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In this paper, the atmospheric circulations on an equatorial beta plane in response to steady tropical heating are investigated by analytically solving a set of linear equations. Special emphasis is placed on the horizontal structure of forced response under the different combinations of momentum damping and thermal damping, as well as the effect of the zonal domain on the forced responses. Two zonal domains are considered: a zonally cyclic domain and a zonally unbounded domain. The linear model is decomposed in terms of the vertical eigenfunctions in a vertically semi-infinite domain. A new feature of the solution is the existence of a continuous spectrum corresponding to energy propagation out the top of the troposphere. The resulting shallow-water equations are then solved using a method similar to that of Gill. Since the zonal decay scale is proportional to the inverse of the square root of the product of the Rayleigh friction rate and the Newtonian cooling rate, the solutions in a zonally un...

In this paper, the three-dimensional structure of the thermally forced atmosphere on an equatorial b plane is investigated. Special emphasis is placed on the relations between the vertical structure of heating and the horizontal structure of the forced response. By solving the vertical eigenvalue‐eigenfunction problem in a vertically semi-infinite domain, the authors obtain a complete set of vertical eigenfunctions that includes a single barotropic (external) mode and a continuous spectrum of baroclinic (internal) modes. These eigenfunctions are used to decompose vertical heating profiles for two types of tropical heating: 1) deep heating representing the convective plume (CP) heating and 2) shallow heating representing mature cloud (MC) cluster heating. By examining the spectral energy density of the heating profile, the contributions of each vertical mode (spectral interval) to the overall structure are explored for each case, and the difference between the responses to these two profiles of heating is discussed. A dry spectral primitive equation model of the atmosphere is employed to verify the analytical results. The results from both the analytical approach and the numerical simulations are consistent in showing that the vertical structure of the heating is fundamental to the structure of the forced response. The CP is deep relative to the MC. Thus, the CP projects onto the vertical eigenfunctions of relatively larger equivalent depth more so than does the MC. As a result, the CP-forced signals propagate away from the heat source much faster than those forced by the MC. Hence, when the atmosphere is subjected to the same linear dampings (Rayleigh friction and Newtonain cooling), the spatial (mainly in the horizontal) decay rate of the CP-forced signals is significantly smaller than that of the MC-forced signals, and the CP-forced signals extend farther. To what extent a shallow-water system of a specified vertical mode (e.g., the Gill model) can approximate the three-dimensional response is also examined. Results show that the effective gravity wave speed of the multimode system varies greatly with location. Hence, it is extremely difficult to select a globally suitable equivalent depth so that a one-mode shallow-water system can approximate the spatially three-dimensional structure of the response to a given heating.

A series of studies are performed to examine the response of the tropical atmosphere to a prescribed steady, large-scale, elevated heat source (i.e., a region of persistent precipitation). Special emphasis is placed on the surface wind response in two idealized cases in which dissipation is achieved exclusively by Rayleigh friction or by Newtonian cooling. Starting from the linearized equations on an equatorial beta plane, theoretical arguments are presented that suggest there are qualitative differences in the solutions of these two models. A dry spectral primitive equation model of the atmosphere is employed and confirms the results obtained from the analytical studies. The results from both the analytical study and the numerical simulations are consistent in showing that Rayleigh friction and Newtonian cooling play totally different roles in the tropical atmosphere. Newtonian cooling homogenizes the atmospheric motion in the vertical direction, and a strong, vertically uniform wind is found below the base of the heat source. When Rayleigh friction dominates, the circulation driven by the heat source is confined to the layer where the heat source is located. It is also shown that a strong Hadley circulation is associated with reasonable strong Rayleigh friction, but not with Newtonian cooling alone. Finally, the numerical solution is found for the case where Newtonian cooling acts uniformly in the vertical and Rayleigh friction is included in the lower atmosphere to mimic crudely the dissipation of momentum in the boundary layer. The introduction of the simple boundary layer dramatically reduces the surface circulation that was supported in the Newtonian cooling alone case. Together these results suggest a significant surface circulation is unlikely to be driven by an elevated heat source if it resides above the top of the boundary layer.

The vertical structure of the low-level atmospheric response to an elevated large-scale, low-frequency heat source in the Tropics is explored using linear tidal theory on an equatorial beta plane. Through the calculation of the projection of a large-scale, low-frequency thermal source onto the meridional eigenfunctions, the contributions from a set of discrete meridional eigenfunctions with positive equivalent depths, and a continuous spectrum of meridional eigenfunctions with negative equivalent depth, are examined. The positive equivalent depth eigenfunctions have been discussed in some literature while the continuous spectrum of the negative equivalent depth eigenfunctions is new. The authors find that, at lower frequencies, the forced response is mainly supported by those continuous modes for which the absolute values of the negative equivalent depths are neither very small nor very large. The implications of these results for thermally driven surface winds are discussed and summarized by Eqs...

The effect of nonlinearities on a previously investigated coupled atmosphere–ocean basin mode is examined. The nonlinearity in the thermodynamic equation for sea surface temperature arises mainly from the dependence of subsurface temperature on the thermocline depth anomaly in the parameterization of entrainment into the mixed layer. This nonlinearity ultimately suppresses the linear growth of the unstable mode and equilibrates it at a finite amplitude. Because this nonlinearity acts differently for warm and cold states, the warm states are enhanced at finite amplitude. It is found that multiple equilibrium states appear as the coupling coefficient increases and as the reflection coefficient of the oceanic Rossby mode at the western boundary decreases. The finite-amplitude warm equilibrium state turns out to be stable, but the finite-amplitude cold state is unstable. The explicit inclusion of the dependence of the coupling strength on the warm and cold sea surface temperature anomalies modulates ...

The fundamental modes of oscillation of a coupled atmosphere–ocean basin system in the presence of a spatially varying oceanic basic state are investigated by formulating and solving an eigenvalue problem, thereby extending the work of Hirst. The model reduces essentially to the linearized Zebiak and Cane model as discussed by Battisti and Hirst. With conventionally chosen basic states, the unstable eigenmode closely resembles the El Nino–Southern Oscillation (ENSO) cycle in these models. It is shown that the unstable low-frequency eigenfunction consists primarily of a Kelvin mode and a gravest equatorial Rossby mode, and the oscillation can be understood in particularly simple term essentially those proposed by Suarez and Schopf and others. The oscillatory nature of the ENSO cycle can be explained by a transition mechanism resulting from the interaction of these two equatorial (but not necessarily propagating) modes. A growing unstable positive wind anomaly in the central Pacific produces a grow...

A model of the convectively driven mixed layer is used to interpret BOMEX Phase III data. It is concluded that the compensating downdrafts and the moisture transports due to trade cumulus must be included explicitly in any theory of the dynamics of the tropical mixed layer.