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Hongyan Zhu and Alan Thorpe

Abstract

Errors in numerical weather forecasts can be attributed to two causes: deficiencies in the modeling system and inaccurate initial conditions. Understanding of the characteristics of the growth of forecast spread related to model uncertainty is less developed than that for initial condition uncertainty. In this research, the authors aim to construct a theoretical basis for describing such forecast error growth resulting from model uncertainty using mostly an empirical modeling approach. Primitive equation models with different vertical discretization and different horizontal resolutions are used to investigate the impacts of model uncertainties on the predictability of extratropical cyclones. Three sets of initial perturbations related to an upper-level trigger, with slightly different amplitudes, are designed for representing the situation when the initial condition uncertainty leads to significant forecast error growth.

Forecast error growth is here estimated by following the properties of a developing cyclone in the simulations. Generally, there are three phases for forecast error growth in the experiments with initial condition and model uncertainties. For the experiments with the structured initial condition uncertainties, the errors grow rapidly at the earlier transient stage, with the growth rate well above the fastest growing normal mode. Afterward the error grows exponentially at approximately the same growth rate as the cyclone, followed by a saturation period, when the growth rate starts to decline. For the experiments with the model uncertainties, the forecast errors are initially zero and increase as time to a power of μ, which is between 0.5 and 3 depending on the strength of the cyclone at the time the simulation is initiated. After a certain time interval, the exponential growth phase and saturation period start as in the initial error experiments. Starting an integration with a stronger initial cyclone, the forecast error associated with the model uncertainty takes a shorter time to reach the exponential growth period and the forecast error grows more rapidly initially with a smaller value of μ. Also, when the initial cyclone is strong enough, then the exponential growth phase may only last for a very short time.

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Alain Joly and Alan J. Thorpe

Abstract

A methodology suitable for assessing the stability of any time-dependent basic state is presented. The equivalent of the normal modes for steady basic states are the eigenvectors of the resolvent matrix; this matrix incorporates the evolution of the large-scale flow, and growth rates are replaced by amplification rates. This method is applied to the three-dimensional stability of two-dimensional fronts undergoing frontogenesis in the presence of latent heat release in a semigeostrophic model. Disturbances developing in this flow are therefore geostrophically balanced. The concepts are first illustrated in a dry time-dependent uniform shear and potential vorticity flow. At any time during the evolution of the basic flow the stability can be compared to that obtained by assuming that the frontogenesis has, at that instant, ceased. Although differences between the results from the two methods exist, general conclusions as to the scales and structure of the modes are not altered; only large-scale waves are unstable. The situation in moist baroclinic waves is dramatically different. Growth rates are enhanced compared to the steady state analysis, but the possibility for frontal waves on the 1000-km scale to amplify most rapidly depends on the rate of development of the parent wave. Such waves dominate the spectrum only when that rate is slow and then only when the frontal ascent takes on a small cross-frontal width and the vorticity maximum penetrates over a deep layer. The short-wave growth is mostly due to latent heat release in the wave. This heating is shown, in a simplified case, to modify the necessary conditions for instability. It is concluded that shearing deformation does not intrinsically inhibit frontal instability, but paradoxically it greatly favors two-dimensional growth in the early stages due to the more rapid frontogenesis in the presence of latent heating. The role of stretching deformation may be substantially different but is not considered here.

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Alain Joly and Alan J. Thorpe

Abstract

The stability of the steady two-dimensional horizontal shear front to geostrophic disturbances in the along-front direction is examined within the framework of semi-geostrophic theory. The basic state corresponds to the geostrophic along-front flow at any time during the nonlinear evolution of a two-dimensional Eady wave. The matrix resulting from the stability analysis can be transformed into a weakly nondiagonal form. Its structure shows that the selection of the most unstable along-front wavenumber is independent of the “intensity” of the front. The growth rate is a linear function of this amplitude. The most unstable along-front mode is a modified Eady mode stationary with respect to the front. It draws a fraction of its energy from the shear. For smaller along-front wavelengths, the solution is dominated by propagating modes near the boundaries. These are also baroclinic, with a larger contribution from the basic kinetic energy and much smaller growth rates. It is apparent that the existence of a vorticity maximum at fronts, however large, is not sufficient to produce the observed small scale of frontal waves. Anomalous potential vorticity at the front is necessary to provide a deep zone of large horizontal shear and hence the reduced horizontal scale of waves.

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Fred Kucharski and Alan J. Thorpe

Abstract

The concept of local extended exergy is here applied to an idealized, dry, and reversible-adiabatic cyclone development. The extended exergy as well as the kinetic energy are decomposed into a mean part, defined by a zonal average, and into a perturbation from the mean. The resulting local energy evolution equations provide an extension of the well-known Lorenz-type available energy equations. A term in the baroclinic conversion rate, connected with static stability anomalies, which is not usually considered, is of significance even in this idealized case study and contributes significantly to the nonlinear equilibration of the baroclinic wave.

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Craig H. Bishop and Alan J. Thorpe

Abstract

It has been shown that lower tropospheric potential vorticity zones formed during moist deformation frontogenesis will support growing waves if at some time the frontogenesis ceases. In this paper, the ways in which these waves are affected by the frontogenetic process are identified.

Observations show that fronts in the eastern Atlantic commonly feature saturated ascent regions characterized by zero moist potential vorticity. Furthermore, in many cases the horizontal temperature gradient in the lowest one to two kilometers of the atmosphere is rather weak. These features are incorporated in an analytical archetype. The dynamical implications of saturated ascent in conditions of zero moist potential vorticity are represented in the model by assuming that adiabatic temperature changes are precisely balanced by diabatic tendencies. The observed small temperature gradient at low levels is represented in the model by taking it to be zero in the lowest two kilometers. Consequently, the forcing of the low-level moist ageostrophic vortex stretching that strengthens the low-level potential vorticity anomaly is confined to middle and upper levels.

A semianalytical initial value solution for the linear development of waves on the evolving low-level potential vorticity anomaly is obtained. The waves approximately satisfy the inviscid primitive equations whenever the divergent part of the perturbation is negligible relative to the rotational part. The range of nonmodal wave developments supported by the front is summarized using RT phase diagrams. This analysis shows that the most dramatic effects of frontogenesis on frontal wave growth are due to (a) the increase in time of the potential vorticity and hence potential instability of the flow and (b) the increase in time of the alongfront wavelength relative to the width of the strip. An optimally growing streamfunction wave is described. Finally, a diagnostic technique suitable for identifying small amplitude frontal waves in observational data is described.

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Craig H. Bishop and Alan J. Thorpe

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In this paper, the role of horizontal deformation and the associated frontogenetic ageostrophic circulation in suppressing the development of nonlinear waves is assessed. Unless linear barotropic frontal waves can become nonlinear, the associated horizontal transports of momentum will not be sufficient to halt frontogenesis or to create nonlinear mixing processes such as vortex roll-up. The analysis of Dritschel et al. suggests that such nonlinear phenomena will not occur if the wave slope remains small. For the linear model described in Part I, a simple relationship between optimal wave slope amplification over a specified time period and the amplification of an initially isolated edge wave is found. Using this relationship, the mechanisms by which strain affects the dependence of optimal wave slope amplification on wavelength and the time of entry of disturbances to the front are investigated. It is found that waves entering the frontal zone when it is intense can experience greater steepening than those appearing earlier in the development of the front. The most rapidly growing waves enter the front with a wavelength about three times the width of the front. As the front collapses, the ratio of wavelength to frontal width rapidly increases. For strain rates greater than 0.6 × 10−5 s−1, the model predicts that wave slope amplification greater than a factor of e is impossible.

The variation of optimal growth with wavenumber and the time of entry of disturbances to the front is explained using diagnostics based on a mathematical model of Bretherton's qualitative description of wave growth in terms of the interaction of counterpropagating edge waves. These diagnostics yield a simple formula for the frontogenesis rate required to completely eliminate wave steepening. For the front considered in Part I, the formula predicts that no amplification is possible for strain rates greater than one-quarter of the Coriolis parameter. Diagnostics of this sort may aid attempts to predict, from the large-scale forcing, the minimum attainable cross-frontal scale of a front.

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Alan J. Thorpe, Hans Volkert, and Dietrich Heimann

Abstract

Observations from the German Front Experiment are presented here that show the existence—in conditions with a dominant flow component parallel to the main Alpine chain—of a mesoscale region to the north of the Alps where the absolute and potential vorticity (PV) are substantially negative. These structures exist before the front arrives to the Alps and appear to be affected little by the passage of the front. A dynamical explanation for these and other mesoscale structures is sought by considering a simple unsheared airflow impinging on the Alps from the west. A linear frictionless model for the steady-state response is used as well as a full nonlinear numerical model with and without friction. A vastly simplified Alpine orography is considered as well as one that adequately describes its mesoscale detail.

The results show that the frictionless linear dynamics lead to a zone north of the Alps with anticyclonic vorticity but with uniform (positive) potential vorticity. With boundary-layer processes included in a nonlinear simulation substantial PV anomalies are produced. This leads to negative PV, and absolute vorticity, north of the Alps and positive PV south of the Alps. The region of PV anomalies in the model bears a suggestive similarity to that in the observations. The PV structures are attributed to frictional processes acting in a boundary layer that acquires a slope due to the sloping mountain sides. This mechanism only operates in this situation.

Other mesoscale aspects of the flow are discussed in regions around the Alps for which we have as yet no detailed observational evidence; for example, there is strong flow retardation immediately downstream of the orography. An important conclusion is that the Alps, in conditions of parallel flow, are a significant source of potential vorticity anomalies in the lower troposphere. These are advected away from the orography and must be an important part of the tropospheric PV budget.

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Douglas J. Parker and Alan J. Thorpe

Abstract

It is shown here that there exists a regime of balanced frontogenesis that is forced almost entirely by the diabatic hating due to convection at a front. This theory is explored in the context of the two-dimensional semigeostrophic equations with an Eady basic state: convection is parameterized to be dependent on the low-level moisture convergence of the cross-frontal ageostrophic flow, in accordance with recent diagnostic studies. The significant result is that the growth rate of the convective frontal system becomes independent of the total wavelength of the domain once the diabatic heating exceeds a relatively large threshold magnitude. In this regime the frontal zone has a width and structure dependent on the heating magnitude but not on the wavelength. The system is described as “solitary” or “isolated” since the dynamics are self-contained and independent of the far field.

The energetics of the system have a diabatic conversion that is an order of magnitude greater than that due to the large-scale alongfront temperature gradient. The large-scale forcing is, however, necessary as a catalyst in maintaining a weak ageostrophic convergence that allows the convective heating to be triggered. The constraint of alongfront geostrophic balance means that convective forcing alone cannot maintain frontogenesis. It is suggested that the dynamics exhibited by the convectively dominated front may also be important in the study of midlatitude squall lines.

The propagation and dynamics of the front are interpreted in terms of the notion of a “diabatic Rossby wave.”

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Kerry A. Emanuel, Maurizio Fantini, and Alan J. Thorpe

Abstract

In the semigeostrophic system, the growth rate of baroclinic waves varies with the inverse square root of the potential vorticity, which acts as the effective static stability. Recent observations in the ascent regions of middle latitude cyclones show that the effective potential vorticity for saturated air is very near zero. In this paper we examine the structure and rate of growth of baroclinic cyclones when the effective potential vorticity is small for upward (saturated) displacements but large in regions of descent. Analytic solutions for two-dimensional disturbances in a two-layer semigeostrophic model and numerical simulations using a multilevel semigeostrophic model show that when the effective potential vorticity is small in regions of upward motion, growth rates are modestly increased and the region of ascent intensifies and collapses onto a thin ascending sheet. In the limit of zero moist potential vorticity the fastest growing wave has a finite growth rate which is about 2.5 times the dry value while the horizontal scale is reduced by a factor of about 0.6 compared to the dry modes. The asymmetry associated with condensation heating leads to frontal collapse first at the surface, rather than at both boundaries as in the dry case. In contrast to the analytic model, the numerical simulations allow the effect of (dry) potential vorticity evolution due to the latent heat release to be included. The anomalies of potential vorticity are advected horizontally through the wave, enhancing the low-level and diminishing the upper-level cyclonic vorticity and static stability in both the saturated and unsaturated regions of the flow.

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