# The vortex gas scaling regime of baroclinic turbulence

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Edited by William R. Young, University of California San Diego, La Jolla, CA, and approved January 15, 2020 (received for review September 20, 2019)

## Significance

Developing a theory of climate requires an accurate parameterization of the transport induced by turbulent eddies. A major source of turbulence in the midlatitude planetary atmospheres and oceans is the baroclinic instability of the large-scale flows. We present a scaling theory that quantitatively predicts the local heat flux, eddy kinetic energy, and mixing length of baroclinic turbulence as a function of the large-scale flow characteristics and bottom friction. The theory is then used as a quantitative parameterization in the case of meridionally dependent forcing in the fully turbulent regime. Beyond its relevance for climate theories, our work is an intriguing example of a successful closure for a fully turbulent flow.

## Abstract

The mean state of the atmosphere and ocean is set through a balance between external forcing (radiation, winds, heat and freshwater fluxes) and the emergent turbulence, which transfers energy to dissipative structures. The forcing gives rise to jets in the atmosphere and currents in the ocean, which spontaneously develop turbulent eddies through the baroclinic instability. A critical step in the development of a theory of climate is to properly include the eddy-induced turbulent transport of properties like heat, moisture, and carbon. In the linear stages, baroclinic instability generates flow structures at the Rossby deformation radius, a length scale of order 1,000 km in the atmosphere and 100 km in the ocean, smaller than the planetary scale and the typical extent of ocean basins, respectively. There is, therefore, a separation of scales between the large-scale gradient of properties like temperature and the smaller eddies that advect it randomly, inducing effective diffusion. Numerical solutions show that such scale separation remains in the strongly nonlinear turbulent regime, provided there is sufficient drag at the bottom of the atmosphere and ocean. We compute the scaling laws governing the eddy-driven transport associated with baroclinic turbulence. First, we provide a theoretical underpinning for empirical scaling laws reported in previous studies, for different formulations of the bottom drag law. Second, these scaling laws are shown to provide an important first step toward an accurate local closure to predict the impact of baroclinic turbulence in setting the large-scale temperature profiles in the atmosphere and ocean.

Oceanic and atmospheric flows are subject to the combined effects of strong density stratification and rapid planetary rotation. On the one hand, these two ingredients add complexity to the dynamics, making the flow strongly anisotropic and inducing waves that modify the characteristics of the turbulent eddies. On the other hand, they permit the derivation of reduced sets of equations that capture the large-scale behavior of the flow: this is the realm of quasigeostrophy (QG). The outcome of this approach is a model that couples two-dimensional (2D) layers of fluid of different density. QG filters out fast-wave dynamics, relaxing the necessity to resolve the fastest timescales of the original system. A QG model with only two fluid layers is simple enough for fast and extensive numerical studies, and yet, it retains the key phenomenon arising from the combination of stable stratification and rapid rotation (1): baroclinic instability, with its ability to induce small-scale turbulent eddies from a large-scale vertically sheared flow. The two-layer quasigeostrophic (2LQG) model offers a testbed to derive and validate closure models for the “baroclinic turbulence” that results from this instability.

In the simplest picture of 2LQG, a layer of light fluid sits on top of a layer of heavy fluid, as sketched in Fig. 1*A*, in a frame rotating at a spatially uniform rate ^{†} These two balances imply that both the flow field and the local thickness of each layer can be expressed in terms of the corresponding stream functions, ^{‡}In our model, the drag term is confined to the lower-layer Eq. **2**. In the case of linear drag, **1** and **2** include hyperviscosity to dissipate filaments of potential vorticity (enstrophy) generated by eddy stirring at small scales.

A more insightful representation arises from the sum and difference of Eqs. **1** and **2**: one obtains an evolution equation for the barotropic stream function—half the sum of the stream functions of both layers—which characterizes the vertically invariant part of the flow, and an evolution equation for the baroclinic stream function—half the difference between the two stream functions—which characterizes the vertically dependent flow. Because in QG the stream function is directly proportional to the thickness of the fluid layer, the baroclinic stream function is also a measure of the height of the interface between the two layers. A region with large baroclinic stream function corresponds to a locally deeper upper layer: there is more light fluid at this location, and we may thus say that on vertical average the fluid is warmer. Similarly, a region of low baroclinic stream function corresponds to a locally shallower upper layer, with more heavy fluid: this is a cold region. Thus, the baroclinic stream function is often denoted as τ and referred to as the local “temperature” of the fluid.

The 2LQG model can be used to study the equilibration of baroclinic instability arising from a prescribed horizontally uniform vertical shear, which represents the large-scale flows maintained by external forcing in the ocean and atmosphere. Denoting the vertical axis as z and the zonal and meridional directions as x and y, the prescribed flow in the upper and lower layers consists of zonal motion ^{§} with a prescribed uniform meridional temperature gradient *A*). This tilt provides an energy reservoir, the available potential energy (APE) (4), that is released by baroclinic instability, acting to flatten the density interface. We denote, respectively, as

Traditionally, these questions have been addressed using descriptions of the flow in spectral space, focusing on the cascading behavior of the various invariants (6). In contrast with this approach, Thompson and Young (7) describe the system in physical space and argue that the barotropic flow evolves toward a gas of isolated vortices. Despite this intuition, Thompson and Young (7) cannot derive the scaling behavior of the quantities mentioned above and resort to empirical fits instead. Focusing on the case of linear drag, they conclude that the temperature fluctuations and meridional heat flux are extremely sensitive to the drag coefficient: they scale exponentially in inverse drag coefficient. This scaling dependence was recently shown by Chang and Held (8) to change if linear drag is replaced by quadratic drag: the exponential dependence becomes a power law dependence on the drag coefficient. However, Chang and Held (8) acknowledge the failure of standard cascade arguments to predict the exponents of these power laws, and they resort to curve fitting as well.

In this work, we supplement the vortex gas approach of Thompson and Young (7) with statistical arguments from point vortex dynamics to obtain a predictive scaling theory for the eddy kinetic energy, the temperature fluctuations, and the meridional heat flux of baroclinic turbulence. The resulting scaling theory captures both the exponential dependence of these quantities on the inverse linear drag coefficient and their power law dependence on the quadratic drag coefficient. Our predictions are thus in quantitative agreement with the scaling laws diagnosed empirically by both Thompson and Young (7) and Chang and Held (8). Following Pavan and Held (9) and Chang and Held (8), we finally show how these scaling laws can be used as a quantitative turbulent closure to make analytical predictions in situations where the system is subject to inhomogeneous forcing at large scale.

## The QG Vortex Gas

Denoting as ⟨⋅⟩ a spatial and time average and as

We follow the key intuition of Thompson and Young (7) that the flow is better described in physical space than in spectral space. In Fig. 1, we provide snapshots of the barotropic vorticity and baroclinic stream function from a direct numerical simulation in the low-drag regime (the numerical methods are in *SI Appendix*): the barotropic flow consists of a “gas” of well-defined vortices, with a core radius substantially smaller than the intervortex distance

A schematic of the resulting idealized vortex gas is provided in Fig. 1*D*: we represent the barotropic flow as a collection of vortices of circulation

The first of these relations is the energy budget: the meridional heat flux corresponds to a rate of release of APE, *SI Appendix*)

The next steps of the scaling theory are common to linear and quadratic drag. As in any mixing-length theory, we will express the diffusion coefficient D as the product of the mixing length and a typical velocity scale. In the vortex gas regime, one can anticipate that the mixing length ℓ scales as the typical intervortex distance **11** below. However, a final relationship for the relevant velocity scale is more difficult to anticipate as we have seen that the various barotropic velocity moments scale differently. The goal is thus to determine this velocity scale through a precise description of the transport properties of the assembly of vortices.

Stirring of a tracer like temperature takes place at scales larger than the stirring rods: in our problem, the vortices of size λ. At scales much larger than λ, the τ-equation [**6**] reduces to (7, 20, 21)**10** is thus that of a passive scalar with an externally imposed uniform gradient **5** and **6**, we solve Eq. **10** with τ replaced by the concentration c of a passive scalar and **10**. We can thus safely build intuition into the behavior of the temperature field by studying Eq. **10**.

A natural first step would be to compute the heat flux associated with a single steady vortex. However, this situation turns out to be rather trivial: the vortex stirs the temperature field along closed circles until it settles in a steady state that has a vanishing projection onto the source term *A*: two vortices of opposite circulations *A*, the meridional velocity is positive between the two vortices and becomes negative at both ends of the dipole. For positive U, this corresponds to a heat source between the vortices and two heat sinks away from the dipole. These heat sources and sinks are positively correlated with the local meridional barotropic velocity so that there is a net meridional heat flux **10** for this moving dipole over a time *C* and *D* shows the resulting temperature field and local flux *SI Appendix* has details). A suite of numerical simulations for such dipole configurations indicates that, at the end time of the numerical integration, the local mixing length and diffusivity obey the scaling relations:**8**] and [**9**] at every time. It is interesting that the velocity scale arising in the diffusivity [**12**] is V and not the rms velocity

The relations [**11**] and [**12**] hold for any passive tracer. However, temperature is an active tracer so that the velocity scale in turn depends on the temperature fluctuations, providing the fourth scaling relation. This relation can be derived through a simple heuristic argument: consider a fluid particle, initially at rest, that accelerates in the meridional direction by transforming potential energy into barotropic kinetic energy by flattening the density interface as a result of baroclinic instability. In line with the standard assumptions of a mixing-length model, we assume that the fluid particle travels in the meridional direction over a distance ℓ before interacting with the other fluid particles. Balancing the kinetic energy gained over the distance ℓ with the difference in potential energy between two fluid columns a distance ℓ apart, we obtain the final barotropic velocity of the fluid element: **13**] holds locally for the heat-carrying fluid elements traveling a distance ℓ instead. The estimate [**13**] is also reminiscent of the “free-fall” velocity estimate of standard upright convection, where the velocity scale is estimated as the velocity acquired during a free fall over one mixing length (23⇓–25). The conclusion is that the typical velocity is directly proportional to the mixing length. The baroclinic instability is sometimes referred to as slantwise convection, and the velocity estimate [**13**] is the corresponding “slantwise free-fall” velocity. To validate [**13**], one can notice that, when combined with [**11**] and [**12**], it leads to the simple relation**14**] above. A relation very close to [**14**] was reported by Larichev and Held (21) using turbulent cascade arguments. Their relation is written in terms of an “energy-containing wavenumber” instead of a mixing length. If this energy-containing wavenumber is interpreted to be the inverse intervortex distance of the vortex gas model, then their relation becomes identical to [**14**].

The four relations needed to establish the scaling theory are [**7**] and [**11**]–[**13**]. In the case of linear drag, their combination leads to **8**] for the dissipation of kinetic energy. It is remarkable that these authors could extract the correct functional dependence of *SI Appendix*. In Fig. 3, we plot **15**] and our numerical data using **14**], which leads to

When linear friction is replaced by quadratic drag, only the energy budget [**7**] is modified. As can be seen in Eq. **9**, the main difference is that quadratic drag operates predominantly in the vicinity of the vortex cores, which has a direct impact on the scaling behaviors of **7**] and [**11**]–[**13**] yields**14**], leads to the diffusivity

## Using These Scaling Laws as a Local Closure

We now wish to demonstrate the skill of these scaling laws as local diffusive closures in situations where the heat flux and the temperature gradient have some meridional variations. For simplicity, we consider an imposed heat flux with a sinusoidal dependence in the meridional direction y. The modified governing equations for the potential vorticities **20**] from [**19**] and dividing by two, has a source term **19**] and [**20**] has no source terms. The goal is to determine the temperature profile associated with the imposed meridionally dependent heat flux. This slantwise convection forced by sources and sinks is somewhat similar to standard upright convection forced by sources and sinks of heat (26, 27). We focus on the statistically steady state by considering a zonal and time average denoted as **19** and **20** leads to**21**] and substituting the scaling law [**18**] for **23**] holds for

In the case of linear drag, we substitute the scaling law [**16**] for **24**] holds for

To test these theoretical predictions, we solved numerically Eqs. **19** and **20** inside a domain **23**] and [**24**] are in excellent agreement with the numerical results for both linear and quadratic drag, and this good agreement holds provided the various length scales of the problem are ordered in the following fashion: **15**] and [**17**], this loss of scale separation occurs for

## Discussion

The vortex gas description of baroclinic turbulence allowed us to derive predictive scaling laws for the dependence of the mixing length and diffusivity on bottom friction and to capture the key differences between linear and quadratic drag. The scaling behavior of the diffusivity of baroclinic turbulence seems more “universal” than that of its purely barotropic counterpart. This is likely because many different mechanisms are used in the literature to drive purely barotropic turbulence. For instance, the power input by a steady sinusoidal forcing (29, 30) strongly differs from that input by forcing with a finite (31) or vanishing (32) correlation time, with important consequences for the large-scale properties and diffusivity of the resulting flow. By contrast, baroclinic turbulence comes with its own injection mechanism—baroclinic instability—and the resulting scaling laws depend only on the form of the drag. We demonstrated the skills of these scaling laws when used as local parameterizations of the turbulent heat transport in situations where the large-scale forcing is inhomogeneous. While this theory provides some qualitative understanding of turbulent heat transport in planetary atmospheres, it should be recognized that the scale separation is at best moderate in Earth atmosphere, where meridional changes in the Coriolis parameter also drive intense jets. However, our firmly footed scaling theory could be the starting point to a complete parameterization of baroclinic turbulence in the ocean, a much-needed ingredient of global ocean models. Along the path, one would need to adapt the present approach to models with multiple layers, possibly going all of the way to a geostrophic model with continuous density stratification or even back to the primitive equations. The question would then be whether the vortex gas provides a good description of the equilibrated state in these more general settings. Even more challenging would be the need to include additional physical ingredients in the scaling theory: the meridional changes in f mentioned above, but also variations in bottom topography and surface wind stress. Whether the vortex gas approach holds in those cases will be the topic of future studies.

### Data Availability.

The data associated with this study are available within the paper and *SI Appendix*.

## Acknowledgments

Our work is supported by Eric and Wendy Schmidt by recommendation of the Schmidt Futures Program and by NSF Grant AGS-6939393. This research is also supported by European Research Council Grant FLAVE 757239.

## Footnotes

↵

^{1}B.G. and R.F. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: basile.gallet{at}cea.fr.

Author contributions: B.G. and R.F. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

↵*Hydrostatic balance is the balance between the upward-directed pressure gradient force and the downward-directed force of gravity.

↵

^{†}Geostrophic balance is the balance between the Coriolis force and lateral pressure gradient forces.↵

^{‡}The Rossby radius of deformation λ is the length scale at which rotational effects become as important as buoyancy or gravity wave effects in the evolution of a flow.↵

^{§}A flow in thermal wind balance satisfies both hydrostatic and geostrophic balance.This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1916272117/-/DCSupplemental.

Published under the PNAS license.

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