## 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.

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 *U* − *Nd*/*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 *N*^{2}(*z*) at the tropopause height *z _{t}* in the MR sounding can act as a (somewhat leaky) waveguide and, if

*U*−

*Nz*/

_{t}*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

*U*−

*Nz*/

_{t}*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).

## 2. Physical model

*h*[where

*h*and

*π*/

*N*as the respective length and time scales, the dimensionless equations (with tildes dropped on the new variables) are

*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 *x*–*z*) 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.

*U*= 0 in (5) and (6),

*ψ*(

*x*,

*z*= 0) = 0, and the initial condition

*x*< 0 and

*x*> 0. Because of the neglect of nonlinear advection in the governing equations, the solutions maintain the form

*f*=

*b*,

*δ*,

*η*,

*ψ*, or

*w*)—the simplest linear vertical mode that satisfies the boundary conditions at

*z*= 0 and 1.

*πz*in (9), are

From the previous paragraph it is clear that for an initial condition *x* < 0 (saturated), the initial tendency of the solution is to remain stationary, but if *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 *ν*∂_{xx}*η* is added to the right-hand side of (5), where *ν* = 0.03.

For the first initial-value problem described in the previous section, the model domain is −*l* ≤ *x* ≤ +*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,

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

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

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

*U*= 0 for the initial condition

*x*< 0); consistently, Fig. 4 shows that

*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

Numerical simulations of dissipative, nonhydrostatic displacement *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

Numerical simulations of dissipative, nonhydrostatic displacement *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

Numerical simulations of dissipative, nonhydrostatic displacement *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

Numerical simulations of dissipative, nonhydrostatic streamfunction

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

Numerical simulations of dissipative, nonhydrostatic streamfunction

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

Numerical simulations of dissipative, nonhydrostatic streamfunction

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

Numerical simulations of dissipative, hydrostatic displacement *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

Numerical simulations of dissipative, hydrostatic displacement *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

Numerical simulations of dissipative, hydrostatic displacement *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

Numerical simulations of dissipative, hydrostatic streamfunction

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

Numerical simulations of dissipative, hydrostatic streamfunction

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

Numerical simulations of dissipative, hydrostatic streamfunction

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. 3–6) 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. 3–6 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. 3–6) 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

*x*–

*t*regions distinguished solely by the sign of

### 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 *P _{s}* (thick solid); and a resaturation boundary (thick dashed). This establishes three distinct solution regions in

*x*–

*t*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 RZ

_{1}–RZ

_{4}. All dashed boundaries coincide with derivative discontinuities.

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, RZ_{1}–RZ_{4}, 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

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, RZ_{1}–RZ_{4}, 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

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, RZ_{1}–RZ_{4}, 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 8–10 show the inviscid, hydrostatic solutions for *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. 3–6. The hydrostatic computations of Figs. 5 and 6 are a perfect matchup to the dissipative smoothing of the discontinuities.

Plots and regions of nondissipative, hydrostatic *t* = 1.5. The left-propagating shock at *x* = −*t*/2 and the derivative discontinuity at *x* = *t* are boundaries between the labeled SA, RZ_{1}, 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 _{1} corresponds to rising motion and increasing

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

Plots and regions of nondissipative, hydrostatic *t* = 1.5. The left-propagating shock at *x* = −*t*/2 and the derivative discontinuity at *x* = *t* are boundaries between the labeled SA, RZ_{1}, 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 _{1} corresponds to rising motion and increasing

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

Plots and regions of nondissipative, hydrostatic *t* = 1.5. The left-propagating shock at *x* = −*t*/2 and the derivative discontinuity at *x* = *t* are boundaries between the labeled SA, RZ_{1}, 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 _{1} corresponds to rising motion and increasing

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

Plots and regions of nondissipative, hydrostatic *t* = 2.5 after the appearance of the resaturation region. The RE boundaries (thin dashed) occur where _{1} zone remains between RE and UN.

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

Plots and regions of nondissipative, hydrostatic *t* = 2.5 after the appearance of the resaturation region. The RE boundaries (thin dashed) occur where _{1} zone remains between RE and UN.

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

Plots and regions of nondissipative, hydrostatic *t* = 2.5 after the appearance of the resaturation region. The RE boundaries (thin dashed) occur where _{1} zone remains between RE and UN.

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

Plots and regions of nondissipative, hydrostatic *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 RZ_{1} from downward motion in RZ_{2}.

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

Plots and regions of nondissipative, hydrostatic *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 RZ_{1} from downward motion in RZ_{2}.

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

Plots and regions of nondissipative, hydrostatic *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 RZ_{1} from downward motion in RZ_{2}.

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

*x*+

*t*and right-going counterparts propagate with constant

*x*−

*t*. The method of characteristics encodes the solution of the wave equation in terms of left-going and right-going Riemann invariants—that is, quantities

*x*±

*t*. The initially unsaturated [

*x*≥ 0] conditions (15) give these invariants as

*x*≥ 0 (subscripts denote the region of evaluation). These invariants then imply the nondispersive wave solutions (13) and (14).

*t*-axis direction. Along such characteristic lines

*t*with a constant rate given by the vertical motion

*x*≤ 0. Note, however, for

*t*> 0, the domains of influence of the unsaturated

*I*

^{−}invariant (17) and the saturated solution (18) overlap for −

*t*≤

*x*≤ 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).

*x*–

*t*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

*R*(

*τ*),

*T*(

*τ*)) is a parameterized curve of discontinuity in the

*x*–

*t*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.

*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),

*I*

^{−}that is incident to the shock. Evaluating the difference quantities at

*x*=

*R*(

*t*) gives

*R*′ = −1/2 indicates that the shock is left propagating but is slower than the characteristic wave speed of −1.

*I*

^{+}is reflected whose value is obtained from the other linear combination of the R–H conditions:

*x*=

*R*(

*t*), anticipating

*ψ*(

*R*,

*t*)|

_{SA}= 0 as above, gives

*R*′ = −1/2 known, this determines the right-going Riemann invariant

_{1}denotes a reflection zone influenced by these shock-reflected characteristics. Since the reflected characteristic line that intersects the shock at (

*R*(

*t*),

*t*) = (

*R*, −2

*R*) has the equation

*x*−

*t*=

*R*− (−2

*R*) = 3

*R*, an explicit formula for

*I*

^{+}is obtained,

*I*

^{−}Riemann invariants in the RZ

_{1}region are just the continuations from the UN region, the solutions for the displacement field in the regions SA, RZ

_{1}, and UN can be written

*t*= 1.5 in Fig. 8 with the regions of (28) identified. The boundary between the SA and RZ

_{1}regions is the shock at

*x*= −

*t*/2. The boundary between the RZ

_{1}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

*x*–

*t*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 RZ_{1}, and its effects on the left-propagating shock will lead to an upstream acceleration that begins at the event labeled *P*_{1} 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

_{1}region. This is labeled as the RE region in Fig. 7. The transition boundary (thick dashed) is obtained as the zero contour of

_{1}equation (28). The resaturation time

*T*(

*x*) is defined by the condition

*P*= (

_{r}*x*,

_{r}*t*), which requires both

_{r}*x*≈ 1.107 and

_{r}*t*≈ 1.981 specific to the initial condition (15).

_{r}*t*≥

*T*(

*x*) are determined by applying continuity across the transition. Following the vertical characteristics,

*t*and takes the RZ

_{1}value at the resaturation boundary

*t*=

*T*(

*x*):

*t*integration of (11):

Values at the resaturation boundary *t* = *T*(*x*) for *x _{s}* ≈ −0.090 where

*x*,

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

Values at the resaturation boundary *t* = *T*(*x*) for *x _{s}* ≈ −0.090 where

*x*,

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

Values at the resaturation boundary *t* = *T*(*x*) for *x _{s}* ≈ −0.090 where

*x*,

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

*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 RZ

_{1}and RE regions for the R–H conditions (19):

*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 *t* = 2.5 > *t _{r}* as representative solutions past resaturation. The locations with

*x*=

*t*characteristic, so that only a sliver of the rightmost branch of the RZ

_{1}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 RZ_{1}

_{1}solution (28) to determine the

*P*. The boundary ends because the

_{s}*x*≤

*x*because of the shadowing of left-going characteristics (carrying the

_{s}*I*

^{−}Riemann invariant) by the resaturation boundary itself. The critical event

*P*= (

_{s}*x*,

_{s}*t*) is defined by the point on the resaturation boundary whose tangent line, as the last unblocked characteristic, has

_{s}*T*′(

*x*) = −1. Mathematically, this translates to

_{s}_{1}, but also in RE by (33)]. Substitution of the RZ

_{1}form for

*P*event is situated in Fig. 7 using numerical values

_{s}*x*≈ −0.090 and

_{s}*t*≈ 2.554 specific to the initial condition (15). Also shown (thin dashed) is the portion of the last unblocked characteristic line [

_{s}_{1}.

### e. Rightward-propagating shock

*P*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

_{s}*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

*I*

^{+}Riemann invariant for the RZ

_{1}reflection (27). The R–H condition is solved numerically as an ODE for

*S*(

*t*) with the initial launch point

*S*(

*t*) =

_{s}*x*. [Note that the starting value of

_{s}*S*′(

*t*) 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

_{s}*P*—but only up to the event

_{s}*P*

_{2}, where the RZ

_{1}equation (27) for the

*I*

^{+}Riemann invariant no longer applies.

### f. Reflection zone RZ_{2}

_{2}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),

*S*′ known from (35) can be solved for

_{1}is sufficient to determine fully the solution in RZ

_{2}.

Figure 10 shows *t* = 4.5 > *t _{s}*, 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 RZ

_{1}from downward motion in RZ

_{2}.

### g. Long time features of the solution

The shock formation event *P _{s}* determines the last characteristic line that reaches the left-propagating shock from the unsaturated initial conditions (

*x*> 0). The point (

*x*

_{1},

*t*

_{1}) ≈ (−2.463, 4.926) on the left-propagating shock marks the transition event

*P*

_{1}, 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

*P*

_{2}= (

*x*

_{3},

*t*

_{3}) ≈ (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

*P*= (

_{s}*x*,

_{s}*t*) are required, and thus define the final two reflection zones RZ

_{s}_{3}and RZ

_{4}. This calculation also provides the shock positions that are the region boundaries shown for

*t*= 9.0 in Figs. 3–6.

*P*

_{∞}in Fig. 7, generates a Riemann invariant whose computed value

*I*

^{−}≈ 0.161. This implies that the shock jump remains finite with

*t*→ ∞. Consequently, the reflected Riemann invariant

_{3}version of (35). A further approximation of (31) using the linear-in-time growth of

*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 log

*t*. 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

*T*(

*x*) ≤

*t*≤

*S*(

*x*), and avoids unboundedness that would contradict the conservation property that the

*x*integral of

## 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 *T _{x}* 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 *t* = 9 (light) with a zero dissipation computation at quadruple resolution (grid spacing Δ*x* = 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.

(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

(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

(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

_{1}–RZ

_{4}). 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 RZ_{1} (−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).

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

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

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

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

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

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

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