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: 
3.0 MODEL PHYSICS IMPROVEMENTS: COMPARING MC2 V3.2 AND V4.0
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:
and
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 ¶qv / ¶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.