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  • View in gallery

    Schematic derivation of the adding formulas.

  • View in gallery

    (top) EP flux for the half-sine heating profile (solid and dashed) and the constant heating profile (dotted and dotted–dashed). The dashed and dotted–dashed curves show the case for β = 0. (bottom) EP flux as a function of equivalent depth for the constant heating profile with N = 0.009 s−1 and cloud-top height at 10 km (solid), N = 0.013 s−1 and cloud-top height at 10 km (dashed), and N = 0.009 s−1 and cloud-top height at 8 km (dotted–dashed).

  • View in gallery

    Flux response for the two-layer atmosphere and the one-layer atmosphere (dashed) with half-sine heating profile with top at 10 km. The values of N2 and interface altitude for the two-layer configurations are indicated in the figure. N = N1 = 0.0009 s−1 in the bottom layer.

  • View in gallery

    Flux response to the half-sine heating profile with cloud top at 10 km as function of equivalent depth for the homogeneous atmosphere (dashed), the two-layer atmosphere (dotted), and the three-layer atmosphere (solid). Response for (top),(middle) the portions of the heating profile between 5 and 10 km and between 0 and 5 km, respectively, and (bottom) the full profile.

  • View in gallery

    (top) The three-level model N profile (solid) and a smoothed version of the same profile (dashed). (bottom) The flux response spectra for the two N profiles shown at top.

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Solutions to the Vertical Structure Equation for Simple Models of the Tropical Troposphere

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  • 1 NorthWest Research Associates, Redmond, Washington
  • | 2 Colorado Research Associates Division, NWRA, Boulder, Colorado
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Abstract

Observation and modeling studies indicate that the wave flux from tropical heating sources that propagates into the lower stratosphere is sensitive to the buoyancy frequency profile N(z) in the troposphere. This sensitivity is explained by examining analytic solutions to the vertical structure equation for various simplified models of the tropical troposphere. An efficient method for obtaining expressions for these analytic solutions when N(z) is piecewise constant is presented. The solution is expressed in terms of reflection and transmission coefficients. It is found that the response to heating for Hough modes with small equivalent depth is quite sensitive to the shape of the heating profile, the magnitude of N(z) within the heating profile, and the internal wave reflections that result from the sharp change in N(z) at the tropopause. The location of the primary peak in the wave response, which occurs where the wavelength is twice the depth of the heating for a constant N(z) profile, is also sensitive to the occurrence of internal wave reflection.

Corresponding author address: David Ortland, NorthWest Research Associates, 4118 148th Ave. NE, Redmond, WA 98052. E-mail: ortland@nwra.com

Abstract

Observation and modeling studies indicate that the wave flux from tropical heating sources that propagates into the lower stratosphere is sensitive to the buoyancy frequency profile N(z) in the troposphere. This sensitivity is explained by examining analytic solutions to the vertical structure equation for various simplified models of the tropical troposphere. An efficient method for obtaining expressions for these analytic solutions when N(z) is piecewise constant is presented. The solution is expressed in terms of reflection and transmission coefficients. It is found that the response to heating for Hough modes with small equivalent depth is quite sensitive to the shape of the heating profile, the magnitude of N(z) within the heating profile, and the internal wave reflections that result from the sharp change in N(z) at the tropopause. The location of the primary peak in the wave response, which occurs where the wavelength is twice the depth of the heating for a constant N(z) profile, is also sensitive to the occurrence of internal wave reflection.

Corresponding author address: David Ortland, NorthWest Research Associates, 4118 148th Ave. NE, Redmond, WA 98052. E-mail: ortland@nwra.com

1. Introduction

Waves forced by tropical convection make a considerable contribution to the dynamics throughout the atmosphere. It is therefore important to accurately estimate the magnitude and structure of latent heating in models. Bergman and Salby (1994) calculated latent heating from high-resolution cloud imagery by relating brightness temperatures to rainfall rates. They then determined the atmospheric wave response to this heating using the Hough mode theory developed in Salby and Garcia (1987). Significant progress has been made in modeling the waves generated by tropical heating through convective parameterizations, but Ricciardulli and Garcia (2000) showed that these parameterizations must be constrained by heating estimates derived from observations. New rainfall rate data and more realistic assumptions about heating profile structure will lead to improvements in these estimates of the tropical heating spectrum.

We have recently performed a similar study of the tropical wave spectrum, presented by Ortland et al. (2011) and Ryu et al. (2011), using a nonlinear time-dependent primitive equation model with zonal mean background winds and temperatures derived from the National Centers for Environmental Prediction (NCEP) reanalysis data (Kalnay et al. 1996) and forced with latent heating estimates derived from the Tropical Rainfall Measuring Mission (TRMM) rainfall rate and cloud-top data products. A question that arises when deriving latent heating profiles from two-dimensional data such as cloud imagery or rainfall rate observations is how to shape the heating profile. Ortland et al. (2011) showed that the wave-flux spectrum was sensitive to the shape and extent of the heating profile. Ryu et al. (2011) compared model results to waves seen in High Resolution Dynamics Limb Sounder (HIRDLS) satellite data. They found that the model conforms well to observations when the rainfall is further classified as convective or stratiform and different profile models for these rain types are used (e.g., Shige et al. 2004; Schumacher et al. 2004). It was also found by Ortland et al. (2011) that the wave-flux spectrum was sensitive to the background zonal mean temperatures used in the model.

To explain the sensitivity of the wave flux to the heating and background temperatures, it is convenient to think in terms of a Hough mode decomposition of the wave spectrum (e.g., Salby and Garcia 1987; Andrews et al. 1987; Ortland et al. 2011). According to this point of view, the wave response to heating is obtained by projecting the Fourier spectrum of the heating onto the horizontal structure Hough functions and then solving the vertical structure equation that results for each mode. Salby and Garcia (1987) defined the vertical projection of the wave mode as the amplitude of the vertical structure solution, which is constant above the heating. They studied the properties of the vertical projection as a function of the wavelength and found that for a constant buoyancy profile N(z) the vertical projection peaks for waves whose wavelength is close to twice the heating depth. This result was found to be independent of the heating profile shape.

However, for inertia–gravity wave modes, the horizontal projection of the heating is largest for modes whose vertical wavelength is much smaller than twice the heating depth. Hence the full wave-flux spectrum is sensitive to the vertical projection response away from the primary peak, as is found in the numerical solutions obtained by Ortland et al. (2011). The study presented here attempts to explain this sensitivity of the vertical projection to both the heating and buoyancy profiles by examining the solution to the vertical structure equation in some detail. Analytic solutions are obtained when N(z) is piecewise constant. This approach provides insight into the solutions through the concepts of wave reflection and transmission, similar to a study of solutions to the one-dimensional Schrödinger equation for piecewise constant potentials (e.g., Bohm 1951). Reframing the problem in terms of reflection and transmission coefficients greatly simplifies the solution process to the extent that, with a little practice, the result can be written down without doing any calculation. The result of this study shows that the wave flux will increase if N(z) is reduced within the altitude range of heating, and it can be greatly enhanced for certain waves because of internal reflections.

Section 2 derives the method of solving the vertical structure equation for piecewise constant N(z) in terms of reflection and transmission coefficients. This method is applied to obtain solutions for several simple models of the tropical troposphere and lower stratosphere in section 3. These solutions are used to derive the flux response to heating as a function of equivalent depth. This section describes the sensitivity of the flux response to the structure of the heating profiles and to the structure of N(z) in the troposphere. The final section presents a summary and discussion.

2. The vertical structure equation

The forced primitive equations linearized about a basic state at rest are solved using the method of separation of variables. Details may be found in Andrews et al. (1987), Salby and Garcia (1987), Ricciardulli and Garcia (2000), or Ortland et al. (2011). The solution expresses the velocity and geopotential fields in terms of a sum of products of horizontal and vertical structure functions. Each of these products, comprising a Hough mode, is indexed by the zonal wavenumber s, frequency ω, and a mode index n. The separation constant is related to the equivalent depth of the Hough mode. The horizontal structure functions for the geopotential are the Hough functions. The vertical structure functions for the velocity and geopotential fields of a Hough mode with equivalent depth h are all related to a vertical structure function Vh that satisfies the equation
e1
where J(z) is the vertical structure function for the component of heating in this mode, z = −H ln(p/p0) is log-pressure altitude with scale height H, N(z) is an altitude-dependent basic-state buoyancy frequency, and g is acceleration of gravity. The vertical structure function satisfies the lower boundary condition
e2
where T0 is the temperature at the surface and R is the gas constant of dry air.
The vertical Eliassen–Palm (EP) flux Fz of a single Hough mode with h is proportional to the invariant wave flux I(h, z) of the vertical structure equation given by
e3
The proportionality factor between Fz and I, which may be found in Ortland et al. (2011), depends only on zonal wavenumber and frequency of the Hough mode. Both I(h, z) and the vertical EP flux satisfy the conservation law
e4
at altitudes z above the heating. In this study, we shall fix a heating profile and examine how the value of I(h, zref) at some reference altitude zref above the heating varies with equivalent depth. The flux does not depend on altitude when z > zref, so the altitude dependence will be suppressed in this case. When viewed in this way as a function of equivalent depth alone, the wave flux I(h) ≡ I(h, zref) will be referred to as the flux response function. We shall study how the response functions depend on the shape of the buoyancy frequency and heating profiles with which they are associated.

a. Formulation

Solutions of the vertical structure equation shall be expressed in the form
e5
where G(z, z′) is the Green’s function. The Green’s function is the solution to Eq. (1) when the forcing is given by the Dirac delta function δ(zz′) in place of the Hough projection of heating. To obtain analytic solutions, we divide the atmosphere into homogeneous layers by assuming that the N(z) profile is stepwise constant:
e6
The altitudes zi mark the layer interfaces. In each layer, the Green’s function is sinusoidal with vertical wavenumber mi given by the dispersion relation
e7
The layer boundaries include z0 = 0 and zN = ∞. The Green’s function has the form
e8
throughout all layers that do not contain the forcing level z′. In fact, the solution to the vertical structure equation for any forcing also has the form Eq. (8) in those layers for which the forcing is zero. We denote the layer containing the forcing level z′ with index I. Within this layer the Green’s function also has the form Eq. (8), but we must use different coefficients denoted above z′ and denoted below z′.
The coefficients in Eq. (8) for each layer are determined by imposing two conditions at each layer boundary, two conditions at the forcing level, and one condition each at the top and bottom boundaries. For each layer interface zi, both G and ∂G/∂z are continuous:
e9
In the case when the heating layer index I = i + 1, this formula holds with replaced by . At the forcing level z′, G is continuous and ∂G/∂z has a unit jump:
e10
The unit jump in ∂G/∂z implies that ∂2G/∂z2 is the delta function. The conditions at the forcing level may be rearranged to be in the form
e11
where
e12
Equations (11) state that the upward propagating wave component in I above the source equals the sum of the wave component propagating upward from below the source and the wave component emanating from the source, as well as a similar relation for downward propagating wave components. Note that are functions of z′. Equations (9) and (11) can be used to express all other coefficients in terms of hence all coefficients are also implicitly functions of z′.
We adapt the convention that, for real values of mi, the upward propagating wave component is given by . The vertical wavenumber satisfies mi > 0 for waves with westward phase speed and mi < 0 for waves with eastward phase speed. If mi is imaginary then we assume that Im(mi) < 0. With these conventions, the top boundary condition in all cases is given by
e13
The bottom boundary condition (2) implies the relation
e14
where α = H−1[RT(0)/gh − ½] and Eq. (14) defines β. We shall refer to β as the surface reflection coefficient since it relates the amplitude of the upward wave in the bottom layer to the amplitude of the downward wave. (Note that if I = 1 then we must replace with ). For propagating waves (m1 is real), the surface reflection coefficient represents a phase shift between the downward propagating wave and its upward reflection from the surface, since |β| = 1 (because α is real). Except for very large h, the value of α is much larger than |m1|, so that to a good approximation β = −1. This means that the bottom boundary condition is essentially given by V(0) = 0.

b. Reflection and transmission coefficients

It has already been mentioned that all coefficients and can be expressed in terms of by using the interface and boundary conditions. In general, the process for obtaining these expressions can be quite tedious. However, there is a simple method for determining the wave coefficients that is derived by formulating the interface conditions in terms of the physical processes of wave reflection and transmission. For a fixed equivalent depth, consider a solution to the vertical structure equation for a set of adjacent homogeneous layers i through j with no wave source in the interior. There are two wave coefficients that define the solution within each layer and two conditions on these coefficients at the layer boundaries. Two conditions remain to be imposed. One way to set these conditions is to assume that an incident upward propagating wave enters the bottom of these layers and that there is no downward propagating wave incident at the top of these layers. The set of linear continuity conditions (9) imply that the solution in the top of the layer may be written as and the solution in the bottom layer may be written as . The factors and so defined are the reflection and transmission (RT) coefficients, respectively, for this composite set of layers. The superscript plus sign on the coefficients indicates that they are related to an upward incident wave, and the pair of indices in the subscript refer to the first and last layer in the composite set. These coefficients have complex values since they relate both the amplitude and phase of the waves emanating from the top and bottom to that of the incident wave at the bottom boundary. We may similarly define RT coefficients and relating the waves emanating from the top and bottom to a downward propagating wave incident on the top layer. The solution to the vertical structure equation will be derived in terms of the RT coefficients.

For the case of only two layers, the RT coefficients will be simply denoted and by using the single index of the bottom layer, which is the same as the index of the interface at zi. Note that for a single homogeneous layer, and . Thus, reflection and partial transmission is associated with the interfaces between the individual layers. The coefficients and are derived from Eq. (9) and the assumption by solving for the remaining wave coefficients in terms of as and . We obtain
e15
Similarly, and are derived from Eq. (9) and the assumption by expressing and :
e16
In the general case, if we specify arbitrary values for both and , then linearity implies that the coefficients for any two adjacent layers satisfy the relations
e17
These relations have a simple interpretation. For example, the first relation states that the downward wave within the bottom layer is the sum of the transmission of the downward wave within the top layer and the reflection of the upward wave within bottom layer.
The RT coefficients for a system of several layers can now be derived inductively, starting from the top two layers. Suppose that , , , and are known. Assume the two conditions and for the system of layers i through N. By the definition of RT coefficients, we then have , , and . Using Eq. (17), we obtain
e18
These relations between RT coefficients, called adding formulas, are analogous to the adding formulas used in multiple scattering radiative transfer theory by Lacis and Hansen (1974). Starting from the bottom, we may also derive an adding formula for as an easy generalization of Eq. (18). The index 0 appears because we may treat the opaque surface as an interface with reflection coefficient given by .
The adding formulas have a physical picture that is illustrated in Fig. 1. The system of layers i + 1 through N is represented in the figure by the two top layers. A wave incident from the bottom is partially transmitted and reflected from the interface created by adding the new bottom layer. An infinite number of reflections can occur in between the new interface and the set of layers above, as illustrated by the multiple reflections in the middle layer of the figure. The sum of these multiply reflected components is represented by a geometric series
e19
that appears in the denominators of Eq. (18). This series does converge since it can be shown that reflection coefficients must always have absolute values less than one. The adding formula for then states that the reflection of a wave incident from the bottom either is a single reflection from the lowest layer interface or results from transmission upward through that interface, at least one downward reflection within the upper layers (represented by ), and finally transmission down through the lowest interface into the bottom layer. Multiple reflections between the interfaces in the top composite layer and the lowest interface are represented by the factor (19). The interpretation of the second adding formula in Eq. (18) is similar.
Fig. 1.
Fig. 1.

Schematic derivation of the adding formulas.

Citation: Journal of the Atmospheric Sciences 68, 9; 10.1175/2011JAS3719.1

c. Solution of the vertical structure equation

Now that we know how to compute RT coefficients, the solution for the Green’s function can be completed. The solution for the wave coefficients for all layers will not be presented, since ultimately we are only interested in calculating the EP flux emanating from the topmost layer. The great advantage of using RT coefficients is that the solution for the topmost layer can be derived without solving the complete system.

Suppose that the RT coefficients , , and have been calculated, where the heating level at z′ is within layer I. The defining relations , (assuming ), and the conditions (11) at the forcing may be solved to obtain
e20
This equation expresses the upward wave coefficient immediately above the forcing in terms of defined by Eq. (12) for the waves emanating from the source, and is also easily interpreted in terms of the physical principles illustrated in Fig. 1. Equation (20) may also be viewed as an adding formula that can easily be derived from the same simple principles illustrated in Fig. 1. Note again that the denominator signals the presence of an infinite number of reflections between the top and bottom layers. The solution for the Green’s function wave coefficient in the top layer, when z′ is in layer I, is now finally given by
e21
The solution to the vertical structure equation will have the form in the topmost layer above the heating. An expression for P is obtained from Eq. (5) by integrating the product of the Green’s function and the heating profile. The integral over z′ must be expressed as a sum of integrals over each of the homogenous layer subintervals in which the heating profile resides, since the analytic expression for the Green’s function depends on I containing z′. All factors not containing z′ may be pulled out of the integral, leaving behind only within the integrals. By combining Eqs. (20) and (21), the expression for P is thus given by
e22
e23
and where the sum is over all layers in which heating occurs. Following Salby and Garcia (1987), we shall refer to P as the vertical projection of the heating profile. However, the reader should keep in mind that the vertical projection of Salby and Garcia pertains to the vertical structure of the horizontal velocity and geopotential fields, whereas the vertical projection given here is closely related to the vertical structure of the vertical velocity field. The relation between all vertical structure functions is given by Eq. (4) in Ortland et al. (2011).

3. Simple models of the tropical troposphere and lower stratosphere

In this section we will consider a simple model atmosphere consisting of three layers and the EP flux response as a function of equivalent depth. According to Eq. (3), I for a layered atmosphere is given in the top layer above the heating in terms of the vertical projection (22) as
e24
when the vertical wavenumber mN is real. If mN is imaginary then I = 0. The functional dependence on equivalent depth is obtained by using the relation (7) to convert all vertical wavenumbers that appear in the analytic expressions obtained in the previous section. For a single homogeneous layer it also makes sense to view the flux response as a function of the vertical wavenumber of the mode.

To understand how various physical parameters affect the shape of the EP flux response, we will successively build up from one to three layers, since the addition of each layer introduces new effects. The physical parameters that we will consider include the heating profile shape and depth, the values of buoyancy frequency in the heating region, the difference in buoyancy frequency at the layer interfaces, and the bottom boundary condition. The heating profile will be either a half-sine shaped profile from surface to cloud top or a constant heating profile from surface to cloud top, as considered by Salby and Garcia (1987), Bergman and Salby (1994), and Ricciardulli and Garcia (2000). There is some uncertainty in the shape and depth of convective heating profiles, so these issues are important for convective parameterization schemes and atmospheric modeling. It is also possible that understanding how the flux response depends on these parameters is important for investigating the effects of climate change, since it is conceivable that climate change may affect the structure of both the tropical temperature profile and the convective heating profiles.

The troposphere and lower stratosphere system with a cold tropopause layer can be modeled with three homogeneous layers for which the heating occurs only in the lower two layers. The solution for the vertical projection of the heating, obtained from Eqs. (20), (21), and (22), is written as a sum of two terms that represent the contributions from the portion of the heating that occurs in either layer 1 or layer 2:
e25
The composite RT coefficients that appear are given by the adding Eqs. (18)
e26
The one- and two-layer models may be viewed as special cases of Eq. (25). If the buoyancy frequency N2 in the second layer is increased until it equals the value in the first layer, then and , hence the composite RT coefficients become , and . The two-layer formula for the vertical projection is therefore given by
e27
where . The two-layer projection (27) also follows directly from Eq. (22). The RT coefficients and in Eq. (27) differ from the coefficients with the same symbols that appear in Eq. (26) because the vertical wavenumber values change as N2 is changed. Similarly, in Eq. (27) have different values than the heating terms that appear in Eq. (25). Hence, the specialization to the two-layer atmosphere brought about by increasing N2 1) changes the heating source term in the second layer, 2) alters the RT properties of the interface between the second and third layers, and 3) eliminates internal reflection from the interface between the first and second layers. All of these changes are reflected in these simple formulas for the vertical projection of the heating. Equation (27) has a simple interpretation along the lines of Fig. 1 in terms of reflection and transmission of the upward and downward waves from the heating source, with the denominator representing multiple reflections between the single layer interface and the surface.
Specializing further to the one-layer case by reducing the buoyancy frequency in the top layer until it equals the value in the lower layer, all reflection coefficients become equal to zero and all transmission coefficients equal to one, hence the vertical projection is given simply by
e28
The heating terms do not change from the two-layer case. Thus, all internal reflection has been eliminated and the only wave reflection occurs from the surface. Equation (28) simply states that the wave solution above the heating consists of the upward wave generated by the source plus the surface reflection of the downward wave generated by the source.

a. Homogeneous model

The homogeneous case is particularly useful for examining the dependence of the wave response on the value of the buoyancy frequency, on the effects of heating profile shape, and on surface reflection. We consider heating profiles that are given by either half-sine or constant-step profiles that are equal to zero above the cloud top at ztop. Below the cloud top, the forcing functions for the vertical structure equation are given, respectively, by
e29
where m0 = π/ztop. The factors of ez/2H that appear here as part of the definition for the vertical structure function J of the heating (see Ortland et al. 2011). These heating profiles have been normalized to have unit column heating rate, so differences in the vertical projection and flux response are due solely to profile shape.
After performing the integration, analytic expressions for the half-sine and constant heating source functions (23) are
e30
e31
The source functions are written in terms of the scale-invariant wavenumber . Notice that these functions, when expressed in terms of , do not depend on N, but they do still depend on the depth of the heating through a factor of m0 in the denominator and a factor multiplying the scale height. The factor of m0 in the denominator implies that the vertical projection amplitude will increase with the depth of the heating. The factor of m0 multiplying the scale height primarily affects the amplitude of the exponential terms. It is easy to show that scale dependence of the vertical projection follows from the vertical structure equation when it is written in terms of the nondimensional altitude coordinate . The oscillatory behavior of these source functions may be viewed as a consequence of constructive and destructive interference that results by combining the delta-function responses from each point within the heating profile.
The flux response is obtained from the vertical projection by squaring the vertical projection, multiplying by the vertical wavelength, and dividing by the equivalent depth. The equivalent depth is expressed in terms of as
e32
Hence, when the flux response is written in terms of there is a factor of m0/N2 out front. Thus, neglecting the effects of terms that involve the product m0H, the flux response for a fixed heating profile shape will increase inversely proportional to the heating depth and to the square of the buoyancy frequency. This is an important relationship between the temperature profile and the magnitude of the wave flux produced by heating, because a large change in the magnitude of the EP flux can result from a change in the buoyancy frequency within the wave source region.

The top panel of Fig. 2 shows the flux response I as a function of . The solid and dotted curves in the top panel show the flux response for the half-sine and constant-step heating profiles, respectively. As shown by Salby and Garcia (1987), both response functions have a primary peak near —that is, when the wavelength is twice the depth of the heating. The response for the constant-step profile has secondary peaks located near wavelengths ⅔, ⅖, etc. of the heating depth. The response for the half-sine profile has secondary peaks whose magnitudes are two orders of magnitude smaller than for the step profile. This is a special property of the half-sine profile for a homogeneous layer. Although the structure of the heating profile has a minor effect on the location and magnitude of the primary peak, as shown by Salby and Garcia (1987), we shall find that the secondary peaks in the flux response are quite sensitive to the structure of both the heating and buoyancy profiles. This sensitivity is important because Ortland et al. (2011) show that this portion of the response occurs at small equivalent depths where the Hough projections of heating onto the inertia–gravity waves have the largest amplitude, and where phase speeds are relevant for the QBO.

Fig. 2.
Fig. 2.

(top) EP flux for the half-sine heating profile (solid and dashed) and the constant heating profile (dotted and dotted–dashed). The dashed and dotted–dashed curves show the case for β = 0. (bottom) EP flux as a function of equivalent depth for the constant heating profile with N = 0.009 s−1 and cloud-top height at 10 km (solid), N = 0.013 s−1 and cloud-top height at 10 km (dashed), and N = 0.009 s−1 and cloud-top height at 8 km (dotted–dashed).

Citation: Journal of the Atmospheric Sciences 68, 9; 10.1175/2011JAS3719.1

The dashed and dotted–dashed curves in the top panel of Fig. 2 show the flux response to the half-sine and step profiles, respectively, for a surface with β = 0 that completely absorbs the wave. This is equivalent to the response for a homogenous layer that extends infinitely downward as well as upward. Surface reflection combines the upward and downward wave propagating from the source. The primary peak occurs because the upward and downward source functions are nearly in phase at . Thus, the vertical projection with a reflecting surface has twice the amplitude of the upward source function and the flux response with a reflecting surface is 4 times larger than the case with no reflection. The pattern of secondary peaks in the flux response will depend on the interference that occurs between the upward and downward source functions. The figure shows that for the step profile only constructive interference occurs everywhere between these two components, because the dotted curve always lies above the dotted–dashed curve. On the other hand, the figure shows that destructive interference mainly occurs beyond the primary peak for the half-sine profile because the solid curve lies mostly beneath the dashed curve.

The bottom panel of Fig. 2 illustrates the flux response for constant step heating for different values of N and ztop as a function of equivalent depth. Although it is natural to view the flux response as a function of for a single homogeneous layer, it is more convenient to view the flux response as a function of equivalent depth for more general N profiles. The flux response as a function of equivalent depth is the form used by Ortland et al. (2011) to construct the EP flux spectrum. The plot transformation is simply accomplished by changing the dependent variable via Eq. (32). The primary peak now appears as the local maximum with largest equivalent depth. Since the transformation (32) depends on m0 and N, the peaks in the flux response patterns become shifted relative to each other.

Comparing the two cases in Fig. 2 with cloud-top height at 10 km but different values of N, the peak response for N = 0.013 s−1 (dashed) is reduced by roughly a factor of 2 relative to the response for N = 0.009 s−1 (solid) because the flux response is proportional to N−2, as discussed above. Comparing the two cases with N = 0.009 s−1, the flux response with the cloud-top height at 8 km (dotted) is increased relative to the case with cloud-top height at 10 km (solid) because the flux response is proportional to m0. The increase is somewhat more than a factor of 1.25 because of terms with m0H.

b. Two-layer model

The two-layer model illustrates how the wave response is affected by the inclusion of a tropopause; it has been considered before by Horinouchi and Yoden (1996). The two-layer models considered here will have a larger buoyancy frequency above the layer interface. A model with two distinct layers introduces the new effect of internal reflection and reduced transmission through the layer interface. From now on only the half-sine heating profile will be considered. The heating for the two-layer model is completely contained in the bottom layer.

Figure 3 compares the flux response for the two-layer (solid) and one-layer (dashed) atmosphere with the same value of N within the heating region. The two-layer examples all have N1 = 0.009 s−1 in the lower layer. The heating profile has the cloud top at 10 km. The flux response in the two-layer models is larger than for the homogeneous model in the neighborhood of two or three distinct values of equivalent depth within the primary peak of the one-layer response. The location of these enhancements depends on the altitude of the layer interface, which are at 14 and 17 km in the examples shown. Note that we can no longer say that the primary response occurs for waves whose wavelength is twice the heating depth.

Fig. 3.
Fig. 3.

Flux response for the two-layer atmosphere and the one-layer atmosphere (dashed) with half-sine heating profile with top at 10 km. The values of N2 and interface altitude for the two-layer configurations are indicated in the figure. N = N1 = 0.0009 s−1 in the bottom layer.

Citation: Journal of the Atmospheric Sciences 68, 9; 10.1175/2011JAS3719.1

The flux increase in the two-layer model is due to multiple reflections between the surface and the layer interface, as will now be shown. Let the upward wave coefficient above the heating in the bottom layer of the two-layer model be denoted by and upward wave coefficient (the vertical projection) for the homogeneous model be denoted by . Since the two models have the same N in the heating layer, it follows from Eq. (20) applied to each model that these coefficients are related by
e33
Using the definition of the transmission coefficient, the upward wave coefficient in the two-layer model is thus related to a+ by
e34
The superfluous lower index is suppressed for the RT coefficients because there is only a single interface. The denominators that appear in these expressions represent the effect of multiple internal wave reflections, as discussed in the previous section. Now consider how this denominator will vary as functions of equivalent depth. The phase of R+, given by the argument of the exponential in Eq. (15), increases indefinitely as h goes to zero. We may assume to good approximation that the surface reflection coefficient remains fixed at β = −1. Thus, as equivalent depth varies, the amplitude of the denominator |1 − βR+| oscillates between the bounds
e35
with the lower bound achieved when is real and positive and the upper bound achieved when is real and negative. Since |T+| = 2m1/(m1 + m2), it follows from Eq. (34) that varies with equivalent depth between the bounds
e36
The equality holds for the discrete set of equivalent depths where the value of R+ is real and negative. The assumption that N2 > N1 implies that m2 > m1; hence, for the equivalent depths where the two-layer flux maximizes, the wave flux satisfies
e37
This result depends on the existence of surface reflection. If the surface reflection coefficient is set to β = 0, then . Thus, for all equivalent depths in this case, it follows from 4m1m2 < (m1 + m2)2 that
e38
According to this argument, a change in the altitude of the layer interface will cause a shift in the phase of the reflection coefficient. This explains why the equivalent depths where the flux enhancements occur depend on the altitude of the interface.
Figure 3 also shows that the magnitude of the peak fluxes increase and the minima between the peaks decrease when N2 is increased while holding the interface altitude fixed. The increase in the peak flux follows from Eq. (37) since m2 increases with N2. At the minima between the peaks, R+ is real and positive, and . Hence, at the minima,
e39
and the minimum value will decrease with an increase in N2.

c. Three-layer model

The three-layer model considered here closely approximates the buoyancy frequency profile determined from NCEP temperatures for May 2006. The values of the buoyancy frequency in each layer are N1 = 0.0134 s−1 below 5 km, N2 = 0.009 s−1 between 5 and 17 km, and N3 = 0.02 s−1 above 17 km. The three-layer model is compared here to one- and two-layer models. The two-layer model is obtained from the three-layer model by increasing N in the middle layer to 0.0134 s−1. The one-layer model has this value of N throughout. The main effect of the transition from two to three layers, or to realistic buoyancy profiles in general, is the enhancement of the secondary peaks in the flux response. This enhancement arises when the heating occurs across two layers. The regular pattern we have seen in the flux response for the one- and two-layer models also becomes more complex because the three-layer model involves multiple-layer RT coefficients. The single interface RT coefficients oscillate in a regular manner as a function of the equivalent depth, while the variability of the multiple-interface coefficients is irregular.

Since the formula for vertical projection of the three-layer model [Eq. (25)] has separate terms for heating in the bottom and middle layer, we will examine the flux response for each separate term and for the full vertical projection. Each separate term also represents the vertical projection for the cases when the heating is given only by the part of the profile that lies in either the bottom or the middle layer. A similar decomposition can be performed for the one- and two-layer models already discussed. The top panel of Fig. 4 displays the flux response for the top half of the half-sine heating profile only (contained in the middle layer), the center panel shows the flux response for the bottom half of the heating profile only (contained in the bottom layer), and the bottom panel shows the flux response for the full heating profile. The three-layer model is shown with a solid curve, the two-layer model with a dotted curve, and the one-layer model with a dashed curve.

Fig. 4.
Fig. 4.

Flux response to the half-sine heating profile with cloud top at 10 km as function of equivalent depth for the homogeneous atmosphere (dashed), the two-layer atmosphere (dotted), and the three-layer atmosphere (solid). Response for (top),(middle) the portions of the heating profile between 5 and 10 km and between 0 and 5 km, respectively, and (bottom) the full profile.

Citation: Journal of the Atmospheric Sciences 68, 9; 10.1175/2011JAS3719.1

For the one- and two-layer models the heating occurs in a layer with a single value of N, hence the flux response for the two-layer atmosphere in all three examples results from modification of the one-layer response by including the effects of internal reflection. The same holds true for the three-layer model only for the bottom half of the heating profile, which is shown in the middle panel. In this case both the two- and three-layer response functions closely follow the pattern of the one-layer response, with the addition of peaks and valleys produced by internal reflection as discussed above. On the other hand, the three-layer response for heating above 5 km has much larger amplitude relative to the other cases because the buoyancy frequency is reduced in the heating layer for the three-layer case. The reason for this increase, as discussed in the section on the homogenous model, is because the flux response to heating is proportional to N−2. Moreover, the decrease in amplitude in the secondary peaks for the one- and two-layer model no longer occurs. The complex internal reflection pattern that likely results because β has been replaced by in Eq. (25) actually gives the second and third peaks from the right larger amplitude than the primary peak.

The peak amplitude in the three-layer flux response, shown in the bottom panel for the full heating profile, is considerably larger than for the one- and two-layer flux responses because of the reduced buoyancy frequency in the middle layer. Notice that the amplitude of the secondary peaks drops several orders of magnitude, off the plot scale, for the one- and two-layer models. Since this drop does not occur in the response shown in the top two panels, it must be the result of destructive interference between the vertical projections of the top and bottom half of the heating profile. The near cancellation of the flux response in the secondary peaks due to destructive interference is a special property of the half-sine profile, as already mentioned, because it does not occur for the constant step heating profile. However, significant amplitude remains in the secondary peaks for the three-layer model because the heating in the middle and bottom layers now occur in regions with different N values, so the cancellation can no longer occur. As already noted above, the occurrence of enhanced secondary peaks in the flux response is important because they occur at small equivalent depths where the Hough projections of heating have large amplitude, and where phase speeds are relevant for the QBO.

The three panels in Fig. 4 provide another illustration of the effects of heating profile shape on the flux response. The bottom two panels essentially compare the flux response patterns for a heating profile with depths of 5 and 10 km. The top and bottom panels compare the response pattern for two heating profiles with depth of 10 km but different shape. The heating profile used for the top panel, where all the heating is confined above 5 km, may be viewed as a crude model of stratiform heating, while the heating profile used for the bottom panel is a model for convective heating. Comparing the three-layer model result from the top and bottom panels, we see that a stratiform heating shape enhances the response, relative to the response for convective profile shape, for waves with short wavelengths, a result also found by Ryu et al. (2011) using a numerical model. This holds true even though the total column heating used to produce the flux response in the top panel is half the value for the total column heating used for the bottom panel.

4. Summary and discussion

We have described an efficient method for deriving an analytic solution to the vertical structure equation for piecewise constant N(z). Although it is fairly straightforward to solve the continuity conditions (9) and (10) for the wave coefficients, either through algebraic manipulation or numerically, very little insight can be gained from the many possible formulas that can result. Our approach shows that the continuity conditions are more simply expressed in terms of RT coefficients. The vertical projection is easily obtained as the simple expression (22) in terms of these coefficients and the convolution (23) of the heating profile with the simple functions . The RT coefficients themselves are calculated from simple formulas and an iterative algorithm.

Use of the RT coefficients and the Green’s function method provides some insight into the physical processes behind the structure of the vertical projection and flux response as a function of equivalent depth. This point of view suggests that the wave solution is a superposition of the responses to impulses along the heating profile, each of which is subject to multiple reflections and transmissions between the layers and the surface. The expression (12) immediately shows that the response is proportional to the wavelength of the excited wave. This response dependence gives rise to the result that the vertical projection amplitude increases with the depth of the heating. The process of wave reflection from the surface is behind the result of Salby and Garcia (1987) that the primary peak in the vertical projection for a homogeneous atmosphere occurs for waves whose wavelength is close to twice the heating depth, because it is at this wavelength that the sum of the upward and downward source responses J+ + βJ always constructively superimpose above the source. Internal reflection from a single-layer interface modifies this constructive superposition through the term (1 − βR+)−1 so that several peak enhancements may occur. Internal reflection from multiple-layer interfaces introduces new terms of the form J+ + RJ, which further alter the response pattern.

The flux response spectra for stepwise constant N profiles can be good approximations to the exact solution obtained through numerical integration of the vertical structure equation for realistic N profiles. Figure 5 provides a demonstration. The solid curve in the top panel gives the N profile used for the three-layer model and the dashed curve is a smoothed version that is similar to the realistic tropical N profile shown in Fig. 1 of Ortland et al. (2011). The bottom panel shows the flux response spectra obtained by solving the vertical structure equation with the half-sine heating profile for each of the N profiles shown in the top panel. (The solid curve in Fig. 5 is the same flux response shown in the bottom panel of Fig. 4 with the solid curve.) These response spectra are very similar for larger vertical depths, while the dashed spectrum becomes a more smoothed version of the solid spectrum as equivalent depth decreases. This indicates that the two N profiles produce a similar amount of reflection for waves with larger vertical wavelength, whereas the smoother N profile produces less reflection as the wavelength decreases.

Fig. 5.
Fig. 5.

(top) The three-level model N profile (solid) and a smoothed version of the same profile (dashed). (bottom) The flux response spectra for the two N profiles shown at top.

Citation: Journal of the Atmospheric Sciences 68, 9; 10.1175/2011JAS3719.1

The presence of the vertical structure functions for both the horizontal and vertical velocities in the definition of the vertical EP flux gives rise to the presence of the equivalent depth in the denominator of the wave flux Eq. (3). As a result, the flux amplitude for a wave with given wavelength within the source layer will increase as the background buoyancy frequency is decreased. This gives rise to a fundamental dependence of the vertical EP flux entering the stratosphere on the structure of the background temperature in the tropical troposphere. A correlation between Kelvin wave amplitude and background temperature has been noted in HIRDLS measurements by Alexander and Ortland (2010).

Acknowledgments

This work was supported by the NASA Atmospheric Composition Aura Science Team Program under Contract NNH08CD37C, by the NASA Geospace Science Program under Contract NNH08CD34C, and by NSF Grant 0121564.

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