# Monthly Archives: March 2011

## Radial profiles of velocity and pressure for condensation-induced hurricanes

Makarieva A.M., Gorshkov V.G. (2011) Radial profiles of velocity and pressure for condensation-induced hurricanes, Physics Letters A, 375, 1053-1058.

Abstract.

The Bernoulli integral in the form of an algebraic equation is obtained for the hurricane air ﬂow as the sum of the kinetic energy of wind and the condensational potential energy.With an account for the eye rotation energy and the decrease of angular momentum towards the hurricane center it is shown that the theoretical profiles of pressure and velocity agree well with observations for intense hurricanes. The previous order of magnitude estimates obtained in pole approximation are confirmed.

In simple words

Maximum wind speeds that arise in hurricanes and tornadoes are developed when the pressure gradient force significantly exceed friction forces. In the absence of friction energy along the streamline (the Bernoulli integral) is conserved, equal to the sum of kinetic and potential energy. Potential energy coincides with air pressure, while the pressure gradient determines the force that makes the air move.

The stream power is equal to the product of force and velocity. The characteristic change of pressure along the vertical is $\Delta p = \beta p_v$, where $p_v$ is partial pressure of water vapor at the Earth’s surface, $\beta$ is the share of water vapor that ultimately condenses in the ascending air. The mean vertical force acting per unit air volume is equal to $\Delta p/h$, where $h$ is the characteristic scale height of the condensation process.

Due to the continuity constraints the vertical power $(\Delta p/h)w$, where $w$ is vertical velocity of the rising air, should be equal to radial power $(\partial p/\partial r) u$, where $u$ is radial velocity directed towards the center of the condensation area, $r$ is radial distance from the center. On the other hand, change $d(2\pi r hu)$ of the radial air flux entering the condensation region via a circular wall of area $2 \pi r h$ is equal to $2\pi rw dr$, which is the vertical air outflow via the ring of radius $r$ and area $2\pi r dr$. This equality relates the vertical and radial velocity by the well-know relationship $w = (h/r) (\partial ur/\partial r)$. Putting this relationship into the equality of the vertical and horizontal powers, we obtain that condensation causes the air pressure to fall along an unusual non-linear law: $p(r) = p_1 + \Delta p \ln ur$.

The smallness of vertical velocity $w$ allows one to neglect it compared to radial $u$ and tangential (rotational) $v$ velocities, $v \approx a/r$, where $a$ is the (partially) conserved angular momentum. This causes the Bernoulli integral for the air flow to take the form of an algebraic equation on radial velocity

$\rho/2 (u^2 + a^2/r^2) + \Delta p \ln ur = \rm const.$

The obtained transparent form of the Bernoulli integral describes the major quantitative hurricane characteristics, including profiles of pressure and velocity and the size of the eye.