An idealized model for the spatial structure of the eddy-driven Ferrel cell in mid-latitudes

Moon, Woosok; Lee, Seung Pyo; Vanderborght, Elian; Manucharyan, Georgy; Dijkstra, Henk

doi:https://doi.org/10.5194/egusphere-2025-1004

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https://doi.org/10.5194/egusphere-2025-1004

Preprints

18 Mar 2025

| 18 Mar 2025

Status: this preprint is open for discussion and under review for Weather and Climate Dynamics (WCD).

An idealized model for the spatial structure of the eddy-driven Ferrel cell in mid-latitudes

Woosok Moon, Seung Pyo Lee, Elian Vanderborght, Georgy Manucharyan, and Henk Dijkstra

Abstract. As global warming intensifies, mid-latitude regions increasingly experience unusual and disruptive weather phenomena, such as extreme heatwaves and devastating floods, posing significant threats to societies. The dynamics of the jet stream largely govern mid-latitude weather patterns, with fluid dynamical instabilities generating baroclinic waves that propagate through the atmosphere before breaking. These waves play a critical role in shaping regional weather by influencing the jet stream’s maintenance and inducing an indirect meridional circulation. Understanding how the life cycle of baroclinic waves maintains the zonal-mean zonal wind and the indirect circulation is essential for improving predictions of mid-latitude weather under global warming. This study introduces a simplified theoretical framework that provides analytic solutions for the steady-state zonal-mean zonal wind and the indirect circulation. By incorporating a parameterization of turbulent eddies into the zonal-mean quasi-geostrophic potential vorticity equation, the model establishes a balance that drives eddy-induced circulations in mid-latitudes, analogous to the Ferrel cell. The meridional temperature structure reveals two key features: (1) a linear decrease in anomalous potential temperature, producing westerly winds, and (2) jet streams generated by eddy momentum fluxes. These jet streams are accompanied by downward (upward) motions on their southern (northern) flanks, further characterizing the eddy-driven circulations. The intensity and extent of the Ferrel cell are found to be governed by the baroclinic wave life cycle, constrained by the meridional temperature gradient, static stability, and boundary conditions influenced by tropical forcing. Under global warming, projected changes in these factors will alter mid-latitude circulation patterns. The theoretical framework proposed in this study offers a robust basis for analyzing and predicting the evolving dynamics of mid-latitude circulations in a warming climate.

Received: 03 Mar 2025 – Discussion started: 18 Mar 2025

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Woosok Moon, Seung Pyo Lee, Elian Vanderborght, Georgy Manucharyan, and Henk Dijkstra

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RC1:
'Comment on egusphere-2025-1004', Anonymous Referee #1, 29 Apr 2025 reply
Summary
This manuscript presents an idealized solution for the midlatitude zonal mean heat and momentum equations by assuming the eddy momentum and heat fluxes can be represented by a simple diffusive parameterization. The first-order solution reproduces the basic qualitative structure of the midlatitude tropospheric circulation, with a Ferrel cell located between two oppositely-oriented cells, a westerly jet at the center of the Ferrel cell, poleward eddy heat flux in the lower troposphere and up-gradient eddy momentum flux. The solution also allows for an examination of the effect of the model parameters on the mean flow properties and structure.
While the solution presented in this manuscript could potentially be of scientific interest to the WCD community, I find the presentation of the manuscript to be overly-complicated, which makes it difficult to evaluate the scientific contribution. The major issues are listed below.
I would recommend that the authors resubmit the manuscript, with a different focus and structure. Currently, I am not able to assess whether this manuscript contributes to new insights not addressed by previous studies.

General comments
The abstract and introduction give a very vague impression of what the manuscript is about, while the actual subject is quite simple. The inclusion of topics such as extreme events, eddy life cycle and the response to climate change in the abstract give the wrong impression that the model relates to these topics, while actually the model solves the equations only for the climatological zonal mean flow. Also, it would be preferred to mention that the parameterization for the eddy fluxes in this model is a simple diffusive parameterization with a constant diffusion coefficient and that the model is a beta-plane model. This give the reader the right context for the presented research. The introduction gives a lengthy overview of the general atmospheric circulation (lines 20-85), while not covering the more specific topics of this manuscript: How are the midlatitude circulation properties affected by general parameters such as Earth’s rotation rate, static stability, the meridional width of the domain, etc. or by the non-dimensional parameters derived from these parameters?; How can the eddy fluxes be represented using simple diffusive parameterizations while capturing their qualitative properties? While previous studies have dealt with these questions, the introduction does not discuss these papers and does not clarify what are the remaining open questions which this manuscript addresses.

Overly-complicated mathematical derivation. The manuscript contains 54 equations, in addition to a few inline equations and 5 additional equations in the appendix. Also, the number of variables, parameters and signs used is very large, which makes it very difficult to follow the derivation. I find this complication unnecessary, as the model itself is quite simple. The derivation could be shortened by beginning with a clear listing of the model assumption and presenting the beta-plane zonal-mean heat and momentum equations, which are quite standard in the literature. Subsequently, the diffusive approximation can be replaced into the equations and the four boundary conditions can be presented. There is no need to develop the Ekman layer equations, as there is no use of them in the model. This would reduce equations 1-26 to around 8 equations. The derivation of the analytical solution in section 3.1 (equations 35-50) is quite messy and could be organized in a more readable manner. Also, there is some redundancy, where some equations are repeated twice. There are too many signs and subscripts, which I don’t think are necessary. The detailed comments below elaborate on this issue.

I think that a slightly different parameterization would be more physically consistent. Several previous studies have used diffusive parameterization for the potential vorticity (PV) flux, instead of the heat and momentum fluxes. Using this approximation and a non-dimensional parameter that describes the ratio between the vorticity and stretching terms in the PV equation (the Burger number) would give a parameterization for the heat and momentum fluxes. As PV is the conserved variable in the absence of friction and diabatic heating, it makes more sense to approximate its flux using a diffusive parameterization.

It is not clear what is the new scientific contribution of this manuscript. Previous studies used diffusive approximations for eddy fluxes, and/or performed comprehensive parameter sweeps investigating the effect of non-dimensional parameters on the properties of the midlatitude circulation, including the Ferrel cell and the eddy-driven jet. Most of these studies are not mentioned in the manuscript. Some are mentioned, but without discussing their outcome. It is not clear what exactly this study adds to the existing knowledge.

The potential for a significant scientific contribution based on the suggested model is not fulfilled. I think that the potential for new insights from this study comes mostly from the relations obtained from the parameter sweep, such as displayed in figure 8b and figure 9. However, these sweeps and their discussion are quite limited in this manuscript, and they are not compared with the results of previous studies. The other results are not really new, and I don’t think they add any new insights for the dynamics of the midlatitude circulation.

Specific comments
Lines 10-12: It is not clear from reading the abstract what these sentences are describing. Only after reading the whole manuscript I understood that this is a description of the solution for the zonal-mean climatological flow in the idealized analytical model, and that the solution is a sum of the two listed features.

Lines 42,45: Instead of “the original theory” I would suggest referring to the Held-Hou theory as the axisymmetric theory.

Lines 51-52: It is not clear in what sense the Ferrel cell is “originating” from the downward motion near the edge of the Hadley cell. Is this sentence suggesting causal relations? The same comment applies to line 288.

Lines 86-93: I think there should be more discussion on previous parameterizations of the heat and momentum fluxes. This could come at the expense of shortening lines 20-85 in the introduction. The following studies explored the properties of the midlatitude circulation using diffusive parameterizations for the eddy fluxes, or using scaling laws based on geostrophic turbulence theory, to name a few (see bibliography below): Held and Larichev (1996) – cited in this manuscript but not in the introduction; Pavan and Held (1996); Lapeyre and Held (2003); Zurita-Gotor (2007); Thompson and Young (2007); Scheider and Walker (2008) - cited in this manuscript but not in the introduction; Jansen and Ferrari (2015); Chen and Plumb (2014). These papers and others should not only be cited, but also there should be given an overview of the current knowledge of how the midlatitude circulation depends on the non-dimensional parameters of the system, and the open questions addressed here should be highlighted.

Section 2.1: The section should begin with the context of the basic equations (where you are trying to get to by examining these equations) and a list of the assumptions used. The section begins immediately with the first equation, which uses a specific decomposition of the flow that is not explained before presenting the equation. It also uses the beta-plane approximation, which is not mentioned or defined in the text before the equation. Additionally, the non-dimensionalization of the equations is not done properly, which makes it very confusing: (a) in equation 1 the variables are dimensional, but the Coriolis parameter is missing on the right-hand-side term. (b) In line 104 it says that S is non-dimensional, but all other variables are dimensional and the definition of S is not given. (c) The method for non-dimensionalizing the equations is not given in an orderly manner. It should be listed clearly how each variable is non-dimensionalized after writing the dimensional equations and before writing the non-dimensional equations. (d) internal and external Rossby numbers are mentioned without defining them. (e) Some constants are written without defining them in the first time (f,g,H,L…). (f) The Coriolis parameter is missing on the left-hand-side of equation 4, and on the left-hand-side of equation 6.

Line 128: I couldn’t understand the definition of Theta_0. Is it really necessary to define the potential temperature using two notations (theta and eta)? I couldn’t understand why you are using the derivative of Q with respect to theta_0 and what it means.

Jumping back and forth between neglecting and not-neglecting the non-conservative terms makes it very confusing. I suggest the authors to plan the order of the equations so the transition between neglecting to not neglecting these terms would be done only once.

The description of the eddy life cycle is repeated too many times in the manuscript. It is enough to describe it once (for example in lines 161-166), and say that the parameterization used here tries to capture the integrated effect of eddies over many life cycles.

In some places the authors mention “previous studies” without citing them explicitly (lines 199-200, 373).

In several places it is argued that the adiabatic heating and cooling by the vertical motion is induced by eddy momentum flux (lines 215-217, 592-593, 633, 637, 647). However, eddy momentum flux is only one of the factors controlling the vertical motion of the overturning circulation, all other terms in the zonal-mean zonal momentum and heat equations also play a role (see various studies which solved the Kuo-Eliassen equation).

Figure 2 – what is the range of latitudes and pressure levels used to calculate the dots in panels a and b?

Section 2.2 leads to equation (11) in an overly complicated way. I suggest that the steps to get to this equation would follow this order: 1. The zonal momentum equation is in steady state and expresses the balance between the Coriolis force and eddy momentum flux convergence (EMFC). 2. The heat equation is in steady state and expresses a balance between adiabatic heating by vertical motion and eddy heat flux convergence (EHFC). 3. Continuity connects the Coriolis force and the adiabatic term, so that together with the momentum and heat equations leads to a relation between EMFC and EHFC. 4. Using diffusive approximations allows to express the EMFC and EHFC in terms of zonal wind and temperature. 5. Use thermal wind balance to turn the equation into an equation with only one variable (temperature).

Figure 3 – I don’t think this figure is necessary. The boundary conditions should be listed clearly in the text, not in the figure.

Equation 13: It is strange that u_0 was regarded as the zonal wind and then it turns out to be one of the components of the total zonal wind (u_T). Specifically, the eddy momentum flux parameterization should be related to the total zonal wind and not just one of its components. If u_s has a meridional shear, it would affect the eddy momentum flux as well.

Lines 297-299: The use of the word “geostrophic” in this paragraph is confusing. Both u_0 and u_s are referred to as “geostrophic”, so why is it mentioned interchangeably that they are geostrophic as if the other term isn’t?

Equations 14-22: I don’t see the added value of deriving the Ekman layer equations here. It is standard to approximate the friction at the boundary layer by a linear drag parameterization, which leads to equation (22) and makes this derivation unnecessary.

Equation 23 repeats equation 6, but without the friction term. The connection should be clarified. Why is the friction term neglected and not neglected interchangeably?

Line 359: Is this a definition of a new parameter c(z)? I don’t see what you need this notation for if it’s not used later.

Equation 33: Why repeat equation 11 again?

I think the discussion of the non-acceleration theorem should be given once, and in the first place where the equations are presented without the non-conservative terms. It is only presented in line 366, but it should be explained earlier.

Section 3.1: This section should begin with a text motivating the analytical solution and explaining where this is going and why. Instead, the analytical solution is presented here without any explanation, justification or description of the model assumptions.

Equation (35): One term is missing, with the derivative of Q with respect to z.

Equations 39-50: This part of the manuscript is super-difficult to follow, and I don’t think it should be. I think the number of variables and parameters could be reduced, to make the derivation more compact. Some of the equations could be moved to an appendix. The reader should be able to follow easily the path to the main equations used later, most specifically equation (43). All the definitions of the parameters in this equation should be easily found in the text.

The use of the functions f(q), g(q) and h(q) in the text and in figure 4 is very confusing. If f(q)=g(q), why do you need two notations for the same function? It is not clear at first from reading the text and the caption of figure 4 what these functions represent in the solution. Is it the meridional or vertical component of the solution? Is it the solution for temperature or zonal wind? Is it just the first-order component of the total solution in equation (43) or is it the total solution? Why do you choose to show only the first mode in the sum in equation (43)?

The choice of parameter values for presenting examples for the solution is not motivated (lines 411).

Equations 45-50: What does the subscript “1” represent?

The discussion in lines 441-489, including the description of figure 5, relates to the solution for eta_0 (the total temperature profile). However, according to equation 43, eta_0 is a sum of an infinite number of modes, while the solution discussed here is only the first mode in this sum (is that correct?). Are you assuming that the first mode is larger or more important than the other modes, or are you simply choosing it as one example? This should be clarified before the beginning of the discussion. Also in section 3.2 the first mode is treated as if it is the general solution.

Figures 4 and 5: it would help the reader if the caption would mention what each parameter represents. Also, the choice of parameter values (line 475) should be explained. Are these parameter values realistic for the atmosphere? Are they based on the GCM simulation?

Lines 500-502: This is repeating what was written in the previous section.

Section 3.3: Previous studies examined how changing the parameters of the flow equations changes the number of jets (or equivalently, the jet width). The conclusions of these studies should be mentioned here and compared with the current results. To name a few: Panetta (1993); Pavan and Held (1996); Esler (2008); Lee (2005); Chemke and Kaspi (2015). Overall, the manuscript does not relate to much of the previous literature, and it is not clear what new contribution arises from these results.

Lines 536-537: This sentence is not clear. What do you mean by “multiple vertical layers”?

Line 546 and equation 53: Why is q1 the smallest positive solution? Again, it is very difficult to follow the equations and the notations.

Lines 551-557: These lines repeat things that were written in previous sections.

Caption of figure 8: I don’t see any black dots in panel a. Also, how are L_1 and the size of the Ferrel cell calculated?

Line 568: Is the meridional shear calculated around the center of the jet? Isn’t it zero there?

Lines 577-582: The subject discussed here disserves more careful attention. What determines the relation between the eddy heat and momentum fluxes? The authors should refer to previous studies addressing this question. My understanding is that this ratio is related to the Burger number (N*H/f L)^2 – see AMS glossary for example. This non-dimensional number gives the ratio between the stretching term and the vorticity term in the QG PV. I think that what the authors call “the structural number” could be expressed in terms of the Burger number. This would help to relate it to previous studies.

Line 594: What is “external heat flux”?

Line 608: Did you mean “non-acceleration” instead of “non-zero acceleration”?

Section 4: Again, it is not clear what is the new contribution of this study. This should be clarified in this section.

Lines 624-625: How does this research highlight that?

Lines 627-630: This paragraph doesn’t add any information. I would suggest to remove it.

The appendix is not referred to in the text. I suggest removing it.

Bibliography
Pavan, V., & Held, I. M. (1996). The diffusive approximation for eddy fluxes in baroclinically unstable jets. Journal of Atmospheric Sciences, 53(9), 1262-1272.

Lapeyre, G., & Held, I. M. (2003). Diffusivity, kinetic energy dissipation, and closure theories for the poleward eddy heat flux. Journal of the atmospheric sciences, 60(23), 2907-2916.

Zurita-Gotor, P. (2007). The relation between baroclinic adjustment and turbulent diffusion in the two-layer model. Journal of the atmospheric sciences, 64(4), 1284-1300.

Jansen, M., & Ferrari, R. (2015). Diagnosing the vertical structure of the eddy diffusivity in real and idealized atmospheres. Quarterly Journal of the Royal Meteorological Society, 141(687), 631-641.

Thompson, A. F., & Young, W. R. (2007). Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. Journal of the Atmospheric Sciences, 64(9), 3214-3231.

Chen, G., & Plumb, A. (2014). Effective isentropic diffusivity of tropospheric transport. Journal of the Atmospheric Sciences, 71(9), 3499-3520.

Panetta, R. L. (1993). Zonal jets in wide baroclinically unstable regions: Persistence and scale selection. Journal of Atmospheric Sciences, 50(14), 2073-2106.

Esler, J. G. (2008). The turbulent equilibration of an unstable baroclinic jet. Journal of Fluid Mechanics, 599, 241-268.

Lee, S. (2005). Baroclinic multiple zonal jets on the sphere. Journal of the atmospheric sciences, 62(7), 2484-2498.

Chemke, R., & Kaspi, Y. (2015). The latitudinal dependence of atmospheric jet scales and macroturbulent energy cascades. Journal of the Atmospheric Sciences, 72(10), 3891-3907.

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Citation: https://doi.org/10.5194/egusphere-2025-1004-RC1

Woosok Moon, Seung Pyo Lee, Elian Vanderborght, Georgy Manucharyan, and Henk Dijkstra

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Short summary

As the climate warms, extreme weather is becoming more frequent in mid-latitudes. A key factor is the jet stream, shaped by atmospheric waves that influence wind and storm patterns. This study presents a simplified model showing how swirling air currents (eddies) maintain the jet stream and impact weather. As global warming alters these patterns, this research helps improve predictions of future weather changes.


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