Equatorial Beta Plane Research Articles

Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane

We study the nonlinear response of the equatorial shallow-water system at rest to a localized long-wave perturbation with small meridional to zonal aspect ratio. An asymptotic theory of such a response (adjustment) for small Rossby numbers is constructed. Possible scenarios of nonlinear adjustment are classified depending on the relation between the Rossby number and the aspect ratio. The calculations show that slow, geostrophically balanced Rossby and Kelvin waves and the fast inertia–gravity waves are dynamically split off. The fast component of motion exerts no drag on the slow one, which is proved by direct computation. Evolution equations are derived for both components confirming earlier results which were obtained by ad hoc filtering of one of the components of motion. A well-defined initialization procedure is developed for each component.Due to the breaking of non-dispersive Kelvin waves, the asymptotic theory has obvious limits of validity. In order to go beyond these limits and to study strongly nonlinear effects during the adjustment process we undertook high-resolution shock-capturing numerical simulations based on recent progress in finite-volume numerical methods. The simulations confirm theoretical results but also reveal new effects such as fission of a strongly nonlinear Rossby-wave packet into a sequence of equatorial modons or jet formation in the wake of a breaking Kelvin wave.

A Climatology of Waves in the Equatorial Region

Abstract Propagating anomalies of moisture and moist deep convection in the Tropics are organized into a variety of large-scale modes. These include (but are not limited to) the so-called intraseasonal oscillations, convectively coupled waves similar to those predicted by shallow water theory on the equatorial beta plane, and tropical-depression-type disturbances. Along with the annual and diurnal cycles, these modes act and interact to control much of the variance of tropical convection. Analyses of 10 yr of outgoing longwave radiation (OLR) and precipitable water (PW) data are carried out to develop comparative climatologies of these wavelike modes. The analysis relaxes the commonly used cross-equatorial symmetry constraints, which allows study of the portions of the wavelike processes that are asymmetric across the equator. Mean background states are found for OLR and for PW as functions of day of the year. Examination of anomalies together with the background reveals much about how the waves are affec...

The Completeness of Eigenfunctions of the Tidal Equation on an Equatorial Beta Plane*

Abstract The approximate tidal theory on an equatorial beta plane has been widely applied to tropical atmospheric dynamics. There are many successful examples of such applications. However, the mathematical and physical origin of the recently discovered continuous spectrum associated with meridional eigenfunctions of negative equivalent depth is yet to be given, and the completeness of the meridional eigenfunctions in the approximate tidal theory remains to be proved. In this note, a proof of the completeness of the meridional eigenfunction is presented. The differential equation is first transformed into an equivalent integral equation that relates the solution of the differential equation to the corresponding Green's function. It is then shown that the Green's function corresponding to the meridional eigenvalue–eigenfunction problem is linear, self-adjoint, completely continuous, and square integrable over the meridional infinite domain under the principle of analytic continuation. Therefore, the eigenf...

Explicit Simulations of the Intertropical Convergence Zone

Abstract The intertropical convergence zone (ITCZ) is one of the most important components of the global circulation. In order to understand the dynamical processes that regulate its formation, latitudinal preference, and structure, explicit two-dimensional numerical modeling of convection on an equatorial beta plane was conducted with a nonhydrostatic cloud-system-resolving model. The model was forced by energy fluxes associated with constant sea surface temperature (SST) and by horizontally homogeneous radiative cooling. Two distinct patterns were identified for the spatial distribution of convective activity in the Tropics. The first was characteristic of enhanced off-equator convection, namely, a double ITCZ-like morphology (one more salient than the other) straddling the equator during the early period of the integration. The second featured enhanced equatorial convection, namely, a single ITCZ-like morphology on the equator during the later quasi-equilibrium period. Diagnostic analysis and two addit...

Time‐dependent response of the tropical atmosphere to a fixed sea surface temperature anomaly

The time‐dependent response of the tropical atmosphere to a fixed, localized sea surface temperature (SST) anomaly is explored using a highly simplified nonlinear shallow water numerical model. The work builds on that by Webster [1972] and Gill [1980]. The present model has three layers and is formulated on an equatorial beta plane in an atmosphere initially in radiative‐convective equilibrium. Observations suggest that the tropical atmosphere is only marginally unstable to moist convection. Accordingly, the convective parameterization in the model assumes that clouds consume the convective available potential energy at approximately the same rate as large‐scale processes generate it. The convective parameterization allows for both shallow nonprecipitating and deep precipitating clouds. The convection scheme allows the shallow convection to condition the atmosphere for further deep convection, which is an important factor controlling deep convection in the tropics. The model calculations show that shallow convection moistens the middle troposphere, providing favorable conditions for the development of deep convection. In contrast, radiative cooling and drying of the middle troposphere act to suppress deep convection. In the model, these competing processes modulate the deep convection over the localized SST anomaly with a period of about 30 days. The convective heating also excites large‐scale, equatorially trapped normal modes. The response ranges from a steady state flow similar to that by Gill to a periodic generation of Rossby‐Kelvin wave couplets and finally to a transition to chaotic behavior depending on the strength of the forcing.

Deleted Residuals, the QR-Factored Newton Iteration, and Other Methods for Formally Overdetermined Determinate Discretizations of Nonlinear Eigenproblems for Solitary, Cnoidal, and Shock Waves

Solitary waves, cnoidal waves, and shock waves can be computed by solving a nonlinear eigenvalue problem, which in discretized form is a system of nonlinear algebraic equations. Unfortunately, many such systems are singular because the solution is not unique until one or more additional constraints are imposed. For example, if the waves are translationally invariant and u( X) is a solution, then so also is u( X+Φ) for arbitrary Φ. To obtain a unique solution, one must impose an additional condition to reduce the one-parameter family of solutions by constraining Φ. We describe five methods for coping with such singular systems: (i) reformulation of the problem, (ii) deleting residuals, (iii) Keller's bordered matrix scheme, (iv) QR-factored, overdetermined Newton iteration, and (v) pseudoinverse-Newton iteration. We illustrate these ideas using the cnoidal waves of the Korteweg–de Vries equation, the traveling shocks of the Korteweg–de Vries-Burgers equation, and the weakly nonlocal solitary waves of the nonlinear equatorial beta-plane equations. Finite difference, Fourier and rational Chebyshev pseudospectral methods, and spectrally upgraded finite differences are applied. Reformulation and deleting residuals are the cheapest strategies, but the QR-factored Newton iteration is needed for the shock waves, which lack the symmetry of the other two wave species.

赤道ベータ面の線型化した浅水方程式系においていろいろな初期擾乱で励起される赤道波のエネルギーの分配について

大規模大気海洋力学においていろいろな初期擾乱に対する(準)地衡風成分と非地衡風成分へのエネルギーの分配の仕方を理解することは重要である(地衡風調節)。この問題は中緯度f面においてよく調べられてきたが、赤道域ではいくつかの赤道波が励起されてそれらへのエネルギーの分配の仕方は必ずしも明らかではなかった。 ここでは赤道ベータ面の線型化した浅水方程式系を用いて、いろいろな初期擾乱に対する赤道波のエネルギーを求め赤道域の地衡風調節を調べた。初期擾乱として、関数exp(-x2/λ2)Dn(y)(n=1, 2)で表される5つの物理量(圧力、東西風、南北風、発散、渦度)を用いた。ここで、x(y)は東西(南北)方向の座標、λは擾乱のx方向のスケール、Dnはn次の放物柱関数である。また励起される赤道波は、準地衡風的な赤道波(ロスビー波、大きいスケールのケルビン波、ロスビー波に近い混合ロスビー重力波)と非地衡風的な赤道波(慣性重力波、小さいスケールのケルビン波、慣性重力波に近い混合ロスビー重力波)の二つに分けた。 圧力と東西風の初期擾乱の場合にはλが大きいほど、また南北風と渦度の初期擾乱の場合には逆にλが小さいほど、準地衡風的な赤道波がより強く励起されることがわかった。発散の初期擾乱の場合には非地衡風的な赤道波がほとんどである。このような赤道ベータ面における地衡風調節は中緯度f面の場合と同じである。これから、いろいろな初期擾乱に対する準地衡風成分と非地衡風成分へのエネルギーの分配の仕方は全球で成り立つ固有の特性であることが確かめられた。

Kelvin waves: rotationally induced circulations

Abstract A linear equatorial beta plane model is used to find the oscillatory modes associated with internal Kelvin waves with acoustic and gravity modes in a stable or neutrally stable stratified compressible fluid. External modes that decay exponentially with height are also identified. Emphasized are the contributions from the Coriolis parameter f′=2Ω cos θ . Requiring that the Kelvin waves be finite at latitudes far from the equator imposes limitations on the frequency. A non-zero upper bound on the frequency is required when f ′ is introduced into the calculations. Furthermore, when f ′ is included, in the absence of any external forcing, the frequency equation can be partitioned such that frequency becomes independent of the vertical wave number. Vertical wave numbers are then defined in terms of these known natural frequencies, provided the zonal wave number, Brunt Vaisalla frequency, and mean density and temperature gradients are all specified constants. The characteristics of 12 natural frequencies are identified and, in the absence of an external forcing, become important only when f 0 ≠0. It is shown that several of the predicted natural frequencies are associated with observed Kelvin waves. Kelvin waves that include non-zero f ′ with zonal wave number one flow have oscillatory periods that range from 3 days to semiannual or longer. Several fall in the Madden–Julian oscillation range. In the ocean, the natural internal Kelvin wave frequencies associated with zonal wave number one flow have a period varying approximately between 4 months and 6 years. In smaller basins, the period is proportionally less.

Thermally Driven Tropical Circulations under Rayleigh Friction and Newtonian Cooling: Analytic Solutions*

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 unbounded domain can be good approximations for the solutions in a zonally cyclic domain only when both Rayleigh friction and Newtonian cooling are large enough. When either Rayleigh friction or Newtonian cooling is very weak, the solutions are essentially zonally uniform regardless of the longitudinal location of the heat source in a zonally cyclic domain except in a very narrow zone along the equator. The characteristic meridional scale of the shallow-water system is the equatorial radius of deformation of the shallow-water system multiplied by the fourth root of the ratio between the Rayleigh friction rate and the Newtonian cooling rate. Therefore, the characteristic meridional scale is very large for the Rayleigh friction–dominant case, and the forced response can extend far outside the heating latitude. In contrast, in the Newtonian cooling–dominant case the characteristic meridional scale is very small and the forced response is confined to the heating latitudes. The implications of these solutions for both the thermally driven surface winds and the zonally uniform low-frequency variation in pressure and temperature in the upper half of the tropical troposphere are also discussed.

Vertical Structure of Convective Heating and the Three-Dimensional Structure of the Forced Circulation on an Equatorial Beta Plane*

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.

An Analysis of Frictional Feedback on a Moist Equatorial Kelvin Mode

Abstract A simple theoretical model of the tropical troposphere is used to study whether boundary layer friction is destabilizing to the Madden–Julian oscillation (MJO) and other convectively coupled moist equatorially trapped Kelvin-like modes. A linear stability analysis is performed on an equatorial beta plane with a continuously stratified atmosphere using a Betts–Miller-like convective parameterization. The troposphere is divided into a frictional boundary layer close to the surface and a frictionless free troposphere. The basic state is horizontally homogeneous and uniformly convecting. The full linear stability problem can be discretized into an eigenvalue problem that is barely computationally tractable. A scaling analysis appropriate for low-frequency, long wavelength modes, such as the MJO, leads to a much simpler eigenvalue problem. Friction is found to be modestly destabilizing for the moist Kelvin mode, increasing its growth rate by 0.03 day−1. It also has a smaller destabilizing effect on th...

Rayleigh Friction, Newtonian Cooling, and the Linear Response to Steady Tropical 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.

Barotropic Vortex Motion Near Large-Scale Mountain Ranges: Case Studies of Hurricane Gilbert (1988) and Tropical Storm Hermine (1980)

A shallow-water model is used to examine the motion of a barotropic vortex on an equatorial beta plane in the presence of a large-scale mountain range that represents the Sierra Madre of Mexico. The model is initialized with the analyzed large-scale winds of two Atlantic storms, Hurricane Gilbert (1988) and Tropical Storm Hermine (1980). The study is focused on understanding the motion of tropical cyclones in the western Gulf of Mexico.

The Interaction of Easterly Waves, Orography, and the Intertropical Convergence Zone in the Genesis of Eastern Pacific Tropical Cyclones

The interaction of an idealized easterly wave with an orographic feature representing the Sierra Madre of Mexico and a large-scale, meridionally sheared zonal flow representing the ITCZ in the eastern North Pacific is examined through numerical integration of the shallow water equations on an equatorial beta-plane channel. The model simulations show that as a wave approaches the orography from the east, the relative vorticity in the lee increases due to the wave being modified by the orography. This relative vorticity maximum increases the northward advection of the ITCZ in the lee of the mountain that occurs due to the influence of the mountain anticyclone. The result of this interaction is the development of a localized region of enhanced cyclonic vorticity that moves parallel to the contours of the model orography. There is a minimal increase in the amplitude of the vorticity occuring during interactions between the wave and orography, ITCZ and orography, and wave and ITCZ. The position of the incident wave upstream of the orography at the time that the disturbance in the lee begins to develop and the subsequent motion of the leeside vorticity maximum are in agreement with an observed case of eastern Pacific tropical cyclogenesis, which is also presented.

Modulation of the mesospheric semiannual oscillation by the quasibiennial oscillation

Recent satellite and radar observations suggest that the semiannual oscillation (SAO) in the mesosphere is modulated by the stratospheric quasibiennial oscillation (QBO). The modulation is only apparent during the SAO easterly phase, which is considerably stronger when QBO winds are westerly than when they are easterly. We use an equatorial beta-plane model to demonstrate how such a modulation could come about through selective damping of the equatorial wave spectrum excited by deep convection. The waves affected most strongly are easterly inertia-gravity waves of phase speeds slower than ∼ −40 m s−1. This is close to the zonal wind speed during the easterly phase of the QBO (−30 to −35 m s−1), so the waves suffer strong thermal damping or even absorption as they propagate through the stratosphere. Because these waves are important for driving the easterly phase of the mesopause SAO in the model, that phase is weaker when the stratospheric QBO winds are easterly. A similar modulation of the westerly phase of the SAO does not occur for two reasons: (1) QBO westerlies are only half as strong as QBO easterlies, and (2) much of the driving of the westerly phase of the SAO is accomplished by Kelvin waves of phase speed ∼40–60 m s−1. As a consequence, the QBO winds have negligible influence on the vertical propagation of waves with westerly phase velocity and hence on the westerly phase of the modeled SAO.

Thermally Forced Surface Winds on an Equatorial Beta Plane*

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 and . In the inviscid case, since the solution associated with the continuous modes with negative equivalent depth is vertically evanescent, it is expected that the vertical energy transfer from the elevated thermal source to the surface is limited. However, in the presence of Newtonian cooling, the continuous modes that contribute significantly to accounting for the large-scale heat source are those modes with moderate values of negative equivalent depth as frequencies goes to zero so that the forced horizontal winds become vertically uniform below the heating. Hence, surface winds can be driven by the elevated heat source in the presence of only linear thermal damping.

High order models for the nonlinear shallow water wave equations on the equatorial beta-plane with application to Kelvin wave frontogenesis

Abstract We describe two new numerical models for solving the nonlinear shallow water wave equations on the equatorial beta-plane. The finite difference code is eighth order in space and 4th order in time. The second model is a four-mode Hermite–Galerkin scheme in latitude combined with a Fourier pseudospectral algorithm in longitude and 4th order time-marching. As a first application, we test the Boyd–Ripa theory for Kelvin wave frontogenesis. The zeroth order strained coordinates approximation is good even for waves of nondimensional unit amplitude. The first order corrections grow. Even so, for small amplitude, the first order analytical approximations encapsulate most of the difference between the zeroth order approximation and the numerical solution if the longitudinal derivatives are interpreted as x -derivatives rather than as derivatives with respect to the strained coordinate ξ . The main discrepancy is that for moderate and large amplitude, the front curves westward with increasing latitude.

Fundamental Moist Modes of the Equatorial Troposphere

Abstract Raymond's diabatic parameterizations are combined with a three-dimensional channel model on an equatorial beta plane. The equatorially symmetric moist modes of this model for an atmosphere initially at rest are “nonlinear WISHE” (wind-induced surface heat exchange) modes. These modes are driven by the enhanced sea–air fluxes of entropy produced by self-generated westerly winds. Two varieties of these modes are found, those which move to the east at 8 m s−1 and those which move to the west at 4 m s−1. In both modes heavy rain occurs in air that has been moistened by convection of low precipitation efficiency. This preconditioning introduces a phase shift of several days between strong surface entropy fluxes into an air column and the heaviest rainfall. This phase shift may be responsible for the relatively low modal propagation speeds compared to those seen in other models. The modes identified in simulations with uniform sea surface temperature play a fundamental role in a simulation with a warm-...

An Analytical Study of Ozone Feedbacks on Kelvin and Rossby–Gravity Waves: Effects on the QBO

An equatorial beta-plane model of the middle atmosphere is used to analytically examine the effects of radiative cooling and ozone heating on the spatial and temporal evolution of the quasi-biennial oscillation (QBO). Under the assumption that the diabatic heating is weak and the background fields of wind, temperature, and ozone are slowly varying, a perturbation analysis yields expressions describing the vertical spatial modulation of Kelvin and Rossby–gravity waves in the presence of ozone. These expressions show that wave-induced changes in the diabatic heating arising from the advection of basic-state ozone reduce the local radiative damping rate by up to 15% below 35 km. In a one-dimensional model of the QBO, eddy ozone heating increases the amplitude of the zonal wind QBO by 1–2 m s−1 and increases the oscillation period by about two months. The significance of these results to the observed QBO is discussed.

The Role of Equatorial Waves Forced by Convection in the Tropical Semiannual Oscillation

Abstract Recent satellite observations suggest that convection over the tropical continents is capable of exciting wave motions over a wide range of spatial and temporal scales. An equatorial beta-plane model was used to investigate the forcing by convective heating of equatorial waves with zonal wavenumbers from 1 to 15 and a wide range of periods, including diurnal oscillations. Also studied are the propagation of these waves in the equatorial middle atmosphere and their role in driving the tropical semiannual oscillation (SAO). Specification of the heating distribution used to force the model is guided by observations and analyses of tropical convection. It was found that intermediate-scale Kelvin and inertia–gravity waves provide between 25% and 50% of the forcing necessary to drive the westerly phase of the SAO near the stratopause, while the remainder is supplied by planetary-scale Kelvin waves. In the mesosphere, intermediate-scale waves account for an even larger fraction of the force required to ...