Total Energy Equation Method for Calculating Hurricane Intensity

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Results	Conclusion	References

Total Energy Equation Method
for Calculating Hurricane Intensity.

Louis. M. Michaud
November, 2000

Summary

Surface pressure reduction in hurricanes is calculated by applying the total energy equation (TEE) to ideal isentropic upflow in a vertical tube. The pressure reduction at the base of the tube, called the intensity , is calculated for three upflow processes: reversible upflow of air approaching equilibrium with the sea at the sea level pressure outside the tube; irreversible upflow of air approaching equilibrium with the sea at the sea level pressure outside the tube; and upflow of air approaching equilibrium with the sea at the reduced surface pressure inside the tube. The sensitivity of intensity to the upflow process and to the sea surface temperature is investigated. Intensities calculated with the TEE are shown to agree with intensities calculated with the hydrostatic method. The TEE method is used to show how work is produced when heat is carried by convection from the sea to the upper troposphere.

1. Introduction

Holland (1997) calculated tropical cyclone surface pressure by integrating the hydrostatic equation downward from the level of neutral buoyancy. He developed an iterative method for calculating minimum hurricane surface pressure which he called Maximum Potential Intensity (MPI). This paper shows that MPI's agreeing with Holland's can be obtained by simply applying the total energy equation to upward flow in a vertical tube. This paper will use the pressure reduction at the surface, called intensity , as the measure of hurricane intensity (HI).

In tropical-oceanic areas, the air near the surface approaches equilibrium with the sea; in light to moderate wind, the air near the surface is typically 1 K cooler than the sea surface temperature (SST) and has a relative humidity (U) of approximately 80%. High winds in hurricanes reduce the approach to equilibrium because the air comes in more intimate contact with the water; Holland (1997) used an air temperature 1 K lower than the SST and a relative humidity of 90%. It is common engineering practice to consider that a gas in close contact with a liquid, for example the air coming out of a cooling tower, approaches equilibrium with the liquid. The relative humidity of surface air at hurricane eyewall is difficult to measure, but can be inferred from minimum hurricane cloud height which is around 300 m, Holland (1997). The lifting condensation level of air with 90% relative humidity is approximately 300 m; therefore the air rising in the eyewall must have a relative humidity of approximately 90%.

Holland pointed out that the release of all latent-heat energy of air at normal surface pressure and humidity can only reduce eyewall pressure by 4 kPa and that intense tropical cyclones must obtain additional energy from air-sea interaction at eyewall pressure. The mixing ratio (r) of air at 101 kPa, 30 ° C, 80% relative humidity is 21.7 g kg^-1. The mixing ratio of air at 101 kPa, 30 ° C, 90% RH is 24.5 g kg^-1. The mixing ratio of air at 85 kPa, 30 ° C, 90% RH is 29.4 g kg^-1. Pressure can have more effect on mixing ratio than relative humidity. Increasing the mixing ratio of the air increases its enthalpy and hurricane intensity.

2. Thermodynamic basis

The three ideal thermodynamic flow processes shown in Fig. 1 will be analyzed. In each case, surface air is raised isentropically in a vertical tube and the flow is restricted at bottom of the tube, see Michaud (2000). Holland's (1997) calculations were based on the rising air having a constant equivalent potential temperature (q_e); the total energy equation method is based on the rising air having a constant entropy (s). The two methods are equivalent since a constant equivalent potential process is an isentropic process. The hydrostatic method of calculating surface pressure reduction implies that the rising air is separated from the environment by a partition such as the wall of the tube in Fig. 1.

Reversible Case A corresponds to true-adiabatic expansion. Reversibility requires that kinetic energy be removed from the system before dissipating. The turbine in Fig. 1a captures the kinetic energy of the jet (w_x) coming out of the opening in the base of the tube. In irreversible Case B the kinetic energy of the jet is not captured and is allowed to revert to heat; process 1-2 changes from a constant entropy process, s₂ = s₁ , to a constant enthalpy process, h₂ = h₁ which is also a constant temperature T₂ = T₁ process. In reversible Case A and irreversible Case B, the air has a relative humidity of 80% and a temperature 1 K lower than SST at P₁ before the start of the expansion. In air-sea interaction Case C, the air has a relative humidity of 90% at and a temperature 1 K lower than SST at reduced pressure P₂ after the initial expansion.

Work in a continuous flow process is given by the total energy equation.

where w is the work given up by the flowing fluid, q is the heat received by the flowing fluid, h is the enthalpy of the air including the enthalpy of its water content, g is the acceleration of gravity, z is the height, and v is the velocity. The deltas (D ) are taken between the inlet and outlet conditions of each process. Eq. (1) is applicable to individual processes 1-2 and 2-3 and to the overall process 1-3. For processes without heat transfer, q = 0 and with negligible velocities, v approaches zero; (1) reduces to:

where m is the static-energy of the air including its water content, where m =h+gz.

The TEE method is based on the realization that the work (w₂₃= 0) during constant entropy upflow process 2-3 approaches zero. The pressure drop due to friction in the upflow process is small compared to the hydrostatic surface pressure reduction. The pressure drop due to friction in a horizontal tube 1 km in diameter by 15 km long at an air velocity of 20 m s^-1 is less than 0.01 kPa. The pressure at the base of the tube, P₂, is therefore the pressure for which the work during process 2-3 is zero, the pressure for which the change in the static-energy of the rising air is zero. The same procedure is used to calculate base pressure in the three cases. A solver is used to calculate the work during upflow process 2-3 (w₂₃) for two P₂ guesses; linear interpolation is then used to calculate the pressure P₂ for which w₂₃ is zero. The base pressure P₂ was subsequently verified independently by downward integration of the hydrostatic equation from the top of the tube to the surface.

Table 1 shows the results for air raised form the surface to the 20 kPa level under typical tropical-oceanic conditions. The geo-potential height of the 20 kPa surface is taken as 12400 m, which is typical for oceanic-tropical areas. The calculations assume that the condensed water does not separate from the air and that the condensed water freezes, true-adiabatic expansion. The intensity would be slightly higher if the air was raised to its level of neutral buoyancy. The level of neutral buoyancy in tropical-oceanic areas is usually above the 20 kPa level. The equations used to calculate the thermodynamic properties of the air are given in Appendix A of Michaud (1995).

Results

3a. Reversible Case A

The conditions of the surface air given in Table 1 are: P₁=101 kPa, T₁=27 ° C, U₁=80%. The MPI is 99.06 kPa, and the intensity is 1.94 kPa. The entropy (s) of the air is 256.7 J kg^-1 in the three states. Intensity is very sensitive to surface temperature and humidity. Increasing the temperature of the surface air by 1 ° C at constant mixing ratio (r) increases intensity by 0.27 kPa. Increasing the relative humidity of the surface air by 1% at constant temperature increases intensity by 0.11 kPa. Increasing the temperature of the surface air by 1 ° C at constant relative humidity increases intensity by 0.90 kPa because both the temperature and the mixing ratio increase. Decreasing the 20 kPa geo-potential height by 100 m by decreasing the average temperature of the environment by » 2 K increases intensity by 1.11 kPa. A small change in air temperature has a large effect on intensity. Decreasing the SST or the temperature of the surface air by 2 ° C at constant relative humidity would decrease the intensity to close to zero.

3b. Irreversible Case B

In irreversible Case B, the MPI is 98.48 kPa, and the intensity is 2.52 kPa. The entropy of the air increases from 256.7 to 264.4 J K^-1kg^-1 as its kinetic energy is dissipated in irreversible process 1-2. Irreversibility increases the intensity and the kinetic energy of the air at the restriction outlet by 30%; the velocity of the air at the restriction outlet increases by 14%. Bister and Emanuel (1998) used a numerical model to show that dissipative heating can increase the maximum wind speed in hurricanes by 20%. The sensitivity of hurricane intensity to surface temperature and humidity is higher than in reversible Case A. Increasing the temperature of the surface air by 1 ° C at constant mixing ratio increases intensity by 0.35 kPa. Increasing the relative humidity of the surface air by 1% increases intensity by 0.16 kPa. Increasing the temperature of the surface air by 1 ° C at constant relative humidity increases intensity by 1.13 kPa.

3c. Air-Sea Interaction Case C

The level of neutral buoyancy for air approaching equilibrium with water at the reduced surface pressure is closer to 10 kPa than to 20 kPa. The height of the 10 kPa geo-potential surface in tropical-oceanic areas is typically 16600 m. Column 3 of Table 1 shows the base pressure when the air is raised to the 20 kPa level, and Table 2 shows the base pressure when the air is raised to the 10 kPa level. For the 28 ° C SST and 90% relative humidity case, the intensity is 4.71 kPa when the air is raised to the 20 kPa level and 7.78 kPa when the air is raised to the 10 kPa level. The level of neutral buoyancy for the Willis island average January sounding used by Holland is »7 kPa.

Table 2 shows that intensity in the air-sea interaction case is extremely sensitive to surface temperature and humidity. Increasing SST by 1 ° C at constant relative humidity increases intensity by 2.93 kPa . Increasing the relative humidity at constant surface air temperature by 1% increases intensity by 0.4 kPa . The sensitivities obtained with the TEE method are very close to the sensitivities obtained by Holland (1997).

Holland (1997) obtained following eye-MPI for a mean January Willis island sounding, see his Table 5:

SST=27 ° C, U=90%, MPI=96.3 kPa,

SST=29 ° C, U=90%, MPI=89.7 kPa,

SST=31 ° C, U=90%, MPI=83.2 kPa ,

SST=29 ° C, U=80%, MPI=93.3 kPa.

Sensitivity of MPI to SST: -3.3 kPa ° C^-1 .

Sensitivity of MPI to U: -0.48 kPa %^-1 .

A 31 ° C SST is sufficient to produce an MPI of 83 kPa and to explains the lowest observed hurricane pressure. Reducing the SST to » 26 ° C reduces the intensity to close to zero. The fact that the minimum SST required for cyclone development is 26.5 ° C is well recognized, Holland (1997). The sensitivity of intensity to SST is 0.90 kPa K^-1 in the reversible case, 1.13 kPa K^-1 in the irreversible case, and 2.93 kPa K^-1 in the air-sea interaction case. The increase in intensity in the air-sea interaction case is mainly due to the fact that for the same temperature and relative humidity air holds more water vapor at lower pressure.

Hurricane upflow takes place in the eyewall, therefore hurricane Intensity should corresponds to the pressure reduction at the eyewall. Eye pressure is lower than eyewall pressure because of the drag exerted by the eyewall on the eye. The eyewall acts like the impeller of a centrifugal pump by reducing the pressure in the eye and drawing the air at the center of the vortex air down. The additional pressure reduction at the eye depends on the tangential velocity profile and is estimated to be only about 20% of eyewall pressure reduction since the tangential velocity decreases rapidly inward from the eyewall. Friction losses tend to make the actual eyewall pressure reduction slightly less than the pressure reduction at the base of the tube. The eyewall friction loss is offset by the eyewall to eye pressure differential tending to make the eye pressure correspond to the pressure at the base of the ideal tube.

The intensity calculation were based on the condensed water freezing and not separating from the air, commonly called true-adiabatic expansion. Separating the water from the air, commonly called pseudo-adiabatic expansion, has little effect on intensity . More mechanical energy is produced in true-adiabatic expansion because more heat is transferred from the condensed water to the air, but the additional energy is used to lift water and does not appreciably increase intensity.

Holland (1997) used a mean Willis Island January sounding, but did not provide the sounding. The total energy equation (TEE) method was applied to a specific sounding, the January 17, 1999, 0000Z Willis island sounding. Fig. 2 shows the ambient pressure and the pressure inside the tube for this particular sounding. The conditions of the rising air in cases A and B are the base sounding conditions: P=100.3 kPa, T=29.4 C, U=78.2%. The base pressure reduction was calculated using the two guesses method assuming a tube top pressure of 10 kPa. The buoyancy of the lifted air at the tube top was then checked and the tube top height adjusted until the tube reached the level of neutral buoyancy. The level of neutral buoyancy is 10 kPa for case A, 9 kPa for Case B, and 7 kPa for case C. The intensity is not very sensitive small changes in tube height because the absolute value of the buoyancy is low close to the level of neutral buoyancy.

The intensity calculated with the TEE method agrees with the results obtain by Holland (1997), but the actual intensity of hurricanes can be reduced by 6-10 kPa because of SST cooling near the eyewall. The SST used in the TEE method must be the temperature of the water with which the air comes in intimate contact, the temperature of the spray or of the top few centimeters of the sea. Holland (1997) pointed out that there are indications that the temperature of the spray; at the eyewall can be 4 ºC lower than the SST. Decreasing spray temperature by a further 3 ºC would decrease intensity by 10 kPa. The passage of a hurricane can reduce SST temperature by 2 to 6 ºC, Schade and Emanuel (1999). The SST reduction is often attributed to entrainment of cold water, Schade (2000), but the SST reduction is mainly the result of heat transfer from to sea to the rising air. The heat received by the rising air in the base case of Table 2 is approximately 10 kJ kg^-1 (h₂-h₁). The heat transfer required to support a column of air rising at 1 m s^-1 is 36 MJ hr^-1 m^-2 enough to cool a column of water 10 m deep by 1 ºC hr^-1. The heat transfer rate between the sea and the air in the eyewall area can be extremely high. The temperature of the spray could be several degrees lower than the temperature one meter below the surface.

The TEE method does not explicitly use efficiency, nevertheless heat flow from the sea surface to the upper troposphere is the source of the energy of hurricanes. Mechanical energy is produced when heat is carried upward because there is heat flow from a hot source, the sea; to a cold source, the upper troposphere. The mechanical energy produced and dissipated in a hurricane is approximately 20% of the heat removed from the sea. The heat to work conversion efficiency can be calculated using the SST for the hot source temperature and the average temperature at which heat would have to be removed from the upper troposphere to restore its initial condition as the cold source temperature. Taking the hot source temperature as 300 K and the cold source as 230 K, the Carnot efficiency is 23%; of which »5% is required to lift water leaving »18% available to produce motion. The cold source temperature is somewhat less than the average temperature of the troposphere because raising air has more warming effect on the upper troposphere than on the lower troposphere, see Michaud (1995). Incremental heating was used by Michaud (2000) to show that 20-25% of the heat transported upward in tropical cyclones is converted to mechanical energy. The upward heat flux and the mechanical energy production rate in a mid size hurricane could be 20 and 4 TW respectively.

The kinetic energy of the air coming out of the restriction is captured in Case A, and dissipated in Cases B and C. The work is dissipated near the earth's surface, as the air converges towards the base of the updraft. The converging flow in hurricanes is restricted to the thin surface layer where centrifugal force is reduced by friction, Michaud (2000). Bister and Emanuel (1998) showed that work dissipation is concentrated near the surface corresponding to Fig. 1, see their Fig. 4.

Emanuel (1991) calculated hurricane surface pressure using thermodynamic efficiency and surface heat transfer and obtained a sensitivity to SST of 0.6 kPa ºC^-1, much lower than the 3.3 kPa ºC^-1obtained by Holland (1997). Calculating the heat transfer from the sea to the air introduced unnecessary difficulties. With the Emanuel method, decreasing the relative humidity of the air increases the heat transfer from the sea and increases intensity and eventually the humidity of the rising air. With the Holland method, increasing the relative humidity of the air increases intensity directly; there is no need to consider heat transfer when the approach to equilibrium is known. The Emanuel method only considers the heat transferred from the sea to the air and does not consider the heat content of the air at ambient condition. Schade and Emanuel (1999) developed an intensity reduction factor to estimate how intensity is reduced by SST reduction. Schade (2000) showed that the results of Emanuel and Holland are equivalent when SST reduction is considered.

The kinetic energy of the air (w_x) and the velocity of the air at the restriction outlet (v_x) are shown in Tables 1 and 2. The double restriction shown in the air-sea interaction case of Fig. 1 represent the fact that the work can be dissipated in stages as the air converges towards the eyewall. Hurricane minimum pressure generally agree with Table 2; the maximum velocities are somewhat lower than those shown in Table 2 because the work is dissipated in stages as the air converges towards the eyewall.

4. Conclusion

The determination of which variables are conserved is a key consideration in applying the total energy equation. Once the conserved variables have been established the missing variables can easily be found. In Case A, entropy and mixing ratio are conserved during process 1-2; in Case B, enthalpy (h) and mixing ratio are conserved during process 1-2. Entropy (s), static energy (m ) and mixing ratio (r) are all conserved during process 2-3 in all three cases. P₂ in Case A can be determined because mixing ratio and entropy at state 2 are known. P₂ in Case B can be determined because mixing ratio and enthalpy at state 2 are known. P₂ in Case C can be determined because relative humidity and temperature in state 2 are known.

Reversible Case A correspond to the customary constant entropy processes, which is an idealization, since in nature work is not removed from the system. For air raised to the 20 kPa level, see Table 1: the work produced in reversible Case A (w₁₂=1710 J kg^-1) corresponds to the CAPE for a true-adiabatic lifting process; the work dissipated in irreversible Case B (w_x=2232 J kg^-1) is 30% higher than produced in reversible Case A; the work dissipated in air-sea interaction Case C (w_x=4230 J kg^-1) is 150% higher than produced in reversible Case A.

For air raised to the 10 kPa level with air-sea interaction, see Table 2: the work is 7074 J kg^-1 for 28 ºC SST and 9909 J kg^-1 for 29 ºC SST. The sensitivity of MPI to surface air temperature and humidity was examined. Michaud (2000) showed that the work dissipated in the atmosphere is approximately 17% of the convective heat flux at the bottom of the atmosphere; the efficiency for hurricanes is higher than this average because the heat is carried higher in the troposphere. It is estimated that 50 to 80% of the energy of hurricanes is due to heat received from air-sea interaction at reduced pressure near the eyewall; and that the remainder is due to heat received from the sea at near normal surface pressure either prior to or during the hurricane.

The intensity of tropical cyclones is primarily a function of SST because the relative humidity of air approaching equilibrium with an agitated sea will inevitably be close to 90%, and because the elevation of geo-potential surfaces near the top of the troposphere in sub-tropical latitudes is essentially constant.

References

Bister, M., and K.A. Emanuel, 1998: Dissipative heating and hurricane intensity. Meteorol. Atmos. Phys., 65 , 233-240.

Emanuel, K.A., 1991: The theory of hurricanes. Annu. Rev. Fluid Mech., 23 , 179-196.

Holland, G.J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54 , 2519-2541.

Michaud, L.M., 1995: Heat to work conversion during upward heat convection. Part I: Carnot engine method. Atmos. Res., 39 , 157-178.

Michaud, L.M., 2000: Thermodynamic cycle of the atmospheric upward heat convection process. Meterol. Atmos. Phys., 72, 29 -46.

Schade, L.R., 2000: Tropical cyclone intensity and sea surface temperature. J. Atmos. Sci., 57 , 3109-3121.

Schade, L.R., and K.A. Emanuel, 1999: The ocean's effect on the intensity of tropical cyclones: Results from a simple atmosphere-ocean model. J. Atmos. Sci., 56 , 642-651.

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Mathcad 5.0 Program used to generate Table 1 and 2 and Fig. 2.

Total Energy Equation Method for Calculating Hurricane Intensity.

Total Energy Equation Method
for Calculating Hurricane Intensity.