MAR 572 Geophysical Simulation

Wed/Fri 2:20-3:30 pmEndeavour 168

Prof. Marat Khairoutdinov Office hours: Tue/Thu 2:00-3:00 pm, or by email appt; Endeavour 121Email: mkhairoutdin@ms.cc.sunysb.edu Class website: http://rossby.msrc.sunysb.edu/~marat/MAR572.htmllogin: MAR572; passwd: coolclouds

Recommended TextbooksFletcher, C. A. J., 1991: Computational Techniques for Fluid Dynamics, Vol. 1. 2nd ed. Springer-Verlag, 401 pp.Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynamics. Springer, 465 pp.Daley, R. 1991: Atmospheric data analysis. Cambridge University Press, 455pp

Grading50% Homework 50% 3 exams (No final)

Grade conversion: A: >90, B+: 86-90, B: 76-85, C+: 70-75, C: <70

Homework PolicyHomework will be handed out weekly and is due in one week. After the due date, homework can be turned in for 50% credit. Homework can be discussed with other students; however, each student is expected to write the solutions independently.

Hands-on experience is the best way to learn numerical methods. Homework will involve writing simple programs and plotting the results. You will need to have access to computers with programming and graphing software. Knowledge of high-level compiled (e.g., Fortran (preferred), C) or scripting (e.g., IDL, Matlab) computer languages is required for this course. It is, however, up to you which programming language or graphing application to use.

Absolutely no cellphone/texting is allowed during the class (except for me)!

Fundamentals of Finite-Difference SchemesDefinitions of consistence, convergence, and stability; First and second order derivatives; Construction of higher order approximations; Numerical solution of nonlinear equations.

Methods for Initial-Value Problems of Linear Partial Differential EquationsLinear computational stability analysis; Classification and canonical forms; Basic numerical schemes for advection and diffusion equations; Upstream and downstream biased schemes; Time-integration schemes; Time-splitting and directional splitting schemes; Implicit and explicit schemes; Numerical diffusion and dispersion; Extension to multiple dimensions; Grid systems.

Methods for Nonlinear Initial-Value ProblemsFourier representation of discrete fields; Nonlinear interaction and instability; Methods to eliminate nonlinear instability; Construction of conservation schemes; Monotonic and positive definite schemes; Barotropic vorticity model; the Arakawa Jacobian; Basic concepts of spectral methods; Semi-Lagrangian and finite-volume methods

Methods to Solve Elliptic EquationsFourier method; Relaxation methods; Multi-grid methods; Tri-diagonal matrix solver Data AnalysisClassical objective analysis; Statistical estimation; Maximum likelihood estimation;Least variance estimation; Kalman filtering; Statistical spatial interpolation; Variational analysis method; Adjoint models; Multivariant analysis

MAR 572 Geophysical Simulation

Top 500 supercomputers

15 FEBRUARY 2003 621K H A I R O U T D I N O V A N D R A N D A L L

FIG. 13. (top to bottom) Uncertainty of the time series of the sim-ulated shaded cloud fraction, precipitable water, and surface precip-itation rate for the ensemble runs (solid lines) and the sensitivity-to-microphysics runs (dashed lines). The uncertainty is defined as amaximum spread among the runs at various times.

km. A relatively good performance of the model withsuch coarse horizontal resolution as 16 km and even 32km was very surprising suggesting that, perhaps, strong-ly forced CRM simulations could be too self-constrainedand may not fully reveal the model deficiencies.Sensitivity to the domain size has been tested for both

2D and 3D CRMs using much shorter 4-day simulationsto accommodate the computationally expensive 3D sim-ulations performed over the domain as small as 256 km� 256 km, and as large as 1024 km � 1024 km. The2D simulations were performed for even wider domainsranging from 512 to 9192 km. Besides the familiar dif-ferences between the 2D and 3D models, we found verylittle sensitivity to the size of the domain of the samedimensionality. Despite much higher expected predict-ability of the 4-day runs compared to the 28-day runs,a strong bifurcation of the precipitable water after astrong precipitation event was found to be similar to theone that occurred in much longer runs.Sensitivity to the parameters that prescribe charac-

teristics of the precipitating hydrometeors, water auto-conversion and ice aggregation rates, cloud ice aggre-gation threshold, and cloud ice sedimentation velocity,was also tested using interactive radiation rather thanprescribed rates. As expected, the hydrometeors mixingratios are strongly affected by the changes to the mi-crophysics scheme as is the simulated cloud fraction,with the latter varying within a factor of two. It is foundthat changes to the autoconversion/accretion rate co-efficients, as well as the increase of the cloud ice ag-gregation threshold, affect the amount of cloud waterand ice most profoundly, and have the strongest effectson the mean dynamical and thermodynamical statistical

properties of the simulated convection. However, theeffects on the predicted mean temperature and moisturebias are found to be relatively modest, most likely, dueto relatively low cloud occurrence frequency and ratherstrong large-scale advective tendencies in this particularcase of continental convection.It is also found that the spread among the time series

of the simulated shaded cloud fraction, precipitable wa-ter, and precipitation rates for different configurationsof the microphysics scheme is mostly within the rangeof the ensemble runs. This rather unexpected resultstrongly suggests that long time series, like the onesused in this study, may be, in general, inadequate toverify the accuracy of microphysics schemes because itmight be difficult to differentiate the changes that oc-curred due to a particular modification to the micro-physics scheme from those that occurred simply bychance. Therefore, it may be useful to estimate the fun-damental uncertainty of CRM simulations using ensem-ble runs before making any definitive conclusions aboutthe sensitivity of CRM simulations to various modelparameters, because some of the revealed sensitivitiescould in fact be statistically insignificant.In conclusion we would like to stress that this study

was not intended to reveal the full range of the modelphysical sensitivities and uncertainies because of therestrictive nature of prescribed external forcing with nofeedbacks. Such sensitivities could be amplified whensuch feedbacks to the large-scale circulation are in-cluded, a subject that we may explore in the future.

Acknowledgments. This research was supported inpart by the U.S. Department of Energy Grant DE-FG03-02ER63323 to Colorado State University as part of theAtmospheric Radiation Measurement Program, and Na-tional Science Foundation Grant ATM-9812384 to Col-orado State University.

APPENDIX A

Model Equations

a. Prognostic equations

The anelastic momentum and scalar conservation andcontinuity equations are written in tensor notation as

�u 1 � � p�i � � (�u u � � ) � � � Bi j i j i3�t � �x �x �j i

�ui� � f (u � U ) � , (A1)i j3 j g j � ��t l.s.

��u � 0, (A2)i�xi

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

Momentum equations (anelastic) Scalar equations

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

�h 1 �L � � (�u h � F )i L h iL�t � �xi

1 �� (L P � L P � L P )c r s s s g� �z

�h �hL L� � , (A3)� � � ��t �trad l.s.

�q�q 1 � pT � � (�u q � F ) �i T q iT � ��t � �x �ti mic

�qT� , and (A4)� ��t l.s.

�q 1 �p � � (�u q � F )i p q ip�t � �xi

�q1 � p� (P � P � P ) � . (A5)r s g � �� �z �t mic

Here, ui(i� 1, 2, 3) are the resolved wind componentsalong the Cartesian x, y, and vertical z directions, re-spectively; � is the air density; p is pressure; hL is liquid/ice water static energy [�cpT � gz � Lc(qc � qr) �Ls(qi � qs � qg)]; qT is total nonprecipitating water(water vapor � cloud water � cloud ice) mixing ratio(�q� � qc � qi � q� � qn); qp is total precipitatingwater (rain � snow � graupel) mixing ratio (�qr � qs� qg); f is Coriolis parameter; Ug is prescribed geostrophic wind; B is buoyancy [��g(��/�) � g(T�/ �T0.608 � qn � qp � p�/ )]; g is gravitational accel-q� p�

eration; cp is specific heat at constant pressure; Lc andLs are latent heat of evaporation and sublimation, re-spectively; �ij is subgrid-scale stress tensor; F , F , andh qL T

F are subgrid-scale scalar fluxes; Pr, Ps, and Pg areqprain, snow, and graupel precipitation fluxes, respective-ly; the subscript ‘‘rad’’ denotes the tendency due toradiative heating; ‘‘mic’’ represents the tendency of pre-cipitating water due to conversion of cloud water/iceand due to evaporation; ‘‘l.s.’’ denotes the prescribedlarge-scale tendency; the overbar and prime representthe horizontal mean and perturbation from that mean,respectively.

b. Subgrid-scale (SGS) modelThe SGS model has been adopted from the LES mod-

el of Khairoutdinov and Kogan (1999), and is similarto the 1.5-order closure model of Deardorff (1980). Themodel has also an option to use a simple first-orderSmagorinsky closure scheme. Both closures define thelocal moist Brunt–Vaisala frequency in terms of themodel thermodynamic variables as

[h � 0.61Tc q � (L � c T )q ]g � L p T p p2N � � �T �z cp

outside cloud, and (A6)

h � (L � c T )(q � q )g � L p T p2N � � � [ ]T �z c � L(�q /�T )p s

inside cloud. (A7)The SGS eddy exchange coefficient is proportional to

the local grid scale squared, which is usually computedin LES models as a geometric mean of all three gridspacings. However, in cloud resolving simulations of deepconvection, it is rather typical (especially near the surface)for the horizontal grid to be much coarser than the verticalgrid, which can lead to high values of the SGS eddyexchange coefficient, and thus, to artificially excessive ver-tical mixing. To prevent this undesirable behavior, onlythe vertical grid spacing is used as the SGS grid lengthscale in the case of highly anisotropic grids.

c. Hydrometeor partitioningThe partitioning among the hydrometeors is assumed

to be as follows:q � � q , (A8)c n n

q � (1 � � )q , (A9)i n n

q � � q , (A10)r p p

q � (1 � � )(1 � � )q , and (A11)s p g p

q � (1 � � )� q , (A12)g p g p

where the partition functions �n, �p, and �g dependonly on temperature:

T � T00m� � max 0, min 1, , (A13)m � �[ ]T � T0m 00m

where, m � n, p, g,so that �m � 0 for T � T00m, �m � 1 for T � T0m, and0 � �m � 1 for T00m � T � T0m.The total cloud condensate (cloud water � cloud ice)

qn is diagnosed from the prognostic thermodynamicalvariables along assuming the so-called all-or-nothingapproach, so that no excess of water vapor with respectto the water vapor saturation mixing ratio is allowed.The latter is defined as a linear combination of the sat-uration mixing ratios over water and ice:

q � � q � (1 � � )q .sat n satw n sati (A14)Given the liquid/ice water static energy hL, total non-

precipitating qT and total precipitating qp water mixingratios, and adopting the partitioning relations (A8)–(A14),one can diagnose the temperature and, consequently, themixing ratio of various hydrometeors using a suitable it-erative procedure. In this model, we use a variant of therapidly converging Newton–Raphson iterative method.

d. Bulk microphysics equationsThe conversion rates among the hydrometeors are

parameterized assuming that a number concentration Nm

Arakawa C-grid

. φ

Δx

Δy

Δz

System for Atmospheric Modeling (SAM) Cloud-Resolving Model (CRM)

Khairoutdinov and Randall (2003)