The Planet Surface Rotational Warming Phenomenon

The Planet Mean Surface Temperature Equation Tmean = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

"Specular reflection from a body of water is calculated by the Fresnel equations.[6] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection."

https://en.wikipedia.org/wiki/Reflectance#Hemispherical_reflectance

Spherical zone height calculation

Surface area of spherical zone

A = 2πrh

r - the radius of a planet

h - the height of a spherical zone

The height  "h.i " of a spherical zone  "i"  at the point of solar flux's angular incidence "θ°i"

 

h.i = [ r*cos θ°i - r*cos θ°(i+1) ] =

= r [ cos θ°i - cos θ°(i+1) ]

The total amount of the specularly reflected portion of solar flux

Our comment:

The planet solar irradiated area from normal point to the terminator of 90o is gradually increasing in dimensions. The further away on the globe's surface from the point of ZENITH INCIDENCE ANGLE to the larger angles of incidence the more extend in dimensions the spherical zones' areas are.

So the larger angles of incidence are accompanied with much larger areas times the much higher the specular reflection portion outgoing to space from them.

Consequently the Φ = 0,47 and the specular reflection of the "waterworld" sphere is expected to be very much comparable.

End of comment.

In order to demonstrate our thought we shall make the effort, and we shall calculate, for the entire solar lit hemisphere, the total amount of the specularly reflected portion of solar flux, based on the above graph's approximate reading data.

We shall divide the by the solar flux lit entire hemisphere in small surface area spherical zones, by proceeding with small 2,5° steps of the angle of incidence variables.

We shall calculate every spherical zone's area (m²) within every and each of this small angle of incidence change 2,5° steps [ θ°(i+1) - θ°i = 2,5° ]

Then we shall read the approximate values on water surface specular reflectivity graph for every chosen for calculation spherical zone (angle of incidence)

(Reflectance of smooth water at 20°C (refractive index 1.333).)

We shall then calculate the product for every small spherical zone area with the related to the same angle of incidence the local reflectance from graph.

Finally we shall summarize all the resulted 2,5° steps (spherical zones * local reflectance from graph) products, and average the sum over the planet's cross-section area, which is perpendicular to solar flux (the area of the perpendicular incidence).

Reflectance of smooth water at 20°C (refractive index 1.333)

Reflectance of smooth water at 20°C (refractive index 1.333).

Water-earth total specular reflection based on the Fresnel reflection

Specular reflection from a body of water is calculated by the Fresnel equations.[6] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

The radius of a planet is r

The height " h.i " of a spherical zone " i " at the point of solar flux's angular incidence θ°i is

In the above scheme we explain the spherical zone's height calculation method:

h.i = [ r*cos θ°i - r*cos θ°(i+1) ] =

= r [ cos θ°i - cos θ°(i+1) ]

Analysis of terms.

S - the incident on the planet solar flux (W/m²), perpendicular to the planet's cross-section

r - planet's radius (m)

πr² - planet's cross-section area perpendicular to the solar flux's beams (m²)

N - the normal to the surface

θ° - angle of solar flux's incidence

θ°i - angle of solar flux's incidence at i point

h.i = r [ cos θ°i - cos θ°(i+1) ] - the i spherical zone area height

A.i - spherical zone area m² at point i

A.i = 2πr * ( r*cos θ°i - r*cos θ°i+1 ) - spherical zone area at i point (m²)

A.i = 2πr² * ( cos θ°i - cos θ°i+1 ) (m²)

...........

Spec.i - ( specular reflectivity at point i ) ( specular reflectivity at point i ) taken from graph for the ( θ°i ) angle of incidence

A.i * Spec.i - the total incident on zone Ai area solar irradiation reflected portion m² *W/m² = W

Σ ( Α.ι* Spec.i ) - the Sum total incident on the entire hemisphere's surface solar irradiation reflected portion (W)

Σ ( Α.ι* Spec.i ) /πr² - the total specular reflection portion of the incident solar flux, averaged on the planet's cross-section disk (W/m²)

When substituting terms in the above sentence we would have:

Σ [ 2πr² * ( cos θ°i - cos θ°i+1 ) (m²) * Spec.i W/m² ] /πr² (m²)

When simplifying by eliminating the πr² term

Σ 2 *( cos θ°i - cos θ°i+1 ) * Spec.i (W/m²) - the sphere's total specular reflection portion of the incident solar flux, averaged on the planet's cross-section disk perpendicular to the incoming solar flux

or

2 * Σ [ ( cos θ°i - cos θ°i+1 ) ] * Spec.i (W/m²) (1)

let's symbolize the ( cos θ°i - cos θ°i+1 ) expression with Δcosθ°i term

So we shall write:

2 * Σ Δcosθ°i * Spec.i (W/m²)

Table of data (by 2,5° steps ) and the product ( Δcosθ°i * Spec.i ) results

Angle of

incidence..................cosθ°i - cosθ°i+1 ..graph data.......product

θ°ι...............cos θ°ι.......... Δcosθ°i........Spec.i....... Δcosθ°i * Spec.i

0°......................1.............0,00095..............0....................0

2,5°................0,99905.........0,00285........0,02...............0,00019 5°............0,99619.........0,004750............0,02...............0,000056

7,5°................0,99144.........0,006637......0,02............... 0,000091

10°..........0,98481..........0,0085117............0,02............ 0,000135

12,5°..............0,97630..........0,0104........0,02................ 0,000165 15°..........0,96593............0,01221...........0,02...............0,00020

17,5°..............0,95372...........0,0140...........0,02............. 0,00024

20°..........0,93969............0,0158...........0,02................0,00028

22,5°..............0,92388..........0,0176...........0,02.............. 0,00031 25°..........0,90631............0,0193...............0,02.............0,00035

27,5°..............0,88701...........0,0210........0,02............... 0,00036

30°..........0,86603...........0,0226.............0,02...............0,00042 32,5°..............0,84339...........0,0242..........0,02...............0,000452 35°..........0,81915...........0,0258..............0,02.................0,000484 37,5°.............0,79335...........0,0273.........0,02...............0,000516 40°..........0,76604...........0,0288.............0,02...............0,000546 42,5°.............0,73728............0,0302..........0,023.............0,000662 45°..........0,70711...........0,0315...............0,025...........0,000755 47,5°.............0,67559............0,0329...........0,031............0,00098 50°..........0,64279...........0,0340.................0,035.............0.00115 52,5°..............0,60876...........0,035.............0,037............0,00126 55°..........0,57358..........0,0361.....................0,040.........0,00141 57,5°..............0,53730...........0,0373............0,055............0,00200 . 60°..........0,5....................0,0383...............0,065...........0,00243 62,5°..............0,46175...........0,0391............0,085............0,00325 65°..........0,42262............0,0399.................0,1..............0,003913 67,5°..............0,38268...........0,0407...............0,17...........0,00679 70°..........0,34202.............0,0413................0,22..............0,00895

72,5°..............0,30071..........0,0419...............0,27...........0,01115

75°..........0,25882.......... 0.0424 ...............0,30.............0,01257 77,5°..............0,21644...........0,0428............0,39............0,01653 80°..........0,17365...........0,0431................0,45................0.01926 82,5°..............0,13053...........0,0434.............0,60..............0,02587 85°..........0,08716...........0,0435...............0,70................0,03036 87,5°..............0,04362..........0,0436..............0,82............0,03570 90°...................0...........................................1.............0,04362

Σ....................................................................................0,217

When summarizing from the Table the Δcosθ°i * Spec.i the product results we shall have

Σ Δcosθ°i * Spec.i = 0,217

and multiplying times 2 according to the equation (1) above

2 * Σ Δcosθ°i * Spec.i = 0,217 * 2 = 0,434

- it is the specularly reflected portion of the incident solar flux It is the sphere's total specular reflection portion of the incident solar flux, which is averaged on the planet's cross-section disk.

When considering an oceanic-like planet Earth total reflected energy the diffuse a*So + specular 0,434 *So =

= 0,3 * 1.362W/m² + 0,434 * 1.362W/m² =

= 0,734 * 1.362 W/m² = 999,71 W/m² REFLECTED

and only 1.362 - 999,71 = 362 W/m² "ABSORBED"

This result (362 W/m² "ABSORBED") is in a satisfactory magnitude accordance with the smooth planet surface the solar incident flux's "absorption"  ( 444 W/m² "ABSORBED").

Φ(1 - a)So = 0,47(1 - 0,306)1362 W/m² =  444 W/m² "ABSORBED"

when compared with the blackbody theory the 1.362 * (1 - 0,306) =

= 945 W/m² "ABSORBED"

The difference is more than twice as much!