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Wayne H. Schubert
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Wayne H. Schubert
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Wayne H. Schubert
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Wayne H. Schubert

Abstract

The geopotential tendency form of semigeostrophic theory is derived and compared with the potential vorticity form. The tendency form is compact and particularly convenient for non-Boussinesq, nonuniform potential vorticity flows.

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Wayne H. Schubert

Abstract

Lilly's model of a horizontally homogeneous cloud-topped mixed layer is studied. The model is closed by taking a weighted (weighting factor or entrainment parameter k) average of Lilly's maximum and minimum entrainment cases. The dependence of steady-state solutions on large-scale divergence, sea surface temperature and entrainment parameter k is investigated. By numerical integration the response of the mixed layer to a diurnally varying radiative flux is investigated. Significant variations in the state of the mixed layer and in the convective fluxes are found.

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Ricardo Prieto and Wayne H. Schubert

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Equilibrium states of initially barotropically unstable polar vortices are predicted using two different approaches: minimum enstrophy and maximum entropy theories, which have been extended to include flows evolving on the surface of a sphere.

Minimum enstrophy theory shows very good agreement with an ensemble of direct numerical integrations of a polar vortex that mixes vorticity mainly on a spherical cap. For the case of a polar vortex with a substantial resistance to vorticity mixing at its core, the maximum entropy prediction shows good consistency with an ensemble of direct numerical integrations.

Maximum entropy theory gives an additional source of information with its density functions, which in a probabilistic sense reveal how the vorticity field (and therefore the mass field) is redistributed in the equilibrium state. The density functions show good skill in predicting several passive tracer distributions in the numerical experiments. Also, from a local point of view, density functions determine the degree of mixing of initially well separated air masses, information that is valuable in tracing atmospheric chemical components.

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Mark DeMaria and Wayne H. Schubert

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The three-layer balanced axisymmetric tropical cyclone model presented by Ooyama is generalized to dimensions and the resultant primitive equations are solved using the spectral (Galerkin) method with Fourier basis functions on a doubly-periodic midlatitude β-plane. The nonlinear terms are evaluated using the transform method where the necessary transforms are performed using FFT algorithms. The spectral equations are transformed so that the dependent variables represent the normal modes of the linearized equations. For the three-layer model, the normal modes correspond to internal or external gravity or rotational modes or to inertial oscillations associated with the constant depth boundary layer. When the governing equations, are written in terms of the normal modes, the linear terms can be evaluated exactly and the application of the nonlinear normal mode initialization scheme proposed by Machenhauer is straightforward.

Results from a simulation with an axisymmetric initial condition on an f-plane are presented and it is shown that the model can produce a vortex similar to tropical cyclones observed in nature. The energy of the gravity modes and rotational modes are calculated for this simulation and it is shown that the gravity mode energy is more than an order of magnitude smaller than the rotational mode energy. The model is then run on the β-plane and it is shown that the variation of the Coriolis parameter with latitude causes the tropical cyclone to move toward the northwest at about 2 m s−1, in agreement with several other studies. It is also shown that the dispersion of the rotational modes causes the tropical cyclone to elongate toward the west and develop sharper geopotential gradients toward the cast. The model is also run with a basic state wind profile and it is shown that the motion of the storm produces asymmetries in the boundary layer convergence field.

The effect of initialization procedures on a tropical cyclone simulation is also studied. The results from linear and nonlinear normal mode initialization procedures and results from applying an initialization procedure based on the nonlinear balance equation are compared. It is shown that the nonlinear normal mode initialization procedure results in much smaller track and intensity forecast errors, and prevents the excitation of spurious gravity waves.

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Gudrun Magnusdottir and Wayne H. Schubert

Abstract

We develop here the isentropic–geostrophic coordinate version of semigeostrophic theory on a midlatitude β-plane. This approach results in a simple mathematical form in which the horizontal ageostrophic velocities are implicit and the entire dynamics reduces to a predictive equation for the potential pseudodensity and an invertibility relation. Linearized versions of the theory lead to a generalized Charney–Stern theorem for combined barotropic–baroclinic instability and to Rossby wave solutions with a meridional structure different from that in quasi-geostrophic theory.

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Gudrun Magnusdottir and Wayne H. Schubert

Abstract

This paper presents the combined isentropic and spherical geostrophic coordinate version of semigeostrophic theory. This is accomplished by first proposing a spherical coordinate generalization of the geostrophic momentum approximation and discussing its associated conservation principles for absolute angular momentum, total energy, potential vorticity and potential pseudodensity. We then show how the use of the spherical geostrophic coordinates allows the equations of the geostrophic momentum approximation to be written in a canonical form that makes ageostrophic advection implicit. This leads to a simple equation for the prediction of the potential pseudodensity. The potential pseudodensity can then be inverted to obtain the associated wind and mass fields. In this way the more general semigeostrophic theory retains the same simple mathematical structure as quasi-geostrophic theory—a single predictive equation which does not explicitly contain ageostrophic advection and an invertibility principle. The combined use of isentropic and spherical geostrophic coordinates is crucial to retaining this simplicity.

In order to demonstrate how the theory applies to problems of barotropic–baroclinic instability and Rossby–Haurwitz wave dispersion, we derive the semigeostrophic generalization of the Charney–Stern theorem and compare the semigeostrophic Rossby–Haurwitz wave frequencies with those of Laplace's tidal equations. The agreement between these frequencies is generally better than 0.5%. Thus, the theory appears to encompass a wide range of meteorological phenomena including both planetary-scale and synoptic-scale waves, along with their finer scale aspects such as fronts and jets.

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Mark DeMaria and Wayne H. Schubert

Abstract

A three-layer, primitive equation tropical cyclone model is used to test the effect of nonlinear normal mode initialization (NNMI) in a tropical cyclone simulation. The model is solved using a spectral method with normal mode basis functions. Results from a six-day tropical cyclone simulation are used as initial data to test the NNMI. It is shown that Machenhauer’s NNMI scheme converges rapidly under tropical cyclone conditions when the dissipative and convective terms are not included in the nonlinear forcing (adiabatic initialization). When these terms are included (diabatic initialization), Machenhauer’s scheme no longer converges, but can still reduce the initial gravity mode time tendencies to an acceptable level. The radial winds produced by the diabatic initialization are qualitatively similar to those in the control simulation, but the positioning of the convective heating with respect to the radius of maximum wind is somewhat different. Because of this, the errors in the storm intensity using the control simulation as a perfect forecast are about the same for the adiabatic and diabatic initializations. This indicates that the usefulness of the standard Machenhauer initialization in tropical cyclone models may be limited. The diabatic initialization scheme used in the ECMWF operational model is also tested. For this case, the time-averaged diabatic forcing is determined from a short model run and then held fixed during the iteration. For this case, the diabatic initialization converges and produces a radial wind field consistent with the control simulation.

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