v]adtj6 edv - atmospheric and oceanic...

36

Upload: haanh

Post on 30-May-2018

214 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

����������� ��������������������������! #" $&%('#)&*(+,)&'.-0/21

354�687:9;6�<=42>?7�@BADC�E�F2<�9(AHG�6JIKF2ILF�ADMNMNOP9;6�4�AHG�6QASR2>?TJ<LF�UL6�EDVWF�7�X�F2MYAZ<L>?E�7�I0E�7[<L4�6�ILF2U\V]ADTJ6�EDV<=426,IL68A�E�U;M?AD@D6JI8^W_`4�6aOSF�I\F�ADMNMNO�R�ULEDb�UL6JILI(VcU\E�deA�UL6Jb�>?E�7fEDV&VcE�U\dSAZ<L>?E�7Q<=E�A�T8E�AZI\<(9g4268UL6,<L4�6JOSADU\6X�>NILIL>NR�AZ<L68XPE�UgR�ADU!<=MNOfUL6ah�68Ta<=6JXji]I\686Kkl>?bnmoa^

p`qcrts u�wv��`x X�68ADM&IL>N7�6yE�UgT8E�I\>?7�6,9(AHG�6JI8^

z I\ILF�d�>?7�b{<L4�AZ<g<=4�6y9(AHG�68I�AZ<`<=426KI\6HAQILF�U!V]ADT86|ADU\6yIL>?d�R�MN6,T8E�I\>?7�6y9(AHG�68IJ^}B~D�D�K���a�������,����� ~D���P~D�����������2���S�c������<=4�6KG�6JU\<=>NTHADM.X�>NI\<�AZ7�T86�VcULE�d�<LULE�F�bD4B<=EfT8U\68I\<�>?I�THADMNM?6JX

<=426:�\4�6J>?b�4�<L�fEDV�AQ9(AHG�6D t9g4�>?T�4¡>NI`<¢9g>?T86|EDVl<L4�6:�\ADdfR2M?>N<LF�X�6J��^(_`4�6�ADd�R�M?>�<=F�X�6y>NI�<L4�6KdSA�£�>Nd�F�dX�>NILR�M?ADT86Jdf6J7�<�EDV¤<=4�6gG�6JU\<=>NTHADM�E�I\T8>NM?MYA�<=E�U!O�d�ED<=>NE�7QEDV¤<L4�60I\F�U\V]AZT86,AZCtEG�6�E�U�Ct6JM?E9�<L4�60d�6HAZ7�9(AZ<=6JUM?6aG�6JM¥^

}B~D�D�K�?�a���D���¦��§�����_`4�6�X�>NI\<�AZ7�T86yVcULEDd¨T8U\68I\<g<LE�TJUL6JI\<0E�U�VcULE�d©<=ULEDF�b�4n<=E{<=U\E�F�b�4¤^}B~D�D�K������ªa�a«Q��¬�����_`4�6KU=ADX2>YAD7 9(AHG�6�7�F�d�Ct6JU�­¯®° >NI`EDV�<=6J7±F2IL68X¤^}B~D�D�²���a«³�¥´8���8µ�����_`4�6(<L>?d�6�C�6J<¢9;686J7{<¢9(EyI\F�T8TJ68ILI\>NGD6�TJUL6JI\<=IlE�U²<LULE�F�bD4�IlR�AZILIL>N7�byA0¶�£26JX{RtED>?7�<H^}B~D�D�W·³«=�L¸8���a�¤¹aº[�N»l����_`4�6|ULADX�>?AD7 VcUL6J¼�F2687�TaO ­¯®½ >NIgEDV�<=6J7±F2IL68X¤^}B~D�D�0���2~�¾�|¾¢�����L�¿��Àt����_`4�6�ILR�686JX¦AZ<�9g4�>NT�4¦ASR�ADU!<=>?TJF�MYAZU0R�AZU\<,EZV�<=4�6�9(AHG�6{R�ADI\IL6JI�AS¶2£�68X

R�E�>?7�<H �9g4�>?T�4±>?IgÁÃÂÅÄ;ÆuÇÅÂÈ»(Æ�É�^Ê «=´��u�˾¢�����L��Ì0�D�?¾¢���a«=¾a�¥´���� x 7:UL6HAZM²E�T868AD7Í ¤9(AHG�68IKADU\6�7�ED<�R�F�U\6�IL>N7�6�9(AGD68I�C�F2<yADUL6{AfILF�dÎEDV

IL>N7�6|9(AHG�6JI�9g>�<=4:AfU=AD7�bD6�EDV�9(AHG�68MN687�bZ<=4�IJ &T8E�U\UL68I\RtED7�X�>?72b Rt6JUL>NE�X�I0AD7�X¿AZdfR�MN>N<LF�X�68IJ^0_`426�b�ULEDF�RILR�686JXÍ �Á(Ïu �AZ<g9g4�>?T�4P<=426|b�U\E�F�Rn<=ULAHG�68MNI0>NI{Ð=ÑÐ³Ò ^�kl>Nb�ÓQ6³£�AZdfR�MN6D^0ÔÖÕ²^�×ØÕ.^�kl>?bPmuÓ�^ÚÙ�o³^�ÔÛIL6J6�Õ.^�×ØÕ²^VcE�U`X�6JUL>�GDA�<=>?ED72Ü¥^

3wAHGD6n687�6JULbDOË<=U=AHGD68M?I�AZ<�<L4�6 b�U\E�F�RÝILR�686JXÍ WU=AZ<L4�68U|<=4�AD7ÞR�4�ADI\6nILR�6868X¤^ x V`<L4�6 b�U\E�F�RÝI\Rt6J68XT�4�AD72b�68I�9g>�<=4S<L4�6,T�4�AD72b�6,EDV&VcU\68¼�F�6872TJO� �9;6yI=AHO�<=4�6�I\O�I\<L68d¨>NI{�\X�>NILR�68U\IL>NGD68�� �Ct6JTHADF2IL6,<=426,687�6JULbDO

>

Page 2: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

>?7Ý<L4�6PI\O�I\<L68d 9g>?MNMQ�\X�>NILR�68U\IL68�¿EG�6JU�<L>?d�6D^w_`4�AZ<�>?I8 W9(AGD68ISAZ<QX�> � 68U\687�<QVcUL6J¼�F2687�TJ>?68IQ<=ULAHG�68MgA�<X�> � 6JUL687�<KILR�686JXÍ^ x 7¦T8E�7�<LU=ADI!<H l>�V�Á(Ï�>?I,<L4�6fILADd�6fA�<�AZM?MlVcU\68¼�F�6872T8>?6JI8 .<=4�6�I\O�I\<L68d >?IyT8ADM?MN68X¿7�E�7��X�>NILR�68ULI\>NGD6D^

}B~D�D�P�a�¤�a«��Dº�^ _²EZ<�ADMW687268ULbZO¿Rt6JU|F�72>N<�ADUL68A±EDV�IL6HA±ILF�U!V]ADT86f>?I�X�6a¶�7�68XwADI��Πi���� ­ oLÆDÓBÂi���� ­ oLÆ��±i��\E�F�M?6JIgd�� ­ oa^

� � ���¢���¤��¾�¾a^�_`426KULAZ<=>NE{EDV� ÆDÄ >NI`THADMNM?6JX±I!<=6J68R�7�6JILIJ^

p`qcr�s u����

�����������! ³$�)&'#"$��$�%�%¯/&('w$&%;'.1*),+ -,*/. *0"�'.+,'�*/�,1*12%!+`%�% 12)3%54 *g)&'.$W+

_`4�6±68¼�F�AZ<L>?E�7�IQEDV�dfEZ<=>?ED7w9(6±X�68U\>NGD68X 6HADU\M?>?6JU{THAD7 4�6JM?R F�I{<=E¦F�7�X�6JULI\<=AD7�X <¢9(EË>?d�RtE�U!<�AD7�<ADI\Rt6JTJ<=I�EDV.<=4�6�E�T868AD7BTJ>?U\T8F�M?AZ<=>NE�7Í�

i]>Yo;<=4�6y<=ULAD7�I\>?687�<�UL6JILR�E�7�I\6|EDV.<=426KE�T868AD7n<=E�T�4�AD7�bD68Ig>?7n<=426KI\F�U\V]AZT86�VcE�ULTJ>?7�b76i]>N>�o�<L4�6�df68AD7PE�TJ6HAD7±T8>NULT8F2MYAZ<L>?E�7 X2UL>NGD687±C�On<=4�6�ILF2U\V]ADTJ6�9g>N7�X�I�AD7�X±C�F�EO�AD7�TJOP6 � 6JTJ<LI8^3Þ6K9g>?MNM&X�>?I\T8F�I\I,CtEZ<=4[EZV <=4�6JIL6�ADILR�68Ta<=I,>N7B<=42>?I�T8E�F�U\IL6Z ¤C�68b�>N7�7�>?72b�9g>�<=4B<L4�6�<=U=AZ7�IL>N687�<,E�T868AD7

UL6JILR�E�7�I\698�9(AHG�68IJ^3Þ6�9g>?MNMl¶�ULI!<�X2>?ILTJF�ILI�4�E9Ø<=4�6�E�T86HAZ7Ë9g>?MNM�UL6JILR�E�7�XË9g42687Þ<L4�68U\6S>?IKA±T�4�AD7�b�6f>?7¿<=4�6QVcE�U\T8>?72b

¶�6JM?XÍ^�_²Ef<=4�>NI06J7�X:AfGZADMNF�ADC�MN6K>N7�IL>Nb�4�<�>N7�<=Ef<L4�6�AZX:�\F�I\<Ldf6J7�<,EDV�<L4�6�E�T868AD7[<=EfT�4�AD7�bD68I,>N7BVcE�U\T8>?72bT8EDdf6JI,VcULE�d API!<=F�X�O:EDV;M?>?726HADU0E�TJ6HAD7¿9�AHGD68I8�<;g68T8ADM?MlVcU\E�d <L4�6{R�U\6JG�>?E�F2IKTJMYADI\I,<=4�A�<y<=4�6=;gE�ILI\C�OAD7�X?>W@�dfAD7 7�F�d�C�68U\I�iA@�B�AD7�X?�DC�oWADU\6�ILdfADMNM�>N7f<L4�6�E�T868AD7 >?7�<L68UL>NE�U8 �AD72XS<=4�60>?7�<=6JUL>NE�U;E�T868AD7 T8AD7C�6fG�>?6a9(6JXÝADI:�!M?>?726HADU\�[AD72XÞ>?7�G�>?I\T8>NXÍ^FE�>?7�TJ6HG(E�U\>?E�MN>?I,VcE�U\T8>N7�b[>NIKGD68U\OË>?d�R�E�U\<=AD7�<y<=E[M?ADULbD6fI\THADMN6d�ED<=>NE�7Í 29;6|9g>NIL4B<LEf>NILE�M?AZ<=6�>�<=Ig6 � 68Ta<H^JI,7�6|ADR�R�U\E�ADT�4B<LES>NILE�M?AZ<=6,<L4�>?Ig6 � 68Ta<,>NIg<LEÝ�\6JM?>Ndf>N7�AZ<=6J��>N<VcULEDd©<=426K6J¼�F�AZ<=>NE�7�I`¶�U\I\<H^W_`4�6J7[I\686�9g4�AZ<g>N<gX2E�6JIg<LE{<=4�6�E�T86HAZ7BC�OPADX2X�>?7�bQ>�<g>?7Í^

_`4�6yT8M?ADILI;9g>?M?M�VcE�T8F�I(E�7S<=426,T8E�72T868R�<=I�AD7�X R24�O�I\>?T8IgULAZ<=4�6JU;<L4�AD7nX�6a<�AD>NM?68XfdSA�<=4Í �<=4�6JUL6JVcEDUL6�9;6E�7�M�O:C�U\>?6Jh2O:b�En<=4�U\E�F�b�4¿<=426fdfAZ<=4268dfAZ<=>NTHADM²R�ULE�T86JX�F�UL6JI|X2F�UL>N7�bBEDF�UyX�>?I\T8F�I\IL>NE�7Í^ k²>?ULI!<H Í9;6�9g>NM?ME�C2<=AD>?7¿<=426fX�>NILR�68U\IL>?ED7ËU\68MYA�<=>?ED7Í .AD7�XÞI\686�9g4�AZ<K<¢O�R�68I|EDV�9(AHG�68I�ADUL6fAHGDAZ>?MYAZC�M?6�>N7ËEDF�U|I\>?d�R�M?>�¶�68XI\O�I!<=68d±^

_`4�6�68¼�F�AZ<L>?E�72I�EDV.dfEZ<=>?ED7 >?7nA�42E�dfEDb�687�6JE�F�IJ �>?7�G�>?I\T8>?XnE�T868AD7nC�On>?bD7�E�UL>N7�b|<L4�6<G(E�U\>?E�MN>?IWVcE�ULTJ6ADU\6D�

K7LKM ÂON m

��PQSRQ7TVU i!mA�o

>N>

Page 3: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

K��KM Â#N m

� PQSRQ�� U i!mHC�o

K��K M  N m

� PQSRQ�� N ��� i!mHTHo

Q LQ!T�

Q �Q���

Q �Q��  ��� i!mHX�o

�&6J<�� I0ADILI\F�df6y<L4�6�68¼�F�>?MN>?C�U\>?F�d I!<�AZ<L6KEZV²<L4�6�E�TJ6HAD7±>?IgAZ<`<L4�6�I\<=AZ<=6�EDVlUL68I!<H^W_`4�AZ<�>NI8�L P  � P  � P  �2 R P  R P i � o³ �{ � P i]TJE�7�I!<�AD7�< 8�4�E�d�E�b�687268E�F�I=oa^k�E�U`ILdfADM?MtRt6JU\<LF�ULC�A�<=>?ED7�I�� U�� U��2U�� AZCtE�F�<`<=4�6�UL6JI\<L>?7�bQI\<=AZ<=6Z �9(6K4�AHG�6Z�L  L P � �nÂ��. �  � P � �  �  �  � P � �  �  R  R P i � o � �  �{ � P^k�E�U�<=4�6�C�AZT�@�bDULE�F�72X[I\<=AZ<=6Z �<=4�6KADC�EG�6�68¼�F�A�<=>?ED7�IgEDVld�ED<L>?E�7nC�68TJE�df6Z�

N m� PQSR PQ�� N �QÂ���� i¥ÓDo

k�E�Ul<=426`Rt6JU\<=F2ULC�AZ<L>?E�72I8 D<LE,<=4�6��W«=¾a��´�«\���a«�ic>?b�7�EDUL6;<L4�6`4�>Nb�4�68U²E�ULX268U <=6JULd�I�o³ �>?7QA,7�ED7��¢U\ED<�AZ<L>?7�bVcU=AZdf6�i��±Â���oa 29;6�4�AHG�6Z�

Q �Q M Â N m

� PQ �Q7T U i���A�o

Q �Q M ÂON m� PQ �Q� U i��DC�o

Q �Q M  N m� PQ �Q�� � i��DTHo

Q �Q7T �

Q �Q�!�Q �Q��  �"� i��DX�o

k�ULE�de<=4�6KADC�EG�6K68¼�F�AZ<L>?E�7�IJ �9(6y9gU\>N<=6KA{IL>N7�b�MN6,68¼�F�AZ<L>?E�7±>?7 � ADMNE�7�6D�Ð$#&%Ð C # � Ð$#'%Ð)( # � Ð$#'%Ð)* #  ��^

+(E�F�72X�ADU\O T8E�7�X2>N<=>NE�7�IJ�, z <Q<=426nE�TJ6HAD7 CtEZ<L<=EDd �  N.-[ W9(6±4�AHG�6 � Â/��^ÅÔ z ILILF2df6nh�AZ<QE�T868AD7 CtED<\<=E�d 9g4�68U\60�Â,T8ED7�I\<=AD7�<LÜ�^

, z <0ILF�U!V]ADT86Z  �  1t �9;6�UL6J¼�F�>?U\6�AD7�Onh�F2>?X±R�ADULTJ68MNI�AZ<�<=426|I\F�U\V]ADTJ6K<LEfU\68dfAD>?7BAZ<g<=426|I\F�U\V]ADTJ6icdSAZ<L68U\>YADMtCtE�F27�X�ADU!OnT8E�72X�>N<L>?E�7�o³^�k�E�U�I\F�T�4Bh�F2>?X±R�ADULTJ68MNI8  � N21Q �QADMNM¤<=4�6y<L>?d�6D^ E�EK i � N212oLÆ KM  �2 �AZ7�X� Â435*376  Ð98Ð)6 i]>?bD7�E�UL6y72E�7�M?>N7�6HAZU(<=6JULd�I�oa^

>N>?>

Page 4: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

, z < �  1�  R  R P � �  �2 �i � 68U\E¿R�UL6JILILF2UL6BAZ<�<=426nE�TJ6HAD7ÝI\F�U\V]ADTJ6D �>?b�72E�UL6�<=4�6nI\F�U\V]ADTJ6AZ<LdfEDILR�4�6JUL>NTyR�UL6JILILF2UL6o³^;_`42687 �  N R P  � P ��1�^E�>N7�T86 1���� -ÎI\EQ9;6KTHAD7P<�AD@Z6 � Â�1�Â��{ILEQ<=4�6�T8ED7�X�>N<L>?E�7n>NI8�z < �  �2  �  � P ��1�^

�0E9� W<=4�6nE�T868AD7j>?I�C�E�F�7�X268X >?7 � X�>NUL68Ta<=>NE�7wC�F2<QEDRt6J7 >N7Þ£jAD72XjOÞX�>?U\68Ta<=>?ED7�I8^ E�E:9(6PT8AD7ADI\ILF�d�6yVcUL686y9(AHG�6|VcEDULd >N7 £±AD7�XnO� �C2F2<�M?6HAHGD6�<L4�6yG�6JU\<=>NTHADM&I!<=U\F�TJ<LF�UL6KF�7�X�6a<=6JULd�>?7�6JXÍ^

z I\ILF�d�6�A�<LU=AHG�6JM?>?72bQ9(AHG�69�¢M?>N@D6�ILEDM?F2<L>?E�7�I`EZV²<L4�6yVcE�ULd±�1�Â�1�P�����i¥É T � � � NÞ» M oa �  ��� i � o�����i�É T ��� � NÞ» M oa E�F�C�I!<=>�<=F2<L>?7�b{<=4268IL6KILEDM?F2<L>?E�7�I;<=EQ<=4�6�IL>N7�b�MN6,68¼�F�AZ<L>?E�7±>?7nRBADMNE�7�6D �9(6�4�AHG�6Z�Ð # %�Ð)* # NÅi¥É ­ ��� ­ o ��� Â��2^_`4�F�I8 ��� i � o� ��������i�� � o � � ������i�� � oa 9g4�68U\6�� ­ Â É ­ � � ­ ^+;O ADR�R2MNO�>?7�b¦C�ED<L<LE�d C�E�F�7�X�AZU\OjT8ED7�X�>N<L>?E�7 i]AZ< �  N.-[  �  �ËAZ7�X <L4�F2I Ð7%Ð)*  �:VcULE�d68¼�F�A�<=>?ED7Þi �DTo\oa �AD72XBI\F�U\V]ADTJ6�CtE�F27�X�ADU!OnT8E�72X�>N<L>?E�7 �  � P ��1nAZ< �  �2 �9(6KE�C2<=AD>?7Í 

� Â��� P 1 P � ���i¥É T � � � NË» M o!� ��"�#�.i � � -no������i���- o � i%$�o

k�ULE�d©68¼�F�AZ<L>?E�7Þi��DTHo;9(6K4�AHG�6

� Â�7��1 P ���� i¥É T � � � NË» M o�������#�.i � � - o»&� ��"�.i��-no � i¥ÙDo

�0E9 MN6J<K68¼�F�AZ<L>?E�72I i'$�oyAD7�X i¥Ù�o,I=A�<=>?I!V�O:C�E�F�7�X�AZU\O¿T8ED7�X�>N<L>?E�7:9;6fX268UL>�G�6JXÝADC�EuGD6 �  Ð98Ð)6 A�<�  �2 29;6|EDC2<�AD>N7 <=4�6�X�>NILR�68U\IL>?ED7±U\68M?AZ<=>NE�7Í�

» ­  ��� M!( ����i���-no)� i�)�oE�686|kl>Nb �{VcEDU�<=4�6�X�>NILR�68ULI\>?E�7PU\68MYA�<=>?ED7nX�>YADbDU=ADd±^

p`qcr�s u�+*Í� 0,>NILR�68ULI\>?E�7nU\68M?AZ<=>NE�7 VcE�U`ILF2U\V]ADTJ6|b�ULAHG�>�<¢OS9(AHG�68IJ^

>NG

Page 5: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

, §�´�������~D�D��¾a^ k�E�UfM?E�72bË9(AGD68InI\F�T�4È<=4�AZ< Ä N N��� ��� - AZ7�X�<=4�F�I �- � � mÂKÂ� �M!( ����i���-no� ��-B^ >W¼�F�A�<=>?ED7Þi%)�o(C�68T8EDdf6JI8�» ­  �"- � ­ ^_`4�F�IJ .<L4�6{9�AHG�6�ILR�686JXwÁ ­  �"-[^�Á(Ï� Ð=ÑÐ �  � N�� �"-B &9g4�>?T�4¦X�E�68I�7�ED<�T�4�AD7�b�6{9g>N<L4VcU\68¼�F�6872TJO� tAD7�Xn<L4�F2I0>NI`7�E�7��¯X�>?I\Rt6JULIL>�G�6Z^� � r � s �¥����� r u��� q ���������t���n��u�Ë� ����� q �J������ q ����� _`4�68I\6{9�AHG�6JIKADUL6|THADMNM?68X[b�U=AHG�>N<¢O9(AHG�68IyC�68T8ADF�I\6�<=4�6J>?U�UL6JI\<=EDUL>?72bfVcE�U\T86�>NIn�\bDU=AHG�>N<¢O���^g_`4�6JOBTHAZ7:R�ULE�R�ADb�AZ<L6|>N7¡ADMNM.X2>?UL6JTJ<L>?E�7�IEDVlA{4�E�U\> � ED7�<=ADMÍR�M?AD7�6Z^�¤«=´�� �L¸8��~D�]�¥´��2¾¦���J���(�¡¹L~D� ¾���±�c�2~D����·H´�«±~D�È�������]�]~D�l���a«³����«Jª�~D���¥´��È��� � �������:´¹��L~D���������«=��¾¢��´����{���[�]«L~D�2¾a�¥�a�����S´��]�¥´��:ª8ºQ«\~��D�]~D�]�����|�D«\~D����]º���~D�D��¾����:~D���Í�D��«=��¹a���¥´��2¾|´ ·�~{��´�«³����´����¥~D�����~D�¤�! W� ·³� �a«��c��� �D«\~D����]º"��~D�D��¾�«\~��D�]~D�¢�`´������&���2~D�¤�?¾;�����nÐÐ)6 �¢�a«³��¾��L¸8�2~D�$#%�¤�c���g´¹��L~D�{«=�L~�¹�����¾~f¾a�¥~D� �Q´ ·K«=��¾a�& ,� ·³�¢�a«{~D�:�������]�]~D�����a«³�]��«Jª�~D�]�¥´��$�;�c����´¹��L~D�¦~��('³��¾a��¾|� ´ ~D�w�L¸8�������]ª8«a�����¨¾a�¯~D�¢��������a«��D«\~D����]º[��� ~¡�¤´��*)¢«=´��¯~D�]�����¦´¹��L~D�+ ±�¨��´´8�¦�(,�~D�����?���?¾��(�f�c��«=´ � ~:ª8���B¾a� ´��¤�f��� ~¹L~D��� ��´����- ¿µÍ�������������]~D�����a«³����«Jª�~D���¥´��.�������W¹L~D��¾�K�D«\~D����]º/��~D�D��¾��¢´n«\~��D�]~D�¢�S´����`��� ¹a��«=¹a�?��¾��~\·³�¢�a«��c����¾����~D�D��¾y«L~��D�]~D�¢�{´����0�������`��´����f�?¾{¹L~D��� ~��~D���+ 1��a«³�]�¥¹L~D�t¾a��«³��¹a����«=�a^ E�>?7�TJ6Si��-no ��� m� 2<L4�687 � ��"��i��-no�NÈm� �AZ7�X

�  ���P91�P � ���i¥É T � � � NÞ» M o U i32Do

Q �Q M ���� iA��o

AD72X±<L4�F2I �  � �"�� M!( � M ^x 7�X268R�687�X�6J7�<,EDV²X�68R�<=4Í^W_`4�6aOnVc6868MÍ<L4�6�CtEZ<L<=EDd EDV²<=4�6�E�T86HAZ7Í^

, � ��´�«³�4��~D�D��¾a^²k�EDU.<L4�6(I\4�E�U\<²9�AHGD68I�9g>N<L4�Ä N5�� ����-ØAD72X�<=4�F�Igi��-no ��� m� Z<L4�6(X2>?ILR�68U\IL>NE�7U\68MYA�<=>?ED7nCt6JT8E�d�68IJ�» ­  � �¤^Á Â Ñ � AD7�X±Á(Ïg Ð�ÑÐ � >?IgA{VcF�7�Ta<=>?ED7±EZV.»�AD7�Xn<=4�F�Ig<=426JO±ADU\6yX�>?I\Rt6JULIL>�G�6Z^+;O IL>?d�R�MN>NV�O�>?7�bK<=4�6J>?U`I\E�M?F2<L>?E�72I`C�OS<=4�6�X�>NILR�68ULI\>?E�7PU\68MYA�<=>?ED7Í �9;6|4�AGD6

�  ��� P 1 P ����i�É T ��� � NÞ» M o(6 � * U i87�o

� Â� ��1 P �����i�É T ��� � NÞ» M o(6 � *» i!m ��o

E�42E�U\<�9�AHGD68I`ADU\6�<LU=ADR2Rt6JXf<=E�<=426�F�R�R�68U�E�T86HAZ7f<=E|A|U\68b�>NE�7�EDVÍX�6JR2<=4 � � � ^�_`4�6JO�X�E|7�ED<WVc686JM<L4�6�E�TJ6HAD7±CtEZ<L<=EDdB^p u�:9 s ���.�;�Î���� r �<� r u��� q �������������D�>=@? q �4Al� q ��� q � �e� ���= ��2� q � r �B�&�J��:CD �FE/=HGJIy���K� �2�.L � �¯u�:9 s ���.�;��I,��������� s CNM��� q �NL � ���K� �u� s �¡�������:L]��� r �u� q �

G

Page 6: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

L � r ���n� ���Þ�H�.� L8L,�J����I q � �j �H�2� q � r �B���J��:C �u�����8��L � r ���������P��u��C � q A²�@����K� �u� s � r u��� q ���Ý���������Q��� q �J� �KL�� q � �u���[���� q0 � M.��� � ���8��� � �.��?��.�-��� �_`4�6�9�AHGD6n687�6JULbDO¦R�U\E�R�ADb�A�<=>?ED7Ë>NI|X�> � 6JUL6J7�<KVcULEDd 9(AZ<L68U�R�ADU\<L>?T8MN6QdfEG�6Jdf6J7�<H^ x 7ÞV]ADTa<H ²<LE<L4�60MNE9(6JI\<;E�ULX�6JU8 �<=426�9(AZ<=6JU;R�ADU\<L>?T8MN68I�VcE�UWIL42E�U\<W9(AGD68I`AZUL6gT8>NULT8F2MYADU�AD7�XQVcE�UWM?ED7�by9�AHG�6JI`ADU\66JM?M?>NR2<=>NTHADM�^�Ô �0686JX¡<=E I\E�MNGD6|<L4�6�I\E�M?F2<L>?E�72I�<LE @�7�E9#<L4�6�X�6a<�AD>NM?I!Ü¥^ x V�9(6�T8E�7�I\>?X�6JU0<L4�6�4�>Nb�4�6JUE�U\X�68U�6 � 68Ta<=I�<=4�6KE�ULC2>N<=I�EDVlR�ADU\<L>?TJM?68I�ADU\6|72ED<�¼�F�>�<=6�T8MNE�IL6JX 6�VcE�U0I\4�E�U!<09(AHG�68I0<=4�6JUL6�>NI�Af7�6a<h�E9 >?7:<=4�6�X�>NUL68Ta<=>NE�7:EDV(<LU=AHG�6JM�AD7�X¿<=4�>NIy7�6J<�<=ULAD7�I\RtE�U!<K>?IyTHAZM?M?6JX � � ´¬���¾f�D«³� ·³�]^���E9(6aG�68UJ <L4�>?I`6 � 6JTJ<0>?I`ILdfADMNM�i�A{I\dSADMNMtVcULADTJ<L>?E�7PEDV.<=4�6�E�U\C�>N<=ADM¤I\Rt6J68X�o³^

������È'3 9��$�)¤/7%!*;+ $W+�4����'.$ �J%!+�" %!+ 17-,$���� * ' 'Þ$l/�'3�

�¯�Ú���g�c·³«\~�¹a���¥´���^ z I,9(6��\F2I\<yX�>NILTJF�ILI\68XÍ �ILdfADMNMlADd�R�MN>N<=F2X�6|MNE�7�bS9(AHG�6JI|AZM?M�<=U=AHGD68M�AZ<�<=4�6�ILADd�6ILR�686JX¿>?7B9(AZ<L68U,EDV�Afb�>NGD687¡X�6JR2<=4 - i�Á  � �"- oa^ +(F2<y>�V <=426�CtEZ<L<=EDdÎX268R2<L4:>?I�T�4�AD7�b�>N7�bf<L4�68>NUX�>NUL68Ta<=>NE�7 EDV�<=U=AHGD68M�dfAHO T�4�AD7�b�6Z^ z I�9�AHGD68Ind�EuGD6¡>?7�<LEÞIL4�ADM?MNEu9 9�AZ<L68US<L4�68>NUfR�68U\>?E�X U\68dfAD>?7�IT8ED7�I\<=AD7�<�C�F2<�Á X�68TJUL6HAZIL68I�AD7�Xf<L4�68U\6JVcE�U\609(AHG�68MN687�bZ<=4nX�6JT8UL68ADIL6JI8^0Ô »j ��ÁQ^ x VÍ»Ý>NI;TJE�7�I!<�AD7�<`AZ7�XÁØX�6JT8UL68ADIL6JI8 �<=4�6J7 �B4�ADI`<=E�>?7�TJUL6HAZIL68I0AD7�XP<=4�F�Ig9(AGD68MN687�bD<L4Ýi]Äj ­¯®� o�X�6JT8UL68ADIL6JI8^ÖÜ x V�AQIL6JUL>N68IgEDVR�ADULADM?MN68M �¢T8U\68I!<=68XP9(AGD68IKADR�R2ULE�ADT�4268IyAZ<,AD7:AD72b�M?6�<=ESASI\<LU=AD>Nb�4�<,IL42E�UL6JM?>?726 i]kl>?b $�o�EGD68UyASI\dfE�ED<L4IL68A¦C�ED<\<=E�d 9g4�>?T�4 IL4�E�ADM?I{b�ULADX�F�AZM?MNOD  <=4�6aOÝR�U\E�b�UL6JILI\>NG�6JMNOÝT�4�AD7�b�6±X�>?U\68Ta<=>?ED7jADI{<=4�6n6J7�X EDV�<=4269(AGD6Q7�6HADU\68U�<LES<=426�IL4�EDUL6D^ z IyA UL6JILF�M�<H t<L4�6�9(AHG�68I�C�68TJE�df6�dfE�U\6�R�AZU=ADMNM?68M&<LES<=4�6{I\4�E�U\6�C�O[<=426<=>Ndf6�<L4�AZ<�<=4�6aOQR�>?MN6gF�R ADI�ILF2U\V ^�µÍ����¹��2~D���������B�D��«=��¹a���¥´��B~�¾�¾´¹a�]~D�¢�L� �����c�f�����|¹��2~D�����y´ ·`¾¢�����L���?¾¹L~D���?�L�n«��c·³«\~�¹a�]�¥´���^�_`4�6{U\6JVcULADTJ<L>?E�7BR�AZ<L<L68U\7¡9g>?M?M�C�6�d�E�UL6|T8E�d�R�M?>NTHAZ<L68XB>NV <L4�6�E�T86HAZ7:CtED<\<=E�d 4�AZIT8EDdfR�MN>?T8AZ<=6JX <=E�R�E�b�ULADR�4�O�^

p`qcr�s u�[��� 3ÞAHG�6KUL6Jb�U=ADTa<=>NE�7Í^

�¯���Ú���g«=�L~�¬������D^ z IQ9�AHG�6JISd�EG�6±>?7��¯IL4�E�U\6±AD7�X ILM?E9 X�E9g7Í �7�ED<�E�72MNOÝX�E�68IQ<=426±9(AHG�68MN687�bZ<=4X�6JT8UL68ADIL6�C�F�<�ADM?I\E�<=4�6,9(AHG�6�4�68>Nb�4�<`T�4�AD72b�68IJ^��³�:~Q¾a�¢�L~��Dº�¾a�¯~D�¢���W�������a�¤�a«��Dº��(� ,:�¯�c���;��«=´8�D��¹a��´ ·

G�>

Page 7: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

��������~D�D���a�¤�a«��Dº,���a«�¾8¸8��~D«=���S�a�¢�a«y�����S��¾y�c�����D«=´���¡¾¢�����L��Á(Ï=�Q�?¾�¹�´��2¾�a«³�D�L�f~�¾��c��� ��~D�D��¾��S´��D����*)¥¾���´�«=���¯���������&�����aº ª8«=�L~�¬���^

>�7�68U\bDOÝh�F�£ ���: ��� ­ Á(ÏJÆ ��Â,T8E�7�I!<�AD7�<8^Ýk�E�U�<=4�6nMNE�7�b¡9(AHG�68IJ `ADIQ>N<�dfEG�6JIQ>?7��¯IL4�E�U\6D �Á(ÏX�6JT8UL68ADIL6JI(AZ7�X{9�AHG�6JM?6J7�bD<=4�X�6JT8UL68ADIL6JI,icIL>N7�T86�VcU\68¼�F�6872TJOQX�E�68I�7�EZ<WT�4�AD7�b�6Hoa �AD7�X{<L4�F2I0 ¨>N7�T8U\6HADI\68IJ^_`4�6JUL6JVcEDUL6D �<L4�6(9(AHG�68IWC�68TJE�df6;I\<L6868R�68UWAZ7�X�I\<L686JRt6JU8^ z I²<=4�6;I\<=6J68R�7268ILI�b�E�68I.<=E�muÆ�muÓ0>?7|UL68ADM2E�T868AD7i!m�� 2�>?7n<=4�6JE�U\O2o³ �<=4�6y9(AGD68I0C�UL68AD@¿i�k²>?b�Ù�o³^

p`qcrts u���Í� 3ÞAHG�6KC�UL68AD@�>?7�b�^

���c�(�D'Þ' � � 1�$W+�4��Þ1 �y+,$ . % 1E�9;68M?M���9�AHGD68Iyb�6J7�68ULAZ<=6JX¡C�O[R�68U\IL>?I!<=6J7�<09g>?7�XBVcE�ULTJ>?7�b�^ ��ADU\b�6�AZdfR�MN>N<LF�X�68I�AD72X¡R�ULEDR�ADb�AZ<L6KVcEDU

A{M?E�72b�X2>?I\<=AD7�TJ6D^_`ILF�7�ADdf>NI8����ED7�b�9(AZ<=6JU �¯9(AHG�6JI;F�I\F�ADMNMNOKb�687268U=A�<=68XQC�O�I\6HAyC�ED<L<LE�d d�EG�6Jdf6J7�<�ADILI\E�TJ>YAZ<L68X�9g>N<L4

6HAZU\<=42¼�F�AD@D6JI8^$E�E�d�6J<L>?d�68I.THADMNM?6JX¡�\I\68>NILd�>?T�IL6HA�9(AHG�6J�76�C�F2<l<L4�6JO|THAZ7{ADM?I\E�C�6(T8ADF�IL6JX{C�O�B6a<=6JE�UL>�<=6D^_`4�6�6 � 6JTJ<LIyADUL6|F�ILF�ADM?M�OP7�ED<�9g>?X�6JILR�U\6HADXÍ ¤C�F2<,_`I\F�7�ADd�>?I0b�687268U=A�<=68X[C�OBIL6J>?I\df>NT�ADTa<=>�G�>�<¢OPE�7¡E�726IL>NX�6�EZV�<=4�6�ÕlADTJ>N¶�T�4�AGD6�THAZF�IL6JX¦X26JGZADI\<=AZ<=>NE�7¡E�7B<=4�6|ED<=4�6JU�I\>?X�6Z ÍADI,>N7B<L4�6�THAZIL6�EDVl<=4�6�<LILF�7�ADdf>b�6J7�68ULAZ<=6JXËC�O¡<L4�6Pm%7 ) � 68ADU\<L4�¼�F�AD@D6�>?7 G(4�>NM?6�9g42>?T�4¦THADF�I\68XËIL68U\>?E�F2IyX�ADdfADb�6�<=4268UL6�AD72XËAZM?ILEP>N7�ADR�AZ7{7�6HADU\MNO{Ó �2  � � �D@�d A9(AHO�^�_`4�6`U\68TJ687�< 0,68T86Jd�C�68U�Ó � ��$�<=I\F�7�ADd�>2EG�6JU z IL>?A�9�AZI�X�6JGZADI!<�AZ<L>?7�b�^

������ �+(/2'3��+,$��V"$��$�%�%¯/& 'w$&%('�1*) 4y'.+,17%¯/& 1�/ ��$l/!%���'34 *�)&'�$W+ ^x 7�<=426�ADC�EG�6D Z9(6;X�>?I\T8F�I\IL68X{<L4�6�9(AHG�6JI�>?7�A�4�E�d�E�b�6J7�68E�F2I8 D>N7�G�>NILTJ>?X�E�T868AD7Í^.3Þ6;E�C2<�AZ>?7Kb�U=AHG�>N<¢O

9(AGD68I¡�\68>�<=4�6JUyILF�U!V]ADT86Q<=ULADR�R�68X��Ëi]I\4�E�U!<K9(AHG�6JI�oa lE�U±�\X�6JR2<=4¦>?7�X268R�687�X�6J7�<=�B9g42E�IL6�Rt6JU\<LF�ULC�A�<=>?ED7>?I(MYADU\b�6yAZ<�<L4�6�ILF�U!V]ADT86�icM?E�72b�9�AHG�6JI�o³^�3Ë6yTHADMNMt<L4�68I\6�9(AHG�6JI �\I\F�U\V]AZT86�b�U=AHG�>N<¢Of9�AHGD68IL�2^ x 7PUL68ADM?>�<¢O� <=426�E�TJ6HAD7:>NI�I!<=U=A�<=>N¶�68X¡ULAZ<=4�6JU�<=4�AD7:4�E�d�E�b�6J7�68E�F2I8^ +;OBT8E�72IL>?X268UL>N7�bS<L4�6¿�\I!<=U=A�<=>N¶�THAZ<L>?E�7���EDVW<=426

G�>N>

Page 8: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

E�T86HAZ7Í �9(6�E�C�<�AD>N7 �!>?7�<=6JUL7�ADM¤b�U=AHG�>N<¢O 9(AHG�6JIL��^A�o z <¢9(E:�¢MYAHOD68UgdfE�X�6JM¥^_`4�6 I\>?d�R�M?6JI\<KTHADI\6f<LE[UL6JR�UL6JIL6J7�<�<=4�6fI\<=ULAZ<=>�¶�68XËE�TJ6HAD7Ë>?I�<=E±G�>N6J9Ø<L4�6SE�T868AD7wAZI�A±<¢9;E �¯MYAHO�6JU

I\O�I!<=68d±^|kl>?b )�^K_`42>?I,>NIn�\U\6HADI\E�7�ADC�MN68�nC�68T8ADF�IL6{<L4�68U\6�>?IyAS9;68MNMld�>�£�68X:I\F�U\V]AZT86{MYAHOD68U,>N7[<=4�6{U\6HADME�T86HAZ7Í �9g4�>?T�4B>?IgIL6JR�ADU=A�<=68XBVcULE�d <L4�6KX�686JR¡E�TJ6HAD7BC�O±AfIL4�ADULRBR�O�TJ7�E�TJM?>N7�6D^���ILF�AZM?MNOD �df> £�68X±MYAHO�6JUX�6JR2<=4 �  m � ��� NÃÓ � ���: gAZ7�X <=426¡X�68R�<=45Ct6J7�6HA�<=4�<=426[<=4�6JULd�E�TJM?>N7�6 ­  ��� � ���:^ _`4�6�\>N7�<L68U!V]ADT86J��>N7n<=4�6y<¢9;EfM?AOD68Ugd�E�X�6JMÍUL6JR�UL6JIL6J7�<LI0<L4�6B�\R�O�TJ7�E�T8M?>N7�68�2^

p`qcr�s u���&� z <¢9;E �¯MYAHO�6JU�d�E�X�68M�^

_`4�>?IWI!O�I!<=6Jd 4�ADI�<¢9(E�G�6JU\<L>?THAZM�dfE�X�68IJ� I,7�6gC�AZULED<LULE�R�>NT�d�E�X�6gAZ7�XQE�7�6gC�ADU\E�TJM?>?72>?T(d�E�X�6D^²_`4�6C�ADU\ED<=U\E�R�>NT dfE�X�6�U\68R�U\68IL6J7�<=I²<L4�6�G�6JU\<=>NTHADMNMNO � AHGD68U=AZb�68X�d�ED<=>NE�7Í^&_`4�6�UL6JI\<=EDUL>?72b�VcE�ULTJ6�>NI&b�U=AHG�>N<¢OP�\b���^_`4�6|C�ADULE�T8MN>?7�>NTydfE�X�6�4�ADIgG�6JU\<L>?THAZM²I\4�6HADU�AD7�XBX�E�6JI072ED<�TJE�7�<=U\>?C�F2<L6K<LEf<L4�6|GD68U\<L>?T8ADM?M�O �¯>?7�<=6Jb�U=A�<=68X<=ULAD7�I\RtE�U!<H^._`4�6�UL68I!<=E�U\>?7�b`VcE�U\T86�>NI��\UL6JX�F�T86JX�b�ULAHG�>�<¢O���  Ï��Ï Â���� � � �2 �AD7�X �� Ï���

# � ���� #

 ��� � ��� � ­ ^x 9g>NM?M¤I\@�>?Rn<=4�6�X�6JUL>�GDA�<=>?ED7nAD7�X2�\F�I!<0bD>NG�6,<L4�6�R�4�ADI\6�ILR�6868X2I�VcE�U;<=4�6y<¢9;Efd�E�X268I8 2UL68I\Rt6JTJ<L>NG�6JMNOD^k�E�U�<=4�6�C�AZULED<LULE�R�>NTydfE�X�6yÁ P Â�� �¤i� � � ­ oa^k�E�U�<=4�6�C�AZULE�T8M?>N7�>?T0dfE�X�6�Á � Â�� � �� � � #� ��� � # ^, k�E�U�<L4�6�C�ADULEZ<=ULEDR�>?T,d�E�X�6D �8  � #� ��� � #�� mD AD72X�� #� � � m�^+(6JTHADF2IL6 ­�� �   1y>?I�T8MNE�IL6W<=E ��AD72X � ­ >NI.TJM?E�I\6�<=E � � ^l_`4�>NI.>NI&9g4�O�9;6(T8ADM?M�>�<0�\C�ADU\ED<=U\E�R�>?Td�E�X268��^ x <�>NIgADR�R�U\E£�>NdSAZ<L68M�OSX�6JR2<=4B>?7�X268R�687�X�6J7�<H^! P� � � ��� � # � � K �#"Â����k�E�U � � ­  $ � � �Dd± &Á P Â©Ó � ���±Æ��^%$W68U!O¡V]ADI!<H^S_`426JO¿THAZ7ÞT8U\E�ILI�<=426SÕ²ADT8>�¶�T{9g>N<=42>?7ËÓX�AHO�I8^ x <g>?I`I\F�U\V]AZT86Kb�U=AHG�>N<¢Of9�AHG�6Z^

, k�E�U�<L4�6�C�ADULE�T8MN>?7�>NT�d�E�X26D  � 8 ÂON ÏÏ � Â#N � � �2^E�E �j N � � � 1�^ x VgI\F�U\V]AZT86SULAD>?I\68IfmST8d±  1�Â�mHTJdB .<=4�6J7Þ<=4�6�<=4�6JULd�E�TJM?>N7�6�9g>?MNM�X�686JRt6J7ÝC�O�Dd'&(&JE�>?7�TJ6�I\F�U\V]ADTJ6�R�68U!<=F�U\C�AZ<=>NE�7±>?I�d�F�T�4¡MN68ILI�<=4�AZ7B<=426|GZADUL>?AZ<=>NE�7PEDV <L4�6:�!<L4�68U\dfE�T8MN>?7�6J�� 9;6{THADMNM.<L4�68I\6�9�AHGD68I[�\>?7�<=6JUL7�AZM²b�ULAHG�>�<¢OB9(AHG�6JIL��^�$W68M?E�T8>�<=>N68IgVcE�U�<=426�C�ADU\E�TJM?>?72>?T�dfE�X�6�4�AGD6

G�>?>N>

Page 9: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u������ +(ADULED<LULE�R2>?T,d�E�X26D^

p`q]r�s ����@M;� +�AZULE�T8M?>N7�>?T0dfE�X�6Z^

E�R2RtE�I\>N<L6,IL>Nb�7�I�>N7 <=4�6,<¢9;EQMYAHO�6JULI8^W_`4�>NI`dfE�X�6Z 24�E9;6JG�6JU8 �X�E�68I`7�EZ<gT8E�7�<=U\>?C�F�<=6�<LEQ<L4�6�<LED<�ADMGD68U\<L>?T8ADM?M�OS>N7�<L68b�ULAZ<=6JX±<LU=AD72ILR�E�U\<8^ P� � � � � � # � � K �  ���x <LI�I\<LULF�Ta<=F�U\6S>NIKGD68U\<L>?T8ADM?M�O:X�68R�6872X�687�<H �I\4�6HAZU|h�E9�^±Á � ¨Ó�N���±Æ f9g4�>?T�4Þ>?IQm � �P<=>?d�68II\M?E9(6JU`<=4�AD7P<=426KI\F�U\V]AZT86�b�U=AHG�>N<¢OS9�AHG�6Z^

C�o z m �­ �¯MYAHO�6JU8 �U\68X�F�TJ68XBb�U=AHG�>N<¢OSdfE�X�68M�^z I|9;6S4�AHG�6fIL6J687jADC�EG�6D lb�U=AHG�>N<¢O:9(AHG�68I�ADILI\E�TJ>YAZ<L68XË9g>�<=4¦<=4�6fC�ADULEZ<=ULEDR�>?TQd�E�X�6�ADU\6�GD68U!OV]ADI!<H^ x Vy9(6:AZUL6[ED7�MNOj>N7�<L68U\68I\<L68XÈ>N7 <=426¡ILMNEu9;68UfC�ADU\E�TJM?>?72>?Tn9�AHGD6D g9(6[9g>NIL4 <LEw68MN>?d�>?7�A�<=6<L4�6BC�ADULED<LULE�R2>?TPdfE�X�6Z^ I,7�6±9(AHOj<=E¦X�EÞI\EÞ>?IQ<=EËADI\ILF�d�6��!UL68X2F�T86JXÈb�U=AHG�>N<¢O��¦T8E�7�X2>N<=>NE�7Í 9g4�>NT�4SAZILILF2df6JI�<L4�AZ<W<=4�6gX�6J68RfE�TJ6HAD7[i]C�68MNEu9 <=4�6gI\F�U\V]ADTJ60d�>�£�6JX�M?AHO�68U=o >?I�d�ED<=>NE�7�M?6JILIJ^l_`4�>NIADI\ILF�d�R2<L>?E�7Å68MN>?d�>?7�AZ<L68IS<L4�6¦C�ADULEZ<=ULEDR�>?T¿d�E�X�6Z �9g4�>NT�4 >?I±X�6JR2<=4��¯>?7�X268R�687�X�6J7�<H^ Ô ��I,_D>`�;g68T8ADM?M2<L4�AZ<W>N7Q<L4�6f�\b�6JE�I\<LULE�R�42>?Tgdf6a<=4�E�X��yI\68TJ<L>?E�7¤ �9;60ADILI\F�df6�A�UL6aVc68UL6J7�T86�M?6aG�6JM ��<=4�6gM?6aG�6JMEDV²7�EQd�ED<=>NE�7Í^l3Ë6KADULb�F268XP<=4�AZ<`<L4�>?IgAZILILF2dfR2<L>?E�7n¶�M�<=6JUL68X±E�F2<`<L4�6�C�ADU\ED<=U\E�R�>?Tyd�E�X�6D^ÖÜ_`4�6JUL6aVcE�UL6Z ²>N7¿A¦m �­ �¯MYAHO�6JUyI\O�I\<L68d± .9;6QE�7�M�O¡4�AHG�6�E�7�6QC�AZULE�T8M?>N7�>?T|dfE�X�6Z^{kl>?b�� E�T�4268dfAZ<=>NT

>�£

Page 10: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

X�>?ADb�ULADd©I\4�E9g>?7�bQ<=426KC�ADULE�T8MN>?7�>NT�d�E�X�6y>N7±APm+�­ �¢MYAHOD68U`I\O�I!<=68d±^

p`qcr�s u���&� _`426KC�ADULE�T8MN>?7�>NT�d�E�X�6y>N7n<=4�6Qm �­ �¯MYAHO�6JU`dfE�X�68M�^

z InX�>?I\T8F�I\IL68XÅ>?7 <=4�6¦Ó �¯MYAHO�6JU d�E�X�6JM¥ �<L4�6:C�ADU\E�TJM?>?72>?T±dfE�X�6¡4�AZInAwMYADU\b�6 �¢ADd�R�M?>�<=F�X�6PR�68U5�<LF�ULC�A�<=>?ED7:>?7¡<=426�<=4�6JULd�E�TJM?>?726D ÍAZ7�X¦API\dSAZM?M �¢ADd�R�M?>�<=F�X�6yR�68U!<=F�U\C�AZ<L>?E�7¿AZ<�<=4�6QIL68AnILF�U!V]ADT86Z^_`4�6JUL6aVcE�UL6Z Í>N<0UL6JR�UL6JIL687�<LIK>N7�<=68U\7�ADM�9�AHGD68I8^K_`4�6¿�!E�7�6�AZTJ<=>�G�6�M?AHO�68U\�nADC�EG�6�<=426:�\>N7�68U!<,X�6J68RE�T868AD7��yUL6JR�UL6JIL6J7�<LI�<L4�6`E�T86HAZ7�>?TQ�!ILF�U!V]ADT86gd�>�£�68X{M?AOD68U\�� �AD7�X{<L4�6`dfEZ<=>?ED7�EDV�<=4�6`d�>�£�6JX{MYAHO�6JUC�ED<L<LE�d UL6JR�UL6JIL687�<LIK<L4�6�d�EuGD68d�687�<�EDVW<=426�<=4�6JULd�E�TJM?>N7�6D^ x 7:<L4�>?I0I\O�I\<L68d± ÍGDAZUL>YA�<=>?ED7[EDVW<=426M?AOD68UK<=4�>NT�@�7268ILI{U\68I\F�MN<LI�R�U\>?dfADUL>NMNO[VcU\E�d <=426S<=4�6JULd�E�TJM?>N7�6QdfED<L>?E�7¤^n_`4�6 C�ADU\E�TJM?>N7�>?TQd�E�X�6I\Rt6J68X[EZVlb�ULAHG�>�<¢O >NI8�&�g � � Þ^

����������'�)¤/21�*0 ���*�/�$l/7% *(+ 1 �È*�121 & ��$ 4 % �,1 *$ 4y' ³*$�S.e$l/7% *(+

x 7±<L4�6|ADCtEGD6D �9;6|4�AHGD6|TJE�7�IL>NX�68U\68X±<=4�6KADX:�\F2I\<=d�687�<�F�7�X268U�b�U=AHG�>N<¢OS>?7±A�72E�7��¢U\ED<�A�<=>?72b{E�TJ6HAD7Í^x 7f<L4�>?IWI!O�I!<=6JdB �AZV�<=6JU�<=4�6g<=ULAD7�IL>N687�<;b�U=AHG�>N<¢O�9(AHG�6JI�U=ADX2>YAZ<L6`E�F2<H �<L4�60E�T868AD7SU\6HADT�4268I�A�68¼�F�>NM?>NC�UL>NF�dI\<=AZ<=6Z�S<=426nI\<�A�<=6PEDV�UL6JI\<H^¦3Þ6P6JM?>?d�>?7�AZ<=6JXË<=426nULED<=AZ<=>NE�7w6 � 68Ta<=IQC�OÝIL6a<L<L>?7�b G(E�U\>?E�MN>?IKR�ADULADdf6a<=6JU�P �2^�|´ �È�?�a� ¾�~������Û·� |ª�~�¹�¬Q�����¢´������{�L¸8��~D�]�¥´��2¾��;~D���Q¾��� �l�2~D�W���`��´��¾y� ´������y¾aºu¾a�¢�a�� 35>N<L4SAKF�7�>NVcEDULdØU\ED<�AZ<L>?E�7Bi��P>NI(ADI\ILF�d�68X�<=EKCt60A|TJE�7�I!<�AD7�<³o³ �<L4�6�68¼�F�AZ<L>?E�7�I�EDV&d�ED<L>?E�7�VcE�UW<=426

F�72VcEDULT86JXÍ �>?7�G�>?I\T8>NX±E�T868AD7BADUL6Z�K7LKM N � � Â N m

��PQ RQ!T U i m�mA�o

K��KM � � L ÂON m

� PQSRQ� U i!m�mHC�o

K��K M  N m

� PQSRQ�� N ��� i m�mHTHo

Q LQ!T �

Q �Q� �

Q �Q��  ��� i!m�mHX�o

�n�H� s CÞ��� q �.�D�i]>Yo �ÍÂ,T8E�7�I!<�AD7�<�i]>�^ 6Z^ <=4�6yV �¯R�MYAD726yADR�R�U\EH£2>NdSA�<=>?ED7�o 6i]>N>�o;4�EDdfE�bD687�6JE�F�IgE�TJ6HAD7�ÂK � �{ � P Â,TJE�7�I\<=AD7�< 6

£

Page 11: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

i]>N>?>�o�4�O�X�ULEDI\<�A�<=>?T�ADR�R�U\E£�>NdSAZ<L>?E�7Þi Ð��Ð$*  NJ� P ��o96Í_`4�>NI,ADR�R2ULEH£�>?dfAZ<=>NE�7[6JM?>?d�>?7�AZ<=6JIg<L4�6¿�\I\4�E�U!<b�ULAG�>�<¢O±9(AHG�6JIL�� &MN6HAHG�6|E�7�MNOn<L4�6�MNE�7�bQ9�AHG�6JIy>?7[<L4�6�I!O�I\<=6JdB ¤ADI�X2>?ILTJF�ILI\68X¿>?7±<=4�6|R�UL6aG�>?E�F�I�T8MYAZILI8^z I`9;6|U\68T8ADM?M� �<=426KMNE�7�b�9(AHG�68I04�AHG�6|A{I\<=U\F�TJ<LF�UL6�<=4�AZ<�>?I��\>N7�X�68R�6872X�687�<�EDV²X�68R2<L4���^

i]>�G�o;h�AZ<�CtED<\<=E�d± 2X268R2<L4 �ÃAZ<`9g4�6JUL6 �  ��^icG2o �&>?7�68ADUL> � 6�ADC�E�F2<�A{UL6JI\<L>?7�b�C�ADI\>?TyI\<=AZ<=6fi L P  � P  � P Â���ok�E�U`ILdfADM?MtRt6JU\<LF�ULC�A�<=>?ED7�I�� U�� U��2U�� AZCtE�F�<`<=4�6�UL6JI\<L>?7�bQI\<=AZ<=6Z �9(6K4�AHG�6Z�L  L P � �nÂ��. �  � P � �  �  �  � P � �  �  R  R P i � o � �  �{ � P ^_²EQ<=426��W«=¾a�`´�«\���a«a �9(6K4�AHG�6Z�

Q �Q M N � � ÂON m

� PQ �Q7T U i muÓDA�o

Q �Q M � � � ÂON m��PQ �Q�$U i!muÓZC�o

Q �Q7T �

Q �Q� �Q �Q��  �"� i muÓZTHo

+;O ADR�R�M�O�>?7�b¦4�O�X2ULE�I!<�AZ<L>?TBADR�R�U\E£�>NdSAZ<L>?E�7 AZ< �  �2  R  R P � �  �  � P ��1Åi]7�EZ<=6D� �>?I�>?72X�68R�687�X2687�<nEDVyX268R2<L4�oa �GD68U\<L>?T8ADM?M�Oj>?7�<=6Jb�U=AZ<L6B6J¼�F�AZ<=>NE�7 i muÓZTHo�AD72X�F�IL>N7�bÞI\F�U\V]AZT86¡C�E�F�7�X�ADU\OT8ED7�X�>N<L>?E�7 �  Ð)8Ð96  �C�ED<L<LE�d C�E�F�7�X�ADU\OnTJE�7�X�>�<=>NE�7 � Â��2 9(6KE�C2<=AD>?7¤�

Q �Q M N � �  ND� Q 1Q!T U i m ��A�oQ �Q M � � �  ND� Q 1Q� U i!m �DC�o

Q 1Q M � wi

Q �Q7T �

Q �Q�� oWÂ���� i m �DTHokl>?bDF�UL6�72^z I >N7 <=426¡7�E�7��¯ULED<=AZ<=>N7�bÞT8ADIL6Z �9;6¡TJE�d�C�>N7�6[<=426:ADC�EuGD6[<=42UL686[68¼�F�AZ<L>?E�7�If<=EÞ9gU\>N<=6¡AÞIL>N7�b�M?6

68¼�F�A�<=>?ED7±>N7!1nADM?ED7�6D^Q ­ 1Q M ­ N � ­�� ­ 1 � �* ��|Â�� U i!m�$�o

9g4�68U\6 � ­  � >NIfR�4�ADIL6BILR�6868X IL¼�F�ADU\68XÍ ��  Ð��Ð C N Ð �Ð9( >?IQT8ADM?MN68XÅ«��a��~D�]���D�±�D´�«³���¥¹a���]º�  �5>NIR�M?AD7�6J<=ADU\OfG�E�U!<=>NT8>N<¢OD �AD7�X�� � �:>?I`THAZM?M?6JXPADC�I\E�M?F2<L6,G�E�U!<=>NT8>N<¢OD^D � G� ���= ��2� q � r �����J� D �P � GJ� kl>?bnm ��^/E�T�4�68dfAZ<=>NT8I`VcE�U`b�ULAHG�>�<¢O 9(AGD68I0U=ADX�>?AZ<=>NE�7Í^���6a<�� IM?E�E�@ AZ<f<=4�6¡I\>?d�R�M?6Z `E�7�69�¢X�>Ndf6J7�IL>NE�7�ADMgTHAZIL6:>?7 T X2>?UL6JTJ<L>?E�7Í^ x VK<L4�68U\6¡>?If7�E �² g68¼�F�A�<=>?ED7#i!m�$�oC�68T8EDdf6JI8 

Ð # 8Ð)6 # N � ­ �Q­ 1�Â���^z I\ILF�d�6 1{Â�1 P � ���i¥É T NÞ» M oa �<L4�687P9;6|E�C�<�AD>N7 VcULE�d <L4�6KADC�EuGD6�68¼�F�AZ<L>?E�7Í�

£�>

Page 12: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u���Í� E�T�4�6JdSAZ<L>?T0X�>YADbDU=ADd I\4�E9g>?7�b�<L4�6,I\F�U\V]ADTJ6,R�68U\<LF�ULC�AZ<=>NE�7fVcE�U�<=4�60IL4�AZM?M?E9È9(AZ<=6JU�68¼�F�A �<L>?E�7¤^

Ng» ­ � � ­ É ­  �2 �AD7�Xn» ­  � ­ É ­  2<=4�F�I»jÂON � É �u^�_`4�AZ<g>?IJ �b�ULAG�>�<¢OS9(AGD68I�U=ADX2>YAZ<L6,E�F2<`>N7�N T AD7�X � T X2>?UL6JTJ<L>?E�7�I;9g>N<=4PA�I\Rt6J68X±EDV�g � �� Ë^ z V�<L68U`<=4268IL6�9(AGD68I0U=ADX�>?AZ<=6yEDF2<H 2<L4�6�IL68A�I\F�U\V]ADTJ6Kh�AZ<=6J7�I�AD7�XP<=4�F�I�1�Â��2^D M G�� �H�2� q � r �����8� D � "Â�� GJ� k�E�U�ED7�6�X�>?d�687�I\>?E�7�ADM¤T8ADIL6Z �6J¼�F�AZ<=>NE�7Ëi!m�$�o(Ct6JT8E�d�68IJ�Ð$#�8Ð)6 # N � ­ Ð$#�8Ð C # � � �Ð��Ð C  ��^z V�<L68U�b�ULAHG�>�<¢O¿9�AHG�6JIQU=ADX�>?AZ<=6fE�F2<8 l<=4�A�<{>?IJ  Ð # 8Ð)6 #  �¡>?7Ë<L4�6nADC�EG�6n6J¼�F�AZ<=>NE�7Í l9;6nE�C2<=AD>?7ÞAD768¼�F�>NM?>NC�UL>NF�d©I!<�AZ<L6D�N � ­ Ð # 8Ð C # � �* Ð �Ð C  �2 �AZ7�X±<L4�F�I8�NJ� Ð98Ð C � � �  �2 �E�U� �  � Ð98Ð C ^ ���y68EDI\<=U\E�R�4�>NTyC�ADM?AD7�T86Z^E�E�MNF2<=>NE�7�I�<=E 1¿Ô VcE�U`X�6JUL>�GDA�<=>?ED7nIL686��y>?M?MH2�^ÚÓ�^�m|Õ`m%7�Ó�Ü�>?IJ�

1� 1�PuÔ NKm � 6 T � i NT( o Ü U � ��� T � � U i muÙDA�o

1QÂ�1�PuÔ�mJN 6 T � iT( o Ü U � ��� T ��� U i!muÙZC�o

9g4�68U\6 ( Â� >?I�THAZM?M?6JXK<L4�60;�E�I\ILC�OKU=ADX2>?F�I�EDV2X�6JVcE�U\dSA�<=>?ED7Í^ z I�9(6;THAD7|IL686�VcULE�d <=4�6 1yILE�MNF2<=>NE�7Í IL68AnM?6JGD68M >NIyX�6JVcEDULd�68X¡9g>N<L4�>?7¡<L4�6{X�>NI\<�AZ7�T86{EDV��LAZ�� &AD7�X¡<L4�6{b�6JE�I\<LULE�R�42>?T{T8F2ULUL6J7�<=IKADUL6{TJE�72¶�7268X9g>N<L4�>?7Q<=426�U=AZ7�b�6�EDV¤<L4�6�;gE�ILI\C�OfULADX�>NF�I8^ I,F2<LIL>NX�60<L4�>?IWU=AZ7�b�6D  � >?I�T8MNE�IL6g<=E �2^�_`4�6JUL6aVcE�UL6Z 7;gE�ILI\C�OU=AZX�>?F�I�>?I�A�7�AZ<=F2U=ADM&I\THADMN6KVcE�Ug<=426|M?ADULb�69�¢I\THADMN6yE�TJ6HAD7BT8>?U\T8F�M?AZ<=>NE�7Í^�Ô k�E�Ug<=426K¶�U\I\<0C�ADU\E�TJM?>?72>?Tyd�E�X�69g>N<L4 � � 8"�{d �ZI8  ( � >?IgAZCtE�F�< � �D@�d�AZ< $�Ù�� � 9g4�68U\6 �P m � �� aÆ �^ Ü�|´ � �(�:¹L~D� ¾���±�c�2~D��� �����������,��¹a��¾:´ ·P«�´��¥~D�]�¥´��$�������:�L¸8�������]ª8«³����� ¾a�¯~D�¢�:´ ·n�����:´¹��L~D�È�?¾P���

�N����´¾a�]«�´������¥¹Kª�~D��~D�¤¹�� ��«\~D�c���a«����2~D�:�c��� �Y¾a�¥~D�¢�{´ ·y«=��¾a� |~�¾K���¡�����K�¤´��*)¢«=´��¯~D�]�����P¹L~�¾�!

£2>N>

Page 13: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u�Þv:=&� E�T�4268dfAZ<=>NTyX�>YADbDU=ADd©IL4�E9g>?72bQ<L4�6�E�TJ6HAD72>?T�ADX:�\F2I\<=d�687�<�>?7n<L4�6�7�E�7��¯ULED<=AZ<=>N7�b{THADI\6D^

3Þ6��\F�I!<�ILA9 <=4�AZ<KU\ED<�AZ<L>?E�7¿T8AD7ÞT�4�AZ7�b�6Q<=4�6Þ�\68¼�F�>NM?>NC�UL>NF�df�SI\<�A�<=6�EDV(<L4�6fE�T868AD7Í .UL6JMYAZ<L>NG�6�<LE<=426�7�E�7��¢ULEZ<�AZ<L>?7�bKTHADI\6D^ �06³£�<H �9;6�9g>NM?M�T8E�72IL>?X268U(42Eu9ÅU\ED<�AZ<L>?E�7�d�E�X�>�¶�68I�<=4�60b�U=AHG�>N<¢OQ9(AHG�68I`>N7S<=426I\O�I!<=68d±^+;OPADI\ILF�d�>?72bQ<L4�6y9�AHGD6|VcE�U\d EDV �

1� 1�P 6 T � ��i�É T � � � NÞ» M o U i m )�A�o�n � P 6 T � ��i�É T ��� � NÞ» M o U i!m )DC�o

�  � P 6 T � �=i¥É T � � � NË» M o U i m )DTHoILF�C2I\<=>�<=F2<L>?7�b�<=4268d >?7�<LEK6J¼�F�AZ<=>NE�7�I�i!m�moa �AD7�XfM?6a<�<=4�6�T8E�6���T8>N687�<;dSAZ<LUL> £�6J¼�F�ADM;���D�� �9;60E�C�<�AD>N7

<=426KX2>?ILR�68U\IL>NE�7PUL68M?AZ<=>NE�7

» ­  � ­ � ­ � � ­ U i!mJ2�o9g4�68U\6�� ­ Â É ­ � � ­ ^i]>Yogk²>?ULI!<H 2<L4�6KM?E�7�b{9(AHG�68I�Ct6JT8E�d�6B�\X�>?I\Rt6JULI\>NG�6J�� �IL>N7�T86KÁ(Ï� Ð=ÑÐ � >NI0A{VcF�7�Ta<=>NE�7BEZV²»5AZ7�X±<L4�F�I

�Í^i]>N>�o/E�68T8ED7�XÍ �<=426,M?E�72b{b�U=AHG�>N<¢O�9�AHGD68I�X�E{7�ED<`4�AGD6B� � 6JULE��{VcUL6J¼�F�687�TJ>?6JI0AZ7�O�d�E�UL6Z^�_`4�68>NUQ�\MNEu9 �

68I!<=�PVcUL6J¼�F2687�TaOË>NI�AZ<B� �t�2 Í9g4�>?T�4¦4�ADIKAnRt6JUL>NE�X i�Ç Â ­¯®Ñ o,EDV(A Vc6J9ØX�AO�IJ^�_`4�AZ<K>?I8 Í<L4�6QM?E�72b�68I!<R�68UL>NE�X¦VcE�U�bDU=AHG�>N<¢O¿9�AHGD68IQ>?I[�\X�AHO�IL��^[_`4�68>NU�Rt6JUL>NE�XËU=AD7�bD6S>?I¡�\IL6JT8E�72X�I58�X�AO�I\��^:_`4�6JOwADUL6fT8E�7��IL>NX�68U\68X �\4�>Nb�4��{VcUL6J¼�F2687�TaOP>?7 �\E�T86HAZ7�E�b�ULADR�4�O��� �IL>N7�T86<;gE�ILI\C�On9(AHG�6JI�4�AHG�6KR�68UL>NE�X2I`EDV d�E�7�<=4�I�<LEO�68ADULIJ^

�����Ø" $&%('�1 %!+ $W+ *g)&'.$W+ ' %¯/!- $Ø%�$ �7&�%!+�"���)��È*;1�1 & 'w$&%;'.1

�0RÝ<=E:72Eu9� �9;6n4�AHG�6nTJE�7�IL>NX�68U\68XÝ<=426PADX:�\F�I!<=d�687�<{F�7�X268U{b�ULAG�>�<¢OÞ>N7jA¡7�E�7��¯ULED<=AZ<=>N7�b[E�T868AD79g>N<L42�  ��  AD7�XÞ>?7ËA[ULED<=AZ<=>N7�b±E�TJ6HAD7¦9g>N<=4 �� ���� M!( � M ^ x 7Þ<L4�68I\6 I\O�I!<=68d�I8 �bDU=AHG�>N<¢O¿9�AHGD68I

£�>?>N>

Page 14: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`q]r�s ��Þv�v�� E�T�4�68dfAZ<L>?T�X�>YAZb�U=ADdeIL42Eu9g>N7�bQ<=4�6�68¼�F�>NM?>NC�UL>NF�d I\<�A�<=6�>?7n<=426KU\ED<�A�<=>?72b{THADI\6D^

6a£�>NI\<�AD7�X <L4�6JOnADUL6,d�E�X�>N¶�6JX C�OnULEZ<�AZ<L>?E�7 >N7S<=426yULED<=AZ<=>N7�b�THAZIL6D^0E�>N7�T86,<L4�6,68ADU\<L4±>NI�ULEZ<�AZ<L>?7�b{AZ7�XG(E�U\>?E�MN>?I(R�ADU=ADd�6J<L68U �±Â Ó�� ������² ��:>?7 V]AZTJ<gGZADUL>N68I�9g>N<L4PMYAZ<L>N<=F2X�6��.^�_`4268UL6aVcE�UL6Z �9(6KTHAD7P9gUL>�<=6�BÂ�� P ��� �  9g4�68U\6 � P >NI�<=4�6FG(E�U\>?E�MN>?I|R�ADU=AZdf6a<=68UQA�<QMYAZ<L>N<=F2X�6�� P  ;ILAO��  $�Ù � �| �AD72X � >?I{d�68UL>NX�>?ED7�ADM

X�>NI\<�AZ7�T86QVcULE�d M?AZ<=>�<=F�X26�w<LE� P ^ x <|>?I�UL6aVc68ULU\68X¦<=E¡AZI|d�>?X��¯MYAZ<L>N<LF�X�6:� � �¢R�M?AD7�6�ADR�R�U\EH£2>NdSA�<=>?ED7���^�P � P >?I`U\6JVc6JULUL6JX±<LESAZI��!V �¢R�M?AD7�6�ADR2R�ULEH£�>?dfAZ<=>NE�7���^kl>?bDF�UL6{m �=E�T�4268dfAZ<=>NTyX�>YADbDU=ADdeVcE�U � �¢R�M?AD7�6Z^�5·H� ��«=�a�?�a��~D����¹�´��¤¹��¢����¾a^ 3Þ6�EDV�<L687±UL6JVc6JU�<=E���¦AZI(����~D�¤�a�¯~D«³º��D´�«a�]�¥¹a���]ºZ �� Р�Ð C N Ð �Ð)( ADIy«=�a��~D�]���D�K�D´�«a�]�¥¹a���]ºZ � � �SAZI�~�ªa¾´������¢���D´�«³���¥¹a���]ºZ �AZ7�X� � �� ADI���´��¢�a�����]~D�&�D´�«³�]�¥¹a����º� �9g4�68U\62 >NI�<=4�6�X�6JR2<=4±EDV.9�AZ<L68UgT8EDM?F�d�7Í^z b�AZ>?7Í �9(6±F�IL6±<=4�6±IL>NdfR2M?6nIL4�ADM?MNEu9©9�A�<=68U�68¼�F�A�<=>?ED7 <LEËX�6JdfED7�I\<LU=AZ<L6±9g4�A�< � X2E�6JI�<LEË<=426

I\O�I!<=68d±^{_`4�6�68¼�F�AZ<L>?E�72I�ADUL6�<L4�6QI=ADd�6QADI,<=42E�IL6Q>?7:<L4�6�V �¢R�M?AD7�6�T8ADIL6QX�>NILTJF�ILI\68XÞ>N7:<=4�6{R�U\6JG�>?EDF�IT8M?ADILIJ �6a£�T86JR2<�<=4�AZ<.�±Â � P ��� � 4�6JUL6KU=AZ<L4�68U�<L4�AD7 �±Â � P  � �"�� M!( � M ^

Q �Q M N � �  ND� Q 1Q!T U i m ��A�oQ �Q M � � �  ND� Q 1Q� U i!m �DC�o

Q 1Q M � wi

Q �Q7T �

Q �Q�� oW���� i m �DTHo

£�>NG

Page 15: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u� v��Í� E�T�4�68dfAZ<L>?T�X�>?ADb�ULADd�IL4�E9g>N7�bn<=4�6�X2>?ILR�68U\IL>NE�7:UL6JMYAZ<L>?E�7±VcE�U�b�ULAHG�>�<¢OB9(AHG�6JIK9g>�<=4¡<=426>N72h�F�6J7�T86KEDV²ULED<=AZ<=>NE�7Í^

3Þ6yTJE�d�C�>N7�6�<L4�6yADC�EG�6,<=42UL686y6J¼�F�AZ<=>NE�7�I�<LE�9gUL>N<L6yA�IL>N7�b�MN6�68¼�F�A�<=>?ED7n>?7 � ADM?ED7�6D 2AD7�XPADI\ILF�d�6R�68UL>NE�X2>?T�9(AHG�6KILE�MNF2<=>NE�7 � N N"6 � � Ñ 6  �9(6KE�C2<=AD>?7¤�

»�� � � » � ­Q ­ �Q7T ­ N � ­P � » � � ­ �

Q �Q!T � � » � ­Q ­ �Q� ­ Â�� U i!m%7�o

9g4�68U\6 �B � �� >?I{R24�ADIL6nI\Rt6J68XjVcE�U{M?ED7�b¡ILF2U\V]ADTJ6±bDU=AHG�>N<¢OÞ9(AHG�6JI8^ z I\ILF�d�>?72b¡9�AHG�6PVcE�U\dB�� N N"6 � � Ò C ��� ( � Ñ 6 � >?7 68¼�F�AZ<L>?E�7Ãi m%7�oQAD7�XÈU\6 � AZULU=AZ7�b�6B<L4�6¡6J¼�F�AZ<=>NE�7Í �9;6¡E�C2<=AD>?7 <L4�6[X2>?ILR�68U\IL>NE�7UL6JMYAZ<L>?E�7¤�

i¥É � �Ó» o ­ � � ­ » ­ N � ­P

� ­ � � ­$Z» ­ � i¥Ó ��o�&6J<�� I�MNE�E�@QA�<(<=426�IL>NdfR2M?6�THADI\6�9g>N<L4 � Â��� 2AD72X 9(6,I\4�ADMNM�I\686,9g4�A�<(9(AGD68I�ADU\6,AHGZAD>?M?ADC�M?6`>N7S<=426I\O�I!<=68d±^W_`4�6KX�>?I\Rt6JULIL>NE�7PUL6JMYAZ<L>?E�7nC�68TJE�df6JI8�

i¥É � �Óu» o ­ » ­ N � ­P

� ­ � � ­$Z» ­ � i¥Ó�muo

i]>Yo�k�E�U�4�>?bD4BVcUL6J¼�F�687�TaO[9(AHG�68I,9g>�<=4BMYADU\b�6�»` �<L4�6|<L68ULd�I09g>�<=4P»5>?7±<=4�6�X2687�E�d�>?7�AZ<=E�UgT8AD7¡C�6>?bD7�E�UL6JXÍ^�_`4�F�I8 

É ­ Â Ñ # � � #� #  2E�U

» ­ Â É ­ � ­ � � ­P�� i¥Ó�Ó�o_`4�68I\6wADU\6:M?E�72b I\F�U\V]AZT86¿b�U=AHG�>N<¢O 9�AHG�6JIBF27�X�68UP<=426¦>N72h�F�6J7�T86¿EDV¡�!V\�:8�<L4�6JO5AZUL6¿ADMNILEjTHADMNM?6JX

�a�¤�a«³�]�]~D���D«\~D�����º���~D�D��¾a^

i]>N>�o(k�E�U`MNEu9ÅVcUL6J¼�F�687�TaOn9�AHGD68I09g>�<=4±ILdfADM?M�»g �9;6KT8AD7±X�ULE�Rn» ­ <=6JULd±^�_`4�687±9(6�4�AHG�6Z�i¥É �

�­¥Ñ o ­ Â

� # Ñ # N

� #� # ^0;g6J9gU\>N<L6y>N<H 

£�G

Page 16: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u�wv *Í� E�T�4�6JdSAZ<L>?TyX�>?ADb�ULADdeVcEDU � �¯R�MYAZ7�6D^

ÉfÂ#N �Óu» � NmÓ � ­» ­ N $ � ­P� ­ Â#N �Óu» i mDN � mJN $ � ­P » ­� ­ � ­

o)� i¥Ó ��o

_²EPVcF�U!<=4�6JU|F�72X�68U\I\<�AZ7�XË<L4�6{9�AHG�6fR�ULEDRt6JU\<=>N68IJ �9;6�7�686JXË<=E±E�C2<=AD>?7¿I\>?d�R�M?>�¶�68X:U\68ILF2MN<=IJ^?E�>?7�TJ6� #� Ñ # # � # >NIgAQI\dSAZM?MÍ7�F�d�C�68Ugb�>�G�6J7P<L4�AZ<`» >?I`ILdfADMNM¥ �<=4�6KADC�EG�6�68¼�F�AZ<L>?E�72I�T8AD7±Ct6KADR�R2ULEH£�>?dfAZ<=6JX±C�O��ÉfÂON �

­¥Ñ Ô mDN � i mJN Ó� #� Ñ #� # # o¢Ü¥^

G(4�E�E�IL>N7�b{<=4�6B�5�¯��I\>?b�7n9;6|4�AGD6D ÉfÂON �

­¥Ñ i!m NÈm � Ó� #� Ñ #� # # oW N Ñ � #�� #  �E�U

»jÂ#N � � ­� ­P ÉQ N ���³É�� i¥Ó�$�o

£�G�>

Page 17: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u� v���� E�T�4�68dfAZ<L>?TSX2>YADb�ULADd IL4�E9g>N7�b¡<=4�6SX�>?I\Rt6JULIL>NE�7wU\68MYA�<=>?ED7ËVcE�U�>N7�68U!<=>?ADMWb�U=AHG�>N<¢O¦9�AHGD68IAZ7�X ;gE�I\ILC�On9�AHGD68I8^

, E�>N7�T86��³Ïn Ð=ÑÐ³Ò Â N � �Q>NI�>?72X�68R�687�X2687�<QEDV`»�  <=4�6JIL6n9(AGD68IfADUL6S7�E�7��¯X�>?I\Rt6JULIL>�G�6PAD72XÝCtED<L4R�4�ADIL6±i%� % Â�Ñ Ò Â N ���ao�AD72X:b�ULE�F2R¡G�68MNE�TJ>N<¢Owi]6J7�68U\bDO+��ÏJo0R�ULEDR�ADb�AZ<L6�9;68I\<¢9(ADU\XÍ^{k�ULE�d <=426X�>NILR�68U\IL>?ED7±U\68M?AZ<=>NE�7 9(6�T8AD7BI\686�<=4�6JIL6�9�AHGD68I,AZUL6D��MNE�7�b ;gE�ILI\C�On9�AHG�6JI8^

, ��E:dSAZ<\<=6JU � >?I�RtE�I\>N<L>NG�6¡i���E�U\<L4�68U\7 ��68d�>?I\R�4�68U\6o|EDU�7�68b�AZ<=>�G�6¦i E�E�F2<=4268UL7 ��68d�>?ILR24�68U\6oa MNE�7�b ;gE�ILI\C�On9(AHG�6JI�R�U\E�R�ADb�A�<=6,9;68I\<¢9(ADU\XÍ tC�ED<L4±VcE�U`6J7�68U\bDOPAD72XBR�4�ADIL6Z^

, �&E�7�b�;�E�I\ILC�OK9�AHG�6`I\Rt6J68X{X�6JT8U\6HADI\68I 9g>N<L4�<=426(>?72T8UL68ADIL6(EDV�MYA�<=>N<LF�X�6Z� �³Ï` � %  N � �(Â#N � #� #�^

z Ig<L4�6KADC�I\E�M?F2<L6,GZADM?F�6yEZV � P >N7�T8U\6HADI\68I�RtE�MN6J9(ADULX¤ !;�EDILILC�On9(AHG�68I0ILR�686JX[X�6JT8UL68ADIL6JI8^G(4�E�E�IL6�<=4�6B� � �QIL>Nb�7±AD7�XP>?bD7�E�UL6,<L4�6�IL6JT8E�7�XBE�ULX268U(» <=68U\dB�

»ÝÂON� É � i¥Ó�Ù�o

0,>?I\Rt6JULIL>�G�6nI\>?7�TJ6 �³Ïn Ð=ÑÐ³Ò Â�Ò # >?IQA[VcF27�TJ<L>?E�7ÝEDV�ÉÝAD7�XÝ<L4�F2I{»`^ � % Â Ñ Ò Â N �

Ò # ^Þ_`4�6JIL69(AGD68I;ADUL6�¾���´�«a� �g´¾�¾8ª8º���~D�D��¾ � µÍ���a��«��D«�´���±�D�a�?´¹a���]º`��«=´��2~��~D�¢��¾��L~�¾a� ��~D«\��ª8�������2~�¾� ��«=´��2~��~D�¢��¾�(��¾a� ��~D«L�- (_`4268IL6|THAD7BCt6�I\68687BVcULE�d <L4�6KX�>?I\Rt6JULIL>NE�7PT8F�U!G�6Z^JE�4�E�U!< ;gE�I\ILC�OP9�AHG�6JIyADU\6�GD68U!OBI\4�E�U!<ILE�<L4�AZ<`d�>�£�>N7�b�ADTa<=I`I!<=ULED7�b�MNOfE�7 <L4�68I\6,9�AHG�6JI8^�_`4�6JOnADUL6,68ADIL>NMNOf@�>?M?MN68XfC�OSdf> £�>?7�b�AD7�X <=4�F�I�GD68U!O4�ADU\X¡<=EPR�U\E�R�ADb�A�<=6�V]ADU,A9(AHO[VcU\E�d <L4�6�VcE�U\T8>?72bnUL68bD>?E�7Í^ z M?ILE2 Í>?7:U\6HADM²E�Rt6J7¿E�TJ6HAD7¡<L4�6JO¿AZUL6�7�EZ<I\<LULE�72b�MNO[6a£�T8>�<=68XËC�O:<=4�6Ë�\M?ADULb�69�¢I\THADMN68� 9g>?72X�I�Ct6JTHADF�I\6�<L4�68>NU,9�AHGD68M?6J7�bD<L4�I�ADU\6{<=E�E±I\dSAZM?M¥^��06HAZU<=42609;68I\<L68U\7nCtE�F27�X�ADU!O� �<=4�6JIL609(AHG�6JIgADUL6�@�>?M?MN68XQC�OQ<=4�6�I\<LULE�7�b�df> £2>N7�by<=4�6JUL60AD7�XS<L4�F�I(TJE�7�<=UL>NC�F2<L6<=E{<L4�6y3�+ G�^

,�� � q �J�H���.��� � � �H�BMH�¿������������u� � s �S� � ���u���f���2 q �2� q0 � ��� Lc���������%�¿� � q � q ���D � G�� q ��.Lc�2� q � s �.� D � GJI����K� �  Р�Ð9( ��� � �����8��� � Â�� �u�����¿�u��� � q �8������ q ���{u�:L]��� q �� q L8L�M���� s �J���u��� q �.��� q �*L r u��� q � �j���������;�

£�G�>N>

Page 18: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

, � �H�BMH�[���������0�2���L � � �¯u�@9 s ���.�;�[���������;���|�.� q ,���� q �.�0�2�� Ç � � � �.�-���;�������2�q �;I q �K�u����� �� q � r �P��u�Ý�2�[���� q0 �²�B�H� ���[��.��� � C �t�u��I � �H�!MH� ���������P� q L L� � M��[� �&� q ��:���

, �|�.� q C � q � �8��� q � Al������� s ��� q � r � �a� �²� ���K�� F�ADIL> �¢b�6JE�I\<LULE�R24�O¿>NI�A±T8E�d�R�MN>?THA�<=68XTJE�7�T86JR2<QAD72Xw4�ADI|R�UL6JT8>?I\6 X�6a¶�7�>N<L>?E�7¤^ +;F2<�9;6S9g>?MNMW7�ED<�TJEG�68U�>N<|4�68U\6D^ ��E9(6aG�68UJ �9(6S9g>NM?MI\4�E9 <=4�6�¼�F�ADI\>?b�6JE�I\<LULE�R24�>?Tyd�ED<=>NE�7nEDV�;�EDILILC�On9(AHG�68I0>?7Pkl>Nb�F�U\6{muÙ�^

p`qcr�s u�Ýv �Í� E�T�4268dfAZ<=>NTyX�>YADbDU=ADd©IL4�E9g>?72b ;gE�ILI\C�On9�AHG�6KR�U\E�R�ADb�A�<=>?ED7Í^

���������'�)¤/21�*0 P1 % 4y' �*0�y+�4y$ �S%!'�1 )¿)&*�$�1�/2$�� � &Î/ ��$W-y-�'34 'w$&%;'�1

£�G�>?>N>

Page 19: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u� v �Í� �&E�7�bD>N<=F2X�6 �¥<=>?d�6±R�M?ED< EDV2E�6HA E�F2U\V]ADTJ6��068>Nb�4�<¦i E E �,o AD72E�dSAZMNO�� IyC2IL68U!G�6JX d�>?X��M?AZ<=>�<=F�X26 ;�EDILILC�On9(AHG�68I0>?7n<=426|Õ²ADT8>�¶�TD^

£�>�£

Page 20: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

_`4�6yR�U\68IL6J7�T86�EDV�M?AZ<=6JU=ADM�C�E�F�7�X�AZU\OfX�F�6,<=E|<=4�6yTJE�7�<=>?72687�<�ADMtdSAZULb�>N7�I;X2>?I\<L>?7�bDF�>?I\4�68I(<L4�6,E�T868AD7T8>NULTJF�MYAZ<L>?E�7ËVcULE�d <=4�6±AZ<LdfE�I\R�4�6JUL>?TST8>?U\T8F�M?AZ<=>NE�7Í^ z I�9;6±9g>NM?M(I\686Z ;<=426nT8E�ADI!<�ADM`E�T868AD7 ADTJ<LI�ADIQA9(AGD68b�F2>?X�6Z^ x 72VcE�U\dSAZ<L>?E�7PT8AD7BC�6�<=U=AZ7�ILR�E�U\<L68XBU=ADR�>NX�MNOfVcULE�d M?E9 M?AZ<=>�<=F�X�6JI�<=E�4�>?bD4BM?AZ<=>�<=F�X�6JI8^ x 7kl>Nb�F�UL6:m%2� �9(6±THAD7 IL6J6FE7E � AD7�E�dfADMN>?68I�4�F2b¦<L4�6P6HADI!<=68U\7 Õ²ADT8>�¶�TPT8E�AZI\<=I�>?7w<L4�6PE � 68¼�F�A�<=E�U\>YADMMYA�<=>N<LF�X�6JI8^

p`qcr�s u�Ýv �t� I,C�IL6JU\GD68X E7E � z ^

A�oDG(E�ADI!<�ADM��y6JMNG�>?7n9(AGD6D^_²E�I\>?d�R�M?>�V�OK<L4�>?72b�I8 D9;6gM?E�E�@|AZ< <=426�I\4�ADMNM?E9j9�AZ<L68U�6J¼�F�AZ<=>NE�7{9g>N<=4 �±Â � P  � �"�� M  �ILE ;gE�ILI\C�O

9(AGD68I0X�EQ7�ED<�6a£�>?I!<�>?7n<=4�6�I!O�I!<=6JdB^+(E�F�72X�ADU\O¿TJE�7�X�>�<=>NE�7�I�ADU\6 �  � U �  �2^{3Þ6{¶�72XËI\E�M?F2<L>?E�72I,9g>N<=4 �  �n6JGD68U\O�9g4�6JUL6Z^n_`4�668¼�F�A�<=>?ED7�IgCt6JT8E�d�6D�

Q �Q M ÂOND� Q 1Q7T U i�Ó�)�A�o

� �SÂ NJ� Q 1Q�$U i¥Ó�)DC�o

Q 1Q M �

Q �Q7T ���� i��)DTHo

+;OPADI\ILF�d�>?72bfA�9(AHG�6JVcEDULd ILE�MNF2<=>NE�7 O�EDFBT8AD7B68ADIL>NMNOSX�68U\>NG�6y<L4�6�X�>?I\Rt6JULIL>NE�7PUL6JMYAZ<L>?E�7PADIJ�» ­ Â É ­ � ­  �9g4�68U\6 �� � �� Ë 2E�U»jÂON � É �u^

£�£

Page 21: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u�Þv �Í� E�T�4268dfAZ<=>NTyX�>YADbDU=ADd©IL4�E9g>?72bQ<L4�6�T8E�AZI\<�AZMÍG�68U!<=>NTHADMÍ9(ADM?M�^

_`4�>?Ig>NI�<=4�6�I=AZdf6�X�>NILR�68ULI\>?E�7PU\68MYA�<=>?ED7SVcE�U��\b�ULAHG�>�<¢O 9(AGD68I0>?7PAQ7�E�7��¯ULED<=AZ<=>N7�b���I!O�I!<=6JdB^x 9g>NM?M¤72ED<�X�68U\>NG�6y<L4�6�ILE�MNF2<=>NE�7n>?7PX�6a<�AD>NMÍ4�68U\6D �C�F�< x � M?Mt9gUL>N<L6y>N<gX�E9g7Í^_`4�6KILE�MNF2<=>NE�7 VcE�U 1 >NI`T8E�d�RtEDIL68X±EDV.<¢9(E�R�>N68T86JI�� AD7�X�����à6����� � i T � � M o U�à� 6����� � ci T N � M o U9g4�68U\6fC�ED<L4 � .AD7�X�� ²ADU\6QADULC2>N<=ULADU\OBVcF�7�Ta<=>?ED7�I8^Q_`4�AZ<K>NI8 &E�7�6{9(AHG�6SR2ULE�R�AZb�AZ<=6JIy9(6JI\<¢9(ADULX

i��|o�AD72Xj<L4�6 ED<L4�68U{ED7�6nR�ULEDR�ADb�AZ<L68I�68ADI\<¢9(ADU\X i���o³^�Ô 0,6J<L68ULd�>?726S9�AHGD6±R2ULE�R�AZb�AZ<=>NE�7wX�>NUL6JTJ<=>NE�7Í�kl>N7�X:T8E�72I\<�AZ7�<�R24�ADIL6{MN>?7�6Z^��  T � � M  ���� M Â4�� &AD7�X��  T N � M  ���� M  �2^ z I,<=>Ndf6b�E�68IgC�O¦i M >?7�TJUL68ADIL6JI�oa  T >?7�TJUL68ADIL6JI0>N7 � T X�>?U\68Ta<=>?ED7PVcEDU�� C�F2<�>?7 N T X�>NUL68Ta<=>NE�7 VcE�U��Q^ Ü

z I � >N7�T8U\6HADI\68I,<LE±>N72¶�7�>�<¢O� ��©Ct6JT8E�d�68Iy>N72¶�7�>�<¢O�^�k�E�U,A Rt6JU\<=F2ULC�AZ<L>?E�7¡<L4�AZ<�>NI,>?7�>�<=>?AZ<=6JX¡7�6HAZU<=426|TJE�ADI\<8 ¤AZdfR�MN>N<LF�X�68I`EZVl<=426K9(AHG�68Iy>N7�T8U\6HADI\6K<LEf>N72¶�7�>�<¢O±AH9�AHOPVcULE�d©<=4�6KT8E�AZI\<H^`_`42>?I�>?I�ADb�AD>N7�I\<<=426K6J7�68U\bDOPT8ED7�IL6JU\GZAZ<=>NE�7±AD7�XP<=4�F�I`<=42>?IgR�>?6JT86yEDVlILE�MNF2<=>NE�7n>?I`>NdfR�E�I\IL>?C2M?6�>N7PUL6HAZMÍE�TJ6HAD7¤^

z I � >N7�T8U\6HADI\68Ig<=EQ>?72¶�7�>N<¢OD ��  �QAD7�XP<=4�F�I�>?I�<L4�6�UL68ADILE�7�ADC�M?6yI\E�M?F�<=>?ED7Í^_`4�68U\6JVcE�U\6D �<=426KI\E�M?F�<=>?ED7n>?I8�

�à� 6 ����� � i T N � M o$� i¥Ó 2�oµÍ�2~D���?¾��f´�����º¿�����f�D«\~D�����º ��~D�D��¾P���2~D����«�´��2~��~D�¢�¡�L~�¾a� ��~D«\� �\´�«/�����c�Ý�c���¡¹�´8~�¾a���¢´¦���c¾n«a�������

���������n�¤´�«³�����a«a� ���a�{�?¾¢�����a«��¢�:~D«=�P�����P«=�L~�¾´���~�ª8�?�f¾´��������¥´��2¾,·H´�«n�c���B¹�´8~�¾a�¯~D� ��~D�D��¾! µÍ���aº¦~D«=�«=�c·H�a«³«��L��� ´S~�¾ �¯¹�´8~�¾a�¥~D��{�a������ ��~D�D��¾ B

_`4�6KX�>?I\Rt6JULIL>NE�7PUL6JMYAZ<L>?E�7 VcE�U�<L4�6�T8E�AZI\<�AZM��y68MNG�>N7P9(AHG�6JI,ADU\6D�

»jÂ É � U i¥Ó���oIL>N7�T86�E�7�M�OS<L4�6�E�7�6�R�ULEDR�ADb�AZ<L68I`9g>N<L4n<=4�6�T8E�ADI\<g<=EQ>�<=I`UL>Nb�4�<g6a£�>?I!<=Ig>?7 � �Ã>N7n<=4�6�I\O�I!<=68d±^

, � ���J�� L��B�:Lc� q �¿��������� � q �8������ q0 �Ý�Z����u���D��� q �,�u���n�8�*CÞ�S���K�u� �8�/L � rBr u��� q ������������ q � �Þ� �4�L �H�2� q � r�� s.q ��� �����-� �2��¡� ����� q �J������ q ���[���K���u��� � �. �.� r �2��

£�£�>

Page 22: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

� q �u� �u���:� ���J�f� �u��� q S qcr �t� D Lc�:���;G q � ��.� ��� D�� � GJ� �|�.� q �J�H s �D� s u��� IW� � �������:I|��u�È�u���²���:� � �u���5� ���J�Þ� q �u�Ø��� � � � L � q � r �H���*Lc� � � IK�f� q �Z� q �¦��.�� �H�BMH� u�*� q�s � � �.� � CÞ�2� q ��� _`4�>?I�>NI�<L4�6QUL68ADILE�7¿VcE�U,<L4�68d <LE±C�6QTHADMNM?6JX �!T8E�ADI!<�ADM�y68M�G�>?7 9(AGD68� C�68T8ADF�I\6K<L4�6JOPAZUL6y<=ULADR�R�68XP<=E{<=4�6�TJE�ADI\<8^

, À;´8~�¾a�¥~D� �{�a������ ��~D�D��¾|¹L~D�¡ªa�����a�¤�a«\~D� �L�fª8º�¾a��«�·J~�¹�� �������`·H´�«=¹a�����n¹a�?´¾�y�¢´{�����,¾���´�«=�����]�]��~D�·H´�«=¹���¾��W«=� �g��¹a���¥´��Þ´ ·�´������a«���~D�D��¾����¤¹a�]���a����´��¦~P¹�´8~�¾a�3

x 7Åkl>?b�F2UL6Þm%2� �<L4�6 E7E � AZ7�E�dfADM?>N68IQ<=4�AZ< 4�F2bj<L4�6:T8EDI\<=In>N7ÈILF�C�<=ULEDR�>?T8ADM �¯df>NX M?AZ<=>�<=F�X�6JInADU\6T8E�ADI\<=ADM �y68MNG�>N7 9�AHG�6JI�b�6J7�68ULAZ<=6JX[C�O <L4�6�>?7�TJ>?X�6J7�<g6J¼�F�AZ<=EDUL>YAZM��y68MNG�>N7P9(AHG�6KIL>Nb�7�ADMNI8^

C�oa^0G(E�7�<L>?7�6J7�<=ADM&IL4268MNV.9(AGD68IJ^354�687:<L4�6�X�6JR2<=4:EDVW<L4�6�E�T86HAZ7:AZ<,<L4�6�T8E�ADI\<,GZADU\>?6JI8 t<L4�6 �y68M�G�>?7[9(AHG�68IKADU\6�d�E�X2>N¶�6JXÍ ¤AZ7�X¿A

7�6a9 T8M?ADILIKEDV(9(AHG�6JI�ADU\>?IL6Z^S_`4�6JIL6�9�AHG�6JI{ADU\6�T8ADM?MN68XÍ� G(E�7�<L>?7�6J7�<=ADMWIL4�6JMNV;9�AHG�6JI8  EDU�<LE�R�E�b�U=AZR�4�>?T;gE�ILI\C�On9(AHG�6JI8^

p`qcr�s u�wv ����

x 7±<L4�>?I`T8ADIL6Z �<=4�6�IL4�ADM?MNEu9Å9(AZ<=6JU�6J¼�F�AZ<=>NE�7�IgE�7±A{V �¢R�M?AD7�6yADU\6D�Q �Q M N � �  ND� Q 1Q!T U i�Ó;7�A�oQ �Q M � � �  ND� Q 1Q� U i¥Ó;7DC�o

Q 1Q M �

Q iA �toQ7T �

Q i� � oQ� Â���� i�Ó;7DTHo

£�£2>N>

Page 23: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u�Þv � M�� E�T�4�6JdSAZ<L>?TyX�>?ADb�ULADd©VcE�U`I\4�68M�V²<LE�R�E�b�U=AZR�4�O�^

x 7¡<=4�>NI0T8ADIL6� Â# � AD7�X[ILE�MNF2<=>NE�7B>NI�d�E�UL6KT8EDdfR�MN>?T8AZ<=6JXB<=EfILE�M�G�6{AZ7�ADMNO�<L>?THAZM?MNOD^JE�ES9;6J�\F�I!<R�U\EuG�>NX�6�ILE�d�6yC�ADI\>?T�X�>?I\T8F�I\IL>NE�7BU\68b�AZULX�>N7�b�<=4�6J>?U`6J7�68U\bDOPX�>?I\Rt6JULI\>?E�7Í^

� ��� q �����t��*L��H���:L �|���������B�² �²� r ���� q � �u�.� � q u���D� q0 � ��� ���J��*L �B�:L�� q � �����t���;I� q ��È��.�¿� ���J�S� jq ��f q]r �t� D Lc� �¥�;G q � � � D�� � GJ��� � �Í� q ���*L � ���8� u� Lc��� r �u�5�H��� L]� q �**=@? �*=��4C � q �u� �NKÓ��±Æ � � �������� I s �KL q �����B�:Lc� q �n����������� �t� q �����t��*L��H���@L0�&������������u�� q �8������ q ���+� ����� CÞ���Z����� q �!C � q �u���. �²� r �2� q �¿����� M�� ���:C �²�a�u�2��:�NMH�¡� ����t� q �*L� u� q � q ���j� �²�8������� q ��Il�f� q ��� q ���  � � ��  � �"�� M���� � q L8L s �J�u�2��:� q � p`qcrts u� �*=4��� �8���� q ��� q �8� q ��I s � � ����@� ��������I��u���5� �H����� q � L|� � q � q � � q �Ë� �.�8���t�:� � Ý� C � q � r������� � L s Cw����

������ �J�,$l/�*$�S%!$���� & / ��$W-y-�'34 'w$&%('�1

� �S� ��:�Þ���� L q ����u���2����Lc�2���u�*LKM ts �K�.��%� q �¦�u�.� �������Ý�����w���D�|�����P������� r�s²q �.��I�f� q ��� r�s.q �����0�u�������������0� �² �.� r �2��Q�*L � r±q � � �|�.���8�{��������� D � ���J�H�*L �B�:Lc� q �[������������K� �H���@L0�����������JGS�2�� �D�u���.���:� ��� �u�.�w� ���J�±��� � �u� s �n����� s JM.�2� q0 �²�P���H� � q �2�H�@�� q ��©�u�����J� ��������� �.�����-�������-� �¯ C �u��� � ���J�-� p4 È� ���J� �B�:L�� q �e��������� IS�u��� q �*CÞ� L q � s ����� �.�����-��� q �u�È���Å� � � L8� q � r �������-�5�8���*L]� � � �8�!MH�5u�*� q�s � (  � � �J� �¯���D�-I�����?¾ � ���2~D�(��~�¬���¾y�����{¹�´8~�¾a�¥~D� �{�a������ ��~D�D��¾��]«L~a�����L��� ´������{¹�´8~�¾a� B �n�����¡�8�.�*L L(�J���[�.���Í� I²�u���[�:9 s �2� ���D��Q���Q�:������� r�s²q ���+� � ���t���{��u�[�u���.���:� �

�u��� �:9 s �2� �M������ s �J� � � � I���.�Ý���2 q �2� q0 � � � � q �u� L]��� q � s ����� I�u�2��.��:�u����� � � � �� ����� s �8�¡�2���u���[�:9 s �2� I �P Ó�� ���� �±Â���� M s � � "Â�� �

£�£�>?>N>

Page 24: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u� ��=&� E�4268MNV.9(AGD68IJ^

�B��u� q �¦�� q �,�-Lc���H�;I¤����� q L8L � s �J��� � s � ��� q �H� s �H� q � r �u�����l� �&� q ����� q �u� ts �y�. � q �4�q � r ��.� CÞ���u�.�:CÞ��� q ��� L��.���� q L]�;� p y�������¿���&��� �.�������+I������ q L8L � q �H� s �H����.� � L L � q � r���8�����D�u�Q���H� � q ����:� � q �u�j�u�.�[�@9 s �2� q �*L L0�Ý�u���.���:� ���������D�

� ������ r � � q �8������ q ��Il�²�.���8�¿�8�����:� D � % G���� � r ts � ���:L � q ��� D ��Ï G��

� � L s � q0 � �Z����u���D�����

� � � C�CÞ��� q � �² ����� ���

� p4 �� q � r �

� � ���]v*� �|�.�B�:9 s ��� q �*L �±�:Lc� q � �����t�

~- ����?¾¢���a«=¾a�¥´��w«=�a��~D�]�¥´��+ �|�.� � q �J������ q0 � u�@Lc�2� q0 � � [�u���Ý�:9 s �2� q �*L �±�:Lc� q � �������q �D�

»jÂ É � U i�� ��o�f�.��u� �{q �{�.M²�� � � q � M²�� �-L q � q � C ���:�8�����:� � � ����.� ������� q � M²�� � � q � I

�� � P  � � Â©Ó � ���±Æ �  $ � � ��� ���|��%� �¯���J�-� p4 [��� �t� q � s ts � L�� �J�u�2� q Al�:� ��������I����j������� ��� M.�� � � q ��C ���j���K� q � Al� q ��Ý� s CNM��� � M²�� �-L q � q � C �.���;� � s �*L8L���I²��.� Al��J��M.�� �-L q � q � C ���¡�����Q�Þ�J�����:� � � �  Ó�N2���±Æ ��n�,��� �����Ý�8���+IÍ��.�f�@9 s �2� q �*L �±�:Lc� q �:�������n�.�����u��� �J�*CÞ� � q �8����Z� q �Þu�:L]��� q �¿���

�u���±� ���J��*L �B�:L�� q �w�������±���K� �����u��� r ���� q ���Þ��������� q � �¿� �4�L �H�2� q � r �ls²q � D �±Â�� GJ�

£�£�>NG

Page 25: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

�����-� ��u�j� �4� � q �8������ q �t�j� q �.��� �³Ï� Ð=ÑÐaÒ Â �Pq � q � ���������K������� � �¯u�:9 s ���²�;��� ��Ï{ � % �� �u� �²�.���8�:���K� r �s ���t�:L � q � � D ���.�� r � G|�² �²� r ����:�����J�³���� ���

��¸8��~D�¢´�«³�]~D� �{�a������ ��~D�D��¾���«=´��2~��~D� ���L~�¾a�8��~D«\�-

p`qcr�s u���&v�� 0,>?I\Rt6JULI\>?E�7nU\68MYA�<=>?ED7nT8F�U!G�6�VcE�U �y68MNG�>N7P9(AHG�6Z^

ª% � ´������]�¥´�� � �����¡� L s � q �.��� �±�@L�� q � �������¡�2����

1� 6 ��� � ## � � ci T N � M o I�P Ï

6 � � � ## � � i T N � M o I

� Â��"��n�¦���È����� �8�����¯ C ��.�È� M ���5� L s � q0 �.�;I �B�:L�� q � ���������Ë�. �.� r �2��È�����J�³���� �� q ��#�5�8�����:� � � � �|�.� q Ë� CÞ� L q � s �����B��u�j�u��� Lc�� r ���a�:�2�:�u��� �:9 s �2� ¦���K� �.�����-�� L]������ � � �������#� q ��� �K�u���Ý�:9 s �2� [� q �� ��� �-� � L � q � r �H���*Lc� � �  � ­

�� �|�.�-�

��u��L q ���j�³� � ���a��*L �±�@L�� q �#���������¿�² �²� r ��� q � r M.��� � �L� � M.��� � �*L � r ��.���:9 s �2� D �f����u� �±Â�� GJI ��� q ��� ��.�[� �u� �`�u���¡�:9 s ��� S���K� �u��� �u������ ��.�[� �s �u��� �8� �u����|� �²�;IÍ��.� �:9 s �2� M��������t��� L q ���S���w��� q Al� q �*L²� ���J�JL q ��� � ������Z�Ý���:C q �J�l��������

� �¥�����I �Ë�.�2� s u�*L L0� ��� s u q � r CË�� q � q0 �.� L Lc��� r �u���8���*L]� q � �u�.�±� � q ��� q �Z�(��  � ­ � D �8����� q L L�v�v$GJ��a� q �:���*L Lc�@� �u��� ���:9 s �2� q �*L�u�*� q�s � � �.� � CÞ�2� q0 � �KI����K� q �[�����*L �r��s �:� ��.�

� �H�!M � u�*� q�s � � ��� � CË�2� q0 � �2� � �L�:9 s �2� q �*L L]�2� q � s ����� ��;��~D���$�;�����Q�L¸8��~D�¢´�«³�]~D����ºS�]«L~a�����L�S��~D�]��«��Q´ · ��~D�D��¾K«=��¾a������¾�·³«=´����c���g��«=��¾�a�¤¹��{´ · � ��· �(�y¾�a�

� Â������f�c���g~�ªa´��D�(¾´��������¥´��2¾��c���a�f�c��� ��~D�D��¾`~D«=�`�¤´����]«L~a�����L�|~D��ºZ�S´�«=��� �g��¹L~D�������2~D���c���g�]«\~a���������´ ·�¹�´8~�¾a�¥~D��{�a������ ��~D�D��¾��?¾|�D���K�¢´��c���`��«=��¾�a�¤¹���´ · � � B p4 f�u�.� Al��J��M.�� �-L q � q � C �.�±� q �u� � � N2���±Æ I ( � N Ó�Ù ��É � �D � q � q ���t�8��;����Q��� � � CÞ�:LF��� �2��� r+ ����� �� q � rtq ����!� � �&�4� CÞ�:� � r�q � I2�f� q ���±�8� �S�;����*L

� q C�P��:C�������� s u�B��� ���4�����*C q �B��� q]r �t�����8� � q �2��:� � q �� �B�:L�� q �j�����������.� s �8��� GJ�

£�£�G

Page 26: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`q]r�s �� �S�&� E�T�4�68dfAZ<=>NT,VcE�U �y68M�G�>N7n9�AHG�6JI8^

¹! � º��{�S�a��«³�¥¹���«=´����a«³��º � �n�,���n�8��� �¥ C �u���n� L s � q0 �²� I �B�:L�� q �Ë���������K��u�n� � C�CÞ���!� q �n� M �s �S�u���[�:9 s �2� � ����� � ��u�:�B� C CÞ��� q �S���������;� � �u� � ��� � 1 ��u�:� � C�CÞ��� q �� M �s �f�u���[�:9 s �2� � �  � �

�- ��.´�«=¹a����� � �{�Ë�u���È�:9 s �2� I��u��� �B�:L�� q � �����t���¦��u� C �a�JL0� r ���.��u�2��:� MH� ��.����.��� r�q � r � q � �.�;� �����2� q � I²���2 q �2� q0 �²� �g� q �K�²��� q �u� � q CÞ����n�H� s CÞ� �u��� ������� q � ���*L8C q � q � q �*L L0��� �������Î�*L L �[� � s �K�.���������J��� L��©� q � �.�

D � q � �.� M L � q � r �¯ C �����J�{� �����J� G|�J����{� M L � �*L � r �u���[�:9 s �2� ��� �s � q L8LW�8����B�:L�� q � �������[�.� s �8�:���H� � q ����:� � q �u� �Þ� q]r � �8��� Lc���t�:L���� �È���������������@���u���� C �-L q ���q � � � q CÞ� Lc� �@��Lc�-�t��[�B���J��:C � ��� q � q � M������ s �J�Þ�����J��� L��È� q �K�²� � q L Lg��� s �8�Þ�����J�³���� �� s �u���t�u� q � q � q � L8L���I{���K� � q L q �5� ���� � �u��� �@9 s �2� È����K�¨� � s L Lf�u��� C � q �� ���� � qcr �t� D L]� �¥�;G q � ��.� � � D�� � GJ� �n�Þ� u��� s Lc� I|�u������ q �¦���2��� � ���t�� r ��� ��u���¡�:9 s �2� n���K�È�� s �Q� s �H�����f�u���[�u���� C �-L q ����� �f�t���� �5��� �Èu� q �8�[�u���:�8����Lc�����@L ������������]������¾±´ · �{�a������ ��~D�D��¾P~D«=�|��«�´���´�«³�]�¥´���~D�;�¢´:����� � ��´���~D�������¢�¥�D«L~D� P´ · �����/�(��¾a�¢�a«³��ºN��������¾� ���a«³����~D���¥´��[�?¾��¤´�����«=´���]���L�a�;

p4 ����:� �B�:L�� q � ���K� � �H�!MH� �����������¯ C � L q �.���� C ���:LF� � q L L0v�v*�]v�v D � q ��:� �¯ C� ���|�����%� v � � � GJ��� ���J�H�*L �±�@L�� q �w���������{��u�[�-L]���� L��Ý�8����� �*L � r �u���B�����J�H���� �������M �s �K����%�Ý� �¥�����u��� �:9 s �2� q � L �B�:L�� q �¦���������K��u q ��� �2�K�u���/M �s �K�.�2J��� �{� Al��J�-IÍ��.�� ���J�� L �B�:Lc� q � ���������B�2��w�H��2�²���:� � �u���j� ���J��M������ s �8�Ý�u���-� �. �.� r �2��5�¥���J� D �2��8�����:� � � GJ� �����������{� q CË� r� ��� MH��I²��.�-� M���� CË�[���t�u���.���:��� s �[� �u�.�±�����J�³���2 �u�*� q �2� q � ���u��� � �H�BMH�¿��������� � � �S�����w�8��� �¯ C ��.��A r�s ����u���2�|�8�����@�.� � � �H�BMH���������� ������u�����8�[� q �� L]��� q � s ��� q �.��������J�+�

� � ��� ��� �|�.�B�:9 s ��� q �*L � �8�!MH� �������+�

~- ���?¾¢���a«=¾a�¥´��[«=�a��~D�]�¥´��+ »jÂON � Ò

Ò # � � � � ­�� � � � I�f�.��u� �  m U Ó �0�0� q �w�u�.� ����5� s CNM��� � �±�� C q ���� s �²�D� q � D �����u� M L q ��;� L q �K������ s �²�D� q � GJ�

p4 �L � r � �H�!M �Ý��������� I É ­ q ��� CË�*L8L ��� �j�u� s ���u�.� � q �J������ q0 � ����� M��n�� q �H����Ý���D�»jÂON cÒ­�� � � �

£�£�G�>

Page 27: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`q]r�s u���#*Í� E�T�4�6JdSAZ<L>?T�X�>?ADb�U=AZd©I\4�E9g>?7�b�68¼�F�AZ<LE�UL>?ADM��y68M�G�>?7 9(AGD68I�VcE�ULTJ68X±C�On9;68I!<=6JULMNOn9g>N7�X�IJ^

Á(ÏgÂ#N ­�� � � q � q � ��������� �����t� � » I����K�±�u� s � L � r � �H�BMH�n������������u��� �4� � q �8������ q �t��I���K� �³Ï� � %  � �"�� M�� �|�.��u� � ���I M �u� �²�.���J� ��� � r ts � ���@L � q ���5�² �²� r ����w�����J�!����2 ��I ���{�u��� L � r � �H�!MH� ��������� q � C q �4� Lc�2� q � s �����

p4 � �  ���±Æ I �¿Âem I;���K� Én ­¯®� P P P Ò�� I Ç Â ­¯®Ñ  ��Ù K ( � I �f� q ��� q �f�*L]� � �²� q �J�H������ q �� �u���:� q]r �����a� �¥��:9 s ���²�;� D �H� ����J������ q0 � G � M²�2 �-L q � q � � �H�!M � ���������;�

p4 �H� � � �H�!MH� ���������;I É ­ q � L]�� r ��S���K� �u� s �;I»ÅÂ�N � Ò � �����Þ�8� CÞ�Þ��� � q �J������ q0 � u�:L]��� q �5���S�u���w�H� � � �H�!MH�5��������� �2� C q �4�

Lc�2� q � s ����� �H�!MH� ���������;� ����� � ��u�N� q �J������ q ���B���������;��³Ï` �

Ò # ���K� � % ÂON �Ò # �� ��´�«³� ��´¾�¾8ª8º/��~D�D��¾y�D«=´��u�w�D�a�?´¹a����ºn�?¾f�L~�¾a�8��~D«\� �gª8��� ���2~�¾�K¾¢�����L�±�?¾ �(��¾a�8��~D«\�- ����� �Å��u�

�³� �8� ��� M��{���w� q ���t�JL��B� �&� q ��:� q �¡�u�.� ������� q �t��� q � �Ãq � q � r �*L]� ���D�u�0�J� � r L�� �#�u�����8���8� �¡���������;� �n��� q �H� s �H�8�:� q �Ã�u��� C q ��� L]�2� q � s ��� � �H�!MH� ���������¿�8���D� q0 ��I�u�����8�È���������w��� q �J� �KL0�Ø�.����j�u���È�����J����� M �s � ����%��� ����� q Ý������ r � q � � q L8Lc�:� MH�C q � q � r ���K���u� s � � C q � r �u��� � � � �

ª% � ´������]�¥´�� � �|�.�|� L s � q0 �²� � ��:9 s �2� q �*L � �8�!MH�n������� q �t� L��t� �±�� C q �� � s �.�D� q0 � ������2� q � I.�u���:� L s � q �.����u�¡���&�²��� ���:� q � � � q u���D� q0 � MH� �B�� C q �H� � s �.�D� q0 ���

� � Â�- � i � ­ �� o�����i¥É T NÞ» M oWÂ Ó � ���¯­

�i � �

� oF6 � � � ## � ����i¥É T NË» M o I

1� ­ Ï � Ó � �ZÔ����� � � # �� ( �Ò � Ñ � � ��

� � � # �� ( �Ò � Ñ Ü ���� i¥É T NË» M o I�� � � ­ Ô����

� � � # �� ( �Ò � Ñ N � �� � � � # �� ( �Ò � Ñ Ü ���� i¥É T NË» M o I

�f�.��u� -�q � �.��u� M L q �Ã�;� L q �K�.���� s �.�D� q � � ���� � I±���K�

�q ��� �B�� C q ��

� L0�&� C q �*L � �.�� � �

£�£�G�>N>

Page 28: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p4 ¦������� � s C M����� � � � � I -�q �¡� � C�CÞ��� q � � p4 �K�Ø� s CNM���Z� � � � �KI -

�q �

���t� q �B�4C�CÞ��� q � �p C �u�����8�w� L s � q �.�;IW���Ë����� �8���Ë��.�2� ~D���������]�2����¾S´ · ��´¾�¾8ª8º/��~D�D��¾{����¹L~Dº���´��?� ��~D«\�

~D��� ´¾¹a������~D� ´�«³º¦���5º � ����� �e��u� �@9 s �2� q �*L L0�Ø�H��2�²���:�W� � q �u� ��� � � � L8� q � r �8���*L]� �(��  � ­ � �� � � � C�CÞ��� q � �. ������� � � � s �8� �B� C CÞ��� q � �² ����u� q ��� � � � D 1 G¡� ��� Al�����B� C CÞ��� q �Þ�. ����� q ��� �Q�����t���;� �����2� q � I q � � � D 1 G±��u� � � C�CÞ��� q �D ����� q � � C�CÞ��� q �@G � M �s �g�u�.�K�:9 s �2� I��u�.���B���|u� �¯��uu�@�±� ��.���8�����������g����� � C�CÞ��� q �D ����� q � � C�CÞ��� q �@G,���������;�

p C �u��� � M ����� L s � q �.�;I q � � � q �¡���t� q �B� C�CË���H q � I �����K� 1

���u� �B� C�C��� q � �

� �H�!M � ���������S����� M�� M �u�È�B� C�CË���H q � S���t� q �B�4C�CÞ��� q � IK���������K� q � r5 � �u�.��� Z� �q � r �J� s �D� s u���;� �B�n�u��� � �� q � r q �Ý�B� C�CÞ��� q � I{�B� C CÞ��� q � � �8�!MH� ���������Ý� q L8L M������� q ��:� � � ���u���.� �� q � r q �S���t� q �B�4C�CÞ��� q � I²�u�.���Å���t� q �B� C�CË���H q � � �8�!MH����������� � q L LM�� r ���.��u�2��:���

� � p4 �� q � r � � r �@9 s �2� q �*L � �H�!M � ���������¡����� M��Ý���Ý� q ���t�JL�� ����� q ��:� MH� ��.����.��� r � � L]�� r �w�J�²�2� q �*L0�H���*Lc�¦� q � �.�;� � ��� p`qcrts u� ��� � � q �.���¦�����a��� L��È� q �K�.�±��� s �8��:9 s �2� q �*L{� �t���� r ���.����� s ��� � q0 L q �¡� � ���D��;I q � �u��� � �:9 s ��� q �*L�u� r�q � � M �u� � � ��� � � ��I � q �t�� r ���²��� q � r ���.��u�2��:� � s �¦� ���2��� � �f�S� ���� �5�u���¿�:9 s �:�� � �������� � u��I����Å�@9 s �2� q �*L �.���������.�:� �u���� C �-L q ��� D �f� q ���Åu���²u���8���t�u�n� qcr �Å�8���Lc�����:L � .M.�� �-L q � q � C ����� G ��� s ��;I,���K� � �:9 s ��� q �*L��H���*L L ���:�Ã��.�� C �-L q ��� q ��. � s ���:� � � q �.���[�u��� � �� q � r � q � �.�Q��u�¿�B� C�CÞ��� q � � M �s �S�u�.�[�@9 s �2� I �KL0� �B� C��CÞ��� q � � �H�!MH� ������������u� r ������u�2�H�@���

� � ��� � � � � � p � � � �u� � p � � � � � � � � � ��� � � � � � � � � � �

���+��� q ���D� �B�:Lc� q �[���������0�. �²� r �2�H�f�����J�³���� � I+M �u�N� y�l�����8�����K� r �s �:���:L � q ������ �¥�����B�:L�� q �:��������� q CÞ� q � r � �t� ��.�������J����� ������� M �s � ����%��I&�²�2u� ���u���f������ r �q �fu� � ���D��:� M²��� � q �t� �u��� ������� q �t��� q0 f��� �-L � r � �8�!MH����������� �KI(� q �.���¿��.��L � r� �H�!M � �����������0������ r � �. �²� r �2�H� �����a�a���� �#��� �Ã�u� s �¡�����#�H� ���j��.�j���.�� r � �����-��¯ C �u���:�����J������M �s �K�.�2J���(�.��� �0�u���:���.�� r �È�. �.� r �2����S� Lc������ ��� q �Þ� ���J��*L�B�:L�� q � ���������;� ������u� � u��Il���� q � M q L q ���w��.�2� q � q � q �2������2�Q�u���¡�:9 s �2� n�������. �.� r �2��� �u�.��C q �4� L]��� q � s ��� 9 s²q � �4L0�j�u�� �s.r ��� ���J�H�*L �B�:L�� q � ���������;�

� r � �H�!MH�¦�����������² �.� r �2�� �����J�³���� � � �n� L � r � �H�!M �Þ��������� q CÞ� q � r � �t� �u���B�����a���Z� ������� M �s �K�.�2J��I��.��� ���u���[������ r � q �Qu� � ���D�H�@� M²��� � q �t� �u��� �������q �t��� q ¡��� �±�@L�� q � ��������� Iy��� �#�²�2u� � q �¡���¿�H� � � �H�!MH� �����t���;I M �u� ���S� q �Z��������|�����J�³���� � r �s �[���:L � q ���n��� �B�� s �g�����±�� ���K��.�|������ r �±�����-� �¯ C �u���K�����a���Z� ������� M �s �K����%��� � q �.���Ý�H� ��� �H�!M � ���������±�2��j�8� �[��� �Ã� L � I(�u��� q B���.�� r � q ��. q CÞ�� q L0� � q L Lc�@� MH� C q � q � r �.����f�u�.�±�����J����� M �s �K����%���� � ���¦�.�����Ë� 9 s ���J� q0 ���{�f� � � �����n���Ë�������Ë� ���a��*L �B�:Lc� q �����������S�u���2� �� ���

�u���[������ r � � Lc������ �È������S�u���B�����a���Z� M ts �K�.��%��� � �K CË�.�{�u���¡�-Lc���H� �� � ��� *�� � 9 s ��� q �*L L0�Þ�u���²���:� q �.��� q �*L r u��� q ��� �����t���

~- ���?¾¢���a«=¾a�¥´��[«=�a��~D�]�¥´��+ » ­ Â É ­ � ­ ��� �Di�Ó�� � muo I

£�£�G�>?>N>

Page 29: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

�f�.��u� �BÂ#m U Ó�� � ��q �,�u��� �.��{� s CNM��� � �±�� C q ���� s �²�D� q0 � D �,�2��$M L q �f�;� L q �4����� � s �.�D� q0 � GJ�

�������8�¡������������u�:��� L8Lc�:� q ����� q � L r u��� q ���j���������;Il�f� q ��� q ��� q C q Lc��{� �u��� q �.��� q �*Lr u��� q ���Ã���������¦�2�.C q ����Lc�2� q � s �.��� �|�.�-� ��u� r ���� q ��� ��������� s � ����w�u��� q � �ls ���.��� �� D � G��2�n�u���Þ�:9 s �2� D C q �4� L]��� q � s ��� GJ� µÍ���aº±~D«=���D�?¾¢���a«=¾a���D� ��~D�D��¾��g~D���P�����a��«S�a�¤�a«��Dº±~D������2~�¾��¹L~D�f��«=´��2~��~D�¢�|ªa´����:�L~�¾a�8��~D«\� ~D��� �(��¾a� ��~D«\�-

����� q NC q � q C s C �¯u�:9 s ���.�;� q �D� »� � �  � � �Di�Ó�� � muo � p4 � Â�m I � � Â4���±Æ Ig���K�É� ­¯®� P P P Ò�� I»

� � �  Ó���� ��� � K ( � � � I���K�Å�u� s �f�u����L � r ���a� ���� q � q ��*�� � ���-��� � r u��� q � � ���������S���� q0 �.�S��u�¦�H� ���

�u���������¯��� ��� �&�;�

ª% � ´��������¥´�� � �����B� L s � q0 �.� � ��:9 s ��� q �*L L0�Ë�u���.���@� q ����u� q �*L r u��� q ���w�����t���{�����t��u���[�J�*CÞ� � C �����u�.�B� L s � q0 �²� � � �8�!MH�Ý���������;� µÍ���aºf~D«��|´¾¹a������~D� ´�«³º{��� � �D��«=��¹a�]�¥´��~D���Ý~D�������n¾8~D�S�±�]���S�[����¹L~Dº±��´��?� ��~D«\�- �|�.�-�e��u� �:9 s �2� q �*L8L�� �H��2�²���:�W� � q ��e���©�-�� L � q � r �H���*Lc� � (��  � ­ � �� � � � C CÞ��� q � �² ����u����� � � s �8� �B� C CÞ��� q � �. ����� q ��� � � � D 1 G:� ��� Al�����B� C CÞ��� q �Þ�. ����� q ��� �Q�����t���;� �����2� q � I q � � � D 1 G±��u� � � C�CÞ��� q �D ����� q � � C�CÞ��� q �@G¡� M �s �j�u�.�Å�:9 s �2� I��u�����©���Åu� �¯���u�:�©� �u�����8� ���������Ý���j�B� C��CÞ��� q � D ���t� q �B�4C�CÞ��� q �@G0���������;� �8����� q � L r u��� q ���w�����t���Q����� M�� M �u���B� C CÞ��� q � ���t� q �B�4C�CÞ��� q � I ���������K� q � r �Ý��.� � Z� q � r �J� s �D� s ����;���B�W�u��� � �� q � r¡q ���B� C�CÞ��� q � I�B� C CÞ��� q � r u��� q ���5���������±� q L L M��Ý� �&� q ��:��� � �K��.��� �� q � r q �P���t� q �B�4C�CÞ��� q � I��u��������t� q �B�4C�CÞ��� q � r u��� q ��� �����t���{� q L L M�� r ������u�2��:� �

� � p4 Z� q � r � � q � �.�:���H� � q �2�H�@� � q �u� � ��� ��� q �j�������u�.��Ë�l����� CÞ������I r�s �J���� q �K�.�{� q �u� ���� q0 �²� �0� �¯��� � �s ��;�

� � ��� � � �|�.��C q �&�:� � �H�!MH� r u��� q ������������ �|���.� q �������

~- ���?¾¢���a«=¾a�¥´��[«=�a��~D�]�¥´��+ ÉfÂON �Ñ � Ñ

��� q �Ý������� q � q �t���u���J� q � r M������ s �J� D q GË��� �¯u�:9 s ���.�;� q � � qcr � D » q � L]�� r � GJIf��.�� q �8������ q0 � M���� CÞ��� É Â Ñ

I��f� q ��� q � �B�:L�� q � ������� �N� � q �J������ q0 �©u�:L]��� q � � D q]q G¡���

�¯u�:9 s ���.�;� q � L � IW�u��� � q �8����Z� q � M���� CÞ��� ɱ N �Ñ I��S� q �Z� q � �8� � � �H�!MH�È�����t� ���� q �8������ q0 �Ãu�:Lc�2� q0 ��� �|�.��u� � u�+I0���.�� r � � q �8������ q0 � ���u� q �P���&��� �|������� M��������t���L q ���B�H� � � �H�!MH� �������:�2� L � �¯u�:9 s ���²�;��I����K� L q ��� �±�:Lc� q � ������� D �S� �8�¡u���J� q � r� ���� q � r u��� q ��� GP���¦� qcr � �¯u�:9 s ���.�;��� � �u�.�-�Ã��u� ���*L8Lc�:� � C q �&�:� � �H�BMH� r u��� q ������������ � I �K���.� q ���������f� q �.��� �K����� q A²Z�a� � q �H� ����u�@� q �-�

� In�2� L � �¯u�:9 s ���²�;��I �|�.���8�Ã�8�����:� � �K����� q ��������� q �j�����J�³���� �>M s � r �s ����:L � q ��� q �������J�³���� ��� �Q�Q� qcr � �¯u�:9 s ���.�;��I �²�.���J�¡�8�����:� � �|���.� q ��������� q �������J�³���� ����K� r ts � ���:L � q ��� q ��� L�� �����J�³���2 ��� Ê «=´��u�¿�D�a�?´¹a���]º ´ ·���~D��~D� ��~D�D��¾,�?¾��L~�¾a� ��~D«\�- ��W�2~�¾�¾¢�����L� ¹L~D� ªa�¦�L~�¾a� ��~D«L� ´�«��(��¾a� ��~D«\� �S���¢���a���D����� ´�� ������·³«=�L¸8���a�¤¹aº �����-� �2�� � q �8����Z� q ������������;� �{s �Q�2�S� q]r � �¯u�:9 s ���²�;� �u��� � ��u�:�-L �8�±� M��:� �4� � q �8������ q �t���

� ��� p`q]r�s u��**=/� �K����� q ������� � q �8����Z� q �¦� s ��� �.� q �8������ q0 �¿u�:L]�2� q � � q � r u�*C �

£�£�>�£

Page 30: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

ª% � ´������]�¥´�� � �����¦� L s � q0 �²��� �K����� q �������¿��u�¿��.� �¦Â � �����8��� n�u�.�¦� L s � q �.� � � �8�!MH�j���������;� �|�.��u� � u�+I

� P Â�- P i � ­ �� o�����i¥É T NË» M o �

� � � � C�CË���H q �[�. ������� � �n�S���Ë����� �8����I � q � �B� C�CË���H q ��� �K���.� q ���������n���K��u� s �����t� q �B�4C�CÞ��� q � � � ��� � � � �K����� q �����������2��K���t� q �B�4C�CÞ��� q �g��������� ���u�.� q �� �.�*L���:L � q ��� � ��� �Å�²u���H� s u� � �2��¿����� q � � C�CÞ��� q �±� M �s �n�u���¦�:9 s �2� I���� � CÞ�� q � q ���*L���:L � q ��� � q �Q�B� C�CË���H q � � M �s �f�u���¡�:9 s �2� �

� � p4 �� q � r � ���8� q L L]��� J� CÞ�� q � q ���*L�� q �K�²� ���t� q ��� � C�CÞ��� q � �²�2u� ��� �.�*L� q �K�.�;�

£�£�£

Page 31: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u� ����� 35>N7�X��¯X�UL>�G�687±68¼�F�AZED<=>NU=ADM��y6JMNG�>?7PAD72X ;gE�ILI\C�On9�AHG�6JI0VcU\E�d¨APm �­ �¢M?AHO�68U`d�E�X�68M�^

£�£�£�>

Page 32: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcr�s u� � �Í� E�T�4268dfAZ<=>NTyX�>YADbDU=ADd VcE�U�<=426�¶�ULI\<�A�Vc6J9 -�^

£�£�£�>N>

Page 33: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`q]r�s �� � �&� I,C�I\68U!G�68XB68¼�F2<LE�UL>?ADM3;�EDILILC�On9(AHG�68I�VcULE�d©I=AZ<L68MNM?>N<L6 E7E �ÃAD72E�dSAZMNO�^

£�£�£�>?>N>

Page 34: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u� � �t� E�T�4�6JdSAZ<L>?TWX�>?ADb�U=AZdfIÍVcE�U�6J¼�F�AZ<=E�U\>YADM �y6JMNG�>?7KAD7�X2;�EDILILC�O�9(AGD68I C�E�F�72X�ADU\O�U\6Jh�6JTJ<L>?E�7Í^

p`qcr�s u� � �Í� E�T�4�6JdSA�<=>?TyX2>YADb�ULADd©VcEDU`T8E�ADI!<�ADM �y6JMNG�>?7 9(AHG�68IJ^

£�£�£�>NG

Page 35: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`q]r�s u��� �Í� 0,>?I\Rt6JULIL>NE�7PUL6JMYAZ<L>?E�7�I;VcE�U`68¼�F�AZ<LE�UL>?ADM��y68M�G�>?7PAD7�XF;gE�ILI\C�On9(AHG�6JI8^

p`qcrts u� **=&��� AD7�AZ>Í9�AHG�6KX�>NILR�68ULI\>?E�7PTJF�U\GD6|E�7±X�>NILR�68ULI\>?E�7PU\68MYA�<=>?ED7 X�>YAZb�U=ADd±^

£�£�£�G

Page 36: V]ADTJ6 EDV - Atmospheric and Oceanic Sciencesstorm.colorado.edu/~whan/ATOC5051/2012spring/Class_Notes/chapter4.pdf3Þ6 9g>?mnml¶ uli!?iltjfili4e 9Ø?mnm

p`qcrts u� *&v�� E�<=U\F�TJ<LF�UL6JI�VcEDU � AD7�AD>¤9(AHG�6JI8^

£�£�£�G�>