GEOPHYSICS

Corresponding Member of the Academy of Sciences of the USSR E. N. BLINOVA

A HYDRODYNAMIC STUDY OF NONSTATIONARY ATMOSPHERIC PROCESSES OF PLANETARY SCALE

In hydrodynamic studies of atmospheric processes of planetary scale and in constructing a hydrodynamic theory of long-range weather forecasting, one should adopt such a formulation of the problem that the principal climatic elements would be described by the solution of the equations itself. This leads to the necessity of introducing the influx of heat due to radiation and thermal conductivity into the energy equation and into the boundary conditions of the problem.

In works \((^{1,2})\) a theory was developed which makes it possible, for a given configuration of continents and oceans and a given influx of heat from the Sun, to determine in its main features the climatic (possessing an annual period) distribution of temperature, pressure, and wind by solving a boundary-value problem of hydrothermodynamics. This theory can be generalized to the problem of long-range weather forecasting, in which processes with an annual period are not singled out but are considered together with nonperiodic processes (depending on the initial data).

Let us take the Earth to be a sphere of radius \(a_0\) and introduce spherical coordinates: \(\theta\) is the colatitude, \(\lambda\) is the longitude of the place; as the vertical coordinate we take, as in \((^{3,4})\), the reduced standard pressure \(\xi=\tilde p(z)/\tilde p(0)\), where \(z\) is the height above sea level and \(\tilde p(z)\) is the standard pressure.

We write the equations of motion for the horizontal velocities approximately in the form:

\[ \frac{\partial v_\theta}{\partial t} +\frac{1}{a_0}\frac{\partial}{\partial\theta}\frac{V^2}{2} +v_z\frac{\partial v_\theta}{\partial z} = -\frac{T}{a_0T_{\mathrm{cp}}}\frac{\partial \Phi}{\partial\theta} +v_\lambda(2\omega\cos\theta+\Omega), \tag{1} \]

\[ \frac{\partial v_\lambda}{\partial t} +\frac{1}{a_0}\frac{\partial}{\partial\lambda}\frac{V^2}{2} +v_z\frac{\partial v_\lambda}{\partial z} = -\frac{T}{a_0T_{\mathrm{cp}}\sin\theta}\frac{\partial \Phi}{\partial\lambda} -v_\theta(2\omega\cos\theta+\Omega). \tag{2} \]

Here \(t\) is time; \(\omega\) is the angular velocity of the Earth’s rotation; \(v_\theta, v_\lambda, v_z\) are the velocity components along the axes \(\theta,\lambda,z\), respectively; \(V^2=v_\theta^2+v_\lambda^2\);

\[ \Omega=\frac{1}{a_0\sin\theta} \left[ \frac{\partial(\sin\theta v_\lambda)}{\partial\theta} -\frac{\partial v_\theta}{\partial\lambda} \right]; \quad T=T(\theta,\lambda,z,t) \]
is the temperature; \(T_{\mathrm{cp}}\) is the mean temperature of the entire atmosphere \((T_{\mathrm{cp}}\approx250^\circ\mathrm{K})\),

\[ \Phi=RT_{\mathrm{cp}}\frac{p'(\theta,\lambda,z,t)}{\tilde p(z)}, \]

where \(p'\) is the deviation of pressure from its standard value; \(R\) is the gas constant. The quantity \(\Phi\) has the dimension of geopotential and may be identified with the deviation of the geopotential height from its standard value.

The quasistatic hypothesis makes it possible to replace the third equation of motion by the relation

\[ T'\approx-\frac{\xi}{R}\frac{\partial\Phi}{\partial\xi}, \tag{3} \]

where \(T'=T'(\theta,\lambda,z,t)\) is the deviation of temperature from its standard value \(\tilde T(z)\).

We shall write the continuity equation in the form

\[ -\frac{1}{a_0\sin\theta}\left[\frac{\partial(v_\theta\sin\theta)}{\partial\theta}+\frac{\partial v_\lambda}{\partial\lambda}\right] =\frac{\partial W}{\partial\xi}, \tag{4} \]

where \(W=\dfrac{g}{p(0)}\tilde{\rho}v_z\); \(g\) is the acceleration of gravity; \(\tilde{\rho}\) is the standard density; \(v_z\) is the vertical component of velocity \(\left(W\text{ has dimension } \dfrac{1}{t}\right)\).

Represent the velocity components \(v_\theta\) and \(v_\lambda\) in the form

\[ v_\theta=-\frac{1}{a_0\sin\theta}\frac{\partial\psi}{\partial\lambda} +\frac{1}{a_0}\frac{\partial\varphi}{\partial\theta}, \qquad v_\lambda=\frac{1}{a_0}\frac{\partial\psi}{\partial\theta} +\frac{1}{a_0\sin\theta}\frac{\partial\varphi}{\partial\lambda}. \tag{5} \]

For brevity, introduce the symbol

\[ (A,B)=\frac{\partial A}{\partial\theta}\frac{\partial B}{\partial\lambda} -\frac{\partial A}{\partial\lambda}\frac{\partial B}{\partial\theta}. \tag{6} \]

From equations (1) and (2) one obtains the vorticity-transfer equation

\[ \begin{aligned} \frac{\partial\Delta\psi}{\partial t} &+\frac{1}{a_0^2\sin\theta}\left(\psi,\Delta\psi+2\omega a_0^2\cos\theta\right) +\frac{1}{a_0^2}\frac{\partial\varphi}{\partial\theta} \frac{\partial(\Delta\psi+2\omega a_0^2\cos\theta)}{\partial\theta} +\\ &+\frac{1}{a_0^2}\frac{1}{\sin^2\theta} \frac{\partial\varphi}{\partial\lambda}\frac{\partial\Delta\psi}{\partial\lambda} -W\frac{\partial\Delta\psi}{\partial\xi} -a_0\frac{\partial W}{\partial\theta}\frac{\partial v_\lambda}{\partial\xi} +\frac{a_0}{\sin\theta}\frac{\partial W}{\partial\lambda}\frac{\partial v_\theta}{\partial\xi} =\\ &=\frac{1}{T_{\mathrm{cp}}\sin\theta}(\Phi,T') -\left(\Delta\psi+2\omega a_0^2\cos\theta\right)\frac{\partial W}{\partial\xi}, \tag{7} \end{aligned} \]

where

\[ \Delta\psi=a_0^2\Omega =\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial\psi}{\partial\theta}\right) +\frac{1}{\sin^2\theta}\frac{\partial^2\psi}{\partial\lambda^2}. \]

The generalized balance equation on the sphere is obtained in the form

\[ \begin{aligned} \frac{\partial\Delta\varphi}{\partial t} &-W\frac{\partial\Delta\varphi}{\partial\xi} -a_0\frac{\partial W}{\partial\theta}\frac{\partial v_\theta}{\partial\xi} -\frac{a_0}{\sin\theta}\frac{\partial W}{\partial\lambda}\frac{\partial v_\lambda}{\partial\xi} +\Delta\Phi+\\ &+2\left[\left(\frac{\partial v_\theta}{\partial\theta}\right)^2 +\left(\frac{\partial v_\lambda}{\partial\theta}\right)^2\right] +\frac{(\Delta\varphi)^2}{a_0^2} -\frac{2}{a_0}\Delta\varphi\frac{\partial v_\theta}{\partial\theta} -\\ &-\frac{2}{a_0}\frac{\partial v_\lambda}{\partial\theta}\Delta\psi +\frac{v_\theta}{a_0}\frac{\partial\Delta\varphi}{\partial\theta} +\frac{v_\lambda}{a_0}\frac{1}{\sin\theta}\frac{\partial\Delta\varphi}{\partial\lambda} +v_\theta^2+v_\lambda^2-\\ &-2\omega\cos\theta\,\Delta\psi +2a_0\omega\sin\theta\,v_\lambda =0^{*} \tag{8} \end{aligned} \]

* The first term on the right-hand side of (7), representing the effect of horizontal baroclinicity, is usually considered small in comparison with the terms on the left-hand side of (7). However, this term cannot be neglected when “long” waves are under discussion. Indeed, by virtue of (8) we may assume that \(\Phi\) is of order \(2\omega\cos\theta\,\psi\) (quasi-geostrophicity), so that in equation (7) we must compare \(\Delta\psi\) from the Jacobian on the left-hand side of (7) with \(2\omega a_0^2\cos\theta\,T'/T_{\mathrm{cp}}\) from the Jacobian on the right-hand side of (7), or, by (3) and (8), compare \(\Delta\psi\) with \(4\omega^2a_0^2\cos^2\theta\,\psi/T_{\mathrm{cp}}R\approx 5\). If we now represent \(\psi\) as a series in spherical functions

\[ \sum_n\sum_m(\psi_n^m\cos m\lambda+\psi_n^{\prime m}\sin m\lambda)P_n^m \]

(with coefficients depending on \(z\) and \(t\)), then \(\Delta\psi=\)

\[ =-n(n+1)(\psi_n^m\cos m\lambda+\psi_n^{\prime m}\sin m\lambda)P_n^m, \]

and thus it is a matter of comparing the terms \(n(n+1)\) and \(4\omega^2a_0^2(\cos^2\theta)_{\mathrm{cp}}/T_{\mathrm{cp}}R\approx 5\). For long waves, when \(n=2\) or \(3\), these terms will be of the same order; only for \(n\geq4\) can the Jacobian of the right-hand side of (7) (horizontal baroclinicity) be neglected. Meanwhile, for the distribution of \(T'\) over the horizontal, the first tendency is toward a stationary long wave caused by nonadiabatic processes: the temperature difference over oceans and continents. These stationary long waves \(T'\) lead to the formation of stationary long waves \(\psi\) satisfying the stationary solution (when \(\partial\Delta\psi/\partial t=0\)). This is how centers of action are described \((^5,^6)\). If in equation (7) we discard the first term on the right-hand side (the Jacobian, the effect of horizontal baroclinicity), then in solving the nonstationary problem we obtain a rapid displacement of the centers of action (long waves), distorting the picture of the phenomenon.

Meteorologists encountered this difficulty already in preparing 24–48-hour forecasts using the barotropic model for the Northern Hemisphere; in practical work it proved necessary to “remove” long waves \((^7,^8)\) or else to introduce empirically selected coefficients into the vorticity-transfer equation in order to stabilize the long waves \((^9)\).

Let us turn to the equation for heat inflow. It can be written in the form \((^{1,2})\)

\[ \begin{aligned} \frac{\partial T'}{\partial t} &+ \frac{v_\theta}{a_0}\frac{\partial T'}{\partial \theta} + \frac{v_\lambda}{a_0 \sin \theta}\frac{\partial T'}{\partial \lambda} - \frac{\gamma_a}{g}\frac{\partial \Phi}{\partial t} + \frac{R T}{g}\,c_p\,(\gamma_a-\gamma)\frac{W}{\xi} \\ &= \frac{1}{N}\left(\frac{dT}{dE}\right)_{cp} \left[ \frac{2}{m^2-1}\frac{\partial^2 E}{\partial \xi^2} -2E + e^{-\xi}\int_0^\xi E e^\zeta d\zeta - e^\xi\int_Z^\xi E e^{-\zeta} d\zeta \right. \\ &\qquad\qquad\qquad\left. + e^{\xi-Z}(E)_{\zeta=Z} + \beta S_0 e^{-\beta \xi} \right] = F . \end{aligned} \tag{9} \]

Here \(\gamma_a\) is the adiabatic or pseudoadiabatic gradient; \(\gamma=-dT/dz\); \(\xi\) is the optical thickness for “gray” long-wave radiation,

\[ \xi=\int_z^\infty a_2\rho_w dz \]

(\(a_2\) is the absorption coefficient of long-wave radiation, \(\rho_w\) is the density of the absorbing substance); \(Z=\int_0^\infty a_2\rho_w dz \simeq 12.6\); \(E=E(T)\) is a function of temperature alone (it has been tabulated \((^2)\)); \(2a_2\rho_w E\) characterizes the amount of radiant energy emitted by an air particle; \((dE/dT)_{cp}\) is the mean value of the derivative of \(E\) with respect to \(T\); \(N\) and \(m\) are parameters which we usually take to be constant:

\[ N=\left(\frac{c_p\rho}{a_2\rho_w}\frac{dT}{dE}\right)_{cp}, \qquad \frac{2}{m^2-1}=\left(\lambda' a_2\rho_w \frac{dT}{dE}\right)_{cp}, \]

\(\lambda'\) is the coefficient of turbulent thermal conductivity of air in the vertical direction, \(c_p\) is the heat capacity of air at constant pressure; \(\beta=a_1/a_2\), where \(a_1\) is the absorption coefficient for short-wave solar radiation; \(S_0=[1-\Gamma(\theta,\lambda,t)]w(\theta,t)\), where \(w(\theta,t)\) is the mean amount of solar radiation received by a unit surface per unit time at the upper boundary of the atmosphere at the given parallel and at the given time of year; \(\Gamma\) is the albedo of the Earth (or of clouds, if at the given point \((\theta,\lambda)\) there is cloudiness at the given moment of time); finally, \((E)_{\zeta=Z}=r\sigma(T^4)_{\zeta=Z}\), where \(\sigma\) is the Stefan–Boltzmann constant, \(T_Z\) is the temperature of the Earth’s surface (one of the unknown quantities), and \(r\) is a quantity close to unity*.

In equation (9), the first term in the square brackets on the right describes the effect of turbulent thermal conductivity due to vertical mixing, the last term gives the inflow of energy from the Sun, and the remaining terms on the right-hand side describe the effect of long-wave radiation of the atmosphere and the Earth’s surface. In contrast to work (1), we do not introduce here into consideration hori-

* To describe the heat inflow from radiation we adopt the Schwarzschild–Emden scheme and regard the radiation as “gray.” It will be seen from the further exposition that this simplification is not fundamental. One may introduce diffuse radiation and consider separately the radiations for water vapor, carbon dioxide, and ozone. We preferred at first to deal with the previous models and parameters tested by us in climate theory. The quantities \(\zeta\) and \(\xi\) are connected by the simple relation. If we take, as in (1), \((1+\zeta)^{5/4}-1=[(1+Z)^{5/4}-1]\cdot 10^{-z/l}\) \((l\simeq 8\cdot 10^3\text{ m})\), then it is easy to obtain that

\[ \ln \xi = \frac{5}{4}\frac{l}{T_\infty \ln 10}\frac{g}{R} \int_Z^\xi \frac{d\zeta}{(1+\zeta)^{5/4}-1} \simeq 0.675 \int_Z^\xi \frac{d\zeta}{(1+\zeta)^{5/4}-1} \]

\((T_\infty=220^\circ\mathrm{K})\). With a high degree of accuracy one may write \(\xi=(\zeta/Z)^{0.540}\).

horizontal turbulent heat conduction on a large scale, since we take fully into account the nonlinear terms*.

The 4 equations (3), (7), (8), and (9) contain 5 functions to be determined: \(\psi, \varphi, W, T', \Phi\). To these equations one must add the continuity equation (4). The equations contain 4 essential differentiations with respect to height (of them 2 with respect to \(\xi\) and 2 with respect to \(\zeta\)).

We shall impose 2 boundary conditions at the upper boundary of the atmosphere:

\[ W=0,\qquad \frac{\partial T'}{\partial t}=0,\qquad \text{for } \xi=\zeta=0 \tag{10} \]

and 2 at the Earth’s surface:

\[ W=0 \qquad \text{for } \xi=1; \tag{11} \]

\[ \frac{2}{m^{2}-1}\frac{\partial E}{\partial \zeta} +\lambda^{*}\frac{\partial T^{*}}{\partial z} = e^{-z}\int_{0}^{z}Ee^{\zeta}d\xi-r\sigma T^{4}z+S_{0}e^{-\beta z} \qquad \text{for } \xi=Z \tag{12} \]

(\(\xi=1\); condition of heat balance)**.

Here \(T^{*}\) and \(\lambda^{*}\) are the temperature and coefficient of thermal conductivity of the underlying surface, respectively (for the ocean, \(\lambda^{*}\) is the coefficient of turbulent thermal conductivity). For the new unknown function \(T^{*}\) we can write the equation

\[ c^{*}\rho^{*}\frac{\partial T^{*}}{\partial t} = \frac{\partial}{\partial z} \left( \lambda^{*}\frac{\partial T^{*}}{\partial z} \right) \tag{13} \]

(\(c^{*}\) and \(\lambda^{*}\) are the heat capacity and density of the underlying surface, respectively) and the boundary conditions

\[ (T^{*})_{z=0}=(T)_{z=0},\qquad \left(\frac{\partial T^{*}}{\partial T}\right)_{z=-\infty}=0. \tag{14} \]

In the next paper \(^{(12)}\) we shall propose a concrete method for solving the system (7), (8), (9), (13) under the boundary conditions (10), (11), (12), and (14).

Unified Meteorological
Computing Center
of the Academy of Sciences of the USSR
and of the Hydrometeorological Service of the USSR

Received
22 VI 1961

CITED LITERATURE

\(^{1}\) E. N. Blinova, Izv. AN SSSR, ser. geogr. i geofiz., 11, No. 1, 3 (1947).
\(^{2}\) E. N. Blinova, Tr. Inst. fiz. atmosfery, No. 2, 5 (1958).
\(^{3}\) E. N. Blinova, DAN, 110, No. 6, 975 (1956).
\(^{4}\) Chzhu Yu h’-t-i, Izv. AN SSSR, ser. geofiz., No. 12, 1807 (1959).
\(^{5}\) E. N. Blinova, DAN, 39, No. 7, 284 (1943).
\(^{6}\) E. N. Blinova, DAN, 92, No. 3, 557 (1953).
\(^{7}\) R. M. Wolff, Monthly Weather Rev., 86, No. 7, 245 (1958).
\(^{8}\) M. B. Galin, DAN, 138, No. 5 (1961).
\(^{9}\) G. P. Cresmann, Monthly Weather Rev., 86, No. 8, 283 (1958).
\(^{10}\) R. D. Thompson, Tellus, 9, No. 1, 69 (1957).
\(^{11}\) E. N. Blinova, DAN, 123, No. 3, 440 (1958).
\(^{12}\) E. N. Blinova, DAN, 140, No. 3 (1961).

* To determine theoretically the climate pattern, we can average both parts of (9) over the totality of a large number of “weather” states, replacing that part with

\[ \frac{v_{\theta}}{a_{0}}\frac{\partial T'}{\partial \theta} + \frac{v_{\lambda}}{a_{0}\sin\theta}\frac{\partial T'}{\partial \lambda}, \]

which will contain products of pulsations, by a single term describing the effect of turbulent heat conduction (horizontally):

\[ -\frac{k}{a_{0}^{2}}\Delta T', \]

where \(k\) is the coefficient of turbulent temperature conductivity. In the forecasting problem one should proceed otherwise: if we wish to predict smoothed quantities, we must describe “turbulence” in another way, introducing correlation moments as new unknown functions (see \(^{(10)},{}^{(11)}\)).

** The influence of orography may be introduced into condition (11). The heat flux going into evaporation may be included in the heat-balance condition (12). We do not dwell on this here.