### Theory of the Periodic Variations of Temperature at Subsurface Depths

Subsurface temperature variations due to processes of heat conduction can be studied by means of the equation (p. 159)

[Equation]

*ρ*is the density and

*A*is the eddy conductivity which, in general, varies with depth and time. When one writes ∂υ/∂

*t*, the local change in temperature, in this form it is supposed that heat conduction takes place in a vertical direction only and that advection can be neglected. The terms “local change” and “advection” are explained on p. 157. An integral of equation (1) is easily found if

*A*/∂ is constant, if the average temperature is a linear function of depth, and if the temperature variations at the surface (

*z*= 0) can be represented by means of a series of harmonic terms:

[Equation]

*σ*= 2π/

*T*and

*T*is the period length of the first harmonic term.

Then

[Equation]

[Equation]

Defant (1932) has shown that at the *Meteor* anchor station no. 288 in latitude 12°38′N, longitude 47°36′W, the diurnal variation of temperature in the upper homogeneous layer indicated a constant eddy conductivity. The amplitudes of the diurnal term at the surface and at 50 m were 0.093 and 0.017 degree, respectively, and the phase difference was 6.5 hours. With *ρ* = 1.024, and *T* = 24 hours, he obtained from both the decrease of the amplitude and the difference in phase angle *A* = 320 g/cm/sec.

In cases in which the annual variation of temperature has been examined, the decrease of amplitude and change of phase give different values of *A*, indicating that *A* is not independent of depth and time, as assumed when performing the integration. Fjeldstad (1933) has developed a method for computing the eddy conductivity if it changes with depth, provided that the periodic temperature variations are known at a number of depths between the surface and a depth, *h,* at which they are supposed to vanish. He arrives at the formula

[Equation]

*a*is the amplitude of the

_{n}*n*th harmonic term and

*α*is the phase angle.

_{n}Fjeldstad has applied the method to the annual temperature variations off the Bay of Biscay, which have been determined by Helland-Hansen (p. 132). He found, with ρ = 1.025,

Depth (m) | 0 | 25 | 50 | 100 |

Amplitude, a_{1}, °C | 3.78 | 3.24 | 1.24 | 0.23 |

Phase angle, a_{1} | 225.1° | 235.2° | 254.7° | 289.3° |

Eddy conductivity, g/cm/sec | 16.4 | 3.2 | 2.1 | 3.8 |

Several features show, however, that the observed temperature variations cannot be accounted for by assuming that the eddy conductivity varies with depth only, and variations with seasons also must be considered. Fjeldstad has examined this question and finds that the conductivity reaches a maximum in spring when the stability is at a minimum, but the values remain small throughout the year.

Fjeldstad#x0027;s method can also be applied to the annual variation of temperature in the Kuroshio, which has been discussed by Koenuma (1939) (fig. 32B, p. 131). It is necessary, however, to make the reservation

*5)*is therefore correct only if the advection term is independent of time and depth. The harmonic constants and the results, with ρ

*=*1.025, are

Depth (m) | 0 | 25 | 50 | 100 | 200 |

Amplitude, a_{1}, °C | 4.26 | 3.97 | 3.49 | 2.09 | 0.71 |

Amplitude, a_{2}, °C | 0.58 | 0.49 | 0.44 | 0.39 | 0.14 |

Phase angle, a_{1} | 250.2° | 253.5° | 258.7° | 271.8° | 289.3° |

Phase angle, a_{2} | 71.4° | 81.0° | 100.0° | 135.5° | 152.6° |

Eddy conductivity, A_{1}, g/cm/sec | 78 | 34 | 23 | 22 | 29 |

Eddy conductivity, A_{2}, g/cm/sec | 58 | 43 | 39 | 32 | 26 |

Both the annual and the semiannual periods have been used for computing the eddy conductivity, and the agreement between the values of *A* derived from them must be considered satisfactory, in view of the small amplitudes of the semiannual variation. The numerical values decrease with depth, but are much greater than off the Bay of Biscay, as might be expected, because the high velocity of the Kuroshio must lead to intense turbulence. A possible annual variation of the eddy conductivity has not been examined.

In the Kuroshio region, where the velocity of the current is great and the turbulence correspondingly intense, the annual variation of temperature becomes perceptible to a depth of about 300 m, but in the Bay of Biscay it is very small at 100 m. It can therefore safely be concluded that below a depth of 300 m the temperature of the ocean is not subject to any annual variation.

The eddy conductivity off the Bay of Biscay and in the Kuroshio region is much smaller than that in the upper homogeneous layer near the Equator. The difference can be ascribed to the facts that in the former localities the density increases with depth and that the eddy conductivity is greatly reduced where this takes place (p. 477).