Mountain-wave-like spurious waves due to inconsistency of horizontal and vertical resolution
Shin-ichi Iga 1 Hirofumi Tomita Masaki Satoh Koji Goto Frontier Reserch Center for Global Change, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama-city, Kanagawa, 236-0001, Japan
1
Full address: Frontier Reserch Center for Global Change, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama-city, Kanagawa, 236-0001, Japan. E-mail:
[email protected]
1
Abstract Life cycle experiments of baroclinic waves (LCEs) are performed using a dynamical core of NICAM (Non-hydrostatic ICosahederal Atmospheric Model) developed at Frontier Research Center for Global Change, focused on the effects of resolution. The smallest horizontal grid interval adopted is 3.5km, and the LCEs are integrated for 12 days. When the ratio of vertical and horizontal grid interval ∆z/∆x is sufficiently small compared with the tilt of the fronts s, the solutions are not largely affected by resolution, although fronts and corresponding vertical flow become sharp and intense as horizontal resolution increases. On the other hand, when ∆z/∆x is not sufficiently small, the solutions are accompanied by spurious gravity waves generated from cold front. We show that a linear theory of mountain waves well explain the mechanism of the spurious waves quantitatively. According to the theory, the distribution of basic wind is a major factor to determine the amplitude and extension of the waves. Existence of the critical level where basic wind velocity and front velocity are equal prohibits these spurious waves from propagating above the level. For some resolutions, these waves are strong, e.g. its maximum vertical velocity is ∼0.4ms−1 for dx=3.5km and dz=600m, enough to affect to cloud generation in the cloud resolving models. The theory indicates an effective way of choosing vertical grid to moderate the amplitudes of spurious waves: small ∆z at levels with strong wind in the frame moving with the front, and large ∆z at levels with weak wind or with ambiguous front. 2
1
Introduction
In recent studies, importance of the consistency between vertical and horizontal resolution of numerical models has been suggested (e.g. Pecnick and Keyser 1989, Lindzen and FoxRabinovitz 1989, Persson and Warner 1991). Pecnick and Keyser (1989) and Persson and Warner (1991) who focused on fronts derived a relationship
∆z ≤ s∆x,
(1)
where ∆x is horizontal grid interval, ∆z is vertical grid interval (in pseudo height coordinate in there cases) and s is tilt of the front. Lindzen and Fox-Rabinovitz (1989) proposed two consistency relationships
∆z ≤
f ∆x N
(2)
∆z ≤
δ ∆x, N
(3)
where N is buoyancy frequency, f is Coriolis parameter and δ is damping rate. (3) is derived from non-equatorial quasi-geostrophic flows based on the concept of the Rossby radius of deformation, and (3) is derived from gravity waves near critical levels in the tropics. According to Persson and Warner (1991), equals of both (1) and (3) produce approximately the same ∆z/∆x ∼ 0.005 − 0.02 and that of (3) does smaller ∆z/∆x.
3
However, these conditions are proposed supposing the large grid interval model representing synoptic scale phenomena, and these conditions are severe for the calculations with horizontally high resolution: i.e. when ∆x = 1km, the conditions (1) and (3) require ∆z to be 0 u(φ, p) = 0 for φ < 0 z˜ − z˜0 π˜ z 1 1 − tanh3 sin F (˜ z) = 2 ∆˜ z0 z˜1 Z φ −1 0 ∂u T (φ, z˜) = −HR (af + 2u tan φ ) dφ0 ∂ z˜
(5) (6) (7)
where z˜ ≡ −H log(p/p0 ) and f = 2Ω sin φ with numerical values of the parameters u0 , z˜0 , ∆˜ z0 , z˜1 , H, p0 , R, a and Ω given in Table 2. T0 (˜ z ) is chosen so that, at each pressure p, the global average of T is identical to the US 1976 Standard Atmospheric Temperature value. They are hydrostatically and geostrophycally balanced. In addition to the basic field, temperature perturbation T (λ, φ) = Tˆ sech 0
2
λ − λ0 2 φ − φ0 sech , α β
(8)
where λ is longitude with Tˆ = 1K is given for all cases, except for GLEVEL=10 and 11 with dz=600m where
T 0 (λ, φ) =
Tˆsech2 λ−λ0 sech2 φ−φ0 α
β
0
for λ ≥ λ0
(9)
for λ < λ0 ,
is adopted instead of (8). Vertical domain is 0-30km with free surface.
3
Solution-convergence tests Before getting into the main issue, we show the results of solution-convergence tests and
compare them to those of Polvani et al. (2004)’s, because NICAM is relatively new model 7
and some kind of verification seems to be required. Following Polvani et al. (2004), in this section, horizontal diffusion is fixed to ν = 7.0 × 105 m2 s−1 . In addition, divergence damping which is usually required for the grid model to run stably is added. Its coefficient κ is shown in Table 3. This effect on the results revealed to be negligible because they acts only on small scales near the grid interval. Vertical grid interval is fixed to 600m for all the cases while horizontal resolution varied from GLEVEL=5 to 8. Vorticity at σ = 0.975 and t = 12days, estimated from the obtained velocity at z = 300m and z = 900m by extrapolation for the sake of comparizon to the results of Polvani et al.(2004), is shown in Fig. 1. To the eye, the solutions converge as horizontal resolution increases and the converged solution is almost the same to Fig. 4 of Polvani et al. (2004), although the amplitudes are slightly weaker in eastside (developing-cyclon region) and slightly larger in westside (decaying-cyclon region). A possible reason of the difference is that NICAM calculate diffusion operator on z plane while Polvani et al. (2004)’s experinemt does on σ plane. To evaluate the solution-convergence more quantitatively, a new indicator
l2 (ζ) ≡
|ζ − ζT |2 |ζt |2
(10)
with
|X|2 ≡
1 4π
Z
π/2
cos φdφ π/2
Z
0
8
2π
dλX
2
1/2
,
ζ is vorticity and ζT is vorticity of GLEVEL 8, is adopted following Polvani et al. (2004). l2 on σ = 0.975 is shown in Fig. 2. This figure indicate that l2 decays in the order of ∆x−2 N , and this is reasonable because NICAM adopts second order horizontal differenciation. Time step dependency is also examined, but revealed that they does not affect to the solution at all.
4
Results with various resolution In this section, we show results with various resolution. Time step, horizontal diffusion
with ∇4 type and divergence damping dependent on horizontal resolution are shown in table 4. Horizontal diffusion term is added not only on the equations of (u, v, w) and T , but also on the equation of ρ in order to damp high-wavenumber computational mode efficiently. Before showing the detailed solutions with various horizontal and vertical resolution, we roughly classify the obtained solutions in two category: those accompanied by spurious waves from cold fronts as predicted in section 1 and those without them. In Fig. 3, the former cases are denoted by crosses and the latter cases are denoted by circles. The figure implies that the spurious waves appears when ∆z/∆x is larger than a certain critical value sc in 0.0107 ∼ 0.0214, which is about 2 ∼ 4 times larger than the obtained tilt of a cold front s = 0.00523 and f /N = 0.00763 at latitude 40◦ . One possibly cause of the difference between the critical value sc obtained by experiments and introduced value s in section 1
9
is that the existence of horizontal viscosity and the error due to the horizontal difference scheme decrease the effective horizontal resolution of the NICAM. Assuming the effective horizontal grid interval
∆xe = 2∆x ∼ 4∆x
(11)
and replacing ∆x to ∆xe , the condition (1) well explains the appearance of the spurious waves. Another kind of distinguished spurious waves are not apparent in our experiment, although true gravity waves obtained are deformed when vertical resolution is not sufficient.
a
The case ∆z/∆x ≤ sc First, we show the results with vertical resolution dz = 150m and horizontal resolution
GLEVEL= 5, 7 and 9 where ∆z/∆x ≤ sc . Figure 4 is their temperature field at z=75m after 9 days. For these resolution, four baroclinic cyclones: A decayed one, B decaying one, C developed one and D developing one from left to right, are apparent and moving eastward. B corresponds to initially ejected disturbance, and A, C and D are secondary cyclones generated by wave propagation. This figure indicates that their synoptic behaviour and propagating speed are roughly independent on horizontal resolution at least for LEVEL ≥ 5. It is consistent to the results of Methoven and Hoskins (1998)’s spectral model where the characteristic of life cycle is unaffected by the increase in resolution from T95 (∆x ∼180km). On the other hand, apparently different points in these resolutions are (i) sharper front
10
for larger GLEVEL (ii) longer swirling of cyclone B for larger GLEVEL. This kind of spirals are also visible in PV contour (not shown) and well discussed by researchers in the past (e.g. Methven and Hoskins (1998)). Focusing on cyclone C, two regions of intense upwelling flows are placed at northern occluded front and south-eastern cold front (Fig. 5). More precisely, they are generated at the reading edge of the fronts (Fig. 6), and they are getting narrower and more intense as GLEVEL increases. In the case of GLEVEL=9 its vertical velocity is beyond 0.06ms−1 and its width is about 50km corresponding to several times of grid interval. Basic field of u ¯d , defined as the horizontal velocity component normal to the front line in the Doppler shifted frame moving with the front, is shown in Fig. 7. Positive (negative) ud region indicate the flow from warm (cold) air region, or from right (left) to left (right) in the figure. The figure shows that the upwelling flows are generated by the warm southerly flow near the ground interfered by the fronts. The strong bottom flow is due to frictionless lower boundary condition, and in the real situation with friction, the bottom flow is weaker and the intense upwelling flow is expected to be moderated. In the case of GLEVEL=9, the upwelling flow generate a secondary gravity wave propagating up-left direction (Fig. 6). The amplitude of the gravity wave decreases in height and the wave disappears. The existence of a critical level where ud = 0 is suspected to be its cause. In upper troposphere, the other gravity waves are also exist along the jets especially at the jet stream exit region (not shown), consistent to the results of O’Sullivan and Dunkerton
11
(1995). Vertical resolution does not seem to affect to the results (e.g. dz=150m and 300m of GLEVEL=8 in Fig. 4) unless ∆z/∆x ≤ sc .
b
The case ∆z/∆x > sc When ∆z/∆x > sc , solutions are accompanied by spurious gravity waves extending up-
ward from the cold front (Fig. 8). These waves appear as the cold front becomes sharp and disappear as the cold front breaks. Maximum vertical wind velocity of these waves shown in table 5 increases as horizontal resolution increases and as vertical resolution decreases. Horizontal scale of these waves are proportional to the horizontal grid interval, and propagating direction becomes upward as horizontal resolution increases. These behavior, especially for GLEVEL=11, resembles to the flow over step orography (Gallus and Klemp 2000). As a generation mechanism of these waves, we propose an theory: (i) temperature field on the front plane is represented like staircase due to insufficient vertical resolution consistency (same as Persson and Warner (1991)) (ii) staircase front plane act as if it were real terrain (iii) a flow parallel to the front plane is affected by the staircase and gravity waves are generated from front (see, Fig. 9) as if they were mountain waves. A point of this theory is that Doppler-shifted wind speed parallel to the front plane determine the strength and the shape of the waves. To proof our theory, we first introduce a 2-dimensional (x-z plane) linear system with
12
Boussinesq approximation:
d¯ u 1 ∂p0 ∂ ∂ u0 + w 0 +u ¯ + ∂t ∂x dz ρ ∂x ∂ ∂ v0 +u ¯ ∂t ∂x ∂ ∂ 1 ∂p0 θ 0 +u ¯ − g w0 + ∂t ∂x ρ ∂z θ ∂ ∂ dθ¯ +u ¯ θ0 + w0 ∂t ∂x dz
= f v 0 − αu0
(12)
= f v 0 − αv 0
(13)
= −αw 0
(14)
= −αθ 0
(15)
∂u0 ∂w 0 + = 0, ∂x ∂z
(16)
where (u, v, w) is velocity of (x, y, z) direction, p is pressure, f is Coriolis parameter, θ is potential temperature and α is damping coefficient which has value only in upper levels. Values with dash indicate anomaly and values with bar indicate basic field. Applying to stationary mountain waves, ∂/∂t is neglected and lower boundary is staircase front plane where w 0 = 0. For simplicity, we focus on a certain step of staircase and define
zb
∆z x 2x ≡ − ∆z , tanh 2 β∆x Rf
(17)
where zb is level of lower boundary, β is smoothing coefficient representing the deference between ∆x and ∆xe and the thickness of the front, and [−0.5Rf : 0.5Rf ] is range of x. An ideal case with ∆xe = ∆x and with delta-function-type front whose thickness is zero corresponds to β = 1, and realistic cases with ∆xe > ∆x and with finite thickness of the front correspond to β > 1. From (11), we conveniently assume β = 4. The second term on left-hand side is added for convenience of Fourier transformation such that zb has no jump at 13
x = ±0.5Rf when horizontally cyclic boundary condition is adopted. Rf is sufficiently large such that the second term does not affect to the significant region near x = 0, where the major amplitude is distributed. The solution for multi-step staircase is the superposition of these solutions. Assuming w 0 = 0 at upper boundary and adding damping in upper levels, waves not-originated from mountains: i.e. waves with downward group velocity and waves increasing exponentially in z, are eliminated. Finally, only the mountain waves remain in lower significant levels. To apply the theory to the results, u ¯ is replaced by u ¯ d . N 2 obtained is about 0.00015 above the front plane below the tropopause. Hence, in the linear analysis, we simply define them as:
u ¯d
2π(z + 300) = 4 + 6 sin 6000 + (z + 300)/1.3
N 2 = 0.00015.
(18) (19)
With above conditions and basic field of (18 and 19), the linear system (12)-(16) is solved and linearized mountain-waves from fronts are quantitatively estimated. In Fig. 10, spurious waves obtained by NICAM and mountain waves obtained by linear analysis with β = 4 at t = 9day are compared. For GLEVEL=10 and 11, linear solutions well represent the spurious waves obtained by NICAM both in shape and in quantity, although spurious waves break in pieces and spread horizontally above 3000m and finally disappear due to the finite horizontal and vertical resolution. For GLEVEL=9, the panel obtained
14
by NICAM is more resemble to linear solution of GLEVEL=10 with β = 4 than that of GLEVEL=9. Because the linear solution of GLEVEL=10 with β = 4 is equivalent to that of GLEVEL=9 with β = 2, β = 2 is more likely in the case of GLEVEL=9. One might think that the spurious waves in the left panel of GLEVEL=9 propagate horizontally rather than vertically. However, we must note that in this case the horizontal wavelength is nearly the same to the run of staircase, and waves generated from other steps are superposed. In these three resolutions, these waves does not propagate above the critical level until the front is broken to peaces at t = 10day when weak secondary gravity waves are generated above the critical level. These type of spurious waves appear on all cyclons (B, C, D) except on A whose front genesis is incomplete.
c
For vertically varying grid interval If we only focus on the spurious waves from the cold front, the mountain-wave theory
indicates an effective way of choosing vertical grid to moderate the amplitudes of spurious waves: small ∆z at levels with large ud and large ∆z at levels with small ud or with ambiguous front. That is, in this case, dense grid points in z¡4000m especially lower level. Incidentally, AGCMs usually adopt such kind of vertical grid: i.e. grid points are located densely in boundary layer and sparsely in higher atmosphere. Hence, an interest is whether spurious waves with these grid appear. As an example, we focus on a vertical grid adopted in GCSS WG4 (Redelsperger et al. 2000). 15
The results using GCSS-WG4-like vertical grid (shown in the Table 6) with GLEVEL=9 and 10 are shown in Fig. 12. For GLEVEL=9, spurious waves do not appears. On the other hand, for GLEVEL=10, spurious waves appear at the 26◦ E with z=2500m and extending above. The waves are generated at the time between t = 8day+12hour and t = 8day+18hour and disappear at least by t = 9day+6hour. Different to the spurious waves described in previous subsection, the waves extend to the top of the model. The waves also seem to be generated by mountain-wave mechanism and their appearance is connected to u ¯ d . At t = 8day+18hour and at the place of waves, u¯d is negative (indicating down sloping flow) at all levels above the front without critical level, causing the wave extension toward upper boundary. At t = 9day, u ¯d changes sign, and after that they disappear. Their vertical velocity is about 0.06ms−1 , three times of that obtained in GLEVEL=10 with dz=600m. The results indicate that the vertical grid density is insufficient near the wave source level z=2500m for GLEVEL=10. At the wave source level z=2500m, dz is about 450m. To eliminate these spurious waves, dz must be less than the half at the level for GLEVEL=10.
5
Discussion and conclusions In this study, we performed life cycle experiments of baroclinic waves varying horizontal
and vertical resolution. When the condition (1) is filled, the results does not largely vary by resolutions, although as horizontal resolution increases (i) front plane becomes thin and
16
sharp (ii) vertical flow generated from leading edge of the fronts becomes strong and narrow, and (iii) contour spiral of decaying cyclones becomes longer. Synoptic features such as location and growth rate of the baroclinic cyclones are almost the same when GLEVEL ≥ 5 (corresponding horizontal resolution dx∼223km) which is consistent to the results of the past studies. Vertical resolution does not affect to the results unless aspect ratio of the vertical and horizontal grid interval is smaller than the tilt of the fronts. When the condition (1) is violated, spurious gravity waves are generated from cold fronts. We explain the generation mechanism of these spurious waves using a linear mountain-wave theory quantitatively: i.e. (i) cold fronts are represented as staircase shape due to insufficient vertical resolution (ii) corner of the staircase cold front generate gravity waves as if it were step-mountain. One may have a question why these kind of spurious waves appears only on the cold front in lower atmosphere and not on another kind of discontinuous plane such as mid-latitudinal tropopause. The mountain-wave theory also answers the question. According to the theory, strength and extent of these waves depend on distribution of u ¯ d , stratification, ambiguity of the front plane and grid interval. At mid-latitudinal tropopause, direction of horizontal wind is zonal and that of front tilt is y-z approximately, indicating zero u ¯ d and no spurious waves. Possibility of the spurious waves in other models with moist or in other phenomena is
17
also an interest. In the case of Persson and Warner (1991), upwelling warm air is sandwiched between motionless environmental air (see, Fig. 11). According to the mountain-wave theory, spurious waves are generated only in warm air because u ¯ d of environmental air is zero, and they are confined in the warm air because upper and lower boundary of the warm air becomes critical levels. This is consistent to the spurious waves obtained in their model, and seems to be more reasonable to explain their spurious waves than their explanation. Similar spurious waves generated by smaller scale phenomena, such as gravity flow or small scale front, may also be possible. In the frame moving with the edge of the gravity flow, motionless atmosphere above the flow is seen to move opposite direction of the gravity flow and the conditions of upward propagating spurious waves in mountain-wave theory are satisfied. For some resolutions of the present experiment, obtained spurious waves are strong, e.g. its maximum vertical velocity is ∼0.4ms−1 for dx=3.5km and dz=600m, enough to affect to cloud generation in the cloud resolving models, although it may be moderated to some extend in the real situation with surface friction. The mountain-wave theory provides an effective way to moderate the amplitudes of these waves: small ∆z at levels with large u d and large ∆z at levels with small ud . If we only need to mind the spurious waves from the cold front, small ∆z is required at lower troposphere especially near the ground and large ∆z is allowed at upper levels.
18
Incidentally, AGCM usually adopts such kind of vertical grid: i.e. grid point densely distributed in boundary layer. In the calculation with vertical grid based on that of GCSS WG4, spurious waves with maximum amplitude of w 0 ∼ 0.06ms−1 appear for GLEVEL=10 (dx=7km) while they do not appear for GLEVEL≥9. The results indicate that the vertical grid density is insufficient at least near the wave source level z=2500m for GLEVEL=10.
Acknowledgment The Earth simulator is used for calculation.
References [Gallus and Klemp(2000)] Gallus, W. A. and J. B. Klemp, 2000: Behavior of flow over step orography. Mon. Weath. Rev., 128, 1153–1164.
[Lindzen and Fox-Rabinovitz(1989)] Lindzen, R. S. and Fox-Rabinovitz, 1989: Consistent vertical and horizontal resolution. Mon. Weath. Rev., 117, 2575–2583.
[Methven and Hoskins(1999)] Methven, J. and B. Hoskins, 1999: The advection of highresolution tracers by low-resolution winds. J. Atmos. Sci., 56, 3262–3285.
[O’Sullivan and Dunkerton(1995)] O’Sullivan, D. and T. J. Dunkerton, 1995: Generation of inertia-gravity waves in a simulated life cycle of baroclinic instability. J. Atmos. Sci., 52, 3695–3716.
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[Pecnick and Keyser(1989)] Pecnick, M. J. and D. Keyser, 1989: The effect of spatial resolution on the simulation of upper-tropospheric frontogenesis using a sigma-coordinate primitive-equation model. Meteor. Atmos. Phys., 40, 137–149.
[Persson and Warner(1991)] Persson, P. O. G. and T. T. Warner, 1991: Model generation of spurious gravity waves due to inconsistency of the vertical and horizontal resolution. Mon. Weath. Rev., 119, 917–935.
[Polvani et al.(2004)] Polvani, L. M., R. K. Scott and S. J. Thomas, 2004: Numerically converged solutions of the global primitive equations for testing the dynamical core of atmospheric gcms. Mon. Weath. Rev., 132, 2539–2552.
[Redelsperger et al.(2000)] Redelsperger, J.-L. et al., 2000: A GCSS model intercomparison for a tropical squall line observed during toga-coare. I: Cloud-resolving models. Q. J. R. Meteorol. Soc., 126, 823–863.
[Thorncroft et al.(1993)] Thorncroft, C. D., B. J. Hoskins and M. E. McIntyre, 1993: Two paradigms of baroclinic-wave life-cycle behavior. Q. J. R. Meteorol. Soc., 119, 17–55.
[Tomita and Satoh(2004)] Tomita, H. and M. Satoh, 2004: A new dynamical framework of nonhydrostatic global model using the icosahedral grid. Fluid Dyn. Res., 34, 357–400.
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[Tomita et al.(2002)] Tomita, H., M. Satoh and K. Goto, 2002: An optimization of the icosahedral grid modified by spring dynamics. J. Comput. Phys., 183, 307–331.
[Tomita et al.(2001)] Tomita, H., M. Tsugawa, M. Satoh and K. Goto, 2001: Shallow water model on a modified icosahedral grid by spring dynamics. J. Comput. Phys., 174, 579–613.
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Table 1: Relation between GLEVEL and equivalent grid interval. GLEVEL 04 05 06 07 08 09 10 11
grid interval 446.7 km 223.4 km 111.7 km 55.84 km 27.92 km 13.96 km 6.979 km 3.489 km
Table 2: Parameter values we used. parameter g a Ω R κ cp H p0
value 9.806 6.371 × 106 7.292 × 10−5 287 2/7 R/κ 7.34 105
unit m/s2 m 1/s J/Kg/K J/Kg/K km Pa
22
parameter u0 z˜0 ∆˜ z0 z1 Tˆ λ0 φ0 α β
value 50 22 5 30 1 0 π/4 1/3 1/6
unit m/s km km km K radians radians
Table 3: Time step and divergence damping rate use on resolution convergency experiments. GLEVEL 05 06 07 08
time step [sec] 1800 900 450 225
damping coefficient [m4 s−1 ] 1.0000 × 1016 1.2500 × 1015 1.5625 × 1014 1.9531 × 1013
Table 4: Time step, horizontal diffusion and divergence damping rate used on super-highresolutoin experiments. GLEVEL 07 08 09 10 11
time step [sec] 450 225 100 50 25
diffusion coefficient [m4 s−1 ] damping coefficient [m4 s−1 ] 3.9063 × 1014 1.5625 × 1014 4.8828 × 1013 1.9531 × 1013 12 6.1035 × 10 2.4414 × 1012 7.6294 × 1011 3.0518 × 1011 10 9.5367 × 10 3.8147 × 1010
Table 5: Maximum vertical velocity [ms−1 ]of computational modes in Fig. 8 GLEVEL 8 9 10 11
dz=600m dz=300m 0.04 0.12 0.04 0.2 0.4 -
Table 6: Elevation of grid used in section 4c . They are based on the calculation of GCSS WG4 (Redelsperger et al. 2000) 35.5 110 193 286 391 512 650 807 986 1186 1410 1660 1935 2237 2566 2922 3307 3719 4159 4626 5120 5640 6184 6753 7343 7955 8586 9233 9896 10572 11258 11952 12650 13350 14050 14750 15450 16150 16850 17550 18250 18950 19650 20350 21120 22044 23153 24483 26080 27996 30295 33054 36365 40338 (m)
23
GLEVEL=5
GLEVEL=6
GLEVEL=7
GLEVEL=8
Figure 1: Vorticity at σ = 0.975 and t = 12days. Contour interval is 10−5 s−1 from −7.5 × 10−5 s−1 to 7.5 × 10−5 s−1 . They are estimated from the results at z = 300m and z = 900m by extrapolation for the sake of comparizon to the results of Polvani et al.(2004).
24
0.45
GLEVEL=5 GLEVEL=6 0.4 GLEVEL=7
0.35 0.3
l2
0.25 0.2 0.15 0.1 0.05 0 0
2
4
6 t (days)
8
10
12
Figure 2: Time development of l2 norm on σ = 0.975 plane.
GLEVEL
=11
10
09
08
07
3.5
7
14 dx (km)
28
56
dz (m)
600
300
150
Figure 3: Horizontal and vertical resolution of the calculations. ’×’ indicate a run accompanied by gravity-wave-like noize while ’◦’ indicate a run not accompanied. Dotted line indicates front tilt and dashed line indicates f/N ratio at lat 40◦ .
25
GLEVEL=5
GLEVEL=7
A
B
GLEVEL=9 C D
Figure 4: Temperature at z = 75m after 9 days.
26
Figure 5: Horizontal velocity (vector) at z = 75m and vertical velocity (contour) at z = 150m around the cyclon C after 9 days.
GLEVEL=9, dz=150m
GLEVEL=7, dz=150m
Figure 6: Potential temperature (dotted contour) and vertical velocity (solid contour) on vertical-longitudinal cross section (lat 40◦ ) at t=9days. Note that the front is not pararell to zonal direction and these panels correspond to the slantwise cross section of the front.
27
Figure 7: Same as Fig. 6 but vertical velocity (dashed contour) and Doppler-shifted horizontal velocity ud toward front line in the frame moving with the front (solid contour for positive and dotted contour for negative value). Thick contour indicate critical level u d = 0.
28
GLEVEL=9, dz=600m
GLEVEL=9, dz=300m
GLEVEL=8, dz=600m
GLEVEL=11, dz=600m
Figure 8: Same as Fig. 6 but for the cases with spurious waves. Note that the longitudinal range for GLEVEL=11 is different because initial condition is different.
29
z
l
Critical leve
x Figure 9: Diagram of a cold front deformed by inconsistency of grid interval. Dots indicate grid points, open arrows indicate basic wind in the moving frame of the front ud , and dark (pale) region indicates cold (warm) air. The stepwise front generates spurious mountain waves (wavy arrows) and propagate until they reaches critical level where u d = 0. For convenience, thickness of the front is assumed to be zero.
30
GLEVEL=11
GLEVEL=10
GLEVEL=09
Figure 10: The zoom of the Fig. 8 with θ = 292K line (leftside) and corresponding solution obtained from the linear system (12)-(16) with β = 4(rightside). Horizontal range of left panels approximately correspond to that of right panels. In all cases, ∆z = 600m. 31
s
les
n tio
mo
el
lev
al itic
cr
t
l ica
cri
ss
le ion
t
mo
el
lev
Figure 11: Simplified diagram of the front in the case of Persson and Warner (1991)
GLEVEL=9,
GLEVEL=10,
Figure 12: Same as Fig. 6 but with vertically varying grid interval.
32
