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- Author or Editor: Joseph Egger x

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## Abstract

Two linear minimum-resolution models of *β*-plane channel flow are presented in analogy to the well-known two-layer model of baroclinic instability in order to see if basic features of barotropic instability can be demonstrated using similarly simple models. Two models are discussed within this pedagogical framework. A spectral model with two wave modes is applied to a cosine jet. Necessary conditions for instability are derived. The stability analysis shows that this simple model captures the shortwave cutoff and the asymmetry of the instability with respect to the direction of the jet quite well. It is demonstrated that the cutoff follows from Fjörtoft’s theorem for wave triads. A gridpoint model with two points in the interior of the channel is discussed as well. An analog to the classical necessary condition for instability is derived. A stability problem in a nonrotating system is discussed where the mean flow velocity is constant near the walls and a linear shear flow is assumed near the channel’s axis. In this case, the stability characteristics of the low-order model come close to those of the full problem where a simple analytic solution is available. Addition of the *β* term stabilizes the flow. A proper choice of the initial conditions always enables short-term growth of the perturbation energy for stable mean flows. A qualitative interpretation of the instability mechanism is presented for both models, which exploits the fact that the locations of corresponding extrema of streamfunction and vorticity need not coincide. It is concluded that low-resolution models are well suited for a discussion of the basic features of barotropic instability.

## Abstract

Two linear minimum-resolution models of *β*-plane channel flow are presented in analogy to the well-known two-layer model of baroclinic instability in order to see if basic features of barotropic instability can be demonstrated using similarly simple models. Two models are discussed within this pedagogical framework. A spectral model with two wave modes is applied to a cosine jet. Necessary conditions for instability are derived. The stability analysis shows that this simple model captures the shortwave cutoff and the asymmetry of the instability with respect to the direction of the jet quite well. It is demonstrated that the cutoff follows from Fjörtoft’s theorem for wave triads. A gridpoint model with two points in the interior of the channel is discussed as well. An analog to the classical necessary condition for instability is derived. A stability problem in a nonrotating system is discussed where the mean flow velocity is constant near the walls and a linear shear flow is assumed near the channel’s axis. In this case, the stability characteristics of the low-order model come close to those of the full problem where a simple analytic solution is available. Addition of the *β* term stabilizes the flow. A proper choice of the initial conditions always enables short-term growth of the perturbation energy for stable mean flows. A qualitative interpretation of the instability mechanism is presented for both models, which exploits the fact that the locations of corresponding extrema of streamfunction and vorticity need not coincide. It is concluded that low-resolution models are well suited for a discussion of the basic features of barotropic instability.

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## Abstract

The numerical solution to the linear advection equation with oscillating forcing is derived in analytic form for two-level schemes and for the leapfrog scheme with an Asselin filter. The numerical solutions are compared to the analytic solution of the advection equation with emphasis on the forced part. A detailed analysis is presented for the trapeze, the backward, and the leapfrog scheme with Asselin filter. Large deviations are found in quasi-resonant situations where the period of forcing and advection are close. Damping schemes fail completely to capture the resonant case. As for amplitude errors, the backward scheme is generally better than the trapeze scheme outside the quasi-resonant domain. However, the backward scheme produces large phase errors while the trapeze solutions are free of such errors. The leapfrog scheme has a resonant solution but generates large-amplitude errors near resonance. On the other hand, phase errors are particularly small in that case. The amplitude of the numerical mode tends to be large if either the forcing period or the advective period are but coarsely resolved. Addition of the Asselin filter removes the numerical high-frequency oscillations but destroys the resonant solution.

## Abstract

The numerical solution to the linear advection equation with oscillating forcing is derived in analytic form for two-level schemes and for the leapfrog scheme with an Asselin filter. The numerical solutions are compared to the analytic solution of the advection equation with emphasis on the forced part. A detailed analysis is presented for the trapeze, the backward, and the leapfrog scheme with Asselin filter. Large deviations are found in quasi-resonant situations where the period of forcing and advection are close. Damping schemes fail completely to capture the resonant case. As for amplitude errors, the backward scheme is generally better than the trapeze scheme outside the quasi-resonant domain. However, the backward scheme produces large phase errors while the trapeze solutions are free of such errors. The leapfrog scheme has a resonant solution but generates large-amplitude errors near resonance. On the other hand, phase errors are particularly small in that case. The amplitude of the numerical mode tends to be large if either the forcing period or the advective period are but coarsely resolved. Addition of the Asselin filter removes the numerical high-frequency oscillations but destroys the resonant solution.

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## Abstract

The spurious numerical generation and/or destruction of various types of entropies in models is investigated. It is shown that entropy *s*
_{
θ
} of dry matter tends to be generated if potential temperature is advected by a damping scheme. There is no mean tendency of entropy if the reversible leapfrog scheme is used. Generalized entropies can be assigned to conserved quantities. In particular, the generalized entropy *s*
_{
ζ
} of the vorticity of two-dimensional nondivergent flow is shown to grow in presence of irreversible diffusive processes. This entropy increases numerically if the vorticity equation is integrated with an upstream scheme. There are weak oscillations of *s*
_{
ζ
} if a leapfrog time step is combined with the Arakawa scheme. Similar results are obtained for an entropy *s*
_{
p
} related to potential vorticity. Information entropy provides a gross measure of the information contained in ensemble forecasts. It is shown that information entropy decreases spuriously if schemes are used that are contracting in phase space. It is argued that the evaluation of entropies provides a useful check of the quality of numerical schemes.

## Abstract

The spurious numerical generation and/or destruction of various types of entropies in models is investigated. It is shown that entropy *s*
_{
θ
} of dry matter tends to be generated if potential temperature is advected by a damping scheme. There is no mean tendency of entropy if the reversible leapfrog scheme is used. Generalized entropies can be assigned to conserved quantities. In particular, the generalized entropy *s*
_{
ζ
} of the vorticity of two-dimensional nondivergent flow is shown to grow in presence of irreversible diffusive processes. This entropy increases numerically if the vorticity equation is integrated with an upstream scheme. There are weak oscillations of *s*
_{
ζ
} if a leapfrog time step is combined with the Arakawa scheme. Similar results are obtained for an entropy *s*
_{
p
} related to potential vorticity. Information entropy provides a gross measure of the information contained in ensemble forecasts. It is shown that information entropy decreases spuriously if schemes are used that are contracting in phase space. It is argued that the evaluation of entropies provides a useful check of the quality of numerical schemes.

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## Abstract

It is proposed to monitor the conservation of “material” volumes in the phase space of a numerical model. In principle, strict volume conservation is required for reversible flow problems. In particular, the phase space volume occupied by the initial distribution of an ensemble forecast should not change in time for such flows. It is not obvious to what extent numerical schemes satisfy this requirement. A corresponding test for volume conservation is designed. Euler, upstream, Lax–Wendroff and Runge–Kutta schemes are exposed to this test as well as a time centered implicit scheme and the leapfrog method. Members of the former group are numerically irreversible, while those of the latter are reversible in time. Tests are performed for gridpoint representations of advection equations and of one-dimensional inviscid shallow-water flow. The numerically irreversible schemes exhibit spurious contraction or expansion. The error is *O*(*C*) for the upstream scheme and *O*(*C*
^{2}) for all the other irreversible schemes except fourth-order Runge–Kutta (*C* is the Courant number). The implicit scheme conserves volume for reversible flow problems. It is a surprising result that the leapfrog scheme conserves volume for any type of equations. Applications to Monte Carlo forecasts are presented. It is shown that clouds of initial states contract or expand indeed as predicted by the test in the early stages of an integration. This affects the quality of the Monte Carlo forecast.

## Abstract

It is proposed to monitor the conservation of “material” volumes in the phase space of a numerical model. In principle, strict volume conservation is required for reversible flow problems. In particular, the phase space volume occupied by the initial distribution of an ensemble forecast should not change in time for such flows. It is not obvious to what extent numerical schemes satisfy this requirement. A corresponding test for volume conservation is designed. Euler, upstream, Lax–Wendroff and Runge–Kutta schemes are exposed to this test as well as a time centered implicit scheme and the leapfrog method. Members of the former group are numerically irreversible, while those of the latter are reversible in time. Tests are performed for gridpoint representations of advection equations and of one-dimensional inviscid shallow-water flow. The numerically irreversible schemes exhibit spurious contraction or expansion. The error is *O*(*C*) for the upstream scheme and *O*(*C*
^{2}) for all the other irreversible schemes except fourth-order Runge–Kutta (*C* is the Courant number). The implicit scheme conserves volume for reversible flow problems. It is a surprising result that the leapfrog scheme conserves volume for any type of equations. Applications to Monte Carlo forecasts are presented. It is shown that clouds of initial states contract or expand indeed as predicted by the test in the early stages of an integration. This affects the quality of the Monte Carlo forecast.

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## Abstract

The prognostic equations for the total angular momentum vector **M** are derived for *f*- and *β*-plane geometries and compared to those of spherical models. It is shown that the omission of the centrifugal effects and the corresponding adjustment of gravity in atmospheric *f*- (*β*-) plane models imply that a torque is exerted in analogy to the spherical case where this torque is caused by the nonspherical shape of the earth. For hydrostatic flow on the *f*- (*β*-) plane, it is only for the vertical component *M*
_{
z
} of angular momentum that a prognostic equation can be derived. If the traditional approximation is introduced, *M*
_{
z
} becomes a conserved quantity on the *f* plane in the absence of orographic and frictional torques while the corresponding component *M̃*
_{
z
} on the sphere is not conserved. The prognostic equation for *M*
_{
z
} on the *β* plane is an approximation to that on the sphere at least for nondivergent flow. The *f*- (*β*-) plane equations for the horizontal components of **M** deviate substantially from those valid on the sphere in the nonhydrostatic case.

Numerical integrations of the shallow water equations are performed in order to illustrate these points. The total angular momentum is evaluated for localized flow structures. It is found that the *β*-plane model captures the most important characteristics of the corresponding changes of *M̃*
_{
z
} on the sphere at least for short times and for initially geostrophic flows. Moreover, *M̃*
_{
z
} is reasonably well conserved for isolated flow structures of small scale as suited for the *f* plane.

## Abstract

The prognostic equations for the total angular momentum vector **M** are derived for *f*- and *β*-plane geometries and compared to those of spherical models. It is shown that the omission of the centrifugal effects and the corresponding adjustment of gravity in atmospheric *f*- (*β*-) plane models imply that a torque is exerted in analogy to the spherical case where this torque is caused by the nonspherical shape of the earth. For hydrostatic flow on the *f*- (*β*-) plane, it is only for the vertical component *M*
_{
z
} of angular momentum that a prognostic equation can be derived. If the traditional approximation is introduced, *M*
_{
z
} becomes a conserved quantity on the *f* plane in the absence of orographic and frictional torques while the corresponding component *M̃*
_{
z
} on the sphere is not conserved. The prognostic equation for *M*
_{
z
} on the *β* plane is an approximation to that on the sphere at least for nondivergent flow. The *f*- (*β*-) plane equations for the horizontal components of **M** deviate substantially from those valid on the sphere in the nonhydrostatic case.

Numerical integrations of the shallow water equations are performed in order to illustrate these points. The total angular momentum is evaluated for localized flow structures. It is found that the *β*-plane model captures the most important characteristics of the corresponding changes of *M̃*
_{
z
} on the sphere at least for short times and for initially geostrophic flows. Moreover, *M̃*
_{
z
} is reasonably well conserved for isolated flow structures of small scale as suited for the *f* plane.

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## Abstract

The mechanism of baroclinic instability in the Eady model is interpreted by explicitly calculating the ageostrophic circulations related to the model’s hyperbolic basic functions. It is advantageous to perform the analysis at the midlevel where the model’s “barotropic” mode provides the streamfunction and the “baroclinic” mode represents the temperature. These modes interact and instability occurs if the horizontal advection of background potential temperature by the barotropic mode dominates over the vertical one because of the same mode at the midlevel. A rather simple picture of the stable as well as the unstable flow configurations emerges. Other interpretations are discussed briefly.

## Abstract

The mechanism of baroclinic instability in the Eady model is interpreted by explicitly calculating the ageostrophic circulations related to the model’s hyperbolic basic functions. It is advantageous to perform the analysis at the midlevel where the model’s “barotropic” mode provides the streamfunction and the “baroclinic” mode represents the temperature. These modes interact and instability occurs if the horizontal advection of background potential temperature by the barotropic mode dominates over the vertical one because of the same mode at the midlevel. A rather simple picture of the stable as well as the unstable flow configurations emerges. Other interpretations are discussed briefly.

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## Abstract

Piecewise potential vorticity inversion (PPVI) is widely accepted as a useful tool in atmospheric diagnostics. This method is thought to quantify the instantaneous interaction at a distance of anomalies of potential vorticity (PV) separated horizontally and/or vertically. Doubts with respect to the dynamical justification of PPVI are formulated. In particular, it is argued that the tendency of the inverted streamfunction must be determined in order to quantify far-field effects. Elementary tests of PPVI are conducted to clarify these points. First, PPVI is performed for the textbook case of linear Rossby waves in a one-dimensional barotropic fluid. Analytic solutions are presented for PPVI and the related tendency problem. It is found that PPVI does not contribute to an understanding of Rossby wave dynamics. On the other hand, PPVI turns out to be more useful when applied to confined PV extrema. Neither the application of PPVI to linear baroclinic waves in zonal shear flow nor the inversions of the related temperature anomalies at the boundaries help to better understand the wave development.

It is concluded that PPVI with additional tendency calculations poses and solves a specific problem by retaining observed PV anomalies in one subdomain and removing them in others. The usefulness of the results with regard to the diagnosis of the observed state depends strongly on the flows considered and on the partitions chosen, which must comply with a simple rule.

## Abstract

Piecewise potential vorticity inversion (PPVI) is widely accepted as a useful tool in atmospheric diagnostics. This method is thought to quantify the instantaneous interaction at a distance of anomalies of potential vorticity (PV) separated horizontally and/or vertically. Doubts with respect to the dynamical justification of PPVI are formulated. In particular, it is argued that the tendency of the inverted streamfunction must be determined in order to quantify far-field effects. Elementary tests of PPVI are conducted to clarify these points. First, PPVI is performed for the textbook case of linear Rossby waves in a one-dimensional barotropic fluid. Analytic solutions are presented for PPVI and the related tendency problem. It is found that PPVI does not contribute to an understanding of Rossby wave dynamics. On the other hand, PPVI turns out to be more useful when applied to confined PV extrema. Neither the application of PPVI to linear baroclinic waves in zonal shear flow nor the inversions of the related temperature anomalies at the boundaries help to better understand the wave development.

It is concluded that PPVI with additional tendency calculations poses and solves a specific problem by retaining observed PV anomalies in one subdomain and removing them in others. The usefulness of the results with regard to the diagnosis of the observed state depends strongly on the flows considered and on the partitions chosen, which must comply with a simple rule.

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## Abstract

Although mountains are generally thought to exert forces on the atmosphere, the related transfers of energy between earth and atmosphere are not represented in standard energy equations of the atmosphere. It is shown that the axial rotation of the atmosphere must be included in the energy budget in order to resolve this issue. The energy transfer resulting from mountains turns out to be closely related to mountain torques. The energetic effects of a changing rotation of the earth are discussed, as well as those of friction torques and those of the nonspherical shape of the earth.

## Abstract

Although mountains are generally thought to exert forces on the atmosphere, the related transfers of energy between earth and atmosphere are not represented in standard energy equations of the atmosphere. It is shown that the axial rotation of the atmosphere must be included in the energy budget in order to resolve this issue. The energy transfer resulting from mountains turns out to be closely related to mountain torques. The energetic effects of a changing rotation of the earth are discussed, as well as those of friction torques and those of the nonspherical shape of the earth.