the barotropic energy spectrum to peak strongly at the halting wavenumber k0 (i.e., k01lmλ; see CH19). Thus, we may take υbt2υbc2 and approximate 〈υbt υbc〉 by the covariance near k0, with υbtυbcυbt̂k0υbĉk0, where ^k0 represents the rms value in the spectral vicinity of k0 (e.g., Smith and Vallis 2002). We may further express υbt̂k0υbĉk0VU. This relation follows from Held and Larichev’s (1996) argument that the rms barotropic and baroclinic eddy velocity near k0 take on the value of V and U, respectively. Note that scaling υbt̂k02V2 is a straightforward deduction from a steep Kolmogorovian-like energy spectrum (i.e., slope ≥ −5/3) in which the bulk of barotropic energy is contained near k0. On the other hand, the scaling relation of υbt̂k0U can be derived from an assumption that the baroclinic potential vorticity flux is diffusive [for details see Eqs. (3.12)–(19) in Smith and Vallis 2002 and Eqs. (3)–(5) in Held and Larichev 1996]. Combining the above then leads to
υ22=υbt22υbtυbc+υbc2V22VU.
Applying the simple correction in (10) reduces the barotropization error. Inserting (10) into (9), we obtain a corrected dissipation
εcorr=μ21/21.2(V22VU)3/2.
Note that, without the correction term (i.e., −2VU), the dissipation is equivalent to εbt in (2) which has been shown to increasingly overestimate the true dissipation ε as the drag strength increases (i.e., εbt/ε in Fig. 2a). With the correction, the dissipation estimates improve significantly. The corrected ratio εcorr/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 μ* approaches 1 (Fig. 1a). The degradation is likely due to the use of a simplified relation υbĉk0U and to representing 〈υbtυbc〉 by only the covariance near k0. Note that the scaling of υbĉk0U derived by Held and Larichev (1996) is strictly valid for scales much greater λ. Such a condition is most likely violated toward μ*1 when the mixing length becomes comparable to 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.

Importantly, the above correction also leads to improvement in the eddy scale predictions. As is done in section 2a, equating the energy production εp with the corrected dissipation εcorr and invoking the cascade rate constraint [i.e., εc/εp = constant in (8)] result in a closed set of equations for V and lm,
(c0Vlm)(U2/λ2)μ21/21.2(V22VU)3/2
lm/Vc1λ/U.
In (12), we have included two proportionality constants of c0 and c1 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 lm without the need of parameter tunning (see below). We take c0 = 0.31 from Fig. 2b and c1 = 1.2 as an averaged time scale ratio in Fig. 2c. Note that these constants are relatively well constrained. TY06 reported c0 = 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.
Substituting the mixing length lm in (12a) by (12b) yields a single equation for the rms barotropic velocity V made dimensionless by the mean flow U
α0μ*2/3(VU)1/3=(VU)2
where α0=(c0c1/1.22)2/3=0.36. Equation (13) is a cubic polynomial which can be easily solved using, for example, the root-finding code in MATLAB. Once V/U is found, we can obtain lm/λ from (12b) and the diffusivity via D/(Uλ)=c0(V/U)(lm/λ). Note in (13) that the correction for the barotropization assumption is represented by the −2 term. When ignoring this term, Eq. (13) reverts back to the Held scaling.

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., μ*>0.1) where using barotropic velocity strongly overestimates the dissipation (e.g., εbt/ε > 2 in Fig. 2a), and the correction reduces the steep drop in V/U.

Fig. 3.
Fig. 3.

Evaluations of eddy scale predictions for (a) rms barotropic velocity V, (b) mixing length lm, and (c) diffusivity D in the case of quadratic drag. In each panel, the thick gray curve indicates the Held theory modified to incorporate partial barotropization [Eqs. (12) and (13)]. The thick black curve is with an additional correction of drag-dependent cascade rate per input [Eqs. (14) and (15)]. The vortex gas scaling in (6) is denoted by the thin dashed line, whereas the original Held scaling in (4) is the thin gray line. In (a), there is an extra curve for the vortex gas prediction. This curve is offset from the best-fit curve by a factor of 5 to illustrate that it concaves downward, opposite to the simulations. In (c), the simulation results from CH19 are included as cross symbols.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

Similar results are found for the mixing length lm: adding the barotropization correction weakens the μ* sensitivity at large drag. Compared with the straight power-law line of Held, the gray curve in Fig. 3b becomes concave upward toward large drag in a manner similar to the simulations. However, the agreement is not as good as the eddy velocity. It is evident in Fig. 3b that the prediction curve using (12) is still too steep when compared with the simulations (see more below).

Since the diffusivity is well described by the mixing length scaling (i.e., D=c0Vlm in Fig. 2b), we expect the results described above to carry over to the diffusivity. Indeed, toward large drag, there is a general agreement between the predicted and simulated diffusivity (gray curve versus filled symbols in Fig. 3c), because adding the barotropization correction allows the predictions to capture the weakened μ* sensitivity. However, one can again notice the overprediction as shown by the steeper slope in the gray curve toward weak drag, which can be traced back to the overprediction in the mixing length.

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., εbt/ε increases with μ* in Fig. 2a; see section 6 for further discussion). This is supported by the result that incorporating partial barotropization into the Held model via (12a) allows the prediction curves to concave up, consistent with the simulations. By the same token, eddies become more barotropized toward the weak-drag limit. At μ*=0.001and0.003, εbt/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., V/U2 in Eq. (13)]. It thus makes sense to see that the slope of the prediction curve using (12) approaches the Held scaling in the weak-drag limit.

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 εcεp, V/U and lm/λ are constrained to have the same functional form, as they only differ by a constant factor c1 in (12b). Yet, simulations do suggest the two to scale differently, with lm/λ responding less sensitively to drag changes (i.e., μ*0.51 versus μ*0.73, consistent with the finding in CH19). We will show below that this different μ* sensitivities can be largely recovered and that the overall performance of the eddy scale predictions can be improved when the cascade rate per energy input εc/εp is allowed to vary with μ*.

2) Adding a cascade correction

The overprediction of μ* sensitivity in the mixing length can be largely removed by considering a correction for the cascade rate. We have seen in Fig. 2c that the ratio of eddy turnover to Eady time scales has a relatively weak but nonnegligible drag dependence. But, as a crude approximation, the ratio is taken as a constant in both (12b) (i.e., c1 = 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
lm/V(c2μ*m)λ/U,
where c2 = 2 and m = 1/7. Physically, we may interpret this positive μ* dependence by first recognizing that μ*λ/μ1 represents the degree of separation between two length scales: the deformation radius 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., lm,Heldμ1 in Eq. (4)]. The drag strength parameter μ* thus also serves as an inverse measure of spectral room for inverse cascade. Combining (14) with (8) yields a relation of εc/εpμ*2m, which has a clear physical meaning: a larger spectral room (i.e., smaller μ*) would correspond to greater energy cascade per input (i.e., larger εc/εp). Analyses of spectral energy fluxes to be shown in the following subsection will provide further support for this drag-dependent cascade fraction.
With the cascade rate correction, the Eqs. (14) and (12a) now form a closed set. Replacing lm in (12a) with (14), we obtain a simple cubic equation for V/U
α1μ*(1m)2/3(VU)1/3=(VU)2,
where α1=(c0c2/1.22)2/30.5. Setting 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 lm/λ via (14).

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, εpεbt, in (3). This leads to relations of V/Uμ*1+mμ*0.86 and lm/λμ*1+2mμ*0.71. When comparing these with numerical data in Fig. 3, we see that applying the cascade correction alone does improve the Held scaling by weakening the drag sensitivity. However, the scaling slopes of −0.86 and −0.71 for eddy velocity and mixing length are still too steep as compared with the empirical fits of −0.73 and −0.51 obtained over the range of μ*0.01. This exercise again points to the importance of considering partial barotropization even when the quadratic drag strength is 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 Te/TEadyc2μ*m or equivalently εc/εpμ*2m used above has been interpreted as a result of decreased spectral room for barotropic inverse cascade (decreased εc/εp) 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 μ*=0.01 is shown in Fig. 4a. One can see that a three-term balance stands out. The energy is injected into the barotropic mode via the baroclinic-to-barotropic conversion [blue; term II in (A15)] and is dissipated through bottom drag (red; term IV). Some energy input is redistributed across scales via the barotropic flux convergence (term I) (see below). The budget is closed reasonably well. The amplitude of the residual is small and only fluctuates around zero (gray).

Fig. 4.
Fig. 4.

(a) An example of spectral budget of barotropic energy for the case with μ*=0.01. The budget terms are defined in Eq. (A15) and are obtained by integrating over a shell of constant total wavenumber K with dK = 1 (K is made dimensionless by domain length L); (b) distribution of barotropic flux normalized by the baroclinic energy input in wavenumber space. The spectral flux is obtained by integrating the flux convergence term in (a) [see Eq. (A16) for definition. The inset is a schematic for spectral flux in the presence of an inertial range. (c) The magnitude of normalized barotropic flux is plotted against μ*. The dashed line indicates the power-law fit of (16). The cross symbols are differences of wavenumber centroids (KC,IIKC,IV) that measure the scale separation between the energy conversion and dissipation spectra (see main text).

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

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., εp=ε=(IV)dK], and its wavenumber distribution, Π(K)/εp, is shown in Fig. 4b for different cases. Taking the μ*=0.01 case as an example (thick black curve), the predominant negative values indicate that the transfer of barotropic energy is upscale as expected for the inverse cascade. From the limit of lowest wavenumber, the convergence term (I) integrates to zero (e.g., Π = 0 at 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).

If the inertial range assumption is valid, one expects the energy input and dissipation to be well separated so that there is a range of wavenumbers over which the barotropic flux is constant, as schematized in the inset of Fig. 4b. Such a condition is clearly not met. In Fig. 4a, there is significant overlap between the conversion and dissipation in wavenumbers. In Fig. 4b, the barotropic flux per input Π/εp 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/εp does increase with decreasing drag strength (filled circles in Fig. 4c), approximately following a power-law relation of
|Π|max/εp=0.1μ*1/4.
This increase is mainly due to an increase in scale separation between conversion and dissipation. We may quantify the scale separation using the difference in wavenumber centroids. In Fig. 4a, the centroids of conversion and dissipation spectra are calculated as their amplitude-weighted wavenumber [e.g., KC,II=(II)KdK/(II)dK] and are denoted by the vertical dashed lines. The scale separation as measured by (KC,IIKC,IV) is plotted against μ* in Fig. 4c (cross symbols). As can be seen, the separation increases monotonically as μ* decreases. A greater separation corresponds to less overlapping between conversion and dissipation in wavenumbers, thereby needing to transfer a greater fraction of energy input upscale and hence greater |Π|max/εp.

Importantly, the power law (16) obtained from direct quantification of barotropic flux is consistent with the empirical relation Te/TEadyc2μ*m. Recognizing that the cascade rate εc represents the magnitude of barotropic flux, we can substitute (16) into (8) to find Te/TEady(|Π|max/εp)1/2μ*1/8. Note that the exponent of 1/8 is quite close to the value of 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

The same corrections described above are applicable to the cases with linear bottom drag. With linear drag, the energy dissipation is ε = κ〈|u2|2〉. Note here that the bottom stresses are parameterized as 2κ|u2|u2, following the formulation and notations in GF20. Assuming horizontal isotropy and applying the correction in (10) give
ε=κ|u2|22κ|υ2|22κ(V22VU).
Replacing the dissipation in RHS of (12a) by (17) and using (12b), we obtain an equation for V/U,
(c0c12)κ*1(VU)=(VU)2,
which can be rearranged to yield the following close-form solution
V/U=2/(1c0c12κ*).
In the case of linear drag, the dimensionless drag strength is measured by κ*κλ/U.
Again, we can incorporate a cascade rate correction by using (14) in the place of (12b), giving an equation for V/U as
(c0c22)κ*(1m)(VU)=(VU)2.
The solution then becomes
V/U=2/(1c0c22κ*1m).
Switching off the cascade correction, we set 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 c0 = 0.29, c1 = 1.2, and c2 = 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 Te/TEady 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 μ*=0.01, V/U takes a value of around 22. This energy level is roughly comparable to V/U of 17 at κ*=0.2. This means that a dynamical range of cascade obtained via varying μ* by a factor of 100 (i.e., μ*=0.011) roughly occurs over a factor-of-5 change in κ* (i.e., κ*=0.21) (see also section 6b). Therefore, Te/TEady (or equivalently εc/εp) ought to be more sensitive to changes in linear drag strength.

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 (κ*=0.1). Possible reasons for the failure of this case will be discussed separately below. Excluding κ*=0.1, the outcomes of the comparisons are similar to the quadratic drag: without the cascade correction, the prediction using (19) quite accurately captures the drop in V/U and the weakened drag sensitivity as κ* approaches O(1) and beyond (gray curve in Fig. 5a). But, the predicted lm/λ using (19) and (12b) is too sensitive to drag changes, as the gray curve in Fig. 5b falls too fast. Adding the cascade correction improves the mixing length representation [using Eqs. (21) and (14)]. The black curve in Fig. 5b is in overall better agreement with the simulations than the gray curve, and the solution captures the relative drag-insensitive state in the vicinity of κ*1.

Fig. 5.
Fig. 5.

As in Fig. 3, but for linear bottom drag. In these panels, only the vortex gas scaling is shown because the original Held model cannot be applied to linear drag due to the degeneration problem. In (a), values of εbt/ε for selected cases are labeled to show that, first, the errors of barotropization approximation increase steeply with drag strength, and second, the case with κ*=0.1 is near a state of complete barotropization (i.e., weak drag limit) so that the proposed correction for partial barotropization in (17) has no use.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

Why does the above theory fail for κ*=0.1? One possible reason is that this case has approached a state of complete barotropization so that the balance constraints in the theory become degenerate. The values of εbt/ε are labeled in Fig. 5a for a number of cases. The κ*=0.1 case has εbt/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 (−2VU) in (17), the energy balance εp = ε becomes (lm/V)κ*(λ/U). This is of the same functional form as the cascade constraint in (12b) or (14), thereby leading to the degeneracy (e.g., Smith and Vallis 2002). The degeneracy suggests that, in the weak-drag limit, the theory may lack representation of certain cascade-halting processes so that eddies in the simulation are equilibrated but the predictions are not. At this time we cannot provide a definite answer (see section 6c for further discussion). However, we will test an alternative hypothesis in section 5. From Fig. 5, the vortex gas scaling by GF20 is applicable toward the weak-drag limit (black dashed curve). A hypothesis to be examined is that the weak-drag limit is better described by vortex interactions, as opposed to turbulent cascade phenomenology.

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., c0, c1, c2) 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., κ*=0.1 in Fig. 5). In this section, we will compare the proposed theory with vortex gas scaling in a quantitative way. The case with κ*=0.1 will be examined to understand the properties of vortex interactions in the weak drag limit.

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 100.1 ≈ 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%.

Table 3

Summary of root-mean-square errors (RMSEs) when the proposed theory and vortex gas scaling are compared against simulations. The RMSE is calculated for log10(variable). Full range refers to 103μ*10 for quadratic drag and 0.1κ*2.5 for linear drag. The errors for full range linear drag are infinite because Eq. (21) cannot be applied to κ*=0.1 (see text). The exact formulations for the vortex gas predictions are lm/λ=2.6μ*0.5, D/(Uλ)=1.9μ*1, and V/U=1.4μ*0.5lm/λ for quadratic drag, and lm/λ=3.2exp(0.36/κ*), D/(Uλ)=1.8exp(0.72/κ*), and V/U=1.5exp(0.36/κ*)lm/λ for linear drag, with the scaling coefficients determined from best fits. For the proposed theory, predictions are obtained from (15) and (14) for quadratic drag and from (21) and (14) for linear drag, with D=c0Vlm. The coefficients, c0, c2, and m, are determined from the experiments, with c0 = 0.31 (0.29), c2 = 2.0 (1.9), and m = 1/7 (0.26) for quadratic (linear) drag.

Table 3

For quadratic drag, we separate the comparisons into two ranges: μ*0.1 and full range. This is because the vortex gas model invoked the barotropization approximation [Eq. (5)] and is suited for low-drag conditions. A fair comparison would thus be to evaluate performance for μ*0.1. Over this range, the RMSEs of the two predictions are generally comparable. The vortex gas scaling is slightly better in representing the mixing length, but the proposed theory works better for 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 μ* sensitivity (i.e., steepened slope) as μ* decreases, which leads to overall lower RMSEs. Over the full range, the proposed theory clearly has higher predictive skills. The RMSEs for V/U and lm/λ are smaller than those of vortex gas by almost an order of magnitude. From Fig. 3, one can see that the higher skills are primarily the result of barotropization correction. Incorporation of partial barotropization allows the theory to better represent the weakened μ* sensitivity over 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 κ*<0.17 [i.e., c0c2/(2κ*1m)<1], leading to essentially infinite errors (Table 3). To compare the RMSEs, we again exclude the κ*=0.1 case temporarily (see below). Unlike the quadratic drag cases, there is no need to further separate the drag into different ranges. The vortex gas represents the eddy scales well into O(1) drag. From Table 3 for κ*0.2, the RMSEs of the proposed theory and vortex gas scaling are comparable for eddy velocity and mixing length, but the vortex gas predictions yield a notably smaller error for the diffusivity. The generally comparable RMSEs are reflected in Fig. 5 where the theory and vortex gas curves are close to each other. The working of vortex gas at 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., εbt/ε = 3.7 and 9.1 for κ*=1and2, respectively (see Fig. 5). However, unlike quadratic drag cases where the misfits between the scaling and simulations increase with μ* over O(1) μ* (see Fig. 3), the vortex gas continues to perform well into O(1) κ*. We have not found a satisfactory explanation for this contrasting behavior.

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 κ*=0.1, whereas the proposed theory fails. Our analysis in section 4 has suggested degeneracy as the main reason for the failure. To understand why the vortex gas scaling works well in the weak-drag limit, we examine the κ*=0.1 case further.

b. Vortex gas properties at weak linear drag

One possible interpretation for the superior performance of vortex gas at κ*=0.1 is that eddy fluxes in the limit of weak linear drag are better described as a result of vortex interactions. To check this, we examine vortex heat transport and velocity structure for κ*=0.1. Detailed examinations are described in appendix B. Main results are summarized here.

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, |ubt|3Viv3(liv/λ) as opposed to Viv3, which weakens the sensitivity to quadratic drag [μ*0.5 versus μ*1 in Eqs. (6) and (4)] and avoids the degeneracy for linear drag. These two assumptions are examined by using a vortex finding algorithm and composite velocity profiles. Note that the examinations only look for general consistency, as these assumptions are better interpreted as guiding principles for the vortex gas theory and shall not be taken at face value.

Overall, the results show no clear indication that eddy transport is controlled by vortex interactions at κ*=0.1. Despite the use of different vortex-finding thresholds and different ranges of vortex influences, the averaged heat fluxes inside and outside vortex pairs are found to be similar for the κ*=0.1 case (Fig. B2a). This is contrary to the qualitative picture of transport enhancement within vortex pairs expected from the vortex gas model. The vortex velocity is also found to deviate from the assumed r−1 structure. We average the velocity of all identified vortices to obtain composite-mean profiles (see Fig. B2c). Within around 0.2lm from the center where vortex velocity is stronger and hence more important to dissipation, the composite velocity is notably flatter than the 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 lm and rms barotropic velocity V, which combines to characterize the meridional heat transport via a diffusivity DVlm. The responses of these scales to varied quadratic and linear drag are studied, with the drag strength covering weak to O(1) drag conditions.

The main motivations are that 1) the Held scaling is known to be oversensitive to the quadratic drag strength μ* (e.g., Figs. 1a,b). The same scaling cannot be applied to linear drag because the balance constraints become degenerate, and 2) existing scalings, including Held and a vortex gas model put forth by GF20, make the barotropization approximation and are thus valid only for low-drag conditions. Yet, our analyses suggest that this approximation is inappropriate even at O(0.1) drag. The use of pure barotropic velocity can overestimate the dissipation by more than a factor of 2 for μ*=0.1, and the error rises steeply with increasing μ* (Fig. 2a). Corrections are incorporated into the Held scaling for its straightforward extension. But quantitative comparisons with the vortex gas are carried out to evaluate the proposed theory.

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 lm/λ and diffusivity D/() (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 κ*<0.2 in Fig. 5). The advantage has motivated us to explore the possibility that eddy transport in this limit may be better described by vortex interactions. Yet, the results do not support this possibility because the spatial structure of heat transport and composite-mean vortex velocity are found to deviate from the guiding assumptions of the vortex gas model (appendix B). The eddy characteristics in the weak linear drag limit needs to be looked into further.

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

a. The degree of eddy barotropization

The simple correction applied to the Held model gives information about the degree of eddy barotropization that can be compared with prior studies. We first recognize that the increase in barotropization errors in Fig. 2a is equivalent to a decrease in the degree of barotropization of the eddy field. Via (9), εbt/ε(υbt2/υ22)3/2(KEbot/KEbt)3/2. An increase in εbt/ε is equivalent to a decrease in KEbot/KEbt, where the ratio of lower-layer kinetic energy (KEbot) 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
KEbot/KEbtυ22/υbt212(V/U)1,
which can be inferred using the solutions of eddy velocity in (13) or in (15) with an additional cascade correction. In Fig. 6a, the KE ratios from the simulations and theories are plotted against μ*. It can be seen that the degree of barotropization as measured by the KE ratio decreases continuously with increasing drag strength (filled gray symbols), as drag preferentially dampens the lower layer to oppose the barotropization. This pattern is well represented by the theories (gray and black curves).
Fig. 6.
Fig. 6.

(a) Degree of barotropization in the eddy field as measured by the bottom to barotropic KE ratio is plotted against the drag strength. The filled symbols are the present experiments. The thick gray curve is the predictions from Eq. (22) and (13), whereas the thick black curve is with an additional cascade rate correction [Eqs. (22) and (15)]. Open symbols are the unstable zonal jet experiments reported in Jansen et al. (2015, denoted as J15), with the red dashed curve denoting their empirical fit. (b) Comparison of the rms barotropic velocity with (filled symbols) and without (open symbols) partial barotropization [see Eqs. (A13) and (A14) in appendix A and the main text].

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

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 μ*λ/μ1 is equivalent to their length scale ratio Ld/Lf (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 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 μ*0.03), approaching the simulation results. The importance of considering partial barotropization is confirmed in separate sets of experiments. In “BT drag 1” and “BT drag 2,” we rewrite the QGPV equations in terms of barotropic and baroclinic modes so that the energy dissipation is calculated using only the barotropic velocity [i.e., replacing |u2| in Eq. (1) by |ubt|; see appendix A]. When the effect of partial barotropization is removed, the concave upward shape in the V/U versus μ* relationship disappears (open versus filled symbols in Fig. 6b). The experiments therefore support that the weakened drag sensitivity toward large drag is mainly a result of decreased barotropization.

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 μ*0.1 in Fig. 3), it is apparent in (22) that the correction term is no longer negligible. That is, the rms lower-layer velocity is significantly weaker than the barotropic velocity. Therefore, as the drag parameter is varied, the weak lower-layer flow buffers the changes in energy dissipation [ε as compared with εbt 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 μ*λ/μ1Ld/(H/CD)CDN/f. For a typical quadratic drag coefficient CD = 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 μ* range is estimated to be 0.1–2.0. Note in Figs. 2a and 3 that this drag range is where the use of εbt 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|u2|. The conversion between κ* and μ* is then κ*=(μ|u2|)λ/U=μ*|u2|/Uμ*2|υ2|/U. As in CH19, we use domain-averaged rms value for |υ2|, and via (10), the conversion becomes κ*μ*2υ22/Uμ*2[(V/U)22(V/U)]1/2. From Fig. 3a, V/U ≈ 5.4 and 2.25 for μ*=0.1and2.0, respectively. Plugging into these numbers gives an estimate of linear drag range κ*=0.62.1 that is equivalent to μ*=0.12.0. Note again in Fig. 5 that the range of κ*=0.62.1 is where the barotropization approximation is erroneous (i.e., εbt/ε for κ*=0.6 is 2.3, similar to the value for μ*=0.1) and where the proposed corrections work reasonably well.

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 κ*=0.1 (or κ*<0.2). The problem is attributed to the fact that eddies in this limit are near a state of complete barotropization so that the correction of partial barotropization has no practical use. Below we will first comment on why the correction is useful to avoid the degeneracy problem for κ*0.2. Then we come back to discuss possible ways to interpret the discrepancy for κ*<0.2.

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., V/lmU/λ). On the other hand, the damping rate can vary with scales for quadratic drag but not for linear drag. This is seen in (11) and (17) that, when making the barotropization approximation, the damping rate R scales as RquadraticμV and Rlinearκ for quadratic and linear drag. Evidently, Rquadratic can increase as cascade proceeds to increase both V and lm, but Rlinear 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 Rlinear,corrκ(1 − 2U/V). It has scale dependency analogous to Rquadratic. As V/U increases, Rlinear,corr also increases because eddies become more barotropized (i.e., smaller −2U/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 κ* decreases, the barotropic energy level (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 κ*=0.1, giving a negligible correction of 1.3%. This means that the damping rate would revert back to the scale-independent form Rlinear,corrRlinearκ. Then, as argued by Smith and Vallis (2002), the inverse cascade at κ*=0.1 cannot be arrested by the linear drag alone. The theory hence predicts continued accumulation of barotropic eddy energy, leading to the observed large discrepancy in Fig. 5.

Physically speaking, the above interpretation has attributed the discrepancy to the possibility that the theory lacks representations of certain cascade-stopping mechanisms. At κ*=0.1, the eddy field in the numerical simulation does equilibrate (filled symbols in Fig. 5), but the theoretical prediction does not. The theory essentially predicts infinitely large eddy scales due to the inability of scale-independent linear damping to stop the cascade (black and gray curves in Fig. 5). Alternatively, one may interpret the discrepancy as departure of the underlying dynamics from a turbulent cascade regime. That is, eddy properties in this limit may be better described by vortex interactions instead. This alternative interpretation is supported by the fact that GF20’s vortex gas theory predicts the eddy scales rather well at κ*=0.1 (thin dashed curve in Fig. 5). However, as summarized in section 5 and detailed in appendix B, further analyses have found evidence that is against this alternative interpretation. By conducting a range of sensitivity calculations using a vortex-finding algorithm, we find the heat transport pattern and composite- averaged vortex velocity to deviate from the guiding assumptions of the vortex gas model. It thus seems reasonable to reject (at least temporarily) the alternative interpretation.

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 κ*=0.1? At the time we cannot offer a definite answer for it. However, some preliminary analyses have led us to speculate that eddies at κ*=0.1 may have begun to feel the arresting effect of domain-size limitation. One indirect evidence for this comes from evaluations of a length sale ratio. TY06 found that the ratio of mixing length lm to a length characterizing the peak of a barotropic energy spectrum (k01) is a good indicator for the domain limitation. This is because, as the drag strength decreases, lm is saturated earlier than k01, thereby causing the ratio to drop abruptly (see TY06’s Fig. 6a). Using their definition of k0|ψ|2/ψ2, we plot lmk0 against κ* for the L002 experiments (not shown). Consistent with TY06, the length scale ratio first increases as κ* decreases. But the ratio exhibits a clear drop at κ*=0.1, suggesting that the mixing length of this case is likely limited by the domain size. Another indirect evidence is obtained by inferring the transition point in the TY06 data. TY06 showed the saturation of lm/λ as κ* decreases for different experiments in their Fig. 6b. We digitized the figure to estimate the critical lm of the domain saturation. Two methods are tested: by visual inspection and by first fitting the converged portion of the curve (i.e., over κ*>0.2) and then finding the point where log10(lm/λ) deviated from the extrapolated fit by 10%. The resulting estimates give fairly consistent values among different experiments: The domain saturation begins when the domain-size to mixing length ratio (i.e., 2π/lm) is around 2.5 to 4. For our cases, the value of this ratio is 3.5 at κ*=0.1and16.1 at κ*=0.2. Clearly, if TY06’s result is applicable to our cases, the κ*=0.1 case is at the range where domain saturation occurs, but κ*=0.2 case is far from the transition. This result again suggests a high likelihood of domain limitation at κ*=0.1.

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 κ*<0.2, it is in fact difficult to envision that the data-theory discrepancy is due only to the domain-size effect. It is possible that our overly simplified representation of variable cascade fraction using (14) contributes to part of the discrepancy (see below). Undoubtedly, further clarification is needed to sort out the eddy behavior in the weak linear drag asymptote.

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/εp as represented by εc/εp) for the absence of an inertial range (see Fig. 4). By recognizing μ*λ/μ1 as an approximate, inverse measure of scale separation between the input and dissipation, we have interpreted the increase in turnover time with μ* as a result of reduced cascade fraction when the scale separation decreases. Such an interpretation is supported by the direct calculations of barotropic flux per input (|Π|max/εp) 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 Te/TEady(lm/λ)x/2. With x = 0.72 from empirical fits and the scaling for mixing length [their Eq. (16)], one arrives at Te/TEadyμ*0.21—the turnover time showing a positive power-law relation with the drag strength that agrees qualitatively with (14). Despite the consistency with CH19, one should be aware of the limitations of our empirical approach. The power laws of (14) and (16) cannot be correct at sufficiently small drag, as the cascade fraction, |Π|max/εp, would exceed one and hence become unphysical. From Fig. 4c, the cascade fraction has not reached this limit at μ*=103. But it is not known at what drag strength the cascade fraction will begin to saturate and hence sets the limit to which the cascade correction may be useful. More fundamentally though, we need a quantitative theory for barotropic cascade when an inertial range is not present.

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

Below we briefly summarize the governing equations used for simulations and the total energy equation for balance diagnostics. The formulations are identical to GF20, and we largely follow their notations. The two-layer quasigeostrophic potential vorticity (QGPV), with imposed zonal mean flow of ±U in equal-thickness layers on an f plane, evolves following
tq1+Uxq1+(U/λ2)xψ1+J(ψ1,q1)=ν8q1,  and
tq2Uxq2(U/λ2)xψ2+J(ψ2,q2)=ν8q2+drag
with QGPV for the layer 1 (upper) and 2 (lower) of
q1=2ψ1+12λ2(ψ2ψ1),q2=2ψ2+12λ2(ψ1ψ2).
In the above equations, ψ1 and ψ2 denote the eddy part of the streamfunction [e.g., (u1, υ1) = (−∂yψ1, ∂xψ1)], 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
drag={μ[x(|ψ2|xψ2)+y(|ψ2|yψ2)]   forquadraticdrag2κ2ψ2forlineardrag,
with μ (unit of m−1) and κ (unit of s−1) being the respective drag coefficients.
Following Arbic and Flierl (2004), the total energy equation in layer form is obtained by multiplying (A1) by −ψ1, (A2) by −ψ2, 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
t12[12(ψ1)2+12(ψ2)2+12λ2(ψ1ψ2)22]dA=U2λ2ψ1xψ2dA12ψ2(drag)dA+ν2ψ18q1+ψ28q2dA.
Using the following definitions for domain averaged quantities,
E=1A12[12(ψ1)2+12(ψ2)2+12λ2(ψ1ψ2)22]dA,
εp=1A(U2λ2ψ1xψ2dA)=[1A(xψτ)dA]Uλ2=ψxτUλ2,
ε=1A[12ψ2(drag)dA]={μ2[1A(u22+υ22)3dA]=μ2|u2|3κ[1A(u22+υ22)2dA]=κ|u2|2,
hyper=1A(ν2ψ18q1+ψ28q2dA)=ν2ψ18q1+ψ28q2,
we obtain the domain-averaged energy equation
tE=εp+ε+hyper,
where the angle brackets  ⟩=A1dA denote a domain (i.e., horizontal) average. In (A6)(A10), the domain-averaged total energy E consists of KE (first and second terms) and PE (third term) components, εp is the baroclinic energy input (see main text), and 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 ≡ (ψ1 + ψ2)/2 and τ ≡ (ψ1ψ2)/2.
The numerical experiments are carried out using the dimensionless version of (A1)(A3). Following GF20, these equations are made dimensionless using the spatial scale L that characterize a square domain length of 2πL 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
t˜q1˜+x˜q1˜+(L2λ2)x˜ψ1˜+J˜(ψ1˜,q1˜)=ν˜˜8q1˜,
t˜q2˜x˜q2˜(L2λ2)x˜ψ2˜+J˜(ψ2˜,q2˜)=ν˜˜8q2˜μ*(Lλ)[x˜(|˜ψ2˜|x˜ψ2˜)+y˜(|˜ψ2˜|y˜ψ2˜)],
with q1˜=˜2ψ1˜+(1/2)(L2/λ2)(ψ2˜ψ1˜) and q2˜=˜2ψ2˜+ (1/2)(L2/λ2)(ψ1˜ψ2˜). In (A11) and (A12), three nondimensional parameters emerge. They are L/λ measuring the domain size relative to the deformation radius and μ* (≡ μℓ) and ν˜ [(≡ ν/(UL7)]) characterizing the strength of bottom drag and hyperviscous damping, respectively. The experimental designs refer to the main text.
To illustrate the influences of barotropization approximation applied to bottom drag dissipation [i.e., approximating ε by εbt in Eq. (2)], we also carry out experiments in which ε is formulated using pure barotropic velocity (i.e., imposing ε = εbt). 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 < εbt; L002) or removed (i.e., ε = εbt; BT drag). For implementation, it is easier to impose ε = εbt in the modal form of QGPV equations:
tqbt+J(ψ,qbt)+J(τ,2τ)+Ux2τ=ν8qbt+drag/2,
tqbc+J(ψ,qbc)+J(τ,2ψ)+Ux2ψ+Uλ2xψ=ν8qbcdrag/2,
where qbt = ∇2ψ and qbc = ∇2ττ/λ2. Then, for BT drag experiments, we simply set +drag/2 → − μ[∂x(|∇ψ|∂xψ) + ∂y(|∇ψ|∂yψ)] 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.
Last, the barotropic energy balance in spectral space is analyzed in section 4c to help interpret the use of a drag-dependent cascade rate per energy input. Following Larichev and Held (1995), the spectral budget can be obtained by multiplying Fourier component of (A13) with the complex conjugate of the streamfunction (ψk,l*). This yields
[(k2+l2)|ψk,l|2/2]/t=Re[ψk,l*Jk,l(ψ,2ψ)](I)+Re[ψk,l*Jk,l(τ,2τ)](II)+URe[ψk,l*(2τ/x)k,l](III)+drag(IV)+hyper(V).
The terms on the right-hand side may be interpreted as convergence of barotropic energy flux (I), baroclinic-to-barotropic conversion (II), interaction with mean shear (III), and dissipation of barotropic energy due to bottom drag (IV) and hyperviscosity (V). Defining −∂Π/∂K = (I) following Scott and Wang (2005), we can obtain the barotropic flux by integrating this expression from a wavenumber K to infinity:
Π(K)=K=k2+l2Re[ψk,l*Jk,l(ψ,2ψ)]dK.

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., |ubt|3Viv3(liv/λ), as opposed to Viv3]. This enhanced dissipation is crucial in weakening the drag sensitivity as compared with the Held scaling for quadratic drag [μ*0.5 versus μ*1 in Eqs. (6) and (4)] and in avoiding the degeneracy problem for linear drag. As is emphasized in the main text, our examinations below only look for general consistency, as the above assumptions are guiding principles for developing a scaling theory and shall not be taken at face value.

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., lm is set by liv in the vortex gas model). Once a pair is found, their range of influences is defined by two radii, lm/2 or lm, from the vortex centers, to evaluate the effects of different ranges. It is known that the vortex finding method applied here is sensitive to OW threshold. To explore this sensitivity, we test different thresholds with OWc = 0.05, 0.02, 0.01, and 0.005. For example, a value of OWc = 0.05 as in TY06 indicates a threshold equal to 5% of the maximum OW magnitude in the field.

In Fig. B1, an example of instantaneous barotropic vorticity ∇2ψ, baroclinic streamfunction τ, and heat fluxes ψxτ is shown. From Fig. B1a, the vortex finding algorithm with OWc = 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 r=lm. By visual inspections, we cannot find clear indications for enhanced heat transport occurring only within regions of vortex interactions. For example, regions indicated by the red arrows show large positive heat fluxes, but these regions appear to be associated with elongated filaments in Figs. B1a and B1b, away from vortex pairs. Note that the snapshots shown in Fig. B1 are not a special case. We inspect all heat flux maps during the steady period and find a consistent pattern: heat fluxes outside regions of vortex interactions look similar to those inside.

Fig. B1.
Fig. B1.

An example of instantaneous structure of (a) barotropic vorticity (∇2ψ, normalized by U/λ), (b) baroclinic streamfunction (τ, normalized by ), and (c) meridional heat flux (ψxτ, normalized by U2λ) for a case with linear drag strength κ*=0.1. This case has approached the weak drag limit where eddies are nearly barotropized (i.e., εbt/ε = 1.01 ≈ 1). This regime is also where the proposed theory fails but the vortex gas scaling works well (see Fig. 5). In each panel, cyclonic and anticyclonic vortices identified by the vortex finding algorithms are denoted by cyan and red circles, with OWc = 0.02. In (c), two ranges of vortex-pair influences defined using a radius of lm/2 and lm are indicated by the solid and dashed yellow contours. The red arrows indicate regions which are away from vortex pairs but exhibit large heat fluxes.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

We can quantify the heat fluxes inside and outside regions of vortex interactions. Since the heat flux is proportional to energy production εp, we calculate the area averaged production as (AψxτU/λ2dA)/(AdA), where the integral area 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 εp and plotted against different choices of OWc in Fig. B2a. The main feature to note is that, despite the use of different OWc 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 OWc and ranges. A larger range gives a larger area fraction (filled and open symbols for r=lm and lm/2, respectively). A lower OWc allows more vortices to be found, leading to a larger area occupation. For example, at OWc = 0.005, the entire surface area is within vortex range of influences. We thus see the values of averaged flux/εp 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/εp 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.

Fig. B2.
Fig. B2.

(a) Conditionally averaged heat fluxes outside (triangles) and within (circles) vortex pairs and (b) area fraction occupied by vortex pairs, plotted against different Okubo–Weiss threshold (OWc) for vortex identification. In (a) and (b), ranges of vortex-pair influences with r=lm/2 and lm (see Fig. B1) are denoted by open and filled symbols, respectively. In (a), the conditional average fluxes are normalized by the domain-averaged value εp. The flux is not shown if the corresponding area fraction is below 0.05. (c),(d) The composite averaged meridional velocity for cyclonic vortices. In (c), the blue shading indicates one standard deviation of the composite average for OWc = 0.02. In (d), the velocity is normalized by the value at r=lm/2 to help evaluate if the averaged vortex velocity follows r−1 profile (i.e., υ/υr=0.5lm(r/0.5lm)1). Two reference slopes are given.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0102.1

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 (x/lm) and then take average. The profiles of cyclones for different OWc are shown in Fig. B2c. One can see that the use of larger OWc picks out stronger vortices, which gives a stronger composite velocity (e.g., red versus orange curves). The shading denoting one standard deviation for OWc = 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).

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=0.5lm (i.e., center of a vortex pair in GF20; vertical dashed line in Fig. B2c). The choice of this reference location is unimportant because we are only after the power-law exponent. For a r−1 profile, we expect υ/υr=0.5lm(r/0.5lm)1. Due to the symmetry, only one side of the profiles is shown. As can be seen in Fig. B2d, the averaged vortex velocity cannot be described by a single slope. Yet, over the region of r/lm=0.030.2 where vortex velocity is stronger (υ/υr=0.5lm=2.57.5) and hence more important for dissipation, the averaged velocity profiles are notably gentler than the −1 slope. For all OWc 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 υ/Viv(r/lm)n. It can be shown that |ubt|2Viv2lm2n2λlmr2n+1dr and |ubt|3Viv3lm3n2λlmr3n+1dr. For n = 1, we recover the vortex gas results, with |ubt|2Viv2ln(lm/λ) and |ubt|3Viv3(lm/λ). However, for n < 1, the vortex gas scaling is significantly modified. With n ≠ 1, the above integral becomes |ubt|2Viv2lm2n2(r2n+2|λlm)/(2n+2). Taking n < 1 and lmλ, this simplifies to an approximate scaling relation of |ubt|2Viv2. Replacing the dissipation in (5) by εκ|ubt|2κViv2, the energy balance constraint becomes lm/Vivκ*(λ/U)—this has the same functional form as the energy conversion constraint [second equation in (5)]. In other words, the vortex gas model may run into the degeneracy problem when the vortex velocity profile is significantly flatter than the assumed r−1 shape.

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Klein, P., and Coauthors, 2019: Ocean-scale interactions from space. Earth Space Sci., 6, 795817, https://doi.org/10.1029/2018EA000492.

    • Search Google Scholar
    • Export Citation
  • Larichev, V. D., and I. M. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr., 25, 22852297, https://doi.org/10.1175/1520-0485(1995)025<2285:EAAFIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417443, https://doi.org/10.1017/S0022112075001504.

  • Salmon, R., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167211, https://doi.org/10.1080/03091928008241178.

    • Search Google Scholar
    • Export Citation
  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Scott, R. B., and F. Wang, 2005: Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr., 35, 16501666, https://doi.org/10.1175/JPO2771.1.

    • Search Google Scholar
    • Export Citation
  • Scott, R. B., and B. K. Arbic, 2007: Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics. J. Phys. Oceanogr., 37, 673688, https://doi.org/10.1175/JPO3027.1.

    • Search Google Scholar
    • Export Citation
  • Smith, K. S., and G. K. Vallis, 2002: The scales and equilibration of midocean eddies: Forced-dissipative flow. J. Phys. Oceanogr., 32, 16991720, https://doi.org/10.1175/1520-0485(2002)032<1699:TSAEOM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Thompson, A. F., and W. R. Young, 2006: Scaling baroclinic eddy fluxes: Vortices and energy balance. J. Phys. Oceanogr., 36, 720738, https://doi.org/10.1175/JPO2874.1.

    • Search Google Scholar
    • Export Citation
  • Thompson, A. F., and W. R. Young, 2007: Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. J. Atmos. Sci., 64, 32143231, https://doi.org/10.1175/JAS4000.1.

    • Search Google Scholar
    • Export Citation
Save
  • Arbic, B. K., and G. R. Flierl, 2004: Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom Ekman friction: Application to midocean eddies. J. Phys. Oceanogr., 34, 22572273, https://doi.org/10.1175/1520-0485(2004)034<2257:BUGTIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Arbic, B. K., and R. B. Scott, 2008: On quadratic bottom drag, geostrophic turbulence, and oceanic mesoscale eddies. J. Phys. Oceanogr., 38, 84103, https://doi.org/10.1175/2007JPO3653.1.

    • Search Google Scholar
    • Export Citation
  • Burns, K. J., G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, 2020: Dedalus: A flexible framework for numerical simulations with spectral methods. Phys. Rev. Res., 2, 023068, https://doi.org/10.1103/PhysRevResearch.2.023068.

    • Search Google Scholar
    • Export Citation
  • Cessi, P., 2008: An energy-constrained parameterization of eddy buoyancy flux. J. Phys. Oceanogr., 38, 18071819, https://doi.org/10.1175/2007JPO3812.1.

    • Search Google Scholar
    • Export Citation
  • Chang, C.-Y., and I. M. Held, 2019: The control of surface friction on the scales of baroclinic eddies in a homogeneous quasigeostrophic two-layer model. J. Atmos. Sci., 76, 16271643, https://doi.org/10.1175/JAS-D-18-0333.1.

    • Search Google Scholar
    • Export Citation
  • Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 10871095, https://doi.org/10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ferrari, R., and C. Wunsch, 2009: Ocean circulation kinetic energy: Reservoirs, sources and sinks. Annu. Rev. Fluid Mech., 41, 253282, https://doi.org/10.1146/annurev.fluid.40.111406.102139.

    • Search Google Scholar
    • Export Citation
  • Gallet, B., and R. Ferrari, 2020: The vortex gas scaling regime of baroclinic turbulence. Proc. Natl. Acad. Sci. USA, 117, 44914497, https://doi.org/10.1073/pnas.1916272117.

    • Search Google Scholar
    • Export Citation
  • Gallet, B., and R. Ferrari, 2021: A quantitative scaling theory for meridional heat transport in planetary atmospheres and oceans. AGU Adv., 2, e2020AV000362, https://doi.org/10.1029/2020AV000362.

    • Search Google Scholar
    • Export Citation
  • Griffies, S. M., and Coauthors, 2000: Developments in ocean climate modelling. Ocean Modell., 2, 123192, https://doi.org/10.1016/S1463-5003(00)00014-7.

    • Search Google Scholar
    • Export Citation
  • Haidvogel, D. B., and I. M. Held, 1980: Homogeneous quasi-geostrophic turbulence driven by a uniform temperature gradient. J. Atmos. Sci., 37, 26442660, https://doi.org/10.1175/1520-0469(1980)037<2644:HQGTDB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., 1999: The macroturbulence of the troposphere. Tellus, 51A, 5970, https://doi.org/10.3402/tellusb.v51i1.16260.

  • Held, I. M., and V. D. Larichev, 1996: A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane. J. Atmos. Sci., 53, 946952, https://doi.org/10.1175/1520-0469(1996)053<0946:ASTFHH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hua, B. L., and D. B. Haidvogel, 1986: Numerical simulations of the vertical structure of quasi-geostrophic turbulence. J. Atmos. Sci., 43, 29232936, https://doi.org/10.1175/1520-0469(1986)043<2923:NSOTVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jansen, M. F., A. J. Adcroft, R. Hallberg, and I. M. Held, 2015: Parameterization of eddy fluxes based on a mesoscale energy budget. Ocean Modell., 92, 2841, https://doi.org/10.1016/j.ocemod.2015.05.007.

    • Search Google Scholar
    • Export Citation
  • Klein, P., and Coauthors, 2019: Ocean-scale interactions from space. Earth Space Sci., 6, 795817, https://doi.org/10.1029/2018EA000492.

    • Search Google Scholar
    • Export Citation
  • Larichev, V. D., and I. M. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr., 25, 22852297, https://doi.org/10.1175/1520-0485(1995)025<2285:EAAFIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417443, https://doi.org/10.1017/S0022112075001504.

  • Salmon, R., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167211, https://doi.org/10.1080/03091928008241178.

    • Search Google Scholar
    • Export Citation
  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Scott, R. B., and F. Wang, 2005: Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr., 35, 16501666, https://doi.org/10.1175/JPO2771.1.

    • Search Google Scholar
    • Export Citation
  • Scott, R. B., and B. K. Arbic, 2007: Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics. J. Phys. Oceanogr., 37, 673688, https://doi.org/10.1175/JPO3027.1.

    • Search Google Scholar
    • Export Citation
  • Smith, K. S., and G. K. Vallis, 2002: The scales and equilibration of midocean eddies: Forced-dissipative flow. J. Phys. Oceanogr., 32, 16991720, https://doi.org/10.1175/1520-0485(2002)032<1699:TSAEOM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Thompson, A. F., and W. R. Young, 2006: Scaling baroclinic eddy fluxes: Vortices and energy balance. J. Phys. Oceanogr., 36, 720738, https://doi.org/10.1175/JPO2874.1.

    • Search Google Scholar
    • Export Citation
  • Thompson, A. F., and W. R. Young, 2007: Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. J. Atmos. Sci., 64, 32143231, https://doi.org/10.1175/JAS4000.1.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Responses of (a) mixing length lm and (b) diffusivity D to varied dimensionless quadratic drag strength μ*. The experiments reported in GF20 are denoted by the filled red squares. The black and gray open circles correspond to GF and L002 in Table 1, with the GF set designed to repeat GF20’s results. The solid and dashed lines denote the vortex gas [Eq. (6)] and Held scaling [Eq. (4)], respectively. (c) An example of eddy energy budget [Eq. (A10)] with μ*=10 in the GF set. The dissipation approximated using pure barotropic velocity [εbt in Eq. (2)] is shown as the orange curve. The steady-state quantities are obtained by averaging over the last 200 time units throughout this work. Steady balances between the baroclinic production εp and dissipation due to bottom drag ε and hyperviscosity for different cases are shown as the cross symbols in (b).

  • Fig. 2.

    Evaluations of three assumptions in the Held model for different drag strength μ*: (a) the barotropization approximation (εεbt), (b) diffusive representation of heat fluxes in terms of mixing length and barotropic velocity (DVlm), and (c) existence of an inertial range [εcεp; Eq. (8)] In (a), a corrected dissipation using Eq. (11) is also plotted as open diamonds for comparison. The cases correspond to the L002 set with ν˜=1017.

  • Fig. 3.

    Evaluations of eddy scale predictions for (a) rms barotropic velocity V, (b) mixing length lm, and (c) diffusivity D in the case of quadratic drag. In each panel, the thick gray curve indicates the Held theory modified to incorporate partial barotropization [Eqs. (12) and (13)]. The thick black curve is with an additional correction of drag-dependent cascade rate per input [Eqs. (14) and (15)]. The vortex gas scaling in (6) is denoted by the thin dashed line, whereas the original Held scaling in (4) is the thin gray line. In (a), there is an extra curve for the vortex gas prediction. This curve is offset from the best-fit curve by a factor of 5 to illustrate that it concaves downward, opposite to the simulations. In (c), the simulation results from CH19 are included as cross symbols.

  • Fig. 4.

    (a) An example of spectral budget of barotropic energy for the case with μ*=0.01. The budget terms are defined in Eq. (A15) and are obtained by integrating over a shell of constant total wavenumber K with dK = 1 (K is made dimensionless by domain length L); (b) distribution of barotropic flux normalized by the baroclinic energy input in wavenumber space. The spectral flux is obtained by integrating the flux convergence term in (a) [see Eq. (A16) for definition. The inset is a schematic for spectral flux in the presence of an inertial range. (c) The magnitude of normalized barotropic flux is plotted against μ*. The dashed line indicates the power-law fit of (16). The cross symbols are differences of wavenumber centroids (KC,IIKC,IV) that measure the scale separation between the energy conversion and dissipation spectra (see main text).

  • Fig. 5.

    As in Fig. 3, but for linear bottom drag. In these panels, only the vortex gas scaling is shown because the original Held model cannot be applied to linear drag due to the degeneration problem. In (a), values of εbt/ε for selected cases are labeled to show that, first, the errors of barotropization approximation increase steeply with drag strength, and second, the case with κ*=0.1 is near a state of complete barotropization (i.e., weak drag limit) so that the proposed correction for partial barotropization in (17) has no use.

  • Fig. 6.

    (a) Degree of barotropization in the eddy field as measured by the bottom to barotropic KE ratio is plotted against the drag strength. The filled symbols are the present experiments. The thick gray curve is the predictions from Eq. (22) and (13), whereas the thick black curve is with an additional cascade rate correction [Eqs. (22) and (15)]. Open symbols are the unstable zonal jet experiments reported in Jansen et al. (2015, denoted as J15), with the red dashed curve denoting their empirical fit. (b) Comparison of the rms barotropic velocity with (filled symbols) and without (open symbols) partial barotropization [see Eqs. (A13) and (A14) in appendix A and the main text].

  • Fig. B1.

    An example of instantaneous structure of (a) barotropic vorticity (∇2ψ, normalized by U/λ), (b) baroclinic streamfunction (τ, normalized by ), and (c) meridional heat flux (ψxτ, normalized by U2λ) for a case with linear drag strength κ*=0.1. This case has approached the weak drag limit where eddies are nearly barotropized (i.e., εbt/ε = 1.01 ≈ 1). This regime is also where the proposed theory fails but the vortex gas scaling works well (see Fig. 5). In each panel, cyclonic and anticyclonic vortices identified by the vortex finding algorithms are denoted by cyan and red circles, with OWc = 0.02. In (c), two ranges of vortex-pair influences defined using a radius of lm/2 and lm are indicated by the solid and dashed yellow contours. The red arrows indicate regions which are away from vortex pairs but exhibit large heat fluxes.

  • Fig. B2.

    (a) Conditionally averaged heat fluxes outside (triangles) and within (circles) vortex pairs and (b) area fraction occupied by vortex pairs, plotted against different Okubo–Weiss threshold (OWc) for vortex identification. In (a) and (b), ranges of vortex-pair influences with r=lm/2 and lm (see Fig. B1) are denoted by open and filled symbols, respectively. In (a), the conditional average fluxes are normalized by the domain-averaged value εp. The flux is not shown if the corresponding area fraction is below 0.05. (c),(d) The composite averaged meridional velocity for cyclonic vortices. In (c), the blue shading indicates one standard deviation of the composite average for OWc = 0.02. In (d), the velocity is normalized by the value at r=lm/2 to help evaluate if the averaged vortex velocity follows r−1 profile (i.e., υ/υr=0.5lm(r/0.5lm)1). Two reference slopes are given.

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