Relative importance of tropopause structure and diabatic heating for baroclinic instability

Misrepresentations of wind shear and stratification around the tropopause in numerical weather prediction models can lead to errors in potential vorticity gradients with repercussions for Rossby wave propagation and baroclinic instability. Using a diabatic extension of the linear quasi-geostrophic Eady model featuring a tropopause, we investigate the influence of such discrepancies on baroclinic instability by varying tropopause sharpness and altitude as well as wind shear and stratification in the lower stratosphere, which can be associated with model or data assimilation errors or a downward extension of a 5 weakened polar vortex. We find that baroclinic development is less sensitive to tropopause sharpness than to modifications in wind shear and stratification in the lower stratosphere, where the latter are associated with a net change in the vertical integral of the horizontal potential vorticity gradient across the tropopause. To further quantify the relevance of these sensitivities, we compare these findings to the impact of including mid-tropospheric latent heating. For representative modifications of wind shear, stratification, and latent heating intensity, the sensitivity of baroclinic instability to tropopause structure is significantly 10 less than that to latent heating of different intensities. These findings indicate that tropopause sharpness is ::::: might :: be : less important for baroclinic development than previously anticipated and that latent heating and the structure in the lower stratosphere :::: could : play a more crucial role, with latent heating being the dominant factor.


Introduction
The tropopause is characterised by sharp vertical transitions in vertical wind shear and stratification, resulting in large horizontal 15 and vertical gradients of potential vorticity (PV) (e.g., Birner et al., 2006;Schäfler et al., 2020). These PV gradients act as wave guides for Rossby waves and are crucial for their propagation (see review by Wirth et al., 2018, and references therein). Hence, the common notion that tropopause sharpness must be important for midlatitude weather and its predictability (e.g., Schäfler et al., 2018). In addition to the potentially important impact from the structure of the tropopause, baroclinic development is also greatly influenced by diabatic heating associated with cloud condensation (e.g., Manabe, 1956;Craig and Cho, 1988;Snyder 20 and Lindzen, 1991). As diabatic heating strongly influences the horizontal scale and intensification of cyclones (e.g., Emanuel et al., 1987;Balasubramanian and Yau, 1996;Moore and Montgomery, 2004), its misrepresentation is a common source for errors in midlatitude weather and cyclone forecasting (Beare et al., 2003;Gray et al., 2014;Martínez-Alvarado et al., 2016).
While the effect of diabatic heating on baroclinic development is relatively well known, few studies have investigated the also indicated that the linear growth phase of the development might respond more to changes in the stratospheric wind if the horizontal PV gradients were further modified. As no previous studies have directly investigated how modifications in the vertical integral of the horizontal PV gradient influences baroclinic development, the importance of preserving the vertical integral of PV gradients remains unclear.
While tropopause sharpness is mainly related to vertical changes across the tropopause, misrepresentations of either stratification or vertical wind shear may also lead to implicit modifications of the altitude of the tropopause itself. Such fluctuations of the tropopause are associated with enhanced analysis and forecast errors (Hakim, 2005) and are often induced by baroclinic waves through vertical and meridional heat transport (Egger, 1995). While some studies argue that baroclinic instability is sensitive to the level of the tropopause (Blumen, 1979;Harnik and Lindzen, 1998), Müller (1991) found that the vertical distance between the waves at the tropopause and at the surface is not very important for baroclinic development. Thus, the net effect on baroclinic instability by altering stratification and wind shear in ways that affect tropopause altitude remains unclear. 70 To evaluate the relative importance of the various aspects of tropopause structure and diabatic heating for baroclinic instability, we use a moist extension of the linear quasi-geostrophic (QG) Eady (1949) model where we vary wind shear and stratification across the tropopause using different heating intensities. While previous idealised studies focused on the impact of abrupt environmental changes across the tropopause (e.g., Blumen, 1979;Müller, 1991;Wittman et al., 2007) and how sharp and smooth transitions across the tropopause affected neutral modes and the longwave cutoff (de Vries and Opsteegh, 2007) 75 as well as wave frequency, energetics, and singular modes (Plougonven and Vanneste, 2010), we systematically investigate the sensitivity of the most unstable baroclinic mode to both changes across the tropopause region as well as different degrees of smoothing. We also include the effect of latent heating and contrast its impact on baroclinic growth to the structure of the tropopause.
2 Model and methods 80

Model setup and solution procedure
Focusing on the incipient stage of baroclinic development, we use a numerical extension of the linear 2D QG model by Eady (1949), formulated similarly to the model of Haualand and Spengler (2019) and Haualand and Spengler (2020), which is based on an analytic version of Mak (1994). We use pressure as the vertical coordinate and assume wavelike solutions in the x direction for the QG streamfunction ψ and vertical motion ω: 85 [ψ, ω] = Re{ ψ (p),ω(p) exp(i(kx − σt))} , where the hat denotes Fourier transformed variables, k is the zonal wavenumber, and σ is the wave frequency. The nondimensionalised ω and potential vorticity (PV) equations can then be expressed as d 2ω dp 2 − Sk 2ω = i2λk 3ψ + k 2Q (uk − σ) d dp 1 S dψ dp − k 2ψ + k d dp where S is the basic-state static stability as defined in Haualand and Spengler (2019), λ is the basic-state vertical wind shear, and u is the basic-state zonal wind. As introduced by Mak (1994) and implemented by Haualand and Spengler (2019), the diabatic heating rate divided by pressure is Q = − ε 2 h(p)ω lhb , where ε is the heating intensity parameter, h(p) is the vertical heating profile defined as 1 between the bottom (p lhb ) and the top of the heating layer (p lht ) and zero elsewhere, and ω lhb is 95 the vertical velocity at the bottom of the heating layer.
Unlike Mak (1994) and Haualand and Spengler (2019), we include an idealised tropopause with a default setup of uniform λ and S in the troposphere and in the stratosphere, separated by a discontinuity at the tropopause. The discontinuity introduces an interface condition for the vertical integral of the PV equation across the tropopause:

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where p * is the pressure at the sharp tropopause interface and p + * and p − * denote locations just below and just above the tropopause, respectively. Following Haualand and Spengler (2019), we refer to ∂ψ/∂p, which is proportional to the negative density perturbation, as temperature. In line with Bretherton (1966), the jump in λ/S is proportional to the vertical integral of ∂q/∂y across the sharp tropopause. Thus, the changes in λ and S across the tropopause introduce a meridional PV gradient at the tropopause, which is positive for the parameter space we explore.

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The set of equations is completed with the boundary conditionsω = 0 at p t and p b , the thermodynamic equation as well as ∂ψ/∂p = 0 at p t , where p t and p b are the pressure at the top and bottom of the domain, respectively. The upper boundary condition is in line with Müller (1991) and Rivest et al. (1992) and prescribes vanishing temperature anomalies. As temperature anomalies at the model boundaries can be interpreted as PV anomalies (e.g., Bretherton, 1966;de Vries et al., 110 2010), this boundary condition is associated with zero PV anomalies at the model top, ensuring that the instability is mainly restricted to the troposphere, where PV anomalies at the tropopause mutually interact with PV anomalies at the surface. Additional tropospheric PV anomalies appear at the top and bottom of the heating layer in the presence of latent heating Q.
The default setup is the same as in Haualand and Spengler (2019) with the following exceptions (summarised in Table   1). The tropopause is at p = p * = 0.25, corresponding to 250 hPa, and the model top is, in accordance with Mak (1998), at 115 p = p t = 0. Furthermore, the wind shear λ reverses sign across the tropopause, from λ tr = 3.5 in the troposphere to λ st = −3.5 in the stratosphere, with the zonal wind profile being defined as which we argue is a good representation of the zonal wind profile in the midlatitudes when compared to observations (e.g., Birner et al., 2006;Houchi et al., 2010;Schäfler et al., 2020). In the stratosphere, the stratification S st = 4 remains the same as 120 that of the full model domain in Mak (1994) and Haualand and Spengler (2019), but is reduced to S tr = 1 in the troposphere, which is a more representative value for the midlatitude troposphere (e.g., Birner, 2006;Grise et al., 2010;Gettelman and Wang, 2015) and is consistent with previous studies (Rivest et al., 1992;de Vries and Opsteegh, 2007;Wittman et al., 2007).
The choice of a weaker tropospheric stratification results in stronger vertical motion and hence a larger scaling of latent heating as well as increased growth rates. To compensate for this, we consistently reduce the heating intensity parameter of ε = 12.5 125 from Haualand and Spengler (2019) to ε = 2, such that the growth rates and the scaling of latent heating remain of the same order of magnitude as in Haualand and Spengler (2019).

Smoothing procedure
To investigate the sensitivity of baroclinic instability to smoothing the tropopause, we substitute the step function of λ/S around the tropopause with a sine function that gradually increases from (λ/S) st in the upper stratosphere to (λ/S) tr in the 135 lower troposphere in a vertical range symmetric around the sharp tropopause interface, i.e., p * − δ/2 ≤ p ≤ p * + δ/2: where τ (p) increases linearly from −π/2 at p = p * − δ/2 to π/2 at p = p * + δ/2 such that sin[τ (p)] ∈ [−1, 1] for p ∈ [p * − δ/2, p * + δ/2], and α =α (λ/S) st (λ/S) tr is the scaling parameter, withα being an offset parameter that shifts (λ/S) st such that the vertical integral of ∂q/∂y around the tropopause region is modified whenα = 1 compared to whenα = 1. We conduct sharp and smooth experiments with p * = 250 hPa,α = 1, and δ = 0 hPa and δ = 150 hPa for the "sharp CTL" and "smooth CTL", respectively (grey and black profiles in Fig. 1). These settings are summarised in Table 1 together with the default setup of λ and S. We further vary δ between 50 hPa and 200 hPa (compare black and red profiles in Fig. 1), p * between 200 hPa and 300 hPa (compare black and blue profiles), andα between 1 and 0.7 (compare black and dashed yellow profiles). Note that due to the finite resolution of the model grid, there is always some smoothing even for the sharp profiles, which results in a finite 145 value of ∂q/∂y in Fig. 1d.
The choices for δ, p * , andα are based on vertical profiles in the midlatitudes from observational studies (Birner et al., 2002;Birner, 2006;Grise et al., 2010;Gettelman and Wang, 2015;Schäfler et al., 2020), where the motivation for varying the offset parameter down toα = 0.7 is based on the finding that some models only capture about 70% of the observed magnitude of the wind shear (Schäfler et al., 2020, see their Fig. 9 c,d). Experiments withα = 1, resulting in a modified vertical integral 150 of the horizontal PV gradient, are labeled "MOD", with the offset parameterα shown in percentage after "MOD", such that "MOD-70" corresponds toα = 0.7 and means that (λ/S) st is reduced to 70% of its original value. In some cases we also refer to experiments withα = 1 and hence an unaltered vertical integral of the horizontal PV gradient as "NO-MOD" experiments to avoid confusion with the MOD experiments.
After smoothing λ/S, we define the smoothed profiles of λ and S (see Fig. 1a,b) by letting 155 λ(p) = λ stα + ∆λ · γ(p) and S(p) = S stα + ∆S · γ(p), where ∆λ = λ tr − λ stα and ∆S = S tr − S stα are the respective differences in λ across the tropopause region and is a factor based on the smoothed profile of λ/S ensuring that the smoothing of λ and S is distributed equally from p = p * −δ/2 to p = p * +δ/2 relative to the total increments ∆λ and ∆S. After defining the smoothed profile λ(p), we set u(p) = p p b λ(p)dp, 160 where we assumed u(p b ) = 0.
Note that if the step function of λ shifts sign at the tropopause, while S is positive everywhere, the zero value of the smoothed profile of λ/S will be located at a higher vertical level than the discontinuity of the original sharp profile at p * . Thus, the maximum vertical gradient of λ and S is, unlike that of λ/S, typically shifted above the tropopause (compare e.g., black lines in Fig. 1a-c).

Energy equations
The relation between baroclinic growth and changes in wind shear and stratification across the tropopause is investigated from the energetics perspective following Lorenz (1955). The tendency of domain averaged eddy available potential energy EAPE is ∂ ∂t

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where C a = − λ S ψ x ψ p is the conversion from basic-state available potential energy (APE) to EAPE, C e = ωψ p is the conversion from EAPE to eddy kinetic energy, and G e = − 1 S Qψ p is the diabatic generation of EAPE. The bar denotes zonal and vertical averages.

Validity of QG assumptions
Although several other studies have implemented discontinuous vertical profiles of λ and/or S around an idealised tropopause 175 in QG models (e.g., Robinson, 1989;Rivest et al., 1992;Juckes, 1994;Plougonven and Vanneste, 2010), Asselin et al. (2016) argued that the quasi-geostrophic approximation is less appropriate near sharp gradients and narrow zones like the tropopause.
Hence, to justify our modelling framework, we tested the validity of the QG approximation by comparing the magnitude of the QG terms in the thermodynamic equation with the magnitude of the nonlinear vertical advection term neglected in the QG framework. As such a quantitative comparison between linear and nonlinear terms requires a scaling of variables (see section by Blumen (1979), de Vries and Opsteegh (2007), and Wittman et al. (2007). The qualitative differences in growth rate and wavelength of the most unstable mode as well as the shortwave and longwave cutoffs between these three experiments are the same as those found by Müller (1991) (see his Fig. 2). We present a more detailed discussion of these findings in subsection 3.2, where we explore the parameter space of λ and S more extensively.
Below the tropopause, the structure of ψ (shading in Fig. 3a) and temperature T (black contours) for the most unstable mode 205 is similar to the structure of the most unstable Eady mode, with ψ tilting westward and T tilting eastward with height. Together with the westward tilt in both ω (Fig. 3a) and meridional wind v = ikψ (not shown, but phase shifted a quarter of a wavelength upstream from ψ), this structure is baroclinically unstable and is consistent with warm air ascending poleward and cold air descending equatorward.
In contrast to the Eady model, where the tropopause is represented by a rigid lid, the inclusion of a tropopause with dis-210 continuous profiles of λ and S introduces nonzero ω at the tropopause interface. Just below the tropopause, this nonzero ω adiabatically cools (warms) the air upstream of the positive (negative) temperature anomaly (compare grey contours and shading in Fig. 3a), thereby weakening the temperature wave as well as accelerating its downstream propagation. This effect is opposed by the meridional temperature advection, which warms (cools) the air upstream of the positive (negative) temperature anomaly just below the tropopause. Thus, with a negative meridional temperature gradient associated with the positive   wind shear λ via the thermal wind relation, meridional temperature advection amplifies the temperature wave and retards its downstream propagation at this level. The net effect is propagation against the zonal wind such that the propagation speed of the temperature wave just below the tropopause matches the propagation speed of the wave at the surface. Only when these propagation speeds are identical, the waves can phase lock and travel together with a common propagation speed that equals the average phase speed of the two waves (de Vries and Opsteegh, 2007). The phase of the temperature wave reverses across the tropopause and does not tilt with height in the entire stratosphere (shading in Fig. 3). Such a barotropic structure is in line with the lack of mutual intensification of PV anomalies in this layer.
There is a monotonic decay of the temperature anomaly toward the top of the model domain related to the upper boundary condition ∂ψ/∂p = 0. Together with the barotropic structure, this decay yields T ∝ −∂ψ/∂p being exactly in phase with −ψ and therefore also exactly 90 degrees out of phase with v = ikψ. Nevertheless, due to the reversal of the wind shear across 225 the tropopause, the meridional temperature advection is still retarding the downstream wave propagation above the tropopause such that the stratospheric part of the wave propagates together with the tropospheric part.
However, due to the 90 degrees phase shift between v and T , meridional advection can no longer amplify the stratospheric part of the temperature wave. Instead, the amplification of the wave in the stratosphere is entirely due to ω, where ω is almost in phase with temperature. Hence, the role of ω on the amplification of the wave reverses across the tropopause.

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The weakening and acceleration of the temperature wave just below the tropopause associated with nonzero ω is in line with a weaker growth rate, higher phase speed, and hence longer wavelength compared to the most unstable Eady mode (compare contour at the black dot with the black contour in Fig. 4). Such effects on baroclinic development were also found in similar experiments by Müller (1991) and partly by de Vries and Opsteegh (2007).

Sensitivity to variations in stratospheric wind shear and/or stratification 235
Varying λ st and S st while holding λ tr and S tr fixed changes ∂q/∂y through its relation to the jump in λ/S across the tropopause (see Eq. (4) and related arguments), which has implications for baroclinic growth through the arguments of mutual intensification by interacting PV anomalies (Hoskins et al., 1985). For the parameter space explored in this study, decreasing λ st relative to λ tr always increases ∂q/∂y, whereas increasing S st relative to S tr increases ∂q/∂y only when λ st is positive and decreases ∂q/∂y when λ st is negative (Fig. 4d).

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The increase in ∂q/∂y for varying λ st and S st yields the observed decrease in phase speed and wavelength (compare pattern of black contours in Fig. 4b-d). As argued by Wittman et al. (2007), the relation between ∂q/∂y, phase speed, and wavelength is in line with the proportionality of the phase speed of Rossby waves to -1/k · ∂q/∂y. Thus, a larger positive ∂q/∂y reduces the phase speed, which can be partly compensated by increasing the wavenumber k. A similar qualitative relation between increasing wavelengths for decreasing ∂q/∂y related to varying λ st and S st was found by Müller (1991) (see his Fig. 2b).

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Müller (1991) also found that decreasing λ st reduces the phase speed for a ratio of static stability of 1.5 across the tropopause (see his Fig. 3a-c), which is confirmed by our results (Fig. 4c). Furthermore, our results also show that this relation between λ and phase speed holds for all investigated configurations of S st .
The sensitivity on the growth rate is less straightforward, with growth rates being largest in the upper right corner of the λ-S parameter space, where the wind shear is uniform and the stratification in the stratosphere is larger than in the troposphere 250 (Fig. 4a). Growth rates decrease from this maximum toward weaker λ st and S st . A similar sensitivity on the growth rate to changes in λ and S was found by Müller (1991), where the growth rate of the most unstable mode also peaked when λ st and S st were large and decreased toward weaker λ st and S st (see his Fig. 2a). While the decrease in growth rates toward the upper left corner of the λ-S parameter space in Fig. 4a can be explained by the absence of a tropopause due to a uniform λ and S   To further understand the changes in growth rate, we consider the conversion of basic-state APE to EAPE (C a ), which is constant with height in the troposphere where PV anomalies mutually intensify (not shown). As this energy conversion term is the main source for EAPE when dry baroclinic waves intensify, it should reflect the observed changes in growth rate. We therefore explore this term by considering the location and amplitude of v ∼ ψ x and T ∼ ψ p .

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Just below the tropopause, v and T are more in phase when λ st is positive (Fig. 5d), which is beneficial for the energy conversion. At the surface, v and T are generally less in phase than just below the tropopause and changes in phase between v and T are small for different λ st and S st (Fig. 5c). Given that C a is constant throughout the troposphere, the different phase relation between v and T at the surface and just below the tropopause are consistent with larger amplitudes of v = ikψ and T at the surface relative to the amplitudes just below the tropopause (Fig. 5a,b). This dominance of v and T at the surface relative 265 to just below the tropopause is strongest when the phase between v and T just below the tropopause and the magnitude of ∂q/∂y are small (compare pattern of Figs. 4d, 5a, and 5d). For positive λ st , we thus argue that the beneficial phase relation between v and T and the larger amplitudes of v and T at the surface favour a larger conversion of basic-state APE to EAPE (C a ) compared to when λ st is negative. With a large source of EAPE, baroclinic growth is expected to intensify.
To justify the argument relating increased growth rates to an increased source of EAPE through C a , we need to understand 270 what sets the phase relation between v and T . Due to the difference condition for temperature across the tropopause in Eq.
(4), where the difference in 1/S · ∂ψ/∂p is proportional to the jump in λ/S and hence the vertical integral of ∂q/∂y across the tropopause, the temperature anomaly typically reverses across the tropopause (see example in Fig. 3a). When the jump in λ/S is large, the temperature difference is also large, such that the temperature anomaly just below the tropopause becomes zonally more aligned with the opposite temperature anomaly just above the tropopause, reducing the freedom for a phase shift 275 to a more beneficial phase relation with the meridional wind. In contrast, when the jump in λ/S is small, the difference in temperature across the tropopause is less constrained such that the temperature anomaly just below the tropopause can more easily be shifted upshear to be more in phase with the meridional wind.
In line with these arguments, the jump in temperature across the tropopause is monotonically increasing with decreasing λ st /λ tr when S st and S tr are constant (not shown). In contrast, as mentioned in the beginning of this subsection, an increase 280 in S st relative to S tr increases the jump in λ/S only when λ st is positive and is therefore not always associated with an increase in the difference of T across the tropopause. Furthermore, as S appears on both sides of the difference condition in Eq. (4), an increase of S st relative to S tr can compensate for a significant part of the changes in the jump of 1/S · ∂ψ/∂p, which would leave the temperature more or less unaltered.
It is also worth noting that increasing S st yields a more dominant omega term in the thermodynamic equation that amplifies 285 the temperature anomaly just above the tropopause (as discussed in section 3.1). For a given difference in temperature across the tropopause, the latter effect allows the temperature wave below the tropopause to move more freely away from its antiphase relation with the wave above the tropopause, thereby improving its correlation with v. The above arguments related to the complex role of S on temperature near the tropopause demonstrate that the phase relation between v and T just below the tropopause is more sensitive to changes in λ than S (as shown in Fig. 5d).

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The arguments related to the beneficial phase relation between v and T for large λ st together with the absence of instability for uniform λ and S, i.e., no tropopause, yield the observed pattern in growth rates (Fig. 4a), with a maximum where λ is uniform and the jump in S is large. Hence, baroclinic growth is not largest when the tropopause is at its most abrupt 13 https://doi.org/10.5194/wcd-2021-13 Preprint. Discussion started: 4 March 2021 c Author(s) 2021. CC BY 4.0 License. configuration (lower right corner around the black dot in Fig. 4), but rather when the linear increase in zonal wind is extended to above the tropopause (upper right corner around the blue dot in Fig. 4).  Figs. 3a and b). Moreover, the sensitivity to λ and S for growth rate, wavelength, phase speed, and ∂q/∂y, as well as the amplitude and phase of v and T remain qualitatively Even though smoothing weakens the maximum of ∂q/∂y by 90% (shading in Fig. 4d), the growth rate, wavelength, and phase speed change by less than ±4% (shading in Fig. 4a-c). In line with a weaker ∂q/∂y and the dispersion relation for Rossby waves (as discussed in Sect. 3.2), smoothing increases the wavelength and the phase speed for most of the investigated 305 configurations of λ st and S st (Fig. 4b,c) and decreases the growth rates by up to 2.9% when λ st is negative and S st is weak (Fig. 4a).
However, when λ st and S st are large, the growth rate increases by up to 0.9% (Fig. 4a). We argue that this enhancement is related to an improved phase relation between v and T compared to the experiments with discontinuous profiles (shading in Fig. 5d), where a smooth tropopause with a wider vertical distribution of ∂q/∂y yields more flexibility in relative location 310 between the temperature anomalies just below and above the tropopause. Such an improved phase relation is associated with enhanced conversion of basic-state APE to EAPE and may overcompensate for the detrimental impact from the weakening of ∂q/∂y. In fact, for the most realistic setup where both λ and S change across the tropopause (around the black dot in Fig.   4a), the sensitivity on the growth rate from smoothing is almost negligible, indicating that the positive impact related to the improved phase relation between v and T is balanced by the detrimental impact from the weakening of ∂q/∂y. This suggests 315 that baroclinic growth is typically not very sensitive to an accurate representation of λ and S around the tropopause.
The perhaps largest qualitative difference from the impact of smoothing on the overall instability analysis is an additional mode at long wavelengths when λ st is negative and S st is large (Fig. 6). The streamfunction structure of this mode features its strongest westward tilt with height within the smoothed tropopause region and decays rapidly above (not shown). This mode exists only due to the additional levels of opposing and nonzero ∂q/∂y in the smoothed tropopause region. We will not focus 320 on these modes at long wavelengths, as we argue that their weak growth rate and long wavelength as well as their westward tilt bound solely to the tropopause region make them less relevant for an assessment for typical midlatitude cyclones.

Sensitivity to vertical extent and altitude of tropopause
Comparing the sensitivity of baroclinic growth to the vertical extent of smoothing, tropopause height, and changes in the vertical integral of ∂q/∂y (see details in Sect. 2.2), the greatest sensitivity is related to the changes in the vertical integral 325 of ∂q/∂y, where the growth rates of the sharp and smooth MOD-70 experiments are similar and increase by 2.7% to 3.5% compared to their NO-MOD counterpart experiments (compare sharp and faint colours in Fig. 6). The increase in growth rate from the NO-MOD experiments to the MOD-70 experiments is associated with a decrease in ∂q/∂y at the tropopause toward a more optimal value that better matches with the ∂q/∂y at the surface (not shown), such that the waves at the tropopause and at the surface can more easily phase lock and travel together with the same phase speed (Blumen, 1979;de Vries and Opsteegh, 330 2007; Wittman et al., 2007). For the NO-MOD experiments, the sensitivity to tropopause height (solid and dashed blue in Fig.   6) and vertical extent of smoothing (solid and dashed red) changes the growth rate by only -0.24% to 0.31% compared to the sharp control experiment (black).
The sensitivity to vertical extent of smoothing and tropopause height is qualitatively the same for both the NO-MOD and the MOD-70 experiments. Lowering (raising) the tropopause weakens (enhances) the growth rate (solid and dashed blue in Fig.   335 6). This can be related to an increased (decreased) vertical average of S from the surface to the tropopause (Fig. 7) S tr = 1 where the stratification is related to the growth rate through the inverse proportionality between the static stability and the maximum Eady growth rate (Lindzen and Farrell, 1980;Hoskins et al., 1985).
In contrast to the sensitivity to tropopause height, increasing the vertical extent of smoothing does not necessarily have a 340 monotonic impact on the growth rate. Deepening the tropopause region from a narrow (solid red in Fig. 6) to an intermediate The changes in growth rate may seem small, but as variables grow nearly exponentially at the incipient stage of development, errors grow quickly with time. Relative to a reference experiment (subscript ref ), the forecast error of the relative wave S Figure 7. Schematic illustrating how the altitude of the tropopause modifies the vertical average of the tropospheric stratification S.
Assuming perfect initial conditions, i.e., A /A ref = 0 at t = 0, the forecast error for the NO-MOD smooth experiments relative 360 to the sharp control experiment is less than +/-1% [2%] during a short-range forecast of 2 days [medium-range forecast of 5 days], while the corresponding error for the MOD-70 experiments is up to 6% [17%] (dashed lines in Fig. 9). In comparison, assuming a relative initial error of 5%, the relative forecast error is down to 4% [3%] after 2 [5] days for the NO-MOD smooth experiments, and up to 12% [22%] for the MOD-70 experiments. The decrease in the relative error for some of the NO-MOD smooth experiments is a result of an underestimate of the growth rate relative to the sharp control experiment, which reduces 365 the initial positive relative error. If the growth rates are compared to the growth rate of a weakly smoothed experiment instead of the sharp reference experiment, the error is more or less unaltered. We therefore let the growth rate of the sharp experiment be the reference for the error growth calculations.
Keeping in mind that these results are based on a highly idealised model, the findings indicate that it is not so important if models fail to accurately represent λ and S around the tropopause. Instead, it is much more important that λ and S are   The importance of representing the lower stratospheric winds is further supported by Rupp and Birner (2021), who found that baroclinic lifecycle experiments are sensitive to changes in the wind structure in the lower stratosphere. Such changes in wind structure are often related to a downward extension of a weak polar vortex after sudden stratospheric warming events (Baldwin and Dunkerton, 2001), which have been shown to significantly alter midlatitude weather in the troposphere (see review by 375 Kidston et al., 2015, and references therein).

Sensitivity to latent heating intensity
Including latent heating in the mid-troposphere does not significantly change the qualitative findings of the sensitivity experiments from section 4.2 (compare Fig. 10 with Fig. 6). Nevertheless, the most unstable mode at shorter wavelengths is associated with dominant diabatic PV anomalies at the heating boundaries (Fig. 11b), which align with the westward tilt of ψ 380 (Fig. 11a). Growth rates peak at shorter wavelengths, which is consistent with the presence of diabatic PV anomalies and hence a shallower effective depth of interacting PV anomalies (Hoskins et al., 1985).
For some of the experiments, the weak and positive growth rates at long wavelengths are split into two modes (Fig. 10). The longest of the two is similar to their adiabatic counterpart mentioned in the end of Sect. 4.1, while the shortest of the two is associated with the increased dominance of the diabatic PV anomalies at the top of the heating layer. Due to the irrelevance for 385 midlatitude cyclones mentioned in section 4.1, these modes are beyond the scope of this study.
In line with the dominance of diabatic PV anomalies in the lower and middle troposphere, latent heating also weakens the relative sensitivity to the modifications of the vertical integral of ∂q/∂y across the tropopause (compare Fig. 10  weakest growth rate dry strongest growth rate dry weakest growth rate moist strongest growth rate moist Figure 9. Evolution of error for the weakest (blue) and strongest (red) maximum growth rates from Fig. 6 (dashed, dry) and Fig. 10 (solid, moist) starting with initial relative errors of 0% and 5% (grey dotted horizontal lines). with growth rates for the MOD-70 experiments increasing by only 1.0-1.1% relative to the NO-MOD counterpart experiment instead of 2.7-3.5% as for the adiabatic experiments. Keeping the idealised context of this study in mind, this finding indicates 390 that the presence of latent heating makes models relatively less vulnerable to an inaccurate representation of λ and S around the tropopause.
Decreasing (increasing) the heating parameter from ε = 2 to ε = 1.5 (ε = 2.5), which corresponds to a 25% decrease (increase) in latent heating and associated precipitation, yields a much larger variation in the maximum growth rate compared to the tropopause sensitivity experiments for a fixed heating parameter (Fig. 12). The change in growth rate relative to the sharp 395 experiment for ε = 2 is between -10.2% (for ε = 1.5) and +14.2% (for ε = 2.5), and the corresponding error after 2 [5] days is between -21% [-44%] (for ε = 1.5) and +38% [+124%] (for ε = 2.5) if there are no initial errors, and a few percent larger if the relative initial error is 5% instead (solid lines in Fig. 9). In comparison, the corresponding numbers for the relative change in growth rate when changing the latent heating intensity ε by only 5% [10%] instead of 25% are between -2.4% [-4.5%] and +2.4% [+4.9%] instead of -10.2% and +14.2%. All aforementioned changes associated to the intensity of the diabatic heating are larger than the relative changes in growth rate for the various tropopause smoothing experiments for a fixed ε = 2 (middle row in Fig. 12), which range between -0.2% and +1.7%. Moreover, these findings remain similar when using smooth vertical profiles of latent heating as in Haualand and Spengler (2019) (see their Fig. 11a), with the relative change in growth rate being between -5.0% and +3.0% when changing the latent heating intensity ε by 5% (not shown). Again, these numbers are all larger than the change in growth rate relative 405 to the experiment with the discontinuous profiles for a fixed ε = 2, which are between -2.1 and +1.9% when using a smooth heating profile. With such a high sensitivity of the forecast error to heating intensity, our results indicate that it is much more important to adequately represent diabatic processes than the sharpness of the tropopause.

Conclusions
Including sharp and smooth transitions of vertical wind shear and stratification across a finite tropopause in a linear QG model 410 extended from the Eady (1949) model, we investigated the relative importance of changes across the tropopause region at different degrees of smoothing on baroclinic development and compared its sensitivity to that of diabatic heating. We found that impacts related to tropopause structure are secondary to diabatic heating related to mid-tropospheric latent heating.
In contrast to the Eady mode, where the tropopause is represented by a rigid lid, the inclusion of an idealised tropopause with abrupt changes in wind shear and/or stratification introduces nonzero vertical motion at the tropopause. The vertical motion 415 leads to adiabatic cooling/warming at the tropopause, which opposes the effect of meridional temperature advection. The adiabatic cooling/warming weakens the amplitude of the wave at the tropopause but accelerates its downstream propagation, resulting in weaker growth rates and higher phase speed than the most unstable Eady mode.
In agreement with the dispersion relation for Rossby waves, increasing (decreasing) ∂q/∂y at the tropopause by varying the stratospheric wind shear and/or stratification is associated with relatively weak (strong) phase speed and short (long) 420 wavelength. In contrast to wavelength and phase speed, the impact from wind shear and stratification on the growth rate is less straight forward, with growth rates being strongest when wind shear is uniform and the increase in stratification is large across the tropopause. The strong growth rates are related to a beneficial phase relation between meridional wind and temperature near the tropopause, which is associated with enhanced conversion of basic-state available potential energy to eddy available potential energy. Thus, baroclinic growth is not strongest when the tropopause is sharpest.    Fig. 8, but including latent heating for three different heating intensity parameters (ε = 1.5, 2.0, 2.5). Note that the colorbar is extended from the one in Fig. 12 but contains the same colours at lower values.
Smoothing the tropopause is associated with a positive effect on baroclinic growth related to a further enhancement of energy conversion through an improved phase relation between meridional wind and temperature, as well as a negative effect related to a weaker maximum gradient of ∂q/∂y in the tropopause region. The positive effect from smoothing dominates when there are no or small changes in wind shear and large changes in stratification across the tropopause, resulting in increased growth rates compared to when the tropopause is sharp. In contrast, the negative effect dominates when there are large changes in wind shear and no or small changes in stratification, yielding weaker growth rates than for a sharp tropopause. For the most realistic configuration, with large changes in both wind shear and stratification across the tropopause, these opposing effects balance each other, resulting in negligible changes in growth rate from smoothing, suggesting that baroclinic growth is not very sensitive to tropopause sharpness.
The effect of smoothing for a realistic configuration of wind shear and stratification remains weak when increasing the 435 vertical extent of smoothing and altering the tropopause altitude, with an error growth for exponentially growing quantities of less than 2% in a medium-range forecast of 5 days. In contrast, modifying the wind shear and stratification above the tropopause, resulting in modifications in vertical integral of the PV gradient relative to a sharp control experiment, has a much more pronounced effect on baroclinic growth than the effects related to smoothing and varying tropopause altitude.
The associated exponentially growing forecast error of any wave amplitude assuming perfect initial conditions is 17% in a 440 medium-range forecast of 5 days when the stratospheric wind shear divided by stratification is reduced to 70% of its original value, which is a reduction actually occurring in operational numerical weather prediction models (Schäfler et al., 2020). The relatively large sensitivity to the lower stratospheric winds on baroclinic development is in line with Rupp and Birner (2021), who also argued that baroclinic growth may be sensitive to modifications in the horizontal PV gradients.
Although the relative impact on baroclinic growth depends on how much the profiles of wind shear and stratification are 445 altered for the different sensitivity experiments, our estimates indicate that it is much more important to maintain the vertical integral of the PV gradient than to accurately represent the abrupt vertical contrasts across the tropopause. Such modifications above the tropopause may represent modelling challenges related to observational errors, vertical resolution, a low model lid, or limitations related to data assimilation techniques, but they can also represent changes in the lower stratospheric winds resulting from downward extensions of a weak polar vortex after a sudden stratopheric warming event.

450
As expected from the strong impact of diabatic heating on baroclinic development, including mid-tropospheric latent heating of moderate intensity increases the growth rate. However, including latent heating does not alter the qualitative findings regarding the impact of tropopause structure on baroclinic development. Nevertheless, modifying the heating intensity by 5-25% has a significantly larger impact on the growth rate than the effects of smoothing tropopause structure, varying tropopause altitude, and maintaining the vertical integral of the PV gradient. This highlights the main finding of this study that baroclinic growth is 455 more sensitive to diabatic heating than tropopause structure.
While this study is the first to quantify the relative effect of tropopause sharpness and latent heating on baroclinic development, it is important to keep in mind the highly idealised character of this study. More realistic simulations with numerical weather prediction models should be performed to test our findings and to further clarify the relative importance of the representation of the tropopause and diabatic forcing on midlatitude cyclones.