Asymmetry in the Seasonal Cycle of Zonal‐Mean Surface Air Temperature

At most latitudes, the seasonal cycle of zonal‐mean surface air temperature is notably asymmetric: the length of the warming season is not equal to the length of the cooling season. The asymmetry varies spatially, with the cooling season being ∼40 days shorter than the warming season in the subtropics and the warming season being ∼100 days shorter than the cooling season at the poles. Furthermore, the asymmetry differs between the Northern Hemisphere and the Southern Hemisphere. Here, we show that these observed features are broadly captured in a simple model for the evolution of temperature forced by realistic insolation. The model suggests that Earth's orbital eccentricity largely determines the hemispheric contrast, and obliquity broadly dictates the meridional structure. Clouds, atmospheric heat flux convergence, and time‐invariant effective surface heat capacity have minimal impacts on seasonal asymmetry. This simple, first‐order picture has been absent from previous discussions of the surface temperature seasonal cycle.

Here, we consider asymmetry in the seasonal cycle of zonal-mean surface air temperature, and in particular we focus on latitudinal variations in asymmetry (i.e., the meridional structure of the asymmetry). We show that the meridional structure of the asymmetry is consistent with idealized model results that describe the local thermodynamic response to the seasonal cycle in insolation, which features substantial departures from the annual harmonic in the high latitudes and subtropics. This model is cast as a simple ordinary differential equation (ODE) forced by realistic insolation; it is adopted from Roach et al. (2022) and applied here to a different question. The predicted asymmetry is found to be mostly insensitive to the inclusion of seasonally varying clouds, atmospheric circulation, and the specification of radiative feedbacks.

Methods
We use the daily 2-m air temperature (T2m) from the ERA5 reanalysis (Hersbach et al., 2020) and smooth it spatially by averaging the 0.25° data onto a coarser 2° grid. Leap years are linearly interpolated to 365 days. We then take the zonal mean and compute the climatology over the first four decades of the satellite era . Even after this spatial coarsening and temporal and spatial averaging, the seasonal cycle of T2m is visibly not smooth, and the computation of the timing of the maximum and the minimum of the seasonal cycle is sensitive to "noise" from internal variability. To further remove this noise, we fit a Fourier series with four harmonics to the zonal-mean climatological seasonal cycle of T2m at each latitude, θ. Specifically, we write where ξ n (θ) and ϕ n (θ) are the Fourier coefficients (results are similar when using 2-10 harmonics, as shown in Figure S1 in Supporting Information S1). Figure 1 shows zonal-mean seasonal cycles at specific example latitudes from ERA5 alongside a Fourier series with one harmonic (T 1 , which contains only the annual harmonic and is symmetric) and a Fourier series with four harmonics, T 4 .
We then compute the days of maximum and minimum temperature from the Fourier fit with four harmonics, T 4 (vertical solid black lines in Figure 1). We define the length of the warming season as the duration from the minimum to the maximum of T 4 . The length of the cooling season is similarly the duration from maximum to minimum, or equivalently 365 days minus the length of the warming season. As a measure of asymmetry, we define δ T as the difference between the lengths of the warming and cooling seasons (as in Roach et al., 2022). Positive δ T values represent a shorter warming season than cooling season, and negative values represent a shorter cooling season than warming season. For example, at 75°N, the date of minimum T 4 is delayed relative to its annual harmonic counterpart, whereas the date of maximum T 4 precedes maximum T 1 (see vertical black and gray-dashed lines in Figure 1b), with a warming season length of 162 days and a cooling season length of 203 days. This indicates a shorter warming season and positive δ T , with δ T = 41 days.
Globally, results are similar using the JRA55 reanalysis (Japan Meteorological Agency, 2013) in place of ERA5, although there are some deviations over the Antarctic continent ( Figure S2 in Supporting Information S1).

Results
To investigate large-scale drivers of asymmetry in the seasonal cycle of T2m, we first consider the broad-scale structure in the zonal-mean seasonal cycle. Zonal variations in observed asymmetry are shown by Donohoe, Dawson, et al. (2020), who emphasize the land-ocean contrast in δ T , which is readily apparent in the maps. Averaging T2m into the zonal mean reveals a striking meridional structure in observed δ T (Figure 2a), which is the focus of the present study. Values of δ T exceed 100 days near the poles, cross zero in the mid-latitudes, and reach −50 days in the tropics. Equatorward of 20°, we do not compute δ T because there are typically two local maxima and minima in T2m each year. δ T crosses zero (δ T = 0 days represents symmetry in the seasonal cycle of T2m) at 61° in the Northern Hemisphere and 37° in the Southern Hemisphere. There is a deviation from this overall pattern in the Southern Hemisphere, with δ T crossing zero again at 64°S. The hemispheric differences are highlighted in Figure 2b by plotting values for the Southern Hemisphere at their equivalent Northern latitudes. Values in the two hemispheres are offset by around 20 days on average, with higher values in the Southern Hemisphere ( Figure 2b).
We can gain understanding of the mechanisms controlling the meridional structure of δ T using the framework of an idealized climate model. Roach et al. (2022) show that a substantial asymmetry in the seasonal cycle of temperature (δ T ≈ 1 month) at high latitudes is obtained when solving a simple ODE in which seasonal variations in insolation are balanced locally by heating and radiative feedbacks. Hence we adopt Equation 7 from Roach et al. (2022): where c w is the heat capacity of the ocean mixed layer, chosen to approximate a 75 m mixed layer depth, T is the surface temperature departure from some reference value, and outgoing longwave radiation is approximated as A + BT (Budyko, 1969;Koll & Cronin, 2018) with A and B constants. Top-of-atmosphere (TOA) insolation S is computed using present-day orbital parameters following Huybers and Eisenman (2006) and Rose (2018), and is scaled by planetary coalbedo a, where a = a 0 − a 2 sin 2 θ with a 0 , and a 2 constants. Initial parameter values are shown in Table S1 in Supporting Information S1. Note that asymmetry in temperature is always insensitive to changes in A. We solve Equation 2 numerically, using 100 simulation years, discretized with 1,000 timesteps per year and 800 gridboxes in one spatial dimension running from pole to pole (0.1° resolution at the equator to 2° at the poles). Importantly, the equation neglects many factors including seasonal variations in atmospheric heat transport and cloud albedo, and spatial variations in radiative feedbacks-we return to the impact of these neglected processes later.
The asymmetry in the seasonal cycle in temperature, δ T , simulated using Equation 2 is shown in Figure 2c. This captures the broad-scale features shown in observations. In the simulations, δ T decreases from 80 days near the poles to −50 days in the tropics. It crosses zero at 51° in the Northern Hemisphere and 37° in the Southern Hemisphere, similar to the observations. Values in the two hemispheres are offset by around 17 days on average, with higher values in the Southern Hemisphere (Figure 2d), as compared with 20 days in observations (Figure 2b). Thus, the simple ODE quantitatively reproduces two essential features of the observed spatial structure of δ T : (a) more positive values in the Southern Hemisphere as compared to the same latitudes in the Northern Hemisphere and (b) positive values in the polar regions and negative values in the subtropics.
We first consider the hemispheric difference in δ T . The eccentricity of Earth's orbit is key to the hemispheric difference in simulated temperature asymmetry. We re-compute TOA insolation using the code from Rose (2018) with eccentricity set to e = 0 rather than its present-day value of e = 0.017. In this case, solving Equation 2 yields identical values for both hemispheres (blue line in Figure 2d).
Second, we consider the meridional structure of δ T , that is, the general pattern of increasing δ T from negative values in the tropics to positive values near the poles, which is retained after setting eccentricity to zero. This general pattern results from the obliquity (or tilt) of Earth's rotation relative to the plane of its orbit, which causes the seasonal cycle to depart from its annual harmonic in ways that are qualitatively different at high and low latitudes (left panels of Figure 3). At high latitudes, the insolation minimum is prolonged and the insolation maximum is larger and more concentrated than its annual harmonic (Figure 3a). In contrast, at subtropical latitudes, the insolation minimum is more peaked while the insolation maximum is more prolonged than the corresponding annual harmonic (Figure 3c).

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To understand how the departures in insolation from the annual harmonic impact δ T , we further simplify Equation 2 by omitting radiative feedbacks, that is, setting B = 0. As discussed in Text S1 of Roach et al. (2022), the dominant balance in the ODE for annual forcing is between insolation and heating. Decomposing the insolation forcing as = + ′ ( ) , where is the annual mean and ′ ( ) ≡ − is the seasonally varying departure, and similarly for temperature, we obtain: In the regime where ′ ≫ ′ , this becomes which is Equation S4 in Roach et al. (2022). In this case, the temperature is in quadrature with the forcing. Therefore, having a narrow summer insolation peak with and a wider winter insolation trough with , as occurs in high latitudes, leads to a brief summer warming period and a longer winter cooling period (Figure 3b). This is discussed equivalently for sea ice in Roach et al. (2022). Conversely, having a wide summer insolation peak with and a narrow winter insolation trough with , as occurs in low latitudes, leads to a long summer warming period and a brief winter cooling period (Figure 3d).
The impact of obliquity on δ T can be recovered when using only the first two harmonics of TOA insolation, to force Equation 2 ( Figure S3 in Supporting Information S1). The solution for temperature in Equation 4 forced by two harmonics in insolation (Equation 5) is the superposition of two temperature harmonics in quadrature phase with the insolation harmonics, Thus, the temperature response to obliquity is essentially a consequence of the relative phasing of the annual and semi-annual components of insolation, which vary by latitude and sum to make the seasonal peaks and troughs of insolation broader or narrower than those of the annual harmonic alone.
While the broad-scale features of δ T can be obtained considering only the remarkably simple Equation 2, we also consider some modifications to this expression. Simulated amplitudes for T can be brought into agreement with observations by tuning c w for each latitude. Changes to c w corresponding to ocean mixed layer depths between 10 and 200 m do not substantially alter δ T ( Figure S4 in Supporting Information S1). As c w increases, the system remains in the regime where ′ ≫ ′ , so the temperature response remains in quadrature with the forcing. Minor differences in asymmetry arise when c w tends toward the heat capacity of land, c l (we use c l = 0.336 W yr m −2 K −1 following North and Coakley (1979)), for example, over the fully land-covered latitudes of the Antarctic continent ( Figure S5 in Supporting Information S1). Equation 2 is conceptualized as a description of the surface energy balance with a parameterized radiative feedback to space forced by downwelling solar radiation at the surface. The modification of TOA insolation by co-albedo a in Equation 2 accounts for the reflection of insolation by clouds, but it assumes that cloud reflection is seasonally invariant. We assess how seasonally evolving cloud cover might impact δ T by forcing Equation 2 instead with surface zonal-mean downwelling shortwave radiation from CERES (Doelling et al., 2013(Doelling et al., , 2016, that is, including the net shortwave effect of clouds instead of using TOA insolation. This has a relatively minor effect, although it amplifies asymmetry near the poles ( Figure S6 in Supporting Information S1). Note that the higher values of δ T here are somewhat closer to those seen in observations. We also modify Equation 2 to account in a simple way for atmospheric heat flux convergence, computed from ERA5 data by Donohoe, Armour, et al. (2020). The impact on δ T is similar to or smaller in magnitude than switching from TOA to surface shortwave; note that it adds more meridional structure to the smooth curve that is obtained from Equation 2 alone ( Figure S7 in Supporting Information S1).
As demonstrated by Donohoe, Dawson et al. (2020) (see also Figure S8a in Supporting Information S1), there are substantial zonal variations in asymmetry across the globe. To examine the role of zonal variations in cloud cover, we force Equation 2 with surface downwelling shortwave radiation from CERES across longitudes λ and solve for T (θ, λ). The impacts are minor in most regions, with the exception of the Indian subcontinent and the North Pacific (Figures S8b and S8f in Supporting Information S1). The seasonal cycle of the fraction of shortwave radiation absorbed by clouds over these two regions is nearly opposite. Over the Indian subcontinent, there are more clouds in the summer/early fall season due to summer monsoon activity, whereas in the North Pacific region, there are more clouds in the winter/spring season due to storm activity. Our simple model accounting for the impact of clouds cannot explain other zonally varying departures from the overall pattern expected based on insolation, including southeast of the South American continent and south of New Zealand.

Conclusions
The seasonal cycle of surface temperature has a large amplitude outside the tropics. Furthermore, the climatological seasonal evolution of temperature can serve as a test of our understanding of the underlying physics of the climate system, as it represents a well-observed response to a large change in solar forcing. The seasonal cycle is shaped by a variety of factors, including the tilt of the Earth's axis, Earth's orbital eccentricity, global atmospheric circulation patterns, and land surface characteristics. A number of previous studies have analyzed the seasonal cycle using the amplitude and phase of the annual harmonic, which is symmetric. However, at most latitudes, the seasonal cycle of temperature is strikingly asymmetric, deviating from symmetry by around −40 days in the subtropics and 100 days at the poles. Donohoe, Dawson et al. (2020) suggested a number of possible mechanisms that could contribute to this asymmetry, but nothing that could describe the broad-scale features.
Here, adopting the approach of Roach et al. (2022), we suggest that the broad meridional structure of asymmetry in the zonal-mean seasonal cycle of surface temperature results directly from variations in top-of-atmosphere insolation. The eccentricity of Earth's orbit generates the difference in asymmetry between the two hemispheres. The obliquity, or axial tilt, of the Earth generates much of the variation in asymmetry from the tropics to the poles, and it dictates the latitude where the seasonal cycle is symmetric. In the reanalysis products analyzed, values of asymmetry are on average 20 days higher in the Southern Hemisphere than the Northern Hemisphere (averaging over latitudes polewards of 20°). Using a simple idealized model of the evolution of surface temperature forced by realistic insolation, we reproduce both effects, with an average hemispheric difference of 17 days. Moreover, both effects can be captured using a simplified variant of the model forced by only two harmonics of the insolation.
We find that the asymmetry of the seasonal cycle of zonal-mean surface temperature arises primarily due to the direct thermodynamic response to insolation forcing. Our results suggest that clouds and atmospheric heat flux convergence do not play a substantial role in modulating the zonal-mean structure of asymmetry, although clouds impact temperature asymmetry in certain regions, specifically the North Pacific and the Indian subcontinent. Seasonal asymmetry may also be influenced by factors not considered here, including seasonal changes in surface effective heat capacity. The simple first-order picture presented here has been absent from previous discussions of the seasonal cycle of surface temperature.