Tropical storms and the First Law of ThermodynamicsATMO 435

SST 22-24 C along path

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dQ = dU + dW

First Law of Thermodynamics

The internal energy dU changes when:1. heat dQ is exchanged between a parcel and its environment2. work is done by a parcel on its environment (or vice-versa)

For a cyclic process in which the internal energy does not change (you end up where you started), then any heating dQ must be balanced by work done to or by the system dW. In this case

This merely says that the total work done by the system is balanced by the total heating done through the cycle.

This cycle has a maximum efficiency, the Carnot engine. This is the physics that sets the speed limit on how intense a tropical storm can be.€

dW = dQ

W total = Qtotal

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Carnot cycle

The temperature returns to its starting point, so there is no change in the internal energy over the course of this cycle. All the work done must be balanced by all the heating that occurs.

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1. Gas is heated (Q > 0), so piston rises to reduce the pressure. This pressure drop would cause the temperature to go down, but heat added compensates exactly so that T is constant at Thigh. This step is isothermal expansion.

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2. The heat source is turned off (Q = 0 now), but the piston keeps moving up. Now without any heat source to balance, the continued decrease in pressure results in the gas cooling (T goes down). This step is adiabatic expansion.

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3. In the reverse of step 1, the a cold source is applied (Q < 0). The piston falls, which would lead to a warming but for the loss of heat by the system. The temperature is fixed at Tlow. This step is isothermal compression.

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4. With the cooling source removed (Q = 0 again), the piston continues to drop back to its starting point. With the increasing pressure and smaller volume comes a rise in temperature. This is adiabatic compression.

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Carnot cycle

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W total = Qtotal = Qhigh − Qlow( )

An ideal hurricane

A natural coordinate system for working with hurricanes is cylindrical coordinates. All equations are written in terms of radius, angle, and height, as opposed to Cartesian coordinates (distance east, distance north, height).

Cartesian coordinates are (x, y, z). Cylindrical coordinates are (r, , z).

An ideal hurricane does not vary azimuthally. That means that it is axisymmetric.That means that all the structure can be viewed in the (r, z) plane.

Real hurricanes are not ideal.

But the strongest ones follow most closely to these constraints. And we’re using this analysis to find out how intense a storm could possibly be.

Deviations from this structure can only be detrimental to a storm’s health.

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An ideal hurricane

The total work done must balance the heat input at the surface:

The heat input at the surface comes from two sources: 1. Fluxes of heat--actually, enthalpy, k--from the ocean2. Dissipative heating from recycled frictional dissipation

Where D is the second source, the dissipative heating from frictional dissipation. This is the same expression as the total work done by the environment (friction at the surface):

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W total =Thigh − Tlow

Thigh

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Qhigh =

Tsurface − Toutflow

Tsurface

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Qinput at surface

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Qinput at surface = Ckρr V

0

ro

∫0

2π

∫ kocean* − k( ) rdrdθ + D

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W total = D = CDρr

V 3

rdrdθ0

ro

∫0

2π

∫

An ideal hurricane

Putting all of this together and solving for the wind speed V, the highest wind speed possible in a hurricane is:

This says that the maximum wind speed possible is a function of the exchange coefficients Ck and CD, the sea surface temperature Ts, the temperature at the top of the convection To, and the thermodynamic disequilibrium between the saturated ocean surface and the overlying marine boundary layer.

Notice that while the ocean surface temperature affects the outcome, the critical thing is not the actual value of the water temperature, but rather the difference between it and the temperature at which convection terminates.

This is a point lost on too many meteorologists. There is not anything magic about water that is 79oF or 26.5oC. This particular isotherm happens to live in the same neighborhood as the boundary where it becomes very difficult for convective motions originating at the surface to puncture through the trade inversion and produce deep convection that penetrates into the upper troposphere.

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rV =

Ck

CD

Ts − To

To

kocean* − k( )

http://wxmaps.org/pix/hurpot.html

An ideal hurricane meets reality

Of course there are many things that inhibit actual storms from reaching the maximum speed permitted by the laws of thermodynamics.

But interestingly, the strongest storms we observe on Earth are the same magnitude as that predicted by these thermodynamic principles, providing an important validation of this theory.

Storms cannot be as strong as they like: they cannot exceed the values set by the relationship between the input temperature, the outflow temperature, and the strength of the surface fluxes. So a hurricane with 600 mph winds is not possible on Earth, given the relationship of these parameters.

Real hurricanes are limited by many factors; these include: incorporating drier air into the circulation (this inhibits convection, raising the outflow temperature), land (reduces the surface fluxes), wind shear (axisymmetry is violated and deep convection is limited), and interactions with the ocean itself.

These actually produce fascinating coupled atmosphere-ocean dynamics, and the result can be seen in cooler sea surface temperatures left in a storm’s wake.

time = 1

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time = 2

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time = 3

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time = 3

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Mixed-layer depth in a hurricane’s wake

meters

SST change in a hurricane’s wake

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