Origins and suppression of bifurcation phenomena in lowerorder monsoon models
 ^{1}Divecha Centre for Climate Change, Indian Institute of Science, Bengaluru, India
 ^{2}Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bengaluru, India
 ^{1}Divecha Centre for Climate Change, Indian Institute of Science, Bengaluru, India
 ^{2}Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bengaluru, India
Abstract. South Asian monsoon rainfall varies rapidly in the paleoclimate record, and this has been interpreted using simple models as arising from tipping points. This study explores a class of simple monsoon models, based on convective quasiequilibrium, and the bifurcations permitted by their mathematical forms. Specifically, loworder models are derived starting from the Quasiequilibrium tropical circulation model (QTCM) to examine the bifurcations present. Previous studies that have pointed to an abrupt transition in loworder monsoon models typically identify a saddle node bifurcation occurring as a result of changes in the radiation budget. The present study shows how such saddle node structures arise across a wide range of modeling assumptions and parameter values, and yet permit a continuous transition into and out of precipitating regimes without any bifurcation being physically manifest. This is because the bifurcation points lie in a regime that is not physically relevant when the dry thermal stratification is sufficiently large. As a result, these loworder models can be interpreted as possessing abrupt transitions that are latent in the equations but do not express themselves physically. However, when the dry thermal stratification is reduced, bifurcations can occur. This paper also shows that these latent saddlenode structures are themselves part of the unfolding of a pitchfork bifurcation. These findings help understand the role of stabilizing phenomena on the general absence of abrupt monsoon transitions despite the presence of nonlinear terms in these models.
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S. Krishna Kumar and Ashwin K. Seshadri
Status: final response (author comments only)

RC1: 'Comment on esd202230', Jacob Schewe, 02 Sep 2022
General comments:
This paper is motivated by previous studies of simple monsoon models, which either found bifurcation behavior or a quasilinear behavior of monsoon strength in response to external forcing. The paper starts with a more complex and comprehensive model of the tropical circulation, the QTCM, and reduces it to obtain a loworder monsoon model comparable to those used in those previous studies. Importantly, the authors derive two separate sets of equations, one each for nonzero precipitation and zero precipitation. They then demonstrate that each of these sets exhibits bifurcation behavior. The physically relevant solution to the full model is comprised of the relevant portions of each of the two separate sets. The authors show that, using a standard set of parameter values, this full solution does not exhibit a bifurcation point in the physically relevant regime, explaining previous results that showed nearlinear behavior of the full loworder model. On the other hand, the underlying bifurcations are still present, and perturbations in terms of parameter values can move bifurcation points into the physically relevant regime; explaining other previous results that showed bifurcation behavior in a similar loworder model.
The paper is well written and highly useful as it reconciles opposing findings of previous studies. It also reveals the underlying dynamical structure of the model across a broader range of parameters, showing how a pitchfork bifurcation emerges when the effects of gross moist stratification and moisture advection on horizontal velocity are no longer assumed to cancel each other. The authors also illustrate, by gradually reducing the dry thermal stratification parameter, the transition from the case with nearlinear physical solution to a case with a physically relevant bifurcation point.
I recommend publication of the paper subject to some technical corrections and consideration of a few comments and suggestions, as listed below.
Specific comments:
 Some more discussion is needed of the change in sign of a_T. What does it mean, why is it necessary, and how is the point at which the sign should change determined? There is some discussion of this in the SI, but it is not clear how arbitrary this choice is and how a deliberate change in sign of one quantity affects the selfconsistency of the overall model.
 Regarding the model equations: The stepbystep derivation in the SI is useful, as is Table 1. I think it could be even more helpful if the authors could provide a bit more interpretation of the different terms in equations 2 and 3 (e.g. horizontal temperature gradient, heat advection, moisture advection etc.). And potentially provide some introduction/references for the concepts of static energy and static stability, which are of central importance.
 An optional, but very interesting addition to the paper would be some discussion of the plausible ranges of the relevant parameters. For instance, the authors show that a physical bifurcation point emerges already well before the dry thermal stratification parameter reaches zero. Can the transition point be pinned down, and how far away is it from realistic values of the parameter for either modern or paleo climates (where those are known)? Similarly, given that the solution structure changes strongly for small deviations from the balance between gross moist stratification and moisture advection, how prevalent or relevant are such deviations for the real largescale circulation? Finally, indicating typical or expected values of the radiation parameter R might help readers interpret the figures with respect to plausible regimes.
 Related to this, and also optional, would be some discussion of what the solutions would look like in terms of a different parameter than the insolation R, such as the moisture at the sea boundary q_s.
Technical corrections:
Main paper:
 Equations 4 and 5: Indices c and q are not explained – do they stand for cubic and quadratic, respectively? Important to clarify, since q could also correspond to moisture.
 line 205: “comparing Fig. 2” – compare to what? Do you mean comparing the two curves in Fig. 2?
 line 277: a_q is not explained, I think it is the notation used by Boos & Storelvmo – please clarify. Perhaps alternative notations (also a_T for the temperature advection coefficient) could be indicated directly in Table 1.
 line 303: “partially nullify” – should this read “fully nullify”? I understand the two terms need to cancel each other exactly in order to remove the cubic term.
 line 357: delete “they occur in”
 Figure 6: Please indicate the (relative) values of M_s chosen for each panel, e.g. in the caption or in a legend.
 Figure 7, caption: “M_qp is reduced from its standard value.” – this sentence seems superfluous given the following sentence. Delete?
SI:
 Figure S1: the grey bar connecting “Galerkin expansion” and “Tailored basis functions” is not explained. It is also unclear which of the blue boxes the expressions “Convective” and “Nonconvective regions” refer to.
 Equations (1): Some symbols are not explained, e.g. f (Coriolis frequency?), or the indices of epsilon (01, 10 etc.). Please explain all symbols used.

RC2: 'Comment on esd202230', Anonymous Referee #2, 23 Sep 2022
This manuscript essentially details the study of a very simple model of monsoons that is the basis of one section of Boos and Storelvmo (2016)’s rebuke of Leverman et al (2009)’s model which produces abrupt transitions between regimes with and without monsoon. The manuscript provides more information on the structure of the dynamical system than what Boos and Storelvmo (2016) provided, and this might be worth publication if developed further. Section 3.3 on the pitchfork bifurcation is the most novel part of the manuscript; its interest lies in relation to Leverman et al (2009)’s study. It is otherwise an essentially mathematical exercise since it relies on breaking the physical consistency of the model. There are also flaws in the original model that should be addressed, a section (3.2) should be shortened and a lot of technical details should be improved before it can be published.
Main comments:
1. This simple model presents monsoons as a largescale sea breezes. It neglects rotation and does not simulate the reversed trade winds. But, from the early definition of monsoons (Ramage 1971) to recent work on global monsoon (e. g., Gadgil 2018, Geen et al. 2020), the reversal of the winds is an inherent part of the monsoon circulations that distinguishes them from breeze circulations. As a result, the amplitude of the meridional wind is about one order of magnitude larger than the observed wind (well, if v refers to the lowlevel wind and not to (minus) v_{1} form the QTCM). I think it would be an improvement on Boos and Storelvmo (2016)’s model to include the effect of rotation by considering an fplane. This would actually not change the number of equilibria, ε_{1} would just have to be substituted by ε_{1} + f^{2} / ε_{1} , but it would change the amplitude of the meridional wind and precipitation response and maybe modify the stability of the equilibria.
2. In the QTCM, by construction, M_{qp} =  <b_{1}V_{1}>, which is imposed by the conservation of water vapor mass by transport (in the absence of phase change): the integral of the terms of transport overt the whole horizontal domain has to be zero. Mathematically, it makes Equation 4 quadratic. There is no pitchfork bifurcation if the physical basis of the model is respected. This should be stated clearly even before the first results. At first, I was wondering why there was no third solution shown for P>0 in figure 2, but that’s because of the equality above, which is not highlighted until Section 3.3.
In Section 3.3, it would be worth providing clarification that making M_{qp} different from  <b_{1}V_{1}> amounts to disregarding mass conservation of moisture. In the current version of the manuscript, lines 276282 do not make clear that this equality is a result of a fundamental law of physics. Lines 298300 refer to a physical “interpretation” of the equality above. It is more than a physical interpretation, it is the expression of watervapor mass conservation by transport. The interest of this section is to clearly show the unphysical assumption in Leverman et al (2009)’s model, this should be investigated further and the most interesting oints in the further analysis mentioned on lines 342343 should be included in this section to enhance its content.
3. The equations could be simplified, clarified, and made easier to interpret physically.
In the supplementary material, the derivation of the reduced model from the QTCM equations is too long and a little confusing. First, the terms of meridional advection of zonal wind and zonal advection of meridional wind are wrong. Second, the symbol v_{1 }is used for both the QTCM variable of firstbaroclinic meridional wind and the fixed vertical profile of wind associated to this mode; in the reference articles on QTCM, the latter is noted V_{1}. In the LSSS geometry, the imposed boundary conditions are incorrect: on boundaries B and D, the presence of zonal gradients of temperature (hence, of geopotential) precludes assuming that the zonal winds u_{B} and u_{D} are equal, and same for v_{B} and v_{D} in the presence of rotation. Actually, I think the LSSS geometry and the whole derivation are not necessary. By assuming zonal symmetry and a flat, constantpressure surface, continuity imposes the barotropic meridional velocity to be constant and therefore zero. Neglecting the nonlinear momentum transport essentially sets the barotropic zonal velocity to zero as well and reduces the equations of baroclinic wind to the MatsunoGill system (Matsuno 1966, Gill 1980). This system is sufficient to simulate the main features of monsoon circulations (Gill 1980, Bellon and Reboredo 2022).
Also, in the main text, the coefficients <a_{1}> and <b_{1}> resulting from vertical averaging should appear, respectively, in front of the time derivatives of T_{1L} and q_{1L }in Equations 2 and 3. The authors should check that they are taken into account in the computations of the stability of the equilibria. Equations 2 and 3 need to be clarified, for ease of undertanding: terms corresponding to the same physical contribution (horizontal transport, vertical transport, diabatic sources) should be factorized as much as possible and regrouped; most parentheses are currently not necessary (and one is not opened in 3b), the diabatic terms Hg/p_{t} , Rg/p_{t }, Eg/p_{t} could be written <H>, <R>, <E> for simplicity, and a notation for R+H (the source of dry static energy) would also simplify the equations.
Finally, it seems to me that the expression of B_{c} after Equation 4 is missing a factor τ_{c }in its second term on the right hand side, and I have doubts about the sign in front of <a_{1}V_{1}> in the first term on the righthand sign.
4. As I understand it, R is imposed and varied systematically, but there is no mention of how H and E are set for the solutions presented in Section 3. And if all the QTCM parameters can be found in the reference article, it would be worth giving the values of these parameters. Also, T_{1s} and q_{1s }should be specified and, if they are set to zero (i.e., the oceanic surface is at the reference state of the QTCM), it could be specified early in the manuscript so as to simplify the equations.
5. If Boos and Storelvmo (2016) listed the physical misconceptions in Leverman et al. (2009), it would be worth mentioning that the model with no stratification does not simulate a nonprecipitating equilibrium for R< 0, which can be considered as winter conditions. Indeed, without adiabatic warming due to subsidence, there is no term that can compensate diabatic cooling. Leverman et al. (2009) considered only horizontal advection in the lower troposphere, which can be a cooling term over the continent for onshore lowlevel flow but can hardy be a warming term ( except if the advection by the returning uppertropospheric flow is included). Arbitrarily changing the sign of <a_{1}V_{1}> is really not physically relevant and does not really show any particularly interesting behavior of the system. I think Section 3.2 should investigate only the sensitivity to M_{s}, which can be considered to depend on the reference stratification of temperature T_{r} and therefore changed to some extent. To better document the sensitivity of the system, the authors could investigate the sensitivity to the profile of temperature perturbation a_{1}(p) (more or less similar to a perturbation of the moist adiabat, similarly to Section 3.c of Bellon and Sobel 2010), which would modify V_{1}(p) and multiple other parameters (<a_{1}>, <a_{1}V_{1}>, M_{sp}, <b_{1}V_{1}>, M_{qp}) in a pÄ¥ysically consistent framework.
Minor edits:
a. Some references are supposed to be in line in the text but appear in parentheses.
b. In many instances, the text and captions refer to thick and thin solid lines in the figures. The figures have obviously been changed since these descriptions have been written.
c. Readers should not have to read Boos and Storelvmo (2016)’s article to know what α_{T} and α_{q} mean.
d. On line 58, moist static energy (MSE) does not increase with altitude. It has a minimum in the middle troposphere. Overall, the gross moist stability is positive because in average the uppertropospheric MSE (above the minimum) is larger than the lower tropospheric MSE.
Additional references:
Bellon, G., & Reboredo, B. (2022). Scale sensitivity of the Gill circulation. Part II: Offequatorial case. Journal of the Atmospheric Sciences, 79(1), 1930.
Gadgil, S. (2018). The monsoon system: Landsea breeze or the ITCZ? Journal of Earth System Science, 127 (1), 1–29.
Geen, R., Bordoni, S., Battisti, D. S., & Hui, K. (2020). Monsoons, ITCZs, and the concept of the global monsoon. Reviews of Geophysics, 58(4), e2020RG000700.
Gill, A. E. (1980). Some simple solutions for heatâinduced tropical circulation. Quarterly Journal of the Royal Meteorological Society, 106(449), 447462.
Matsuno, T. (1966). Quasigeostrophic motions in the equatorial area. Journal of the Meteorological Society of Japan. Ser. II, 44(1), 2543.
Ramage, C. S. (1971). Monsoon meteorology, Academic Press.
S. Krishna Kumar and Ashwin K. Seshadri
S. Krishna Kumar and Ashwin K. Seshadri
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