1. Introduction
The variation of the asymmetry parameter with warming was investigated in a previous study by O’Gorman et al. (2018) using an idealized aquaplanet GCM in which large changes in climate and in the extent of the nonlinearity of the flow can be simulated relatively easily. While the asymmetry factor λ increased strongly with warming for the most unstable modes of moist baroclinic instability, the asymmetry increased only slightly with warming in fully nonlinear simulations (O’Gorman et al. 2018, their Fig. 3b). This distinction is significant for our dynamical understanding since the atmosphere is in a macroturbulent state more akin to that of the fully nonlinear simulations, even if insights into cyclogenesis can be obtained from the study of unstable modes. Here, macroturbulence refers to the turbulence of large-scale eddies in the troposphere following Held (1999). We will refer to the small-amplitude unstable modes as the “modal regime” and the fully nonlinear simulations at statistical equilibrium as the “macroturbulent regime” from here on. The slight increase in λ with warming in the macroturbulent regime of the idealized GCM over a wide range in climates is also consistent with what has been found for projected changes under the representative concentration pathway 8.5 emission scenario with the MPI-ESM-LR model (Tamarin-Brodsky and Hadas 2019).
In O’Gorman et al. (2018), the calculations of moist baroclinic instability differed from the fully nonlinear simulations by having small-amplitude disturbances but also by assuming that upward motion is saturated and by only taking into account moist diabatic tendencies from the large-scale condensation scheme and not from the moist convection scheme. To exclude the differences in the representation of moist processes as a cause for the different behavior of the asymmetry with warming, the authors performed a second set of simulations in which both large-scale condensation and moist convection schemes were turned off and the effects of latent heating were parameterized simply by reducing the dry static stability in the region of ascent by a factor 0 < r ≤ 1 in the spirit of simple moist dynamical theories (e.g., Emanuel et al. 1987; Zurita-Gotor 2005). Here, r = 1 corresponds to a fully dry simulation and r → 0 corresponds to an increasingly warm and moist climate with weak moist static stability. The mean state of the simulations was held close to that of a control simulation by using a strong relaxation. From here on, we will refer to these simulations as “reduced stability simulations” to distinguish them from the “global warming simulations” that include convection and large-scale condensation schemes. Even with this greatly simplified representation of moist physics, a similar distinction between modal and macroturbulent regimes emerged: as r → 0, λ in the modal regimes increases toward one corresponding to highly asymmetric vertical velocities, but λ in the macroturbulent regime increases only slightly before equilibrating to a much lower value of about λ = 0.71 (O’Gorman et al. 2018, their Fig. 9). The different representation of moist physics are thus not a likely contributor to the different behavior of λ. Instead, the authors concluded that nonlinear equilibration to a macroturbulent state leads to a significant reduction of λ compared to small-amplitude modes particularly in warm climates.
While simple theoretical scalings laws for the asymmetry of moist baroclinic waves exist (Emanuel et al. 1987; Zurita-Gotor 2005; Pendergrass and Gerber 2016), these do not carry over to the macroturbulent regime (O’Gorman et al. 2018, their Fig. 9), making it desirable to understand why modal theory fails and to develop a theory for the value of λ that is reached in the macroturbulent regime.
To this end, in section 2 we place ourselves in the framework of moist quasigeostrophic (QG) theory and more specifically a moist QG omega equation in which the effects of latent heating are represented as an internal rather than external process. We show that the moist QG omega equation captures the behavior of λ in the idealized GCM simulations of O’Gorman et al. (2018) when the dynamical forcing of vertical motion by the balanced motion is taken as given from the output of the idealized GCM. We go on to show that changes in λ in the modal regime with warming or decreasing r are related to changes in both the moist static stability and the dynamical forcing, while changes in λ in the macroturbulent regime are primarily related to changes in moist static stability with the dynamical forcing not becoming very skewed. This leads to a smaller asymmetry in the macroturbulent phase compared with the modes for warm climates or at small values of r.
In section 3, we use a two-layer moist quasigeostrophic model to better understand the role of the moist static stability and dynamical forcing in setting λ. We show how a feedback between the dynamical forcing in the moist omega equation and the vertical velocity leads to an increase in asymmetry of the vertical velocity field in the modal regime. We then distill the insights from the macroturbulent inversions in section 2 into a simple toy model of the moist omega equation in the macroturbulent phase that is solved for a given moist static stability and wavenumber of the dynamical forcing. In contrast to moist baroclinic theory, we show that the toy model better reproduces the slow increase of the asymmetry with climate warming in the idealized GCM simulations.
In section 4, we apply moist baroclinic theory and our simple toy model to the change of asymmetry seen over the seasonal cycle in reanalysis. The seasonal cycle forms a useful test ground for asymmetry theories, since the moist static stability decreases a lot from winter to summer, particularly in the Northern Hemisphere. We show that while moist baroclinic theory overpredicts the increase in asymmetry from winter to summer, the toy model does better at capturing the slow change of the asymmetry seen over the seasonal cycle.
In section 5, we summarize our results and discuss future work.
2. Moist QG omega equation inversions applied to the idealized GCM simulations
The goal of this section is to understand the different sensitivity of λ to warming between modal and macroturbulent regimes by applying a moist QG omega equation to the idealized GCM output from O’Gorman et al. (2018). The advantage of studying the vertical velocity through the framework of the moist omega equation rather than looking at the vertical velocity from the GCM output directly is that the moist QG omega equation allows us to tease apart the contributions to the vertical velocity and its asymmetry coming from dynamical forcing of vertical motion by the horizontal balanced flow versus thermodynamic contributions from the thermal stratification. The perspective of the moist omega equation then allows us to identify the causes of differences between λ in the modal and macroturbulent regimes.
a. The moist QG omega equation
b. Details of the idealized GCM simulations
We briefly describe the GCM and simulations to which the moist omega equation is applied here but further details can be found in O’Gorman et al. (2018). The idealized GCM is based on a spectral version of the Geophysical Fluid Dynamics Laboratory (GFDL) atmospheric dynamical core (see Frierson et al. 2006; Frierson 2007; O’Gorman and Schneider 2008). The resolution is T85 with 30 vertical levels, and this resolution is used for both the global warming and reduced stability simulations. A thermodynamic mixed layer ocean of depth 0.5 m forms the lower boundary condition and no horizontal ocean heat transport or sea ice is included. Moist convection is parameterized using the scheme of Frierson (2007). Longwave radiation is modeled using a two-stream gray scheme. There are no seasonal or diurnal cycles of shortwave radiation and no cloud or water vapor radiative effects.
For the global warming simulations, the climate is varied by multiplying the longwave optical thickness by a factor α, where α = 1.0 corresponds to the control climate with global mean surface temperature of 288 K. The simulations were run for a period of 300 days after spinup. The most unstable modes were calculated through repeated rescaling of perturbations to small amplitude, assuming upward motion to be saturated, for a period of 300 days and using a basic state equal to the zonal and time mean of a fully nonlinear simulation for that climate (but with mean meridional and vertical winds set to zero). We emphasize that these instability calculations are hence different from baroclinic life cycle experiments, in which perturbations are not periodically rescaled to keep them at small amplitude.
For the reduced stability simulations, latent heating was parameterized by reducing the static stability in ascending air, and the radiation, large-scale condensation and moist convection schemes were turned off. The reduction factor r in ascending air was specified as a constant in the troposphere and smoothly transitioned as a function of pressure to a value of one in the stratosphere following the vertical profile r + 0.5(1 − r){1 − tanh[(p − pu)/pw]}, where pu = 200 hPa is the nominal uppermost pressure level for the reduction in static stability, and pw = 50 hPa is the width of the transition region about this level (O’Gorman et al. 2018). Hence, the reduction factor r no longer follows Eq. (6) in the reduced stability simulations. The mean state of the simulations was held close to that of the control simulation (α = 1.0) by using a strong relaxation. The simulations were run for 300 days after spinup. The most unstable mode was calculated under the reduced stability parameterization following the same small-amplitude approach as was used for the global warming simulations.
c. Methods: Numerical approach to inverting the moist omega equation
We use the output from the idealized GCM simulations of O’Gorman et al. (2018) for the moist static stability and the dynamical forcing at every time step to invert Eq. (2) for ω, thus leaving the time evolution of the flow to the higher-order dynamics of the GCM. The nonlinearity in latent heating enters through the factor
While σ can in general be a function of the horizontal and vertical, we have found it useful for numerical stability to average T and θ horizontally before calculating σ. Hence, the background stratification that enters Eq. (2) for our inversions does not vary in the horizontal, although it is recalculated for each time step and so can vary in time. Because the moist static stability is a product of
Horizontal winds, vertical velocity, and temperature from the GCM output were interpolated from sigma to pressure coordinates and replaced with NaN wherever the interpolated pressure was below the surface pressure. The pressure levels span from 989 to 3 hPa. The lower boundary condition was imposed at the lowest pressure level where no NaN values were encountered in the domain at each instant in time, which is typically 928 hPa. The geostrophic component of the wind was calculated as the rotational part of the full horizontal wind field, to minimize the influence of gravity waves (Nielsen-Gammon and Gold 2008), by inverting the relative vorticity on a global spherical grid in pressure coordinates.
Inversions with random initial guesses were also tried and the solutions were found to be insensitive to the choice of the initial guess but take longer to converge. We have found it necessary to include ω = ωGCM as the lower boundary condition to better capture the macroturbulent values of λ in the global warming simulations, which were underestimated with the simpler boundary condition ω = 0. We use the same lower boundary condition for the modal inversions and for the macroturbulent inversions in the reduced stability simulations for consistency, even though it did not substantially improve the agreement in these cases. Stricter convergence criteria rms < 10−4 have also been experimented with but the solutions and values of λ were visually indistinguishable. Although the GCM domain is periodic in the zonal direction, the ω = 0 boundary condition in the zonal direction has been adopted for implementational simplicity since the solver was developed from a preexisting code with Dirichlet boundary conditions used in Li and O’Gorman (2020). Since we are interested in the statistics of λ and are considering averages over a large domain, the statistics are expected to be insensitive to what happens near the horizontal boundaries. The goodness of the agreement between the inverted and GCM vertical velocities and their asymmetry λ calculated over the domain (see section 2d) give us confidence that the periodic boundary effects can be neglected for the purpose of this study.
d. Results of the inversions
We begin by comparing the GCM and inverted vertical velocity field at 500 hPa for the most unstable mode and macroturbulence regime of the global warming simulation at a global-mean surface air temperature of 288 K (the reference simulation that is most similar to the current climate) at a single instant in time. A midtroposphere level is chosen because that is roughly where the vertical velocity is strongest. Two-dimensional fields are shown in Fig. 1 and cross sections at 50°N latitude in Fig. 2. Focusing on the modes, we observe that inverted and GCM vertical velocity field are in near perfect agreement, except close to the boundaries where a different boundary condition was implemented (see edges of the domain in Fig. 2a). Focusing on the macroturbulent fields, we observe that the agreement between inverted and GCM vertical velocity field is less good, and this is as expected since the GCM flow is in a larger Rossby number state and does not assume upward motion to be saturated. Nevertheless, the inverted vertical velocity is able to capture most of the large-scale ascent and descent patterns well, as confirmed by the cross section shown in Fig. 2b.2 Similar results were found in the reduced stability simulations (not shown).
Comparison of the instantaneous GCM vertical velocity field (−ω; red indicates upward motion) to the inverted vertical velocity field obtained from inversion of Eq. (2), at 500 hPa. Results are shown for (a),(b) the unstable mode and (c),(d) the macroturbulent regime of the global warming simulations of O’Gorman et al. (2018) at a global-mean surface air temperature of 288 K. The modes were calculated by O’Gorman et al. (2018) through repeated rescaling of the equations to small amplitude, and hence, their magnitude is arbitrary. The time instant chosen for comparison was arbitrary.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
Cross section of the GCM (black) and inverted (blue) vertical velocity fields shown in Fig. 1 at latitude 50° for (a) the mode and (b) the macroturbulent regime. The amplitude of the mode is arbitrary. Please note that instead of periodic boundary conditions used in the zonal direction in the GCM, Dirichlet conditions with ω = 0 have been used in the inversions, and hence, agreement is not expected at the boundaries (see further discussion in section 2c).
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
We now compare the statistics of the asymmetry parameter λ for inverted and GCM vertical velocities in both the reduced stability and global warming simulations (Fig. 3). The value of λ was calculated between 40° and 60° latitude for the global warming simulations and between 25° and 65° in the reduced stability simulations and then averaged in time, meridionally over the latitude band and vertically over the troposphere. Following O’Gorman et al. (2018), a wider latitude band is chosen for the reduced stability simulations because the unstable modes are not necessarily localized in the 40°–60° latitude band in this case. The tropopause was defined as the highest level at which the domain (40°–60° latitude band) and time mean lapse rate is greater than 2 K km−1. To facilitate comparison between the global warming and reduced-stability simulations, Fig. 4 shows the reduction factor r, calculated from Eq. (6), versus global mean surface temperature in the global warming simulations at 500 hPa and averaged vertically up to the tropopause. The reduction factor r decreases as the climate warms and the midlatitude stratification approaches moist adiabatic.
Comparison of the asymmetry parameter λ for GCM vertical velocities (solid lines) and the QG inverted vertical velocities calculated from the inversion of Eq. (2) (dashed lines). Results are shown for the modal (red) and macroturbulent (blue) regimes in the (a),(b) reduced stability simulations and (c),(d) global warming simulations from O’Gorman et al. (2018). In (a) and (c)
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
Reduced stability parameter r vs global mean surface temperature in the idealized global warming simulations. The r value was calculated using Eq. (6) and is shown both at 500 hPa (blue line) and averaged vertically up to the tropopause (black line). The tropopause was defined as the highest level at which the domain (40°–60° latitude band) and time mean lapse rate for a given climate simulation is greater than 2 K km−1.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
The basic behavior of the idealized simulations that we are trying to capture and understand is that in response to increasing global-surface temperature or decreasing reduction factor r, λ increases strongly for the most unstable modes but increases only moderately in the macroturbulent regime (Figs. 3a,c; solid red vs solid blue line). We first discuss results for full inversions that include latent heating through
We focus first on the results of the full inversion that includes latent heating through
Focusing next on the inversions in which
3. Understanding asymmetry behavior using two-layer moist QG
We next use a two-layer moist QG framework to better understand the asymmetry behavior in the modal and macroturbulent regimes. We begin by developing an understanding for why the dynamical forcing is skewed in the modal regime in section 3a, before distilling the insights of the 3D inversions into a toy model for λ in the macroturbulent regime in section 3b.
a. Behavior of the dynamical forcing for moist unstable modes
Equations (11)–(13) have been studied in quasigeostrophic (Zurita-Gotor 2005) and semigeostrophic (Emanuel et al. 1987) form to analyze the effects of latent heating on the growth rate and length scale of the most unstable modes of baroclinic instability. It was found that latent heating increases the growth rate and shifts the most unstable mode to smaller length scales. Here, we focus on the effect of latent heating on the asymmetry of the vertical velocity of the most unstable moist modes.
1) Methods: Calculation of most unstable mode in two-layer model
2) Results
An example vertical velocity profile of the most unstable moist mode in this system at r = 0.1 is shown in Fig. 5a. The solution consists of a periodic wave whose ascent length is reduced compared to the descent length. This is consistent with the structure of the most unstable mode that was found in the idealized GCM calculations (see Fig. 2a).
(a) Vertical velocity profile of the most unstable mode of 1D moist baroclinic theory at r = 0.1. (b) Comparison of the asymmetry parameter λ for the most unstable modes predicted by 1D moist baroclinic theory, and for the most unstable modes calculated using a reduced stability parameterization for the GCM simulations of O’Gorman et al. (2018). The λ of the modes in the reduced stability GCM simulations is averaged over the troposphere and was also shown in Fig. 3a.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
We repeat the calculation for different values of r and compare the asymmetry of the most unstable mode in this two-layer system to the asymmetry for the modes of the reduced stability GCM simulations (see Fig. 5b). The reduced stability simulations are chosen for ease of comparison to our two-layer model, since a constant reduction factor is applied throughout the troposphere in these simulations. In the global warming simulations, r varies with altitude which is more difficult to capture in a two-layer setting. Because the vertical velocity field is a function of x only, we will refer to the predictions of this two-layer moist QG model in the small-amplitude regime as 1D modal theory (to be distinguished from the 1D toy model for the macroturbulent phase introduced in the next section).
Looking at Fig. 5b, we see that the asymmetry of the most unstable modes of the 1D theory agrees remarkably well with that found in the idealized GCM experiments given the simplicity of the two-layer setup. The modes become very skewed as r → 0, which can also be confirmed by looking at the w profile of the most unstable mode from the 1D theory at r = 0.1 (Fig. 5a). The ascent length is greatly reduced compared to the descent length, in line with the results of Emanuel et al. (1987) and Zurita-Gotor (2005). Adv is markedly skewed in the two-layer theory with a skewness of −2.1, where the skewness is calculated as
(a) Vertical velocity profiles predicted by the 1D toy model at k = 1.7 for different values of r. (b) Asymmetry parameter λ for the vertical velocity field predicted by the 1D toy model [Eq. (17)] for different wavenumbers k as a function of the reduction factor r. (c) Asymmetry parameter λ for the vertical velocity field predicted by the 1D toy model for different values of the reduction factor r as a function of k.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
b. Toy model for moist macroturbulence
1) Methods: Solution of the toy model and diagnosis of k
Comparison of the asymmetry parameter λ for the vertical velocity field predicted by the 1D toy model [Eq. (17); red lines], to the asymmetry found in the reduced-stability GCM simulation (blue solid line; as in Fig. 3a) and the 3D omega equation inversions applied to the reduced stability GCM simulations (blue dashed line, as in Fig. 3a). The red solid line shows the 1D toy model prediction using a wavenumber k which was calculated from the centroid of the w spectrum of the GCM simulation. The red dotted line shows the 1D toy model prediction where the wavenumber was alternatively calculated from the centroid of the Adv spectrum.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
2) Results
Vertical velocity profiles from the toy model are shown in Fig. 6a for a range of values of r and for a fixed value of k = 1.7 which is the k value that was diagnosed from the reduced stability GCM simulation at r = 0.01. Values of λ as a function of both r and k are shown in Figs. 6b and 6c. Focusing on the vertical velocity profile at k = 1.7 (Fig. 6a), we see that although the ascent length shrinks as r becomes smaller, the ascent length does not collapse and the vertical velocity profiles do not become very asymmetric, especially compared to the profile of the mode predicted by 1D theory at r = 0.1 (Fig. 5a). Looking at the corresponding behavior of λ(r) at k = 1.7 in Fig. 6b, we find that the toy model has a value of λ = 0.75 at r = 0.01 and asymptotes to λ ≃ 0.78 as r → 0 (not shown). By comparison the 1D modal theory has a value of λ = 0.95 at r = 0.01 (Fig. 5b) and asymptotes to λ = 1 as r → 0 (this limit is known from the theory of moist baroclinic modes, which predicts vanishing updraft length as r → 0; Emanuel et al. 1987; Zurita-Gotor 2005). Thus, the asymmetries are much higher for modal theory than for the toy model.
In addition to predicting that λ increases as r decreases, the toy model also predicts that λ increases with increasing wavenumber k, and the sensitivity to k is greatest for low values of r (Figs. 6b,c). When r = 0.5, for instance, the asymmetry is already converged at k ≈ 2 but when r = 0.01 the asymmetry is converged only at k ≈ 6. The dependence of λ on k can be understood by considering the left-hand side of the toy model Eq. (17). Increasing k leads to smaller length scales and thus increases the importance of the first term on the left-hand side compared to the second term (because of the Laplacian operator in the first term), and it is the first term that is the root cause of asymmetry through
As shown in Fig. 7, the toy model prediction for λ is in reasonably good agreement with the reduced stability GCM simulations given the simplicity of the toy model (solid red vs solid blue line). Very similar results are found when k is calculated from the centroid of the Adv rather than w spectrum (red dotted line in Fig. 7), even though k is 1.7 times larger in this case averaged over all r values and reaches values nearly twice as large at r = 0.01 (k = 3.2 instead of k = 1.7). This is because λ from the toy model theory is not strongly sensitive to k except for very low r values (Fig. 6c). The underestimate of λ as r → 1 by the toy model is likely a result of the simplified nature of the dynamical forcing which is completely unskewed and represented by a simple sinusoid, since the full 3D moist QG omega inversion does a better job at reproducing the GCM asymmetry for r → 1 (see dashed line in Fig. 7). The assumption of a one-dimensional dynamical forcing also contributes to underestimating λ: while the GCM vertical velocity fields include linear frontal bands, they are not purely one-dimensional, and including this two-dimensionality would increase the effective k and thus increase λ. The overestimate of λ at low r values by the toy model points to a deficiency of the moist QG omega framework at capturing all the controls on the vertical velocity field, since the full 3D moist QG omega inversions also overestimate λ at r = 0.01.
Overall, the toy model helps to explain why λ increases less rapidly as r decreases compared to moist baroclinic modes, and it also demonstrates a weak sensitivity of λ to the wavenumber of the dynamical forcing in the QG omega equation.
4. Applying the toy model to the seasonal cycle of λ in reanalysis
We next apply the toy model to the seasonal cycle of λ observed in the current climate and contrast it with the predictions from moist baroclinic modes. The seasonal cycle forms a useful test ground for asymmetry theories, since the moist static stability is smaller in summer than in winter, particularly in the Northern Hemisphere. We compare the theoretical predictions from the 1D modal theory and the 1D toy model to the seasonal cycle of λ found in ERA5 at 500 hPa in both the Northern Hemisphere (NH) and Southern Hemisphere (SH) (Fig. 8). We use ERA5 as a modern global reanalysis in which the dynamics were evolved at the highest available resolution, but earlier reanalyses can show quite different results as discussed in appendix B.
(a) Comparison of the seasonal cycle of the asymmetry parameter λ at 500 hPa in the Northern Hemisphere (NH) from ERA5 (blue line) to the predictions from the 1D toy model (black dash–dotted line) and 1D modal theory (black dashed line). Six-hourly fields were used for all reanalysis data spanning years 2009 to 2018. ERA5 fields were coarse grained from a native grid spacing of 0.25° to 1.5°. The asymmetry parameter has been calculated over the latitude band of 30°–70°. (b) As in (a), but for the Southern Hemisphere (SH).
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
a. Methods: Calculation of λ and evaluation of theories based on reanalysis
The vertical velocity and temperature data used are 6-hourly fields spanning the latitude band 30°–70° and years 2009–18. The fields have been coarse grained from a native ERA5 grid spacing of 0.25° to 1.5° to make it more comparable to the GCM grid spacing (1.4°) and because the omega equation has been found to remain applicable only down to horizontal scales of roughly 140 km (Battalio and Dyer 2017). We show and discuss the results for λ at the native ERA5 grid spacing in appendix B.
The asymmetry parameter λ in reanalysis has been calculated based on a horizontal average over the latitude band and then averaged over the 10 years for each month. For the theoretical predictions, r was calculated from the temperature field at 500 hPa using Eq. (6), and then averaged over the latitude band 30°–70° and over the 10 years for each month. The wavenumber k was calculated again as the centroid wavenumber of the 1D zonal power spectrum of w in ERA5 at 500 hPa using Eq. (18), and then averaged over the latitude band and over the 10-yr period for each month. Again, we have also tried estimating k from the centroid of spectrum of Adv rather than the spectrum of w, but the toy model predictions were nearly identical, because λ is not strongly sensitive to the choice of k for the values of r considered, and thus, these results are not shown. The wavenumber k was then nondimensionalized by the deformation radius
b. Seasonal cycle of λ
Looking at Fig. 8, we see that λ in ERA5 has a seasonal cycle that peaks during summer in each hemisphere. This is as expected given that r as shown in Fig. 9a decreases as the stratification becomes closer to moist adiabatic in summer (Stone and Carlson 1979), and it is also consistent with the result that λ is larger in summer compared to winter in extratropical cyclones (Tamarin-Brodsky and Hadas 2019). The seasonal cycle is more pronounced in the NH, varying between values of λ = 0.62 and λ = 0.69, than in the SH where it varies between λ = 0.62 and λ = 0.65.
Seasonal cycle of (a) the static stability reduction factor r and (b) the nondimensional wavenumber of the dynamical forcing k in the Northern (solid) and Southern (dashed) Hemispheres at 500 hPa in ERA5. ERA5 fields have been coarse grained from a native grid spacing of 0.25° to 1.5°. The reduction factor and wavenumber of the dynamical forcing have been averaged over latitudes 30°–70° and years 2009–18. r = 1 corresponds to a dry atmosphere, and r = 0 corresponds to a moist atmosphere with a moist adiabatic lapse rate. In both hemispheres, r is smallest during the summer, but the seasonal cycle is more pronounced in the Northern Hemisphere.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
c. Toy model versus modal prediction for the seasonal cycle
Given that r undergoes large variations between winter and summer months (0.10 < r < 0.45 in the NH and 0.28 < r < 0.40 in the SH) as shown in Fig. 9a, the magnitude of the seasonal cycle of λ is surprisingly small in both hemispheres from the point of view of modal theory. Indeed, 1D modal theory consistently overestimates λ in both hemispheres reaching peak values of λ = 0.83 in the NH and λ = 0.74 in the SH, and overestimates the size of the seasonal range of λ by a factor of 2.6 in the NH and 2.4 in the SH. This is in line with the results of the idealized GCM simulations which showed that variations of λ in the macroturbulent state with warming are considerably smaller than what moist modal theory predicts.
In comparison to the modal theory, the seasonal cycle range and overall values in reanalysis are better captured by the 1D toy model. The toy model still overestimates the size of the seasonal cycle by 1.6 in the NH and 1.1 in the SH, but these overestimates are considerably smaller than what was found from modal theory (2.6 and 2.4, respectively). According to the toy model, the seasonal cycle in λ is primarily from the variations in r because variations of k (between 2.5 and 3.2 in the NH, and between 2.3 and 2.5 in the SH, see Fig. 9b) would only substantially affect λ at smaller values of r than are found in the seasonal cycle (compare the variations of λ with k predicted by the toy model at r = 0.01 and r = 0.5 in Fig. 6c). This was further checked by calculating λ from the toy model keeping k fixed over the seasonal cycle which did not affect the results in Fig. 8. This is also the reason why the toy model predictions remain virtually unaffected when k is taken from the centroid of the Adv spectrum rather than the w spectrum. Hence, from the point of view of the toy model, it is the weak seasonality of r in the SH that is reason for weak seasonality of λ in SH and it is the stronger seasonality of r in the NH that is the reason for a stronger seasonality of λ in the NH.
In conclusion, despite the simplicity and rough approximations of the toy model, we argue that it better captures the moderate variation of λ that is observed over the seasonal cycle as compared to modal theory.
5. Conclusions
Idealized GCM simulations of moist macroturbulence show that the asymmetry of the vertical velocity distribution is considerably smaller than what moist baroclinic instability theory predicts in warm climates or low values of r. This is significant given that the atmosphere is constantly in a state more akin to that of the macroturbulent simulations than the moist baroclinic modes that our theoretical understanding is based on. This makes the development of a theory for the asymmetry in the macroturbulent state desirable.
To bridge this gap in understanding, we have applied inversions of a moist QG omega equation [Eq. (2)] to the idealized GCM output to identify why the asymmetry is larger in the modal compared to macroturbulent regime (section 2). The inversions showed that while dynamical forcing of the omega equation is very skewed in the modal regime, it contributes negligibly to the asymmetry in the macroturbulent regime which is almost entirely determined by the reduction in static stability in ascending air on the left-hand side of the moist omega equation (Fig. 3). Hence, the asymmetry in the macroturbulent regime of the simulations is lower than for the modes.
A two-layer moist QG framework was then used to understand asymmetry behavior (section 3). We showed that in the modal regime, a feedback between the dynamical forcing and the vertical velocity [deduced from Eqs. (14) and (15)] leads to large asymmetries contributed by the dynamical forcing, consistent with what was found for the modes in the GCM. Such a feedback is not expected for the macroturbulent phase because of advective nonlinearities which disrupt the modal structure and lower the skewness of the dynamical forcing of the moist omega equation.
We then distilled the insights from the moist omega inversions and, in particular, the unskewed dynamical forcing in the macroturbulent regime into a simple 1D toy model of the moist QG omega equation by replacing Adv by an unskewed function [Eq. (17)]. The toy model was solved for a given wavenumber k of the dynamical forcing on the right-hand side of the omega equation and for a given static stability reduction factor r. Compared to the 1D moist modal theory, the toy model was able to better reproduce the weak increase of the asymmetry with warming that has been observed in the macroturbulent regime of the idealized GCM simulations (Figs. 6, 7). However, both toy model and the full 3D moist QG omega equation inversions overpredict the asymmetry found in the idealized GCM simulations in warm climates or at low values of the reduction factor r pointing to a deficiency of the moist QG omega framework at capturing all the controls on λ in these limits.
We went on to study the seasonal cycle of λ in reanalysis which forms a useful test ground for asymmetry theories, since the moist static stability varies a lot between seasons, particularly in the NH (section 4). We showed that moist baroclinic modal theory considerably overpredicts the increase in λ from winter to summer, whereas the toy model better reproduces λ over the seasonal cycle in ERA5 (Fig. 8). The interpretation is once again that asymmetry changes in macroturbulent flows in response to changes in moist static stability are much smaller than what moist unstable baroclinic modes suggest. While the seasonal cycle of λ is somewhat similar in ERA5 and the earlier ERAI, it is completely absent in the NCEP2 reanalysis (appendix B), and further studies of the sensitivity of λ to resolution and reanalysis product would be useful.
The dynamical forcing assumed in the toy model is highly idealized, but we argue that it is nonetheless useful to illustrate the controls on the asymmetry implied by the moist QG omega equation. Our toy model theory is not closed since it takes as given a single wavenumber for the dynamical forcing, although the sensitivity to this wavenumber is relatively weak. A more complete theory would not use just one wavenumber but rather take as input the power spectrum of the dynamical forcing, and understanding what sets this power spectrum remains an important outstanding problem for future work.
Finally, our toy model predicts that high λ is still possible in a macroturbulent state even with unskewed dynamical forcing of the omega equation provided that 1) r is sufficiently low, 2) k is large meaning that the length scale of the dynamical forcing is small compared to the dry deformation radius, and 3) the Rossby number is low so that the omega equation remains valid despite the high k and low r. It would thus be interesting to investigate such states and their asymmetries further by running simulations of the two-layer moist QG systems and comparing them to simulations of the moist primitive equations at a range of Rossby numbers, and we plan to report results on this in future work.
If the mean ω is zero, then the asymmetry parameter λ and the negative skewness
For inversions in the reference climate shown here, it is frequently the case that the inversion underestimates the strongest updrafts, even when the full horizontal winds, rather than the geostrophic winds, are used in the inversion. However, in warmer climates this is no longer the case and the inversion can overestimate the peak updraft strength. This is consistent with the overprediction in asymmetry in warmer climates in Fig. 3.
Discretizing the continuous thermodynamic equation leads to a deformation radius involving N at the midtropospheric level rather than a reduced gravity.
Equations (14) and (15) have the property that for r = 1 and a given solution w and Adv, there is also a solution −w and −Adv, and thus, there is nothing to break the up–down symmetry and Adv must remain unskewed.
Note the extra factor of 2 in the definition of the deformation radius because H in our two-layer theory is the layer height and not the tropopause height.
Acknowledgments.
We thank Ziwei Li for providing the code for solving the dry omega equation which formed the basis of our moist omega equation solver. We acknowledge helpful discussions with Ziwei Li, Stephan Pfahl, Kerry Emanuel, and Glenn Flierl. We acknowledge support from NSF AGS 2031472 and the mTerra Catalyst Fund.
Data availability statement.
ERA5 data are available from ECMWF (https://cds.climate.copernicus.eu). ERAI data are available from ECMWF (https://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=pl/). NCEP2 data are available from NOAA (https://psl.noaa.gov/data/gridded/data.ncep.reanalysis2.html). Code for the inversions, instability calculations, and processing of reanalysis datasets is available on GitHub (https://github.com/matthieukohl/Asymmetry_Paper).
APPENDIX A
Derivation of the Moist QG Omega Equation
APPENDIX B
Discussion of Differences in λ for Different Reanalysis Products and Resolutions
We compare the seasonal cycle of λ based on different reanalysis products and with and without coarse graining of ω. Figure B1 shows results for ERA5 data coarse grained to a grid spacing of 1.5° as was used in our main analysis, ERA5 data at their native grid spacing of 0.25°, ERAI data which have grid spacing of 0.75° (Dee et al. 2011), and NCEP2 reanalysis which has a grid spacing of 2.5° (Kanamitsu et al. 2002). The coarse graining we applied to ERA5 can be seen to have modestly decreased λ with the biggest effect in NH summer. Comparing ERA5, ERAI, and NCEP2 on their native grids, we find that the lower-resolution ERAI has somewhat smaller values of λ than ERA5 but overall remains quite similar, whereas NCEP2 shows much smaller values and almost no seasonal cycle. This is true despite the fact that the seasonal cycle of r is close to identical in all reanalysis products (not shown) and follows the profile shown in Fig. 9a. The centroid wavenumber k is lower for NCEP2 than for the other reanalyses but we find this is not sufficient to explain the absence of a seasonal cycle according to the toy model. The absence of a seasonal cycle in NCEP2 is also not just due to the resolution of ω: a seasonal cycle for ERA5 persists even when it is coarse grained to the 2.5° grid spacing of NCEP2 (not shown). Similarly, we observe that λ for ERA5 coarse grained to 1.5° is larger than what is found from ERAI despite the fact that the ERAI grid spacing is 0.75°. This suggests that the differences in λ between reanalysis products are not just a result of the resolution of ω but are also affected by the different dynamics and other aspects of the models.
(a) Seasonal cycle of the asymmetry parameter λ at 500 hPa in the NH in ERA5 with grid spacing of 0.25° (dashed blue line), in ERA5 coarse grained to a grid spacing of 1.5° (blue solid line), in ERAI with grid spacing of 0.75° (red solid line), and in NCEP2 reanalysis which has a grid spacing of 2.5° (green solid line). The 6-hourly fields were used for all reanalysis data spanning the years 2009–18 and a latitude band 30°–70°. (b) As in (a), but for the SH.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0128.1
The question arises whether future reanalysis products with substantially higher resolution than ERA5 could show an even larger seasonal cycle of λ. We do not anticipate this to be the case because Booth et al. (2015) found that vertically averaged λ did not increase from a grid spacing of 50 to 3.125 km in simulations of a moist baroclinic life cycle, and λ at z = 5 km only increased slightly over this range of grid spacings. The QG omega equation is not a good tool to use at such short length scales, and it will be interesting to analyze the question of λ at very high resolution using other theoretical approaches in future work.
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