Generalized Asymmetric Holland Model - Revision history

Taylorgasher: /* Derivation of the GAHM */ fixed missing phi in math equation and a formatting issue

2024-06-11T19:38:03Z

Derivation of the GAHM: fixed missing phi in math equation and a formatting issue

← Older revision		Revision as of 15:38, 11 June 2024
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	<center> <math> V_g(r) = \sqrt{V_{max}^2(1+1/R_o)e^{1-(R_{max}/r)^{B_g}}(R_{max}/r)^B_g + (\frac{rf}{2})^2} - \frac{rf}{2} \quad </math> (13) </center>		<center> <math> V_g(r) = \sqrt{V_{max}^2(1+1/R_o)e^{\varphi(1-(R_{max}/r)^{B_g})}(R_{max}/r)^{B_g} + (\frac{rf}{2})^2} - \frac{rf}{2} \quad </math> (13) </center>

	Influence of the Coriolis force on the radial pressure and wind profiles are evidenced by the presence of <math> R_o </math> and <math> \varphi </math> in (12)&(13). A special case scenario is when we set <math> f=0 </math>, which corresponds to an infinitely large <math> R_o </math>, then (12)&(13) in the GAHM reduce to (6)&(7) in the HM. However，for a hurricane with a relatively small <math> R_o </math>, the influence of the Coriolis force can only be addressed by the GAHM. It meets our expectation that the GAHM’s solution approaches to that of the HM’s when the influence of Coriolis force is small, but departs from it when the Coriolis force plays an important role in the wind system.		Influence of the Coriolis force on the radial pressure and wind profiles are evidenced by the presence of <math> R_o </math> and <math> \varphi </math> in (12)&(13). A special case scenario is when we set <math> f=0 </math>, which corresponds to an infinitely large <math> R_o </math>, then (12)&(13) in the GAHM reduce to (6)&(7) in the HM. However，for a hurricane with a relatively small <math> R_o </math>, the influence of the Coriolis force can only be addressed by the GAHM. It meets our expectation that the GAHM’s solution approaches to that of the HM’s when the influence of Coriolis force is small, but departs from it when the Coriolis force plays an important role in the wind system.

Taylorgasher: added categories

2020-05-12T17:11:50Z

added categories

← Older revision		Revision as of 13:11, 12 May 2020
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	== References ==		== References ==
	<references />		<references />

			[[Category:physics]]
			[[Category:meteorology]]

Luckygirl001: /* Derivation of the GAHM */

2020-04-16T19:03:02Z

Derivation of the GAHM

← Older revision		Revision as of 15:03, 16 April 2020
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	To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.		To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.

	[[File:~~Figure~~ 3. GAHM.png\|center~~\|750px~~\|Normalized gradient wind profiles]]		[[File:Fig 3. GAHM.png\|center\|Normalized gradient wind profiles]]
	<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>		<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>

	<!-- ******************* chapter 3 ***************** -->		<!-- ******************* chapter 3 ***************** -->

	== Calculation of the Radius to the Maximum Wind ==		== Calculation of the Radius to the Maximum Wind ==

Luckygirl001 at 19:00, 16 April 2020

2020-04-16T19:00:09Z

Show changes

Luckygirl001: /* The AHM vs. the GAHM */

2020-04-14T05:59:48Z

The AHM vs. the GAHM

← Older revision		Revision as of 01:59, 14 April 2020
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	Since the AHM is an advanced version of the HM, here we only use model results from the AHM for comparisons with the GAHM. First, the single-isotach approach was evaluated using Hurricane Irene (2011) as an example. Figure 5 gives the comparison of radial wind profiles of Hurricane Irene (2011) between the AHM and the GAHM using the single-isotach approach at three snapshots, each representing the developing (top panels), mature (middle panels), and dissipating (bottom panels) stages of the hurricane.		Since the AHM is an advanced version of the HM, here we only use model results from the AHM for comparisons with the GAHM. First, the single-isotach approach was evaluated using Hurricane Irene (2011) as an example. Figure 5 gives the comparison of radial wind profiles of Hurricane Irene (2011) between the AHM and the GAHM using the single-isotach approach at three snapshots, each representing the developing (top panels), mature (middle panels), and dissipating (bottom panels) stages of the hurricane.

	[[File:Fig 7. GAHM.png\|center~~\|800px~~\|Radial wind profiles of Irene (2011) at three different stages.]]		[[File:Fig 7. GAHM.png\|center\|Radial wind profiles of Irene (2011) at three different stages.]]

	<center> Figure 5. Comparison of radial wind profiles of Irene (2011) at three different stages between the AHM and the GAHM. </center>		<center> Figure 5. Comparison of radial wind profiles of Irene (2011) at three different stages between the AHM and the GAHM. </center>
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	{\| border="0" cellpadding="5" cellspacing="0" align="center"		{\| border="0" cellpadding="5" cellspacing="0" align="center"
	\|-		\|-
	\| align="center"\|[[File:Fig 8.1. GAHM.png\|~~600px~~]]		\| align="center"\|[[File:Fig 8.1. GAHM.png\|800px]]
	\|-		\|-
	\| align="center"\|[[File:Fig 8.2. GAHM.png\|~~600px~~]]		\| align="center"\|[[File:Fig 8.2. GAHM.png\|800px]]
	\|-		\|-
	\| align="center"\|Figure 6. Comparison of the modeled and “Best Track” maximum winds (upper two panels), and the modeled and “Best Track” <math> R_{max} </math> (lower two panels) between the AHM and the GAHM based on all seven hurricanes.		\| align="center"\|Figure 6. Comparison of the modeled and “Best Track” maximum winds (upper two panels), and the modeled and “Best Track” <math> R_{max} </math> (lower two panels) between the AHM and the GAHM based on all seven hurricanes.

Luckygirl001: /* Evaluation of the GAHM's Composite Wind */

2020-04-14T05:58:05Z

Evaluation of the GAHM's Composite Wind

← Older revision		Revision as of 01:58, 14 April 2020
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	<center> Figure 7. 3D plot of Irene’s radial wind profiles (left) and interpolated spatial wind fields (right) by the single-isotach approach (upper panels) and the multiple-isotach approach (lower panels). </center>		<center> Figure 7. 3D plot of Irene’s radial wind profiles (left) and interpolated spatial wind fields (right) by the single-isotach approach (upper panels) and the multiple-isotach approach (lower panels). </center>

	=== The GAHM's ~~Composite~~ Wind ===		=== Evaluation of GAHM's Composite Wind Field ===

			* '''Comparison of Radial Wind Profiles'''

			Using the multiple-isotach approach, radial wind profiles generated from the GAHM were compared to those from the AHM, the SLOSH, HWinds, and OWI winds, shown in Figure 8 at three different stages of Irene (2011), with the left panels showing the SW to NE cross-section winds, and the right showing the NW to SE cross-section winds (same as in Figure 5). It is evident that the GAHM's composite radial wind profiles matched all available storm isotachs at all time, while the rest of the models failed most of the time. The SLOSH model, which is also a parametric wind model, does not take the isotach information to construct its wind field, so generally did a bad job matching any given isotaches. More detailed wind structures can be observed from the radial profiles extracted from the HWind and OWI winds than those from the parametric wind profiles, as expected. However, profiles from different models in general did not match each other due to different mechanisms involved in each model.

			[[File:Picture12 GAHM.png\|center\|Comparison of radial wind profiles of Irene (2011)]]
			<center> Figure 8. Comparison of radial wind profiles of Irene (2011) at three different stages among the AHM, the GAHM, SLOSH, H*Wind and OWI winds. </center>


			* '''Evaluation of Model Results at Radii to Given Isotaches'''
			<!-- statistics -->
			[[File:Picture13 GAHM.png\|center\|Comparison of specified isotachs and modeled winds]]
			<center> Figure 9. Evaluation of modeled wind at radii to given isotaches based on all seven selected storms. </center>

			Quantitative evaluation of the GAHM's performance at radii to given isotaches were given in Figure 9, with statistics given in Table 3, based on all seven selected hurricanes. The GAHM had almost perfect match to each of the 34-, 50-, and 64-kt isotaches, with a standard deviation around 0.1 kt, which was very impressive. The AHM matched the 64-kt isotach reasonably well with a mean of 63.3 kts, but failed at the lower isotaches. The SLOSH did not take any isotach information to construct its wind fields, thus behaved poorly in matching any given isotaches. Its data also had the largest spread compared to other models, with standard deviation ranging from 11 to 16 kts. For the H*Wind and the OWI winds, the means of modeled winds were close to each specified isotach, generally within ±3 kts, but the spreads of data were also large compared to that of the AHM and the GAHM, with a standard deviation greater than 7 kts.

			{\| class="wikitable" style="text-align:center" style="margin: 1em auto 1em auto" width = "75%"
			\|+'''Table 3. Statistical analysis of modeled winds at radii to specified isotaches based on all seven storms'''
			\|
			! colspan="3" \| MEAN (kt)
			! colspan="3" \| Standard Deviation (kt)
			\|-
			\|
			\| Iso-34 \|\| Iso-50 \|\| Iso-64 \|\| Iso-34 \|\| Iso-50 \|\| Iso-64
			\|-
			\| '''AHM''' \|\| 26.9 \|\| 44.7 \|\| 63.3 \|\| 7.96 \|\| 7.09 \|\| 2.55
			\|-
			\| '''GAHM''' \|\| 34.0 \|\| 50.0 \|\| 64.0 \|\| 0.10 \|\| 0.12 \|\| 0.10
			\|-
			\| '''SLOSH''' \|\| 30.0 \|\| 48.9 \|\| 69.5 \|\| 11.00 \|\| 14.11 \|\| 16.08
			\|-
			\| '''OWI''' \|\| 33.1 \|\| 48.3 \|\| 61.3 \|\| 7.65 \|\| 8.71 \|\| 9.42
			\|-
			\| '''H*Wind''' \|\| 34.9 \|\| 49.2 \|\| 61.2 \|\| 7.55 \|\| 8.88 \|\| 8.89
			\|}

			* '''Comparison of Spatial Wind fields'''
			<!-- spatial plots among 5 models -->

			Comparisons of spatial wind fields among the AHM, the GAHM using the multiple-isotach approach, the SLOSH, HWind and OWI winds (in rows) were given by Figure 10 at the same snapshots (in columns) as in Figure 8. The 34-, 50-, and 64-kt contour lines were shown in each wind snapshot. In general, the AHM and GAHM winds shared a lot of similarities. During weaker periods of Irene, the differences between the AHM and the GAHM's spatial wind fields were mostly observed in the inner region of the hurricane due to differences in calculated <math> R_{max} </math>. More specifically, the calculated <math> R_{max} </math> in the AHM was under-predicted (thus is closer to the storm center) than that in the GAHM, resulted from the faulty cyclostrophic balance assumption made in the HM and AHM. During stronger periods of Irene, however, the differences were mostly observed in the outer region of the hurricane due to the usage of the multiple-isotach approach in the GAHM. Without taking into account storm information from lower isotaches, the AHM winds tended to die off too quickly away from the storm center. The SLOSH winds did not show much similarity to the AHM and GAHM winds. It used an azimuthally constant <math> R_{max} </math> from the "best track" file to generate its vortex winds, and a distance-weighted translational speed to account for storm asymmetry, which was not rendered properly when the storm was strong. The winds fields of HWind and OWI winds were much more complex than those of the simple parametric models, but the GAHM did a relatively good job matching the spatial patterns of the hurricane. Although it is unlikely that the parametric winds constructed over a minimum set of data would match the level of details and complexity in the re-analysis H*Wind and the numerical OWI winds, the ability of a the GAHM to produce reasonable estimates of surface wind fields in a timely manner was highly desirable for real-time forecasting.


			[[File:Fig 10. GAHM.png\|center\|Three snapshots (in columns) of Irene’s two-dimensional wind fields by the AHM, GAHM, SLOSH, H*Wind and OWI winds]]
			<center> Figure 10. Comparison of Irene’s spatial wind fields by the AHM, GAHM, SLOSH, H*Wind and OWI winds at three different stages </center>

Luckygirl001: /* Derivation of the GAHM */

2020-04-10T07:59:34Z

Derivation of the GAHM

← Older revision		Revision as of 03:59, 10 April 2020
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	To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.		To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.

	[[File:~~Figure~~ 3. GAHM.png\|center~~\|750px~~\|Normalized gradient wind profiles]]		[[File:Fig 3. GAHM.png\|center\|Normalized gradient wind profiles]]
	<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>		<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>

	<!-- ******************* chapter 3 ***************** -->		<!-- ******************* chapter 3 ***************** -->

	== Calculation of the Radius to the Maximum Wind ==		== Calculation of the Radius to the Maximum Wind ==

Luckygirl001 at 07:56, 10 April 2020

2020-04-10T07:56:52Z

Show changes

Luckygirl001: /* The HM vs. the GAHM */

2020-04-10T07:49:43Z

The HM vs. the GAHM

@@ Line 193: / Line 193: @@
 |}
-=== The HM vs. the GAHM ===
+=== The AHM vs. the GAHM ===
 === The GAHM's Composite Wind ===

Luckygirl001: /* Derivation of the GAHM */

2020-04-09T05:35:55Z

Derivation of the GAHM

← Older revision		Revision as of 01:35, 9 April 2020
Line 75:		Line 75:
	To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.		To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.

	[[File:Fig 3. GAHM.png\|center\|Normalized gradient wind profiles]]		[[File:Fig 3. GAHM.png\|center\|750px\|Normalized gradient wind profiles]]
	<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>		<center> Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at <math> \log_{10}R_o =0, 1, 2 </math> (or correspondingly <math> R_o =1, 10, 100 </math>). </center>

	<!-- ******************* chapter 3 ***************** -->		<!-- ******************* chapter 3 ***************** -->

	== Calculation of the Radius to the Maximum Wind ==		== Calculation of the Radius to the Maximum Wind ==