1. Introduction

Orography can force stationary waves by obstructing the atmospheric flow or by creating anomalies in the diabatic heating field. The physical obstruction of the low-level flow is referred to as mechanical forcing and the diabatic heating is termed thermal forcing. Mechanical forcing is strongly linked to the low-level atmospheric flow; if the surface flow changes, the mechanical forcing changes. Since thermal forcing will modify the horizontal flow, it has the potential to indirectly modify the mechanical forcing.

This study is intended as a step toward understanding the coupling between mechanical and thermal forcing. Several recent works have suggested that the interaction between mechanical and thermal forcing is of considerable importance in the Northern Hemisphere midlatitudes in both the winter and summer seasons. Chen and Trenberth (1988b) found that the net diabatic heating in the Northern Hemisphere winter significantly modifies the mechanical forcing and upper-level stationary wave patterns. In a simulation of the Northern Hemisphere summer, M. Ting (1994) found that only after accounting for mechanical forcing produced by the thermally induced response could a diagnostic linear model reproduce the observed stationary wave train arcing over the North Pacific. Similarly, Hoskins and Rodwell (1995) found that the interaction between mechanical and thermal forcing is essential for the realistic simulation of the low-level Asian monsoon winds. Cumulatively, these works suggest that the interaction between mechanical and thermal forcing is important for understanding both the low-level flow and the upper-level stationary wave response.

In order to understand the extent to which the interaction between these forcings is important and how the total response differs from the sum of the individual responses, the problem must be analyzed within a nonlinear framework. The atmosphere responds to the total forcing due to the orography and diabatic heating, not to each individual forcing separately as is assumed in standard linear theory. Determining the conditions when the atmospheric response is well approximated by the sum of the individual responses is one goal of this work.

Classical linear theory approximates the mechanical forcing as [u](∂h/∂x), the zonal-mean zonal wind times the zonal gradient in orography. This approximation neglects the eddy zonal wind interacting with the orography, as well as the meridional wind interacting with the orography. We refer to the portion of the mechanical forcing that the classical linear theory neglects as the nonlinear mechanical forcing. Attempts have been made to account for nonlinear mechanical forcing without solving the fully nonlinear equations. Chen and Trenberth (1988a) used a linear set of equations that retained the interaction between the wind and the orography while neglecting the nonlinear advection terms. M. Ting (1994) accounted for the interaction in a manner similar to that proposed by Saltzman and Irsch (1972). After finding the linear response to diabatic heating, she computed a correction to the mechanical forcing that would be generated by the thermally induced response interacting with the orography. In both of these works the assumption is made that nonlinear advection is negligible so that the linearly balanced flow is representative of the nonlinearly balanced flow.

While many mechanisms are involved in generating the diabatic heating anomalies used in the above cited works, here we are mainly interested in near-surface sensible heating in the extratropics. Our focus on low-level heating stems from the fact that this type of heating makes up a significant portion of the net diabatic heating and produces a response that can readily alter the mechanical forcing. Also, the modification of the low-level heating field over orography occurs on the same spatial scale as the orography. For large features such as those considered here, the scale of the heating is large enough to force planetary waves. Upper-level heating, which is dominated by condensational heating, generally occurs on smaller space scales and may be less effective in forcing stationary waves. Observations indicate that low-level thermal forcing often coincides with the presence of the large-scale orography. Chen and Trenberth (1988b) found that when the heating is distributed closer to the ground, the modification of the mechanical forcing by the thermally induced response is more significant.

Evidence suggesting that the interaction between surface heating and mechanical forcing may be important is obtained from Fig. 5b and from Ringler and Cook (1997, RC hereafter). Near the surface, the nonlinear response to mechanical forcing exhibits weak horizontal gradients in entropy (effective potential temperature) along the mountain’s equatorward slope. If a small amount of heating (cooling) is added to the system, it must be balanced by either the advection of low (high) entropy air into the region or by dissipation. Given that very weak gradients in entropy exist due to the nonlinear response to the mechanical forcing, large changes in the flow could result when diabatic sources are added to the system.

In this work we address several questions that will lead to a better understanding of the interaction between the responses to mechanical and thermal forcing. First, given geophysically realizable conditions (orographic height, winds, heating rates), is the interaction important? Is the nonlinear response to combined mechanical forcing and thermal forcing different from the sum of the individual linear responses? If so, in what ways is it different and why? To what extent is the observed seasonality of stationary waves over Northern Hemisphere orography the result of the interaction between the mechanical and thermal forcing? In RC, the portions of the parameter space for which linear theory is valid were determined when only mechanical forcing was present. Here, that analysis is extended to include thermal forcing.

In section 2 observations of the low-level atmospheric flow and diabatic heating in the vicinity of the Rockies, Tibetan Plateau, and Greenland are discussed. Section 3 contains a summary of the model (discussed more fully in RC), the assumptions made, and the limitations of our framework. Linear theory is used in section 4 to give a first-order explanation of how and why the interaction between mechanical and thermal forcing is important. Balanced nonlinear responses for idealized heating/mechanical forcing and cooling/mechanical forcing are presented in section 5. In section 6 we come back to the observations by comparing them to model simulations to better understand the observed summertime and wintertime responses. Conclusions and areas of future work are discussed in section 7.

2. Observations

Recent reanalysis projects [Goddard Earth Observing System (GEOS), Schubert et al. 1993; National Centers for Environmental Prediction (NCEP), Kalnay et al. 1996] provide an excellent opportunity for associating stationary waves with large-scale orography, atmospheric conditions, and forcing functions. Here we primarily use the GEOS-4D dataset for years 1985–92. Comparisons to other datasets such as the NCEP reanalysis and European Centre for Medium-Range Weather Forecasts data were made whenever possible to ensure consistency.

The observed low-level circulation is examined to characterize the stationary wave response to the orography of the Rockies, Tibetan Plateau, and Greenland. Diabatic heating fields are also examined to assist in the setting of the model’s thermal forcing function. Unfortunately, since diabatic heating rates are derived quantities of the reanalysis, they are susceptible to model bias and can be used only for guidance. We have taken into account direct, on-site measurements of surface heat budgets in setting diabatic heating rates, but these measurements are not of sufficient spatial extent or temporal length to accurately determine the climatological heating rates.

Figures 1a,b show the eddy streamfunction observed on the σ = 0.9 surface for summer [June–August (JJA)] and winter [December–February (DJF)], respectively, in the vicinity of the Rocky Mountains. The term, eddy, is defined here as the deviation from the zonal mean. The summer pattern consists of a low positioned between two subtropical highs. In the vicinity of the low, the GEOS data show a surface heat flux on the order of 100 W m⁻² that results in low-level heating on the order of 3 K day⁻¹. The surface low and the heat source both decay rapidly in the vertical with characteristic scales of approximately 2 km, suggesting that the response is primarily produced by the heating. At high latitudes (60°N) the structure of the response changes, with a surface high in phase with the orography. In contrast to the summer pattern, the winter (Fig. 1b) is characterized by a surface high located over the majority of the Rocky Mountain region. The GEOS data show low-level cooling on the order 1–2 K day⁻¹ at elevations above 1 km.

Figures 2a,b show the low-level eddy streamfunction in the vicinity of the Tibetan Plateau for the summer and winter, respectively. The summertime response is characterized by a surface (monsoon) low with a spatial scale much larger than the orography. Although large amounts of latent heating occur in this region, the Tibetan Plateau is relatively dry; the GEOS data show surface heat fluxes and low-level diabatic heating rates on the order of 80 W m⁻² and 3 K day⁻¹, respectively.

The wintertime pattern is dominated by a high situated poleward of the maximum orography. Along with the low to the south, the pattern resembles a north–south-oriented dipole, which is suggestive of a nonlinear mechanical response to orography (see RC). The GEOS data indicate near-surface cooling on the order of 1–2 K day⁻¹ at elevations above of 2 km.

An independent check of the GEOS diabatic heating rates in the vicinity of the Tibetan Plateau is obtained from station data gathered by the Chinese National Project on Meteorology of the Tibetan Plateau (I. Ting 1994). Analysis of the collected data reveals that the Tibetan Plateau heats the atmosphere at an average rate of 1.5 K day⁻¹ during the summer and cools it at an average rate of 1.2 K day⁻¹ during the winter. The vertical distribution of this heating generally has a maximum at the surface.

In contrast to the Rockies and the Tibetan Plateau, the summer and winter patterns near Greenland are similar (Figs. 3a,b); in both seasons, a surface high is located atop the ice sheet with a region of cutoff anticyclonic circulation. Low-level diabatic cooling occurs in both the summer and winter at a rate of ∼2 K day⁻¹ due to the presence of the ice sheet. Throughout the year, the zonal-mean flow is characterized by weak surface westerlies and weak meridional temperature gradients. The stationary response is as strong as the zonal-mean basic state with southerly (northerly) winds occurring along Greenland’s west (east) coast.

In summary, seasonal variations in the low-level stationary wave response and diabatic heating are observed over the Rockies and Tibetan Plateau, but not over Greenland. The purpose of this work is to determine and understand the essential dynamics responsible for producing these low-level circulations by considering the stationary response to mechanical and thermal forcing over orography.

3. Model description

A detailed description of the model and numerical methods is given in RC and only a brief summary is provided here.

A nonlinear, stationary wave model based on the quasigeostrophic equations is forced by orography and diabatic heating. Conservation of potential vorticity governs the interior flow, and the potential temperature equation serves as the upper and lower boundary condition. Orography enters the system at the lower boundary as a forcing in the potential temperature equation, while diabatic forcing appears in both the potential temperature and potential vorticity equation.

The lower boundary potential temperature equation can be written as

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where the advection and mechanical forcing are on the lhs. The rhs includes the effects of diabatic heating (Q), Ekman pumping ψ, N², f₀, g, and θ (α) and dissipatation (D). Standard quasigeostrophic notation is used with denoted streamfunction, Brunt–Väïsälä frequency, Coriolis parameter, gravity and potential temperature. (see RC for full derivation).

When orography is present but heating and dissipation are absent or negligible, a quantity equal to the sum of the potential temperature and the potential temperature anomaly created by the orography’s presence is conserved at the surface following the geostrophic flow. Multiplying (1) by f₀H_s/R, this quantity can be expressed as

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where Θ_e is the parcel’s potential temperature minus the reference potential temperature, Θ(z = 0). We refer to this quantity as the surface entropy and will use it to determine the effect of near-surface diabatic heating on the large-scale flow. [See Tung (1983) and RC’s Eq. (10) for further discussion regarding surface entropy.]

The model generates balanced nonlinear solutions in response to imposed forcing. For a given forcing, the nonlinear response is obtained by iterating until the error becomes negligible compared to the forcing. Large amplitude, nonlinear responses are obtained by beginning with small amplitude forcing and slowly increasing the magnitude of the forcing into the nonlinear regime.

The quasigeostrophic equations are an approximation to the primitive equations and have well-known strengths and weaknesses. In addition to the limitations discussed in RC, diabatic heating introduces several others. In the quasigeostrophic system, static stability is prescribed as a function of the vertical coordinate only. Since the static stability cannot vary in time or in the horizontal directions, the response does not alter the static stability. Thus, if the response to an imposed forcing produces potential temperature anomalies that would create an unstable temperature profile, the quasigeostrophic equations will be invalid. In the real atmosphere, this hydrostatically unstable condition would be restored to neutral stability by transient motion. Since this model neglects both transient motion and the modification of the static stability by the response, we check all of our results to ensure that the total response (basic state + eddy response) is statically stable. (The possibility of generating statically unstable responses exists in any stationary wave model but is generally not addressed.)

The structure of the imposed diabatic heating is Q(x, y) exp(−z/H_q) where H_q is a constant-scale height. The heating at the surface, Q(x, y), and the orography, h(x, y), are Gaussian in form. The shapes of the forcings are realistic enough to obtain a first-order understanding of the system but are not sufficiently detailed to reproduce the observations with exactness.

4. Limits of linear theory

In RC the concept of a critical mountain height is used to distinguish between linear and nonlinear regimes when only mechanical forcing is present. In that case, the linear forcing grows linearly with orographic height, O(h), while the error grows as O(h²) [see RC’s Eq. (13)]. The linear error is estimated by computing post facto the terms that linear theory neglects. The critical height is defined as some fraction, say one-third, of the orographic height at which the error becomes as large as the forcing.

The same concept can be applied when both mechanical and thermal forcing are present. The linear forcing at the surface, L, is defined as

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Here L(x, y) includes the effects of orography obstructing the atmospheric flow (first term on rhs) and forcing by diabatic heating. (For a full derivation of the linear and nonlinear quasigeostrophic equations, see RC.) The linear response to (3), ψ′, is the sum of the individual responses, ψ^′₀ and ψ^′_Q, where ψ^′₀ is the response to mechanical forcing and ψ^′_Q is the response to thermal forcing. An estimate of the error incurred by using the linearized equations can be obtained by computing the terms that are neglected in the linear approximation post facto as

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The linear approximation neglects the advection of perturbation potential temperature by perturbation winds and the interaction between the perturbation wind and orography. When both mechanical forcing and thermal forcing are present, various terms in the linear error estimate (4) grow as O(h²), O(h)O(Q), and O(Q²). If we think of the heating, Q, as being specified, then ψ^′_Q is fixed and terms in (4) are only a function of orography. First, define h(x, y) = H · h̃(x, y) where H is the maximum amplitude of the orography and h̃(x, y) is the shape of the orography normalized to have a maximum amplitude of 1 m. We can then define ψ^′₀ = Hψ̃^′₀ where ψ̃^′₀ is the response to mechanical forcing with H = 1 m. Substituting these expressions into (4) yields

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where

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The purpose of this analysis is to understand which neglected terms are most important and to isolate the interaction between mechanical and thermal forcing, so the magnitude of E is approximated as the sum of magnitudes of the terms on the rhs of (5) as

EH²aHbc(7)

Approximating the linear error as the residual of the linearized equations [Eq. (4) and approximating the magnitude of (4) as the sum of the magnitudes of the individual terms [Eq. (7)] are potential shortcomings of this analysis.

Figures 4a–c show h_crit with mechanical forcing alone (no thermal forcing), for mechanical forcing and a maximum heating is 1.5 K day⁻¹, and for mechanical forcing and a maximum heating is 4.5 K day⁻¹, respectively, for a range of basic state winds. The orography is Gaussian in shape with both the zonal and meridional half-widths equal to 1500 km. The heating distribution has the same horizontal shape as the orography at the surface and the location of maximum heating coincides with the maximum orography. The heating decays as exp(−z/H_q) where H_q = 2 km. The winds are meridionally uniform with the surface wind, u_s, ranging from 0.01 to 9.0 m s⁻¹ and the surface wind shear, du_s/dz, ranging from 0.0 to 36 (m s⁻¹)/H_s. The vertical profile of the wind is similar to that shown in Fig. 2 of RC; the winds increase nearly linearly from the surface to 1.2 times the atmospheric scale height, H_s, where a maximum is obtained with winds decaying to weak easterlies above.

According to Fig. 4a and as discussed in RC, the meridional temperature gradient is the dominant factor controlling h_crit when only mechanical forcing is present. Comparing Fig. 4a to Figs. 4b,c, it is apparent that the overall effect of diabatic heating is to reduce the critical height so that, for a given basic state, the presence of low-level heating causes the response to become nonlinear at a lower orographic height. What is responsible for this decrease in the critical height? Stated alternatively, what is responsible for the increase of nonlinearity with the addition of heating?

The differences between these figures are the result of changes in the thermal forcing. Since the contribution to the linear error by (6a) is solely dependent on the amplitude and structure of the mechanically forced response, (6a) cannot explain any of the differences in Fig. 4. At these heating rates, the error associated with the linear thermal response, (6c), is small compared to (6a) and (6b). Thus, the interaction between the mechanically forced response and the thermally forced response, (6b), is responsible for the decrease in the critical height.

Term (6b) is composed of the nonlinear advection of potential temperature, J(ψ^′_Q, ∂ψ̃^′₀/∂z) + J(ψ̃^′₀, ∂ψ^′_Q/∂z), and the effect of the thermal response interacting with the orography, J(ψ^′_Q, N²h̃/f₀). Throughout most of the (u_s, du_s/dz) parameter space, nonlinear temperature advection is more important than the indirect mechanical forcing (not shown). The result implies that accounting for the nonlinear temperature advection is as, or more, important than accounting for the thermally induced mechanical forcing.

The physical reasoning of why the nonlinear temperature advection is important is as follows. Throughout most of the (u_s, du_s/dz) parameter space and at these levels of mechanical and thermal forcing, the amplitudes ψ^′₀ and ψ^′_Q are the same order of magnitude (although ψ^′_Q is generally weaker then ψ^′₀). Here ψ^′_Q is not as sensitive to the low-level winds as would be expected from scaling analysis (Held and Ting 1991) due to the presence of Ekman pumping and linear damping. While the amplitudes of ψ^′_Q and ψ^′₀ are comparable, ∂ψ^′_Q/∂z > ∂ψ^′₀/∂z since ψ^′_Q changes on the scale of H_q (2 km), whereas ψ^′₀ changes on the scale of H_s (8.5 km). Physically, this means that low-level heating tends to produce larger potential temperature anomalies than mechanical forcing. This suggests the importance of the advection of the thermally forced potential temperature anomaly by the mechanically forced winds, J(ψ^′₀, ∂ψ^′_Q/∂z), for determining the critical height.

The above analysis is meant to shed light on why the linear equations “breakdown” with the classical lower boundary condition and low-level heating present. Lower boundary conditions other than the classical lower boundary condition have been proposed as improvements. For example, the wave-coupled boundary condition developed by Chen and Trenberth (1988a) improves on traditional linear forcing by including the effect of the response interacting with the orography. However, the effects of nonlinear advection of potential temperature are omitted in their formulation. The results discussed above highlight the importance of the nonlinear temperature advection at the lower boundary and suggest that any lower boundary condition that neglects nonlinear advection may not be significantly more accurate than the classical lower boundary condition.

The critical height is somewhat dependent on the vertical scale of the low-level heating. This is explored in the linear model by imposing the same vertically integrated heating but changing the maximum heating at the surface to 3 K day⁻¹ (H_q = 1 km) and 1 K day⁻¹ (H_q = 3 km) (not shown). Throughout most of the (u_s, du_s/dz) space, the critical height is smaller when the heating is distributed over 1 km than when it is distributed over 3 km. The differences are most noticeable at moderate to large shears [du_s/dz > (20 m s⁻¹)/H_s] with small surface winds. In that portion of the parameter space, the linear response to heating with H_q = 1 km is a factor of 2 stronger than when H_q = 3 km. This increase in amplitude with decreasing H_q is consistent with the scaling argument of Hoskins and Karoly (1981, their [Eq.(3.2)]. With small surface winds and large vertical wind shear, the vertical scale of the wind, H_u = u/u_z, is less than H_q, so meridional advection balances the heating and the response is proportional to the maximum heating, Q_max. In addition, the vertical scale of the forced response is the same as the scale of the thermal forcing. Therefore, the potential temperature response, which scales as ψ^′_q/H_q, is about a factor of 6 stronger when H_q = 1 km than when H_q = 3 km. This leads to all the terms in (6b) being larger when H_q = 1 km than when H_q = 3 km and, thus, a smaller critical height.

5. Nonlinear response

Figures 5a,b show the full streamfunction and surface entropy with total wind vectors, respectively, in response to mechanical forcing alone over a 3-km mountain. The shaded regions in Fig. 5 denote areas of rising and sinking motion. The basic-state wind is meridionally uniform with u_s = 3 m s⁻¹ and du_s/dz = (15 m s⁻¹)/H_s. The critical height analysis done in section 4 gives a critical mountain height of 3 km with this basic state (Fig. 4a). The nonlinear response to mechanical forcing alone is characterized by a surface high (low) located to the northwest (southeast) of the mountain center in the Northern Hemisphere. The strongest surface wind and strongest surface entropy gradient are located on the poleward mountain slope, whereas the equatorward slope is characterized by weak wind and weak surface entropy gradient.

Figures 5c,d show the same quantities as Figs. 5a,b except that thermal forcing has been added to the system. The thermal forcing is maximum at the surface at 1.5 K day⁻¹, has the same horizontal structure as the orography, and decays as exp(−z/H_q) with H_q = 3 km [the critical mountain height is 1.5 km (Fig. 4b)]. When surface heating is present in addition to mechanical forcing, the surface flow is characterized by a surface cyclone situated directly atop the orography. The low is strong enough to produce a region of cutoff circulation (Fig. 5c). Since the surface low is coincident with the region of maximum surface entropy (Fig. 5d), the surface low is a warm-core low and decays rapidly in the vertical direction. The effect of the heating is to block much of the incident flow from going over top the orography.

Figures 5e,f show the surface streamfunction and surface entropy with total winds, respectively, when both a 3-km mountain and thermal forcing of −1.5 K day⁻¹ are present. The effect of the cooling is to amplify the“mechanical only” response shown in Figs. 5a,b. The surface response is dominated by an anticyclone that acts to draw the warm, equatorward air poleward and force it over the orography.

How does the response forced by both mechanical and thermal forcing (Figs. 5c,d) evolve with increasing surface heating from the response to mechanical forcing alone (Figs. 5a,b)? A Lagrangian perspective of the flow along the lower boundary states the presence of heating at the surface forces parcels across isentropes toward higher entropy. In the Eulerian framework, rewriting (1) in terms of surface entropy, S, gives

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Neglecting the effects of Ekman pumping and dissipation (second and third terms on rhs), for advection to balance the heating, strong horizontal gradients of S must exist or be generated by the response. In the response to mechanical forcing alone (Fig. 5b), a relatively large meridional gradient of entropy is generated on the poleward slope of the mountain. When heating is added to the system, it is balanced on the mountain’s poleward slope by the equatorward advection of low-entropy air. This low-entropy (cold air) advection is associated partly with the meridional temperature gradient and partly with the adiabatic effect of flow moving up the mountain slope. The anticyclonic turning of this flow repositions the ridge to the west.

The same mechanism cannot operate on the equatorward slope of the mountain because the entropy gradient in the mechanical-only solution is weaker there (Fig. 5b). The only source of low-entropy air is located on the poleward side of the orography, but the system cannot tap into that air because after the flow crosses the mountain top the winds become downslope, which produces adiabatic warming and negates the cold air advection. Dissipation in the form of Ekman pumping and linear dissipation balances the heating on the equatorward slope. Ekman pumping produces rising motion to partially balance the heating (thus, quasigeostrophic dynamics imply that motion must be cyclonic). Linear damping also helps balance the heating since the eddy potential temperature is positive. The two forms of dissipation constructively add to balance the forcing on the equatorward slope. The presence of a warm-core low on the equatorward slope is a signature of this balance.

The surface entropy balance is much different when low-level diabatic cooling is present (Figs. 5e,f) than when diabatic heating is present (Figs. 5c,d). When both mechanical forcing and diabatic cooling are present, the cooling occurring on the west slope is balanced by the advection of higher potential temperature air, which is located equatorward of the mountain center. This cross-isentropic flow is most evident on the southwest mountain slope in Fig. 5f. The response produces a warm eddy potential temperature off the west slope (2.5 K) and a cold eddy temperature off the east slope (−8.0 K), allowing the zonal advection of potential temperature to partially balance diabatic cooling on the west slope. Comparing Figs. 5e and 5a shows that cooling tends to strengthen, broaden, and shift eastward the surface high. The anticyclonic circulation associated with the ridge allows Ekman pumping to partially balance the cooling, as well. On the east slope, the cooling is partially balanced by both linear damping and downslope winds.

The nonlinear response to mechanical forcing alone tends to produce closed isentropes of locally high entropy air along the equatorward mountain slope. In addition, heating forces air toward higher levels of entropy. The combination of the local maximum of entropy produced by mechanical forcing and the forcing of air up the entropy gradient by the diabatic heating tends to enlarge the region of closed isentropes. Since a potential temperature anomaly at the surface can be viewed as a potential vorticity anomaly, the positive potential temperature anomaly induces cyclonic circulation, which leads to the cutoff flow at the surface. Cooling, on the other hand, moves air parcels down the gradient of entropy. Cooling acts to reduce, or even eliminate, the local maximum in entropy, which is created by mechanical forcing alone.

Comparing Figs. 5a,b to Figs. 5c,d and Figs. 5e,f it is apparent that modest amounts of thermal forcing can substantially alter the surface response. While the thermal forcing leads to large changes in the surface flow, Fig. 5 does not indicate how or why the thermal forcing is important. The thermal forcing may be important in two ways. First, the direct (linear) effect of the thermal forcing produces a stationary response that will alter the total response. This linear response will have opposite signs in Figs. 5c,d and Figs. 5e,f and could possibly explain the differences. Second, the indirect (nonlinear) effect of the thermal forcing interacting with the mechanical forcing could be the mechanism producing the changes in the surface response.

In order to determine how the thermal forcing is important, Figs. 6a,b show the linear response in terms of eddy streamfunction to the 3-km mountain and to the +1.5 K day⁻¹ thermal forcing, respectively. (For the complete derivation and discussion of the linearized equations see RC.) These are typical linear responses to extratropical orography and surface heating. In the case of mechanical forcing (Fig. 6a), the surface high is shifted slightly west due the effects of Ekman pumping. In the case of thermal forcing, an anticyclone and cyclone straddle the region of heating in order to advect cold air equatorward and balance the heating.

Figures 6c and 6d show the linear and nonlinear response to combined mechanical and positive thermal forcing. Figure 6c is the sum of the linear responses to mechanical forcing and thermal forcing, while Fig. 6d is the same as Fig. 5c with the zonal-mean streamfunction removed. Figures 6e,f are the same as Figs. 6c,d except the thermal forcing is −1.5 K day⁻¹. If the system is linear, then the total linear responses (Figs. 6c,e) will be identical to the nonlinear response (Figs. 6d,f). The nonlinear response to combined mechanical forcing and diabatic heating differs significantly from the linear prediction; the linear response produces an anticyclone that is too strong and mispositioned. The nonlinear response to combined mechanical forcing and diabatic cooling differs somewhat from the linear prediction, but not as dramatically as the previous comparison. The linear response has the correct amplitude and nearly the correct phase but does not capture the equatorward bias of wave propagation.

Figures 7a,b show the nonlinear, upper-level, eddy streamfunction response and horizontal wave-flux vectors (Plumb 1985) for mechanical forcing alone and for heating alone, respectively. The stationary wave response to heating alone is about an order of magnitude smaller than the response to mechanical forcing alone. The response at this heating rate is quite linear, so the response to heating alone or cooling alone is the same except for a sign change.

Figures 7c and 7d show the nonlinear, upper-level stationary wave response to combined mechanical forcing and heating and combined mechanical forcing and cooling, respectively. It is clear that these responses are not the sum of the individual responses shown in Figs. 7a,b. Heating (Fig. 7c) decreases the amplitude of the stationary wave response and splits the single wave train into two wave trains. Cooling has the opposite effect; it amplifies the stationary wave response and produces a wave train that propagates more zonally.

The differences between Figs. 7a and 7c and between Figs. 7a and 7d can be traced to differences in the mechanical forcing. The presence of heating alters the flow at the surface, and therefore, the structure and amplitude of the mechanical forcing. The structure of the forced vertical velocity is indicated by the shading in Figs. 5a,c, and e. When orography alone is present (Fig. 5a), the mechanically forced vertical motion resembles a dipole oriented northwest to southeast. The addition of heating (Fig. 5c) tilts the dipole toward a more north–south orientation and creates a secondary region of ascent on the mountain’s west slope. The addition of cooling (Fig. 5e) produces rising and sinking motion with strong east–west structure

Figures 8a–c show the power spectrum of the mechanical forcing, v · ∇h, when forced by orography alone, orography combined with diabatic heating, and orography combined with diabatic cooling, respectively. Also shown on each figure is the stationary wavenumber, K_s, for this basic-state flow. For a vertical wind profile very similar to the one used in these computations (RC, Fig. 2), Held et al. (1985) found the equivalent barotropic height to be z_e = 0.68H_s. The wavenumber of the external mode is then defined as K_s = [β/u(z_e)]^1/2. The wavenumber of the external mode is approximately 1.15 × 10⁻⁶ m⁻¹ (or nondimensional wavenumber 3.25 with a domain length of 20 000 km) and is shown as the dashed semicircle in Figs. 8a–c. Since the external mode is the resonant mode, the amplitude of the response should be correlated to the amplitude of the forcing in the vicinity of the stationary wavenumber rather than the overall amplitude of the forcing. An analysis of the upper-level responses shown in Figs. 7a–d indicates that the distances between ridges and troughs are approximately 3400 km in all cases, which is equal to nondimensional wavenumber 3.

Figure 8a shows a single center of forcing at (k, l) = (+2, −2), which happens to be close to the resonant wavenumber 3. The initial direction of propagation of the forced external mode can be found by drawing a vector from the origin of the figure to the location along the dashed line where the forcing is a maximum. The power spectrum shown in Fig. 8a indicates that the mechanical forcing will generate a wave that propagates southeast. This agrees with the stationary wave pattern shown in Fig. 7a. The power spectrum of the upper-level eddy streamfunction (not shown) also displays a maximum at (+2, 2), which is consistent with both the forcing and the wavenumber of the external mode.

By comparing Figs. 8a and 8b, it is readily apparent that the effect of heating is to weaken the pattern and to produce a second maxima at (+5, +3). Analysis of the power spectrum of the upper-level eddy streamfunction (not shown) shows two centers of propagation, one at (+2, −2) representing the equatorward propagating wave train and one at (+3, +2) representing the poleward propagating wave train. By comparison of the power spectra in Fig. 8a to 8b, it is determined that the heating reduces the power of the forcing by about 50%. The amplitude of the response when heating is included (Fig. 7c) is reduced by 25% when measured by eddy streamfunction amplitude and 50% when measured by wave-flux vector magnitudes as compared with the orography-only case (Fig. 7a). (It is best to compare the power of the mechanically forced vertical velocity to the magnitude of the wave-flux vectors since both are quadratic quantities.)

By comparing Figs. 8a and 8c, it is seen that the effect of the cooling is to amplify the mechanical. The power of the forcing has increased by 100% in Fig. 8c over Fig. 8a. The amplitude of the response has increased by 60% using eddy streamfunction and 160% when using wave-flux vector magnitudes. Comparison of Fig. 7a to Fig 7d indicates that the cooling combined with the mechanical forcing produces a more zonally oriented wave train.

Changes in the far-field responses can be explained in terms of the modification of mechanical forcing. Figures 8a–c show that the changes in the mechanical forcing produced by the diabatic sources are largely responsible for the changes in the far-field stationary wave response. Given how complicated the flow is in the vicinity of the forcing, it is somewhat surprising that linear stationary wave theory is as useful as it is in explaining the far-field response. We have checked these conclusions by taking the mechanical forcing produced in each of the three cases and finding the linear response to that forcing. The respective linear responses show a stationary wave train of the same amplitude and structure as shown in Figs. 7a, c, and d. Therefore, as far as the far-field response is concerned, the diabatic source is important for how it alters the mechanical forcing. The direct diabatic forcing and the asymmetry of the interior winds that it produces are of secondary importance.

6. Understanding the observed seasonality of orographically forced stationary waves

The preceding analysis demonstrates that the realistic low-level heating or cooling rates that occur in the vicinity of large-scale orography have the potential for causing profound changes in the stationary response. In this section we apply a nonlinear model to understand the observed seasonality of stationary waves over the Rockies, Tibetan Plateau, and Greenland.

The GEOS 4D assimilated dataset (Schubert et al. 1993) is used for guidance in selecting wind profiles and diabatic heating rates. Climatological diabatic heating rates are an analysis product that has only recently become available. The amplitude and vertical structure of the heating rates will certainly be dependent upon the analysis algorithm used in the assimilation process. Because of this uncertainty, we have chosen to use the GEOS diabatic heating rates for guidance only, and to specify heating distributions with simple shapes well within the range of heating rates that are geophysically realizable.

The summertime and wintertime basic-state wind profiles used for both the Rockies and Tibetan Plateau simulations are shown in Figs. 9a and 9b, respectively. The summertime basic state has a jet of 19 m s⁻¹ located at 45° lat (45° corresponds to y = 0 in Fig. 9). Surface westerlies attain maximum strength of 3 m s⁻¹ at 45° lat and surface easterlies have a maximum of 1.5 m s⁻¹ at what would be 10° lat. Although the easterlies are relatively weak, equatorward of 30° lat they extend through the depth of the troposphere and will likely inhibit wave propagation toward the equator. The wintertime basic state has a jet strength of 40 m s⁻¹ located at about 35° lat. The surface westerlies have a maximum value of 4 m s⁻¹ at 45° lat and the surface easterlies attain a maximum of 3.5 m s⁻¹ at 15° lat. The basic-state winds are tapered to zero in the meridional direction to meet the periodic boundary conditions necessary for an accurate spectral representation. While these basic states are not exact reproductions of the zonal-mean winds for the summer and winter, they are representative of seasonality in Northern Hemisphere winds and serve as a starting point for understanding the summer and winter stationary wave responses.

7. Conclusions

The observed seasonality of stationary waves cannot be explained without the inclusion of diabatic effects. The reanalysis products (GEOS and NCEP) indicate that low-level heating (cooling) occurs above the Rockies and Tibet in the summer (winter). Using a simple nonlinear quasigeostrophic model that includes both mechanical and thermal forcing, we are able to qualitatively reproduce the seasonality of the low-level stationary response to the Rockies, Tibet, and the Greenland ice sheet. While seasonal changes in the zonal-mean wind and shear would be expected to cause changes in the response, the observations are not consistent with changes solely in the basic-state winds.

This work has demonstrated that the nonlinear response to combined mechanical forcing and thermal forcing differs significantly from the sum of the individual linear responses. The seasonal variability of the low-level stationary response can be understood only by analyzing the combined effects of mechanical and thermal forcing. These findings are in agreement with the hypothesis of M. Ting, (1994) and the results of Hoskins and Rodwell (1995). Since the system described in this work is nonlinear, attempting to understand the total response by analyzing each part separately is not, in general, helpful and can often be misleading. The surface heating or surface cooling forces a response that alters the mechanical forcing, which, in turn, alters the total response.

When moderate heating rates are present and closed cyclonic circulations develop at the surface, dissipation becomes important within those cutoff regions where advection cannot balance the heating. While we have chosen values of dissipation that are representative of the atmosphere (linear damping = 0.1 day⁻¹, Ekman pumping = 150 m), the true atmosphere will surely be different. Also, the low-level thermal transients that are neglected here may act as thermal dissipation (Held and Ting 1990). Whatever its form, dissipation will be important whenever closed circulations exist. Adding heating to the mechanically forced system tends to diminish the amplitude of the far-field stationary wave response. In contrast, adding cooling to the system tends to amplify the far-field response. As discussed above, these changes can be traced to changes in the mechanical forcing.

The differences between the far-field responses with and without a diabatic heat source (Fig. 7) can be attributed to differences in the mechanical forcing (Fig. 8). Thus, the thermal forcing is most important for the manner in which it alters the surface flow and thereby alters the mechanical forcing. In the case of combined mechanical forcing and low-level heating, the majority of the flow is blocked and forced to circumvent the orography, thus reducing the mechanical forcing. In contrast, low-level cooling acts to amplify the mechanical forcing by advecting warm air poleward up and over the orography. It is important to note that although the far-field stationary wave train is nearly the linear response to this altered mechanical forcing, in order to obtain the correct mechanical forcing the full nonlinear equations are needed. (It should also be noted that we have analyzed a specific profile of diabatic heating that is located near the surface and decays with increasing height. Other diabatic heating profiles may yield more substantial far-field responses than those used here.)

Historically, approximations to the lower boundary condition have been made to attempt to simplify the system. One such approximation is the classical linear boundary condition where the mechanical forcing is approximated by the zonal-mean flow interacting with the orography (eddy wind–orography interaction and nonlinear potential temperature advection are neglected). Figure 5 clearly shows that the classical lower boundary condition is not an accurate approximation when both orography and heating are present. In fact, RC showed that the classical linear boundary condition is often inaccurate when just mechanical forcing is present. The analysis of the linear error (section 4) demonstrates the importance of nonlinear potential advection at the lower boundary. This suggests, but does not prove, that any approximation to the exact lower boundary condition that does not account for nonlinear potential temperature advection will have a limited range of validity.

The framework of these simulations is limited in that heating rates are specified. In the atmosphere surface sensible heat fluxes will be dependent upon the low-level flow. To analyze the fully coupled mechanical–diabatic heating system, the diabatic sources should be computed directly or parameterized. Since the heating rates specified are quite modest and are certainly realizable in the real atmosphere, we feel our findings will hold true after a study of the fully coupled system is complete. The analysis of the fully coupled system should bring to light dynamics that we have not captured in this work.

We did investigate a fully coupled mechanically and thermally forced system by adding an equation for water vapor to predict latent heating. We were able to obtain balanced nonlinear solutions that predicted realistic precipitation fields. Our results suggested that the coupling was weak because the precipitation occurred over a limited spatial extent and, therefore, did not alter the large-scale flow significantly. While this result is interesting, we are concerned about its robustness given the simplicity of the water vapor model and the lack of any convective modeling.

This work has implications not only for climatological stationary wave patterns, but also for interannual variability. Many mechanisms, such as soil moisture, vegetation type, snow cover, and snowmelt, can alter the year-to-year distribution of low-level heating. Since modest amounts of heating or cooling (±2 K day⁻¹) can produce dramatic changes in both the near-field and far-field response, variability at the surface–atmosphere boundary could be responsible for a portion of the interannual atmospheric variability both at the surface and in the atmosphere’s upper levels.

This modeling framework has provided a means of answering some fundamental questions concerning the role of nonlinearity in large-scale atmospheric dynamics. The approximations made in the quasigeostrophic equations impose limitations on this model’s use. For example, it will not aid in the study of systems in the deep Tropics or when very large heating rates are present. While a powerful framework would be a nonlinear stationary wave model based on the primitive equations, problems with obtaining convergent solutions have limited its utility. One approach may be to use a transient, dry primitive equation model possibly coupled to a planetary boundary layer model to further understand the interaction between the earth’s large-scale orography and low-level thermal forcing.