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 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 less than that to latent heating of different intensities. These findings indicate that tropopause sharpness 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.

The tropopause is characterised by sharp vertical transitions in vertical wind shear and stratification, resulting in large horizontal and vertical gradients of potential vorticity (PV)

The initialisation of the tropopause in weather and climate prediction models is based on a sparse observational network of satellites and radiosondes, resulting in large estimates of analysis errors and analysis error variance in the tropopause regions

Even if these sharp structures were well represented at the initial state, forecast errors at the tropopause have been shown to quickly develop in a few days

Another challenge influencing the forecast skill related to structures near the tropopause is the chosen altitude of the top of the atmospheric model, because it affects how artefacts from the upper boundary imprint themselves at the tropopause. Lifting the model lid has been shown to significantly improve the medium-range forecast of the stratosphere

While the modelling challenges related to the model lid, model resolution, data assimilation techniques, and observations typically lead to a smoothing of the sharp PV gradients around the tropopause, they may also contribute to misrepresentations of wind

While tropopause sharpness is mainly related to vertical changes

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)

Setup of sharp CTL and smooth CTL experiments, as described in the text.

Focusing on the incipient stage of baroclinic development, we use a numerical extension of the linear 2D QG model by

Unlike

The set of equations is completed with the boundary conditions

Vertical profiles of

The default setup is the same as in

Equations (

To investigate the sensitivity of baroclinic instability to smoothing the tropopause, we substitute the step function of

The choices for

After smoothing

Note that if the step function of

The relation between baroclinic growth and changes in wind shear and stratification across the tropopause is investigated from the energetics perspective following

Although several other studies have implemented discontinuous vertical profiles of

For the sharp CTL experiment, where profiles are discontinuous across the tropopause, the non-linear vertical advection term is less than 0.25 of the dominant QG term in the thermodynamic equation at all grid points in the baroclinic wave apart from the tropopause interface (not shown). Given the discontinuity at the tropopause due to the jump in wind shear and stratification, the temperature is a priori undefined at this level. Evaluating the thermodynamic equation with an arbitrary definition of temperature at this interface would therefore be inconsistent.

For the smooth CTL experiment, where profiles are smoothed across the tropopause, the vertical advection term is also less than 0.25 of the dominant QG term at most grid points, though near the tropopause this ratio becomes up to 7.5 (4.7) [3.3] when the vertical extent of the tropopause is 100 (150) [200] hPa. Thus, there are grid points where the non-linear vertical advection term becomes dominant. With the uncertain implications of such a dominance for our findings, the validity of the QG framework should be further tested in more comprehensive models accounting for the non-linear vertical advection term. Nevertheless, that we obtained qualitatively similar solutions for all smoothing ranges, including the sharp experiment, indicates the suitability of the QG framework to explore the sensitivity to the sharpness of the tropopause.

Introducing the effect of variations in

Growth rate vs. wavelength for the sharp CTL (black), CTL-

Below the tropopause, the structure of

Structure of

Unlike the structure of the most unstable Eady mode, the maximum in the amplitude of the streamfunction at the tropopause is weaker than the maximum at the surface. With such a secondary maximum around the tropopause, the wave structure resembles that of the “Charney+” mode studied by

In further contrast to the Eady model, where the tropopause is represented by a rigid lid, the inclusion of a tropopause with discontinuous profiles of

Just below the tropopause, the nonzero

The phase of the temperature wave reverses across the tropopause and does not tilt with height in the entire stratosphere (shading in Fig.

However, due to the 90

The weakening and acceleration of the temperature wave just below the tropopause associated with nonzero

Growth rate, wavelength, and phase speed of the most unstable mode together with absolute value of

Varying

The increase in

The sensitivity on the growth rate is less straightforward, with growth rates being largest in the upper right corner of the

To further understand the changes in growth rate, we consider the conversion of basic-state APE to EAPE (

Just below the tropopause,

Same as Fig.

To justify the argument relating increased growth rates to an increased source of EAPE through

In line with these arguments, the jump in temperature across the tropopause is monotonically increasing with decreasing

It is also worth noting that increasing

The arguments related to the beneficial phase relation between

Smoothing the vertical profiles of

Even though smoothing weakens the maximum of

However, when

The perhaps largest qualitative difference from the impact of smoothing on the overall instability analysis is an additional mode at long wavelengths when

Comparing the sensitivity of baroclinic growth to the vertical extent of smoothing, tropopause height, and changes in the vertical integral of

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.

Schematic illustrating how the altitude of the tropopause modifies the vertical average of the tropospheric stratification

Change in growth rate (shading and numbers) for various smooth experiments relative to the CTL experiment with the same discontinuous profiles of

In contrast to the sensitivity to tropopause height, increasing the vertical extent of smoothing does not necessarily have a monotonic impact on the growth rate.
Deepening the tropopause region from a narrow (solid red in Fig.

The maximum in growth rate for some intermediate degree of smoothing is associated with an intermediate

Changes in growth rate relative to the sharp CTL experiment are summarised in Fig.

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 amplitude

Evolution of error for the weakest (blue) and strongest (red) maximum growth rates from Fig.

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

Including latent heating in the mid-troposphere does not significantly change the qualitative findings of the sensitivity experiments from Sect.

Same as Fig.

For some of the experiments, the weak and positive growth rates at long wavelengths are split into two modes (Fig.

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

To compare the sensitivity of baroclinic growth to modifications in heating intensity with the sensitivity to modifications in tropopause structure, we decrease (increase) the heating parameter from

Same as Fig.

All aforementioned changes associated with the intensity of the diabatic heating are larger than the relative changes in growth rate for the various tropopause smoothing experiments for a fixed

Including sharp and smooth transitions of vertical wind shear and stratification across a finite tropopause in a linear QG model extended from the

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 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)

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

The effect of smoothing for a realistic configuration of wind shear and stratification remains weak when increasing the 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 d. In contrast, modifying the wind shear and stratification above the tropopause, resulting in modifications in the 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 medium-range forecast of 5 d 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

Although the relative impact on baroclinic growth depends on how much the profiles of wind shear and stratification are 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 stratospheric warming event.

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 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, which limits the focus of the study to the incipient stage of development. 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.

The version of the model used to produce the results in this paper is archived on Zenodo (

KFH and TS designed the experiments and KFH carried them out. KFH developed the model code, and KFH and TS analysed the model output. KFH prepared the manuscript with support from TS.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Michael Reeder for his valuable input on an earlier version of the manuscript and Vicky Meulenberg for her preliminary work on tropopause sharpness during her internship in Bergen, which motivated this study.

This work has been supported by the the Research Council of Norway project UNPACC (RCN project number 262220).

This paper was edited by Juliane Schwendike and reviewed by two anonymous referees.