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Basics of Fock’s Theory – the Hartree-Fock Approach to Evaporation Ducts

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Discover the essentials of Fock’s Theory and its application in the Hartree-Fock method for modeling evaporation ducts. This article delves into the principles of quantum mechanics, explaining how Fock’s approach enhances our understanding of atmospheric phenomena.

Learn about the significance of the Hartree-Fock theory in environmental science and its role in predicting evaporation duct behavior.

Basics of Fock’s Theory

In this section we briefly reproduce the basic results obtained by Fock for the vertical dipole in the case of normal refraction, in order to have a reference model for the further studies described in this category.

Following Fock, we introduce a non-dimensional distance and height ξ, h, respectively:

ξ=mxa,    h=kzm              Eq. 1

where x = aθ and z = r–a, the same physical coordinates as defined in Parabolic Approximation to a Wave Equation in a Stratified Troposphere Filled with Turbulent Fluctuations of the Refractive Index“Parabolic Approximation to the Wave Propagation”. The relation between the Debye’s potential U and the slowly varying attenuation factor (envelope) V is given by:

U=exp jkaθa2θ sin θVξ, h1, h2, q             Eq. 2

where h1 and h2 are the non-dimensional heights of the receiver and transmitter, respectively, above the earth’s surface, ξ is the distance between them along the earth’s surface. A normalised impedance q is defined by:

q=jmη+1.             Eq. 3

The attenuation factor V is given by a contour integral in a plane of complex parameter t:

Vξ, h1, h2, q=exp jπ4ξπCdt exp jξt F t, h1, h2, q.             Eq. 4

The contour C comes from infinity in the second quadrant of the t-plane passing around and below all the poles of the integrand and then goes to infinity in the first quadrant of the t-plane. In the case of h2 > h1, the function F is given by:

Ft, h1, h2, q=j2w1th2w2th1w2tqw2tw1tqw1tw1th1.             Eq. 5
In the case h1 > h2 we need to swap the h1 and h2 in the expression for F above. Three characteristic regions can be selected in the space of ξ, h1, h2 or, over the distance ξ, given fixed values of h1, h2. The first region is a line-of-sight region with
ξ<h1+h2,
where the field is composed by superposition of the direct wave and the wave reflected from the earth’s surface. The second region is a shadow region at
ξ<h1+h2,
where the electromagnetic field propagates only due to a diffraction mechanism and, finally, the third region is a transition region with
ξh1+h2,
which separates the line-of-sight region from the shadow region. These regions are shown schematically in Figure 1. The transition region, also called a “shade cone”, is filled with dots in the figure.
Wave propagation
Fig. 1 Geometry of wave propagation

Fock obtained an asymptotical expansion of the integral (4) for all three regions. We reproduce his results for both the line-of-sight and the shadow regions.

In the shadow region the function F can be written in the following form:

Ft, h1, h2, q=w1th2fh1, tw1tqw1t                Eq. 6

where:

fh1, t=w1tqw1t νth1vtqvt w1th1.                Eq. 7

With h1 = 0 the function f and its derivative have the values:

f0, t=0, df/dh1=q.                Eq. 8

The above equations can be obtained using the representation the Airy function w1 via the Airy functions u and ν, see Appendix 1***. From Eq. (8) one can find that if t = tn is a root of the equation:

w1tqw1t=0, t=t1, t2, ...                Eq. 9

the value of the function f is given by:

fh1, ts=fsh1=w1tsh1w1ts               Eq. 10

and can be regarded as a height gain function of the mode with number s. In two limiting cases q = 0 and q = ∞ the roots ts can be estimated from asymptotic expressions valid for large s:

ts,q=03π2s342/3exp jπ3,ts,q=3π2s142/3exp jπ3,               Eq. 11

and the first roots are as follows:

  • t1,q=0=1,01879ejπ/3;
  • t1,q=∞=2,33811ejπ/3.

The integral (4) can then be calculated as a sum of the residues in the poles of the integrand given by Eq. (9):

Vξ, h1, h2, q=exp jπ4 2πξs=1ejξt1tsq2w1tsh1wtsw1tsh2wts.

Equation (11) represents the series of the normal modes of the diffracted waves converging rapidly in a shadow region.

Consider the field in the line-of-sight interference region,
ξ<h1+h2.

In that region we have to obtain the reflection formula corresponding to the reflection from the spherical surface. The integral (4) can be represented by the sum of two terms:

Vξ, h1, h2, q=VdVr,
Vd=12 exp jπ4ξπCdt exp jξt w1th2 w2th1,            Eq. 12
Vr=12 exp jπ4ξπCdt exp jξt w2tqw2tw1tqw1tw1th1 w1th2.            Eq. 13

Assuming that the major contribution to integrals (12) and (13) comes from the interval of large and negative t we may use the asymptotic formulas for the Airy functions w1 and w2. Then for direct wave Vd we obtain:

Vd=12 exp jπ4ξπCdtexpjφth1th2t1/4.            Eq. 14

The phase of the integrand φ(t) is given by:

φt=ξt+23h2t3/223h1t3/2.

The stationary value of ts is determined from φ(ts) = 0. The stationary value of the phase we denote as φ ≡ φ(ts) and it is given by:

φ=h1h224ξ+12ξh1+h2ξ312.               Eq. 15

The value of φ has a simple geometric meaning of the difference between the phase of the direct wave along the distance R between the correspondents and the distance x = aθ along the earth’s surface, Figure 2:

Traces of the rays
Fig. 2 Ray traces in the line-of-sight region
φ=kRx.               Eq. 16

Application of the stationary phase method to integral (14) leads to the following expression for the direct wave:

Vd=expjφ.               Eq. 17

Consider the term Vr. Using a similar asymptotic expression for the Airy function in the integrand we obtain:

Vd=12 exp jπ4ξπCdtexp jψth1th2t1/4 qjtq+jt               Eq. 18

and:

ψt=ξt+23h2t3/2+23h1t3/243t3/2.               Eq. 19

The root of the equation

ψt=0

we define as t = –p2, where p > 0. The stationary point in terms of p will be the root of the equation:

h1+p2+h2+p2=2p+x.               Eq. 20

The stationary value of the phase can then be defined in terms of the parameter p:

ψ=3p2ξ+2ph1+h2ξ2+ξh1+h2ξ33.               Eq. 21

The application of the stationary phase method to Eq. (18) then results in the following expression for reflected wave:

Vr=qjpq+jpA exp jψ              Eq. 22

with:

A=px3px+x2h1h2.              Eq. 23
The formula (22) has the following geometric meaning. The parameter p is p = m cos (γ), where γ is an incidence angle as shown in Figure 2. The factor q-jp/q+jp is a Fresnel coefficient of reflection taken with the opposite sign. The parameter
A
is a correction term accounting for divergence of the beam after reflection multiplied by a facto R/r1, where r1 is the distance along the ray from the source to the reflection point, Figure 2. The phase ψ can be written in a form similar to Eq. (16):
ψ=kr1+r2x.                Eq. 24

Combining both terms in V we obtain the reflection formula:

V=exp jφqjpq+jpA exp jψ.                Eq. 25

The actual stationary value of p is given by:

p=12xh1+h212ξ2+4ρ2 sin2 α3               Eq. 26

where:

ρ2=13ξ2+2h1+2h2;     sin α =ξh1h2ρ2;       π2<α<π2.              Eq. 27

The reflection formula (25) is valid for large enough p, practically it gives a good approximation for p > 2.

As will be discussed further, experimental results at frequencies above 1 GHz suggest little difference in the propagation of the vertically and horizontally polarised fields. As may be seen from the definition and values of m and η, parameter q is actually large for sea water at frequencies above 1 GHz, which suggests that, at least in the case of long range propagation, i. e. at distances beyond the opti- cal horizon, the impedance boundary conditions can be approximated by conditions for a vertically polarised field with q → ∞, practically suitable for both polarizations.

Fock’s Theory of the Evaporation Duct

In the case of the super-refraction associated with the evaporation duct the M-profile has one minimum at some height Zs above the sea surface, called the height of the evaporation duct. Considering the case where q = ∞, the attenuation factor V(ξ, h1, h2, ∞) is given by Eq. (4) with the integrand F(t, h1, h2, ∞) given by:

Ft, h1, h2=j2f1t, h2f2t, h1f20, tf10, tf1t, h1.             Eq. 28

and the height-gain functions f1, f2 are the solutions to the equation:

d2fdh2+Uhf=tf.             Eq. 29

The function U(h) is assumed to be an analytical function of its argument. We concentrate further on h and t where the equation U(h)-t = 0 has two roots: h = b1 and h = b2. For real t in the interval (U(Hs) < t ≤ U(0)), the above roots are also real.

Consider an asymptotical integration of Eq. (29) for height-gain functions which is valid for all values h and t, including t = U(Hs), i. e. the point of the minimum of the U-profile. The asymptotical solution in this case should be built upon an “etalon” equation, which behaves similarly to Eq. (29), i. e. has a single minimum, and, on the other hand, has a known analytical solution. The “etalon” equation can be expressed in the form of an equation for functions of a parabolic cylinder:

d2gdς2+14ς2+ν g=0.             Eq. 30

The relation between Eqs. (30) and (29) is established by a transformation of h into ς in a form which ensures that two conditions are met:

  1. The parameter U(h)-t in Eq. (29) has the same roots as 1/4 ς2, and;
  2. With large values of the above parameters, either Eq. (29) or Eq. (30) results in the same asymptotic expression for their respective solutions f and g.

Such conditions are satisfied with the following substitution:

b1hUhtdh=122jνςς2+4νdς             Eq. 31

with parameter ν bounded by following equation:

b1b2Uhtdh=122jν2jνς2+4νdς.             Eq. 32

The value integral in Eq. (30) results in the relationship between roots b1, b2, and ν:

jπν=b1b2U(h)tdh.

Expanding the right-hand side of Eq. (32) into a Taylor series in the vicinity of the minimum of U(h) we obtain ν as a holomorphic function of t:

ν=UHst2UHs+ ...            Eq. 33

Since U′′(HS) > 0, we can read formula (32) in the following way: ν > 0 when t < U(Hs) and ν < 0 when t > U(Hs); with U(Hs) < t < U(0) we then have:

ν=1πb1b2tUhdh.            Eq. 34

Then, introducing the notations for the phase terms:

S=0hUhtdh,   S0=120b1Uhtdh+120b2Uhtdh,             Eq. 35

the substitution (31), which in fact defines the relation between h and ς, can be expressed in the form:

ShS0=122jνςς2+4νdς=14ςς2+4ν+ν ln ς+ς2+4νν2 ln 4ν.             Eq. 36

The respective solutions of Eqs. (29) and (30) are related as follows:

f=dhdςg             Eq. 37

where function g, the solution to Eq. (30), can be expressed via a function of a parabolic cylinder Dn(z), given by the series:

Dnz=2n21Γnez24 m=0 Γmn2Γm+1 2m2 zm.             Eq. 38

Function Dn(z) will obey Eq. (30) when n = jν-1/2 and z = ςejπ/4. The appropriate solutions to Eq. (29) are then given by:

f1h, t=c1ν2dhdςg1ς,f2h, t=c2ν2dhdςg2ς              Eq. 39

and:

g1ς=Djν1/2ςejπ/4,g2ς=Djν1/2ςejπ/4,              Eq. 40
c1ν=exp πν4+jπ8+j12ν12ν ln νS0,c2ν=c1ν*=exp πν4jπ8j12ν12ν ln νS0.              Eq. 41

The coefficients c1(ν) and c2(ν) are defined by Eq. (42) with the aim being to obtain a correct asymptotic expression for f1 (h, t), f2 (h, t) with large values of the argument h and holomorphic functions (40) in the vicinity of ν = 0.

With large positive ς, that corresponds to large heights h, well above Hs, the height gain functions take the following asymptotic forms:

f1h, t=eπ/4U(h)t4 exp jS2S0,f2h, t=ejπ/4Uht4 exp jS2S0.              Eq. 42

Inside the evaporation duct and, strictly speaking, far below the inversion height Hs when ς is large and negative, the asymptotic representations for the height-gain functions take the form:

f1h, t=χ1νejπ/4Uht4 exp jS2S0+eπνejπ/4Uht4 exp jS,f2h, t=χ2νejπ/4Uht4 exp jS2S0+eπνejπ/4Uht4 exp jS              Eq. 43

where:

χ1ν=2πΓ12jv exp πν2+jνν ln ν,χ2ν=2πΓ12+jv exp πν2jνν ln ν.              Eq. 44

The asymptotic representation of the integrand (28) is then given by:

Ft, h1, h2=j21Uh1t4·Uh2t4·ejSh1ΛejSh·ejSh2ejSh21Λ              Eq. 45

with Hs > h1 > h2 , where:

Λ=jeπν+2jS0χ1ν.              Eq. 46

The poles in the integral (4) with the integrand in the form (45) will be given by the roots of the characteristic equation:

1Λ=0.              Eq. 47

In many practical cases the evaporation duct parameters are such that the values of parameter ς are of the order of 1, or even less, in Eqs. (45) and (47). Therefore, some doubts can be raised as to the validity of the asymptotic formulas (46) and (47) in this case. Fock calculated the propagation constants given by Eq. (45) and the exact equation:

g10, t=0.              Eq. 48

expressed via functions of a parabolic cylinder. A sample of the calculations of propagation constants for the hyperbolic profile:

Uh=hHsh+hl,              Eq. 49

where hl is a parameter, is presented in table.

Comparison of the calculated propagation constants using the asymptotic Eq. (47) and the exact formula (48)
Hs+hlAsymptotic t1Exact t1
23,11-0,085+j0,466-0,107+j0,443
25,24-0,148+j0,262-0,158+j0,269
48,07-0,104+j0,224-0,113+j0,227

As observed, the agreement between the asymptotic formulas and the exact solution is satisfactory, especially with regard to the imaginary part of the propagation constants, Im(tn). The relation between the attenuation rate of the nth mode, γn, and Im(tn) is given by:

γn=0,434km2 Im tn, dB/km.             Eq. 50

Consider Eq. (47) for trapped modes which have a small imaginary component of the propagation constant tn and are in the interval: U(Hs) ≤ Re tn < U(0). The respective values of parameter v are negative. For v < 0 we define ln(ν) = ln(-ν)+jπ and from Eq. (35) we obtain:

S0=0b1Uhtdhjπ2νS1jπ2ν.             Eq. 51

Equation (47) then takes the form:

je2jS1=χν1.             Eq. 52

For large negative ν, χ(ν) → 1, and we obtain:

S10b1Uhtdh=m14π,    m=1, 2, ...            Eq. 53

For either large positive ν or complex ν with positive Re ν, it follows that Re t < U(Hs). In this case the phase S1 can be defined as:

S1=S0+jπ2ν            Eq. 54

and Eq. (47) will again be truncated to Eq. (53).

A basic conclusion followed from the study of Eq. (47) and its truncated form (53) (performed by Fock) is that the attenuation rate of the field in the evaporation duct is affected not only by the height Zs and the M-deficit of the evaporation duct but also by a curvature of the M-profile at the level of inversion height Zs. The attenuation rate γm of the mth mode from Eq. (53) can be estimated as follows:

γm=2πm14 103ZsMZsβ2M0MZsΘ=2πλm106 Zs2MZsΘ            Eq. 55

where:

Θ=λmλIm tm;            Eq. 56
λm=β 103 Zsm1/42M0MZs

is a critical wavelength of the mth mode trapped in the evaporation duct formed by a given M-profile, λ is the wavelength of the radiated field. The parameter β is introduced by:

β=01qzdz;   q0=4;   q1=0;              Eq. 57
qzZs=4MzMZsM0MZs.              Eq. 58

For most M-profiles the value of β is close to 1.

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Literature
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  2. Fock, V.A. Distribution of currents excited by a plane wave on the surface of conductor, J. Exp. Theor. Phys., (Nauka), 1945, 15 (12), 693–702.
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  10. Andrianov, V.A. Diffraction of UHF in a bilinear model of the troposphere over the Earth’s surface, Radiophys. Quantum Electron., 1977, 22 (2), 212–222.
  11. Wait, J.R. Electromagnetic Waves in Stratified Media, Pergamon Press, Oxford, 1962, 372 pp.
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