Model mass balance - Revision history

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-20T15:41:51Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-20T14:57:38Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:57:44Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:56:30Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:51:10Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:46:50Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:46:08Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:41:44Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:25:26Z

subroutine nexttimestep_ebm

RosDeath: /* subroutine nexttimestep_ebm */

2007-11-13T17:25:08Z

subroutine nexttimestep_ebm

@@ Line 113: / Line 113: @@
 where <math>\rho\,</math> is density, d is the depth of the thermally-active layer and c is the specific heat capacity (we currently assume snow throughout).  ''A'', ''B'' and ''C'' are free parameters. <math>\alpha\,^{'}</math> is the co-albedo of the surface, ''I'' is the insolation at the surface; the 2-m air temperature (<math>T_{2m}</math>) is based on the daily mean and amplitude value supplied. All temperatures are in K.<br>
-<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 Wm<sup>2</sup>. <math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon. <br>
+<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 Wm<sup>-2</sup>. <math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon. <br>
 We use an implicit time-marching scheme,

@@ Line 113: / Line 113: @@
 where <math>\rho\,</math> is density, d is the depth of the thermally-active layer and c is the specific heat capacity (we currently assume snow throughout).  ''A'', ''B'' and ''C'' are free parameters. <math>\alpha\,^{'}</math> is the co-albedo of the surface, ''I'' is the insolation at the surface; the 2-m air temperature (<math>T_{2m}</math>) is based on the daily mean and amplitude value supplied. All temperatures are in K.<br>
-<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 W<math>m^{2}</math>. <math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon. <br>
+<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 Wm<sup>2</sup>. <math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon. <br>
 We use an implicit time-marching scheme,

@@ Line 116: / Line 116: @@
 We use an implicit time-marching scheme,
-<math>T^{n}-T^{n-1}=\frac{\delta\,t}{\rho\,cd}\left({C\alpha\,^{'}I^{n}+BT_{2m}^{n}+\frac{A}{2}\left({T^{n}-T^{n-1}}\right)}\right)</math>
+<math>T^{n}-T^{n-1}=\frac{\delta\,t}{\rho\,cd}\left({C\alpha\,^{'}I^{n}+BT_{2m}^{n}+\frac{A}{2}\left({T^{n}-T^{n-1}}\right)}\right)</math>.
+In these subroutines, time through the day is in radians.  If the new temperature is above the melt point then we estimate an amount of melt by converting energy in the active layer into an equivalent melt depth.  We use the trapezium rule to do this.  There are two cases.
 #The first is where the preceeding temperature is at the melt point also; this is easy.
 #The second in when the preceeding temperature is below the melt point, in which case we need to estimate the fraction of the time step during which temperature was above the melt point (using linear interpolation).

@@ Line 113: / Line 113: @@
 where <math>\rho\,</math> is density, d is the depth of the thermally-active layer and c is the specific heat capacity (we currently assume snow throughout).  ''A'', ''B'' and ''C'' are free parameters. <math>\alpha\,^{'}</math> is the co-albedo of the surface, ''I'' is the insolation at the surface; the 2-m air temperature (<math>T_{2m}</math>) is based on the daily mean and amplitude value supplied. All temperatures are in K.<br>
-<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 W<math>m^{2}</math>. .<math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon
+<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 W<math>m^{2}</math>. <math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon. <br>
 We use an implicit time-marching scheme

@@ Line 113: / Line 113: @@
 where <math>\rho\,</math> is density, d is the depth of the thermally-active layer and c is the specific heat capacity (we currently assume snow throughout).  ''A'', ''B'' and ''C'' are free parameters. <math>\alpha\,^{'}</math> is the co-albedo of the surface, ''I'' is the insolation at the surface; the 2-m air temperature (<math>T_{2m}</math>) is based on the daily mean and amplitude value supplied. All temperatures are in K.<br>
-<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 W<math>m^{2}</math>.    Appropriate sets are taken to ensure that the sun does not sun when it is below the horizon.
+<math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)and <math>I_{0}</math> is the insolation received at the top of the atmosphere, and is given a constant value of 1371 W<math>m^{2}</math>. .<math>\theta_{d}\,</math> is the fraction of the day expressed in radians and relative to midday. Appropriate sets are taken to ensure that there is no insolation when the sun is below the horizon
 We use an implicit time-marching scheme

@@ Line 112: / Line 112: @@
-where   is the surface’s aspect and   its slope (in radians);   is the fraction of the day expressed in radians and relative to midday; A, B and C are free parameters;  is the co-albedo of the surface; d is the depth of the thermally-active layer,  its density and c its specific heat capacity (we currently assume snow throughout); the 2-m air temperature (T2m) is based on the daily mean and amplitude value supplied; and temperatures are in K.  Appropriate sets are taken to ensure that the sun does not sun when it is below the horizon.
+where <math>\rho\,</math> is density, d is the depth of the thermally-active layer and c is the specific heat capacity (we currently assume snow throughout).  ''A'', ''B'' and ''C'' are free parameters. <math>\alpha\,^{'}</math> is the co-albedo of the surface, ''I'' is the insolation at the surface; the 2-m air temperature (<math>T_{2m}</math>) is based on the daily mean and amplitude value supplied. All temperatures are in K.<br>
+xwhere <math>\beta\,</math> is the surface’s aspect, <math>\phi\,</math> is the slope (in radians)      Appropriate sets are taken to ensure that the sun does not sun when it is below the horizon.
 We use an implicit time-marching scheme

@@ Line 100: / Line 100: @@
 <math>\rho\,cd\frac{dT}{dt}=C\alpha\,^{'}I+BT_{2m}+AT </math><br>
 ::<math>I =I_{0}\left({\cos\,z\cos\phi\,+\sin\,z\sin\,\phi\,\cos\,\left({\psi\,-\beta\,}\right)}\right)</math><br>
 :<math>cos\,z=z^{'}_{1}+z^{'}_{2}\cos\,\theta_{d}</math><br>
 :<math>cos\,\psi\,=\frac{\psi\,^{'}_{1}\cos\,\theta\,_{d}-\psi\,^{'}_{2}}{\cos\,z}</math>

← Older revision		Revision as of 17:25, 13 November 2007
Line 103:		Line 103:
	::<math>I =I_{0}\left({\cos\,z\cos\phi\,+\sin\,z\sin\,\phi\,\cos\,\left({\psi\,-\beta\,}\right)}\right)</math><br>		::<math>I =I_{0}\left({\cos\,z\cos\phi\,+\sin\,z\sin\,\phi\,\cos\,\left({\psi\,-\beta\,}\right)}\right)</math><br>

−	:<math>cos\,z=z^{'}_{1}+z^{'}_{2}\cos\,\theta_{d}</math>	+	:<math>cos\,z=z^{'}_{1}+z^{'}_{2}\cos\,\theta_{d}</math><br>
		+
		+	:<math>cos\,\psi\,=\frac{\psi\,^{'}_{1}\cos\,\theta\,_{d}-\psi\,^{'}_{2}}{\cos\,z}</math>