Equations of Motion and Force Balances

Equations of motion and force balance

The momentum equation for an incompressible viscous fluid is \rho \frac{D\mathbf{V}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{V} + \mathbf{B}, where the left side is the inertial term, -\nabla p is the pressure gradient force, \mu\nabla^2\mathbf{V} is the viscous force, and \mathbf{B} is the body force per unit volume. Dimensional analysis is a reliable way to check any term of such an equation. The dynamic viscosity \mu has dimensions [M][L]^{-1}[T]^{-1}, the Laplacian \nabla^2 has [L]^{-2}, and \mathbf{V} has [L][T]^{-1}, so the viscous term carries [M][L]^{-2}[T]^{-2}. This is the same as \rho D\mathbf{V}/Dt and \nabla p, so the equation is dimensionally consistent.

On a rotating Earth, the Coriolis acceleration -2\boldsymbol{\Omega}\times\mathbf{V} is an apparent force. Its horizontal magnitude is 2\Omega V\sin\phi = fV, where f = 2\Omega\sin\phi is the Coriolis parameter. It depends only on latitude and speed, not on longitude, and it vanishes at the equator. The Coriolis force is always perpendicular to the velocity, so its work \mathbf{F}_c\cdot\mathbf{V} is zero. It changes the direction of motion but never the speed. On a horizontal plane, a freely moving body is therefore deflected into a circle of radius R = V/f, called the inertial circle. This is why a moving body curves to the right in the Northern Hemisphere.

Advection changes the local temperature by \partial T/\partial t = -\mathbf{V}\cdot\nabla T. For a zonal wind this is \partial T/\partial t = -u\,\partial T/\partial x, so if temperature increases toward the east, an eastward wind cools a fixed point. The change over an interval is the tendency multiplied by the elapsed time.

The relative importance of inertia and Coriolis force is measured by the Rossby number Ro = U/(fL), where U is a characteristic speed and L a length scale. For a typical synoptic-scale system Ro is about 0.1, so Coriolis force dominates and the flow is close to geostrophic balance. In a liquid in solid-body rotation about a vertical axis, the isobaric surfaces are paraboloids, z = \omega^2 r^2/(2g) + C, obtained by combining hydrostatic balance with the centrifugal term. The height of an isobar therefore rises with the square of the radial distance from the axis.

Geostrophic balance

Geostrophic balance is the state in which the pressure gradient force is balanced exactly by the Coriolis force, with no curvature term. Its speed is fV_g = \frac{1}{\rho}\left|\frac{\partial p}{\partial n}\right|, so V_g = \frac{1}{\rho f}\frac{\partial p}{\partial n}, where n is the direction normal to the isobars. The wind blows parallel to the isobars, with low pressure on the left in the Northern Hemisphere.

As an example, take \rho = 1.2 kg m^{-3} at 60°N, where f \approx 1.26\times10^{-4} s^{-1}. A pressure difference of 300 Pa over 300 km gives \partial p/\partial n = 10^{-3} Pa m^{-1} and a geostrophic speed of about 6.6 m s^{-1}. Because f is proportional to \sin\phi, the geostrophic speed for a given gradient is larger at higher latitudes, and the approximation fails at the equator where f vanishes.

Along a straight isobar at a fixed latitude, both f and the normal pressure gradient are uniform, so the geostrophic speed does not change along it. Cyclostrophic speed, by contrast, changes with the radius of curvature of the isobar. A steady, frictionless flow along straight, parallel isobars has zero curvature, so the centrifugal term vanishes. The Coriolis and pressure gradient forces then balance, and the acceleration of the flow is close to zero.

The three basic balances are distinguished by which forces appear. Geostrophic flow involves only pressure gradient and Coriolis forces. Gradient flow around a cyclone in the Northern Hemisphere adds the centrifugal force to that pair. Cyclostrophic flow, as in a tornado, involves only the pressure gradient and centrifugal forces.

Thermal wind and barotropic–baroclinic flow

A barotropic fluid is one in which density depends on pressure alone, so isopycnals (surfaces of constant density) are parallel to isobars. In that case the geostrophic wind is the same at every height and there is no vertical shear of the geostrophic wind.