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2. The Interaction of Waves and Convection in the Tropics. Summary
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Summary Interest in tropical waves and their interaction with convection has been rekindled in recent years by the discovery, using satellite infrared data to track high cloud, that such waves closely display the dispersive properties of linear, inviscid wave theory for an atmosphere with a resting basic state and equivalent depths between 10 and 100 m. While several current approaches focus on internal modes in the atmosphere, this is inconsistent with the absence of internal modes in the atmosphere which is characterized by a single isolated eigenmode and a continuous spectrum. It will be shown, using a triggering type approach to convection, that the observed properties of waves are consistent with a continuous spectrum. The approach assumes that the total convection is determined by mean evaporation, but that the convection is patterned by zero averaged perturbations to triggering energy following the recent approach of Mapes. The observed convection associated with the migrating semidiurnal tide is used to calibrate the timescale for the convective response to patterning. This time scale is more representative of tropical mesoscale systems than of convection itself. It is shown that this time scale leads to the observed phase lead of low level convergence in tropical waves vis a vis the convective heating. Finally, it is shown that this phase is sensitive to the equivalent depth which it is suggested is the basis for the selection of equivalent depth. Reasonable simulations of observed waves are readily obtained.
Waves in a stratified spherical fluid Linearized equations of motion lead to a separable second order partial differential equation (in altitude and latitude) with solutions of the form: where
The vertical structure equation is characteristically of the form where, Essentially, the equivalent depth is a measure of the vertical wavenumber, and depending on the relation of hn to zonal wavenumber, frequency, etc., the wave will essentially be an internal gravity or Rossby wave or some combination of the two.
Free v. forced waves Forced waves: We are given w and s. Laplace’s Tidal Equation is solved for hn and Qn, where hn is an eigenvalue and Qn is an eigenfunction. The forcing is expanded in these eigenfunctions (known as Hough Functions), and the vertical structure equation is solved for the response to each component of the forcing. Free waves: The vertical structure equation is solved in the absence of forcing. The eigenvalues are the equivalent depths of the fluid system. For shallow water, the only eigenvalue is the depth of the fluid. For a stably stratified liquid with a top, the equivalent depths correspond to an infinite set of vertical modes. For the atmosphere, there is generally only a single eigenvalue,
corresponding to a Lamb mode with an equivalent depth of about 10 km. There is also a continuous spectrum. For each equivalent depth and zonal wavenumber, Laplace’s Tidal Equation is solved for the eigenfrequencies, wn, and the associated Hough Functions. For each Hough Function, one obtains a relation between frequency and wavenumber. The name ‘equivalent depth’ was chosen by analogy with the shallow water case where the equivalent depth was the actual depth of the fluid. Despite the fact that the atmosphere generally is found to have only a single equivalent depth (10 km), many tropical waves are observed to behave as though they had a relatively unique equivalent depth of around 12-60m.
Wheeler-Kiladis (1999) analysis of OLR The new ‘equivalent depth’ is generally assumed to arise from the interaction of waves with convection. Note that MJO is not characterized by a particular equivalent depth.
It should be noted that the Wheeler-Kiladis results implicitly demonstrate the relevance of simple linearized tropical wave theory. It should also be noted that recent attempts to interpret the Wheeler-Kiladis results in terms of internal normal modes (Mapes, 2000, Majda and Schefter, 2001, Emanuel et al, 1994) are inconsistent with the spectral properties of the vertical structure equation for realistic atmospheres without lids.
Various approaches to interaction between large scale motion system and convection. 1. Charney and Eliassen (1964) and early Wave-CISK (Yamasaki, 1969, Lindzen, 1974): Effective cumulus heating drives large scale motion which includes low level convergence (due to Ekman pumping in CISK, and vertically propagating wave in Wave-CISK) which lifts air to Lifting Condensation Level (ca 500m). Problems: Eliassen critique: Trade wind boundary layer is turbulent so that there is no need to lift air to LCL. N.B. Early wave-CISK studies always got equivalent depths of about 10 m corresponding to vertical wavelengths of about 2 km (corresponding to a quarter wavelength equal to the height of the lifting condensation level).
2. Moisture budget (Tiedke parameterization, Lindzen (1981), later Wave-CISK: Stevens and Lindzen (1978) ): Convection is a response to moisture budget below about 2 km (Trade Inversion). Problem ( Stevens and Lindzen (1978)) : Wave-CISK doesn't provide instability. (Note, however, observed equals , suggesting that observed waves equilibrate by exhausting moisture supply.) N.B. An equivalent depth of about 30m corresponds to a vertical wavelength of about 8 km (and a quarter wavelength to the height of the trade inversion).
3. Wave triggering: In an unstable environment any perturbation will trigger patterning of existing convection. In general, convective elements are randomly distributed. In mass budget approach, Mc responds to directly determined convergence below trade inversion. In triggering approach, convection automatically provides self-consistent low level convergence, but perturbations to convergence determine pattern of convection. Of course, if the perturbation provides more convergence, the convection will not have to provide as much. (Basically analogous to Benard convection.) Note, that the present work uses w as a measure of triggering, but, in fact, any variable could be used without fundamentally altering the formalism.
In the present approach, waves do not change the total amount of convective activity. Rather, the low level fields of the wave perturbation biases the random breakdowns of CIE produced by boundary layer turbulence so as to pattern the convection which would occur anyway. Thus, the mean amount of convection is essentially determined by the mean evaporation. While there is evidence that squall systems play an important role in the convection itself, other systems ranging from gravity waves to easterly and Kelvin waves to the Hadley and Walker circulations serve primarily to pattern the convection that would otherwise exist. Moreover, all the sources of patterning can simultaneously coexist.
As noted by Reed & Recker (1971) and many since, the amplitude in precipitation of tropical waves tends to be approximately the mean precipitation -- suggesting a reorganization of existing precipitation rather than the production of additional precipitation.
Cumulus Heating Cumulus convection does not directly heat the ambient atmosphere. However, that portion of the mean vertical velocity which is carried in cumulus towers also does not contribute to adiabatic cooling. Thus, we must subtract this part from the adiabatic cooling: i.e., the adiabatic cooling term becomes The term constitutes an effective cumulus heating term. The patterning of the convection gives rise to a contribution to the effective cumulus heating in the form of the pattern. The contribution of the patterning to the mean is, however, zero.
Modes of cumulus-large-scale flow interaction (regardless of interaction): a) Self-excitation where, for example, motion systems forced by effective cumulus heating provide low level convergence that, in turn, triggers the convective pattern that forces the wave. b) Motion systems that have their origin in processes separate from effective cumulus heating provide low level convergence with the resulting effective cumulus heating modifying the motion system. § Examples of (a) are the tropical waves analyzed by Wheeler and Kiladis (1999) (except for the MJO). § Examples of (b) are the solar semidiurnal tide, and, possibly, the MJO. We will begin with the second case.
Madden-Julian Oscillation (MJO) Baroclinic instability of subtropical jet leads to wave 1-2 40 day westerly waves which are strongly coherent with the tropical MJO (Straus and Lindzen, 2001). Unfortunately, it is not obvious how such instabilities would effectively penetrate to near equatorial regions.
Unfortunately, prevailing easterlies prevent propagation. Tunneling, however, remains a possibility. Also, in reanalyses, qy is negative below 900 mb.
Case of the solar semidiurnal tide: Forcing is primarily due to insolation absorption by ozone (Butler and Small, 1963) and water vapor (Siebert, 1961). Amplitude okay, but phase is about one hour off (0900, 2100 instead of 1000, 2200). By contrast, the diurnal component of the rainfall does not have a uniform phase in local time. Lindzen (1978) and Hamilton (1981) showed that observed semidiurnal component of rainfall provided additional forcing that would correct discrepancy. However, tidal convergence was one order of magnitude less than needed. This implies that triggering rather than direct forcing of convective pattern is involved. Moreover, tidal component of rainfall is only a fifth of mean rainfall. The reason, we suggest, is that tidal time scale (12 hours/2p) is short compared with convective response to triggering.
The following equation roughly describes how we expect to behave: For convenience, we will let , so that the above equation becomes Although will, of course, be distorted from a sine wave, its impact on the wave will be associated with its projection on the sinusoidal w component.
The quantity a/w determines the amplitude and phase of . For semidiurnal tide, we know that . This, in turn, implies a/w » 0.16, or a-1»11.94 hours. The phase lag is about 81.8o, which is what is needed to correct the tide. For most tropical waves, adjustment time is relatively short.
Self-excited tropical waves. Self excitation with triggering only requires that triggering be in phase approximately with heating required to force wave. Procedure: Using characteristic heating profile, calculate low level w as a function of equivalent depth, and see how phase varies with equivalent depth. N.B. In contrast to most earlier studies, we are not associating observed waves with instabilities. Rather, we are looking for consistency between triggering and wave forcing. Waves for which these are inconsistent will self destruct (though the rate of self-destruction depends on the degree of inconsistency).
In point of fact, the calculation of waves is subject to considerable complexity and uncertainty -- especially due to such factors as cumulus friction and trade wind boundary layer turbulence. However, there is a simple test of triggering which bypasses these complexities. The patterning time of about 11.9 hours implies that the low level convergence should lead the upper level heating by this amount in order for the two to be compatible.
The following simplified approach illustrates how one ought to approach the problem of determining the preferred equivalent depth. The approach is probably too simple to take the results as being anything other than a very rough approach. They are, nonetheless, remarkably successful.
We consider the linearized equation for vertical structure of the ‘vertical velocity' in log p coordinates, w*, weighted by , (where , p=pressure, and ps=surface pressure): where and h is the equivalent depth, is the basic state temperature, g is the acceleration of gravity, , R = the gas constant for air, cp is the heat capacity of air at constant pressure, and Q(x) is the vertical distribution of effective convective heating.
N.B. Given the oversimplified physics, it doesn’t pay to take the precise phases too seriously. More important is the fact that phase varies substantially with h, allowing for the possibility of selection. Nonetheless, in these cases the proper phase seems to occur for h’s that are too large.
Mapes (Mapes, B.E. (2000) Convective Inhibition, Subgrid-Scale Triggering Energy, and Stratiform Instability in a Toy Tropical Wave Model. J. Atmos. Sci., 57, 1515-1535) pointed out the ubiquitous importance of shallow heating from cumulus congestus that is shallower, and somewhat smaller than cumulonimbus heating, and precedes it by several hours.
The addition of a relatively small amount of shallow heating at low levels does, indeed, correct the phase appropriately. Interestingly, the effect does not depend much on the phase lead of the shallow convection. For example, we might add the following to our Q For K, we will try 0.2. For φ, we will try 0 and π/6.
We see that for widely different basic states, and small congestus source, equivalent depths between around 12-60m will either be self-sustaining or slow to self-destruct, while outside this range, self-destruction will be more rapid.
The vertical structure of waves for a single h displays the characteristics that are observed. The relatively small stratospheric leakage may be an artifact of the choice for the vertical structure of the thermal forcing. However, the fact that the solution follows the heating in the heating region, and looks like a vertically propagating wave above this region is almost certainly robust and corresponds to observations.
Even for the present basic state, there is more leakage for longer vertical wavelengths or, equivalently, larger equivalent depths.
What is happening is that when the vertical scale of F is larger than half the vertical wavelength given by l, then the particular solution dominates, and the deeper F is relative to the half wavelength, the less is the leakage of the wave into regions where F is small. However, what does leak out, behaves like an homogeneous solution: ie, a radiating wave.
As noted in Stevens, Lindzen and Shapiro (1977), solutions of simple inviscid linear theory for tropical waves suffer from one major drawback: for observed values of rainfall (i.e., effective cumulus heating), amplitudes of temperature and horizontal velocity oscillations are too large. Models commonly replicate tropical easterly waves, but with reduced wave components of rainfall. SLS showed that these problems could readily be eliminated by the inclusion of a simple model for cumulus momentum transport (Schneider and Lindzen, 1980). Since then, there has been much interest and controversy over the form or even the existence of so-called ‘cumulus friction.' Sardeshmukh and Hoskins (1987), for example, argued that there was no evidence for any such phenomenon. Tung and Yanai (2002a,b), however, have recently presented evidence to the contrary. Until these issues are resolved, the present approach must be considered simply a suggestion for how simple tropical wave physics can select the observed preferred equivalent depth. The dispersion relation observed by Wheeler and Kiladis (1999) suggests that such physics, nevertheless, remains relevant.
Despite uncertainty, the primary test of the patterning hypothesis for equatorial waves was met. The phase lead observed by Straub & Wheeler (2002) is essentially what is called for by the patterning time derived from the semidiurnal tide. This, in turn, implies quite a lot about both tropical waves and convection: namely, convection is triggered and waves are forced by the resulting ‘effective cumulus heating.’ In addition, it was easy, within the context of classical wave theory to simulate both the observed distribution of equivalent depths and the relatively complex observed vertical structure.
In summary: 1. Using the semidiurnal tide as an example of patterning, one gets an estimate for the time scale for patterning. This time scale is longer than the time scales characteristic of convective elements. It is more likely appropriate for squalls which organize most convection. (Note that, for simplicity, we have taken the adjustment rate to be independent of the spatial scale of the perturbation. More to the point, we don’t know what influence such spatial scales should have.)
2. Patterning requires only a consistent phase in the trade wind boundary layer. This talk has simply shown that phase varies substantially with equivalent depth, hovering around acceptable phases for the observed equivalent depths -- especially when small heating from cumulus congestus is included. This serves to preferentially select these equivalent depths over other equivalent depths which don't present consistent phases. However, given the uncertainty as to how triggering is actually to be formulated, this remains speculative.
3. While the precise calculated phase should indeed depend on aspects of the physics that were treated casually at best, the observed phase lead for tropical waves is, indeed, what is implied by the tidal estimate for patterning time. Note that the fact that the congestus clouds may play an important role in the cumulonimbus response to dynamic triggering, and that the inclusion of congestus heating is important (in the present formulation) in order to achieve consistency in phase between triggering and cumulus heating at the observed equivalent depth suggests that the interaction of waves and convection may be more subtle than anticipated.
