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<br />1078
<br />
<br />and
<br />
<br />JOURNAL OF CLIMATE AND APPLIED METEOROLOGY
<br />
<br />t
<br />VOLUMJ 22
<br />
<br />Po = PGFR + PGACW + PGACR + PGSUB
<br />
<br />+ (1 - {j2)PSACR + (1 - {j3)P1ACR + PGMLT,
<br />. P's = PSFW + PSACW + {j2PSACR + {j3PIACR
<br />+ {jtPSDEP + (1 - {jt)PSSUB + PSMLT. (56)
<br />
<br />In (53), Too is a reference temperature and 4>' is related
<br />to entropy and equivalent potential temperature.
<br />Additionally, {j is 1.0 for T < ooe and 0 otherwise,
<br />and U R, U G , Us represent terminal velocities for rain,
<br />hail and snow, respectively, and k is the unit vector
<br />in the z-direction.
<br />The first two terms on the right-hand side of (53)
<br />represent the advection and turbulent mixing effects.
<br />The third term shows the heating effect when freezing
<br />liquid water, or a cooling effect when melting, or part
<br />of the sublimational cooling effect when hail and
<br />snow are outside a cloudy environment. The fourth
<br />term indicates the energy needed to warm the melted
<br />hail and snow from ooe to the ambient temperature.
<br />The last two terms represent the energy changes due
<br />to the various hydro meteors coming into thermal
<br />equilibrium with the environment as they move
<br />through a temperature gradient. For subsaturated
<br />conditions, the wet-bulb temperat4re is actually more
<br />appropriate for rain and melting snow and hail than
<br />the environmental temperature used here (Kinzer
<br />and Gunn, 195 I).
<br />
<br />4. Boundary and initial conditions and numerical
<br />techniques
<br />
<br />a. Boundary conditions
<br />
<br />Since the domain in the cloud model is limited,
<br />boundary conditions must be specified along all sides
<br />of the model' domain. The treatment of boundary
<br />conditions can have a significant impact on model
<br />results, as is indicated by the considerable amount of
<br />interest and research being devoted to the problem.
<br />The boundary conditions applied in this model will
<br />now be described.
<br />The top boundary is assumed to be rigid with all
<br />. variables held constant. The vorticity, vertical veloc-
<br />ity, rain, snow, hail, cloud water, and cloud ice are
<br />all set to zero. The stream function, entropy, and
<br />water vapor mixing ratio are maintained undisturbed
<br />at their init~al values.
<br />At the lower boundary, the vertical velocity, vor-
<br />ticity and streamfunction are set to zero. Evaporation
<br />and heating rates at the surface are prescribed. Heat
<br />and water vapor are allowed to diffuse into the lower
<br />boundary. The cloud is not permitted to form at the
<br />surface, but precipitation is allowed to fall through
<br />the surface level.
<br />
<br />(55)
<br />
<br />At the lateral boundaries, the treatment is different
<br />from an earlier treatment of lateral boundary con-
<br />ditions in Orville and Kopp (1977). In the old treat-
<br />ment, the horizontal gradients were set equal to zero
<br />at the lateral boundaries as follpws:
<br />
<br />X(1, K) =,x(2, K) and X(97, K) = X(96, K)
<br />
<br />so that actually
<br />
<br />ax = 0 at J = lilt, 961/2,
<br />ax
<br />
<br />where X(J, K) is a variable on the grid point (J, K),
<br />J and K vary from I to 97; J = K = 1 is the lower
<br />left-hand corner of the grid, and J = K = 97 is the
<br />upper right-hand corner of the grid. The new treat-
<br />ment is modified to keep the physics intact by using
<br />fictitious points whenever information beyond the
<br />model domain is required. In other words, we estab-
<br />lish fictitious points, (0, K) and (98, K), and let
<br />
<br />X(O, K) = X(2, K) and X(98, K) = X(96, K),
<br />
<br />so that
<br />
<br />ax = 0 J 97
<br />ax at = I, ,
<br />
<br />i.e., the centered horizontal derivative of all variables
<br />at a lateral boundary is assumed to be zero. The ver-
<br />tical velocity along the lateral boundary is zero be-
<br />cause a1/;/ax = 0 there. Boundary conditions for the
<br />advection calculations are slightly different. For in-
<br />flow boundaries the second derivative is also zero.
<br />That is to say
<br />
<br />X(l, K) = X(2, K) and X(97, K) = X(96, K),
<br />
<br />as well as the fictitious points described above. For
<br />outflow boundaries upstream differencing is used for
<br />advection. After the advection calculations are com-
<br />pleted, the model prognostic equations are solved at
<br />the lateral boundary grid points.
<br />
<br />b. Initial conditions
<br />
<br />Radiosonde sounding data oftemperature, humid-
<br />ity and pressure are used as input data. The horizontal
<br />wind in the direction of motion of the storm is re-
<br />duced to allow the storm to remain in the domain.
<br />Dynamically, this approach is not on firm ground,.
<br />and it would be better to subtract a mean wind from
<br />the actual winds and thus allow the domain to move.
<br />This approach will be incorporated in future work.
<br />A warm moist bubble is used to initiate the con-
<br />vection; i.e., temperature and water vapor perturba-
<br />tions are prescribed in the boundary layer with max-
<br />imum excesses at 30e and 2 g kg-I, respectively. The
<br />perturbations take the form of a sine curve distri-
<br />bution with its maximum at the central point of each
<br />modified horizontal layer, i.e.,
<br />
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