3.1 Changes to the model physics

Compared with MC2 version 3.2, version 4.0 has some significant changes in boundary-layer and turbulence parameterization schemes within the model physics package (note that version 3.1 of the physics is used in MC2 v3.2 and that version 3.5.3 of the physics is used in MC2 v4.0). These changes include:

3.2 Description of the boundary-layer and turbulence parameterization schemes in MC2 V4.0 versus V3.2

3.2.1 Surface layer formulation
We found significant differences between MC2 v4.0 and v3.2 boundary-layer and turbulence parameterizations in the surface layer formulation. Both in stable and unstable conditions, the stability functions are modified to new formulations (Delage and Girard, 1992; Delage, 1997).

a. Stability functions
Stability functions are defined for u and v (fM) and for qv and q (fT). In v4.0, for unstable convective conditions ( z / L < 0 ):   where c = 40. For static stability (Ri 0):   Ri and L are defined as follows:  


where U is the horizontal wind vector. L and Ri are related by:
    Previously, in v3.2, different stability functions were used. They were:

for Ri < 0,   for Ri 0,    
The change in stability functions to (1) in v4.0 solves a problem with the free convective limit when using (3). The use of equation (2) has been shown to more closely match observed stability behaviour (Delage and Girard, 1992; Delage, 1997).

b. Surface fluxes
Virtual potential temperature flux is generally defined as:   and the kinematic shear stress as:  

For v4.0, formulation (1) is used for the unstable condition. As a consequence, the heat flux can be represented as:   which presents the sum of two positive contributions: one from mechanically generated turbulence, the other from turbulence generated by buoyancy that is unaffected by u* even if u* Þ 0. In version 3.2, the heat flux was zero when u* Þ 0.

c. Surface-layer exchanges
Surface fluxes can be obtained from Monin-Obukhov similarity theory for the surface layer (a thin turbulent region above the surface where the vertical fluxes are quasi-constant with height). This allows us to integrate from z = 0 to z = z a (within the surface layer):   where y is u, v, q or qv.

For the surface layer, equilibrium is assumed, and   Then (6) is rewritten as:  

where the Cy , are integrated transfer coefficients.

3.2.2 Boundary-layer parameterization

a. Vertical diffusion equation
Vertical transfers due to turbulence are parameterized as a form of vertical diffusion. This effect is important, especially close to the surface but is present over the entire atmosphere. In one dimension the governing equation would be:
    where r is the density. The symbol gy represents a counter-gradient term. Coriolis terms are added in the momentum equations.

b. Vertical diffusion coefficients
The vertical diffusion coefficients Ky are variable and reflect the intensity of the turbulent exchanges. The Ky are expressed as:   where E is the turbulent kinetic energy, c is a constant (= 0.516), l is a mixing length for the statically neutral case, and fy are stability functions. These stability functions are the same as those used in (1) and (2).

The mixing length is determined by   where k = 0.4, le = 200 m is the asymptotic value. The momentum roughness length, z0m, is provided directly to the model by the user as an initial geophysical field. The thermal roughness length, z0h, is calculated as follows:    

c. Height of the boundary layer
The height of boundary layer, h, is calculated from a relaxation equation:
 where h- is the value of h at t - Dt, he is an equilibrium value and t is a relaxation constant
(= 1.5 hours). In version 4.0 of the model, h is used in surface layer calculations for more accurate fluxes and winds. See Delage (1997) for details of these calculations.

For the unstable PBL, the equilibrium value he is diagnosed from the virtual potential temperature profile qv. The equilibrium value is taken as the height of the middle of the first stable model layer (i.e., where &para;qv / &para;z 0 ), starting from the surface.

For the stable PBL, the equilibrium value he is given by:

where f is the absolute value of the Coriolis parameter (with a lower bound of 7 x 10-5s-1), u* is the friction velocity and L is the Obukhov length.

d. Turbulent kinetic energy
The TKE is calculated by a predictive equation (Mailhot and Benoit 1982; Benoit et al. 1989):

The left-hand-side neglects advection of the E term. The right-hand-side terms represent the source-sink term from mechanical shear and buoyancy, the viscous dissipation and the redistribution term. MC2 uses a time-splitting technique to solve the prognostic equations and the TKE equation is solved by a fractional step method:  

The first step (15a) can be done analytically (assuming B and C to be independent of  time). For the diffusive step (15b), the boundary conditions are vanishing fl.uxes at the top and bottom of the modelled atmosphere. Details of the solution of these equations can be found in Mailhot (1994). A vertical filter is also applied:
    where v = 0.1. A lower bound of 10-4 m2s-2 is imposed on E.

3.3 Analysis of selected fields illustrating differences between MC2 V 3.2 and V4.0

a. Wind Field
The wind field patterns from both versions are quite similar through the entire modelling period. However, there is a significant difference in the wind speed time series. In the period when the lake breezes are developing (at 1600Z), 10 m wind speeds for both versions are very similar (Figure 3.1). As lake breezes become well-developed on each lake shore by 2000Z (Figure 3.2), v4.0 10 m wind speeds are 30-40% higher than those of v3.2. This difference can also be found at higher levels. Figure 3.3 shows that at 2000Z, the 200 m wind speeds of v4.0 are again about 40% higher than those of v3.2. The v4.0 wind field also exhibits more spatial and temporal variablility than that of v3.2.

b. Vertical motion
Vertical motion fields generated by MC2 versions 3.2 and 4.0 are shown in Figure 3.4. Both fields are shown for 200m (in the lower half of the lake breeze circulation) at 2000Z. The v4.0 solution has more intense upward and downward vertical velocities not only near lake breeze fronts but inland as well. As illustrated in Figure 3.4, v4.0 maximum upward velocities (along the lake shore) are much larger than that of v3.2 with the maximum v4.0 values almost three times the values of v3.2. Also, v4.0 generates stronger vertical motions than v3.2 over the mountainous region south of Lake Ontario.

c. Temperature
Surface (1.5 m) air temperature fields are shown in Figures 3.5a and 3.5b at 2000Z (lake breezes fully developed). In Figure 3.5a from v3.2, the coolest air is over eastern Lake Ontario, and the warmer regions are located on northeast and southeast sides of the Lake. In contrast, v4.0 (Figure 3.5b) shows the cooler air is along southern edge of the lake and the warmer air on the northeast and west sides of the Lake. In general, version v4.0 temperatures are slightly cooler over both land and water.

d. Dew point temperature and relative humidity
In Figures 3.6a and 3.6b, dew point temperature fields at 60m, valid at 2000Z, are shown to have significant differences between v3.2 and v4.0. In Figure 3.6a (v3.2), the highest values occur along both the northeast and southeast shores of Lake Ontario. The higher dew point temperature region is confined to the lake area. In Figure 3.6b (v4.0), the highest value of dew point temperature occurs along the south side of Lake Ontario, with the maximum value higher than that of v3.2. On the other hand, the area of lower dew point temperature in v4.0 is much larger than in v3.2. Even more interesting is that the higher value area is not only over the lake but spills over to the southeast of Lake Ontario moving inland tens of kilometers. This point is made clearer by showing relative humidity. Figure 3.7a shows the v3.2 relative humidity at 60m valid at 2000Z. The high moisture area is confined to the area over Lake Ontario with much drier air inland. Figure 3.7b shows v4.0 relative humidity at 60m valid at 2000Z. The higher humidity area obviously affects much of the south shore region of Lake Ontario. Note also that the relative humidity over land is significantly lower for v4.0 than for v3.2.

The surface dew point temperature and relative humidity fields are not used in this analysis since, as was mentioned in the previous section, problems with the v4.0 model code for the first model level resulted in unrealistically high dew point temperatures and relative humidities. The problem was found to be negligible above the first level.

e. Divergence
Figures 3.8a and 3.8b show the divergence field at 10m valid at 2000Z for v3.2 and v4.0. It is clear that divergence / convergence is much stronger for v4.0 than for v3.2. The v4.0 results show the strongest convergence in a line along the lakeshore associated with the lake breeze front. There is also much more divergence / convergence inland. It appears that v4.0 is able to simulate stronger vertical motions and generate more vorticity.

f. Pressure
The sea level pressure patterns generated by v3.2 and v4.0 are quite different. Figure 3.9a shows that the sea level pressure generated by v3.2 is higher only over the lakes. However, v4.0 results in Figure 3.9b show not only high pressure over the lakes, but lots of pressure waves / noise over land.  

3.4 Summary

The greatest differences between v3.2 and v4.0 are found in the lowest model levels. Thus, changes to the physics package were found to have a significant effect. For example, in version 4.0, there are a couple of new schemes to improve vertical motion and free convection such as a new free convection formula, shallow convection scheme, vertical diffusion coefficient option and reduction of horizontal diffusion coefficient. These changes make version 4.0 more sensitive to turbulence effects, and thus cause wind speeds and directions to change, vertical motion to increase, and humidity distribution differences. Version 4.0 does produce some unrealistic results since there are apparently still some bugs in the physics package. In general, however, we find that version 4.0 generates more realistic physical fields.

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