## 2.3.3. Solar Zenith Angle[¶](#solar-zenith-angle "Permalink to this headline")
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The CLM uses the same formulation for solar zenith angle as the Community Atmosphere Model. The cosine of the solar zenith angle \\(\\mu\\) is

(2.3.76)[¶](#equation-3-76 "Permalink to this equation")\\\[\\mu =\\sin \\phi \\sin \\delta -\\cos \\phi \\cos \\delta \\cos h\\\]

where \\(h\\) is the solar hour angle (radians) (24 hour periodicity), \\(\\delta\\) is the solar declination angle (radians), and \\(\\phi\\) is latitude (radians) (positive in Northern Hemisphere). The solar hour angle \\(h\\) (radians) is

(2.3.77)[¶](#equation-3-77 "Permalink to this equation")\\\[h=2\\pi d+\\theta\\\]

where \\(d\\) is calendar day (\\(d=0.0\\) at 0Z on January 1), and \\(\\theta\\) is longitude (radians) (positive east of the Greenwich meridian).

The solar declination angle \\(\\delta\\) is calculated as in [Berger (1978a,b)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#berger1978a) and is valid for one million years past or hence, relative to 1950 A.D. The orbital parameters may be specified directly or the orbital parameters are calculated for the desired year. The required orbital parameters to be input by the user are the obliquity of the Earth \\(\\varepsilon\\) (degrees, \\(-90^{\\circ } <\\varepsilon <90^{\\circ }\\) ), Earth’s eccentricity \\(e\\) (\\(0.0<e<0.1\\)), and the longitude of the perihelion relative to the moving vernal equinox \\(\\tilde{\\omega }\\) (\\(0^{\\circ } <\\tilde{\\omega }<360^{\\circ }\\) ) (unadjusted for the apparent orbit of the Sun around the Earth ([Berger et al. 1993](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bergeretal1993))). The solar declination \\(\\delta\\) (radians) is

(2.3.78)[¶](#equation-3-78 "Permalink to this equation")\\\[\\delta =\\sin ^{-1} \\left\[\\sin \\left(\\varepsilon \\right)\\sin \\left(\\lambda \\right)\\right\]\\\]

where \\(\\varepsilon\\) is Earth’s obliquity and \\(\\lambda\\) is the true longitude of the Earth.

The obliquity of the Earth \\(\\varepsilon\\) (degrees) is

(2.3.79)[¶](#equation-3-79 "Permalink to this equation")\\\[\\varepsilon =\\varepsilon \*+\\sum \_{i=1}^{i=47}A\_{i} \\cos \\left(f\_{i} t+\\delta \_{i} \\right)\\\]

where \\(\\varepsilon \*\\) is a constant of integration ([Table 2.3.8](#table-orbital-parameters)), \\(A\_{i}\\), \\(f\_{i}\\), and \\(\\delta \_{i}\\) are amplitude, mean rate, and phase terms in the cosine series expansion ([Berger (1978a,b)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#berger1978a), and \\(t=t\_{0} -1950\\) where \\(t\_{0}\\) is the year. The series expansion terms are not shown here but can be found in the source code file shr\_orb\_mod.F90.

The true longitude of the Earth \\(\\lambda\\) (radians) is counted counterclockwise from the vernal equinox (\\(\\lambda =0\\) at the vernal equinox)

(2.3.80)[¶](#equation-3-80 "Permalink to this equation")\\\[\\lambda =\\lambda \_{m} +\\left(2e-\\frac{1}{4} e^{3} \\right)\\sin \\left(\\lambda \_{m} -\\tilde{\\omega }\\right)+\\frac{5}{4} e^{2} \\sin 2\\left(\\lambda \_{m} -\\tilde{\\omega }\\right)+\\frac{13}{12} e^{3} \\sin 3\\left(\\lambda \_{m} -\\tilde{\\omega }\\right)\\\]

where \\(\\lambda \_{m}\\) is the mean longitude of the Earth at the vernal equinox, \\(e\\) is Earth’s eccentricity, and \\(\\tilde{\\omega }\\) is the longitude of the perihelion relative to the moving vernal equinox. The mean longitude \\(\\lambda \_{m}\\) is

(2.3.81)[¶](#equation-3-81 "Permalink to this equation")\\\[\\lambda \_{m} =\\lambda \_{m0} +\\frac{2\\pi \\left(d-d\_{ve} \\right)}{365}\\\]

where \\(d\_{ve} =80.5\\) is the calendar day at vernal equinox (March 21 at noon), and

(2.3.82)[¶](#equation-3-82 "Permalink to this equation")\\\[\\lambda \_{m0} =2\\left\[\\left(\\frac{1}{2} e+\\frac{1}{8} e^{3} \\right)\\left(1+\\beta \\right)\\sin \\tilde{\\omega }-\\frac{1}{4} e^{2} \\left(\\frac{1}{2} +\\beta \\right)\\sin 2\\tilde{\\omega }+\\frac{1}{8} e^{3} \\left(\\frac{1}{3} +\\beta \\right)\\sin 3\\tilde{\\omega }\\right\]\\\]

where \\(\\beta =\\sqrt{1-e^{2} }\\). Earth’s eccentricity \\(e\\) is

(2.3.83)[¶](#equation-3-83 "Permalink to this equation")\\\[e=\\sqrt{\\left(e^{\\cos } \\right)^{2} +\\left(e^{\\sin } \\right)^{2} }\\\]

where

(2.3.84)[¶](#equation-3-84 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {e^{\\cos } =\\sum \_{j=1}^{19}M\_{j} \\cos \\left(g\_{j} t+B\_{j} \\right) ,} \\\\ {e^{\\sin } =\\sum \_{j=1}^{19}M\_{j} \\sin \\left(g\_{j} t+B\_{j} \\right) } \\end{array}\\end{split}\\\]

are the cosine and sine series expansions for \\(e\\), and \\(M\_{j}\\), \\(g\_{j}\\), and \\(B\_{j}\\) are amplitude, mean rate, and phase terms in the series expansions ([Berger (1978a,b)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#berger1978a)). The longitude of the perihelion relative to the moving vernal equinox \\(\\tilde{\\omega }\\) (degrees) is

(2.3.85)[¶](#equation-3-85 "Permalink to this equation")\\\[\\tilde{\\omega }=\\Pi \\frac{180}{\\pi } +\\psi\\\]

where \\(\\Pi\\) is the longitude of the perihelion measured from the reference vernal equinox (i.e., the vernal equinox at 1950 A.D.) and describes the absolute motion of the perihelion relative to the fixed stars, and \\(\\psi\\) is the annual general precession in longitude and describes the absolute motion of the vernal equinox along Earth’s orbit relative to the fixed stars. The general precession \\(\\psi\\) (degrees) is

(2.3.86)[¶](#equation-3-86 "Permalink to this equation")\\\[\\psi =\\frac{\\tilde{\\psi }t}{3600} +\\zeta +\\sum \_{i=1}^{78}F\_{i} \\sin \\left(f\_{i} ^{{'} } t+\\delta \_{i} ^{{'} } \\right)\\\]

where \\(\\tilde{\\psi }\\) (arcseconds) and \\(\\zeta\\) (degrees) are constants ([Table 2.3.8](#table-orbital-parameters)), and \\(F\_{i}\\), \\(f\_{i} ^{{'} }\\), and \\(\\delta \_{i} ^{{'} }\\) are amplitude, mean rate, and phase terms in the sine series expansion ([Berger (1978a,b)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#berger1978a))). The longitude of the perihelion \\(\\Pi\\) (radians) depends on the sine and cosine series expansions for the eccentricity \\(e\\)as follows:

(2.3.87)[¶](#equation-3-87 "Permalink to this equation")\\\[\\begin{split}\\Pi =\\left\\{\\begin{array}{lr} 0 & \\qquad {\\rm for\\; -1}\\times {\\rm 10}^{{\\rm -8}} \\le e^{\\cos } \\le 1\\times 10^{-8} {\\rm \\; and\\; }e^{\\sin } = 0 \\\\ 1.5\\pi & \\qquad {\\rm for\\; -1}\\times {\\rm 10}^{{\\rm -8}} \\le e^{\\cos } \\le 1\\times 10^{-8} {\\rm \\; and\\; }e^{\\sin } < 0 \\\\ 0.5\\pi & \\qquad {\\rm for\\; -1}\\times {\\rm 10}^{{\\rm -8}} \\le e^{\\cos } \\le 1\\times 10^{-8} {\\rm \\; and\\; }e^{\\sin } > 0 \\\\ \\tan ^{-1} \\left\[\\frac{e^{\\sin } }{e^{\\cos } } \\right\]+\\pi & \\qquad {\\rm for\\; }e^{\\cos } <{\\rm -1}\\times {\\rm 10}^{{\\rm -8}} \\\\ \\tan ^{-1} \\left\[\\frac{e^{\\sin } }{e^{\\cos } } \\right\]+2\\pi & \\qquad {\\rm for\\; }e^{\\cos } >{\\rm 1}\\times {\\rm 10}^{{\\rm -8}} {\\rm \\; and\\; }e^{\\sin } <0 \\\\ \\tan ^{-1} \\left\[\\frac{e^{\\sin } }{e^{\\cos } } \\right\] & \\qquad {\\rm for\\; }e^{\\cos } >{\\rm 1}\\times {\\rm 10}^{{\\rm -8}} {\\rm \\; and\\; }e^{\\sin } \\ge 0 \\end{array}\\right\\}.\\end{split}\\\]

The numerical solution for the longitude of the perihelion \\(\\tilde{\\omega }\\) is constrained to be between 0 and 360 degrees (measured from the autumn equinox). A constant 180 degrees is then added to \\(\\tilde{\\omega }\\) because the Sun is considered as revolving around the Earth (geocentric coordinate system) ([Berger et al. 1993](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bergeretal1993))).

Table 2.3.8 Orbital parameters[¶](#id14 "Permalink to this table")  
| Parameter
 |  |
| --- | --- |
| \\(\\varepsilon \*\\)

 | 23.320556

 |
| \\(\\tilde{\\psi }\\) (arcseconds)

 | 50.439273

 |
| \\(\\zeta\\) (degrees)

 | 3.392506

 |
