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.

It has been known that the Held scaling cannot be applied to linear drag because the two balances in (3) reduce to the same functional form (e.g., Smith and Vallis 2002). This is referred to as the degeneracy problem in this study. For linear drag, the dissipation under the barotropization approximation scales as ε_bt ∼ κV². Substituting it into the energy balance yields $({\mathcal{l}}_m/V)\sim {\kappa }_{*}(\lambda /U)$, where ${\kappa }_{*}$ (≡ 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.

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, ${\mu }_{*}$ (≡ μℓ), and $\tilde{\nu }\hspace{0.17em}$ (≡ 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 $\tilde{\nu }$ 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 ${\mu }_{*}$ (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), $\tilde{\nu }={10}^{-13}$, and ${\mu }_{*}=0.003\text{–}1$ as in GF20. The set “hyper” is to find a suitable value of $\tilde{\nu }$ that yields negligible hyperviscous dissipation as compared with bottom drag. A $\tilde{\nu }$ value of 10⁻¹⁷, much lower than GF20’s, is found (see section 3c below). Setting $\tilde{\nu }={10}^{-17}$, 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 ${\mu }_{*}$ values overlapped with L002. These sets are to ensure the robustness of numerical solutions and to expand ${\mu }_{*}$ range. The combination of L001, L002, and L004 gives a wide ${\mu }_{*}$ range of 10⁻³–10 for quadratic drag and ${\kappa }_{*}$ 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 (${\mu }_{*}$ and ${\kappa }_{*}$ are for quadratic and linear drag, respectively). See main text and appendix A for the definitions of the three dimensionless parameters.

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b. Eddy scale definitions and model validation

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 ${\mathcal{l}}_m$ and D are shown for different bottom drag strength ${\mu }_{*}$. 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.

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Responses of (a) mixing length ${\mathcal{l}}_m$ and (b) diffusivity D to varied dimensionless quadratic drag strength ${\mu }_{*}$. 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 ${\mu }_{*}=10$ in the GF set. The dissipation approximated using pure barotropic velocity [ε_bt 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 ε_p 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

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Since our focus is to examine the barotropization approximation across different drag strength, that is, approximating ε by ε_bt in (2) for different ${\mu }_{*}$, 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 ${\mu }_{*}=0.1$ 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 (ε_bt; orange) overestimates the true dissipation by more than a factor of 1.5 (see more in section 4). The accuracy of dissipation estimates is also confirmed for other cases. We evaluate the steady-state energy balance by averaging the budget terms over the last 200 dimensionless time units. This is equivalent to 200 T_Eady, where T_Eady ≡ λ/U. In Fig. 1b, the steady-state values of −ε/(ε_p + hyper) are close to one for all cases (cross symbols), again suggesting a closed three-way balance in (1) and supporting the accuracy of the dissipation estimates.

c. Reducing hyperviscosity

A problem revealed by the energy budget in Fig. 1c is that the hyperviscous dissipation is not negligible in the GF cases. As described in section 2, both vortex gas and Held models assume hyperviscosity to be negligible as compared with bottom drag in (1). This limit is not yet reached with the value of $\tilde{\nu }={10}^{-13}$ 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 $\tilde{\nu }$ 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 $\tilde{\nu }$ is reduced to 10⁻¹⁷. The mixing length and barotropic eddy velocity, both normalized by the values with $\tilde{\nu }={10}^{-13}$, also show sign of convergence.

Table 2

Sensitivity tests aiming to find suitable value of $\tilde{\nu }$ that yields negligible hyperviscous energy dissipation as compared with that due to bottom drag [i.e., small hyper/(ε + hyper)]. These tests have ${\mu }_{*}=0.03$.

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Quite surprisingly, though, the error of neglecting hyperviscosity has a minor effect on the eddy scales. In Figs. 1a and 1b, when we use $\tilde{\nu }$ that is four orders of magnitude smaller ($\tilde{\nu }={10}^{-17}$; gray circles), the dependence of mixing length and diffusivity on ${\mu }_{*}$ do not show significant changes. This is because the relative contribution of hyperviscous dissipation is insensitive to drag changes. For ${\mu }_{*}$ 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 $\tilde{\nu }={10}^{-17}$ or lower. With a sufficiently low $\tilde{\nu }$, we have confirmed that there is a tight balance between production ε_p and bottom drag dissipation ε at the steady state for all cases.

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 $\epsilon =-(\mu /2)⟨{\left|{\mathbf{u}}_2\right|}^3⟩$, whereas the approximation in (2) has ${\epsilon }_{\text{bt}}=-(\mu /2)⟨{\left|{\mathbf{u}}_{\text{bt}}\right|}^3⟩$. If the barotropization assumption is valid, we expect the ratio of ε_bt/ε to be roughly a constant among different cases.

In Fig. 2a, the dissipation ratio ε_bt/ε is plotted against ${\mu }_{*}$ (gray circles). Inconsistent with the barotropization assumption, the ratio is not approximately constant across different drag strength. It increases steeply from 1.06 at ${\mu }_{*}=0.001$ to 3.7 at ${\mu }_{*}=0.03$. This means that, at ${\mu }_{*}=0.001$, the error of making the barotropization approximation is negligible at only 6%. However, toward O(0.1) drag with ${\mu }_{*}=0.1\text{and}0.3$, the error becomes a strong overestimation of ε by a factor of 2.2 and 3.7, respectively. Note that this strong ${\mu }_{*}$ dependence was not considered in the Held and vortex gas scalings as both simply took ε ∼ ε_bt in (3) and (5).

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Evaluations of three assumptions in the Held model for different drag strength ${\mu }_{*}$: (a) the barotropization approximation (ε ≈ ε_bt), (b) diffusive representation of heat fluxes in terms of mixing length and barotropic velocity ($D\sim V{\mathcal{l}}_m$), and (c) existence of an inertial range [ε_c ∼ ε_p; 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 $\tilde{\nu }={10}^{-17}$.

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

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

We may estimate the influence of barotropization error on the scaling theories. The barotropization error refers to the degree of ε_bt/ε deviated from one. For simplicity, we represent the ${\mu }_{*}$ dependence as a power law ${\epsilon }_{\text{bt}}/\epsilon \sim {\mu }_{*}^n$, where the exponent n is obtained by power-law fits in Fig. 2a. For the Held scaling, when combining ${\epsilon }_{\text{bt}}/\epsilon \sim {\mu }_{*}^n$ with ε ≈ ε_p in (1) and ε_bt ∼ μV³ and ${\mathcal{l}}_m/V\sim \lambda /U$ in (3), we obtain a scaling that incorporates the barotropization error for the mixing length as ${\mathcal{l}}_m/\lambda \sim {\mu }_{*}^{-(1-n)}$. Similarly, for vortex gas, the modified scaling becomes ${\mathcal{l}}_m/\lambda \sim {\mathcal{l}}_{\text{iv}}/\lambda \sim {\mu }_{*}^{-0.5(1-n)}$. 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 ${\mu }_{*}=0.001\text{–}0.03$ and ${\mu }_{*}=0.03\text{–}0.3$, 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 ${\mu }_{*}=0.001\text{–}0.03$. 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 ε_p and an assumption of inertial range for barotropic inverse cascade. These assumptions are examined as well.

In (3), the heat flux 〈ψ_xτ〉 in the baroclinic energy production ε_p is expressed as downgradient diffusive transport, with a diffusivity $D\sim V{\mathcal{l}}_m$. This diffusive representation is well supported by the simulations. In Fig. 2b, we compare the proportionality coefficient for the diffusive transport, $⟨{\psi }_x\tau ⟩/(V{\mathcal{l}}_{m}U)$, 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 ${\mu }_{*}$ 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).

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

$\epsilon =\frac{\mu }{2}⟨{\left|{\mathbf{u}}_2\right|}^3⟩\approx \frac{\mu }{2}{2}^{3/2}⟨{\left|{\upsilon }_2\right|}^3⟩\approx \mu 2^{1/2}1.2{(\sqrt{⟨{\upsilon }_2^2⟩})}^3.$(9)

In (9), the factor of 2^3/2 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)2^3/2 = 1.2. Calculating the factor directly in the experiments yields a value of around 1.3. For the purpose of developing an approximate theory, we simply use a constant value of 1.2.

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 $⟨{\upsilon }_2^2⟩=⟨{\upsilon }_{\text{bt}}^2⟩-2⟨{\upsilon }_{\text{bt}}{\upsilon }_{\text{bc}}⟩+⟨{\upsilon }_{\text{bc}}^2⟩$. 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., $k_0^{-1}\sim {\mathcal{l}}_m\gg \lambda$; see CH19). Thus, we may take $⟨{\upsilon }_{\text{bt}}^2⟩\gg ⟨{\upsilon }_{\text{bc}}^2⟩$ and approximate 〈υ_bt υ_bc〉 by the covariance near k₀, with $⟨{\upsilon }_{\text{bt}}{\upsilon }_{\text{bc}}⟩\approx {\widehat{{\upsilon }_{\text{bt}}}}_{k0}{\widehat{{\upsilon }_{\text{bc}}}}_{k0}$, where ${\widehat{}}_{k0}$ represents the rms value in the spectral vicinity of k₀ (e.g., Smith and Vallis 2002). We may further express ${\widehat{{\upsilon }_{\text{bt}}}}_{k0}{\widehat{{\upsilon }_{\text{bc}}}}_{k0}\sim VU$. 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 ${\widehat{{\upsilon }_{\text{bt}}}}_{k0}^2\sim V^2$ 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 ${\widehat{{\upsilon }_{\text{bt}}}}_{k0}\sim U$ 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

$⟨{\upsilon }_2^2⟩=⟨{\upsilon }_{\text{bt}}^2⟩-2⟨{\upsilon }_{\text{bt}}{\upsilon }_{\text{bc}}⟩+⟨{\upsilon }_{\text{bc}}^2⟩\approx V^2-2VU.$(10)

Applying the simple correction in (10) reduces the barotropization error. Inserting (10) into (9), we obtain a corrected dissipation

${\epsilon }_{\text{corr}}=\mu 2^{1/2}1.2{(V^2-2VU)}^{3/2}.$(11)

Note that, without the correction term (i.e., −2VU), the dissipation is equivalent to ε_bt in (2) which has been shown to increasingly overestimate the true dissipation ε as the drag strength increases (i.e., ε_bt/ε in Fig. 2a). With the correction, the dissipation estimates improve significantly. The corrected ratio ε_corr/ε as denoted by diamond symbols in Fig. 2a is now mostly near 1, suggesting that the biases are largely removed.

While the above correction has significantly improved the dissipation estimates, its performance degrades as ${\mu }_{*}$ approaches 1 (Fig. 1a). The degradation is likely due to the use of a simplified relation ${\widehat{{\upsilon }_{\text{bc}}}}_{k0}\sim U$ and to representing 〈υ_btυ_bc〉 by only the covariance near k₀. Note that the scaling of ${\widehat{{\upsilon }_{\text{bc}}}}_{k0}\sim U$ derived by Held and Larichev (1996) is strictly valid for scales much greater λ. Such a condition is most likely violated toward ${\mu }_{*}\sim 1$ 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 ε_p with the corrected dissipation ε_corr and invoking the cascade rate constraint [i.e., ε_c/ε_p = constant in (8)] result in a closed set of equations for V and ${\mathcal{l}}_m$,

$(c_{0}V{\mathcal{l}}_m)(U^2/{\lambda }^2)\approx \mu 2^{1/2}1.2{(V^2-2VU)}^{3/2}$(12a)

${\mathcal{l}}_m/V\approx c_1\lambda /U.$(12b)

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 ${\mathcal{l}}_m$ 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 ${\mathcal{l}}_m$ in (12a) by (12b) yields a single equation for the rms barotropic velocity V made dimensionless by the mean flow U

${\alpha }_0{\mu }_{*}^{-2/3}{\left(\frac{V}{U}\right)}^{1/3}=\left(\frac{V}{U}\right)-2$(13)

where ${\alpha }_0={\left(c_0{c}_1/1.2\sqrt{2}\right)}^{2/3}=0.36$. 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 ${\mathcal{l}}_m/\lambda$ from (12b) and the diffusivity via $D/(U\lambda )=c_0(V/U)({\mathcal{l}}_m/\lambda )$. 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., ${\mu }_{*}>0.1$) where using barotropic velocity strongly overestimates the dissipation (e.g., ε_bt/ε > 2 in Fig. 2a), and the correction reduces the steep drop in V/U.

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Evaluations of eddy scale predictions for (a) rms barotropic velocity V, (b) mixing length ${\mathcal{l}}_m$, 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

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Similar results are found for the mixing length ${\mathcal{l}}_m$: adding the barotropization correction weakens the ${\mu }_{*}$ 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., $D=c_{0}V{\mathcal{l}}_m$ 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 ${\mu }_{*}$ 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., ε_bt/ε increases with ${\mu }_{*}$ 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 ${\mu }_{*}=0.001\text{and}0.003$, ε_bt/ε has a value of 1.06 and 1.24, not too far from one. The correction for partial barotropization in (12a) then becomes less important [i.e., $V/U\gg 2\hspace{0.17em}$ 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 ε_c ∼ ε_p, V/U and ${\mathcal{l}}_m/\lambda$ 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 ${\mathcal{l}}_m/\lambda$ responding less sensitively to drag changes (i.e., ${\mu }_{*}^{-0.51}$ versus ${\mu }_{*}^{-0.73}$, consistent with the finding in CH19). We will show below that this different ${\mu }_{*}$ 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 ε_c/ε_p is allowed to vary with ${\mu }_{*}$.

3) Cascade from spectral energy budgets

The empirical relation of $T_e/T_{\text{Eady}}\approx c_2{\mu }_{*}^m$ or equivalently ${\epsilon }_c/{\epsilon }_p\sim {\mu }_{*}^{-2m}$ used above has been interpreted as a result of decreased spectral room for barotropic inverse cascade (decreased ε_c/ε_p) 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 ${\mu }_{*}=0.01$ 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).

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(a) An example of spectral budget of barotropic energy for the case with ${\mu }_{*}=0.01$. 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 ${\mu }_{*}$. The dashed line indicates the power-law fit of (16). The cross symbols are differences of wavenumber centroids (K_C,II − K_C,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

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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., ${\epsilon }_p=\epsilon =\int (\text{IV})dK$], and its wavenumber distribution, Π(K)/ε_p, is shown in Fig. 4b for different cases. Taking the ${\mu }_{*}=0.01$ 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/ε_p 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/ε_p does increase with decreasing drag strength (filled circles in Fig. 4c), approximately following a power-law relation of

${\left|\text{Π}\right|}_{\text{max}}/{\epsilon }_p=0.1{\mu }_{*}^{-1/4}.$(16)

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., $K_C{}_{,\text{II}}=\int (\text{II})KdK/\int (\text{II})dK$] and are denoted by the vertical dashed lines. The scale separation as measured by $(K_C{}_{,\text{II}}-K_{C,\text{IV}})$ is plotted against ${\mu }_{*}$ in Fig. 4c (cross symbols). As can be seen, the separation increases monotonically as ${\mu }_{*}$ 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/ε_p.

Importantly, the power law (16) obtained from direct quantification of barotropic flux is consistent with the empirical relation $T_e/T_{\text{Eady}}\approx c_2{\mu }_{*}^m$. Recognizing that the cascade rate ε_c represents the magnitude of barotropic flux, we can substitute (16) into (8) to find $T_e/T_{\text{Eady}}\sim {({\left|\text{Π}\right|}_{\text{max}}/{\epsilon }_p)}^{-1/2}\sim {\mu }_{*}^{1/8}$. 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

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 (${\kappa }_{*}=0.1$). Possible reasons for the failure of this case will be discussed separately below. Excluding ${\kappa }_{*}=0.1$, 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 ${\kappa }_{*}$ approaches O(1) and beyond (gray curve in Fig. 5a). But, the predicted ${\mathcal{l}}_m/\lambda$ 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 ${\kappa }_{*}\sim 1$.

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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 ε_bt/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 ${\kappa }_{*}=0.1$ 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

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Why does the above theory fail for ${\kappa }_{*}=0.1$? 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 ε_bt/ε are labeled in Fig. 5a for a number of cases. The ${\kappa }_{*}=0.1$ case has ε_bt/ε= 1.01, meaning that using the barotropic velocity to estimate dissipation has a negligible error of only 1%. Neglecting the correction for partial barotropization (−2VU) in (17), the energy balance ε_p = ε becomes $({\mathcal{l}}_m/V)\sim {\kappa }_{*}(\lambda /U)$. 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.

a. Root-mean-square error

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

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 ${10}^{-3}\le {\mu }_{*}\le 10$ for quadratic drag and $0.1\le {\kappa }_{*}\le 2.5$ for linear drag. The errors for full range linear drag are infinite because Eq. (21) cannot be applied to ${\kappa }_{*}=0.1$ (see text). The exact formulations for the vortex gas predictions are ${\mathcal{l}}_m/\lambda =2.6{\mu }_{*}^{-0.5}$, $D/(U\lambda )=1.9{\mu }_{*}^{-1}$, and $V/U=1.4{\mu }_{*}^{-0.5}\sqrt{{\mathcal{l}}_m/\lambda }$ for quadratic drag, and ${\mathcal{l}}_m/\lambda =3.2\text{exp}(0.36/{\kappa }_{*})$, $D/(U\lambda )=1.8\text{exp}(0.72/{\kappa }_{*})$, and $V/U=1.5\text{exp}(0.36/{\kappa }_{*})\sqrt{{\mathcal{l}}_m/\lambda }$ 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 $D=c_{0}V{\mathcal{l}}_m$. 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.

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For quadratic drag, we separate the comparisons into two ranges: ${\mu }_{*}\le 0.1$ 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 ${\mu }_{*}\le 0.1$. 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 ${\mu }_{*}$ sensitivity (i.e., steepened slope) as ${\mu }_{*}$ 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 ${\mathcal{l}}_m/\lambda$ 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 ${\mu }_{*}$ 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 ${\kappa }_{*}<0.17$ [i.e., $c_0{c}_2/(2{\kappa }_{*}^{1-m})<1$], leading to essentially infinite errors (Table 3). To compare the RMSEs, we again exclude the ${\kappa }_{*}=0.1$ 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 ${\kappa }_{*}\ge 0.2$, 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., ε_bt/ε = 3.7 and 9.1 for ${\kappa }_{*}=1\text{and}2$, respectively (see Fig. 5). However, unlike quadratic drag cases where the misfits between the scaling and simulations increase with ${\mu }_{*}$ over O(1) ${\mu }_{*}$ (see Fig. 3), the vortex gas continues to perform well into O(1) ${\kappa }_{*}$. 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 ${\kappa }_{*}=0.1$, 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 ${\kappa }_{*}=0.1$ case further.

b. Vortex gas properties at weak linear drag

One possible interpretation for the superior performance of vortex gas at ${\kappa }_{*}=0.1$ 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 ${\kappa }_{*}=0.1$. 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, $⟨{\left|{\mathbf{u}}_{\text{bt}}\right|}^3⟩\sim V_{\text{iv}}^3({\mathcal{l}}_{\text{iv}}/\lambda )$ as opposed to $V_{\text{iv}}^3$, which weakens the sensitivity to quadratic drag [${\mu }_{*}^{-0.5}$ versus ${\mu }_{*}^{-1}$ 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 ${\kappa }_{*}=0.1$. 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 ${\kappa }_{*}=0.1$ 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 $0.2{\mathcal{l}}_m$ 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.

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), ${\epsilon }_{\text{bt}}/\epsilon \approx {(⟨{\upsilon }_{\text{bt}}^2⟩/⟨{\upsilon }_2^2⟩)}^{3/2}\approx {({\text{KE}}_{\text{bot}}/{\text{KE}}_{\text{bt}})}^{-3/2}$. An increase in ε_bt/ε is equivalent to a decrease in KE_bot/KE_bt, where the ratio of lower-layer kinetic energy (KE_bot) to barotropic KE serves as a measure for the degree of barotropization (i.e., a ratio of 1 means complete barotropization). The theory described in section 4c allows for a theoretical estimate for the KE ratio. Using (10), the KE ratio is

${\text{KE}}_{\text{bot}}/{\text{KE}}_{\text{bt}}\approx ⟨{\upsilon }_2^2⟩/⟨{\upsilon }_{\text{bt}}^2⟩\approx 1-2{(V/U)}^{-1},$(22)

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 ${\mu }_{*}$. 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).

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(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

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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 ${\mu }_{*}\equiv {\lambda /\mu }^{-1}$ is equivalent to their length scale ratio L_d/L_f (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 ${\mu }_{*}\ge 0.03$), 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 |u_bt|; see appendix A]. When the effect of partial barotropization is removed, the concave upward shape in the V/U versus ${\mu }_{*}$ 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.