Internal Gravity Waves in a Saturated Moist Neutral Atmosphere

David J. Muraki Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada

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Richard Rotunno National Center for Atmospheric Research, Boulder, Colorado

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Abstract

This work is motivated by an unusual feature associated with the start-up of a moist nearly neutral atmospheric flow over a mountain ridge that was previously observed in a full-physics numerical model. In that study, the upstream propagation of a wave of subsidence precluded the establishment of upward-displaced and saturated flow that might be expected upstream of the topography. This phenomenon was hypothesized to be a consequence of the peculiar property of saturated moist neutral flow: an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. In the present study, this hypothesis is confirmed within numerical solutions to a reduced system of equations that incorporates the saturated-atmosphere property in a particularly simple manner. The relatively uncomplicated nature of these solutions motivates the numerical solution of a further simplified initial-value problem for both nonhydrostatic and hydrostatic flow. Exact analytic solutions are developed for the latter hydrostatic case, which explains the upstream-propagating wave of subsidence as a shock phenomenon.

Corresponding author address: David J. Muraki, Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada. E-mail: muraki@math.sfu.ca

Abstract

This work is motivated by an unusual feature associated with the start-up of a moist nearly neutral atmospheric flow over a mountain ridge that was previously observed in a full-physics numerical model. In that study, the upstream propagation of a wave of subsidence precluded the establishment of upward-displaced and saturated flow that might be expected upstream of the topography. This phenomenon was hypothesized to be a consequence of the peculiar property of saturated moist neutral flow: an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. In the present study, this hypothesis is confirmed within numerical solutions to a reduced system of equations that incorporates the saturated-atmosphere property in a particularly simple manner. The relatively uncomplicated nature of these solutions motivates the numerical solution of a further simplified initial-value problem for both nonhydrostatic and hydrostatic flow. Exact analytic solutions are developed for the latter hydrostatic case, which explains the upstream-propagating wave of subsidence as a shock phenomenon.

Corresponding author address: David J. Muraki, Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada. E-mail: muraki@math.sfu.ca

1. Introduction

This study was motivated by a curious feature of several numerical simulations of saturated moist neutral flow over an idealized two-dimensional mountain ridge (Miglietta and Rotunno 2005, hereafter MR). In that study standard procedures were followed to establish the steady solution over the ridge by introducing it into a uniform flow and then waiting for transients to decay. However, attempts to maintain a saturated flow upstream of the ridge were thwarted by an upstream-propagating wave of subsidence originating from the lee side of the ridge at the initial time (Fig. 1). Herein we seek to clarify the origin and nature of this disturbance.

Fig. 1.
Fig. 1.

Evolving flow at t = 5 h after an impulsive start of saturated nearly moist neutral flow over a mountain ridge showing (a) the vertical velocity w (contour interval = 0.1 m s−1), and (b) the cloud water mixing ratio qc where qc < 0.01 g kg−1 (white), 0.01 < qc < 0.1 g kg−1 (light gray), 0.1 < qc < 0.5 g kg−1 (medium gray), and qc > 0.5 g kg−1 (dark gray). (Adapted from MR.)

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

The introduction of a ridge into an incompressible airstream immediately forces upward motion on the windward side and downward motion on the leeward side of the ridge. From the discussion in Barcilon et al. (1979), it follows that in a moist neutral atmosphere that is just at saturation (i.e., with no liquid water), an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. MR reasoned qualitatively that the positive buoyancy anomaly occurring initially on the lee side of the ridge would subsequently produce a vorticity distribution that induces downward flow in the surrounding saturated air, causing desaturation there and thus an upstream and downstream propagation of the zone of desaturated air. MR took the vertical scale of the initial zone of desaturation d as the relevant scale for calculation of the hydrostatic wave speed UNd/π (<0) of the upstream-propagating disturbances, where U is the (positive) ambient wind speed and N is the dry Brunt–Väisälä frequency. Keller et al. (2012) observed that the sharp increase of N2(z) at the tropopause height zt in the MR sounding can act as a (somewhat leaky) waveguide and, if UNzt/π is less than zero, then upstream-propagating modes bearing a resemblance to those of MR occur even without the above-described effects of moisture on buoyancy. Keller et al. (2012) noted, however, that in the saturated moist neutral case, the upstream-propagating wave has an upstream speed slower than that expected based on UNzt/π. Another difference between the dry and moist upstream-propagating modes, which we will emphasize here, is that the leading edge is characterized by updraft in the dry case but by downdraft in the moist case (Keller et al. 2012, cf. their Figs. 11 and 12).

To develop an understanding of this contrast between the dry and moist flow dynamics, we present solutions here to a simplified system of equations that includes the basic nonlinearity inherent in the dependence of buoyancy on the vertical displacement of saturated air in a moist neutral environment, namely,
e1
where is the buoyancy and is the vertical displacement field. Using (1) in a Boussinesq two-dimensional nonhydrostatic numerical model, we compute the evolution in a vertically bounded atmosphere of the flow over a mountain ridge. We find a very good qualitative and quantitative correspondence of the upstream-propagating disturbance in the present idealized model with that found by MR using a full-physics mesoscale numerical model. As shown in Fig. 1, the upstream-propagation disturbance leaves behind it a trailing zone of subsided (unsaturated) air. To isolate and clarify the physics of this type of motion, we will present a two-dimensional nonhydrostatic numerical solution to a simplified initial-value problem that best illustrates what is in essence internal gravity wave propagation in a moist neutral atmosphere. Numerical solution of this same problem but with the hydrostatic approximation retains the salient features of the nonhydrostatic solution—in particular, the upstream-propagating wave of (desaturating) subsidence. Finally, we present analytical solutions to the simplified hydrostatic initial-value problem that explain the behavior of the hydrostatic numerical solutions, and by extension, important aspects of our simplified nonhydrostatic simulations and the full-physics solutions of MR.

2. Physical model

Restricting attention to Boussinesq and anelastic flow (Emanuel 1994, section 1.3) and neglecting nonlinear advection, the equation governing the two-dimensional field of motion in the horizontal–vertical plane is
e2
where is the -directed component of vorticity and is the environmental flow. Defining the streamfunction through , the requirement for incompressible flow is satisfied and the velocity can be inferred from the vorticity through solution of
e3
The solution of (3) gives the velocity field from which the vertical displacement can be computed through the definition
e4
A system is formed from (1)(4) in the four unknowns , and . Herein, we present solutions of (1)(4) on the domain , for given initial distributions of the prognostic variables and . Based on Keller et al. (2012), we identify the channel depth h [where ] with the height of the tropopause.
Basing a nondimensionalization on h and π/N as the respective length and time scales, the dimensionless equations (with tildes dropped on the new variables) are
e5
e6
e7
and
e8
on the domain −∞ < x < +∞, 0 ≤ z ≤ 1.

In this paper we consider numerical solutions to two different initial-value problems: the first is the evolution of a two-dimensional (in xz) topographic flow and the second is an idealization to the lateral dynamics (in x) of a single vertical mode. The first is the analog to MR's calculation, which can be thought of as the impulsive acceleration from the rest of an obstacle of height [H(x)] to the steady speed (−U). The initial condition ψ(x, z, t = 0) is obtained by solving (7) with η(x, z, t = 0) = 0 subject to terrain and lid conditions: ψ(x, z = 0) = −UH(x) and ψ(x, z = 1) = 0. The initial condition on displacement is δ(x, z, t = 0) = 0. The potential flow produced by this initial condition has upward motion on the windward side and downward motion on the leeward side of the obstacle producing through (6) a corresponding pattern of upward and downward displacement; a buoyancy distribution is produced through (8) and then vorticity evolves according to (5). Specifically, the leeside downward-displaced air produces a positive buoyancy anomaly whereas the windward side upward-displaced air has none; the vorticity produced by the positive buoyancy anomaly on the leeside induces downward motion on its lateral edges, and hence the region of downward unsaturated air spreads both upstream and downstream. This tendency for the disturbance in the leeside unsaturated air to propagate toward the windward side saturated air motivates the second initial-value problem.

Since the essential mathematical difficulty is the nonlinear dependence of buoyancy on displacement (1), we also consider a further idealization that incorporates key aspects of the dynamics and yet allows more in-depth analysis. In this second problem, we take U = 0 in (5) and (6), ψ(x, z = 0) = 0, and the initial condition
e9
where we will consider an initial distribution of displacement such that for x < 0 and for x > 0. Because of the neglect of nonlinear advection in the governing equations, the solutions maintain the form (where f = b, δ, η, ψ, or w)—the simplest linear vertical mode that satisfies the boundary conditions at z = 0 and 1.
This reduced system motivates further simplification with the hydrostatic approximation by the neglect of the nonhydrostatic, first term in (7). For the analysis of section 4, the hydrostatic approximation of (5) and (6) takes the explicit form
e10
e11
where the vorticity (7) has simplified to and the buoyancy (8) is rescaled to
e12
The initial conditions, implied by the coefficients of sinπz in (9), are and . In preparation for interpreting the numerical solutions to (10)(12) in the next section, it useful to consider the initial condition (9) first in the extremes of single-signed displacement: (i) and (ii) . In nonnegative case (i), it is obvious by inspection of (5)(8) or (10)(12) that the solution is steady. In nonpositive case (ii), (10)(12) revert to the classic nondispersive wave equation with the solution
e13
e14
(Whitham 1974, 5–6), indicating that the initial field of downward displacement divides into left- and right-propagating replicas of itself, each with one-half the original amplitude. Numerical solutions to the nonhydrostatic equations (5)(8) for case (ii) (not shown) indicate that as long as the horizontal scale of is large compared to the vertical scale, (13) and (14) describe the basic solution except for the smaller amplitude trailing dispersive waves modified slightly by the nonlinear dependence (8).

From the previous paragraph it is clear that for an initial condition for x < 0 (saturated), the initial tendency of the solution is to remain stationary, but if for x > 0 (unsaturated), then there is the initial tendency for left (and right) propagation of downward displacement. Therefore, at x = 0, there is a conflict between the (saturated) stationary tendency and the (unsaturated) left-propagating tendency. The resolution of this conflict and its consequences are the subject of the following.

3. Numerical solutions

For the numerical solution of (5)(8), the derivatives are discretized using second-order-accurate forms on a nonstaggered grid. The solution is advanced in time using the leapfrog method, and the Poisson equation (7) is solved using the National Center for Atmospheric Research library routine POIS. Although inviscid solutions of (5)(8) can be obtained (appendix A) for the present computations, a dissipative term νxxη is added to the right-hand side of (5), where and is the dimensional dissipation coefficient; for the present calculations ν = 0.03.

For the first initial-value problem described in the previous section, the model domain is −lx ≤ +l, 0 ≤ z ≤ 1, where l = 100 is sufficiently large to avoid boundary artifacts. Setting U = 0.45, the numerical solution to (5)(8) for w and δ at t = 50 is shown in Fig. 2 for −20 ≤ x ≤ +20. The qualitative correspondence of this solution with that of MR (Fig. 1) is apparent, as there is downward vertical motion at the leading edge of an upstream-propagating wave of desaturation. A good quantitative comparison results when we let N = 0.01 s−1 and h = 7000 m, since in that case, , , and is approximately the position of the leading edge of the wave of desaturation seen in Fig. 1. The averaged intrinsic propagation speed of this disturbance, , which is slower than −1, the (dimensionless) speed of a left-propagating unsaturated hydrostatic disturbance given by (13) and (14).

Fig. 2.
Fig. 2.

Numerical solution of (5)(8) for the evolving flow at t = 50 after an impulsive start of flow over a mountain ridge showing (a) vertical velocity w contours ±(0.001 25, 0.0025, 0.005, 0.01, 0.015, and 0.02) and (b) displacement δ intervals where δ < −0.005 (white), −0.005 < δ < 0 (lighter gray), and 0 < δ (darker gray), with solid contours (0.01, 0.05).

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

As described in the previous section (and in MR's section 5) the upstream-propagating wave of desaturation originates in the leeside region of downward parcel displacement. To focus on the pure wave-propagation problem, we consider the solution of (5)(8) with U = 0 for the initial condition
e15
as illustrated in the bottom curves of Figs. 3 and 4. Figure 3 shows the evolution of displacement and confirms our expectation that the unsaturated region expands into the initially saturated region (x < 0); consistently, Fig. 4 shows that at the leading edge. Figures 3 and 4 also show that there is wave of depression of lesser amplitude than the original that moves to the right at a speed c ≃ 1 as expected from the second terms on the right-hand sides of (13) and (14). We note that during the initial phases (t ≃ 0−4.5), the left-propagating disturbance is considerably slower and has a more complex structure compared with the right-propagating depression. And at later times (t ≃ 2.5–9), a zone of resaturated air develops from the approximate location of the initial zone of maximum downward displacement.
Fig. 3.
Fig. 3.

Numerical simulations of dissipative, nonhydrostatic displacement for times t = 0.0, 1.5, 2.5, 4.5, and 9.0. The region boundaries (vertical dividers) are reference values obtained from the inviscid, hydrostatic analysis of section 4. Unsaturated air appears following the left-propagating front.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Fig. 4.
Fig. 4.

Numerical simulations of dissipative, nonhydrostatic streamfunction for times and with region boundaries identical to the displacement plot of Fig. 3. The left-propagating front corresponds to a downdraft.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

A clearer picture of the above-described evolution emerges from the numerical solution of the hydrostatic system (10)(12). Figure 5 shows the formation of an unambiguous near discontinuity in at the leading edge of the left-propagating disturbance that can now be seen clearly as a wave of depression; this wave is associated with a similarly sharp increase in (Fig. 6), which corresponds to a localized spike of . Also clear from Fig. 5 is less dispersion of the right-propagating wave of depression, but only a single zone of resaturation. In short, the hydrostatic solution captures the salient features of the nonhydrostatic solution. (The oscillations seen at later times in Figs. 3 and 4 are nonhydrostatic waves between the leading edge of the left-propagating disturbance and the left edge of the resaturation zone; for further elaboration, see appendix A.)

Fig. 5.
Fig. 5.

Numerical simulations of dissipative, hydrostatic displacement for times t = 0.0, 1.5, 2.5, 4.5, and 9.0. The correspondence with the nondissipative solutions of section 4 is exact up to smoothing of shocks and rounding of derivative discontinuities at region boundaries (vertical dividers). The basic solution structure at t = 9.0 differs significantly from the nonhydrostatic case only in the dispersive oscillations found in the region identified as the reflection zone.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Fig. 6.
Fig. 6.

Numerical simulations of dissipative, hydrostatic streamfunction for times identical to the displacement plot of Fig. 5. The left-propagating front appears as a sharp downdraft.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Comparing the evolution of the left-propagating disturbance in the simplified initial-value problem (Figs. 36) with that of the solution for the impulsively started obstacle (Figs. 1 and 2), we note that in both there is a left-propagating wave of depression with a very sharp leading edge of subsidence that is followed by a trailing zone of desaturated air. An important datum is that the left-propagating disturbance in Figs. 36 propagates at a speed of c ≃ −1/2 (dimensionally −1/2 × Nh/π) during the earlier interval. As noted above this slower propagation speed is consistent with MR's solution (Fig. 1) and the present idealization thereof (Fig. 2). Since the left-propagating disturbance in Figs. 1 and 2 propagates away from the mountain ridge toward level terrain, the analogy between it and the simplified disturbance (Figs. 36) is expected to be good. On the other hand, near and in the lee of the mountain top, Figs. 1 and 2 show the continuous mountain forcing that has no analogy in our simplified initial-value problem; however, there is a leeside zone of resaturated air found above desaturated air in the steady state (Figs. 1b and 2b). This leeside resaturated feature is suggestive of the resaturated zone that appears in the idealized solutions (Figs. 3 and 5).

In the following section, analytical solutions are developed for the hydrostatic case that allow a clearer picture of both the mathematical and physical consequences of the nonlinear dependence of buoyancy on displacement (1).

4. Analysis of the hydrostatic case

Mathematically, the inviscid, hydrostatic equations (10)(12) are a system of hyperbolic conservation laws, and thus they are solvable by the method of characteristics (Lax 1973). The piecewise linear nature of the buoyancy dependence on displacement (12),
e16
has the simplifying feature that the partial differential equations (PDEs) (10)(12) are linear in xt regions distinguished solely by the sign of in the solution. Hence, for this analysis, the method of characteristics consists of the piecewise construction of linear solutions where the nonlinearity of the flux term manifests only in a free-boundary determination of each region. Details specific to hyperbolic systems with piecewise linear flux are little addressed in the literature (Holden and Risebro 2002; Correia et al. 2001; LeFloch and Mohammadian 2008); however, Frierson et al. (2004) have discussed very closely related systems in their fast relaxation limit for tropical fronts. Here we focus on explaining the solution structure arising from the particular initial conditions (15) in the context of the numerical results presented in section 3.

a. Summary of analytical results

The main results of the hydrostatic solution by characteristics are summarized in the space–time regime diagram shown in Fig. 7. The boundaries between saturated and unsaturated flow are defined by three dominant features: a left-propagating shock that initially emanates from x = 0 (thick solid); a right-propagating shock appearing from the event labeled Ps (thick solid); and a resaturation boundary (thick dashed). This establishes three distinct solution regions in xt space: an upstream saturated region (SA; shaded), a resaturated region (RE; shaded), and an unsaturated region (UN; unshaded) that is partitioned by thin dashed lines into zones labeled UN and RZ1–RZ4. All dashed boundaries coincide with derivative discontinuities.

Fig. 7.
Fig. 7.

Space–time regions for the nondissipative, hydrostatic solution from initial conditions (15). Boundaries between saturated (shaded) and unsaturated (unshaded) regions are shown as thick curves, with shocks solid and derivative discontinuities dashed. Thin dashed lines indicate partitions of unsaturated solutions by characteristics carrying derivative discontinuities. Unsaturated solutions within the reflection zones, RZ1–RZ4, involve characteristics reflected from shocks. The point labeled P is the source for the left-going characteristic line that is the asymptote of the left-propagating shock.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Figures 810 show the inviscid, hydrostatic solutions for and as derived by the method of characteristics for illustrative times t = 1.5, 2.5, and 4.5. Along with these displacement and streamfunction fields are labeled intervals identifying the space–time regions of Fig. 7 that are associated with features of the solution structure. The boundaries of these regions are shown by vertical dividers corresponding to shocks (thin solid) or derivative discontinuities (thin dashed). For comparison, these same intervals are indicated in the plots from the nonhydrostatic and hydrostatic computations of Figs. 36. The hydrostatic computations of Figs. 5 and 6 are a perfect matchup to the dissipative smoothing of the discontinuities.

Fig. 8.
Fig. 8.

Plots and regions of nondissipative, hydrostatic and at t = 1.5. The left-propagating shock at x = −t/2 and the derivative discontinuity at x = t are boundaries between the labeled SA, RZ1, and UN regions. Consistent with Fig. 7, a solid vertical line indicates a shock boundary and a dashed line indicates a derivative discontinuity. The negative slope of in RZ1 corresponds to rising motion and increasing .

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Fig. 9.
Fig. 9.

Plots and regions of nondissipative, hydrostatic and at t = 2.5 after the appearance of the resaturation region. The RE boundaries (thin dashed) occur where and correspond to derivative discontinuities in the solutions. Only a thin sliver of the downstream branch of the RZ1 zone remains between RE and UN.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Fig. 10.
Fig. 10.

Plots and regions of nondissipative, hydrostatic and at t = 4.5 after the upwind boundary of the resaturation region has become a right-propagating shock. The rightmost RE boundary and the leftmost UN edge are now virtually indistinguishable. The weak derivative discontinuity closely following the left-propagating shock is the boundary that separates upward motion in RZ1 from downward motion in RZ2.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Moreover, both of the key observed behaviors are established within this hydrostatic analysis. First is the initiation of a left-propagating shock that leaves behind a region of downward-displaced, unsaturated air upstream from the initial disturbance. Second is the later appearance (t ≥ 1.981 in our example) of a region of resaturated air that appears downstream of the initial disturbance. Highlights of the analysis follow in this section.

b. Leftward-propagating shock

Within unsaturated regions, where , there are two families of independent characteristic lines along which left-going signals propagate with constant x + t and right-going counterparts propagate with constant xt. The method of characteristics encodes the solution of the wave equation in terms of left-going and right-going Riemann invariants—that is, quantities that are constant along characteristics that are lines with constant x ± t. The initially unsaturated [ for x ≥ 0] conditions (15) give these invariants as
e17
where the inequalities are the restriction to characteristics launched from x ≥ 0 (subscripts denote the region of evaluation). These invariants then imply the nondispersive wave solutions (13) and (14).
Saturated regions, where , represent the degenerate situation where a double-zero wave speed corresponds to a characteristic line in the t-axis direction. Along such characteristic lines is constant, but varies linearly in t with a constant rate given by the vertical motion . Since the initial conditions (15) have no vertical motion,
e18
and both fields are constant along vertical characteristic lines of constant x ≤ 0. Note, however, for t > 0, the domains of influence of the unsaturated I invariant (17) and the saturated solution (18) overlap for −tx ≤ 0. This collision of characteristic trajectories (as shown at point a in Fig. B1) necessitates the formation of a shock discontinuity at (x, t) = (0, 0).
Weak solutions of conservation laws require that a Rankine–Hugoniot (R–H) condition be satisfied along xt curves of discontinuity (Evans 2010). For the usual (second derivative) dissipative regularization, the R–H conditions for difference quantities across a shock (denoted by Δ) specific to the system (10)(12) are
e19
where (R(τ), T(τ)) is a parameterized curve of discontinuity in the xt plane, and the prime denotes differentiation by τ (where τ might even be x or t). These two R–H equations with (16) are sufficient to determine both the shock propagation and the reflected signal.
Introducing a time-parameterized curve (R(t), t) for the shock, an expression for the shock speed R′(t) is constructed with the particular linear combination of the R–H conditions (19),
e20
that involves only the left-going invariant quantity I that is incident to the shock. Evaluating the difference quantities at x = R(t) gives
e21
and substituting the known solutions (17) and (18) gives the shock ordinary differential equation (ODE)
e22
The solution is verified by direct substitution to be the unexpectedly simple
e23
whenever the initial condition has odd symmetry, . The speed R′ = −1/2 indicates that the shock is left propagating but is slower than the characteristic wave speed of −1.
Also at the shock, a right-going Riemann invariant I+ is reflected whose value is obtained from the other linear combination of the R–H conditions:
e24
Evaluation at the shock x = R(t), anticipating ψ(R, t)|SA = 0 as above, gives
e25
but now with R′ = −1/2 known, this determines the right-going Riemann invariant
e26
where RZ1 denotes a reflection zone influenced by these shock-reflected characteristics. Since the reflected characteristic line that intersects the shock at (R(t), t) = (R, −2R) has the equation xt = R − (−2R) = 3R, an explicit formula for I+ is obtained,
e27
where odd symmetry of is again applied to obtain the above final form. This rightmost expression represents an inverted and attenuated signal (per the prefactor −⅓), and a Doppler reflection that is red shifted (per the factor of −⅓ in the argument) as expected from a comoving reflector. As the I Riemann invariants in the RZ1 region are just the continuations from the UN region, the solutions for the displacement field in the regions SA, RZ1, and UN can be written
e28
Similar formulas can obtained for , and these fields are plotted at t = 1.5 in Fig. 8 with the regions of (28) identified. The boundary between the SA and RZ1 regions is the shock at x = −t/2. The boundary between the RZ1 and UN is the x = t characteristic, which corresponds to a derivative discontinuity induced by the first reflected characteristic. These features, the shock (thick solid) and the derivative discontinuity (thin dashed), are the earliest that appear in the xt plane of Fig. 7.

Note that the shock ODE (22) that produces R(t) = −t/2 is valid only when the left-going unsaturated characteristic corresponds to the initial-value Riemann invariant (17). However, since the left-propagating shock is slower than the left-going wave speed, downstream events can affect its later motion. In the next step of this analysis, this is shown to happen via a resaturation that occurs in RZ1, and its effects on the left-propagating shock will lead to an upstream acceleration that begins at the event labeled P1 in Fig. 7. To this end, it is noted that the shock solution R(t) = −t/2 that follows from the odd-symmetric initial condition is a particularly fortuitous outcome that permits a complete geometrical analysis and the production of Fig. 7.

c. Resaturation

The solution profile in Fig. 8 illustrates that the right-going reflected signal from the shock leads to upward motion , and hence increasing displacement . This increase is sufficient to achieve eventual resaturation—that is, the appearance of a new region of positive within the unsaturated RZ1 region. This is labeled as the RE region in Fig. 7. The transition boundary (thick dashed) is obtained as the zero contour of from the RZ1 equation (28). The resaturation time T(x) is defined by the condition
e29
The earliest point at which resaturation occurs is the event Pr = (xr, tr), which requires both and . This event is situated in Fig. 7 using numerical values xr ≈ 1.107 and tr ≈ 1.981 specific to the initial condition (15).
Resaturated solutions for tT(x) are determined by applying continuity across the transition. Following the vertical characteristics, is thus independent of t and takes the RZ1 value at the resaturation boundary t = T(x):
e30
Curves of , along with the corresponding , can be computed by numerical root finding of (29) and are shown in Fig. 11. The displacement is obtained by t integration of (11):
e31
starting from (29) as the initial value.
Fig. 11.
Fig. 11.

Values at the resaturation boundary t = T(x) for (solid) and (dashed). The left edge of the plot is xs ≈ −0.090 where . For large values of x, decays exponentially.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

Although this construction produces continuous and across the transition boundary (x, T(x)), their derivatives are discontinuous. By the conservation law property, it is required to demonstrate that these discontinuities are consistent with derivative R–H conditions. Since the PDEs are piecewise linear, the derivative R–H conditions share the same form as (19) except that the Δ differencing involves x-derivative values. Consider then, differences at (x, T(x)) between the RZ1 and RE regions for the R–H conditions (19):
e32
e33
where (10)(12) are used to introduce t derivatives. As all of the total derivative quantities are zero by continuity, the constructed RE solutions (30) and (31) indeed satisfy both R–H conditions (though they are not determined using them!).

Figure 9 shows and at t = 2.5 > tr as representative solutions past resaturation. The locations with demarcate the edges of the newly formed RE region, and display discontinuous derivatives for both and . By this time, the right-going edge of the resaturation zone has nearly caught up to the x = t characteristic, so that only a sliver of the rightmost branch of the RZ1 zone remains. This (apparent) motion of a region boundary that is faster than the characteristic speeds (1/|T′| > 1) is a phenomenon that was also observed in the system of Frierson et al. (2004)—this quite unusual behavior is briefly elaborated upon in appendix B.

d. Reflection zone RZ1

The resaturation solutions (30) and (31) require the validity of the RZ1 solution (28) to determine the boundary contour. In Fig. 7, however, the resaturation boundary curve is seen to terminate abruptly at the event labeled Ps. The boundary ends because the equation (28) loses validity for xxs because of the shadowing of left-going characteristics (carrying the I Riemann invariant) by the resaturation boundary itself. The critical event Ps = (xs, ts) is defined by the point on the resaturation boundary whose tangent line, as the last unblocked characteristic, has T′(xs) = −1. Mathematically, this translates to and [in RZ1, but also in RE by (33)]. Substitution of the RZ1 form for from (28) into the second condition above gives
e34
The Ps event is situated in Fig. 7 using numerical values xs ≈ −0.090 and ts ≈ 2.554 specific to the initial condition (15). Also shown (thin dashed) is the portion of the last unblocked characteristic line [, by exact inversion of (34)] that represents a boundary edge of the zone RZ1.

e. Rightward-propagating shock

The event Ps marks a location precisely at the meeting of saturated and unsaturated air, and where the vertical motion is zero on the resaturated side [by the tangency condition and (32)]. This essentially mirrors the initial situation at x = 0, except that the saturated air is now downstream, and hence there is initiated a right-propagating shock. Parameterizing this shock curve as (S(t), t), the analogous R–H condition of (21) for the wave speed S′(t) is
e35
which involves only known solution quantities (30) and (31) and in particular, the incident I+ Riemann invariant for the RZ1 reflection (27). The R–H condition is solved numerically as an ODE for S(t) with the initial launch point S(ts) = xs. [Note that the starting value of S′(ts) involves a subtle limit evaluation with two invocations of the l'Hôpital rule.] This produces the shock curve (thick solid) in Fig. 7 emanating from the event Ps—but only up to the event P2, where the RZ1 equation (27) for the I+ Riemann invariant no longer applies.

f. Reflection zone RZ2

The solution in RZ2 requires left-going Riemann invariants I that are reflected from the right-propagating shock. The Riemann invariant value at the right-propagating shock satisfies an analogous R–H condition to (25),
e36
and with S′ known from (35) can be solved for . As before, combining the implied value of with the continuations of right-going Riemann invariants (24) from RZ1 is sufficient to determine fully the solution in RZ2.

Figure 10 shows and at t = 4.5 > ts, where the left edge of the resaturation zone has become a shock discontinuity. Additionally, the weak derivative discontinuity closely following the left-propagating shock is the boundary that separates upward motion in RZ1 from downward motion in RZ2.

g. Long time features of the solution

The shock formation event Ps determines the last characteristic line that reaches the left-propagating shock from the unsaturated initial conditions (x > 0). The point (x1, t1) ≈ (−2.463, 4.926) on the left-propagating shock marks the transition event P1, where the incident left-going characteristics switch from being launched from the initial conditions to the right-propagating shock. This effects a change in the wave speed R′ of the shock where (22) must be reworked with the Riemann invariants from the right-propagating shock (36). Likewise, the (slight) redirection of the left-propagating shock also sends out reflected characteristics that in turn affect the right-propagating shock after the event P2 = (x3, t3) ≈ (0.075, 8.146). Although the details are omitted, these redirected shock motions have been computed from reworked ODEs and are as shown in Fig. 7. The geometry of the shocks is such that only two reflections of the characteristic from Ps = (xs, ts) are required, and thus define the final two reflection zones RZ3 and RZ4. This calculation also provides the shock positions that are the region boundaries shown for t = 9.0 in Figs. 36.

Since the saturated and values (18) ahead of the left-propagating shock tend to zero (15), the shock speed gradually accelerates to the characteristic wave speed −1. The asymptote is thus a left-going characteristic line, and its launch point, labeled as event P in Fig. 7, generates a Riemann invariant whose computed value I ≈ 0.161. This implies that the shock jump remains finite with and as t → ∞. Consequently, the reflected Riemann invariant and has diminishing influence on the right-propagating shock by the RZ3 version of (35). A further approximation of (31) using the linear-in-time growth of reduces this shock ODE to
e37
for which exact integration gives
e38
This result is confirmed against the numerical solution of the shock position S(t) with the suggestive value of K ≈ 1.002. Since ψ(x)|RE, as also seen in Fig. 11, is exponentially decaying for large x, the shock at x = S(t) drifts leisurely downstream as logt. This continued motion of the right-propagating shock guarantees that it eventually intersects with all characteristics of the resaturated zone, which restricts the linear-in-time growth of (31) to the interval T(x) ≤ tS(x), and avoids unboundedness that would contradict the conservation property that the x integral of be constant in time.

5. Conclusions

The present attempt to understand the basic character of internal gravity wave propagation in a saturated moist-neutral atmosphere was motivated by a curious feature in the impulsive start-up of the flow of a moist nearly neutral atmosphere over a two-dimensional mountain ridge (Miglietta and Rotunno 2005). That feature, an upstream propagating wave of subsidence (Fig. 1), has now been identified as a manifestation of the dynamics explored in the simplified models discussed herein.

Fundamental to the dynamics of a saturated moist-neutral atmosphere is the asymmetric property (1) that upward parcel displacements (saturated) produce zero buoyancy but that downward displacements (unsaturated) produce positive buoyancy. We have shown here that this property has a profound effect on the character of internal gravity wave propagation. In particular we have shown that an initial disturbance characterized by a zone of upward displacement adjacent to a zone of downward displacement (such as in the start-up flow over a mountain ridge) will evolve such that the (unsaturated) downward displacement zone encroaches upon the initial zone (saturated) upward displacement. Exact analytical solutions for the initial conditions (15) in the hydrostatic case show that this encroachment has a discontinuous form and initially propagates at one-half the characteristic dry wave speed.

In closing, we note that the asymmetry between rising, saturated and descending, unsaturated motion is a central feature of several prevalent atmospheric circulations. Analytical modeling of frontal circulations (Emanuel 1985) and baroclinic waves (Emanuel et al. 1987; Fantini 1999) model the moisture effect as a step change in static stability that depends on whether the air is rising or sinking. In these problems, the mathematical effect comes through the static stability in the inversion of an elliptic equation for the circulation in the vertical plane [Emanuel 1985, see (21)]. Closer to the system considered here are the representations of the effects of moisture on static stability by Bretherton (1987) and Pauluis and Schumacher (2010, 2011). In the latter studies of moist convection, the effects of mixing between saturated and unsaturated air are central features of the flow and the step change in static stability is conditioned on variables that take mixing into account [Bretherton 1987, see (52)]. The studies closest to the present one involve wave propagation in a conditionally unstable atmosphere such as Frierson et al. (2004), Stechmann and Majda (2004), and Dias and Pauluis (2010). In their particular limit of fast convective dynamics, the saturation is conditioned on vertical motion (rather than displacement); and consequently, shock discontinuities are observed in w and the temperature gradient Tx variables. We believe the method of characteristic analysis presented here offers a useful blueprint for understanding the dynamics of similar systems governed by hyperbolic conservation laws involving nonlinear flux with discontinuous gradients.

Acknowledgments

Support for DJM was provided by NSERC Grant RGPIN-238928. DJM also thanks the MMM Division of NCAR for their intellectual hospitality during the course of this work. The authors are grateful to Marcello Miglietta, Olivier Pauluis, and an anonymous reviewer for their thorough reading of and constructive recommendations on the original submission.

APPENDIX A

Nonhydrostatic Solutions with Zero Dissipation

Figure A1a shows a comparison of the nonhydrostatic case shown in Fig. 3 for at t = 9 (light) with a zero dissipation computation at quadruple resolution (grid spacing Δx = 0.025/4, dark). In the absence of explicit (ν = 0) or numerical dissipation, the interval between the left-propagating wave of subsidence at x ≃ −5 and the left edge of the resaturation zone at x ≃ 1 is seen to contain large-amplitude and small-wavelength oscillations. These are plotted on an expanded scale in Fig. A1b. Initially, we had concerns that these oscillations might be associated with the appearance of infinitely small scales in the inviscid limit (after finite time). To verify that this was not the case, we confirmed convergence of the spatial oscillations with increased numerical resolution for fixed times up to t = 15. Based on these tests, we conclude that these oscillations are indeed finite-scale features of the nonhydrostatic solution with zero dissipation.

Fig. A1.
Fig. A1.

(a) Fully resolved, zero-dissipation limit of the nonhydrostatic displacement (black line) compared to the weakly dissipative case (gray line) of Fig. 3. (b) Details of the oscillations near x = 0.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

We attribute these small-scale waves as the phenomenon known as a dispersive shock—similar to that arising in the weak dispersion limit of the Korteweg–de Vries equation (Gurevich and Pitaevskii 1974). For our particular problem, dispersion is introduced through the nonhydrostatic term in the vorticity equation (5). Integration by parts shows that single vertical mode solutions (9) conserve the integral
ea1
from which a subtle mathematical argument guarantees that shocks cannot occur from continuous initial conditions (Evans 2010). However, this regularization of shocks by dispersion (as originally analyzed by Lax and Levermore 1983; Venakides 1985) leads to a wake of oscillations whose wavelengths are set at a finite dispersion scale. As there is no wave propagation in saturated regions, dispersive oscillations are restricted to unsaturated flow trailing both the left- and right-propagating shocks. Time-dependent nonhydrostatic solutions, as typified by Fig. 7, are thus consistent with a wave pattern further complicated by the constructive–destructive interference of two short-scale wave trains in the reflection zones (RZ1–RZ4). Finally, as the small scales of these oscillations make them susceptible to elimination by even slight amounts of viscous dissipation, we believe the that computations such as Fig. 3 retain the most important spatial features of the atmospheric process.

APPENDIX B

A Comment on the Speed of Discontinuities

In Frierson et al. (2004), several PDE models with a switching linear term similar to (16) display discontinuities whose propagation violates the Lax entropy condition (Lax 1973; LeVeque 1992). For the solution constructed in section 4, the resaturation boundary is also in violation since its apparent motion has a speed, 1/|T′| > 1, lying outside the characteristic speed range of the unsaturated equations. This technical point is noteworthy, as discontinuities in violation of the Lax condition have the usual expectation that they are unstable (albeit proved only for continuous flux functions; Conway and Smoller 1973). As the derivative discontinuities here are observed to propagate stably, some comments on this unusual occurrence are warranted.

We first note that the left-propagating shock has the usual adherence to the Lax entropy condition. Figure B1 highlights the geometry of the characteristic lines for events on the left-propagating shock at a and on the resaturation boundary at b. At events such as a, the entropy condition is satisfied since the shock speed R is intermediate to the wave speeds of colliding characteristic lines from the same family. These lines are the left-going in RZ1 (−1 < R′) and the left-going in SA (R′ < 0). The bracketing inequality, −1 < R′ < 0, then implies the familiar geometry whereby the shock path is a time-like curve with respect to left-going characteristics that approach from different sides of the shock (Whitham 1974).

Fig. B1.
Fig. B1.

Space–time geometry of characteristic lines at a, the left-propagating shock, and b, the resaturation boundary, as in Fig. 7. On the shock at a, the collision of a left-going unsaturated characteristic line with a (degenerate) saturated double-characteristic (double arrow) results in a right-going reflection. On the resaturation boundary at b, the meeting of left- and right-going characteristic lines at continues as a saturated double-characteristic line.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-051.1

This contrasts with event at b on the left-going part of the resaturation boundary, where the speed of discontinuity (1/T′) relates to both the left- and right-going characteristic wave speeds by the inequality, 1/T′ < −1 < 0 < 1. Thus, the Lax entropy condition is violated for both left- and right-going families. A similar violation applies on the right-going part of the resaturation boundary. These inequalities now imply a geometry with the boundary being a space-like curve, so that only one characteristic from each family is inbound from UN.

Our interpretation of the Lax violation begins from the observation that the resaturation boundary is a discontinuity that does not develop in the usual collision of same-family characteristic lines. Rather, the derivative discontinuity is induced by the discontinuous wave speed at , and hence it should not be identified with a shock in the usual sense. As the geometry of the level curve is not controlled by characteristic speed considerations, there is really no violation of the propagation speed limit. (Intuitively then, one would expect that perturbative smoothing of flux derivatives would be accompanied by smoothing of the solution.) Nonetheless, as noted in Frierson et al. (2004), all discontinuities, including those associated with the wave-speed jump, must still satisfy a R–H condition in keeping with the mathematical notion of a weak solution. This certainly is the case for the solution in section 4 where the construction uses the R–H condition, not as an ODE for the space–time location of the discontinuity, but as the derivative jump of the solution at the resaturation time. Finally, a simple illustration of a stable violation of the Lax entropy condition is found in LeVeque (2002), in which a scalar hyperbolic equation has a jump in wave-speed coefficient that describes a two-piece conveyor belt.

REFERENCES

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  • Lax, P. D., 1973: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 11, SIAM, 48 pp.

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  • LeFloch, P. G., and M. Mohammadian, 2008: Why many theories of shock waves are necessary: Kinetic functions, equivalent equations, and fourth-order models. J. Comput. Phys., 227, 41624189.

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    • Search Google Scholar
    • Export Citation
  • Pauluis, O., and J. Schumacher, 2011: Self-aggregation of clouds in conditionally unstable moist convection. Proc. Natl. Acad. Sci. USA, 108, 12 62312 628.

    • Search Google Scholar
    • Export Citation
  • Stechmann, S. N., and A. J. Majda, 2004: The structure of precipitation fronts for finite relaxation time. Theor. Comput. Fluid Dyn., 20, 377404.

    • Search Google Scholar
    • Export Citation
  • Venakides, S., 1985: The zero dispersion limit of the Korteweg-de Vries equation with non-trivial reflection coefficient. Commun. Pure Appl. Math., 38, 125155.

    • Search Google Scholar
    • Export Citation
  • Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley, 636 pp.

Save
  • Barcilon, A., J. C. Jusem, and P. G. Drazin, 1979: On the two-dimensional hydrostatic flow of a stream of moist air over a mountain ridge. Geophys. Astrophys. Fluid Dyn., 13, 125140.

    • Search Google Scholar
    • Export Citation
  • Bretherton, C. S., 1987: A theory for nonprecipitating moist convection between two parallel plates. Part I: Thermodynamics and linear solutions. J. Atmos. Sci., 42, 18091827.

    • Search Google Scholar
    • Export Citation
  • Conway, E. D., and J. A. Smoller, 1973: Shocks violating Lax's condition are unstable. Proc. Amer. Math. Soc., 39, 353356.

  • Correia, J., P. G. LeFloch, and M. D. Thanh, 2001: Hyperbolic systems of conservation laws with Lipschitz continuous flux-functions: The Riemann problem. Bull. Braz. Math. Soc., 32, 271301.

    • Search Google Scholar
    • Export Citation
  • Dias, J., and O. Pauluis, 2010: Impacts of convective lifetime on moist geostrophic adjustment. J. Atmos. Sci., 67, 29602971.

  • Emanuel, K. A., 1985: Frontal circulations in the presence of small moist symmetric stability. J. Atmos. Sci., 42, 10621071.

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 592 pp.

  • Emanuel, K. A., M. Fantini, and A. J. Thorpe, 1987: Baroclinic instability in an environment of small stability to slantwise moist convection. Part I: Two-dimensional models. J. Atmos. Sci., 44, 15591573.

    • Search Google Scholar
    • Export Citation
  • Evans, L. C., 2010: Partial Differential Equations. 2nd ed. American Mathematical Society, 749 pp.

  • Fantini, M., 1999: Evolution of moist-baroclinic normal modes in the nonlinear regime. J. Atmos. Sci., 56, 31613166.

  • Frierson, D. M. W., A. J. Majda, and O. M. Pauluis, 2004: Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit. Commun. Math. Sci., 2, 591626.

    • Search Google Scholar
    • Export Citation
  • Gurevich, A. V., and L. P. Pitaevskii, 1974: Nonstationary structure of a collisionless shock wave. Sov. J. Exp. Theor. Phys., 38, 291297.

    • Search Google Scholar
    • Export Citation
  • Holden, H., and N. H. Risebro, 2002: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences, Vol. 152, Springer, 361 pp.

    • Search Google Scholar
    • Export Citation
  • Keller, T. L., R. Rotunno, M. Steiner, and R. D. Sharman, 2012: Upstream-propagating wave modes in moist and dry flow over topography. J. Atmos. Sci., 69, 30603076.

    • Search Google Scholar
    • Export Citation
  • Lax, P. D., 1973: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 11, SIAM, 48 pp.

  • Lax, P. D., and C. D. Levermore, 1983: The small dispersion limit of the Korteweg-de Vries equation. I. Commun. Pure Appl. Math., 36, 253290.

    • Search Google Scholar
    • Export Citation
  • LeFloch, P. G., and M. Mohammadian, 2008: Why many theories of shock waves are necessary: Kinetic functions, equivalent equations, and fourth-order models. J. Comput. Phys., 227, 41624189.

    • Search Google Scholar
    • Export Citation
  • LeVeque, R. J., 1992: Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich, Birkhaüser Verlag, 214 pp.

  • LeVeque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Vol. 31, Cambridge University Press, 558 pp.

  • Miglietta, M. M., and R. Rotunno, 2005: Simulations of moist nearly neutral flow over a ridge. J. Atmos. Sci., 62, 14101427.

  • Pauluis, O., and J. Schumacher, 2010: Idealized moist Rayleigh-Benard convection with piecewise linear equation of state. Commun. Math. Sci., 8, 295319.

    • Search Google Scholar
    • Export Citation
  • Pauluis, O., and J. Schumacher, 2011: Self-aggregation of clouds in conditionally unstable moist convection. Proc. Natl. Acad. Sci. USA, 108, 12 62312 628.

    • Search Google Scholar
    • Export Citation
  • Stechmann, S. N., and A. J. Majda, 2004: The structure of precipitation fronts for finite relaxation time. Theor. Comput. Fluid Dyn., 20, 377404.

    • Search Google Scholar
    • Export Citation
  • Venakides, S., 1985: The zero dispersion limit of the Korteweg-de Vries equation with non-trivial reflection coefficient. Commun. Pure Appl. Math., 38, 125155.

    • Search Google Scholar
    • Export Citation
  • Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley, 636 pp.

  • Fig. 1.

    Evolving flow at t = 5 h after an impulsive start of saturated nearly moist neutral flow over a mountain ridge showing (a) the vertical velocity w (contour interval = 0.1 m s−1), and (b) the cloud water mixing ratio qc where qc < 0.01 g kg−1 (white), 0.01 < qc < 0.1 g kg−1 (light gray), 0.1 < qc < 0.5 g kg−1 (medium gray), and qc > 0.5 g kg−1 (dark gray). (Adapted from MR.)

  • Fig. 2.

    Numerical solution of (5)(8) for the evolving flow at t = 50 after an impulsive start of flow over a mountain ridge showing (a) vertical velocity w contours ±(0.001 25, 0.0025, 0.005, 0.01, 0.015, and 0.02) and (b) displacement δ intervals where δ < −0.005 (white), −0.005 < δ < 0 (lighter gray), and 0 < δ (darker gray), with solid contours (0.01, 0.05).

  • Fig. 3.

    Numerical simulations of dissipative, nonhydrostatic displacement for times t = 0.0, 1.5, 2.5, 4.5, and 9.0. The region boundaries (vertical dividers) are reference values obtained from the inviscid, hydrostatic analysis of section 4. Unsaturated air appears following the left-propagating front.

  • Fig. 4.

    Numerical simulations of dissipative, nonhydrostatic streamfunction for times and with region boundaries identical to the displacement plot of Fig. 3. The left-propagating front corresponds to a downdraft.

  • Fig. 5.

    Numerical simulations of dissipative, hydrostatic displacement for times t = 0.0, 1.5, 2.5, 4.5, and 9.0. The correspondence with the nondissipative solutions of section 4 is exact up to smoothing of shocks and rounding of derivative discontinuities at region boundaries (vertical dividers). The basic solution structure at t = 9.0 differs significantly from the nonhydrostatic case only in the dispersive oscillations found in the region identified as the reflection zone.

  • Fig. 6.

    Numerical simulations of dissipative, hydrostatic streamfunction for times identical to the displacement plot of Fig. 5. The left-propagating front appears as a sharp downdraft.

  • Fig. 7.

    Space–time regions for the nondissipative, hydrostatic solution from initial conditions (15). Boundaries between saturated (shaded) and unsaturated (unshaded) regions are shown as thick curves, with shocks solid and derivative discontinuities dashed. Thin dashed lines indicate partitions of unsaturated solutions by characteristics carrying derivative discontinuities. Unsaturated solutions within the reflection zones, RZ1–RZ4, involve characteristics reflected from shocks. The point labeled P is the source for the left-going characteristic line that is the asymptote of the left-propagating shock.

  • Fig. 8.

    Plots and regions of nondissipative, hydrostatic and at t = 1.5. The left-propagating shock at x = −t/2 and the derivative discontinuity at x = t are boundaries between the labeled SA, RZ1, and UN regions. Consistent with Fig. 7, a solid vertical line indicates a shock boundary and a dashed line indicates a derivative discontinuity. The negative slope of in RZ1 corresponds to rising motion and increasing .

  • Fig. 9.

    Plots and regions of nondissipative, hydrostatic and at t = 2.5 after the appearance of the resaturation region. The RE boundaries (thin dashed) occur where and correspond to derivative discontinuities in the solutions. Only a thin sliver of the downstream branch of the RZ1 zone remains between RE and UN.

  • Fig. 10.

    Plots and regions of nondissipative, hydrostatic and at t = 4.5 after the upwind boundary of the resaturation region has become a right-propagating shock. The rightmost RE boundary and the leftmost UN edge are now virtually indistinguishable. The weak derivative discontinuity closely following the left-propagating shock is the boundary that separates upward motion in RZ1 from downward motion in RZ2.

  • Fig. 11.

    Values at the resaturation boundary t = T(x) for (solid) and (dashed). The left edge of the plot is xs ≈ −0.090 where . For large values of x, decays exponentially.

  • Fig. A1.

    (a) Fully resolved, zero-dissipation limit of the nonhydrostatic displacement (black line) compared to the weakly dissipative case (gray line) of Fig. 3. (b) Details of the oscillations near x = 0.

  • Fig. B1.

    Space–time geometry of characteristic lines at a, the left-propagating shock, and b, the resaturation boundary, as in Fig. 7. On the shock at a, the collision of a left-going unsaturated characteristic line with a (degenerate) saturated double-characteristic (double arrow) results in a right-going reflection. On the resaturation boundary at b, the meeting of left- and right-going characteristic lines at continues as a saturated double-characteristic line.

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