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## Abstract

We consider the frictionless, axisymmetric, balanced flow occurring in a thermally forced vortex on an *f*-plane. Following Eliassen (1952) we derive the diagnostic equation for the forced secondary circulation. This equation contains the spatially varying coefficients *A* (static stability), *B* (baroclinity), *C* (inertial stability), and the thermal forcing *Q*. Assuming that *A* is a constant, *B* = 0, and that *C* and *Q* are piecewise constant functions of radius, we obtain analytical solutions for the forced secondary circulation. The solutions illustrate the following points. 1) For a given *Q* an increase in inertial stability leads to a decrease in the forced secondary circulation and a change in the radial distribution of local temperature change, with enhanced ∂θ/∂*t*; in the region of high inertial stability. 2) Lower tropospheric tangential wind accelerations are larger inside the radius of maximum wind, which leads to a collapse of the radius of maximum wind. 3) The fraction of *Q* which ends up as ∂θ/∂*t*; increases during the tropical cyclone development, particularly if the horizontal extent of *Q* is small and close to the region of high inertial stability. 4) One can regard the formation of an eye as a process which tends to stabilize the vortex since it removes *Q* from the protected, highly stable inner region.

## Abstract

We consider the frictionless, axisymmetric, balanced flow occurring in a thermally forced vortex on an *f*-plane. Following Eliassen (1952) we derive the diagnostic equation for the forced secondary circulation. This equation contains the spatially varying coefficients *A* (static stability), *B* (baroclinity), *C* (inertial stability), and the thermal forcing *Q*. Assuming that *A* is a constant, *B* = 0, and that *C* and *Q* are piecewise constant functions of radius, we obtain analytical solutions for the forced secondary circulation. The solutions illustrate the following points. 1) For a given *Q* an increase in inertial stability leads to a decrease in the forced secondary circulation and a change in the radial distribution of local temperature change, with enhanced ∂θ/∂*t*; in the region of high inertial stability. 2) Lower tropospheric tangential wind accelerations are larger inside the radius of maximum wind, which leads to a collapse of the radius of maximum wind. 3) The fraction of *Q* which ends up as ∂θ/∂*t*; increases during the tropical cyclone development, particularly if the horizontal extent of *Q* is small and close to the region of high inertial stability. 4) One can regard the formation of an eye as a process which tends to stabilize the vortex since it removes *Q* from the protected, highly stable inner region.

## Abstract

Using an axisymmetric primitive tropical cyclone model, we first illustrate the way in which nonlinear processes contribute to the development of an atmospheric vortex. These numerical experiment show that nonlinearities allow a given diabatic beat source to induce larger tangential wind (and kinetic energy) changes as the vortex develops and the inertial stability becomes large. In an attempt to gain a deeper theoretical understanding of this process, we consider the energy cycle in the balanced vortex equations of Eliassen. The temporal behavior of the total potential energy *P* is governed by *dP*/*dt*=*H*−*C* where *H* is the rate of generation of total potential energy by diabatic heating, and *C* is the rate of conversion to kinetic energy. We define a time-dependent system efficiency parameter as η¯(*t*)=*C*/*H*. Then, using the dynamical simplifications of balanced vortex theory, we express η¯(*t*) as a weighted average of a dynamic efficiency factor η(*r*, *z*, *t*). The dynamic efficiency factor is a measure of the efficacy of diabatic heating at any point in generating kinetic energy and can be determined by solving a second-order partial differential equation whose coefficients and right-hand side depend only on the instantaneous vortex structure. The diagnostic quantities η¯(*t*) and η(*r*, *z*, *t*) are utilized in the analysis of several balanced numerical experiments with different vertical and radial distributions of a diabatic heat source.

## Abstract

Using an axisymmetric primitive tropical cyclone model, we first illustrate the way in which nonlinear processes contribute to the development of an atmospheric vortex. These numerical experiment show that nonlinearities allow a given diabatic beat source to induce larger tangential wind (and kinetic energy) changes as the vortex develops and the inertial stability becomes large. In an attempt to gain a deeper theoretical understanding of this process, we consider the energy cycle in the balanced vortex equations of Eliassen. The temporal behavior of the total potential energy *P* is governed by *dP*/*dt*=*H*−*C* where *H* is the rate of generation of total potential energy by diabatic heating, and *C* is the rate of conversion to kinetic energy. We define a time-dependent system efficiency parameter as η¯(*t*)=*C*/*H*. Then, using the dynamical simplifications of balanced vortex theory, we express η¯(*t*) as a weighted average of a dynamic efficiency factor η(*r*, *z*, *t*). The dynamic efficiency factor is a measure of the efficacy of diabatic heating at any point in generating kinetic energy and can be determined by solving a second-order partial differential equation whose coefficients and right-hand side depend only on the instantaneous vortex structure. The diagnostic quantities η¯(*t*) and η(*r*, *z*, *t*) are utilized in the analysis of several balanced numerical experiments with different vertical and radial distributions of a diabatic heat source.

## Abstract

We consider the axisymmetric balanced flow occurring in a thermally forced vortex in which the frictional inflow is confined to a thin boundary layer. Above the boundary layer the absolute angular momentum ½*fR*
^{2}=*rv*+½*fr*
^{2} is conserved. We refer to *R* as the potential radius, i.e., the radius to which a particle must be moved (conserving absolute angular momentum) in order to change its tangential component *v* to zero. Using *R* as one of the dependent variables we review the equations of the Eliassen balanced vortex model.

We next reverse the roles of the actual radius *r* and the potential radius *R*, i.e., we treat *R* as an independent variable and *r* as a dependent variable. Introducing transformed components (*u*
^{*}, *w*
^{*}) of the transverse circulation we obtain the transformed Eliassen balanced vortex equations, which differ from the original equations in the following respects: 1) the radial coordinate is *R* which results in a stretching of positive relative vorticity regions and a shrinking of negative relative vorticity regions; 2) the thermodynamic equation contains only the transverse circulation component *w*
^{*}, the coefficient of which is the potential vorticity *q*; 3) the equation for *r* contains only the transverse circulation component *u*
^{*}; 4) the transverse circulation equation contains only two vortex structure functions, the potential vorticity *q* and the inertial stability *s*, where *pq*=(ζ/*f*)(*g*/θ_{0})(∂θ/∂*Z*) and ρ*s*=*f*
^{2}
*R*
^{4}/*r*
^{4}.

The form of the transverse circulation equation leads naturally to a generalized Rossby radius proportional to (*q*/*s*)^{½}. A typical distribution Of (*q*/*s*)^{½} is calculated using the composite tropical cyclone data of Gray. The fundamental dynamical role of (*q*/*s*)^{½} is then illustrated with a simple analytical example.

## Abstract

We consider the axisymmetric balanced flow occurring in a thermally forced vortex in which the frictional inflow is confined to a thin boundary layer. Above the boundary layer the absolute angular momentum ½*fR*
^{2}=*rv*+½*fr*
^{2} is conserved. We refer to *R* as the potential radius, i.e., the radius to which a particle must be moved (conserving absolute angular momentum) in order to change its tangential component *v* to zero. Using *R* as one of the dependent variables we review the equations of the Eliassen balanced vortex model.

We next reverse the roles of the actual radius *r* and the potential radius *R*, i.e., we treat *R* as an independent variable and *r* as a dependent variable. Introducing transformed components (*u*
^{*}, *w*
^{*}) of the transverse circulation we obtain the transformed Eliassen balanced vortex equations, which differ from the original equations in the following respects: 1) the radial coordinate is *R* which results in a stretching of positive relative vorticity regions and a shrinking of negative relative vorticity regions; 2) the thermodynamic equation contains only the transverse circulation component *w*
^{*}, the coefficient of which is the potential vorticity *q*; 3) the equation for *r* contains only the transverse circulation component *u*
^{*}; 4) the transverse circulation equation contains only two vortex structure functions, the potential vorticity *q* and the inertial stability *s*, where *pq*=(ζ/*f*)(*g*/θ_{0})(∂θ/∂*Z*) and ρ*s*=*f*
^{2}
*R*
^{4}/*r*
^{4}.

The form of the transverse circulation equation leads naturally to a generalized Rossby radius proportional to (*q*/*s*)^{½}. A typical distribution Of (*q*/*s*)^{½} is calculated using the composite tropical cyclone data of Gray. The fundamental dynamical role of (*q*/*s*)^{½} is then illustrated with a simple analytical example.

## Abstract

A nonlinear shallow-water model on the sphere is used to study barotropic aspects of the formation of twin tropical disturbances by Madden–Julian oscillation (MJO) convection.

In the model, the effect of MJO convection upon the lower-tropospheric tropical circulation was simulated by an eastward moving, meridionally elongated mass sink straddling the equator. The intensity and propagation speed of the mass sink were chosen to simulate observations that MJO convection intensifies while nearly stationary in the eastern equatorial Indian Ocean, weakens while moving eastward over the Maritime Continent, again intensifies once it reaches the west Pacific Ocean, and finally becomes stationary and dies off near the date line. This mass sink produced twin cyclones in the two regions where it was stationary, namely, where it was initially turned on and where it was turned off. In addition, the mass sink produced two zonally elongated cyclonic potential vorticity anomalies straddling the equator in the region where it propagated eastward.

It is proposed that MJO convection produces twin tropical disturbances in the two regions where it is nearly stationary, namely, its region of formation in the eastern Indian Ocean and its region of decay near the date line. Additional tropical disturbances may arise from the breakdown of the elongated shear regions produced by the eastward propagating MJO convection.

In addition, a series of initial value experiments was performed to determine the conditions under which twin cyclones become so strongly coupled that they propagate directly eastward as a cyclone pair. Apparently, such movement requires the cyclones to be so close together that the situation rarely, if ever, occurs in nature.

## Abstract

A nonlinear shallow-water model on the sphere is used to study barotropic aspects of the formation of twin tropical disturbances by Madden–Julian oscillation (MJO) convection.

In the model, the effect of MJO convection upon the lower-tropospheric tropical circulation was simulated by an eastward moving, meridionally elongated mass sink straddling the equator. The intensity and propagation speed of the mass sink were chosen to simulate observations that MJO convection intensifies while nearly stationary in the eastern equatorial Indian Ocean, weakens while moving eastward over the Maritime Continent, again intensifies once it reaches the west Pacific Ocean, and finally becomes stationary and dies off near the date line. This mass sink produced twin cyclones in the two regions where it was stationary, namely, where it was initially turned on and where it was turned off. In addition, the mass sink produced two zonally elongated cyclonic potential vorticity anomalies straddling the equator in the region where it propagated eastward.

It is proposed that MJO convection produces twin tropical disturbances in the two regions where it is nearly stationary, namely, its region of formation in the eastern Indian Ocean and its region of decay near the date line. Additional tropical disturbances may arise from the breakdown of the elongated shear regions produced by the eastward propagating MJO convection.

In addition, a series of initial value experiments was performed to determine the conditions under which twin cyclones become so strongly coupled that they propagate directly eastward as a cyclone pair. Apparently, such movement requires the cyclones to be so close together that the situation rarely, if ever, occurs in nature.

## Abstract

Through the use of a zonal balance model we investigate the response of the mean meridional circulation to a specified diabatic forcing for both resting and nonresting zonal flows. The use of a potential latitude coordinate and transformed meridional circulation components results in a simplified meridional circulation equation in which the variable coefficients are the normalized potential vorticity and inertial stability. Solutions of this equation illustrate how latent heat release away from the equator forces a winter hemisphere Hadley cell that is more intense than the summer hemisphere cell. This asymmetric response is due primarily to the anisotropy associated with the spatial variation of the inertial stability field. Despite the sensitivity of the meridional circulation to the location and breadth of the forcing, the low latitude thermodynamic response is for the most part insensitive as long as the total latent heat release remains the same.

Numerical solutions of the zonal balance model result in evolving zonal wind and temperature fields that modify the potential vorticity and inertial stability fields. In the vicinity of the ITCZ, the potential vorticity becomes large in the lower troposphere and small in the upper troposphere which, in addition to modifying the response of the meridional circulation, generates the necessary dynamical conditions for wave instability. Since the inertial stability is only slightly modified, however, the basic anisotropy in the response of the meridional circulation remains. At the same time, the evolving zonal wind and temperature fields result in an increasing dynamic efficiency of latent heat release, which leads to an accelerated growth of zonal kinetic energy, especially when the ITCZ is located poleward of 10 degrees latitude.

## Abstract

Through the use of a zonal balance model we investigate the response of the mean meridional circulation to a specified diabatic forcing for both resting and nonresting zonal flows. The use of a potential latitude coordinate and transformed meridional circulation components results in a simplified meridional circulation equation in which the variable coefficients are the normalized potential vorticity and inertial stability. Solutions of this equation illustrate how latent heat release away from the equator forces a winter hemisphere Hadley cell that is more intense than the summer hemisphere cell. This asymmetric response is due primarily to the anisotropy associated with the spatial variation of the inertial stability field. Despite the sensitivity of the meridional circulation to the location and breadth of the forcing, the low latitude thermodynamic response is for the most part insensitive as long as the total latent heat release remains the same.

Numerical solutions of the zonal balance model result in evolving zonal wind and temperature fields that modify the potential vorticity and inertial stability fields. In the vicinity of the ITCZ, the potential vorticity becomes large in the lower troposphere and small in the upper troposphere which, in addition to modifying the response of the meridional circulation, generates the necessary dynamical conditions for wave instability. Since the inertial stability is only slightly modified, however, the basic anisotropy in the response of the meridional circulation remains. At the same time, the evolving zonal wind and temperature fields result in an increasing dynamic efficiency of latent heat release, which leads to an accelerated growth of zonal kinetic energy, especially when the ITCZ is located poleward of 10 degrees latitude.

## Abstract

A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high.

When the basic state possesses positive relative vorticity in an inner region, the energy partition can be substantially modified, and cloud-cluster-scale heating can become considerably more efficient.

The energy partition results have important implications for the lateral boundary condition used in tropical cyclone models. Faced with the fact that a perfect non-reflecting condition is possible but impractical to implement, one is forced to use an approximate condition which causes some reflection of gravity-inertia waves and hence some distortion of the geostrophic adjustment process. The distortion can be kept small by the use of a suitable radiation condition.

## Abstract

A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high.

When the basic state possesses positive relative vorticity in an inner region, the energy partition can be substantially modified, and cloud-cluster-scale heating can become considerably more efficient.

The energy partition results have important implications for the lateral boundary condition used in tropical cyclone models. Faced with the fact that a perfect non-reflecting condition is possible but impractical to implement, one is forced to use an approximate condition which causes some reflection of gravity-inertia waves and hence some distortion of the geostrophic adjustment process. The distortion can be kept small by the use of a suitable radiation condition.