Validation of mantle convection code of spherical shell geometry

Our numerical results were compared with those by Stemmer et al. (2006), for steady-state flow patterns with weakly temperature-dependent viscosity. Data summarized on July 5, 2007 by Masanori Kameyama.

Tetrahedral patterns Cubic patterns

r_η=1 r_η=20 r_η=1 r_η=30

3-D distribution of temperature (green for T > 0.4; blue for T < 0.1)

Distribution of temperature at mid-depth r=(r_max+r_min)/2 (contour interval of 0.1)

3-D distribution of lateral temperature anomalies (yellow to red for δT > +0.1; blue for δT < -0.1)

Distribution of lateral temperature anomalies at mid-depth r=(r_max+r_min)/2 (contour interval of 0.05)

Distribution of radial velocity at mid-depth r=(r_max+r_min)/2 (contour interval of 10; positive upward)

Plots of lateral average of temperature at given r

Comparison of global (Nusselt numbers Nu, RMS velocities V_rms and average temperature T_av) and local (maximum/minimum of the interior temperature T_i,max/min and of the interior radial velocity v_i,max/min at mid-depth) of tetrahedral steady-state flow patterns with weakly temperature-dependent viscosity and Ra_1/2=7000.
r_η	Model	Nodes	r×(θ×φ)	Nu	V_rms	T_av	T_i,max	T_i,min	V_i,max	V_i,min
1	St06 (FV)	Ext.		3.4949	32.6234	0.21560	0.89326	0.01731	115.157	-29.616
	KKS (FV)	24576	16×(2×16×48)	3.4090	31.9466	0.22944	0.80282	0.02644	99.8727	-31.686
		196608	32×(2×32×96)	3.4726	32.4842	0.22054	0.86010	0.02030	109.866	-30.560
		1572864	64×(2×64×192)	3.4898	32.6024	0.21705	0.88409	0.01807	113.655	-29.906
		12582912	128×(2×128×384)	3.4945	32.6308	0.21597	0.89061	0.01754	114.700	-29.702
10	St06 (FV)	Ext.		3.2475	27.2878	0.23438	0.90423	0.02115	140.188	-16.970
	KKS (FV)	24576	16×(2×16×48)	3.2100	26.7867	0.24733	0.82195	0.02848	117.059	-18.450
		196608	32×(2×32×96)	3.2410	27.1853	0.23889	0.87555	0.02331	131.975	-17.598
		1572864	64×(2×64×192)	3.2468	27.2696	0.23566	0.89442	0.02171	137.423	-17.204
		12582912	128×(2×128×384)	3.2482	27.2892	0.23469	0.90137	0.02134	139.342	-17.067
20	St06 (FV)	Ext.		3.1526	25.7600	0.24155	0.90861	0.02329	148.063	-14.876
	KKS (FV)	24576	16×(2×16×48)	3.1326	25.4115	0.25361	0.83384	0.02947	124.441	-15.993
		196608	32×(2×32×96)	3.1514	25.6981	0.24576	0.88155	0.02503	139.196	-15.282
		1572864	64×(2×64×192)	3.1532	25.7495	0.24276	0.89981	0.02375	145.085	-15.024
		12582912	128×(2×128×384)	3.1534	25.7607	0.24185	0.90598	0.02343	147.076	-14.936

Comparison of global (Nusselt numbers Nu, RMS velocities V_rms and average temperature T_av) and local (maximum/minimum of the interior temperature T_i,max/min and of the interior radial velocity v_i,max/min at mid-depth) of cubic steady-state flow patterns with weakly temperature-dependent viscosity and Ra_1/2=7000.
r_η	Model	Nodes	r×(θ×φ)	Nu	V_rms	T_av	T_i,max	T_i,min	V_i,max	V_i,min
1	St06 (FV)	Ext.		3.6090	31.0709	0.21583	0.85902	0.02714	109.464	-34.330
	KKS (FV)	24576	16×(2×16×48)	3.5225	30.3245	0.23427	0.73863	0.04006	94.0879	-37.226
		196608	32×(2×32×96)	3.5885	30.9281	0.22274	0.81629	0.03024	104.164	-35.804
		1572864	64×(2×64×192)	3.6044	31.0486	0.21793	0.84559	0.02798	107.779	-34.793
		12582912	128×(2×128×384)	3.6083	31.0741	0.21639	0.85439	0.02745	108.867	-34.438
20	St06 (FV)	Ext.		3.3530	25.0268	0.25087	0.88352	0.03970	134.321	-17.808
	KKS (FV)	24576	16×(2×16×48)	3.3088	24.5016	0.26620	0.76767	0.05022	103.102	-19.335
		196608	32×(2×32×96)	3.3435	24.9239	0.25627	0.84225	0.04210	123.401	-18.398
		1572864	64×(2×64×192)	3.3507	25.0070	0.25239	0.87068	0.04046	130.899	-17.970
		12582912	128×(2×128×384)	3.3525	25.0622	0.25119	0.88027	0.03984	133.454	-17.841
30	St06 (FV)	Ext.		3.2976	24.2435	0.25704	0.88740	0.04223	137.773	-16.234
	KKS (FV)	24576	16×(2×16×48)	3.2648	23.8120	0.27190	0.78712	0.05303	108.540	-17.585
		196608	32×(2×32×96)	3.2915	24.1675	0.26221	0.85126	0.04463	127.386	-16.750
		1572864	64×(2×64×192)	3.2959	24.2300	0.25844	0.87495	0.04293	134.180	-16.369
		12582912	128×(2×128×384)	3.2969	24.2435	0.25728	0.88439	0.04231	136.982	-16.255

Reference

K. Stemmer, H. Harder, and U. Hansen A new method to simulate convection with strongly temperature- and pressure-dependent viscosity in a spherical shell: Applications to the Earth's mantle. Phys. Earth Planet. Inter., 157(3-4):223-249, 2006.