Navier-Stokes equation

The fundamental equation governing the motion of air in a planetary atmosphere is the Navier-Stokes equation. Consider a parcel of air of volume SV = Sx Sy Sz and density p. From Newton's Second Law, F = ma and thus

where V is the velocity vector. There are three main forces that may act on air parcels in a non-rotating planetary atmosphere: gravity, the pressure gradient force, and friction. Incorporating these forces leads to the Navier-Stokes equation in an inertial frame d V 1 7 2 , N

dt p p where 7 is the viscosity; and g;- is the gravitational acceleration, excluding centrifugal forces. In reality, of course, planets rotate and thus to describe atmospheric motion we need to re-express this equation in a form suitable for a rotating frame of reference. A vector A in frame SR, which is rotating at an angular velocity H with respect to a stationary inertial frame Sj will have a component of motion H x A in the stationary frame Sj. Hence, differentiating A with respect to time in the two frames leads to dA\ idA\ „ A ,

and putting A = dr/dt we have

Combining the two we obtain the expression 'dVA (dV

which may be substituted into the Navier-Stokes equation to give dV** -- Vp + 2VR x n + g; - n x(n x r)+- V2VR (5.7)

dt JsR P P

or more conveniently dV 1 n dV = g -+ 2V x n + n V2V (5.8)

where the velocity and differentiation are assumed to be with respect to the rotating frame, and where g = g; — n x(n x r) is the local effective gravity, which includes the centrifugal force.

Equation (5.8) can be seen to be almost identical to the equation in the inertial frame with the exception that g is modified as just described and also the new equation has an additional term 2(V x n) that arises due to the so-called Coriolis force, an apparent force experienced by objects moving in a rotating frame, which since it acts at right-angles to the velocity gives rise to circular motion. Near the equator, or at all latitudes on a slowly rotating planet such as Venus, the Coriolis force is small and in such conditions the primary circulation for a planet with low obliquity responding to solar heating tends to be a simple Hadley cell circulation whereby air rises at the equator, moves polewards at high altitude before descending and then returning to the equator at low altitude. For slowly rotating planets such as Venus the descending part of the circulation is at the poles themselves and the Hadley cell is thus global in extent. However, for the Earth and the giant planets, the Coriolis force becomes significant quite close to the equator and thus air is deflected eastwards as it moves towards the poles at high altitude, and deflected westwards as it returns to the equator at low altitudes. Coriolis force modification to the terrestrial equatorial Hadley cell, which extends from the equator to latitudes of approximately ±30°, thus forces the low-altitude, equatorward air to be deflected towards the west, giving rise to the easterly "trade winds" at subequatorial latitudes. While the terrestrial Hadley cell breaks down at latitudes of approximately ±30°, the corresponding critical latitude limit for the far more rapidly rotating giant planets is substantially smaller.

To solve the Navier-Stokes equation for planetary atmospheres where the Coriolis force is important, which is clearly the case in the rapidly rotating giant planet atmospheres, we first need to split up V into its three components (u, v, w), where u is the velocity east/west in the x-direction, v is the velocity north/south in the y-direction, and w is the vertical velocity in the z-direction. Using these components we find that for a spherical planet, following the approach of Houghton (1986), dV fdu uv tan ó uw\. Id v u2 tan ó vw\. l dw u2 + v2\ .

where 0 is the latitude; and R is the planetary radius. Similarly, the Coriolis force term may be rewritten as

2V x H = 2Q(u sin 0 — w cos 0)i — 2Qu sin 0j + 2Qu cos 0k. (5.10)

For the significantly oblate giant planets, Equation (5.10) must be expressed in terms of the planetographic latitude 0g (introduced earlier in Chapter 2), not the planetocentric latitude. Modifications to Equation (5.9) for the oblate giant planets are more complicated, but fortunately the differences are in terms that are usually neglected since it is found to be very difficult to solve the Navier-Stokes equation explicitly. Instead, a number of reasonable and justifiable approximations are made. First of all, it is found to be a good approximation to assume that the magnitude of the winds is much less than the radius of the planet, and thus that the 1 /R terms in Equation (5.9) may be ignored. Second, it is assumed that the vertical wind speeds are very much less than the horizontal wind speeds. This is found to be a good approximation since the action of gravity in the vertical direction keeps departures from hydrostatic equilibrium small and frictional forces are usually negligible meaning that horizontal winds, once initiated, tend to blow relatively freely. Finally, it is found that the vertical component of 2V x H, and the vertical component of friction may both be neglected compared with the gravitational acceleration g. Making these approximations to the Navier-Stokes equation and resolving into components we derive the momentum equations:

dt pdz where f = 2Q sin 0g is called the Coriolis parameter and where we have written the components of friction acting in the x-direction and y-direction explicitly. In most cases the vertical accelerations of the winds are also negligible compared with local gravitational acceleration and thus Equation (5.13) may be well approximated by the hydrostatic equation dP = —gp. (5.14)

Further approximations may be made to these equations, including the geostrophic approximation, which is relevant to giant planet dynamics.

For large-scale motion away from the planetary surface (always obeyed by the giant planets!) the frictional forces are to a first approximation negligible. Furthermore, for steady motion with small curvature, dV/dt is also negligible and hence the horizontal momentum equations reduce to:

These are the geostrophic equations and lead to the familiar situation for the Earth's atmosphere of winds blowing along isobars, with (for prograde-spinning planets) anticlockwise motion about centers of low pressure (cyclones) in the northern hemisphere, and clockwise motion about cyclones in the southern hemisphere. To make this geostrophic approximation we need to ensure that the acceleration terms in the momentum equations are much less than the Coriolis terms and we define the Rossby number Ro to be the ratio of the acceleration to Coriolis terms, for which an approximate expression is

where U is the mean wind speed; and L is the typical horizontal dimension of motion. The geostrophic approximation is then valid, and thus Coriolis forces are dominant, if Ro ^ 1. For the giant planets, the Rossby number is estimated to be of the order of 10~2 (Chamberlain and Hunten, 1987) and thus to a first approximation the atmospheric flows of these planets should be geostrophic.

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