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The Mechanism of Formation of Mesoscale Pulsating Jet Streams in the Earth’s Atmosphere

Abstract

The formation mechanism of local pulsatile jet streams is considered. It may in general be reduced to the following sequence: (i) a breaking gravity wave produces a turbulent patch that “moves” with it; (ii) adiabatic mixing within the “moving” turbulent patch maintains a near-adiabatic lapse rate; and (iii) just after turbulence withdrawal, the buoyant forces initiate developing a pulsatile jet stream. The base of the associated mathematical model is a second-order nonlinear ordinary differential equation obtained from the set of the governing hydrodynamic equations under the Boussinesq approximation. It is demonstrated that such phenomena as ripples in airglow images and extremely large winds in the lower thermosphere may be related to local pulsatile jet streams, which, in turn, appear in the lower thermosphere as a result of gravity-wave breaking.

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Correspondence to A. N. Belyaev.

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This paper was prepared on the basis of an oral report presented at the All-Russian Conference “Intrinsic Radiation, Structure, and Dynamics of the Middle and Upper Atmosphere” (Moscow, November 22–23, 2021).

APPENDIX

APPENDIX

We will look for a solution to the Cauchy problem for system of differential equations (2), (5), (11) with initial conditions (12)–(14), assuming that, in buoyancy functions \(b(x,z)\) and vertical velocity \(w(x,z)\), variables x and z can be divided as follows:

$$b(x,z) = zB({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C}),$$
(A1)
$$w(x,z) = zW({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C}),$$
(A2)

while horizontal velocity function \(u(x)\) is represented using the similarity relation

$$u(x) \equiv CU({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C}).$$
(A3)

The last relation follows from the joint consideration of Eqs. (A2) and (5). Here and below, functions \(B({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C})\), \(W({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C})\), and \(U({x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C})\) are functions of one variable \(\tau = {x \mathord{\left/ {\vphantom {x C}} \right. \kern-0em} C}\).

After substituting relations (A1), (A2), and (A3) into system of Eqs. (2), (5) and (11), we arrive at the following system:

$$U\frac{{dB}}{{d\tau }} + {{N}^{2}}W = - \alpha B,$$
(A4)
$$U\frac{{dW}}{{d\tau }} = B,$$
(A5)
$$\frac{{dU}}{{d\tau }} + W = 0,$$
(A6)

with the basic conditions

$$B(0) = - {{N}^{2}},$$
(A7)
$$W(0) = 0,$$
(A8)
$$U(0) = 1.$$
(A9)

Substituting (A6) into (A5) leads to the following relation between functions\(B(\tau )\) and \(U(\tau )\):

$$B = - U\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}}.$$
(A10)

Taking into account (A10) and (A6), equation (A4) is transformed into an ordinary nonlinear differential equation of the third order:

$$\frac{d}{{d\tau }}\left( {U\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}}} \right) + \alpha {\kern 1pt} {\kern 1pt} \frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}} + {{N}^{2}}\frac{1}{U}\frac{{dU}}{{d\tau }} = 0.$$
(A11)

Integrating (A11), lower the order of this differential equation

$$U\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}} + \alpha {\kern 1pt} {\kern 1pt} \frac{{dU}}{{d\tau }} + {{N}^{2}}\ln U + {{c}_{1}} = 0,$$
(A12)

where \({{c}_{1}}\) is the integration constant:

$${{c}_{1}} = B(0) + W(0) - {{N}^{2}}\ln U(0) = - {{N}^{2}}.$$
(A13)

Substituting the expression for \({{c}_{1}}\) in (P12), we transform this equation to the final form:

$$U\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}} + \alpha {\kern 1pt} {\kern 1pt} \frac{{dU}}{{d\tau }} + {{N}^{2}}\ln \left( {\frac{U}{e}} \right) = 0,$$
(A14)

with the initial condition (A9) supplemented by the condition

$${{\left. {\frac{{dU}}{{d\tau }}} \right|}_{{\tau = 0}}} = - W(0) = 0.$$
(A15)

The validity of Eq. (A14) is determined by the validity of condition (9) for its solutions. In order to transform condition (9) into a form convenient for such a check, we substitute \(U\) and \(W\) from (A3) and (A2) into (9):

$${{C}^{2}}\left| {U\frac{{dU}}{{d\tau }}} \right| \gg {{z}^{2}}\left| {W\frac{{dW}}{{d\tau }}} \right|.$$
(A16)

Using (A6), we rewrite (A16) as

$$\frac{{{{C}^{2}}}}{{{{z}^{2}}}} \gg \left| {\frac{1}{U}\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}}} \right|.$$
(A17)

In accordance with the law of conservation of air flow through an arbitrary vertical section of a plane jet, we can write \(z = {{{{z}_{0}}} \mathord{\left/ {\vphantom {{{{z}_{0}}} {U(\tau )}}} \right. \kern-0em} {U(\tau )}}\), where \(z = z{}_{0}\) at \(\tau = 0\). Equation (A17) can then be reduced to the form

$$\frac{{{{C}^{2}}}}{{z_{0}^{2}}} \gg \left| {\frac{1}{{{{U}^{3}}}}\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}}} \right|.$$
(A18)

Since \(\left| {{{z}_{0}}} \right| \leqslant {{h}_{0}}\), the use of Eq. (A14) will be justified if the following condition is met:

$$\frac{{{{C}^{2}}}}{{{{h}_{0}}^{2}}} \gg \left| {\frac{1}{{{{U}^{3}}}}\frac{{{{d}^{2}}U}}{{d{{\tau }^{2}}}}} \right|.$$
(A19)

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Belyaev, A.N. The Mechanism of Formation of Mesoscale Pulsating Jet Streams in the Earth’s Atmosphere. Izv. Atmos. Ocean. Phys. 58, 295–301 (2022). https://doi.org/10.1134/S0001433822030033

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  • DOI: https://doi.org/10.1134/S0001433822030033

Keywords:

  • breaking/saturated gravity wave
  • turbulent patch
  • local plane jet stream