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
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 εbt ∼ κV2. Substituting it into the energy balance yields
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.
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,
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),
A summary of numerical experiments designing to investigate the responses of eddy scales to varied drag strength (
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
Since our focus is to examine the barotropization approximation across different drag strength, that is, approximating ε by εbt in (2) for different
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
Sensitivity tests aiming to find suitable value of
Quite surprisingly, though, the error of neglecting hyperviscosity has a minor effect on the eddy scales. In Figs. 1a and 1b, when we use
4. Partial barotropization and modifications of held scaling
With the diagnostics laid out in sections 2 and 3 and the experiments that satisfy εp ≈ ε at a steady state, we are in a position to explore the scaling theories further. Below we will illustrate the problem of barotropization approximation and modify the Held model by allowing for partial barotropization. An additional correction of drag-dependent cascade rate per energy input and application to linear drag will also be considered.
a. Problem of the barotropization approximation
We first illustrate the problem of the barotropization approximation by comparing the ratio of true and approximate dissipation among the L002 cases. The true dissipation defined in (1) is
In Fig. 2a, the dissipation ratio εbt/ε is plotted against
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
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
c. Proposed corrections and eddy scale predictions
1) Partial barotropization
While the above correction has significantly improved the dissipation estimates, its performance degrades as
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.,
Similar results are found for the mixing length
Since the diffusivity is well described by the mixing length scaling (i.e.,
A tentative conclusion we may draw from the comparisons in Fig. 3 is that the weakening of drag sensitivity toward large drag is largely attributable to a decrease in the degree of barotropization (i.e., εbt/ε increases with
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 εc ∼ εp, V/U and
2) Adding a cascade correction
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, εp ≈ εbt, in (3). This leads to relations of
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
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.,
Importantly, the power law (16) obtained from direct quantification of barotropic flux is consistent with the empirical relation
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 (
Why does the above theory fail for
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.,
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%.
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
For quadratic drag, we separate the comparisons into two ranges:
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
Focusing back to the low-drag condition, an obvious advantage of the vortex gas scaling over the proposed theory is that it remains applicable toward the weak-drag asymptote: in Fig. 5, the vortex gas describes the eddy scales well at
b. Vortex gas properties at weak linear drag
One possible interpretation for the superior performance of vortex gas at
As reviewed in section 2, the vortex gas model was constructed based on two main assumptions: 1) a vortex dipole is sufficient to represent the bulk property of turbulent transport (see GF20’s Fig. 2) and 2) the vortex velocity scales with r−1 like a point vortex. The latter is crucial because it leads to enhanced dissipation,
Overall, the results show no clear indication that eddy transport is controlled by vortex interactions at
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
The main motivations are that 1) the Held scaling is known to be oversensitive to the quadratic drag strength
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
When compared with the vortex gas scaling, the theory with proposed corrections can extend the eddy scale predictions to O(1) quadratic drag and generally has comparable skills for linear drag (Figs. 3 and 5 and Table 3). However, the vortex gas scaling has an advantage in that it remains applicable in the limit of weak linear drag (i.e., dashed and solid curves diverge for
The analyses described in this study have a number of implications and unresolved issues that warrant further discussion:
a. The degree of eddy barotropization
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
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
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
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
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
c. Unresolved issues
There are a number of unresolved issues raised in this study. The most apparent one is the large discrepancy toward the limit of weak linear drag: the proposed theory (black and gray curves in Fig. 5) cannot represent the eddy scales for
It was pointed out by Smith and Vallis (2002) and demonstrated in section 4d that the Held scaling cannot be applied to linear drag because the balance constraints are degenerate. This has led Smith and Vallis (2002) to argue that linear drag alone cannot stop the inverse cascade. Such an argument may be understood from the perspective of scale dependency in the drag form. Without β, inverse cascade may be arrested at scales near which the damping rate exceeds eddy turnover rate that sets the cross-scale energy transfer. In the Held scaling, the turnover rate is an imposed quantity via (3) and is thus held fixed as the barotropic energy is transferred and accumulated toward large scales (i.e.,
The inclusion of partial barotropization changes the above picture. With the correction in (17), the damping rate for linear drag becomes Rlinear,corr ∼ κ(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
Physically speaking, the above interpretation has attributed the discrepancy to the possibility that the theory lacks representations of certain cascade-stopping mechanisms. At
Following the above reasoning, if we are to proceed with a turbulent cascade model like the one proposed here, we will need to face a lingering question: What are the missing cascade-stopping mechanisms that may help explain the data-theory discrepancy at
As emphasized above, the evidence we provide for domain limitation is only suggestive. To properly evaluate this effect, one will need to conduct additional experiments by increasing the domain size (i.e., expanding the range of λ/L) and further reducing the drag strength. It can be anticipated that, by removing the domain size limitation, both barotropic energy and mixing length in the simulation would be allowed to grow, thereby reducing the difference between numerical data and predictions in Fig. 5. However, given the steepness of the prediction curves for
Finally, the consideration of drag dependency in the eddy turnover time in (14) is largely empirical. The main idea there is to incorporate a variable cascade fraction (i.e., |Π|max/εp as represented by εc/εp) for the absence of an inertial range (see Fig. 4). By recognizing
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
APPENDIX B
Examinations of Vortex Gas Properties at Weak Linear Drag
Here we examine two guiding assumptions of the vortex gas model using a vortex finding algorithm. The assumptions are that 1) a vortex dipole is sufficient to capture the qualitative transport properties in turbulent flow. The heat flux is thus expected to be preferentially enhanced within vortex pairs in a manner similar to GF20’s Fig. 2; 2) the vortex velocity scales with r−1 like a point vortex. The increase of velocity toward a vortex core enhances the vortex-averaged dissipation [i.e.,
For the above item 1, we compare the heat fluxes inside and outside vortex pairs, with an expectation that the former would be preferentially enhanced. To check this picture, we apply the vortex finding algorithm of TY06 to the barotropic vorticity field. A vortex center is identified to have magnitude of Okubo–Weiss (OW) parameter greater than a threshold value and magnitude of vorticity greater than its surrounding points (see appendix B in TY06 for details). Vortex pairs are found as vortices whose intervortex distance is shorter than the mixing length (i.e.,
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
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
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 (
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
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
REFERENCES
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, 2257–2273, https://doi.org/10.1175/1520-0485(2004)034<2257:BUGTIT>2.0.CO;2.
Arbic, B. K., and R. B. Scott, 2008: On quadratic bottom drag, geostrophic turbulence, and oceanic mesoscale eddies. J. Phys. Oceanogr., 38, 84–103, https://doi.org/10.1175/2007JPO3653.1.
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.
Cessi, P., 2008: An energy-constrained parameterization of eddy buoyancy flux. J. Phys. Oceanogr., 38, 1807–1819, https://doi.org/10.1175/2007JPO3812.1.
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, 1627–1643, https://doi.org/10.1175/JAS-D-18-0333.1.
Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 1087–1095, https://doi.org/10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.
Ferrari, R., and C. Wunsch, 2009: Ocean circulation kinetic energy: Reservoirs, sources and sinks. Annu. Rev. Fluid Mech., 41, 253–282, https://doi.org/10.1146/annurev.fluid.40.111406.102139.
Gallet, B., and R. Ferrari, 2020: The vortex gas scaling regime of baroclinic turbulence. Proc. Natl. Acad. Sci. USA, 117, 4491–4497, https://doi.org/10.1073/pnas.1916272117.
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.
Griffies, S. M., and Coauthors, 2000: Developments in ocean climate modelling. Ocean Modell., 2, 123–192, https://doi.org/10.1016/S1463-5003(00)00014-7.
Haidvogel, D. B., and I. M. Held, 1980: Homogeneous quasi-geostrophic turbulence driven by a uniform temperature gradient. J. Atmos. Sci., 37, 2644–2660, https://doi.org/10.1175/1520-0469(1980)037<2644:HQGTDB>2.0.CO;2.
Held, I. M., 1999: The macroturbulence of the troposphere. Tellus, 51A, 59–70, 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, 946–952, https://doi.org/10.1175/1520-0469(1996)053<0946:ASTFHH>2.0.CO;2.
Hua, B. L., and D. B. Haidvogel, 1986: Numerical simulations of the vertical structure of quasi-geostrophic turbulence. J. Atmos. Sci., 43, 2923–2936, https://doi.org/10.1175/1520-0469(1986)043<2923:NSOTVS>2.0.CO;2.
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, 28–41, https://doi.org/10.1016/j.ocemod.2015.05.007.
Klein, P., and Coauthors, 2019: Ocean-scale interactions from space. Earth Space Sci., 6, 795–817, https://doi.org/10.1029/2018EA000492.
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, 2285–2297, https://doi.org/10.1175/1520-0485(1995)025<2285:EAAFIA>2.0.CO;2.
Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417–443, https://doi.org/10.1017/S0022112075001504.
Salmon, R., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167–211, https://doi.org/10.1080/03091928008241178.
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, 1650–1666, https://doi.org/10.1175/JPO2771.1.
Scott, R. B., and B. K. Arbic, 2007: Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics. J. Phys. Oceanogr., 37, 673–688, https://doi.org/10.1175/JPO3027.1.
Smith, K. S., and G. K. Vallis, 2002: The scales and equilibration of midocean eddies: Forced-dissipative flow. J. Phys. Oceanogr., 32, 1699–1720, https://doi.org/10.1175/1520-0485(2002)032<1699:TSAEOM>2.0.CO;2.
Thompson, A. F., and W. R. Young, 2006: Scaling baroclinic eddy fluxes: Vortices and energy balance. J. Phys. Oceanogr., 36, 720–738, https://doi.org/10.1175/JPO2874.1.
Thompson, A. F., and W. R. Young, 2007: Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. J. Atmos. Sci., 64, 3214–3231, https://doi.org/10.1175/JAS4000.1.